3.1. Inverse kinematics of the relative motion

Before starting the analysis, we introduce three reference frames (Fig. 2). The first frame, $Oxyz$ , is attached to the base. Its $Oz$ axis coincides with the axis of the circular rail and points upward. The $Ox$ axis points to the left branch of the upper module. The second frame, $O_1x_1y_1z_1$ , is attached to the moving platform of the upper module. Its axes are parallel to the corresponding axes of the $Oxyz$ frame when the platform is horizontal. The third frame, $O_2x_2y_2z_2$ , is attached to the moving platform of the lower module. Its $O_2x_2$ and $O_2y_2$ axes are parallel to the corresponding axes of the $Oxyz$ frame when the platform is not rotated and not tilted. The $O_2z_2$ axis passes through point $F$ , where the axes of the central branch joints intersect. The origins of all the reference frames can be selected arbitrarily, ensuring that the conditions stated above are satisfied.

Fig. 2.

Kinematic schemes of the upper (a) and lower (b) modules and their reference frames.

Now, we can specify the relative configuration of the modules using transformation matrix $\mathbf{T}$ , which defines the configuration of frame $O_1x_1y_1z_1$ relative to $O_2x_2y_2z_2$ :

(1) \begin{align} \mathbf{T} = \begin{bmatrix} \mathbf{R} &\quad \mathbf{p} \\[3pt] \mathbf{0}^{\mathrm{T}} &\quad 1 \end{bmatrix}, \end{align}

where $\mathbf{R}$ is a rotation matrix that defines the orientation of frame $O_1x_1y_1z_1$ relative to $O_2x_2y_2z_2$ ; $\mathbf{p}$ is a position vector of point $O_1$ relative to $O_2x_2y_2z_2$ ; $\mathbf{0}$ is a $3 \times 1$ vector of zeros.

The discretization approach of the workspace analysis implies sampling parameters $\mathbf{p}$ and $\mathbf{R}$ . For example, if we determine the translation workspace, we set constant rotation matrix $\mathbf{R}$ and sample the components of vector $\mathbf{p}$ . If we determine the orientation workspace, we set constant position vector $\mathbf{p}$ and sample the components of rotation matrix $\mathbf{R}$ . In this case, it is convenient to parameterize the rotation matrix by three parameters (e.g., roll-pitch-yaw angles [Reference Cao, Huang, Zhou and Ji49] or tilt-and-torsion angles [Reference Bonev and Gosselin50]), simplifying the workspace representation.

Other types of workspaces can be determined by simultaneous discretization of the position and orientation parameters [Reference Zhou, Niu, Liu and Zhang47]. In any case, we assume that position vector $\mathbf{p}$ and rotation matrix $\mathbf{R}$ have some predefined values, for which we compute displacements in the joints and check the constraints. This computation involves solving the inverse kinematic problem for each module. Thus, we should first determine configurations of the modules relative to the base. We can specify these configurations by transformation matrices $\mathbf{T}_1$ and $\mathbf{T}_2$ that define the configuration of frames $O_1x_1y_1z_1$ and $O_2x_2y_2z_2$ relative to $Oxyz$ , respectively (Fig. 2):

(2) \begin{align} \mathbf{T}_1 = \begin{bmatrix} \mathbf{R}_y(\varphi ) &\quad \mathbf{p}_{O_1} \\[3pt] \mathbf{0}^{\mathrm{T}} &\quad 1 \end{bmatrix}, \qquad \mathbf{T}_2 = \begin{bmatrix} \mathbf{I}_3 &\quad l_{OF}\hat {\mathbf z} \\[3pt] \mathbf{0}^{\mathrm{T}} &\quad 1 \end{bmatrix} \begin{bmatrix} \mathbf{R}_z(\alpha ) &\quad \mathbf{0} \\[3pt] \mathbf{0}^{\mathrm{T}} &\quad 1 \end{bmatrix} \begin{bmatrix} \mathbf{R}_x(\beta ) &\quad \mathbf{0} \\[3pt] \mathbf{0}^{\mathrm{T}} &\quad 1 \end{bmatrix} \begin{bmatrix} \mathbf{I}_3 &\quad l_{FO_2}\hat {\mathbf z} \\[3pt] \mathbf{0}^{\mathrm{T}} &\quad 1 \end{bmatrix}, \end{align}

where $\varphi$ is the tilt angle of the upper platform; $\alpha$ and $\beta$ are the pan and tilt angles of the lower platform, respectively; $\mathbf{R}_x(\! * \!)$ , $\mathbf{R}_y(\! * \!)$ , and $\mathbf{R}_z(\! * \!)$ are the rotation matrices about the corresponding axes; $\mathbf{p}_{O_1}$ is a position vector of point $O_1$ in frame $Oxyz$ ; $l_{OF}$ and $l_{FO_2}$ are the distances between the corresponding points; $\mathbf{I}_3$ is a $3 \times 3$ identity matrix; $\hat {\mathbf z} = [0\,0\,1]^{\mathrm{T}}$ .

Matrix $\mathbf{T}_1$ shows that the upper module configuration depends on four parameters: angle $\varphi$ and three components of vector $\mathbf{p}_{O_1}$ . The configuration of the lower module depends on angles $\alpha$ and $\beta$ : the left and right transformation matrices in the expression of $\mathbf{T}_2$ are constant and correspond to the displacements along the $Oz$ and $O_2z_2$ axes (Fig. 2). Thus, these are angles $\alpha$ , $\beta$ , and $\varphi$ and vector $\mathbf{p}_{O_1}$ that should be determined for specified matrix $\mathbf{T}$ . The transformation matrices are related by the following equation:

(3) \begin{align} \mathbf{T}_1 = \mathbf{T}_2 \mathbf{T}. \end{align}

First, we consider the rotational part of Eq. (3). Substituting Eqs. (1) and (2) into this equation and rearranging the terms, we obtain:

(4) \begin{align} \mathbf{R} &= \mathbf{R}_x^{\mathrm{T}}(\beta ) \mathbf{R}_z^{\mathrm{T}}(\alpha ) \mathbf{R}_y(\varphi ) \end{align}

\begin{align*} = \begin{bmatrix} \phantom {-}\cos (\alpha )\cos (\varphi ) &\ \ \phantom {-}\sin (\alpha ) &\ \ \phantom {-}\cos (\alpha )\sin (\varphi ) \\[3pt] -\sin (\alpha )\cos (\beta )\cos (\varphi ) - \sin (\beta )\sin (\varphi ) &\ \ \phantom {-}\cos (\alpha )\cos (\beta ) &\ \ -\sin (\alpha )\cos (\beta )\sin (\varphi ) + \sin (\beta )\cos (\varphi ) \\[3pt] \phantom {-}\sin (\alpha )\sin (\beta )\cos (\varphi ) - \cos (\beta )\sin (\varphi ) &\ \ -\cos (\alpha )\sin (\beta ) &\ \ \phantom {-}\sin (\alpha )\sin (\beta )\sin (\varphi ) + \cos (\beta )\cos (\varphi ) \end{bmatrix}. \end{align*}

