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

Zhumadil Baigunchekov, Giuseppe Carbone, Med Amine Laribi, Rustem Kaiyrov, Li Qian, Zhadyra Zhumasheva

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Zhumadil Baigunchekov, Giuseppe Carbone, Med Amine Laribi, Rustem Kaiyrov, Li Qian, et al. Ge- ometry, Kinematics, Workspace, and Singularities of a Novel 3-PRRS Parallel Manipulator. Robotics, 2025, 15 (1), pp.10. 10.3390/robotics15010010 . hal-05491605

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Article

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

Zhumadil Baigunchekov ¹ID, Giuseppe Carbone ²ID, Med Amine Laribi ³ID, Rustem Kaiyrov <fnref n="4" />,* ID, Li Qian ⁵ID
and Zhadyra Zhumasheva ¹ID

## 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.

**Keywords:** parallel manipulators; tripod; geometry; matrices of links and kinematic pairs; direct and inverse kinematics; pose of moving platform; workspace; singularity

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Academic Editor: Xinjun Liu

Received: 6 November 2025

Revised: 22 December 2025

Accepted: 24 December 2025

Published: 29 December 2025

**Copyright:** © 2025 by the authors.
Licensee MDPI, Basel, Switzerland.
This article is an open access article
distributed under the terms and
conditions of the Creative Commons
Attribution (CC BY) license.

## 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

<footnote n="1">¹ Faculty of Mechanics and Mathematics, Department of Mechanics, Al-Farabi Kazakhstan National University, 71/23 Al-Farabi Avenue, Almaty 050040, Kazakhstan; bzh47@mail.ru (Z.B.)</footnote>

<footnote n="2">² School of Engineering, Department of Mechanical, Energy and Management Engineering (DIMEG), University of Calabria, Via Bucci Cubo 45C, 87036 Rende, Italy</footnote>

<footnote n="3">³ Department of GMSC, Prime Institute CNRS, ENSMA, University of Poitiers, UPR 3346, Poitiers 86000, France</footnote>

<footnote n="4">⁴ Laboratory of Applied Mechanics and Robotics, Karaganda Buketov University, Karaganda 100001, Kazakhstan</footnote>

<footnote n="5">⁵ Institute of Automation, Shandong Academy of Sciences, Jinan 250000, China</footnote>

<footnote>* Correspondence: kairov.rustem@mail.ru</footnote>
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 [9–11]. 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 [47–53] are devoted to the investigation of the kinematics, workspace, and singularities of PMs.

In this paper, we propose a 3-RRS 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.

<figure><img src="image_3.png" /><figcaption>**Figure 1.** 3D CAD model of the 3-PRRS-type PM (a) and the kinematic diagram of one leg (b).</figcaption></figure>

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
$$G_{12} = G_{12}(a_{12}, -90, 0, 0, 0, 0) = \begin{bmatrix} 1 & 0 & 0 & 0 \\ a_{12} & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & -1 & 0 \end{bmatrix}, \qquad (3)$$

$$G_{23} = G_{23}(a_{23}, 0, 0, 0, 0, 0) = \begin{bmatrix} 1 & 0 & 0 & 0 \\ a_{23} & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}, \qquad (4)$$

$$G_{34} = G_{34}(a_{34}, 0, 0, 0, 0, 0) = \begin{bmatrix} 1 & 0 & 0 & 0 \\ a_{34} & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}, \qquad (5)$$

$$\mathbf{P}_{1}^{P} = \mathbf{P}_{1}^{P}(0, 0, 0, 0, c_{11}, 0) = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ s_{1} & 0 & 0 & 1 \end{bmatrix}, \qquad (6)$$

$$\mathbf{P}_{2}^{R} = \mathbf{P}_{2}^{R}(0, 0, 0, 0, b_{12}, \gamma_{22}) = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & c\theta_{2} & -s\theta_{2} & 0 \\ 0 & s\theta_{2} & c\theta_{2} & 0 \\ b_{12} & 0 & 0 & 1 \end{bmatrix}, \qquad (7)$$

$$\mathbf{P}_{3}^{R} = \mathbf{P}_{3}^{R}(0, 0, 0, \beta_{33}, 0, 0) = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & c\theta_{3} & -s\theta_{3} & 0 \\ 0 & s\theta_{3} & c\theta_{3} & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}, \qquad (8)$$

where $\alpha_{01} = 90^{\circ}, c_{01} = OO_{1}, \gamma_{01}, a_{12} = O'_{1}O_{2}, \alpha_{12} = -90^{\circ}, c_{22} = O_{2}O'_{2}, a_{23} = O'_{2}O_{3}$, and $a_{34} = O'_{3}O_{4}$ are constant parameters; $c_{11} = O_{1}O'_{1} = s_{1}, \gamma_{22} = \theta_{2}$, 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 $PX_{P}Y_{P}Z_{P}$, 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 $OU_{0}V_{0}W_{0}$ are written through the legs $OABCD$:

$$[U_{O_{4,i}}, V_{O_{4,i}}, W_{O_{4,i}}]^{T} = G_{01}P_{1i}^{P}G_{12}P_{2i}^{R}G_{23}P_{3i}^{R}G_{34}[1, 0, 0, 0]^{T}, \qquad (9)$$

where the coordinates of the joints $O_{4,i}$ are known and are expressed through the coordinate system $PX_{P}Y_{P}Z_{P}$ by the following equations:

$$[U_{O_{4,i}}, V_{O_{4,i}}, W_{O_{4,i}}]^{T} = T_{OP}G_{P4}[1, 0, 0, 0]^{T}. \qquad (10)$$
Expanding Equation (9) for the given values of the coordinates of the centers of the spherical joints, we obtain:

$$ 
\left. 
\begin{aligned}
U_{O_{4,i}} &= -b_i c \gamma_i + s_i s \gamma_i - f_i s \gamma_i s \theta_{2,i} - g_i s \gamma_i s \theta_{23,i} \\
V_{O_{4,i}} &= -b_i s \gamma_i - s_i c \gamma_i + f_i c \gamma_i s \theta_{2,i} + g_i c \gamma_i s \theta_{23,i} \\
W_{O_{4,i}} &= c_i + a_i + f_i c \theta_{2,i} + g_i c \theta_{23,i}
\end{aligned}
\right\} , \qquad (11)
$$

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:

$$ A_i s^2 \theta_{2,i} - B_i s \theta_{2,i} + C_i = 0, \qquad (12) $$

where $A_i$, $B_i$, and $C_i$ are known parameters

$$ 
\begin{aligned}
A_i &= (s_i - K_i)^2 + M_i^2, \quad B_i = 2(f + gc\theta_{3,i})(s_i - K_i), \\
C_i &= (f + gc\theta_{3,i})^2 - M_i^2, \quad K_i = \frac{U_{O_{4,i}} + bc\gamma_i}{s\gamma_i}, \quad M_i = W_{O_{4,i}} - c - a.
\end{aligned}
\qquad (13)
$$

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:

$$ 
\begin{aligned}
\theta_{2,i} &= s^{-1} \frac{B_i \pm \sqrt{B_i^2 - 4A_i C_i}}{2A_i}, \quad \theta_{2,i} = \pi - s^{-1} \frac{B_i \pm \sqrt{B_i^2 - 4A_i C_i}}{2A_i}, \\
\theta_{3,i} &= \pm c^{-1} \frac{(s_i - K_i)^2 + M_i^2 - f^2 - g^2}{2fg}.
\end{aligned}
\qquad (14)
$$

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 $X_p, Y_p, \varphi$ and the independent parameters $\psi, \theta, Z_p$ 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:

$$ 
\left. 
\begin{aligned}
(U_{O_{4,1}} - U_{O_{4,2}})^2 + (V_{O_{4,1}} - V_{O_{4,2}})^2 + (W_{O_{4,1}} - W_{O_{4,2}})^2 &= d^2 \\
(U_{O_{4,1}} - U_{O_{4,3}})^2 + (V_{O_{4,1}} - V_{O_{4,3}})^2 + (W_{O_{4,1}} - W_{O_{4,3}})^2 &= d^2 \\
(U_{O_{4,2}} - U_{O_{4,3}})^2 + (V_{O_{4,2}} - V_{O_{4,3}})^2 + (W_{O_{4,2}} - W_{O_{4,3}})^2 &= d^2
\end{aligned}
\right\} , \qquad (15)
$$
where d represents the distances between the spherical joints.

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

$$ \sum_{k=0}^{16} P_k t_{31}^k = 0, \qquad (16) $$

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: $a_i = 15$, $b_i = 8$, $c_i = 7$, $f_i = 60$, $g_i = 70$, $\gamma_1 = \pi/2 + \xi$, $\gamma_2 = 7\pi/6 + \xi$, $\gamma_3 = \xi - \pi/6$, $h = 43$, $d = h \cdot \sqrt{3}$, $\xi = a \sin(b/h)$. For the input parameters $s_1 = s_2 = s_3 = 60$ and $\theta_{2,1} = -40^\circ$, $\theta_{2,2} = -45^\circ$, $\theta_{2,3} = -35^\circ$, four positions of the moving platform were obtained (Figure 2).

<figure><img src="image_4.png" /><figcaption>Figure 2. Four positions of the moving platform of the 3-PRRS-type PM.</figcaption></figure>

Table A1 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 $\theta_{2,1}$, $-\pi/4$, $-\pi/5$, $-3\pi/20$, $-\pi/10$, $-\pi/20$, 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 $O_0U_0V_0W_0$ are determined (Tables A2–A4 in Appendix A).
Then, the coordinates of the point P on the moving platform in the absolute coordinate system $O_0U_0V_0W_0$ can be determined by the following equations:

$$X_P = \frac{1}{3} \cdot \sum_{i=1}^{n} U_{O_4,i}, \quad Y_P = \frac{1}{3} \cdot \sum_{i=1}^{n} V_{O_4,i}, \quad Z_P = \frac{1}{3} \cdot \sum_{i=1}^{n} W_{O_4,i}. \qquad (17)$$

Table A5 in Appendix A presents the obtained values of the coordinates of point P in the absolute coordinate system $O_0U_0V_0W_0$, 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.

