Geometry, Kinematics, Workspace, and Singularities of a Novel 3-PRRS Parallel Manipulator

Abstract

“Experiments were conducted at DIMEG, University of Calabria, located in the main campus in Arcavacata di Rende, Italy.” This article focuses on the study of the geometry, direct and inverse kinematics, workspace, and singularity of a novel 3-PRRS parallel manipulator (PM) with a redundantly actuated architecture. The PM consists of three active revolute joints and three passive prismatic redundant input joints, all located on a fixed platform. The constant and variable parameters characterizing the PM’s geometry and kinematics are determined. The direct kinematics problem is formulated as a 16th-degree polynomial, while the inverse kinematics problem is solved in closed form. A comparison of the direct and inverse kinematics is provided, and the correctness of the solutions is validated through numerical examples. The equations of motion for the moving platform are derived, and the PM’s workspace is defined based on the inverse kinematics. This work demonstrates how the passive prismatic input joints, specifically included in the design, contribute to an enlarged workspace—particularly in the vertical direction—compared to traditional 3-RRS PM architecture.

1. Introduction

Parallel robots are most commonly 6-DOF (Degrees of Freedom) parallel manipulators (PMs) with six legs (hexapods) [1,2]. Depending on the application of parallel robots, the number of legs and DOFs of PMs can be reduced. Three-legged PMs, or tripods, have been extensively researched in the literature. The simplest form of a tripod is a planar 3-RRR (R stands for revolute joint) PM with 3-DOF, which was investigated by Gosselin and Angeles in [3]. The moving platform has three planar DOFs, which are two translations along the x and y axes and a rotation about the axis perpendicular to the OXY plane. An example of a special 3-RRR-type tripod is a spherical PM, which was investigated by Gosselin and Angeles [4], Gosselin, St-Pierre, and Gagné [5], Liu, Jin, and Gao [6]. In this type of PM, the axes of rotation of all revolute kinematic pairs intersect at a single point. The moving platform of this PM has only orientation DOFs with respect to the base. There are known PMs with 3-DOF in which the moving platform has complex motions. Carretero et al. [7] developed a new 3-DOF PM with one rotation and two translations, described using three Euler angles. Liu and Bonev [8] developed a PM with two translations of the moving platform, where azimuth and tilt angles are employed. Such mechanisms are denoted by 3-[PP]S ones (S denotes for spherical joints), and they are referred to as zero-torsion mechanisms by Bonev [9,10,11]. The moving platform of these PMs is attached to three legs via spherical joints, which move in vertical planes intersecting at a common line. Extensive literature on 3-[PP]S PMs has been published. A 3-RPS-type PM proposed by Hunt [12] has been studied by K.-M. Lee and Arjunan [13], Kim and Tsai [14], Liu and Cheng [15], Fang and Huang [16], and Sokolov and Xirouchakis [17]. A 3-RRS PM was analyzed by J. Lee et al. [18,19]. A 3-PRS-type tripod was studied by Y. Li and Xu [20], Carretero et al. [21], and Tsai et al. [22]. Based on this type of PM, a new Sprint Z3 machining tool head was developed by Wahl [23].

One of the most well-known parallel manipulators with three translational DOFs is the DELTA robot proposed by Clavel [24]. Its moving platform is attached to the base via three legs with PR(Ps) chains (Ps denotes the spatial four-bar parallelogram with four spherical joints), in which the R joints are actuated. Other PMs with three translational DOFs are Tsai’s manipulator [25] and the StarLike manipulator [26]. Tsai’s manipulator has three PR(Pa)R legs, where (Pa) denotes a planar four-bar parallelogram with four revolute joints. The StarLike manipulator was designed by Hervé based on group theory. DELTA and Tsai’s PMs have the advantages of high velocity and high acceleration. Some 3-DOF translational PMs have two universal joints in their legs; for example, the Tsai’s [27] and the Park SNU (Seoul National University) [28] 3-UPU-type PMs. A review of the literature on lower-mobility PMs of the 3-UPU or 3-URU types is provided by Di Gregorio [29].

In 3-DOF PM’s, replacing single-DOF active joints with two 2-DOF active joints or adding additional single-DOF joints in the legs allows the formation of 6-DOF PMs. Using universal joints as active kinematic pairs in each leg, Cleary and Uebel [30] developed a 3-URS PM with 6 DOF. In the works [31,32] by Baigunchekov et al. a 3-CCC PM (C denotes a cylindrical joint) with 6 DOF was developed. Based on the application of a 2-DOF planar kinematic pair E as the active kinematic pair, Romiti and Soreli [33], and Soreli et al. [34], developed a 3-EPS PM with 6-DOF. Tsai and Tahmasebi [35], Ben-Horin and Shoham [36], and Byun and Cho [37] developed the 3-ESR, 3-ERS, and 3-ESP types of PMs with 6 DOF, respectively. A 3-RES type PM with 6 DOF was developed by Collins and Long [38], Mimura and Funabashi [39], and Ebert and Gosselin [40]. In these PMs, four-bar and five-bar mechanisms are used as equivalents of planar kinematic pairs. By adding additional active prismatic and revolute kinematic pairs to 3-DOF PMs, Behi [41], Kohli et al. [42], Zlatanov et al. [43], and Alizade, Tagiyev, and Duffy [44] developed PRPS, RPRS, and RRPS types of PMs. Kohli et al. [42] referred to the 3-RPRS-type PM as a manipulator with Rotary-Linear (R-L) actuators. The study [45] also considers a six-degree-of-freedom PM with the simplest solution to both direct and inverse kinematics, featuring a U-shaped base and the absence of serial singularities. Shanzeng L. [46] proposed a novel 3-CRS PM architecture with six DOFs, and derived the kinematic relationships based on screw theory. The studies [47,48,49,50,51,52,53] are devoted to the investigation of the kinematics, workspace, and singularities of PMs.

In this paper, we propose a 3-PRRS parallel manipulator that, in contrast to the classical 3-RRS PM, additionally incorporates three passive prismatic kinematic pairs located on the fixed platform. These passive prismatic joints introduce three additional translational motions that actuate the three active revolute input joints. This architectural modification represents an extension of the conventional 3-RRS PM, in which each leg is driven by a single active revolute joint.

The introduction of passive translational actuators leads to several key breakthroughs. First, it enables a significant enlargement of the reachable workspace without increasing the number of active DOF. Second, it alters the kinematic structure of the manipulator, resulting in extended Jacobian matrices that fundamentally differ from the classical 3 × 3 Jacobian matrix of standard 3-RRS PM. This, in turn, allows a deeper investigation of velocity relationships, redundancy effects, and singular configurations.

The main results obtained in this work can be summarized as follows: the proposal of a novel 3-PRRS PM architecture with redundant prismatic actuators located on the fixed platform; a detailed investigation of the manipulator geometry, including the determination of constant geometric parameters of the links and variable parameters describing the relative motions of the kinematic pairs; a closed-form solution of the inverse kinematics problem, as well as the derivation of the direct kinematics problem using a polynomial-based approach, enabling the determination of all possible assembly modes of the PM; the determination of the workspace based on the inverse kinematics solution, taking into account constraint equations and the influence of redundant actuators on singularity avoidance and workspace enlargement; the derivation of extended Jacobian matrices based on the constraint equations, allowing the analysis of both the direct and inverse velocity problems for a redundantly actuated PM; numerical validation of the theoretical results, confirming the correctness of the solutions for both the direct and inverse kinematics of position and velocity.

The obtained results provide a theoretical foundation for future developments in dynamic modeling, control design, and the practical implementation of the proposed PM. An invention patent No. 36,849 of the Kazakhstan [54] has been granted for the 3-PRRS-type PM.

2. Geometry and Kinematic Model

Figure 1 shows a 3D CAD model of the 3-PRRS-type PM and the kinematic diagram of one of its legs.

In this PM, each leg is actuated by two actuators: a linear actuator 1 (or the prismatic pair A) and a revolute actuator B. The axis of translational motion of the linear actuator 1 is perpendicular to the axis of rotation of the revolute actuator B. Therefore, each leg of this PM performs a single rotational motion relative to the fixed base 0. Consequently, the moving platform 4 of the 3-PRRS-type PM has 3 DOFs and belongs to the class of redundantly actuated, kinematically defective PMs. Three revolute actuators are called active input joints, while three redundant prismatic joints are called passive input joints. The three passive input joints are used to enlarge the workspace and to avoid singular configurations.

To describe the geometry and analyze the kinematics of the PM, two Cartesian coordinate systems $O_j{U}_j{V}_j{W}_j$ and $O_k{X}_k{Y}_k{Z}_k$ are attached to the corresponding elements of each kinematic pair. The $W_j$ and $Z_k$ axes of these coordinate systems are aligned with the rotational and translational motion axes of the kinematic pair elements, while the $U_j$ and $X_k$ axes are directed along the perpendicular $t_{jk}$ drawn from the $W_j$ axis to the $Z_k$ axis. The transformation matrix between the $O_j{U}_j{V}_j{W}_j$ and $O_k{X}_k{Y}_k{Z}_k$ coordinate systems has the following form:

$$T_{jk}(a_{jk},{\alpha }_{jk},b_{jk},{\beta }_{jk},c_{jk},{\gamma }_{jk})=\left[\begin{array}{cccc}1& 0& 0& 0\\ \begin{array}{l}{a}_{jk}c{\gamma }_{jk}\\ +b_{jk}s{\gamma }_{jk}s{\alpha }_{jk}\end{array}& \begin{array}{l}c{\gamma }_{jk}c{\beta }_{jk}\\ -s{\gamma }_{jk}c{\alpha }_{jk}s{\beta }_{jk}\end{array}& \begin{array}{l}-c{\gamma }_{jk}s{\beta }_{jk}\\ -s{\gamma }_{jk}c{\alpha }_{jk}c{\beta }_{jk}\end{array}& s{\gamma }_{jk}s{\alpha }_{jk}\\ \begin{array}{l}{a}_{jk}s{\gamma }_{jk}\\ -b_{jk}c{\gamma }_{jk}s{\alpha }_{jk}\end{array}& \begin{array}{l}s{\gamma }_{jk}c{\beta }_{jk}\\ +c{\gamma }_{jk}c{\alpha }_{jk}s{\beta }_{jk}\end{array}& \begin{array}{l}c{\gamma }_{jk}c{\alpha }_{jk}c{\beta }_{jk}\\ -s{\gamma }_{jk}s{\beta }_{jk}\end{array}& -c{\gamma }_{jk}s{\alpha }_{jk}\\ c_{jk}+b_{jk}c{\alpha }_{jk}& s{\alpha }_{jk}s{\beta }_{jk}& s{\alpha }_{jk}c{\beta }_{jk}& c{\alpha }_{jk}\end{array}\right],$$

(1)

where c and s denote the cosine and sine functions, respectively.

The matrix ${\mathbf{T}}_{jk}$ has the following six parameters:

• $a_{jk}$ is the distance between the axes $W_j$ and $Z_k$ measured along the direction of the common normal $t_{jk}$ between these axes;
• ${\alpha }_{jk}$ is the angle between the positive directions of the axes $W_j$ and $Z_k$ measured counterclockwise relative to the positive direction of the $t_{jk}$;
• $b_{jk}$ is the distance from the direction $t_{jk}$ to the direction of the axis $X_k$ measured along the positive direction of the axis $Z_k$;
• ${\beta }_{jk}$ is an angle between the positive directions of the $t_{jk}$ and axis $X_k$ measured counterclockwise relative to the positive direction of the axis $Z_k$;
• ${\mathrm{c}}_{jk}$ is the distance from the direction of the axis $U_j$ to the direction of the $t_{jk}$ measured along the positive direction of the axis $W_j$;
• ${\gamma }_{jk}$ is the angle between the positive directions of the axis $U_j$ and $t_{jk}$ measured counterclockwise relative to the positive direction of the axis $W_j$. Figure 1b shows the kinematic diagram of the PM with the chosen coordinate systems, where the coordinate systems $O{U}_0{V}_0{W}_0$ and $P{X}_P{Y}_P{Z}_P$ are attached to the fixed and moving platforms, respectively.