We can use Eq. (4) to find angles $\alpha$ , $\beta$ , and $\varphi$ for specified rotation matrix $\mathbf{R}$ . Let $r_{kl}$ be the $(k, l)$ -th element of this matrix. Following the method from ref. [Reference Lynch and Park51, app. B], we compute two sets of solutions for these angles:

(5) \begin{align} \begin{aligned} \alpha &= \textrm {atan2}\Bigl (r_{12}, \sqrt {r_{22}^2 + r_{32}^2}\Bigr ), &&&\alpha &= \textrm {atan2}\Bigl (r_{12}, -\sqrt {r_{22}^2 + r_{32}^2}\Bigr ), \\ \beta &= \textrm {atan2}(\! -r_{32}, r_{22}), &\quad \text{and}\qquad &&\beta &= \textrm {atan2}(r_{32}, -r_{22}), \\ \varphi &= \textrm {atan2}(r_{13}, r_{11}), &&&\varphi &= \textrm {atan2}(\! -r_{13}, -r_{11}), \end{aligned} \end{align}

where $\textrm {atan2}(*,*)$ is the two-argument arctangent function [Reference Lynch and Park51, ch. 6].

We select the solution set for which angle $\beta \in [-90^\circ , 90^\circ ]$ , where the lower platform is not flipped. Note that Eq. (5) holds as long as $\alpha \neq \pm 90^\circ$ , which corresponds to a singular configuration of the robot. In this configuration, the tilt axes of the upper and lower modules become parallel, and the robot loses one rotational DOF in the relative motion; each module, however, does not meet a singularity in this case. At the same time, there is an infinite number of combinations $(\beta , \varphi )$ satisfying the considered relative configuration of the modules.

Angles $\alpha$ and $\beta$ allow us to compute matrix $\mathbf{T}_2$ using Eq. (2). Next, we consider the position part of Eq. (3), from where we find vector $\mathbf{p}_{O_1}$ :

(6) \begin{align} \mathbf{p}_{O_1} = l_{OF}\hat {\mathbf z} + \mathbf{R}_z(\alpha ) \mathbf{R}_x(\beta ) \left (l_{FO_2}\hat {\mathbf z} + \mathbf{p}\right ). \end{align}

Thus, we have determined matrices $\mathbf{T}_1$ and $\mathbf{T}_2$ that specify configurations of the upper and lower modules relative to the base. We can now proceed with the independent analysis of each module and its mechanical constraints.

3.2. Constraints of the upper module

The mechanical constraints of the upper module are due to the limited displacements of its carriages and limited inclination angles of the rods in its spherical joints. Next, we discuss how to compute these displacements and inclination angles.

Let $i = 1, \ldots , 4$ be the branch number of the upper module, and let points $A_i$ and $B_i$ indicate the centers of the spherical joints in the $i$ -th branch (Fig. 2a; for branches with the parallelogram sub-chains, we can select these points in the middle of the parallelogram short sides). Vectors $\mathbf{p}_{A_i}$ and $\mathbf{p}_{B_i}$ define the positions of these points in frame $Oxyz$ :

(7) \begin{align} \mathbf{p}_{A_i} = \begin{bmatrix} \boldsymbol{\mathrm{\rho }}_{A_i} \\[3pt] h_i \end{bmatrix}, \qquad \begin{bmatrix} \mathbf{p}_{B_i} \\[3pt] 1 \end{bmatrix} = \mathbf{T}_1 \begin{bmatrix} \mathbf{r}_{B_i} \\[3pt] 1 \end{bmatrix}, \end{align}

where $h_i$ is the carriage displacement; vector $\boldsymbol{\mathrm{\rho }}_{A_i}$ defines the $x$ and $y$ coordinates of point $A_i$ in frame $Oxyz$ ; vector $\mathbf{r}_{B_i}$ defines the coordinates of point $B_i$ in frame $O_1x_1y_1z_1$ . Parameters $\boldsymbol{\mathrm{\rho }}_{A_i}$ and $\mathbf{r}_{B_i}$ have constant values that depend on the robot geometry.

The distance between points $A_i$ and $B_i$ is constant and equal to rod length $l_{AB}$ , which we assume is identical for each branch. Thus, we can write the following expression:

(8) \begin{align} \|\mathbf{p}_{A_i} - \mathbf{p}_{B_i}\|^2 = l_{AB}^2. \end{align}

Substituting Eq. (7) into Eq. (8), we compute parameter $h_i$ :

(9) \begin{align} h_i = \pm \sqrt {l_{AB}^2 - \left (x_{B_i} - x_{A_i}\right )^2 - \left (y_{B_i} - y_{A_i}\right )^2} + z_{B_i}, \end{align}

where $[x_{A_i}\,y_{A_i}]^{\mathrm{T}} = \boldsymbol{\mathrm{\rho }}_{A_i}$ and $[x_{B_i}\,y_{B_i}\,z_{B_i}]^{\mathrm{T}} = \mathbf{p}_{B_i}$ .

The $\pm$ sign in Eq. (9) corresponds to the carriage position above or below the moving platform. We select the plus sign for each branch according to the robot design. This design imposes the following constraints on the carriage displacement:

(10) \begin{align} h_{\mathrm{min}} \leq h_i \leq h_{\mathrm{max}}, \end{align}

where $h_{\mathrm{min}}$ and $h_{\mathrm{max}}$ are the minimum and maximum values of displacement $h_i$ , which we assume identical for each branch.

Next, we consider the constraints on the rod inclination in the spherical joints. We start with the carriage spherical joint, centered at point $A_i$ . To analyze how this joint constrains the rod motion, we represent the spherical joint as a combination of two revolute joints with intersecting and orthogonal axes (Fig. 3a). We ignore the third rotation of the rod (about line $A_iB_i$ ) because it does not affect the platform motion.

Fig. 3.

Equivalent representation of the spherical joints in the $i$ -th branch of the upper module: (a) spherical joint of the carriage; (b) spherical joint of the moving platform.

Let $\psi _{A_i}$ and $\eta _{A_i}$ be the rotational angles in these revolute joints. We introduce the $A_ix_{A_i}y_{A_i}z_{A_i}$ reference frame, whose axes $A_ix_{A_i}$ and $A_iy_{A_i}$ are parallel to the $Oxy$ plane and coincident with the axes of the revolute joints when $\psi _{A_i} = \eta _{A_i} = 0$ . Constant rotation matrix $\mathbf{R}_{A_i}$ determines the orientation of this frame relative to $Oxyz$ . According to the robot design, we can write:

(11) \begin{align} \mathbf{R}_{A_1} = \mathbf{R}_z(\pi /2), \qquad \mathbf{R}_{A_2} = \mathbf{R}_{A_3} = \mathbf{I}_3, \qquad \mathbf{R}_{A_4} = \mathbf{R}_z(\! -\pi /2). \end{align}

Let $\hat {\mathbf s}_i$ be a unit vector directed from point $B_i$ to point $A_i$ , expressed in the $Oxyz$ frame. Figure 3a shows that this vector depends on angles $\psi _{A_i}$ and $\eta _{A_i}$ by the following equation:

