{"title": "Fast Neural Network Emulation of Dynamical Systems for Computer Animation", "book": "Advances in Neural Information Processing Systems", "page_first": 882, "page_last": 888, "abstract": null, "full_text": "Fast Neural Network Emulation of Dynamical \n\nSystems for Computer Animation \n\nRadek Grzeszczuk 1 \n\nDemetri Terzopoulos 2 \n\nGeoffrey Hinton 2 \n\n1 Intel Corporation \n\nMicrocomputer Research Lab \n2200 Mission College Blvd. \nSanta Clara, CA 95052, USA \n\n2 University of Toronto \n\nDepartment of Computer Science \n\n10 King's College Road \n\nToronto, ON M5S 3H5, Canada \n\nAbstract \n\nComputer animation through the numerical simulation of physics-based \ngraphics models offers unsurpassed realism, but it can be computation(cid:173)\nally demanding. This paper demonstrates the possibility of replacing the \nnumerical simulation of nontrivial dynamic models with a dramatically \nmore efficient \"NeuroAnimator\" that exploits neural networks. Neu(cid:173)\nroAnimators are automatically trained off-line to emulate physical dy(cid:173)\nnamics through the observation of physics-based models in action. De(cid:173)\npending on the model, its neural network emulator can yield physically \nrealistic animation one or two orders of magnitude faster than conven(cid:173)\ntional numerical simulation. We demonstrate NeuroAnimators for a va(cid:173)\nriety of physics-based models. \n\n1 \n\nIntroduction \n\nAnimation based on physical principles has been an influential trend in computer graphics \nfor over a decade (see, e.g., [1, 2, 3]). This is not only due to the unsurpassed realism \nthat physics-based techniques offer. In conjunction with suitable control and constraint \nmechanisms, physical models also facilitate the production of copious quantities of real(cid:173)\nistic animation in a highly automated fashion. Physics-based animation techniques are \nbeginning to find their way into high-end commercial systems. However, a well-known \ndrawback has retarded their broader penetration--compared to geometric models, physical \nmodels typically entail formidable numerical simulation costs. \n\nThis paper proposes a new approach to creating physically realistic animation that differs \n\n\fEmulation for Animation \n\n883 \n\nradically from the conventional approach of numerically simulating the equations of mo(cid:173)\ntion of physics-based models. We replace physics-based models by fast emulators which \nautomatically learn to produce similar motions by observing the models in action. Our \nemulators have a neural network structure, hence we dub them NeuroAnimators. \n\nOur work is inspired in part by that of Nguyen and Widrow [4]. Their \"truck backer-upper\" \ndemonstrated the neural network based approximation and control of a nonlinear kinematic \nsystem. We introduce several generalizations that enable us to tackle a variety of complex, \nfully dynamic models in the context of computer animation. Connectionist approximations \nof dynamical systems have been also been applied to robot control (see, e.g., [5,6]). \n\n2 The NeuroAnimator Approach \n\nOur approach is motivated by the following considerations: Whether we are dealing with \nrigid [2], articulated [3], or nonrigid [I] dynamic animation models, the numerical sim(cid:173)\nulation of the associated equations of motion leads to the computation of a discrete-time \ndynamical system of the form StHt = ~[St, Ut, ft ]. These (generally nonlinear) equations \nexpress the vector St+8t of state variables of the system (values of the system's degrees of \nfreedom and their velocities) at time t + r5t in the future as a function ~ of the state vector \nSt, the vector Ut of control inputs, and the vector ft of external forces acting on the system \nat time t. \n\nPhysics-based animation through the numerical simulation of a dynamical system requires \nthe evaluation of the map ~ at every timestep, which usually involves a non-trivial compu(cid:173)\ntation. Evaluating ~ using explicit time integration methods incurs a computational cost of \nO(N) operations, where N is proportional to the dimensionality of the state space. Unfor(cid:173)\ntunately, for many dynamic models of interest, explicit methods are plagued by instability, \nnecessitating numerous tiny timesteps r5t per unit simulation time. Alternatively, implicit \ntime-integration methods usually permit larger timesteps, but they compute ~ by solving a \nsystem of N algebraic equations, generally incurring a cost of O( N 3 ) per timestep. \n\nIs it possible to replace the conventional numerical simulator by a significantly cheaper \nalternative? A crucial realization is that the substitute, or emulator, need not compute \nthe map ~ exactly, but merely approximate it to a degree of precision that preserves the \nperceived faithfulness of the