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Research Papers

Trajectory Tracking via Independent Solutions to the Geometric and Temporal Tracking Subproblems

[+] Author and Article Information
Satyajit Ambike1

Department of Mechanical Engineering, The Ohio State University, Columbus, OH 43210ambike.1@osu.edu

James P. Schmiedeler

Aerospace and Mechanical Engineering, University of Notre Dame, Notre Dame, IN 46556schmiedeler.4@nd.edu

Michael M. Stanišić

Aerospace and Mechanical Engineering, University of Notre Dame, Notre Dame, IN 46556stanisic.4@nd.edu

Ordinary singularities are configurations of the robot in which the Jacobian drops rank by 1 and the path tangent does not lie in the robot’s tangent space. The path variable rate is necessarily zero (5,10). The desired trajectory intersecting the workspace boundary nontangentially is a typical example.

At a nonordinary singularity or bifurcation, $T̂∊col[J]$(28). The term “bifurcation” represents the intersection of multiple solution branches in the joint space. These are discussed in Appendix .

1

Corresponding author.

J. Mechanisms Robotics 3(2), 021008 (Apr 07, 2011) (12 pages) doi:10.1115/1.4003272 History: Received August 16, 2010; Revised November 24, 2010; Published April 07, 2011; Online April 07, 2011

Abstract

Trajectory tracking is accomplished by obtaining separate solutions to the geometric path-tracking problem and the temporal tracking problem. A methodology enabling the geometric tracking of a desired planar or spatial path to any order with a nonredundant manipulator is developed. In contrast to previous work, the equations are developed using one of the manipulator’s joint variables as the independent parameter in a fixed global frame rather than a configuration-dependent canonical frame. Both these features provide significant practical advantages. Furthermore, a strategy for determining joint velocities and accelerations at singular configurations is provided, which allows the manipulator to approach and/or move out of a singular configuration with finite joint velocities without sacrificing the geometric fidelity of tracking. An example shows a spatial six-revolute robot tracking a trajectory using the developed method in conjunction with resolved-acceleration feedback control.

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Figures

Figure 1

A six-revolute robot. Reference frames 3–5 are centered at the wrist. They are drawn separately for clarity. The wrist center is used as a control point, so Ψ=0 and d6=0.

Figure 2

The three control points on the EE form a triangular lamina. The initial, final, and an intermediate position of the lamina are indicated. The orientation of the EE must be constant during the motion, so all control points follow identical paths.

Figure 3

Position error for the three control points, as shown here, are close to each other. The maximum error is 2.6×10−5 units.

Figure 4

Joint velocities for the tracking task. The velocities are bounded throughout the task. The slope of the profiles indicates the joint accelerations. The accelerations are high toward the end of the motion for θ2, θ3, and θ4 .

Errata

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