||This dissertation describes the development of techniques to automate the creation of physically-based, goal-directed motion of synthetic creatures, More specifically, the research concentrates on developing an underlying mathematical representation for constraints and objectives imposed on a creature's motion, and explores possibilities for user assistance in the numerical solution of the optimization problem thus defined. The constraint equations and objectives are functions of space and time, or spacetime, and thus impose relationships across or between points in time.; In addition to the development of an interactive user interface to spacetime constraints, three other primary innovations are introduced, (1) The subdivision of spacetime into discrete pieces, or spacetime windows, over which subproblems can be formulated and solved. (2) The use of approximation techniques for the time dependent functions of a creature's degrees us freedom. (3) The use of hybrid symbolic/numerical methods to solve the constrained optimization problem. Other innovations over earlier work include the ability to specify inequality and conditional constraints.; Creatures, within the context of this dissertation, are restricted to assemblies of rigid links connected by joints defining a set of degrees of freedom. (The problem of collision detection and response are not addressed. The special structure of the problems related to physically based modelling of such creatures, as well as the recent development of symbolic algebraic languages, make it possible to develop hybrid symbolic and numeric techniques to solve the complex constrained optimization problems that arise in this work. Symbolic reduction of the degrees optimization to obtain a solution. A language based interface to both the symbolic and numerical computation is described These processes are linked through a graphical user interface that provides both high and low-level interaction with and inspection of the optimization process and the resulting animation. Implementation issues related to the development of the Spacetime Windows system are discussed. Experiments with a variety of simulated creatures, constraint types, and objectives are also reported.