Optimization design using a global and comprehensive performance index and angular constraints in a type of parallel manipulator

Abstract

The kinematic optimization of a type of parallel manipulator is addressed. Based on the kinematic analysis, the pressure angles within a limb and among the limbs are introduced, which have definite physical and geometrical meanings. In particular, a type of new pressure angle among the limbs (referred to as the second type of pressure angle among the limbs) is defined and considered to be one of the pressure angle constraints to ensure the kinematic performance. In the kinematic optimization phase, a global and comprehensive performance index, which combines the conditioning number of Jacobian matrix and pressure angles with the volume of workspace, is formulated as an objective function for minimization. One optimal kinematic design example that determines the dimensional parameters is provided to clarify the availability of the proposed approach. The analytical results indicate that good kinematic and dynamic performance can be guaranteed with the suggested design approach. Furthermore, the mutual effect between the dexterity and new pressure angle is analyzed. The effects of the pressure angle constraints on the minimum and maximum conditioning numbers and global dynamic performance indices through the entire workspace are discussed. The proposed research provides a method for the kinematic optimization design of parallel manipulators.

Introduction

In light industries such as electronics, foodstuffs, pharmaceuticals, and everyday chemicals, pick-and-place operations are often required to inspect, carry, package, and sort the products. Generally, the operated objects have low weights and small volumes, and the pick-and-place operations are accomplished at notably high speeds and accelerations to achieve the shortest cycle time. Recently, parallel robots driven by rotary or prismatic joints have been rapidly developing to satisfy these industrial applications. Except for special surroundings that require only pure translations, the selective compliance assembly robot arm (SCARA) or Schoenflies motions^1,2 for three translations and one rotation are often applied to the most pick-and-place operations.

Stimulated by the application requirements, a great amount of effort has been devoted to the problems of the types of mechanical topology, kinematics, dynamics, and control of parallel manipulators. Among these aspects, the kinematic design or dimensional synthesis is important in the evolution of the parallel manipulators ³ because the performance of a system with respect to the workspace, velocity, accuracy, and rigidity is significantly controlled by its geometrical features. In other words, it is the basis for subsequent design and analysis. The kinematic design primarily involves the determination of a group of design parameters, which can be fulfilled by optimizing a nonlinear objective function satisfied with a set of suitable constraints. It is universally acknowledged that the methods in the previous works to study the kinematic design of the parallel manipulator can be grouped into two types: the local optimum design,^4–6 which concentrates on seeking the required parameter locus to generate a group of isotropy; and the global optimization design,^7,8 which attempts to optimize a global conditioning index that satisfies the system performance. It is well known that the most applicable local performance index between the joint variables and the end-effector is the conditioning number of the Jacobian matrix.^9–11 The representative global condition index in the kinematic optimization design is the mean value of the reciprocal of the conditioning number of the Jacobian index. ¹² Next, this value has been expanded into various improved versions regarding the kinematic design of parallel manipulators,^13–15 which further improves the kinematic performance.

Although the kinematic design of this type of parallel manipulator has been extensively examined, previous studies mainly focused on the conditioning index obtained from the Jacobian matrix or its improved form. Nevertheless, Huang et al. ³ found that the singular configuration of the 2-degree-of-freedom (DOF) parallel manipulator actuated by rotary joints remained even if the conditioning number of Jacobian matrix was 1. Therefore, with the maximum pressure angle being defined as an additional constraint, a comprehensive index that combined the area of the workspace and the conditioning number of the Jacobian matrix was used for the kinematic optimization design. In the comprehensive index analysis phase, a global fluctuation index based on the kinematic condition number was proposed by Ni et al. ¹⁶ This index was used to provide a statement of fluctuation of the velocity transmission feature, which acted as a part of the operating performance index.

The problems for optimization design of parallel robots have been discussed a lot in recent research work. Russoa et al., ¹⁷ for example, performed the multi-objective optimization using workspace volume, manipulator dexterity, static efficiency, and stiffness, which can decide the geometrical parameters of the parallel manipulator. Huang et al. ¹⁸ optimized their parallel mechanism with the genetic algorithm considering the workspace and global performance indexes of stiffness as well as the dexterity as the performance indices. Brinker et al. ¹⁹ aggregated the transmission and constraint characteristics into a single index, and the validity and applicability for kinematic performance optimization was proven. Another 2-DOF parallel mechanism was analyzed in the study by Huo et al., ²⁰ where non-dominated sorting genetic algorithm II was utilized to optimize the design dimensional parameters by means of three global indices. Besides, optimal design of the Schoenflies-motion parallel robot had been implemented by Wu ²¹ to optimize kinematic performances, wherein the motion/force transmission indices corresponding to two pressure angles were taken into account as the performance evaluation criterion subject to the rotation capability, cuboid workspace size, and geometric constraints.

Several properties and conclusions can be generalized to the parallel manipulators with similar structures and configurations. In general, the ability of force/motion transmission in a mechanical structure can be determined based on the pressure angles and the kinematic performance of the manipulators can be represented by the condition number. Moreover, little research has been conducted to analyze other performances after the kinematic optimization design, such as the dynamic performance.

Based on previous studies, this article attempts to address the kinematic optimization issues of the design of a 4-DOF parallel manipulator under the limit of a group of geometrical constraints and pressure angle constraints. The remainder of the article is organized as follows. First, with the demonstrated kinematic models and Jacobian matrix, the pressure angles within a limb and among the limbs including the first and second types of pressure angles are proposed. Second, based on the constraint conditions of pressure angles and feasible geometric dimensions, a global and comprehensive performance index composed of the conditioning number and pressure angles is first formulated as an objective function for minimization. An optimization design example is provided to illustrate the validity of the proposed approach. Furthermore, the mutual influence between dexterity and the second type of pressure angle among the limbs is analyzed. And the effects of the pressure angle constraints on the minimum and maximum conditioning numbers and global dynamic performance indices through the entire workspace are discussed in depth. Finally, the conclusions are presented.

