Virtual constraint generators for motion control of robots with degree of underactuation one

Virtual holonomic constraints (vhcs) are relations among the configuration variables of a mechanical control system that can be rendered invariant via feedback. A regular vhc is a vhc with the property that the accelerations imparted by the control forces are everywhere transverse to the constraint surface. This paper proposes a methodology for the design of regular vhcs for mechanical systems with degree of underactuation one based on the notion of a virtual constraint generator, a control system on the configuration manifold of the mechanical system whose orbits are all the regular vhcs for the mechanical system. Designing a regular vhc corresponds to designing controllers for the virtual constraint generator. We show how to design regular vhcs inducing two kinds of motions for the constrained dynamics: traversal of the vhc in one direction or oscillation back and forth along the vhc. We propose an optimal control problem for the virtual constraint generator whose solutions are regular vhcs inducing the kind of motions just described and, at the same time, meeting geometric requirements such as obstacle avoidance. We demonstrate the utility of our technique by designing a constraint for an overhead crane making the trolley move from left to right with bounded speed while ensuring that the cable angle remains within fixed limits, and a constraint for a cart-acrobot making the cart oscillate back and forth underneath an obstacle placed overhead while ensuring that the acrobot avoids the obstacle and remains as close as possible to being in the upright position.(c) 2023 Elsevier Ltd. All rights reserved.