Modeling and control of a differential wrist with joint compliance

The dynamics of a differential gear train employed on the compliant wrist drive of the CESAR manipulator was modeled mathematically. Expressions for the kinetic and potential energies of the compliant wrist drive train were developed and Lagrange's equations were applied to derive the equations of motion.

Three methods were used to control the compliant wrist drive train: pole-placement, Pl-compensated pole-placement, and LQR design with asymptotic properties. The pole-placed design assumed the system was fourth-order, linearized, and decoupled. Feedback gains for the pole-placement design chosen so that the closed-loop characteristic equation corresponded to that of the binomial standard filter. The PI, pole-placement design involved choosing the PI gains so that the closed-loop characteristic roots were near the inner-loop characteristic roots. Inner-loop roots were then pole-placed so that the outer-loop characteristic equation would correspond to the binomial standard filter. For the LQR design, the compliant wrist drive train was assumed to be a linearized, coupled, eighth-order system. Using the method developed by Harvey and Stein, a set of weighting matrices was found so that the result of linear quadratic optimization would be a control structure with desired modal properties.

Simulation was performed using the coupled, nonlinear equations of motion with feedback gains determined by the three linear control techniques. Feedback gains for the pole-placement and PI designs were updated as position changed, while the gains for the LQR design were constants. Results showed that all three designs could achieve zero overshoot and minmial coupling under no load, but all failed with 50% of maximum load. The designs were then repeated assuming 50% of maximum load (15 lb) as a point mass on the wrist; pole-placement design still failed, while PI and LQR designs were able to keep overshoot to less than 5% while still minimizing coupled response.