Optimization-based animation

Abstract

of a doctoral dissertation at the University of Miami.Dissertation supervised by Professor Victor J. Milenkovic. No. of pages in text: 162A new paradigm for rigid body simulation is presented and analyzed. Current techniques for rigid body simulation run slowly on scenes with many bodies in close proximity. Each time two bodies collide or make or break a static contact, the simulator must interrupt the numerical integration of velocities and accelerations. Even for simple scenes, the number of discontinuities per frame time can rise to the millions. An efficient optimization-based animation (OBA) algorithm is presented which can simulate scenes with many convex threedimensional bodies settling into stacks and other "crowded" arrangements. This algorithm simulates Newtonian (second order) physics and Coulomb friction, and it uses quadratic programming (QP) to calculate new positions, momenta, and accelerations strictly at frame times. The extremely small integration steps inherent to traditional simulation techniques are avoided.Contact points are synchronized at the end of each frame. Resolving contacts with friction is known to be a difficult problem. Analytic force calculation can have ambiguous or non-existing solutions. Purely impulsive techniques avoid these ambiguous cases, but still require an excessive and computationally expensive number of updates in the case of many simultaneous contacts. It is shown informally that even taking into account advances in stiff integration techniques, penalty force methods cannot overcome this issue of running time in highly crowded scenes. New algorithms are presented that calculate simultaneous impulses to resolve collisions and static contacts under the Coulomb friction model. The simultaneous impulses are the solution to a QP.In addition, the algorithms apply "bouncing at distance" and "freezing of bodies" to further speed up the simulation. These new QP algorithms are hybridized with a traditional priority queue momentum update scheme to allow sequential impulses when they are required for realism, such as in the office toy pendulum. When added to the implementation of OBA, these new algorithms increase the speed of the simulation by a factor of up to 30.The position update has been hybridized with retroactive detection (RD) to prevent fast and thin bodies from passing through each other. Due to the modular design of the OBA simulator, the described techniques can be used as components in any existing simulator that follows a modular design of position update, finding contacts, and resolving contacts. Non-convex bodies are simulated as unions of convex bodies. Links and joints are simulated with bi-directional constraints. Analysis of the algorithm and discussion of example simulations are provided. DedicationIn loving memory of my father.iii

A constraint‐stabilized time‐stepping approach for rigid multibody dynamics with joints, contact and friction

“…Using the Mean Value Theorem and our bounds on the first derivatives, we once again invoke Proposition 6.2 and also get 36) which shows that m ∈ E 2 (q (l) ) .…”

“…Milenkovic and Schmidl [36] solve a quadratic program at each step for their optimization based animation technique in the attempt to avoid the extremely small stepsize that can occur in the traditional classical integrate-detect-restart simulation methods. Unfortunately, this method must still fall prey to the small stepsizes that are inevitable when many bodies collide almost simultaneously.…”

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“…Ong's work [22], [23] can be considered as one of the earliest attempts. The optimization-based method using a quadratic objective function can be regarded as implicitly computing PD g [24]. Ortega et al [25] presented a method to locally minimize the kinetic distance between the configurations of a haptic probe and its proxy using constraintbased dynamics and continuous collision detection.…”

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