<figure><img src="image_5.png" /><figcaption>Figure 3. Simulated trajectory of point P. The colored "*" denote the positions of the moving platform center corresponding to different solutions of the direct kinematics problem.</figcaption></figure>

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 $\theta_{2,1} = \theta_{2,2} = \theta_{2,3} = -45^\circ$, 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 Tables A1–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 (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 cγᵢ and the second equations by sγᵢ, we obtain the equations of the planes along which the O'₂ᵢO₃ᵢO₄ᵢ dyads (PM legs) move:

$$c\gamma_i U_{O_4,i} + s\gamma_i V_{O_4,i} + b_i = 0, (i = 1, 2, 3). \quad (18)$$

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

$$\left. \begin{aligned} g_i s(\theta_{23,i}) &= s_i - \frac{U_{O_4,i} + bc\gamma_i}{s\gamma_i} - f_i s\theta_{2,i} \\ g_i c(\theta_{23,i}) &= W_{O_4,i} - c_i - a_i - f_i c\theta_{2,i} \end{aligned} \right\}. \quad (19)$$

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

$$(W_{O_4,i} - c_i - a_i - f_i c\theta_{2,i})^2 + (s_i - \frac{U_{O_4,i} + b_i c\gamma_i}{s\gamma_i} - f_i s\theta_{2,i})^2 - g_i^2 = 0. \quad (20)$$

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

$$(X_{2,i} - f_i c\theta_{2,i})^2 + (Y_{2,i} - f_i s\theta_{2,i})^2 - g_i^2 = 0, \quad (21)$$

where $X_{2,i} = W_{O_4,i} - c_i - a_i$, $Y_{2,i} = s_i - U_{O_4,i}s\gamma_i + V_{O_4,i}c\gamma_i$. Here, $X_{2,i}, Y_{2,i}, Z_{2,i}$ are the coordinates of the absolute coordinate system relative to the local coordinate systems $O_2,iX_{2,i}Y_{2,i}Z_{2,i}$. $Z_{2,i} = -(c\gamma_i \cdot U_{O_4,i} + s\gamma_i \cdot V_{O_4,i} + b_i)$ is equal to zero according to Equation (18).

Expanding Equation (21) yields:

$$X_{2,i}c\theta_{2,i} + Y_{2,i}s\theta_{2,i} - \frac{X_{2,i}^2 + Y_{2,i}^2 + f_i^2 - g_i^2}{2f_i} = 0. \quad (22)$$

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

$$\left( \frac{X_{2,i}^2 + Y_{2,i}^2 + f_i^2 - g_i^2}{2 \cdot f_i} \right)^2 - (X_{2,i}^2 + Y_{2,i}^2) \le 0. \quad (23)$$

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

$$X_{2,i}^2 + Y_{2,i}^2 = (g_i - f_i)^2 \quad (24)$$

and

$$X_{2,i}^2 + Y_{2,i}^2 = (g_i + f_i)^2. \quad (25)$$

The circles (24) and (25) represent the inner and outer boundaries of the workspaces of the dyads $O_2,iO_3,iO_4,i$ (PM legs). These equations are expressed with respect to the absolute coordinate system $O_0U_0V_0W_0$ as follows:

$$(U_0 - O_{2,ix})^2 + (V_0 - O_{2,iy})^2 + (W_0 - O_{2,iz})^2 = (g_i - f_i)^2 \quad (26)$$
and

$$ (U_0 - O_{2,ix})^2 + (V_0 - O_{2,iy})^2 + (W_0 - O_{2,iz})^2 = (g_i + f_i)^2, \quad (27) $$

where $O_{2,ix}, O_{2,iy}, O_{2,iz}$ 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:

$$ \left. \begin{aligned} U_P &= U_{O_{4,i}} - U_{O_{4,p_i}} s\gamma_i s\theta_{23,i} - V_{O_{4,p_i}} s\gamma_i c\theta_{23,i} - W_{O_{4,p_i}} c\gamma_i \\ V_P &= V_{O_{4,i}} + U_{O_{4,p_i}} c\gamma_i s\theta_{23,i} + V_{O_{4,p_i}} c\gamma_i c\theta_{23,i} - W_{O_{4,p_i}} s\gamma_i \\ W_P &= W_{O_{4,i}} + U_{O_{4,p_i}} c\theta_{23,i} - V_{O_{4,p_i}} s\theta_{23,i} \end{aligned} \right\} , \quad (28) $$

where $U_{O_{4,p_i}}, V_{O_{4,p_i}}, W_{O_{4,p_i}}$ 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 $s\gamma_i$ and the second equations by $-c\gamma_i$, and then adding the first and second equations, we obtain:

$$ \left. \begin{aligned} (U_P - U_{O_{4,i}})s\gamma_i - (V_P - V_{O_{4,i}})c\gamma_i &= -U_{O_{4,p_i}}s\gamma_i s\theta_{23,i} - V_{O_{4,p_i}}s\gamma_i c\theta_{23,i} \\ W_P - W_{O_{4,i}} &= U_{O_{4,p_i}}c\theta_{23,i} - V_{O_{4,p_i}}s\theta_{23,i} \end{aligned} \right\} . \quad (29) $$

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

$$ U_{O_{4,p_i}} = [(V_P - V_{O_{4,i}})c\gamma_i - (U_P - U_{O_{4,i}})s\gamma_i]s\theta_{23,i} + (W_P - W_{O_{4,i}})c\theta_{23,i}. \quad (30) $$

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

$$ V_{O_{4,p_i}} = [(V_P - V_{O_{4,i}})c\gamma_i - (U_P - U_{O_{4,i}})s\gamma_i]c\theta_{23,i} - (W_P - W_{O_{4,i}})s\theta_{23,i}. \quad (31) $$

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

$$ W_{O_{4,p_i}} = -(U_P - U_{O_{4,i}})c\gamma_i - (V_P - V_{O_{4,i}})s\gamma_i. \quad (32) $$

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 $O_{4,i}, X_{4,i}, Y_{4,i}, Z_{4,i}$:

$$ U_{O_{4,p_i}}^2 + V_{O_{4,p_i}}^2 + W_{O_{4,p_i}}^2 = h^2. \quad (33) $$

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

$$ (U_P - U_{O_{4,i}})^2 + (V_P - V_{O_{4,i}})^2 + (W_P - W_{O_{4,i}})^2 = h^2. \quad (34) $$

The equations of these three spheres describe the motion of the center P of the moving platform relative to the absolute coordinate system $O_0U_0V_0W_0$. 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><img src="image_6.png" /><figcaption>Figure 4. Calculated workspace of the PM.</figcaption></figure>

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.

<figure><img src="image_7.png" /><figcaption>Figure 5. Influence of the stroke lengths of the linear actuators on the workspace.</figcaption></figure>

## 8. Jacobian Matrices

Let's derive the closed loop $O_0O_{1,i}O'_{1,i}O'_{2,i}O_{2,i}O_{3,i}O_{4,i}P$ equations of the leg:

$$ \mathbf{r}_P = \mathbf{r}_{O_{1,i}} + \mathbf{s}_i + \mathbf{a}_{1'2',i} + \mathbf{b}_{2'2',i} + \mathbf{f}_i + \mathbf{g}_i - \mathbf{h}_i, \quad (35) $$
where

$$ \begin{bmatrix} 1 \\ \mathbf{r}_P \end{bmatrix} = \mathbf{T}_{OP} \begin{bmatrix} 1 \\ 0 \\ 0 \\ 0 \end{bmatrix} = \begin{bmatrix} 1 \\ a_{OP}c\gamma_{OP} + b_{OP}s\gamma_{OP}s\alpha_{OP} \\ a_{OP}s\gamma_{OP} - b_{OP}c\gamma_{OP}s\alpha_{OP} \\ c_{OP} + b_{OP}c\alpha_{OP} \end{bmatrix}. \quad (36) $$

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:

$$ \mathbf{r}_P = [U_P, V_P, W_P]^T = {}^0\tau_P = \tau(a_{OP}, b_{OP}, c_{OP}, \alpha_{OP}, \beta_{OP}, \gamma_{OP}) \quad (37) $$

and

$$ \mathbf{r}_{O_{1,i}} = [U_{O_{1,i}}, V_{O_{1,i}}, W_{O_{1,i}}]^T = {}^0\tau_i = \tau(a_{01,i}, b_{01,i}, c_{01,i}, \alpha_{01,i}, \beta_{01,i}, \gamma_{01,i}). \quad (38) $$

The coordinate of other vectors in the absolute coordinate system $O_0U_0V_0W_0$ are determined using rotation submatrices:

$$ \begin{align} \mathbf{f}_i &= \overline{O_{2i}O_{3i}} = {}^0\mathbf{R}_{2i} {}^{2i}\mathbf{f}_i = {}^0\mathbf{R}_{1i} {}^{1i}\mathbf{R}_{2i}\mathbf{f}_i \nonumber \\ &= \mathbf{R}(\alpha_{01}, \beta_{01}, \gamma_{01})\mathbf{R}(\alpha_{12,i}, \beta_{12,i}, \gamma_{12,i}) {}^{2i}\mathbf{f}_i \nonumber \\ &= \begin{bmatrix} c\gamma_i & 0 & s\gamma_i \\ s\gamma_i & 0 & -c\gamma_i \\ 0 & 1 & 0 \end{bmatrix} \begin{bmatrix} 0 & 0 & -1 \\ c\theta_2 & -s\theta_2 & 0 \\ -s\theta_2 & -c\theta_2 & 0 \end{bmatrix} \begin{bmatrix} f_i \\ 0 \\ 0 \end{bmatrix} = \begin{bmatrix} -f_i s\gamma_i s\theta_2 \\ f_i c\gamma_i s\theta_2 \\ f_i c\theta_2 \end{bmatrix}. \tag{39} \end{align} $$

