The Kinematic Investigation of the Stephenson-III Spherical Mechanism with a Spherical Slider Containing a Spherical Prismatic Pair

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The methodology proposed in this paper studies the mobility of a multi-loop spherical mechanism with a spherical slider containing a spherical prismatic pair. By analyzing the motion of each closed kinematic chain that the mechanism consists of, not only can the proposed methodology thoroughly and accurately identify the mechanism’s singularity configurations, but the method can also demonstrate the mechanism’s range of motion. Therefore, the proposed method ultimately identifies the mechanism’s motion continuity and smoothness, which are two of the most essential characteristics a multi-loop spherical mechanism possesses. Hence, the proposed methodology can help mechanism designers create multi-loop mechanisms with spherical sliders without motion deficiencies by avoiding the mechanisms’ singularity configurations.

Abstract

Multi-loop spherical mechanisms are extremely beneficial for creating versatile mechanical devices, including robotic joints and surgical tools, since multi-loop spherical mechanisms possess unique capabilities to operate in spatial situations with relatively simple movement. Nevertheless, the research on multi-loop spherical mechanisms with spherical sliders containing spherical prismatic pairs is lacking. Therefore, the main innovation of this paper is to propose the Stephenson-III two-loop spherical mechanism that possesses a spherical slider containing a spherical prismatic pair and to analyze the proposed spherical mechanism’s motion characteristics. An algebraic approach was employed to obtain the branch graphs of the proposed spherical mechanism with a spherical slider. The branch graphs were categorized into two types, according to whether branch points existed. With the algebraic approach, loop equations of the two spherical kinematic chains inside the proposed spherical mechanism were established to identify the input–output curves and singularity curves, with which the branch graphs were obtained. With the branch graphs, the joint rotation spaces (JRSs) of the proposed mechanism were recognized and so were the dead center positions, branches, sub-branches, and branch points. The results from the mathematical analysis were simulated and verified by three-dimensional (3D) models of the proposed spherical mechanism. The analytical results demonstrate that the spherical prismatic pair diversifies the motion of the proposed spherical mechanism by producing rotational sliding movement, which can cover the entire circumference of a specific greater circle on the proposed mechanism’s sphere.

1. State-of-the-Art in Studying Spherical Mechanisms

Multi-loop spherical mechanisms are the backbone of many mechanical structures, such as surgical devices [1] and rehabilitation devices [2]. Compared with planar mechanisms, multi-loop spherical mechanisms are able to operate in three-dimensional spaces while maintaining relatively simple movement. Similar to a planar link of a multi-loop planar mechanism, a spherical link of a multi-loop spherical mechanism has at most three degrees of freedom (DOF); similar to a planar kinematic pair of a multi-loop planar mechanism, a spherical kinematic pair of a multi-loop spherical mechanism has at most two degrees of constraint. Therefore, the algebraic method to analyze the motion of a multi-loop planar mechanism’s closed kinematic chains is similar to the algebraic method to study the motion characteristics of a multi-loop spherical mechanism’s closed kinematic chains. A spherical mechanism, which consists of one closed loop which is also called a closed kinematic chain, or multiple closed loops, has multiple spherical links and kinematic pairs. For instance, a single-DOF single-loop four-bar spherical mechanism with only revolute pairs has four spherical links and four revolute pairs, which together form a closed four-bar kinematic chain. On the basis of the N-bar rotatability laws [3,4], the concept of joint rotation space (JRS) [5], and the discriminant method [6], input–output curves, singularity curves, dead center positions, branches, sub-branches, and branch points are crucial elements to understanding a multi-loop spherical mechanism’s motion. The input–output curve of a single-DOF multi-loop spherical mechanism describes the relationship between the input and output joints. Dead center positions, where the single-DOF multi-loop spherical mechanism’s motion is impeded, sometimes appear on input–output curves. A singularity curve is a collection of all the singularity configurations of a closed spherical kinematic chain; a branch is the range of motion where a multi-loop spherical mechanism operates continuously without impediments; branch points are the intersection points of the input–output curves and singularity curves or the intersection points of singularity curves. Singularity analysis [7,8] and branch identification [9,10] are some of the most efficient approaches to identify input–output curves, singularity curves, dead center positions, branches, sub-branches, and branch points of multi-loop spherical mechanisms with closed kinematic chains.

