Acceleration analysis and optimal design of a 3-degree-of-freedom co-axis parallel manipulator for pick-and-place applications

Abstract

Robot acceleration is the bottleneck for high-speed pick-and-place operations. In this article, the extreme acceleration at null velocity is proposed to characterize the acceleration capability for a planar parallel manipulator having a common actuated axis, the V3 robot. Detailed dynamic model of the V3 robot is developed for computing the extreme acceleration. Since each subchain of the robot is comparable to a cantilever beam structure, its bending deflection can be considerable and thus leads to manipulation failure. Characterized by the bending deflection, the bending stiffness is thus proposed. Combining the acceleration, the bending stiffness and kinematic constraints of velocity, accuracy, and dexterity, an optimal design problem is formulated to maximize the dextrous workspace having specified desired performance. The optimized structure shows greatly improved performance in workspace, velocity, acceleration, and accuracy.

Introduction

Thanks to their low moving mass/inertia and high acceleration, parallel manipulators have been widely recognized as prospective candidates for high-speed pick and placement of light-weight workpieces typically in pharmaceutical, electronics, food, and consumer goods industries. Great success has been achieved in commercialization since 1990s, for example, the Delta robot ¹ by ABB Robotics and the Quattro robot by Adept Technology.^2,3 In order to fundamentally improve the limited workspace, which is regarded as the major drawback of parallel manipulators, common-axis (co-axis for short) design was proposed for actuated joints at base. Using the parallelogram structure to eliminate rotation mobility, Brogårdh ⁴ first proposed a 3-degree-of-freedom (DoF) purely translational co-axis parallel mechanism in 2000. Later, it was patented ⁵ and named the Tau parallel robot with detailed analysis.^6–8 Fully co-axis versions were presented by Brogårdh and Kock. ⁹ In 2010, a co-axis variation with a different arm structure was proposed for 3-DoF pure translation by Huang et al. ¹⁰ Lou et al. ¹¹ proposed co-axis parallel mechanisms for 3-DoF planar motion in 2009. Similar structures were presented by Isaksson ¹² in 2011.

The co-axis design almost fully eliminates the limited workspace drawback of usual parallel manipulators since the whole manipulator is able to rotate about a common actuated axis, which greatly enlarges the workspace to a similar level to that of serial ones. In order to further take advantage of parallel manipulators with co-axis structure, optimal design is an inevitable step since the kinematic and dynamic performances depend very much on their geometric parameters. Since 1980s, the kinematically optimal design has been extensively studied based on workspace, dexterity, manipulability, singularity, isotropy, accuracy, stiffness, force/velocity transmission, and so on, while the dynamics based optimal design remains a difficult problem for parallel manipulators. Some performance indices were proposed only based on the mass matrix in the dynamic equation, for example, the dynamic manipulability, ¹³ the dynamic dexterity, ¹⁴ and the dynamic isotropy.^15,16 Kozak et al. ¹⁷ proposed to use the linearized natural frequency as a performance measure in dynamically optimal design of parallel manipulators. The linearized natural frequency is derived based on not only the mass matrix but also the stiffness matrix in the linearized dynamics. The first-order natural frequency is applied for optimal design of a planar 2-DoF parallel manipulator with redundant actuation. ¹⁸ For better design, more complete information in the dynamic model is used.^19,20 In the dynamic design of a Delta robot, Zhao ²¹ proposed to minimize the maximum input torque and power given the required maximum speed and acceleration. The normalized inertial and centrifugal/Coriolis torques of single actuated joint were taken into account in optimal design of a 2-DoF translational pick-and-place parallel robot ²² and a novel redundantly actuated parallel manipulator. ²³

A commercial parallel manipulator for pick-and-place applications, taking Adept Quattro as an example, performs that its maximum velocity and acceleration can achieve 10 m/s and 150 m/s², respectively, as shown in datasheet of Adept Quattro s650H Robot. While in a typical pick-and-place task (25/305/25, in mm) with the shortest cycle time, the peak velocity of its prototype only needs to reach about 5 m/s, and the peak acceleration must be over 15g³, where its acceleration reaches the peak value while its real motion velocity is much smaller than the peak velocity value. On the contrary, the acceleration usually reaches the extremum. Nabat et al. ²⁴ also presented a Par4 design capable of a cycle time of 0.28 s, where a large acceleration (130 m/s²) was needed while the velocity was relatively small, 3.8 m/s. Overshoot even appeared in the experiments because the actuator cannot continuously provide required torque, neither required acceleration. The R4 redundantly actuated parallel manipulator even achieved an ultrahigh acceleration of more than 100g in the pick-and-place experiment. ²⁵ Therefore, to achieve higher efficiency, robot acceleration is the bottleneck for high-speed pick-and-place operations. However, few researches have directly applied it as a performance index in optimal design of a parallel robot because of its complexity. To obtain higher acceleration, there are fundamentally two solutions, increasing actuation torque and reducing mass/inertia of moving parts. Taking cost into consideration, the latter one is more preferable.

In this article, by taking into account both kinematic and dynamic requirements for high-speed pick-and-place operations, we propose to optimally design the V3 robot, a novel $3-\underset{¯}{\mathcal{R}}\mathcal{R}\mathcal{R}$ parallel mechanism (see Figure 1), where $\mathcal{R}$ represents a passive revolute joint, while $\underset{¯}{\mathcal{R}}$ denotes an actuated revolute joint. The key design in the V3 robot is that the three $\underset{¯}{\mathcal{R}}$ joints are made to have a common axis of rotation, as shown in Figure 1(b), which leads to that the robotic manipulator is able to rotate ${360}^{\mathrm{o}}$ about the common axis. Like a serial robot with a small footprint, the position space around the common axis of rotation becomes the position workspace of the mechanism and thus, its workspace is much larger than the traditional non-co-axis $3-\underset{¯}{\mathcal{R}}\mathcal{R}\mathcal{R}$ parallel mechanisms. Furthermore, the axes of the three passive revolute pairs on the moving platform are parallel but not co-axis, which makes its end-effector capable of unlimited rotation. Besides the indices of velocity and dexterity, the acceleration, the key index for high-speed motion, is taken as the design objective to be maximized so that the shortest possible cycle time can be reached. The transmission of velocity is related to the lengths of links, while the transmission of acceleration is largely due to mass matrix which depends on the lengths and the cross-sectional areas of links simultaneously. Due to the limitations of the velocity and the torque of the actuated joint, the properties of the end-effector cannot be required blindly. In general, in order to achieve high acceleration, the masses of the moving parts are reduced. However, since the V3 robot architecture consists of a group of cantilever beams, the link deformation may become unneglectable. The bending stiffness of the robot is analyzed and applied as a design constraint. Detailed information in kinematics and dynamics of the robot is utilized for the specific application. The design problem is formulated as an optimization problem, which is solved by the differential evolution (DE) algorithm. In this optimization design, all the geometric parameters of links are discussed to improve the potential of the V3 robot for high-speed pick-and-place operations in dextrous workspace. It is a significant prerequisite of control strategy. Although this formulation is proposed for the V3 robot, it is valid and applicable for general robot mechanism designs for pick-and-place operations.

Figure 1.

(a) The 3D model of the V3 robot and (b) side view.

The remainder of this article is organized as follows. The kinematic model and loop-closure equations are presented in section “Geometric description.” The dynamic model and the bending deflection are derived and analyzed in sections “Dynamic modeling” and “Bending deflection and bending stiffness.” The optimal design problem formulation and the optimization result are presented and discussed in section “Optimal design.” Finally, conclusions are drawn in section “Conclusion.”

Geometric description

A structural description of the V3 parallel robot is shown in Figure 2. It consists of $3-\underset{¯}{\mathcal{R}}\mathcal{R}\mathcal{R}$ planar subchains, where $\mathcal{R}$ denotes a revolute joint and the underline means “actuated.” The rotational axes of all nine revolute joints in the robot are parallel. The three actuated revolute joints are fixed to the base and have a common rotational axis. A fixed coordinate frame OXYZ is attached with the first subchain $A_1{B}_1{C}_1$ being in the XOY plane and its origin $O$ being coincident with the pivot point of the actuated joint $A_1$ . The Z-axis is coincident with the common actuated joint axis and the +X-axis passes through point $C_1$ at home configuration.