$$ \begin{align} \mathbf{g}_i &= \overline{O_{3i}O_{4i}} = {}^0\mathbf{R}_{3i} {}^{2i}\mathbf{g}_i = {}^0\mathbf{R}_{1i} {}^{1i}\mathbf{R}_{2i} {}^{2i}\mathbf{R}_{3i} {}^{3i}\mathbf{g}_i \nonumber \\ &= \mathbf{R}(\alpha_{01,i}, \beta_{01,i}, \gamma_i)\mathbf{R}(\alpha_{12,i}, \beta_{12,i}, \gamma_{12,i})\mathbf{R}(\alpha_{23,i}, \beta_{23,i}, \gamma_{23,i}) {}^{3i}\mathbf{g}_i \nonumber \\ &= \begin{bmatrix} c\gamma_i & 0 & s\gamma_i \\ s\gamma_i & 0 & -c\gamma_i \\ 0 & 1 & 0 \end{bmatrix} \begin{bmatrix} 0 & 0 & -1 \\ c\theta_{2i} & -s\theta_{2i} & 0 \\ -s\theta_{2i} & -c\theta_{2i} & 0 \end{bmatrix} \begin{bmatrix} c\theta_{3i} & -s\theta_{3i} & 0 \\ s\theta_{3i} & c\theta_{3i} & 0 \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} g_i \\ 0 \\ 0 \end{bmatrix} \tag{40} \\ &= \begin{bmatrix} -g_i s\gamma_i s(\theta_{2i} + \theta_{3i}) \\ g_i c\gamma_i s(\theta_{2i} + \theta_{3i}) \\ g_i c(\theta_{2i} + \theta_{3i}) \end{bmatrix}. \nonumber \end{align} $$

$$ \mathbf{h}_i = \overline{PO_{4i}} = {}^0\mathbf{R}_P {}^P\mathbf{h}_i = \mathbf{R}(\alpha_i, \beta_i, \gamma_i) \begin{bmatrix} h_i c\varphi_i \\ h_i s\varphi_i \\ 0 \end{bmatrix}, \quad (41) $$

where $\varphi_1 = 0, \varphi_2 = \frac{2\pi}{3}, \varphi_3 = -\frac{2\pi}{3}$.

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

$$ \dot{\mathbf{r}}_P = \dot{\mathbf{r}}_{O_{1,i}} + \dot{\mathbf{s}}_i + \dot{\mathbf{a}}_{1/2',i} + \dot{\mathbf{b}}_{2/2,i} + \dot{\mathbf{f}}_i + \dot{\mathbf{g}}_i - \dot{\mathbf{h}}_i \quad (42) $$

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

$$ \dot{\mathbf{r}}_P = \dot{s}_i \mathbf{e}_{1,i} + \dot{\theta}_{2,i} \times \mathbf{f}_i + \mathbf{\omega}_i \times \mathbf{g}_i - \mathbf{\omega}_P \times \mathbf{h}_i, \quad (43) $$

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, $\dot{\theta}_{2,i} = \dot{\theta}_{2,i} \mathbf{e}_{2,i}$ 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:

$$ \mathbf{g}_i^T \dot{\mathbf{r}}_P = \dot{s}_i \mathbf{g}_i^T \mathbf{e}_{1,i} + \mathbf{g}_i^T (\dot{\theta}_{2,i} \times \mathbf{f}_i) + \mathbf{g}_i^T (\mathbf{\omega}_i \times \mathbf{g}_i) - \mathbf{g}_i^T (\mathbf{\omega}_P \times \mathbf{h}_i) \quad (44) $$
or taking into consideration the relative positions of the vectors, yields:

$$ \mathbf{g}_i^T \dot{\mathbf{r}}_p = \dot{s}_i (\mathbf{g}_i^T \mathbf{e}_{1i}) + \dot{\theta}_{2i} \mathbf{e}_{2i}^T (\mathbf{f}_i \times \mathbf{g}_i) - \mathbf{\omega}_p^T (\mathbf{h}_i \times \mathbf{g}_i). \quad (45) $$

Let us introduce the following notations:

$$ \dot{\mathbf{x}} = [\dot{\mathbf{r}}_p^T, \mathbf{\omega}_p^T], \dot{\mathbf{q}} = [\dot{s}_1, \dot{\theta}_{21}, \dot{s}_2, \dot{\theta}_{22}, \dot{s}_3, \dot{\theta}_{23}]^T \quad (46) $$

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

$$ \mathbf{J}_x \dot{\mathbf{x}} = \mathbf{J}_q \dot{\mathbf{q}}, \quad (47) $$

where $\mathbf{J}_x$ and $\mathbf{J}_q$ are the Jacobian matrices connecting the input and output links velocities:

$$ \mathbf{J}_x = \begin{bmatrix} \mathbf{g}_1^T (\mathbf{h}_1 \times \mathbf{g}_1)^T \\ \mathbf{g}_2^T (\mathbf{h}_2 \times \mathbf{g}_2)^T \\ \mathbf{g}_3^T (\mathbf{h}_3 \times \mathbf{g}_3)^T \end{bmatrix}, \quad (48) $$

$$ \mathbf{J}_q = \begin{bmatrix} \mathbf{g}_1^T \mathbf{e}_{11} & \mathbf{e}_{21}^T (\mathbf{f}_1 \times \mathbf{g}_1) & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{bmatrix}. \quad (49) $$

## 9. Singular Configurations

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

The first type of singularity arises in the following conditions:

$$ \mathbf{g}_i^T \mathbf{e}_{1,i} = 0, \quad (50) $$

and

$$ \mathbf{e}_{2,i}^T (\mathbf{f}_i \times \mathbf{g}_i) = 0. \quad (51) $$

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:

$$ \theta_{23,i} = \theta_{2,i} + \theta_{3,i} = 0 \text{ or } \theta_{23,i} = 2\pi. \quad (52) $$

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

<figure><img src="image_8.png" /><figcaption>Figure 6. The first type of singularity when $\theta_{23,i} = 2\pi$.</figcaption></figure>
<figure><img src="image_9.png" /><figcaption>Figure 7. The first type of singularity when $\theta_{23,i} = 0$.</figcaption></figure>

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).

<figure><img src="image_10.png" /><figcaption>Figure 8. Conditions for the fulfillment of the first type singularity.</figcaption></figure>

The second type of singularity occurs when the matrix $\mathbf{J}_x$ 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 $\mathbf{J}_x$ in the following form:

$$ \mathbf{J}_x = \begin{bmatrix} \underline{g}_{1x} | \underline{g}_{1y} | \underline{g}_{1z} | \underline{n}_{1x} | \underline{n}_{1y} | \underline{n}_{1z} \\ \underline{g}_{2x} | \underline{g}_{2y} | \underline{g}_{2z} | \underline{n}_{2x} | \underline{n}_{2y} | \underline{n}_{2z} \\ \underline{g}_{3x} | \underline{g}_{3y} | \underline{g}_{3z} | \underline{n}_{3x} | \underline{n}_{3y} | \underline{n}_{3z} \end{bmatrix}, \quad (53) $$

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 $\mathbf{n}_i = \mathbf{h}_i \times \mathbf{g}_i$:

$$ \mathbf{g}_i = [g_{ix}, g_{iy}, g_{iz}]^T, \mathbf{n}_i = [n_{ix}, n_{iy}, n_{iz}]^T, i = 1, 2, 3. \quad (54) $$

Since vectors $\mathbf{h}_i$ cannot be zero, it is impossible for all elements of individual rows of the matrix $\mathbf{J}_x$ 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 $\mathbf{J}_x$. This means that there are simultaneously non-zero numbers $\lambda_1$ and $\lambda_2$ for which:

$$ \lambda_1[\mathbf{g}_j^T, \mathbf{n}_j^T] + \lambda_2[\mathbf{g}_k^T, \mathbf{n}_k^T] = 0, j \neq k, \quad (55) $$
from here:

$$[(\lambda_1 \mathbf{g}_j + \lambda_2 \mathbf{g}_k)^T, (\lambda_1 \mathbf{n}_j + \lambda_2 \mathbf{n}_k)^T]^T = 0, j \neq k. \quad (56)$$

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

$$\lambda_1 \mathbf{g}_j + \lambda_2 \mathbf{g}_k = \mathbf{0} \rightarrow \lambda_1 \mathbf{g}_j = -\lambda_2 \mathbf{g}_k \rightarrow \frac{\mathbf{g}_j}{\mathbf{g}_k} = \left| \frac{\lambda_2}{\lambda_1} \right| = \mu, \quad (57)$$

$$\lambda_1 \mathbf{n}_j + \lambda_2 \mathbf{n}_k = \mathbf{0} \rightarrow \lambda_1 \mathbf{n}_j = -\lambda_2 \mathbf{n}_k \rightarrow \frac{\mathbf{n}_j}{\mathbf{n}_k} = \left| \frac{\lambda_2}{\lambda_1} \right| = \mu. \quad (58)$$

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 $\mathbf{J}_x$. Indeed, let us assume for definiteness that $\mathbf{n}_j = 0$, $\mathbf{n}_k \neq 0$. 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 $\mathbf{n}_j \neq 0$, $\mathbf{n}_k \neq 0$. Then, according to conditions (57) and (58), there must be $\mathbf{g}_j || \mathbf{g}_k$ and $\mathbf{n}_j || \mathbf{n}_k$, and with the same proportionality coefficients $\mu$. Moreover, the specified requirements also correspond to the fulfillment of the following conditions:

$$\text{if } \mathbf{g}_j \uparrow \downarrow \mathbf{g}_k, \text{ it must be } \mathbf{n}_j \uparrow \downarrow \mathbf{n}_k, \quad (59)$$

$$\text{if } \mathbf{g}_j \uparrow \uparrow \mathbf{g}_k, \text{ it must be } \mathbf{n}_j \uparrow \uparrow \mathbf{n}_k. \quad (60)$$

