Controllability and Observability of an n-Link Underactuated Planar Robot with Different Actuator-Sensor Configurations: Active Intermediate Joint or Joints

This paper considers an n-link underactuated revolute planar robot with all the links moving in the same vertical plane. This paper studies two open problems of the linear controllability and observability of the robot with only an active intermediate joint or only active intermediate joints around the upright equilibrium point (UEP), where all the links are in the upright position. First, when the robot only has a single active intermediate joint or only has multiple active intermediate nonadjacent joints with the corresponding joint angle(s) being measured, this paper shows via an illustrative example that there always exists a set of mechanical parameters that renders the robot linearly uncontrollable and unobservable around the UEP. This, together with an existing result, shows that an n-link planar robot with an actuator and an encoder is linearly controllable and observable around the UEP, regardless of its mechanical parameters, if and only if the first or last link, or the last joint, with the corresponding angle is measured. Second, when the robot only has two intermediate adjacent joints with the corresponding joint angles being measured, this paper proves that the robot is linearly controllable and observable around the UEP regardless of its mechanical parameters. When neither the first nor last joint of the robot is active, this paper shows that the robot is linearly controllable and observable around the UEP if and only if there are at least two active adjacent joints of 1/2 intermediate joints and the corresponding joint angles are measured.