Singularly perturbed feedback linearization with linear attitude deviation dynamics realization

A new approach for feedback linearization of attitude dynamics for rigid gas jet-actuated spacecraft control is introduced. The approach is aimed at providing global feedback linearization of the spacecraft dynamics while realizing a prescribed linear attitude deviation dynamics. The methodology is based on nonuniqueness representation of underdetermined linear algebraic equations solution via nullspace parametrization using generalized inversion. The procedure is to prespecify a stable second-order linear time-invariant differential equation in a norm measure of the spacecraft attitude variables deviations from their desired values. The evaluation of this equation along the trajectories defined by the spacecraft equations of motion yields a linear relation in the control variables. These control variables can be solved by utilizing the Moore–Penrose generalized inverse of the involved controls coefficient row vector. The resulting control law consists of auxiliary and particular parts, residing in the nullspace of the controls coefficient and the range space of its generalized inverse, respectively. The free null-control vector in the auxiliary part is projected onto the controls coefficient nullspace by a nullprojection matrix, and is designed to yield exponentially stable spacecraft internal dynamics, and singularly perturbed feedback linearization of the spacecraft attitude dynamics. The feedback control design utilizes the concept of damped generalized inverse to limit the growth of the Moore–Penrose generalized inverse, in addition to the concepts of singularly perturbed controls coefficient nullprojection and damped controls coefficient nullprojection to disencumber the nullprojection matrix from its rank deficiency, and to enhance the closed loop control system performance. The methodology yields desired linear attitude deviation dynamics realization with globally uniformly ultimately bounded trajectory tracking errors, and reveals a tradeoff between trajectory tracking accuracy and damped generalized inverse stability. The paper bridges a gap between the nonlinear control problem applied to spacecraft dynamics and some of the basic generalized inversion-related analytical dynamics principles.     

Inverse Dynamics of Servo-Constraints Based on the Generalized Inverse

The acceleration form of constraint equations is utilized in this paper to solve for the inverse dynamics of servo-constraints. A condition for the existence of control forces that enforce servo-constraints is derived. For overactuated dynamical systems, the generalized Moore-Penrose inverse of the constraint matrix is used to parameterize the solutions for these control forces in terms of free parameters that can be chosen to satisfy certain requirements or optimize certain criterions. In particular, these free parameters can be chosen to minimize the Gibbsian (i.e., the acceleration energy of the dynamical system), resulting in ideal control forces (those satisfying the principle of virtual work when the virtual displacements satisfy the servo-constraint equations). To achieve this, the nonminimal nonholonomic form recently derived by the authors in the context of Kanes method is used to determine the accelerations of the system, and hence to determine the forces to be generated by the redundant manipulators. Finally, an extension to inverse dynamics of servo-constraints involving control variables is made. The procedures are illustrated by two examples.     

Development and experimental investigation of a Quadrotor’s robust generalized dynamic inversion control system

Nonlinear Dynamics - The development of a two-loops Quadrotor’s robust generalized dynamic inversion (RGDI)-based control system is presented. The outer (position) loop utilizes PD position...