Since multi-loop spherical mechanisms can generate spatial motion that multi-loop planar mechanisms cannot create, many scholars around the world have designed novel multi-loop spherical mechanisms of closed kinematic chains with only revolute pairs and introduced different methods to study the mechanisms’ motion characteristics. Some of them have also created various practical mechanical structures with those spherical mechanisms. With the concept of JRS and the discriminant method, Nie et al. [11] identified and analyzed motion characteristics, branches, and singularity configurations of the two-DOF two-loop spherical seven-bar mechanism with only spherical revolute pairs that contain two closed five-bar spherical kinematic chains. Hsu et al. [12] put forward a modular approach for analyzing kinematic and mechanical errors of serial spherical mechanisms that can be decomposed into predefined modules. Essomba et al. [13] presented a new spherical decoupled mechanism, which is a new five-bar spherical parallel linkage that generates a remote center of motion of two angular decoupled degrees of freedom, and they conducted kinematic analysis of the proposed mechanism via kinematic and velocity models. A robotic manipulator [14] based on the spherical decoupled mechanism in [13] was invented to perform neuro-endoscopy for brain exploration. Essomba et al. [15] created a spherical mechanism for a robotic manipulator to perform craniotomy, which is a novel 3-RRR (R represents the revolute pair) spherical parallel mechanism whose workspace and kinematic performance were systematically studied. A screw theory approach [16] was created for the computation of the instantaneous rotation axes of indeterminate spherical linkages. Sun et al. [17] put forward a wavelet feature parameters method for the non-periodic design requirements in the path synthesis of a spherical four-bar linkage mechanism. Danaei et al. [18] established the dynamic modeling and base inertial parameter determination of a general 5R two-DOF spherical parallel manipulator via a new geometric approach where inverse and forward kinematic problems were transformed to the problem of determining the intersection of two cones with a common vertex. Wu et al. [19] redesigned the 3-RRR parallel manipulator with reconfigurability. The new design can be used directly for applications, such as flight simulation; the workspace, dexterity, and singularity of the new design were investigated. Palpacelli et al. [20] developed and validated an error kinematic model of a mini spherical robot, which is a spatial five-bar linkage with two rotational degrees of freedom. Elgolli et al. [21] developed an analytical model to solve the dynamic problem of a 3-RRR spherical parallel manipulator. As per the abovementioned mechanisms, multi-loop spherical mechanisms’ versatility is demonstrated by their various industrial and medical applications. Multi-loop spherical mechanisms’ versatility is generated by the fact that the mechanism has a special structure where rotation axes of all its links intersect at the center of the mechanism’s sphere, thus making the mechanisms’ workspace wider and increasing the mechanisms’ stability.

Other scholars have invented other types of spherical mechanisms and other spatial mechanisms that either include spherical kinematic chains or generate spherical motion without spherical links, and they analyzed the mechanisms’ motion characteristics. Some of the mechanisms have valuable industrial usage. Hhan et al. [22] analyzed the dynamic property of the parallel spherical manipulators with three degrees of freedom and presented determination of acceleration input links, oscillations, and the problem of control. Ma et al. [23] presented two novel metamorphic linkages as the spherical-planar 6R metamorphic linkage and the Bennett-spherical 6R metamorphic linkage, both of which have three various distinguished motion branches, and they revealed the conditions of various motion branches and a set of transformations for switching motion branches after establishing the close-loop equations of the spherical-planar 6R metamorphic linkages. Wang et al. [24] presented a geometric method to establish static balancing of mechanisms constructed using spherical chain units. The method was proven with a Bennett-Plano-Spherical hybrid linkage and a 3-RRS (S represents the spherical pair) parallel mechanism. Zhang et al. [25] presented the crease pattern of a Kirigami model, which is kinematically equivalent to a Bennett Plano-spherical linkage, and introduced a new reconfigurable parallel manipulator with three hybrid kinematic limbs, each of which consists of a closed-loop sub-chain, a Bennett Plano-spherical linkage, and a R(RR) serial chain. Chen et al. [26] designed a novel spherical parallel manipulator, which consists of three identical limbs, each of which is formed by a planar parallelogram linkage, a universal joint, and a revolute one successively, and they analyzed the manipulator’s mobility via the reciprocal screws. Wu et al. [27] proposed a novel 6-DOF parallel manipulator, which is in the form of a five-bar spatial linkage with spherical-universal-universal-spherical architecture, where actuation comes from two spherical joints. The manipulator’s active spherical joint is a spherical 3-RRR parallel mechanism. This new design has only two sub-chains, which bring about kinematic advantages, such as large workspace and small footprint ratio. Wang et al. [28] proposed a construction approach from Origami to create multiple spherical integrated mechanisms and built a novel quadruped robot with the kinematotropic metamorphic 8R mechanism as the reconfigurable trunk. Kumar et al. [29] invented a novel mechanism active ankle that consists of three degrees of freedom and operates in an almost spherical work modality. Bai, S. et al. [30] reviewed the state-of-the-art of kinematics, dynamics, and design optimization of spherical mechanisms and novel spherical mechanisms with emerging applications. Some of the aforementioned spherical and spatial mechanisms only contain revolute pairs, while the rest only possess revolute and spherical pairs. None of them contains prismatic pairs.

Many researchers invented non-spherical spatial mechanisms with prismatic pairs. For instance, Chen, X. et al. [31] designed a four-universal joint-prismatic joint-spherical joint/universal joint-prismatic joint-universal joint (4UPS-UPU) spatial parallel mechanism to validate the proposed dynamic modeling method for a spatial parallel mechanism with multi-spherical joint clearances based on the Lagrange multiplier method. Wang et al. [32] designed a four-UPS/1-RPS parallel mechanism, which has five prismatic pairs, and they turned it into a grinding robot by equipping it with a constant force actuator. Hence, prismatic pairs are an indispensable element in creating spatial mechanisms, as they have diversified spatial mechanisms’ motion.

