M. Buehler Planning and Control of Center for Intelligent Machines Mechanical Engineering Department Robotic Juggling and McGill University Montréal, Québec, Canada H3A 2A7 Tasks E-mail: [email protected] Catching D. E. Koditschek Electrical Engineering and Computer Science Department University of Michigan Ann Arbor, Michigan 48109 P. J. Kindlmann Center for Systems Science Department of Electrical Engineering Yale University New Haven, Connecticut 06520-1968 Abstract trajectory for the robot controller via a carefully chosen nonlinear function. Thus, the robot is programmed using A new class of control algorithms—the "mirror algorithms"— a mathematical formula rather than an gives rise to experimentally observed juggling and catching expert system or some other means, an we have behavior in a planar robotic mechanism. The simplest of these &dquo;syntactic&dquo; approach come to call robot It succeeds algorithms (on which all the others are founded) is provably &dquo;geometric programming.&dquo;’ correct with respect to a simplified model of the robot and its over a wide range of initial puck locations and recovers environment. This article briefly reviews the physical setup and gracefully from unexpected perturbations of the puck underlying mathematical theory. It discusses two significant states during flight. extensions of the fundamental algorithm to juggling two objects This article reviews the experimental setup and abstract and We data catching. provide from successful empirical verifi- theory we have developed. It describes the geometric con- cations these control and on the of strategies briefly speculate structs underlying the mathematical formulas that make larger implications for the field of robotics. up the robot’s &dquo;program.&dquo; It presents raw data as well as statistical summaries from extensive experiments attesting to the physical validity of this new class of algorithms 1. Introduction that we call &dquo;mirror&dquo; laws. Beyond the level of simple visceral pleasure afforded by machine juggling, we be- We have built a one-degree-of-freedom robot capable lieve that the experiments and mathematical reasoning of juggling two pucks falling freely on a frictionless here offer the rudiments of a plane inclined into the earth’s gravitational field. The presented general approach to many other classes of robotic tasks. It seems worth robot responds sensibly to distinct circumstances. When pausing to motivate such claims before proceeding with in the middle of juggling two pucks, if suddenly one the puck is fixed and held in place, the robot will continue subject proper. to juggle the other. When the first puck is again released, the robot will to restore adjust its hits symmetry between I.]. Geometric Robot Programming the two pucks’ motions. The juggling algorithm works on the principles of feedback theory and implements A central theme of this article (and, indeed, of our gen- what might be called &dquo;visual servoing&dquo;: the sensor-based eral program of research in robotics [Koditschek 1986, is the of algorithm translates puck states into an on-line reference 1987; Koditschek and Rimon 1990]) desirability translating abstract user-defined goals into phase space 1. There is no relationship with the geometric programming technique for solving algebraic nonlinear programming problems. 101 geometry for purposes of task encoding and control. A have as yet no better claim to analytical origins than any number of advantages arise from the absence of logic old computer program. But, by the same measure, their implemented in some more or less formal syntax. First, generation has been no more arcane than writing code in physical robots and the environments within which they any new computer language. must operate are dynamic systems. Their coupling via We do not seriously expect that all robot tasks at any functional relationships-in this case, the mirror algo- level can or should be forced into the geometric formal- rithm-admits some possibility of correctness proofs (as ism developed here. However, we feel that this approach evidenced below), while the recourse to syntactic pre- is particularly suited to robotics in an intermittent dy- scriptions all but eliminates that hope (for example, see namic environment. the related discussion by Andersson [1989]). Second, there is good reason to expect that careful attention to 1.2. Intermittent Dynamic Environments the (provable) geometric invariants of a particular task domain will reveal general properties required of any There is a large and important range of robotic prob- successful controller. These would need merely be &dquo;in- lems requiring interaction with physical objects governed stantiated&dquo; by the appropriate change of coordinates (for by independent kinematics and dynamics whose char- example, as in Koditschek and Rimon [1990]), thereby acteristics change subject to the robot’s actions. The solving an entire range of problems with one controller first systematic work in this task domain has been the structure. Although properly modular software is reusable, pioneering research of Raibert (1986), whose careful ex- one is hard pressed to imagine a careful study of the code perimental studies verify the correctness of his elegant itself revealing which modules are essential. Furthermore, control strategies for legged locomotion. McGeer (1990) we have consistently experienced less brittle modes of has successfully used local linearized analysis to build failure and decreased sensitivity to modeling errors in a passive (unpowered) walking robot and believes that experiments using geometrically expressive control al- similarly tractable analysis should suffice for controlling gorithms when compared with experiments with more running machines as well (McGeer 1989). Wang (1989a) syntactically expressive laws. The insensitivity to noise has proposed using the same local techniques for studying and unexpected perturbations and the strong stability open-loop robot control strategies in intermittent dy- properties of our juggling algorithms are apparent from namical environments, although his ideas remain to be the experimental data presented in Sections 2.4 and 3. empirically verified. Research by Aboaf et al. (1989) on Finally, the geometry is intrinsic to the problem and does juggling suggests that task level learning methods may not commit the controller to a particular computational relieve dynamics-based (or any other parametric) con- model. Logical statements, in contrast, are intimately troller synthesis methods of the need to achieve precise wedded to a discrete symbolic model of computation that performance requirements once a basically functioning best fits a digital computer equipped with a computer system has been ensured. Thus, increasing numbers of re- language. However, the contemporary hegemony in in- searchers have begun to explore the problems of robotics formation processing of digital computers may represent in intermittent dynamic environments with increasingly a brief interlude in the history of technology. Moreover, successful results. those roboticists who look to biologic systems for in- Our work is principally inspired by Raibert’s success spiration (or who, more radically, treat their robots as in tapping the natural dynamics of the environment to plausibility models of biologic organization) will surely achieve a task. We have previously shown via an analysis not be content with the grip of logic and syntax on their similar to that reviewed in this article (Koditschek and field. Buehler 1991) that a greatly simplified version of Raib- The apparent disadvantage of geometric task encoding ert’s hopping algorithm (Raibert 1986) is correct. Thus, relative to syntactic prescriptions is a dramatic reduc- convinced of its value, we have borrowed Raibert’s idea tion in ease of expression. Regardless of whether the of servoing around a mechanical energy level to produce robot’s and environment’s dynamics will &dquo;understand,&dquo; a stable limit cycle and will demonstrate later that this at least we think we know what we mean when we write procedure accounts for the success of the fundamental down if-then-else statements in our favorite program- mirror algorithm as well. Its extension to the problem of ming language. Thus, a central aim in the presentation juggling two bodies simultaneously may, in turn, have that follows is the demonstration that even complicated significance with respect to problems of gait in legged goals involving some combinatorial component (as does locomotion. Presumably, our robot &dquo;settles down&dquo; to a the two-juggle in Section 3.1) may be readily expressed characteristic steady state juggling pattern, because that via the appropriate geometric formalism. The intuitively pattern is an attracting periodic orbit of the closed-loop generated extensions to the fundamental mirror algorithm robot-environment dynamics. Very likely, similar &dquo;nat- of Section 2.4 described and tested in Section 3.1 and 3.2 ural&dquo; control mechanisms would make good candidates 102 for gait regulation. We have proven only that this pre- sumption is correct for the case of a single puck on our juggling plane. The proof of the two-puck case is the log- ical next step. Establishing the formal connection to gait mechanisms will obviously require more work. Furthermore, we believe that the successful control by a one-degree-of-freedom robot of a two- and a four- degree-of-freedom intermittent