3.1. Inverse kinematic problems

For inverse kinematic problems, the positon of the end effector $\boldsymbol{x} [\begin{array}{l@{\quad}l@{\quad}l@{\quad}l} x & y & z & \psi \end{array}]^{\mathrm{T}}$ is given, and the input variables $\boldsymbol{q} [\begin{array}{l@{\quad}l@{\quad}l@{\quad}l} q_{1} & q_{2} & \theta _{1} & \theta _{2} \end{array}]^{\mathrm{T}}$ need to be computed.

Squaring both sides of the first two equations in Eq. (10) yields

(12) \begin{equation} g_{1}=g_{2}\cos \theta _{1}-g_{3}\sin \theta _{1} \end{equation}

where $g_{1}=l_{11}^{2}+l_{13}^{2}+l_{14}^{2}+2l_{11}l_{13}+x^{2}+y^{2}-l_{12}^{2}, g_{2}=2(l_{11}x+l_{13}x+l_{14}y), g_{3}=2(-l_{14}x+l_{11}y+l_{13}y)$ .

Hence, the input variable $\theta _{1}$ can be derived as:

(13) \begin{equation} \theta _{1}=2\arctan \left(\frac{-g_{3}\pm \sqrt{-g_{1}^{2}+g_{2}^{2}+g_{3}^{2}}}{g_{1}+g_{2}}\right) \end{equation}

Similarly, by squaring both sides of the first and the last equations in Eq. (11), an equation is obtained:

(14) \begin{equation} s_{1}=s_{2}\cos \theta _{2}-s_{3}\sin \theta _{2} \end{equation}

where $s_{1}=l_{21}^{2}+l_{23}^{2}+x^{2}+z^{2}+(p\psi )^{2}-l_{22}^{2}+2pz\psi -2l_{23}x, s_{2}=2l_{21}(p\psi +z), s_{3}=2l_{21}(l_{23}-x)$ .

Then, the input variable $\theta _{2}$ is computed:

(15) \begin{equation} \theta _{2}=2\arctan \left(\frac{s_{3}\pm \sqrt{-s_{1}^{2}+s_{2}^{2}+s_{3}^{2}}}{s_{1}+s_{2}}\right) \end{equation}

Since the two inputs of two prismatic joints can be obtained directly due to partial decoupling motion, the input parameters can be found as follows:

(16) \begin{equation} \left\{\begin{array}{l} q_{1}=-z \\[3pt] q_{2}=y \\[3pt] \theta _{1}=f_{1}\left(x, y\right) \\[3pt] \theta _{2}=f_{2}\left(x, z, \psi \right) \end{array}\right. \end{equation}

3.2. Direct kinematic problems

For direct kinematic problems, the input parameters $\boldsymbol{q} [\begin{array}{l@{\quad}l@{\quad}l@{\quad}l} q_{1} & q_{2} & \theta _{1} & \theta _{2} \end{array}]^{\mathrm{T}}$ are known, and the aim is to determine the coordinates of the end effector $\boldsymbol{x} [\begin{array}{l@{\quad}l@{\quad}l@{\quad}l} x & y & z & \psi \end{array}]^{\mathrm{T}}$ .

Squaring both sides of the first two equations in Eq. (10) yields

(17) \begin{equation} m_{1}=-x^{2}+m_{2}x \end{equation}

where $m_{1}=l_{11}^{2}+l_{13}^{2}+2l_{11}l_{13}+l_{14}^{2}+y^{2}-l_{12}^{2}-2l_{14}y\cos \theta _{1}+2(l_{11}+l_{13})y\sin \theta _{1}$ , $m_{2}=2[(l_{11}+l_{13})$ $\cos \theta _{1}+l_{14}\sin \theta _{1}]$ .

Consequently, the position along the x-coordinate can be derived as:

(18) \begin{equation} x=\frac{1}{2}\left(m_{2}\pm \sqrt{-4m_{1}+m_{2}^{2}}\right) \end{equation}

Squaring both sides of the first and the last equations in Eq. (11) yields

(19) \begin{equation} n_{1}p\psi +\left(p\psi \right)^{2}=n_{2} \end{equation}

where $n_{1}=2z-2l_{21}\cos \theta _{2}, n_{2}=l_{22}^{2}+2l_{23}x-l_{21}^{2}-l_{23}^{2}-x^{2}-z^{2}-2l_{21}(l_{23}-x)\sin \theta _{2}+2l_{21}z\cos \theta _{2}$ .