Kinematic analysis

The studied type of parallel manipulator consists of a base, a moving platform, and four groups of kinematic chains. Each kinematic chain contains active and passive segments, which can achieve three translations and one rotation around the vertical axis normal to the plane of the moving platform in space. To date, the available branched structures that satisfy the requirements can be roughly divided into two types: one type is driven by rotary joints and the other is driven by prismatic joints, as shown in Figure 1. Obviously, the passive link between them adopts the parallelogram branched chain. All articulations of the transformable parallelogram use ball-and-socket joints.

Figure 1.

Two types of kinematic chains of the parallel manipulator: (a) driven by rotary joints and (b) driven by prismatic joints.

Because the four kinematic chains have almost identical structures, the reference frame is defined as shown in Figure 2, which simplifies the moving platform as point P. As shown in Figure 2, the coordinate frame O-xyz is attached to the static platform; therefore, the closed-loop position vector equation in the ith kinematic chain is expressed as

$$\begin{array}{l}r=e_i+a_{i}b+l_1{u}_i+l_2{w}_i,i=1,2,3,4\\ e_i=e{\left(\begin{array}{ccc}\cos {\gamma }_i& \sin {\gamma }_i& 0\end{array}\right)}^{\mathrm{T}},{\gamma }_i=\left(i-1\right)\pi /2\\ u_i={\left(\begin{array}{ccc}\cos {\gamma }_i\cos {\theta }_i& \sin {\gamma }_i\cos {\theta }_i& -\sin {\theta }_i\end{array}\right)}^T\end{array}$$

(1)

where e denotes the inscribed circle radius difference between the fixed base and the moving platform; l₁, l₂, u _i and w _i are the lengths and unit vectors of the ith active link and passive link; r , a _i and b are vector OP , length of E _i A _i and unit vector E _i A _i ; γ_i and θ_i are the structure angle of the static platform and position angle of the ith active link; the joint A_i can be a rotary or prismatic joint.

Figure 2.

Vector diagram of a kinematic chain.

Jacobian analysis

Taking the derivative of equation (1) with respect to time yields

$$\stackrel{·}{\mathit{r}}={\stackrel{·}{a}}_i\mathit{b}+l_1{\stackrel{·}{\theta }}_i({\mathit{v}}_i\times {\mathit{u}}_i)+l_2{\mathbf{\omega }}_i\times {\mathit{w}}_i,i=1,2,3,4$$

(2)

where $\stackrel{·}{\mathit{r}}$ is the linear velocity of point P; ${\stackrel{·}{\theta }}_i$ is the angular velocity of the ith active link; ω _i is the angular velocity vector of the ith passive link; ν _i is the unit vector normal to the plane of the span of e _i and u _i and along the orientation of the angular speed of A_iB_i; thus

$${\mathit{v}}_i={\left(\begin{array}{ccc}-\sin {\gamma }_i\hfill & \cos {\gamma }_i\hfill & 0\hfill \end{array}\right)}^T$$

(3)

If joint A_i is a rotary joint, equation (2) can be written as

$$\stackrel{·}{\mathit{r}}=l_1{\stackrel{·}{\theta }}_i({\mathit{v}}_i\times {\mathit{u}}_i)+l_2{\mathbf{\omega }}_i\times {\mathit{w}}_i$$

(4)

If joint A_i is a prismatic joint, equation (2) can be written as

$$\stackrel{·}{\mathit{r}}={\stackrel{·}{a}}_i\mathit{b}+l_2{\mathbf{\omega }}_i\times {\mathit{w}}_i$$

(5)

Taking the dot product of ${\mathit{w}}_i^T$ on both sides of equations (4) and (5) yields

$$l_1{\stackrel{·}{\theta }}_i{\mathit{w}}_i^T({\mathit{v}}_i\times {\mathit{u}}_i)={\mathit{w}}_i^T\stackrel{·}{\mathit{r}}$$

(6)

$${\stackrel{·}{a}}_i{\mathit{w}}_i^T\mathit{b}={\mathit{w}}_i^T\stackrel{·}{\mathit{r}}$$

(7)

We rewrite equations (6) and (7) in the matrix form as

$${\mathit{J}}_{\mathrm{qr}}\stackrel{·}{\mathbf{\theta }}={\mathit{J}}_{\mathrm{xr}}\stackrel{·}{\mathit{r}}$$

(8)

$${\mathit{J}}_{\mathrm{qp}}\stackrel{·}{\mathit{a}}={\mathit{J}}_{\mathrm{xp}}\stackrel{·}{\mathit{r}}$$

(9)

where

$$\begin{array}{c}{\mathit{J}}_{\mathrm{qr}}=\mathrm{diag}\left[l_1{\mathit{w}}_i^T({\mathit{v}}_i\times {\mathit{u}}_i)\right],\stackrel{·}{\mathbf{\theta }}={\left[\begin{array}{cccc}{\stackrel{·}{\theta }}_1\hfill & {\stackrel{·}{\theta }}_2\hfill & {\stackrel{·}{\theta }}_3\hfill & {\stackrel{·}{\theta }}_4\hfill \end{array}\right]}^{\mathrm{T}}\hfill \\ {\mathit{J}}_{\mathrm{xr}}={\left[\begin{array}{cccc}{\mathit{w}}_1\hfill & {\mathit{w}}_2\hfill & {\mathit{w}}_3\hfill & {\mathit{w}}_4\hfill \end{array}\right]}^T,{\mathit{J}}_{\mathrm{qp}}=\mathrm{diag}\left[{\mathit{w}}_i^T\mathit{b}\right]\hfill \\ \stackrel{·}{\mathit{a}}={\left[\begin{array}{cccc}{\stackrel{·}{a}}_1\hfill & {\stackrel{·}{a}}_2\hfill & {\stackrel{·}{a}}_3\hfill & {\stackrel{·}{a}}_4\hfill \end{array}\right]}^T,{\mathit{J}}_{\mathrm{xp}}={\left[\begin{array}{cccc}{\mathit{w}}_1\hfill & {\mathit{w}}_2\hfill & {\mathit{w}}_3\hfill & {\mathit{w}}_4\hfill \end{array}\right]}^T\hfill \end{array}$$

