N. Harris McClamroch

Geometric tracking control of a quadrotor UAV on SE(3)

This paper provides new results for the tracking control of a quadrotor unmanned aerial vehicle (UAV). The UAV has four input degrees of freedom, namely the magnitudes of the four rotor thrusts, that are used to control the six translational and rotational degrees of freedom, and to achieve asymptotic tracking of four outputs, namely, three position variables for the vehicle center of mass and the direction of one vehicle body-fixed axis. A globally defined model of the quadrotor UAV rigid body dynamics is introduced as a basis for the analysis. A nonlinear tracking controller is developed on the special Euclidean group SE(3) and it is shown to have desirable closed loop properties that are almost global. Several numerical examples, including an example in which the quadrotor recovers from being initially upside down, illustrate the versatility of the controller.

Planar, time-optimal, rest-to-rest slewing maneuvers of flexible spacecraft

The control problem of time-optimal, rest-to-rest slewing of a flexible spacecraft through a large angle is considered. The flexible spacecraft is modeled as a linear, elastic, undamped, nongyroscopic system suitable for analysis of planar rotational maneuvers. Minimum-time open-loop planar maneuvers are studied. The control histories are found to be bang-bang with multiple switches in each control variable. The optimal control history is shown to have an important time symmetry property. The switching times, final time, and costates at midmaneuver satisfy a system of nonlinear algebraic equations that can be solved using a homotopy method. An upper bound on attitude error due to control spillover is obtained. This helps to determine, a priori, the number of vibrational modes that need to be actively suppressed at the final time such that a prespecified pointing accuracy is guaranteed after the maneuver has been completed. A time-optimal slewing example is discussed to demonstrate the applicability of the results.

Attitude stabilization of a rigid spacecraft using two control torques: a nonlinear control approach based on the spacecraft attitude dynamics

The attitude stabilization problem of a rigid spacecraft using control torques supplied by gas jet actuators about only two of its principal axes is considered. If the uncontrolled principal axis of the spacecraft is not an axis of symmetry, then the complete spacecraft dynamics are small time locally controllable. However, the spacecraft cannot be asymptotically stabilized to any equilibrium attitude using time-invariant continuous feedback. A discontinuous stabi- lizing feedback control strategy is constructed which stabilizes the spacecraft to any equilibrium attitude. If the uncontrolled principal axis of the spacecraft is an axis of symmetry, the complete spacecraft dynamics are not even assessible. However, the spacecraft dynamics are strongly accessible and small time locally controllable in a reduced sense. The reduced spacecraft dynamics cannot be asymptotically stabilized to any equilibrium attitude using time-invariant continuous feedback, but again a discon- tinuous stabilizing feedback control strategy is constructed. In both cases, the discontinuous feedback controllers are constructed by switching between several feedback functions which are selected to accomplish a sequence of spacecraft maneuvers. The results of the paper show that although standard nonlinear control techniques are not applicable, it is possible to construct a nonlinear discontinuous control law based on the dynamics of the particular physical system.

Attitude stabilization of a rigid spacecraft using two momentum wheel actuators

It is well known that three momentum wheel actuators can be used to control the attitude of a rigid spacecraft and that arbitrary reorientation maneuvers of the spacecraft can be accomplished using smooth feedback. If failure of one of the momentum wheel actuators occurs, we demonstrate that two momentum wheel actuators can be used to control the attitude of a rigid spacecraft and that arbitrary reorientation maneuvers of the spacecraft can be accomplished. Although the complete spacecraft equations are not controllable, the spacecraft equations are controllable under the restriction that the total angular momentum vector of the system is zero. The spacecraft dynamics under such a restriction cannot be asymptotically stabilized to any equilibrium attitude using a timeinvariant continuous feedback control law, but discontinuous feedback control strategies are constructed that stabilize any equilibrium attitude of the spacecraft in finite time. Consequently, reorientation of the spacecraft can be accomplished using discontinuous feedback control.

