Planning and Control for Planar Batting and Hopping

Manipulation by batting and locomotion by hopping share several common features. We look at planning and control methods for these types of dynamic underactuated manipulation using predictive model optimization. This generates optimal control sequences for reaching a desired goal state. Results are given for disk batting simulations. Current work is focused on experimental implementation. 1. Overview Manipulation by batting and locomotion by hopping share several common features. Hopping can be viewed as a type of “self manipulation,” and the relationship between batting and hopping has been studied in depth by Koditschek and colleagues [1,2,5,12]; see also (M’Closkey and Burdick [8]; Spong [14]; Vakakis et al. [17]). Batting and hopping are both examples of intermittent dynamical systems. Most previous work on controlling such systems has focused on empirically derived control laws for regulating cyclic batting to a fixed point [1,2,5,12] and (Schaal and Atkeson [13]; Zumel and Erdmann [20]) and hopping with a fixed height and forward speed (Raibert [10]). These controllers have been studied experimentally [1,2,10,12] and (Hodgins and Raibert [4]) and their local and global stability properties have been analyzed [5,8,11,17,20]. Our work is aimed at deriving control laws to transfer the system from a given initial state (of the batted object or hopper) to a desired goal state. In the case of batting, a desired goal state might be to bat a ball into a hoop or into another robot's workspace; in the case of a hopper, the goal could be to land on a stepping stone or to perform a flip. Because external control forces can only be applied during the brief contact phase, generally more than one bat or one hop will be required to carry the system to the goal state, requiring the control law to have a form of “look ahead” embedded in it. Interesting problems include determining the number of impact events (bats or hops) necessary to reach the goal state; characterizing the geometry of the system state space accessible from the initial state; and deriving a suitable control law. Ideally the control law would find a control sequence to take the system to the goal state when it is reachable, and reduce to a (nearly) globally convergent control law when the goal state is reachable from successive impact events (as with the mirror law for planar disk juggling [1,2]). We have built a one joint robot which bats planar parts floating on an air table in a gravitational field (Figure 1). 60 Hz vision feedback is provided by an overhead camera. This system is quite similar to the planar juggler of Bühler and Koditschek [1,2]. The goal is to derive a controller to bat parts from a given initial state to a desired state. This paper describes our preliminary work toward this goal. Our controller uses a model of planar impact dynamics in a nonlinear optimization to find sequences of bats that, in simulation, transfer parts to specified goal states. Our current work is aimed at validating the approach on our experimental setup. Our work is motivated in part by Brown and Zeglin's [19] bow leg planar hopping robot. The robot consists of a low mass fiberglass leg and a relatively high mass robot body, so that the leg can be positioned during flight without affecting the body's motion. The bow leg can be cocked by retracting a string attached to the end of the bow, storing energy in the bow and setting the restitution when the robot impacts with the ground. The orientation of the robot's body is passively stabilized by the placement of the leg pivot just above the body's center of mass. Zeglin and Brown have studied the problem of finding a sequence of hops that land on stepping stones [19]. The robot's two controls at each impact are the leg angle and the cocking of the leg (impact restitution). These control variables are analogous to the two controls in a bat by a one joint robot: the angle and angular velocity of the robot at impact. Batting is a form of underactuated manipulation (Lynch and Mason [6,7]). In underactuated manipulation, a low-degree-offreedom robot controls more part freedoms through dynamic coupling. The part necessarily moves relative to the manipulator (e.g., slipping, rolling, and other forms of nonprehensile manipulation). When the robot maintains continuous contact with the part, the infinite-dimensional robot trajectory can be varied to control the part's freedoms. In the case of planar batting, where we have discrete impact events, the part has three configuration variables and six state variables, and only two controls are available at each impact. Therefore, in general, at least three bats are required to transfer the part to a specific state. 1.1 Batting a Disk in the Plane We examine manipulation by planar batting by a one joint robot in a gravitational field. The robot is dubbed 1JAG, for one joint and gravity. Our goal is to show that a simple batting robot can control all six state variables of an object in the plane by a sequence of bats. We use a predictive impact model to find the state of the object after a bat or sequence of bats, then compare the “distance” of the final state to some predefined goal state. We use nonlinear optimization on the control space to minimize this final distance. This approach results in nearly globally convergent bat juggling to a stable batting cycle when the controls correspond to a single bat (the one-bat). The approach has also been extended to a three-bat scheme that can bat an object from nearly any initial state to nearly any final goal state. While these are simulation results, the control computations are performed in real-time, and we are currently implementing the controller on our experimental setup.