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Constraint control of nonholonomic mechanical systems. (English) Zbl 1387.70027
Summary: We derive an optimal control formulation for a nonholonomic mechanical system using the nonholonomic constraint itself as the control. We focus on Suslov’s problem, which is defined as the motion of a rigid body with a vanishing projection of the body frame angular velocity on a given direction \(\xi\). We derive the optimal control formulation, first for an arbitrary group, and then in the classical realization of Suslov’s problem for the rotation group \(\mathrm{SO}(3)\). We show that it is possible to control the system using the constraint \(\xi (t)\) and demonstrate numerical examples in which the system tracks quite complex trajectories such as a spiral.

MSC:
70Q05 Control of mechanical systems
70E17 Motion of a rigid body with a fixed point
70B10 Kinematics of a rigid body
65L10 Numerical solution of boundary value problems involving ordinary differential equations
65P99 Numerical problems in dynamical systems
65-04 Software, source code, etc. for problems pertaining to numerical analysis
49K15 Optimality conditions for problems involving ordinary differential equations
37J60 Nonholonomic dynamical systems
70F25 Nonholonomic systems related to the dynamics of a system of particles
93B05 Controllability
93B40 Computational methods in systems theory (MSC2010)
93C10 Nonlinear systems in control theory
93C15 Control/observation systems governed by ordinary differential equations
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