New Approach of the Variable Fractional-Order Model of a Robot Arm

Abstract

This paper proposes a simple mathematical model based on the variable fractional-order difference equation of a robot arm. The model of the described arm does not consider the impact of the movement of the mobile platform, it was assumed that all degrees of freedom would be taken away from it. The implementation of the task was divided into two stages. First, a mechanical model was developed. In order to estimate the torques of nodal propulsion motors, a description of the components of the Lagrange equation for the considered system, i.e., energy, power, and external interactions, and derivation of the equations of motion of the tested manipulator based on the Lagrange equation was made. An additional criterion was also considered in the selection of drives in the kinematic nodes of the links, which was to set the manipulator in a vertical position at a specific time. Processing the measured data of a robot arm, model parameters were selected, and the order function was chosen. The second stage was a simulation, whose results were compared with the collected data.

1. Introduction

Robots are becoming more and more popular every day. It is hard to find an industry where robots are not used. The most popular form of commonly used robots are robotic manipulators, which are employed in the industry to perform repetitive tasks, search and rescue missions [1], and inspections [2]. In the case of robot arms, the most popular approaches are derived from Euler–Lagrange equations [3]. Their complexity, expressed by many relevant parameters, is often difficult to estimate, forcing a search for new solutions. Another new approach to modelling dynamics is based on fractional calculus.

The concept of fractional calculus is not new and has been known since 1695, but it has recently gained popularity as a tool generalizing the commonly known calculus. It is becoming more and more popular in different research areas, ranging from electrical engineering, electromagnetism, electrochemistry, electronics, mechanics, and rheology to biophysics and economy [4,5,6]. Fractional calculus is also used to solve real-life problems, like control of autonomous vehicles [7], or to model and predict new cases of the COVID-19 [8]. In many instances fractional calculus allows modeling of the dynamics of more complex processes than their integer counterparts [9], for example, in the description of the transient behavior of a robot arm. The fractional system perspective in the study of the PRR (planar parallel robot) with three DOFs (degrees of freedom) was presented in [10]. Fractional calculus also has a generalization; when order varies with time, it is called variable fractional-order (VFO) calculus [11,12].

The concept of variable fractional-order integrals, and differentials was first proposed in 1993 [13]. Some complex physical phenomena show variable fractional order integrators, or differentiator properties [14]. Variable fractional-order calculus was implemented in many different fields. The increasing attention on theoretical analysis and physical modeling using VFO can be noticed. The experimental study of variable fractional-order differentiators and integrators was examined in [14]. Additionally, the viscoelasticity behavior of rubbery state polymers was modeled using variable fractional-order calculus [15]. The comparison of the prediction of the proposed VFO model and those previously published in the literature demonstrated the higher accuracy of the former. VFO calculus is top-rated in control problems. A study was conducted comparing variable fractional-order PID (VFOPID) controllers for different types of unstable order derivatives [16]. The VFOPID controllers opened new possibilities for shaping its transient characteristics [17,18,19]. The simulation results of the VFOPID controller applied to plants with delay showing its advantage over both traditional and fractional-order PID controllers [20]. The VFOPID also provides additional flexibility during the designing phase. Another version of controllers which can be enhanced by VFO is fuzzy PID controllers [21]. The variable fractional-order fuzzy PID (VFOFPID) controller was applied along with an active tuned mass damper to improve building structures’ performance when subjected to horizontal ground motions [22].

In the paper, we propose a simple model of a robot arm using a variable fractional-order difference equation. In the previous studies, the variable fractional-order discrete-time PD controller for this arm was proposed [16,23] along with the analysis of order function selection [24]. The variable fractional-order backward difference is introduced in Section 2. Next, the description of the robot arm is provided along with a discussion concerning its mathematical models. Then, some rules for the choice of the differentiation order functions are given. Finally, measured and simulated step responses of the robot arm are presented.

2. Mathematical Preliminaries

This Section presents essential information concerning the essential mathematical tool used in mathematical model building [3]. The derivative of the fractional order derivative called further fractional derivative is defined as an approximation of the following series [25,26,27].

$$\begin{gathered} \dots ,{\int }_a^t{\int }_a^{{\tau }_1}{\int }_a^{{\tau }_2}f\left({\tau }_3\right)d{\tau }_{3}d{\tau }_{2}d{\tau }_1, {\int }_a^t{\int }_a^{{\tau }_1}f\left({\tau }_2\right)d{\tau }_{2}d{\tau }_1, {\int }_a^{t}f\left({\tau }_1\right)d{\tau }_1 \\ f\left(t\right), \end{gathered}$$

(1)

$$\frac{df\left(t\right)}{dt}, \frac{d^{2}f\left(t\right)}{d{t}^2}, \frac{d^{3}f\left(t\right)}{d{t}^4}, \dots$$

In a Davis notation for any real function, $f\left(t\right)$ the fractional derivative is denoted as

$${}_{t_0}{}^{}D{}_t^{\left(\nu \right)}f\left(t\right),$$

(2)

where

• $t_0,t$—terminals of fractional differentiation,
• $\nu ϵ\mathrm{R}$—fractional order.
• The ${}_{t_0}{}^{}D{}_t^{\left(-\nu \right)}f\left(t\right)$ notation represents an integral of fractional order.

