Upper-Limb Robotic Exoskeleton for Early Cardiac Rehabilitation Following an Open-Heart Surgery—Mathematical Modelling and Empirical Validation

Abstract

Robotic exoskeletons have the potential to enhance the quality of life of patients undergoing cardiac rehabilitation. Recent studies found that the use of such devices was associated with significant improvements in physical function, mobility, and overall well-being for individuals recovering from a cardiac event. These improvements were seen across a range of measures, including cardiovascular fitness, muscle strength, and joint range of motion. In addition, the use of robotic exoskeletons may help to accelerate the rehabilitation process, allowing patients to make faster progress towards their goals. This article proposes a new robotic exoskeleton structure with 12 DOFs (6 DOFs on each arm) in a symmetrical construction for upper limbs intended to be used in the early rehabilitation of cardiac patients following open-heart surgery or a major cardiac event. The mathematical modelling and empirical validation of the robotic exoskeleton prototype are described. The matrix exponential algorithm, kinetic energy, and generalized forces were employed to overcome the problem of high complexity regarding the kinematic and dynamic model of the robotic exoskeleton. The robotic exoskeleton prototype was empirically validated by assessing its functionalities in a lab and medical environment.

1. Introduction

An exoskeleton is “defined as a wearable device that augments, allows, aids, or enhances mobility, posture, or physical activity”, as stated by the ASTM International Technical Committee on Exoskeletons and Exosuits (ASTM F48) [1,2]. Exoskeletons are designed to be controlled by the user, typically through the use of sensors and motors that respond to movements or signals from the user [3,4].

Cardiac rehabilitation programs typically include exercise, education, and lifestyle modifications, and have been shown to reduce morbidity and mortality [5,6]. However, traditional rehabilitation methods can be limited by a lack of motivation, difficulty in accessing rehabilitation facilities, and low adherence rates. To address these challenges, upper-limb robotic exoskeletons have been developed to enhance the rehabilitation process and increase patient engagement [7,8]. These devices provide feedback, resistance, and support to the patient’s limbs, helping to improve the outcomes of rehabilitation [9,10]. Additionally, the use of robotic exoskeletons in cardiac rehabilitation after a major cardiac event can help reduce hospitalization times, leading to lower healthcare costs and improved quality of life of patients.

To obtain these clinical benefits of upper-limb robotic exoskeletons, their design [11] also plays a crucial role in their effectiveness and acceptance by patients [12,13,14,15,16,17]. A well-designed exoskeleton should provide comfort, easy to adjust, and ease of use, while also being lightweight and durable [18]. The design should also consider the patient’s range of motion, limb size, individual cardiac rehabilitation goals, and a deep understanding of the kinematic and dynamic behavior of human limb movement [19,20]. Based on that, the design team must develop the kinematic and dynamic models used to design and evaluate the exoskeleton’s performance. In the design process, the kinematic and dynamic models of the exoskeleton are used to optimize the mechanical design, control algorithms, and sensor systems [21]. Thus, the kinematic model is used to determine the optimal placement of actuators and sensors, while the dynamic model [22] is used to design the control algorithms for providing resistance and support to the patient’s limb. Consequently, the kinematic and dynamic models of a robotic exoskeleton for the upper limb play a crucial role in the design process and are essential for optimizing its performance and effectiveness in cardiac rehabilitation. The kinematic and dynamic models provide a theoretical foundation for understanding the relationships between the exoskeleton and the human limb, allowing for the design of more effective and efficient rehabilitation devices [23]. The kinematic and dynamic models are critical for the design and evaluation of the exoskeleton’s performance, considering the patient’s range of motion and rehabilitation goals. By using these models, the mechanical design, control algorithms, and sensor systems can be optimized, leading to improved rehabilitation outcomes and to reduced hospitalization periods [24,25,26,27,28,29].

Recent studies have demonstrated the feasibility and safety of using upper-limb robotic exoskeletons in cardiac rehabilitation, with promising results [30,31]. However, there is a need for further research to identify and develop new robotic structures that fulfill the needs of early cardiac rehabilitation and fully understand the impact of these devices on patient outcomes, as well as to identify any barriers to their widespread adoption [32,33,34].

Figure 1 highlights the proposed design flowchart that provides a systematic and iterative approach of designing a robotic exoskeleton for the upper limb intended to be used in early cardiac rehabilitation. The approach ensures that the design is safe and effective and meets the needs of the target population, while also being feasible and affordable.

Thus, this paper presents a novel conceptual robotic exoskeleton design for early cardiac rehabilitation following an open-heart surgery or a major cardiac event, together with a mathematical model used to estimate its performance. Additionally, tests were performed to validate the mathematical model. The conceptual exoskeleton is designed to augment the cardiac patient’s upper limbs’ rehabilitation to accelerate cardiac recovery by using stiff linkages driven by six motors per arm. The mathematical model using the matrix exponential algorithm highlights the kinematic and dynamic behavior of the CardioVR-ReTone exoskeleton. This mathematical algorithm was chosen due to its simplicity, flexibility, and efficiency and for the advantages compared with other models, including linearity, matrix representation, computational efficiency, and analytical solutions. The CardioVR-ReTone robotic exoskeleton prototype was manufactured and empirically validated by assessing its functionalities in a lab and medical environment. Medical benefits resulting from using such equipment in cardiac rehabilitation will be highlighted in future articles.

2. Materials and Methods

2.1. Exoskeleton Robotic System Design Overview

The design and development of the CardioVR-ReTone robotic exoskeleton system for cardiac rehabilitation followed a comprehensive design approach that considered multiple key components and steps. This section outlines the process of designing such a system, including the requirements and specifications, kinematic and dynamic modeling, actuation system, sensors and feedback systems, control system, human–machine interface, and power and energy management [35].

The first step in the design process was to define the functional requirements and specifications for the exoskeleton, including the goals of the rehabilitation program, the range of motion (ROM) required, and the expected loads and forces that the exoskeleton will encounter. Next, some of the functional requirements regarding the CardioVR-ReTone exoskeleton will be highlighted: (a) should be safe to operate and not injure the patients by forcing the joints into nonanatomic positions; (b) should support the patients in their rehabilitation exercises; (c) should function in passive, full-assist, and active-assist modes; (d) should be a fixed one used for the early rehabilitation of cardiac patients; (e) should be compatible with wheelchairs commonly used in hospitals; (f) should facilitate arm and trunk exercises, such as raising the arms, reaching forward, and flexing the arms; (g) should facilitate bilateral arm rehabilitation; and (h) should facilitate safety and emergency stop of the exoskeleton and physical limitation of movements in each joint.

The performance parameters of the prototype result from the functional requirements of the robotic exoskeleton. Thus, the performance parameters of the robotic exoskeleton are:

▪

Safety: this is perhaps the most important metric, as the device must be safe for the user to operate. Metrics to evaluate safety might include the frequency and severity of accidents or malfunctions during use, the occurrence of adverse events or injuries, and the device’s ability to prevent falls or other accidents.

▪

Ease of use: the exoskeleton should be user-friendly and easy to operate by the specialized personnel, even for individuals with limited mobility or physical impairments. Metrics to evaluate usability include the time it takes for users to learn how to operate the device, the ease with which they can adjust settings or controls, and the level of comfort and convenience the device provides during use.

▪

Range of motion: the exoskeleton should be capable of assisting users in performing a wide range of movements and exercises that are relevant to cardiac rehabilitation. Metrics to evaluate the range of motion include the device’s ability to assist with movements of the upper limb such different body types and sizes.

▪

Strength and endurance: the exoskeleton should be capable of providing sufficient resistance and support to help users build strength and endurance as they progress through their rehabilitation program. Metrics to evaluate strength and endurance might include the amount of weight the device can support, the duration and intensity of exercise sessions, and the device’s ability to track and monitor progress over time.

▪

Biomechanics: the exoskeleton should be designed to optimize the user’s biomechanics and movement patterns to promote efficient and effective rehabilitation. Metrics to evaluate biomechanics include the device’s ability to adjust to the user’s gait, posture, and other movement characteristics, as well as its ability to provide feedback and guidance to help users maintain proper form and technique (this performance parameter will be evaluated in the medium term in another article).

The information generated in the first step was then used to guide the design of the CardioVR-ReTone exoskeleton, which was made using CATIA software, and to initiate the selection of the appropriate components and technologies for the exoskeleton. The result is highlighted in Figure 2—virtual prototype of the robotic exoskeleton for the upper limb. CardioVR-ReTone is a fixed exoskeleton with two symmetrical arms, each with 6 degrees of freedom (DoFs), which includes only rotational joints. The symmetric design of CardioVR-ReTone facilitates motions that are beneficial to the user’s upper extremities while reducing the risk of injuries.

The shoulder mechanism was designed as a rotational joint to create a highly versatile and adaptable exoskeleton system that can meet the needs of patients undergoing early cardiac rehabilitation. This design approach allowed for a wide range of movement and flexibility, while also providing the stability and support needed for safe and effective movements. The joint geometry is another important factor in the design. The joint geometry determines the range of motion and the freedom of movement of the shoulder mechanism, and it is crucial to consider the design of the pivot point, the orientation of the joint axes, and the positioning of the actuators. The J1 rotation joint (Figure 2) simulates shoulder elevation and flexion, whereas the J2–J3–J4 triangle replicates shoulder protraction and retraction. By placing the joints in a triangle, the kinematic differences and joint misalignments that would occur if only one rotation joint were utilized for protraction and retraction on the rear of the shoulder are eliminated. Mechanical linkages also play an important role in connecting the actuators to the joints and providing stability and support for the exoskeleton. These linkages can include brackets, gears, and other mechanical components.

The forearm mechanism comprises two rotational joints positioned at a 90-degree angle, which were selected to increase the range of motion while avoiding mechanical issues and hindrances in upper-arm movement. Joint J4, which is used in the shoulder mechanism, aligns with the vertical Cartesian axis, while axis J5 aligns with the X-axis, as shown in Figure 2 and Figure 3.

The elbow mechanism consists of two revolute joints, one of which is powered by a motor (Figure 2, joint J6 with 2 joints) to facilitate the movement of flexion and extension in the elbow. This joint will be taken into account when constructing the kinematic and dynamic model. The CardioVR-ReTone exoskeleton’s two robotic arms are mounted on a frame that can be adjusted for height and width, allowing it to be customized to fit patients of different sizes (Figure 2).

Once the requirements and specifications have been established, and the design solution generated, the next step was to develop the kinematic and dynamic model of the exoskeleton. This mathematical representation of the system’s motion and behavior was used to predict how the exoskeleton will respond to different inputs and loads and was critical for the effective control and operation of the system. The kinematic and dynamic model has been carefully developed and validated to ensure that it accurately represents the system and provides the necessary information for the control system. For this, the matrix exponential algorithm was used. The CardioVR-ReTone robotic exoskeleton kinematic structure is highlighted in Figure 3.

2.2. CardioVR-ReTone Robotic Exoskeleton—Geometric, Kinematic, and Dynamic Modelling

In order to mathematically model a mechanical system having n degrees of freedom (DoFs), a series of dedicated algorithms were developed and discussed in the specialized literature, usually involving an impressive volume of matrix or differential calculations. The development of mathematical formalisms, such as algorithms, allows a detailed analysis, in numerical and/or graphic form, regarding the geometry, kinematics, and dynamics of an analyzed structure, regardless of its type and complexity. Unlike the classical algorithms for modelling the geometry, kinematics, and dynamics of a system, the application of matrix exponential functions (ME) [36] has some advantages, such as precision, flexibility, ease of implementation, and efficiency, in terms of determining the equations’ mathematical model [37].

2.2.1. Geometrical Modelling of the Exoskeleton Robotic System Using Matrix Exponential Algorithm (MEG)

In this section, based on functional analysis, the CardioVR-ReTone exoskeleton—6R robot structure—will be geometrically modelled using an algorithm based on matrix calculation with exponential functions [36]. The entire mathematical formalism highlighted below will be developed for one (6R robot) of the arms of the CardioVR-ReTone exoskeleton. The algorithm serves as the foundation for two essential tasks: first, establishing consistent transformations in the geometric model, and second, determining the Jacobian matrix and its time derivative. The matrix exponential algorithm will be applied to compute the exponentials of the location matrices between the different systems $\left\{0\right\}\to \left\{7\right\}$. The matrices are characterizing the position and orientation (situation) of the end effector in relation to the system $\left\{0\right\}$ attached to the fixed base of the robot. Thus, according to the matrix exponential in geometry algorithm (abbreviated MEG), to establish the end-effector position and orientation further are described the steps, where MEGp ($p=1\to 8$) represents a personal notation expressing the phases of the algorithm to be perform for geometrical modeling. Hence, the steps are:

MEG1. There is introduced the syntagma “matrix of nominal geometry” noted $M_{vn}^{\left(0\right)}$. The “matrix of nominal geometry”, having the dimension $(n+1)\times 6$, contains the input data corresponding to the initial configuration of the robot (presented in Figure 3 and

Table 1. The geometrical characteristics of $\mathrm{CardioVR}-\mathrm{ReTone}6\mathrm{R}\mathrm{robot}$.

Joint $\left(\mathit{i}=1\to 6\right)$	Joint Type $\left\{\mathit{R};\mathit{T}\right\}$		${\overline{\mathit{k}}}_{\mathit{i}}^{\left(0\right)\mathit{T}}$			${\overline{\mathit{p}}}_{\mathit{i}}^{\left(0\right)\mathit{T}}$			${\overline{\mathit{V}}}_{\mathit{i}}^{\mathit{T}}$
			${\mathit{k}}_{\mathit{i}\mathit{x}}^{\left(0\right)\mathit{T}}$	${\mathit{k}}_{\mathit{i}\mathit{y}}^{\left(0\right)\mathit{T}}$	${\mathit{k}}_{\mathit{i}\mathit{z}}^{\left(0\right)\mathit{T}}$	${\mathit{x}}_{\mathit{i}}^{\left(0\right)\mathit{T}}$	${\mathit{y}}_{\mathit{i}}^{\left(0\right)\mathit{T}}$	${\mathit{z}}_{\mathit{i}}^{\left(0\right)\mathit{T}}$
1	$R$		0	1	0	$0$	$l_0$	$l_1$	$-l_1$	$0$	$0$
2	$R$		0	0	1	$0$	$-l_2$	$l_3$	$-l_2$	$0$	$0$
3	$R$		0	0	1	$l_4$	$-l_5$	$0$	$-l_5$	$-l_4$	$0$
4	$R$		0	0	1	$0$	$l_6$	$0$	$l_6$	$0$	$0$
5	$R$		1	0	0	$l_7$	$0$	$-l_8$	$0$	$-l_8$	$0$
6	$R$		1	0	0	$-l_9$	$0$	$-l_{10}$	$0$	$-l_{10}$	$0$
7	-		1	0	0	$0$	$l_{11}$	$0$	-	-	-

), and according to [37], it is:

$$M_{vn}^{\left(0\right)}={\left\{\left[\begin{array}{cc}{\overline{k}}_i^{\left(0\right)T}& {\overline{p}}_i^{\left(0\right)T}\end{array}\right] i=1\to \left(n+1\right)\right\}}^T$$

(1)

where

• ${\overline{p}}_i^{\left(0\right)}$ represents the position vector of the geometric center of each joint with respect to the $\left\{O_0\right\}$ reference frame and in the initial configuration.
• ${\overline{k}}_i^{\left(0\right)}$ is a column vector corresponding to each driving joint with respect to the same $\left\{O_0\right\}$ reference frame in the initial configuration. In $M_{vn}^{\left(0\right)}$, the cell corresponding to the rotation around the axis $\left\{x_i,y_i,z_i\right\}$ of the considered joint ($i=1\to 6$) is completed with 1 as: ${\overline{k}}_i^{\left(0\right)}=\left\{\begin{array}{c}{\left(\begin{array}{ccc}1& 0& 0\end{array}\right)}^T\to {\overline{x}}_i\\ {\left(\begin{array}{ccc}0& 1& 0\end{array}\right)}^T\to {\overline{y}}_i\\ {\left(\begin{array}{ccc}0& 0& 1\end{array}\right)}^T\to {\overline{z}}_i\end{array}\right\}$
• $(n+1)$ represents the characteristic point TCP (tool center point), having attached the reference system $\left\{Pnsa\right\}$ [37].

MEG2. The data contained in $M_{vn}^{\left(0\right)}$ can be included in a table, having the following form (see the highlighted columns 3 and 4):

where

• ${\Delta }_i=\left(\left\{1, if i=R\right\} and \left\{0, if i=T\right\}\right)$ is an operator that highlights the type of driving joint in mathematical modelling.
• The parameters contained in the column vector ${\overline{v}}_i^T={\left[{\Delta }_i\cdot \left\{{\overline{p}}_i^{\left(0\right)}\times \right\}\cdot {\overline{k}}_i^{\left(0\right)}+\left(1-{\Delta }_i\right)\cdot {\overline{k}}_i^{\left(0\right)}\right]}^T$ are named homogeneous coordinates.
• $\left\{{\overline{p}}_i^{\left(0\right)}\times \right\}$ represents the antisymmetric matrix attached to the vector ${\overline{p}}_i^{\left(0\right)}$.

MEG3. An outward loop is opened for $\left(i=1\to n\right)$, and the next steps will be completed.

MEG4. The matrix $A_i$ is established, considering [38] as:

$$A_i=\left\{\left[\begin{array}{cc}\left\{{\overline{k}}_i^{\left(0\right)}\times \right\}\cdot {\Delta }_i& {\overline{v}}_i^{\left(0\right)}\\ \begin{array}{ccc}0& 0& 0\end{array}& 0\end{array}\right]\right\}\equiv \left[\begin{array}{cc}\left\{{\overline{k}}_i^{\left(0\right)}\times \right\}\cdot {\Delta }_i& \left\{{\overline{p}}_i^{\left(0\right)}\times \right\}\cdot {\overline{k}}_i^{\left(0\right)}\cdot {\Delta }_i+\left(1-{\Delta }_i\right)\cdot {\overline{k}}_i^{\left(0\right)}\\ \begin{array}{ccc}0& 0& 0\end{array}& 0\end{array}\right]$$

(2)

where $\left\{{\overline{k}}_i^{\left(0\right)}\times \right\}$ represents the antisymmetric matrix attached to the vector ${\overline{k}}_i^{\left(0\right)}$.

MEG5. The term $R\left({\overline{k}}_i ; q_i\cdot {\Delta }_i\right)$ is introduced, denoting “exponential of rotation matrix”, as:

$$\begin{array}{l}\left\{R\left({\overline{k}}_i ; q_i\cdot {\Delta }_i\right)\equiv e^{\left\{{\overline{k}}_i^{\left(0\right)}\times \right\} \cdot q_i\cdot {\Delta }_i}\equiv {}_i{}^{i-1}\left[R\right]=R_{ii-1}^{}\right\}=\\ =\left\{\begin{array}{c}\left\{ I_3+ \left\{ {\overline{k}}_i^{\left(0\right)}\times \right\}\cdot \sin \left(q_i\cdot {\Delta }_i\right)+ {\left\{ {\overline{k}}_i^{\left(0\right)}\times \right\}}^2\cdot \left[1- \cos \left(q_i\cdot {\Delta }_i\right)\right]\right\}\\ \left\{ I_3\cdot \cos \left(q_i\cdot {\Delta }_i\right)+ \left\{ {\overline{k}}_i^{\left(0\right)}\times \right\}\cdot \sin \left(q_i\cdot {\Delta }_i\right)+ {\overline{k}}_i^{\left(0\right)}\cdot {\overline{k}}_i^{\left(0\right)T}\left[1- \cos \left(q_i\cdot {\Delta }_i\right)\right]\right\} \end{array}\right\}\end{array}$$

(3)

MEG6. A column vector ${\overline{b}}_i$ is defined, established with the following:

$$\left\{\begin{array}{c}{\overline{b}}_i=\left\{l_3\cdot \mathit{sin}{q}_i+\left\{k_i^{\left(0\right)}\times \right\}\cdot \left[1-\mathit{cos}\left(q_i\cdot {\Delta }_i\right)\right]+{\left\{k_i^{\left(0\right)}\times \right\}}^2\cdot \left[q_i-\mathit{sin}\left(q_i\cdot {\Delta }_i\right)\right]\right\}\cdot {\overline{v}}_i^{\left(0\right)}=\\ =\left\{l_3+\mathit{sin}{q}_i\cdot \left\{k_i^{\left(0\right)}\times \right\}\cdot \left[1-\mathit{cos}\left(q_i\cdot {\Delta }_i\right)\right]+{\overline{k}}_i^{\left(0\right)}\cdot {\overline{k}}_i^{\left(0\right)T}\cdot \left[q_i-\mathit{sin}\left(q_i\cdot {\Delta }_i\right)\right]\right\}\cdot {\overline{v}}_i^{\left(0\right)}\end{array}\right\}$$

(4)

MEG7. Another matrix exponential, having a great significance for locating transformation, shows as:

$$e^{A_i\cdot q_i}=\mathit{exp}\left\{\left[\begin{array}{cc}\left\{{\overline{k}}_i^{\left(0\right)}\times \right\}& {\overline{v}}_i^{\left(0\right)}\\ 000& 0\end{array}\right]\cdot q_i\right\}$$

(5)

or:

$$e^{A_i\cdot q_i}=\left[\begin{array}{cc}\mathit{exp}\left\{\left\{{\overline{k}}_i^{\left(0\right)}\times \right\}\cdot q_i\cdot {\Delta }_i\right\}& {\overline{b}}_i\\ \begin{array}{ccc}0& 0& 0\end{array}& 1\end{array}\right]$$

(6)

MEG8. The exponential expression that characterizes the locating matrices, for defining the position and orientation of $\left\{n\right\}$ and $\left\{n+1\right\}$ with respect to fixed frame $\left\{0\right\}$, is obtained as follows:

$$\left\{\begin{array}{c}{\displaystyle \prod }_{i=1}^n\left(e^{A_i\cdot q_i}\right)\cdot T_{x0}^{\left(0\right)}=\mathit{exp}\left\{{\displaystyle \sum }_{i=1}^n{A}_i\cdot q_i\right\}\cdot T_{x0}^{\left(0\right)}\\ \mathrm{where}{R}_{x0}=\mathit{exp}\left\{{\displaystyle \sum }_{i=1}^n\left\{{\overline{k}}_i^{\left(0\right)}\times \right\}\cdot q_i\cdot {\Delta }_i\right\}\cdot R_{x0}^{\left(0\right)};{\delta }_x=\left\{\left\{0;x=n\right\};\left\{1;x=n+1\right\}\right\}\\ \text{and }\overline{p}={\displaystyle \sum }_{i=1}^n\left\{\mathit{exp}\left\{{\displaystyle \sum }_{j=1}^n\left\{{\overline{k}}_j^{\left(0\right)}\times \right\}\cdot q_j\cdot {\Delta }_j\right\}\right\}\cdot {\overline{b}}_i+\mathit{exp}\left\{{\displaystyle \sum }_{i=1}^n\left\{{\overline{k}}_i^{\left(0\right)}\times \right\}\cdot q_i\cdot {\Delta }_i\right\}\cdot {\overline{p}}^{\left(0\right)}\cdot {\delta }_x\end{array}\right\}$$

(7)

As an important remark, the matrix exponential algorithm in geometry can be applied for any robot structure due to computational advantages. Further, on the basis of previous assumptions, the steps MEG1–MEG8 will be applied for each kinematic joint of the $\mathrm{CardioVR}-\mathrm{ReTone}6\mathrm{R}\mathrm{robot}$ presented in Figure 3, with the remark that the following notations will be used: $\mathrm{s}{q}_i=\sin q_i,$ and $\mathrm{c}{q}_i=\cos q_i$ for each $i=1\to n$ joint.

