Comparison of Hybrid and Parallel Architectures for Two-Degrees-of-Freedom Planar Robot Legs

Abstract

A comparative analysis between a hybrid and a parallel manipulator, to study the influence of their architecture on performance, is presented in this paper. The two manipulators are modifications of a serial 2-R manipulator and a five-bar manipulator, respectively. They are altered in a way that they both share the same arrangement of the links, while having a distinction only in the actuator arrangement. Indices of performance, such as the measure of manipulability, local conditioning index, and global conditioning index are used to compare their performance.

Appendix: Velocity Jacobian

The analytical expressions for the velocity Jacobian matrices of the hybrid and parallel manipulators are given by:

Fig. 6

Geometric parameters and references to measure link angles in the hybrid and parallel manipulators

$$\begin{aligned}&\varvec{J} _p = \frac{1}{l_{2e}s_{23}} \left[ \begin{array}{cc} -l_1(l_2 + l_{2e}) s_1 s_2 c_3 + l_1 (l_2 c_1 s_2 + l_{2e} s_1 c_2) s_3 &{} l_2 l_4 s_2 (c_{4} s_3 - c_{3} s_{4}) \\ -l_1 (l_2 + l_{2e}) c_1 c_2 s_3 + l_1 (l_2 s_1 c_2 + l_{2e} c_1 s_2) c_3 &{} l_2 l_4 c_2 (c_3 s_4 - c_4 s_3) \\ \end{array} \right] , \\&\varvec{J} _h = \frac{1}{l_{2e}s_{23}} \left[ \begin{array}{cc} -(l_1 (l_2 + l_{2e}) s_1 + l_2 l_4 s_{14})s_2 c_3 + l_1 l_{2e} s_1 c_2 s_3 \\ + l_2 (l_1 c_1 + l_4 c_{14}) s_2 s_3 &{} l_2 l_4 s_2 (c_{14} s_3 - c_{3} s_{14} ) \\ - (l_1 (l_2 + l_{2e}) c_1 + l_2 l_4 c_{14})c_2 s_3 + l_1 l_{2e} c_1 s_2 c_3 \\ + l_2 (l_1 s_1 + l_4 s_{14})c_2 c_3 &{} l_2 l_4 c_2 (s_{14} c_3 - s_{3} c_{14} ) \\ \end{array} \right] , \end{aligned}$$

where $\varvec{J} _p$ and $\varvec{J} _h$ are the velocity Jacobian matrices of the parallel and hybrid manipulators, respectively, and $l_i$’s are the lengths of the links as indicated in Fig. 6, $c_i$ and $s_i$ represent the cosine and sine of the angles $\theta _i$, $\phi _i$ respectively, $s_{23}$ indicates the sine of the compound angle $\phi _2 - \phi _3$, and $c_{14}$ and $s_{14}$ indicate the cosine and sine of the compound angle $\theta _1 + \theta _4$, respectively. It should be noted that the orientation of link $l_4$ is measured in an absolute sense in the parallel manipulator, whereas it is measured in a relative sense in the hybrid manipulator. Thus, $\theta _4$ in the parallel manipulator is equal to $\theta _1 + \theta _4$ of the hybrid manipulator for a given configuration. The two Jacobian matrices possess the following structure:

$$\begin{aligned}&\varvec{J} _p = \left[ \begin{array}{cc} a &{} b \\ c &{} d \\ \end{array} \right] \,,&\varvec{J} _h = \left[ \begin{array}{cc} a + b &{} b \\ c + d &{} d \\ \end{array} \right] \,, \end{aligned}$$

where a, b, c, and d are substitutes for the elements of the matrices. This structure of the Jacobian matrices arises due to the use of the compound angle $\theta _1 + \theta _4$ to describe the angle made by the link $l_4$ with the $\varvec{X}$-axis of the base co-ordinate frame in the hybrid manipulator as opposed to just $\theta _4$ in the parallel manipulator.

The velocity Jacobian matrices for the simplified models of the manipulators (shown in Fig. 3) are:

$$\begin{aligned}&\varvec{J} _p = \left[ \begin{array}{cc} -l_1 \sin \theta _1 &{} -l_2 \sin \theta _4 \\ l_1 \cos \theta _1 &{} l_2 \cos \theta _4 \\ \end{array} \right] \,, \\&\varvec{J} _h = \left[ \begin{array}{cc} -l_1\sin \theta _1 - l_2 \sin (\theta _1 + \theta _4) &{} - l_2 \sin (\theta _1 + \theta _4) \\ l_1\cos \theta _1 + l_2\cos (\theta _1 + \theta _4) &{} l_2\cos (\theta _1 + \theta _4) \\ \end{array} \right] \,. \end{aligned}$$