A minimum attention control law for ball catching

Abstract

Digital implementations of control laws typically involve discretization with respect to both time and space, and a control law that can achieve a task at coarser levels of discretization can be said to require less control attention, and also reduced implementation costs. One means of quantitatively capturing the attention of a control law is to measure the rate of change of the control with respect to changes in state and time. In this paper we present an attention-minimizing control law for ball catching and other target tracking tasks based on Brockett's attention criterion. We first highlight the connections between this attention criterion and some well-known principles from human motor control. Under the assumption that the optimal control law is the sum of a linear time-varying feedback term and a time-varying feedforward term, we derive an LQR-based minimum attention tracking control law that is stable, and obtained efficiently via a finite-dimensional optimization over the symmetric positive-definite matrices. Taking ball catching as our primary task, we perform numerical experiments comparing the performance of the various control strategies examined in the paper. Consistent with prevailing theories about human ball catching, our results exhibit several familiar features, e.g., the transition from open-loop to closed-loop control during the catching movement, and improved robustness to spatiotemporal discretization. The presented control laws are applicable to more general tracking problems that are subject to limited communication resources.

On the Existence and Computation of Minimum Attention Optimal Control Laws

“…It is interesting to observe that the feedforward term initially dominates and eventually approaches zero, while the feedback term initially starts out small but rapidly increases toward the latter part of the motion. This is consistent with observations from the human motor control literature [2], in which human arm motions initially are dominated by feedforward terms, and toward the latter part of the motion when precise positioning is required, feedback terms dominate.…”

“…Referring to Algorithm 1, for initialization we generate an initial reference trajectory and control (x * , u * ) using, e.g., the linear quadratic regulator method of [14]. The initial timevarying feedback gain and feedforward (K (0) (t), v (0) (t)) are then derived from the linearized optimal control perturbed from (x * , u * ) [2]:…”

“…The parameter α adjusts the ratio of the cost placed on the feedback term relative to that of the feedforward term. Some interesting fundamental connections have been pointed out between the attention functional and existing theories for human motor control and learning [2], [3], [4], [5], [6], [7]. New insights and perspectives on the control of soft robots [8] have also been obtained via the minimum attention paradigm.…”

“…Assuming this class of controls, some results on minimum attention trajectories have been obtained for simple kinematic models of wheeled mobile robots [9]. Additional assumptions on the attention functional [10], i.e., setting the terminal cost as the mean square state error, and weighting the attentional functional probabilistically, make it possible to approximate the problem as a linear quadratic regulator (LQR), for which efficient solutions are readily obtainable [2]. It is important to note that these existing works only offer approximate solutions under restricted settings, with no proof guaranteeing the existence of a solution in the general case.…”

Controlling Soft Robots: Balancing Feedback and Feedforward Elements

“…After repeating the reaching trials many times, the subject is able to restore the initial behavior. solution of the MAC problem for a robotic ballcatching example is described in [17]. While the general MAC prob lem is very complex and a comprehensive solution has yet to be found, it does suggest a model of how a progressively bet ter learned feedforward/anticipative action can relieve the need of a strong feedback/reactive action to achieve fast and accurate movements.…”