For the first kinematic joint $i=1$, the following are established:

$$A_1=\left[\begin{array}{cc}\left\{{\overline{k}}_1^{\left(0\right)}\times \right\}& {\overline{v}}_1^{\left(0\right)}\\ \begin{array}{ccc}0& 0& 0\end{array}& 0\end{array}\right]\equiv \left[\begin{array}{cccc}0& 0& 1& -l_1\\ 0& 0& 0& 0\\ -1& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]$$

(8)

$$\left\{\begin{array}{c}{e}^{\left\{{\overline{k}}_1^{\left(0\right)}\times \right\}\cdot q_1\cdot {\Delta }_1}\\ \mathit{exp}\left\{\left\{{\overline{k}}_1^{\left(0\right)}\times \right\}\cdot q_1\cdot {\Delta }_1\right\}\end{array}\right\}\equiv \left[\begin{array}{ccc}c{q}_1& 0& s{q}_1\\ 0& 1& 0\\ -s{q}_1& 0& c{q}_1\end{array}\right]$$

(9)

$$\left\{\begin{array}{c}{\overline{b}}_1\\ I_3\cdot q_1\cdot {\overline{v}}_1^{\left(0\right)}\end{array}\right\}=\left\{\begin{array}{c}\left\{I_3\cdot s{q}_1+\left\{{\overline{k}}_1^{\left(0\right)}\times \right\}\cdot \left(1-c{q}_1\right)+{\left\{{\overline{k}}_1^{\left(0\right)}\times \right\}}^2\cdot \left(q_1-s{q}_1\right)\right\}\cdot {\overline{v}}_1^{\left(0\right)}\equiv \\ \equiv \left\{I_3\cdot s{q}_1+\left\{{\overline{k}}_1^{\left(0\right)}\times \right\}\cdot \left(1-c{q}_1\right)+{\overline{k}}_1^{\left(0\right)}\cdot {\overline{k}}_1^{\left(0\right)T}\cdot \left(q_1-s{q}_1\right)\right\}\cdot {\overline{v}}_1^{\left(0\right)}\end{array}\right\}=\left[\begin{array}{c}-l_1\cdot s{q}_1\\ 0\\ -l_1\cdot \left(s{q}_1-1\right)\end{array}\right]$$

(10)

$$\left\{\begin{array}{c}{e}^{A_1\cdot {\Delta }_1}\\ \mathit{exp}\left(\left[\begin{array}{cc}\left\{{\overline{k}}_1^{\left(0\right)}\times \right\}& {\overline{v}}_1^{\left(0\right)}\\ \begin{array}{ccc}0& 0& 0\end{array}& 0\end{array}\right]\cdot q_1\right)\end{array}\right\}=\left[\begin{array}{cc}R\left({\overline{k}}_1^{\left(0\right)};q_1\right)& {\overline{b}}_1\\ \begin{array}{ccc}0& 0& 0\end{array}& 1\end{array}\right]=\left[\begin{array}{cccc}c{q}_1& 0& s{q}_1& -l_1\cdot s{q}_1\\ 0& 1& 0& 0\\ -s{q}_1& 0& c{q}_1& -l_1\cdot \left(c{q}_1-1\right)\\ 0& 0& 0& 1\end{array}\right]$$

(11)

According to Figure 3, for the second joint $\left(i=2\right)$, the following are determined:

$$A_2=\left[\begin{array}{cc}\left\{{\overline{k}}_2^{\left(0\right)}\times \right\}& {\overline{v}}_2^{\left(0\right)}\\ 000& 0\end{array}\right]\equiv \left[\begin{array}{cccc}0& -1& 0& -l_2\\ 1& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]$$

(12)

$$\left\{\begin{array}{c}{e}^{\left\{{\overline{k}}_2^{\left(0\right)}\times \right\}\cdot q_2\cdot {\Delta }_2}\\ \mathit{exp}\left\{\left\{{\overline{k}}_2^{\left(0\right)}\times \right\}\cdot q_2\cdot {\Delta }_2\right\}\end{array}\right\}=\left[\begin{array}{ccc}c{q}_2& -s{q}_2& 0\\ s{q}_2& c{q}_2& 0\\ 0& 0& 1\end{array}\right];\left\{\begin{array}{c}{\overline{b}}_2\\ l_3\cdot q_2\cdot {\overline{v}}_2^{\left(0\right)}\end{array}\right\}=\left[\begin{array}{c}-l_2\cdot s{q}_2\\ l_2\cdot \left(c{q}_2-1\right)\\ 0\end{array}\right]$$

(13)

$$\left\{\begin{array}{c}{e}^{A_2\cdot q_2}=\mathit{exp}\left\{A_2\cdot q_2\right\}\\ \mathit{exp}\left(\left[\begin{array}{cc}\left\{{\overline{k}}_2^{\left(0\right)}\times \right\}& {\overline{v}}_2^{\left(0\right)}\\ \begin{array}{ccc}0& 0& 0\end{array}& 0\end{array}\right]\cdot q_2\right)\end{array}\right\}=\left[\begin{array}{cc}R\left({\overline{k}}_2^{\left(0\right)};q_2\right)& {\overline{b}}_2\\ \begin{array}{ccc}0& 0& 0\end{array}& 1\end{array}\right]=\left[\begin{array}{cccc}c{q}_2& -s{q}_2& 0& -l_2\cdot s{q}_2\\ s{q}_2& c{q}_2& 0& l_2\cdot \left(c{q}_2-1\right)\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]$$

(14)

For the third joint, the following expressions are established:

$$A_3=\left[\begin{array}{cc}\left\{{\overline{k}}_3^{\left(0\right)}\times \right\}& {\overline{v}}_3^{\left(0\right)}\\ 000& 0\end{array}\right]\equiv \left[\begin{array}{cccc}0& -1& 0& -l_5\\ 1& 0& 0& -l_4\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]$$

(15)

$$\left\{\begin{array}{c}{e}^{\left\{{\overline{k}}_3^{\left(0\right)}\times \right\}\cdot q_3}\\ \mathit{exp}\left\{\left\{{\overline{k}}_3^{\left(0\right)}\times \right\}\cdot q_3\right\}\end{array}\right\}\equiv l_3+\left\{{\overline{k}}_3^{\left(0\right)}\times \right\}\cdot \mathrm{s}{q}_3+{\left\{{\overline{k}}_3^{\left(0\right)}\times \right\}}^2\cdot \left(1-\mathrm{c}{q}_3\right)=\left[\begin{array}{ccc}\mathrm{c}{q}_3& -s{q}_3& 0\\ \mathrm{s}{q}_3& \mathrm{c}{q}_3& 0\\ 0& 0& 1\end{array}\right]$$

(16)

$${\overline{b}}_3=\left\{\begin{array}{c}\left\{I_3\cdot q_3+\left\{{\overline{k}}_3^{\left(0\right)}\times \right\}\cdot \left(1-\mathrm{c}{q}_3\right)+{\left\{{\overline{k}}_3^{\left(0\right)}\times \right\}}^2\cdot \left(q_3-\mathrm{s}{q}_3\right)\right\}\cdot {\overline{V}}_3^{\left(0\right)}\equiv \\ \equiv \left\{I_3\cdot \mathrm{s}{q}_3+\left\{{\overline{k}}_3^{\left(0\right)}\times \right\}\cdot \left(1-\mathrm{c}{q}_3\right)+{\overline{k}}_3^{\left(0\right)}\cdot {\overline{k}}_3^{\left(0\right)T}\cdot \left(q_3-\mathrm{s}{q}_3\right)\right\}\cdot {\overline{V}}_3^{\left(0\right)}\end{array}\right\}=\left[\begin{array}{c}-l_4\cdot \left(\mathrm{c}{q}_3-1\right)-l_5\cdot \mathrm{s}{q}_3\\ l_5\cdot \left(\mathrm{c}{q}_3-1\right)-l_4\cdot \mathrm{s}{q}_3\\ 0\end{array}\right]$$

(17)

$$\left\{\begin{array}{c}{e}^{A_3\cdot q_3}=\mathit{exp}\left\{A_3\cdot q_3\right\}\\ \mathit{exp}\left(\left[\begin{array}{cc}\left\{{\overline{k}}_3^{\left(0\right)}\times \right\}& {\overline{v}}_3^{\left(0\right)}\\ \begin{array}{ccc}0& 0& 0\end{array}& 0\end{array}\right]\cdot q_3\right)\end{array}\right\}=\left[\begin{array}{cc}R\left({\overline{k}}_3^{\left(0\right)};q_3\right)& {\overline{b}}_3\\ \begin{array}{ccc}0& 0& 0\end{array}& 1\end{array}\right]=\left[\begin{array}{cccc}c{q}_3& -s{q}_3& 0& -l_4\cdot \left(c{q}_3-1\right)-l_5\cdot s{q}_3\\ s{q}_3& c{q}_3& 0& l_5\cdot \left(c{q}_3-1\right)-l_4\cdot s{q}_3\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]$$

(18)

For $\left(i=4\right)$, the following expressions are expressed:

$$A_4=\left[\begin{array}{cc}\left\{{\overline{k}}_4^{\left(0\right)}\times \right\}& {\overline{v}}_4^{\left(0\right)}\\ 000& 0\end{array}\right]\equiv \left[\begin{array}{cccc}0& -1& 0& l_6\\ 1& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]$$

(19)

$$\left\{\begin{array}{c}{e}^{\left\{{\overline{k}}_4^{\left(0\right)}\times \right\}\cdot q_4}\\ \mathit{exp}\left\{\left\{{\overline{k}}_4^{\left(0\right)}\times \right\}\cdot q_4\right\}\end{array}\right\}\equiv l_3+\left\{{\overline{k}}_4^{\left(0\right)}\times \right\}\cdot s{q}_4+{\left\{{\overline{k}}_4^{\left(0\right)}\times \right\}}^2\cdot \left(1-c{q}_4\right)=\left[\begin{array}{ccc}c{q}_4& -s{q}_4& 0\\ s{q}_4& c{q}_4& 0\\ 0& 0& 1\end{array}\right]$$

(20)

$${\overline{b}}_4=\left\{\begin{array}{c}\left\{I_3\cdot q_4+\left\{{\overline{k}}_4^{\left(0\right)}\times \right\}\cdot \left(1-c{q}_4\right)+{\left\{{\overline{k}}_4^{\left(0\right)}\times \right\}}^2\cdot \left(q_4-s{q}_4\right)\right\}\cdot {\overline{v}}_4^{\left(0\right)}\equiv \\ \equiv \left\{I_3\cdot s{q}_4+\left\{{\overline{k}}_4^{\left(0\right)}\times \right\}\cdot \left(1-c{q}_4\right)+{\overline{k}}_4^{\left(0\right)}\cdot {\overline{k}}_4^{\left(0\right)T}\cdot \left(q_4-s{q}_4\right)\right\}\cdot {\overline{v}}_4^{\left(0\right)}\end{array}\right\}=\left[\begin{array}{c}{l}_6\cdot s{q}_4\\ -l_6\cdot \left(c{q}_4-1\right)\\ 0\end{array}\right]$$

(21)

$$\left\{\begin{array}{c}{e}^{A_4\cdot q_4}=\mathit{exp}\left\{A_4\cdot q_4\right\}\\ \mathit{exp}\left(\left[\begin{array}{cc}\left\{{\overline{k}}_4^{\left(0\right)}\times \right\}& {\overline{v}}_4^{\left(0\right)}\\ \begin{array}{ccc}0& 0& 0\end{array}& 0\end{array}\right]\cdot q_4\right)\end{array}\right\}=\left[\begin{array}{cc}R\left({\overline{k}}_4^{\left(0\right)};q_4\right)& {\overline{b}}_4\\ \begin{array}{ccc}0& 0& 0\end{array}& 1\end{array}\right]=\left[\begin{array}{cccc}c{q}_4& -s{q}_4& 0& l_6\cdot s{q}_4\\ s{q}_4& c{q}_4& 0& -l_6\cdot \left(c{q}_4-1\right)\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]$$

(22)

For joint $\left(i=5\right)$, the following expressions are established:

$$A_5=\left[\begin{array}{cc}\left\{{\overline{k}}_5^{\left(0\right)}\times \right\}& {\overline{v}}_5^{\left(0\right)}\\ 000& 0\end{array}\right]\equiv \left[\begin{array}{cccc}0& 0& 0& 0\\ 0& 0& -1& -l_8\\ 0& 1& 0& 0\\ 0& 0& 0& 0\end{array}\right]$$

(23)

$$\left\{\begin{array}{c}{e}^{\left\{{\overline{k}}_5^{\left(0\right)}\times \right\}\cdot q_5}\\ \mathit{exp}\left\{\left\{{\overline{k}}_5^{\left(0\right)}\times \right\}\cdot q_5\right\}\end{array}\right\}\equiv l_3+\left\{{\overline{k}}_5^{\left(0\right)}\times \right\}\cdot s{q}_5+{\left\{{\overline{k}}_5^{\left(0\right)}\times \right\}}^2\cdot \left(1-c{q}_5\right)=\left[\begin{array}{ccc}1& 0& 0\\ 0& c{q}_5& -s{q}_5\\ 0& s{q}_5& c{q}_5\end{array}\right]$$

(24)

$${\overline{b}}_5=\left\{\begin{array}{c}\left\{I_3\cdot q_5+\left\{{\overline{k}}_5^{\left(0\right)}\times \right\}\cdot \left(1-c{q}_5\right)+{\left\{{\overline{k}}_5^{\left(0\right)}\times \right\}}^2\cdot \left(q_5-s{q}_5\right)\right\}\cdot {\overline{v}}_5^{\left(0\right)}\equiv \\ \equiv \left\{I_3\cdot s{q}_5+\left\{{\overline{k}}_5^{\left(0\right)}\times \right\}\cdot \left(1-c{q}_5\right)+{\overline{k}}_5^{\left(0\right)}\cdot {\overline{k}}_5^{\left(0\right)T}\cdot \left(q_5-s{q}_5\right)\right\}\cdot {\overline{v}}_5^{\left(0\right)}\end{array}\right\}=\left[\begin{array}{c}0\\ -l_8\cdot s{q}_5\\ l_8\cdot \left(c{q}_5-1\right)\end{array}\right]$$

(25)

$$\left\{\begin{array}{c}{e}^{A_5\cdot q_5}=\mathit{exp}\left\{A_5\cdot q_5\right\}\\ \mathit{exp}\left(\left[\begin{array}{cc}\left\{{\overline{k}}_5^{\left(0\right)}\times \right\}& {\overline{v}}_5^{\left(0\right)}\\ \begin{array}{ccc}0& 0& 0\end{array}& 0\end{array}\right]\cdot q_5\right)\end{array}\right\}=\left[\begin{array}{cc}R\left({\overline{k}}_5^{\left(0\right)};q_5\right)& {\overline{b}}_5\\ \begin{array}{ccc}0& 0& 0\end{array}& 1\end{array}\right]=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& c{q}_5& -s{q}_5& l_8\cdot s{q}_5\\ 0& s{q}_5& c{q}_5& -l_8\cdot \left(c{q}_5-1\right)\\ 0& 0& 0& 1\end{array}\right]$$

(26)

For the last joint $\left(i=6\right)$, according to the algorithm, the following are obtained:

$$A_6=\left[\begin{array}{cc}\left\{{\overline{k}}_6^{\left(0\right)}\times \right\}& {\overline{v}}_6^{\left(0\right)}\\ 000& 0\end{array}\right]\equiv \left[\begin{array}{cccc}0& 0& 0& 0\\ 0& 0& -1& -l_{10}\\ 0& 1& 0& 0\\ 0& 0& 0& 0\end{array}\right]$$

(27)

$$\left\{\begin{array}{c}{e}^{\left\{{\overline{k}}_6^{\left(0\right)}\times \right\}\cdot q_6}\\ \mathit{exp}\left\{\left\{{\overline{k}}_6^{\left(0\right)}\times \right\}\cdot q_6\right\}\end{array}\right\}\equiv l_3+\left\{{\overline{k}}_6^{\left(0\right)}\times \right\}\cdot s{q}_6+{\left\{{\overline{k}}_6^{\left(0\right)}\times \right\}}^2\cdot \left(1-c{q}_6\right)=\left[\begin{array}{ccc}1& 0& 0\\ 0& c{q}_6& -s{q}_6\\ 0& s{q}_6& c{q}_6\end{array}\right]$$

(28)

$${\overline{b}}_6=\left\{\begin{array}{c}\left\{I_3\cdot q_6+\left\{{\overline{k}}_6^{\left(0\right)}\times \right\}\cdot \left(1-c{q}_6\right)+{\left\{{\overline{k}}_6^{\left(0\right)}\times \right\}}^2\cdot \left(q_6-s{q}_6\right)\right\}\cdot {\overline{v}}_6^{\left(0\right)}\equiv \\ \equiv \left\{I_3\cdot s{q}_6+\left\{{\overline{k}}_6^{\left(0\right)}\times \right\}\cdot \left(1-c{q}_6\right)+{\overline{k}}_6^{\left(0\right)}\cdot {\overline{k}}_6^{\left(0\right)T}\cdot \left(q_6-s{q}_6\right)\right\}\cdot {\overline{v}}_6^{\left(0\right)}\end{array}\right\}=\left[\begin{array}{c}0\\ -l_{10}\cdot s{q}_6\\ l_{10}\cdot \left(c{q}_6-1\right)\end{array}\right]$$

(29)

$$\left\{\begin{array}{c}{e}^{A_6\cdot q_6}=\mathit{exp}\left\{A_6\cdot q_6\right\}\\ \mathit{exp}\left(\left[\begin{array}{cc}\left\{{\overline{k}}_6^{\left(0\right)}\times \right\}& {\overline{v}}_6^{\left(0\right)}\\ \begin{array}{ccc}0& 0& 0\end{array}& 0\end{array}\right]\cdot q_6\right)\end{array}\right\}=\left[\begin{array}{cc}R\left({\overline{k}}_6^{\left(0\right)};q_6\right)& {\overline{b}}_6\\ \begin{array}{ccc}0& 0& 0\end{array}& 1\end{array}\right]=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& c{q}_6& -s{q}_6& -l_{10}\cdot s{q}_6\\ 0& s{q}_6& c{q}_6& l_{10}\cdot \left(c{q}_6-1\right)\\ 0& 0& 0& 1\end{array}\right]$$

(30)

$$\left\{T_{70}^{\left(0\right)}\equiv \left[\begin{array}{cc}{R}_{70}^{\left(0\right)}& {\overline{p}}^{\left(0\right)}\\ 000& 1\end{array}\right]\right\}=\left[\begin{array}{cccc}\left(\begin{array}{c}{n}_x^{\left(0\right)}\\ n_y^{\left(0\right)}\\ n_z^{\left(0\right)}\end{array}\right)& \left(\begin{array}{c}{s}_x^{\left(0\right)}\\ s_y^{\left(0\right)}\\ s_z^{\left(0\right)}\end{array}\right)& \left(\begin{array}{c}{a}_x^{\left(0\right)}\\ a_y^{\left(0\right)}\\ a_z^{\left(0\right)}\end{array}\right)& \begin{array}{c}{p}_x^{\left(0\right)}\\ p_y^{\left(0\right)}\\ p_z^{\left(0\right)}\end{array}\\ 0& 0& 0& 1\end{array}\right]$$

(31)

In keeping with the algorithm, the homogeneous transformation matrix (see [38] or [39]), which expresses the position and orientation of the end effector, is:

$${}_7{}^0\left[T\right]=\left\{\begin{array}{c}{}_7{}^0\left[R\right]\\ \left[\begin{array}{ccc}\overline{n}& \overline{s}& \overline{a}\end{array}\right]\end{array}\right\}=\left[\begin{array}{ccc}\left(\begin{array}{c}{n}_x\\ n_y\\ n_z\end{array}\right)& \left(\begin{array}{c}{s}_x\\ s_y\\ s_z\end{array}\right)& \left(\begin{array}{c}{a}_x\\ a_y\\ a_{{}_z}\end{array}\right)\end{array}\right]=\left\{\exp {\left\{\left\{{\overline{k}}_6^{\left(0\right)}\times \right\}\cdot q_6\right\}}^{}\right\}\cdot R_{70}^{\left(0\right)}$$

(32)

$$\left\{\begin{array}{c}\overline{p}\equiv {\overline{b}}_1+\exp {\left\{\left\{{\overline{k}}_1^{\left(0\right)}\times \right\} q_1\right\}}^{}\cdot {\overline{b}}_2+{\displaystyle \prod _{i=1}^2\exp {\left\{{\left\{{\overline{k}}_i^{\left(0\right)}\times \right\}}^{}\cdot q_i\right\}}^{}\cdot {\overline{b}}_3+}\\ +{\displaystyle \prod _{i=1}^3\exp {\left\{{\left\{{\overline{k}}_i^{\left(0\right)}\times \right\}}^{}\cdot q_i\right\}}^{}\cdot {\overline{b}}_4+}{\displaystyle \prod _{i=1}^4\exp {\left\{{\left\{{\overline{k}}_i^{\left(0\right)}\times \right\}}^{}\cdot q_i\right\}}^{}\cdot {\overline{b}}_5+}{\displaystyle \prod _{i=1}^5\exp {\left\{{\left\{{\overline{k}}_i^{\left(0\right)}\times \right\}}^{}\cdot q_i\right\}}^{}\cdot {\overline{p}}^{\left(0\right)}}\end{array}\right\}$$

(33)

On the basis of (32) and (33), according to [37], the expression of the column vector of operational variables is obtained as:

$${}^0\overline{X}=\left[\begin{array}{c} \overline{p} \\ \overline{\theta } \end{array}\right]=\left[\begin{array}{c}{\left(\begin{array}{ccc}{p}_x& p_y& p_z\end{array}\right)}^T\\ {\left(\begin{array}{ccc}{\alpha }_x& {\beta }_y& {\gamma }_z\end{array}\right)}^{{}^T}\end{array}\right]$$

(34)

According to [39], to establish the orientation angles, ${\alpha }_x,{\beta }_y,{\gamma }_z$ for exact determination of the values, the trigonometric function $A\tan 2$ is used, defined by:

$$x=A\tan 2\left(\sin \alpha ; \cos \alpha \right)=\left\{\begin{array}{c}\left\{\alpha ;\left[\sin \alpha \ge 0;\cos \alpha >0\right] \right\};\\ \left\{\pi /2+\alpha ;\left[\sin \alpha >0;\cos \alpha <0\right] \right\}\\ \left\{\pi +\alpha ;\left[\sin \alpha <0;\cos \alpha <0\right] \right\};\\ \left\{-\pi /2+\alpha ;\left[\sin \alpha <0;\cos \alpha \ge 0\right] \right\}\end{array}\right\}$$

(35)

Hence, in keeping with (35), the results are:

$${\alpha }_x=\left\{\frac{\pi }{2},-\frac{\pi }{2}\right\},{\beta }_y=\left\{\pi ,0\right\},{\gamma }_z=\left\{\frac{\pi }{2},-\frac{\pi }{2}\right\}$$

(36)

Defined by (34), the column vector of the operational coordinate for $\mathrm{CardioVR}-\mathrm{ReTone}6\mathrm{R}\mathrm{robot}$ becomes:

$${}^0\overline{X}={\left[\begin{array}{cccc}{\overline{p}}_6& \raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$2$}\right.& \pi & \raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$2$}\right.\end{array}\right]}^T\mathrm{or}{}^0\overline{X}={\left[\begin{array}{cccc}{\overline{p}}_6& \raisebox{1ex}{$-\pi $}\!\left/ \!\raisebox{-1ex}{$2$}\right.& 0& \raisebox{1ex}{$-\pi $}\!\left/ \!\raisebox{-1ex}{$2$}\right.\end{array}\right]}^T$$

(37)

Equation (36) defines the direct geometric model of the 6R robotic exoskeleton under investigation. The parameters presented by Equation (36) are describing the position and orientation of the TCP, relative to the fixed reference system that is attached to the base of the robot.

