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On the motion planning of rolling surfaces. (English) Zbl 1044.93015
This paper addresses the problem of motion planing for a pair of 2-dimensional Riemannian manifolds rolling on each other without slipping or spinning. Conceptually, the paper refers to the following fundamental contributions to the subject: V. Jurdjevic, “The geometry of the plate-ball problem” [Arch. Ration. Mech. Anal. 124, 305–328 (1993; Zbl 0809.70005)], [A. A. Agrachev and Y. L. Sachkov, “An intrinsic approach to the control of rolling bodies” [Proc. IEEE CDC, Vol. 1 (1999)], and A. Marigo and A. Bicchi, “Rolling bodies with regular surfaces: controllability theory and applications” [IEEE Trans. Autom. Control 45, 1586–1599 (2000; Zbl 0986.70002)]. A conceptualization of the problem is equivalent to that proposed by Agrachev and Sachkov, and leads to a 5-dimensional state space being a circle bundle over a product of the rolling manifolds. The nonslipping and nonspinning conditions are expressed as a preservation of the Riemannian lengths of trajectories on the rolling manifolds, and as a suitable balance of angular velocities at the contact point. Denoting the state of the rolling manifolds $$M_1$$, $$M_2$$ as $$(c_1,c_2,R)$$, where $$c_i\in M_i$$, the equations of rolling in local coordinates are given the form of a driftless control system \begin{aligned} \dot{c}_1&=u_1X_1^1+u_2X_2^1\\ \dot{c}_2&=u_1(X^2R)_1+u_2(X^2R)_2\\ \dot{R}R^{-1}&=u_1(\omega_1(X_1^1)-\omega_2(X^2R)_1)+u_2(\omega_1(X_2^1)-\omega_2(X^2R)_2), \end{aligned} \tag{1} with $$X^1=(X_1^1,X_2^1)$$, $$X^2=(X^2_1,X^2_2)$$ denoting suitable orthonormal moving frames on $$M_1$$ and $$M_2$$, where $$u_1,u_2$$ are controls. Using geodesic coordinates on $$M_1$$,$$M_2$$, a controllability result for system (1) by Agrachev and Sachkov is recovered (Theorem 1). This result states in particular that a necessary and sufficient condition for controllability is nonisometry of the rolling manifolds. The main contribution of this paper lies in elicitating approaches to the motion planning problem of rolling manifolds. The first approach uses the theory of Liouville control systems, the second one relies on an application of the continuation or homotopy method. The Liouville systems may be called partially differentially flat, what means that it is possible to compute Liouville system trajectories by quadratures from suitably chosen linearizing outputs and their finitely many derivatives. For a representation of (1) in geodesic coordinates it is proved that this system is not differentially flat (Proposition 2), and that if one of the rolling manifolds has a symmetry of revolution then system (1) is Liouvillean (Proposition 3). In the special case when either of the rolling manifolds is a plane, the linearizing outputs are found explicitly (Proposition 4). The continuation method solution of the motion planning problem is based on solving a so-called Ważewski or Path Lifting Equation in the Hilbert space of system controls. The main difficulties with applying this method result from the existence of singularities, and a possibility of explosion of a solution to the Path Lifting Equation.
In the paper a sufficient condition for successful application of the continuation method is provided (Proposition 6). This condition is satisfied e.g. for the plate-ball problem. A specific, continuation-based planning strategy is derived.
Although addressed primarily to mathematicians, this paper should also be of remarkable interest for mathematically oriented people of the mechanics and the robotics communities.

##### MSC:
 93B29 Differential-geometric methods in systems theory (MSC2000) 93C85 Automated systems (robots, etc.) in control theory 70E55 Dynamics of multibody systems 93B05 Controllability
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