Since $\mathbf{n}_j || \mathbf{n}_k$, then $\mathbf{h}_j x \mathbf{g}_j || \mathbf{h}_k x \mathbf{g}_k$. Let's assume that $\mathbf{g}_j || \mathbf{g}_k$ and check the parallelism of the vectors $\mathbf{n}_j$ and $\mathbf{n}_k$ with the components:

$$\mathbf{h}_j = [h_{jX}, h_{jY}, 0]^T, \mathbf{h}_k = [h_{kX}, h_{kY}, 0]^T, \quad (61)$$

$$\mathbf{g}_j = [g_{jX} \ g_{jY} \ g_{jZ}]^T, \mathbf{g}_k = [g_{kX} \ g_{kY} \ g_{kZ}]^T. \quad (62)$$

and

$$\mathbf{n}_j = (\mathbf{h}_j \times \mathbf{g}_j) = [h_{jY}g_{jZ}; -h_{jX}g_{jZ}; h_{jX}g_{jY} - h_{jY}g_{jX}]^T, \quad (63)$$

$$\mathbf{n}_k = (\mathbf{h}_k \times \mathbf{g}_k) = [h_{kY}g_{kZ}; -h_{kX}g_{kZ}; h_{kX}g_{kY} - h_{kY}g_{kX}]^T. \quad (64)$$

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

$$(\mathbf{n}_j \times \mathbf{n}_k) = \left[ \frac{-h_{jX}g_{jZ}(h_{kX}g_{kY} - h_{kY}g_{kX}) + h_{kX}g_{kZ}(h_{jX}g_{jY} - h_{jY}g_{jX})}{g_{jZ}g_{kZ}(h_{jX}h_{kY} - h_{jY}h_{kX})} \right]. \quad (65)$$

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

$$\left. \begin{array}{l} -h_{jX}g_{jZ}(h_{kX}g_{kY} - h_{kY}g_{kX}) + h_{kX}g_{kZ}(h_{jX}g_{jY} - h_{jY}g_{jX}) = 0 \\ -h_{jY}g_{jZ}(h_{kX}g_{kY} - h_{kY}g_{kX}) + h_{kY}g_{kZ}(h_{jX}g_{jY} - h_{jY}g_{jX}) = 0 \\ g_{jZ}g_{kZ}(h_{jX}h_{kY} - h_{jY}h_{kX}) = 0 \end{array} \right\}, \quad (66)$$

where

$$(h_{kX}g_{kY} - h_{kY}g_{kX}) = \begin{vmatrix} h_{kX} & h_{kY} \\ g_{kX} & g_{kY} \end{vmatrix} \neq 0, \text{ since } \mathbf{h}_k \text{ и } \mathbf{g}_k \text{ are not parallel,} \quad (67)$$

$$(h_{jX}g_{jY} - h_{jY}g_{jX}) = \begin{vmatrix} h_{jX} & h_{jY} \\ g_{jX} & g_{jY} \end{vmatrix} \neq 0, \text{ since } \mathbf{h}_j \text{ и } \mathbf{g}_j \text{ are not parallel,} \quad (68)$$
$$ (h_{jX}h_{kY} - h_{jY}h_{kX}) = \begin{vmatrix} h_{jX} & h_{jY} \\ h_{kX} & h_{kY} \end{vmatrix} \neq 0, \text{ since } \mathbf{h}_j \text{ and } \mathbf{h}_k \text{ are not parallel.} \quad (69) $$

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

$$ g_{jZ} = 0, g_{kZ} = 0. \quad (70) $$

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

$$ \mathbf{n}_j = [0; 0; h_{jX}g_{jY} - h_{jY}g_{jX}]^T, \quad (71) $$

$$ \mathbf{n}_k = [0; 0; h_{kX}g_{kY} - h_{kY}g_{kX}]^T, \quad (72) $$

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

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

$$ \frac{|\mathbf{n}_j|}{|\mathbf{n}_k|} = \frac{n_j}{n_k} = \frac{h_j \cdot g_j \cdot \sin(\Omega_j)}{h_k \cdot g_k \cdot \sin(\Omega_k)} = \mu, \quad (73) $$

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:

$$ h_j \cdot \sin(\Omega_j) = h_k \cdot \sin(\Omega_k). \quad (74) $$

that is, the projection of the segments $PO_{4,j} = h_j$ and $PO_{4,k} = h_k$ (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 $P$. Therefore, 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 $PQ_k$. 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 $PQ_k$, 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 $\mathbf{J}_x$ 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.
<figure><img src="image_11.png" /><figcaption>Figure 9. The second type of singularity: (a) singular configuration; (b,c) nonsingular configurations with parallel arrangement of the links.</figcaption></figure>

## 10. Direct Velocity Analysis

The Jacobian matrices $J_x$ and $J_q$ 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 $\mathbf{q}$ is known. To accomplish this, additional equations are required. Let us consider the following loop-closure vector equations

$$ \mathbf{O}_{4,i} = \mathbf{O}\mathbf{P} + \mathbf{h}_i. \qquad (75) $$

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

$$ \mathbf{V}_{O_{4,i}} = \dot{\mathbf{r}}_p + \omega_p \times \mathbf{h}_i. \qquad (76) $$

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

$$ \mathbf{e}_{2,i} \cdot \dot{\mathbf{r}}_p + \mathbf{e}_{2,i} \cdot (\omega_p \times \mathbf{h}_i) = 0. \qquad (77) $$

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

$$ \mathbf{e}_{2,i} \cdot \dot{\mathbf{r}}_p + (\mathbf{h}_i \times \mathbf{e}_{2,i}) \cdot \omega_p = \mathbf{0}. \qquad (78) $$

Equation (78) can be expressed in the matrix form

$$ \mathbf{J}_g \cdot \dot{\mathbf{x}} = \mathbf{0}, \qquad (79) $$

where,

$$ \mathbf{J}_g = \begin{bmatrix} \mathbf{e}_{2,1}^T & (\mathbf{h}_1 \times \mathbf{e}_{2,1})^T \\ -\mathbf{e}_{2,2}^T & (\mathbf{h}_2 \times \mathbf{e}_{2,2})^T \\ -\mathbf{e}_{2,3}^T & (\mathbf{h}_3 \times \mathbf{e}_{2,3})^T \end{bmatrix}. \qquad (80) $$
Combining Equations (47) and (79), and augmenting the matrix $J_q$, we obtain

$$J_a \cdot \dot{x} = J_e \cdot \dot{q}, \qquad (81)$$

where

$$J_a = \begin{bmatrix} -g_{1x} & -g_{1y} & -g_{1z} & -n_{1x} & -n_{1y} & -n_{1z} \\ -g_{2x} & -g_{2y} & -g_{2z} & -n_{2x} & -n_{2y} & -n_{2z} \\ g_{3x} & g_{3y} & g_{3z} & n_{3x} & n_{3y} & n_{3z} \\ -c\gamma_1 & -s\gamma_1 & 0 & h_{1z} \cdot s\gamma_1 & -h_{1z} \cdot c\gamma_1 & h_{1y} \cdot c\gamma_1 - h_{1x} \cdot s\gamma_1 \\ -c\gamma_2 & -s\gamma_2 & 0 & h_{2z} \cdot s\gamma_2 & -h_{2z} \cdot c\gamma_2 & h_{2y} \cdot c\gamma_2 - h_{2x} \cdot s\gamma_2 \\ -c\gamma_3 & -s\gamma_3 & 0 & h_{3z} \cdot s\gamma_3 & -h_{3z} \cdot c\gamma_3 & h_{3y} \cdot c\gamma_3 - h_{3x} \cdot s\gamma_3 \end{bmatrix}, \qquad (82)$$

$$J_e = \begin{bmatrix} -g \cdot s\theta_{23,1} & f \cdot g \cdot s\theta_{3,1} & 0 & 0 & 0 & 0 \\ 0 & 0 & -g \cdot s\theta_{23,2} & f \cdot g \cdot s\theta_{3,2} & 0 & 0 \\ 0 & 0 & 0 & 0 & -g \cdot s\theta_{23,3} & f \cdot g \cdot s\theta_{3,3} \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{bmatrix}. \qquad (83)$$

Now the direct velocity problem is solved as follows

$$\dot{x} = J_a^{-1} \cdot J_e \cdot \dot{q}. \qquad (84)$$

## 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

$$\left\{ \begin{aligned} (X_P + h \cdot t_{11})c\gamma_1 + (Y_P + h \cdot t_{21})s\gamma_1 + b = 0 \\ (X_P + \frac{h}{2}(\sqrt{3}t_{12} - t_{11}))c\gamma_2 + (Y_P + \frac{h}{2}(\sqrt{3}t_{22} - t_{21}))s\gamma_2 + b = 0 \\ (X_P - \frac{h}{2}(\sqrt{3}t_{12} + t_{11}))c\gamma_3 + (Y_P - \frac{h}{2}(\sqrt{3}t_{22} + t_{21}))s\gamma_3 + b = 0 \end{aligned} \right\}, \qquad (85)$$

where

$$\begin{aligned} t_{11} &= c\theta c\phi + s\psi s\theta s\phi, & t_{12} &= -c\theta s\phi + s\psi s\theta c\phi, & t_{13} &= c\psi s\theta, \\ t_{21} &= c\psi s\phi, & t_{22} &= c\psi c\phi, & t_{23} &= -s\psi, \\ t_{31} &= s\theta s\phi + s\psi c\theta c\phi, & t_{32} &= -s\theta c\phi + s\psi c\theta s\phi, & t_{33} &= c\psi c\theta, \end{aligned} \qquad (86)$$

Here $\psi, \theta, \phi$ are the Euler angles that describe the orientation of the local coordinate system $PX_P Y_P Z_P$, and they are defined as follows

$$\mathbf{R}_{OP} = R_y(\theta)R_x(\psi)R_z(\phi) = \begin{bmatrix} t_{11} & t_{12} & t_{13} \\ t_{21} & t_{22} & t_{23} \\ t_{31} & t_{32} & t_{33} \end{bmatrix}. \qquad (87)$$

By subtracting the third equation from the second in the system of Equation (85), after dividing them by $s\gamma_2$ and $s\gamma_3$ respectively, we can determine $X_P$ as follows

$$X_P = \frac{h}{2} [t_{11} + (3 - 4c^2\xi)t_{22}] + hs(2\xi)t_{12} - 2bs\xi, \qquad (88)$$
where $\xi = s^{-1}(b/h)$ is the angle between the axis $U_0$ of the absolute coordinate system and the vector of the first prismatic actuator $s_1$.