2.1. Grünwald–Letnikov Definition

For a given function $f\left(t\right)ϵ\mathrm{R}$ of a real variable $t,$ the first-order left derivative has the form

$$f^{\prime }\left(t\right)=\frac{df\left(t\right)}{dt}=\underset{h\to 0}{lim}\frac{f\left(t\right)-f\left(t-h\right)}{h},$$

(3)

The second-order left derivative is as follows

$$\begin{gathered} f^{″}\left(t\right)=\frac{d^{2}f\left(t\right)}{d{t}^2}=\underset{h\to 0}{lim}\frac{f^{\prime }\left(t\right)-f^{\prime }\left(t-h\right)}{h} \\ =\underset{h\to 0}{lim}\frac{\frac{f\left(t\right)-f\left(t-h\right)}{h}-\frac{f\left(t-h\right)-f\left(t-2h\right)}{h}}{h} \\ =\underset{h\to 0}{lim}\frac{f\left(t\right)-2f\left(t-h\right)+f\left(t-2h\right)}{h^2} \\ =\underset{h\to 0}{lim}\frac{1}{h^2}[1\hspace{1em}-2\hspace{1em}1]\left[\begin{array}{c}f\left(t\right)\\ f\left(t-h\right)\\ f\left(t-2h\right)\end{array}\right], \end{gathered}$$

(4)

For any order n, one has

$$f^{\left(n\right)}\left(t\right)=\frac{d^{n}f\left(t\right)}{d{t}^n}=\underset{h\to 0}{lim}\frac{1}{h^n}\left[\begin{array}{cccc}{a}_0^{\left(n\right)}& a_1^{\left(n\right)}& \cdots & a_n^{\left(n\right)}\end{array}\right] \left[\begin{array}{c}f\left(t\right)\\ f\left(t-h\right)\\ ⋮\\ f\left(t-nh\right)\end{array}\right],$$

(5)

or

$$f^{\left(n\right)}\left(t\right)=\frac{d^{n}f\left(t\right)}{d{t}^n}=\underset{h\to 0}{lim}\frac{1}{h^n}{\displaystyle \sum }_{i=0}^n{a}_i^{\left(n\right)}f\left(t-hi\right),$$

(6)

where the coefficients $a_i^{\left(n\right)}$ are defined as follows

$$a_i^{\left(n\right)}=\{\begin{array}{ccc}1& for& i=0\\ {(-1)}^i\frac{n\left(n-1\right)\left(n-2\right)\cdots \left(n-i+1\right)}{i!}& for& i=1,2,3,\cdots \end{array}$$

(7)

or

$$a_i^{\left(n\right)}=a_{i-1}^{\left(n\right)}\left(1-\frac{n+1}{i}\right),$$

(8)

The above formulas are valid for any order including fractional ones. Then, one has

$${}_{t_0}{}^{}D{}_t^{-n}f\left(t\right)=\underset{h\to 0}{lim}{h}^{-n}{\sum }_{i=0}^{\frac{t-t_0}{h}}{a}_i^{\left(-n\right)}f\left(t-hi\right)=\frac{1}{\left(n-1\right)!} {\int }_{t_0}^t{\left(t-\tau \right)}^{n-1}f\left(\tau \right)d\tau ,$$

(9)

The variable fractional-order backward difference (VFOBD) of a bounded discrete function $f_k$ is defined as follows:

$${}_0{}^{GL}\mathsf{\Delta }{}_k^{\left({\nu }_k\right)}{f}_k=a_k^{\left({\nu }_k\right)}\ast f_k={\sum }_{i=0}^k{a}_i^{\left({\nu }_k\right)}{f}_{k-i},$$

(10)

where

$$a_k^{\left({\nu }_k\right)}=\{\begin{array}{ccc}0& \mathit{for}& k<0\\ 1& \mathit{for}& k=0\\ {\left(-1\right)}^k\left(\begin{array}{c}{\nu }_k\\ k\end{array}\right)& \mathit{for}& k\ge 1\end{array},$$

(11)

$$\begin{gathered} \left(\begin{array}{c}{\nu }_k\\ k\end{array}\right)=\frac{{\nu }_k\left({\nu }_k-1\right)\left({\nu }_k-2\right)\cdots \left({\nu }_k-k+1\right)}{k!}, \\ {v}_k-order of the fractional difference in the time step k. \end{gathered}$$

(12)