2.2.2. Kinematical Modeling of Exoskeleton Robotic System Using Matrix Exponential Algorithm (MEK Algorithm)

One can use matrix exponential functions to express the matrix transfer equations for any mechanical structure within a kinematic chain [40,41,42]. To compute the Jacobian matrix, also known as the velocity transfer matrix, the matrix exponential in kinematics algorithm (abbreviated as MEK) is used. To accomplish this, the inputs used are including results derived from the matrix exponential in geometry algorithm.

Establishing the expression of the Jacobian matrix and its derivative

According to [38], the kinematic equations for any structure can be expressed by the Jacobian matrix and its time derivative, included in the following expression:

$$\left[\begin{array}{c}{}^0\dot{\overline{X}}\\ {}^0\ddot{\overline{X}}\end{array}\right]=\left[\begin{array}{l}{\left[\begin{array}{cc}{}^0\overline{V}{}_n^T& {}^0\overline{\omega }{}_n^T\end{array}\right]}^{ T}\\ {\left[\begin{array}{cc}{}^0\dot{\overline{V}}{}_n^T& {}^0\dot{\overline{\omega }}{}_n^T\end{array}\right]}^{ T}\end{array}\right]=\left[\begin{array}{c}\begin{array}{cc} \left[0\right]& {}^{0}J{}_n\left(\overline{\theta }\right)\end{array}\\ \begin{array}{cc}{}^{0}J{}_n\left(\overline{\theta }\right)& {}^0\dot{J}{}_n\left(\overline{\theta }\right)\end{array}\end{array}\right]\cdot \left[\begin{array}{c}\ddot{\overline{\theta }}\\ \dot{\overline{\theta }}\end{array}\right]$$

(38)

where

• ${}^0\overline{V}{}_n^T$ and ${}^0\overline{\omega }{}_n^T$ are representing the transpose of the linear and angular velocities’ column vector; ${}^0\dot{\overline{V}}{}_n^T$ and ${}^0\dot{\overline{\omega }}{}_n^T$ are representing the transpose of the linear and angular accelerations column vector for joint “n” with respect to {0} frame;
• ${}^{0}J{}_n\left(\overline{\theta }\right)$ and ${}^0\dot{J}{}_n\left(\overline{\theta }\right)$ are the Jacobian matrix and its time derivative with respect to {0} frame for each column vector:

  $$\underset{n\times 1}{\overline{\theta }}=\left(q_i, i=1\to n\right)$$

  (39)

In accordance with [22,23,38,43], in this article, the expressions necessary for establishing the Jacobian matrix and its derivative for any mechanical robot structure will be presented. These components are crucial for mathematical modeling in the advanced mechanics of robot systems. To establish the end-effector position and orientation further, the steps are described, where MEKr ($r=1\to 11$) represents a personal notation expressing the phases of the algorithm to be performed for kinematical modeling. Hence, the steps are:

MEK1. The general expression of the Jacobian matrix using matrix exponentials from the expression (38) is determined as:

$$\underset{\left(6\times 1\right)}{{}^{0}J{}_i}\equiv \left[ME\left\{J_{i1}\right\}\cdot ME\left\{J_{i2}\right\}\cdot ME\left\{J_{i3}\right\}\cdot M_{iV\omega }\right], i=1\to n$$

(40)

where $\left\{ME\left\{J_{iN}\right\}, N=1,2,3\right\}$, and $M_{iv\omega }$ are representing personal notations for matrix exponentials, defined in the next steps.

MEK2. The exponential matrix from the previous expression is determined by:

$$\underset{\left(6\times 6\right)}{ME}\left(J_{i1}\right)=\left[\begin{array}{cc}ME\left\{V_{i1}\right\}& \left[0\right]\\ \left[0\right]& ME\left\{V_{i1}\right\}\end{array}\right]$$

(41)

$$\underset{\left(6\times 6\right)}{ME}\left(J_{i1}\right)=\left[\begin{array}{cc}ME\left\{V_{i1}\right\}& \left[0\right]\\ \left[0\right]& ME\left\{V_{i1}\right\}\end{array}\right]$$

(42)

$$\underset{\left(6\times 6\right)}{ME}\left(J_{i1}\right)=\left[\begin{array}{cc}ME\left\{V_{i1}\right\}& \left[0\right]\\ \left[0\right]& ME\left\{V_{i1}\right\}\end{array}\right]$$

(43)

MEK3. The matrices previously presented in exponential form are:

$$\underset{\left(3\times 3\right)}{ME}\left(V_{i1}\right)=\exp \left\{{\displaystyle \sum _{j=0}^{i-1}\left\{{\overline{k}}_j^0\times \right\}\cdot q_j\cdot {\Delta }_j}\right\}$$

(44)

$$\underset{\left(3\times 6\right)}{ME}\left(V_{i2}\right)=\left[\begin{array}{cc}{I}_3& {\Delta }_i\cdot \left\{{\overline{k}}_i^{\left(0\right)}\times \right\}\end{array}\right]$$

(45)

$$\underset{\left\{6\times \left[9+3\cdot \left(n-i\right)\right]\right\}}{ME\left(V_{i3}\right)}=\left[\begin{array}{ccc}{l}_3& \left[0\right]& \left[0\right]\\ \left[0\right]& \left\{\begin{array}{l}\left[\exp \left\{{\displaystyle \sum _{m=1-1}^{k-1}\begin{array}{l}\left\{{\overline{k}}_m^{\left(0\right)}\times \right\}\cdot q_m\cdot {\delta }_m\cdot {\Delta }_m\\ \\ where: k=i\to 3\end{array}}\right\}\right]\\ and, {\delta }_m=\left\{\left\{0;m=i-1\right\};\left\{1;m\ge i\right\}\right\}\end{array}\right\}& \exp \left\{{\displaystyle \sum _{k=1}^n\left\{{\overline{k}}_k^{\left(0\right)}\times \right\}\cdot q_k\cdot {\Delta }_k}\right\}\end{array}\right]$$

(46)

MEK4. The matrix (41)–(43) from MEK1 are included in the expression:

$$\underset{\left\{9\times \left[12+3\cdot \left(n-i\right)\right]\right\}}{ME\left(J_i\right)}=ME\left\{J_{i1}\right\}\cdot ME\left\{J_{i2}\right\}\cdot ME\left\{J_{i3}\right\}$$

(47)

and represents the matrix exponential for the “i” joint, with respect to the $\left\{0\right\}$reference frame, which will be conducted to establish the Jacobian matrix (see MEK6).

MEK5. The vector $M{E}_{iv\omega }$ is determined, having the components:

$$\underset{\left\{\left[12+3\cdot \left(n-i\right)\times 1\right]\right\}}{M{E}_{iV\omega }}={\left[\begin{array}{cccc}{\overline{V}}_i^{\left(0\right)T}& {\left[\begin{array}{cc}{\overline{b}}_k;& k=i\to n\end{array}\right]}^T& {\overline{p}}_n^{\left(0\right)T}& {\Delta }_i\cdot {\overline{k}}_i^{\left(0\right)T}\end{array}\right]}^T$$

(48)

MEK6. The expression for the column $\left(i\right)$ of the Jacobian matrix is:

$$\underset{\left(6\times 1\right)}{{}^{0}J{}_i}\equiv ME\left\{J_{i1}\right\}\cdot ME\left\{J_{i2}\right\}\cdot ME\left\{J_{i3}\right\}\cdot M_{iV\omega }=ME\left\{{}^{0}J{}_i\right\}\left.M_{iV\omega }\right\}$$

(49)

where

$$ME\left\{J_{i1}\right\}\cdot ME\left\{J_{i2}\right\}\cdot ME\left\{J_{i3}\right\}=\left[\begin{array}{cc}ME\left\{V_{i1}\right\}& \left[0\right]\\ \left[0\right]& ME\left\{V_{i1}\right\}\end{array}\right]\cdot \left[\begin{array}{cc}ME\left\{V_{i2}\right\}& \left[0\right]\\ \left[0\right]& I_3\end{array}\right]\cdot \left[\begin{array}{cc}ME\left\{V_{i3}\right\}& \left[0\right]\\ \left[0\right]& I_3\end{array}\right]$$

MEK7. The time derivative of the Jacobian matrix is established as [37]:

$$\underset{\left(6\times 1\right)}{{}^0\dot{J}{}_i}=M\left\{ {\dot{J}}_{i1}\right\}\cdot M\left\{ {\dot{J}}_{i2}\right\}\cdot M\left\{ {\dot{J}}_{i3}\right\}\cdot M\left\{ {\dot{J}}_{i4}\right\}$$

(50)

where $\left\{M\left\{ {\dot{J}}_{iM}\right\}, M=1,2,3,4\right\}$ are representing personal notations for matrix exponential derivatives, defined in the next steps.

MEK8. The matrices from (50) are:

$$\underset{\left(6\times 18\right)}{M\left\{ {\dot{J}}_{i1}\right\}}=\left[\begin{array}{ccc} ME\left\{ {\dot{J}}_{i1}\right\}& ME\left\{ J_{i1}\right\}& ME\left\{ J_{i1}\right\} \end{array}\right]$$

(51)

$$\underset{\left(18\times 27\right)}{M\left\{ {\dot{J}}_{i2}\right\}}=\left[\begin{array}{ccc}ME\left\{J_{i2}\right\}& \left[0\right]& \left[0\right]\\ \left[0\right]& ME\left\{J_{i2}\right\}& \left[0\right]\\ \left[0\right]& \left[0\right]& ME\left\{J_{i2}\right\}\end{array}\right]$$

(52)

$$\underset{\left\{27\times 9\cdot \left[4+\left(n-i\right)\right]\right\}}{M\left\{ {\dot{J}}_{i3}\right\}}=\left[\begin{array}{ccc}ME\left\{J_{i3}\right\}& \left[0\right]& \left[0\right]\\ \left[0\right]& ME\left\{ {\dot{J}}_{i3}\right\}& \left[0\right]\\ \left[0\right]& \left[0\right]& ME\left\{J_{i3}\right\}\end{array}\right]$$

(53)

$$\underset{\left\{9\cdot \left[4+\left(n-i\right)\right]\times 1\right\}}{M\left\{{\dot{J}}_{i4}\right\}}=\left[\begin{array}{c}{M}_{iV\omega }\\ M_{iV\omega }^{*}\\ {\dot{M}}_{iV\omega }\end{array}\right]=\left[\begin{array}{c}\begin{array}{c}{\left[\begin{array}{cccc}{\overline{V}}_i^{\left(0\right)T}& {\left[{\overline{b}}_k;k=i\to n\right]}^T& {\overline{p}}^{\left(0\right)T}& {\left\{{\overline{k}}_i^{\left(0\right)}\cdot {\Delta }_i\right\}}^T\end{array}\right]}^T\end{array}\\ \begin{array}{c}{\left[\begin{array}{cccc}{\overline{V}}_i^{\left(0\right)T}& {\left[{\overline{b}}_k;k=i\to n\right]}^T& {\overline{p}}^{\left(0\right)T}& {\overline{0}}^T\end{array}\right]}^T\end{array}\\ {\left[\begin{array}{cccc}{\overline{0}}^T& {\left[{\dot{\overline{b}}}_k;k=i\to n\right]}^T& {\overline{0}}^T& {\overline{0}}^T\end{array}\right]}^T\end{array}\right]$$

(54)

where

$${\dot{\overline{b}}}_k^T=\left\{\begin{array}{c}\left\{ I_3+\left\{{\overline{k}}_k^{\left(0\right)}\times \right\} s \left(q_k\cdot {\Delta }_{ k}\right)+{\left\{{\overline{k}}_k^{\left(0\right)}\times \right\}}^2\left[1-c \left(q_k\cdot {\Delta }_{ k}\right)\right]\right\}\cdot {\overline{V}}_k^{\left(0\right)}\cdot {\dot{q}}_k\\ \left\{ \exp \left\{ \left\{ {\overline{k}}_k^{\left(0\right)}\times \right\} q_k\cdot {\Delta }_{ k}\right\} \right\} {\overline{V}}_k^{\left(0\right)}{\dot{q}}_k=\left\{R \left({\overline{k}}_k^{\left(0\right)};q_k\right)\cdot {\Delta }_{ k}+I_3\cdot \left(1-{\Delta }_k\right)\right\} \cdot {\overline{V}}_k^{\left(0\right)}\cdot {\dot{q}}_k\end{array}\right\}$$

(55)

MEK9. The time derivative matrix contained in (51)–(53) can be expressed as:

$$\underset{\left(6\times 6\right)}{ME\left\{ {\dot{J}}_{i1}\right\}}=\left[\begin{array}{cc}ME\left\{ {\dot{V}}_{i1}\right\}& \left[0\right]\\ \left[0\right]& ME\left\{ {\dot{V}}_{i1}\right\}\end{array}\right]$$

(56)

$$\underset{\left\{9\times 3\cdot \left[4+\left(n-i\right)\right]\right\}}{ME\left\{ {\dot{J}}_{i3}\right\}}=\left[\begin{array}{cc}ME\left\{ {\dot{V}}_{i3}\right\}& \left[0\right]\\ \left[0\right]& I_3\end{array}\right]$$

(57)

MEK10. The matrices (56) and (57) are determined as:

$$\underset{\left(3\times 3\right)}{ME\left\{ {\dot{V}}_{i1}\right\}}=\frac{d}{dt}\left\{\exp \left\{{\displaystyle \sum _{j=0}^{i-1}\left\{ {\overline{k}}_j^{\left(0\right)}\times \right\} q_j{\Delta }_j}\right\}\right\}$$

(58)

$$\underset{6\times 3\cdot \left[3+\left(n-i\right)\right]}{ME\left\{ {\dot{V}}_{i3}\right\}}=\left\{\left[\begin{array}{ccc}{I}_3& \left[0\right]& \left[0\right]\\ \left[0\right]& \left[\begin{array}{c}\frac{d}{dt}\left\{\exp \left\{{\displaystyle \sum _{m=i-1}^{k-1} \left\{ {\overline{k}}_m^{\left(0\right)}\times \right\} q_m{\delta }_m{\Delta }_m}\right\}\right\}\\ k=i\to n\end{array}\right]& \frac{d}{dt}\left\{\exp \left\{{\displaystyle \sum _{k=i}^n\left\{{\overline{ k}}_k^{\left(0\right)}\times \right\} q_k{\Delta }_k}\right\}\right\}\end{array}\right]\right\}$$

(59)

MEK11. According to the algorithm, the previous expressions will be applied for $i=1\to 6$ joints, hence resulting in each column of Jacobian matrix time derivative defined with (50):

$$\underset{\left(6\times 1\right)}{{}^0\dot{J}_i}=\left\{\begin{array}{l}\left[\begin{array}{cc}ME\left\{{\dot{V}}_{i1}\right\}& \left[0\right]\\ \left[0\right]& ME\left\{{\dot{V}}_{i1}\right\}\end{array}\right]\cdot \left[\begin{array}{cc}ME\left\{V_{i2}\right\}& \left[0\right]\\ \left[0\right]& I_3\end{array}\right]\cdot \left[\begin{array}{cc}ME\left\{V_{i3}\right\}& \left[0\right]\\ \left[0\right]& I_3\end{array}\right]\cdot M_{iv\omega }+\\ +\left[\begin{array}{cc}ME\left\{V_{i1}\right\}& \left[0\right]\\ \left[0\right]& ME\left\{V_{i1}\right\}\end{array}\right]\cdot \left[\begin{array}{cc}ME\left\{V_{i2}\right\}& \left[0\right]\\ \left[0\right]& I_3\end{array}\right]\cdot \left[\begin{array}{cc}ME\left\{{\dot{V}}_{i3}\right\}& \left[0\right]\\ \left[0\right]& I_3\end{array}\right]\cdot M_{iv\omega }^{*}+\\ +\left[\begin{array}{cc}ME\left\{V_{i1}\right\}& \left[0\right]\\ \left[0\right]& ME\left\{V_{i1}\right\}\end{array}\right]\cdot \left[\begin{array}{cc}ME\left\{V_{i2}\right\}& \left[0\right]\\ \left[0\right]& I_3\end{array}\right]\cdot \left[\begin{array}{cc}ME\left\{V_{i3}\right\}& \left[0\right]\\ \left[0\right]& I_3\end{array}\right]\cdot {\dot{M}}_{iv\omega }\end{array}\right\}$$

(60)

The previous expressions (49) and (60) are expressing the direct kinematic equations (linear and angular velocities). They complement the equations used in describing the end-effector movement of any type of mechanical structure, known as direct kinematics equations (characteristic to effector operating velocities and accelerations).

Further, for the CardioVR-ReTone robotic exoskeleton structure, the steps presented in MEK1–MEK11 will be applied.