Nevertheless, very few researchers have incorporated spherical sliders, which contain spherical prismatic pairs, in designing spherical mechanisms. Enferadi, J. et al. [33] created a novel three-DOF spherical parallel mechanism with arc prismatic pairs whose singularity analysis was conducted. Saiki, N. et al. [34] built a two-DOF curved biaxial swing spherical mechanism with two active arc sliders and solved the forward and inverse kinematics for posture control of the mechanism’s end effector. The spherical mechanism has rotational degrees of freedom in two directions around a fixed center. Saiki, N. et al. [35] theoretically confirmed the influence that the arc sliders’ friction had on the workspace of the mechanism proposed in [34] and used roller guides to reduce friction created by the arc sliders. The two-DOF curved biaxial swing spherical mechanism with two active arc sliders can be utilized in automobiles such as stabilization platforms and acceleration impact reduction mechanisms. Hou, Y. et al. [36] presented a three-DOF spherical parallel mechanism for the shoulder complex of the humanoid robot and chose 3-PSS/S as the original configuration for the shoulder complex. With the genetic algorithm, the optimization of the 3-PSS/S spherical parallel mechanism was performed, and the orientation workspace of the prototype mechanism was enlarged. Some scholars focused on designing the single-DOF single-loop spherical four-bar crank-slider mechanism [37,38] and studied its output motion. However, the abovementioned spherical mechanisms with prismatic pairs are not multi-loop spherical mechanisms that only consist of closed spherical kinematic chains.

To investigate whether or not spherical prismatic pairs are capable of diversifying multi-loop spherical mechanisms’ motion, the Stephenson-III spherical mechanism with a spherical slider containing a spherical prismatic pair, which has two closed spherical kinematic chains, is proposed. An algebraic approach will be employed to establish the loop equations of the two closed spherical kinematic chains. Afterwards, the mechanism’s input–output curves and singularity curves can be obtained via the discriminant method mentioned in [6,11]; JRS, branches, dead center positions, sub-branches, and branch points of the two spherical kinematic chains can be identified via the branch graphs of the proposed mechanism. Later, the branch graphs of the proposed mechanism will be categorized into two types according to whether or not branch points exist. With the obtained dead center positions and branch points, the precise displacement of the spherical prismatic joint, which is the spherical slider, and revolute joints, can also be obtained. Finally, the results from the mathematical analysis will be simulated and verified by the three-dimensional (3D) models of the proposed mechanism.

2. The Loop Equation of the Single-DOF Spherical Kinematic Chain with Only Revolute Pairs

The loop equation is an effective method to study a spherical kinematic chain’s motion. The loop equation considers each spherical link as a spatial vector. With the loop equation, the relationship between the input and output joints of the spherical kinematic chain can be established. To demonstrate how the loop equation is established, the four-bar spherical mechanism is utilized. Three steps are involved to establish the loop equation of a single DOF spherical kinematic chain with only revolute pairs.

In step one, a three-dimensional Cartesian coordinate system at every vertex (or at every spherical revolute pair), where two spherical links meet, is established. As is shown in Figure 1, a spherical four-bar kinematic chain, with four spherical revolute pairs, has four three-dimensional Cartesian coordinate systems, each of which has a Z_i-axis (i = 1, 2, 3, 4) that goes from the center of the kinematic chain’s sphere to one of the four vertices.

The yellow spherical link in Figure 1 is fixed. Each of the three remaining blue spherical links in the spherical four-bar kinematic chain has two types of rotation, axis rotation Z_i (i = 1, 2, 3, 4), which is represented by a matrix and side rotation S_ij (i = 1, 2, 3, j = 2, 3, 4), which is also a matrix. Since the Z_i-axis of the three-dimensional coordinate system on each revolute pair i (i = 1, 2, 3, 4) is designated as a rotation axis around which the two spherical links connected by revolute pair i rotate, the X_i- and Y_i-axes of the same three-dimensional coordinate system can be obtained with the right-hand rule, which makes the right thumb aim towards the positive direction of the Z_i-axis, and it makes the curling motion of the other four fingers of the same right hand move from the positive direction of the X_i-axis to that of the Y_i-axis. The axis rotation Z_i is the rotation of spherical link i(i + 1) (i = 1, 2, 3). In Figure 1, each of the four axis rotations, Z₁, Z₂, Z₃, and Z₄, is a red dotted line with an arrow. Axis rotation Z_i is depicted by Equation (1).

$${{\rm Z}}_i=\left[\begin{array}{ccc}\cos {\theta }_i& -\sin {\theta }_i& 0\\ \sin {\theta }_i& \cos {\theta }_i& 0\\ 0& 0& 1\end{array}\right]$$

(1)

The value of Z_i depends on the rotation angle θ_i (i = 1, $\cdots$, 4), which is the angular displacement of a particular spherical link relative to the adjacent spherical link. The connecting rotation between any two adjacent axis rotations is a side rotation S_ij which is described in Equation (2).

$$S_{ij}=\left[\begin{array}{ccc}1& 0& 0\\ 0& \cos {\alpha }_{ij}& -\sin {\alpha }_{ij}\\ 0& \sin {\alpha }_{ij}& \cos {\alpha }_{ij}\end{array}\right]$$

(2)

As the radius of the four-bar kinematic chain’s sphere is predetermined to have one unit, S_ij is determined by the arc length of a particular link in the kinematic chain, which is the central angle α_ij (i = 1, 2, 3, j = 2, 3, 4), whose two sides are two adjacent rotation axes shown in Figure 1. The side rotation S_ij is normally constant for a spherical mechanism, with the exception that the mechanism has spherical prismatic pairs, which is further discussed in Section 3. Therefore, the movement of a spherical link can be divided into axis and side rotations, which appear alternately.