On the basis of the above equations, one derives a general formula

$$\begin{gathered} \left[\begin{array}{ccccc}{a}_0^{\left(n\right)}& a_1^{\left(n\right)}& \cdots & a_{n-1}^{\left(n\right)}& a_n^{\left(n\right)}\end{array}\right]=\left[\begin{array}{ccccc}{a}_0^{\left(n-m\right)}& a_1^{\left(n-m\right)}& \cdots & a_{n-1}^{\left(n-m\right)}& a_n^{\left(n-m\right)}\end{array}\right] \\ \left[\begin{array}{cccccccc}{a}_0^{\left(m\right)}& a_1^{\left(m\right)}& a_2^{\left(m\right)}& a_3^{\left(m\right)}& \cdots & 0& 0& 0\\ 0& a_0^{\left(m\right)}& a_1^{\left(m\right)}& a_2^{\left(m\right)}& \cdots & 0& 0& 0\\ 0& 0& a_0^{\left(m\right)}& a_1^{\left(m\right)}& \cdots & 0& 0& 0\\ 0& 0& 0& a_0^{\left(m\right)}& \cdots & 0& 0& 0\\ ⋮& ⋮& ⋮& ⋮& & ⋮& ⋮& ⋮\\ 0& 0& 0& 0& \cdots & a_0^{\left(m\right)}& a_1^{\left(m\right)}& a_2^{\left(m\right)}\\ 0& 0& 0& 0& \cdots & 0& a_0^{\left(m\right)}& a_1^{\left(m\right)}\\ 0& 0& 0& 0& \cdots & 0& 0& a_0^{\left(m\right)}\end{array}\right], \end{gathered}$$

(13)

Equation (13) denotes a discrete convolution and ${\nu }_k>0$ is the differentiation order function. For negative order functions, Equation (10) defines variable fractional-order integration (discrete summation). One should realize that for integer order ${\mathbb{Z}}_{+}\ni {\nu }_k=n>0,$ one immediately obtains a classical backward difference.

$${}_0{}^{GL}\mathsf{\Delta }{}_k^{\left(n\right)}{f}_k=a_k^{\left(n\right)}\ast f_k={\sum }_{i=0}^k{a}_i^{\left(n\right)}{f}_{k-i}={\mathsf{\Delta }}^{\left(n\right)}{f}_k,$$

(14)

The above difference may serve as an approximation of a left derivative.

$$\frac{d^{n}f\left(t\right)}{d{t}^n}={\frac{d^{n}f\left(t\right)}{d{t}^n}|}_{t=hk}\approx \frac{{}_0{}^{GL}\mathsf{\Delta }{}_k^{\left(n\right)}{f}_{kh}}{h^n}=\frac{1}{h^n}{\sum }_{i=0}^k{a}_i^{\left(n\right)}{f}_{kh-ih}=\frac{{\mathsf{\Delta }}^{\left(n\right)}{f}_k}{h^n},$$

(15)

2.2. The Fractional Order Integrator

As in classical integer order dynamical systems case, on a base of derivatives, one can build differential equations. The simplest one is an equation describing the fractional integrator. So in the fractional systems case,

$$y\left(t\right)=T_u{}_0{D}_t^{-v}u\left(t\right),$$

(16)

where y(t) and u(t) such that u(t) for t < 0 denote the system output and input signals, respectively. $\nu$ is an order of the integrator whereas $0<T_u\in R$ is a time constant. For a zero initial condition, the one-sided Laplace transform is of the form which can be further represented by its transfer function.

$$G_{FI}\left(s,T_u\right)=\frac{Y\left(s\right)}{U\left(s\right)}=\frac{1}{T_u{s}^{\nu }} ,$$

(17)

The impulse response is described as

$$g_D\left(t,T_u\right)={ℒ}^{-1}\left\{G_{FI}\left(s,T_u\right)\right\}={ℒ}^{-1}\left\{\frac{1}{T_u{s}^{\nu }}\right\}=\frac{1}{T_u}\frac{t^{\nu -1}}{\Gamma \left(\nu \right)},$$

(18)

where $\Gamma \left(\nu \right)$ is the Euler gamma function. The unit step response is described as follows

$$h_D^{\left(\nu \right)}I\left(t\right)={ℒ}^{-1}\left\{\frac{1}{s}{G}_{FI}^{\left(\nu \right)}\left(s,T_u\right)\right\}={ℒ}^{-1}\left\{\frac{1}{T_u\left(s^{\nu +1}\right)}\right\}=\frac{t^{-\nu }}{\Gamma \left(1-\nu \right)},$$

(19)

The fractional integrator response to any signal can be described by an integral

$$y\left(t\right)={\int }_0^t\frac{{\theta }^{-\nu -1}}{\Gamma \left(-\nu \right)}u\left(t-\theta \right)d\theta ={\int }_0^t\frac{{(t-\theta )}^{-\nu -1}}{\Gamma \left(-\nu \right)}u\left(\theta \right)d\theta ,$$

(20)

2.3. Fractional Derivative

In this subsection, one considers the formula

$$\begin{gathered} f_h^{\left(\nu \right)}\left(t\right)=\frac{1}{h^{\nu }}{\sum }_{i=0}^{\frac{t-t_0}{h}}{a}_i^{\left(\nu \right)}f\left(t-hi\right)= \\ \frac{1}{h^{\nu }}\left[\begin{array}{ccccc}{a}_0^{\left(\nu \right)}& a_1^{\left(\nu \right)}& \cdots & a_{n-1}^{\left(\nu \right)}& a_n^{\left(\nu \right)}\end{array}\right]\left[\begin{array}{c}f\left(t\right)\\ f\left(t-1h\right)\\ ⋮\\ f\left(t-nh+h\right)\\ f\left(t-nh\right)\end{array}\right], \end{gathered}$$

(21)

where $\nu \in {ℝ}_{+}$, $h>0$ and $n=E\left(\frac{t-t_0}{k}\right)$ is an integer part of a number. Based on the above formula, one defines a fractional order derivative. It is called the Grünwald–Letnikov fractional order derivative [16].

