~ ComputerGraphics, Volume 22, Number 4, August 1988 l, Spacetime Constraints Andrew Witkin Michael Kass Schlumberger Palo Alto Research 3340 Hillview Avenue, Palo Alto, CA 9~30~ Abstract Siggraph '87 [12]. Although £uzo, Jr. showed us that the team of animator, keyframe system, and renderer can be Spacetime constraints are a new method for creating char- a powerful one, the responsibility for defining the motion acter animation. The animator specifies what the char- remains almost entirely with the animator. acter has to do, for instance, "jump from here to there, Some aspects of animation--personality and appeal, clearing a hurdle in between;" how the motion should be for example---will surely be left to the animator's artistry performed, for instance "don't waste energy," or "come and skill for a long time to come. However, many of the down hard enough to splatter whatever you land on;" the principles of animation are concerned with making the character's physical structure--the geometry, mass, con- character's motion look real at a basic mechanical level nectivity, etc. of the parts; and the physical resources, that ought to admit to formal physical treatment. Con- available to the character to accomplish the motion, for sider for example a jump exhibiting anticipation, squash- instance the character's muscles, a floor to push off from, and-stretch, and follow-through. Any creature--human or etc. The requirements contained in this description, to- lamp--can only accelerate its own center of mass by push- gether with Newton's laws, comprise a problem of con- ing on something else. In jumping, the opportunity to con- strained optimization. The solution to this problem is a trol acceleration only exists during contact with the floor, phyJically valid motion satisfying the "what" constraints because while airborne there is nothing to push on. Antici- and optimizing the "how" criteria. We present as exam- pation prior to takeoff is the phase in which the needed mo- pies a Luxo lamp performing a variety of coordinated mo- mentum is acquired by squashing then stretching to push tions. These realistic motions conform to such principles of off against the floor. Follow-through is the phase in which traditional animation as anticipation, squash-and-stretch, the momentum on landing is absorbed. follow-through, and timing. Such physical arguments make nice poJt hoe explana- tions, but can physics be brought to bear in creating the Keywords -- Animation, Constraints complex active motions of characters like Luxo? If so, how much of what we regard as "nice" motion follows directly I. Introduction from first principles, and how much is really a matter of Computer animation has made enormous strides in the style and convention? past several years. In particular, Pixar's Luzo, Jr. [13] This paper presents a physically-based approach to marked a turning point as perhaps the first computer- character animation in which coordinated, active motion generated work to compete seriously with works of tra- is created automatically by specifying: ditional animation on every front. Key among the reasons • What the character has to do, for instance "jump from for Luzo, .)'r. 's success is that it was made by a talented here to there." animator who adapted the principles of traditional anima- • How the motion should be performed, for instance tion to the computer medium. Luzo, Jr., in large measure, "don't waste energy," or "come down hard enough is a work of traditional animation that happens to use a to splatter whatever you land on." computer to render and to interpolate between keyframes. John Lasseter speUed this out clearly in his presentation to • What the character's physical structure is--what the pieces are shaped llke, what they weigh, how they're Permission to copy without fee all or part of this material is granted provided that the copies are not made or distributed for direct connected, etc. commercial advantage, the ACM copyright notice and the title of the • What physical resources are available to the charac- publication and its date appear, and notice is given that copying is by ter to accomplish the desired motion, for instance the permission of the Association for Computing Machinery. To copy otherwise, or to republish, requires a fee and/or specific permission. character's muscles (or whatever an animate lamp has in place of muscles,) a floor to push off from, etc. @1988 ACM-0-89791-275-6/88/008/0159 $00.75 "Luxo" is a trademark of Jac Jacobsen Industries AS. 159 SIGGRAPH '88, Atlanta, August 1-5, 1988 Our initial experiments with this approach have aimed or completely constrained, are called two-point boundar!l at making a Luxo lamp execute a convincing jump just by problems, requiring more elaborate solution methods than telling it where to start mad where to end. The results forward simulation.