Thus, the output rotation angle is calculated by:

(20) \begin{equation} \psi =\frac{\pm n_{1}+\sqrt{n_{1}^{2}+4n_{2}}}{2p} \end{equation}

In summary, based on the above analysis, the position coordinates and the orientation of the end effector can be obtained as:

(21) \begin{equation} \left\{\begin{array}{l} x=F_{1}\left(q_{2}, \theta _{1}\right)\\[3pt] y=q_{2}\\[3pt] z=-q_{1}\\[3pt] \psi =F_{2}\left(q_{1}, q_{2}, \theta _{1}, \theta _{2}\right) \end{array}\right. \end{equation}

3.3. Velocity analysis

Based on the forgoing kinematic analysis, differentiation on both sides of Eq. (16) with respect of time yields

(22) \begin{equation} \boldsymbol{K}_{x}\dot{\boldsymbol{x}}=\boldsymbol{K}_{q}\dot{\boldsymbol{q}} \end{equation}

The Jacobian J can be obtained by:

(23) \begin{equation} \boldsymbol{J}=\boldsymbol{K}_{\mathrm{q}}^{-1}\boldsymbol{K}_{\mathrm{x}} \end{equation}

where

\begin{align*} \dot{\boldsymbol{x}} & =[\begin{array}{l@{\quad}l@{\quad}l@{\quad}l} \dot{x} & \dot{y} & \dot{z} & \dot{\psi } \end{array}]^{\mathrm{T}}, \dot{\boldsymbol{q}}=[\begin{array}{l@{\quad}l@{\quad}l@{\quad}l} \dot{\theta }_{1} & \dot{\theta }_{2} & \dot{q}_{1} & \dot{q}_{2} \end{array}]^{\mathrm{T}}, \boldsymbol{K}_{\mathrm{x}}=\left[\begin{array}{c@{\quad}c@{\quad}c@{\quad}c} a_{11} & a_{12} & 0 & 0\\ a_{21} & 0 & a_{23} & a_{24}\\ 0 & 1 & 0 & 0\\ 0 & 0 & -1 & 0 \end{array}\right],\\ & \boldsymbol{K}_{\mathrm{q}}=\left[\begin{array}{c@{\quad}c@{\quad}c@{\quad}c} b_{11} & 0 & 0 & 0\\ 0 & b_{22} & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1 \end{array}\right]\end{align*}

with

$a_{11}=2x-2\cos \theta _{1}(l_{11}+l_{13})-2\sin \theta _{1}l_{14}, a_{12}=2y-2\cos \theta _{1}l_{14}+2\sin \theta _{1}(l_{11}+l_{13})$ ,

$a_{21}=-2x+2l_{23}+2l_{21}\cos \theta _{2}, a_{23}=2p\psi -2z-2l_{21}\sin \theta _{2}, a_{24}=2pz+2pl_{21}\sin \theta _{2}-2p\psi$ ,

$b_{11}=2((l_{11}+l_{13})y-l_{14}x)\cos \theta _{1}+2((l_{11}+l_{13})x+l_{14}y)\sin \theta _{1}$ ,

$b_{22}=2p\psi l_{21}\cos \theta _{2}+2l_{21}\left(l_{23}-x\right)\sin \theta _{2}-2l_{21}z\cos \theta _{2}$

Thus, the velocity equation of the proposed manipulator can be obtained as:

(24) \begin{equation} \boldsymbol{J}\dot{\boldsymbol{x}}=\dot{\boldsymbol{q}} \end{equation}

4.1. Jacobian and singularity analysis

The Jacobian matrix plays a critical role in analyzing the singular configurations of a parallel manipulator by connecting the end effector velocity with the joint velocity. In this study, screw theory-based method [Reference Tsai45] is used to solve the Jacobian matrix.

Singular configurations within the workspace of a parallel mechanism can limit its performance, as these configurations may cause a change in DOF of the mechanism, allowing the end effector to move even though all actuators are locked. This is undesirable in practical applications, and it can affect manipulability and control accuracy. Therefore, a manipulator should not only avoid singular configurations but also maintain a safe distance from the region close to singularity to ensure better kinematic performance and control. When the manipulator approaches a singular configuration, the relationship between the input motion and output motion distorts, leading to difficulty in control. Thus, singularity analysis is important for better kinematic performance and control of a manipulator.

The screws associated with the joints in the mechanism are presented in Fig. 7. The instantaneous twist of the end effector can be given as $\boldsymbol{\$ }_{p}=[\begin{array}{l@{\quad}l} \boldsymbol{w}^{\mathrm{T}} & \boldsymbol{v}^{\mathrm{T}} \end{array}]^{\mathrm{T}}$ , where $\boldsymbol{w} = [\begin{array}{l@{\quad}l@{\quad}l} 0 & 0 & w_{z} \end{array}]^{\mathrm{T}}$ and $\boldsymbol{v} = [\begin{array}{l@{\quad}l@{\quad}l} v_{x} & v_{y} & v_{z} \end{array}]^{\mathrm{T}}$ are referred to the angular velocity and linear velocity of the end effector, respectively. The detailed expression can be obtained by using the linear combination of all joints in each limb:

(25) \begin{equation} \boldsymbol{\$ }_{P}=\dot{q}_{1}\boldsymbol{\$ }_{11}+\dot{\theta }_{1}\boldsymbol{\$ }_{12}+w_{13}\boldsymbol{\$ }_{13}+\dot{\psi }\boldsymbol{\$ }_{24} \end{equation}

(26) \begin{equation} \boldsymbol{\$ }_{P}=\dot{q}_{2}\boldsymbol{\$ }_{21}+\dot{\theta }_{2}\boldsymbol{\$ }_{22}+w_{23}\boldsymbol{\$ }_{23}+w_{24}\boldsymbol{\$ }_{24}+\dot{\psi }\boldsymbol{\$ }_{25} \end{equation}

in which $\boldsymbol{\$ }_{ij}$ indicates the unit screw associated with the jth joint in the ith limb, $\dot{q}_{i}$ is the linear velocity of the actuated prismatic joints, while $\dot{\theta }_{i}, w_{ij}$ and $\dot{\psi }$ denote the angular velocity of the actuated revolute joints, passive revolute joints, and screw joint, respectively.

