Analytic Proportional-Derivative Control for Precise and Compliant Motion Brian F. Allen, Michael Neff and Petros Faloutsos Abstract— Precise control with proportional-derivative (PD) control generally requires stiffness. The proposed method deter- mines critically damped PD control trajectories that precisely obtain target position and velocity constraints for arbitrary initial conditions. An analytic solution provides the PD control parameters, thereby determining the required impedance. The resulting controller precisely interpolates the target state by solving the full boundary-value problem. Control parameters are time-invariant, and need only be recomputed if the system diverges from the computed trajectory due to unexpected forces or noise. The resulting method provides control with automatically determined compliance, yielding natural response to perturbation. NOMENCLATURE (θ(t),ω(t)) Position and velocity at time t m(t) Moment of inertia at time t (θ 0 ,ω 0 ,t 0 ) Initial state of the system at time t 0 (θ d ,ω d ,t d ) Desired state for the target time t d τ (t) Computed control torque for time t k Computed stiffness of PD controller γ Computed damping of PD controller I. INTRODUCTION Proportional-derivative (PD) control is one of the most commonly used forms of feedback control. This popularity stems from its simplicity, global stability, robustness and broad-applicability[1]. Despite many years of wide-spread use, practical implementations often resort to one of a variety of tuning methods to determine the stiffness and damp- ing constants. One popular example is the Ziegler-Nichols method[2] for tuning proportional-integral-derivative (PID) controllers. A wide variety of automatic tuning methods have been proposed in the robotics and control literature[3]. Our approach considers the use of PD control for precisely interpolating future state. Whereas traditional practice might use a PD controller solely to track an interpolating spline, we propose to eliminate the tracked spline. In it’s place, our system solves for PD control parameters that will result in a critically damped trajectory that will attain the target state precisely. With this method, both the target position and the target velocity of the desired state can satisfied. The calculation of B. F. Allen and P. Faloutsos are with the Department of Computer Science, University of California, Los Angeles, USA. vector@cs.ucla.edu and pfal@cs.ucla.edu M. Neff is with the Department of Computer Science, University of California, Davis, USA. neff@cs.ucdavis.edu This research was funded in part by U.S. Army Medical Research & Materiel Command’s Telemedicine and Advanced Technology Research Center and the UCLA Center for Advanced Surgical and Interventional Technology. this critically damped trajectory is made possible by using an analytic solution for the PD control parameters. Thus, the proposed method finds the unique trajectory curve that solves the full boundary-value problem (BVP) of controlling the system from an initial state (θ 0 ,ω 0 ,t 0 ) to a given future state (θ d ,ω d ,t d ). By chaining a sequence of BVPs, complex trajectories interpolating a series of desired states are made possible. There are several advantages to formulating control as the trajectory of a critically damped PD controller. First, since the result of the proposed method is simply a set of constant control parameters, the method’s stability and robustness are well understood[3]. Second, PD trajectories may be more similar to muscle-generated trajectories and therefore may appear less “robotic” than trajectories based on, for example, cubic splines[4]. Third, the stiffness of the system is implicit in the trajectory. That is, the system has a stiffness that is uniquely determined by the BVP being solved, and so the system responds to perturbations and disturbances with natural compliance. Thus the system will have many of the properties observed in human muscle control, such as an increase in stiffness with increased torque[5]. II. OVERVIEW To begin, section III introduces a recent analytic so- lution [6] of critically damped PD control parameters to robotics. This solution provides a means to determine con- stant PD control parameters that drive the system through a trajectory starting at the system’s initial state (θ 0 ,ω 0 ,t 0 ) and precisely interpolating the target position at the target time (θ d ,t d ). Following on, section IV provides a method for also honoring a desired target velocity ω d at the target time t d , i.e., ω(t d )= ω d . This is shown to be possible, despite the analytic solution of section III already being fully constrained, leaving no (mathematical) degrees-of-freedom to allow control over final velocity. Our approach is to introduce an invertible transform, called the f -adjustment. As the magnitude of the f -adjustment varies, so does the velocity at the target time ω(t d ) change. Once the f -adjustment that satisfies ω(t d )= ω d is found, the system is transformed back to the original coordinates. The resulting solution satisfies the full boundary-value problem, i.e., the analytically calculated trajectory satisfies both the initial conditions (θ(t 0 ),ω(t 0 )) = (θ 0 ,ω 0 ) and the target conditions (θ(t d ),ω(t d )) = (θ d ,ω d ). Section V describes the implementation, section VI presents the results of the simulation of the control of a single