Spline joints for multibody dynamics

ABSTRACT

Spline joints are a novel class of joints that can model general scleronomic constraints for multibody dynamics based on the minimal-coordinates formulation. The main idea is to introduce spline curves and surfaces in the modeling of joints: We model 1-DOF joints using splines on SE(3), and construct multi-DOF joints as the product of exponentials of splines in Euclidean space. We present efficient recursive algorithms to compute the derivatives of the spline joint, as well as geometric algorithms to determine optimal parameters in order to achieve the desired joint motion. Our spline joints can be used to create interesting new simulated mechanisms for computer animation and they can more accurately model complex biomechanical joints such as the knee and shoulder.

REFERENCES

• 1: Marc Alexa, Linear combination of transformations, ACM Transactions on Graphics (TOG), v.21 n.3, July 2002
• 2: Alan H. Barr , Bena Currin , Steven Gabriel , John F. Hughes, Smooth interpolation of orientations with angular velocity constraints using quaternions, Proceedings of the 19th annual conference on Computer graphics and interactive techniques, p.313-320, July 1992
• 3: Delp, S. L., Loan, J. P., Hoy, M. G., Zajac, F. E., Topp, E. L., and Rosen, J. M. 1990. An interactive graphics-based model of the lower extremity to study orthopaedic surgical procedures. IEEE Transactions on Biomedical Engineering 37, 8 (Aug.), 757--767.
• 4: Roy Featherstone , Kluwer Academic Publishers, Robot Dynamics Algorithm, Kluwer Academic Publishers, Norwell, MA, 1987
• 5: Gabriel, S., and Kajiya, J. 1985. Spline interpolation in curved space. In SIGGRAPH 85 Course Notes for "State of the Art in Image Synthesis".
• 6: Kapandji, I. 1974. The Physiology of the Joints. Churchill Livingstone, Edinburgh.
• 7: Danny M. Kaufman , Timothy Edmunds , Dinesh K. Pai, Fast frictional dynamics for rigid bodies, ACM Transactions on Graphics (TOG), v.24 n.3, July 2005
• 8: Myoung-Jun Kim , Myung-Soo Kim , Sung Yong Shin, A general construction scheme for unit quaternion curves with simple high order derivatives, Proceedings of the 22nd annual conference on Computer graphics and interactive techniques, p.369-376, September 1995 [doi>10.1145/218380.218486]
• 9: Paul G. Kry , Dinesh K. Pai, Continuous contact simulation for smooth surfaces, ACM Transactions on Graphics (TOG), v.22 n.1, p.106-129, January 2003 [doi>10.1145/588272.588280]
• 10: Sung-Hee Lee , Demetri Terzopoulos, Heads up!: biomechanical modeling and neuromuscular control of the neck, ACM Transactions on Graphics (TOG), v.25 n.3, July 2006
• 11: Anderson Maciel , Luciana Porcher Nedel , Carla M. Dal Sasso Freitas, Anatomy-Based Joint Models for Virtual Human Skeletons, Proceedings of the Computer Animation, p.220, June 19-21, 2002
• 12: Richard M. Murray , S. Shankar Sastry , Li Zexiang, A Mathematical Introduction to Robotic Manipulation, CRC Press, Inc., Boca Raton, FL, 1994
• 13: F. C. Park , Bahram Ravani, Smooth invariant interpolation of rotations, ACM Transactions on Graphics (TOG), v.16 n.3, p.277-295, July 1997 [doi>10.1145/256157.256160]
• 14: F. C. Park , J. E. Bobrow , S. R. Ploen, A Lie group formulation of robot dynamics, International Journal of Robotics Research, v.14 n.6, p.609-618, Dec. 1995 [doi>10.1177/027836499501400606]
• 15: Ravi Ramamoorthi , Alan H. Barr, Fast construction of accurate quaternion splines, Proceedings of the 24th annual conference on Computer graphics and interactive techniques, p.287-292, August 1997 [doi>10.1145/258734.258870]
• 16: Reuleaux, F. 1876. Kinematics of Machinery: Outlines of a Theory of Machines. MacMillan and Co, London.
• 17: Wei Shao , Victor Ng-Thow-Hing, A general joint component framework for realistic articulation in human characters, Proceedings of the 2003 symposium on Interactive 3D graphics, April 27-30, 2003, Monterey, California [doi>10.1145/641480.641486]
• 18: Ken Shoemake, Animating rotation with quaternion curves, Proceedings of the 12th annual conference on Computer graphics and interactive techniques, p.245-254, July 1985
• 19: Tändl, M., and Kecskeméthy, A. 2007. A comparison of B-spline curves and Pythagorean hodograph curves for multibody dynamics simulation. Proceedings of Twelfth World Congress in Mechanism and Machine Science (June), 380--387.
• 20: Demetri Terzopoulos , Hong Qin, Dynamic NURBS with geometric constraints for interactive sculpting, ACM Transactions on Graphics (TOG), v.13 n.2, p.103-136, April 1994 [doi>10.1145/176579.176580]

INDEX TERMS

Primary Classification:
  I. Computing Methodologies
  I.3 COMPUTER GRAPHICS
      I.3.7 Three-Dimensional Graphics and Realism
          Subjects: Animation

Keywords:
biological joints, multibody dynamics, scleronomic joints, splines