To facilitate the analysis, we combine equations (8) and (9)

$${\mathit{J}}_q\stackrel{·}{\mathit{q}}={\mathit{J}}_x\stackrel{·}{\mathit{r}}$$

(10)

where $\stackrel{·}{\mathit{q}}$ is the vector related to the active joint rate; $\stackrel{·}{\mathit{r}}$ is the velocity vector of the moving platform; J _q and J _x are the direct and indirect Jacobians, respectively. Thus, a further calculation can be obtained

$$\stackrel{·}{\mathit{q}}={\mathit{J}}_q^{-1}{\mathit{J}}_x\stackrel{·}{\mathit{r}}=\mathit{J}\stackrel{·}{\mathit{r}}$$

(11)

where J is the Jacobian associated with the specific pose of the end-effector. Apparently, the Jacobian matrices of the low-mobility parallel robots have similar forms. ²² However, the Jacobian matrix being analyzed is singular in equation (11), which causes many inconveniences for further research. We know that the different Jacobian matrices are obtained through the diverse structures of the moving platform.^23–25 Based on the invention patent of Tianjin University, ²⁶ the following study will be performed, which centers on the moving platform (including the primary platform and secondary platform) composed of a screw-nut-guide assembly for angular conversion. Then, the Jacobian matrix can be written as

$$\begin{array}{c}J=J_q^{-1}{J}_x\\ J_q=diag\left[l_1{w}_i^T\left(v_i\times u_i\right)\right]\\ J_x={\left[\begin{array}{cccc}{w}_1& w_2& w_3& w_4\\ 0& w_2^T\widehat{z}& 0& w_4^T\widehat{z}\end{array}\right]}^T\end{array}$$

where $\hat{\mathit{z}}$ is the unit vector of the z-axis in the coordinate frame O-xyz. Obviously, the indirect Jacobian J _x is dimensionless, whereas the direct Jacobian J _q has a unit length; therefore, the Jacobian matrix is dimensionally inhomogeneous. The rotation angle of the moving platform around the z-axis can be transformed into displacement through the screw pair. The condition number of the Jacobian matrix, which is considered an important evaluation index of the kinematic performance, is only meaningful when the Jacobian matrix is dimensionally homogeneous, since the condition number is dimensionless. The most commonly used methods to standardize the Jacobian matrix can be segmented into the characteristic length^27,28 and weighting factor ²⁹ methods. Hosseini and Daniali ³⁰ used both methods for the Tricept parallel manipulator and concluded that the characteristic length method might be preferable. Hence, the characteristic length L is introduced to homogenize the Jacobian matrix as follows

$${\mathit{J}}_h=L{\mathit{J}}_q^{-1}{\mathit{J}}_x={({\mathit{J}}_q/L)}^{-1}{\mathit{J}}_x={\mathit{J}}_{\mathrm{qh}}^{-1}{\mathit{J}}_x$$

(13)

where L = l₁ based on equation (12). The new Jacobian matrix can be expressed as

$$J_h=J_{qh}^{-1}{J}_x=diag{\left[w_i^T\left(v_i\times u_i\right)\right]}^{-1}{\left[\begin{array}{cccc}{w}_1& w_2& w_3& w_4\\ 0& w_2^T\widehat{z}& 0& w_4^T\widehat{z}\end{array}\right]}^T$$

(14)

In summary, the condition number can be obtained using the dimensionally homogeneous Jacobian matrix provided by equation (14).

Pressure angle analysis

To establish the performance constraints in a visible form, some analyses will be performed. In general, the ideally accessible transmission paths can be divided into two categories: within a limb and among the limbs. Any failure of the first type of transmission will introduce the direct singularity, which signifies that the system loses the DOF. A failure of the second type will cause the indirect singularity, which implies that the system becomes uncontrollable. Therefore, two types of pressure angles can be formulated to describe the force/motion transmission abilities of the investigated parallel manipulator.

Pressure angle α_i within a limb

With the above Jacobian analysis, the determinant of the direct Jacobian of the 4-DOF parallel manipulator is formulated as

$$det\left({\mathit{J}}_q\right)=l_1^4\underset{i=1}{\overset{4}{\Pi }}{\mathit{w}}_i^T({\mathit{v}}_i\times {\mathit{u}}_i)$$

(15)

Then, pressure angle α_i within a limb can be defined by

$${\alpha }_i=\arccos \left({\mathit{w}}_i^T({\mathit{v}}_i\times {\mathit{u}}_i)\right),i=1,2,3,4$$

(16)

The physical meaning of α_i can be interpreted as the pressure angle between the velocity (along ν _i  ×  u _i ) of point B_i and the force (along w _i ) applied by the active link to the passive link at the same point, which signifies the force/motion transmission ability within the limb.

Pressure angle γ/β_i among the limbs

The determinant of the indirect Jacobian can be obtained from equation (12)

$$det\left({\mathit{J}}_x\right)={({\mathit{w}}_{13}\times {\mathit{w}}_{24})}^T\hat{\mathit{z}}$$

(17)

where w ₁₃ =  w ₁ ×  w ₃ and w ₂₄ =  w ₂ ×  w ₄.

