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Geometric Tracking Control of the Attitude Dynamics of a Rigid Body on SO(3)

Taeyoung Lee

arXiv:1010.1725v1math.OC

TL;DR

Attitude control on SO(3) faces challenges from its nonlinear geometry and from attitude representations that can introduce singularities or ambiguities. This paper develops a geometric tracking controller with an error function designed for large initial attitude errors, and reports fixed-command stabilization without inertia-matrix knowledge.

  • Problem

    Attitude control must address the nonlinear geometry of SO(3), while common attitude representations can involve singularities or ambiguities.

    Source: §I · p. 1

  • Method

    The paper develops a geometric SO(3) tracking controller using a new configuration error function.

    Source: A. Attitude Error Function · p. 2, §V · p. 7

  • Results

    The proposed controller provides uniform tracking performance for large initial attitude errors, and fixed-command stabilization is possible without knowing the inertia matrix.

    Source: §I · p. 1, §V · p. 7

  • Takeaways & Limitations

    The controller is expressed directly on SO(3) to avoid singularities and ambiguities inherent to other attitude representations.

    Source: §V · p. 7

Abstract

from arXiv · show

This paper provides new results for a tracking control of the attitude dynamics of a rigid body. Both of the attitude dynamics and the proposed control system are globally expressed on the special orthogonal group, to avoid complexities and ambiguities associated with other attitude representations such as Euler angles or quaternions. By selecting an attitude error function carefully, we show that the proposed control system guarantees a desirable tracking performance uniformly for nontrivial rotational maneuvers involving a large initial attitude error. In a special case where the desired attitude command is fixed, we also show that the attitude dynamics can be stabilized without the knowledge of an inertia matrix. These are illustrated by numerical examples.

I. INTRODUCTION

Rigid-body attitude control must account for motion on the nonlinear manifold SO(3), where global stabilization and common attitude representations present challenges. The paper proposes a geometric tracking controller whose performance remains uniform for large initial attitude errors.

I. INTRODUCTION

  • No continuous feedback controller asymptotically stabilizes attitude globally on SO(3), a nonlinear configuration manifold.Attitude dynamics arise in applications including aerial and underwater vehicles, robotics, and spacecraft.Source: §I · p. 1
  • Minimal attitude representations have local singularities, while quaternions can introduce antipodal convergence and unwinding behavior.Quaternion representations double-cover SO(3), and their control systems can be discontinuous when applied to actual attitude dynamics.Source: §I · p. 1
  • Geometric control uses manifold properties to avoid singularities and ambiguities associated with local coordinates, and prior rigid-body controllers achieved almost-global stability,,,,.The cited controllers are not desirable for large initial errors because their control-input magnitude approaches zero near a 180° attitude error,,.Source: §I · p. 1
  • The proposed SO(3) controller tracks attitude and angular velocity uniformly for large initial errors; fixed-command stabilization is also possible without knowing the inertia matrix.The stated region of attraction almost covers SO(3), and initial angular-velocity error can be arbitrarily large if the controller gain is sufficiently large.Source: §I · p. 1

II. ATTITUDE DYNAMICS OF A RIGID BODY

The section models a fully actuated rigid body's rotational attitude using a rotation matrix on SO(3), with dynamics expressed using body-frame quantities.

II. ATTITUDE DYNAMICS OF A RIGID BODY

  • The rigid body's orientation relative to an inertial frame is represented by R∈SO(3), the group of 3×3 orthogonal matrices with determinant one.The body-fixed frame is centered at the rigid body's mass center.Source: §II · p. 1
  • The section introduces the equations of motion for the fully actuated rigid body's rotational attitude dynamics.The equations themselves are referenced but not reproduced in the supplied passage.Source: §II · p. 1
  • J denotes the body-frame inertia matrix, while Ω and u denote body-frame angular velocity and control moment, respectively.Source: §II · p. 2

III. GEOMETRIC TRACKING CONTROL ON SO(3)

The tracking setup uses a smooth desired attitude command on SO(3) and its associated desired angular velocity.

III. GEOMETRIC TRACKING CONTROL ON SO(3)

  • The controller is designed to follow a smooth desired attitude command Rd(t)∈SO(3).Source: §III · p. 2
  • The desired attitude command has a kinematics equation.Source: §III · p. 2
  • Ωd denotes the desired angular velocity associated with the attitude command.Source: §III · p. 2

A. Attitude Error Function

The section develops a configuration error function for SO(3) because the choice of error function shapes geometric controller performance. The proposed error vector tracks rotation angle proportionally, improving performance for large initial attitude errors.