Figure 2.

Notations and the coordinate system.

The variables and parameters are depicted in Figures 2 and 3 and Table 1. The V3 parallel robot undergoes the same planar motion type, 2-DoF translation in a plane and 1-DoF rotation about the axis perpendicular to the plane, as a generic planar $3-\mathcal{R}\mathcal{R}\mathcal{R}$ parallel mechanism. ²⁶ In this article, workspace of the V3 robot is parameterized by the coordinate variable of point $C_1$ and the orientation angle $\varphi$ of the end-effector. To kinematically analyze a parallel robot, first, its the loop-closure equations should be derived. For the V3 parallel robot, the position relation for each subchain can be obtained as follows

$$\begin{array}{c}{x}_i=l_{i1}\cos {\theta }_i+l_{i2}\cos ({\theta }_i-\mathrm{sgn}({\gamma }_i))\hfill \\ y_i=l_{i1}\sin {\theta }_i+l_{i2}\sin ({\theta }_i-\mathrm{sgn}({\gamma }_i))\hfill \end{array}$$

(1)

where $\mathrm{sgn}({\gamma }_i)=+{\gamma }_i$ , $i=1,3$ and $\mathrm{sgn}({\gamma }_i)=-{\gamma }_i$ , $i=2$ . Note $(x_1,y_1)=(x,y)$ .

Figure 3.

(a) An XOY projection of the V3 parallel robot and (b) the end-effector design.

Table 1.

Nomenclature.

Notations	Meaning
OXYZ	The fixed coordinate frame
$l_{\mathrm{ij}}$	Length of the $j\mathrm{th}$ link in the $i\mathrm{th}$ subchain
$(x,y)$	Cartesian coordinate of point $C_1$
$(x_i,y_i)$	Cartesian coordinate of point $C_i$
$\varphi$	The included angle from $+X$ direction to the vector $H_1{H}_2$
${\theta }_i$	The actuated joint angle variable of the $i\mathrm{th}$ subchain
${\gamma }_i$	The passive joint angle variable of the $i\mathrm{th}$ subchain
$b_1$	The length of the bar $H_1{H}_2$ in the crank
$b_2$	The length of the bar $H_3{H}_4$ in the crank

The coordinates $(x_2,y_2)$ and $(x_3,y_3)$ can be gotten from the coordinate of $C_1$ , $(x,y)$ , the lengths of $H_1{H}_2$ (and $H_3{H}_4$ ), and the orientation angle $\varphi$ , as follows. Detailed kinematic derivation can be found in our previous paper ²⁷

$$\left[\begin{array}{c}{x}_2\hfill \\ \multicolumn{1}{c}{y_2}\end{array}\right]=\left[\begin{array}{c}x\hfill \\ \multicolumn{1}{c}{y}\end{array}\right]+\left[\begin{array}{cc}\cos \varphi \hfill & -\sin \varphi \hfill \\ \multicolumn{1}{c}{\sin \varphi }& \cos \varphi \hfill \end{array}\right]\left[\begin{array}{c}{b}_1\hfill \\ \multicolumn{1}{c}{0}\end{array}\right]$$

(2)

$$\left[\begin{array}{c}{x}_3\hfill \\ \multicolumn{1}{c}{y_3}\end{array}\right]=\left[\begin{array}{c}x\hfill \\ \multicolumn{1}{c}{y}\end{array}\right]+\left[\begin{array}{cc}\cos \varphi \hfill & -\sin \varphi \hfill \\ \multicolumn{1}{c}{\sin \varphi }& \cos \varphi \hfill \end{array}\right]\left[\begin{array}{c}{b}_1-b_2\hfill \\ \multicolumn{1}{c}{0}\end{array}\right]$$

(3)

Acceleration analysis and dynamic modeling

In this section, the extreme acceleration capability is analyzed for optimal design of the V3 robot. Its dynamic model is then derived in detail.

Acceleration analysis

For an n-DoF non-redundant manipulator, its Jacobian matrix $\mathit{J}\in {\mathbb{R}}^{n\times n}$ maps the actuated joint velocity vector $\stackrel{·}{\mathbf{\Theta }}\in {\mathbb{R}}^n$ to the end-effector velocity vector $\stackrel{·}{\mathit{X}}\in {\mathbb{R}}^n$

$$\stackrel{·}{\mathit{X}}=\mathit{J}(\mathit{X},\mathbf{\Theta })\stackrel{·}{\mathbf{\Theta }}$$

(4)

where $\mathbf{\Theta }=({\theta }_1,\dots ,{\theta }_n)\in {\mathbb{R}}^n$ is the vector of actuated joint variables and $\mathit{X}=(x_1,\dots ,x_n)\in {\mathbb{R}}^n$ is the vector of end-effector position.

Differentiating equation (4) with respect to time, the relation between the end-effector acceleration vector $\stackrel{··}{\mathit{X}}\in {\mathbb{R}}^n$ and the joint velocity vector $\stackrel{·}{\mathbf{\Theta }}$ and the acceleration vector $\stackrel{··}{\mathbf{\Theta }}\in {\mathbb{R}}^n$ is obtained

$$\stackrel{··}{\mathit{X}}=\mathit{J}\stackrel{··}{\mathbf{\Theta }}+\stackrel{·}{\mathit{J}}\stackrel{·}{\mathbf{\Theta }}$$

(5)

For a planar parallel manipulator like the V3 robot, the dynamic model can be expressed in joint space as follows^28,29

$$\mathit{M}(\mathbf{\Theta })\stackrel{··}{\mathbf{\Theta }}+\mathit{C}(\mathbf{\Theta },\stackrel{·}{\mathbf{\Theta }})\stackrel{·}{\mathbf{\Theta }}=\mathbf{\tau }+{\mathit{A}}^T\mathbf{\lambda }$$

(6)

where $\mathbf{\tau }\in {\mathbb{R}}^n$ is the vector of actuated joint torques, $\mathit{M}(\mathbf{\Theta })\in {\mathbb{R}}^{n\times n}$ is the mass matrix, $\mathit{C}(\mathbf{\Theta },\stackrel{·}{\mathbf{\Theta }})\in {\mathbb{R}}^{n\times n}$ is the Coriolis matrix, $\mathbf{\lambda }\in {\mathbb{R}}^n$ is the vector of Lagrange multipliers, and the matrix $\mathit{A}(\mathbf{\Theta })$ is derived by differentiating the loop-closure equation and satisfies the Pfaffian constraint ³⁰

$$\mathit{A}(\mathbf{\Theta })\stackrel{·}{\mathbf{\Theta }}=0,\forall \mathbf{\Theta }\in {\mathbb{R}}^n$$

(7)

The matrix $\mathit{A}$ has the property

$${\mathit{J}}^{-T}{\mathit{A}}^T\lambda =0,\forall \mathbf{\lambda }\in {\mathbb{R}}^n$$

(8)

Substituting equations (4) and (5) into equation (6) leads to the dynamics in task space relating the actuator torque to the end-effector acceleration and velocity

$$\mathit{M}{\mathit{J}}^{-1}\stackrel{··}{\mathit{X}}+(\mathit{C}{\mathit{J}}^{-1}-\mathit{M}{\mathit{J}}^{-1}\stackrel{·}{\mathit{J}}{\mathit{J}}^{-1})\stackrel{·}{\mathit{X}}=\mathbf{\tau }+{\mathit{A}}^T\mathbf{\lambda }$$

(9)

By left multiplying ${\mathit{J}}^{-T}$ to both sides of equation (9) and applying equation (8), we obtain

$${\mathit{J}}^{-T}\mathit{M}{\mathit{J}}^{-1}\stackrel{··}{\mathit{X}}+{\mathit{J}}^{-T}(\mathit{C}{\mathit{J}}^{-1}-\mathit{M}{\mathit{J}}^{-1}\stackrel{·}{\mathit{J}}{\mathit{J}}^{-1})\stackrel{·}{\mathit{X}}={\mathit{J}}^{-T}\mathbf{\tau }$$

(10)

Given $\mathbf{\tau }$ , the acceleration $\stackrel{··}{\mathit{X}}$ in task space depends not only on the pose $\mathit{X}$ , but also on the current velocity $\stackrel{·}{\mathit{X}}$ . In this article, we investigate the acceleration capability in equation (10) with a null end-effector velocity, ³¹ that is, the starting acceleration from rest. This end-effector acceleration can be written as

$$\stackrel{··}{\mathit{X}}=\mathit{H}(\mathit{X})\mathbf{\tau }$$

(11)

where $\mathit{H}=\mathit{J}{\mathit{M}}^{-1}$ . This acceleration may be at the starting point of a pick-and-place trajectory where a high acceleration is specifically required.