In step two, the loop equation is established and is shown Equation (3),

$${\mathbf{Z}}_1{\mathbf{S}}_{12}{\mathbf{Z}}_2{\mathbf{S}}_{23}\dots {\mathbf{Z}}_{k-1}{S}_{\left(k-1\right)k}{\mathbf{Z}}_k{S}_{k\left(k+1\right)}=\mathbf{I}(k=1,2,3\cdots n),$$

(3)

where the identity matrix, which is named I, equals $\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$. Equation (3) demonstrates that matrix multiplication order follows how Z_i and S_ij are sequenced.

In step three, Equation (3) is simplified. With z equaling [0 0 1]^T, (4) and (5) can be obtained.

z^T Z_i = z^T

(4)

Z_iz = z

(5)

Equation (3) is post-multiplied by the matrices S_k(k+1)^T and z consecutively on both sides, and then it is pre-multiplied by the matrix z^T on both sides. Thereafter, Equation (3) can be converted into Equation (6), which can be further simplified into Equation (7).

$${\mathbf{z}}^{\mathrm{T}} {\mathbf{Z}}_1{\mathbf{S}}_{12}{\mathbf{Z}}_2\cdots {\mathbf{Z}}_{k-1}{S}_{\left(k-1\right)k}{\mathbf{Z}}_k{S}_{k\left(k+1\right)}{S_{k\left(k+1\right)}}^{\mathrm{T}}z=z^{\mathrm{T}}{{\mathbf{S}}_{k\left(k+1\right)}}^{\mathrm{T}}z$$

(6)

$${\mathbf{z}}^{\mathrm{T}}{\mathbf{S}}_{12}{\mathbf{Z}}_2\cdots {\mathbf{Z}}_{k-1}{S}_{\left(k-1\right)k}z=z^{\mathrm{T}}{{\mathbf{S}}_{k\left(k+1\right)}}^{\mathrm{T}}z$$

(7)

In Equation (6), S_k(k+1), which equals $\left[\begin{array}{ccc}1& 0& 0\\ 0& \cos {\alpha }_{k\left(k+1\right)}& -\sin {\alpha }_{k\left(k+1\right)}\\ 0& \sin {\alpha }_{k\left(k+1\right)}& \cos {\alpha }_{k\left(k+1\right)}\end{array}\right]$, post-multiplied by S_k(k+1)^T, which equals $\left[\begin{array}{ccc}1& 0& 0\\ 0& \cos {\alpha }_{k\left(k+1\right)}& \sin {\alpha }_{k\left(k+1\right)}\\ 0& -\sin {\alpha }_{k\left(k+1\right)}& \cos {\alpha }_{k\left(k+1\right)}\end{array}\right]$, equals $\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$. In the left side of Equation (7), the two axis rotations Z₁ and Z_k have been removed, and, therefore, the variables θ₁ and θ_k are eliminated.

3. Kinematics of the Stephenson-III Spherical Mechanism with a Spherical Slider Containing a Spherical Prismatic Pair

3.1. The Loop Equation

A single-DOF two-loop Stephenson-III spherical mechanism with a spherical slider that contains a spherical prismatic pair, which contains two closed kinematic chains, A₀ABB₀ and B₀BDCC₀, is shown in Figure 2.

In Figure 2, the yellow solid spherical line is the fixed spherical link; the blue solid spherical lines are the unfixed spherical links. C represents both the spherical prismatic pair C_P and the revolute pair C_R; CC₀ is the blue dotted spherical line that represents the arc distance the spherical slider C with prismatic pair C_P reaches on the fixed track that coincides with CC₀. Therefore, α₅₆ is a variable that represents the arc distance that the spherical slider C reaches; θ₅ is a constant, since arc distance CC₀ coincides with the fixed track. Input and output joints are spherical revolute joints B₀ and B, respectively; the input angle is θ₄, and the output angle is θ₃. The fixed yellow spherical link A_oB_oC_o consists of two separate spherical sub-links that are at an angle, β, which is a constant. The triangular spherical link ABD consists of three spherical sub-links, two of which, AB and BD, are at an angle, λ, which is also a constant. The loop equation of the spherical four-bar kinematic chain A₀ABB₀ is shown in Equation (8).

Z₂S₂₃Z₃S₃₄Z₄S₄₁Z₁S₁₂ = I

(8)

The loop equation of the spherical five-bar kinematic chain B₀BDCC₀ is shown in (9).

Z₆S₅₆Z₅S₄₅Z_4′S₃₄Z_3′S₇₃Z₇S₆₇ = I

(9)

In Figure 2, the rotation angles θ_i of Z₃, Z₄, and the rotation angles θ_i of Z_3′ and Z_4′ in Equations (8) and (9), are different; Z₃ and Z₄ correspond to θ₃ and θ₄, respectively; however, Z_3′ and Z_4′ correspond to (λ$-$θ₃) and (θ₄$-$β), respectively. According to Equations (3), (6) and (7), Equations (8) and (9) can be simplified into Equations (10) and (11), respectively, by eliminating the variables θ₁, θ₂, θ₆, and θ₇.