Based on the transformation matrix ${\mathbf{T}}_{jk}$, the link matrices ${\mathbf{G}}_{jk}$ and the kinematic pair matrices ${\mathbf{P}}_j^k$ have been derived. These matrices contain the constant geometric parameters of the links and the variable parameters describing the relative motion of the kinematic pair elements, respectively. Each leg of the PM (Figure 1b) is associated with the following link and kinematic pair matrices:

$$G_{01}=G_{01}(0,{\alpha }_{01},0,0,c_{01},{\gamma }_{01})=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& \mathrm{c}{\gamma }_{01}& 0& \mathrm{s}{\gamma }_{01}\\ 0& \mathrm{s}{\gamma }_{01}& 0& -\mathrm{c}{\gamma }_{01}\\ c_{01}& 0& 1& 0\end{array}\right],$$

(2)

$$G_{12}=G_{12}(a_{12},-90,0,0,0,0)=\left[\begin{array}{cccc}1& 0& 0& 0\\ a_{12}& 1& 0& 0\\ 0& 0& 0& 1\\ 0& 0& -1& 0\end{array}\right],$$

(3)

$$G_{23}=G_{23}(a_{23},0,0,0,0,0)=\left[\begin{array}{cccc}1& 0& 0& 0\\ a_{23}& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right],$$

(4)

$$G_{34}=G_{34}(a_{34},0,0,0,0,0)=\left[\begin{array}{cccc}1& 0& 0& 0\\ a_{34}& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right],$$

(5)

$${\mathbf{P}}_1^{\mathit{P}}={\mathbf{P}}_1^{\mathit{P}}(0,0,0,0,c_{11},0)=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ s_1& 0& 0& 1\end{array}\right],$$

(6)

$${\mathbf{P}}_2^{\mathit{R}}={\mathbf{P}}_2^{\mathit{R}}(0,0,0,0,b_{12},{\gamma }_{22})=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& \mathrm{c}{\theta }_2& -\mathrm{s}{\theta }_2& 0\\ 0& \mathrm{s}{\theta }_2& \mathrm{c}{\theta }_2& 0\\ b_{12}& 0& 0& 1\end{array}\right],$$

(7)

$${\mathbf{P}}_3^{\mathit{R}}={\mathbf{P}}_3^{\mathit{R}}(0,0,0,{\beta }_{33},0,0)=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& \mathrm{c}{\theta }_3& -\mathrm{s}{\theta }_3& 0\\ 0& \mathrm{s}{\theta }_3& \mathrm{c}{\theta }_3& 0\\ 0& 0& 0& 1\end{array}\right],$$

(8)

where ${\alpha }_{01}={90}^0,c_{01}=O{O}_1,{\gamma }_{01,}{a}_{12}=O_1^{\prime }{O}_2,{\alpha }_{12}=-{90}^0,c_{22}=O_2{O}_2^{\prime },a_{23}=O_2^{\prime }{O}_3,$ and $a_{34}=O_3^{\prime }{O}_4$ are constant parameters; $c_{11}=O_1{O}_1^{\prime }=s_1,{\gamma }_{22}={\theta }_2,\mathrm{and} {\beta }_{33}={\theta }_3$ represent variable parameters, among which $s_1$ and ${\theta }_2$ are input parameters, and ${\theta }_3$ is an output parameter.

3. Inverse Kinematics

Since the 3-PRRS type PM has three identical legs, the variable parameters are denoted as $s_i,{\theta }_{2,i},{\theta }_{3,i}(i=1,2,3)$, while the constant parameters are denoted as $a=a_{12},b=b_{12},c=c_{01},f=a_{23},g=a_{34},\gamma ={\gamma }_{01}$.

In inverse kinematics, the pose (position and orientation) of the coordinate system $P{X}_P{Y}_P{Z}_P$, attached to the moving platform, is given. It is necessary to determine the input parameters $s_i$ and ${\theta }_i$. To do this, the equations for determining the coordinates of the centers of the spherical joints $O_{4,i}$ in the absolute coordinate system $O{U}_0{V}_0{W}_0$ are written through the legs OABCD:

$${[U_{O_{4,i}},V_{O_{4,i}},W_{O_{4,i}}]}^T={\mathbf{G}}_{01}{\mathbf{P}}_{1\mathit{i}}^{\mathit{P}}{\mathbf{G}}_{12}{\mathbf{P}}_{2\mathit{i}}^{\mathit{R}}{\mathbf{G}}_{23}{\mathbf{P}}_{3\mathit{i}}^{\mathit{R}}{\mathbf{G}}_{34}{[1,0,0,0]}^T,$$

(9)

where the coordinates of the joints $O_{4,i}$ are known and are expressed through the coordinate system $P{X}_P{Y}_P{Z}_P$ by the following equations:

$${[U_{O_{4,i}},V_{O_{4,i}},W_{O_{4,i}}]}^T={\mathbf{T}}_{OP}{\mathbf{G}}_{P4}{[1,0,0,0]}^T.$$

(10)

Expanding Equation (9) for the given values of the coordinates of the centers of the spherical joints, we obtain:

$$\left.\begin{array}{l}{U}_{O_{4,i}}=-b_{i}c{\gamma }_i+s_{i}s{\gamma }_i-f_{i}s{\gamma }_{i}s{\theta }_{2,i}-g_{i}s{\gamma }_{i}s{\theta }_{23,i}\\ V_{O_{4,i}}=-b_{i}s{\gamma }_i-s_{i}c{\gamma }_i+f_{i}c{\gamma }_{i}s{\theta }_{2,i}+g_{i}c{\gamma }_{i}s{\theta }_{23,i}\\ W_{O_{4,i}}=c_i+a_i+f_{i}c{\theta }_{2,i}+g_{i}c{\theta }_{23,i}\end{array}\right\},$$

(11)

where ${\theta }_{23,i}={\theta }_{2,i}+{\theta }_{3,i}$.

An analysis of the system of Equation (11) shows that the first two equations are linearly dependent. We transform the first and third equations of system (11) into the following form:

$$A_i{\mathrm{s}}^2{\theta }_{2,i}-B_i\mathrm{s}{\theta }_{2,i}+C_i=0,$$

(12)

where $A_i,\hspace{0.17em}{B}_i,\hspace{0.17em}\mathrm{and}\hspace{0.17em}{C}_i$ are known parameters

$$\begin{gathered} A_i={(s_i-K_i)}^2+M_i^2, B_i=2(f+gc{\theta }_{3,i})(s_i-K_i), \\ {C}_i={(f+gc{\theta }_{3,i})}^2-M_i^2, K_i={\displaystyle \frac{U_{O_{4,i}}+b\mathrm{c}{\gamma }_i}{\mathrm{s}{\gamma }_i}}, M_i=W_{O_{4,i}}-c-a. \end{gathered}$$

(13)

From Equation (12), and also from the first and third equations of system (11), two possible values of the parameters ${\theta }_{2,i}$ and ${\theta }_{3,i}$ can be determined:

$$\begin{array}{l}{\theta }_{2,i}=s^{-1}{\displaystyle \frac{B_i\pm \sqrt{B_i^2-4A_i{C}_i}}{2A_i}},{\theta }_{2,i}=\pi -s^{-1}{\displaystyle \frac{B_i\pm \sqrt{B_i^2-4A_i{C}_i}}{2A_i}},\\ {\theta }_{3,i}=\pm {\mathrm{c}}^{-1}{\displaystyle \frac{{(s_i-K_i)}^2+M_i^2-f^2-g^2}{2fg}}.\end{array}$$

(14)

which correspond to two assembly configurations of each PM leg.

Since the tripod has only three DOF, it is difficult to specify the coordinates of the centers of the spherical joints through the three planes along which they must move. Therefore, in our work [55], based on the constraint equations, a relationship was established between the dependent parameters $X_P,Y_P,\phi$ and the independent parameters $\psi ,\theta ,{\mathit{Z}}_P$ which determine the position and orientation of the moving platform.

In addition, the relationships between the independent parameters $\psi ,\theta ,Z_P$ and the coordinates of the centers of the spherical joints $U_{O_{4,i}},V_{O_{4,i}},W_{O_{4,i}}$ were determined. This formulation of the inverse kinematics problem is convenient for comparison with the direct problem, since in solving the forward kinematics problem the angles of the intermediate links ${\theta }_{3,i}$ are determined.

The positions of the passive linear actuators $s_i$ are not determined when solving the inverse kinematics problem; they are assigned arbitrarily, since the tripod has only three DOF. Naturally, they can be calculated within the framework of the inverse problem, but this requires fixing all three active revolute actuators.

4. Direct Kinematics

In the direct kinematics of the 3-PRRS-type PM, it is necessary to determine the pose of the moving platform based on the given values of the input parameters $s_i$ and ${\theta }_{2,i}$. To achieve this, the coordinates of the spherical joints are determined using Equation (9), where the unknown variables are the angles ${\theta }_{3,i}$, which are part of the kinematic pair matrix ${\mathbf{P}}_{3,i}^R$. To determine the angles ${\theta }_{3,i}$, the following system of equations has been derived:

$$\left.\begin{array}{l}{(U_{O_{4,1}}-U_{O_{4,2}})}^2+{(V_{O_{4,1}}-V_{O_{4,2}})}^2+{(W_{O_{4,1}}-W_{O_{4,2}})}^2=d^2\\ {(U_{O_{4,1}}-U_{O_{4,3}})}^2+{(V_{O_{4,1}}-V_{O_{4,3}})}^2+{(W_{O_{4,1}}-W_{O_{4,3}})}^2=d^2\\ {(U_{O_{4,2}}-U_{O_{4,3}})}^2+{(V_{O_{4,2}}-V_{O_{4,3}})}^2+{(W_{O_{4,2}}-W_{O_{4,3}})}^2=d^2\end{array}\right\},$$

(15)

where d represents the distances between the spherical joints.

By introducing the following notation ${{\theta }^{\prime }}_{3,i}={\theta }_{2,i}+{\theta }_{3,i},$ $t_{3,i}=\tan ({{\theta }^{\prime }}_{3,i}/2)$, $s{{\theta }^{\prime }}_{3,i}=2t_{3,i}/1+t_{3,i}^2,$ $c{{\theta }^{\prime }}_{3,i}=1-t_{3,i}^2/1+t_{3,i}^2,(i=1,2,3)$, and applying the Sylvester method [46], the system of Equation (15) is reduced to a 16th-degree polynomial in a single variable $t_{3,1}$:

$${\displaystyle \sum _{k=0}^{16}{P}_k{t}_{31}^k}=0,$$

(16)

which yields a maximum of 16 values of ${\theta }_{3,i}$ for a single assembly configuration of the PM leg; that is, for one value of ${\theta }_{2,i}$, up to 16 corresponding values of ${\theta }_{3,i}$ can be obtained, representing 16 possible positions of the moving platform. A detailed analysis of the forward kinematics of the considered PM can be found in [56].

5. Numerical Examples of Direct Kinematics

The following constant parameters of the 3-PRRS-type tripod are given: $a_i=15, b_i=8,$$c_i=7,$$f_i=60,$$g_i=70,$${\gamma }_1=\pi /2+\xi ,$${\gamma }_2=7\pi /6+\xi ,$${\gamma }_3=\xi -\pi /6,$$h=43,\hspace{0.17em}\hspace{0.17em}d=h\cdot \sqrt{3},\hspace{0.17em}\hspace{0.17em}\xi =a\sin (b/h)$. For the input parameters $s_1=s_2=s_3=60$ and ${\theta }_{2,1}=-40^0,{\theta }_{2,2}=-{45}^0,{\theta }_{2,3}=-35^0$, four positions of the moving platform were obtained (Figure 2).

Table A1. Calculated values of the angles ${\theta }_{3,1},{\theta }_{3,2},{\theta }_{3,3}$. 

k	${\theta }_{2,1}$	${\theta }_{3,1}$	${\theta }_{3,2}$	${\theta }_{3,3}$
1	$-{\displaystyle \frac{\pi }{4}}$	1.9714	2.5518	2.3137
		2.2974	2.8269	2.8562
		2.5953	2.0203	2.2978
		2.8565	2.3016	2.8091
2	$-{\displaystyle \frac{\pi }{5}}$	1.7111	2.4516	2.2484
		2.1030	2.8145	2.8768
		2.6020	2.0306	2.2723
		2.8699	2.3431	2.8393
3	$-{\displaystyle \frac{3\pi }{20}}$	1.4626	2.3653	2.2078
		1.9157	2.8016	2.8927
		2.5883	2.0407	2.2549
		2.8699	2.3781	2.8622
4	$-{\displaystyle \frac{\pi }{10}}$	1.2202	2.2936	2.1857
		1.7311	2.7877	2.9062
		2.5581	2.0511	2.2409
		2.8614	2.4108	2.8813
5	$-{\displaystyle \frac{\pi }{20}}$	0.9802	2.2365	2.1769
		1.5466	2.7723	2.9180
		2.5113	2.0625	2.2287
		2.8458	2.4434	2.8984
6	0	0.7384	2.1934	2.1764
		1.3602	2.7547	2.9285
		2.4452	2.0756	2.2173
		2.8234	2.4777	2.9142

in Appendix A presents the obtained values of the angles ${\theta }_{3,1},{\theta }_{3,2},{\theta }_{3,3}$ for k = 6 given values of ${\theta }_{2,1},$ $-{\displaystyle \frac{\pi }{4}},$ $-{\displaystyle \frac{\pi }{5}},$ $-{\displaystyle \frac{3\pi }{20}},$ $-{\displaystyle \frac{\pi }{10}},$ $-{\displaystyle \frac{\pi }{20}}$, 0 at the input parameter value $s_i=60$, ${\theta }_{2,2}=-\pi /4$, ${\theta }_{2,3}=-\pi /3$.

