1 Complete Controllability of the Rolling n-Sphere - A Constructive Proof M. Kleinsteuber 1 , K. H¨ uper 1 and F. Silva Leite 2 1 National ICT Australia, Canberra Research Laboratory, SEACS Program, Locked Bag 8001, Canberra ACT 2601, Australia and Department of Information Engineering, Research School of Information Sciences and Engineering, The Australian National University, Canberra ACT 0200, Australia. {knut.hueper, martin.kleinsteuber}@nicta.com.au 2 Department of Mathematics, University of Coimbra, 3001-454 Coimbra, Portugal and Institute of Systems and Robotics, University of Coimbra - Polo II, 3030-290 Coimbra, Portugal. fleite@mat.uc.pt Summary. A constructive proof for complete controllability of the rolling n−sphere is presented. By rolling we mean rolling along the hyperplane tangent to the sphere at a particular point, without twisting or slipping. The proof can essentially be reduced to showing that any twist or slip motion can be performed by rolling without twist or slip along a closed path in the hyperplane. We get insight for the general case from a detailed study of the 2−sphere. Keywords Complete controllability, nonholonomy, kinematic equations, rolling maps, Cartan decomposition. 1.1 Introduction The problem of finding a path along which a sphere has to roll (without slipping or twisting) in order to achieve a given position and a given orien- tation is a famous example in control theory. The question of how a shortest path must look like in the case of a 2-dimensional sphere has independently been answered in (Arthurs and Walsh, 1986) and (Jurdjevic, 1993). While A.M. Arthurs and G.R. Walsh formulate the problem in terms of quaternions, V. Jurdjevic considers this as an optimal control problem on the Lie group R 2 × SO 3 . It turns out that the optimal path can be described in terms of elliptic integrals. A rather theoretic treatment of the optimal control problem in n dimensions can be found in (Zimmerman, 2005).