Circular orbit spacecraft control at the L4 point using Lyapunov functions

The objective of this work is to demonstrate the utility of Lyapunov functions in control synthesis for the purpose of maintaining and stabilizing a spacecraft in a circular orbit around the L4 point in the circular restricted three body problem (CRTBP). Incorporating the requirements of a fixed radius orbit and a desired angular momentum, a Lyapunov function is constructed and the requisite analysis is performed to obtain a controller. Asymptotic stability is proved in a defined region around the L4 point using LaSalle's principle.Comment: Accepted for presentation at European Control Conference 201

The Principal Fiber Bundle Structure of the Gimbal-Spacecraft System

The gimbal-spacecraft system, that consists of a variable speed control moment gyro (VSCMG) mounted inside a spacecraft, has been employed as an actuator for the attitude control of a spacecraft and has been much studied in the aerospace control community. Employing a Newtonian approach, the equations of motion are derived, and further study focusses on singularity issues and control law synthesis. While the geometric mechanics community has studied many mechanical systems of engineering interest, including spinning rotors (or momentum wheels) that are used as actuators, there has not been a particular effort to model and control the gimbal-spacecraft system in a geometric framework. This article serves two purposes: it presents the gimbal-spacecraft system in a geometric mechanics framework, and in particular, highlights the connection form, that could form the basis for future control design, and secondly, the exposition is of a tutorial nature whereby the willing reader, with minimal prerequisites, is introduced to the tools of differential geometry in this context.Comment: 1 figur

Geometric approach to tracking and stabilization for a spherical robot actuated by internal rotors

This paper presents tracking control laws for two different objectives of a nonholonomic system - a spherical robot - using a geometric approach. The first control law addresses orientation tracking using a modified trace potential function. The second law addresses contact position tracking using a $right$ transport map for the angular velocity error. A special case of this is position and reduced orientation stabilization. Both control laws are coordinate free. The performance of the feedback control laws are demonstrated through simulations

Almost-global tracking for a rigid body with internal rotors

Almost-global orientation trajectory tracking for a rigid body with external actuation has been well studied in the literature, and in the geometric setting as well. The tracking control law relies on the fact that a rigid body is a simple mechanical system (SMS) on the $3-$dimensional group of special orthogonal matrices. However, the problem of designing feedback control laws for tracking using internal actuation mechanisms, like rotors or control moment gyros, has received lesser attention from a geometric point of view. An internally actuated rigid body is not a simple mechanical system, and the phase-space here evolves on the level set of a momentum map. In this note, we propose a novel proportional integral derivative (PID) control law for a rigid body with $3$ internal rotors, that achieves tracking of feasible trajectories from almost all initial conditions

Discrete Optimal Control of Interconnected Mechanical Systems

This article develops variational integrators for a class of underactuated mechanical systems using the theory of discrete mechanics. Further, a discrete optimal control problem is formulated for the considered class of systems and subsequently solved using variational principles again, to obtain necessary conditions that characterize optimal trajectories. The proposed approach is demonstrated on benchmark underactuated systems and accompanied by numerical simulations

Lyapunov-like functions for attitude control via feedback integrators

The notion of feedback integrators permits Euclidean integration schemes for dynamical systems evolving on manifolds. Here, a constructive Lyapunov function for the attitude dynamics embedded in an ambient Euclidean space has been proposed. We then combine the notion of feedback integrators with the proposed Lyapunov function to obtain a feedback law for the attitude control system. The combination of the two techniques yields a domain of attraction for the closed loop dynamics, where earlier contributions were based on linearization ideas. Further, the analysis and synthesis of the feedback scheme is carried out entirely in Euclidean space. The proposed scheme is also shown to be robust to numerical errors.Comment: Submitted to CDC 201

Structure-preserving discrete-time optimal maneuvers of a wheeled inverted pendulum

The Wheeled Inverted Pendulum (WIP) is a nonholonomic, underactuated mechanical system, and has been popularized commercially as the {\it Segway}. Designing optimal control laws for point-to-point state-transfer for this autonomous mechanical system, while respecting momentum and torque constraints as well as the underlying manifold, continues to pose challenging problems. In this article we present a successful effort in this direction: We employ geometric mechanics to obtain a discrete-time model of the system, followed by the synthesis of an energy-optimal control based on a discrete-time maximum principle applicable to mechanical systems whose configuration manifold is a Lie group. Moreover, we incorporate state and momentum constraints into the discrete-time control directly at the synthesis stage. The control is implemented on a WIP with parameters obtained from an existing prototype; the results are highly encouraging, as demonstrated by numerical experiments

Discrete-time optimal attitude control of spacecraft with momentum and control constraints

This article solves an optimal control problem arising in attitude control of a spacecraft under state and control constraints. We first derive the discrete-time attitude dynamics by employing discrete mechanics. The orientation transfer, with initial and final values of the orientation and momentum and the time duration being specified, is posed as an energy optimal control problem in discrete-time subject to momentum and control constraints. Using variational analysis directly on the Lie group SO(3), we derive first order necessary conditions for optimality that leads to a constrained two point boundary value problem. This two point boundary value problem is solved via a novel multiple shooting technique that employs a root finding Newton algorithm. Robustness of the multiple shooting technique is demonstrated through a few representative numerical experiments

Robust Attitude Tracking for Aerobatic Helicopters: A Geometric Approach

This paper highlights the significance of the rotor dynamics in control design for small-scale aerobatic helicopters, and proposes two singularity free robust attitude tracking controllers based on the available states for feedback. 1. The first, employs the angular velocity and the flap angle states (a variable that is not easy to measure) and uses a backstepping technique to design a robust compensator (BRC) to \textbf{\textit{actively}} suppress the disturbance induced tracking error. 2. The second exploits the inherent damping present in the helicopter dynamics leading to a structure preserving, \textbf{\textit{passively}} robust controller (SPR), which is free of angular velocity and flap angle feedback. The BRC controller is designed to be robust in the presence of two types of uncertainties: structured and unstructured. The structured disturbance is due to uncertainty in the rotor parameters, and the unstructured perturbation is modeled as an exogenous torque acting on the fuselage. The performance of the controller is demonstrated in the presence of both types of disturbances through numerical simulations. In contrast, the SPR tracking controller is derived such that the tracking error dynamics inherits the natural damping characteristic of the helicopter. The SPR controller is shown to be almost globally asymptotically stable and its performance is evaluated experimentally by performing aggressive flip maneuvers. Throughout the study, a nonlinear coupled rotor-fuselage helicopter model with first order flap dynamics is used

A simple proof of the discrete time geometric Pontryagin maximum principle on smooth manifolds

We establish a geometric Pontryagin maximum principle for discrete time optimal control problems on finite dimensional smooth manifolds under the following three types of constraints: a) constraints on the states pointwise in time, b) constraints on the control actions pointwise in time, c) constraints on the frequency spectrum of the optimal control trajectories. Our proof follows, in spirit, the path to establish geometric versions of the Pontryagin maximum principle on smooth manifolds indicated in [Cha11] in the context of continuous-time optimal control.Comment: 11 pages. arXiv admin note: text overlap with arXiv:1708.0441