Similarly, by subtracting the third equation from the second in the system of Equation (85) and dividing them by $c\gamma_2$ and $c\gamma_3$ respectively, $\gamma_P$ can be determined as follows

$$ \gamma_P = \frac{h}{2} [t_{21} + (1 - 4c^2\xi)t_{12}] - hs(2\xi)t_{22} - 2bc\xi. \quad (89) $$

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

$$ (t_{11} + t_{22})s\xi + (t_{12} - t_{21})c\xi = \frac{2b}{h}. \quad (90) $$

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

$$ A c \varphi + B s \varphi - \frac{2b}{h} = 0, \quad (91) $$

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

$$ \varphi_{1,2} = \pm c^{-1} \left( \frac{C}{r} \right) + \alpha, \quad (92) $$

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

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 $X_P$, $\gamma_P$, 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.

$$ \dot{\mathbf{x}} = \mathbf{J}_{\mathbf{r}} \cdot \dot{\mathbf{X}}, \quad (93) $$

where

$$ \mathbf{J}_{\mathbf{r}} = \begin{bmatrix} \frac{\partial X_P}{\partial Z_P} & \frac{\partial X_P}{\partial \psi} + \frac{\partial X_P}{\partial \varphi} \cdot \frac{\partial \varphi}{\partial \psi} & \frac{\partial X_P}{\partial \theta} + \frac{\partial X_P}{\partial \varphi} \cdot \frac{\partial \varphi}{\partial \theta} \\ - \frac{\partial Y_P}{\partial Z_P} & - \frac{\partial Y_P}{\partial \psi} + \frac{\partial Y_P}{\partial \varphi} \cdot \frac{\partial \varphi}{\partial \psi} & - \frac{\partial Y_P}{\partial \theta} + \frac{\partial Y_P}{\partial \varphi} \cdot \frac{\partial \varphi}{\partial \theta} \\ - \frac{1}{Z_P} & - \frac{0}{\psi} & - \frac{0}{\varphi} \\ 0 & 1 & 0 \\ 0 & - \frac{0}{\varphi} & - \frac{1}{\psi} \end{bmatrix}. \quad (94) $$

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

$$ \dot{\mathbf{q}} = \mathbf{J}_{\mathbf{q}}^{-1} \cdot \mathbf{J}_{\mathbf{x}} \cdot \mathbf{J}_{\mathbf{r}} \cdot \dot{\mathbf{X}}. \quad (95) $$

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 $O_0U_0V_0W_0$

$$ \left. \begin{aligned} \omega_{P_{U_0}} &= \dot{\psi} \cdot c\theta + \dot{\phi}s\theta c\psi \\ \omega_{P_{V_0}} &= \dot{\theta} - \dot{\phi}s\psi \end{aligned} \right\} , \qquad (96) $$

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

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

$$ \dot{\theta} = \frac{k_{21} \cdot \omega_{P_{U_0}} - k_{11} \cdot \omega_{P_{V_0}}}{k_{12} \cdot k_{21} - k_{22} \cdot k_{11}}, \quad \dot{\psi} = \frac{k_{22} \cdot \omega_{P_{U_0}} - k_{12} \cdot \omega_{P_{V_0}}}{k_{11} \cdot k_{22} - k_{21} \cdot k_{12}}, \qquad (97) $$

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

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

$$ \omega_{W_0} = -\dot{\psi}s\theta + \dot{\phi}c\theta c\psi. \qquad (98) $$

Now, instead of the matrix $J_x$, the matrix $J_a$ can be used. The augmented part of the matrix $J_a$ does not affect the analysis of singular configurations. Since the matrix $J_a$ 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: $b = 0, h = 50, k = \sin(b/h)$, $a = 15, c = 7, L_1 = 50, L_2 = 70, \gamma_1 = \pi/2 + k, \gamma_2 = 7\pi/6 + k, \gamma_3 = k - \pi/6$ with $s_1 = s_2 = s_3 = 40$, when checking the extended Jacobian matrix $J_x$ 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 $\psi = -9.3^\circ, \theta = -12^\circ$. 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.

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 $J_x$ 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 $\psi = -9.3^\circ, \theta = -12^\circ$. 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.

<figure><img src="image_12.png" /><figcaption>Figure 10. Expansion of the workspace without singular configurations: (a) singular configuration; (b) exit from the singular configuration.</figcaption></figure>
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: $a_i = 15, b_i = 8, c_i = 7, f_i = 50, g_i = 70, \gamma_1 = \pi/2 + \zeta, \gamma_2 = 7\pi/6 + \zeta, \gamma_3 = \zeta - \pi/6, a = h \cdot \sqrt{3}, h = 50, \zeta = s^{-1}(b_i/h)$.

In solving the direct velocity problem, the independent parameters $\psi, \theta$ and $Z_p$ are specified, and the dependent parameters $\varphi, X_p$ and $Y_p$ are determined (Table A7). The input parameters of the prismatic kinematic pairs $s_1, s_2$ 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). The direct velocity problem is then solved using Equation (84).

Table A9 presents the specified velocities of the six input links $\dot{\mathbf{q}} = [\dot{s}_1, \dot{\theta}_{2,1}, \dot{s}_2, \dot{\theta}_{2,2}, \dot{s}_3, \dot{\theta}_{2,3}]^T$. Table A10 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 $\mathbf{V} = [V_{PU_0}, V_{PV_0}, V_{PW_0}, \omega_{PU_0}, \omega_{PV_0}, \omega_{PW_0}]^T$.

## 13. Numerical Examples of Solving the Inverse Velocity Problem

The following constant parameters are specified: $a_i = 15, b_i = 8, c_i = 7, f_i = 50, g_i = 70, \gamma_1 = \pi/2 + \zeta, \gamma_2 = 7\pi/6 + \zeta, \gamma_3 = \zeta - \pi/6, a = h \cdot \sqrt{3}, h = 50, \zeta = s^{-1}(b_i/h)$.

Table A11 presents the positions and orientations of the moving platform of the parallel manipulator. Table A12 specifies the components of the instantaneous velocities of the moving platform with respect to absolute coordinate system $O_0U_0V_0W_0$, determined from the solution of the direct kinematics problem. Table A13 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.

**Author Contributions:** Conceptualization, Z.B. and R.K.; methodology, R.K. and M.A.L.; software, Z.B. and Z.Z.; validation, R.K., G.C. and L.Q.; formal analysis, R.K. and M.A.L.; investigation, Z.B., Z.Z. and L.Q.; resources, G.C. and M.A.L.; data curation, Z.Z. and L.Q.; writing—original draft preparation, Z.B. and R.K.; writing—review and editing, R.K., G.C. and M.A.L.; visualization, Z.B. and R.K.; supervision, Z.B. and G.C.; project administration, Z.B.; funding acquisition, Z.B. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work is funded by the Committee of Science of the Ministry of Science and Higher Education of Kazakhstan (Grant No. AP 23488130, "Development and research of a domestic social humanoid robot").

**Data Availability Statement:** The data supporting the findings of this study are available from the corresponding author upon reasonable request.

**Conflicts of Interest:** The authors declare no conflict of interest.

## Appendix A

<table><caption>Table A1. Calculated values of the angles $\theta_{3,1}$, $\theta_{3,2}$, $\theta_{3,3}$.</caption><thead><tr><th>k</th><th>θ<sub>2,1</sub></th><th>θ<sub>3,1</sub></th><th>θ<sub>3,2</sub></th><th>θ<sub>3,3</sub></th></tr></thead><tbody><tr><td rowspan="4">1</td><td rowspan="4">-π/4</td><td>1.9714</td><td>2.5518</td><td>2.3137</td></tr><tr><td>2.2974</td><td>2.8269</td><td>2.8562</td></tr><tr><td>2.5953</td><td>2.0203</td><td>2.2978</td></tr><tr><td>2.8565</td><td>2.3016</td><td>2.8091</td></tr><tr><td rowspan="4">2</td><td rowspan="4">-π/5</td><td>1.7111</td><td>2.4516</td><td>2.2484</td></tr><tr><td>2.1030</td><td>2.8145</td><td>2.8768</td></tr><tr><td>2.6020</td><td>2.0306</td><td>2.2723</td></tr><tr><td>2.8699</td><td>2.3431</td><td>2.8393</td></tr><tr><td rowspan="4">3</td><td rowspan="4">-3π/20</td><td>1.4626</td><td>2.3653</td><td>2.2078</td></tr><tr><td>1.9157</td><td>2.8016</td><td>2.8927</td></tr><tr><td>2.5883</td><td>2.0407</td><td>2.2549</td></tr><tr><td>2.8699</td><td>2.3781</td><td>2.8622</td></tr></tbody></table>
<table><caption>Table A1. Cont.</caption><tr><td></td><td></td><td>1.2202</td><td>2.2936</td><td>2.1857</td></tr><tr><td rowspan="3">4</td><td rowspan="3">-π/10</td><td>1.7311</td><td>2.7877</td><td>2.9062</td></tr><tr><td>2.5581</td><td>2.0511</td><td>2.2409</td></tr><tr><td>2.8614</td><td>2.4108</td><td>2.8813</td></tr><tr><td rowspan="3">5</td><td rowspan="3">-π/20</td><td>0.9802</td><td>2.2365</td><td>2.1769</td></tr><tr><td>1.5466</td><td>2.7723</td><td>2.9180</td></tr><tr><td>2.5113</td><td>2.0625</td><td>2.2287</td></tr><tr><td rowspan="3">6</td><td rowspan="3">0</td><td>2.8458</td><td>2.4434</td><td>2.8984</td></tr><tr><td>0.7384</td><td>2.1934</td><td>2.1764</td></tr><tr><td>1.3602</td><td>2.7547</td><td>2.9285</td></tr><tr><td rowspan="2"></td><td rowspan="2"></td><td>2.4452</td><td>2.0756</td><td>2.2173</td></tr><tr><td>2.8234</td><td>2.4777</td><td>2.9142</td></tr></table>

