A Symbolic Approach to Compute a Null-Space Basis in the Projection Method
We present a hybrid symbolic-numeric approach for the so-called projection method for solving the parameterized differential-algebraic constraint equations associated with multibody mechanical systems. A primary problem in this approach is computing a null-space basis of a matrix of multivariate rational functions, the Jacobian of the symbolic constraint matrix. A purely symbolic approach is untenable in terms of the sheer size of the output, whereas a purely numerical approach does not offer the flexibility of leaving some or all parameters unspecified. Instead we propose a hybrid approach, which does a symbolic preconditioning, followed by representing the null-space basis by straight-line C code, i.e., a black-box null-space basis. We do this in a numerically sensitive way, and show that our black box is numerically robust at almost all parameter settings. This is verified by experimental results on inputs from typical multibody models.
The authors thank Dr. Jürgen Gerhard, Maplesoft Inc., for the motivating problem and his many helpful remarks. This work was supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC) and MITACS Canada.
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