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Research Papers

Multitask Constrained Motion Control Using a Mass-Weighted Orthogonal Decomposition

[+] Author and Article Information
Vincent De Sapio1

Department of Scalable Modeling and Analysis, Sandia National Laboratories, Livermore, CA 94550vdesap@sandia.gov

Jaeheung Park

Department of Intelligent Convergence Systems, Seoul National University, Suwon 443-270, Koreapark73@snu.ac.kr

1

Corresponding author.

J. Appl. Mech 77(4), 041004 (Apr 09, 2010) (10 pages) doi:10.1115/1.4000907 History: Received November 19, 2008; Revised December 10, 2009; Published April 09, 2010

This paper presents an approach to formulating task-level motion-control for holonomically constrained multibody systems based on a mass-weighted orthogonal decomposition. The basis for this approach involves the formation of a recursive null space for constraints and motion-control tasks onto which subsequent motion-control tasks are projected. The recursive null space arises out of the process of orthogonalizing individual task Jacobian matrices. This orthogonalization process is analogous to the Gram–Schmidt process used for orthogonalizing a vector basis. Based on this mass-weighted orthogonal decomposition, recursive algorithms are developed for formulating the overall motion-control equations. The natural symmetry between task-level dynamics and the dynamics of constrained systems is exploited in this approach. An example is presented to illustrate the practical application of this methodology.

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Copyright © 2010 by American Society of Mechanical Engineers
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Figures

Grahic Jump Location
Figure 1

(Left) A chain with a single task defined. The task space vector x=(x,y,z) describes the Cartesian position of the terminal point of the chain. The Jacobian J corresponds to this task. The task space force is denoted by f. (Right) A branching chain with two tasks defined. The task space vectors x1=(x1,y1,z1) and x2=(x2,y2,z2) describe the Cartesian positions of the two independent terminal points.

Grahic Jump Location
Figure 4

(Left) Parallel mechanism consisting of serial chains with loop closures. The three elbow joints are actively controlled while the remaining joints are passive. (Right) The position of the platform is commanded to move to a target while its orientation is uncontrolled (uncontrolled null space motion). In this case nq=9, mT=2, mC=6, and k=3.

Grahic Jump Location
Figure 5

The position of the platform is commanded to move to a target while its orientation is uncontrolled. (Left) Time response of the platform position showing linear critically damped behavior to the target. The control gains are Kp=100 and Kv=20. (Right) Time response of the platform orientation showing undamped null space oscillation due to the uncontrolled null space.

Grahic Jump Location
Figure 6

The position of the platform is commanded to move to a target while its orientation is uncontrolled. Time response of the control torques during goal movement. Zero control torque (numerical error at the order of 10−14) is produced at the passive joints τ1, τ3, and τ5 due to the imposition of the passivity requirement in the controller. The control gains are Kp=100 and Kv=20.

Grahic Jump Location
Figure 2

A multibody system with holonomic constraints in the form of loop constraints. The task space vector x=(x,y,z) describes the Cartesian position of a point on one of the links. The objective is to control the system using task-level commands in the presence of the mechanism constraints.

Grahic Jump Location
Figure 3

The configuration space constrained motion manifold Qp defined by the constraint equations ϕ(q)=0. All constraint consistent virtual variations δq lie in the tangent space of Qp and are orthogonal to the constraint forces.

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