[6] we present in this paper show that such properties as an- Character animation poses a still more difficult prob- ticipation, follow-through, squash-and-stretch, and timing lem. Animals move by using their muscles to exert forces indeed emerge from a bare description of the motion's pur- that vary as a function of time. Calculating the motion by pose and the physical context in which it occurs. Moreover, simulation is straightforward once these tlme-dependent simple changes to the goals of the motion or to the phys- force functions are known, but the difficult problem is to ical model give rise to interesting variations on the basic calculate force functions that achieve the goals of the mo- motion. For example, doubling (or quadrupling) the mass tion. Specifying these functions by hvaad would be hope- of Luxo's base creates amusingly exaggerated motion in less, equivalent to making a robot move gracefuLly by man- which the base 1oo~ heavy. ually varying its motor torques. OvLr method entails the numerical solution of laxge In an effort to reconcile the advaaatages of simula- constrained optimization problems, for which a variety of tion with the need for control, several researchers [2, 10] standard algorithms exist. These algorithms, while rela- have proposed methods for blending positional constraints tively expensive, spend most of their time solving sparse with dynamic simulations. The idea behind these meth- linear systems, and are therefore amenable to accelera- ods is to treat kinematic constraints as the consequences tion by array processors and other commonly available of unknown "constraint forces," solve for the forces, then hardware. The greatest difficulty arises not in comput- add them into the simulation, exactly canceling that com- ing the numerical solution, but in setting up the intricate ponent of the applied forces that fights against the con- sparse matrix equations that drive the solution process. To straints. address this problem we implemented an object-oriented Constraint force methods permit parts, such as a symbolic algebra system that automates this difficult task character's hands or feet, to be moved along predefined almost entirely. We therefore believe the method described keyframed trajectories, but provide no help in defining here can become a practical animation tool requiring no the trajectories, which is the central problem in creat- more mathematical sophistication of the end user than do ing character animation. While allowing a character to current keyframlng systems. be dragged around manually like a marionette, constraint The remainder of the paper is organized as follows: forces sidestep the central issue of deciding how the char- the following section discusses the previous use of physical acter should move. methods in animation. The spacetime method is then in- These shortcomings led us to adopt a new formula- troduced using a moving particle as a toy example. Next, tion of the constraint problem, whose central characteris- our extension of the method to complex problems is dis- tic is that we solve for the character's motion and time- cussed. Finally, the Luxo model and the results obtained varying muscle forces over the entire time interval of in- with it are described. terest, rather than progressing sequentially through time. Because we extend the model through time as well as space, II. Background and Motivation we call the formulation spacetim¢ constraints. The spacetime formulation permits the imposition of Recently, there has been considerable interest in incorpo- constraints throughout the time course of the motion, rating physics into animation using simulation methods. with the effects of constraints propagating freely back- [10, 17, 18, 2, 16, 7, 9] The appeal of physical simulation ward as well as forward in time. Constraints on initial, as an animation technique lies in its promise to produce re- final, or intermediate positions and velocities directly en- alistic motion automatically by applying the same physical code the goal~ of the motion, while constraints limiting laws that govern real objects' behavior. muscle forces or preventing interpenetration define prop- Unfortunately, the realism of simulation comes at the erties of the physical situation. Additionally, Newtouian expense of control. Simulation methods solve initial value physics provides a constraint relating the force and po- problems: the course of a simulation is completely deter- sition functions that must hold at every instant in time. mined by the objects' initial positions and velocities, and Subject to these constraints we optimize functions that by the forces applied to the objects along the way. An an- specify how the motion should be performed, in terms of imator, however, is usually concerned as much with where efficiency, smoothness, etc. Solving this constrained opti- the objects end up and how they get there as where they mization problem yields optimal, physically valid motion begin. Problems cast in this form are not initial value that achieves the goals specified by the animator.