<table><caption>Table A2. Calculated coordinates of the spherical joint O<sub>4,1</sub>.</caption><thead><tr><th>k</th><th>θ<sub>2,1</sub></th><th>U<sub>O<sub>4,1</sub></sub></th><th>V<sub>O<sub>4,1</sub></sub></th><th>W<sub>O<sub>4,1</sub></sub></th></tr></thead><tbody><tr><td rowspan="4">1</td><td rowspan="4">-π/4</td><td>38.3763</td><td>-0.8755</td><td>90.6979</td></tr><tr><td>33.4671</td><td>-1.8050</td><td>68.5344</td></tr><tr><td>35.3065</td><td>-1.4567</td><td>47.8432</td></tr><tr><td>41.7807</td><td>-0.2308</td><td>30.8433</td></tr><tr><td rowspan="4">2</td><td rowspan="4">-π/5</td><td>34.3408</td><td>-1.6396</td><td>103.3579</td></tr><tr><td>26.6316</td><td>-3.0993</td><td>77.2571</td></tr><tr><td>31.8213</td><td>-2.1167</td><td>43.0953</td></tr><tr><td>41.2162</td><td>-0.3377</td><td>27.0288</td></tr><tr><td rowspan="4">3</td><td rowspan="4">-3π/20</td><td>29.6504</td><td>-2.5277</td><td>113.7845</td></tr><tr><td>18.9741</td><td>-4.5493</td><td>84.2738</td></tr><tr><td>28.4385</td><td>-2.7572</td><td>39.0912</td></tr><tr><td>40.6844</td><td>-0.4384</td><td>23.9019</td></tr><tr><td rowspan="4">4</td><td rowspan="4">-π/10</td><td>24.5216</td><td>-3.4989</td><td>122.2393</td></tr><tr><td>10.6921</td><td>-6.1175</td><td>89.7869</td></tr><tr><td>24.8862</td><td>-3.4298</td><td>35.4180</td></tr><tr><td>40.1464</td><td>-0.5403</td><td>21.0662</td></tr><tr><td rowspan="4">5</td><td rowspan="4">-π/20</td><td>19.2295</td><td>-4.5009</td><td>128.8561</td></tr><tr><td>2.0118</td><td>-7.7612</td><td>93.8796</td></tr><tr><td>20.9347</td><td>-4.1781</td><td>31.8606</td></tr><tr><td>39.5715</td><td>-0.6491</td><td>18.3165</td></tr><tr><td rowspan="3">6</td><td rowspan="3">0</td><td>14.1445</td><td>-5.4638</td><td>133.7668</td></tr><tr><td>-6.8187</td><td>-9.4333</td><td>96.6268</td></tr><tr><td>16.3261</td><td>-5.0507</td><td>28.2959</td></tr><tr><td></td><td></td><td>38.9275</td><td>-0.7711</td><td>15.5125</td></tr></tbody></table>
<table><caption>Table A3. Calculated coordinates of the spherical joint $O_{4,2}$.</caption><thead><tr><th>k</th><th>θ<sub>2,1</sub></th><th>U<sub>O<sub>4,2</sub></sub></th><th>V<sub>O<sub>4,2</sub></sub></th><th>W<sub>O<sub>4,2</sub></sub></th></tr></thead><tbody><tr><td rowspan="4">1</td><td rowspan="4">−π/4</td><td>−15.9630</td><td>30.8067</td><td>50.8172</td></tr><tr><td>−20.0582</td><td>35.5642</td><td>32.6796</td></tr><tr><td>−17.6435</td><td>32.7591</td><td>87.4987</td></tr><tr><td>−15.1594</td><td>29.8733</td><td>68.2404</td></tr><tr><td rowspan="4">2</td><td rowspan="4">−π/5</td><td>−15.2994</td><td>30.0359</td><td>57.7563</td></tr><tr><td>−19.8044</td><td>35.2693</td><td>33.4561</td></tr><tr><td>−17.4904</td><td>32.5811</td><td>86.8149</td></tr><tr><td>−15.0955</td><td>29.7990</td><td>65.3402</td></tr><tr><td rowspan="4">3</td><td rowspan="4">−3π/20</td><td>−15.0935</td><td>29.7967</td><td>63.7835</td></tr><tr><td>−19.5487</td><td>34.9723</td><td>34.2636</td></tr><tr><td>−17.3451</td><td>32.4123</td><td>86.1439</td></tr><tr><td>−15.1026</td><td>29.8073</td><td>62.8889</td></tr><tr><td rowspan="4">4</td><td rowspan="4">−π/10</td><td>−15.1810</td><td>29.8983</td><td>68.8041</td></tr><tr><td>−19.2795</td><td>34.6596</td><td>35.1433</td></tr><tr><td>−17.1998</td><td>32.2436</td><td>85.4496</td></tr><tr><td>−15.1598</td><td>29.8736</td><td>60.6035</td></tr><tr><td rowspan="4">5</td><td rowspan="4">−π/20</td><td>−15.4182</td><td>30.1739</td><td>72.7831</td></tr><tr><td>−18.9889</td><td>34.3220</td><td>36.1304</td></tr><tr><td>−17.0466</td><td>32.0656</td><td>84.6886</td></tr><tr><td>−15.2653</td><td>29.9963</td><td>58.3260</td></tr><tr><td rowspan="4">6</td><td rowspan="4">0</td><td>−15.6954</td><td>30.4959</td><td>75.7714</td></tr><tr><td>−18.6706</td><td>33.9521</td><td>37.2615</td></tr><tr><td>−16.8769</td><td>31.8684</td><td>83.8074</td></tr><tr><td>−15.4288</td><td>30.1862</td><td>55.9355</td></tr></tbody></table>

<table><caption>Table A4. Calculated coordinates of the spherical joint $O_{4,3}$.</caption><thead><tr><th>k</th><th>θ<sub>2,1</sub></th><th>U<sub>O<sub>4,3</sub></sub></th><th>V<sub>O<sub>4,3</sub></sub></th><th>W<sub>O<sub>4,3</sub></sub></th></tr></thead><tbody><tr><td rowspan="4">1</td><td rowspan="4">−π/4</td><td>−22.4664</td><td>−40.0022</td><td>72.9707</td></tr><tr><td>−22.0580</td><td>−38.8345</td><td>35.4773</td></tr><tr><td>−22.5790</td><td>−40.3241</td><td>74.0271</td></tr><tr><td>−21.8258</td><td>−38.1708</td><td>38.7014</td></tr><tr><td rowspan="4">2</td><td rowspan="4">−π/5</td><td>−22.9649</td><td>−41.4274</td><td>77.2815</td></tr><tr><td>−22.1748</td><td>−39.1684</td><td>34.0835</td></tr><tr><td>−22.7720</td><td>−40.8760</td><td>75.7184</td></tr><tr><td>−21.9688</td><td>−38.5794</td><td>36.6323</td></tr><tr><td rowspan="4">3</td><td rowspan="4">−3π/20</td><td>−23.3216</td><td>−42.4472</td><td>79.9112</td></tr><tr><td>−22.2717</td><td>−39.4455</td><td>33.0095</td></tr><tr><td>−22.9116</td><td>−41.2750</td><td>76.8607</td></tr><tr><td>−22.0907</td><td>−38.9279</td><td>35.0748</td></tr></tbody></table>
<table><caption>Table A4. Cont.</caption><tr><td></td><td></td><td>24.5216</td><td>-3.4989</td><td>122.2393</td></tr><tr><td rowspan="3">4</td><td rowspan="3">-π/10</td><td>10.6921</td><td>-6.1175</td><td>89.7869</td></tr><tr><td>24.8862</td><td>-3.4298</td><td>35.4180</td></tr><tr><td>40.1464</td><td>-0.5403</td><td>21.0662</td></tr><tr><td rowspan="3">5</td><td rowspan="3">-π/20</td><td>19.2295</td><td>-4.5009</td><td>128.8561</td></tr><tr><td>2.0118</td><td>-7.7612</td><td>93.8796</td></tr><tr><td>20.9347</td><td>-4.1781</td><td>31.8606</td></tr><tr><td rowspan="4">6</td><td rowspan="4">0</td><td>39.5715</td><td>-0.6491</td><td>18.3165</td></tr><tr><td>-23.6213</td><td>-43.3042</td><td>81.9128</td></tr><tr><td>-22.5106</td><td>-40.1286</td><td>30.6074</td></tr><tr><td>-23.2354</td><td>-42.2009</td><td>79.3032</td></tr><tr><td></td><td></td><td>-22.4115</td><td>-39.8451</td><td>31.5667</td></tr></table>

