Elsevier

Systems & Control Letters

Volume 39, Issue 1, 28 January 2000, Pages 63-70
Systems & Control Letters

A topological obstruction to continuous global stabilization of rotational motion and the unwinding phenomenon

https://doi.org/10.1016/S0167-6911(99)00090-0Get rights and content

Abstract

We show that a continuous dynamical system on a state space that has the structure of a vector bundle on a compact manifold possesses no globally asymptotically stable equilibrium. This result is directly applicable to mechanical systems having rotational degrees of freedom. In particular, the result applies to the attitude motion of a rigid body. In light of this result, we explain how attitude stabilizing controllers obtained using local coordinates lead to unwinding instead of global asymptotic stability.

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 m-dimensional manifold M. Under this correspondence, the dynamics of the physical system give rise to a dynamical system on M. The manifold M models the physical state space, while the dynamical system on M models the dynamics of the physical system. In such a case, we call M the state space of the system. Physically relevant quantities such as angular displacement, angular velocity, etc., yield local coordinates on M. These local coordinates can be used to locally represent the dynamical system on M as a set of differential equations defined on an open subset of Rm, even though M might be globally quite different from an open subset of Rm. 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 M=SO(3)×R3, where SO(3) is the group of rotation matrices, that is, 3×3 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 M 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 M, no continuous vector field on M 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 Rn for some n [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 M. Indeed, Theorem 1 in Section 2 states that if M has the structure of a vector bundle over a compact manifold Q, then no continuous vector field on M 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 SO(3), 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 M=S1×R, where S1 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 M of the system of interest is related to another manifold N, called a covering manifold, through a covering map p:NM, which is onto and a local diffeomorphism everywhere but globally many to one. Since p is a local diffeomorphism everywhere, a given control system Σ on M may be uniquely lifted to a control system Σ̂ on N that is locally equivalent to Σ. This property of covering manifolds is particularly useful for modeling since a covering manifold of M may have a simpler structure and thus may be more easily coordinatized than M itself. For instance, the set of unit quaternions S3 is a covering manifold for the set of rotation matrices SO(3) and the corresponding covering map provides a globally nonsingular parametrization of SO(3) in terms of unit quaternions. This parametrization is widely used for modeling attitude dynamics because S3 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 Σ on M may not be globally equivalent to the lifted control system Σ̂ on the covering manifold N. Under certain feedback controllers, two solutions of Σ̂ may project onto two distinct curves in M passing through the same initial condition. Such feedback controllers will give a family of motions on M 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 M=SO(3)×R3, it is convenient to represent the problem in terms of unit quaternions on the covering manifold N=S3×R3. A controller that is defined in terms of quaternions, that is, on N, need not determine a well-defined control law in terms of attitude, that is, on M. In other words, such a controller may assign more than one control value to a point in M. In such a case, the closed loop does not give rise to a well-defined dynamical system on M. Thus, not only does such a controller fail to yield global asymptotic stability, but in fact leads to unwinding on M. 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 M.

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 M be a manifold of dimension m and consider a continuous vector field f on M with the property that for every x∈M, there exists a unique right maximally defined integral curve of f starting at x, and, furthermore, every right maximally defined integral curve of f is defined on [0,∞). In this case, the integral curves of f 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 r-dimensional manifold Q, the configuration manifold of the system, where r is the number of degrees of freedom of the mechanical system. The state space M 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 S1={z∈C:|z|=1} in the complex plane with z=1 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 θ and the

Covering manifolds and unwinding

A smooth surjective (onto) map p:NM, where N and M are manifolds, is called a covering map if every point x∈M has an open neighborhood UM such that p−1(U) is a disjoint union of open sets Vk and, for each k,p restricted to Vk is a diffeomorphism. If p:NM is a covering map, then N is called a covering manifold of M, and N and M are locally diffeomorphic everywhere. For instance, N=R2 is a covering manifold of M=S1×R with a covering map p:R2→S1×R given byp(θ,ω)=(eiθ,ω),where i=−1 [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 Q=SO(3). The Lie group structure of SO(3) makes it possible to write the equations of motion on M=SO(3)×R3 instead of the tangent bundle TSO(3) of SO(3). The equations of motion are given by [16]Ṙ(t)=−(ω(t)×)R(t),Jω̇(t)=−(ω(t)×)Jω(t)+u(t),where R∈SO(3) 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.

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