Equation (11) provides a linear relation to compute possible end-effector accelerations depending on different configurations. Suppose that the torques are identical for all actuated joints and they are normalized by

$$|{\tau }_i|\le 1,i=1,\dots ,n$$

(12)

where ${\tau }_i$ is the $i\mathrm{th}$ element of the torque vector $\mathbf{\tau }$ . Therefore, given a configuration, we can compute acceleration component in task space

$${\stackrel{··}{\mathrm{x}}}_i=\sum _{j=1}^n{H}_{\mathrm{ij}}{\tau }_j$$

(13)

where $H_{\mathrm{ij}}$ , $i,j=1,\dots ,n$ is the entry at the $i\mathrm{th}$ row and the $j\mathrm{th}$ column in the matrix $\mathit{H}$ . The extreme translation acceleration $a_e$ and the extreme rotation acceleration ${\alpha }_e$ can be computed using the corresponding axis components

$$a_e=\underset{{\tau }_i\in [-1,1]}{\max }\sqrt{{\stackrel{··}{\mathrm{x}}}_1^2+\cdots +{\stackrel{··}{\mathrm{x}}}_m^2}$$

(14)

$${\alpha }_e=\underset{{\tau }_i\in [-1,1]}{\max }\sqrt{{\stackrel{··}{\mathrm{x}}}_{m+1}^2+\cdots +{\stackrel{··}{\mathrm{x}}}_n^2}$$

(15)

where ${\stackrel{··}{\mathrm{x}}}_1,\dots ,{\stackrel{··}{\mathrm{x}}}_m$ and ${\stackrel{··}{\mathrm{x}}}_{m+1},\dots ,{\stackrel{··}{\mathrm{x}}}_n$ are translational acceleration components and orientation acceleration components, respectively. Those two quantities represent the extreme acceleration capability respectively in translation and rotation, which are the potential maximum translation acceleration and the potential maximum rotation acceleration of the end-effector, when the robot is stationary at a configuration. They can thus be used as acceleration indices in optimal design to evaluate the potentials of the accelerations.

In order to calculate $a_e$ and ${\alpha }_e$ , the mass matrix $\mathit{M}$ and the Jacobian matrix $\mathit{J}$ should first be derived. Detailed derivation of the Jacobian matrix can be found in our previous paper. ²⁷ The mass matrix is derived by the dynamic modeling in the next subsection.

Dynamic modeling

The modeling procedure for Delta robot ³² is applied to calculate the mass matrix of the V3 robot. According to Lagrangian formulation, the mass matrix of a robot can be computed from its kinetic energy. The kinetic energy of a rigid body $i$ can be written as

$$T_i=\frac{1}{2}[m_i{\mathit{v}}_i^T{\mathit{v}}_i+{\mathbf{\omega }}_i^T{\mathit{I}}_i{\mathbf{\omega }}_i]$$

(16)

where ${\mathbf{v}}_i$ is the translational velocity vector of the mass center of the rigid body, ${\mathbf{\omega }}_i$ is the angular velocity vector, $m_i$ is the mass of the body, and ${\mathit{I}}_i$ is the inertia matrix. The kinetic energy of the whole robot is the summation of the kinetic energies of all bodies

$$T=\sum _{i=1}^p{T}_i$$

(17)

where $p$ denotes the total number of moving bodies. Suppose that the Jacobian matrices ${\mathit{J}}_{v,i}$ and ${\mathit{J}}_{\omega ,i}$ map the joint velocity respectively to the mass center translational velocity and the angular velocity

$$\begin{array}{c}{\mathit{v}}_i={\mathit{J}}_{v,i}\stackrel{·}{\mathbf{\Theta }}\hfill \\ {\mathbf{\omega }}_i={\mathit{J}}_{\omega ,i}\stackrel{·}{\mathbf{\Theta }}\hfill \end{array}$$

(18)

Substituting equations (16) and (18) into equation (17) leads to

$$T=\frac{1}{2}{\stackrel{·}{\mathbf{\Theta }}}^T\mathit{M}(\mathbf{\Theta })\stackrel{·}{\mathbf{\Theta }}$$

(19)

where $\mathit{M}$ is the robot mass matrix, defined by

$$\mathit{M}=\sum _{i=1}^p(m_i{\mathit{J}}_{v,i}^T{\mathit{J}}_{v,i}+{\mathit{J}}_{\omega ,i}^T{\mathit{I}}_i{\mathit{J}}_{\omega ,i})$$

(20)

From this equation, the mass matrix is formulated based on the Jacobian matrices. Thus, the main task is seeking the Jacobian matrices of the robot.

The velocity of each body can be calculated by the velocities at both of its extremities. As shown in Figure 4, the kinetic energy of the rigid bar can thus be obtained by $v_1$ and $v_2$ at the extremities ³²

$$\begin{array}{c}T=\frac{1}{2}[\frac{1}{3}m({\mathit{v}}_1^T{\mathit{v}}_1+{\mathit{v}}_1^T{\mathit{v}}_2+{\mathit{v}}_2^T{\mathit{v}}_2)]\hfill \end{array}$$

(21)

Figure 4.

Velocities along a rigid bar.

Introducing the Jacobians ${\mathit{J}}_1$ and ${\mathit{J}}_2$ , the mass matrix of the bar is obtained

$${\mathit{M}}_{\mathrm{bar}}=\frac{1}{3}m({\mathit{J}}_1^T{\mathit{J}}_1+{\mathit{J}}_1^T{\mathit{J}}_2+{\mathit{J}}_2^T{\mathit{J}}_2)$$

(22)

The V3 robot consists of seven moving rigid bodies, three actuated arms (the mass matrix ${\mathit{M}}_{\mathrm{AB}}$ ), three passive arms (the mass matrix ${\mathit{M}}_{\mathrm{BC}}$ ), and the end-effector (the mass matrix ${\mathit{M}}_{\mathrm{end}}$ ). The robot mass matrix is thus the summation of the contributions of all bodies

$$\mathit{M}={\mathit{M}}_{\mathrm{AB}}+{\mathit{M}}_{\mathrm{BC}}+{\mathit{M}}_{\mathrm{end}}$$

(23)

The contribution of the actuated arms can be readily derived as follows

$${\mathit{M}}_{\mathrm{AB}}={\mathit{I}}_{\mathrm{AB}}=\left[\begin{array}{ccc}{I}_1\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& I_2\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & I_3\hfill \end{array}\right]$$

(24)

where $I_i=I_m+l_{i1}^2(m_{A_i{B}_i}/3+m_c)$ , $i=1,2,3$ , $I_m$ is the inertia of the motor (if there exists a gear box, the inertia must be the combined contribution), $m_{A_i{B}_i}$ is the mass of the actuated arm of the $i\mathrm{th}$ subchain, $R_i$ is the radius of the cylindrical bar, and $m_c$ is the elbow mass.

