Assembly problems require that a robot with fewer actuated degrees of freedom manipulate an environment containing a greater number of unactuated degrees of freedom. From the perspective of control theory, these problems hold considerable interest because they are characterized by the presence of non-holonomic constraints that preclude the possibility of feedback stabilization. In this sense they necessitate the introduction of a hierarchical controller. This paper explores these issues in the simple instance when all of the pieces to be assembled are constrained to lie on a line. A hierarchical controller is devised for this problem and is shown to be correct: the closed loop system achieves any desired final assembly from all initial configurations that lie in its connected component in configuration space; the generated sequence of motions never causes collisions between two pieces. Further examination of this approach interprets the controller's mediation of conflicting subgoals as promoting an M-player game amongst the pieces to be assembled.