Tracking in nonlinear differential-algebraic control systems with applications to constrained robot systems

We consider the design of a feedback control law for control systems described by a class of nonlinear differential-algebraic equations so that certain desired outputs track given reference inputs. The nonlinear differential-algebraic control system being considered is not in state variable form. Assumptions are introduced and a procedure is developed such that an equivalent state realization of the control system described by nonlinear differential-algebraic equations is expressed in a familiar normal form. A nonlinear feedback control law is then proposed which ensures, under appropriate assumptions, that the tracking error in the closed loop differential-algebraic system approaches zero exponentially. Applications to simultaneous contact force and position tracking in constrained robot systems with rigid joints, constrained robot systems with joint flexibility, and constrained robot systems with significant actuator dynamics are discussed.

Dynamics of a Closed Chain Manipulator

In many manipulator configurations, where the end effector of the manipulator is in contact with a fixed object, a complete mathematical model for the manipulator dynamics should include the effects of the resulting contact force between the end effector and the fixed object. Equations for such a closed chain manipulator are developed, where the end effector constraint is defined by a smooth manifold. These equations are shown to be complete in the sense that the direct dynamics problem and the inverse dynamics problem are well-posed. This formulation suggests a new approach to planning and tracking control for manipulators in a constrained environment.

Switched mode feedback control laws for nonholonomic systems in extended power form

Abstract A class of nonholonomic control systems in extended power form is studied. It is demonstrated that under appropriate assumptions Lagranges equations, including classical nonholonomic constraints, can be transformed into the extended power form. A switched mode feedback controller is used to obtain global convergence of the states of the extended power form to the origin. This feedback controller can be interpreted as a hybrid system consisting of a high level discrete event supervisor and a family of low level feedback controllers. The closed loop system exhibits finite-time responses.

Nonlinear Dynamics of the 3D Pendulum

A 3D pendulum consists of a rigid body, supported at a fixed pivot, with three rotational degrees of freedom. The pendulum is acted on by a gravitational force. 3D pendulum dynamics have been much studied in integrable cases that arise when certain physical symmetry assumptions are made. This paper treats the non-integrable case of the 3D pendulum dynamics when the rigid body is asymmetric and the center of mass is distinct from the pivot location. 3D pendulum full and reduced models are introduced and used to study important features of the nonlinear dynamics: conserved quantities, equilibria, relative equilibria, invariant manifolds, local dynamics, and presence of chaotic motions. The paper provides a unified treatment of the 3D pendulum dynamics that includes prior results and new results expressed in the framework of geometric mechanics. These results demonstrate the rich and complex dynamics of the 3D pendulum.

Stability and Stabilization of Relative Equilibria of Dumbbell Bodies in Central Gravity

A dumbbell-shaped rigid body can be used to represent certain large spacecraft or asteroids with bimodal mass distributions. Such a dumbbell body is modeled as two identical mass particles connected by a rigid, massless link. Equations of motion for the five degrees of freedom of the dumbbell body in a central gravitational field are obtained. The equations of motion characterize three orbit degrees of freedom, two attitude degrees of freedom, and the coupling between them. The system has a continuous symmetry due to a cyclic variable associated with the angle of right ascension of the dumbbell body. Reduction with respect to this symmetry gives a reduced system with four degrees of freedom. Relative equilibria, corresponding to circular orbits, are obtained from these reduced equations of motion; the stability of these relative equilibria is assessed. It is shown that unstable relative equilibria can be stabilized by suitable attitude feedback control of the dumbbell. Nomenclature er = unit vector along local vertical (radial) direction ex = unit vector along longitudinal axis of dumbbell ey, ez = orthogonal unit vectors spanning plane perpendicular to dumbbell axis eλ

Aggressive Longitudinal Aircraft Trajectory Tracking Using Nonlinear Control

Flight-control system designs are complicated if the aircraft dynamics are nonlinear and nonminimum phase. The nonminimum phase property can result from the choice of output vector and coupling between the moment generating actuators and the aerodynamic forces on the aircraft. In this paper we study a flight-control problem for a conventional aircraft longitudinal dynamic model that explicitly includes the coupling between the moment generating actuators and the aerodynamic forces. In particular, we study the execution of a maneuver for which the aircraft is intended to track a given motion in a vertical plane. We formulate the problem as a nonlinear tracking control problem. Controllers are developed for an aggressive maneuver that requires the use of a two-degrees-of-freedom controller design. We demonstrate the value of this control architecture in order to achieve aggressive maneuvering with good tracking performance. Our approach throughout is to make use of nonlinear control theory. Our analysis is complicated by the nonminimum phase characteristics of the flight model.