Figure 7.

Screws in the mechanism.

The screws can be given as:

(27) \begin{align} \boldsymbol{\$ }_{i1} & =\left[\begin{array}{l@{\quad}l} 0^{\mathrm{T}} & \boldsymbol{s}_{i1}^{\mathrm{T}} \end{array}\right]^{\mathrm{T}}\nonumber\\[3pt]\boldsymbol{\$ }_{i2} & =\left[\begin{array}{l@{\quad}l} \boldsymbol{s}_{i2}^{\mathrm{T}} & \boldsymbol{o}\boldsymbol{A}_{i}^{\mathrm{T}}\times \boldsymbol{s}_{i2}^{\mathrm{T}} \end{array}\right]^{\mathrm{T}}\nonumber\\[3pt]\boldsymbol{\$ }_{i3} & =\left[\begin{array}{l@{\quad}l} 0^{\mathrm{T}} & \boldsymbol{s}_{i3}^{\mathrm{T}} \end{array}\right]^{\mathrm{T}}\nonumber\\[3pt]\boldsymbol{\$ }_{14} & =\left[\begin{array}{l@{\quad}l} \boldsymbol{s}_{14}^{\mathrm{T}} & 0 \end{array}\right]^{\mathrm{T}}\nonumber\\[3pt]\boldsymbol{\$ }_{24} & =\left[\begin{array}{l@{\quad}l} \boldsymbol{s}_{24}^{\mathrm{T}} & \boldsymbol{o}{\boldsymbol{C}^{\mathrm{T}}_{2}}\times \boldsymbol{s}_{24}^{\mathrm{T}} \end{array}\right]^{\mathrm{T}} \end{align}

in which $\boldsymbol{s}_{11}=\boldsymbol{s}_{12}=\boldsymbol{s}_{14}=[\begin{array}{l@{\quad}l@{\quad}l} 0 & 0 & 1 \end{array}]^{\mathrm{T}}, \boldsymbol{s}_{21}=\boldsymbol{s}_{22}=\boldsymbol{s}_{24}=[\begin{array}{l@{\quad}l@{\quad}l} 0 & 1 & 0 \end{array}]^{\mathrm{T}}$ ,

$\boldsymbol{s}_{13}=\left[\begin{array}{l@{\quad}l@{\quad}l} -\dfrac{\cot \left(\phi _{1}+\theta _{1}\right)}{\sqrt{1+\cot ^{2}\left(\phi _{1}+\theta _{1}\right)}} & \dfrac{1}{\sqrt{1+\cot ^{2}\left(\phi _{1}+\theta _{1}\right)}} & 0 \end{array}\right]^{\mathrm{T}}$ ,

$\boldsymbol{s}_{23}=\left[\begin{array}{l@{\quad}l@{\quad}l} \dfrac{\cot \left(\phi _{2}+\theta _{2}\right)}{\sqrt{1+\cot ^{2}\left(\phi _{2}+\theta _{2}\right)}} & 0 & \dfrac{1}{\sqrt{1+\cot ^{2}\left(\phi _{2}+\theta _{2}\right)}} \end{array}\right]^{\mathrm{T}}$ .

To eliminate certain passive joint screws, we use the theory of reciprocal screws. Specifically, when locking the actuated prismatic in the first limb, a screw denoted as $\boldsymbol{\$ }_{{r}11}$ can be identified that is reciprocal to all screws but $\boldsymbol{\$ }_{11}$ . This screw is parallel to the axes of $\boldsymbol{\$ }_{12}$ and $\boldsymbol{\$ }_{14}$ and simultaneously perpendicular to the axis of $\boldsymbol{\$ }_{13}$ :

(28) \begin{equation} \boldsymbol{\$ }_{{r}11}=\left[\boldsymbol{s}_{{r}11}^{\mathrm{T}} \boldsymbol{o}{\boldsymbol{C}^{\mathrm{T}}_{1}}\times \boldsymbol{s}_{{r}11}^{\mathrm{T}}\right]^{\mathrm{T}} \end{equation}

After locking the actuated revolute joint in the first limb, a screw can be found as $\boldsymbol{\$ }_{{r}12}$ that is reciprocal to all screws besides $\boldsymbol{\$ }_{12}$ . This screw intersects with $\boldsymbol{\$ }_{14}$ and is parallel to the normal direction of the plane determined by the vectors $\boldsymbol{s}_{11}$ and $\boldsymbol{s}_{13}$ :

(29) \begin{equation} \boldsymbol{\$ }_{{r}12}=\left[\boldsymbol{s}_{{r}12}^{\mathrm{T}}0\right]^{\mathrm{T}} \end{equation}

Taking the orthogonal products of both sides of Eq. (25) with (28) and (29) in turn yields