$${}_{t_0}{}^{}D{}_t^{\left(\nu \right)}f\left(t\right)=\underset{h\to 0}{lim} f_h^{\left(\nu \right)}\left(t\right)=\underset{h\to 0}{lim} \frac{1}{h^{\nu }}{\displaystyle \sum }_{i=0}^{\frac{t-t_0}{h}}{a}_i^{\left(\nu \right)}f\left(t-hi\right),$$

(22)

Fractional Derivative Selected Properties

Fractional derivative linearity property for two differentiable functions $f_1\left(t\right), f_2\left(t\right)$ and constants ${\mathrm{a}}_1,{\mathrm{a}}_2,$ there is

$${}_{t_0}{}^{}D{}_t^{\nu }\left[a_1{f}_1\left(t\right)+a_2{f}_2\left(t\right)\right]=a_1{}_{t_0}{}^{}D{}_t^{\nu }\left[f_1\left(t\right)\right]+a_2{}_{t_0}{}^{}D{}_t^{\nu }\left[f_2\left(t\right)\right],$$

(23)

The fractional derivative superposition property for any

$${}_{t_0}{}^{}D{}_t^{{\nu }_1+{\nu }_2}\left[f\left(t\right)\right]={}_{t_0}{}^{}D{}_t^{{\nu }_1}\left\{{}_{t_0}{}^{}D{}_t^{{\nu }_2}\left[f\left(t\right)\right]\right\}={}_{t_0}{}^{}D{}_t^{{\nu }_2}\left\{{}_{t_0}{}^{}D{}_t^{{\nu }_1}\left[f\left(t\right)\right]\right\},$$

(24)

3. The Robot Arm Description

The presented solution is an element of building a comprehensive simulation environment for the robot’s actuator in the form of an arm. The study of the system properties is primarily associated with the possibly precise reconstruction of the actual course of the arm’s work. Mathematical models are used for this purpose. The model of the described arm does not consider the impact of the movement of the mobile platform, and it was assumed that all degrees of freedom would be taken away from it. The motion system of the presented arm is a simple open kinematic chain, in which the adjacent members relate to each other by means of a kinematic pair. The movement of individual links is independent, performed using dedicated harmonic gears with the ability to maintain a specific position. The mechanical structure of the robot arms allows for their further expansion. The modularity of the structure allows it to quickly change its configuration with no necessity to modify the operation of the entire system. The observation head can be replaced with a gripper or a specialized tool or supplemented with further links. The assumed possibilities of further expansion of the mechanical structure allow for its adaptation to the requirements of the future user, both in the defense and civil operations zone. The whole construction is controlled by a microprocessor system with implanted algorithms using original control procedures with controllers with variable non-integer orders of differentiation. They are a generalization of classical PD regulators. Regulators of this type can be included in the so-called adaptive regulators, where the variables are not so-called controller settings but differentiation and summation orders. Before commencing the work regulation process, appropriate executive systems had to be selected. During the numerical simulations, the following values of the manipulator parameter were assumed:

• m1 = 1.0 [kg]—the weight of the drive systems;
• m2 = 7.0 [kg]—total weight of the manipulator;
• m = 0.5 [kg]—weights of the manipulator links;
• l = 0.5 [m]—the length of the manipulator links;
• M₀(t), M₁(t)—driving torque.

Selected ranges of displacements φ₁, and φ₂ of the manipulator links in the given time interval Δt were subjected to numerical analysis in the context of selecting the values and time courses of the drive torques required to complete the given trajectory of the manipulator links. Two basic cases of the initial positions of both links were considered. The calculations assumed the same time course, presented in Figure 1, for changes in the moments M₀(t), M₁(t), where:

• M₀, M₁—fixed values of driving torque;
• Δt₀—start-up phase, an increase of drive torques to the assumed fixed values;
• Δt₁—phase of steady drive torques;
• Δt₂—phase of reduction of drive torques to zero;
• Δt₃—stop phase drive torques equal zero;
• l = 0.5 [m]—the length of the manipulator links.

During the analysis, two prominent cases were studied (Figure 2a,b), mainly setting manipulators in the horizontal and vertical orientation. The first analyzed case is to set the manipulator in vertical position φ₁ = π/2, φ₂ = π/2 (Figure 2b) starting from horizontal position φ₁ = 0, φ₂ = π (Figure 2a) in time Δt = 1 [s].

From the application point of view, it is important to achieve the required position of the links with the minimum possible values of velocities and accelerations to stop the manipulator. In order to achieve the optimal effect, the following links position and speed functions were considered during numerical simulations:

$$f\left(x\right)={\left({\phi }_1-\frac{\pi }{2}\right)}^2+{\left({\phi }_2-\frac{\pi }{2}\right)}^2,$$

(25)

$$f\left(v\right)={\dot{\phi }}_1^2+{\dot{\phi }}_2^2,$$

(26)

When a function (25) reaches the minimum value of zero, the vertical position of the links is realized (Figure 2b). On the other hand, the minima of the function (26) indicate the lowest possible velocities of the manipulator links after a given time Δt. Figure 3a,b show three-dimensional projections of surfaces representing the functions f(x) and f(v) over the two-dimensional space of parameters M₀, and M₁, while Figure 4a,b shows (in red) areas of these parameters, where functions (25) and (26) reach minimum values. The calculations were carried out by numerically integrating the equations of motion using the 4th order Runge–Kutta method. The time values of the drive parameters (Figure 1) are Δt₀ = 0.05 [s], Δt₁ = 0.45 [s], Δt₂ = 0.30 [s], and Δt₃ = 0.20 [s].