As can be seen from Table A1 in Appendix A, for each value of the angle ${\theta }_{2,1}$, four values of the angles ${\theta }_{3,1},{\theta }_{3,2},{\theta }_{3,3}$ are determined. Using the obtained values of the angles ${\theta }_{3,1},{\theta }_{3,2},{\theta }_{3,3}$, the coordinates of the spherical joint centers of the moving platform in the absolute coordinate system $O_0{U}_0{V}_0{W}_0$ are determined (

Table A2. Calculated coordinates of the spherical joint  $O_{4,1}$. 

k	${\theta }_{2,1}$	$U_{O_{4,1}}$	$V_{O_{4,1}}$	$W_{O_{4,1}}$
1	$-{\displaystyle \frac{\pi }{4}}$	38.3763	−0.8755	90.6979
		33.4671	−1.8050	68.5344
		35.3065	−1.4567	47.8432
		41.7807	−0.2308	30.8433
2	$-{\displaystyle \frac{\pi }{5}}$	34.3408	−1.6396	103.3579
		26.6316	−3.0993	77.2571
		31.8213	−2.1167	43.0953
		41.2162	−0.3377	27.0288
3	$-{\displaystyle \frac{3\pi }{20}}$	29.6504	−2.5277	113.7845
		18.9741	−4.5493	84.2738
		28.4385	−2.7572	39.0912
		40.6844	−0.4384	23.9019
4	$-{\displaystyle \frac{\pi }{10}}$	24.5216	−3.4989	122.2393
		10.6921	−6.1175	89.7869
		24.8862	−3.4298	35.4180
		40.1464	−0.5403	21.0662
5	$-{\displaystyle \frac{\pi }{20}}$	19.2295	−4.5009	128.8561
		2.0118	−7.7612	93.8796
		20.9347	−4.1781	31.8606
		39.5715	−0.6491	18.3165
6	0	14.1445	−5.4638	133.7668
		−6.8187	−9.4333	96.6268
		16.3261	−5.0507	28.2959
		38.9275	−0.7711	15.5125

,

Table A3. Calculated coordinates of the spherical joint  $O_{4,2}$.

k	${\theta }_{2,1}$	$U_{O_{4,2}}$	$V_{O_{4,2}}$	$W_{O_{4,2}}$
1	$-{\displaystyle \frac{\pi }{4}}$	−15.9630	30.8067	50.8172
		−20.0582	35.5642	32.6796
		−17.6435	32.7591	87.4987
		−15.1594	29.8733	68.2404
2	$-{\displaystyle \frac{\pi }{5}}$	−15.2994	30.0359	57.7563
		−19.8044	35.2693	33.4561
		−17.4904	32.5811	86.8149
		−15.0955	29.7990	65.3402
3	$-{\displaystyle \frac{3\pi }{20}}$	−15.0935	29.7967	63.7835
		−19.5487	34.9723	34.2636
		−17.3451	32.4123	86.1439
		−15.1026	29.8073	62.8889
4	$-{\displaystyle \frac{\pi }{10}}$	−15.1810	29.8983	68.8041
		−19.2795	34.6596	35.1433
		−17.1998	32.2436	85.4496
		−15.1598	29.8736	60.6035
5	$-{\displaystyle \frac{\pi }{20}}$	−15.4182	30.1739	72.7831
		−18.9889	34.3220	36.1304
		−17.0466	32.0656	84.6886
		−15.2653	29.9963	58.3260
6	0	−15.6954	30.4959	75.7714
		−18.6706	33.9521	37.2615
		−16.8769	31.8684	83.8074
		−15.4288	30.1862	55.9355

and

Table A4. Calculated coordinates of the spherical joint  $O_{4,3}$. 

k	${\theta }_{2,1}$	$U_{O_{4,3}}$	$V_{O_{4,3}}$	$W_{O_{4,3}}$
1	$-{\displaystyle \frac{\pi }{4}}$	−22.4664	−40.0022	72.9707
		−22.0580	−38.8345	35.4773
		−22.5790	−40.3241	74.0271
		−21.8258	−38.1708	38.7014
2	$-{\displaystyle \frac{\pi }{5}}$	−22.9649	−41.4274	77.2815
		−22.1748	−39.1684	34.0835
		−22.7720	−40.8760	75.7184
		−21.9688	−38.5794	36.6323
3	$-{\displaystyle \frac{3\pi }{20}}$	−23.3216	−42.4472	79.9112
		−22.2717	−39.4455	33.0095
		−22.9116	−41.2750	76.8607
		−22.0907	−38.9279	35.0748
4	$-{\displaystyle \frac{\pi }{10}}$	24.5216	−3.4989	122.2393
		10.6921	−6.1175	89.7869
		24.8862	−3.4298	35.4180
		40.1464	−0.5403	21.0662
5	$-{\displaystyle \frac{\pi }{20}}$	19.2295	−4.5009	128.8561
		2.0118	−7.7612	93.8796
		20.9347	−4.1781	31.8606
		39.5715	−0.6491	18.3165
6	0	−23.6213	−43.3042	81.9128
		−22.5106	−40.1286	30.6074
		−23.2354	−42.2009	79.3032
		−22.4115	−39.8451	31.5667

in Appendix A).

Then, the coordinates of the point P on the moving platform in the absolute coordinate system $O_0{U}_0{V}_0{W}_0$ can be determined by the following equations:

$$X_P={\displaystyle \frac{1}{3}}\cdot {\displaystyle \sum _{i=1}^n{U}_{O_{4,i}}},Y_P={\displaystyle \frac{1}{3}}\cdot {\displaystyle \sum _{i=1}^n{V}_{O_{4,i}}},Z_P={\displaystyle \frac{1}{3}}\cdot {\displaystyle \sum _{i=1}^n{W}_{O_{4,i}}}.$$

(17)

Table A5. Calculated coordinates of the point P. 

$i$	$X_P$	$Y_P$	$Z_P$
1	−0.0177	−3.3570	71.4953
	−1.3078	−4.3437	79.4652
	−2.9215	−5.0594	85.8264
	−4.7299	−5.5481	90.7885
	−6.6017	−5.8724	94.5070
	−8.3907	−6.0907	97.1503
2	−2.8830	−1.6917	45.5638
	−5.1158	−2.3328	48.2655
	−7.6154	−3.0075	50.5156
	−10.3152	−3.7169	52.3445
	−13.1382	−4.4530	53.7731
	−16.0000	−5.2032	54.8319
3	−1.6387	−3.0072	69.7897
	−2.8137	−3.4705	68.5428
	−3.9394	−3.8733	67.3653
	−5.1138	−4.2647	66.2124
	−6.4152	−4.6742	65.0379
	−7.9287	−5.1277	63.8022
4	1.5984	−2.8427	45.9283
	1.3839	−3.0393	43.0005
	1.1636	−3.1863	40.6219
	0.9281	−3.3043	38.4813
	0.6659	−3.4008	36.4214
	0.3623	−3.4766	34.3382

in Appendix A presents the obtained values of the coordinates of point P in the absolute coordinate system $O_0{U}_0{V}_0{W}_0$, and Figure 3 shows a graph of the motion (trajectory) of point P. As can be seen from the graph, point P moves along an arbitrary curve lying on the sphere.

The number of real roots of polynomial (16) depends on the arrangement of the active kinematic pairs. In some configurations, the number of real roots reaches 16, but most commonly it ranges from 4 to 8. In the symmetric configuration of the active rotational actuators, when ${\theta }_{2,1}={\theta }_{2,2}={\theta }_{2,3}=-45^0$, and the active prismatic actuators are fixed at $s_1=s_2=s_3=60$, four positions corresponding to the first four solutions were determined (Figure 2). Based on this, it can be concluded that the robot has four working modes, which are separated by serial singularities.

6. Numerical Examples of Solving the Inverse Kinematics Problem

For a numerical example of the inverse kinematics of the 3-PRRS-type PM, we use the obtained coordinates of the spherical joints from the direct kinematics. For instance, we take the first values of the spherical joint coordinates from Table A1, Table A2 and Table A3 (in Appendix A): $U_{O_{4,1}}=38.3763,$ $V_{O_{4,1}}=-0.8755,$ $W_{O_{4,1}}=90.6979,$ $U_{O_{4,2}}=-15.9630,$ $V_{O_{4,2}}=30.8067,$ $W_{O_{4,2}}=90.6979,$ $U_{O_{4,3}}=-22.4664,$ $V_{O_{4,3}}=-40.0022,$ $W_{O_{4,3}}=72.9707$, and determine two values of the angles ${\theta }_{2,i}$ and ${\theta }_{3,i}$, which are presented in

Table A6. Calculated values of the angles  ${\theta }_{2,i}$  and  ${\theta }_{3,i}$. 

${\theta }_{2,1}$	−0.7853	1.4172	${\theta }_{3,1}$	1.9714	−1.9714
${\theta }_{2,2}$	−0.7853	0.8790	${\theta }_{3,2}$	2.5518	−2.5518
${\theta }_{2,3}$	−1.0471	1.5283	${\theta }_{3,3}$	2.3137	−2.3137

(in Appendix A).

The obtained values of the angles ${\theta }_{2,i}$ and ${\theta }_{3,i}$ correspond to their given values in the direct kinematics. Furthermore, the results of the inverse kinematics were verified using the direct kinematics results for other positions of the moving platform.

7. Workspace

The workspace of the proposed 3-PRRS-type PM is determined by its inverse kinematics. To define the workspace, it is necessary to derive equations that limit the movement of the moving platform and establish a relationship between the dependent and independent parameters that define the pose of the moving platform. Then, the pose of the moving platform should be determined, taking into account these constraints. All valid poses of the moving platform, for which solutions to the inverse kinematics problem exist, are included in the workspace.

It is known that, due to the constraints of the revolute kinematic pairs, the legs of the 3-PRRS-type PM move within specific planes. By summing the first and second equations of system (11), after multiplying the first equations by $\mathrm{c}{\gamma }_i$ and the second equations by $s{\gamma }_i$, we obtain the equations of the planes along which the ${O^{\prime }}_{2,i}{O}_{3,i}{O}_{4,i}$ dyads (PM legs) move:

$$c{\gamma }_i{U}_{O_{4,i}}+s{\gamma }_i{V}_{O_{4,i}}+b_i=0,(i=1,2,3).$$

(18)

To define the trajectory of the moving platform center, the first and third equations of system (11) are reduced to the following form:

$$\left.\begin{array}{l}{g}_{i}s({\theta }_{23,i})=s_i-{\displaystyle \frac{U_{O_{4,i}}+bc{\gamma }_i}{s{\gamma }_i}}-f_{i}s{\theta }_{2,i}\\ g_{i}c({\theta }_{23,i})=W_{O_{4,i}}-c_i-a_i-f_{i}c{\theta }_{2,i}\end{array}\right\}.$$

(19)

Adding the squares of the two equations of system (19) yields:

$${(W_{O_{4,i}}-c_i-a_i-f_{i}c{\theta }_{2,i})}^2+{(s_i-{\displaystyle \frac{U_{O_{4,i}}+b_{i}c{\gamma }_i}{s{\gamma }_i}}-f_{i}s{\theta }_{2,i})}^2-g_i^2=0.$$

(20)

Determining $b_i$ from Equation (18) and substituting it into Equation (20), we obtain the following equations:

$${(X_{2,i}-f_{i}c{\theta }_{2,i})}^2+{(Y_{2,i}-f_{i}s{\theta }_{2,i})}^2-g_i^2=0,$$

(21)

where $X_{2,i}=W_{O_{4,i}}-c_i-a_i,Y_{2,i}=s_i-U_{O_{4,i}}s{\gamma }_i+V_{O_{4,i}}\mathrm{c}{\gamma }_i.$ Here, $X_{2,i},Y_{2,i},Z_{2,i}$ are the coordinates of the absolute coordinate system relative to the local coordinate systems $O_{2,i}{X}_{2,i}{Y}_{2,i}{Z}_{2,i}.$ $Z_{2,i}=-(c{\gamma }_i\cdot U_{O_{4i}}+s{\gamma }_i\cdot V_{O_{4i}}+b_i)$ is equal to zero according to Equation (18).

Expanding Equation (21) yields:

$$X_{2,i}c{\theta }_{2,i}+Y_{2,i}s{\theta }_{2,i}-{\displaystyle \frac{X_{2,i}^2+Y_{2,i}^2+f_i^2-g_i^2}{2f_i}}=0.$$

(22)

The solution of Equation (22) exists under the following condition [47]:

$${\left({\displaystyle \frac{X_{2,i}^2+Y_{2,i}^2+f_i^2-g_i^2}{2\cdot f_i}}\right)}^2-(X_{2,i}^2+Y_{2,i}^2)\le 0.$$

(23)

Equating Equation (23) to zero, we obtain the equations of the following circles:

$$X_{2,i}^2+Y_{2,i}^2={(g_i-f_i)}^2$$

(24)

and

$$X_{2,i}^2+Y_{2,i}^2={(g_i+f_i)}^2.$$

(25)

The circles (24) and (25) represent the inner and outer boundaries of the workspaces of the dyads $O_{2,i}{O}_{3,i}{O}_{4,i}$ (PM legs). These equations are expressed with respect to the absolute coordinate system $O_0{U}_0{V}_0{W}_0$ as follows:

$${(U_0-O_{2,ix})}^2+{(V_0-O_{2,iy})}^2+{(W_0-O_{2,iz})}^2={(g_i-f_i)}^2$$

(26)

and

$${(U_0-O_{2,ix})}^2+{(V_0-O_{2,iy})}^2+{(W_0-O_{2,iz})}^2={(g_i+f_i)}^2,$$

(27)

where $O_{2,ix},O_{2,\mathit{i}\mathit{y}},O_{2,\mathit{i}\mathit{z}}$ are the coordinates of the centers of the circles.