(30) \begin{equation} \boldsymbol{\$ }_{r11}^{\mathrm{T}}\boldsymbol{\$ }_{P}=\dot{q}_{1}\boldsymbol{\$ }_{r11}^{\mathrm{T}}\boldsymbol{\$ }_{11} \end{equation}

(31) \begin{equation} \boldsymbol{\$ }_{r12}^{\mathrm{T}}\boldsymbol{\$ }_{P}=\dot{\theta }_{1}\boldsymbol{\$ }_{r12}^{\mathrm{T}}\boldsymbol{\$ }_{12} \end{equation}

When locking the actuated prismatic joint in the second limb, a screw denoted as $\boldsymbol{\$ }_{{r}21}$ can be identified that is reciprocal to all screws apart from $\boldsymbol{\$ }_{21}$ . This screw is parallel to the axes of $\boldsymbol{\$ }_{22}$ and $\boldsymbol{\$ }_{24}$ and simultaneously perpendicular to $\boldsymbol{\$ }_{23}$ :

(32) \begin{equation} \boldsymbol{\$ }_{{r}21}=\left[\boldsymbol{s}^{\mathrm{T}}_{{r}21} \boldsymbol{o}{\boldsymbol{C}^{\mathrm{T}}_{2}}\times \boldsymbol{s}^{\mathrm{T}}_{{r}21}\right]^{\mathrm{T}} \end{equation}

After locking the actuated revolute joint in the second limb, a screw can be found as $\boldsymbol{\$ }_{{r}22}$ that is reciprocal to all screws but $\boldsymbol{\$ }_{22}$ . This screw intersects with $\boldsymbol{\$ }_{24}$ and $\boldsymbol{\$ }_{25}$ simultaneously and is parallel to the normal direction of the plane determined by the vectors $\boldsymbol{s}_{21}$ and $\boldsymbol{s}_{23}$ :

(33) \begin{equation} \boldsymbol{\$ }_{{r}22}=\left[\boldsymbol{s}_{{r}22}^{\mathrm{T}} \boldsymbol{o}{\boldsymbol{C}^{\mathrm{T}}_{2}}\times \boldsymbol{s}_{{r}22}^{\mathrm{T}}\right]^{\mathrm{T}} \end{equation}

Taking the orthogonal products of both sides of Eq. (26) with (32) and (33) in turn yields

(34) \begin{equation} \boldsymbol{\$ }_{r21}^{\mathrm{T}}\boldsymbol{\$ }_{P}=\dot{q}_{2}\boldsymbol{\$ }_{r21}^{\mathrm{T}}\boldsymbol{\$ }_{21} \end{equation}

(35) \begin{equation} \boldsymbol{\$ }_{r22}^{\mathrm{T}}\boldsymbol{\$ }_{P}=\dot{\theta }_{2}\boldsymbol{\$ }_{r22}^{\mathrm{T}}\boldsymbol{\$ }_{22} \end{equation}

Based on the above analysis, Eqs. (30), (31), (34), and (35) can be rewritten in matrix form:

(36) \begin{equation} \left[\begin{array}{l} \boldsymbol{\$ }_{r11}^{\mathrm{T}}\\[5pt] \boldsymbol{\$ }_{r12}^{\mathrm{T}}\\[5pt] \boldsymbol{\$ }_{r21}^{\mathrm{T}}\\[5pt] \boldsymbol{\$ }_{r22}^{\mathrm{T}} \end{array}\right]\boldsymbol{\$ }_{P}=\left[\begin{array}{l@{\quad}l@{\quad}l@{\quad}l} \boldsymbol{\$ }_{r11}^{\mathrm{T}}\boldsymbol{\$ }_{11} & & & \\[5pt] & \boldsymbol{\$ }_{r12}^{\mathrm{T}}\boldsymbol{\$ }_{12} & & \\[5pt] & & \boldsymbol{\$ }_{r21}^{\mathrm{T}}\boldsymbol{\$ }_{21} & \\[5pt] & & & \boldsymbol{\$ }_{r22}^{\mathrm{T}}\boldsymbol{\$ }_{22} \end{array}\right]\left[\begin{array}{l} \dot{q}_{1}\\[5pt] \dot{\theta }_{1}\\[5pt] \dot{q}_{2}\\[5pt] \dot{\theta }_{2} \end{array}\right] \end{equation}

Equation (36) can also be expressed in a general form as:

(37) \begin{equation} \boldsymbol{J}_{x}\boldsymbol{\$ }_{P}=\boldsymbol{J}_{q}\dot{\boldsymbol{q}} \end{equation}

where $\boldsymbol{J}_{{x}}=\left[\begin{array}{l} \boldsymbol{\$ }_{{r}11}^{\mathrm{T}}\\[5pt] \boldsymbol{\$ }_{{r}12}^{\mathrm{T}}\\[5pt] \boldsymbol{\$ }_{{r}21}^{\mathrm{T}}\\[5pt] \boldsymbol{\$ }_{{r}22}^{\mathrm{T}} \end{array}\right], \boldsymbol{J}_{{q}}=\left[\begin{array}{l@{\quad}l@{\quad}l@{\quad}l} \boldsymbol{\$ }_{{r}11}^{\mathrm{T}}\boldsymbol{\$ }_{11} & 0 & 0 & 0\\[5pt] 0 & \boldsymbol{\$ }_{{r}12}^{\mathrm{T}}\boldsymbol{\$ }_{12} & 0 & 0\\[5pt] 0 & 0 & \boldsymbol{\$ }_{{r}21}^{\mathrm{T}}\boldsymbol{\$ }_{21} & 0\\[5pt] 0 & 0 & 0 & \boldsymbol{\$ }_{{r}22}^{\mathrm{T}}\boldsymbol{\$ }_{22} \end{array}\right], \dot{\boldsymbol{q}} = \left[\begin{array}{l} \dot{q}_{1}\\[5pt] \dot{\theta }_{1}\\[5pt] \dot{q}_{2}\\[5pt] \dot{\theta }_{2} \end{array}\right]$