resulting animation to the simulated dynamics of the physical \nmodel. Neural networks offer a general mechanism for approximating complex maps in \nhigher dimensional spaces [7].1 Our premise is that, to a sufficient degree of accuracy and \nat significant computational savings, trained neural networks can approximate maps ~ not \njust for simple dynamical systems, but also for those associated with dynamic models that \nare among the most complex reported in the graphics literature to date. \n\nThe NeuroAnimator, which uses neural networks to emulate physics-based animation, \nlearns an approximation to the dynamic model by observing instances of state transitions, \nas well as control inputs and/or external forces that cause these transitions. By generalizing \nfrom the sparse examples presented to it, a trained NeuroAnimator can emulate an infinite \nvariety of continuous animations that it has never actually seen. Each emulation step costs \nonly O(N2) operations, but it is possible to gain additional efficiency relative to a numer(cid:173)\nical simulator by training neural networks to approximate a lengthy chain of evaluations \nof the discrete-time dynamical system. Thus, the emulator network can perform \"super \nI Note that q, is in general a high-dimensional map from RS+u+ f t---7 RS, where s, u, and f denote \n\nthe dimensionalities of the state, control, and external force vectors. \n\n\f884 \n\nR. Grzeszczuk, D. Terzopoulos and G. E. Hinton \n\ntimesteps\" b.t = n6t, typically one or two orders of magnitude larger than 6t for the com(cid:173)\npeting implicit time-integration scheme, thereby achieving outstanding efficiency without \nserious loss of accuracy. \n\n3 From Physics-Based Models to NeuroAnimators \n\nOur task is to construct neural networks that approximate

, 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 [st, Ut, ftl. The trained \nemulator network N takes as input the state of the model, its control inputs, and the \nexternal forces acting on it at time t, and produces as output the state of the model at \ntime t + t1t by evaluating the network. The emulation process is a sequence of these \nevaluations. After each evaluation, the network control and force inputs receive new values, \nand the network state inputs receive the emulator outputs from the previous evaluation. \nSince the emulation step is large compared with the numerical simulation step, we res ample \nthe motion trajectory at the animation frame rate, computing intermediate states through \nlinear interpolation of states obtained from the emulation. \n\n3.1 Network Input/Output Structure \n\nFig. lea) illustrates different emulator input/output structures. The emulator network has \na single set of output variables specifying St+~t. In general, for a so-called active model, \nwhich includes control inputs, under the influence of unpredictable applied forces, we em(cid:173)\nploy a full network with three sets of input variables: St. Ut. and ft. as shown in the figure. \nFor passive models, the control Ut = 0 and the network simplifies to one with two sets of \ninputs, St and ft. In the special case when the forces ft are completely determined by the \nstate of the system St. we can suppress the ft inputs, allowing the network to learn the ef(cid:173)\nfects of these forces from the state transition training data, thus yielding a simpler emulator \nwith two input sets St and Ut. The simplest type of emulator has only a single set of in(cid:173)\nputs St. This emulator suffices to approximate passive models acted upon by deterministic \nexternal forces. \n\n3.2 \n\nInput and Output Transformations \n\nThe accurate approximation of complex functional mappings using neural networks can \nbe challenging. We have observed that a simple feedforward neural network with a single \nlayer of sigmoid units has difficulty producing an accurate approximation to the dynamics \nof physical models. In practice, we often must transform the emulator to ensure a good \napproximation of the map as a network N ~ which is trained \n\n\fEmulation/or Animation \n\n-\"-\n\nIlJ \n\n., ! \n\nI ... ~ \n....!L. : \n\n-\n\nr------------------------------------------------------. \n\n;;~ \n! ____________________________________________________ ~_~J \n1- ------ - - - --- ---- -------------------------------------I \n\nG \n\nN4 \n~ \n\nu \n, \n\nY \n\n, \nI \n\nI \n\n885 \n\n;;~ N' ~ \n\nUt I \nNcJ): \n: \nt ___ __ _ ____________________________ _ _ ______ _ __________ __ J \n\ny : \n\n~ \n\nx \n\ny \n\nT' \nX \n\n~ \nX \n\nN\" \n1$ \n\nT'J \nY \n\nT' \nY \n\n, _____ _ ______ __ _ __ __ ___ __ _ _ _ __ __ _ ____ _______ _ _ __ _ _ _ _ _ __ _ J \n\nI \n\nI. \nI''''' \ny: \nT\" \nNcJ): \n\n-I \n\nI, I \nI t \nu, I \n: \n\n(a) \n\n(b) \n\nFigure 1: (a) Different types of emulators. (b) Transforming a