According to Liu et al., ²⁵ pressure angle γ among the limbs of the 4-DOF parallel manipulator under consideration is defined by equation (18). In the following discussion, pressure angle γ is referred to as the first type of pressure angle among the limbs. Based on the calculation and analysis, a type of new pressure angle β_i among the limbs is proposed by equation (19) as the second type of pressure angle among the limbs

$$\gamma =\arccos \frac{{({\mathit{w}}_{13}\times {\mathit{w}}_{24})}^T}{|{\mathit{w}}_{13}\times {\mathit{w}}_{24}|}\hat{\mathit{z}}$$

(18)

On one hand, the geometric meaning of γ can be explained as the angle between the z-axis and the intersecting line of planes π ₁₃ and π ₂₄, where π ₁₃ ( π ₂₄) is spanned by vectors w ₁ ( w ₂) and w ₃ ( w ₄). On the other hand, the physical explanation of angle γ can be construed as the angle between the velocity (along w ₂₄ =  w ₂ ×  w ₄) of point P and the forces (along w ₁ and w ₃) applied by the passive links in limbs 1 and 3 to the same point P, when we assume that the driven joints in limbs 2 and 4 are temporarily locked.

Drawing an analogy with the pressure angle among the limbs of Delta robot, ³¹ the second type of pressure angle β_i among the limbs can be formulated as

$$\begin{array}{c}{\beta }_1=\arccos \left(\frac{{({\mathit{w}}_1+{\mathit{w}}_2)}^T({\mathit{w}}_4\times {\mathit{w}}_3)}{|{\mathit{w}}_1+{\mathit{w}}_2||{\mathit{w}}_4\times {\mathit{w}}_3|}\right)\hfill \\ {\beta }_2=\arccos \left(\frac{{({\mathit{w}}_2+{\mathit{w}}_3)}^T({\mathit{w}}_1\times {\mathit{w}}_4)}{|{\mathit{w}}_2+{\mathit{w}}_3||{\mathit{w}}_1\times {\mathit{w}}_4|}\right)\hfill \\ {\beta }_3=\arccos \left(\frac{{({\mathit{w}}_3+{\mathit{w}}_4)}^T({\mathit{w}}_2\times {\mathit{w}}_1)}{|{\mathit{w}}_3+{\mathit{w}}_4||{\mathit{w}}_2\times {\mathit{w}}_1|}\right)\hfill \\ {\beta }_4=\arccos \left(\frac{{({\mathit{w}}_4+{\mathit{w}}_1)}^T({\mathit{w}}_3\times {\mathit{w}}_2)}{|{\mathit{w}}_4+{\mathit{w}}_1||{\mathit{w}}_3\times {\mathit{w}}_2|}\right)\hfill \end{array}$$

(19)

To explain the physical meaning of β_i, we use β₁ as an example (as illustrated in Figure 3). We assume that the driven joints in limbs 3 and 4 can be locked for the present. Then, β₁ can be interpreted as the pressure angle between the velocity (along w ₄₃ =  w ₄ ×  w ₃) of point P and the forces (along ( w ₁+ w ₂)) imposed by the passive links in limbs 1 and 2 to the same point P, which also represents the force/motion transmission ability among the limbs. These explanations can be made for the cases where the driven joints in limbs 1 and 4, limbs 2 and 1, and limbs 3 and 2 are hypothetically locked.

Figure 3.

Schematic diagram of the second type of pressure angle β₁ among the limbs.

To provide a geometrical explanation of angle β_i, and according to the indirect Jacobian obtained by equation (9) with the singular form, we can define the following submatrices J _xi (i = 1–4) by intercepting matrix J _x

$$\begin{array}{c}{\mathit{J}}_{x1}={\left[\begin{array}{ccc}{\mathit{w}}_2\hfill & {\mathit{w}}_3\hfill & {\mathit{w}}_4\hfill \end{array}\right]}^T,{\mathit{J}}_{x2}={\left[\begin{array}{ccc}{\mathit{w}}_1\hfill & {\mathit{w}}_3\hfill & {\mathit{w}}_4\hfill \end{array}\right]}^T\hfill \\ {\mathit{J}}_{x3}={\left[\begin{array}{ccc}{\mathit{w}}_1\hfill & {\mathit{w}}_2\hfill & {\mathit{w}}_4\hfill \end{array}\right]}^T,{\mathit{J}}_{x4}={\left[\begin{array}{ccc}{\mathit{w}}_1\hfill & {\mathit{w}}_2\hfill & {\mathit{w}}_3\hfill \end{array}\right]}^T\hfill \end{array}$$

(20)

Similar to the definition of the pressure angles among the limbs of the Delta robot and combining with the 4-DOF parallel manipulator under consideration, we can formulate