(12) \begin{align} \hat {\mathbf s}_i = \mathbf{R}_{A_i} \mathbf{R}_x(\psi _{A_i}) \mathbf{R}_y(\eta _{A_i}) \hat {\mathbf z}, \end{align}

which we rewrite as below:

(13) \begin{align} \mathbf{R}_{A_i}^{\mathrm{T}} \hat {\mathbf s}_i = \mathbf{R}_x(\psi _{A_i}) \mathbf{R}_y(\eta _{A_i}) \hat {\mathbf z} = \begin{bmatrix} \sin (\eta _{A_i}) \\[2pt] -\sin (\psi _{A_i})\cos (\eta _{A_i}) \\[2pt] \phantom {-}\cos (\psi _{A_i})\cos (\eta _{A_i}) \end{bmatrix}. \end{align}

The left side of Eq. (13) is known because vector $\hat {\mathbf s}_i = (\mathbf{p}_{A_i} - \mathbf{p}_{B_i}) / l_{AB}$ , and it can be computed using Eq. (7). Let $\mathbf{R}_{A_i}^{\mathrm{T}} \hat {\mathbf s}_i = [s_i^x \, s_i^y \, s_i^z]^{\mathrm{T}}$ , then we can find the angles using the following expressions:

(14) \begin{align} \psi _{A_i} = \textrm {atan2}(\! -s_i^y, s_i^z), \qquad \eta _{A_i} = \textrm {asin}(s_i^x), \end{align}

which are valid because $|\eta _{A_i}|$ is usually much less than $90^\circ$ in physical spherical joints.

Angles $\psi _{A_i}$ and $\eta _{A_i}$ are limited by the robot design:

(15) \begin{align} \psi _{A\,\mathrm{min}} \leq \psi _{A_i} \leq \psi _{A\,\mathrm{max}}, \qquad \eta _{A\,\mathrm{min}} \leq \eta _{A_i} \leq \eta _{A\,\mathrm{max}}, \end{align}

where $\psi _{A\,\mathrm{min}}$ , $\eta _{A\,\mathrm{min}}$ , $\psi _{A\,\mathrm{max}}$ , and $\eta _{A\,\mathrm{max}}$ are the minimum and maximum values of angles $\psi _{A_i}$ and $\eta _{A_i}$ , and we assume the minimum and maximum values are identical for each branch.

Constraints in the spherical joints of the moving platform are analyzed in the same way. We represent the spherical joint in the $i$ -th branch as a combination of two revolute joints, whose axes intersect orthogonally at point $B_i$ (Fig. 3b). Let $\psi _{B_i}$ and $\eta _{B_i}$ be the rotational angles in these revolute joints. We define the $B_ix_{B_i}y_{B_i}z_{B_i}$ reference frame, whose axes $B_ix_{B_i}$ and $B_iy_{B_i}$ are parallel to the $O_1x_1y_1$ plane and coincident with the axes of the revolute joints when $\psi _{B_i} = \eta _{B_i} = 0$ . Constant rotation matrix $\mathbf{R}_{B_i}$ determines the orientation of this frame relative to $O_1x_1y_1$ . We place each frame $B_ix_{B_i}y_{B_i}z_{B_i}$ in such a way that $\mathbf{R}_{B_i} = \mathbf{R}_{A_i}$ for each $i = 1, \ldots , 4$ .

To find angles $\psi _{B_i}$ and $\eta _{B_i}$ , we follow the same approach as for the carriage spherical joint. We can compose an equation similar to Eq. (12). Its right side should include rotation matrix $\mathbf{R}_y(\varphi )$ , which corresponds to the tilt of the upper platform. Therefore, we obtain the following expression:

(16) \begin{align} \hat {\mathbf s}_i = \mathbf{R}_y(\varphi ) \mathbf{R}_{B_i} \mathbf{R}_x(\psi _{B_i}) \mathbf{R}_y(\eta _{B_i}) \hat {\mathbf z}, \end{align}

which we rewrite as below:

(17) \begin{align} \mathbf{R}_{B_i}^{\mathrm{T}} \mathbf{R}_y^{\mathrm{T}}(\varphi ) \hat {\mathbf s}_i = \mathbf{R}_x(\psi _{B_i}) \mathbf{R}_y(\eta _{B_i}) \hat {\mathbf z} = \begin{bmatrix} \sin (\eta _{B_i}) \\[2pt] -\sin (\psi _{B_i})\cos (\eta _{B_i}) \\[2pt] \phantom {-}\cos (\psi _{B_i})\cos (\eta _{B_i}) \end{bmatrix}. \end{align}

The right side of Eq. (17) is similar to Eq. (13), and we compute angles $\psi _{B_i}$ and $\eta _{B_i}$ using expressions similar to Eq. (14). These angles are limited by the robot design:

(18) \begin{align} \psi _{B\,\mathrm{min}} \leq \psi _{B_i} \leq \psi _{B\,\mathrm{max}}, \qquad \eta _{B\,\mathrm{min}} \leq \eta _{B_i} \leq \eta _{B\,\mathrm{max}}, \end{align}

where $\psi _{B\,\mathrm{min}}$ , $\eta _{B\,\mathrm{min}}$ , $\psi _{B\,\mathrm{max}}$ , and $\eta _{B\,\mathrm{max}}$ are the minimum and maximum values of angles $\psi _{B_i}$ and $\eta _{B_i}$ , and we assume the minimum and maximum values are identical for each branch.

Thus, we have analyzed all mechanical constraints of the upper module. Before proceeding to the lower module, we should make a couple of remarks about the performed analysis:

1. 1. To analyze the constraints in the spherical joints, we represent each joint as a combination of two revolute joints with intersecting and orthogonal axes. This representation is not unique. For example, in Fig. 3a, we could assume that the axis of the second revolute joint coincides with the $A_iz_{A_i}$ axis when $\psi _{A_i} = 0$ , instead of the $A_iy_{A_i}$ axis. In this case, multiplier $\mathbf{R}_y(\eta _{A_i}) \hat {\mathbf z}$ in the right side of Eq. (12) should be replaced with $-\mathbf{R}_z(\eta _{A_i}) \hat {\mathbf y}$ . Equations (13) and (14) change too, as well as limits $\psi _{A\,\mathrm{min}}$ and $\psi _{A\,\mathrm{max}}$ in Eq. (15).

2. 2. The branches with the parallelogram sub-chains include two rods ending with the spherical joints. Each rod in such a parallelogram sub-chain inclines by the same angles. Therefore, angles $\psi _{A_i}$ , $\eta _{A_i}$ , $\psi _{B_i}$ , and $\eta _{B_i}$ in these branches are computed correctly with our assumption that points $A_i$ are $B_i$ are the middle points of the short sides of the parallelograms.

3.3. Constraints of the lower module

The mechanical constraints of the lower module include the limited rotational angles in its revolute joints and the possible collision of its carriages. Next, we show how to find these angles and check the collision.