Z^TS₂₃Z₃S₃₄Z₄S₄₁z= z^TS₁₂^Tz

(10)

z^TS₅₆Z₅S₄₅Z_4′S₃₄Z_3′S₇₃z = z^TS₆₇^Tz

(11)

Equations (10) and (11) have three unknown variables α₅₆, θ₃, and θ₄; α₅₆ is the arc distance of the spherical slider. The tangent half-angle formulas and discriminant method are utilized to eliminate θ₄ and α₅₆. Firstly, the equations $\cos {\theta }_3$ = $\frac{1-l^2}{1+l^2}$ and $\sin {\theta }_3$ = $\frac{2l}{1+l^2}$, where l equals $\tan \frac{{\theta }_3}{2}$, are employed in Equation (10). Secondly, the equations $\cos {\alpha }_{56}$ = $\frac{1-m^2}{1+m^2}$ and $\sin {\alpha }_{56}$ = $\frac{2m}{1+m^2}$, where m equals $\tan \frac{{\alpha }_{56}}{2}$, are employed in Equation (11). Hence, Equations (10) and (11) can be, respectively, simplified into Equations (12) and (13).

a₁(θ₄)l² + b₁(θ₄)l + c₁(θ₄) = 0

(12)

a₂(θ₃, θ₄)m² + b₂(θ₃, θ₄)m + c₂(θ₃, θ₄) = 0

(13)

In Equation (12), which can be seen as a quadratic equation whose sole variable is l, a₁, b₁, and c₁ are the coefficients of variable l; a₂, b₂, and c₂ are the three coefficients of the quadratic equation shown in Equation (13), whose sole variable is m. The requirement that Equation (12) needs to meet to have valid results for l is Equation (14), which is the discriminant of Equation (12) and only contains variable θ₄.

Δ₁ = b₁² − 4a₁c₁ ≥ 0

(14)

When spherical four-bar kinematic chain A₀ABB₀ encounters singularity configurations, the precise values of θ₄ can be obtained via Δ₁ equaling 0. Thereafter, the precise values of θ₃ can be obtained via Equation (15) or Equation (16).

$$l_1=\frac{-b_1-\sqrt{{\Delta }_1}}{2a_1}, {\theta }_3=2\arctan l_1$$

(15)

$$l_2=\frac{-b_1+\sqrt{{\Delta }_1}}{2a_1}, {\theta }_3=2\arctan l_2$$

(16)

The requirement that Equation (13) needs to meet to have valid results for m is Equation (17), which is the discriminant of Equation (13) and only contains variables θ₃ and θ₄.

Δ₂ = b₂² − 4a₂c₂ ≥ 0

(17)

With the values of the variables θ₃ and θ₄ acquired, the value of α₅₆ can be obtained by Equations (18) and (19), which are the results of Equation (13).

$$m_1=\frac{-b_2-\sqrt{{\Delta }_2}}{2a_2}, {\alpha }_{56}=2\arctan m_1$$

(18)

$$m_2=\frac{-b_2+\sqrt{{\Delta }_2}}{2a_2}, {\alpha }_{56}=2\arctan m_2$$

(19)

3.2. The Branch Graph of the Proposed Spherical Mechanism

The branch graph of the proposed mechanism, which clearly describes the motion characteristics of the mechanism, consists of the input–output curves of A₀ABB₀, which are created by Equation (12), and the singularity curves of B₀BDCC₀, which are created by Equation (17). Dead center positions, where the motion of the closed four-bar kinematic chain, A₀ABB₀, is impeded, may appear on the input–output curves of A₀ABB₀. A singularity curve of B₀BDCC₀ is the collection of dead center positions at which the motion of B₀BDCC₀ is impeded. The intersection points of the input–output curves of A₀ABB₀ and the singularity curves of B₀BDCC₀ are branch points, each of which is a set of input and output values where the motion of the mechanism’s closed five-bar kinematic chain, B₀BDCC₀, is impeded. As per whether or not the branch points exist, the branch graphs of the proposed mechanism can be classified into two categories.

3.2.1. The Type-One Branch Graph

In the first category, branch points do not exist.

Table 1. Parameters of the proposed mechanism that has the type-one branch graph.

parameters	α41	α12	α23	α34	α45	α67	α73	θ5	β	λ
values	54.1°	22.5°	25°	100°	30°	140°	50°	0°	55°	95°

enumerates the parameters for the proposed mechanism that has the type-one branch graph.

With the given parameters, the branch graph is shown in Figure 3a. The three-dimensional model of the proposed mechanism with the type-one branch is shown in Figure 3b.

In Figure 3a, the red input–output curves of A₀ABB₀ and the blue singularity curves of B₀BDCC₀ do not intersect; there are no common solutions for Equations (12) and (17). Hence, no branch point exists. The two red curves are identical and repeated with an interval of 2π. The two blue curves are different. Each of the two blue curves is also repeated with an interval of 2π. Since the branch graph shown in Figure 3a emphasizes both the two red curves and the two blue curves, the repeated appearance of the two blue curves is also clearly presented in Figure 3a. JRS-2, which is the light gray area surrounded by blue curves, is the JRS of B₀BDCC₀; JRS-2 is determined by Δ₂ $\ge$ 0. The red input–output curves are completely covered by JRS-2. Therefore, the two branches are the two red input–output curves. Each point on the two branches satisfies the equation, Δ₂ $>$ 0. When Δ₁ equals 0, A₀ABB₀ encounters singularity configurations. The set of input and output values, where A₀ABB₀ encounters a singularity configuration, is a dead center position. In Figure 3a, the four dead center positions are indicated by numbers 1 to 4, which are enumerated in

Table 2. Dead center positions.