The contribution of each passive link to the robot mass matrix can be obtained based on equation (22)

$${\mathit{M}}_{B_i{C}_i}=\frac{1}{3}{m}_{B_i{C}_i}({\mathit{J}}_{B_i}^T{\mathit{J}}_{B_i}+{\mathit{J}}_{B_i}^T{\mathit{J}}_{C_i}+{\mathit{J}}_{C_i}^T{\mathit{J}}_{C_i})$$

(25)

The Jacobians are presented in detail in the next section. Then, the mass matrix contributed by the three passive arms is the summation of ${\mathit{M}}_{B_i{C}_i}$ , $i=1,\dots ,3$

$${\mathit{M}}_{\mathrm{BC}}=\sum _{i=1}^3{\mathit{M}}_{B_i{C}_i}$$

(26)

By direct application of equation (20) to the end-effector, its mass matrix is

$${\mathit{M}}_{\mathrm{end}}=m_{\mathrm{end}}{\mathit{J}}_v^T{\mathit{J}}_v+{\mathit{J}}_{\omega }^T{I}_{\mathrm{end}}{\mathit{J}}_{\omega }$$

(27)

where $m_{\mathrm{end}}$ is the mass of both the end-effector and the payload, and $I_{\mathrm{end}}$ is their inertia. They are the summation of the contribution of each part (see Figure 5)

$$m_{\mathrm{end}}=\sum _{i=1}^5{m}_i+m_{\mathrm{load}}$$

(28)

$$I_{\mathrm{end}}=\sum _{i=1}^5{I}_i+I_{\mathrm{load}}$$

(29)

where

$$\begin{array}{c}{I}_1=\frac{1}{2}{m}_1{r}^2\hfill \\ \multicolumn{1}{c}{I_2=\frac{1}{12}{m}_2(b_1^2+3r^2)+m_2{\left(\frac{b_1}{2}\right)}^2}\\ \multicolumn{1}{c}{I_3=\frac{1}{2}{m}_3{r}^2+m_3{b}_1^2}\\ \multicolumn{1}{c}{I_4=\frac{1}{12}{m}_4(b_2^2+3r^2)+m_4{(b_1-\frac{b_2}{2})}^2}\\ \multicolumn{1}{c}{I_5=\frac{1}{2}{m}_5{r}^2+m_5{(b_1-b_2)}^2}\end{array}$$

Figure 5.

Masses and inertias of the end-effector.

It is supposed that the end-effector is made up of cylinder bars with an identical radius $r$ . The payload inertia $I_{\mathrm{load}}$ depends on its material and geometric shape.

By combining equations (24), (26) and (27), the mass matrix of V3 manipulator is obtained as follows

$$\begin{array}{cc}\multicolumn{1}{c}{\mathit{M}=}& {\mathit{I}}_{\mathrm{AB}}+m_{\mathrm{end}}{\mathit{J}}_v^T{\mathit{J}}_v+{\mathit{J}}_{\omega }^T{I}_{\mathrm{end}}{\mathit{J}}_{\omega }\hfill \\ \multicolumn{1}{c}{}& +\sum _{i=1}^3\frac{1}{3}{m}_{B_i{C}_i}({\mathit{J}}_{B_i}^T{\mathit{J}}_{B_i}+{\mathit{J}}_{B_i}^T{\mathit{J}}_{C_i}+{\mathit{J}}_{C_i}^T{\mathit{J}}_{C_i})\hfill \end{array}$$

(30)

Since our robot is a planar manipulator, the potential energy keeps constant during motion. Therefore, the Coriolis matrix is dependent only on the robot mass matrix. Based on Lagrangian formulation, it can be derived as follows

$$C_{\mathrm{ij}}=\frac{1}{2}\sum _{k=1}^3(\frac{\partial M_{\mathrm{ij}}}{\partial {\theta }_k}+\frac{\partial M_{\mathrm{ik}}}{\partial {\theta }_j}-\frac{\partial M_{\mathrm{kj}}}{\partial {\theta }_i}){\stackrel{·}{\theta }}_k$$

(31)

Substituting equations (30) and (31) into equation (6), the dynamic model of the V3 robot is thus derived.

Jacobian matrices

For the end-effector of the V3 robot, assume its translational velocity $\mathit{v}=[\begin{array}{ccc}\stackrel{·}{x}\hfill & \stackrel{·}{y}\hfill & 0\hfill \end{array}{]}^T$ and its rotational velocity $\mathbf{\omega }=[\begin{array}{ccc}0\hfill & 0\hfill & \stackrel{·}{\varphi }\hfill \end{array}{]}^T$ . For the passive arms, the velocity at $B_i$ can be calculated as follows

$${\mathit{V}}_{B_i}=\left[\begin{array}{c}-l_{i1}{\stackrel{·}{\theta }}_i\sin {\theta }_i\hfill \\ \multicolumn{1}{c}{l_{i1}{\stackrel{·}{\theta }}_i\cos {\theta }_i}\\ \multicolumn{1}{c}{0}\end{array}\right]i=1,\dots ,3$$

(32)

which can be written in the form of matrix multiplication with $\stackrel{·}{\mathbf{\Theta }}=[{\theta }_1,{\theta }_2,{\theta }_3{]}^T$

$${\mathit{V}}_{B_1}=\left[\begin{array}{ccc}-l_{11}\sin {\theta }_1\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{l_{11}\cos {\theta }_1}& 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill \end{array}\right]\stackrel{·}{\mathbf{\Theta }}={\mathit{J}}_{B_1}\stackrel{·}{\mathbf{\Theta }}$$

(33)

$${\mathit{V}}_{B_2}=\left[\begin{array}{ccc}0\hfill & -l_{21}\sin {\theta }_2\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& l_{21}\cos {\theta }_2\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill \end{array}\right]\stackrel{·}{\mathbf{\Theta }}={\mathit{J}}_{B_2}\stackrel{·}{\mathbf{\Theta }}$$

(34)

$${\mathit{V}}_{B_3}=\left[\begin{array}{ccc}0\hfill & 0\hfill & -l_{31}\sin {\theta }_3\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & l_{31}\cos {\theta }_3\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill \end{array}\right]\stackrel{·}{\mathbf{\Theta }}={\mathit{J}}_{B_3}\stackrel{·}{\mathbf{\Theta }}$$

(35)

where we identify the Jacobians as ${\mathit{J}}_{B_1}$ , ${\mathit{J}}_{B_2}$ , and ${\mathit{J}}_{B_3}$ , respectively.

Before calculating the velocity of the end $C_i$ that links to the end-effector, let us define two matrices, ${\mathit{J}}_v\in {\mathbb{R}}^{3\times 3}$ with the third row being $0_{1\times 3}$ and ${\mathit{J}}_{\omega }\in {\mathbb{R}}^{3\times 3}$ with the first two rows being both $0_{1\times 3}$ , satisfying $\mathit{J}={\mathit{J}}_v+{\mathit{J}}_{\omega }$ . The velocity ${\mathit{V}}_{C_i}$ can be calculated as follows

$${\mathit{V}}_{C_1}=\mathit{v}={\mathit{J}}_v\stackrel{·}{\mathbf{\Theta }}={\mathit{J}}_{C_1}\stackrel{·}{\mathbf{\Theta }}$$

(36)

$$\begin{array}{cc}\multicolumn{1}{c}{{\mathit{V}}_{C_2}}& =\mathit{v}+\mathbf{\omega }\times {\mathit{H}}_1{\mathit{H}}_2=\left[\begin{array}{c}\stackrel{·}{x}+(-b_1\stackrel{·}{\varphi }\sin \varphi )\hfill \\ \multicolumn{1}{c}{\stackrel{·}{y}+b_1\stackrel{·}{\varphi }\cos \varphi }\\ \multicolumn{1}{c}{0}\end{array}\right]\hfill \\ \multicolumn{1}{c}{}& =[{\mathit{J}}_v+\left[\begin{array}{ccc}0\hfill & 0\hfill & -b_1\sin \varphi \hfill \\ \multicolumn{1}{c}{0}& 0\hfill & b_1\cos \varphi \hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill \end{array}\right]{\mathit{J}}_{\omega }]\stackrel{·}{\mathbf{\Theta }}={\mathit{J}}_{C_2}\stackrel{·}{\mathbf{\Theta }}\hfill \end{array}$$

(37)

$$\begin{array}{cc}\multicolumn{1}{c}{{\mathit{V}}_{C_3}}& =v+\mathbf{\omega }\times ({\mathit{H}}_1{\mathit{H}}_2+{\mathit{H}}_3{\mathit{H}}_4)\hfill \\ \multicolumn{1}{c}{}& =\left[\begin{array}{c}\stackrel{·}{x}+(-(b_1-b_2)\stackrel{·}{\varphi }\sin \varphi )\hfill \\ \multicolumn{1}{c}{\stackrel{·}{y}+(b_1-b_2)\stackrel{·}{\varphi }\cos \varphi }\\ \multicolumn{1}{c}{0}\end{array}\right]\hfill \\ \multicolumn{1}{c}{}& =[{\mathit{J}}_v+\left[\begin{array}{ccc}0\hfill & 0\hfill & -(b_1-b_2)\sin \varphi \hfill \\ \multicolumn{1}{c}{0}& 0\hfill & (b_1-b_2)\cos \varphi \hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill \end{array}\right]{\mathit{J}}_{\omega }]\stackrel{·}{\mathbf{\Theta }}\hfill \\ \multicolumn{1}{c}{}& ={\mathit{J}}_{C_3}\stackrel{·}{\mathbf{\Theta }}\hfill \end{array}$$

(38)

where we identify the Jacobians as ${\mathit{J}}_{C_1}$ , ${\mathit{J}}_{C_2}$ , and ${\mathit{J}}_{C_3}$ , respectively. The derived Jacobian matrices ${\mathit{J}}_{C_i}$ and ${\mathit{J}}_{B_i}$ , $i=1,\dots ,3$ , are used in the last section.