We start with the central branch that has two revolute joints (Fig. 2b). Angle $\alpha$ in the first joint is not constrained: the lower moving platform can rotate unlimitedly about the axis of the circular rail. On the other hand, angle $\beta$ is limited by the robot design:

(19) \begin{align} \beta _{\mathrm{min}} \leq \beta \leq \beta _{\mathrm{max}}, \end{align}

where $\beta _{\mathrm{min}}$ and $\beta _{\mathrm{max}}$ are the minimum and maximum values of angle $\beta$ .

Next, we consider the lateral branches of the lower module. Let $j = 1, 2$ be the branch number. For the $j$ -th branch, we define three points, $C_j$ , $D_j$ , and $E_j$ , as follows. Point $E_j$ is the intersection of the axes of the two revolute joints near the moving platform. Points $C_j$ and $D_j$ stay in the vertical plane passing through point $E_j$ and belong to the axes of two other revolute joints, as shown in Figs. 2b and 4.

Vectors $\mathbf{p}_{C_j}$ and $\mathbf{p}_{E_j}$ define the positions of points $C_j$ and $E_j$ in frame $Oxyz$ :

(20) \begin{align} \mathbf{p}_{C_j} = \begin{bmatrix} R_C\cos (\theta _j) \\[3pt] R_C\sin (\theta _j) \\[3pt] z_C \end{bmatrix}, \qquad \begin{bmatrix} \mathbf{p}_{E_j} \\[3pt] 1 \end{bmatrix} = \mathbf{T}_2 \begin{bmatrix} \mathbf{r}_{E_j} \\[3pt] 1 \end{bmatrix}, \end{align}

where $\theta _j$ is the carriage angular displacement; parameters $R_C$ and $z_C$ define the radial and vertical position of point $C_j$ in frame $Oxyz$ , and we assume these parameters are identical for each branch; vector $\mathbf{r}_{E_j}$ defines the coordinates of point $E_j$ in frame $O_2x_2y_2z_2$ . Parameters $R_C$ , $z_C$ , and $\mathbf{r}_{E_j}$ have constant values that depend on the robot geometry.

Fig. 4.

Rotational angles in the joints of the $j$ -th lateral branch of the lower module: (a) location of frame $C_jx_{C_j}y_{C_j}z_{C_j}$ ; (b) planar view of the $j$ -th branch.

Points $C_j$ and $E_j$ belong to the same vertical plane; therefore, we can compute angle $\theta _j$ as follows:

(21) \begin{align} \theta _j = \textrm {atan2}(y_{E_j}, x_{E_j}), \end{align}

where $x_{E_j}$ and $y_{E_j}$ are the first two components of vector $\mathbf{p}_{E_j}$ .

To avoid the collision of the carriages, we should limit angles $\theta _j$ . Let $\theta _{\mathrm{min}} \gt 0$ be the minimum angular distance between the carriages, which is defined by the robot design. Since the image of the $\textrm {atan2}$ function in Eq. (21) is $[\! -\pi , \pi ]$ , the following inequalities ensure that the carriages do not collide:

(22) \begin{align} \theta _{\mathrm{min}} \leq |\theta _1 - \theta _2| \leq 2\pi - \theta _{\mathrm{min}}. \end{align}

Next, we analyze the constraints in two revolute joints, whose axes pass through points $C_j$ and $D_j$ . Let $\theta _{C_j}$ and $\theta _{D_j}$ be the rotational angles in these joints (Fig. 4a). To find these angles, we first introduce reference frame $C_jx_{C_j}y_{C_j}z_{C_j}$ attached to the $j$ -th carriage. Its axis $C_jy_{C_j}$ is parallel to the $Oz$ axis, and axis $C_jx_{C_j}$ is opposite to the $Ox$ axis when $\theta _j = 0$ . Rotation matrix $\mathbf{R}_{C_j}$ defines the orientation of this frame relative to $Oxyz$ and can be written as follows:

(23) \begin{align} \mathbf{R}_{C_j} = \begin{bmatrix} -\mathbf{R}_z(\theta _j)\hat {\mathbf x} &\quad \hat {\mathbf z} &\quad \hat {\mathbf z} \times \left (\mathbf{R}_z(\theta _j)\hat {\mathbf x}\right ) \end{bmatrix} = \begin{bmatrix} -\cos (\theta _j) &\quad 0 &\quad -\sin (\theta _j) \\[3pt] -\sin (\theta _j) &\quad 0 &\quad \phantom {-}\cos (\theta _j) \\[3pt] 0 &\quad 1 &\quad 0\end{bmatrix}, \end{align}

and transformation matrix $\mathbf{T}_{C_j}$ between frames $C_jx_{C_j}y_{C_j}z_{C_j}$ and $Oxyz$ has the following form:

(24) \begin{align} \mathbf{T}_{C_j} = \begin{bmatrix} \mathbf{R}_{C_j} &\quad \mathbf{p}_{C_j} \\[3pt] \mathbf{0}^{\mathrm{T}} &\quad 1 \end{bmatrix}. \end{align}

Matrix $\mathbf{T}_{C_j}$ depends only on angle $\theta _j$ , which we computed in Eq. (21). Now, we can consider the 2-DOF planar mechanism located in plane $C_jx_{C_j}y_{C_j}$ (Fig. 4b). Let $\boldsymbol{\mathrm{\rho }}_{E_j} = [x_j\,y_j]^{\mathrm{T}}$ be the position vector of point $E_j$ in frame $C_jx_{C_j}y_{C_j}$ . This vector is related to vector $\mathbf{p}_{E_j}$ by transformation matrix $\mathbf{T}_{C_j}$ :

(25) \begin{align} \begin{bmatrix} \mathbf{p}_{E_j} \\[3pt] 1 \end{bmatrix} = \mathbf{T}_{C_j} \begin{bmatrix} \boldsymbol{\mathrm{\rho }}_{E_j} \\[3pt] 0 \\[3pt] 1 \end{bmatrix}, \end{align}

from where we determine vector $\boldsymbol{\mathrm{\rho }}_{E_j}$ :

(26) \begin{align} \begin{bmatrix} \boldsymbol{\mathrm{\rho }}_{E_j} \\[3pt] 0 \\[3pt] 1 \end{bmatrix} = \mathbf{T}_{C_j}^{-1} \begin{bmatrix} \mathbf{p}_{E_j} \\[3pt] 1 \end{bmatrix} = \begin{bmatrix} \mathbf{R}_{C_j}^{\mathrm{T}} & -\mathbf{R}_{C_j}^{\mathrm{T}} \mathbf{p}_{C_j} \\[3pt] \mathbf{0}^{\mathrm{T}} & 1 \end{bmatrix} \begin{bmatrix} \mathbf{p}_{E_j} \\[3pt] 1 \end{bmatrix}. \end{align}

To compute angles $\theta _{C_j}$ and $\theta _{D_j}$ , we follow the approach from [Reference Lynch and Park51, sect. 6] and introduce three auxiliary angles, $\sigma _j$ , $\xi _{1j}$ , and $\xi _{2j}$ (Fig. 4b), which can be found from the following expressions:

(27) \begin{align} \sigma _j = \textrm {atan2}(y_j, x_j), \;\; \xi _{1j} = \textrm {acos}\left (\frac {\|\boldsymbol{\mathrm{\rho }}_{E_j}\|^2 + l_{CD}^2 - l_{DE}^2}{2\|\boldsymbol{\mathrm{\rho }}_{E_j}\|l_{CD}}\right ), \;\; \xi _{2j} = \textrm {acos}\left (\frac {l_{CD}^2 + l_{DE}^2 - \|\boldsymbol{\mathrm{\rho }}_{E_j}\|^2}{2l_{CD}l_{DE}}\right ), \end{align}

where $l_{CD}$ and $l_{DE}$ are the lengths of links $C_jD_j$ and $D_jE_j$ , respectively, which we assume identical for each branch.

