Spacetime constraints are a new method for creating character animation. The animator specifies what the character must do, how the motion should be performed, the character's physical structure, and the physical resources available. These requirements, along with Newton's laws, form a constrained optimization problem. The solution is a physically valid motion that satisfies the "what" constraints and optimizes the "how" criteria. The paper presents examples of a Luxo lamp performing coordinated motions, which conform to traditional animation principles like anticipation, squash-and-stretch, follow-through, and timing. The paper discusses the use of physical methods in animation, introducing the spacetime method using a moving particle as an example. It then extends the method to complex problems, describing the Luxo model and results. The spacetime method allows for the imposition of constraints throughout the motion, with effects propagating backward and forward in time. Constraints on positions and velocities encode motion goals, while constraints on muscle forces or interpenetration define physical properties. Newtonian physics provides a constraint relating force and position functions. Solving this constrained optimization problem yields optimal, physically valid motion. The paper describes a spacetime particle example, formulating a problem of making the particle fly from a starting point to a destination in a fixed time with minimal fuel consumption. The problem is solved using a variant of Sequential Quadratic Programming (SQP), which computes a second-order Newton-Raphson step for the objective function and a first-order step for the constraints. The solution involves solving linear systems, with the choice of method being critical due to the sparsity of the matrices. The paper also discusses the extension of the method to complex models, introducing a system that automatically performs symbolic differentiation and simplification of tensor forms, generating optimized code for evaluations. The system includes a runtime environment that dynamically composes functions and sparse matrices, and an SQP solver. The method is applied to a Luxo lamp model, which consists of rigid bodies connected by frictionless joints, with muscles modeled as angular springs. The model's kinetic energy and muscle forces are derived using Lagrangian Dynamics. The paper presents results showing that spacetime methods can produce realistic, complex, and coordinated motion with minimal kinematic constraints. The methods allow for the specification of motion goals and optimization criteria, leading to natural and efficient motion. The results include a jump example with variations in mass, contact force, and hurdle clearance, demonstrating the method's ability to generate realistic motion with physical constraints. The paper concludes that spacetime methods offer new forms of motion control, allowing for global optimization criteria and more efficient animation.Spacetime constraints are a new method for creating character animation. The animator specifies what the character must do, how the motion should be performed, the character's physical structure, and the physical resources available. These requirements, along with Newton's laws, form a constrained optimization problem. The solution is a physically valid motion that satisfies the "what" constraints and optimizes the "how" criteria. The paper presents examples of a Luxo lamp performing coordinated motions, which conform to traditional animation principles like anticipation, squash-and-stretch, follow-through, and timing. The paper discusses the use of physical methods in animation, introducing the spacetime method using a moving particle as an example. It then extends the method to complex problems, describing the Luxo model and results. The spacetime method allows for the imposition of constraints throughout the motion, with effects propagating backward and forward in time. Constraints on positions and velocities encode motion goals, while constraints on muscle forces or interpenetration define physical properties. Newtonian physics provides a constraint relating force and position functions. Solving this constrained optimization problem yields optimal, physically valid motion. The paper describes a spacetime particle example, formulating a problem of making the particle fly from a starting point to a destination in a fixed time with minimal fuel consumption. The problem is solved using a variant of Sequential Quadratic Programming (SQP), which computes a second-order Newton-Raphson step for the objective function and a first-order step for the constraints. The solution involves solving linear systems, with the choice of method being critical due to the sparsity of the matrices. The paper also discusses the extension of the method to complex models, introducing a system that automatically performs symbolic differentiation and simplification of tensor forms, generating optimized code for evaluations. The system includes a runtime environment that dynamically composes functions and sparse matrices, and an SQP solver. The method is applied to a Luxo lamp model, which consists of rigid bodies connected by frictionless joints, with muscles modeled as angular springs. The model's kinetic energy and muscle forces are derived using Lagrangian Dynamics. The paper presents results showing that spacetime methods can produce realistic, complex, and coordinated motion with minimal kinematic constraints. The methods allow for the specification of motion goals and optimization criteria, leading to natural and efficient motion. The results include a jump example with variations in mass, contact force, and hurdle clearance, demonstrating the method's ability to generate realistic motion with physical constraints. The paper concludes that spacetime methods offer new forms of motion control, allowing for global optimization criteria and more efficient animation.