Let us define the coordinates of point P relative to the absolute coordinate system through the coordinates of the spherical joints using the following equation:

$$\left.\begin{array}{l}{U}_P=U_{O_{4,i}}-U_{O_{4,Pi}}\mathrm{s}{\gamma }_i\mathrm{s}{\theta }_{23,i}-V_{O_{4,Pi}}\mathrm{s}{\gamma }_i\mathrm{c}{\theta }_{23,i}-W_{O_{4,Pi}}\mathrm{c}{\gamma }_i\\ V_P=V_{O_{4,i}}+U_{O_{4,Pi}}\mathrm{c}{\gamma }_i\mathrm{s}{\theta }_{23,i}+V_{O_{4,Pi}}\mathrm{c}{\gamma }_i\mathrm{c}{\theta }_{23,i}-W_{O_{4,Pi}}\mathrm{s}{\gamma }_i\\ W_P=W_{O_{4,i}}+U_{O_{4,Pi}}\mathrm{c}{\theta }_{23,i}-V_{O_{4,Pi}}\mathrm{s}{\theta }_{23,i}\end{array}\right\},$$

(28)

where $U_{O_{4,Pi}},V_{O_{4,Pi}},W_{O_{4,Pi}}$ are the coordinates of the center of the moving platform relative to the local coordinate systems $O_{4,i}{X}_{4,i}{Y}_{4,i}{Z}_{4,i}$.

Multiplying the first equations of system (28) by $\mathrm{s}{\gamma }_i$ and the second equations by $-\mathrm{c}{\gamma }_i$, and then adding the first and second equations, we obtain:

$$\left.\begin{array}{l}(U_P-U_{O_{4,i}})\mathrm{s}{\gamma }_i-(V_P-V_{O_{4,i}})\mathrm{c}{\gamma }_i=-U_{O_{4,Pi}}\mathrm{s}{\gamma }_i\mathrm{s}{\theta }_{23,i}-V_{O_{4,Pi}}\mathrm{s}{\gamma }_i\mathrm{c}{\theta }_{23,i}\\ W_P-W_{O_{4,i}}=U_{O_{4,Pi}}\mathrm{c}{\theta }_{23,i}-V_{O_{4,Pi}}\mathrm{s}{\theta }_{23,i}\end{array}\right\}.$$

(29)

Multiplying the first equations of system (29) by $\mathrm{s}{\theta }_{23,i}$ and the second equations by $-\mathrm{c}{\theta }_{23,i}$, and then adding them, yields:

$$U_{O_{4,Pi}}=\left[(V_P-V_{O_{4,i}})\mathrm{c}{\gamma }_i-(U_P-U_{O_{4,i}})\mathrm{s}{\gamma }_i\right]\mathrm{s}{\theta }_{23,i}+(W_P-W_{O_{4,i}})\mathrm{c}{\theta }_{23,i}.$$

(30)

To determine $V_{O_{4,Pi}}$, we also add the two equations of system (29), previously multiplying the first equations by $\mathrm{c}{\theta }_{23,i}$ and the second equations by $\mathrm{s}{\theta }_{23,i}$, and we obtain:

$$V_{O_{4,Pi}}=\left[(V_P-V_{O_{4,i}})\mathrm{c}{\gamma }_i-(U_P-U_{O_{4,i}})\mathrm{s}{\gamma }_i\right]\mathrm{c}{\theta }_{23,i}-(W_P-W_{O_{4,i}})\mathrm{s}{\theta }_{23,i}.$$

(31)

To determine $W_{O_{4,Pi}}$, we add the first and second equations of system (28), previously multiplying the first equation by $\mathrm{c}{\gamma }_i$ and the second equation by $\mathrm{s}{\gamma }_i$, and we obtain:

$$W_{O_{4,Pi}}=-(U_P-U_{O_{4,i}})\mathrm{c}{\gamma }_i-(V_P-V_{O_{4,i}})\mathrm{s}{\gamma }_i.$$

(32)

Since the distances from the center P of the moving platform to the centers of the spherical joints are equal to h, the following equations are true in the local coordinate systems $O_{4,i}{X}_{4,i}{Y}_{4,i}{Z}_{4,i}$:

$$U_{O_{4,Pi}}^2+V_{O_{4,Pi}}^2+W_{O_{4,Pi}}^2=h^2.$$

(33)

Substituting Equations (30)–(32) into Equation (33), we obtain the equations of three spheres in the absolute coordinate system along which point P moves:

$${(U_P-U_{O_{4,i}})}^2+{(V_P-V_{O_{4,i}})}^2+{(W_P-W_{O_{4,i}})}^2=h^2.$$

(34)

The equations of these three spheres describe the motion of the center P of the moving platform relative to the absolute coordinate system $O_0{U}_0{V}_0{W}_0$. Therefore, the legs of the PM move along arcs of circles relative to the absolute coordinate system, and the center of the moving platform moves relative to the spherical joints $O_{4,i}$ along arcs of circles belonging to the sphere, in accordance with Equation (34). Therefore, the legs of the PM move along arcs of circles relative to the absolute coordinate system, and the center of the moving platform moves relative to the spherical joints $O_{4,i}$ along an arbitrary curve lying on the sphere, in accordance with Equation (34).

With the following values of constant parameters: $a_i=15, b_i=8, c_i=5, f_i=60,$ $g_i=0, h=43$, we define the workspace of the considered PM based on the inverse kinematics problem. In Figure 4, with fixed strokes of linear actuators $s_1=s_2=s_3=50$, and when changing $W_P$ from 0 to 170 with a step of 5, and changing $\psi$ and $\theta$ from $-\pi /2$ to $\pi /2$ with a step of $\pi /50$, the inverse kinematics problem is solved. If there is a solution to the inverse kinematics problem, the computer program places a point in space. In Figure 4, the red mark “o” and blue mark “*” indicate two orientations of the moving platform in space.

Figure 5 shows the workspaces of the PM for two values of the stroke lengths of the linear actuators, $s_1=s_2=s_3=40$ and $s_1=s_2=s_3=60$. From this figure, it can be seen that the change in the stroke lengths of the linear actuators affects the workspace in the vertical direction. When the stroke lengths of the linear actuators decrease, the workspace in the upper part of the PM increases, and when the stroke lengths increase, the workspace expands in the lower part of the PM.

8. Jacobian Matrices

Let’s derive the closed loop $O_0{O}_{1,i}{O^{\prime }}_{1,i}{O^{\prime }}_{2,i}{O}_{2,i}{O}_{3,i}{O}_{4,i}P$ equations of the leg:

$${\mathbf{r}}_P={\mathbf{r}}_{O_{1,i}}+{\mathbf{s}}_i+{\mathbf{a}}_{1^{\prime }{2}^{\prime },i}+{\mathbf{b}}_{2^{\prime }2,i}+{\mathbf{f}}_i+{\mathbf{g}}_i-{\mathbf{h}}_i,$$

(35)

where

$$\left[\begin{array}{c}1\\ {\mathbf{r}}_P\end{array}\right]={\mathbf{T}}_{OP}\left[\begin{array}{c}1\\ 0\\ 0\\ 0\end{array}\right]=\left[\begin{array}{c}1\\ a_{OP}\mathrm{c}{\gamma }_{OP}+b_{OP}\mathrm{s}{\gamma }_{OP}\mathrm{s}{\alpha }_{OP}\\ a_{OP}\mathrm{s}{\gamma }_{OP}-b_{OP}\mathrm{c}{\gamma }_{OP}\mathrm{s}{\alpha }_{OP}\\ c_{OP}+b_{OP}\mathrm{c}{\alpha }_{OP}\end{array}\right].$$

(36)

As can be seen from (36), the radius vectors of the origins of the coordinate systems can also be determined using the shift submatrices of the matrix (1) in the following forms:

$${\mathbf{r}}_P={\left[U_P,V_P,W_P\right]}^T={}^0\mathbf{\tau }_P=\mathbf{\tau }(a_{OP},b_{OP},c_{OP},{\alpha }_{OP},{\beta }_{OP},{\gamma }_{OP})$$

(37)

and

$${\mathbf{r}}_{O_{1,i}}={\left[U_{O_{1,i}},V_{O_{1,i}},W_{O_{1,i}}\right]}^T={}^0\mathbf{\tau }_i=\mathbf{\tau }(a_{01,i},b_{01,i},c_{01,i},{\alpha }_{01,i},{\beta }_{01,i},{\gamma }_{01,i}).$$

(38)

The coordinate of other vectors in the absolute coordinate system $O_0{U}_0{V}_0{W}_0$ are determined using rotation submatrices:

$$\begin{array}{l}{\mathbf{f}}_i=\overline{O_{2i}{O}_{3i}}={}^0\mathbf{R}_{2i}{}^{2i}\mathbf{f}_i={}^0\mathbf{R}_{1i}{}^{1i}\mathbf{R}_{2i}{\mathbf{f}}_i\\ =\mathbf{R}({\alpha }_{01},{\beta }_{01},{\gamma }_{01})\mathbf{R}({\alpha }_{12,i},{\beta }_{12,i},{\gamma }_{12,i}){}^{2i}\mathbf{f}_i\\ =\left[\begin{array}{ccc}\mathrm{c}{\gamma }_i& 0& \mathrm{s}{\gamma }_i\\ \mathrm{s}{\gamma }_i& 0& -\mathrm{c}{\gamma }_i\\ 0& 1& 0\end{array}\right]\left[\begin{array}{ccc}0& 0& -1\\ \mathrm{c}{\theta }_2& -\mathrm{s}{\theta }_2& 0\\ -\mathrm{s}{\theta }_2& -\mathrm{c}{\theta }_2& 0\end{array}\right]\left[\begin{array}{c}{f}_i\\ 0\\ 0\end{array}\right]=\left[\begin{array}{c}-f_i\mathrm{s}{\gamma }_i\mathrm{s}{\theta }_{2i}\\ f_i\mathrm{c}{\gamma }_i\mathrm{s}{\theta }_{2i}\\ f_i\mathrm{c}{\theta }_{2i}\end{array}\right].\end{array}$$

(39)

$$\begin{array}{l}{\mathbf{g}}_i=\overline{O_{3i}{O}_{4i}}={}^0\mathbf{R}_{3i}{}^{2i}\mathbf{g}_i={}^0\mathbf{R}_{1i}{}^{1i}\mathbf{R}_{2i}{}^{2i}\mathbf{R}_{3i}{}^{3i}\mathbf{g}_i\\ =\mathbf{R}({\alpha }_{01,i},{\beta }_{01,i},{\gamma }_i)\mathbf{R}({\alpha }_{12,i},{\beta }_{12,i},{\gamma }_{12,i})\mathbf{R}({\alpha }_{23,i},{\beta }_{23,i},{\gamma }_{23,i}){}^{3i}\mathbf{g}_i\\ =\left[\begin{array}{ccc}\mathrm{c}{\gamma }_i& 0& \mathrm{s}{\gamma }_i\\ \mathrm{s}{\gamma }_i& 0& -\mathrm{c}{\gamma }_i\\ 0& 1& 0\end{array}\right]\left[\begin{array}{ccc}0& 0& -1\\ \mathrm{c}{\theta }_{2i}& -\mathrm{s}{\theta }_{2i}& 0\\ -\mathrm{s}{\theta }_{2i}& -\mathrm{c}{\theta }_{2i}& 0\end{array}\right]\left[\begin{array}{ccc}\mathrm{c}{\theta }_{3i}& -\mathrm{s}{\theta }_{3i}& 0\\ \mathrm{s}{\theta }_{3i}& \mathrm{c}{\theta }_{3i}& 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{c}{\mathrm{g}}_i\\ 0\\ 0\end{array}\right]\\ =\left[\begin{array}{c}-{\mathrm{g}}_i\mathrm{s}{\gamma }_i\mathrm{s}({\theta }_{2i}+{\theta }_{3i})\\ {\mathrm{g}}_i\mathrm{c}{\gamma }_i\mathrm{s}({\theta }_{2i}+{\theta }_{3i})\\ {\mathrm{g}}_i\mathrm{c}({\theta }_{2i}+{\theta }_{3i})\end{array}\right].\end{array}$$

(40)

$${\mathbf{h}}_i=\overline{P{O}_{4i}}={}^0\mathbf{R}_P{}^P\mathbf{h}_i=\mathbf{R}({\alpha }_i,{\beta }_i,{\gamma }_i)\left[\begin{array}{c}{h}_i\mathrm{c}{\phi }_i\\ h_i\mathrm{s}{\phi }_i\\ 0\end{array}\right],$$

(41)

where ${\phi }_1=0,{\phi }_2={\displaystyle \frac{2\pi }{3}},{\phi }_3=-{\displaystyle \frac{2\pi }{3}}$.