Using these angles, we determine two sets of solutions for angles $\theta _{C_j}$ and $\theta _{D_j}$ :

(28) \begin{align} \begin{aligned} \theta _{C_j} &= \sigma _j - \xi _{1j}, \\ \theta _{D_j} &= \pi - \xi _{2j}, \end{aligned}\qquad \text{and}\qquad \begin{aligned} \theta _{C_j} &= \sigma _j + \xi _{1j}, \\ \theta _{D_j} &= \xi _{2j} - \pi . \end{aligned} \end{align}

The first solution set matches the “elbow-down” case when $\theta _{D_j} \gt 0$ ; the second solution set matches the “elbow-up” case when $\theta _{D_j} \lt 0$ . We choose the second set according to the lower module design (Fig. 1). This design imposes the following constraints on angles $\theta _{C_j}$ and $\theta _{D_j}$ :

(29) \begin{align} \theta _{C\,\mathrm{min}} \leq \theta _{C_j} \leq \theta _{C\,\mathrm{max}}, \qquad \theta _{D\,\mathrm{min}} \leq \theta _{D_j} \leq \theta _{D\,\mathrm{max}}, \end{align}

where $\theta _{C\,\mathrm{min}}$ , $\theta _{D\,\mathrm{min}}$ , $\theta _{C\,\mathrm{max}}$ , and $\theta _{D\,\mathrm{max}}$ are the minimum and maximum values of angles $\theta _{C_j}$ and $\theta _{D_j}$ , and we assume the minimum and maximum values are identical for each branch.

Rotation in the revolute joint between the coupler and the moving platform is not constrained according to the robot design. Thus, it remains to check the rotational angle $\theta _{E_j}$ in the revolute joint whose axis passes through point $E_j$ . We measure this angle as the angle between two unit vectors, $\hat {\mathbf u}_j$ and $\hat {\mathbf v}_j$ , directed along line $D_jE_j$ and the axis of the platform revolute joint, respectively (Fig. 5). We compute these unit vectors using the following expressions:

(30) \begin{align} \hat {\mathbf u}_j = \frac {\mathbf{p}_{E_j} - \mathbf{p}_{D_j}}{l_{DE}}, \qquad \begin{bmatrix} \hat {\mathbf v}_j \\[3pt] 0 \end{bmatrix} = \mathbf{T}_2 \begin{bmatrix} \hat {\mathbf e}_j \\[3pt] 0 \end{bmatrix}, \end{align}

where $\mathbf{p}_{D_j}$ is the position vector of point $D_j$ in frame $Oxyz$ ; $\hat {\mathbf e}_j$ is a unit vector along the axis of the platform revolute joint, expressed in frame $O_2x_2y_2z_2$ . Vector $\hat {\mathbf e}_j$ is constant and defined by the geometry of the lower module. Vector $\mathbf{p}_{D_j}$ can be computed similarly to Eq. (25), where we replace vector $\boldsymbol{\mathrm{\rho }}_{E_j}$ with vector $\boldsymbol{\mathrm{\rho }}_{D_j} = [l_{CD}\cos (\theta _{C_j})\;\;l_{CD}\sin (\theta _{C_j})]^{\mathrm{T}}$ .

Fig. 5.

Measuring angle $\theta _{E_j}$ : (a) vectors $\mathbf{k}_j$ and $\hat {\mathbf t}_j$ have the same directions; (a) vectors $\mathbf{k}_j$ and $\hat {\mathbf t}_j$ have opposite directions.

To set the constraints on angle $\theta _{E_j}$ correctly, we must ensure that it is measured in the same sense, disregarding the directions of vectors $\hat {\mathbf u}_j$ and $\hat {\mathbf v}_j$ . For this purpose, we introduce two auxiliary vectors. The first one is $\mathbf{k}_j = \hat {\mathbf u}_j \times \hat {\mathbf v}_j$ , which is orthogonal to both vectors $\hat {\mathbf u}_j$ and $\hat {\mathbf v}_j$ . The second auxiliary vector is $\hat {\mathbf t}_j$ parallel to the $C_jz_j$ axis and equal to the third column of rotation matrix $\mathbf{R}_{C_j}$ in Eq. (23) (Fig. 5). Using these vectors, we can compute angle $\theta _{E_j}$ as follows:

(31) \begin{align} \theta _{E_j} = \begin{cases} \textrm {acos}(\hat {\mathbf u}_j \cdot \hat {\mathbf v}_j), & \text{if}\quad \mathbf{k}_j \cdot \hat {\mathbf t}_j \geq 0, \\[3pt] 2\pi - \textrm {acos}(\hat {\mathbf u}_j \cdot \hat {\mathbf v}_j), & \text{if}\quad \mathbf{k}_j \cdot \hat {\mathbf t}_j \lt 0. \end{cases} \end{align}

In the first case, vectors $\mathbf{k}_j$ and $\hat {\mathbf t}_j$ have the same directions, and angle $\theta _{E_j} \in [0^\circ , 180^\circ ]$ (Fig. 5a). In the second case, vectors $\mathbf{k}_j$ and $\hat {\mathbf t}_j$ have opposite directions, and angle $\theta _{E_j} \in (180^\circ , 360^\circ )$ (Fig. 5b). We can now define the constraints on angle $\theta _{E_j}$ :

(32) \begin{align} \theta _{E\,\mathrm{min}} \leq \theta _{E_j} \leq \theta _{E\,\mathrm{max}}, \end{align}

where $\theta _{E\,\mathrm{min}}$ and $\theta _{E\,\mathrm{max}}$ are the minimum and maximum values of angle $\theta _{E_j}$ , and we assume the minimum and maximum values are identical for each branch. This concludes the analysis of the mechanical constraints of the lower module.

3.4. Interference between the upper and lower modules

Satisfying the mechanical constraints considered in Sections 3.2 and 3.3 does not guarantee that the upper and lower modules will not interfere with each other. This interference includes collisions between the links of the modules. It affects the workspace of the hybrid robot and should be checked for correct workspace evaluation.