<table><caption>Table A5. Calculated coordinates of the point P.</caption><thead><tr><th><i>i</i></th><th><i>X</i><sub>P</sub></th><th><i>Y</i><sub>P</sub></th><th><i>Z</i><sub>P</sub></th></tr></thead><tbody><tr><td rowspan="5">1</td><td>-0.0177</td><td>-3.3570</td><td>71.4953</td></tr><tr><td>-1.3078</td><td>-4.3437</td><td>79.4652</td></tr><tr><td>-2.9215</td><td>-5.0594</td><td>85.8264</td></tr><tr><td>-4.7299</td><td>-5.5481</td><td>90.7885</td></tr><tr><td>-6.6017</td><td>-5.8724</td><td>94.5070</td></tr><tr><td rowspan="5">2</td><td>-8.3907</td><td>-6.0907</td><td>97.1503</td></tr><tr><td>-2.8830</td><td>-1.6917</td><td>45.5638</td></tr><tr><td>-5.1158</td><td>-2.3328</td><td>48.2655</td></tr><tr><td>-7.6154</td><td>-3.0075</td><td>50.5156</td></tr><tr><td>-10.3152</td><td>-3.7169</td><td>52.3445</td></tr><tr><td rowspan="5">3</td><td>-13.1382</td><td>-4.4530</td><td>53.7731</td></tr><tr><td>-16.0000</td><td>-5.2032</td><td>54.8319</td></tr><tr><td>-1.6387</td><td>-3.0072</td><td>69.7897</td></tr><tr><td>-2.8137</td><td>-3.4705</td><td>68.5428</td></tr><tr><td>-3.9394</td><td>-3.8733</td><td>67.3653</td></tr><tr><td rowspan="4">4</td><td>-5.1138</td><td>-4.2647</td><td>66.2124</td></tr><tr><td>-6.4152</td><td>-4.6742</td><td>65.0379</td></tr><tr><td>-7.9287</td><td>-5.1277</td><td>63.8022</td></tr><tr><td>1.5984</td><td>-2.8427</td><td>45.9283</td></tr><tr><td></td><td>1.3839</td><td>-3.0393</td><td>43.0005</td></tr><tr><td></td><td>1.1636</td><td>-3.1863</td><td>40.6219</td></tr><tr><td></td><td>0.9281</td><td>-3.3043</td><td>38.4813</td></tr><tr><td></td><td>0.6659</td><td>-3.4008</td><td>36.4214</td></tr><tr><td></td><td>0.3623</td><td>-3.4766</td><td>34.3382</td></tr></tbody></table>
<table><caption>Table A6. Calculated values of the angles $\theta_{2,i}$ and $\theta_{3,i}$.</caption><tr><td>θ<sub>2,1</sub></td><td>-0.7853</td><td>1.4172</td><td>θ<sub>3,1</sub></td><td>1.9714</td><td>-1.9714</td></tr><tr><td>θ<sub>2,2</sub></td><td>-0.7853</td><td>0.8790</td><td>θ<sub>3,2</sub></td><td>2.5518</td><td>-2.5518</td></tr><tr><td>θ<sub>2,3</sub></td><td>-1.0471</td><td>1.5283</td><td>θ<sub>3,3</sub></td><td>2.3137</td><td>-2.3137</td></tr></table>

<table><caption>Table A7. Independent and dependent parameters determining the position and orientation of the moving platform.</caption><thead><tr><th></th><th>ψ</th><th>θ</th><th>Z<sub>P</sub></th><th>φ</th><th>X<sub>P</sub></th><th>Y<sub>P</sub></th></tr></thead><tbody><tr><td colspan="7">s<sub>1</sub> = s<sub>2</sub> = s<sub>3</sub> = 40</td></tr><tr><td>1</td><td>0</td><td>0</td><td>90</td><td>0</td><td>0</td><td>0</td></tr><tr><td>2</td><td>0</td><td>0</td><td>90</td><td>0</td><td>0</td><td>0</td></tr><tr><td>3</td><td>0</td><td>0</td><td>90</td><td>0</td><td>0</td><td>0</td></tr><tr><td>4</td><td>0</td><td>0</td><td>90</td><td>0</td><td>0</td><td>0</td></tr><tr><td colspan="7">s<sub>1</sub> = s<sub>2</sub> = s<sub>3</sub> = 45</td></tr><tr><td>5</td><td>-0.1396</td><td>0.1221</td><td>90</td><td>-0.0099</td><td>-0.0757</td><td>0.4210</td></tr><tr><td>6</td><td>0.1570</td><td>-0.1396</td><td>100</td><td>-0.0128</td><td>-0.1038</td><td>0.5381</td></tr><tr><td>7</td><td>0.1745</td><td>0.1396</td><td>100</td><td>0.0102</td><td>0.3257</td><td>-0.5268</td></tr><tr><td>8</td><td>-0.1745</td><td>-0.1396</td><td>120</td><td>0.0102</td><td>0.3257</td><td>-0.5268</td></tr><tr><td colspan="7">s<sub>1</sub> = s<sub>2</sub> = s<sub>3</sub> = 50</td></tr><tr><td>9</td><td>-0.1221</td><td>-0.2268</td><td>120</td><td>0.0112</td><td>-0.2057</td><td>-0.7961</td></tr><tr><td>10</td><td>-0.1221</td><td>-0.2268</td><td>120</td><td>0.0112</td><td>-0.2057</td><td>-0.7961</td></tr><tr><td>11</td><td>0.2094</td><td>0.1919</td><td>80</td><td>0.0169</td><td>0.4112</td><td>-0.9066</td></tr><tr><td>12</td><td>-0.2094</td><td>-0.1919</td><td>80</td><td>0.0169</td><td>0.4112</td><td>-0.9066</td></tr></tbody></table>

<table><caption>Table A8. Positions of the input and output revolute kinematic pairs.</caption><thead><tr><th></th><th>θ<sub>2,1</sub></th><th>θ<sub>2,2</sub></th><th>θ<sub>2,3</sub></th><th>θ<sub>3,1</sub></th><th>θ<sub>3,2</sub></th><th>θ<sub>3,3</sub></th></tr></thead><tbody><tr><td colspan="7">s<sub>1</sub> = s<sub>2</sub> = s<sub>3</sub> = 40</td></tr><tr><td>1</td><td>-1.3640</td><td>-1.3640</td><td>-1.3640</td><td>1.9649</td><td>1.9649</td><td>1.9649</td></tr><tr><td>2</td><td>-1.3640</td><td>-1.3640</td><td>-1.3640</td><td>1.9649</td><td>1.9649</td><td>1.9649</td></tr><tr><td>3</td><td>-1.3640</td><td>-1.3640</td><td>-1.3640</td><td>1.9649</td><td>1.9649</td><td>1.9649</td></tr><tr><td>4</td><td>-1.3640</td><td>-1.3640</td><td>-1.3640</td><td>1.9649</td><td>1.9649</td><td>1.9649</td></tr><tr><td colspan="7">s<sub>1</sub> = s<sub>2</sub> = s<sub>3</sub> = 45</td></tr><tr><td>5</td><td>-1.3970</td><td>-1.3555</td><td>-1.1342</td><td>2.1016</td><td>2.0385</td><td>1.7799</td></tr><tr><td>6</td><td>-1.0124</td><td>-1.0819</td><td>-1.2823</td><td>1.5971</td><td>1.6793</td><td>1.9818</td></tr><tr><td>7</td><td>-1.2446</td><td>-0.9300</td><td>-1.2016</td><td>1.9090</td><td>1.4968</td><td>1.8478</td></tr><tr><td>8</td><td>-0.6479</td><td>-0.9617</td><td>-0.7090</td><td>1.0280</td><td>1.5443</td><td>1.1217</td></tr><tr><td colspan="7">s<sub>1</sub> = s<sub>2</sub> = s<sub>3</sub> = 50</td></tr><tr><td>9</td><td>-0.4914</td><td>-0.9033</td><td>-0.7528</td><td>0.8679</td><td>1.5418</td><td>1.2604</td></tr><tr><td>10</td><td>-0.4914</td><td>-0.9033</td><td>-0.7528</td><td>0.8679</td><td>1.5418</td><td>1.2604</td></tr><tr><td>11</td><td>-1.5544</td><td>-1.1278</td><td>-1.4758</td><td>2.3732</td><td>1.8983</td><td>2.2719</td></tr><tr><td>12</td><td>-1.2308</td><td>-1.5565</td><td>-1.3268</td><td>1.9917</td><td>2.4566</td><td>2.0976</td></tr></tbody></table>
<table><caption>Table A9. Values of the instantaneous actuator velocities.</caption><thead><tr><th></th><th>dot s<sub>1</sub></th><th>dot s<sub>2</sub></th><th>dot s<sub>3</sub></th><th>dot θ<sub>2,1</sub></th><th>dot θ<sub>2,2</sub></th><th>dot θ<sub>2,3</sub></th></tr></thead><tbody><tr><td colspan="7">s<sub>1</sub> = s<sub>2</sub> = s<sub>3</sub> = 40</td></tr><tr><td>1</td><td>0</td><td>0</td><td>0</td><td>30</td><td>30</td><td>30</td></tr><tr><td>2</td><td>-30</td><td>-30</td><td>-30</td><td>0</td><td>0</td><td>0</td></tr><tr><td>3</td><td>0</td><td>0</td><td>0</td><td>-50</td><td>50</td><td>0</td></tr><tr><td>4</td><td>500</td><td>0</td><td>0</td><td>-500</td><td>0</td><td>0</td></tr><tr><td colspan="7">s<sub>1</sub> = s<sub>2</sub> = s<sub>3</sub> = 45</td></tr><tr><td>5</td><td>-5</td><td>-15</td><td>-5</td><td>-10</td><td>20</td><td>-12</td></tr><tr><td>6</td><td>-10</td><td>-10</td><td>-10</td><td>-10</td><td>10</td><td>10</td></tr><tr><td>7</td><td>0</td><td>0</td><td>0</td><td>0</td><td>80</td><td>0</td></tr><tr><td>8</td><td>80</td><td>80</td><td>0</td><td>0</td><td>0</td><td>0</td></tr><tr><td colspan="7">s<sub>1</sub> = s<sub>2</sub> = s<sub>3</sub> = 50</td></tr><tr><td>9</td><td>-40</td><td>-40</td><td>-40</td><td>0</td><td>0</td><td>0</td></tr><tr><td>10</td><td>40</td><td>40</td><td>40</td><td>0</td><td>0</td><td>0</td></tr><tr><td>11</td><td>0</td><td>0</td><td>0</td><td>20</td><td>10</td><td>30</td></tr><tr><td>12</td><td>0</td><td>0</td><td>0</td><td>20</td><td>10</td><td>30</td></tr></tbody></table>