Differentiating Equation (35) with respect to time, we obtain:

$${\dot{\mathbf{r}}}_P={\dot{\mathbf{r}}}_{O_{1,i}}+{\dot{\mathbf{s}}}_i+{\dot{\mathbf{a}}}_{1^{\prime }{2}^{\prime },i}+{\dot{\mathbf{b}}}_{2^{\prime }2,i}+{\dot{\mathbf{f}}}_i+{\dot{\mathbf{g}}}_i-{\dot{\mathbf{h}}}_i$$

(42)

or taking into consideration that the vectors ${\mathbf{r}}_{O_{1,i}}\mathbf{,}{\mathbf{a}}_{1^{\prime }{2}^{\prime },i},{\mathbf{b}}_{2^{\prime }2,i}$ are constants, yields:

$${\dot{\mathbf{r}}}_P={\dot{s}}_i{\mathbf{e}}_{1,i}+{\dot{\theta }}_{2,i}\times {\mathbf{f}}_i+{\mathbf{\omega }}_i\times {\mathbf{g}}_i-{\mathbf{\omega }}_P\mathbf{\times }{\mathbf{h}}_i,$$

(43)

where ${\dot{\mathbf{r}}}_P={[{\dot{U}}_{OP},{\dot{V}}_{OP},{\dot{W}}_{OP}]}^T={[V_{P_{U_0}},V_{P_{V_0}},V_{P_{W_0}}]}^T$ is a linear velocity vector of the moving platform point P; ${\dot{\mathbf{s}}}_i={\dot{s}}_i\cdot {\mathbf{e}}_{1,i}$ is a linear generalized velocity vector, ${\dot{\theta }}_{2,i}={\dot{\theta }}_{2,i}{e}_{2,i}$ is an angular generalized velocity vector, ${\mathbf{\omega }}_i$ is an angular velocity of the i-th intermediate link; ${\mathbf{\omega }}_P={[{\omega }_{P_{U_0}},{\omega }_{P_{V_0}},{\omega }_{P_{W_0}}]}^T$ is an angular velocity vector of a moving platform.

Scalar multiplying of both sides of Equation (43) by the vector ${\mathbf{g}}_i^T$ of the intermediate link, we obtain:

$${\mathbf{g}}_i^T{\dot{\mathbf{r}}}_P={\dot{s}}_i{\mathbf{g}}_i^T{\mathbf{e}}_{1,i}+{\mathbf{g}}_i^T({\dot{\mathbf{\theta }}}_{2,i}\times {\mathbf{f}}_i)+{\mathbf{g}}_i^T({\mathbf{\omega }}_i\times {\mathbf{g}}_i)-{\mathbf{g}}_i^T({\mathbf{\omega }}_P\times {\mathbf{h}}_i)$$

(44)

or taking into consideration the relative positions of the vectors, yields:

$${\mathbf{g}}_i^T{\dot{\mathbf{r}}}_P={\dot{s}}_i({\mathbf{g}}_i^T{\mathbf{e}}_{1i})+{\dot{\theta }}_{2i}{\mathbf{e}}_{2i}^T({\mathbf{f}}_i\times {\mathbf{g}}_i)-{\mathbf{\omega }}_P^T({\mathbf{h}}_i\times {\mathbf{g}}_i).$$

(45)

Let us introduce the following notations:

$$\dot{\mathbf{x}}=\left[{\dot{\mathbf{r}}}_P^T,{\mathbf{\omega }}_P^T\right],\dot{\mathbf{q}}={\left[{\dot{s}}_1,{\dot{\theta }}_{21},{\dot{s}}_2,{\dot{\theta }}_{22},{\dot{s}}_3,{\dot{\theta }}_{23}\right]}^T$$

(46)

and rewrite Equation (45) in the following matrix form:

$${\mathbf{J}}_{\mathbf{x}}\dot{\mathbf{x}}={\mathbf{J}}_{\mathbf{q}}\dot{\mathbf{q}},$$

(47)

where ${\mathbf{J}}_{\mathbf{x}}$ and ${\mathbf{J}}_{\mathbf{q}}$ are the Jacobian matrices connecting the input and output links velocities:

$${\mathbf{J}}_{\mathbf{x}}=\left[\begin{array}{cc}{\mathbf{g}}_1^T& {\mathbf{(}{\mathbf{h}}_1\times {\mathbf{g}}_1)}^T\\ {\mathbf{g}}_2^T& {\mathbf{(}{\mathbf{h}}_2\times {\mathbf{g}}_1)}^T\\ {\mathbf{g}}_3^T& {\mathbf{(}{\mathbf{h}}_3\times {\mathbf{g}}_3)}^T\end{array}\right],$$

(48)

$${\mathbf{J}}_{\mathbf{q}}=\left[\begin{array}{cccccc}{\mathbf{g}}_1^T{\mathbf{e}}_{11}& {\mathbf{e}}_{21}^T({\mathbf{f}}_1\times {\mathbf{g}}_1)& 0& 0& 0& 0\\ 0& 0& {\mathbf{g}}_2^T{\mathbf{e}}_{12}& {\mathbf{e}}_{22}^T({\mathbf{f}}_2\times {\mathbf{g}}_2)& 0& 0\\ 0& 0& 0& 0& {\mathbf{g}}_3^T{\mathbf{e}}_{13}& {\mathbf{e}}_{23}^T({\mathbf{f}}_3\times {\mathbf{g}}_3)\end{array}\right].$$

(49)

9. Singular Configurations

Singular configurations of the PM 3-PRRS arise when the determinants of one or both of the matrices ${\mathbf{J}}_{\mathbf{x}}$ and ${\mathbf{J}}_{\mathbf{q}}$ (48) and (49) are equal to zero. The singularity at det(${\mathbf{J}}_{\mathbf{q}}$) = 0 and det(${\mathbf{J}}_{\mathbf{x}}$) = 0 belong to singularities of the first and second types, respectively.

The first type of singularity arises in the following conditions:

$${\mathbf{g}}_i^T{\mathbf{e}}_{1,i}=0,$$

(50)

and

$${\mathbf{e}}_{2,i}^T({\mathbf{f}}_i\times {\mathbf{g}}_i)=0.$$

(51)

The first condition (50) takes place when, i.e., the vector ${\mathbf{g}}_i$, determining the shortest distance between the axis of the passive revolute and spherical kinematic pairs, is perpendicular to the axis of the active prismatic pair. This is possible in the cases:

$${\theta }_{23,i}={\theta }_{2,i}+{\theta }_{3,i}=0\mathrm{or}{\theta }_{23,i}=2\pi .$$

(52)

The corresponding locations of the leg links in the cases ${\theta }_{23,i}=0$ and ${\theta }_{23,i}=2\pi$ are shown in Figure 6 and Figure 7.

At the same time, the conditions (51) show, that the three vectors ${\mathbf{e}}_{2,i},{\mathbf{f}}_i$ and ${\mathbf{g}}_i$ must lie either in parallel planes or in the same plane (this plane is denoted by ${\Omega }_j$) as shown in (Figure 8).

The second type of singularity occurs when the matrix ${\mathbf{J}}_{\mathbf{x}}$ experiences a rank deficiency, i.e., when the rows or columns of this matrix are linearly dependent. This type of singularity, unlike the first one, is located inside the workspace. Let us rewrite the matrix ${\mathbf{J}}_{\mathbf{x}}$ in the following form:

$${\mathbf{J}}_{\mathbf{x}}=\left[\begin{array}{cccccc}{g}_{1x}& g_{1y}& g_{1z}& n_{1x}& n_{1y}& n_{1z}\\ g_{2x}& g_{2y}& g_{2z}& n_{2x}& n_{2y}& n_{2z}\\ g_{3x}& g_{3y}& g_{3z}& n_{3x}& n_{3y}& n_{3z}\end{array}\right],$$

(53)

where the elements of the first three columns are components of one group of vectors ${\mathbf{g}}_i$, and the elements of the last three columns are components of the second group of vectors ${\mathbf{n}}_i={\mathbf{h}}_i\times {\mathbf{g}}_i$:

$${\mathbf{g}}_i={[g_{ix},g_{iy},g_{iz}]}^T,{\mathbf{n}}_i={[n_{ix},n_{iy},n_{iz}]}^T,i=1,2,3.$$

(54)

Since vectors ${\mathbf{h}}_i$ cannot be zero, it is impossible for all elements of individual rows of the matrix ${\mathbf{J}}_{\mathbf{x}}$ to be equal to zero. Therefore, the second type of singularity occurs if there is a linear relationship between the rows of this matrix.

Let us assume that the linear dependence of any two of its rows with numbers j and k leads to the degeneracy of the matrix ${\mathbf{J}}_{\mathbf{x}}$. This means that there are simultaneously non-zero numbers ${\lambda }_1$ and ${\lambda }_2$ for which:

$${\lambda }_1[{\mathbf{g}}_j^T,{\mathbf{n}}_j^T]+{\lambda }_2[{\mathbf{g}}_k^T,{\mathbf{n}}_k^T]=0,j\ne k,$$

(55)

from here:

$${[{({\lambda }_1{\mathbf{g}}_j+{\lambda }_2{\mathbf{g}}_k)}^T,{({\lambda }_1{\mathbf{n}}_j+{\lambda }_2{\mathbf{n}}_k)}^T]}^T=0,j\ne k.$$

(56)

Then for the corresponding configuration the following conditions must be met:

$${\lambda }_1{\mathbf{g}}_j+{\lambda }_2{\mathbf{g}}_k=\mathbf{0}\to {\lambda }_1{\mathbf{g}}_j=-{\lambda }_2{\mathbf{g}}_k\to {\displaystyle \frac{{\mathbf{g}}_j}{{\mathbf{g}}_k}}=\left|{\displaystyle \frac{{\lambda }_2}{{\lambda }_1}}\right|=\mu ,$$

(57)

$${\lambda }_1{\mathbf{n}}_j+{\lambda }_2{\mathbf{n}}_k=\mathbf{0}\to {\lambda }_1{\mathbf{n}}_j=-{\lambda }_2{\mathbf{n}}_k\to {\displaystyle \frac{{\mathbf{n}}_j}{{\mathbf{n}}_k}}=\left|{\displaystyle \frac{{\lambda }_2}{{\lambda }_1}}\right|=\mu .$$

(58)

Analysis of conditions (57) and (58) shows that the equality of one of the vectors ${\mathbf{n}}_j$ and ${\mathbf{n}}_k$ to the zero vector excludes the linear dependence of the corresponding rows of the matrix ${\mathbf{J}}_{\mathbf{x}}$. Indeed, let us assume for definiteness that ${\mathbf{n}}_j=0$, ${\mathbf{n}}_k\ne 0$. Then from (58) it follows that ${\lambda }_2=0$. But in ${\lambda }_2=0$ and vector ${\mathbf{g}}_j$ cannot be zero, from (57) we obtain that, i.e., the numbers ${\lambda }_1$ and ${\lambda }_2$ are simultaneously equal to zero, therefore, the considering rows cannot be linearly dependent.