The time intervals of the driving torques were selected based on numerical analysis, which was carried out at the fixed values M₀ = 9.0 [Nm], M₁ = 48.0 [Nm], and the constant on start-up phase Δt₀ = 0.05 [s]. However, the parameters Δt₁ and Δt₂ changed during analysis (Δt₃ is the result of select the remaining intervals). The results of this numerical experiment are presented in Figure 5a,b, where the quantities f(x) and f(v) are given as a function of the time intervals Δt₁ and Δt₂. It can be seen that the possibly low-speed values (green areas in the bottom left of the diagram) in Figure 5b coincide with the range where the manipulator reaches the vertical position (green stripe in Figure 5a).

In the last phase of movement, Δt₃ the drive is switched off. Such control of drive torques allows us to achieve the lowest possible values of velocities and angular accelerations when the manipulator links reach the vertical position. This fact is confirmed by the time courses of displacements, velocities, and accelerations presented in Figure 6. The determined driving torques are M₀ = 9.46 [Nm] and M₁ = 49.44 [Nm]. These moments deviate slightly from the optimal values resulting from Figure 4a,b (areas marked in red). The reason for this slight discrepancy may be a different method (Euler’s algorithm) or the step of integration of the equations of motion in the case of the diagrams shown in Figure 6. It can be seen that at Δt = 1.0 [s] the cells occupy the required vertical position φ₁ = φ₂ = π/2 at relatively low values of velocities and accelerations (Figure 6).

Figure 7 shows the results of the numerical analysis analogous to the one shown in Figure 6, but taking into account the trajectory limitation of the links by the presence of the base. Contact with the ground means that the first link cannot occupy positions defined by a negative angle φ₁. Hence, when integrating equations, an additional condition was introduced in the form “if φ₁ < 0 then φ₁ = 0, ${\dot{\phi }}_1$ = 0”. The application of this condition allowed us, for the assumed Δt and the vertical position of the links (Figure 7), to obtain the optimal values (Figure 7) at slightly lower values for the drive torques M₀ (8.89 Nm) and M₁ (48.27 Nm).

The second of the considered cases is shown in Figure 8. The task is to set the manipulator in a vertical position φ₁ = π/2, φ₂ = π/2 (Figure 2b) from the initial horizontal position φ₁ = 0, φ₂ = 0 of the links (Figure 8).

Numerical experiments have shown that an attempt to place the manipulator with positively directed moments M₀(t), M₁(t) (Figure 8) requires the application of a maximum torque of at least M₀ ≈ 90 [Nm] in node “0”, both in the case of operation of both drives, as well as in the case of applying only the M₀(t) moment with the locked node “1”. On the other hand, the driving power demand is much lower when lifting the manipulator in stages, i.e., moving the link 1 from the position φ₂ = π/2 (Figure 8) to the position φ₂ = 0 in the first stage, using only the drive M₁(t), and then setting the links in a vertical position in accordance with the case 1 described above, using both drives.

Setting the link 2 in a vertical position (with the locked link 1) with a driving torque of a fixed value M₁ = 42.4 [Nm] in time Δt = 1.0 [s] is shown in Figure 9. The time intervals of the torque M₁(t) are: Δt₀ = 0.05 [s], Δt₁ = 0.75 [s], Δt₂ = 0.20 [s], Δt₃ = 0.0 [s].

Figure 10 presents a block diagram of the mathematical model of the robot arm. The arm work is modeled as a simple first-order integrator with negative feedback generated by the moment caused by the gravitational force of the robot arm. Thanks to the use of harmonic gears in the electromagnetic brake in the kinematic nodes, the dynamics of motion can be simplified to the presented model. The correctness of adopting the simplified model can be proven by the presented transient states of the actuator of the described object.

Connecting to the inverter positive feedback with a one-step delay block causes the PWM inverter to work as a discrete-time integrator. In order to protect the system against doubled integrating action in the robot arm angular velocity control loop, one introduces additional negative feedback with the discrete differentiation to obtain a closed-loop system treated. Figure 11 presents the block diagram of the first robot arm angular velocity control system.

4. The Robot Arm’s Mathematical Models

To design optimal manipulator control algorithms, it is necessary to prepare a correct mathematical model describing it [28]. The trade-off between model accuracy and its complexity should also be taken into consideration. In this section two mathematical models of the robot arm are analyzed. The first one is derived from the well-known Newton—Euler formalism whereas in the second the Fractional Calculus is applied. To simplify the calculation model, it was assumed that the mobile platform on which the arm is located is stationary. In addition, the horizontal position of the trolley was assumed, as well as a non-deformable suspension.

4.1. Classical Mathematical Model (CM) of the Robot Arm

The theoretical model of a robot arm is presented in Figure 12 [6,10].