For $i=1$, according to MEK2–MEK3, the following matrix expressions are obtained:

$$\underset{\left(3\times 3\right)}{ME}\left(V_{11}\right)=\left.\left\{\exp \left\{{\displaystyle \sum _{j=0}^{1-1}\left\{{\overline{k}}_1^{\left(0\right)}\times \right\}{q}_1\cdot {\Delta }_1}\right\}\equiv e^{\left[0\right]}\right.\right\}\equiv I_3$$

(61)

$$\underset{\left(6\times 6\right)}{ME}\left\{J_{11}\right\}=\left\{\left[\begin{array}{cc}ME\left\{V_{11}\right\}& \left[0\right]\\ \left[0\right]& ME\left\{V_{11}\right\}\end{array}\right]\right\}=\left[\begin{array}{cc}{I}_3& \left[0\right]\\ \left[0\right]& I_3\end{array}\right]$$

(62)

$$\underset{\left(3\times 6\right)}{ME}\left(V_{12}\right)=\left\{\left[\begin{array}{cc}{I}_3& {\Delta }_1\cdot \left\{{\overline{k}}_1^{\left(0\right)}\times \right\}\end{array}\right]\right\}=\left[\begin{array}{cc}{I}_3& \left[0_3\right]\end{array}\right]$$

(63)

$$\underset{\left(6\times 9\right)}{ME}\left(J_{12}\right)=\left\{\left[\begin{array}{cc}ME\left\{V_{12}\right\}\equiv \left[I_3\left\{{\overline{k}}_1^{\left(0\right)}\times \right\}\right]& \left[0\right]\\ \left[0\right]& I_3\end{array}\right]\right\}\equiv \left[\begin{array}{cc}{I}_3& \underset{3\times 6}{\left[0\right]}\\ \underset{3\times 6}{\left[0\right]}& \stackrel{}{I_3}\end{array}\right]$$

(64)

$$\begin{array}{c}\underset{\left(6\times 24\right)}{ME}\left(V_{13}\right)=\left[\begin{array}{ccc}{I}_3& \left[0\right]& \left[0\right]\\ \left[0\right]& \left\{\begin{array}{l}\left[\begin{array}{c}{\displaystyle \sum _{e^{m=1-1}}^{k-1}\left\{{\overline{k}}_m^{\left(0\right)}\times \right\}\cdot q_m\cdot {\delta }_m\cdot {\Delta }_m}\\ k=1\to 3\end{array}\right]\\ {\delta }_m=\left\{\left\{0;m=i-1\right\};\left\{1;m\ge i\right\}\right\}\end{array}\right\}& e^{{\displaystyle \sum _{k=1}^6\left\{{\overline{k}}_k^{\left(0\right)}\times \right\}\cdot q_{{}_k\cdot }{\Delta }_k}}\end{array}\right]=\\ =\left[\begin{array}{cc}{I}_6& \begin{array}{ccccc}\left[0\right]& \left[0\right]& \underset{}{\left[0\right]}& \left[0\right]& \left[0\right]\\ & & c{q}_1+c{q}_2+c{q}_3+c{q}_4+2& -s{q}_2-s{q}_3-s{q}_4& s{q}_1\\ I_3& I_3& s{q}_2+s{q}_3+s{q}_4& c{q}_2+c{q}_3+c{q}_4+c{q}_5+c{q}_6+1& -s{q}_5-s{q}_6\\ & & -s{q}_1& s{q}_5+s{q}_6& c{q}_1+c{q}_5+c{q}_6+3\end{array}\end{array}\right]\end{array}$$

(65)

$$\underset{\left(9\times 27\right)}{ME}\left\{J_{13}\right\}=\left\{\begin{array}{c}\left[\begin{array}{cc}ME\left\{V_{13}\right\}& \left[0\right]\\ \left[0\right]& I_3\end{array}\right]=ME\left\{\begin{array}{c}\exp \left\{{\displaystyle \sum _{m=0}^{k-1}\left\{{\overline{k}}_m^{\left(0\right)}x\right\}{q}_m\cdot {\delta }_m\cdot {\Delta }_m}\right\};\exp \left\{{\displaystyle \sum _{m=0}^{k-1}\left\{{\overline{k}}_m^{\left(0\right)}\times \right\}{q}_m\cdot {\Delta }_m}\right\}\\ {\delta }_m=\left\{\left\{0;m=0\right\};\left\{1;m\ge 1\right\}\right\}\end{array}\right\}\\ \left[\begin{array}{cccccc}{I}_3& & \left[0\right]& & \left[0\right]& \left[0\right]\\ \left[0\right]& I_3& I_3& I_3& \begin{array}{ccc}c{q}_1+c{q}_2+c{q}_3+c{q}_4+2& -s{q}_2-s{q}_3-s{q}_4& s{q}_1\\ s{q}_2+s{q}_3+s{q}_4& c{q}_2+c{q}_3+c{q}_4+c{q}_5+c{q}_6+1& -s{q}_5-s{q}_6\\ -s{q}_1& s{q}_5+s{q}_6& c{q}_1+c{q}_5+c{q}_6+3\end{array}& \left[0\right]\\ \left[0\right]& & \left[0\right]& & \left[0\right]& I_3\end{array}\right]\end{array}\right\}$$

(66)

The expressions contained in (62), (64), and (66) are characterizing the first-column Jacobian matrix. Through carrying out the calculations, the transfer matrix corresponding to the first column of the Jacobian matrix is obtained as:

$$\underset{\left(6\times 27\right)}{ME}\left(J_1\right)=\left\{\begin{array}{c}ME\left\{J_{11}\right\}\cdot ME\left\{J_{12}\right\}\cdot ME\left\{J_{13}\right\}\\ \left[\begin{array}{cc}ME\left\{V_{11}\right\}& \left[0\right]\\ \left[0\right]& ME\left\{V_{11}\right\}\end{array}\right]\cdot \left[\begin{array}{cc}ME\left\{V_{12}\right\}& \left[0\right]\\ \left[0\right]& I_3\end{array}\right]\cdot \left[\begin{array}{cc}ME\left\{V_{13}\right\}& \left[0\right]\\ \left[0\right]& I_3\end{array}\right]\end{array}\right\}=\left[\begin{array}{cc}{I}_{3_{{}_{{}_{{}_{{}_{{}_{{}_{{}_{}}}}}}}}}& \underset{}{\left[0\right]}\\ \underset{}{\left[0\right]}& I_3\end{array}\right]$$

(67)

The correspondent column vector from MEK5 is:

$$\underset{\left\{ 27 \times 1 \right\}}{M_{1V\omega }}={\left[\begin{array}{cccc} {\overline{V}}_1^{\left(0\right)T}& {\left[{\overline{b}}_k ; k=1\to 6\right]}^T& {\overline{p}}_6^{\left(0\right)T}& {\Delta }_1\cdot {\overline{k}}_1^{\left(0\right)T} \end{array}\right]}^T$$

(68)

By performing the matrix product between (67) and (68), the first column of the Jacobian matrix $\underset{\left(6\times 1\right)}{{}^{0}J{}_1}$ there is obtained as:

$$\left\{\begin{array}{c}\underset{\left(6\times 1\right)}{{}^{0}J_1}={\left[\begin{array}{cc}{}^{0}J_{1v}^T& {}^{0}J_{1\Omega }^T\end{array}\right]}^T=\left\{ME\left\{J_{11}\right\}\cdot ME\left\{J_{12}\right\}\cdot ME\left\{J_{13}\right\}\cdot M_{1v\omega }\equiv ME\left\{{}^{0}J_1\right\}\cdot M_{1v\omega }\right\}\equiv \\ \equiv \left[\begin{array}{cc}ME\left\{V_{11}\right\}& \left[0\right]\\ \left[0\right]& ME\left\{V_{11}\right\}\end{array}\right]\cdot \left[\begin{array}{cc}ME\left\{V_{12}\right\}& \left[0\right]\\ \left[0\right]& I_3\end{array}\right]\cdot \left[\begin{array}{cc}ME\left\{V_{13}\right\}& \left[0\right]\\ \left[0\right]& I_3\end{array}\right]\cdot \left[\begin{array}{c}{\overline{v}}_1^{\left(0\right)}\\ {\left[{\overline{b}}_k;k=1\to 6\right]}^T\\ {\overline{p}}_6^{\left(0\right)}\\ {\Delta }_1\cdot {\overline{k}}_1^{\left(0\right)}\end{array}\right]\end{array}\right\}$$

(69)

The time derivative of $\underset{\left(6\times 1\right)}{{}^{0}J{}_1}$ according to MEK7 is:

$$\underset{\left(6\times 1\right)}{{}^0\dot{J}{}_1}=M\left\{ {\dot{J}}_{11}\right\}\cdot M\left\{ {\dot{J}}_{12}\right\}\cdot M\left\{ {\dot{J}}_{13}\right\}\cdot M\left\{ {\dot{J}}_{14}\right\}$$

(70)

where

$$\underset{\left(6\times 18\right)}{M\left\{ {\dot{J}}_{11}\right\}}=\left[\begin{array}{ccc} ME\left\{ {\dot{J}}_{11}\right\}& ME\left\{ J_{11}\right\}& ME\left\{ J_{11}\right\} \end{array}\right]$$

(71)

$$\underset{\left(18\times 27\right)}{M\left\{ {\dot{J}}_{12}\right\}}=\left[\begin{array}{ccc}ME\left\{J_{12}\right\}& \left[0\right]& \left[0\right]\\ \left[0\right]& ME\left\{J_{12}\right\}& \left[0\right]\\ \left[0\right]& \left[0\right]& ME\left\{J_{12}\right\}\end{array}\right]$$

(72)

$$\underset{\left(27\times 81\right)}{M\left\{ {\dot{J}}_{13}\right\}}=\left[\begin{array}{ccc}ME\left\{J_{13}\right\}& \left[0\right]& \left[0\right]\\ \left[0\right]& ME\left\{ {\dot{J}}_{13}\right\}& \left[0\right]\\ \left[0\right]& \left[0\right]& ME\left\{J_{13}\right\}\end{array}\right]$$

(73)

$$\underset{\left(81\times 1\right)}{M\left\{{\dot{J}}_{14}\right\}}=\left[\begin{array}{c}{M}_{1v\omega }\\ M_{1v\omega }^{*}\\ {\dot{M}}_{1v\omega }\end{array}\right]=\left[\begin{array}{c}\begin{array}{c}{\left[\begin{array}{cccc}{\overline{v}}_1^{\left(0\right)T}& {\left[{\overline{b}}_k;k=1\to 6\right]}^T& {\overline{p}}_6^{\left(0\right)T}& {\left\{{\overline{k}}_1^{\left(0\right)}\cdot {\Delta }_1\right\}}^T\end{array}\right]}^T\end{array}\\ \begin{array}{c}{\left[\begin{array}{cccc}{\overline{v}}_1^{\left(0\right)T}& {\left[{\overline{b}}_k;k=1\to 6\right]}^T& {\overline{p}}_6^{\left(0\right)T}& {\overline{0}}^T\end{array}\right]}^T\end{array}\\ {\left[\begin{array}{cccc}{\overline{0}}^T& {\left[{\dot{\overline{b}}}_k;k=1\to 6\right]}^T& {\overline{0}}^T& {\overline{0}}^T\end{array}\right]}^T\end{array}\right]$$

(74)

In (71) there is:

$$\underset{\left(6\times 6\right)}{ME\left\{ {\dot{J}}_{11}\right\}}=\left[\begin{array}{cc}ME\left\{ {\dot{V}}_{11}\right\}& \left[0\right]\\ \left[0\right]& ME\left\{ {\dot{V}}_{11}\right\}\end{array}\right]$$

(75)

where

$$\underset{\left(3\times 3\right)}{ME\left\{ {\dot{V}}_{11}\right\}}=\frac{d}{dt}\left\{\exp \left\{{\displaystyle \sum _{j=0}^{1-1}\left\{ {\overline{k}}_j^{\left(0\right)}\times \right\} q_j\cdot {\Delta }_j}\right\}\right\}$$

(76)

In (73) is written:

$$\underset{9\times 27}{ME\left\{ {\dot{J}}_{13}\right\}}=\left[\begin{array}{cc}ME\left\{ {\dot{V}}_{13}\right\}& \left[0\right]\\ \left[0\right]& I_3\end{array}\right]$$

(77)

where

$$\underset{\left\{6\times 24\right\}}{ME\left\{ {\dot{V}}_{13}\right\}}=\left\{\left[\begin{array}{ccc}{I}_3& \left[0\right]& \left[0\right]\\ \left[0\right]& \left[\begin{array}{c}\frac{d}{dt}\left\{\exp \left\{{\displaystyle \sum _{m=1-1}^{k-1} \left\{ {\overline{k}}_m^{\left(0\right)}\times \right\} q_m{\delta }_m{\Delta }_m}\right\}\right\}\\ k=1\to 6\end{array}\right]& \frac{d}{dt}\left\{\exp \left\{{\displaystyle \sum _{k=i}^6\left\{{\overline{ k}}_k^{\left(0\right)}\times \right\} q_k{\Delta }_k}\right\}\right\}\end{array}\right]\right\}$$

(78)

Hence, the time derivative of the Jacobian matrix (70) for the first kinematic joint is:

$$\underset{\left(6\times 1\right)}{{}^0\dot{J}_1}=\left\{\begin{array}{l}\left[\begin{array}{cc}ME\left\{{\dot{V}}_{11}\right\}& \left[0\right]\\ \left[0\right]& ME\left\{{\dot{V}}_{11}\right\}\end{array}\right]\cdot \left[\begin{array}{cc}ME\left\{V_{12}\right\}& \left[0\right]\\ \left[0\right]& I_3\end{array}\right]\cdot \left[\begin{array}{cc}ME\left\{V_{13}\right\}& \left[0\right]\\ \left[0\right]& I_3\end{array}\right]\cdot M_{1v\omega }+\\ +\left[\begin{array}{cc}ME\left\{V_{11}\right\}& \left[0\right]\\ \left[0\right]& ME\left\{V_{11}\right\}\end{array}\right]\cdot \left[\begin{array}{cc}ME\left\{V_{12}\right\}& \left[0\right]\\ \left[0\right]& I_3\end{array}\right]\cdot \left[\begin{array}{cc}ME\left\{{\dot{V}}_{13}\right\}& \left[0\right]\\ \left[0\right]& I_3\end{array}\right]\cdot M_{1v\omega }^{*}+\\ +\left[\begin{array}{cc}ME\left\{V_{11}\right\}& \left[0\right]\\ \left[0\right]& ME\left\{V_{11}\right\}\end{array}\right]\cdot \left[\begin{array}{cc}ME\left\{V_{12}\right\}& \left[0\right]\\ \left[0\right]& I_3\end{array}\right]\cdot \left[\begin{array}{cc}ME\left\{V_{13}\right\}& \left[0\right]\\ \left[0\right]& I_3\end{array}\right]\cdot {\dot{M}}_{1v\omega }\end{array}\right\}$$

(79)

For $\left(i=2\right)$, the following will be established:

$$\underset{\left(3x3\right)}{ME}\left(V_{21}\right)=\left\{\exp \left\{{\displaystyle \sum _{j=0}^1{\left\{ {\overline{k}}_j^{\left(0\right)}\times \right\}}^{}\cdot q_j\cdot {\Delta }_j}\right\} \right\}=\left[\begin{array}{ccc}\mathrm{c}{q}_1& 0& \mathrm{s}{q}_1\\ 0& 1& 0\\ -\mathrm{s}{q}_1& 0& \mathrm{c}{q}_1\end{array}\right]$$

(80)

$$\begin{array}{l}\underset{(6\times 6)}{ME}\left\{J_{21}\right\}=\left[\begin{array}{cc}ME\left\{V_{21}\right\}& \left[0\right]\\ \left[0\right]& ME\left\{V_{21}\right\}\end{array}\right]\equiv \left[\begin{array}{cc}\exp \left\{\left\{{\overline{k}}_2^{\left(0\right)}\times \right\}\cdot q_2\cdot {\Delta }_2\right\}& \left[0\right]\\ \left[0\right]& \exp \left\{\left\{{\overline{k}}_2^{\left(0\right)}\times \right\}\cdot q_2\cdot {\Delta }_2\right\}\end{array}\right]=\\ \equiv \left[\begin{array}{cccccc}\mathrm{c}{q}_1& 0& \mathrm{s}{q}_1& 0& 0& 0\\ 0& 1& 0& 0& 0& 0\\ -\mathrm{s}{q}_1& 0& \mathrm{c}{q}_1& 0& 0& 0\\ 0& 0& 0& \mathrm{c}{q}_1& 0& \mathrm{s}{q}_1\\ 0& 0& 0& 0& 1& 0\\ 0& 0& 0& -\mathrm{s}{q}_1& 0& \mathrm{c}{q}_1\end{array}\right]\end{array}$$

(81)

$$\underset{\left(3\times 6\right)}{ME}\left(V_{22}\right)=\left\{\left[\begin{array}{cc} I_3& {\Delta }_2\cdot \left\{ {\overline{k}}_2^{\left(0\right)}\times \right\} \end{array}\right]\right\}\equiv \left[\begin{array}{cc} I_3& \left[0_3\right] \end{array}\right]$$

(82)

$$\underset{\left(6\times 9\right)}{ME}\left(J_{22}\right)=\left\{\left[\begin{array}{cc}ME\left\{V_{22}\right\}\equiv \left[I_3\left\{{\overline{k}}_2^{\left(0\right)}\times \right\}\right]& \left[0\right]\\ \left[0\right]& I_3\end{array}\right]\right\}\equiv \left[\begin{array}{cc}{I}_3& \underset{3\times 6}{\left[0\right]}\\ \underset{3\times 6}{\left[0\right]}& \stackrel{}{I_3}\end{array}\right]$$

(83)

$$\underset{\left(6\times 21\right)}{ME}\left(V_{23}\right)=\left[\begin{array}{ccc}{I}_3& \left[0\right]& \left[0\right]\\ \left[0\right]& \left\{\begin{array}{c}\left[\begin{array}{c}\exp \left\{{\displaystyle \sum _{m=1}^{k-1}\left\{{\overline{k}}_m^{\left(0\right)}\times \right\}{q}_m\cdot {\delta }_m\cdot {\Delta }_m}\right\}\\ k=2\to 3\end{array}\right]\\ {\delta }_m=\left\{\left\{0;m=1\right\};\left\{1;m\ge 2\right\}\right\}\end{array}\right\}& \exp \left\{{\displaystyle \sum _{k=2}^6\left\{{\overline{k}}_k^{\left(0\right)}\times \right\}{q}_k\cdot {\Delta }_k}\right\}\end{array}\right]=\left[\begin{array}{cc}{}^{{}^{{}^{{}^{}}}}{I}_3{}_{{}_{{}_{{}_{{}_{{}_{{}_{}}}}}}}& \underset{}{\left[0\right]}\\ {}^{{}^{{}^{{}^{}}}}{\left[0\right]}_{{}_{{}_{{}_{{}_{}}}}}& \begin{array}{cc}{I}_3& {}_3^2\left[R\right]\end{array}\end{array}\right]$$

(84)

$$\begin{array}{c}\underset{\left(9\times 24\right)}{ME}\left\{J_{23}\right\}=\left[\begin{array}{cc}ME\left\{V_{23}\right\}& \left[0\right]\\ \left[0\right]& I_3\end{array}\right]\equiv \\ \equiv ME\left\{\begin{array}{c}\exp \left\{{\displaystyle \sum _{m=1}^{k-1}\left\{ {\overline{k}}_m^{\left(0\right)}\times \right\} q_m\cdot {\delta }_m\cdot {\Delta }_m}\right\} ; \exp \left\{{\displaystyle \sum _{k=2}^6\left\{{\overline{k}}_k^{\left(0\right)}\times \right\} q_k\cdot {\Delta }_k}\right\}\\ \\ {\delta }_m = \left\{\left\{0; m=1\right\} ; \left\{1; m\ge 2\right\}\right\}\end{array}\right\}\end{array}$$

(85)

By performing the product between the relations (81), (83), and (85), the transfer matrix corresponding to the second column of Jacobian matrix is obtained as:

$$\underset{\left(6\times 24\right)}{ME}\left(J_2\right)=\left\{\begin{array}{c}ME\left\{J_{21}\right\}\cdot ME\left\{J_{22}\right\}\cdot ME\left\{J_{23}\right\}\\ \left[\begin{array}{cc}ME\left\{V_{21}\right\}& \left[0\right]\\ \left[0\right]& ME\left\{V_{21}\right\}\end{array}\right]\cdot \left[\begin{array}{cc}ME\left\{V_{22}\right\}& \left[0\right]\\ \left[0\right]& I_3\end{array}\right]\cdot \left[\begin{array}{cc}ME\left\{V_{23}\right\}& \left[0\right]\\ \left[0\right]& I_3\end{array}\right]\end{array}\right\}=\left[\begin{array}{cc}{I}_{3_{{}_{{}_{{}_{{}_{{}_{{}_{{}_{}}}}}}}}}& \underset{}{\left[0\right]}\\ \underset{}{\left[0\right]}& I_3\end{array}\right]$$

(86)

The procedure outlined in the step MEK5 is used to compute the column vector corresponding to the linear component of the Jacobian matrix. Consequently, by performing the computation, the column vector is obtained:

$$\underset{\left\{ 24 \times 1 \right\}}{M_{2v\omega }}={\left[\begin{array}{cccc} {\overline{v}}_2^{\left(0\right)T}& {\left[{\overline{b}}_k ; k=2\to 6\right]}^T& {\overline{p}}_6^{\left(0\right)T}& {\Delta }_2\cdot {\overline{k}}_2^{\left(0\right)T} \end{array}\right]}^T$$

(87)

By multiplying the matrix functions (84) and (85) as previously shown, the resulting second column of the Jacobian matrix $\underset{\left(6\times 1\right)}{{}^{0}J{}_2}$ can be obtained:

$$\left\{\begin{array}{c}\underset{\left(6\times 1\right)}{{}^{0}J_2}={\left[\begin{array}{cc}{}^{0}J_{2v}^T& {}^{0}J_{2\Omega }^T\end{array}\right]}^T=\left\{ME\left\{J_{21}\right\}\cdot ME\left\{J_{22}\right\}\cdot ME\left\{J_{23}\right\}\cdot M_{2v\omega }\equiv ME\left\{{}^{0}J_2\right\}\cdot M_{2v\omega }\right\}\equiv \\ \equiv \left[\begin{array}{cc}ME\left\{V_{21}\right\}& \left[0\right]\\ \left[0\right]& ME\left\{V_{21}\right\}\end{array}\right]\cdot \left[\begin{array}{cc}ME\left\{V_{22}\right\}& \left[0\right]\\ \left[0\right]& I_3\end{array}\right]\cdot \left[\begin{array}{cc}ME\left\{V_{23}\right\}& \left[0\right]\\ \left[0\right]& I_3\end{array}\right]\cdot \left[\begin{array}{c}{\overline{v}}_2^{\left(0\right)}\\ {\left[{\overline{b}}_k;k=2\to 6\right]}^T\\ {\overline{p}}_6^{\left(0\right)}\\ {\Delta }_2\cdot {\overline{k}}_2^{\left(0\right)}\end{array}\right]\end{array}\right\}$$

(88)

The time derivative is:

$$\underset{\left(6\times 1\right)}{{}^0\dot{J}{}_2}=M\left\{ {\dot{J}}_{21}\right\}\cdot M\left\{ {\dot{J}}_{22}\right\}\cdot M\left\{ {\dot{J}}_{23}\right\}\cdot M\left\{ {\dot{J}}_{24}\right\}$$

(89)

where

$$\underset{\left(6\times 18\right)}{M\left\{ {\dot{J}}_{21}\right\}}=\left[\begin{array}{ccc} ME\left\{ {\dot{J}}_{21}\right\}& ME\left\{ J_{21}\right\}& ME\left\{ J_{21}\right\} \end{array}\right]$$

(90)

$$\underset{\left(18\times 27\right)}{M\left\{ {\dot{J}}_{i2}\right\}}=\left[\begin{array}{ccc}ME\left\{J_{22}\right\}& \left[0\right]& \left[0\right]\\ \left[0\right]& ME\left\{J_{22}\right\}& \left[0\right]\\ \left[0\right]& \left[0\right]& ME\left\{J_{22}\right\}\end{array}\right]$$

(91)

$$\underset{\left\{27\times 72\right\}}{M\left\{ {\dot{J}}_{23}\right\}}=\left[\begin{array}{ccc}ME\left\{J_{23}\right\}& \left[0\right]& \left[0\right]\\ \left[0\right]& ME\left\{ {\dot{J}}_{23}\right\}& \left[0\right]\\ \left[0\right]& \left[0\right]& ME\left\{J_{23}\right\}\end{array}\right]$$

