Abstract

We compare three solution approaches that use the index 3 set of differential algebraic equations (DAEs) to solve the constrained multibody dynamics problem through straight discretization via an implicit time integrator. The first approach is described in a companion paper and dwells on the connection between the orientation matrix and the SO(3) group. Its salient point is that the orientation matrix A is a problem unknown, directly computed without resorting to the use of other position-level generalized coordinates such as Euler angles or Euler parameters. The second approach employs Euler angles as part of the position-level generalized coordinates, and uses them to subsequently evaluate the orientation matrix A. The third approach replaces the Euler angles with Euler parameters (quaternions). The numerical integration method of choice in this contribution is first order implicit Euler. We report a similar number of iterations for convergence for all solution implementations (called rA, rε, and rp); we also observed an approximately twofold speedup of rA over rp and rε. The tests were carried out in conjunction with three models: simple pendulum, slider crank, and four-link mechanism. These simulation results were obtained using two Python simulation engines that were developed independently as part of this formulation comparison undertaking. The codes are available in a GitHub public repository and were developed to provide two different perspectives on the formulation performance issue. The improvements in simulation speed are traced back to a simpler form of the equations of motion and more concise Jacobians that enter the numerical solution. It remains to investigate whether these speed gains carry to higher order integration formulas, where the underlying Lie-group structure of SO(3) brings additional complexity in the rA solution.

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