Simplified Linear Multivariable Control of Robots

NASA 's Jet Propulsion Laboratory, Pasadena, California

A simplified method has been developed to design a control system that would make the joints of a robot follow reference trajectories. The generic design includes independent multivariable feedforward and feedback controllers (see figure). The feedforward controller is based on the inverse of the linearized mod el of the dynamics of the robot and implements a control law that contains only proportional and first and second derivatives of the reference trajectories with respect to time. The feedback controller, which implements a control law of proportional, first-derivative, and integral terms, makes tracking errors converg e toward zero as time passes. The control theory is based on a robot of n rotary joints, the incremental angula r positions of which are described by an n x 1 vector
. The linearized behavior of the robot is expressed by
where A, B, and C are the n x n matrices of the coupled dynamics of the n degrees of freedom and T is the n x 1 vector that represents the incremental control torques applied to the joints. The problem is to make the actual trajectories of the joints,
, follow or converge toward the reference trajectori es
(where t = time). The n x n feedforward controller Q(s) is chosen to be based on the minimal- order inverse of the robot model; that is, to process the input signals
accordi ng to the control law
where s is the Laplace-transform complex frequency. This choice enables the robot to track any reference trajectory. The contribution of this controller to t he overall control law is
. The n x n feedback controller K(s) is intended to produce a stable closed-loop system with desired pole positions in the s plane; that is, with desired frequenc ies of oscillation and damping of the tracking errors
. The feedback transfer funct ion is chosen to be
where
and
represent the n x n proportional, derivative, and integral feedback gains, respectively. This controller contributes
to the overall control law. It is possible to choose the feedback gains so that the system exhibits the desired transient response of tracking errors and so that the tracking error in each joint angle can be controlled independently. This is done by working "backward". The designer first specifies the desired damping factor
and undamped natural frequency
for each angle
and uses these values to determine the elements of the n x n diagonal matrices
that would yield the decoupled error dynamics according to
. The required gains are then found from the following simple equations:

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

Telerobotics Program page

Please email the site webmaster with any comments, criticisms or corrections for this page.
Maintained by: Dave Lavery
Last updated: May 10, 1996