Adaptive Linear Quadratic Regulator-Based Stabilizing Control for Underactuated Double-Gimbal Control Moment Gyroscopes

This paper presents an adaptive linear quadratic regulator (ALQR)-based stabilizing control strategy for underactuated double-gimbal control moment gyroscopes (UDGCMGs). First, for the first time a state-dependent state-space model is derived for the UDGCMGs by systematically reducing their nonlinear Euler-Lagrange dynamics via partial linearization. Then, based upon the model, the ALQR-based control strategy is designed by extending the classical LQR framework via the state-dependent algebraic Riccati equation (SDARE) method tailored for the UDGCMGs. An iterative algorithm is developed to solve for the SDARE to update the state-dependent feedback gain matrix for the control law in real time. The proposed ALQR control strategy for stabilizing the reference tracking of the UDGCMG is validated against two state-of-the-art nonlinear control strategies, i.e., a model-based feedback linearization control (MBFLC) strategy and an interconnection and damping assignment passivity-based control (IDA-PBC) strategy, by experimental tests on a three-degree-of-freedom UDGCMG platform under constant reference tracking with an initial error and varying reference tracking with start-up conditions, which represents the first hardware validation of a stabilizing control algorithm for the UDGCMGs. For the constant reference tracking with an initial underactuated axis position error of 10.35° (ALQR), 2.20° (MBFLC), and 2.32° (IDA-PBC), the ALQR stabilized the underactuated axis without oscillation and achieved a 3.83 s settling time and 0° steady-state error; while both the MBFLC and the IDA-PBC failed to stabilize the underactuated axis. For the time-varying reference, the ALQR stabilized the underactuated axis and attained a 0° minimum error and a 3.37° maximum error; whereas neither the MBFLC nor the IDA-PBC was able to stabilize the system during start-up.