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On the integration of singularity-free representations of \(\varvec{SO(3)}\) for direct optimal control

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Abstract

In this paper we analyze the performance of different combinations of: (1) parameterization of the rotational degrees of freedom (DOF) of multibody systems, and (2) choice of the integration scheme, in the context of direct optimal control discretized according to the direct multiple-shooting method. The considered representations include quaternions and Direction Cosine Matrices, both having the peculiarity of being non-singular and requiring more than three parameters to describe an element of the Special Orthogonal group \(\textit{SO}(3)\). These representations yield invariants in the dynamics of the system, i.e., algebraic conditions which have to be satisfied in order for the model to be representative of physical reality. The investigated integration schemes include the classical explicit Runge–Kutta method, its stabilized version based on Baumgarte’s technique, which tends to reduce the drift from the underlying manifold, and a structure-preserving alternative, namely the Runge–Kutta Munthe-Kaas method, which preserves the invariants by construction. The performances of the combined choice of representation and integrator are assessed by solving thousands of planning tasks for a nonholonomic, underactuated cart-pendulum system, where the pendulum can experience arbitrarily large 3D rotations. The aspects analyzed include success rate, average number of iterations and CPU time to convergence, and quality of the solution. The results reveal how structure-preserving integrators are the only choice for lower accuracies, whereas higher-order, non-stabilized standard integrators seem to be the computationally most competitive solution when higher levels of accuracy are pursued. Overall, the quaternion-based representation is the most efficient in terms of both iterations and CPU time to convergence, albeit at the cost of lower success rates and increased probability of being trapped by higher local minima.

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Acknowledgements

This work was partially supported by Grant No. 645599 ”SoMa” (Soft-bodied intelligence for Manipulation) within the H2020-ICT-2014-1 program. Support by the EU via ERC-HIGHWIND (259 166), ITN-TEMPO (607957) and ITN-AWESCO (642 682), by DFG via Research Unit FOR 2401, and by the German BMWi via the project eco4wind is also gratefully acknowledged. Marco Gabiccini wishes to thank the colleagues Luca Greco and Paolo Mason from the Laboratoire des Signaux & Systèmes, CentraleSupélec for having initially disclosed the benchmark mechanical system investigated in this paper.

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Correspondence to Silvia Manara.

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Manara, S., Gabiccini, M., Artoni, A. et al. On the integration of singularity-free representations of \(\varvec{SO(3)}\) for direct optimal control. Nonlinear Dyn 90, 1223–1241 (2017). https://doi.org/10.1007/s11071-017-3722-8

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Keywords

  • Special orthogonal group \(\textit{SO}(3)\)
  • Rotation parameterization
  • Direction Cosine Matrix
  • Quaternions
  • Lie group integrators
  • Direct optimal control
  • Spherical pendulum
  • Unicycle