The problem of optimal control of a Chaplygin ball by internal rotors
The author of this interesting paper studies the problem of optimal control of a Chaplygin ball on a plane by three internal rotors. In 1983, V. V. Kozlov introduced a new model of mechanical system with nonintegrable constraints [Mosc. Univ. Mech. Bull. 37, No. 3--4, 27--34 (1982; Zbl 0558.70012); ibid. 37, No. 3--4, 74--80 (1982; Zbl 0508.70012); libid. 38, No. 3, 40--51 (1983; Zbl 0558.70013)]. In contrast to the classical nonholonomic mechanics for this model, called the vaconomic mechanics, Hamilton's variational principle holds, and trajectories of the system are extremals of the constraint variational problem. The equations of vaconomic mechanics can be used as an approximation of the motions of bodies in an ideal fluid under certain limit relations between masses and moments of inertia, which leads to nonintegrable constraints. These equations appear also in the geometric control theory. The control system on \({\mathcal{M}}=SO(3)\times \mathbb{R}^2\) has the form \[ \dot{q}=\mathbf{f}(q,\pmb{\omega }), \;\;q =( \pmb{\alpha }, \pmb{\beta }, \pmb{\gamma } , \mathbf{r} ) \in\mathcal{M}, \] where \(\pmb{\alpha , \beta , \gamma }\) are the basic vectors represented in the moving frame, \(\pmb{\omega }\in \mathbb{R}^3\) is the vector of angular velocity, the orientation of the ball is given by the orthogonal matrix \((\pmb{ \alpha , \beta , \gamma })\in \mathrm{SO}(3)\), and the position of the center of the ball is expressed by \( \mathbf{r} \in\mathbb{R}^2\); \(\mathcal{M}\) is the configuration space; \(\pmb{\omega }=\pmb{\omega }(t)\in \mathbb{R}^3\) is the control. The vector field \(\mathbf{f}\) on \(\mathcal{M}\) is given by \[ \dot{\alpha }=\pmb{\alpha }\times\pmb{\omega }, \;\;\dot{\pmb{\beta }} =\pmb{\beta }\times\pmb{\omega } \;\;\dot{\pmb{\gamma }}=\pmb{\gamma }\times\pmb{\omega } \] \[ \dot{x} = R\langle\pmb{\beta } ,\pmb{\omega } \rangle, \;\;\dot{y} = - R\langle\pmb{\alpha } ,\pmb{\omega } \rangle. \] The optimal control problem is to move the ball from position \(q_0\in\mathcal{M}\) to position \(q_1\in\mathcal{M}\) in the time interval \(0\leq t\leq \tau \), minimizing some cost functional \({\mathcal{A}}=\int\limits_{0}^{\tau }{\mathcal{L}}(\gamma (t), \pmb{\omega }(t))dt\), where \(\mathcal{L}\) is the Lagrangian. The extremal of \(\mathcal{A}\) may be of two types: normal and abnormal (corresponding to zero Lagrange multiplier). By Pontryagin's maximum principle, to find normal extremals one may compute Pontryagin's Hamiltonian on \(T^{\ast }\mathcal{M}\), that is, \({\mathcal{H}}(q,p)= \max\limits_{u\in\Gamma_{q}} (\langle p,u \rangle -{\mathcal{L}}(q,u))\), \(p\in T_{q}^{\ast }{\mathcal{M}}\). Here \(\Gamma_{q}\) is a \(k\)-dimensional vector subspace of the tangent space \(T_{q}\mathcal{M}\), and \(\dot{q}\in \Gamma_{q}\). If \(\mathcal{L}\) is strictly convex and superlinear in velocity, then \(\mathcal{H}\) is smooth. If \((q(t),p(t))\) is a solution of the Hamiltonian system, then \(q(t)\) is an extremal of the action functional. Further, it is considered the spherically symmetric mass distribution, or more generally the inertia tensor \(J\) is a scalar. Then the vaconomic model of the rolling ball can be integrated in elliptic functions. Another investigation concerns the motion of the center of the ball, which is described by a Newton type equation in \(\mathbb{R}^2\) with the Lorentz type force. It turns out that it can be integrated in elliptic functions as well.
scientific article; zbMATH DE number 6148384
scientific article