The Jacobian relating the end-effector velocity vector $\stackrel{·}{\mathit{X}}=(\stackrel{·}{x},\stackrel{·}{y},\stackrel{·}{\varphi }{)}^T$ and the joint velocity vector $\stackrel{·}{\mathbf{\Theta }}=({\stackrel{·}{\theta }}_1,{\stackrel{·}{\theta }}_2,{\stackrel{·}{\theta }}_3{)}^T$ are as follows

$${\mathit{J}}_x\stackrel{·}{X}={\mathit{J}}_{\theta }\stackrel{·}{\mathbf{\Theta }}$$

(39)

where

$$\begin{array}{c}{\mathit{J}}_x=\left[\begin{array}{cc}{\mathit{B}}_1{\mathit{C}}_1·\mathit{E}\hfill & (0\times {\mathit{B}}_1{\mathit{C}}_1)·z\hfill \\ \multicolumn{1}{c}{{\mathit{B}}_2{\mathit{C}}_2·\mathit{E}}& ({\mathit{H}}_1{\mathit{H}}_2\times {\mathit{B}}_2{\mathit{C}}_2)·z\hfill \\ \multicolumn{1}{c}{{\mathit{B}}_3{\mathit{C}}_3·\mathit{E}}& (({\mathit{H}}_1{\mathit{H}}_2+{\mathit{H}}_3{\mathit{H}}_4)\times {\mathit{B}}_3{\mathit{C}}_3)·z\hfill \end{array}\right]\hfill \\ {\mathit{J}}_{\theta }=\left[\begin{array}{ccc}J{\theta }_1\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& J_{{\theta }_2}\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & J_{{\theta }_3}\hfill \end{array}\right]\hfill \end{array}$$

and

$$\begin{array}{c}\multicolumn{1}{c}{J_{{\theta }_i}=({\mathit{A}}_i{\mathit{B}}_i\times {\mathit{B}}_i{\mathit{C}}_i)·z,i=1,\dots ,3}\\ \multicolumn{1}{c}{\mathit{E}=\left[\begin{array}{cc}1\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 1\hfill \\ \multicolumn{1}{c}{0}& 0\hfill \end{array}\right],z=\left[\begin{array}{c}0\hfill \\ \multicolumn{1}{c}{0}\\ \multicolumn{1}{c}{1}\end{array}\right]}\end{array}$$

For convenience of representation, it is assumed that ${\mathit{A}}_i{\mathit{B}}_i$ , ${\mathit{B}}_i{\mathit{C}}_i$ , $i=1,2,3$ , ${\mathit{H}}_1$ ${\mathit{H}}_2$ and ${\mathit{H}}_3$ ${\mathit{H}}_4$ are three-dimensional row vectors. The last elements in the vectors are all zeros since the vectors are indeed planar vectors. The overall Jacobian matrix can be obtained by $\mathit{J}={{\mathit{J}}_x}^{-1}{\mathit{J}}_{\theta }$ . The detailed derivation and singularity analysis can be found in our previous paper. ²⁷

Bending deflection and bending stiffness

Since each subchain of the V3 robot is comparable to a cantilever beam structure, especially when the subchain is in its extremely stretched configuration. The bending deformation due to gravity (robot itself and payload) may become so large that the rotation axis of the end-effector would not be perpendicular to the XY plane, which can lead to manipulation failure. In this section, the deflection of the end of each subchain is derived and is included in design.

Instead of investigating the deflection in the whole workspace, the extreme configurations are used as the worst cases to characterize the deformation. In those configurations, the bending deflections are maximum. Figure 6 shows the subchain bending model, where $(E{I}_z{)}_{\mathrm{ij}}(i=1,\dots ,3,j=1,2)$ is the flexural rigidity of the $j\mathrm{th}$ link in the $i\mathrm{th}$ subchain, E is Young’s modulus, $I_z$ is the cross-sectional moment of inertia, $G_{\mathrm{ij}}$ is the gravity of the jth link in the ith subchain, ${\delta }_i$ is the deflection of the free end of $i\mathrm{th}$ subchain, and $F_i$ is the equivalent force at $C_i$ due to the gravity of the end-effector and the payload. The deflection of the free end $C_i$ can be derived by considering it in two separate cases (Figures 7 and 8).

• Link $A_i{B}_i$ is rigid while link B_iC_i is deformed. The force F_i produces a deflection

$${\delta }_{F_i}=\frac{F_i{l}_{i2}^3}{3{(E{I}_z)}_{i2}}$$

Figure 6.

The subchain bending model.

Figure 7.

The bending model of link $B_i{C}_i$ .

Figure 8.

The bending model of link $A_i{B}_i$ .

The deflection due to the gravity $G_{i2}$ of the link $B_i{C}_i$ is

$${\delta }_{G_{i2}}=\frac{G_{i2}{\left(\frac{l_{i2}}{2}\right)}^2}{6{(E{I}_z)}_{i2}}(3l_{i2}-\frac{l_{i2}}{2})=\frac{5G_{i2}{l}_{i2}^3}{48{(E{I}_z)}_{i2}}$$

The total deflection in this case is summation of the above two parts

$${\delta }_{i,1}={\delta }_{F,i}+{\delta }_{G_{i2}}$$

(40)

• Link $A_i{B}_i$ is deformed while link $B_i{C}_i$ is rigid. This case is more complicated than the case above. Let us first investigate the deflection at the point $B_i$ . The self weight $G_{i1}$ produces deflection and angle deflection ${\vartheta }_{G_{i1}}$ as follows

$$\begin{array}{c}{\delta }_{G_{i1}}=\frac{G_{i1}{\left(\frac{l_{i1}}{2}\right)}^2}{6{(E{I}_z)}_{i1}}(3l_{i1}-\frac{l_{i1}}{2})=\frac{5G_{i1}{l}_{i1}^3}{48{(E{I}_z)}_{i1}}\hfill \\ {\vartheta }_{G_{i1}}=\frac{G_{i1}{\left(\frac{l_{i1}}{2}\right)}^2}{2{(E{I}_z)}_{i1}}\hfill \end{array}$$

Furthermore, the weight of the second link $G_{i2}$ and the equivalent force $F_i$ also produce deflection. The corresponding force and moment at $B_i$ are $N_i=G_{i2}+F_i$ and $M_i=G_{i2}(l_{i2}/2)+F_i{l}_{i2}$ , respectively. The force $N_i$ produces deflection and angle deflection as follows

$$\begin{array}{c}{\delta }_{N_i}=\frac{N_i{l}_{i1}^3}{3{(E{I}_z)}_{i1}}\hfill \\ {\vartheta }_{N_i}=\frac{N_i{l}_{i1}^2}{2{(E{I}_z)}_{i1}}\hfill \end{array}$$

The deflection and the angle deflection due to the moment $M_i$ are respectively

$$\begin{array}{c}{\delta }_{M_i}=\frac{M_i{l}_{i1}^2}{2{(E{I}_z)}_{i1}}\hfill \\ {\vartheta }_{M_i}=\frac{M_i{l}_{i1}}{{(E{I}_z)}_{i1}}\hfill \end{array}$$

Therefore, the total deflection on the free end $C_i$ in this case is

$${\delta }_{i,2}={\delta }_{G_{i1}}+{\delta }_{N_i}+{\delta }_{M_i}+{\delta }_{l,i}$$

(41)

where ${\delta }_{l,i}$ is the deflection at $C_i$ due to angle deflections

$${\delta }_{l,i}=({\vartheta }_{G_{i1}}+{\vartheta }_{N_i}+{\vartheta }_{M_i})l_{i2}$$

(42)

Note that the assumption of small angle deflection is applied implicitly in the above equation. In the worst case, the maximum bending deflection of each subchain is the summation of contributions of those two cases

$${\delta }_i={\delta }_{i,1}+{\delta }_{i,2}$$

(43)

The quantity ${\delta }_i$ is used to characterize the bending stiffness of the V3 robot. A large value of ${\delta }_i$ may lead to improper manipulation of the robot.

