0
RESEARCH PAPERS

# Unified Solving Jacobian∕Hessian Matrices of Some Parallel Manipulators With $n$ SPS Active Legs and a Passive Constrained Leg

Yi Lu and Bo Hu
[+] Author and Article Information
Yi Lu

School of Mechanical Engineering, Yanshan University, Qinhuangdao, Hebei 066004, Chinaluyi@ysu.edu.cn

Bo Hu

School of Mechanical Engineering, Yanshan University, Qinhuangdao, Hebei 066004, China

J. Mech. Des 129(11), 1161-1169 (Nov 15, 2006) (9 pages) doi:10.1115/1.2771572 History: Received August 05, 2006; Revised November 15, 2006

## Abstract

Some parallel manipulators with $n$ spherical joint-prismatic joint-spherical joint (SPS)-type active legs and a passive constrained leg possess a larger capability of load bearing and are simple in structure of the active leg. In this paper, a unified and simple approach is proposed for solving Jacobian∕Hessian matrices and inverse∕forward velocity and acceleration of this type of parallel manipulators. First, a general parallel manipulator with $n$ SPS-type active legs and one passive constrained leg in various possible serial structure is synthesized, and some formulae for solving the poses of constrained force∕torque and active∕constrained force matrix are derived. Second, the formulae for solving extension of active legs, the auxiliary velocity∕acceleration equation are derived. Third, the formulae for solving inverse∕forward velocity and acceleration and a Jacobian matrix without the first-order partial differentiation and a Hessian matrix without the second-order partial differentiation are derived. Finally, the procedure is applied to three parallel manipulators with four and five SPS-type active legs and one passive constrained leg in different serial structures and to illustrate.

<>

## Figures

Figure 1

A parallel manipulator with n SPS active legs: a constrained leg

Figure 2

A 4-SPS∕PS parallel manipulator and its force situation

Figure 3

A 5-SPS∕UPU parallel manipulator and its force situation

Figure 4

The independent pose parameters (α,β,λ,ro) of m, the extensions of ri, inverse∕forward, velocity, and acceleration for the 4-SPS∕PS parallel manipulator

## Discussions

Some tools below are only available to our subscribers or users with an online account.

### Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections