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Appendix: Reduced Dynamics Algorithm


With the introduction of holonomic constraints, such as the contact of legs with the ground, it is possible to construct a set of reduced unconstrained dynamics of dimension equal to the number of degrees of freedom, N, minus the number of constraints, m. In this section, we outline our approach to calculating the independent generalized accelerations of the reduced set of dynamics. The novelty of this approach is that it does not require the explicit construction of the reduced dynamics. It will be shown how one may extract the solution of the reduced dynamics from the solution of the Contact Algorithm and the solution of the forward dynamics of the entire system. One main advantage of using a reduced unconstrained dynamical model is that optimization programs which require integration of the dynamics will encounter less numerical difficulties.

In order to satisfy the constraint condition (gif), the generalized velocities tex2html_wrap_inline826 must belong to the null space of the constraint Jacobian, tex2html_wrap_inline828 . If the columns of X represents a basis for tex2html_wrap_inline832 , then there exists a representation of tex2html_wrap_inline826 with respect to X denoted here by tex2html_wrap_inline838 , tex2html_wrap_inline840 . Substituting tex2html_wrap_inline842 into the dynamical equations and multiplying on the left by tex2html_wrap_inline844 will give us the reduced dynamics,


where tex2html_wrap_inline846 , tex2html_wrap_inline848 , tex2html_wrap_inline850 , and tex2html_wrap_inline852 .

If tex2html_wrap_inline742 represents the generalized coordinates of the system, then it is possible to choose N-m independent coordinates tex2html_wrap_inline858 and m dependent coordinates tex2html_wrap_inline862 such that tex2html_wrap_inline864 may be used as an alternative expression for (gif). This approach was made in [12], and it leads to an obvious choice for X,


An advantage of making this choice for the basis X is that, as will be shown in the Reduced Dynamics Algorithm, the reduced accelerations are simply represented as tex2html_wrap_inline870 . Our goal here is to show that the solution of the contact algorithm may be used to obtain a solution of the reduced forward dynamics problem. Then an optimization routine performing numerical integration need only integrate on the independent coordinates tex2html_wrap_inline872 . We first give a lemma before the algorithm is presented.



Reduced Dynamics Algorithm

  1. Beginning with an independent set of coordinates tex2html_wrap_inline938 , solve via inverse kinematics for tex2html_wrap_inline862 from tex2html_wrap_inline858 . Similarly solve for tex2html_wrap_inline944 from tex2html_wrap_inline946 using the algebraic relation tex2html_wrap_inline948 .
  2. Given a set of torque inputs u, one may solve for tex2html_wrap_inline732 with the contact algorithm. Simple algebraic manipulation shows that this solution satisfies (a) of Lemma gif.
  3. Using Equation (gif), it follows that tex2html_wrap_inline870 , and it satisfies the reduced dynamics (gif).

This algorithm thus yields the reduced forward dynamics mapping tex2html_wrap_inline958 .


next up previous
Next: Appendix: Reduced Dynamics for Up: Obtaining Minimum Energy Biped Previous: Appendix: Contact and Collision

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