Following the Euler–Lagrange formalism, two classical differential second-order equations describing the dynamic system are as follows

$$D\left[q\left(t\right)\right]\ddot{q}\left(t\right)+C\left[q\left(t\right),\dot{q}\left(t\right)\right]\dot{q}\left(t\right)+g\left[q\left(t\right)\right]=\tau \left(t\right)$$

(27)

where

$$\tau \left(t\right)=\left[\begin{array}{c}{\tau }_1\left(t\right)\\ {\tau }_2\left(t\right)\end{array}\right]– \mathrm{drive} \mathrm{moments},$$

$$q\left(t\right)=\left[\begin{array}{c}{q}_1\left(t\right)\\ q_2\left(t\right)\end{array}\right]– \mathrm{coordinates} \mathrm{of} \mathrm{the} \mathrm{manipulator} \mathrm{links} \mathrm{position},$$

$$\begin{gathered} D\left[q\left(t\right)\right]=D=\left[\begin{array}{cc}{I}_1+I_2& I_2\\ I_2& I_2\end{array}\right]–\mathrm{moments} \mathrm{of} \mathrm{inertia} \mathrm{matrix} \\ \mathrm{of} \mathrm{the} \mathrm{driving} \mathrm{system}, \end{gathered}$$

$$C\left[q\left(t\right),\dot{q}\left(t\right)\right]=\left[\begin{array}{cc}h\left(t\right){\dot{q}}_2\left(t\right)& h\left(t\right){\dot{q}}_2\left(t\right)+h\left(t\right){\dot{q}}_1\left(t\right)\\ -h\left(t\right){\dot{q}}_1\left(t\right)& 0\end{array}\right]=$$

$$\begin{gathered} =m_2{l}_1{l}_{c2}sin\left[q_2\left(t\right)\right]\left[\begin{array}{cc}-{\dot{q}}_2\left(t\right)& -{\dot{q}}_2\left(t\right)-{\dot{q}}_1\left(t\right)\\ {\dot{q}}_1\left(t\right)& 0\end{array}\right], \\ h\left(t\right)=-m_2{l}_1{l}_{c2}sin\left[q_2\left(t\right)\right], \\ g\left[q\left(t\right)\right]=g\left[\begin{array}{c}\left(m_1{l}_{c1}+m_2{l}_1\right)cos\left[q_1\left(t\right)\right]+m_2{l}_{c2}cos\left[q_1\left(t\right)+q_2\left(t\right)\right]\\ m_2{l}_{c2}cos\left[q_1\left(t\right)+q_2\left(t\right)\right]\end{array}\right], \end{gathered}$$

• $\mathrm{g}$—acceleration of the Earth,
• $l_1,l_2$—lengths of the robot arm,
• $l_{c1},l_{c2}$—distance from the precedent joint to the arm centroid.

Now one assumes that the second arm is immobilized. This means that ${\ddot{q}}_2\left(t\right)=0$,

${\dot{q}}_2\left(t\right)=0$, and $q_2\left(t\right)=q_2=\mathrm{const}$, ${\tau }_2\left(t\right)=0$. From Equation (27), one obtains

$$\begin{gathered} \left[\begin{array}{cc}{I}_1+I_2& I_2\\ I_2& I_2\end{array}\right]\ddot{q}\left(t\right)+{\dot{q}}_1\left(t\right)m_2{l}_1{l}_{c2}sin\left[q_2\right]\left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right]\dot{q}\left(t\right) \\ +g\left[\begin{array}{c}\left(m_1{l}_{c1}+m_2{l}_1\right)cos\left[q_1\left(t\right)\right]+m_2{l}_{c2}cos\left[q_1\left(t\right)+q_2\right]\\ m_2{l}_{c2}cos\left[q_1\left(t\right)+q_2\right]\end{array}\right]=\left[\begin{array}{c}{\tau }_1\left(t\right)\\ 0\end{array}\right] \end{gathered}$$

(28)

Pre-multiplication of both sides of Equation (28) by a matrix

$${\left[\begin{array}{cc}{I}_1+I_2& I_2\\ I_2& I_2\end{array}\right]}^{-1}=\left[\begin{array}{cc}\frac{1}{I_1}& -\frac{1}{I_1}\\ -\frac{1}{I_1}& \frac{1}{I_1}+\frac{1}{I_2}\end{array}\right]$$

(29)

yields

$$\begin{gathered} \ddot{q}\left(t\right)+{\dot{q}}_1\left(t\right)m_2{l}_1{l}_{c2}sin\left[q_2\right]\left[\begin{array}{cc}0& \frac{1}{I_1}\\ \frac{1}{I_2}& -\frac{1}{I_1}\end{array}\right]\dot{q}\left(t\right) \\ +g\left[\begin{array}{cc}\frac{1}{I_1}& -\frac{1}{I_1}\\ -\frac{1}{I_1}& \frac{1}{I_1}+\frac{1}{I_2}\end{array}\right]\left[\begin{array}{c}\left(m_1{l}_{c1}+m_2{l}_1\right)cos\left[q_1\left(t\right)\right]+m_2{l}_{c2}cos\left[q_1\left(t\right)+q_2\right]\\ m_2{l}_{c2}cos\left[q_1\left(t\right)+q_2\right]\end{array}\right] \\ =\left[\begin{array}{c}{\tau }_1\left(t\right)\\ 0\end{array}\right] \end{gathered}$$

