Linear Controllability and Observability of n-Link Underactuated Planar Revolute Robot Moving in Constantly Rotating Frame in Horizontal Plane

For an n-link underactuated planar revolute robot connected to a fixed base in a horizontal plane, in this paper, we study an open problem: Under what actuator configuration is the robot moving in constantly rotating frame with all its links stretching straight out linearly controllable and observable? Different from the existing result for the robot with one unactuated joint, we prove that such linear controllability and observability are guaranteed for all its possible physical parameters, if and only if its first joint is active (actuated). Furthermore, when its first joint is passive (unactuated), we show that for any of its physical parameters, the robot is linearly uncontrollable and unobservable regardless of whether any of its rest joint(s) is active or not. We validate the above results via a physical 3-link underactuated planar revolute robot and a 4-link planar model of a gymnast on the high bar in existing references. This paper presents new insight of the linear controllability and observability for an n-link underactuated planar revolute robot moving in constantly rotating frame in a horizontal plane.