Over the past decade, physics-based simulation has become a key enabling technology for variety of applications. It has taken a front seat role in computer games, animation of virtual worlds and robotic simulation. New applications are still emerging and physics is becoming an integral part of many new technologies that might have been thought of not being directly related to physics. For example, physics has been recently used to explain and recover the motion of the subject from video (Vondrak et al., 2008). Unfortunately, despite the availability of various simulation packages, the level of expertise required to use physical simulation correctly is quite high. The goal of this chapter is thus to establish sufficiently strong grounds that would allow the reader to not only understand and use existing simulation packages properly but also to implement their own solutions if necessary. We choose to model world as a set of constrained rigid bodies as this is the most commonly used approximation to real world physics and such a model is able to deliver predictable high quality results in real time. To make sure bodies, affected by various forces, move as desired, a mechanism for controlling motion through the use of constraints is introduced. We then apply the approach to the problem of physics-based animation (control) of humanoid characters. We start with a review of unconstrained rigid body dynamics and introduce the basic concepts like body mass properties, state parameterization and equations of motion. The derivations will follow (Baraff et al., 1997) and (Erleben, 2002), using notation from (Baraff, 1996). For background information, we recommend reading (Eberly, 2003; Thornton et al., 2003; Bourg, 2002). We then move to Lagrangian constrained rigid body dynamics and show how constraints on body accelerations, velocities or positions can be modeled and incorporated into simpler unconstrained rigid body dynamics. Various kinds of constraints are discussed, including equality constraints (required for the implementation of “joint motors”), inequality constraints (used for the implementation of “joint angle limits”) and bounded equality constraints (used for implementation of motors capable of generating limited motor forces). We then reduce the problem of solving for constraint forces to the problem of solving linear complementarity problems. Finally, we show how this method can be used to enforce body non-penetration and implement a contact model, (Trinkle et al., 1997; Kawachi et al., 1997). Source: Motion Control, Book edited by: Federico Casolo, ISBN 978-953-7619-55-8, pp. 580, January 2010, INTECH, Croatia, downloaded from SCIYO.COM