<table><caption>Table A10. Linear and angular instantaneous velocities of the moving platform center.</caption><thead><tr><th></th><th>V<sub>PU0</sub></th><th>V<sub>PV0</sub></th><th>V<sub>PW0</sub></th><th>ω<sub>PU0</sub></th><th>ω<sub>PV0</sub></th><th>ω<sub>PW0</sub></th></tr></thead><tbody><tr><td colspan="7">s<sub>1</sub> = s<sub>2</sub> = s<sub>3</sub> = 40</td></tr><tr><td>1</td><td>0</td><td>0</td><td>1119.4245</td><td>0</td><td>0</td><td>0</td></tr><tr><td>2</td><td>0</td><td>0</td><td>20.5656</td><td>0</td><td>0</td><td>0</td></tr><tr><td>3</td><td>0</td><td>0</td><td>0</td><td>32.315</td><td>55.9712</td><td>0</td></tr><tr><td>4</td><td>0</td><td>0</td><td>0</td><td>3.9578</td><td>6.8552</td><td>0</td></tr><tr><td colspan="7">s<sub>1</sub> = s<sub>2</sub> = s<sub>3</sub> = 45</td></tr><tr><td>5</td><td>92.6927</td><td>-69.1369</td><td>113.9910</td><td>2.7297</td><td>-24.8426</td><td>1.8560</td></tr><tr><td>6</td><td>-67.026</td><td>-55.2530</td><td>246.0229</td><td>-15.5704</td><td>-5.6285</td><td>-1.3893</td></tr><tr><td>7</td><td>202.2327</td><td>-238.1622</td><td>1376.9457</td><td>48.9479</td><td>28.1426</td><td>-1.9494</td></tr><tr><td>8</td><td>-4.4166</td><td>1.5809</td><td>37.8293</td><td>0.8433</td><td>0.0018</td><td>0.0705</td></tr><tr><td colspan="7">s<sub>1</sub> = s<sub>2</sub> = s<sub>3</sub> = 50</td></tr><tr><td>9</td><td>0.0475</td><td>1.1907</td><td>23.3358</td><td>0.1215</td><td>0.1423</td><td>0.0088</td></tr><tr><td>10</td><td>-0.0475</td><td>-1.1907</td><td>-23.3358</td><td>-0.1215</td><td>-0.1423</td><td>-0.0088</td></tr><tr><td>11</td><td>-87.0485</td><td>-3.1741</td><td>1134.9011</td><td>-11.0659</td><td>5.2406</td><td>1.7211</td></tr><tr><td>12</td><td>123.0215</td><td>-104.3032</td><td>1003.4725</td><td>-21.9508</td><td>-6.7957</td><td>-1.8615</td></tr></tbody></table>
<table><caption>Table A11. Specified positions and orientations of the moving platform.</caption><thead><tr><th></th><th>ψ</th><th>θ</th><th>Z<sub>P</sub></th><th>φ</th><th>X<sub>P</sub></th><th>Y<sub>P</sub></th></tr></thead><tbody><tr><td colspan="7" style="text-align: center;">s<sub>1</sub> = s<sub>2</sub> = s<sub>3</sub> = 40</td></tr><tr><td>1</td><td>-0.1047</td><td>0.1570</td><td>100</td><td>-0.0082</td><td>-0.1674</td><td>0.4101</td></tr><tr><td>2</td><td>0.1047</td><td>-0.1745</td><td>105</td><td>-0.0091</td><td>-0.2386</td><td>0.4559</td></tr><tr><td>3</td><td>-0.2268</td><td>-0.2094</td><td>100</td><td>0.0239</td><td>0.1224</td><td>-1.1666</td></tr><tr><td>4</td><td>-0.1047</td><td>0.1745</td><td></td><td>-0.0091</td><td>-0.2386</td><td>0.4559</td></tr><tr><td colspan="7" style="text-align: center;">s<sub>1</sub> = s<sub>2</sub> = s<sub>3</sub> = 50</td></tr><tr><td>5</td><td>-0.2617</td><td>0.2617</td><td>110</td><td>-0.0346</td><td>0.0580</td><td>1.6736</td></tr><tr><td>6</td><td>0.1396</td><td>0.2443</td><td>110</td><td>0.0171</td><td>-0.4847</td><td>-0.8501</td></tr><tr><td>7</td><td>-0.1396</td><td>0.0872</td><td>110</td><td>-0.0061</td><td>0.1500</td><td>0.3023</td></tr><tr><td>8</td><td>-0.1745</td><td>0.1745</td><td>110</td><td>-0.0153</td><td>0.0115</td><td>0.7537</td></tr><tr><td colspan="7" style="text-align: center;">s<sub>1</sub> = s<sub>2</sub> = s<sub>3</sub> = 60</td></tr><tr><td>9</td><td>0.1221</td><td>0.1570</td><td>85</td><td>0.0096</td><td>-0.1168</td><td>-0.4777</td></tr><tr><td>10</td><td>0.1221</td><td>0.1570</td><td>85</td><td>0.0096</td><td>-0.1168</td><td>-0.4777</td></tr><tr><td>11</td><td>-0.1221</td><td>-0.1570</td><td>90</td><td>0.0096</td><td>-0.1168</td><td>-0.4777</td></tr><tr><td>12</td><td>-0.1396</td><td>0.1221</td><td>95</td><td>-0.0085</td><td>0.0605</td><td>0.4235</td></tr></tbody></table>

<table><caption>Table A12. Components of the instantaneous velocities of the moving platform determined based on the direct and inverse velocity problems.</caption><thead><tr><th></th><th colspan="2">V<sub>P<sub>W<sub>0</sub></sub></sub></th><th colspan="2">ω<sub>P<sub>U<sub>0</sub></sub></sub></th><th colspan="2">ω<sub>P<sub>V<sub>0</sub></sub></sub></th></tr></thead><tbody><tr><td colspan="7" style="text-align: center;">s<sub>1</sub> = s<sub>2</sub> = s<sub>3</sub> = 40</td></tr><tr><td>1</td><td>5</td><td>4.8630</td><td>20</td><td>19.9872</td><td>20</td><td>20.0209</td></tr><tr><td>2</td><td>-50</td><td>-49.9721</td><td>-3</td><td>-3.0037</td><td>-2</td><td>-1.9974</td></tr><tr><td>3</td><td>46</td><td>46.0010</td><td>-0.5</td><td>-0.5007</td><td>-0.7</td><td>-0.6988</td></tr><tr><td>4</td><td>100</td><td>99.9999</td><td>0</td><td>0.0000</td><td>0</td><td>0.0000</td></tr><tr><td colspan="7" style="text-align: center;">s<sub>1</sub> = s<sub>2</sub> = s<sub>3</sub> = 50</td></tr><tr><td>5</td><td>72</td><td>71.6454</td><td>2</td><td>2.0024</td><td>3</td><td>3.0384</td></tr><tr><td>6</td><td>4</td><td>3.9408</td><td>12</td><td>11.9867</td><td>15</td><td>15.0137</td></tr><tr><td>7</td><td>0</td><td>-0.0030</td><td>2</td><td>2.0005</td><td>2</td><td>2.0017</td></tr><tr><td>8</td><td>-5</td><td>-4.8723</td><td>-5</td><td>-4.9987</td><td>-5</td><td>-5.0173</td></tr><tr><td colspan="7" style="text-align: center;">s<sub>1</sub> = s<sub>2</sub> = s<sub>3</sub> = 60</td></tr><tr><td>9</td><td>10</td><td>9.9689</td><td>10</td><td>9.9915</td><td>10</td><td>10.0099</td></tr><tr><td>10</td><td>10</td><td>9.9689</td><td>10</td><td>9.9915</td><td>10</td><td>10.0099</td></tr><tr><td>11</td><td>20</td><td>20.0024</td><td>0.5</td><td>0.5005</td><td>0.5</td><td>0.4998</td></tr><tr><td>12</td><td>-30</td><td>-30.0340</td><td>2</td><td>2.0006</td><td>3</td><td>3.0066</td></tr></tbody></table>
<table><caption>Table A13. Input parameter velocities obtained from the solution of the inverse velocity problem.</caption><thead><tr><th></th><th>˙s<sub>1</sub></th><th>˙s<sub>2</sub></th><th>˙s<sub>3</sub></th><th>θ̇<sub>2,1</sub></th><th>θ̇<sub>2,2</sub></th><th>θ̇<sub>2,3</sub></th></tr></thead><tbody><tr><td colspan="7" style="text-align: center;">s<sub>1</sub> = s<sub>2</sub> = s<sub>3</sub> = 40</td></tr><tr><td>1</td><td>0</td><td>0</td><td>0</td><td>-14.0396</td><td>23.2052</td><td>-7.8316</td></tr><tr><td>2</td><td>-132.4230</td><td>411.5116</td><td>-16.5129</td><td>0</td><td>0</td><td>0</td></tr><tr><td>3</td><td>0</td><td>0</td><td>0</td><td>1.4787</td><td>0.3922</td><td>0.8112</td></tr><tr><td>4</td><td>-145.6813</td><td>-165.6036</td><td>-180.0590</td><td>0</td><td>0</td><td>0</td></tr><tr><td colspan="7" style="text-align: center;">s<sub>1</sub> = s<sub>2</sub> = s<sub>3</sub> = 50</td></tr><tr><td>5</td><td>0</td><td>0</td><td>0</td><td>-0.2430</td><td>3.3399</td><td>1.0959</td></tr><tr><td>6</td><td>587.7873</td><td>-1876.6572</td><td>338.8137</td><td>0</td><td>0</td><td>0</td></tr><tr><td>7</td><td>0</td><td>0</td><td>0</td><td>-1.3726</td><td>1.9778</td><td>-0.7436</td></tr><tr><td>8</td><td>-216.8399</td><td>436.5792</td><td>-198.8104</td><td>0</td><td>0</td><td>0</td></tr><tr><td colspan="7" style="text-align: center;">s<sub>1</sub> = s<sub>2</sub> = s<sub>3</sub> = 60</td></tr><tr><td>9</td><td>0</td><td>0</td><td>0</td><td>-4.9403</td><td>12.1003</td><td>-3.4742</td></tr><tr><td>10</td><td>237.1875</td><td>-758.1083</td><td>188.7878</td><td>0</td><td>0</td><td>0</td></tr><tr><td>11</td><td>0</td><td>0</td><td>0</td><td>-0.1264</td><td>0.5508</td><td>0.1994</td></tr><tr><td>12</td><td>109.9661</td><td>-90.5956</td><td>48.5400</td><td>0</td><td>0</td><td>0</td></tr></tbody></table>

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