Joint	θ3	θ4
Dead center position 1	165.735°	167.039°
Dead center position 2	194.265°	192.961°
Dead center position 3	165.735°	$-$192.961°
Dead center position 3	194.265°	$-$167.039°

.

The values of θ₄ can be obtained from Δ₁ = 0; Equations (15) and (16) are utilized to obtain the values of θ₃. Dead center positions 1 and 2 divide branch 1 into two sub-branches, (1-2) and (2-1). On (1-2), which starts from dead center positions 1 to 2 clockwise, Equation (15) can be utilized to obtain the values of θ₃; on (2-1), which starts from dead center positions 2 to 1 clockwise, Equation (16) can be utilized to obtain the values of θ₃. Dead center positions 3 and 4 divide branch 2 into two sub-branches, (3-4) and (4-3). On (3-4), which starts from dead center positions 3 to 4 clockwise, Equation (15) can be utilized to obtain the values of θ₃; on (4-3), which starts from dead center positions 4 to 3 clockwise, Equation (16) can be utilized to obtain the values of θ₃. With the values of the variables θ₃ and θ₄ acquired, the value of α₅₆ can be obtained by Equations (18) and (19). Inside a sub-branch, the mechanism’s motion is continuous and smooth. Equation (10) can be rewritten as Z₃S₃₄Z₄S₄₁Z₁S₁₂Z₂S₂₃ = I, which can be simplified into z^TS₃₄Z₄S₄₁Z₁S₁₂z= z^TS₂₃^Tz, where θ₁ is the only unknown variable. Therefore, the value of θ₁ can be obtained. Equation (10) can also be rewritten as Z₄S₄₁Z₁S₁₂Z₂S₂₃Z₃S₃₄ = I, which can be simplified into z^TS₄₁Z₁S₁₂Z₂S₂₃z = z^TS₃₄^Tz, where θ₂ is the only unknown variable. Therefore, the value of θ₂ can be obtained. Equation (11) can be rewritten as Z₅S₄₅Z_4′S₃₄Z_3′S₇₃Z₇S₆₇Z₆S₅₆ = I, which can be simplified into z^TS₄₅Z_4′S₃₄Z_3′S₇₃Z₇S₆₇z = z^TS₅₆^Tz, where θ₇ is the only unknown variable. Therefore, the value of θ₇ can be obtained. Equation (11) can also be rewritten as Z_4′S₃₄Z_3′S₇₃Z₇S₆₇Z₆S₅₆Z₅S₄₅ = I, which can be simplified into z^TS₃₄Z_3′S₇₃Z₇S₆₇Z₆S₅₆z = z^TS₄₅^Tz, where θ₆ is the only unknown variable. Therefore, the value of θ₆ can be obtained. Figure 4a–d are the configurations of the proposed mechanism at dead center positions 1, 2, 3, and 4, respectively.

In Figure 4, the dotted lines with arrows OA₀, OA, and OB are coplanar; the four values of α₅₆, each of which is represented by a blue dotted spherical line, are all highlighted in blue. The three-dimensional model of the mechanism shown in Figure 3b simulates the mechanism’s motion and verifies each joint’s displacement and the configurations in Figure 4. Since curves in the branch graph shown in Figure 3a do not intersect, Figure 3a does not have branch points, and the proposed mechanism with the type-one branch has decoupled motion, which is created by the situation where a two-loop spherical mechanism’s motion is entirely decided by one of the two loops. For the two-loop Stephenson-III spherical mechanism shown in Figure 3b with the dimensions given in Table 1, the mechanism’s decoupled motion is completely decided by the spherical kinematic chain A₀ABB₀, which is one of the two loops (or spherical kinematic chains) in the mechanism. Branch 1 has the input range for θ₄, (167.039°, 192.961°) and the output range for θ₃, (160.428°, 200.700°). Branch 2 has the input range for θ₄, ($-$192.961°, $-$167.039°) and the output range for θ₃, (160.428°, 200.700°). The minimal and maximal arc distances reached by the spherical slider in branch 1 is shown in Figure 5a,b; α₅₆∈(71.620°, 105.310°).

The minimal and maximal arc distances reached by the spherical slider in branch 2 is shown in Figure 6a,b; α₅₆∈($-$3.736°, 25.382°).

In Figure 5 and Figure 6, the values of α₅₆, each of which is represented by a blue dotted spherical line, are all highlighted in blue. The three-dimensional model of the mechanism shown in Figure 3b simulates the mechanism’s motion and verifies each joint’s displacement and the configurations in Figure 5 and Figure 6. The three-dimensional simulation conducted by the three-dimensional model verified the decoupled motion of the proposed spherical mechanism. According to Figure 5 and Figure 6, the fixed track is a greater circle of the proposed mechanism’s sphere; the spherical slider produces the rotational sliding motion and is not able to travel the entire circumference of the fixed track. The spherical slider is only capable of moving within the range of 71.620° to 105.310° and $-$3.736° to 25.382° on the fixed track.

3.2.2. The Type-Two Branch Graph

With the parameters in

Table 3. Parameters of the mechanism with the type-two branch graph.

parameters	α41	α12	α23	α34	α45	α67	α73	θ5	β	λ
values	30°	60°	50°	80°	35°	130°	50°	0°	55°	95°

, the type-two branch graph can be obtained.

