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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
##### Keywords:
Suslov’s problem; nonholonomic mechanics; optimal control
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