, with a single layer \nof sigmoidal hidden units, to predict future states using super time steps b.t = n6t while \ncontaining the approximation error so as not to appreciably degrade the physical realism of \nthe resulting animation. The basic emulation step is St+~t = N * of \nthe map CP-, we would like ideally to sample q> as uniformly as possible over its domain, \nwith randomly chosen initial conditions among all valid state, external force, and control \ncombinations. However, we can make better use of computational resources by sampling \nthose state, force, and control inputs that typically occur as a physics-based model is used \nin practice. \n\nWe employ a neural network simulator called Xerion which was developed at the Univer(cid:173)\nsity of Toronto. We begin the off-line training process by initializing the weights of N~ \nto random values from a uniform distribution in the range [0, 1J (due to the normalization \nof inputs and outputs). Xerion automatically terminates the backpropagation learning al(cid:173)\ngorithm when it can no longer reduce the network approximation error significantly. We \nuse the conjugate gradient method to train networks of small and moderate size. For large \nnetworks, we use gradient descent with momentum. We divide the training examples into \nmini-batches, each consisting of approximately 30 uncorrelated examples, and update the \nnetwork weights after processing each mini-batch. \n\n4 Results \n\nWe have successfully constructed and trained several NeuroAnimators to emulate a vari(cid:173)\nety of physics-based models (Fig. 2). We used SDIFAST (a rigid body dynamics simu(cid:173)\nlator marketed by Symbolic Dynamics, Inc.) to simulate the dynamics of the rigid body \n\n\fEmulation/or Animation \n\n887 \n\n(a) \n\n(b) \n\n(c) \n\n(d) \n\nFigure 2: NeuroAnimators used in our experiments. (a) Emulator of a physics-based model \nof a planar multi-link pendulum suspended in gravity, subject to joint friction forces, exter(cid:173)\nnal forces applied on the links, and controlled by independent motor torques at each of the \nthree joints. (b) Emulator of a physics-based model of a truck implemented as a rigid body, \nsubject to friction forces where the tires contact the ground, controlled by rear-wheel drive \n(forward and reverse) and steerable front wheels. (c) Emulator of a physics-based model of \na lunar lander, implemented as a rigid body subject to gravitational forces and controlled by \na main rocket thruster and three independent attitude jets. (d) Emulator of a biomechanical \n(mass-spring-damper) model of a dolphin capable of swimming in simulated water via the \ncoordinated contraction of 6 independently controlled muscle actuators which deform its \nbody, producing hydrodynamic propulsion forces. \n\nand articulated models, and we employ the simulator developed in [10] to simulate the \ndeformable-body dynamics of the dolphin. \n\nIn our experiments we have not attempted to minimize the number of network weights re(cid:173)\nquired for successful training. We have also not tried to minimize the number of sigmoidal \nhidden units, but rather used enough units to obtain networks that generalize well while not \noverfitting the training data. We can always expect to be able to satisfy these guidelines in \nview of our ability to generate sufficient training data. \n\nAn important advantage of using neural networks to emulate dynamical systems is the \nspeed at which they can be iterated to produce animation. Since the emulator for a dynam(cid:173)\nical system with the state vector of size N never uses more than O(N) hidden units, it can \nbe evaluated using only O(N2) operations. By comparison, a single simulation timestep \nusing an implicit time integration scheme requires O(N3) operations. Moreover, a forward \npass through the neural network is often equivalent to as many as 50 physical simulation \nsteps, so the efficiency is even more dramatic, yielding performance improvements up to \ntwo orders of magnitude faster than the physical simulator. A NeuroAnimator that predicts \n100 physical simulation steps offers a speedup of anywhere between 50 and 100 times \ndepending on the type of physical model. \n\n5 Control Learning \n\nAn additional benefit of the NeuroAnimator is that it enables a novel, highly efficient ap(cid:173)\nproach to the