A report presents two general methods for reducing the mathematical complexities of the state equations of robot-arm dynamics. The robot arms may contain both rotary and linear joints. Both methods start with homogenous coordinates and the Lagrangian formulation of mechanics, which are briefly summarized in the report. The first method uses matrix-analysis techniques; the second, vector-analysis techniques. The vector-analysis method includes a new differential-vector representation of centripetal and Coriolis forces. Any of the customary Lagrangian, Newton-Euler, or Hamiltonian methods can be used to derive the equations of motion of rigid-body robot arms. Generally, however, closed form, analytical solutions of the resulting set of complex, nonlinear, coupled, ordinary differential equations do not exist. Formulating efficient computer algorithms for generating torque and force values for a given set of motion variables and constant geometric and inertia parameters of robot arms is one way to attack the mathematical complexities. Such algorithms do not show dynamical details explicitly. A second approachQ namely, formulating explicit state equations for robot-arm dynamicsQmay exhibit too many insignificant details. Both methods require lengthy computation. The report presents both general and case-specific methods of reducing the computational complexity in the state- equation approach. Five general and three specific model reductions or simplifications are derived by matrix methods. They are based in part on exploiting symmetry properties of "dynamic-projection functions." which represent inertial properties and dynamic-coupling coefficients and which are the sums of the traces of certain matrix products appearing in the formulations. One of the general reductions corresponds to the physical observation that the centripetal force of a rotary joint is not felt by the motor of the same rotary joint. The case-specific reductions apply to arms with specified combinations of types of joints. The report section on model reduction by vector analysis introduces differential vectors representing all of the dynamic- projection functions, including those for centripetal and Coriolis forces. Four of the general reductions from the matrix- analysis section are rederived through vector analysis. These general reductions decrease the number of projection functions to about one third of the original number. Seven specific reductions are also derived. The final section of the report considers three further approximate reductions. These simplifications in the dynamic-projection functions are obtained by neglecting terms that are expected to be negligibly small when certain arm-geometry conditions exist. An examples of the simplification of one projection function for an actual robot arm is quoted: A 16-line expression was reduced to a single line. In this case, the worst-case error introduced was less than 8 percent; the average error was about 4 percent. The reduction in computational length was greater than 90 percent.
Point of Contact:
Antal K. Bejczy
Mail Stop 198-219
Jet Propulsion Laboratory
4800 Oak Grove Drive
Pasadena CA 91109
818-354-4568
bejczy@telerobotics.jpl.nasa.gov
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Last updated: May 10, 1996