Performance Evaluation of Parallel Mechanisms

Abstract

This chapter concerns several new performance evaluation indices available for parallel mechanisms as well as introducing the use of conventional but the most frequently used indices. The performance evaluation of parallel mechanisms is the foundation of optimal design and also a challenging task. Some Jacobian-based indices such as condition number and manipulability are no longer suitable for those parallel mechanisms with rotation-translation coupling. In this case, it is necessary to address several better performance indices. Therefore, in addition to commonly used indices for evaluating the kinematic performances such as the workspace, stiffness, and accuracy of a parallel mechanism, a series of novel frame-free indices in terms of motion/force transmissibility, including input transmission index (ITI), output transmission index (OTI), constraint transmission index (CTI), and local singularity index (LSI), are presented and illustrated by several parallel mechanisms.

Appendix: LTI of the Planar 5R and PRRRP Parallel Mechanisms

The planar 5R parallel mechanism is shown in Fig. 7.15. The transmission wrench of each leg is the pure force along the coupler link of the corresponding leg. For example, the transmission wrench of leg 1 is a pure force along link AP. With respect to local coordinate system O-xy, the unit transmission wrenches of legs 1 and 2 can be represented by

and

where ${{\boldsymbol{f}}_1}$ and ${{\boldsymbol{f}\!\!}_2}$ represent the unit vectors along coupler links AP and CP, respectively; b and d stand for the vectors from origin O to the center of joints A and C, respectively.

Revolute joints O ₁ and O ₂ are the input joints; thus, the two input twists can be represented by

and

respectively. According to Eq. (7.35), the reciprocal product between $ _T1 and $ _I1 can be obtained as

Thus, on the basis of Eq. (7.49), the input transmission index of leg 1 can be obtained as follows:

When input revolute joint C is fixed and joint A is actuated, the transmission force of leg 2 becomes a constraint force for the end-effector. In this case, end-effector P can move only along the direction perpendicular to coupler link $DP$. Given that the position vector of joint D can be expressed by

$$ \overrightarrow{OD}=\left[ {\begin{array}{*{20}{c}} {{D_x}} \\ {{D_y}} \\ {{D_z}} \\ \end{array}} \right]=\left[ {\begin{array}{*{20}{c}} {-{r_1}+{r_2}\cos {\theta_1}-{r_3}\cos ({\theta_1}+{\gamma_1})+{r_3}\cos ({\theta_1}+{\gamma_1}+\beta )} \\ {{r_2}\sin {\theta_1}-{r_3}\sin ({\theta_1}+{\gamma_1})+{r_3}\sin ({\theta_1}+{\gamma_1}+\beta )} \\ 0 \\ \end{array}} \right], $$

the output twist can be represented by

According to Eq. (7.52), therefore, the reciprocal product between $ _T1 and $ _O1 can be obtained as

Using Eq. (7.57) as basis, we can derive the output transmission index of leg 1 thus:

Similar to the input and output transmission indices of leg 1, those of leg 2 can be obtained as

$$ {\lambda_2}=\left| {\sin {\gamma_2}} \right| $$

and

$$ {\eta_2}=\left| {\sin \mu } \right|, $$

respectively.

Thus, the LTI of the planar 5R parallel manipulator can be expressed by

$$ \gamma =\min \left\{ {{\lambda_1},\;{\lambda_2},\;{\eta_1},\;{\eta_2}} \right\}=\min \left\{ {\left| {\sin {\gamma_1}} \right|,\;\left| {\sin {\gamma_2}} \right|,\;\left| {\sin \mu } \right|} \right\}. $$

Consequently, $\mu$ is defined as the forward transmission angle, and ${\gamma_1}$ and ${\gamma_2}$ are referred to as the inverse transmission angles of the planar 5R parallel manipulator.

The planar PRRRP parallel mechanism is shown in Fig. 7.16. The transmission wrench of each leg is the pure force along the coupler link of the corresponding leg. For example, the transmission wrench of leg 1 is a pure force along link AO’. With respect to global coordinate system O-xy, the unit transmission wrenches of legs 1 and 2 can be represented by

and

where ${{\boldsymbol{f}}_1}$ and ${{\boldsymbol{f}\!\!}_2}$ represent the unit vectors along coupler links AO’ and BO’, respectively; a and b stand for the vectors from origin O to the center of revolute joints A and B, respectively; and ${d_1}$ and ${d_2}$ stand for the distances from origin O to coupler links AO’ and BO’, respectively.

The two prismatic joints are the input joints; hence, the two input twists can be represented by

According to Eq. (7.35), the reciprocal product between $ _T1 and $ _I1 can be obtained as

Thus, on the basis of Eq. (7.49), the input transmission index of leg 1 can be obtained as follows:

When the input prismatic joint in leg 2 is fixed and that in leg 1 is actuated, the transmission force of leg 2 becomes a constraint force for the end-effector. In this case, end-effector P can move only along the direction perpendicular to coupler link BO’, and the output twist can be represented by

According to Eq. (7.52), the reciprocal product between $ _T1 and $ _O1 can be obtained as

Using Eq. (7.49) as basis, we can obtain the output transmission index of leg 1 as follows:

Similar to the input and output transmission indices of leg 1, those of leg 2 can be obtained as

$$ {\lambda_2}=\left| {\sin {\gamma_2}} \right| $$

and

$$ {\eta_2}=\left| {\sin \mu } \right|, $$

respectively.

Thus, the LTI of the planar PRRRP parallel manipulator can be expressed by

$$ \gamma =\min \left\{ {{\lambda_1},\;{\lambda_2},\;{\eta_1},\;{\eta_2}} \right\}=\min \left\{ {\left| {\sin {\gamma_1}} \right|,\;\left| {\sin {\gamma_2}} \right|,\;\left| {\sin \mu } \right|} \right\}. $$

For the planar PRRRP parallel manipulator, therefore, $\mu$ is defined as the forward transmission angle, and ${\gamma_1}$ and ${\gamma_2}$ are referred to as the inverse transmission angles.