The branch graph is shown in Figure 7a, where the red curves are the input–output curves of A₀ABB₀. The three-dimensional model of the proposed mechanism in Figure 7b was built with the parameters shown in Table 3 to verify the analytical results.

In Figure 7a, the input–output curves of A₀ABB₀ are the red curves, and singularity curves of B₀BDCC₀ are the blue curves. JRS-2 is the JRS of B₀BDCC₀ represented by the light-shaded area surrounded by blue curves. A branch is a section of the red input–output curve that is covered by JRS-2. When the input and output joints are within the input and output ranges of a branch, the proposed mechanism’s motion is continuous. The intersection points of the input–output curves and singularity curves in Figure 7a are branch points which are listed in

Table 4. Branch points.

Branch Points	1	2	3	4	5	6
θ3	221.227°	195.604°	168.374°	144.566°	87.359°	83.134°
θ4	119.570°	56.249°	317.060°	251.795°	159.475°	146.084°
α56	115.107°	184.274°	$-$115.680°	$-$19.022°	$-$26.700°	$-$47.842°
θ1	124.618°	162.090°	$-$164.635°	$-$132.634°	34.663°	$-$18.100°
θ2	$-$96.229°	$-$47.922°	36.762°	88.517°	115.903°	118.988°
θ6	90°	90°	$-$90°	$-$90°	90°	90°
θ7	143.729°	16.977°	168.762°	$-$65.032°	159.474°	181.533°

. At each branch point, Equations (12) and (17) are satisfied, and Δ₂ equals 0, and the motion of the proposed spherical mechanisms is impeded. The precise arc distance, α₅₆, is also enumerated in Table 4 via Equations (17) and (18).

In Figure 7a, the branches are curves A (2-7-6), B (1-8-3), and C (4-5). The dead center positions are indicated by numbers 7 and 8, where θ₃ equals 180°, θ₄ equals 0° and 360°, respectively, and Δ₁ equals 0. Dead center position 7 divides branch A into two sub-branches, curves (2-7) and (6-7); dead center position 8 divides branch B into two sub-branches, curves (1-8) and (3-8). On (2-7) and (1-8), (15) can be utilized to obtain the values of θ₃; on (6-7), (4-5), and (3-8), (16) can be utilized to obtain the values of θ₃. Since Figure 7 has branch points and branches, Figure 7 shows that the Stephenson-III spherical mechanism shown in Figure 7b, with the dimensions shown in Table 3, which have coupled motion, which is decided by the interaction of the two spherical kinematic chains, A₀ABB₀ and B₀BDCC₀. The three-dimensional model was utilized to identify and verify the two dead center positions by identifying the proposed spherical mechanism’s configuration at them, which is shown in Figure 8.

In Figure 8, the dotted lines with arrows, OA₀, OB₀, OA, and OB, are coplanar. The three-dimensional model was also utilized to identify and verify the six branch points enumerated in Table 4 by demonstrating the singularity configuration of the proposed mechanism at each branch point, which are shown in Figure 9.

In Figure 8 and Figure 9, the values of α₅₆, each of which is represented by a blue dotted spherical line, are all highlighted in blue. The three-dimensional simulation conducted by the three-dimensional model verified each joint’s displacement at each branch point shown in Table 4. At branch points 1, 2, 5, and 6, θ₆ $=$ 90°; At branch points 3 and 4, θ₆ $=-$90°. In branch A, the input joint θ₄ is within (0°, 146.084°); the output joint θ₃ is within (83.134°, 195.604°). In branch B (5–6), the output joint θ₃ is within (87.359°, 144.566°); the input joint θ₄ is within (119.570°, 360°). In branch C, the input joint θ₄ is within (159.475°, 251.795°); the output joint θ₃ is within (87.359°, 144.566°). According to Figure 10, where one value of α₅₆, which is represented by a blue dotted spherical line, is highlighted in blue, in branch B, when the output joint θ₃ is at 183.619°, and the input joint θ₄ is at 357.270°, and the spherical slider reaches the farthest arc distance at 180° or $-$180°.

Therefore, in branch B, the spherical slider is able to travel the entire circumference of the fixed track with the given parameters in Table 3.

3.2.3. The Special Type-One Branch Graph

The parameters of the proposed mechanism with the special type-one branch graph are given in

Table 5. Parameters of the proposed spherical mechanism with the type-two branches.

parameters	α41	α12	α23	α34	α45	α67	α73	θ5	β	λ
values	34°	63°	67°	77°	100°	85°	50°	0°	35°	105°

.

Based on the given parameters above, the special type-one branch graph of the proposed mechanism is shown in Figure 11a. The three-dimensional model of the mechanism is shown in Figure 11b.

In Figure 11a, the red curves are the input–output curve of A₀ABB₀; JRS-2, which is the light gray area with blue singularity curves, is the JRS of B₀BDCC₀, and blue curves are B₀BDCC_0′s singularity curves. The entirety of A₀ABB_0′s two input–output curves are the branches of the mechanism. Since the two input–output curves of A₀ABB₀ do not intersect, A₀ABB₀ does not encounter dead center positions. On the upper red curve in Figure 11a, (15) can be utilized to obtain the values of θ₃ and θ₄; on the lower curve in Figure 11a, (16) can be utilized to obtain the values of θ₃ and θ₄. Branch 1 has two sub-branches. Since, in both sub-branches of branch 1, the displacement range of the input joint θ₄ is 0° to 360°and the displacement range of the output joint θ₃ is 76.481° to 151.078°, as is shown in Figure 12, the two sub-branches cannot be shown in Figure 11a.