(92)

$$\underset{\left\{72\times 1\right\}}{M\left\{{\dot{J}}_{24}\right\}}=\left[\begin{array}{c}{M}_{2v\omega }\\ M_{2v\omega }^{*}\\ {\dot{M}}_{2v\omega }\end{array}\right]=\left[\begin{array}{c}\begin{array}{c}{\left[\begin{array}{cccc}{\overline{v}}_2^{\left(0\right)T}& {\left[{\overline{b}}_k;k=2\to 6\right]}^T& {\overline{p}}_6^{\left(0\right)T}& {\left\{{\overline{k}}_2^{\left(0\right)}\cdot {\Delta }_2\right\}}^T\end{array}\right]}^T\end{array}\\ \begin{array}{c}{\left[\begin{array}{cccc}{\overline{v}}_2^{\left(0\right)T}& {\left[{\overline{b}}_k;k=2\to 6\right]}^T& {\overline{p}}_6^{\left(0\right)T}& {\overline{0}}^T\end{array}\right]}^T\end{array}\\ {\left[\begin{array}{cccc}{\overline{0}}^T& {\left[{\dot{\overline{b}}}_k;k=2\to 6\right]}^T& {\overline{0}}^T& {\overline{0}}^T\end{array}\right]}^T\end{array}\right]$$

(93)

The matrix expressions from (90) and (92) are expressed as:

$$\underset{\left(6\times 6\right)}{ME\left\{ {\dot{J}}_{21}\right\}}=\left[\begin{array}{cc}ME\left\{ {\dot{V}}_{21}\right\}& \left[0\right]\\ \left[0\right]& ME\left\{ {\dot{V}}_{21}\right\}\end{array}\right]$$

(94)

$$\underset{\left\{9\times 24\right\}}{ME\left\{ {\dot{J}}_{23}\right\}}=\left[\begin{array}{cc}ME\left\{ {\dot{V}}_{23}\right\}& \left[0\right]\\ \left[0\right]& I_3\end{array}\right]$$

(95)

where

$$\underset{\left(3\times 3\right)}{ME\left\{ {\dot{V}}_{21}\right\}}=\frac{d}{dt}\left\{\exp \left\{{\displaystyle \sum _{j=0}^1\left\{ {\overline{k}}_j^{\left(0\right)}\times \right\} q_j\cdot {\Delta }_j}\right\}\right\}$$

(96)

$$\underset{6\times 21}{ME\left\{ {\dot{V}}_{23}\right\}}=\left\{\left[\begin{array}{ccc}{I}_3& \left[0\right]& \left[0\right]\\ \left[0\right]& \left[\begin{array}{c}\frac{d}{dt}\left\{\exp \left\{{\displaystyle \sum _{m=1}^{k-1} \left\{ {\overline{k}}_m^{\left(0\right)}\times \right\} q_m{\delta }_m{\Delta }_m}\right\}\right\}\\ k=2\to 6\end{array}\right]& \frac{d}{dt}\left\{\exp \left\{{\displaystyle \sum _{k=2}^6\left\{{\overline{ k}}_k^{\left(0\right)}\times \right\} q_k\cdot {\Delta }_k}\right\}\right\}\end{array}\right]\right\}$$

(97)

The time derivative of the Jacobian matrix defined by (50) is:

$$\underset{\left(6\times 1\right)}{{}^0\dot{J}_2}=\left\{\begin{array}{l}\left[\begin{array}{cc}ME\left\{{\dot{V}}_{21}\right\}& \left[0\right]\\ \left[0\right]& ME\left\{{\dot{V}}_{21}\right\}\end{array}\right]\cdot \left[\begin{array}{cc}ME\left\{V_{22}\right\}& \left[0\right]\\ \left[0\right]& I_3\end{array}\right]\cdot \left[\begin{array}{cc}ME\left\{V_{23}\right\}& \left[0\right]\\ \left[0\right]& I_3\end{array}\right]\cdot M_{2v\omega }+\\ +\left[\begin{array}{cc}ME\left\{V_{21}\right\}& \left[0\right]\\ \left[0\right]& ME\left\{V_{21}\right\}\end{array}\right]\cdot \left[\begin{array}{cc}ME\left\{V_{22}\right\}& \left[0\right]\\ \left[0\right]& I_3\end{array}\right]\cdot \left[\begin{array}{cc}ME\left\{{\dot{V}}_{23}\right\}& \left[0\right]\\ \left[0\right]& I_3\end{array}\right]\cdot M_{2v\omega }^{*}+\\ +\left[\begin{array}{cc}ME\left\{V_{21}\right\}& \left[0\right]\\ \left[0\right]& ME\left\{V_{V_{21}}\right\}\end{array}\right]\cdot \left[\begin{array}{cc}ME\left\{V_{22}\right\}& \left[0\right]\\ \left[0\right]& I_3\end{array}\right]\cdot \left[\begin{array}{cc}ME\left\{V_{23}\right\}& \left[0\right]\\ \left[0\right]& I_3\end{array}\right]\cdot {\dot{M}}_{2v\omega }\end{array}\right\}$$

(98)

For $i=3$, according to MEK2–MEK3, the following expressions will be established:

$$\underset{\left(3\times 3\right)}{ME}\left(V_{31}\right)=\left.\left\{\exp \left\{{\displaystyle \sum _{j=0}^2\left\{{\overline{k}}_j^{\left(0\right)}\times \right\}{q}_j\cdot {\Delta }_j}\right\}\right.\right\}$$

(99)

$$\underset{\left(6\times 6\right)}{ME}\left\{J_{31}\right\}=\left\{\left[\begin{array}{cc}ME\left\{V_{31}\right\}& \left[0\right]\\ \left[0\right]& ME\left\{V_{31}\right\}\end{array}\right]\right\}$$

(100)

$$\underset{\left(3\times 6\right)}{ME}\left(V_{32}\right)=\left\{\left[\begin{array}{cc}{I}_3& {\Delta }_3\cdot \left\{{\overline{k}}_3^{\left(0\right)}\times \right\}\end{array}\right]\right\}$$

(101)

$$\underset{\left(6\times 9\right)}{ME}\left(J_{32}\right)=\left\{\left[\begin{array}{cc}ME\left\{V_{32}\right\}& \left[0\right]\\ \left[0\right]& I_3\end{array}\right]\right\}\equiv \left[\begin{array}{cc}{I}_3& \underset{3\times 6}{\left[0\right]}\\ \underset{3\times 6}{\left[0\right]}& \stackrel{}{I_3}\end{array}\right]$$

(102)

$$\underset{\left(6\times 18\right)}{ME}\left(V_{33}\right)=\left[\begin{array}{ccc}{I}_3& \left[0\right]& \left[0\right]\\ \left[0\right]& \left\{\begin{array}{l}\left[\begin{array}{c}{\displaystyle \sum _{e^{m=1-1}}^{k-1}\left\{{\overline{k}}_m^{\left(0\right)}\times \right\}\cdot q_m\cdot {\delta }_m\cdot {\Delta }_m}\\ k=1\to 3\end{array}\right]\\ {\delta }_m=\left\{\left\{0;m=i-1\right\};\left\{1;m\ge i\right\}\right\}\end{array}\right\}& e^{{\displaystyle \sum _{k=1}^6\left\{{\overline{k}}_k^{\left(0\right)}\times \right\}\cdot q_{{}_k\cdot }{\Delta }_k}}\end{array}\right]$$

(103)

$$\underset{\left(9\times 21\right)}{ME}\left\{J_{33}\right\}=\left[\left[\begin{array}{cc}ME\left\{V_{33}\right\}& \left[0\right]\\ \left[0\right]& I_3\end{array}\right]=ME\left\{\begin{array}{c}\exp \left\{{\displaystyle \sum _{m=0}^{k-1}\left\{{\overline{k}}_m^{\left(0\right)}\times \right\}{q}_m\cdot {\delta }_m\cdot {\Delta }_m}\right\};\exp \left\{{\displaystyle \sum _{m=0}^{k-1}\left\{{\overline{k}}_m^{\left(0\right)}\times \right\}{q}_m\cdot {\Delta }_m}\right\}\\ {\delta }_m=\left\{\left\{0;m=0\right\};\left\{1;m\ge 1\right\}\right\}\end{array}\right\}\right]$$

(104)

By performing the product between the relations (100), (102) and (104) there is obtained the transfer matrix correspondent to the second column of Jacobian matrix, as:

$$\underset{\left(6\times 21\right)}{ME}\left(J_3\right)=\left\{\begin{array}{c}ME\left\{J_{31}\right\}\cdot ME\left\{J_{32}\right\}\cdot ME\left\{J_{33}\right\}\\ \left[\begin{array}{cc}ME\left\{V_{31}\right\}& \left[0\right]\\ \left[0\right]& ME\left\{V_{31}\right\}\end{array}\right]\cdot \left[\begin{array}{cc}ME\left\{V_{32}\right\}& \left[0\right]\\ \left[0\right]& I_3\end{array}\right]\cdot \left[\begin{array}{cc}ME\left\{V_{33}\right\}& \left[0\right]\\ \left[0\right]& I_3\end{array}\right]\end{array}\right\}$$

(105)

The step MEK5 describes the process for determining the column vector corresponding to the rotational Jacobian matrix component. Performing the calculation, the column vector is obtained:

$$\underset{\left\{ 21 \times 1 \right\}}{M_{3v\omega }}={\left[\begin{array}{cccc} {\overline{v}}_3^{\left(0\right)T}& {\left[{\overline{b}}_k ; k=3\to 6\right]}^T& {\overline{p}}_6^{\left(0\right)T}& {\Delta }_3\cdot {\overline{k}}_3^{\left(0\right)T} \end{array}\right]}^T$$

(106)

By multiplying the matrix functions (103) and (104), as shown earlier, the third column of the Jacobian matrix $\underset{\left(6\times 1\right)}{{}^{0}J{}_3}$ is obtained:

$$\left\{\begin{array}{c}\underset{\left(6\times 1\right)}{{}^{0}J_3}={\left[\begin{array}{cc}{}^{0}J_{3v}^T& {}^{0}J_{3\Omega }^T\end{array}\right]}^T=\left\{ME\left\{J_{31}\right\}\cdot ME\left\{J_{32}\right\}\cdot ME\left\{J_{33}\right\}\cdot M_{3v\omega }\equiv ME\left\{{}^{0}J_3\right\}\cdot M_{3v\omega }\right\}\equiv \\ \equiv \left[\begin{array}{cc}ME\left\{V_{31}\right\}& \left[0\right]\\ \left[0\right]& ME\left\{V_{31}\right\}\end{array}\right]\cdot \left[\begin{array}{cc}ME\left\{V_{32}\right\}& \left[0\right]\\ \left[0\right]& I_3\end{array}\right]\cdot \left[\begin{array}{cc}ME\left\{V_{33}\right\}& \left[0\right]\\ \left[0\right]& I_3\end{array}\right]\cdot \left[\begin{array}{c}{\overline{v}}_3^{\left(0\right)}\\ {\left[{\overline{b}}_k;k=3\to 6\right]}^T\\ {\overline{p}}_6^{\left(0\right)}\\ {\Delta }_3\cdot {\overline{k}}_3^{\left(0\right)}\end{array}\right]\end{array}\right\}$$

(107)

The time derivative of the Jacobian matrix described by MEK7 is:

$$\underset{\left(6\times 1\right)}{{}^0\dot{J}{}_3}=M\left\{ {\dot{J}}_{31}\right\}\cdot M\left\{ {\dot{J}}_{32}\right\}\cdot M\left\{ {\dot{J}}_{33}\right\}\cdot M\left\{ {\dot{J}}_{34}\right\}$$

(108)

where

$$\underset{\left(6\times 18\right)}{M\left\{ {\dot{J}}_{31}\right\}}=\left[\begin{array}{ccc} ME\left\{ {\dot{J}}_{31}\right\}& ME\left\{ J_{31}\right\}& ME\left\{ J_{31}\right\} \end{array}\right]$$

(109)

$$\underset{\left(18\times 27\right)}{M\left\{ {\dot{J}}_{32}\right\}}=\left[\begin{array}{ccc}ME\left\{J_{32}\right\}& \left[0\right]& \left[0\right]\\ \left[0\right]& ME\left\{J_{32}\right\}& \left[0\right]\\ \left[0\right]& \left[0\right]& ME\left\{J_{32}\right\}\end{array}\right]$$

(110)

$$\underset{\left(27\times 63\right)}{M\left\{ {\dot{J}}_{33}\right\}}=\left[\begin{array}{ccc}ME\left\{J_{33}\right\}& \left[0\right]& \left[0\right]\\ \left[0\right]& ME\left\{ {\dot{J}}_{33}\right\}& \left[0\right]\\ \left[0\right]& \left[0\right]& ME\left\{J_{33}\right\}\end{array}\right]$$

(111)

$$\underset{\left(63\times 1\right)}{M\left\{{\dot{J}}_{34}\right\}}=\left[\begin{array}{c}{M}_{3v\omega }\\ M_{3v\omega }^{*}\\ {\dot{M}}_{3v\omega }\end{array}\right]=\left[\begin{array}{c}\begin{array}{c}{\left[\begin{array}{cccc}{\overline{v}}_3^{\left(0\right)T}& {\left[{\overline{b}}_k;k=1\to 6\right]}^T& {\overline{p}}_6^{\left(0\right)T}& {\left\{{\overline{k}}_3^{\left(0\right)}\cdot {\Delta }_3\right\}}^T\end{array}\right]}^T\end{array}\\ \begin{array}{}\begin{array}{c}{\left[\begin{array}{cccc}{\overline{v}}_3^{\left(0\right)T}& {\left[{\overline{b}}_k;k=1\to 6\right]}^T& {\overline{p}}_6^{\left(0\right)T}& {\overline{0}}^T\end{array}\right]}^T\end{array}\end{array}\\ {\left[\begin{array}{cccc}{\overline{0}}^T& {\left[{\dot{\overline{b}}}_k;k=1\to 6\right]}^T& {\overline{0}}^T& {\overline{0}}^T\end{array}\right]}^T\end{array}\right]$$

(112)

In (108), there is:

$$\underset{\left(6\times 6\right)}{ME\left\{ {\dot{J}}_{31}\right\}}=\left[\begin{array}{cc}ME\left\{ {\dot{V}}_{31}\right\}& \left[0\right]\\ \left[0\right]& ME\left\{ {\dot{V}}_{31}\right\}\end{array}\right]$$

(113)

where

$$\underset{\left(3\times 3\right)}{ME\left\{ {\dot{V}}_{31}\right\}}=\frac{d}{dt}\left\{\exp \left\{{\displaystyle \sum _{j=0}^2\left\{ {\overline{k}}_j^{\left(0\right)}\times \right\} q_j\cdot {\Delta }_j}\right\}\right\}$$

(114)

In the relation (111), there can be expressed:

$$\underset{9\times 21}{ME\left\{ {\dot{J}}_{33}\right\}}=\left[\begin{array}{cc}ME\left\{ {\dot{V}}_{33}\right\}& \left[0\right]\\ \left[0\right]& I_3\end{array}\right]$$

(115)

where

$$\underset{\left\{6\times 18\right\}}{ME\left\{ {\dot{V}}_{33}\right\}}=\left\{\left[\begin{array}{ccc}{I}_3& \left[0\right]& \left[0\right]\\ \left[0\right]& \left[\begin{array}{c}\frac{d}{dt}\left\{\exp \left\{{\displaystyle \sum _{m=2}^{k-1} \left\{ {\overline{k}}_m^{\left(0\right)}\times \right\} q_m{\delta }_m{\Delta }_m}\right\}\right\}\\ k=1\to 6\end{array}\right]& \frac{d}{dt}\left\{\exp \left\{{\displaystyle \sum _{k=3}^6\left\{{\overline{ k}}_k^{\left(0\right)}\times \right\} q_k\cdot {\Delta }_k}\right\}\right\}\end{array}\right]\right\}$$

(116)

Hence, the time derivative of the Jacobian matrix for the third kinematic joint is:

$$\underset{\left(6\times 1\right)}{{}^0\dot{J}{}_3}=\left\{\begin{array}{l}\left[\begin{array}{cc}ME\left\{ {\dot{V}}_{31}\right\}& \left[0\right]\\ \left[0\right]& ME\left\{ {\dot{V}}_{31}\right\}\end{array}\right]\cdot \left[\begin{array}{cc}ME\left\{V_{32}\right\}& \left[0\right]\\ \left[0\right]& I_3\end{array}\right]\cdot \left[\begin{array}{cc}ME\left\{V_{33}\right\}& \left[0\right]\\ \left[0\right]& I_3\end{array}\right]\cdot M_{3v\omega }+\\ +\left[\begin{array}{cc}ME\left\{V_{31}\right\}& \left[0\right]\\ \left[0\right]& ME\left\{V_{31}\right\}\end{array}\right]\cdot \left[\begin{array}{cc}ME\left\{V_{32}\right\}& \left[0\right]\\ \left[0\right]& I_3\end{array}\right]\cdot \left[\begin{array}{cc}ME\left\{ {\dot{V}}_{33}\right\}& \left[0\right]\\ \left[0\right]& I_3\end{array}\right]\cdot M_{3v\omega }^{*}+\\ +\left[\begin{array}{cc}ME\left\{V_{31}\right\}& \left[0\right]\\ \left[0\right]& ME\left\{V_{31}\right\}\end{array}\right]\cdot \left[\begin{array}{cc}ME\left\{V_{32}\right\}& \left[0\right]\\ \left[0\right]& I_3\end{array}\right]\cdot \left[\begin{array}{cc}ME\left\{V_{33}\right\}& \left[0\right]\\ \left[0\right]& I_3\end{array}\right]\cdot {\dot{M}}_{3v\omega }\end{array}\right\}$$

(117)

For $i=4$, the following matrix expressions will be established:

$$\underset{\left(3\times 3\right)}{ME}\left(V_{41}\right)=\left.\left\{\exp \left\{{\displaystyle \sum _{j=0}^3\left\{{\overline{k}}_j^{\left(0\right)}\times \right\}{q}_j\cdot {\Delta }_j}\right\}\right.\right\}$$

(118)

$$\underset{\left(6\times 6\right)}{ME}\left\{J_{41}\right\}=\left\{\left[\begin{array}{cc}ME\left\{V_{41}\right\}& \left[0\right]\\ \left[0\right]& ME\left\{V_{41}\right\}\end{array}\right]\right\}$$

(119)

$$\underset{\left(3\times 6\right)}{ME}\left(V_{42}\right)=\left\{\left[\begin{array}{cc}{I}_3& {\Delta }_4\cdot \left\{{\overline{k}}_4^{\left(0\right)}\times \right\}\end{array}\right]\right\}$$

(120)

$$\underset{\left(6\times 9\right)}{ME}\left(J_{42}\right)=\left[\begin{array}{cc}ME\left\{V_{42}\right\}& \left[0\right]\\ \left[0\right]& I_3\end{array}\right]$$

(121)

$$\underset{\left(6\times 15\right)}{ME}\left(V_{43}\right)=\left[\begin{array}{ccc}{I}_3& \left[0\right]& \left[0\right]\\ \left[0\right]& \left\{\begin{array}{l}\left[\begin{array}{c}{\displaystyle \sum _{e^{m=1-1}}^{k-1}\left\{{\overline{k}}_m^{\left(0\right)}\times \right\}\cdot q_m\cdot {\delta }_m\cdot {\Delta }_m}\\ k=1\to 3\end{array}\right]\\ {\delta }_m=\left\{\left\{0;m=i-1\right\};\left\{1;m\ge i\right\}\right\}\end{array}\right\}& e^{{\displaystyle \sum _{k=1}^6\left\{{\overline{k}}_k^{\left(0\right)}\times \right\}\cdot q_{{}_k\cdot }{\Delta }_k}}\end{array}\right]$$

(122)

$$\underset{\left(9\times 18\right)}{ME}\left\{J_{43}\right\}=\left[\left[\begin{array}{cc}ME\left\{V_{43}\right\}& \left[0\right]\\ \left[0\right]& I_3\end{array}\right]=ME\left\{\begin{array}{c}\exp \left\{{\displaystyle \sum _{m=0}^{k-1}\left\{{\overline{k}}_m^{\left(0\right)}\times \right\}{q}_m\cdot {\delta }_m\cdot {\Delta }_m}\right\};\exp \left\{{\displaystyle \sum _{m=0}^{k-1}\left\{{\overline{k}}_m^{\left(0\right)}\times \right\}{q}_m\cdot {\Delta }_m}\right\}\\ {\delta }_m=\left\{\left\{0;m=0\right\};\left\{1;m\ge 1\right\}\right\}\end{array}\right\}\right]$$

(123)

The relations (119), (121) and (123) are used in establishing of forth column of the Jacobian matrix. By performing the product between the previous expressions, the transfer matrix corresponding to the fourth column of the Jacobian matrix is:

$$\underset{\left(6\times 18\right)}{ME}\left(J_4\right)=\left\{\begin{array}{c}ME\left\{J_{41}\right\}\cdot ME\left\{J_{42}\right\}\cdot ME\left\{J_{43}\right\}\\ \left[\begin{array}{cc}ME\left\{V_{41}\right\}& \left[0\right]\\ \left[0\right]& ME\left\{V_{41}\right\}\end{array}\right]\cdot \left[\begin{array}{cc}ME\left\{V_{42}\right\}& \left[0\right]\\ \left[0\right]& I_3\end{array}\right]\cdot \left[\begin{array}{cc}ME\left\{V_{43}\right\}& \left[0\right]\\ \left[0\right]& I_3\end{array}\right]\end{array}\right\}$$

(124)

According to MEK5, the following is obtained:

$$\underset{\left\{ 18 \times 1 \right\}}{M_{4v\omega }}={\left[\begin{array}{cccc} {\overline{v}}_4^{\left(0\right)T}& {\left[{\overline{b}}_k ; k=4\to 6\right]}^T& {\overline{p}}_6^{\left(0\right)T}& {\Delta }_4\cdot {\overline{k}}_4^{\left(0\right)T} \end{array}\right]}^T$$

(125)

Performing the calculus between (124) and (125), there is obtained:

$$\left\{\begin{array}{c}\underset{\left(6\times 1\right)}{{}^{0}J_4}=\left\{ME\left\{J_{41}\right\}\cdot ME\left\{J_{42}\right\}\cdot ME\left\{J_{43}\right\}\cdot M_{4v\omega }\equiv ME\left\{{}^{0}J_4\right\}\cdot M_{4v\omega }\right\}\equiv \\ \equiv \left[\begin{array}{cc}ME\left\{V_{41}\right\}& \left[0\right]\\ \left[0\right]& ME\left\{V_{41}\right\}\end{array}\right]\cdot \left[\begin{array}{cc}ME\left\{V_{42}\right\}& \left[0\right]\\ \left[0\right]& I_3\end{array}\right]\cdot \left[\begin{array}{cc}ME\left\{V_{43}\right\}& \left[0\right]\\ \left[0\right]& I_3\end{array}\right]\cdot \left[\begin{array}{c}{\overline{v}}_4^{\left(0\right)}\\ {\left[{\overline{b}}_k;k=4\to 6\right]}^T\\ {\overline{p}}_6^{\left(0\right)}\\ {\Delta }_4\cdot {\overline{k}}_4^{\left(0\right)}\end{array}\right]\end{array}\right\}$$