$$\begin{array}{c}{\beta \prime }_1=max\left(\begin{array}{cc}\arccos \left(\frac{{\mathit{w}}_1^T({\mathit{w}}_4\times {\mathit{w}}_3)}{|{\mathit{w}}_4\times {\mathit{w}}_3|}\right)\hfill & \arccos \left(\frac{{\mathit{w}}_2^T({\mathit{w}}_4\times {\mathit{w}}_3)}{|{\mathit{w}}_4\times {\mathit{w}}_3|}\right)\hfill \end{array}\right)\hfill \\ {\beta \prime }_2=\max \left(\begin{array}{cc}\arccos \left(\frac{{\mathit{w}}_2^T({\mathit{w}}_1\times {\mathit{w}}_4)}{|{\mathit{w}}_1\times {\mathit{w}}_4|}\right)\hfill & \arccos \left(\frac{{\mathit{w}}_3^T({\mathit{w}}_1\times {\mathit{w}}_4)}{|{\mathit{w}}_1\times {\mathit{w}}_4|}\right)\hfill \end{array}\right)\hfill \\ {\beta \prime }_3=\max \left(\begin{array}{cc}\arccos \left(\frac{{\mathit{w}}_3^T({\mathit{w}}_2\times {\mathit{w}}_1)}{|{\mathit{w}}_2\times {\mathit{w}}_1|}\right)\hfill & \arccos \left(\frac{{\mathit{w}}_4^T({\mathit{w}}_2\times {\mathit{w}}_1)}{|{\mathit{w}}_2\times {\mathit{w}}_1|}\right)\hfill \end{array}\right)\hfill \\ {\beta \prime }_4=\max \left(\begin{array}{cc}\arccos \left(\frac{{\mathit{w}}_4^T({\mathit{w}}_3\times {\mathit{w}}_2)}{|{\mathit{w}}_3\times {\mathit{w}}_2|}\right)\hfill & \arccos \left(\frac{{\mathit{w}}_1^T({\mathit{w}}_3\times {\mathit{w}}_2)}{|{\mathit{w}}_3\times {\mathit{w}}_2|}\right)\hfill \end{array}\right)\hfill \end{array}$$

(21)

It is clearly demonstrated that if the maximum of the corresponding angle range $(\beta {\prime }_i)$ satisfies the requirements, all angles of the angle range (including β_i) can also guarantee the transmission performance. To better analyze the second type of pressure angle β_i among the limbs, we substitute equation (19) for equation (21) in the theory, which will also make the physical meaning more explicit and guarantee the identical transmission capacity among the limbs. The second type of pressure angle among the limbs is suitable for almost all types of 4-DOF parallel manipulators. However, according to the definition of the parallel mechanism, this type of pressure angle among the limbs may also be applicable to other types of high-speed parallel manipulators with lower mobility.

Equation (16) shows that the direct singularity appears in α_i = 90°, which causes det( J _q ) = 0, whereas equation (18) indicates that the indirect singularity appears in γ = 90°, which causes det( J _x ) = 0. Equation (19) shows that the indirect singularity also occurs at β_i = 90°. Thus, these pressure angles can be used to adequately describe the direct and indirect singularities of parallel manipulators at a particular position.

Optimal kinematic design

In accordance with the foregoing analysis, the optimization design of the 4-DOF parallel manipulator will be conducted. The dimensional parameters e, l₁, l₂, and H′ can be determined to achieve good performance throughout the specified cylindrical workspace with radius R and height H as illustrated in Figure 4.

Figure 4.

Structure diagram of the 4-DOF parallel manipulator.

Objective function

The conditioning number of the Jacobian matrix is universally used as the local performance index to evaluate the velocity, accuracy, and rigidity mapping features between the joint variables and the moving platform. ³² The conditioning number κ is defined as

$$1\le \kappa =\frac{{\sigma }_2}{{\sigma }_1}\le \infty$$

(22)

where σ₁ and σ₂ are the minimum and maximum singular values of the dimensionally homogeneous Jacobian matrix related to the specified pose.

Considering that κ varies with the configuration of the parallel manipulator, a conditioning index ³³ similar to that devised by Gosselin and Angeles ¹² serves as one of the global performance indices in the dimension synthesis. In the reference, a global conditioning index was introduced by the mean value of the conditioning number of the dimensionally homogeneous Jacobian matrix in the entire workspace. This index for the particular problem can be expressed by

$$\overline{\kappa }=\frac{{\int }_0^{W_{\lambda }}{\kappa }_{i}d{W}_i}{{\int }_0^{W_{\lambda }}d{W}_i}=\frac{1}{\pi R^{2}H}\underset{-H-H^{\prime }}{\overset{-H^{\prime }}{\int }}\underset{0}{\overset{2\pi }{\int }}\underset{0}{\overset{R}{\int }}\kappa \mathrm{dV}$$

(23)

where H and R are the height and radius of the cylindrical workspace W_λ, respectively; H′ is the perpendicular distance between the top surface of the cylindrical workspace and the center of the static platform that is the coordinate origin in Figure 4.

The index $\overline{\kappa }$ cannot comprehensively describe the global kinematic performance, since the pressure angle within a limb violently changes even if a better conditioning number has been achieved. A new global optimization performance index will be introduced to evaluate the kinematic performance of the disquisitive parallel manipulator. In this process, since the pressure angles differ at each location in the entire workspace, a global pressure angle ε₂ within a limb in equation (25) is represented. It is the ratio of the maximum value of the mean pressure angle α_i within a limb of every branched chain among four chains to the maximum pressure angle α_i within a limb in the workspace. Analogously, a global pressure angle ε₃ among the limbs is defined, which is the ratio of the mean value of the maximum value of the second type of pressure angle β_i among the limbs at every position to the maximum value of β_i in the entire workspace in equation (26).

