Volume xx (200y), Number z, pp. 1–25 Interactive Simulation of Rigid Body Dynamics in Computer Graphics Jan Bender 1 , Kenny Erleben 2 and Jeff Trinkle 3 1 Graduate School CE, TU Darmstadt, Germany 2 Department of Computer Science, University of Copenhagen, Denmark 3 Department of Computer Science, Rensselaer Polytechnic Institute, USA Abstract Interactive rigid body simulation is an important part of many modern computer tools, which no authoring tool nor game engine can do without. Such high performance computer tools open up new possibilities for changing how designers, engineers, modelers and animators work with their design problems. This paper is a self contained state-of-the-art report on the physics, the models, the numerical methods and the algorithms used in interactive rigid body simulation all of which have evolved and matured over the past 20 years. Furthermore, the paper communicates the mathematical and theoretical details in a pedagogical manner. This paper is not only a stake in the sand on what has been done, it also seeks to give the reader deeper insights to help guide their future research. Keywords: Rigid Body Dynamics, Contact Mechanics, Articulated Bodies, Jointed Mechanisms, Contact Point Generation, Iterative Methods. Categories and Subject Descriptors (according to ACM CCS): Computer Graphics [I.3.5]: Computational Geometry and Object Modeling—Physically-based modeling; Computer Graphics [I.3.7]: Three-Dimensional Graphics and Realism—Animation; Mathematics of Computing [G.1.6]: Numerical Analysis—Nonlinear programming 1. Motivation and Perspective on Interactive Rigid Body Simulation Rigid body dynamics simulation is an integral and important part of many modern computer tools in a wide range of ap- plication areas such as computer games, animation software for digital production, including special effects in film and animation movies, robotics validation, virtual prototyping, and training simulators, just to mention a few. In this paper, we focus on interactive rigid body dynamics simulation (as shown in Figure 1), a subfield that has evolved rapidly over the past 10 years and moved the frontier of run- time simulation to applications in areas where, until recently, only off-line simulation was possible. As a consequence, this changes the computer tools humans use and has great eco- nomical impact on society as a whole. The term “interactive” implies a loop closed around a hu- man and a simulation tool. For applications such as games where the feedback is simply animation on a screen, a rea- sonable goal is that the simulation delivers 60 frames per second (fps). For haptic rendering, the simulation would be part of a feedback loop running at 1000Hz, where this rate is needed to display realistic forces to the user [LO08]. Rigid body dynamics has a long history in computer graphics [AG85,MW88,Hah88,Bar89,BBZ91] and a wealth of work exists on the topic. In this state-of-the-art paper, we will cover the important work over the past 20 years on interactive rigid body simulation since the last state- of-the-art report [Bar93] on the subject. In his STAR pa- per, Baraff discussed penalty- and constraint-based meth- ods which use an acceleration-level nonlinear complemen- tarity problem (NCP) formulation. He did not cover many details on solving the complementarity problem. Not un- til 1994, when Baraff published his version of a direct method based on pivoting, was it feasible to compute so- lutions for Baraff’s complementarity problem formulation. For years this solution was the de-facto standard method of rigid body dynamics and it was used in both Maya and submitted to COMPUTER GRAPHICS Forum (7/2013).