(30)

Further transformations lead to

$$\begin{gathered} \left[\begin{array}{c}{\ddot{q}}_1\left(t\right)\\ 0\end{array}\right]+m_2{l}_1{l}_{c2}sin\left(q_2\right)\left[\begin{array}{c}0\\ \frac{{\dot{q}}_1^2\left(t\right)}{I_2}\end{array}\right] \\ +g\left[\begin{array}{c}\frac{\left(m_1{l}_{c1}+m_2{l}_1\right)cos\left[q_1\left(t\right)\right]}{I_1}\\ \frac{\left(m_1{l}_{c1}+m_2{l}_1\right)cos\left[q_1\left(t\right)\right]}{I_1}+\frac{m_2{l}_{c2}cos\left[q_1\left(t\right)+q_2\right]}{I_2}\end{array}\right]=\left[\begin{array}{c}{\tau }_1\left(t\right)\\ 0\end{array}\right] \end{gathered}$$

(31)

From the first row, one has

$${\ddot{q}}_1\left(t\right)+\frac{g\left(m_1{l}_{c1}+m_2{l}_1\right)cos\left[q_1\left(t\right)\right]}{I_1}={\tau }_1\left(t\right)$$

Additionally, the second row must be satisfied

$$\begin{gathered} \frac{m_2{l}_1{l}_{c2}sin\left(q_2\right)}{I_2}{\dot{q}}_1^2\left(t\right)+\frac{g\left(m_1{l}_{c1}+m_2{l}_1\right)cos\left[q_1\left(t\right)\right]}{I_1} \\ +\frac{g{m}_2{l}_{c2}cos\left[q_1\left(t\right)+q_2\right]}{I_2}=0 \end{gathered}$$

(32)

As a discrete time, mathematical model, one takes an approximation of the derivative by a difference

$$\begin{gathered} \frac{q_1\left(kh\right)-2q_1\left(kh-h\right)+q_1\left(kh-2h\right)}{h^2}+\frac{g\left(m_1{l}_{c1}+m_2{l}_1\right)cos\left[q_1\left(kh\right)\right]}{I_1} \\ ={\tau }_1\left(kh\right) \end{gathered}$$

(33)

In further considerations to verify adopted assumptions, only the first link was the subject of the analysis. Mechanical friction and backlash are omitted. Now one denotes:

$$K=\frac{g\left(m_1{l}_{c1}+m_2{l}_1\right)}{I_1}$$

(34)

Hence, under the assumptions mentioned above, in the model there is only one parameter to evaluate.

4.2. Variable Fractional-Order Mathematical Model (VFOM) of the Robot Arm

The VFOBD may will be applied to model the robot arm. From the general form of a linear, time-invariant variable fractional-order difference equation:

$${\displaystyle \sum }_{i=0}^n{A}_i{}_0{}^{GL}\mathsf{\Delta }{}_k^{\left({\nu }_k\right)}{y}_k={\displaystyle \sum }_{i=0}^m{B}_i{}_0{}^{GL}\mathsf{\Delta }{}_k^{\left({\mu }_k\right)}{u}_k$$

(35)

where

• $A_i,B_j$—constant coefficients with $A_n=1$,
• ${\nu }_{n,k}>{\nu }_{n-1,k}>\cdots >{\nu }_{1,k}>{\nu }_{0,k}=0$—order functions,
• ${\mu }_{m,k}>{\mu }_{m-1,k}>\cdots >{\mu }_{1,k}>{\mu }_{0,k}\ge 0$—order functions,
• $u_k,y_k$—the robot arm input and output functions.

The robot arm possesses the integration element properties. Hence, in the first approach one considers (14) with $n=1,m=0$ and ${\mu }_k=0$. Then, from (35) one gets the equation:

$${}_0{}^{GL}\mathsf{\Delta }{}_k^{\left({\nu }_k\right)}{y}_k=B_0{u}_k$$

(36)

describing a variable fractional-order integrator (VFO-I) [25]. In such simple model structure, there are two parameters to select: a coefficient $B_0$ and an order function ${\mathsf{\nu }}_{\mathrm{k}}$. Note that this is a generalization of the FO discrete integrator, which in turn is a generalization of the classical (first-order) integrator. By (10) and (11), a response of the VFO-I is of the form:

$$y_k=-\left[\begin{array}{cccc}{a}_1^{\left({\nu }_k\right)}& a_2^{\left({\nu }_k\right)}& \cdots & a_k^{\left({\nu }_k\right)}\end{array}\right]\left[\begin{array}{c}{y}_{k-1}\\ y_{k-2}\\ ⋮\\ y_0\end{array}\right]+B_0{u}_k$$

(37)

It is easy to prove that Equation (37) can be transformed to an equivalent form

$$y_k=B_0{}_0{}^{GL}\mathsf{\Delta }{}_k^{\left({\mu }_k\right)}{u}_k$$

(38)

where

$${\mu }_k=-{\nu }_k$$

Because one may expect non-linear behavior, one takes that $B_0\left(u_k\right)$. For a better modelling elasticity, one also assumes that the integration order depends on the input signal. Hence, the final model structure has the form:

$$y_k=B_0\left(u_k\right){}_0{}^{GL}\mathsf{\Delta }{}_k^{\left[{\mu }_k\left(u_k\right)\right]}{u}_k$$

(39)

The block diagram of the Wiener VFOM is presented in Figure 13.