difficult problem of controlling physics-based models to synthesize motions \nthat satisfy prescribed animation goals. The neural network approximation to the physical \nmodel is differentiable; hence, it can be used to discover the causal effects that control force \ninputs have on the actions of the models. Outstanding efficiency stems from exploiting the \ntrained NeuroAnimator to compute partial derivatives of output states with respect to con(cid:173)\ntrol inputs. The efficient computation of the approximate gradient enables the utilization of \nfast gradient-based optimization for controller synthesis. \n\n\f888 \n\nR. Grzeszczuk, D. Terzopoulos and G. E. Hinton \n\nNguyen and Widrow's [4] \"truck backer-upper\" demonstrated the neural network based \napproximation and control of a nonlinear kinematic system. Our technique offers a new \ncontroller synthesis algorithm that works well in dynamic environments with changing \ncontrol objectives. See [8, 9] for the details. \n\n6 Conclusion \n\nWe have introduced an efficient alternative to the conventional approach of producing phys(cid:173)\nically realistic animation through numerical simulation. Our approach involves the learning \nof neural network emulators of physics-based models by observing the dynamic state tran(cid:173)\nsitions produced by such models in action. The emulators approximate physical dynamics \nwith dramatic efficiency, yet without serious loss of apparent fidelity. Our performance \nbenchmarks indicate that the neural network emulators can yield physically realistic ani(cid:173)\nmation one or two orders of magnitude faster than conventional numerical simulation of the \nassociated physics-based models. Our new control learning algorithm, which exploits fast \nemulation and the differentiability of the network approximation, is orders of magnitude \nfaster than competing controller synthesis algorithms for computer animation. \n\nAcknowledgements \n\nWe thank Zoubin Ghahramani for valuable discussions leading to the idea of the rotation and transla(cid:173)\ntion invariant emulator, which was crucial to the success of this work. We are indebted to Steve Hunt, \nJohn Funge, Alexander Reshetov, Sonja Jeter and Mike Gendimenico at Intel, and Mike Revow, Drew \nvan Camp and Michiel van de Panne at the University of Toronto for their assistance. \n\nReferences \n\n[1] D. Terzopoulos, 1. Platt, A. Barr, K. Fleischer. Elastically deformable models. In M.e. Stone, \n\ned., Computer Graphics (SIGGRAPH '87 Proceedings), 21 , 205-214, July 1987. \n\n[2] J.K. Hahn: Realistic animation of rigid bodies. In J. Dill, ed., Computer Graphics (SIGGRAPH \n\n'88 Proceedings), 22, 299-308, August 1988. \n\n[3] J.K. Hodgins, w.L. Wooten, D.e. Brogan, J.F. O' Brien. Animating human athletics. In R. Cook, \n\ned., Proc. of ACM SIGGRAPH 95 Conf, 71-78, August, 1995. \n\n[4] D. Nguyen, B. Widrow. The truck backer-upper: An example of self-learning in neural net(cid:173)\n\nworks. In Proc. Inter. Joint Conf Neural Networks , 357-363. IEEE Press, 1989. \n\n[5] M. I. Jordan. Supervised learning and systems with excess degrees of freedom. Technical \n\nReport 88-27, Univ. of Massachusetts, Comp.& Info. Sci. , Amherst, MA, 1988. \n\n[6] K. S. Narendra, K. Parthasarathy. Gradient methods for the optimization of dynamical systems \n\ncontaining neural networks. IEEE Trans. on Neural Networks, 2(2):252-262, 1991. \n\n[7] G. Cybenko. Approximation by superposition of sigmoidal function. Math. of Control Signals \n\n& Systems, 2(4):303-314, 1989. \n\n[8] R. Grzeszczuk. NeuroAnimator: Fast Neural Network Emulation and Control of Physics-Based \n\nModels . PhD thesis, Dept. of Compo Sci., Univ. of Toronto, May 1998. \n\n[9] R. Grzeszczuk, D. Terzopoulos, G. Hinton. NeuroAnimator: Fast neural network emulation \nand control of physics-based models. In M. Cohen, ed., Proc. of ACM SIGGRAPH 98 Conf, \n9-20, July 1998. \n\n[10] X. Th, D. Terzopoulos. Artificial fishes: Physics, locomotion, perception, behavior. In A. Glass(cid:173)\n\nner, ed., Proc. of ACM SIGGRAPH 94 Conf , 43- 50. July 1994. \n\n\f", "award": [], "sourceid": 1562, "authors": [{"given_name": "Radek", "family_name": "Grzeszczuk", "institution": null}, {"given_name": "Demetri", "family_name": "Terzopoulos", "institution": null}, {"given_name": "Geoffrey", "family_name": "Hinton", "institution": null}]}*