Dynamics and control of a class of underactuated mechanical systems

This paper presents a theoretical framework for the dynamics and control of underactuated mechanical systems, defined as systems with fewer inputs than degrees of freedom. Control system formulation of underactuated mechanical systems is addressed and a class of underactuated systems characterized by nonintegrable dynamics relations is identified. Controllability and stabilizability results are derived for this class of underactuated systems. Examples are included to illustrate the results; these examples are of underactuated mechanical systems that are not linearly controllable or smoothly stabilizable

Control with transverse functions and a single generator of underactuated mechanical systems

The control of a class of underactuated mechanical systems on Lie groups is addressed, with the objective of stabilizing, in a practical sense, any (possibly non-admissible) reference trajectory in the configuration space. The present control design method extends a previous result by the authors to systems underactuated by more than one control. For example, it allows to address the case of a 3D-rigid body immersed in a perfect fluid with only three control inputs. The choice of the control parameters is also discussed in relation to the system's zero-dynamics

ATTITUDE CONTROL ON SO(3) WITH PIECEWISE SINUSOIDS

This dissertation addresses rigid body attitude control with piecewise sinusoidal signals. We consider rigid-body attitude kinematics on SO(3) with a class of sinusoidal inputs. We present a new closed-form solution of the rotation matrix kinematics. The solution is analyzed and used to prove controllability. We then present kinematic-level orientation-feedback controllers for setpoint tracking and command following. Next, we extend the sinusoidal kinematic-level control to the dynamic level. As a representative dynamic system, we consider a CubeSat with vibrating momentum actuators that are driven by small $\epsilon$-amplitude piecewise sinusoidal internal torques. The CubeSat kinetics are derived using Newton-Euler\u27s equations of motion. We assume there is no external forcing and the system conserves zero angular momentum. A second-order approximation of the CubeSat rotational motion on SO(3) is derived and used to derive a setpoint tracking controller that yields order O(ε2) closed-loop error. Numerical simulations are presented to demonstrate the performance of the controls. We also examine the effect of the external damping on the CubeSat kinetics. In addition, we investigate the feasibility of the piecewise sinusoidal control techniques using an experimental CubeSat system. We present the design of the CubeSat mechanical system, the control system hardware, and the attitude control software. Then, we present and discuss the experiment results of yaw motion control. Furthermore, we experimentally validate the analysis of the external damping effect on the CubeSat kinetics

3D Maneuvers For Asymmetric Under-Actuated Rigid Body

Most spacecraft are designed to be maneuvered to achieve pointing goals. This is generally accomplished by designing a three-axis control system. This work explores new maneuver strategies when only two control inputs are available: (i) sequential single-axis maneuvers and (ii) three-dimensional (3D) coupled maneuvers. The sequential single-axis maneuver strategies are established for torque, time, and fuel minimization applications. The resulting control laws are more complicated than the equivalent results for three-axis control because of the highly nonlinear control switch-times. Classical control approaches lead to optimal, but discontinuous control profiles. This problem is overcome by introducing a torque-rate penalty for the torque minimization case. Alternative approaches are also considered for achieving smooth continuous control profiles by introducing a cubic polynomial multiplicative control switch smoother for the time and fuel minimization cases. Numerical and analytical results are presented to compare optimal maneuver strategies for both nominal and failed actuator cases. The 3D maneuver strategy introduces a homotopy algorithm to achieve optimal nonlinear maneuvers minimizing the torque. Two cases are considered: (i) one of the three-axis control actuators fails and (ii) two control actuators fail among four control actuators. The solution strategy first solves the case when all three actuators are available. Then, the failed actuator case is recovered by introducing a homotopy embedding parameter, ε, into the nonlinear dynamics equation. By sweeping ε, a sequence of neighboring optimal control problems is solved that starts with the original maneuver problem and arrives at the solution for the under-actuated case. As ε approaches 1, the designated actuator no longer provides control inputs to the spacecraft, effectively modeling the failed actuator condition. This problem is complex for two reasons: (i) the governing equations are nonlinear and (ii) ε fundamentally alters the spacecraft’s controllability. Davidenko’s method is introduced for developing an ordinary differential equation for the costate variable as a function of ε. For each value of ε, the costate initial conditions are iteratively adjusted so that the terminal boundary conditions for the 3D maneuver are achieved. Optimal control applications are presented for both rest-to-rest and motion-to-rest cases that demonstrate the effectiveness of the proposed algorithm