5. The Robot Arm Tests

The examined system is part of a larger project, which is the development of a comprehensive design of the mobile platform. Figure 14 shows the executive part of the robot with an observation head. The drive in this mechanism is carried out by means of a DC electric drive motor with harmonic gear powered by a single-phase PWM inverter. Positive feedback to the inverter occurs with a delay of one trial period.

To the robot arm electrical drive input, several constant voltages were supplied. Zero initial conditions were assumed although the initial angle is nonzero. This is caused by the robot construction. Hence, there were measured step responses of the robot arm. The input constant voltages are collected in a set $\mathbb{U}$:

$$\mathbb{U}=\left\{3,4,5,6,7,8,9,10,15,20,25\right\} \left[\mathrm{V}\right]$$

(40)

The responses are presented in Figure 15. It can be seen in Figure 15 that higher input voltages cause the faster responses of the robot arm.

6. The VFOM and CM Parameters Selection

In this Section, both models’ parameter selection methods are described. The main fitting criterion is a minimum of the square error defined as a difference between measured real system responses $y_{m,k}$ and simulated $y_{s,k}$ ones

$$I_{CM}=\underset{K,\left[0 k\right]}{\min }{\left|y_{m,k}-y_{s,k}\left(K\right)\right|}^2$$

(41)

$$I_{VFOM}=\underset{B_0,{\nu }_k,\left[0 k\right]}{\min }{\left|y_{m,k}-y_{s,k}\left(B_0,{\nu }_k\right)\right|}^2$$

(42)

First, the coefficient $B_0$ in formula (36) is evaluated. Here, one initially assumes that ${\nu }_k=1$. Then, for $u_k=u_0{1}_k$ ($1_k$ denotes the unit discrete step function) one gets

$$y_{j,k}=B_0{\sum }_{i=0}^k{u}_i=B_0{u}_{j,0}kh for u_{j,0}\in \mathbb{U}$$

(43)

where $h$ denotes a sampling time. For the chosen time instants (for which the responses achieve for the first time its maximal value) from (43) one gets coefficients

$$B_{j,0}\left(u_{j,0}\right)=\frac{y_{k_{max}}}{u_{j,0}{k}_{max}h} for u_{j,0}\in \mathbb{U}$$

(44)

Plots of $B_{j,0}\left(u_{j,0}\right)y_{j,k}$ are presented in Figure 16. The coefficients (44) are plotted in Figure 17.

This enables us to evaluate the static characteristic of the model. It is presented in Figure 18. To check the response shapes, the time coefficients are added. The coefficients $w_j\left(u_{j,0}\right)$ are selected in order to get the same transient times. Comparison of all such responses revealed its changes due to the input signal value. In Figure 18 all mentioned responses are presented.

As the order functions, one takes

$${\nu }_k=-1+c\left(u_k\right)e^{dk}$$

(45)

with $0<c\left(u_k\right)<1,d<0$. For $c\left(u_k\right)$ and $d$ parameters, selection of an ISE (integrated square error) criterion was applied.

$$I\left[c\left(u_k\right),d\right]={\displaystyle \sum }_{i=0}^{k_{max}}{e}_i^2$$

(46)

where

$$e_k=y_k-y_{s,k}=y_k-B_0\left(u_k\right){}_0{}^{GL}\mathsf{\Delta }{}_k^{\left[-1+c\left(u_k\right)e^{dk}\right]}{u}_k$$

(47)

For $0<c\left(u_k\right)<1,$ a plot of coefficients $c\left(u_k\right)$ vs. $u_k$ is presented in Figure 19.

The responses that were measured and simulated for $d=-0.001$ are presented in Figure 20. Figure 20 also shows the response obtained by simulation of the classical model where $\frac{g\left(m_1{l}_{c1}+m_2{l}_1\right)}{I_1}=0.6$ and $K=0.1787$ are added (in blue).

In Figure 21, similar responses for different input signals are compared.

To compare the simulation results, two performance criteria were used: IAE and ISE. The first error is defined by formula (45) and the second is defined as $e_k=y_k-q_1\left(\mathrm{kh}\right)$ where $h=0.0215$ denotes the simulation step. The obtained IAE and ISE criteria values are given in Figure 22.

7. Conclusions

In this article, two mathematical models of a robot arm were analyzed: one derived from the well-known Newton–Euler formalism, and the second described by variable fractional-order difference equations. Both models were compared with measured values of a robot arm. Based on the two criteria used for comparison (IAE and ISE), it was shown that the model described by variable fractional-order difference equations gave better result. The proposed variable fractional-order difference equation-based model had more parameters than the classical one, which led to a better model of the actual behavior of the modeled object. The open question is the selection of the right order function, which can be related to the measured real-time signals in the system. The next step is also to investigate the impact of platform movement on the tested system. In the real model, the actuator in the form of an arm is mounted on a six-wheel mobile platform with independent suspension of each wheel. We intend to develop the model with elements of the dynamics of the platform’s movement (change in speed and direction of movement, impact of ground shape, etc.) on the robot arm.