By analyzing the obtained Jacobian matrix, it is possible to identify singular configurations.

4.1.1. Inverse kinematic singularity

If any of the diagonal elements of the matrix $\boldsymbol{J}_{q}$ are zero, it can lead to the occurrence of the inverse kinematic singularity in the mechanism. After performing calculations, the equation $\boldsymbol{\$ }_{{r}11}^{\mathrm{T}}\boldsymbol{\$ }_{11}=\boldsymbol{\$ }_{{r}21}^{\mathrm{T}}\boldsymbol{\$ }_{21}=1$ can be obtained. Therefore, the inverse kinematic singularity occurs when $\boldsymbol{\$ }_{{r}12}^{\mathrm{T}}\boldsymbol{\$ }_{12}=0$ or $\boldsymbol{\$ }_{{r}22}^{\mathrm{T}}\boldsymbol{\$ }_{22}=0$ . Then the following relations can be obtained:

(38) \begin{equation} \boldsymbol{\$ }_{{r}12}^{\mathrm{T}}\boldsymbol{\$ }_{12}=\boldsymbol{s}_{12}^{\mathrm{T}}\left(\boldsymbol{o}{\boldsymbol{A}^{\mathrm{T}}_{1}}\times \boldsymbol{s}_{{r}12}^{\mathrm{T}}\right) \end{equation}

(39) \begin{equation} \boldsymbol{\$ }_{{r}22}^{\mathrm{T}}\boldsymbol{\$ }_{22}=\boldsymbol{s}_{22}^{\mathrm{T}}\left(\boldsymbol{s}_{{r}22}^{\mathrm{T}}\times \boldsymbol{C}_{2}{\boldsymbol{A}_{2}}^{\mathrm{T}}\right) \end{equation}

If these three vectors $\boldsymbol{s}_{12}, \boldsymbol{o}\boldsymbol{A}_{1}$ , and $\boldsymbol{s}_{\mathrm{r}12}$ are coplanar, which means that the projection of $\mathrm{oA}_{1}$ and $\mathrm{B}_{1}\mathrm{C}_{1}$ on the XOY plane are parallel to each other, and the mechanism moves to the singular configuration. This is the case illustrated in Fig. 8(a). When the three vectors $\boldsymbol{s}_{22}, \boldsymbol{C}_{2}\boldsymbol{A}_{2}$ , and $\boldsymbol{s}_{\mathrm{r}22}$ are coplanar, that is, the long rod and the short rod in the parallelogram joint form a right angle, Eq. (39) is equal to zero. The singular configuration is depicted in Fig. 8(b).

Figure 8.

Inverse kinematic singular configurations: (a) case 1 and (b) case 2.

4.1.2. Forward kinematic singularity

From Eq. (37), we can obtain

(40) \begin{equation} \boldsymbol{J}_{{x}}=\left[\begin{array}{c@{\quad}c} \boldsymbol{s}_{{r}11}^{\mathrm{T}} & \boldsymbol{o}{\boldsymbol{C}^{\mathrm{T}}_{1}}\times \boldsymbol{s}_{{r}11}^{\mathrm{T}}\\[5pt] \boldsymbol{s}_{{r}12}^{\mathrm{T}} & 0\\[5pt] \boldsymbol{s}^{\mathrm{T}}_{{r}21} & \boldsymbol{o}{\boldsymbol{C}^{\mathrm{T}}_{2}}\times \boldsymbol{s}^{\mathrm{T}}_{{r}21}\\[5pt] \boldsymbol{s}_{{r}22}^{\mathrm{T}} & \boldsymbol{o}{\boldsymbol{C}^{\mathrm{T}}_{2}}\times \boldsymbol{s}_{{r}22}^{\mathrm{T}} \end{array}\right] \end{equation}

When the matrix $\boldsymbol{J}_{{x}}$ is rank-deficient, the forward kinematic singularity occurs. There is a zero element in this matrix. If there are two or more zero elements in the matrix, the matrix is rank-deficient. First, when $\boldsymbol{o}{\boldsymbol{C}^{\mathrm{T}}_{1}}\times \boldsymbol{s}_{{r}11}^{\mathrm{T}}$ is equal to zero, the vectors $\boldsymbol{o}\boldsymbol{C}_{1}$ and $\boldsymbol{s}_{{r}11}$ coincide with each other. According to the structural limitations, this case does not exist. Second, when $\boldsymbol{o}{\boldsymbol{C}^{\mathrm{T}}_{2}}\times \boldsymbol{s}^{\mathrm{T}}_{{r}21}$ is equal to zero, the vector $\boldsymbol{o}\boldsymbol{C}_{2}$ should coincide with $\boldsymbol{s}_{{r}21}$ . Similarly, due to the structural limitations, this case does not exist, either. Finally, $\boldsymbol{o}{\boldsymbol{C}^{\mathrm{T}}_{2}}\times \boldsymbol{s}_{{r}22}^{\mathrm{T}}$ vanishes, which means that the vector $\boldsymbol{o}\boldsymbol{C}_{2}$ coincides with $\boldsymbol{s}_{{r}22}$ . This is the case illustrated in Fig. 9.

