A topological obstruction to continuous global stabilization of rotational motion and the unwinding phenomenon
Introduction
It is convenient to think of the evolution of a physical system in terms of a dynamical system evolving on the physical state space, which is simply the set of all states of the physical system. We assume that the states of the system can be placed into a one-to-one correspondence with the points of an abstract -dimensional manifold . Under this correspondence, the dynamics of the physical system give rise to a dynamical system on . The manifold models the physical state space, while the dynamical system on models the dynamics of the physical system. In such a case, we call the state space of the system. Physically relevant quantities such as angular displacement, angular velocity, etc., yield local coordinates on . These local coordinates can be used to locally represent the dynamical system on as a set of differential equations defined on an open subset of , even though might be globally quite different from an open subset of . While local coordinates are convenient and indeed sufficient for analyzing the local properties of a dynamical system, questions of a global nature require global analysis for satisfactory solutions.
One instance where the above observation is relevant is the problem of stabilizing the attitude of a rigid body. In this case, the state space for the attitude dynamics can be identified with , where is the group of rotation matrices, that is, orthogonal matrices with determinant 1. In Section 6, we consider a continuous feedback controller designed to globally asymptotically stabilize a desired rest attitude of a rigid body. However, a closer examination reveals that this controller, which was designed using local coordinates, is not globally well defined on and, in fact, leads to the unwinding phenomenon where the body may start at rest arbitrarily close to the desired final attitude and yet rotate through large angles before coming to rest in the desired attitude. Indeed, it has been observed in [17] that due to the global topology of , no continuous vector field on has a globally asymptotically stable equilibrium. In this paper, we generalize and expand upon this observation by identifying a large class of systems for which the global topology of the state space precludes the existence of globally asymptotically stable equilibrium points under continuous dynamics. Besides the rigid body dynamics in terms of both rotation matrices and quaternions, this class of systems also includes any mechanical system that has a rotational degree of freedom. We also outline a general class of situations in which the use of local coordinates to achieve global objectives leads to unwinding.
It is well known that the domain of attraction of an asymptotically stable equilibrium is homeomorphic to for some [5, Theorem V.3.4]. This fact indicates that unlike local asymptotic stability, global asymptotic stability of an equilibrium depends strongly on the global topology of the state space . Indeed, Theorem 1 in Section 2 states that if has the structure of a vector bundle over a compact manifold , then no continuous vector field on has a globally asymptotically stable equilibrium. This result is applicable to mechanical systems having rotational degrees of freedom. When applied to the rigid body attitude stabilization problem where the state space is a vector bundle over the compact manifold , Theorem 1 leads to the observation made in [17] that rigid body attitude cannot be globally stabilized through continuous feedback.
Theorem 1 follows easily from elementary concepts in differential topology. However, given the substantial literature on the global stabilization of rigid body attitude [15], [20], [21], [26], [27], [28], we feel that it would be useful to understand the ramifications of this simple but fundamental result.
In Section 4 we illustrate the unwinding phenomenon in the special case of a rigid body rotating about a fixed axis. For this case, continuous globally stabilizing feedback controllers that are designed using local coordinates turn out to be multiple-valued on the state space , where is the unit circle in the complex plane. Consequently, the closed-loop system exhibits unwinding wherein the body may start at rest in the desired orientation and yet rotate several times before eventually coming to rest in the initial orientation
Often the state space of the system of interest is related to another manifold , called a covering manifold, through a covering map , which is onto and a local diffeomorphism everywhere but globally many to one. Since is a local diffeomorphism everywhere, a given control system on may be uniquely lifted to a control system on that is locally equivalent to . This property of covering manifolds is particularly useful for modeling since a covering manifold of may have a simpler structure and thus may be more easily coordinatized than itself. For instance, the set of unit quaternions is a covering manifold for the set of rotation matrices and the corresponding covering map provides a globally nonsingular parametrization of in terms of unit quaternions. This parametrization is widely used for modeling attitude dynamics because is easily parametrized in terms of four parameters subject to one constraint, while a rotation matrix contains nine parameters satisfying six constraints. However, one drawback of using covering manifolds to model control systems is that since a covering map may be many to one, the control system on may not be globally equivalent to the lifted control system on the covering manifold . Under certain feedback controllers, two solutions of may project onto two distinct curves in passing through the same initial condition. Such feedback controllers will give a family of motions on that exhibit unwinding. These ideas are explained in greater detail in Section 5.
In Section 6, the above ideas are used to explain how the unwinding phenomenon can arise in the case of attitude stabilizing controllers that are designed using quaternions. Although the state space for the attitude stabilization problem is , it is convenient to represent the problem in terms of unit quaternions on the covering manifold . A controller that is defined in terms of quaternions, that is, on , need not determine a well-defined control law in terms of attitude, that is, on . In other words, such a controller may assign more than one control value to a point in . In such a case, the closed loop does not give rise to a well-defined dynamical system on . Thus, not only does such a controller fail to yield global asymptotic stability, but in fact leads to unwinding on . Strictly speaking, global asymptotic stability is not defined (in terms of attitude) for a controller that does not define a dynamical system on the state space .
Section snippets
A topological obstruction to global stability
We note that by a manifold we mean a smooth, positive dimensional, connected manifold without boundary.
Let be a manifold of dimension and consider a continuous vector field on with the property that for every , there exists a unique right maximally defined integral curve of starting at , and, furthermore, every right maximally defined integral curve of is defined on . In this case, the integral curves of are jointly continuous functions of time and initial condition [13,
Mechanical systems with compact configuration manifolds
The configuration space of a mechanical system is the set of all configurations of the system, where each configuration refers to a particular arrangement of the various particles constituting the mechanical system. The configuration space of a mechanical system can often be identified with an -dimensional manifold , the configuration manifold of the system, where is the number of degrees of freedom of the mechanical system. The state space of a mechanical system with the configuration
The unwinding phenomenon
To illustrate the unwinding phenomenon, consider a mechanical system consisting of a rigid body rotating about a fixed axis under the action of a control torque. The configuration space of the system can be identified with the unit circle in the complex plane with representing a reference configuration. Consequently, by Theorem 1, the rigid body cannot be globally asymptotically stabilized to a rest position using a continuous state feedback. The angular position and the
Covering manifolds and unwinding
A smooth surjective (onto) map , where and are manifolds, is called a covering map if every point has an open neighborhood such that is a disjoint union of open sets and, for each restricted to is a diffeomorphism. If is a covering map, then is called a covering manifold of , and and are locally diffeomorphic everywhere. For instance, is a covering manifold of with a covering map given bywhere [11, p. 150].
The unwinding phenomenon in attitude control
The configuration space for the attitude dynamics of a rigid body can be identified with SO(3). Hence the configuration manifold for the attitude dynamics of a rigid body is the three dimensional compact Lie group . The Lie group structure of SO(3) makes it possible to write the equations of motion on instead of the tangent bundle SO(3) of SO(3). The equations of motion are given by [16]where transforms the components of a
Conclusions
Global properties of a dynamical system, such as global asymptotic stability of an equilibrium, depend strongly on the global topology of the underlying state space. For instance, mechanical systems with rotational degrees of freedom cannot be globally asymptotically stabilized to a rest configuration. Locally stabilizing controllers that are designed using local coordinates lead to unwinding when applied globally in a continuous manner. Since unwinding can be highly undesirable in spacecraft
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This research was supported in part by the Air Force Office of Scientific Research under grant F49620-98-1-0037.