(126)

The time derivative for the fourth column of the Jacobian matrix is:

$$\underset{\left(6\times 1\right)}{{}^0\dot{J}{}_4}=M\left\{ {\dot{J}}_{41}\right\}\cdot M\left\{ {\dot{J}}_{42}\right\}\cdot M\left\{ {\dot{J}}_{43}\right\}\cdot M\left\{ {\dot{J}}_{44}\right\}$$

(127)

where

$$\underset{\left(6\times 18\right)}{M\left\{ {\dot{J}}_{41}\right\}}=\left[\begin{array}{ccc} ME\left\{ {\dot{J}}_{41}\right\}& ME\left\{ J_{41}\right\}& ME\left\{ J_{41}\right\} \end{array}\right]$$

(128)

$$\underset{\left(18\times 27\right)}{M\left\{ {\dot{J}}_{42}\right\}}=\left[\begin{array}{ccc}ME\left\{J_{42}\right\}& \left[0\right]& \left[0\right]\\ \left[0\right]& ME\left\{J_{42}\right\}& \left[0\right]\\ \left[0\right]& \left[0\right]& ME\left\{J_{42}\right\}\end{array}\right]$$

(129)

$$\underset{\left(27\times 54\right)}{M\left\{ {\dot{J}}_{43}\right\}}=\left[\begin{array}{ccc}ME\left\{J_{43}\right\}& \left[0\right]& \left[0\right]\\ \left[0\right]& ME\left\{ {\dot{J}}_{43}\right\}& \left[0\right]\\ \left[0\right]& \left[0\right]& ME\left\{J_{43}\right\}\end{array}\right]$$

(130)

$$\underset{\left(54\times 1\right)}{M\left\{{\dot{J}}_{44}\right\}}=\left[\begin{array}{c}{M}_{4v\omega }\\ M_{4v\omega }^{*}\\ {\dot{M}}_{4v\omega }\end{array}\right]=\left[\begin{array}{c}\begin{array}{c}{\left[\begin{array}{cccc}{\overline{v}}_4^{\left(0\right)T}& {\left[{\overline{b}}_k;k=4\to 6\right]}^T& {\overline{p}}_6^{\left(0\right)T}& {\left\{{\overline{k}}_4^{\left(0\right)}\cdot {\Delta }_4\right\}}^T\end{array}\right]}^T\end{array}\\ \begin{array}{c}{\left[\begin{array}{cccc}{\overline{v}}_4^{\left(0\right)T}& {\left[{\overline{b}}_k;k=4\to 6\right]}^T& {\overline{p}}_6^{\left(0\right)T}& {\overline{0}}^T\end{array}\right]}^T\end{array}\\ {\left[\begin{array}{cccc}{\overline{0}}^T& {\left[{\dot{\overline{b}}}_k;k=4\to 6\right]}^T& {\overline{0}}^T& {\overline{0}}^T\end{array}\right]}^T\end{array}\right]$$

(131)

In (128), there is:

$$\underset{\left(6\times 6\right)}{ME\left\{ {\dot{J}}_{41}\right\}}=\left[\begin{array}{cc}ME\left\{ {\dot{V}}_{41}\right\}& \left[0\right]\\ \left[0\right]& ME\left\{ {\dot{V}}_{41}\right\}\end{array}\right]$$

(132)

where

$$\underset{\left(3\times 3\right)}{ME\left\{ {\dot{V}}_{41}\right\}}=\frac{d}{dt}\left\{\exp \left\{{\displaystyle \sum _{j=0}^2\left\{ {\overline{k}}_j^{\left(0\right)}\times \right\} q_j\cdot {\Delta }_j}\right\}\right\}$$

(133)

In (130), it can be written:

$$\underset{9\times 18}{ME\left\{ {\dot{J}}_{43}\right\}}=\left[\begin{array}{cc}ME\left\{ {\dot{V}}_{43}\right\}& \left[0\right]\\ \left[0\right]& I_3\end{array}\right]$$

(134)

where

$$\underset{\left\{6\times 15\right\}}{ME\left\{ {\dot{V}}_{43}\right\}}=\left\{\left[\begin{array}{ccc}{I}_3& \left[0\right]& \left[0\right]\\ \left[0\right]& \left[\begin{array}{c}\frac{d}{dt}\left\{\exp \left\{{\displaystyle \sum _{m=1-1}^{k-1} \left\{ {\overline{k}}_m^{\left(0\right)}\times \right\} q_m{\delta }_m{\Delta }_m}\right\}\right\}\\ k=4\to 6\end{array}\right]& \frac{d}{dt}\left\{\exp \left\{{\displaystyle \sum _{k=4}^6\left\{{\overline{ k}}_k^{\left(0\right)}\times \right\} q_k{\Delta }_k}\right\}\right\}\end{array}\right]\right\}$$

(135)

The time derivative of the fourth column of the Jacobian matrix is:

$$\underset{\left(6\times 1\right)}{{}^0\dot{J}_4}=\left\{\begin{array}{l}\left[\begin{array}{cc}ME\left\{{\dot{V}}_{41}\right\}& \left[0\right]\\ \left[0\right]& ME\left\{{\dot{V}}_{41}\right\}\end{array}\right]\cdot \left[\begin{array}{cc}ME\left\{V_{42}\right\}& \left[0\right]\\ \left[0\right]& I_3\end{array}\right]\cdot \left[\begin{array}{cc}ME\left\{V_{43}\right\}& \left[0\right]\\ \left[0\right]& I_3\end{array}\right]\cdot M_{4v\omega }+\\ +\left[\begin{array}{cc}ME\left\{V_{41}\right\}& \left[0\right]\\ \left[0\right]& ME\left\{V_{41}\right\}\end{array}\right]\cdot \left[\begin{array}{cc}ME\left\{V_{42}\right\}& \left[0\right]\\ \left[0\right]& I_3\end{array}\right]\cdot \left[\begin{array}{cc}ME\left\{{\dot{V}}_{43}\right\}& \left[0\right]\\ \left[0\right]& I_3\end{array}\right]\cdot M_{4v\omega }^{*}+\\ +\left[\begin{array}{cc}ME\left\{V_{41}\right\}& \left[0\right]\\ \left[0\right]& ME\left\{V_{41}\right\}\end{array}\right]\cdot \left[\begin{array}{cc}ME\left\{V_{42}\right\}& \left[0\right]\\ \left[0\right]& I_3\end{array}\right]\cdot \left[\begin{array}{cc}ME\left\{V_{43}\right\}& \left[0\right]\\ \left[0\right]& I_3\end{array}\right]\cdot {\dot{M}}_{4v\omega }\end{array}\right\}$$

(136)

For $i=5$, according to the algorithm, the following are established:

$$\underset{\left(3\times 3\right)}{ME}\left(V_{51}\right)=\left.\left\{\exp \left\{{\displaystyle \sum _{j=0}^4\left\{{\overline{k}}_j^{\left(0\right)}\times \right\}{q}_j\cdot {\Delta }_j}\right\}\right.\right\}$$

(137)

$$\underset{\left(6\times 6\right)}{ME}\left\{J_{51}\right\}=\left\{\left[\begin{array}{cc}ME\left\{V_{51}\right\}& \left[0\right]\\ \left[0\right]& ME\left\{V_{51}\right\}\end{array}\right]\right\}$$

(138)

$$\underset{\left(3\times 6\right)}{ME}\left(V_{52}\right)=\left\{\left[\begin{array}{cc}{I}_3& {\Delta }_5\cdot \left\{{\overline{k}}_5^{\left(0\right)}\times \right\}\end{array}\right]\right\}$$

(139)

$$\underset{\left(6\times 9\right)}{ME}\left(J_{52}\right)=\left[\begin{array}{cc}ME\left\{V_{52}\right\}& \left[0\right]\\ \left[0\right]& I_3\end{array}\right]$$

(140)

$$\underset{\left(6\times 12\right)}{ME}\left(V_{53}\right)=\left[\begin{array}{ccc}{I}_3& \left[0\right]& \left[0\right]\\ \left[0\right]& \left\{\begin{array}{l}\left[\begin{array}{c}{\displaystyle \sum _{e^{m=1-1}}^{k-1}\left\{{\overline{k}}_m^{\left(0\right)}\times \right\}\cdot q_m\cdot {\delta }_m\cdot {\Delta }_m}\\ k=i\to 3\end{array}\right]\\ {\delta }_m=\left\{\left\{0;m=i-1\right\};\left\{1;m\ge i\right\}\right\}\end{array}\right\}& e^{{\displaystyle \sum _{k=1}^6\left\{{\overline{k}}_k^{\left(0\right)}\times \right\}\cdot q_{{}_k\cdot }{\Delta }_k}}\end{array}\right]$$

(141)

$$\underset{\left(9\times 15\right)}{ME}\left\{J_{53}\right\}=\left[\left[\begin{array}{cc}ME\left\{V_{53}\right\}& \left[0\right]\\ \left[0\right]& I_3\end{array}\right]=ME\left\{\begin{array}{c}\exp \left\{{\displaystyle \sum _{m=0}^{k-1}\left\{{\overline{k}}_m^{\left(0\right)}\times \right\}{q}_m\cdot {\delta }_m\cdot {\Delta }_m}\right\};\exp \left\{{\displaystyle \sum _{m=0}^{k-1}\left\{{\overline{k}}_m^{\left(0\right)}\times \right\}{q}_m\cdot {\Delta }_m}\right\}\\ {\delta }_m=\left\{\left\{0;m=0\right\};\left\{1;m\ge 1\right\}\right\}\end{array}\right\}\right]$$

(142)

The expressions (138)–(142) are used in establishing of fifth column of Jacobian matrix. According to MEK4, there is obtained:

$$\underset{\left(6\times 15\right)}{ME}\left(J_5\right)=\left\{\begin{array}{c}ME\left\{J_{51}\right\}\cdot ME\left\{J_{52}\right\}\cdot ME\left\{J_{53}\right\}\\ \left[\begin{array}{cc}ME\left\{V_{51}\right\}& \left[0\right]\\ \left[0\right]& ME\left\{V_{51}\right\}\end{array}\right]\cdot \left[\begin{array}{cc}ME\left\{V_{52}\right\}& \left[0\right]\\ \left[0\right]& I_3\end{array}\right]\cdot \left[\begin{array}{cc}ME\left\{V_{53}\right\}& \left[0\right]\\ \left[0\right]& I_3\end{array}\right]\end{array}\right\}$$

(143)

According to MEK5, it is found that:

$$\underset{\left\{ 15 \times 1 \right\}}{M_{5v\omega }}={\left[\begin{array}{cccc} {\overline{v}}_5^{\left(0\right)T}& {\left[{\overline{b}}_k ; k=5\to 6\right]}^T& {\overline{p}}_6^{\left(0\right)T}& {\Delta }_5\cdot {\overline{k}}_5^{\left(0\right)T} \end{array}\right]}^T$$

(144)

By performing the product between (143) and (144), the column $\underset{\left(6\times 1\right)}{{}^{0}J{}_5}$ is:

$$\left\{\begin{array}{c}\underset{\left(6\times 1\right)}{{}^{0}J_5}=\left\{ME\left\{J_{51}\right\}\cdot ME\left\{J_{52}\right\}\cdot ME\left\{J_{53}\right\}\cdot M_{5v\omega }\equiv ME\left\{{}^{0}J_5\right\}\cdot M_{5v\omega }\right\}\equiv \\ \equiv \left[\begin{array}{cc}ME\left\{V_{51}\right\}& \left[0\right]\\ \left[0\right]& ME\left\{V_{51}\right\}\end{array}\right]\cdot \left[\begin{array}{cc}ME\left\{V_{52}\right\}& \left[0\right]\\ \left[0\right]& I_3\end{array}\right]\cdot \left[\begin{array}{cc}ME\left\{V_{53}\right\}& \left[0\right]\\ \left[0\right]& I_3\end{array}\right]\cdot \left[\begin{array}{c}{\overline{v}}_5^{\left(0\right)}\\ {\left[{\overline{b}}_k;k=5\to 6\right]}^T\\ {\overline{p}}_6^{\left(0\right)}\\ {\Delta }_5\cdot {\overline{k}}_5^{\left(0\right)}\end{array}\right]\end{array}\right\}$$

(145)

According to MEK7, the time derivative corresponding to J = 5 is established as:

$$\underset{\left(6\times 1\right)}{{}^0\dot{J}{}_5}=M\left\{ {\dot{J}}_{51}\right\}\cdot M\left\{ {\dot{J}}_{52}\right\}\cdot M\left\{ {\dot{J}}_{53}\right\}\cdot M\left\{ {\dot{J}}_{54}\right\}$$

(146)

where

$$\underset{\left(6\times 18\right)}{M\left\{ {\dot{J}}_{51}\right\}}=\left[\begin{array}{ccc} ME\left\{ {\dot{J}}_{51}\right\}& ME\left\{ J_{51}\right\}& ME\left\{ J_{51}\right\} \end{array}\right]$$

(147)

$$\underset{\left(18\times 27\right)}{M\left\{ {\dot{J}}_{52}\right\}}=\left[\begin{array}{ccc}ME\left\{J_{52}\right\}& \left[0\right]& \left[0\right]\\ \left[0\right]& ME\left\{J_{52}\right\}& \left[0\right]\\ \left[0\right]& \left[0\right]& ME\left\{J_{52}\right\}\end{array}\right]$$

(148)

$$\underset{\left(27\times 45\right)}{M\left\{ {\dot{J}}_{53}\right\}}=\left[\begin{array}{ccc}ME\left\{J_{53}\right\}& \left[0\right]& \left[0\right]\\ \left[0\right]& ME\left\{ {\dot{J}}_{53}\right\}& \left[0\right]\\ \left[0\right]& \left[0\right]& ME\left\{J_{53}\right\}\end{array}\right]$$

(149)

$$\underset{\left(45\times 1\right)}{M\left\{{\dot{J}}_{54}\right\}}=\left[\begin{array}{c}{M}_{5v\omega }\\ M_{5v\omega }^{*}\\ {\dot{M}}_{5v\omega }\end{array}\right]=\left[\begin{array}{c}\begin{array}{c}{\left[\begin{array}{cccc}{\overline{v}}_5^{\left(0\right)T}& {\left[{\overline{b}}_k;k=5\to 6\right]}^T& {\overline{p}}_6^{\left(0\right)T}& {\left\{{\overline{k}}_5^{\left(0\right)}\cdot {\Delta }_5\right\}}^T\end{array}\right]}^T\end{array}\\ \begin{array}{c}{\left[\begin{array}{cccc}{\overline{v}}_5^{\left(0\right)T}& {\left[{\overline{b}}_k;k=5\to 6\right]}^T& {\overline{p}}_6^{\left(0\right)T}& {\overline{0}}^T\end{array}\right]}^T\end{array}\\ {\left[\begin{array}{cccc}{\overline{0}}^T& {\left[{\dot{\overline{b}}}_k;k=5\to 6\right]}^T& {\overline{0}}^T& {\overline{0}}^T\end{array}\right]}^T\end{array}\right]$$

(150)

In (147), it can be written:

$$\underset{\left(6\times 6\right)}{ME\left\{ {\dot{J}}_{51}\right\}}=\left[\begin{array}{cc}ME\left\{ {\dot{V}}_{51}\right\}& \left[0\right]\\ \left[0\right]& ME\left\{ {\dot{V}}_{51}\right\}\end{array}\right]$$

(151)

where

$$\underset{\left(3\times 3\right)}{ME\left\{ {\dot{V}}_{51}\right\}}=\frac{d}{dt}\left\{\exp \left\{{\displaystyle \sum _{j=0}^1\left\{ {\overline{k}}_j^{\left(0\right)}\times \right\} q_j\cdot {\Delta }_j}\right\}\right\}$$

(152)

In relation (149):

$$\underset{9\times 15}{ME\left\{ {\dot{J}}_{53}\right\}}=\left[\begin{array}{cc}ME\left\{ {\dot{V}}_{53}\right\}& \left[0\right]\\ \left[0\right]& I_3\end{array}\right]$$

(153)

where

$$\underset{\left\{6\times 15\right\}}{ME\left\{ {\dot{V}}_{53}\right\}}=\left\{\left[\begin{array}{ccc}{I}_3& \left[0\right]& \left[0\right]\\ \left[0\right]& \left[\begin{array}{c}\frac{d}{dt}\left\{\exp \left\{{\displaystyle \sum _{m=1-1}^{k-1} \left\{ {\overline{k}}_m^{\left(0\right)}\times \right\} q_m{\delta }_m{\Delta }_m}\right\}\right\}\\ k=5\to 6\end{array}\right]& \frac{d}{dt}\left\{\exp \left\{{\displaystyle \sum _{k=5}^6\left\{{\overline{ k}}_k^{\left(0\right)}\times \right\} q_k{\Delta }_k}\right\}\right\}\end{array}\right]\right\}$$

(154)

The time derivative for the fifth column of the Jacobian matrix is:

$$\underset{\left(6\times 1\right)}{{}^0\dot{J}_5}=\left\{\begin{array}{l}\left[\begin{array}{cc}ME\left\{{\dot{V}}_{51}\right\}& \left[0\right]\\ \left[0\right]& ME\left\{{\dot{V}}_{51}\right\}\end{array}\right]\cdot \left[\begin{array}{cc}ME\left\{V_{52}\right\}& \left[0\right]\\ \left[0\right]& I_3\end{array}\right]\cdot \left[\begin{array}{cc}ME\left\{V_{53}\right\}& \left[0\right]\\ \left[0\right]& I_3\end{array}\right]\cdot M_{5v\omega }+\\ +\left[\begin{array}{cc}ME\left\{V_{51}\right\}& \left[0\right]\\ \left[0\right]& ME\left\{V_{51}\right\}\end{array}\right]\cdot \left[\begin{array}{cc}ME\left\{V_{52}\right\}& \left[0\right]\\ \left[0\right]& I_3\end{array}\right]\cdot \left[\begin{array}{cc}ME\left\{{\dot{V}}_{53}\right\}& \left[0\right]\\ \left[0\right]& I_3\end{array}\right]\cdot M_{5v\omega }^{*}+\\ +\left[\begin{array}{cc}ME\left\{V_{51}\right\}& \left[0\right]\\ \left[0\right]& ME\left\{V_{51}\right\}\end{array}\right]\cdot \left[\begin{array}{cc}ME\left\{V_{52}\right\}& \left[0\right]\\ \left[0\right]& I_3\end{array}\right]\cdot \left[\begin{array}{cc}ME\left\{V_{53}\right\}& \left[0\right]\\ \left[0\right]& I_3\end{array}\right]\cdot {\dot{M}}_{5v\omega }\end{array}\right\}$$

(155)

For the last kinematic joint, $i=6$, the following expressions are established:

$$\underset{\left(3\times 3\right)}{ME}\left(V_{61}\right)=\left.\left\{\exp \left\{{\displaystyle \sum _{j=0}^4\left\{{\overline{k}}_j^{\left(0\right)}\times \right\}{q}_j\cdot {\Delta }_j}\right\}\right.\right\}$$

(156)

$$\underset{\left(6\times 6\right)}{ME}\left\{J_{61}\right\}=\left\{\left[\begin{array}{cc}ME\left\{V_{61}\right\}& \left[0\right]\\ \left[0\right]& ME\left\{V_{61}\right\}\end{array}\right]\right\}$$

(157)

$$\underset{\left(3\times 6\right)}{ME}\left(V_{62}\right)=\left\{\left[\begin{array}{cc}{I}_3& {\Delta }_6\cdot \left\{{\overline{k}}_6^{\left(0\right)}\times \right\}\end{array}\right]\right\}$$

(158)

$$\underset{\left(6\times 9\right)}{ME}\left(J_{62}\right)=\left[\begin{array}{cc}ME\left\{V_{62}\right\}& \left[0\right]\\ \left[0\right]& I_3\end{array}\right]$$

(159)

$$\underset{\left(6\times 9\right)}{ME}\left(V_{63}\right)=\left[\begin{array}{ccc}{I}_3& \left[0\right]& \left[0\right]\\ \left[0\right]& \left\{\begin{array}{l}\left[\begin{array}{c}{\displaystyle \sum _{e^{m=1-1}}^{k-1}\left\{{\overline{k}}_m^{\left(0\right)}\times \right\}\cdot q_m\cdot {\delta }_m\cdot {\Delta }_m}\\ k=i\to 3\end{array}\right]\\ {\delta }_m=\left\{\left\{0;m=i-1\right\};\left\{1;m\ge i\right\}\right\}\end{array}\right\}& e^{{\displaystyle \sum _{k=1}^6\left\{{\overline{k}}_k^{\left(0\right)}\times \right\}\cdot q_{{}_k\cdot }{\Delta }_k}}\end{array}\right]$$

(160)

$$\underset{\left(9\times 12\right)}{ME}\left\{J_{63}\right\}=\left[\left[\begin{array}{cc}ME\left\{V_{63}\right\}& \left[0\right]\\ \left[0\right]& I_3\end{array}\right]=ME\left\{\begin{array}{c}\exp \left\{{\displaystyle \sum _{m=0}^{k-1}\left\{{\overline{k}}_m^{\left(0\right)}\times \right\}{q}_m\cdot {\delta }_m\cdot {\Delta }_m}\right\};\exp \left\{{\displaystyle \sum _{m=0}^{k-1}\left\{{\overline{k}}_m^{\left(0\right)}\times \right\}{q}_m\cdot {\Delta }_m}\right\}\\ {\delta }_m=\left\{\left\{0;m=0\right\};\left\{1;m\ge 1\right\}\right\}\end{array}\right\}\right]$$

(161)

Performing the product between (157), (159) and (161) is obtained:

$$\underset{\left(6\times 12\right)}{ME}\left(J_6\right)=\left\{\begin{array}{c}ME\left\{J_{61}\right\}\cdot ME\left\{J_{62}\right\}\cdot ME\left\{J_{63}\right\}\\ \left[\begin{array}{cc}ME\left\{V_{61}\right\}& \left[0\right]\\ \left[0\right]& ME\left\{V_{61}\right\}\end{array}\right]\cdot \left[\begin{array}{cc}ME\left\{V_{62}\right\}& \left[0\right]\\ \left[0\right]& I_3\end{array}\right]\cdot \left[\begin{array}{cc}ME\left\{V_{63}\right\}& \left[0\right]\\ \left[0\right]& I_3\end{array}\right]\end{array}\right\}$$