The collision detection problem of the robot links relates directly to analyzing collisions between the rigid bodies representing these links. There are various techniques developed to solve this problem [Reference Jiménez, Thomas and Torras52, Reference Kockara, Halic, Iqbal, Bayrak and Rowe53]. In this paper, we apply the familiar Gilbert–Johnson–Keerthi (GJK) algorithm [Reference Gilbert, Johnson and Keerthi54]. This computationally efficient algorithm allows us to find the distance between convex bodies and, therefore, to check whether they collide or not. The major idea of the GJK algorithm relies on computing the Minkowski difference of two convex subsets of the Euclidean space and evaluating support functions over this difference – we omit the mathematical foundations here, which can be found in the original paper and elsewhere [Reference Van den Bergen55–Reference Montaut, Lidec, Petrik, Sivic and Carpentier57]. Next, we show how to adapt this algorithm to the considered hybrid robot.

According to the robot design (Fig. 1), its upper moving platform should operate above the lower moving platform. Therefore, we consider the following collisions, which are possible between the links of the modules (Fig. 6a):

1. 1. Collision between the upper platform and the lower platform.

2. 2. Collision between the upper platform and the couplers of the lower module.

3. 3. Collision between the upper platform and the upper bars of the lower module.

Fig. 6.

Collision analysis of the upper and lower modules: (a) convex bodies enveloping the links that can collide; (b) parallelepiped around the upper platform; (c) cylinder around the lower platform; (d) cylinder around the $j$ -th coupler; (e) parallelepiped around the $j$ -th upper bar.

In this paper, we follow the original GJK algorithm [Reference Gilbert, Johnson and Keerthi54], where the analyzed rigid bodies are not arbitrary convex objects but convex polytopes. Thus, we should represent each link mentioned above as a convex polytope. The following sections describe this representation.

3.4.1. Upper platform

For the upper platform, we define a rectangular parallelepiped that envelops the moving platform and its spherical joints (Fig. 6b). The parallelepiped is determined by the coordinates of its vertices. Let $M_0$ be the parallelepiped center and $M_k$ , $k = 1, \ldots , 8$ , be its vertices, where $M_1, \ldots , M_4$ and $M_5, \ldots , M_8$ correspond to the top and bottom parallelepiped faces, respectively. We can write coordinates $\mathbf{r}_{M_k}$ of these points in frame $O_1x_1y_1z_1$ :

(33) \begin{align} \mathbf{r}_{M_k} = \mathbf{r}_{M_0} + \frac {1}{2}\begin{bmatrix} \varepsilon _k L \\ \delta _k W \\ \zeta _k H \end{bmatrix}, \end{align}

\begin{align*} \varepsilon _k = \begin{cases} \phantom {-}1, & \text{if}\quad k = 1, 2, 5, 6, \\ -1, & \text{if}\quad k = 3, 4, 7, 8, \end{cases}\quad \delta _k = \begin{cases} \phantom {-}1, & \text{if}\quad k = 1, 4, 5, 8, \\ -1, & \text{if}\quad k = 2, 3, 6, 7, \end{cases}\quad \zeta _k = \begin{cases} \phantom {-}1, & \text{if}\quad k = 1, 2, 3, 4, \\ -1, & \text{if}\quad k = 5, 6, 7, 8, \end{cases} \end{align*}

where $\mathbf{r}_{M_0}$ is the position vector of point $M_0$ in frame $O_1x_1y_1z_1$ ; $L$ , $W$ , and $H$ are the length, width, and height of the parallelepiped. All these parameters have constant values depending on the robot design.

3.4.2. Lower platform

We represent the lower moving platform as a right circular cylinder with radius $R$ and height $S$ (Fig. 6c). To convert the cylinder into a convex polytope, we sample each circular boundary into $N$ points. Let $P_0$ be the center of the top cylinder face and $P_k$ , $k = 1, \ldots , 2N$ , be the sample points, where $P_1, \ldots , P_N$ and $P_{N + 1}, \ldots , P_{2N}$ correspond to the top and bottom cylinder faces, respectively. We can write coordinates $\mathbf{r}_{P_k}$ of these points in frame $O_2x_2y_2z_2$ :

(34) \begin{align} \mathbf{r}_{P_k} = \begin{cases} \mathbf{r}_{P_0} + \begin{bmatrix} R \cos (2\pi (k - 1) / N) \\ R \sin (2\pi (k - 1) / N) \\ 0 \end{bmatrix}, & \text{if}\quad k = 1, \ldots , N, \\ \mathbf{r}_{P_{k - N}} - S\hat {\mathbf z}, & \text{if}\quad k = N + 1, \ldots , 2N, \end{cases} \end{align}

where $\mathbf{r}_{P_0}$ is the position vector of point $P_0$ in frame $O_2x_2y_2z_2$ . Parameters $R$ , $S$ , and $\mathbf{r}_{P_0}$ depend on the platform design and have known and constant values.

3.4.3. Couplers

We also use a right circular cylinder to represent each coupler (Fig. 6d). According to the robot design and Section 3.3, the axis of the $j$ -th cylinder, $j = 1, 2$ , passes through point $E_j$ with coordinates $\mathbf{r}_{E_j}$ , and it is directed along vector $\hat {\mathbf e}_j$ (in frame $O_2x_2y_2z_2$ ). Let $r$ be the cylinder radius and $s_1$ and $s_2$ be the distances from point $E_j$ to the top and bottom cylinder faces, respectively. Because the cylinder is symmetrical above its axis, we can ignore the coupler rotation relative to the moving platform. Therefore, parameters $\mathbf{r}_{E_j}$ , $\hat {\mathbf e}_j$ , $r$ , $s_1$ , and $s_2$ fully define the cylinder geometry in frame $O_2x_2y_2z_2$ . All these parameters have known values depending on the robot design.

To convert the cylinder into a convex polytope, we follow the same approach and sample each circular boundary into $n$ points. Let $U_{j, k}$ , $k = 1, \ldots , 2n$ , be the sample points, where $U_{j, 1}, \ldots , U_{j, n}$ and $U_{j, n + 1}, \ldots , U_{j, 2n}$ correspond to the top and bottom faces of the $j$ -th cylinder, respectively. These points have coordinates $\mathbf{r}_{U_{j, k}}$ in frame $O_2x_2y_2z_2$ :

(35) \begin{align} \mathbf{r}_{U_{j, k}} = \begin{cases} \mathbf{r}_{E_j} + s_1\hat {\mathbf e}_j + r (\! \cos (2\pi (k - 1) / n)\hat {\mathbf a}_j + \sin (2\pi (k - 1) / n)\hat {\mathbf b}_j), & \text{if}\quad k = 1, \ldots , n, \\[3pt] \mathbf{r}_{U_{j, k - n}} - (s_1 + s_2)\hat {\mathbf e}_j, & \text{if}\quad k = n + 1, \ldots , 2n, \end{cases} \end{align}

where $\hat {\mathbf a}_j$ and $\hat {\mathbf b}_j$ are two orthonormal vectors orthogonal to vector $\hat {\mathbf e}_j$ , which can be found as follows:

(36) \begin{align} \textrm {span}\big (\{\hat {\mathbf a}_j, \hat {\mathbf b}_j\}\big ) = \textrm {null}\big (\hat {\mathbf e}_j^{\mathrm{T}}\big ). \end{align}

Equations (33)–(35) are sufficient to analyze the collisions between the upper platform, the lower platform, and the couplers. For this purpose, we just need to transform coordinates $\mathbf{r}_{M_k}$ from frame $O_1x_1y_1z_1$ to frame $O_2x_2y_2z_2$ using transformation matrix $\mathbf{T}$ . Thus, these collisions are analyzed in frame $O_2x_2y_2z_2$ with no need to transform coordinates $\mathbf{r}_{P_k}$ and $\mathbf{r}_{U_{j, k}}$ .