In the computer implementation, the global conditioning index $\overline{\kappa }({\epsilon }_1)$ , global pressure angle ε₂ within a limb, and global pressure angle ε₃ among the limbs can be numerically calculated by

$${\epsilon }_1=\overline{\kappa }=\frac{{\int }_0^{W_{\lambda }}{\kappa }_{i}d{W}_i}{{\int }_0^{W_{\lambda }}d{W}_i}=\frac{1}{\pi R^{2}H}\sum _{m=1}^M\sum _{n=1}^N\sum _{g=1}^G\left({\kappa }_{\mathrm{mng}}\Delta W\right)$$

(24)

$$\begin{array}{c}{\epsilon }_2=\frac{1}{\underset{r\in W_{\lambda }}{max}\underset{i=1-4}{max}\left({\alpha }_i\right)}\underset{i=1-4}{max}\left(\frac{{\int }_0^{W_{\lambda }}{\alpha }_{i}d{W}_i}{{\int }_0^{W_{\lambda }}d{W}_i}\right)\hfill \\ =\frac{1}{\underset{r\in W_{\lambda }}{max}\underset{i=1-4}{max}\left({\alpha }_i\right)}\underset{i=1-4}{max}\hfill \\ \left(\frac{1}{\left(\pi R^{2}H\right)}\sum _{m=1}^M\sum _{n=1}^N\sum _{g=1}^G\left({\alpha }_{\mathrm{mng}\left(i\right)}\Delta W\right)\right)\hfill \end{array}$$

(25)

$$\begin{array}{c}{\epsilon }_3=\left(\frac{{\int }_0^{W_{\lambda }}\underset{i=1-4}{max}\left({\beta }_i\right)d{W}_i}{\underset{r\in W_{\lambda }}{max}\underset{i=1-4}{max}\left({\beta }_i\right){\int }_0^{W_{\lambda }}d{W}_i}\right)\hfill \\ =\left(\frac{1}{\underset{r\in W_{\lambda }}{max}\underset{i=1-4}{max}\left({\beta }_i\right)\left(\pi R^{2}H\right)}\sum _{m=1}^M\sum _{n=1}^N\sum _{g=1}^G\left({\beta }_{\mathrm{mng}(max(i))}\Delta W\right)\right)\hfill \end{array}$$

(26)

where κ_mng is the value of κ, α_mng(i) is the value of α_i, and β_mng(max(i)) is the maximum value of β_i at node (m n g) when W_λ is divided evenly into (M – 1) × (N – 1) × (G – 1) copies; ΔW is the unit volume; $\underset{r\in W_{\lambda }}{max}\underset{i=1-4}{max}({\alpha }_i)$ is the maximum value of the pressure angle α_i within a limb; and $\underset{r\in W_{\lambda }}{max}\underset{i=1-4}{max}({\beta }_i)$ is the maximum value of the second type of pressure angle among the limbs over a provided workspace.

As a result, a global and comprehensive performance index can be formulated by the objective function for minimization. The index is expressed as

$$\epsilon =\sqrt{{\epsilon }_1^2+{\epsilon }_2^2+{\epsilon }_3^2}\to min$$

(27)

where ε is a hopefully nondimensional parameter. The indicator can be considered an improved version of the global condition number.

Constraints

In the optimal kinematic design of the parallel manipulator under consideration, a set of applicable constraint conditions for the force/motion transmission behavior and workspace/machine volume ratio should be considered.

Dimensional constraints

In general, the geometrical constraint condition related to the installation space of servomotors is set to offset e by

$$e\ge e_{min}$$

(28)

In addition, to satisfy the required workspace, the volume of the manipulator should be as compact as possible. Hence, the other constraint condition to consider is the workspace/machine volume ratio, which is expressed as ²⁹

$$\delta =\frac{D}{2(e+l_1)}=1.0$$

(29)

Simultaneously, according to the trilateral relations of the triangle, the following dimensional constraint inequalities can be formulated to enable the mechanism to be assembled, when point P is located at the lower and upper bounds of the workspace

$$\begin{array}{c}\sqrt{{(H^{\prime }+H)}^2+{(D/2+e)}^2}-l_2-l_1<0\hfill \\ l_2-l_1-H^{\prime }<0\hfill \end{array}$$

(30)

Pressure angle constraints

From the above analysis, the pressure angle α_i within a limb and pressure angle γ/β_i among the limbs can denote the singularity of the mechanism and the force/motion transmission capabilities of the parallel manipulator within a limb and among the limbs. Therefore, to guarantee these above two aspects throughout the entire workspace, the following angular constraints are set:

1. Pressure angle constraint within a limb

   $$\underset{r\in W_{\lambda }}{max}\underset{i=1-4}{max}\left({\alpha }_i\right)\le \left[\alpha \right]$$

   (31)

2. Pressure angle constraint among the limbs

   $$\underset{r\in W_{\lambda }}{max}\left(\gamma \right)\le \left[\gamma \right]$$

   (32)
   $$\underset{r\in W_{\lambda }}{max}\underset{i=1-4}{max}\left({\beta }_i\right)\le \left[\beta \right]$$

   (33)

Thus, the maximum values of $\underset{i=1-4}{max}({\alpha }_i)$ , γ, and $\underset{i=1-4}{max}({\beta }_i)$ should not exceed their prescribed allowable pressure angles [α], [γ], and [β] throughout the entire task workspace. It can ensure good force/motion transmission capabilities within and among the limbs of the parallel manipulator under consideration.

The optimization design procedure can be summed up as illustrated in Figure 5. The detailed introduction will be provided below. Furthermore, the effects of pressure angle constraints on system performance will be analyzed at great length through an example.

Figure 5.

Optimization design process.

