1. Physically-Based Motion Synthesis in Computer Graphics
Efficient Generation of Motion Transitions Using Spacetime Constraints
C. Rose, B. Guenter, B. Bodenheimer, M. Cohen
SIGGRAPH 1996
Interactive Manipulation of Rigid Body Simulations
J. Popovic, S. Seitz, M. Erdmann, Z. Popovic, A. Witkin
SIGGRAPH 2000
Svetlana Lazebnik
October 16, 2001

2. Overview
Efficient Generation of Motion Transitions Using Spacetime Constraints
Goal: to generate natural transitions between pre-existing “basis motions” and to create cyclic motions from basis motions (running, walking, etc.)
Model: kinematic tree with 44 degrees of freedom
Method: inverse kinematics and spacetime constraints
Interactive Manipulation of Rigid Body Simulations
Goal: give animator intuitive, interactive control over motions of rigid bodies
Model: generalized state vector consisting of positions and velocities of all the bodies in the system
Method: user interface + fast differential updates + rigid body simulator

3. Efficient Generation of Motion Transitions Using Spacetime Constraints
Human Body Model y z x
Root (6 DOF)
Support points
44 DOF: 38 (joint angles) + 6 (position and orientation)
State space: joint angle function
Efficient Generation of Motion Transitions Using Spacetime Constraints

4. Efficient Generation of Motion Transitions Using Spacetime Constraints
Key Techniques
Inverse kinematics for highly articulated figures [Zhao, Badler 94]
Easy problem: compute the position of the body given all the joint angles
Hard problem: compute the joint angles to accomplish a desired configuration of the body
In particular, we wish to compute joint angles to keep the feet on the floor in the desired positions
Method: solve a high-dimensional nonlinear optimization problem
Spacetime constraints [Witkin, Kass 88]
Problem structure:
Goals – constraints on initial, final, intermediate positions
Optimality criteria – smoothness, efficiency, etc.
Physical constraints – Newton’s laws, collision prevention, etc.
Physical resources – aspects of the system capable of exerting forces
Spacetime formulation: solve for the motion and the forces over the whole time interval of interest, allow constraints to propagate forward and backward in time
Efficient Generation of Motion Transitions Using Spacetime Constraints

5. Generating Transitions
Root motion
Linearly interpolate between initial and final velocities, integrate velocities over desired time interval to get position
Support limb motion (inverse kinematics)
Minimize deviations of constrained coordinate frames from their desired positions
BFGS optimization algorithm is used to find joint angle functions that minimize R
Optimizing the transition (spacetime constraints, inverse dynamics)
Joint torques are used to approximate metabolic energy of the body:
Torques are a function of joint angle functions, and their first and second derivatives
Computing the gradient of e requires finding angular velocities, accelerations, etc. – O(n2) partial derivatives at each iteration
Recursive inverse dynamics method [Balafoutis 91] used to find required physical quantities in O(n) time
Efficient Generation of Motion Transitions Using Spacetime Constraints

6. Efficient Generation of Motion Transitions Using Spacetime Constraints
Results
Beginning and ending points for a transition
Generated transition
Running time – .6 seconds,
Computation time – 72 seconds
Effect of constraints on support links
Interpolation vs. spacetime constraints
Efficient Generation of Motion Transitions Using Spacetime Constraints

7. Interactive Manipulation of Rigid Body Simulations
Framework
Generalized state vector q(t)
Includes each body’s position, orientation, linear velocity, angular velocity: q  R3  SO(3)  R3  R3
Control vector u
Shapes, masses, surface normals, coefficients of elasticity, etc.
Evolution of the system:
Simulation function:
Collision function:
Animator specifies an incremental change to the state vector, and the system approximates the corresponding update to the control vector:
Interactive Manipulation of Rigid Body Simulations

8. Manipulation Without Discontinuities
Local linearization of the simulation function:
Need to find all its partial derivatives with respect to components of the control vector: Decompose S into differential components related to the path of the object: S(tfinal, u) = Spost-collision (tfinal, u) ○ Scollision (tcollision, u) ○ Spre-collision (tcollision, u)
Use automatic differentiation to find derivatives of each component and compose them into a single Jacobian using chain rule
Differential update: solve linear system above for  u
If system is underconstrained, solve a quadratic minimization problem:
Types of user-imposed constraints:
Fixed constraints, expression constraints, floating constraints
Interactive Manipulation of Rigid Body Simulations

9. Manipulation With Discontinuities
In general, the simulation function will have discontinuities with respect to the control parameters (due to polygonal approximation of objects)
Smooth component: set of control vectors for which the simulation function is continuously differentiable
Discontinuities restrict the set of states attainable by the system
Use additional control parameters to vary surface normals
Use finer meshes or curvature-dependent approximations
Basic system will converge only if there exists a path between initial and desired configurations within a single smooth component
With discontinuities present, system must search different smooth components
When simulation goes off the edge of one smooth component, system samples nearby control vectors to find the most promising new component
User has to guide the system between distant smooth components (e.g. different collision sequences)
Interactive Manipulation of Rigid Body Simulations

10. Interactive Manipulation of Rigid Body Simulations
Results
Eggs collide and drop into their respective buckets
Table and tetrahedron collide, table lands on its legs,
Tetrahedron lands on table
Mug tumbles across the floor without tipping over
Scissors bounce, flip, and land on the rack
Animations available at
Interactive Manipulation of Rigid Body Simulations

11. Compare and Contrast Motion Transitions Interactive Manipulation
Animating the human body (kinematic tree)
44 DOF
Length of animations: less than 1 sec.
Requires pre-existing motions
Interactive
Not real time (in 1996)
Converges to local minumum (maybe)
Interactive Manipulation
Animating a collection of interacting rigid bodies
DOF depends on number of bodies, control parameters
Length of animations: a few seconds
Generates motions from scratch
Real time
Converges to local minimum (maybe)

12. Discussion Differences from traditional motion planning
Dynamic constraints (Newton’s laws, etc.)
Motion generation is not fully automatic
Emphasis on interaction, “subjective” optimality criteria
Efficient Generation of Motion Transitions Using Spacetime Constraints
Needs pre-existing motions as input
Model of human body dynamics is not accurate
Can the techniques be extended to generate complex, long motions from scratch?
Interactive Manipulation of Rigid Body Simulations
Has convergence problems in the presence of discontinuities
Does not perform discrete search between different motion sequences
Interactivity requirement limits complexity of systems, motions
Is not appropriate for some multi-body problems