Next, assume that ${\mathbf{n}}_j\ne 0$, ${\mathbf{n}}_k\ne 0$. Then, according to conditions (57) and (58), there must be ${\mathbf{g}}_j\left|\right|{\mathbf{g}}_k$ and ${\mathbf{n}}_j\left|\right|{\mathbf{n}}_k$, and with the same proportionality coefficients $\mu$. Moreover, the specified requirements also correspond to the fulfillment of the following conditions:

$$\mathrm{if}\hspace{0.17em}{\mathbf{g}}_j\uparrow \downarrow {\mathbf{g}}_k,\hspace{0.17em}\mathrm{it} \mathrm{must} \mathrm{be}\hspace{0.17em}{\mathbf{n}}_j\uparrow \downarrow {\mathbf{n}}_k,$$

(59)

$$\mathrm{if}\hspace{0.17em}{\mathbf{g}}_j\uparrow \uparrow {\mathbf{g}}_k,\mathrm{it} \mathrm{must} \mathrm{be}\hspace{0.17em}{\mathbf{n}}_j\uparrow \uparrow {\mathbf{n}}_k.$$

(60)

Since ${\mathbf{n}}_j\parallel {\mathbf{n}}_k$, then ${\mathbf{h}}_{j}x{g}_j\parallel {\mathbf{h}}_{k}x{\mathbf{g}}_k$. Let’s assume that ${\mathbf{g}}_j\left|\right|{\mathbf{g}}_k$ and check the parallelism of the vectors ${\mathbf{n}}_j$ and ${\mathbf{n}}_k$ with the components:

$${\mathbf{h}}_j={[h_{jX},h_{jY},0]}^T,{\mathbf{h}}_k={[h_{kX},h_{kY},0]}^T,$$

(61)

$${\mathbf{g}}_j={[g_{jX}\hspace{0.17em}{g}_{jY}\hspace{0.17em}{g}_{jZ}]}^T,\hspace{0.17em}{\mathbf{g}}_k={[g_{kX}\hspace{0.17em}{g}_{kY}\hspace{0.17em}{g}_{kZ}]}^T.$$

(62)

and

$${\mathbf{n}}_j=\left({\mathbf{h}}_j\times {\mathbf{g}}_j\right)={[h_{jY}{g}_{jZ};-h_{jX}{g}_{jZ};h_{jX}{g}_{jY}-h_{jY}{g}_{jX}]}^T,$$

(63)

$${\mathbf{n}}_k=\left({\mathbf{h}}_k\times {\mathbf{g}}_k\right)={[h_{kY}{g}_{kZ};-h_{kX}{g}_{kZ};h_{kX}{g}_{kY}-h_{kY}{g}_{kX}]}^T.$$

(64)

Let’s also define the vector product of the vectors ${\mathbf{n}}_j$ and ${\mathbf{n}}_k$:

$$\left({\mathbf{n}}_j\times {\mathbf{n}}_k\right)=\left[\begin{array}{c}-h_{jX}{g}_{jZ}(h_{kX}{g}_{kY}-h_{kY}{g}_{kX})+h_{kX}{g}_{kZ}(h_{jX}{g}_{jY}-h_{jY}{g}_{jX})\\ -h_{jY}{g}_{jZ}(h_{kX}{g}_{kY}-h_{kY}{g}_{kX})+h_{kY}{g}_{kZ}(h_{jX}{g}_{jY}-h_{jY}{g}_{jX})\\ g_{jZ}{g}_{kZ}(h_{jX}{h}_{kY}-h_{jY}{h}_{kX})\end{array}\right].$$

(65)

Since (${\mathbf{n}}_j\left|\right|{\mathbf{n}}_k$)$\to {\mathbf{n}}_j\times {\mathbf{n}}_k=\mathbf{0}$, it follows from (65):

$$\left.\begin{array}{l}-h_{jX}{g}_{jZ}(h_{kX}{g}_{kY}-h_{kY}{g}_{kX})+h_{kX}{g}_{kZ}(h_{jX}{g}_{jY}-h_{jY}{g}_{jX})=0\\ -h_{jY}{g}_{jZ}(h_{kX}{g}_{kY}-h_{kY}{g}_{kX})+h_{kY}{g}_{kZ}(h_{jX}{g}_{jY}-h_{jY}{g}_{jX})=0\\ g_{jZ}{g}_{kZ}(h_{jX}{h}_{kY}-h_{jY}{h}_{kX})=0\end{array}\right\},$$

(66)

where

$$(h_{kX}{g}_{kY}-h_{kY}{g}_{kX})=\left|\begin{array}{cc}{h}_{kX}& h_{kY}\\ g_{kX}& g_{kY}\end{array}\right|\ne 0,\mathrm{since} {\mathbf{h}}_kи{\mathbf{g}}_k \mathrm{are} \mathrm{not} \mathrm{parallel},$$

(67)

$$(h_{jX}{g}_{jY}-h_{jY}{g}_{jX})=\left|\begin{array}{cc}{h}_{jX}& h_{jY}\\ g_{jX}& g_{jY}\end{array}\right|\ne 0,\mathrm{since} {\mathbf{h}}_jи{\mathbf{g}}_j\mathrm{are} \mathrm{not} \mathrm{parallel},$$

(68)

$$(h_{jX}{h}_{kY}-h_{jY}{h}_{kX})=\left|\begin{array}{cc}{h}_{jX}& h_{jY}\\ h_{kX}& h_{kY}\end{array}\right|\ne 0,\mathrm{since} {\mathbf{h}}_j \mathrm{and} {\mathbf{h}}_k\mathrm{are} \mathrm{not} \mathrm{parallel}.$$

(69)

Then, taking into account (67)–(69) from the system of Equation (66) we obtain:

$$g_{jZ}=0,g_{kZ}=0.$$

(70)

Substituting Equation (70) into Equations (63) and (64), yields:

$${\mathbf{n}}_j={[0;0;h_{jX}{g}_{jY}-h_{jY}{g}_{jX}]}^T,$$

(71)

$${\mathbf{n}}_k={[0;0;h_{kX}{g}_{kY}-h_{kY}{g}_{kX}]}^T,$$

(72)

that is, the vectors ${\mathbf{n}}_j$ and ${\mathbf{n}}_k$ must be parallel to the $P{Z}_P$ axis.

Thus, the obtained conditions (70)–(72) are necessary conditions for the linear dependence of the considered row vectors of the matrix ${\mathbf{J}}_{\mathbf{x}}$, and they are not enough for degeneracy, since it is also necessary to require the fulfillment of condition (58):

$${\displaystyle \frac{\left|{\mathbf{n}}_j\right|}{\left|{\mathbf{n}}_k\right|}}={\displaystyle \frac{n_j}{n_k}}={\displaystyle \frac{h_j\cdot g_j\cdot \sin ({\Omega }_j)}{h_k\cdot g_k\cdot \sin ({\Omega }_k)}}=\mu ,$$

(73)

where ${\Omega }_j=({h_j}^{\wedge }{g}_j),{\Omega }_k=({h_k}^{\wedge }{g}_k)$. From where, taking into account (56), we obtain the following condition:

$$h_j\cdot \sin ({\Omega }_j)=h_k\cdot \sin ({\Omega }_k).$$

(74)

that is, the projection of the segments $P{O}_{4,j}=h_j$ and $P{O}_{4,k}=h_k$ (magnitudes of the projection of the vectors ${\mathbf{h}}_j$ and ${\mathbf{h}}_k$) onto the direction perpendicular to the vectors ${\mathbf{g}}_j$ and ${\mathbf{g}}_k$ must be equal. In addition, they are parallel, and according to (70) lie in the same plane with the platform pole P. Therefor, first we determine those directions onto which the indicated segments are equally projected. It can be shown that such directions are directions perpendicular to the diagonals of a parallelogram constructed on the vectors ${\mathbf{h}}_j$ and ${\mathbf{h}}_k$. Accordingly, various options for the location of the vectors ${\mathbf{g}}_j$ and ${\mathbf{g}}_k$ are subject to consideration, determined by the directions of the diagonals themselves (Figure 9): along the straight line $O_{4j}{O}_{4k}$ and parallel to the straight line $P{Q}_k$. However, the analysis shows that conditions (59) and (60) are satisfied only by the variants of the location of the links along the straight line $O_{4j}{O}_{4k}$, one of which is shown in Figure 9a. Examples of the arrangement of links parallel to the second diagonal $P{Q}_k$, but not generating singular configurations, are shown in Figure 9b,c.

In our paper [57] with the coordinate of the center of the moving platform $Z_P$ fixed at a certain height, the rotation angles of the moving platform around the $U_0$ and $V_0$ axes of the absolute coordinate system were determined while operating the robot outside singular configurations within the framework of the four forward kinematics solutions presented in Figure 2. In [57], the Jacobian matrix ${\mathbf{J}}_{\mathbf{x}}$ was extended to a 6 × 6 dimension based on the constraint equations in order to study its determinant. A program was developed to identify the configurations at which the extended Jacobian matrix becomes singular.

10. Direct Velocity Analysis

The Jacobian matrices ${\mathbf{J}}_{\mathbf{x}}$ and ${\mathbf{J}}_{\mathbf{q}}$ in Equations (48) and (49) have dimensions of 3 × 6. In the analysis of singular configurations, inversion of these matrices is not required. However, to verify the theoretical results, analyze their determinants, and solve the direct and inverse velocity problems, it is necessary to extend these matrices.

The solution of the direct velocity problem consists of determining the linear and angular velocities of the moving platform for the given velocities of the driving links. In other words, from the matrix relationship (47), it is necessary to determine the vector $\dot{\mathbf{x}}$ when $\dot{\mathbf{q}}$ is known. To accomplish this, additional equations are required. Let us consider the following loop-closure vector equations

$${\mathbf{O}}_{4,i}=\mathbf{O}\mathbf{P}+{\mathbf{h}}_i.$$

(75)

By differentiating Equation (75) with respect to time, we obtain

$${\mathbf{V}}_{O_{4,i}}={\dot{\mathbf{r}}}_P+{\mathbf{\omega }}_P\times {\mathbf{h}}_i.$$

(76)

Due to the constraints of the revolute kinematic pairs, the velocity directions of the spherical joints are always perpendicular to the unit vectors of the rotation axes of these pairs, ${\mathbf{e}}_{2,i}$ and ${\mathbf{e}}_{3,i}$. By taking the scalar product of both sides of Equation (76) with the unit vector of the rotation axis ${\mathbf{e}}_{2,i}$, we obtain

$${\mathbf{e}}_{2,i}\cdot {\dot{\mathbf{r}}}_P+{\mathbf{e}}_{2,i}\mathbf{\cdot }({\omega }_P\times {\mathbf{h}}_i)=0.$$

(77)

Depending on the relative arrangement of the vectors, we obtain the following equations

$${\mathbf{e}}_{2,i}\cdot {\dot{\mathbf{r}}}_P+({\mathbf{h}}_i\times {\mathbf{e}}_{2,i})\cdot {\mathbf{\omega }}_P=\mathbf{0}.$$

(78)

Equation (78) can be expressed in the matrix form

$${\mathbf{J}}_{\mathbf{g}}\cdot \dot{\mathbf{x}}=\mathbf{0},$$

(79)

where,

$${\mathbf{J}}_{\mathbf{g}}=\left(\begin{array}{cc}{\mathbf{e}}_{2,1}^T& {({\mathbf{h}}_1\times {\mathbf{e}}_{2,1})}^T\\ {\mathbf{e}}_{2,2}^T& {({\mathbf{h}}_2\times {\mathbf{e}}_{2,2})}^T\\ {\mathbf{e}}_{2,3}^T& {({\mathbf{h}}_3\times {\mathbf{e}}_{2,3})}^T\end{array}\right).$$

(80)

Combining Equations (47) and (79), and augmenting the matrix ${\mathbf{J}}_{\mathbf{q}}$, we obtain

$${\mathbf{J}}_{\mathbf{a}}\cdot \dot{\mathbf{x}}={\mathbf{J}}_{\mathbf{e}}\cdot \dot{\mathbf{q}},$$

(81)

where

$${\mathbf{J}}_{\mathbf{a}}=\left[\begin{array}{cccccc}{g}_{1x}& g_{1y}& g_{1z}& n_{1x}& n_{1y}& n_{1z}\\ g_{2x}& g_{2y}& g_{2z}& n_{2x}& n_{2y}& n_{2z}\\ g_{3x}& g_{3y}& g_{3z}& n_{3x}& n_{3y}& n_{3z}\\ -\mathrm{c}{\gamma }_1& -\mathrm{s}{\gamma }_1& 0& h_{1z}\cdot \mathrm{s}{\gamma }_1& -h_{1z}\cdot \mathrm{c}{\gamma }_1& h_{1y}\cdot \mathrm{c}{\gamma }_1-h_{1x}\cdot \mathrm{s}{\gamma }_1\\ -\mathrm{c}{\gamma }_2& -\mathrm{s}{\gamma }_2& 0& h_{2z}\cdot \mathrm{s}{\gamma }_2& -h_{2z}\cdot \mathrm{c}{\gamma }_2& h_{2y}\cdot \mathrm{c}{\gamma }_2-h_{2x}\cdot \mathrm{s}{\gamma }_2\\ -\mathrm{c}{\gamma }_3& -\mathrm{s}{\gamma }_3& 0& h_{3z}\cdot \mathrm{s}{\gamma }_3& -h_{3z}\cdot \mathrm{c}{\gamma }_3& h_{3y}\cdot \mathrm{c}{\gamma }_3-h_{3x}\cdot \mathrm{s}{\gamma }_3\end{array}\right],$$

(82)

$${\mathbf{J}}_{\mathbf{e}}=\left[\begin{array}{cccccc}-g\cdot \mathrm{s}{\theta }_{23,1}& f\cdot g\cdot \mathrm{s}{\theta }_{3,1}& 0& 0& 0& 0\\ 0& 0& -g\cdot \mathrm{s}{\theta }_{23,2}& f\cdot g\cdot \mathrm{s}{\theta }_{3,2}& 0& 0\\ 0& 0& 0& 0& -g\cdot \mathrm{s}{\theta }_{23,3}& f\cdot g\cdot \mathrm{s}{\theta }_{3,3}\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\end{array}\right].$$

(83)

Now the direct velocity problem is solved as follows

$$\dot{\mathbf{x}}={\mathbf{J}}_{\mathbf{a}}^{-1}\cdot {\mathbf{J}}_{\mathbf{e}}\cdot \dot{\mathbf{q}}.$$

(84)

11. Inverse Velocity Analysis

The objective of solving the inverse velocity problem is to determine the required instantaneous velocities of the six actuators that ensure the prescribed instantaneous velocities of the moving platform.

