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TECHNICAL PAPERS

# Control of Macro-Micro Manipulators Revisited

[+] Author and Article Information
Kourosh Parsa1

Centre for Intelligent MachinesKourosh.Parsa@space.gc.ca

Jorge Angeles

Department of Mechanical Engineering, McGill University, 817 Sherbrooke St. W., Montreal, QC, H3A 2K6, Canadaangeles@cim.mcgill.ca

Arun K. Misra

Department of Mechanical Engineering, McGill University, 817 Sherbrooke St. W., Montreal, QC, H3A 2K6, Canadaarun.misra@mcgill.ca

In this paper, “micro base” refers to the link number zero of the micromanipulator. It is the very same link that would be installed on the ground if the micro were mounted on a fixed base. When the micro is put on top of the macro, this link will be kinematically indistinguishable from the tip of the macro.

A self-motion of a kinematically redundant manipulator is a joint-space motion of the manipulator that does not result in any change in the end-effector pose, twist, or twist rate.

The compensability measure is defined as the inverse of the product of the singular values of a Jacobian matrix that maps the flexural generalized velocities to the required compensatory micromanipulator joint rates. Recently, it has been argued (10) that such performance indices do not really show how well-conditioned a certain configuration is.

No matter how accurately the macromanipulator dynamics is modeled, the system dynamics model can be cast in the form of Eq. 7 as long as the flexible-link material is linearly elastic and the link deformation can be represented using flexural coordinates.

Dynamics equations of this form have been derived by many other researchers as well. For example, one can refer to (23), a review paper with 105 references.

The generalized inverse $Z†$ of $Z$ need not be calculated explicitly, the solution 20 being efficiently calculated from Eq. 19 using an orthogonalization procedure (25).

For a more general form of proportional damping, see (27).

Properly speaking, the dynamics expressed by Eq. 29 is both time-varying and configuration dependent. In consequence, none of the results reported for time-varying linear systems in (28-30) are directly applicable.

Because both $Mff$ and $K$ are real, symmetric matrices, all the generalized eigenvalues of the system are real.

Similarly, because of the symmetry of $Mff$, the eigenvectors are also mutually orthogonal with respect to $K$, and $TTKT$ is, hence, diagonal as well.

In this case, the total energy of the system is a Lyapunov function (24) for the system.

As mentioned at the very end of Sec. 6, the proper selection of initial joint angles is very critical for the stability of the flexural motion. We chose similar mass and length for all four links to be able to find proper initial conditions for a wider range of motion speed and amplitude.

As it turns out, this has also been suggested in (5).

1

Corresponding author. Currently, with Spacecraft Engineering, Canadian Space Agency, 6767 Route de l'Aeroport, Longueuil, QC J3Y8Y9, Canada

J. Dyn. Sys., Meas., Control 127(4), 688-699 (Mar 03, 2004) (12 pages) doi:10.1115/1.1870039 History: Received April 30, 2003; Revised March 03, 2004

## Abstract

A full-pose, position and orientation, trajectory-tracking control of kinematically redundant flexible-macro-rigid-micro manipulators is proposed. Redundancy is resolved such that the forces exciting the lowest-frequency “modal coordinates” of the system are minimized, while imposing a proportional damping on the flexural dynamics, rendering it stable under certain conditions. The simulation results show that the initial posture of the manipulator plays an important role in satisfying the foregoing stability conditions.

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## Figures

Figure 1

The end-effector Cartesian trajectory

Figure 2

(a) Joint angles, (b) flexural coordinates, (c) actuation torques, and (d) end-effector trajectory errors, for R=0.6m, Kp=250s−2, and Kt=150s−1

Figure 3

Joint speeds for R=0.6m, Kp=250s−2, and Kt=150s−1

Figure 4

The flexural generalized velocities R=0.6m, Kp=250s−2, and Kt=150s−1

Figure 5

(a) Initial and (b) final postures of the manipulator, for R=0.6m, Kp=250s−2, and Kt=150s−1

Figure 11

(a) The flexural coordinates and (b) the tracking error, for Kp=250s−2 and Kt=150s−1

Figure 12

(a) The flexural coordinates and (b) the tracking error, for Kp=300s−2 and Kt=150s−1

Figure 13

(a) The flexural coordinates and (b) the tracking error, for Kp=250s−2 and Kt=100s−1

Figure 14

(a) The flexural coordinates and (b) the tracking error, for Kp=250s−2 and Kt=200s−1

Figure 6

(a) Initial and (b) final postures of the manipulator, for R=0.4m

Figure 7

Results for R=0.4m

Figure 8

(a) Initial and (b) final postures of the manipulator, for R=0.8m

Figure 9

Results for R=0.8m

Figure 10

(a) The flexural coordinates and (b) the tracking error, for Kp=100s−2 and Kt=150s−1

## Errata

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