Physically Based Animation - Computer Animation: Algorithms and Techniques - page 248

The second-order step to minimize R , irrespective of the constraints, is taken by solving the linear
system of equations shown in Equation 7.102 . A first-order step to drive the C j s to zero is taken by
solving the linear system of equations shown in Equation 7.103 . The final update is DS j ¼ S j þ S j .
The algorithm reaches a fixed point when the constraints are satisfied, and any further step that min-
imizes R would violate one or more of the constraints.

X

@ H ij S j

@S j ðRÞ¼

(7.102)

j

X

J ij ð S j þ S j Þ C i ¼ (7.103)

j

Although one of the more complex methods presented here, space-time constraints are a powerful
and useful tool for producing realistic motion while maintaining given constraints. They are particu-
larly effective for creating the illusion of self-propelled objects whose movement is subject to user-
supplied time constraints.

7.7 Chapter summary
If complex realistic motion is needed, the models used to control the motion can become complex and
mathematically expensive. However, the resulting motion is oftentimes more natural looking and believ-
able than that produced using kinematic (e.g. interpolation) techniques. While modeling physics is a non-
trivial task, oftentimes it is worth the investment. It should be noted that there has been much work in this
area that has not been included in this chapter. Some other physically based techniques are covered in
later chapters that address specific animation tasks such as the animation of clothes and water.

References
[1] Baraff D. Rigid Body Simulation. In: Course Notes, SIGGRAPH 92. Chicago, Ill.; July 1992.
[2] Blinn J. Light Reflection Functions for Simulation of Clouds and Dusty Surfaces. In: Computer Graphics.
Proceedings of SIGGRAPH 82, vol. 16(3). Boston, Mass.; July 1982. p. 21-9.
[3] Cohen M. Interactive Spacetime Control for Animation. In: Catmull EE, editor. Computer Graphics. Pro-
ceedings of SIGGRAPH 92, vol. 26(2). Chicago, Ill.; July 1992. p. 293-302. ISBN 0-201-51585-7.
[4] Ebert D, Carlson W, Parent R. Solid Spaces and Inverse Particle Systems for Controlling the Animation of
Gases and Fluids. Visual Computer March 1994;10(4):179-90.
[5] Featherstone R, Orin D. Robot Dynamics: Equations and Algorithms.
[6] Frautschi S, Olenick R, Apostol T, Goodstein D. The Mechanical Universe: Mechanics and Heat, Advanced
Edition. Cambridge: Cambridge University Press; 1986.
[7] Gill P, Hammarling S, Murray W, Saunders M, Wright M. User's Guide for LSSOL: A Fortran Package for
Constrained Linear Least-Squares and Convex Quadratic Programming, Technical Report Sol 84-2. Systems
Optimization Laboratory, Department of Operations Research, Stanford University; 1986.
[8] Gill P, Murray W, Saunders M, Wright M. User's Guide for NPSOL: A Fortran Package for Nonlinear Pro-
gramming, Technical Report Sol 84-2. Systems Optimization Laboratory. Department of Operations
Research, Stanford University; 1986.
[9] Gill P, Murray W, Saunders M, Wright M. User's Guide for QPSOL: A Fortran Package for Quadratic
Programming, Technical Report Sol 84-6. Systems Optimization Laboratory, Department of Operations
Research, Stanford University; 1984.