(162)

According to MEK5, the results are:

$$\underset{\left\{ 12 \times 1 \right\}}{M_{6v\omega }}={\left[\begin{array}{cccc} {\overline{v}}_6^{\left(0\right)T}& {\left[{\overline{b}}_k ; k=6\right]}^T& {\overline{p}}_6^{\left(0\right)T}& {\Delta }_6\cdot {\overline{k}}_6^{\left(0\right)T} \end{array}\right]}^T$$

(163)

Performing the product between (162) and (163), there is obtained the sixth column of Jacobian matrix, as:

$$\left\{\begin{array}{c}\underset{\left(6\times 1\right)}{{}^{0}J_6}=\left\{ME\left\{J_{61}\right\}\cdot ME\left\{J_{62}\right\}\cdot ME\left\{J_{63}\right\}\cdot M_{5v\omega }\equiv ME\left\{{}^{0}J_6\right\}\cdot M_{6v\omega }\right\}\equiv \\ \equiv \left[\begin{array}{cc}ME\left\{V_{61}\right\}& \left[0\right]\\ \left[0\right]& ME\left\{V_{61}\right\}\end{array}\right]\cdot \left[\begin{array}{cc}ME\left\{V_{62}\right\}& \left[0\right]\\ \left[0\right]& I_3\end{array}\right]\cdot \left[\begin{array}{cc}ME\left\{V_{63}\right\}& \left[0\right]\\ \left[0\right]& I_3\end{array}\right]\cdot \left[\begin{array}{c}{\overline{v}}_6^{\left(0\right)}\\ {\left[{\overline{b}}_k;k=6\right]}^T\\ {\overline{p}}_6^{\left(0\right)}\\ {\Delta }_6\cdot {\overline{k}}_6^{\left(0\right)}\end{array}\right]\end{array}\right\}$$

(164)

In concordance with MEK7, it results in the following:

$$\underset{\left(6\times 1\right)}{{}^0\dot{J}{}_6}=M\left\{ {\dot{J}}_{61}\right\}\cdot M\left\{ {\dot{J}}_{62}\right\}\cdot M\left\{ {\dot{J}}_{63}\right\}\cdot M\left\{ {\dot{J}}_{64}\right\}$$

(165)

where

$$\underset{\left(6\times 18\right)}{M\left\{ {\dot{J}}_{61}\right\}}=\left[\begin{array}{ccc} ME\left\{ {\dot{J}}_{61}\right\}& ME\left\{ J_{61}\right\}& ME\left\{ J_{61}\right\} \end{array}\right]$$

(166)

$$\underset{\left(18\times 27\right)}{M\left\{ {\dot{J}}_{62}\right\}}=\left[\begin{array}{ccc}ME\left\{J_{62}\right\}& \left[0\right]& \left[0\right]\\ \left[0\right]& ME\left\{J_{62}\right\}& \left[0\right]\\ \left[0\right]& \left[0\right]& ME\left\{J_{62}\right\}\end{array}\right]$$

(167)

$$\underset{\left(27\times 36\right)}{M\left\{ {\dot{J}}_{63}\right\}}=\left[\begin{array}{ccc}ME\left\{J_{63}\right\}& \left[0\right]& \left[0\right]\\ \left[0\right]& ME\left\{ {\dot{J}}_{63}\right\}& \left[0\right]\\ \left[0\right]& \left[0\right]& ME\left\{J_{63}\right\}\end{array}\right]$$

(168)

$$\underset{\left(36\times 1\right)}{M\left\{{\dot{J}}_{64}\right\}}=\left[\begin{array}{c}{M}_{6v\omega }\\ M_{6v\omega }^{*}\\ {\dot{M}}_{6v\omega }\end{array}\right]=\left[\begin{array}{c}\begin{array}{c}{\left[\begin{array}{cccc}{\overline{v}}_6^{\left(0\right)T}& {\left[{\overline{b}}_6\right]}^T& {\overline{p}}_6^{\left(0\right)T}& {\left\{{\overline{k}}_6^{\left(0\right)}\cdot {\Delta }_6\right\}}^T\end{array}\right]}^T\end{array}\\ \begin{array}{c}{\left[\begin{array}{cccc}{\overline{v}}_6^{\left(0\right)T}& {\left[{\overline{b}}_6\right]}^T& {\overline{p}}_6^{\left(0\right)T}& {\overline{0}}^T\end{array}\right]}^T\end{array}\\ {\left[\begin{array}{cccc}{\overline{0}}^T& {\left[{\dot{\overline{b}}}_6\right]}^T& {\overline{0}}^T& {\overline{0}}^T\end{array}\right]}^T\end{array}\right]$$

(169)

In relation (166):

$$\underset{\left(6\times 6\right)}{ME\left\{ {\dot{J}}_{61}\right\}}=\left[\begin{array}{cc}ME\left\{ {\dot{V}}_{61}\right\}& \left[0\right]\\ \left[0\right]& ME\left\{ {\dot{V}}_{61}\right\}\end{array}\right]$$

(170)

where

$$\underset{\left(3\times 3\right)}{ME\left\{ {\dot{V}}_{61}\right\}}=\frac{d}{dt}\left\{\exp \left\{{\displaystyle \sum _{j=0}^1\left\{ {\overline{k}}_j^{\left(0\right)}\times \right\} q_j\cdot {\Delta }_j}\right\}\right\}$$

(171)

In relation(168):

$$\underset{\left(9\times 12\right)}{ME\left\{ {\dot{J}}_{63}\right\}}=\left[\begin{array}{cc}ME\left\{ {\dot{V}}_{63}\right\}& \left[0\right]\\ \left[0\right]& I_3\end{array}\right]$$

(172)

where

$$\underset{\left(6\times 9\right)}{ME\left\{ {\dot{V}}_{63}\right\}}=\left\{\left[\begin{array}{ccc}{I}_3& \left[0\right]& \left[0\right]\\ \left[0\right]& \left[\frac{d}{dt}\left\{\exp \left\{{\displaystyle \sum _{m=1-1}^5 \left\{ {\overline{k}}_m^{\left(0\right)}\times \right\} q_m{\delta }_m{\Delta }_m}\right\}\right\}\right]& \frac{d}{dt}\left\{\exp \left\{{\overline{ k}}_6^{\left(0\right)}\times \right\} q_6\cdot {\Delta }_6\right\}\end{array}\right]\right\}$$

(173)

Hence, the time derivative for the sixth kinematic joint is obtained as:

$$\underset{\left(6\times 1\right)}{{}^0\dot{J}_6}=\left\{\begin{array}{l}\left[\begin{array}{cc}ME\left\{{\dot{V}}_{61}\right\}& \left[0\right]\\ \left[0\right]& ME\left\{{\dot{V}}_{61}\right\}\end{array}\right]\cdot \left[\begin{array}{cc}ME\left\{V_{62}\right\}& \left[0\right]\\ \left[0\right]& I_3\end{array}\right]\cdot \left[\begin{array}{cc}ME\left\{V_{63}\right\}& \left[0\right]\\ \left[0\right]& I_3\end{array}\right]\cdot M_{6v\omega }+\\ +\left[\begin{array}{cc}ME\left\{V_{61}\right\}& \left[0\right]\\ \left[0\right]& ME\left\{V_{61}\right\}\end{array}\right]\cdot \left[\begin{array}{cc}ME\left\{V_{62}\right\}& \left[0\right]\\ \left[0\right]& I_3\end{array}\right]\cdot \left[\begin{array}{cc}ME\left\{{\dot{V}}_{63}\right\}& \left[0\right]\\ \left[0\right]& I_3\end{array}\right]\cdot M_{6v\omega }^{*}+\\ +\left[\begin{array}{cc}ME\left\{V_{61}\right\}& \left[0\right]\\ \left[0\right]& ME\left\{V_{61}\right\}\end{array}\right]\cdot \left[\begin{array}{cc}ME\left\{V_{62}\right\}& \left[0\right]\\ \left[0\right]& I_3\end{array}\right]\cdot \left[\begin{array}{cc}ME\left\{V_{63}\right\}& \left[0\right]\\ \left[0\right]& I_3\end{array}\right]\cdot {\dot{M}}_{6v\omega }\end{array}\right\}$$

(174)

The relations (164) and (174), substituted in (39), are leading to determination of the direct kinematics equations for CardioVR-ReTone robotic exoskeleton (6R robot).

The matrix exponential algorithm has several advantages given by the compact form, the simple geometric visualization of the exponential functions. Although the application of the mathematical algorithm is challenging, the exponentials matrix has an essential role in the analysis of robotic exoskeleton geometry and direct kinematics; geometric, kinematic, and dynamic control functions; and kinematic and dynamic precision. The formalism based on matrix exponentials leads to the evaluation of the performances for the robot structure, regardless of its complexity level [41].

3. Results

The acquired results, based on matrix exponential algorithms, are essential to the optimal design, from both the dimensional and energy perspective, but furthermore, for mechanical structure kinematic and dynamic behavior simulation of the proposed exoskeleton.

3.1. CardioVR-ReTone—6R Robot Differential Motion Equations

Based on the Lagrange–Euler formalism, differential equations of motion can be established for nonconservative mechanical systems with holonomic links, as indicated by references [38,39]. In this paper, the equations of motion for the CardioVR-ReTone 6R robot’s two driving joints, J2 and J3, will be derived. These two joints are emphasized in Figure 4.

To derive the motion expressions for the analyzed links, the driving moments $Q_m^i, for i=2,3$will be determined as:

$$Q_m^i=\begin{array}{cc}{Q}_{iF}^i+Q_g^i+Q_{SU}^i,& i=2, 3\end{array}$$

(175)

In previous expressions are included $\left\{Q_{iF}^i;Q_g^i;Q_{SU}^i\right\}$, representing the generalized inertia forces, generalized gravitational forces, and generalized external forces [43].

The equations for generalized inertia forces, where $i=2,3$, can be obtained based on kinetic energy as follows:

$$Q_{iF}^i=\frac{d}{dt}\left[\frac{\partial E_C\left(\overline{\theta };\dot{\overline{\theta }}\right)}{\partial {\dot{q}}_i}\right]-\frac{\partial E_C\left(\overline{\theta };\dot{\overline{\theta }}\right)}{\partial q_i}$$

(176)

The determination of the kinetic energy for the 6R structure considered is performed according to [38] using the following expression:

$$E_C\left(\overline{\theta }, \dot{\overline{\theta }}\right)={\displaystyle \sum _{i=1}^6\left\{\frac{1}{2}\cdot M_i\cdot {}^i\overline{v}_{Ci}^T\cdot {}^i\overline{v}_{Ci}+\frac{1}{2} \cdot {}^i\overline{\omega }_i^T\cdot {}^{i}I_i^{*}\cdot {}^i\overline{\omega }_i\right\}}$$

(177)

where $M_i$ is the mass, ${}^i\overline{v}_{C_i}$ and ${}^i\overline{\omega }_i$ representing the linear and angular velocity of the mass center for each joint ${}^{i}I_i^{*}$ the kinetic joint’s inertial axial-centrifugal tensor $\left(i\right)$, which is determined based on the frame used at the joint’s mass center.

Using the SolidWorks© software application, the mass parameters for the CardioVR-ReTone structure have been acquired. Figure 5 displays the results for joints 2 and 3. The generalized gravitational forces are defined by the following:

$$Q_g^i\left(\overline{\theta }\right)={}^{0}J{}_i{\left(\overline{\theta }\right)}^T \cdot {}^{0}F{}_{x_i}\left(\overline{\theta }\right)$$

(178)

where ${}^{0}J{}_i{\left(\overline{\theta }\right)}^T$ represents the line $\left(i\right)$ of the transposed Jacobian matrix defined in (40), and ${}^{0}F{}_{x_i}\left(\overline{\theta }\right)$ is the resultant force-moment vector of gravitational loads [37].

The generalized handling forces are expressed as:

$$Q_{SU}^i\left(\theta \right)={}^{0}J{}_i^T\left(\theta \right)\cdot {}^{0}F{}_X\left(\theta \right) , i=4,5$$

(179)

where ${}^{0}F{}_X\left(\theta \right)$ is the vector of the resultant force-load handling moment [37].

Determining the forces and moments with precision is crucial for designing a kinematic axis correctly. Therefore, it is essential to examine the type and kinematic chain of motion transmission of each motor coupling [44]. By inserting the expressions for the generalized gravitational, manipulation, and inertia forces, which were previously derived, into the definition expression (173), the analytical form of the generalized driving forces acting on the output shafts of the drive systems can be obtained. The resulting form of these driving forces is presented below:

$$Q_m^2=0.421257\cdot {\ddot{q}}_{2jk}+0.0032355456$$

(180)

$$Q_m^3=0.3897\cdot {\ddot{q}}_{3jk}+0.00278$$

(181)

Achieving precise determination of forces and moments is crucial for designing a kinematic axis accurately. To accomplish this, the mode and kinematic chain of motion transmission for each motor coupling must be taken into consideration [20]. The analytical form of the generalized driving forces acting on the output shafts of the drive systems can be obtained by substituting the expressions for the previously derived generalized gravitational, manipulation, and inertia forces into the definition expression (173), as demonstrated in Equations (178) and (179). However, it should be noted that these relationships do not account for frictions.

Accurately determining the driving moments is crucial in selecting the appropriate motor sizes for operating a kinematic axle or related braking systems. This is necessary to avoid any critical situations that could potentially harm the mechanical structure. To achieve this, the generalized variables representing the movements of the mechanical system joints will be substituted with polynomial time functions that illustrate the robot’s working process, as explained in the following paragraph.

3.2. Model the Motion of Kinematic Joints Employing (3n)-Type Polynomial Functions

As per the working process, the motion trajectory in the configuration space should pass through all the points corresponding to the moments $\left({\tau }_i\left(i=0\to n\right)\right)$. Additionally, the movement path must maintain control over position, speed, and acceleration at times ${\tau }_0$ and ${\tau }_n$ based on the initial conditions and ensure continuity of speed and acceleration at $\left[{\tau }_k\left(k=1\to n-1\right)\right]$. To accomplish this, cubic spline functions are used for the interpolation of each segment of the trajectory $k=1\to 6$. These functions are determined by generating a linear function, with respect to time, for the generalized accelerations of each driving joint as described in [38,45]: The working sequences of kinetic links i = 2,3, in the context of j = 1→3, are analyzed by dividing each sequence into three segments denoted as k = 1→3. To study the variation of kinematic parameters and driving moments, the process is divided into three segments as recommended by [38,45]. The motion trajectory in the configuration space for each working sequence is interpolated using cubic spline functions, while complying with the restrictions imposed by the driving and control systems of the structure. The interpolation cubic splines of (3n)-type polynomial functions are established by generating linear time functions for the generalized accelerations of each driving joint.

The motion trajectory in the configuration space should pass through all the points corresponding to the moments $\left[{\tau }_i\left(i=0\to n\right)\right]$, as per the working process. Moreover, the movement path must maintain control over position, speed, and acceleration at times ${\tau }_0$ and ${\tau }_n$ based on the initial conditions and ensure the continuity of speed and acceleration at $\left[{\tau }_k\left(k=1\to n-1\right)\right]$. To accomplish this, cubic spline functions are used for the interpolation of each segment of the trajectory k = 1→6. These functions are determined by generating a linear function, with respect to time, for the generalized accelerations of each driving joint, as described in [38,45]:

$${\ddot{q}}_{ji}\left(\tau \right)=\frac{{\tau }_i-\tau }{t_i}\cdot {\ddot{q}}_{ji}\left({\tau }_{i-1}\right)+\frac{\tau -{\tau }_{i-1}}{t_i}\cdot {\ddot{q}}_{ji}\left({\tau }_i\right)$$

(182)

where the time needed to run each segment (i = 1→3) of the trajectory is represented by $t_i={\tau }_i-{\tau }_{i-1}$. By solving the differential Equation (180) via integration, the following functions can be derived:

$${\dot{q}}_{ji}\left(\tau \right)=-\frac{{\left({\tau }_i-\tau \right)}^2}{2\cdot t_i}\cdot {\ddot{q}}_{ji-1}+\frac{{\left(\tau -{\tau }_{i-1}\right)}^2}{2\cdot t_i}\cdot {\ddot{q}}_{ji}+a_{ji1}$$

(183)

$$q_{ji}\left(\tau \right)=\frac{{\left({\tau }_i-\tau \right)}^3}{6\cdot t_i}\cdot {\ddot{q}}_{ji-1}+\frac{{\left(\tau -{\tau }_{i-1}\right)}^3}{6\cdot t_i}\cdot {\ddot{q}}_{ji}+a_{ji1}\cdot \tau +a_{ji2}$$

(184)

Table 2. CardioVR-ReTone joints j2, j3—coordinates and running times.

Link i	Seq. J = 2,3	Configuration k = 4→9	Joint Rotation Reported to Previous Position [°]	Coordinates Values ${\mathit{q}}_{\mathit{i}\mathit{j}\mathit{k}} \left[\mathit{r}\mathit{a}\mathit{d}\right]$	Duration ${\mathit{t}}_{\mathit{i}} 〈\mathit{s}〉$	$\mathbf{Time}{\mathit{\tau }}_{\mathit{j}\mathit{k}} 〈\mathit{s}〉$
2	2	3	0	0	0	1.50012
		4	46.667	0.81448698	0.6482	2.14832
		5	93.334	1.628973969	0.6482	2.79652
		6	140	2.443460953	0.6482	3.44472
3	3	6	0	0	0	3.44472
		7	80	1.396263402	1.1112	4.55592
		8	160	2.792526803	1.1112	5.66712
		9	240	4.188790205	1.1112	6.77832

presents the numerical values for the coordinates and running times of the intervals used in the study of the working process.

Based on the preceding considerations, the expressions that describe the positions, velocities, and accelerations for each segment (k= 1 to 3) of the sequence are presented in

Table 3. The positions, velocities, and acceleration expressions for joints 2 and 3.

Sequence J = 2,3	Interval K = 1→3	Coordinate	Generalized Positions, Velocities, and Accelerations Equations
			${\mathit{q}}_{\mathit{i}\mathit{j}\mathit{k}} \left[\mathit{r}\mathit{a}\mathit{d}\right]$	${\dot{\mathit{q}}}_{\mathit{i}\mathit{j}\mathit{k}} \left[\frac{\mathit{r}\mathit{a}\mathit{d}}{\mathit{s}}\right]$	${\ddot{\mathit{q}}}_{\mathit{i}\mathit{j}\mathit{k}} [\frac{\mathit{r}\mathit{a}\mathit{d}}{{\mathit{s}}^2}]$
2	1	$q_2$	$1.495\cdot {\left(\tau -1.5\right)}^3-5.42\cdot {10}^{-20}\cdot \tau +39.43$	$4.485\cdot {\left(\tau -1,5\right)}^2-5.42\cdot {10}^{-20}$	$8.969\cdot \tau -13.454$
	2		$-2.991\cdot {\tau }^3+22.1795\cdot {\tau }^2-52.01\cdot \tau +39.425$	$-8.971\cdot {\tau }^2+44.359\cdot \tau -52.001$	$-17.942\cdot \tau +44.359$
	3		$1.495\cdot {\left(\tau -3.445\right)}^3+2.444$	$4.486\cdot {\left(\tau -3.445\right)}^2$	$8.972\cdot \tau -30.904$
3	1	$q_3$	$0.509\cdot {\left(\tau -3.445\right)}^3-5.422\cdot {10}^{-20}\cdot \tau +1.9\cdot {10}^{-19}$	$1.527\cdot {\left(\tau -3.445\right)}^2-5.42\cdot {10}^{-20}$	$3.053\cdot \tau -10.517$
	2		$-1.018\cdot {\tau }^3+15.605\cdot {\tau }^2-76.938\cdot \tau +123.55$	$-3.053\cdot {\tau }^2+31.21\cdot \tau -76.938$	$31.21\cdot \tau -6.106$
	3		$0.509\cdot {\left(\tau -6.779\right)}^3+4.189$	$1.527\cdot {\left(\tau -6.779\right)}^2$	$3.053\cdot \tau -20.694$

.

Based on the information presented in Table 3 and using the expressions (178) and (179), Figure 6 and Figure 7 depicted the changes in the driving moments.

Expressions for the generalized coordinates, speeds, and accelerations are shown in Figure 8, with the run time, duration, and starting and ending coordinates for each trajectory segment serving as input data.

3.3. CardioVR-ReTone Robotic Exoskeleton Actuation and Controlling System

The selection of motors for the exoskeleton actuation system is a crucial step in its design. A thorough analysis of the kinematic and dynamic model of the exoskeleton was necessary to determine the requirements for the motors. The model provided information about the forces and torques that the motors must generate, as well as the joint ranges of motion, which establish the type and size of motors needed.

Based on the methodology highlighted in [46,47], the joint torque required for a specific range of motion was calculated, taking into consideration the weight of the exoskeleton and the patient upper limbs. An example of joint 2 torque calculation is highlighted in

Table 4. Exoskeleton joint 2 necessary torque.

Segment	Human Body/ Exoskeleton	Weight (kg)	Length/ Motor Diameter * (mm)	Center of Mass (%)	FG (N)	Torque Related to the Joint j2 (Nm)
Hand and Joystick	B	1.00	101.25	51.5	9.81	7.55
Forearm	B	2.06	283.75	41.7	20.21	11.17
Elbow to Hand Link	E	0.923	221	15.87	9.05	11.39
Elbow Joint: EC60Flat + CPU-17	E	1.29	79 *	50.0	12.65	14.41
Upper Arm	B	3.73	294.25	50.7	36.59	10.58
Shoulder Link 5 to Elbow	E	1.209	372		11.86	10.71
Shoulder Joint 5 EC60Flat + CPU-17	E	1.29	79 *	50.0	12.65	9.82
Shoulder Link 4 to Shoulder Link 5	E	0.292	173	-	2.86	1.84
Shoulder Joint 4 EC60Flat + CPU-20	E	1.8	98 *	50.0	17.66	7.16
Shoulder Link 3 to Shoulder Link 4	E	0.847	356.3		8.31	2.94
Shoulder	B	3.47	140	50.0	34.04	2.38
Shoulder Joint 3 EC60Flat + CPU-25	E	2.55	116 *	50.0	25.02	6.4
Shoulder Link 3 to Shoulder Link 2	E	0.427	198		4.19	0.34
Shoulder Joint 3 EC60Flat + CPU-25	E	2.55	116 *	50.0	25.02	5.2
Shoulder Link 2 to Shoulder Link 1	E	0.3	150	50.0	2.94	0.22
Joint 2 equilibrium torque						118.34
Oversizing coefficient						1.25
Torque required by joint 2 actuator						147.95

where “*”—is the diameter of the chosen motor.