An example

Determine the dimensional parameters

The aforementioned optimal kinematic design will be fulfilled to identify the dimensional parameters of the 4-DOF type of parallel manipulator with a cylindrical task workspace of D = 1.0 m in diameter and H = 0.25 m in height, as shown in Figure 4. With e_min = 0.125 m for the servomotor installation, we have l₁ = 0.375 m based on equation (29). In accordance with the above analysis, we minimize the global and comprehensive performance index ε in equation (27) and limit the boundaries of the workspace of end-effector by equation (30) with the prescribed allowable pressure angles [α] = 45°, [γ] = 40°, and [β] = 65°, which results in $l_2^{*}=0.950\mathrm{m}$ and $H\prime *=0.726\mathrm{m}$ . The optimal results are shown in Figure 6. The minimum and maximum conditioning numbers of the entire workspace can be obtained by

$${\kappa }_{min}=2.299,{\kappa }_{max}=3.285$$

(34)

The maximum pressure angle within a limb $\underset{i}{max}{\alpha }_i$ , first type of pressure angle among the limbs γ, and second type of pressure angle among the limbs $\underset{i}{max}{\beta }_i$ throughout the entire workspace are

$$\begin{array}{c}\underset{r\in W_{\lambda }}{max}\underset{i=1-4}{max}\left({\alpha }_i\right)=46.{380}^{\circ },\underset{r\in W_{\lambda }}{max}\left(\gamma \right)=37.{920}^{\circ }\hfill \\ \underset{r\in W_{\lambda }}{max}\underset{i=1-4}{max}\left({\beta }_i\right)=69.{077}^{\circ }\hfill \end{array}$$

(35)

Figure 6.

Variations of (a) the conditioning number κ, (b) the pressure angle within a limb $\underset{i}{max}{\alpha }_i$ , (c) the first type of pressure angle among the limbs γ, and (d) the second type of pressure angle among the limbs $\underset{i}{max}{\beta }_i$ throughout the entire workspace: (1) z₁ = –(H′+H), (2) z₂ = –(H′+H/2), and (3) z₃ = –H′.

These analysis results show that the proposed optimization method can guarantee the kinematic performance of the parallel manipulator.

It has been well acknowledged that the effective global dynamic performance indices were proposed by Huang and coworkers. ²⁵ The global performances related to the inertial torque and centrifugal/Coriolis torque can be evaluated by the indices τ _aG and τ _vG , respectively

$${\mathbf{\tau }}_{\mathrm{aG}}=\underset{r\in W_t}{\max }\underset{i}{max}\left({\sigma }_{\mathrm{ai}max}\left({\mathit{G}}_i\right)\right),i=1,2,3,4$$

(36)

$${\mathbf{\tau }}_{\mathrm{vG}}=\underset{r\in W_t}{\max }\underset{i}{max}\left({\sigma }_{\mathrm{vi}max}\left(I_A{\mathit{L}}_i^T{\mathit{H}}_i{\mathit{L}}_i\right)\right),i=1,2,3,4$$

(37)

These indices can be used to validate the effectiveness of the optimization method. The variations of $\underset{i}{max}({\sigma }_{\mathrm{ai}max}({\mathit{G}}_i))$ and $\underset{i}{max}({\sigma }_{\mathrm{vi}max}(I_A{\mathit{L}}_i^T{\mathit{H}}_i{\mathit{L}}_i))$ in the entire workspace are shown in Figure 7, and the two global dynamic performance indices can be derived after optimization. The dynamic performance of the parallel manipulator can be commendably pledged.

$$\begin{array}{c}{\mathbf{\tau }}_{\mathrm{aG}}=\underset{r\in W_t}{\max }\underset{i}{max}\left({\sigma }_{\mathrm{ai}max}\left({\mathit{G}}_i\right)\right)=3.6707\hfill \\ {\mathbf{\tau }}_{\mathrm{vG}}=\underset{r\in W_t}{\max }\underset{i}{max}\left({\sigma }_{\mathrm{vi}max}\left(I_A{\mathit{L}}_i^T{\mathit{H}}_i{\mathit{L}}_i\right)\right)=3.4514\hfill \end{array}$$

(38)

In summary, the proposed design approach is feasible; it can ensure the kinematic and dynamic performance of the parallel manipulator under consideration.

Figure 7.

Variations of (a) $\underset{i}{max}({\sigma }_{\mathrm{ai}max}({\mathit{G}}_i))$ and (b) $\underset{i}{max}({\sigma }_{\mathrm{vi}max}(I_A{\mathit{L}}_i^T{\mathit{H}}_i{\mathit{L}}_i))$ throughout the entire workspace: (1) z₁ = –(H′+H), (2) z₂ = –(H′+H/2), and (3) z₃ = –H′.

Angular analysis

The effects of the pressure angle constraints on the optimization results will be analyzed from two aspects: kinematics and dynamics. On one hand, the effect of the pressure angle constraints on the kinematic performance is as follows. To illustrate the effect of the defined second type of pressure angle among the limbs, the following simulation is performed as shown in Table 1.

Table 1.

Minimum and maximum conditioning numbers in terms of different constraints.

Constraints	Dimensional constraints, [α] = 45° and [γ] = 40°	Dimensional constraints, [α] = 45° and [β] = 65°	Dimensional constraints, [α] = 45°, [γ] = 40°, and [β] = 65°
Conditioning number	κ min = 2.299 κmax = 3.291	κ min = 2.299 κmax = 3.283	κ min = 2.299 κmax = 3.285

Table 1 is obtained by computing the optimization index in equation (27) with different angle constraints, respectively. A comparison of the optimization results shows that the first and second types of pressure angles among the limbs as a constraint have almost identical effects on the optimization of the objective function considering the improvement of the conditioning number. Then, for the pressure angle constraints of the optimization design, the first type of pressure angles among the limbs can be completely replaced by the second type of pressure angles among the limbs. As a result, the newly defined second type of pressure angle among the limbs has a certain action on the optimal kinematic design in this article.