Since the moving platform possesses only three degrees of freedom, it is necessary to derive the equations that relate its dependent and independent parameters.

Let us rewrite the system of Equation (18) in the following form

$$\left.\begin{array}{l}(X_P+h\cdot t_{11})\mathrm{c}{\gamma }_1+(Y_P+h\cdot t_{21})\mathrm{s}{\gamma }_1+b=0\\ \left(X_P+{\displaystyle \frac{h}{2}}(\sqrt{3}{t}_{12}-t_{11})\right)\mathrm{c}{\gamma }_2+\left(Y_P+{\displaystyle \frac{h}{2}}(\sqrt{3}{t}_{22}-t_{21})\right)\mathrm{s}{\gamma }_2+b=0\\ \left(X_P-{\displaystyle \frac{h}{2}}(\sqrt{3}{t}_{12}+t_{11})\right)\mathrm{c}{\gamma }_3+\left(Y_P-{\displaystyle \frac{h}{2}}(\sqrt{3}{t}_{22}+t_{21})\right)\mathrm{s}{\gamma }_3+b=0\end{array}\right\},$$

(85)

where

$$\begin{array}{l}{t}_{11}=\mathrm{c}\theta \mathrm{c}\phi +\mathrm{s}\psi \mathrm{s}\theta \mathrm{s}\phi ,t_{12}=-\mathrm{c}\theta \mathrm{s}\phi +\mathrm{s}\psi \mathrm{s}\theta \mathrm{c}\phi ,t_{13}=\mathrm{c}\psi \mathrm{s}\theta ,\\ t_{21}=\mathrm{c}\psi \mathrm{s}\phi ,t_{22}=\mathrm{c}\psi \mathrm{c}\phi ,t_{23}=-\mathrm{s}\psi ,\\ t_{31}=\mathrm{s}\theta \mathrm{s}\phi +\mathrm{s}\psi \mathrm{c}\theta \mathrm{c}\phi ,t_{31}=-\mathrm{s}\theta \mathrm{c}\phi +\mathrm{s}\psi \mathrm{c}\theta \mathrm{s}\phi ,t_{33}=\mathrm{c}\psi \mathrm{c}\theta ,\end{array}$$

(86)

Here $\psi ,\theta ,\phi$ are the Euler angles that describe the orientation of the local coordinate system $P{X}_P{Y}_P{Z}_P$, and they are defined as follows

$${\mathbf{R}}_{OP}=R_y(\theta )R_x(\psi )R_z(\phi )=\left[\begin{array}{ccc}{t}_{11}& t_{12}& t_{13}\\ t_{21}& t_{22}& t_{23}\\ t_{31}& t_{32}& t_{33}\end{array}\right].$$

(87)

By subtracting the third equation from the second in the system of Equation (85), after dividing them by $\mathrm{s}{\gamma }_2$ and $\mathrm{s}{\gamma }_3$ respectively, we can determine $X_P$ as follows

$$X_P={\displaystyle \frac{h}{2}}\left[t_{11}+(3-4{\mathrm{c}}^2\xi )t_{22}\right]+hs(2\xi )t_{12}-2b\mathrm{s}\xi ,$$

(88)

where $\xi ={\mathrm{s}}^{-1}\left(b/h\right)$ is the angle between the axis U₀ of the absolute coordinate system and the vector of the first prismatic actuator ${\mathit{s}}_1$.

Similarly, by subtracting the third equation from the second in the system of Equation (85) and dividing them by $\mathrm{c}{\gamma }_2$ and $\mathrm{c}{\gamma }_3$ respectively, $Y_P$ can be determined as follows

$$Y_P={\displaystyle \frac{h}{2}}\left[t_{21}+(1-4{\mathrm{c}}^2\xi )t_{12}\right]-h\mathrm{s}(2\xi )t_{22}-2b\mathrm{c}\xi .$$

(89)

From the first Equation of the system (85), taking into account expressions (88) and (89), the following equation is obtained

$$(t_{11}+t_{22})\mathrm{s}\xi +(t_{12}-t_{21})\mathrm{c}\xi ={\displaystyle \frac{2b}{h}}.$$

(90)

From Equation (85), taking into account Equation (86), the following relationship is obtained

$$A\mathrm{c}\phi +B\mathrm{s}\phi -{\displaystyle \frac{2b}{h}}=0,$$

(91)

where $A=(\mathrm{c}\psi +\mathrm{c}\theta )\mathrm{s}\xi +\mathrm{s}\psi \mathrm{s}\theta \mathrm{c}\xi ,\hspace{0.17em}\hspace{0.17em}B=\mathrm{s}\psi \mathrm{s}\theta \mathrm{s}\xi -(\mathrm{c}\psi +\mathrm{c}\theta )\mathrm{c}\xi$. From Equation (91), we obtain two solutions corresponding to the two possible assembly configurations of each leg

$${\phi }_{1,2}=\pm c^{-1}\left({\displaystyle \frac{C}{r}}\right)+\alpha ,$$

(92)

where $\alpha ={\tan }^{-1}\left({\displaystyle \frac{B}{A}}\right),r=\sqrt{A^2+B^2}$.

The solution corresponding to the right assembly configuration of the RRS dyad is used. Thus, by specifying the independent variables $Z_P,\psi$ and $\theta$, the dependent variables $X_P,Y_P$, and f are determined from Equations (88), (89) and (92).

To solve the inverse velocity problem, let us introduce the following notation $\dot{\mathbf{X}}=[{\dot{Z}}_P,\dot{\psi },\dot{\theta }]$. Then, taking into account Equations (88), (89) and (92), we obtain the following expression.

$$\dot{\mathbf{x}}={\mathbf{J}}_{\mathbf{r}}\cdot \dot{\mathbf{X}},$$

(93)

where

$${\mathbf{J}}_{\mathbf{r}}=\left[\begin{array}{ccc}{\displaystyle \frac{\mathsf{\partial }{X}_P}{\mathsf{\partial }{Z}_P}}& {\displaystyle \frac{\mathsf{\partial }{X}_P}{\mathsf{\partial }\psi }}+{\displaystyle \frac{\mathsf{\partial }{X}_P}{\mathsf{\partial }\phi }}\cdot {\displaystyle \frac{\mathsf{\partial }\phi }{\mathsf{\partial }\psi }}& {\displaystyle \frac{\mathsf{\partial }{X}_P}{\mathsf{\partial }\theta }}+{\displaystyle \frac{\mathsf{\partial }{X}_P}{\mathsf{\partial }\phi }}\cdot {\displaystyle \frac{\mathsf{\partial }\phi }{\mathsf{\partial }\theta }}\\ {\displaystyle \frac{\mathsf{\partial }{Y}_P}{\mathsf{\partial }{Z}_P}}& {\displaystyle \frac{\mathsf{\partial }{Y}_P}{\mathsf{\partial }\psi }}+{\displaystyle \frac{\mathsf{\partial }{Y}_P}{\mathsf{\partial }\phi }}\cdot {\displaystyle \frac{\mathsf{\partial }\phi }{\mathsf{\partial }\psi }}& {\displaystyle \frac{\mathsf{\partial }{Y}_P}{\mathsf{\partial }\theta }}+{\displaystyle \frac{\mathsf{\partial }{Y}_P}{\mathsf{\partial }\phi }}\cdot {\displaystyle \frac{\mathsf{\partial }\phi }{\mathsf{\partial }\theta }}\\ 1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\\ {\displaystyle \frac{\mathsf{\partial }\phi }{\mathsf{\partial }{Z}_P}}& {\displaystyle \frac{\mathsf{\partial }\phi }{\mathsf{\partial }\psi }}& {\displaystyle \frac{\mathsf{\partial }\phi }{\mathsf{\partial }\theta }}\end{array}\right].$$

(94)

From the matrix relation (47), taking into account expression (94), we obtain

$$\dot{\mathbf{q}}={\mathbf{J}}_{\mathbf{q}}^{-1}\cdot {\mathbf{J}}_{\mathbf{x}}\cdot {\mathbf{J}}_{\mathbf{r}}\cdot \dot{\mathbf{X}}.$$

(95)

In Equation (47), the matrix ${\mathbf{J}}_{\mathbf{q}}$ has a dimension of 3 × 6. To solve the inverse velocity problem, during the motion of the three rotational actuators, the three prismatic actuators are fixed, and vice versa—during the motion of the three prismatic actuators, the three rotational actuators are fixed. In this case, by removing the zero columns, the dimension of the ${\mathbf{J}}_{\mathbf{q}}$ matrix becomes 3 × 3.

When solving the inverse velocity problem, the independent parameters of the instantaneous velocity of the moving platform $V_{P_{W_0}},{\omega }_{P_{U_0}},{\omega }_{P_{V_0}}$ are prescribed, and the corresponding instantaneous velocities of the generalized coordinates are determined. This is achieved by projecting the elementary angular velocity vectors $\dot{\psi }$ and $\dot{\theta }$ onto the absolute coordinate system $O_0{U}_0{V}_0{W}_0$

$$\left.\begin{array}{l}{\omega }_{P_{U_0}}=\dot{\psi }\cdot \mathrm{c}\theta +\dot{\phi }\mathrm{s}\theta \mathrm{c}\psi \\ {\omega }_{P_{V_0}}=\dot{\theta }-\dot{\phi }s\psi \end{array}\right\},$$

(96)

where $\dot{\phi }={\displaystyle \frac{\mathsf{\partial }\phi }{\mathsf{\partial }\psi }}\dot{\psi }+{\displaystyle \frac{\mathsf{\partial }\phi }{\mathsf{\partial }\theta }}\dot{\theta }$.

The system of Equation (93) is solved with respect to the variables $\psi$ and $\theta$, and the following result is obtained

$$\dot{\theta }={\displaystyle \frac{k_{21}\cdot {\omega }_{P_{U_0}}-k_{11}\cdot {\omega }_{P_{V_0}}}{k_{12}\cdot k_{21}-k_{22}\cdot k_{11}}},\hspace{0.17em}\dot{\psi }={\displaystyle \frac{k_{22}\cdot {\omega }_{P_{U_0}}-k_{12}\cdot {\omega }_{P_{V_0}}}{k_{11}\cdot k_{22}-k_{21}\cdot k_{12}}},$$

(97)

where $k_{11}=\mathrm{c}\theta +\mathrm{s}\theta \mathrm{c}\psi {\displaystyle \frac{\mathsf{\partial }\phi }{\mathsf{\partial }\psi }},k_{12}=\mathrm{s}\theta \mathrm{c}\psi {\displaystyle \frac{\mathsf{\partial }\phi }{\mathsf{\partial }\theta }},k_{21}=-\mathrm{s}\psi {\displaystyle \frac{\mathsf{\partial }\phi }{\mathsf{\partial }\psi }},k_{22}=1-\mathrm{s}\psi {\displaystyle \frac{\mathsf{\partial }\phi }{\mathsf{\partial }\theta }}$.

The component of the instantaneous velocity vector with respect to the $W$ axis is determined as follows

$${\omega }_{W_0}=-\dot{\psi }\mathrm{s}\theta +\dot{\phi }\mathrm{c}\theta \mathrm{c}\psi .$$

(98)

Now, instead of the matrix ${\mathbf{J}}_{\mathbf{x}}$, the matrix ${\mathbf{J}}_{\mathbf{a}}$ can be used. The augmented part of the matrix J_a does not affect the analysis of singular configurations. Since the matrix ${\mathbf{J}}_{\mathbf{a}}$ has a determinant, this property is employed for numerical analysis and for solving the direct velocity problem.

Numerical calculations show that for the following values of the constant parameters: $b=0,h=50,k=asin(b/h),\hspace{0.17em}\hspace{0.17em}a=15,c=7,L_1=50,L_2=70,{\gamma }_1=\pi /2+k,\hspace{0.17em}{\gamma }_2=7\pi /6+k,$ ${\gamma }_3=k-\pi /6$ with $s_1=s_2=s_3$ varying from 40 to 70 in steps of 5, $Z_P$ varying from 75 to 130 in steps of 5, and $\psi$ and $\theta$ varying in two cycles from $-\pi /18$ to $+\pi /18$ in steps of $\pi /200$, no type II singular configurations were found at which the extended Jacobian matrix ${\mathbf{J}}_{\mathbf{x}}$ becomes singular.

In addition, the calculations demonstrate the influence of the passive prismatic kinematic pairs on the expansion of the workspace beyond singular configurations. For example, at $Z_P=75$ and $s_1=s_2=s_3=40$, when checking the extended Jacobian matrix ${\mathbf{J}}_{\mathbf{x}}$ for singularity and varying $\psi$ and $\theta$ in two cycles from $-\pi /15$ to $+\pi /15$ in steps of $\pi /200$, the program identifies type II singular configurations at $\psi =-9.3^0,\hspace{0.17em}\theta =-12^0$. The corresponding configuration is shown in Figure 10a. However, this singular configuration disappears when $s_1=s_2=s_3=60$, with the corresponding configuration illustrated in Figure 10b.