. The motors must be able to generate sufficient torque to overcome these loads, and also provide smooth and controlled motion. The motion must also be accurate and repeatable, as any errors or variability will affect the quality of the rehabilitation. In addition to the technical requirements, the cost and availability of the motors were also considered. A trade-off may need to be made between the optimal technical solution and the practical constraints of the project.

The actuator selection began at joint j6 and continued until joint j1. This strategy was essential because the actuators chosen for a specific joint will necessitate higher torque than actuators chosen for prior joint.

Table 5. Actuator selection based on calculated torque [46].

Joint	Necessary Calculated Torque Tj (Nm)	Actuators
		Type	Weight (Kg)	Repeated Peak Torque (Nm)	Average Torque (Nm)
6	8.91	CPU-17	1.29	54	39
5	40.07	CPU-17	1.29	54	39
4	51.15	CPU-20	1.8	82	49
3	100.30	CPU-25	2.45	157	108
2	144.66	CPU-25	2.45	157	108
1	150.80	CPU-25	2.45	157	108

summarizes the computed torque and safety factor, along with the actuator choice for each joint of the exoskeleton [46].

Once the motors have been selected, their control system has been designed and implemented. The control system must ensure that the motors operate in a safe, efficient, and reliable manner, and provide feedback on the position, velocity, and torque of the joints. The choice of motors, and their control system, will have a significant impact on the performance and functionality of the exoskeleton, and must be carefully considered in the design process. The data presented in Table 4 indicates that, with the exception of joints 1 and 2, the required torque for all other joints is below the average torque output of the selected actuator. This implies that the actuator is capable of providing adequate torque to operate the exoskeleton and its attached body parts. Joint 4 has a torque value that is close to the average, but it exhibits a lower peak torque value upon repetition. On the other hand, joints 1 and 2 require a torque that is greater than the average but less than the maximum torque upon repetition. Despite the fact that the more reliable CPU-32 actuator was available, the decision to use CPU-25 was made due to its smaller size, lighter weight, and lower cost. Once the actuators were chosen and the segments connecting the motors were established (using aluminum 7075—Al–Mg–Si–Cu Alloy), the appropriate actuator controller needed to be identified based on technical and application requirements. In this case, the Maxon EPOS4 15/50 controller with a CANopen fieldbus interface was selected to control each actuator motor. With the EPOS4 15/50 controller, it is possible to control the loops for speed, position, and torque. Additionally, the controller can be seamlessly integrated into high-level programming languages or other PC-based software using manufacturer-provided plug-ins, in addition to these control functions.

Figure 9 highlights the detailed control architecture for the CardioVR-ReTone exoskeleton control strategy, which outlines the steps of the controlling algorithm and provides a high-level description of the control system’s logic. It provides a road map for the implementation of the control algorithm and can help to visualize how the different components of the control system interact. In this architecture, the kinematic and dynamic models are used to calculate the joint torques based on the desired joint angles and velocities. The joint torques are then used to compute the control signals that are sent to the actuators. The kinematic and dynamic models are continuously updated based on sensor data.

The control architecture for the CardioVR-ReTone exoskeleton facilitates multiple control modes, such as assistive, partially assistive, and resistive modes. To achieve this goal, the primary control unit chosen was an open industrial controller, Siemens CPU 1515SP PC2. This controller was equipped with a CANopen communication module, enabling communication with each actuator’s EPOS4 controller (Figure 10), and several IO modules that interfaced with the exoskeleton’s joysticks and mechanical safety limit switches. In the event of a malfunction, particularly during positioning, the mechanical safety limit switches would activate the safe torque off functionality of the actuators. Simulink was hosted on the open controller’s operating system, which executed the kinematic algorithms and transmitted position, speed, and torque data to each joint actuator controller.

The Simulink kinematics algorithm received feedback from each actuator controller. Muscle activity information was collected from the human arms using surface electromyography (EMG) signals, which were transmitted via Bluetooth to the open controller, processed, and visualized. The control architecture operated using a distributed control approach, with the preferred option for implementing control-related functions being the ROS (Robot Operating System) platform.

The selected control architecture and open controller enabled the implementation of multiple control modes and facilitated effective communication and coordination between the various exoskeleton components. Due to the necessity to connect 12 motor controllers on the CANopen fieldbus and the technological limitations of the EPOS4 motor controller, only a reduced number of data packets may be exchanged cyclically between the PLC and EPOS4 controllers at a rate of 1 Mbit/s. Data transferred cyclically are as follows: from EPOS4 to PLC—status word, actual position, and actual operating mode, and from PLC to EPOS4—control word, position setpoint, and desired operating mode. To achieve the desired operating modes of the exoskeleton, additional data need to be transferred to and from EPOS4 controllers. These data can only be transferred on request in order not to overload the fieldbus data transfer performances. Therefore, every 100 ms, motor speed and torque actual values are read from EPOS4 and new motor speed and torque setpoints are sent to EPOS4. Figure 10 presents the control logic, which reads from every EPOS4 controller by using a function block developed by the PLC-vendor, namely, LCan_SDORead function, nodeId is an input parameter to this function, and a variable named Left_EPOS is modified by the PLC logic from 1 to 12 in order to read specific data from every EPOS4 motor controller.

The integration of mechanical safety limit switches and surface EMGs for torque adjustment provided the safety and comfort of the user. In addition, the use of distributed control and the ROS platform provides a versatile and scalable framework for the implementation of control-related functions.

3.4. Technical Feasibility Evaluation of the CardioVR-ReTone Robotic Exoskeleton

The CardioVR-ReTone exoskeleton was manufactured based on the findings of this research and the information presented in this article. Figure 11 illustrates the prototype of the CardioVR-ReTone robotic exoskeleton (technology readiness level—TRL 4 [48]).

Testing the equipment functionality in the laboratory conditions was made on 17 volunteers (Figure 11 and Figure 12). Approval for the study’s ethics was granted by the Etic Committee of the University of Medicine and Pharmacy Cluj-Napoca (registration number 350/02.21.2019). The exoskeleton test involved the following stages: (1) Assembly and calibration: the exoskeleton was assembled, and the actuators and sensors were calibrated to ensure that the exoskeleton can accurately perform the necessary movements required by the rehabilitation therapy. (2) Safety testing: the safety of the exoskeleton was tested to ensure that the device will not cause harm to the user during operation. This includes testing the exoskeleton’s emergency stop systems and stroke limiter mechanisms and assessing the risk of falls or other accidents. (3) Basic functionality testing: the basic functionality of the exoskeleton was tested, including movement tracking and response time. The exoskeleton’s ability to support and augment the user’s movements was also assessed. (4) User experience testing: the user experience of the exoskeleton was evaluated to assess the comfort and usability of the device. This included evaluating the exoskeleton’s fit, adjustability, and ease of use. (5) Performance testing: the performance of the exoskeleton was tested to assess its ability to support the user’s movements during established exercises. This task involved the manipulation and lifting of the human arm and the addition of diverse weights, such as 2, 1.5, 1 kg, with the repetition of the established movement of 10 times.

The laboratory testing was run on 17 healthy volunteers, aged between 21 and 28.5 years. All subjects received training on how to operate the CardioVR-ReTone exoskeleton and the necessary sensors (e.g., EMG electrodes were mounted according to http://www.seniam.org/ (accessed on 3 October 2022)). The test lasted approximately 20 min for each participant. During this period, the exoskeleton was adjusted for each participant based on their vocal feedback until they were comfortable with it and no additional adjustments were required. Two unblinded, devoted personnel with experience in cardiovascular rehabilitation provided them with training. An OptiTrack V100:R2 camera was used to capture the exoskeleton’s motion. To evaluate the volunteers’ muscular activity, EMG signals were wirelessly captured (e.g., Delsys Trigno^TM Delsys Inc., Boston, MA, USA). The workout’s overall duration and the number of repetitions were recorded. Blood pressure (TAS, TAD), cardiac frequency (FC), and SaO₂ were measured and recorded (Figure 12) throughout the activities. The volunteers’ responses about discomfort or pressure were continuously recorded. Pain severity was assessed every 5 min during the test using a numeric rating scale. After the examination, the “quality” of pain was assessed using the Short Form McGill Pain Questionnaire version 2 [47].

The design of the CardioVR-ReTone exoskeleton underwent a series of optimizations based on the lab tests results. To ensure optimal functionality, the team ran a Matlab and CATIA simulation to identify how the structure could be improved. One key area of focus was the end effector and the last link, which was adjusted based on volunteers’ observations, and the simulation results. Additionally, the team also made changes to the motors’ control, fine-tuning the system to ensure that it was operating at peak performance, adjustable length of segments, and elbow support. These optimizations were validated through further lab tests, which confirmed that the exoskeleton was functioning as desired. The adjustments made to the structure and control of the exoskeleton allowed it to be more effective and efficient, making it suitable for testing in a medical environment.

3.5. Empirical Test of CardioVR-ReTone Robotic Exoskeleton in a Medical Environment

In this section, the exoskeleton functionalities within the medical environment will be evaluated. Approval for the study’s ethics was obtained from the Etic Committee of the Institute of Cardiovascular Diseases Timisoara (registration number 8536/26.09.2022). The medical benefits of this type of rehabilitation and the degree of its acceptance by cardiac patients will be detailed in future articles. The CardioVR-ReTone exoskeleton testing in a medical environment was performed as follows: Phase 1—patient selection: a group of 14 patients who had undergone a major cardiac event, or an open-heart surgery were selected from the Institute of Cardiovascular Diseases Timisoara, Romania. Each patient underwent a physical medical assessment (

Table 6. Biological parameters of the selected cardiac patients.

Cardiac Patients Demographic Data	Biological Parameters
Age (years)	61.57+/−9.56
Sex (% men)	50% men
Occupation	64% retirees
Education level	14% higher education 28% high school studies 57% secondary education
BMI (kg/mp)    Overweight    Obesity degree I    Obesity degree II	28.33+/−5.25 36% 21% 7%
Smoking history	36% former smokers
SBP (mmHg)	125.93+/−10.38
DBP (mmHg)	73.71+/−8.74
CF (b/min)	74.43+/−7.85
SaO2 (%)	97.29+/−1.73
Graft type    VSI (internal saphenous vein)    AMI (internal mammary artery)	75% 25%

) to determine their baseline physical capabilities. This included tests for range of motion, strength, and endurance. Patients were provided with information about the study, and they signed an informed consent form before participating.

Phase 2—training and familiarization: the patients were trained on how to use the exoskeleton by qualified personal from the project, who monitored the process throughout. They were given time, approximately 10 min, to familiarize themselves with the equipment and how to control the system. The exoskeleton was fitted to each patient, and initial training was provided to ensure that the patient can use the device safely and effectively. The patient’s response to the device was also assessed during this time. Phase 3—baseline test: each patient underwent a test to determine their physical capabilities without the exoskeleton. This included tests for range of motion, strength, and endurance—for 5 min. Phase 4—exoskeleton test: the patients were asked to perform a series of exercises in three sessions for the upper limb, specific to cardiac rehabilitation early after open-heart surgery or major cardiac event as stated in [47], using the exoskeleton (Figure 13). The sessions were designed to gradually increase in intensity and difficulty over time, and the exercises were similar to the baseline test. The patients were monitored for vital signs (SBP (mmHg), DBP (mmHg), CF (b/min), SaO₂ (%)), and any signs of discomfort or pain.

Based on the functional testing of the robotic exoskeleton performed on a group of 14 cardiac patients, the results showed that the patients were highly satisfied with the device. The exoskeleton proved to be a valuable tool in their cardiac rehabilitation, providing the necessary support for their upper limbs while allowing them to perform various exercises specific for cardiac rehabilitation after a major cardiac event. The patients found the device easy to use and comfortable to wear, which is essential for long-term adherence to the rehabilitation program.

Moreover, the exoskeleton’s closed-loop control system and safety features ensured that the patients were protected from any adverse effects. The system was designed to respond quickly to any unexpected movements or forces and adjust the torque accordingly to ensure the patient’s safety.

Overall, the results of this study suggest that the robotic exoskeleton is a promising robotic system for upper-limb cardiac rehabilitation, with potential for further development and refinement. The positive feedback from patients indicates that the device is well tolerated and can provide a valuable adjunct to traditional rehabilitation methods.

4. Discussion

This study focuses on the design and initial implementation of the upper-limb CardioVR-ReTone exoskeleton, a new robotic system for early cardiac rehabilitation following a major cardiac event or an open-heart surgery. This study reflects that such an equipment facilitates the immediate start of cardiac rehabilitation, which creates the conditions for an accelerated cardiac recovery of the patient, as evidenced by the references [12,26]. The results highlighted in this paper pointed out the successful development of the CardioVR-ReTone robotic exoskeleton. With 6 degrees of freedom on each arm, the robotic exoskeleton creates the premises for a comprehensive cardiac rehabilitation program.

In the development of the CardioVR-ReTone robotic exoskeleton, the use of the matrix exponential algorithm allowed for a more comprehensive understanding of the dynamics and kinematics of the system [37]. The algorithm provided valuable insights into the mechanical and electrical structures of the exoskeleton, which were then optimized accordingly to ensure optimal performance [36]. The optimization of the control architecture was also made possible through the use of the matrix exponential algorithm, which allowed for a better understanding of the interactions between the exoskeleton’s components [40,49,50].

The control architecture for the robotic exoskeleton was designed based on its kinematic and dynamic model, and suitable electrical motors were carefully selected. After thorough consideration, the EC60 flat BLDC electric motor was chosen for its compact size, low backlash, ease of control, and good torque-to-weight ratio. To satisfy the calculated torque requirement for the exoskeleton arms and economic factors, such as discounts for bulk orders, the following actuators were chosen: two AC-CPU17-100, one AC-CPU20-100, and three AC-CPU25-100. Each actuator comes with an incremental encoder for precise position control and feedback. The EC60 flat BLDC electric motor was preferred due to its high efficiency, low maintenance, and quiet operation, among other advantages. The chosen actuators deliver ample torque and are cost-effective, making them ideal for large-scale applications, such as exoskeletons. Overall, the control architecture, electrical motors, and actuators were chosen to offer the necessary power and control to perform a broad range of movements while being budget friendly.

The need for multiple control modes, including assistive, partially assistive, and resistive modes, drove the development of the control architecture for the CardioVR-ReTone exoskeleton. To achieve this, the primary control unit selected was Siemens CPU 1515SP PC2, an industrial open controller equipped with a CANopen communication module for communicating with each actuator EPOS4 controller and several IO modules for interfacing with the exoskeleton joysticks and mechanical safety limit switches on the exoskeleton structure. The mechanical safety limit switches were crucial in activating the safe torque off functionality of the actuators during malfunction, especially during positioning. The Simulink, which executed the kinematic algorithms and transmitted position, speed, and torque data to each joint actuator controller, was hosted on the open controller’s operating system. Feedback from each actuator controller was sent back to the Simulink kinematics algorithm. Muscle activity information was gathered by collecting surface electromyography (EMG) signals from the human arms. The collected surface electromyography (EMG) signals from the human arms were sent via Bluetooth to the open controller, where they were processed and visualized. The control architecture of the CardioVR-ReTone exoskeleton used a distributed control approach, with the ROS (Robot Operating System) platform being the preferred option for implementing control-related functions. This control architecture, along with the open controller, allowed for the implementation of various control modes and ensured efficient communication and coordination among the exoskeleton’s components. The inclusion of mechanical safety limit switches and the use of surface EMGs for torque adjustment ensured user safety and comfort. Furthermore, adopting a distributed control approach and utilizing ROS provided a flexible and scalable platform for implementing control-related functions.

The empirical validation of the prototype was crucial in assessing its performance and effectiveness in a lab and in a real-world setting. The results of the validation demonstrate its potential for use in cardiac rehabilitation. The combination of mathematical modeling and empirical validation has demonstrated technical feasibility of the structure that can be relied upon to support patients in their cardiac rehabilitation journey. In the empirical validation of the CardioVR-ReTone robotic exoskeleton, both lab testing and medical environment testing were performed to assess its functionalities.

First, the basic functionalities of the robot, including actuation, joint movement, and power supply, were tested in laboratory conditions. This allowed the researchers to verify that the exoskeleton met the required specifications and functioned as intended. Additionally, the kinematic and dynamic models were verified through motion tracking and impedance control experiments. This provided valuable insights into the robot’s behavior and helped to refine the control architecture for improved performance. Finally, the robustness and stability of the exoskeleton were assessed by subjecting it to various disturbances and unexpected events. The results from these tests allowed any necessary adjustments to be made to the robot before it is tested in a medical environment.

Medical environment testing, on the other hand, involves evaluating the performance of the exoskeleton in a real-world setting with patients. This testing stage involves a more comprehensive evaluation of the robot, including its ease of use, comfort, and safety. The medical environment testing stage also includes evaluating the robot’s ability to help patients perform exercises that are important for their cardiac rehabilitation.

The results of both lab and medical environment testing showed that the CardioVR-ReTone robotic exoskeleton (TRL 4) was safe to be used, and it creates the premises for further development and testing necessary for higher levels of technology (e.g., TRL 8, 9). The results also indicated that patients readily accepted the exoskeleton, which was deemed comfortable and easy to use. These findings further validated the design and optimization of the exoskeleton using the matrix exponential algorithm.

The study’s findings emphasize the significance of incorporating mathematical modeling and empirical validation in the creation of robotic exoskeletons intended for medical use. The development of the CardioVR-ReTone exoskeleton serves as a prime illustration of the advantages of this method and underscores the potential of such devices to enhance the well-being of patients who require rehabilitation.

The necessity of the current robotic exoskeleton for upper-limb cardiac rehabilitation depends on a variety of factors specific to each individual case. It is important to carefully evaluate the available options in the market and choose the one that is best suited to the individual’s needs and circumstances. While there are several different models of robotic exoskeletons available on the market and in labs, the choice of which model to use for cardiac rehabilitation should be based on several factors, including the patient’s specific needs, the severity of their condition, and the level of customization and adjustability required for the exoskeleton.

One example of a robotic exoskeleton that has shown promise in cardiac rehabilitation is EksoNR, which is a wearable robotic exoskeleton designed to support the upper body and provide assistance with arm movements [51,52,53,54]. EksoNR can be adjusted to fit a range of body types and can be customized to provide varying levels of support and assistance. The Bimeo exoskeleton [54], which is highly customizable to meet the patient’s specific needs, provides support and assistance for the upper limbs, and the amount of assistance could be adjusted as needed. Haptic Master [54] is a robotic exoskeleton that provides sensory feedback to the patient during rehabilitation exercises. It can be used for upper-limb rehabilitation and can be customized to fit the patient’s specific needs. Kinova Jaco [54] is a robotic exoskeleton that provides assistance with arm movements. It is highly customizable and can be adjusted to fit the patient’s specific needs.

While there are few cases in which exoskeletons have been specifically used for cardiac rehabilitation following a major cardiac event, there is growing interest in their potential use for this purpose [11]. In conclusion, there are some limitations to this study. Although the experiments demonstrated the practicality of the robotic exoskeleton, there were some limitations, including the small number of volunteers and patients tested and the need for further research to determine the long-term effects of early use of the exoskeleton in cardiac rehabilitation on patient outcomes. The initial experiments allowed us to demonstrate the technological feasibility of the system, and more extensive testing on patients will be conducted to validate the medical viability of the system. Future efforts will focus on refining the ease of use of the CardioVR-ReTone exoskeleton, improving control, acceleration of cardiac rehabilitation, and expanding the range of exercises. In addition, a risk analysis will be conducted in a future paper in order to determine the risks connected with the CardioVR-ReTone exoskeleton. Nonetheless, this study represents a promising step forward in the use of a robotic exoskeleton for cardiac rehabilitation, and further research in this area has the potential to help patients in need of early cardiac rehabilitation following an open-heart surgery or a major cardiac event.

5. Conclusions

In conclusion, the design, development, and validation of the upper-limb CardioVR-ReTone robotic exoskeleton for cardiac rehabilitation was presented. The robotic exoskeleton has 6 degrees of freedom on each arm and is intended to be used in early-stage cardiac rehabilitation. The robot’s mechanical and electrical structure, as well as its control architecture, was optimized based on the proposed kinematic and dynamic model. The matrix exponential algorithm was used to develop the kinematic and dynamic models. The prototype was validated empirically by assessing its functionalities in both a lab and a medical environment, including an assessment of its safety and patient acceptance. The testing resulted in several optimizations, including changes to the end effector and motor control adjustments.

The study found that mathematical techniques such as the matrix exponential algorithm allow for the establishment of direct geometry equations and geometric control functions in direct geometric modelling. It should be made clear that the static assumption is dropped from the computation, but the assumption of rigidity remains; as a result, the column vectors of the generalized and operational coordinates are time functions. To calculate the geometric and kinematic control functions of a robot, it is important to employ some mathematical techniques that establish the relationship between the elements that determine the end effector’s location in Cartesian space and the speeds and accelerations of motor torques. The steps of kinematic modelling are therefore covered, and the equations that characterize the movement of a robot’s end effector may be simply created by employing certain matrix and differential approaches based on matrix exponentials.

It is clear from this assessment that there are numerous benefits to employing algorithmic approaches to mathematically model the robot structures. These advantages include: the sequence of steps, which leads to a simple geometric visualization of the characteristics of the different parameters considered; a compact form that is simple to comprehend when writing equations; a high degree of generalization, so the use of algorithmizing can solve an intractable problem regardless of the system’s complexity; and a high degree of simplicity.

Overall, the results demonstrate that the CardioVR-ReTone robotic exoskeleton has the potential to improve the efficacy and efficiency of cardiac rehabilitation by providing a safe, controlled, and patient-friendly environment. More study is required to properly comprehend the effect of this exoskeleton on cardiac rehabilitation, including bigger clinical trials, long-term monitoring, and the development of more complex control algorithms.