simple feedforward neural \nnetwork Net> into a practical emulator network N4, that is easily trained to emulate physics(cid:173)\nbased models. The following operators perform the appropriate pre- and post-processing: \nT~ transforms inputs to local coordinates, T~ normalizes inputs, T~ unnormalizes outputs, \nT~ transforms outputs to global coordinates, T~ converts from a state change to the next \nstate (see text and [8] for the details). \n\nto emulate the change in the state vector ~St for given state, external force, and control \ninputs, followed by an operator T~ that computes St+t>.t = St + ~St to recover the next \nstate. \n\nWe can further improve the approximation power of the emulator network by exploiting \nnatural invariances. In particular, since the map !f> is invariant under rotation and transla(cid:173)\ntion, we replace N~ with an operator T~ that converts the inputs from the world coordinate \nsystem to the local coordinate system of the model, a network N~ that is trained to emulate \nstate changes represented in the local coordinate system, and an operator T~ that converts \nthe output of N~ back to world coordinates (Fig. I (b) (center\u00bb. \n\nSince the values of state, force, and control variables can deviate significantly, their effect \non the network outputs is uneven, causing problems when large inputs must have a small \ninfluence on outputs. To make inputs contribute more evenly to the network outputs, we \nnormalize groups of variables so that they have zero means and unit variances. With nor(cid:173)\nmalization, we can furthermore expect the weights of the trained network to be of order \nunity and they can be given a simple random initialization prior to training. Hence, in \nFig. l(b)) (bottom) we replace N~ with an operator T~ that normalizes its inputs, a net(cid:173)\nwork N4, that assumes zero mean, unit variance inputs and outputs, and an operator T~ \nthat unnormalizes the outputs to recover their original distributions. \n\nAlthough the final emulator in Fig. 1 (b) is structurally more complex than the standard \nfeed forward neural network Net> that it replaces, the operators denoted by T are completely \ndetermined by the state of the model and the distribution of the training data, and the \nemulator network N4, is much easier to train. \n\n3.3 Hierarchical Networks \n\nAs a universal function approximator, a neural network should in principle be able to ap(cid:173)\nproximate the map !f> for any dynamical system, given enough sigmoid hidden units and \n\n\f886 \n\nR. Grzeszczuk. D. Terzopoulos and G. E. Hinton \n\ntraining data. In practice, however, the number of hidden layer neurons needed and the \ntraining data requirements grow quickly with the size of the network, often making the \ntraining of large networks impractical. To overcome the \"curse of dimensionality,\" we have \nfound it prudent to structure NeuroAnimators for all but the simplest physics-based models \nas hierarchies of smaller networks rather than as large, monolithic networks. The strategy \nbehind a hierarchical representation is to group state variables according to their dependen(cid:173)\ncies and approximate each tightly coupled group with a subnet that takes part of its input \nfrom a parent network. \n\n3.4 Training NeuroAnimators \n\nTo arrive at a NeuroAnimator for a given physics-based model, we train the constituent \nneural network(s) through backpropagation on training examples generated by simulating \nthe model. Training requires the generation and processing of many examples, hence it \nis typically slow, often requiring several CPU hours. However, once a NeuroAnimator is \ntrained offline, it can be reused online to produce an infinite variety of fast animations. \nThe important point is that by generalizing from the sparse training examples, a trained \nNeuroAnimator will produce an infinite variety of extended, continuous animations that it \nhas never \"seen\". \n\nMore specifically, each training example consists of an input vector x and an output vector \ny. In the general case, the input vector x = [s6', rl, u6'V comprises the state of the \nmodel, the external forces, and the control inputs at time t = O. The output vector y = SLl.t \nis the state of the model at time t = 6.t, where 6.t is the duration of the super timestep. \nTo generate each training example, we could start the numerical simulator of the physics(cid:173)\nbased model with the initial conditions So, ro, and uo, and run the dynamic simulation \nfor n numerical time steps M such that flt = nl5t. In principle, we could generate an \narbitrarily large set of training examples {XT; yT}, T = 1,2, ... , by repeating this process \nwith different initial conditions. To learn a good neural network approximation 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}]}