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Appendix: Contact and Collision Algorithms


We give now a brief summary of the Contact and Collision Algorithms. If the tip contact constraint (free foot touching the ground) is given holonomically as tex2html_wrap_inline784 , then by taking time derivatives we also obtain


where tex2html_wrap_inline786 . Multiplying (gif) by tex2html_wrap_inline726 and substituting for tex2html_wrap_inline790 using (gif), one obtains an operator expression for tex2html_wrap_inline728 .


tex2html_wrap_inline794 is a square matrix of dimension equal to the number of constraints, and it is a quantity related to the Khatib operational space inertia. tex2html_wrap_inline796 are the free generalized accelerations without the influence of the contact force in the dynamics. The final expression for tex2html_wrap_inline728 is expressed in terms of the constrained components of the spatial acceleration tex2html_wrap_inline800 , where tex2html_wrap_inline802 . The quantity tex2html_wrap_inline804 likewise is composed of the constrained components of the linear and angular velocities for the various links in the multibody system.

The true angle accelerations are the sum of tex2html_wrap_inline796 and a correction term tex2html_wrap_inline808 which results from the contact forces propagating throughout the body. These correction accelerations can be calculated from tex2html_wrap_inline728 by the relationship


A very similar algorithm exists for calculating the change in velocities due to an inelastic collision with the ground. The change in the generalized velocities will depend on the leg tip velocities at the moment of contact with the ground, tex2html_wrap_inline812 . One solves for the impulse force tex2html_wrap_inline814 ,


One may solve for tex2html_wrap_inline816 in tex2html_wrap_inline818 to obtain the generalized velocities after collision tex2html_wrap_inline820 . The Contact and Collision Algorithms are discussed at greater length in [1], while recursive algorithms for the explicit calculation of the previously defined quantities in general tree-structured multibody systems are presented in [6].


Michael W. Hardt
Mon Oct 11 17:19:43 MET DST 1999