Optimal design

In this section, important performance indices for a pick-and-place manipulator are presented and a design problem is formulated and solved.

Performance indices

For pick-and-place manipulations, velocity is the ultimate goal to shorten the cycle time. Due to the limited motion distance and thus the very short acceleration time, however, the velocity usually would not reach the speed upper bound imposed by actuators. The acceleration capability becomes the bottleneck to improve production efficiency. Velocity and acceleration are the crucial performance indices to characterize efficiency. Accuracy and dexterity are two additional important indices so that the manipulator is able to accomplish the task correctly and dextrously. For the special structure of the V3 robot, the bending stiffness index of each subchain is utilized to reflect manipulation properness.

The acceleration index and the bending stiffness index are presented in section “Dynamic modeling” and “Bending deflection and bending stiffness.” Brief derivation of velocity, accuracy, and dexterity indices is given as follows.

Velocity

Suppose that the velocity of each actuated joint is bounded identically

$$|{\stackrel{·}{\theta }}_i|\le 1,i=1,\dots ,n$$

(44)

Note that the bounds for actuator velocity are normalized. Given a configuration, the extreme (or supremum) velocity can be defined as

$${\stackrel{·}{x}}_i=\sum _{j=1}^n{J}_{\mathrm{ij}},{\stackrel{·}{\theta }}_j$$

(45)

where $J_{\mathrm{ij}}$ , $i,j=1,\dots ,n$ is the element at the ith row and the jth column in the Jacobian matrix $\mathit{J}$ . This quantity represents the independent axis velocity in the Cartesian coordinate frame. The extreme translation velocity $v_e$ and the extreme rotation velocity ${\omega }_e$ can be computed using the corresponding axis components

$$v_e=\underset{{\stackrel{·}{\theta }}_i\in [-1,1]}{\max }\sqrt{{\stackrel{·}{\mathrm{x}}}_1^2+\cdots +{\stackrel{·}{\mathrm{x}}}_m^2}$$

(46)

$${\omega }_e=\underset{{\stackrel{·}{\theta }}_i\in [-1,1]}{\max }\sqrt{{\stackrel{·}{\mathrm{x}}}_{m+1}^2+\cdots +{\stackrel{·}{\mathrm{x}}}_n^2}$$

(47)

Accuracy

Jacobian matrix $\mathit{J}$ not only establishes a relation between the actuated joint velocity and the end-effector velocity, but also maps the actuated joint error $\Delta \mathbf{\Theta }$ to the end-effector error $\Delta \mathit{X}$ , that is, $\Delta \mathit{X}=\mathit{J}\Delta \mathbf{\Theta }$ . Assume that the actuated joint errors are bounded and normalized

$$|\Delta {\theta }_i|\le 1,i=1,\dots ,n$$

(48)

similar to equation (45), the independent axis error in Cartesian space can be calculated for each axis as follows

$$\Delta {\mathrm{x}}_i=\sum _{j=1}^n{J}_{\mathrm{ij}}\Delta {\theta }_j$$

(49)

The extreme translation error $p_e$ and the extreme rotation error ${\varphi }_e$ can be computed using the corresponding axis components

$$\Delta p_e=\underset{\Delta {\theta }_i\in [-1,1]}{\max }\sqrt{\Delta {\mathrm{x}}_1^2+\cdots +\Delta {\mathrm{x}}_m^2}$$

(50)

$$\Delta {\varphi }_e=\underset{\Delta {\theta }_i\in [-1,1]}{\max }\sqrt{\Delta {\mathit{x}}_{m+1}^2+\cdots +\Delta {\mathit{x}}_n^2}$$

(51)

Dexterity

Dexterity index is utilized to characterize the effectiveness of the workspace. As frequently used in the literatures, the inverse condition number of the kinematic Jacobian matrix is taken to measure the dexterity of a manipulator. Since the velocities of the V3 robot have both translational and rotational components, not all the units of element in the Jacobian are the same. In order to compute the inverse condition number, a characteristic length $L$ is applied to homogenize the Jacobian. ³³ The forward kinematic Jacobian matrix ${\mathit{J}}_x$ is modified to ${\mathit{J}}_x^{*}$

$${\mathit{J}}_x^{*}=\left[\begin{array}{cc}{\mathit{B}}_1{\mathit{C}}_1·\mathit{E}\hfill & {(0\times {\mathit{B}}_1{\mathit{C}}_1)}^{T}z/L\hfill \\ \multicolumn{1}{c}{{\mathit{B}}_2{\mathit{C}}_2·\mathit{E}}& {({\mathit{H}}_1{\mathit{H}}_2\times {\mathit{B}}_2{\mathit{C}}_2)}^{T}z/L\hfill \\ \multicolumn{1}{c}{{\mathit{B}}_3{\mathit{C}}_3·\mathit{E}}& {(({\mathit{H}}_1{\mathit{H}}_2+{\mathit{H}}_3{\mathit{H}}_4)\times {\mathit{B}}_3{\mathit{C}}_3)}^{T}z/L\hfill \end{array}\right]$$

and the kinematics Jacobian matrix becomes ${\mathit{J}}^{*}=({\mathit{J}}_x^{*}{)}^{-1}{\mathit{J}}_{\theta }$ . The dexterity index can be defined as the inverse condition number of ${\mathit{J}}^{*}$

$$\kappa ({\mathit{J}}^{*})=\frac{{\sigma }_{\min }({\mathit{J}}^{*})}{{\sigma }_{\max }({\mathit{J}}^{*})}$$

(52)

where ${\sigma }_{\min }(·)$ and ${\sigma }_{\max }(·)$ are respectively the minimal and maximal singular values of a matrix. In this article, the characteristic length is chosen from the lengths of the end-effector $L=b_1-b_2$ , since it is the velocity transformation ratio from rotation to translation at the end-effector.

Note that the normalized bounds in velocity, acceleration, and accuracy indices are applied for the selection in practice. When the bounds are enlarged, the increased scales of the indices are the same. One example is in the case study.

Problem formulation

The rotational symmetry of the V3 robot architecture leads to a rotational symmetry of its performance. Its characteristics only change in radial direction. Therefore, it is not necessary to carry out performance optimization in the whole workspace. We need only to investigate the performance radially in the workspace. Without loss of generality, performances are analyzed along the radius on the $+X$ axis instead of in the whole workspace.

In general pick-and-place applications, the robot is required to manipulate workpieces in all orientations (excepting singular configurations). Therefore, we focus ourselves on the dextrous workspace, the set of positions with ${360}^{\circ }$ rotation capability of the end-effector, to evaluate the workspace size and quality. The dextrous workspace of V3 manipulator is a torus, whose inner radius $r_d$ and outer radius $R_d$ can be derived as follows ²⁷

$$\begin{array}{l}{r}_d=\max \left\{\left|l_{11}-l_{12}\right|,\left|l_{21}-l_{22}\right|+b_1,\left|l_{31}-l_{32}\right|+\left|b_1-b_2\right|\right\}\\ R_d=\min \left\{l_{11}+l_{12},l_{21}+l_{22}-b_1,l_{31}+l_{32}-|b_1-b_2|\right\}\end{array}$$

The dextrous workspace of the V3 robot is also rotationally symmetric about the common axis. A radial workspace (e.g. along the +X-axis) is taken as the representative, which is discretized for optimization computation. When the link lengths are given, the dextrous workspace is defined. The average dexterity at a position point on the radial workspace is defined as the average of the inverse condition number of the homogenized Jacobian in full 360° orientation. Similarly, the average extreme velocity, the average extreme acceleration, and the average extreme error are defined to characterize the extreme velocity, the extreme acceleration, and the extreme error at the position point. The bending stiffness index is calculated based on the materials and geometry parameters. Given a position point on a radius in the dextrous workspace, all concerned indices are evaluated. If they all lie within given bounds, the position point is called a good point. The design objective is to maximize the number of good points, or in other words, to maximize the area in dextrous workspace having specified good performances.

