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RESEARCH PAPERS

# An Explicit Method for Determining the Force-Moment Capabilities of Redundantly Actuated Planar Parallel Manipulators

[+] Author and Article Information
Alp Zibil, Flavio Firmani

Robotics and Mechanisms Laboratory, Department of Mechanical Engineering, University of Victoria, P.O. Box 3055, Victoria, BC V8W 3P6, Canada

Scott B. Nokleby

Faculty of Engineering and Applied Science, University of Ontario Institute of Technology, 2000 Simcoe Street North, Oshawa, ON L1H 7K4, Canada

Ron P. Podhorodeski1

Robotics and Mechanisms Laboratory, Department of Mechanical Engineering, University of Victoria, P.O. Box 3055, Victoria, BC V8W 3P6, Canadapodhoro@me.uvic.ca

An underlined joint in a branch type indicates an actuated joint.

The reciprocal product of screw $A={apao}$ and screw $B={bpbo}$ is $A∗B=B∗A=ap∙bo+ao∙bp$, where $∗$ denotes the reciprocal product operator.

For the planar case, a wrench $F=[fx,fy,fz;mx,my,mz]T=[fx,fy,0;0,0,mz]T$. Therefore, for simplification, the zero elements ($fz$, $mx$, and $my$) are omitted and the wrench is denoted in a $3×1$ screw as $F=[fx,fy;mz]T$.

Right Moore–Penrose of matrix $A$ is defined as $[A]+=AT(AAT)−1$.

The $p$ norm of a vector is defined as $∥x∥p=(∑i=1n∣xi∣p)1∕p$$p⩾1$(21).

Note that $r$ is the degree of redundancy (DOR) of the system. For a PPM, $r=k−3$, where $k$ is the total number of actuators and $k⩾3$.

The unconstrained minimization function (fminunc) of the Optimization Toolbox in MATLAB ® (23) is used as the optimization algorithm. This function uses the Broyden-Fletcher-Goldfarb-Shanno quasi-Newton optimization technique.

$nC$, total number of combinations; $nA$, maxed out actuator combinations; $nS$, sign combinations.

1

Corresponding author.

J. Mech. Des 129(10), 1046-1055 (Oct 17, 2006) (10 pages) doi:10.1115/1.2756084 History: Received June 24, 2006; Revised October 17, 2006

## Abstract

A new explicit methodology for the determination of the force-moment capabilities of nonredundantly and redundantly actuated planar parallel manipulators (PPMs) is presented. This methodology is based on properly adjusting the actuator outputs to their maximum capabilities. As a result, the wrench to be applied or sustained is maximized. For a nonredundantly actuated PPM, one actuator can be maximized, while for a redundantly actuated PPM, one actuator, beyond the one of the nonredundant case, may be maximized for every degree of redundancy added to the mechanism. This methodology is compared to a previous work that required an optimization algorithm. The new method yields more accurate and reliable results and is considerably more efficient. Four studies of force-moment capabilities are considered: maximum force with prescribed moment, maximum applicable force, maximum moment with a prescribed force, and maximum applicable moment. The methodology is used to generate the force-moment capabilities of an existing PPM throughout its workspace.

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## Figures

Figure 3

Force polygon (nonredundant case)

Figure 4

Force polygon (pseudoinverse case)

Figure 5

Force polygon (optimization case)

Figure 6

Actuator torques, p=10

Figure 7

Actuator torques, p=106

Figure 8

Actuator torques with the explicit method

Figure 9

Minimum pure forces (N)

Figure 10

Maximum pure forces (N)

Figure 11

Minimum applicable force with an associated moment (N)

Figure 12

Associated moments of the minimum applicable forces found in Fig. 1 (N m)

Figure 13

Maximum applicable forces with an associated moment (N)

Figure 14

Associated moments of the maximum applicable forces found in Figure 1 (N m)

Figure 15

Maximum pure moments (N m)

Figure 16

Maximum applicable moments with an associated force (N m)

Figure 1

Force capability polygon

Figure 2

Schematic diagram of a 3-RRR PPM

Figure 17

Associated forces of the maximum applicable moments found in Fig. 1 (N)

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