Figure 9.

Forward kinematic singular configuration.

Furthermore, the inverse and forward kinematic singular surfaces in the joint space and workspace are presented in Figs. 10 and 11 utilizing the structural parameters that were previously introduced.

Figure 10.

Inverse kinematic singular surfaces: (a) in the joint space and (b) in the workspace.

Figure 11.

Forward kinematic singular surfaces: (a) in the joint space and (b) in the workspace.

4.2. Workspace analysis

The workspace of a manipulator is a critical evaluation metric used to assess the ability of the moving platform to achieve different poses, which significantly influences the manipulator’s performance. One common limitation of parallel mechanisms is their limited workspace. Generally, among manipulators with the same DOF, a larger workspace is preferable. In this section, the reachable workspace is analyzed. The primary factors that influence the manipulator’s workspace include the length limit of links, the angle limit of rotational joints, and interference issues between different links and the moving platform. It is essential to analyze the reachable workspace to optimize the manipulator’s design and ensure that it can achieve the desired performance.

The stroke of the first and second active prismatic joints are set to $-500\,\mathrm{mm}\leq q_{1}\leq 350\,\mathrm{mm}$ and $150\,\mathrm{mm}\leq q_{2}\leq 1000\,\mathrm{mm}$ , respectively, and the rotational angles of the first and second active rotational joints are in the range of $-50\deg \leq \theta _{1}\leq 35\deg$ and $-35\deg \leq \theta _{2}\leq 50\deg$ , respectively.

According to the structural characteristics, the rotational angles $\phi _{1}$ and $\phi _{2}$ are calculated in detail to avoid the interference between links:

(41) \begin{equation} \phi _{1}=\arccos \left(\boldsymbol{b}_{1}\boldsymbol{c}_{1}\cdot \boldsymbol{v}\right) \end{equation}

(42) \begin{equation} \phi _{2}=\arccos \left(\boldsymbol{b}_{2}\boldsymbol{c}_{2}\cdot \boldsymbol{u}\right) \end{equation}

in which $\boldsymbol{b}_{1}\boldsymbol{c}_{1}$ and $\boldsymbol{b}_{2}\boldsymbol{c}_{2}$ are normalized vectors of vectors $\boldsymbol{B}_{1}\boldsymbol{C}_{1}$ and $\boldsymbol{B}_{2}\boldsymbol{C}_{2}$ , respectively, and u and v are unit vectors along the X- and Y-axis, respectively. The expressions for these vectors can be given by:

(43) \begin{equation} \boldsymbol{b}_{1}\boldsymbol{c}_{1}=\frac{\boldsymbol{B}_{1}\boldsymbol{C}_{1}}{\left| \boldsymbol{B}_{1}\boldsymbol{C}_{1}\right| }, \boldsymbol{b}_{2}\boldsymbol{c}_{2}=\frac{\boldsymbol{B}_{2}\boldsymbol{C}_{2}}{\left| \boldsymbol{B}_{2}\boldsymbol{C}_{2}\right| }, \boldsymbol{u}=\left[\begin{array}{l@{\quad}l@{\quad}l} 1 & 0 & 0 \end{array}\right]^{\mathrm{T}}, \boldsymbol{v}=\left[\begin{array}{l@{\quad}l@{\quad}l} 0 & 1 & 0 \end{array}\right]^{\mathrm{T}} \end{equation}

Therefore, the rotational angle $\phi _{1}$ between the long rod of the parallelogram joint and the Y-axis should be limited to $45\deg \leq \phi _{1}\leq 100\deg$ , and $\phi _{2}$ between the long rod of the parallelogram joint and the X-axis should be in the range of $-25\deg \leq \phi _{2}\leq 70\deg$ .

For a parallel manipulator, the reachable workspace comprises the set of points that can be reached by the end effector. Hence, based on the structure parameters in Table I, the inverse kinematic solution is employed to calculate the reachable workspace. Then, according to the limitations of $q_{1}, q_{2}, \theta _{1}, \theta _{2}, \phi _{1}$ and $\phi _{2}$ obtained above, the points meeting these constraints can be selected. The reachable workspaces of the proposed mechanism are studied in the space $(\begin{array}{l@{\quad}l@{\quad}l} x & y & z \end{array})$ with the orientation $\psi =0$ and the space $(\begin{array}{l@{\quad}l@{\quad}l} x & \psi & z \end{array})$ with the coordinate y = 400 mm, respectively. The workspaces are depicted in Fig. 12 using Mathematica software. According to the kinematic analysis, the y- and z-coordinate are related to the input variables $q_{2}$ and $q_{1}$ , respectively. As shown in Fig. 12(a), the 3D workspace can be interpreted as extruding the xoy plane along the z-axis, effectively illustrating the characteristic decoupling motion in the z direction. The kinematic solution indicates that the mechanism also possesses the characteristic of decoupling motion in the y direction. However, we acknowledge that Fig. 12(a) reveals a coupling relationship between the displacement in the x direction and the displacement in the y direction. This coupling is a consequence of the interaction between the actuated parameter $q_{2}$ and the position along the x-coordinate.

Table I.

Structural parameters.

Figure 12.

Workspace: (a) in the space $(\begin{array}{l@{\quad}l@{\quad}l} x & y & z \end{array})$ and (b) in the space $(\begin{array}{l@{\quad}l@{\quad}l} x & \psi & z \end{array})$ .

Additionally, Figs. 10(b) and 11(b) plot all kinematic solutions leading to singular configurations within the manipulator’s workspace. The relationship between the reachable workspace and the inverse kinematic singularity inside the reachable workspace is proposed in Fig. 13(a), while the relationship between the reachable workspace and the forward singularity inside the reachable workspace is depicted in Fig. 13(b). The green surface in both figures represents the singular surface within the reachable workspace. To enhance the manipulator’s performance, avoiding singular loci is crucial. In future work, the avoidance of singular loci will be investigated by planning the manipulator’s trajectory to expand its workspace.

Figure 13.

Inverse kinematic singular surface (a) and forward kinematic singular surface (b) within the reachable workspace.

4.3. Manipulability

The kinematic manipulability index is used to evaluate the capability of velocity transmission of a manipulator and can also be considered as an equivalent measure of the dexterity [Reference Merlet46]. According to the definition in [Reference Yoshikawa47], the kinematic manipulability is the square root of the determinant of $\boldsymbol{J}\boldsymbol{J}^{\mathrm{T}}$ . The detailed expression can be given by:

(44) \begin{equation} \omega =\sqrt{\det \left(\boldsymbol{J}\boldsymbol{J}^{\mathrm{T}}\right)} \end{equation}

It should be noted that the Jacobian matrix is posture-dependent, and as a result, the kinematic manipulability is a local performance index. However, it provides a clear indication of whether the manipulator is in a singular configuration. When the value of $\omega$ is close to or equal to 0, the manipulator is in or near a singular configuration. Hence, the value of $\omega$ is desired to be greater than 0 to avoid the singular configurations.

Based on the structure parameters, the kinematic manipulability for various z-value and rotational angles within the workspace can be calculated. The results are presented in Fig. 14. As can be observed, the value of kinematic manipulability is close to 0 when the end effector moves to the boundary along the x-axis or y-axis.

Figure 14.

Kinematic manipulability for various z-value and rotational angles within the workspace.

4.4. Dexterity analysis

The dexterity of a manipulator is a critical measure of its kinematic performance in pick-and-place applications, defined as the ability to move to any position or direction and apply forces or torques in any direction [Reference Li and Xu48]. The condition number of the Jacobian matrix is used to evaluate the dexterity of a manipulator [Reference Gosselin and Angeles49,Reference Merlet50], with a lower value indicating better dexterity. In particular, the inverse of the condition number, which ranges from 0 to 1, is commonly used as a measure. Some studies [Reference Tannous, Caro and Goldsztejn51–Reference Briot and Bonev55] have also used dexterity to evaluate the accuracy and sensitivity of a manipulator. Specifically, a higher dexterity tends to lead to higher accuracy. Although manipulability is a related concept, dexterity is a more appropriate measure for evaluating accuracy in many cases.

The kinetostatic conditioning index (KCI) is a posture-independent index. As defined in [Reference Angeles56], the expression can be given by:

(45) \begin{equation} \mathrm{KCI}=\frac{1}{\kappa _{\min }}\times 100\% \end{equation}

where $\kappa _{\min }$ is the minimum condition number. When $\kappa _{\min }$ is equal to 1, the KCI is 100%. Then, the manipulator has the istropic characteristics, which means that the manipulator possesses good motion transmission performance.

In order to evaluate the dexterity within the whole workspace, the global conditioning index (GCI) is proposed [Reference Gosselin and Angeles49]. The expression can be given by:

(46) \begin{equation} \mathrm{GCI}=\frac{\int _{W}\frac{1}{\kappa }dV}{V} \end{equation}

in which W is the whole workspace and V is the volume of the workspace.

However, different types of degrees of freedom (i.e., three translational motions and one rotation) presents the challenge for unifying the units of each element in the Jacobian matrix and representing the physical meaning of the manipulator with the condition number. To solve this problem, Altuzarra [Reference Altuzarra, Hernandez, Salgado and Angeles57] proposed a method to normalize the nonhomogeneous Jacobian matrix using the concept of characteristic length. The detailed expression of the homogeneous Jacobian matrix $\boldsymbol{J}_{h}$ can be defined as:

(47) \begin{equation} \boldsymbol{J}_{\mathrm{h}}=\left(\begin{array}{l@{\quad}l} \boldsymbol{A} & \frac{1}{L}\boldsymbol{B} \end{array}\right) \end{equation}

where L is the characteristic length, $\boldsymbol{A}$ represents the $4\times 3$ submatrix formed by the first three columns of $\boldsymbol{J}$ , and $\boldsymbol{B}$ denotes the $4\times 1$ submatrix formed by the fourth column of $\boldsymbol{J}$ .