In Figure 12, the values of α₅₆, each of which is represented by a blue dotted spherical line, are all highlighted in blue. The two sub-branches are shown by different displacements of the spherical slider. In sub-branch 1 of branch 1, the maximal positive displacement of α₅₆ is 55.004°, which is shown in Figure 13a, and the maximal negative displacement of α₅₆ is $-$66.750°, which is shown in Figure 13b.

In sub-branch 2 of branch 1, the maximal negative displacement of α₅₆ is $-$257.498°, which is shown in Figure 14a, and the minimal negative displacement of α₅₆ is $-$155.558°, which is shown in Figure 14b.

According to Figure 13 and Figure 14, where the values of α₅₆, each of which is represented by a blue dotted spherical line, are all highlighted in blue, the spherical slider is not able to travel the entire circumference of the fixed track and is only capable of moving within the range of $-$66.750° to 55.004° and $-$257.498° to $-$155.558° on the fixed track.

Branch 2 also has two sub-branches. Since in both sub-branches of branch 2, the displacement range of the input joint θ₄ is 0° to 360° and the displacement range of the output joint θ₃ is 208.729° to 283.503°, as is shown in Figure 15, the two sub-branches cannot be shown in Figure 11a either.

In Figure 15, the values of α₅₆, each of which is represented by a blue dotted spherical line, are all highlighted in blue. The two sub-branches are shown by different displacements of the spherical slider as well. In sub-branch 1 of branch 2, the maximal positive displacement of α₅₆ is 44.290°, which is shown in Figure 16a, and the maximal negative displacement of α₅₆ is $-$73.339°, which is shown in Figure 16b.

In sub-branch 2 of branch 2, the maximal negative displacement of α₅₆ is $-$242.945°, which is shown in Figure 17a, and the minimal negative displacement of α₅₆ is $-$120.035°, which is shown in Figure 17b.

According to Figure 16 and Figure 17, where the values of α₅₆, each of which is represented by a blue dotted spherical line, are all highlighted in blue, the spherical slider is not able to travel the entire circumference of the fixed track and is only capable of moving within the range of $-$73.339° to 44.290°and $-$242.945° to $-$120.035° on the fixed track.

4. Conclusions and Discussions

According to the literature review in the first section, past research on multi-loop spherical mechanisms rarely included spherical sliders. However, prismatic pairs are frequently used in spatial mechanisms to diversify spatial mechanisms’ motion. Therefore, it is necessary to investigate whether or not spherical sliders containing spherical prismatic pairs can diversify the motion of multi-loop spherical mechanisms. This paper proposed the single-DOF two-loop Stephenson-III spherical mechanism with a spherical slider that possesses a spherical prismatic pair. The approach to study the proposed mechanism’s motion is algebraic. The loop equations of the two spherical kinematic chains of the proposed spherical mechanism were established to identify the input–output curves of the closed four-bar kinematic chain A₀ABB₀ and the singularity curves of the closed five-bar kinematic chain B₀BDCC₀ before the branch graphs of the mechanism could be obtained. The branch graphs of the proposed spherical mechanism were categorized into two types according to if branch points existed or not. With the branch graph, the dead center positions, branches, and branch points could be identified. With the obtained dead center positions and branch points, the precise range of the arc distance reached by the spherical slider was obtained. Therefore, whether or not the spherical slider can travel the entire circumference of the fixed track could be determined. The mathematical investigation of the proposed spherical mechanism was paired with three-dimensional simulation, where the three-dimensional models of the proposed spherical mechanism simulated the mechanism’s motion and were utilized to verify the results of the mathematical investigation.

As per the three-dimensional simulation of the proposed mechanism’s motion with different parameters, spherical prismatic pairs are capable of producing rotational sliding motion that a spherical link with only revolute pairs cannot not achieve, thus diversifying the motion of the two-loop spherical mechanism proposed in this paper. The spherical slider containing the prismatic pair with specific parameters for the proposed mechanism is able to move along the entirety of the fixed track’s circumference, which is a specific greater circle on the surface of the mechanism’s sphere.

In this paper, much focus is on incorporating the spherical prismatic pair into multi-loop spherical mechanisms, as well as the algebraic approach and three-dimensional simulation to study the kinematics of the proposed spherical mechanism. Less focus is on how the proposed mechanism can be adopted in real-life situations, which is one of the limitations of this paper. In future research, how the incorporated prismatic pairs advance the performance of devices based on spherical mechanisms should be thoroughly studied. The next limitation is that the proposed mechanism only has one degree of freedom and incorporates only one spherical prismatic pair. If a spherical mechanism has two degrees of freedom and possesses one or more spherical prismatic pairs, the mechanism’s motion can be further diversified. Therefore, in future research, the focus should also be on the two-DOF two-loop seven-bar spherical mechanism with one and two spherical prismatic pairs. The final limitation is that the proposed mechanism only has two closed kinematic chains. Whether or not the algebraic approach mentioned in this paper can be extended to study the kinematics of a spherical mechanism with more than three closed kinematic chains should be the topic of future research, as well.