3.4.4. Upper bars

Finally, we use a rectangular parallelepiped to represent the geometry of each upper bar (Fig. 6e). Let $V_{j, k}$ , $k = 1, \ldots , 8$ , be its vertices, where $V_{j, 1}, \ldots , V_{j, 4}$ and $V_{j, 5}, \ldots , V_{j, 8}$ correspond to the top and bottom faces of the $j$ -th parallelepiped, respectively ( $j = 1, 2$ ). To find the coordinates of these vertices, we first introduce reference frame $D_jx_{D_j}y_{D_j}z_{D_j}$ attached to the $j$ -th upper bar. Its $D_jx_{D_j}$ axis coincides with line $D_jE_j$ , and the $D_jz_{D_j}$ axis has the same direction as axis $C_jz_{C_j}$ . We can write coordinates $\mathbf{r}_{V_{j, k}}$ of the parallelepiped vertices in frame $D_jx_{D_j}y_{D_j}z_{D_j}$ :

(37) \begin{align} \mathbf{r}_{V_{j, k}} = \begin{cases} \begin{bmatrix} -d \\ \zeta _k h/2 \\ \varepsilon _k w/2 \end{bmatrix}, & \text{if}\quad k = 1, 4, 5, 8, \\ \mathbf{r}_{V_{j, k - 4}} + l \hat {\mathbf x}, & \text{if}\quad k = 2, 3, 6, 7, \\ \end{cases} \end{align}

where $l$ , $w$ , and $h$ are the length, width, and height of the parallelepiped; $d$ is a distance from point $D_j$ to the left face of the parallelepiped; $\zeta _k$ and $\varepsilon _k$ have the same meaning as in Eq. (33). All these parameters have constant values depending on the robot design.

To analyze the collision between the upper platform and the $j$ -th upper bar, we should express coordinates $\mathbf{r}_{M_k}$ and $\mathbf{r}_{V_{j, k}}$ in the same reference frame, for example, in base frame $Oxyz$ . Let $\mathbf{p}_{M_k}$ and $\mathbf{p}_{V_{j, k}}$ denote these coordinates in this frame. We can compute these coordinates using transformation matrices $\mathbf{T}_1$ and $\mathbf{T}_{D_j}$ :

(38) \begin{align} \begin{bmatrix} \mathbf{p}_{M_k} \\[3pt] 1 \end{bmatrix} = \mathbf{T}_1 \begin{bmatrix} \mathbf{r}_{M_k} \\[3pt] 1 \end{bmatrix}, \qquad \begin{bmatrix} \mathbf{p}_{V_{j, k}} \\[3pt] 1 \end{bmatrix} = \mathbf{T}_{D_j} \begin{bmatrix} \mathbf{r}_{V_{j, k}} \\[3pt] 1 \end{bmatrix}, \end{align}

where $\mathbf{T}_{D_j}$ is a transformation matrix between frames $D_jx_{D_j}y_{D_j}z_{D_j}$ and $Oxyz$ :

(39) \begin{align} \mathbf{T}_{D_j} = \mathbf{T}_{C_j} \begin{bmatrix} \mathbf{R}_z(\theta _{C_j}) & \mathbf{0} \\[3pt] \mathbf{0}^{\mathrm{T}} & 1 \end{bmatrix} \begin{bmatrix} \mathbf{I}_3 & l_{DE}\hat {\mathbf x} \\[3pt] \mathbf{0}^{\mathrm{T}} & 1 \end{bmatrix} \begin{bmatrix} \mathbf{R}_z(\theta _{D_j}) & \mathbf{0} \\[3pt] \mathbf{0}^{\mathrm{T}} & 1 \end{bmatrix}. \end{align}

Thus, we have represented all the considered links as convex polytopes and derived the explicit expressions for the coordinates of their vertices. This information is sufficient to implement the GJK algorithm and check if the links collide. This concludes the workspace analysis of the hybrid robot. The next section considers numerical examples.

4.1. Translation workspace

To analyze the translation workspace, we sample the $x$ , $y$ , and $z$ components of position vector $\mathbf{p}$ in ranges $[\! -200, 200]$ mm, $[\! -250, 250]$ mm, and $[0, 400]$ mm, respectively, with a step size of 5 mm for each component. In the first example, we consider the zero relative orientation of the modules: $\psi = \theta = \sigma = 0^\circ$ . Figure 7 shows the results. The translation workspace has a symmetrical shape that meets the symmetrical design of the robot. The workspace envelops the rectangular parallelepiped with the sides of 250 mm, 250 mm, and 300 mm along the $x$ , $y$ , and $z$ directions, respectively. Note how the lower part of the analyzed space ( $z \lt 30$ mm) is discarded because of the link interference.

Fig. 7.

Translation workspace and its projections for $\psi = \theta = \sigma = 0^\circ$ .

The second example considers the tilted configuration: $\psi = 0^\circ$ , $\theta = -30^\circ$ , $\sigma = 0^\circ$ . In this case, the upper platform tilts about the $O_2y_2$ axis, while the lower platform keeps zero orientation according to the robot design. Figure 8 shows that the workspace has been reduced by more than a half, and the points with $x \gt 0$ mm are unreachable with this orientation. The points with $z \lt 75$ mm are also unreachable because of the link interference between the modules.

Fig. 8.

Translation workspace and its projections for $\psi = 0^\circ$ , $\theta = -30^\circ$ , $\sigma = 0^\circ$ .

In the third example, we consider the same tilt angle but alter the direction of the tilt axis: $\psi = 30^\circ$ , $\theta = -30^\circ$ , $\sigma = 0^\circ$ . Now, both the upper and lower moving platforms should rotate to achieve this relative orientation. The results in Fig. 9 look similar to Fig. 8, but the whole workspace becomes inclined, reflecting the changes of the tilt axis.

Fig. 9.

Translation workspace and its projections for $\psi = 30^\circ$ , $\theta = -30^\circ$ , $\sigma = 0^\circ$ .

Finally, we add the non-zero torsion to the previous example: $\psi = 30^\circ$ , $\theta = -30^\circ$ , $\sigma = 45^\circ$ . Figure 10 shows the computed translation workspace. Although the workspace still has a large size in the $z$ direction (about 300 mm), it has an irregular shape and is quite limited in the $x$ and $y$ directions, while the points with $x \gt 0$ mm remain unreachable.

Fig. 10.

