2022/10/20 by Vladimir Dragović, Dragović, Vladimir, Borislav Gajić +3
Computer Science · Engineering · #37J35 #37J60 #70E40 #70F25 #Control and Dynamics of Mobile Robots #Dynamical Systems (math.DS) #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Robotic Path Planning Algorithms
paper · pdf · doi:10.48550/arxiv.2210.11586
openalex publication_date 2022/10/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider the nonholonomic systems of n homogeneous balls \mathbf B1,…,\mathbf Bn with the same radius r that are rolling without slipping about a fixed sphere \mathbf S0 with center O and radius R. In addition, it is assumed that a dynamically nonsymmetric sphere \mathbf S with the center that coincides with the center O of the fixed sphere \mathbf S0 rolls without slipping in contact to the moving balls \mathbf B1,…,\mathbf Bn. The problem is considered in four different configurations. We derive the equations of motion and prove that these systems possess an invariant measure. As the main result, for n=1 we found two cases that are integrable in quadratures according to the Euler-Jacobi theorem. The obtained integrable nonholonomic models are natural extensions of the well-known Chaplygin ball integrable problems. Further, we explicitly integrate the planar problem consisting of n homogeneous balls of the same radius, but with different masses, that roll without slipping over a fixed plane Σ0 with a plane Σ that moves without slipping over these balls.