A. Attitude Error Function

  • A configuration error function is a smooth positive-definite measure of the difference between current and desired configurations, and its choice directly affects control-system performance.Error vectors are defined from derivatives of the function and used in Lyapunov-based controller design.Source: A. Attitude Error Function · p. 2
  • The usual attitude error vector approaches zero as the attitude approaches a 180° rotation, weakening performance for large initial errors.The paper introduces a new configuration error function to address this drawback.Source: A. Attitude Error Function · p. 2
  • Within L2, Ψ is positive definite about R=Rd and locally quadratic, with ∥eR∥²≤Ψ(R,Rd)≤2∥eR∥².The error vector is well defined in the sublevel set Ψ<2, which contains only one critical point, Rd.Source: A. Attitude Error Function · p. 2
  • The proposed error-vector magnitude is proportional to the rotation angle about the Euler axis, improving tracking for large initial attitude errors.The passage contrasts this behavior with the error function used in other literature.Source: A. Attitude Error Function · p. 3
  • The proposed error function is designed so that the attitude error vector reflects rotation angle, avoiding the near-zero error-vector magnitude that compromises the prior choice near 180°.The paper links the prior behavior to reduced convergence rate as initial attitude error approaches 180°.Source: A. Attitude Error Function · pp. 2–3

B. Attitude Error Dynamics

The section derives the attitude-error dynamics by differentiating the configuration error and error vector along the attitude kinematics. It also characterizes the matrix used in the error-vector derivative.

B. Attitude Error Dynamics

C. Attitude Tracking

The section defines a feedback control system for a desired attitude command and establishes exponential stability of the tracking errors. Its estimated region of attraction almost covers SO(3), while sufficiently large gain permits arbitrarily large initial angular-velocity error.

C. Attitude Tracking

  • The tracking controller combines proportional attitude and angular-velocity error feedback with rigid-body and desired-command terms.The control law is given for a time-varying attitude command.Source: C. Attitude Tracking · p. 4
  • The attraction-region estimate depends on the maximum eigenvalue of the inertia matrix.Source: C. Attitude Tracking · p. 4
  • The proof uses an invariant sublevel set Ψ(R, Rd) < 2, where the attitude error vector is well defined and Ψ has only one critical point.Source: C. Attitude Tracking · p. 4
  • A Lyapunov function and gain conditions establish the stability bound for the tracking errors.The proof constructs a quadratic Lyapunov function and bounds its time derivative.Source: C. Attitude Tracking · p. 4
  • The proof selects a positive constant so the relevant matrices are positive definite, making the Lyapunov function quadratic.Source: C. Attitude Tracking · p. 4
  • The attitude and angular-velocity tracking-error equilibrium is exponentially stable under the defined control system.Source: C. Attitude Tracking · p. 4
  • The estimated attraction region almost covers SO(3), and sufficiently large gain allows arbitrarily large initial angular-velocity error.The attitude region excludes only the stated two-dimensional subset of SO(3).Source: C. Attitude Tracking · p. 4

D. Attitude Stabilization Without Inertia Matrix

With a fixed desired attitude, the controller stabilizes attitude and angular-velocity errors exponentially without requiring the inertia matrix in its control input. The proof also establishes an invariant set and provides a region-of-attraction estimate, while noting a convergence-rate trade-off.

D. Attitude Stabilization Without Inertia Matrix

E. Properties

The controller is designed directly on SO(3), avoiding representation-related singularities and ambiguities. Its attitude-error function is selected to maintain effectiveness for large attitude errors.

E. Properties

  • Direct rotation-matrix control on SO(3) avoids coordinate singularities and quaternion ambiguities, including issues linked to antipodal convergence, noise sensitivity, and unwinding.Source: E. Properties · p. 5
  • The new attitude-error function makes the error vector proportional to its value, supporting uniform effectiveness at larger attitude errors.Source: E. Properties · pp. 5–6
  • Fig. 2 identifies the proposed controller as red solid, the controller from (28) as blue dashed, and the command as black dotted.Source: Figure 2 · p. 6

IV. NUMERICAL EXAMPLES

Numerical simulations compare the proposed controller with a controller based on (28) in, across tracking and stabilization cases. The proposed controller shows more desirable convergence for complex maneuvers with large initial attitude errors.

IV. NUMERICAL EXAMPLES

  • The examples cover tracking with full knowledge of J and stabilization without knowledge of J, using controllers (19) and (27), respectively.Source: §IV · p. 6
  • Both cases start with an attitude error of 0.999π (179.82°); the tracking case uses a nontrivial rotational command.Source: §IV · p. 6
  • A Lie group variational integrator is used to preserve the geometric structure of SO(3) during simulation.Source: §IV · p. 6
  • The proposed controller shows more desirable convergence than the controller based on (28) in, for complex maneuvers with large initial attitude errors.Source: §IV · p. 6

V. CONCLUSION

The paper concludes that its SO(3)-based controller avoids representation-related singularities and ambiguities, provides uniform tracking performance for large initial errors, and stabilizes a fixed desired attitude without full inertia-matrix knowledge.

V. CONCLUSION

  • The SO(3)-based controller avoids singularities and ambiguities of other attitude representations, tracks uniformly for large initial errors, and requires no full inertia-matrix knowledge for fixed-command stabilization.Source: §V · p. 7
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