Translation workspace and its projections for $\psi = 30^\circ$ , $\theta = -30^\circ$ , $\sigma = 45^\circ$ .

4.2. Orientation workspace

To analyze the orientation workspace, we represent it in a polar coordinate system similar to paper [Reference Bonev and Gosselin50]. For each considered value of torsion angle $\sigma$ , we make a planar plot in polar axes where the radial coordinate is proportional to tilt angle $\theta$ and the angular coordinate is equal to azimuth angle $\psi$ .

We specify all these angular parameters as follows. First, we sample tilt angle $\theta$ in range $[0^\circ , 90^\circ ]$ with a step of $2.5^\circ$ . Then, we sample azimuth angle $\psi$ in range $[0^\circ , 360^\circ )$ , and number $p$ of its samples depends on the value of angle $\theta$ . If $\theta = 0^\circ$ , the azimuth angle is not defined, and we set $\psi = 0^\circ$ . If $\theta \neq 0^\circ$ , we compute number $p$ as follows:

(41) \begin{align} p = p_{\mathrm{max}} \Bigl \lceil \dfrac {\theta }{\theta _{\mathrm{max}}}\Bigr \rceil , \end{align}

where $\theta _{\mathrm{max}}$ is the maximum considered value of angle $\theta$ , which is equal to $90^\circ$ in our examples; $p_{\mathrm{max}}$ is a number of samples of angle $\psi$ when $\theta = \theta _{\mathrm{max}}$ , and we set $p_{\mathrm{max}} = 120$ ; $\lceil *\rceil$ is the ceiling function that gives the lowest integer greater or equal to its argument.

Equation (41) provides us with a uniform sampling of the polar coordinate space and avoids dense sampling near the origin, unlike the regular (independent) sampling of parameters $\psi$ and $\theta$ . Finally, we consider four different values of torsion angle $\sigma$ : $0^\circ$ , $20^\circ$ , $40^\circ$ , and $60^\circ$ .

In the first example, we examine the orientation workspace for the point near the geometric center of the translation workspace depicted in Fig. 7: $x = y = 0$ mm, $z = 150$ mm. Figure 11 illustrates the results. For each value of torsion angle $\sigma$ , the upper platform can tilt relative to the lower platform for $20^\circ$ in any direction. When azimuth angle $\psi$ is near $90^\circ$ or $270^\circ$ , the upper platform can tilt further: tilt angle $\theta$ reaches $40^\circ$ and even $60^\circ$ with $\sigma = 40^\circ$ .

Fig. 11.

Orientation workspace for $x = y = 0$ mm, $z = 150$ mm.

The second example considers the orientation workspace for the point at the same height but displaced along the $x$ direction: $x = 100$ mm, $y = 0$ mm, $z = 150$ mm. Figure 12 shows that the upper platform can tilt only in a limited sector. For example, if we look at the plots for $\sigma = 0^\circ$ and $\sigma = 20^\circ$ , we observe that tilt angle $\theta$ reaches $40^\circ$ but the tilts are possible only for angles $\psi$ in a range of about $[\! -90^\circ , 90^\circ ]$ . The orientation workspaces decreases as torsion angle $\sigma$ takes higher values.

Fig. 12.

Orientation workspace for $x = 100$ mm, $y = 0$ mm, $z = 150$ mm.

In the third example, we include the same displacement in the $y$ direction: $x = 100$ mm, $y = 100$ mm, $z = 150$ mm. Figure 13 shows that the tilt sector (the range of angle $\psi$ ) decreases even more. The higher the torsion angle, the narrower this sector, but tilt angle $\theta$ can reach $60^\circ$ for each considered value of angle $\sigma$ .

Fig. 13.

Orientation workspace for $x = 100$ mm, $y = 100$ mm, $z = 150$ mm.

The previous examples analyzed the orientation workspace when the upper platform stayed beyond the lower one, such that the links of the upper and lower modules did not interfere. In the fourth example, we consider the case when the platforms are near each other, and the height of the examined point is twice lower: $x = y = 0$ mm, $z = 75$ mm. Figure 14 illustrates the obtained orientation workspace. The regions of the link interference (the red areas in each plot) constrain the tilt angle: angle $\theta$ is limited by $40^\circ$ , but it could reach $60^\circ$ and even higher values if we ignore the link collisions between the modules.

Fig. 14.

Orientation workspace for $x = y = 0$ mm, $z = 75$ mm.

4.3. CAD simulations

To validate the theoretical results obtained in the preceding sections, we performed a series of CAD simulations in the KOMPAS 3D software using the computer model from Fig. 1. For this purpose, we selected several points in the workspaces presented in Figs. 7–14. Each point corresponds to a specific relative configuration of the modules, and we checked if the robot could reach the considered configuration. Note that in the CAD software, it is challenging to accurately place the moving platforms relative to each other, especially when they are tilted. To overcome this issue, we first solved the inverse kinematic problem for each selected point and then positioned the carriages of the modules according to the obtained solution. This approach ensures that the relative configuration of the moving platforms meets the selected point.

This section illustrates the results for eight points $G_1, \ldots , G_8$ , and Table I presents the values of position parameters $x$ , $y$ , and $z$ and rotation parameters $\psi$ , $\theta$ , and $\sigma$ corresponding to these points. The considered points are selected in the following way:

• • Points $G_1$ and $G_2$ belong to the top boundary surface of the translation workspace in Fig. 7.

• • Points $G_3$ and $G_4$ belong to the front boundary edge of the translation workspace in Fig. 10.

• • Points $G_5$ and $G_6$ belong to the boundaries of the orientation workspace in Fig. 12.

• • Points $G_7$ and $G_8$ belong to the boundaries of the orientation workspace in Fig. 14.

Table II shows the solution to the inverse kinematic problem for each considered point. We used these data to position the carriages in the developed computer model of the robot. For each point, the obtained relative configuration of the modules met the expected one specified in Table I. Figures 15–22 show the 3D view, front view, and side view of the robot in these configurations, where the right lateral branch of the upper module in the side view is hidden for clarity. The red notations in the figures match the preceding sections and indicate the constraints that limit the workspace in the considered configuration.

As we observe from the figures, these are constraints in the spherical joints that limit the workspace in most cases, that is, the constraints on angles $\eta _{A_i}$ and $\eta _{B_i}$ specified in Eqs. (15) and (18).Footnote ² In Fig. 15, the carriages of the two central branches of the upper module reach their highest positions, satisfying Eq. (10); a similar situation is observed in Figs. 16 and 17 for the lateral branches. These results correspond to our selection of points $G_1$ , $G_2$ , and $G_3$ on the top surface of the workspace (Figs. 7 and 10). Figures 18, 21, and 22 show that the moving platforms of the modules are about to collide with each other. This situation agrees with the selection of points $G_4$ , $G_7$ , and $G_8$ on the boundaries of the green and red areas in Figs. 10 and 14. It is interesting that the link interference is the only constraint limiting the workspace in Fig. 21: other joint constraints are not violated in this configuration. This result indicates that the link interference of the modules should not be ignored for the correct evaluation of the workspace.