Bilateral time-scaling for control of task freedoms of a constrained nonholonomic system

We explore the control of a nonholonomic robot subject to additional constraints on the state variables. In our problem, the user specifies the path of a subset of the state variables (the task freedoms X_P), i.e. a curve X_P(s) where s∈[0,1] is a parameterization that the user chooses. We control the trajectory of the task freedoms by specifying a bilateral time-scaling s(t) which assigns a point on the path for each time t. The time-scaling is termed bilateral because there is no restriction on s(t), the task freedoms are allowed to move backwards along the path. We design a controller that satisfies the user directive and controls the remaining state variables (the shape freedoms X_R) to satisfy the constraints. Furthermore, we attempt to reduce the number of control switchings, as these result in relatively large errors in our system state. If a constraint is close to being violated (at a switchings point), we back up X_P along the path for a small time interval and move X_S to an open region. We show that there are a finite number of switching points for arbitrary task freedom paths. We implement our control scheme on the Mobipulator and discuss a generalization to arbitrary systems satisfying similar properties.