According to the kinematics and dynamics, the design variable vector is $\mathbf{\eta }=(l_{11},l_{12},l_{21},l_{22},l_{31},l_{32},b_1,b_2,r_1,r_2)$ which consists of the link lengths and the radii of cross sections of the actuated and the passive links, $r_1$ and $r_2$ , respectively. Here, it is assumed that the three actuated links are cylindrical bars with an identical cross section. This is also the case for the three passive links. The radius of the bars of the end-effector is a given constant for practical design reasons. The optimal design problem is thus translated into the following optimization problem

$$\begin{array}{cc}\underset{\mathbf{\eta }}{\max }\hfill & \Delta R\left(\mathbf{\eta }\right)\hfill \\ \multicolumn{1}{c}{\mathrm{subject}\mathrm{to}}& \overline{\kappa }(\mathbf{\eta },\mathbf{X})\ge {\kappa }_{\min }\hfill \\ \multicolumn{1}{c}{}& {\overline{v}}_e(\mathbf{\eta },\mathbf{X})\ge v_{\min }\hfill \\ \multicolumn{1}{c}{}& {\overline{\omega }}_e(\mathbf{\eta },\mathbf{X})\ge {\omega }_{\min }\hfill \\ \multicolumn{1}{c}{}& {\overline{a}}_e(\mathbf{\eta },\mathbf{X})\ge a_{\min }\hfill \\ \multicolumn{1}{c}{}& {\overline{\alpha }}_e(\mathbf{\eta },\mathbf{X})\ge {\alpha }_{\min }\hfill \\ \multicolumn{1}{c}{}& \Delta \overline{p}(\eta ,\mathbf{X})\le \Delta p_{\max }\hfill \\ \multicolumn{1}{c}{}& \Delta \overline{\varphi }(\eta ,\mathbf{X})\le \Delta {\varphi }_{\max }\hfill \\ \multicolumn{1}{c}{}& {\delta }_i\le {\delta }_{\max },i=1,\dots ,3\hfill \\ \multicolumn{1}{c}{}& l_{i1}+l_{i2}+b_1+b_2\le l,i=1,\dots ,3\hfill \end{array}$$

where $\overline{\kappa }$ is the average inverse condition number of the homogenized Jacobian characterizing dexterity; ${\overline{v}}_e$ and ${\overline{\omega }}_e$ are respectively the average extreme translational velocity and the average extreme rotational velocity at each position point; ${\overline{a}}_e$ and ${\overline{\alpha }}_e$ are respectively the average extreme translational acceleration and the average extreme rotational acceleration at each point; $\Delta \overline{p}$ and $\Delta \overline{\varphi }$ are respectively the average extreme translational error and the average extreme rotational error at each point; ${\delta }_i$ is the maximum bending deflection of the $i\mathrm{th}$ subchain; ${\kappa }_{\min }$ , $v_{\min }$ , ${\omega }_{\min }$ , ${\alpha }_{\min }$ , ${\beta }_{\min }$ , $\Delta p_{\max }$ , $\Delta {\varphi }_{\max }$ , ${\delta }_{\max }$ are bounds for dexterity, velocity, acceleration, accuracy, and bending deflection, respectively. The manipulator size is constrained by imposing the last inequality, the length of each subchain, where $l$ is a given number. The objective function $\Delta R(\eta )=R_o-R_i$ , characterizing the good-performance workspace, is the difference of the inner and the outer radii. For pick-and-place operations, the larger size of the good-performance workspace there exists, the better effect the manipulator will make.

Optimization and results

In real implementation, the manipulator size bound is given by setting $l=1\mathrm{m}$ , according to the real position workspace requirement. Assume that the peak velocity for each actuator and gear box system is $1\mathrm{rad}/\mathrm{s}$ , the maximum input error for actuators is $1\times {10}^{-4}\mathrm{rad}$ and the peak torque for each actuator and gear box system is $1\mathrm{N}\mathrm{m}$ . Based on the cycle time requirement, motion planning and the actual position and angular error requirements of the task, and after gear ratios and link lengths converting, we give the lower bound of the translational velocity $v_{\min }=0.5\mathrm{m}/\mathrm{s}$ , and the upper bound of error $\Delta p_{\max }=0.15\mathrm{mm}$ . The bound for rotational velocity is ${\omega }_{\min }=2\pi \mathrm{rad}/\mathrm{s}$ , and the orientation error is $\Delta {\varphi }_{\max }=0.002\mathrm{rad}$ . The lower bound for the dexterity is ${\kappa }_{\min }=0.2$ . The material of actuated links is chosen stainless steel. Its density is $7.9\times {10}^3\mathrm{kg}{\mathrm{m}}^3$ and Young’s modulus is 207 GPa. The same material is used for the end-effector. The material of the passive links is chosen carbon fiber. Its density is $1.8\mathrm{kg}{\mathrm{m}}^2$ and Young’s modulus is 150 GPa. The acceleration bounds are given as $a_{\min }=1\mathrm{m}/{\mathrm{s}}^2$ and ${\alpha }_{\min }=4\pi \mathrm{rad}/{\mathrm{s}}^2$ . The maximum bending deflection is set to be 0.1 mm. The radius of the bars in the end-effector is given as 5 mm. Since the payload varies depending on applications, its mass and inertia are ignored in computation. The intervals for design variables are $l_{i1},l_{i2}\in [0,1],i=1,\dots ,3$ , $b_1,b_2\in [0,0.5]$ , $r_1\in [0,0.05]$ , and $r_2\in [0,0.03]$ (m). Thus, the optimal design problem has been presented.

The problem is solved using the DE algorithm, ³⁴ a global optimization technique. In this article, all design variables are independent in the optimization problem to achieve the best performance of the V3 robot. Since there are 10 design parameters to optimize, it is time-consuming to directly carry out the search. The optimization is thus implemented in two steps. In the first step, it is assumed that the first links of all subchains are identical, that is, $l_{11}=l_{21}=l_{31}:=l_1$ . This is also the case for the second links, that is, $l_{12}=l_{22}=l_{32}:=l_2$ . It leads to six independent design variables, $l_1,l_2,b_1,b_2,r_1,r_2$ . In the first step, the initial points were generated randomly and uniformly in the parameter intervals. Once a point satisfies all the constraints, it is regarded as an effective initial point and is used to initialize the search. Once the optimum is found in this step, it serves as the initial point for the second-step optimization, which takes all 10 parameters as independent ones. In other words, the first step is simply applied to generate a good initial point to accelerate the optimization process.

Figure 9 shows computation convergence processes in the first and the second optimizations, where $R_o$ and $R_i$ are respectively the outer and inner radii of the workspace satisfying the constraints. The final (the second-step) optimization results present that the radii of the good-performance area are respectively 0.268 and 0.781 m, which can be computed using the optimal parameters.

Figure 9.

The convergence of the optimizing process: (a) the first optimization and (b) the second optimization.

Table 2 shows the initial and the optimal parameters generated by the first-step optimization. Table 3 shows the final optimization results for the design parameters. It is indicated in Table 3 that the lengths of the second links in subchains tend to be identical. For the first links, however, this is not the case. The first links in the second and third subchains are longer than the first link in the first subchain, and the first links in the second subchain are the longest one. The radius of the first links is close to the double of the second ones. It also shows that the link $H_3{H}_4$ vanishes in the optimal design, $b_2=0$ . The resulting end-effector becomes a structure, as shown in Figure 10.

Table 2.

The initial and the optimal parameters generated by the first-step optimization (m).

Parameters	$l_1$	$l_2$	$b_1$	$b_2$	$r_1$	$r_2$
Initial	0.221	0.387	0.028	0.083	0.032	0.008
Optimal	0.321	0.570	0.070	0	0.025	0.012

Table 3.

The optimal parameters generated by the second-step optimization (m).

$l_{11}$	$l_{21}$	$l_{31}$	$b_1$	$b_2$
0.306	0.396	0.385	0.077	0
$l_{12}$	$l_{22}$	$l_{32}$	$r_1$	$r_2$
0.542	0.524	0.537	0.020	0.012

Figure 10.

The resulting end-effector structure in the optimal design.

By the initial, the first-step and the second-step optimal parameters, it is able to compute individual dextrous workspace for each design. The corresponding inner and outer radii are respectively 0.221 and 0.553 m, 0.319 and 0.821 m, 0.236 and 0.843 m.Figure 11 shows the dextrous workspaces for the three designs, where the tori enclosed by the red solid circles, the black dashed circles, and the blue dotted circles are respectively the dextrous workspaces generated by the three resulting manipulators with the second-step, the first-step, and the initial parameters. It shows that the area of the dextrous workspace is increased by 123% and 155% by the first and the second optimization, respectively. This clearly shows the effectiveness of the optimization. Finally, the good-performance area is 82% of the dextrous workspace.

Figure 11.

The dextrous workspace generated by the initial, the first-step, and the second-step optimized parameters.

Let us evaluate the dexterity of the optimal design radially. The configurations at a fixed point in the radial (X-axis) direction with full orientation of $\varphi$ are equivalent to the constant-orientation configurations obtained by the rotation of the manipulator about the common axis. We can thus visualize the dexterity performance in the constant-orientation workspace instead of in the radial axis.Figures 12–14 show the inverse condition number contour plots generated respectively by the initial, the first-step optimal, and the second-step optimal designs in the position workspace to represent the plots along the X-axis. It shows that the workspace area of $\kappa \ge 0.2$ is consistently increased by the first and the second optimization. Furthermore, the dexterity performance (inverse condition number) is better in Figure 14 than in Figure 13.

Figure 12.

The inverse condition number plot of the V3 robot with the initial parameters at $\varphi =0$ .

Figure 13.

The inverse condition number plot of the V3 robot with the first-step optimal parameters at $\varphi =0$ .

Figure 14.

The inverse condition number plot of the V3 robot with the second-step optimal parameters at $\varphi =0$ .

Figures 15 and 16 present the average extreme translational and rotational velocities and simultaneously the average extreme translational and rotational errors of each point in the good-performance radial workspace. The minimum values of the average extreme velocities are larger than the lower bounds. Since the computation expressions are identical, see equations (45) and (49), the velocity plot and the error plot should be the same shape except a multiplication of the maximum input velocity or the maximum input error. It is shown that the average extreme translational error plot touches the upper bound, which indicates that the accuracy constraints are effective. The figures show that the maximum velocities and the maximum errors appear at the same position, which coincides with our knowledge that speed and accuracy are two directly conflicting indices, that is, high speed leads to low accuracy. The maximum translational velocity/error position is at radius of 0.547 m, and the maximum rotational velocity/error is at radius of 0.519 m. Figures 17 and 18 show the average extreme translational and rotational accelerations along the good-performance radial workspace. The maximum translational acceleration position is at radius of 0.743 m, and the maximum rotational acceleration is at radius of 0.694 m.

Figure 15.

The average extreme translational velocity and error of the end-effector.

Figure 16.

The average extreme rotational velocity and error of the end-effector.

Figure 17.

The average extreme translational acceleration of the end-effector.

Figure 18.

The average extreme rotational acceleration of the end-effector.

Verification by a case study

In this section, a real actuator consisting of a motor and a gear reducer is used to compute maxima of representative indices, velocity, acceleration, and accuracy, in the constraints to verify suitability of constraint bounds. Let us consider a typical path in pick-and-place application, finishing 300 mm translation and π rad within 0.15 s. Using a triangular velocity planning scheme (Figure 19), we can compute its peak translational velocity of $4\mathrm{m}/\mathrm{s}$ , its acceleration of $53.33\mathrm{m}/{\mathrm{s}}^2$ , and peak rotational velocity of $41.89\mathrm{rad}/\mathrm{s}$ , rotational acceleration of $558.51\mathrm{rad}/{\mathrm{s}}^2$ .

Figure 19.

The triangular velocity planning scheme.

Taking a typical actuator, Yaskawa 750W motor and a gear box with a gear ratio of 35, whose major specifications are shown in Table 4. For the actuated joint, the rated joint angular velocity is $(3000\times 2\pi )/60/35=8.98\mathrm{rad}/\mathrm{s}$ , and the rated joint torque is $2.39\times 35=83.65\mathrm{N}\mathrm{m}$ , where the transmission loss of gear box is neglected. Applying the quantities to the optimal design problem, the physical bounds in constraints can be computed as follows:

• Velocity. According to equations (45)–(47), where the actuated joint velocity is physically changed to ${\stackrel{·}{\theta }}_i\in [-8.98,8.98]$ from ${\stackrel{·}{\theta }}_i\in [-1,1]\mathrm{rad}/\mathrm{s}$ . Since in the optimal design problem, the lower bounds of the translational and rotational velocities are respectively $0.5$ and $2\pi \mathrm{m}/\mathrm{s}$ , the lower bound of the translational velocity in this case is $v_{\min }=8.98\times 0.5=4.49\mathrm{m}/\mathrm{s}$ , and the lower bound of the rotational velocity becomes ${\omega }_{\min }=8.98\times 2\pi =56.42\mathrm{rad}/\mathrm{s}$ . They are both a bit larger than corresponding translational and rotational velocity determined from triangular velocity planning, that is, 4 m/s and $41.89\mathrm{rad}/\mathrm{s}$ and thus satisfy the requirements for pick-and-place application.

• Acceleration. Following equations (13)–(15), the actuated torque becomes ${\tau }_i\in [-83.65,83.65]$ from ${\tau }_i\in [-1,1]\mathrm{N}\mathrm{m}$ by multiplying by the real rated torque of 2.39 N m by the gear ratio 35. Therefore, the lower bounds of the translational and rotational accelerations in the optimal problem, $1\mathrm{m}/{\mathrm{s}}^2$ and $4\pi \mathrm{rad}/{\mathrm{s}}^2$ , are changed to $a_{\min }=83.65\times 1=83.65\mathrm{m}/{\mathrm{s}}^2$ and ${\alpha }_{\min }=83.65\times 4\pi =1.05\times {10}^3\mathrm{rad}/{\mathrm{s}}^2$ , respectively. They are both larger than the required values, $53.33\mathrm{m}/{\mathrm{s}}^2$ and $558.51\mathrm{rad}/{\mathrm{s}}^2$ . Since the acceleration index evaluates the extreme acceleration when the velocity vanishes, the computed acceleration bounds are much larger than the required ones so that the manipulator can provide sufficiently large acceleration when the velocities are relatively large, 4 m/s and 41.89 rad/s in the triangular velocity planning.

• Accuracy. The error of actuated joint is largely dependent on the error of the gear box. Here, the maximum error of the gear box is taken as the actuated joint error. In the optimal design problem, the bounds corresponding to maximum actuated joint error of $1\times {10}^{-4}\mathrm{rad}$ are $0.15\mathrm{mm}$ and $0.002\mathrm{rad}$ , respectively. Since the maximum error of the gear box is 3 arcmins ( $3/{60}^{\circ }\approx 8.73\times {10}^{-4}\mathrm{rad}$ ), the upper bounds for errors are $\Delta p_{\max }=8.73\times 0.15=1.31\mathrm{mm}$ and $\Delta {\varphi }_{\max }=8.73\times 0.002=1.75\times {10}^{-2}\mathrm{rad}=1^{\circ }$ , according to equations (49)–(51). This accuracy performance is acceptable in most pick-and-place applications in pharmaceutical, electronics, food and consumer goods industries, and so on.

Table 4.

The parameters of 750W servo motor made by YASKAWA and gear box.

Rated speed	Rated torque	Gear ratio	Maximum error
3000 r/min	2.39 N m	35	3 arcmins

Through the case study above, the constraint bound settings in the optimal design problem are appropriate for high-speed pick-and-place operations.

Conclusion

In this article, the dynamic model of the V3 robot is derived for acceleration optimization. Since each subchain is comparable to a cantilever beam structure, its bending deflection is proposed to characterize the properness when manipulating workpieces. Taking the indices of velocity, acceleration, dexterity, accuracy, and bending stiffness as design constraints, the optimal design problem is formulated to maximize the dextrous workspace having specified good performance. The optimal structure shows great improvement of the good-performance dextrous workspace. It provides a good start for prototype development.