3-Link Planar Robot with Passive First Joint

Abstract

This chapter concerns the swing-up control for a 3-link gymnastic planar robot in the vertical plane with its first joint being passive (unactuated) and the rest being active (actuated). The goal of this chapter is to design and analyze a controller under which the robot can be brought into any arbitrarily small neighborhood of the upright equilibrium point, where all three links of the robot remain in the upright position. To tailor the energy-based control approach to achieve the above objectives, first, this chapter considers the links 2 and 3 as a virtual composite link and proposes a coordinate transformation of the angles of active joints. Second, it constructs a novel Lyapunov function based on the transformation, and devises an energy-based swing-up controller. Third, it carries out a global analysis of the motion of the robot under the controller, and establishes some conditions on control parameters for achieving the swing-up control objective. To validate the theoretical results obtained, this chapter provides simulation results for a 3-link robot with its mechanical parameters being obtained from a human gymnast.