We know that the dexterity has an important effect on the kinematic performance of the manipulator. The dexterity of a manipulator can be considered the ability of the manipulator to arbitrarily change its position/orientation or apply forces/torques in arbitrary directions. ³⁴ The condition number of Jacobian matrix, which is considered an important evaluation index of dexterity, has been widely applied. To clearly present the mutual effect between the dexterity and the second type of pressure angle among the limbs, the optimized results referring to the condition number and the second type of pressure angle among the limbs are analyzed at different poses as shown in Table 2. According to Figure 6, the dexterity of the boundary in the workspace is relatively poor, which will be considered when we select the pose. The height of the cylindrical workspace is grouped into three layers: z₁ = –(H′+H), z₂ = –(H′+H/2), and z₃ = –H′, and the points (corresponding to the poses) on the outermost circle divided equally into eight parts and the center point P_i0 (i = 1,2,3) are selected on each layer. The condition numbers and the second type of pressure angles among the limbs are calculated, respectively, on the aforementioned poses. It is not difficult to note that the second type of pressure angle among the limbs and condition number have similar distributions. They have identical changing trends at different heights. The condition number and the second type of pressure angle among the limbs at the lower boundary of the workspace are relatively larger, that is, the dexterity is poor. Instead, the value at the center of the workspace is smaller. In other words, the closer the points get to the z-axis, the smaller these values are, and have better kinematic performance. However, these indices that correspond to the disparate poses of the outermost boundary at identical heights do not significantly change. Therefore, the second type of pressure angle among the limbs and condition number have similar functions in evaluating the dexterity of the manipulator.

Table 2.

Conditioning numbers and the second type of pressure angles among the limbs at different poses.

z 1	P 10	P 11	P 12	P 13	P 14	P 15	P 16	P 17	P 18
κ maxβi (°)	2.493 49.631	3.287 66.504	3.191 68.695	3.287 66.504	3.191 68.695	3.287 66.504	3.191 68.695	3.287 66.504	3.191 68.695
z 2	P 20	P 21	P 22	P 23	P 24	P 25	P 26	P 27	P 28
κ maxβi (°)	2.300 47.204	2.800 59.186	2.750 61.959	2.800 59.186	2.750 61.959	2.800 59.186	2.750 61.959	2.800 59.186	2.750 61.959
z 3	P 30	P 31	P 32	P 33	P 34	P 35	P 36	P 37	P 38
κ maxβi (°)	2.352 47.888	2.670 53.155	2.647 56.664	2.670 53.155	2.647 56.664	2.670 53.155	2.647 56.664	2.670 53.155	2.647 56.664

It should be pointed out that achieving a better kinematic performance must be at the cost of decreasing the static and dynamic rigidity of the system. ²⁵ In addition, Figure 6 shows that the minimum value with respect to the conditioning number and the second type of pressure angle among the limbs appear in the upper plane of the workspace, whereas the minimum value with respect to the pressure angle within a limb and the first type of pressure angle among the limbs occur in the lower plane of the workspace. Therefore, to achieve better and more comprehensive kinematic performance, these parameters must be compromised. Next, we carefully analyze the effects of the allowable pressure angles within a limb and among the limbs on the conditioning number for the parallel manipulator under consideration. Figure 8 shows that the minimum and maximum conditioning numbers of the entire workspace decrease with the increase in [α] = 40°–50° but increase with the increase in [β] = 60°–70°, whereas there are notably few changes with the increase in [γ] = 35°–45°. Thus, to obtain better mapping features between the joint variables and the end-effector, one must reasonably consider the pressure angle constraints, when we assume that the specified dimensional constraints are satisfied in the optimal kinematic design.

Figure 8.

Variations of the minimum and maximum conditioning numbers through the entire workspace versus (a) [α] = 40°–50°, [γ] = 40°, [β] = 65°; (b) [α] = 45°, [γ] = 35°–45°, [β] = 65°; and (c) [α] = 45°, [γ] = 40°, [β] = 60°–70°.

On the other hand, the variations of the optimization results are discussed when the pressure angle constraints are changed for the global dynamic performance indices (as shown in Figure 9). Similarly, we use cyclic optimization to study the 4-DOF type of parallel manipulator considered. A similar conclusion can be drawn, where the first type of pressure angle among the limbs as a constraint does not significantly affect the optimization results. The effects of the pressure angle within a limb and the second type of pressure angle among the limbs on the dynamic optimization indices have opposite trends. In particular, index τ _vG , which is related to the centrifugal/Coriolis torque, more obviously changes, which can be explained by the above numerical analysis. Moreover, with the comprehensive consideration of the dynamic performance, the pressure angle constraints must be addressed in a compromising manner.

Figure 9.

Variations of the global dynamic performance indices τ _aG and τ _vG versus (a) [α] = 40°–50°, [γ] = 40°, [β] = 65°; (b) [α] = 45°, [γ] = 35°–45°, [β] = 65°; and (c) [α] = 45°, [γ] = 40°, [β] = 60°–70°.

Conclusion

This article introduces a method for the optimization design using a global and comprehensive performance index and pressure angle constraints. The subject is a type of parallel manipulator with three translations and one rotation. The conclusions are as follows:

1. Based on the kinematic analysis, two types of pressure angles for the 4 DOF parallel manipulator are defined, which enables the physical and geometrical meanings to be clearly connected with the geometry of system. Therefore, they can be used to obviously describe the motion/force transmission behaviors.

2. By minimizing the global and comprehensive performance index under the limit of a group of geometrical constraints and pressure angle constraints within and among the limbs, the dimensional parameters can be determined, and the preferable kinematic and dynamic performance can be realized over the entire workspace.

3. By comparing the variation of the condition number of the dimensionally homogeneous Jacobian matrix and the second type of pressure angle among the limbs in the workspace, we can conclude that they have similar functions in evaluating the dexterity of manipulator. The angular analysis shows that the pressure angles within a limb and among the limbs have different effects on the conditioning number and global dynamic performance indices, that is, we must reasonably consider the pressure angle constraints in the kinematic optimization design.

4. The proposed optimization method in this article is also applicable to other parallel manipulators. More comprehensive tests will be performed on this type of parallel manipulator and introduced in detail in the future article.