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TECHNICAL PAPERS

# Control of Closed Kinematic Chains Using A Singularly Perturbed Dynamics Model

[+] Author and Article Information
Zhiyong Wang

Department of Mechanical Engineering and Material Science, Rice University, Houston, TX, 77005

Fathi H. Ghorbel

Department of Mechanical Engineering and Material Science, Rice University, Houston, TX, 77005ghorbel@rice.edu

The notation $Bv$ is used to denote a compact ball centered at $v=0$, $v∊Rp$, $p$ interger, that is, $Bv={v∊Rp:∥v∥⩽ρυ,ρυpositivereal}⊂Rp$.

J. Dyn. Sys., Meas., Control 128(1), 142-151 (Nov 30, 2005) (10 pages) doi:10.1115/1.2171440 History: Received April 03, 2005; Revised November 30, 2005

## Abstract

In this paper, we propose a novel approach to the control of closed kinematic chains (CKCs). This method is based on a recently developed singularly perturbed model for CKCs. Conventionally, the dynamics of CKCs are described by differential-algebraic equations (DAEs). Our approach transfers the control of the original DAE system to the control of an artificially created singularly perturbed system in which the slow dynamics corresponds to the original DAE when the perturbation parameter tends to zero. Compared to control schemes that rely on solving nonlinear algebraic constraint equations, the proposed method uses an ordinary differential equation (ODE) solver to obtain the dependent coordinates, hence, eliminates the need for Newton-type iterations and is amenable to real-time implementation. The composite Lyapunov function method is used to show that the closed-loop system, when controlled by typical open kinematic chain schemes, achieves asymptotic trajectory tracking. Simulations and experimental results on a parallel robot, the Rice planar Delta robot, are also presented to illustrate the efficacy of our method.

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## Figures

Figure 1

The structure of the controller

Figure 2

Control scheme implementation

Figure 3

Pictures of RPDR

Figure 4

Parameters of the RPDR and desired end effector trajectory

Figure 5

Simulated transient behavior in trajectory tracking (q1 and q̇1): ϵ=0.05,0.01,0.005

Figure 6

Simulated transient behavior in trajectory tracking (constraint error w1): ϵ=0.05,0.01,0.005

Figure 7

Trajectory tracking: Simulations and experimental results (q1 and q2): ϵ=0.01

Figure 8

Trajectory tracking: Simulations and experimental results (q̇1 and q̇2): ϵ=0.01

Figure 9

Trajectory tracking: Simulations and experimental results (constraint error and end-effector trajectory): ϵ=0.01

Figure 10

Trajectory tracking: varying ϵ in experiments (q1 and q2): ϵ=0.1,0.01,0.001

Figure 11

Trajectory tracking: varying ϵ in experiments (q̇1 and q̇2): ϵ=0.1,0.01,0.001

Figure 12

Trajectory tracking: varying ϵ in experiments (control torques): ϵ=0.1,0.01,0.001

Figure 13

Trajectory tracking: varying ϵ in experiments (computational error in z and constraint error): ϵ=0.1,0.01,0.001

Figure 14

Trajectory tracking: varying ϵ in experiments (end-effector trajectory and performance index)

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