At the upper boundaries of the workspace without singular configurations, with $Z_P=130$ and $s_1=s_2=s_3=40$, and with $\psi$ and $\theta$ varying over two cycles from $-\pi /13$ to $+\pi /13$ in steps of $\pi /200$, the PM operates without singular configurations. However, for the same position with $s_1=s_2=s_3=60$, the inverse kinematics solution of the tripod does not exist. This confirms that the passive prismatic actuators influence the expansion of the workspace free of singular configurations.

12. Numerical Examples of Solving the Direct Velocity Problem

The following constant parameters are specified: $a_i=15,b_i=8,$ $c_i=7,$ $f_i=50,$ $g_i=70,$ ${\gamma }_1=\pi /2+\xi ,$${\gamma }_2=7\pi /6+\xi ,$${\gamma }_3=\xi -\pi /6,$ $a=h\cdot \sqrt{3},\hspace{0.17em}\hspace{0.17em}h=50,$ $\xi =s^{-1}(b_i/h)$.

In solving the direct velocity problem, the independent parameters $\psi ,\hspace{0.17em}\theta$ and $Z_P$ are specified, and the dependent parameters $\phi ,\hspace{0.17em}{X}_P$ and $Y_P$ are determined (

Table A7. Independent and dependent parameters determining the position and orientation of the moving platform. 

	$\psi$	$\theta$	$Z_P$	$\phi$	$X_P$	$Y_P$
	$s_1=s_2=s_3=40$
1	0	0	90	0	0	0
2	0	0	90	0	0	0
3	0	0	90	0	0	0
4	0	0	90	0	0	0
	$s_1=s_2=s_3=45$
5	−0.1396	0.1221	90	−0.0099	−0.0757	0.4210
6	0.1570	−0.1396	100	−0.0128	−0.1038	0.5381
7	0.1745	0.1396	100	0.0102	0.3257	−0.5268
8	−0.1745	−0.1396	120	0.0102	0.3257	−0.5268
	$s_1=s_2=s_3=50$
9	−0.1221	−0.2268	120	0.0112	−0.2057	−0.7961
10	−0.1221	−0.2268	120	0.0112	−0.2057	−0.7961
11	0.2094	0.1919	80	0.0169	0.4112	−0.9066
12	−0.2094	−0.1919	80	0.0169	0.4112	−0.9066

). The input parameters of the prismatic kinematic pairs $s_1,\hspace{0.17em}{s}_2$ and $s_3$ are arbitrarily defined, and the corresponding positions of the parallel manipulator are first determined by solving the inverse kinematics problem (

Table A8. Positions of the input and output revolute kinematic pairs. 

	${\theta }_{2,1}$	${\theta }_{2,2}$	${\theta }_{2,3}$	${\theta }_{3,1}$	${\theta }_{3,2}$	${\theta }_{3,3}$
	$s_1=s_2=s_3=40$
1	−1.3640	−1.3640	−1.3640	1.9649	1.9649	1.9649
2	−1.3640	−1.3640	−1.3640	1.9649	1.9649	1.9649
3	−1.3640	−1.3640	−1.3640	1.9649	1.9649	1.9649
4	−1.3640	−1.3640	−1.3640	1.9649	1.9649	1.9649
	$s_1=s_2=s_3=45$
5	−1.3970	−1.3555	−1.1342	2.1016	2.0385	1.7799
6	−1.0124	−1.0819	−1.2823	1.5971	1.6793	1.9818
7	−1.2446	−0.9300	−1.2016	1.9090	1.4968	1.8478
8	−0.6479	−0.9617	−0.7090	1.0280	1.5443	1.1217
	$s_1=s_2=s_3=50$
9	−0.4914	−0.9033	−0.7528	0.8679	1.5418	1.2604
10	−0.4914	−0.9033	−0.7528	0.8679	1.5418	1.2604
11	−1.5544	−1.1278	−1.4758	2.3732	1.8983	2.2719
12	−1.2308	−1.5565	−1.3268	1.9917	2.4566	2.0976

). The direct velocity problem is then solved using Equation (84).

Table A9. Values of the instantaneous actuator velocities. 

	${\dot{s}}_1$	${\dot{s}}_2$	${\dot{s}}_3$	${\dot{\theta }}_{2,1}$	${\dot{\theta }}_{2,2}$	${\dot{\theta }}_{2,3}$
	$s_1=s_2=s_3=40$
1	0	0	0	30	30	30
2	−30	−30	−30	0	0	0
3	0	0	0	−50	50	0
4	500	0	0	−500	0	0
	$s_1=s_2=s_3=45$
5	−5	−15	−5	−10	20	−12
6	−10	−10	−10	−10	10	10
7	0	0	0	0	80	0
8	80	80	0	0	0	0
	$s_1=s_2=s_3=50$
9	−40	−40	−40	0	0	0
10	40	40	40	0	0	0
11	0	0	0	20	10	30
12	0	0	0	20	10	30

presents the specified velocities of the six input links $\dot{\mathbf{q}}={\left[{\dot{s}}_1,{\dot{\theta }}_{2,1},{\dot{s}}_2,{\dot{\theta }}_{2,2},{\dot{s}}_3,{\dot{\theta }}_{2,3}\right]}^T$.

Table A10. Linear and angular instantaneous velocities of the moving platform center. 

	$V_{P_{U_0}}$	$V_{P_{V_0}}$	$V_{P_{W_0}}$	${\omega }_{P_{U_0}}$	${\omega }_{P_{V_0}}$	${\omega }_{P_{W_0}}$
	$s_1=s_2=s_3=40$
1	0	0	1119.4245	0	0	0
2	0	0	20.5656	0	0	0
3	0	0	0	32.315	55.9712	0
4	0	0	0	3.9578	6.8552	0
	$s_1=s_2=s_3=45$
5	92.6927	−69.1369	113.9910	2.7297	−24.8426	1.8560
6	−67.026	−55.2530	246.0229	−15.5704	−5.6285	−1.3893
7	202.2327	−238.1622	1376.9457	48.9479	28.1426	−1.9494
8	−4.4166	1.5809	37.8293	0.8433	0.0018	0.0705
	$s_1=s_2=s_3=50$
9	0.0475	1.1907	23.3358	0.1215	0.1423	0.0088
10	−0.0475	−1.1907	−23.3358	−0.1215	−0.1423	−0.0088
11	−87.0485	−3.1741	1134.9011	−11.0659	5.2406	1.7211
12	123.0215	−104.3032	1003.4725	−21.9508	−6.7957	−1.8615

shows the results of solving the direct velocity problem, namely, the components of the linear and angular velocity vector of the vector moving platform center $\mathbf{V}={\left[V_{P_{U_0}},V_{P_{V_0}},V_{P_{W_0}},{\omega }_{P_{U_0}},{\omega }_{P_{V_0}},{\omega }_{P_{W_0}}\right]}^T.$

13. Numerical Examples of Solving the Inverse Velocity Problem

The following constant parameters are specified: $a_i=15,b_i=8,$$c_i=7,$$f_i=50,$$g_i=70,$${\gamma }_1=\pi /2+\xi ,$${\gamma }_2=7\pi /6+\xi ,$${\gamma }_3=\xi -\pi /6,$$a=h\cdot \sqrt{3},\hspace{0.17em}\hspace{0.17em}h=50,$ $\xi =s^{-1}(b_i/h)$.

Table A11. Specified positions and orientations of the moving platform. 

	$\psi$	$\theta$	$Z_P$	$\phi$	$X_P$	$Y_P$
	$s_1=s_2=s_3=40$
1	−0.1047	0.1570	100	−0.0082	−0.1674	0.4101
2	0.1047	−0.1745	105	−0.0091	−0.2386	0.4559
3	−0.2268	−0.2094	100	0.0239	0.1224	−1.1666
4	−0.1047	0.1745		−0.0091	−0.2386	0.4559
	$s_1=s_2=s_3=50$
5	−0.2617	0.2617	110	−0.0346	0.0580	1.6736
6	0.1396	0.2443	110	0.0171	−0.4847	−0.8501
7	−0.1396	0.0872	110	−0.0061	0.1500	0.3023
8	−0.1745	0.1745	110	−0.0153	0.0115	0.7537
	$s_1=s_2=s_3=60$
9	0.1221	0.1570	85	0.0096	−0.1168	−0.4777
10	0.1221	0.1570	85	0.0096	−0.1168	−0.4777
11	−0.1221	−0.1570	90	0.0096	−0.1168	−0.4777
12	−0.1396	0.1221	95	−0.0085	0.0605	0.4235

presents the positions and orientations of the moving platform of the parallel manipulator.

Table A12. Components of the instantaneous velocities of the moving platform determined based on the direct and inverse velocity problems. 

	$V_{P_{W_0}}$		${\omega }_{P_{U_0}}$		${\omega }_{P_{V_0}}$
	$s_1=s_2=s_3=40$
1	5	4.8630	20	19.9872	20	20.0209
2	−50	−49.9721	−3	−3.0037	−2	−1.9974
3	46	46.0010	−0.5	−0.5007	−0.7	−0.6988
4	100	99.9999	0	0.0000	0	0.0000
	$s_1=s_2=s_3=50$
5	72	71.6454	2	2.0024	3	3.0384
6	4	3.9408	12	11.9867	15	15.0137
7	0	−0.0030	2	2.0005	2	2.0017
8	−5	−4.8723	−5	−4.9987	−5	−5.0173
	$s_1=s_2=s_3=60$
9	10	9.9689	10	9.9915	10	10.0099
10	10	9.9689	10	9.9915	10	10.0099
11	20	20.0024	0.5	0.5005	0.5	0.4998
12	−30	−30.0340	2	2.0006	3	3.0066

specifies the components of the instantaneous velocities of the moving platform with respect to absolute coordinate system $O_0{U}_0{V}_0{W}_0$, determined from the solution of the direct kinematics problem.

Table A13. Input parameter velocities obtained from the solution of the inverse velocity problem. 

	${\dot{s}}_1$	${\dot{s}}_2$	${\dot{s}}_3$	${\dot{\theta }}_{2,1}$	${\dot{\theta }}_{2,2}$	${\dot{\theta }}_{2,3}$
	$s_1=s_2=s_3=40$
1	0	0	0	−14.0396	23.2052	−7.8316
2	−132.4230	411.5116	−16.5129	0	0	0
3	0	0	0	1.4787	0.3922	0.8112
4	−145.6813	−165.6036	−180.0590	0	0	0
	$s_1=s_2=s_3=50$
5	0	0	0	−0.2430	3.3399	1.0959
6	587.7873	−1876.6572	338.8137	0	0	0
7	0	0	0	−1.3726	1.9778	−0.7436
8	−216.8399	436.5792	−198.8104	0	0	0
	$s_1=s_2=s_3=60$
9	0	0	0	−4.9403	12.1003	−3.4742
10	237.1875	−758.1083	188.7878	0	0	0
11	0	0	0	−0.1264	0.5508	0.1994
12	109.9661	−90.5956	48.5400	0	0	0

shows the results of solving the inverse velocity problem, namely, the instantaneous velocity values required to achieve the specified motion of the moving platform.

Table A12 presents two sets of instantaneous velocity components of the moving platform, corresponding to the solutions of the direct and inverse velocity problems, respectively.

Outside the singular configurations, the relative error between the results of the direct and inverse velocity problems does not exceed 5%.

14. Conclusions

This paper proposes a novel 3-PRRS type PM architecture, which features three active revolute joints and three passive prismatic input joints. The design is protected by an invention patent granted in Kazakhstan. The three passive prismatic input joints (linear actuators) are located on a fixed platform and drive three active revolute input joints. The direct and inverse kinematics problems of the PM are solved, and the workspace as well as the singular configurations are calculated. When solving the direct kinematics problem, a 16th-degree polynomial was derived, enabling the determination of all possible configurations of the PM. For given input parameter values, a maximum of 16 positions of the moving platform can be obtained for each assembly mode of the mechanism. The inverse kinematics problem is solved in closed form. Numerical results for both the direct and inverse kinematics problems are presented. All results from the direct kinematics problem were verified against those from the inverse kinematics problem. The workspace of the PM is defined based on the solution of the inverse kinematics problem. All valid poses of the moving platform, for which a solution to the inverse kinematics problem exists, are included within the PMs workspace. It is demonstrated that the legs of the PM move along circular arcs relative to the absolute coordinate system, while the center of the moving platform moves along an arbitrary curve lying on a sphere. Numerical results for the workspace definition are presented, and the influence of the linear actuator stroke lengths on the vertical expansion of the workspace is shown. The results obtained indicate that the proposed PM can provide an expanded workspace. The Jacobian matrices have been derived, and the singular configurations of the first type and the second type have been identified. To verify the theoretical results of the singularity analysis, the Jacobian matrices were extended based on the constraint equations. The extended Jacobian matrices were obtained for both the direct and inverse velocity problems. The obtained numerical results confirm the correctness of the theoretical conclusions, and the introduction of passive prismatic input actuators contributes to an enlargement of the workspace without singular configurations.