Multivariable PID Controller for Robotic Manipulator

NASA 's Jet Propulsion Laboratory, Pasadena California

A conceptual multivariable controller for a robotic manipulator includes a proportional/derivative (PD) controller in an inner feedback loop, and a proportional/integral/derivative (PID) controller in an outer feedback loop. The PD controller places the poles of the transfer function (in Laplace-transform domain ) of the control system for the linearized mathematical model of the dynamics of the robot. The PID controller tracks the trajectory and decouples the input from the output. The mathematical model of the dynamics of an n-joint robotic manipulator can be represented by a set of coupled nonlinear differential equations of the general form
where
and
are the n x 1 vectors of joint angles, velocities, and acceleratio ns respectively; T(t) is the n x 1 vector of joint torques;
is the n x n inertia m atrix,
is the n x 1 Coriolis-and-centrifugal-torque vector; G(
) is the n x 1 gravity-loadi ng vector; and H (
) is the n x 1 frictional-torque vector. This set of equations i s highly nonlinear in
and
. The problem is to obtain a control scheme that generates the joint torques T(t) required to ensure that the joint angles
track the reference trajectories
as closely as possible. The multivariable controller, which is designed to do this, is based on linear multivariable control theory. The theory, in turn, requires a linearized mathematical model of the dynamics of the robot. The model is, in effect, piecewise-linear because the equations of motion are treated as linear fo r small perturbations during small intervals of time. The controller derived from the linearized equations (see figure) includes the stabilizing (inner-loop) controller K(s) and the tracking (outer-loop) controller Q(s) where K and Q are transfer-function matrices and s is the Laplace-transform complex frequency. The stabilizing controller implements the PD feedback control law
or, taking the Laplace transform,


where T is the torque applied to the robotic manipulator,
is the torque due to the tracking controller,
are constant n x n position and velocity-feedback galn matrices respectively, and
is the n x n transfer-function matrix of the multivariable PD controller. This control law provides, in effect, full state fee dback for the linearized mathematical model and it is therefore both necessary and sufficient for placement of all 2n poles of the transter function of the system a t arbitrary locations in the complex plane. To stabilize the robot and obtain acceptable transient responses, the feedback gains
and
are chosen to place the closed-loop poles at some desired locations in the left half of the complex plane. By providing for decoupling of inputs from outputs, the tracking controller ensur es that the reference trajectory for each joint angle will affect only that joint an gle and that there will be robust steady-state tracking for a class of reference trajectories and torque disturbances. The tracking controller implements the control law

or

where
is the n x I joint angle error vector; and
are proportional, integral, and derivative gain matrices, respectively, and are related to the dynamics by a set of equations. Two approaches to the tuning of the conceptual controller have been proposed. In the first approach, the matrices of the controller are updated during operatio n to compensate for the variations in the matrices of the mathematical model of the robot during gross motion or changes in the payload. In the second approach, high controller gains are used to obtain desirable performance during gross motion and changes in the payload. In the high gain approach, uncertainties in the mathematical model of the dynamics of the robot have negligible effect on the performance of the closed-loop system. The results of the numerical simulation of a controller for a two-link manipulato r show that satisfactory performance is obtained even when the robot is subjected to large changes in payload and torque disturbances. High controller gains do not produce excessive torques or saturation.

More details can be found in:

1. Tarokh, M., and Seraji, H.: RA multivariable control scheme for robot manipulators,S Journal of Robotic Systems, 1991, 8(1), pp. 1-19.

Point of Contact:
Homayoun Seraji,
Mail Stop 198-219
Jet Propulsion Laboratory
4800 Oak Grove Drive
Pasadena, CA 91109
seraji@telerobotics.jpl.nasa.gov

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Last updated: May 10, 1996