After normalizing the nonhomogeneous Jacobian matrix, the Jacobian matrix $\boldsymbol{J}_{\mathrm{h}}$ can attain dimensionally homogeneous. The isotropy condition can be given by:

(48) \begin{equation} \boldsymbol{J}_{\mathrm{h}}^{\mathrm{T}}\boldsymbol{J}_{\mathrm{h}}=\left[\begin{array}{l@{\quad}l} \boldsymbol{A}^{\mathrm{T}}\boldsymbol{A} & \dfrac{1}{L}\boldsymbol{A}^{\mathrm{T}}\boldsymbol{B}\\[11pt] \dfrac{1}{L}\boldsymbol{B}^{\mathrm{T}}\boldsymbol{A} & \dfrac{1}{L^{2}}\boldsymbol{B}^{\mathrm{T}}\boldsymbol{B} \end{array}\right]=\sigma ^{2}\boldsymbol{I} \end{equation}

where $\sigma$ is a nondimensional scalar and $\boldsymbol{I}$ denotes the $4\times 4$ identity matrix. The manipulator is considered isotropic when the homogeneous Jacobian matrix becomes isotropic. If there is no solution to the isotropy condition, then the manipulator is nonisotropic.

From Eq. (48), we can obtain the following relationships:

(49) \begin{equation} \mathrm{tr}\left(\boldsymbol{A}^{\mathrm{T}}\boldsymbol{A}\right)=3\sigma ^{2} \end{equation}

(50) \begin{equation} \frac{1}{L}\boldsymbol{A}^{\mathrm{T}}\boldsymbol{B}=\mathbf{0} \end{equation}

(51) \begin{equation} \frac{1}{L^{2}}\boldsymbol{B}^{\mathrm{T}}\boldsymbol{B}=\sigma ^{2} \end{equation}

Then, the characteristic length can be expressed as:

(52) \begin{equation} L=\sqrt{\frac{3\boldsymbol{B}^{\mathrm{T}}\boldsymbol{B}}{\mathrm{tr}\left(\boldsymbol{A}^{\mathrm{T}}\boldsymbol{A}\right)}} \end{equation}

The condition number $\kappa$ of the homogeneous Jacobian matrix $\boldsymbol{J}_{\mathrm{h}}$ can be calculated by:

(53) \begin{equation} \kappa =\left\| \boldsymbol{J}_{\mathrm{h}}\right\| \left\| \boldsymbol{J}_{\mathrm{h}}^{-1}\right\| \end{equation}

in which $\| \cdot \|$ indicates the norm of the matrix argument. It should be noted that Frobenius norm is frame-invariant and analytic. Hence, the calculation of the condition number by means of the Frobenius norm can be expressed as:

(54) \begin{equation} \kappa =\frac{1}{4}\sqrt{\mathrm{tr}\left(\boldsymbol{J}_{\mathrm{h}}^{\mathrm{T}}\boldsymbol{J}_{\mathrm{h}}\right)\mathrm{tr}\left[\left(\boldsymbol{J}_{\mathrm{h}}^{\mathrm{T}}\boldsymbol{J}_{\mathrm{h}}\right)^{-1}\right]} \end{equation}

Based on the structure parameters, the value of the characteristic length can be calculated as L = 0.2171 m. The KCI is depicted in Fig. 15 with various z-values and rotational angles. According to the analytical results, the maximum KCI over the whole workspace is observed when the end effector is located at x = 0.65 m and y = 0.5 m, indicating that the manipulator has better performance in this position. Additionally, smaller values of KCI correspond to smaller z-coordinate magnitudes and larger rotational angles. Conversely, the value of KCI approaches zero at the boundary of the workspace, suggesting that the manipulator is less dexterous in these postures, and this region should be avoided.

Figure 15.

Reciprocal of Jacobian matrix condition number.

However, to the best of our knowledge, the Jacobian matrix is configuration-dependent. The local conditioning number varies with different postures and provides a local indication of dexterity of a manipulator. To obtain a global measure of dexterity within the entire workspace, the GCI is calculated. The expression in Eq. (46) is too complicated to compute the value of GCI. Furthermore, the value of $1/\kappa$ can be zero in some postures, indicating that singularity occurs in these postures. Hence, a simplified numerical approach in ref. [Reference Li and Xu48] is used to approximate the GCI value:

(55) \begin{equation} \mathrm{GCI}=\frac{1}{N}\sum _{i}^{N}\frac{1}{\kappa _{i}} \end{equation}

where the workspace is discretized into a set of N points and $\kappa _{i}$ is the value of $\kappa$ at the ith point. Therefore, the results can be calculated by researching the points within the workspace. The larger the value of GCI, the higher the global dexterity. The GCI of the proposed manipulator can be obtained as $\mathrm{GCI}=0.2754$ .