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Geometric Fixed-Time Sliding Mode Control for Constrained Attitude Tracking on $\mathrm{SO}(3)$
Saumitra Barman, Shashi Ranjan Kumar, Rohit Gupta
TL;DR
The paper addresses constrained spacecraft attitude tracking on SO(3) with multiple pointing constraints and matched disturbances. It constructs an intrinsic potential and fixed-time geometric sliding-mode controller, showing admissible-set invariance and convergence to a small neighborhood of the desired equilibrium within a prescribed fixed time.
Problem
The paper addresses spacecraft attitude tracking that must maintain multiple pointing constraints on SO(3) in the presence of matched external disturbances.
Method
The paper combines an intrinsic SO(3) attitude potential with a nonsingular fixed-time geometric sliding manifold and control law.
Results
The closed-loop trajectory remains in the admissible attitude subset for every admissible initial attitude and reaches a sufficiently small neighborhood of the desired equilibrium within a prescribed fixed time.
Takeaways & Limitations
The simulations show that constraints reshape the feasible attitude path while preserving the final tracking objective.
Abstract
from arXiv · showhide
This paper studies constrained spacecraft attitude tracking on the Riemannian configuration manifold $\mathrm{SO}(3)$ in the presence of multiple attitude pointing constraints and matched external disturbances. To address this, an attitude potential function is proposed intrinsically on $\mathrm{SO}(3)$, and its key properties are established using intrinsic geometric analysis. Under mild conditions, the potential function is shown to admit a unique nondegenerate minimum at the desired attitude over the admissible subset of $\mathrm{SO}(3)$, defined by excluding the forbidden attitude regions as well as a measure-zero set, thereby ensuring a well-posed constrained attitude tracking problem. A Riemannian Hessian analysis shows that the Hessian of the potential function is locally uniform positive definite in an open neighborhood of the desired attitude, thereby establishing local strong convexity. A nonsingular fixed-time geometric sliding manifold is proposed using the Riemannian gradient of the potential function, leading to a geometric fixed-time sliding-mode-based constrained attitude control law. It is shown that, for every initial attitude in the admissible subset, the closed-loop state trajectory evolves on $\mathrm{SO}(3)\times\mathbb{R}^3$, with the attitude remaining in the admissible subset throughout the maneuver, while the state converges to a sufficiently small compact neighborhood of the desired equilibrium in a prescribed fixed time. Numerical simulations validate the proposed control approach and illustrate the theoretical results.
I. INTRODUCTION
The paper addresses constrained spacecraft attitude tracking on SO(3), where pointing constraints must be maintained despite limitations of existing planning and Euclidean or quaternion formulations. It develops an intrinsic geometric sliding-mode controller with fixed-time convergence and admissible-set invariance.
- Motivation: Pointing constraints must prevent body-fixed instruments from entering undesirable directions while tracking a commanded spacecraft attitude.Examples include avoiding bright celestial objects to protect instruments and preserve mission functionality.
- Related Work: Trajectory-planning approaches explicitly incorporate pointing constraints but generally require substantial computational resources.
- Related Work: Standard sliding-mode controllers are robust to uncertainties and disturbances but do not generally guarantee attitude-constraint satisfaction.Artificial-potential-based sliding-mode approaches address constraints, but the cited formulations use Euclidean-space attitude parameterizations.
- Geometric Formulation: Euler-angle and modified-Rodrigues-parameter formulations can suffer coordinate singularities, while unit-quaternion formulations introduce double-cover ambiguity and possible unwinding.Direct tracking on SO(3) avoids these parameterization issues and the unwinding phenomenon.
- Research Gap: Constrained attitude tracking on SO(3) has received limited attention, and prior work had not reported sliding-mode-based constrained spacecraft tracking on this manifold.
- Contributions: The proposed controller combines an intrinsic SO(3) attitude potential with a nonsingular fixed-time sliding manifold on SO(3) × R3.It requires no offline or online optimization and handles multiple pointing constraints and matched external disturbances.
- Contributions: The attitude potential has a unique nondegenerate minimum at the desired attitude over the admissible subset, which excludes forbidden regions and a measure-zero set.Riemannian Hessian analysis establishes local strong convexity near the desired attitude.
- Contributions: For every admissible initial attitude, the closed-loop trajectory remains admissible and converges to a sufficiently small neighborhood of the desired equilibrium within a prescribed fixed time.
A. Notations
The notation defines the Riemannian and matrix structures used for analysis on SO(3), including tangent spaces, volume measure, norms, and the Lie algebra so(3).
- Riemannian Notation: A Riemannian manifold assigns a Riemannian metric ⟨·,·⟩x to each tangent space TxM.
- Matrix Groups: SO(3) is the set of 3 × 3 rotation matrices satisfying RᵀR = I3×3 and det R = 1.
- Matrix Groups: The Lie algebra so(3) consists of skew-symmetric 3 × 3 matrices, and TRSO(3) consists of tangent matrices RX with Xᵀ = −X.
- Riemannian Notation: The bi-invariant Riemannian metric on SO(3) is defined by ⟨M,N⟩R = tr(MᵀN).
B. Definitions and Lemmas
This section extends fixed-time stability concepts to Riemannian manifolds and provides a Lyapunov-style criterion for attraction to a compact target set.
- Definitions: The paper considers autonomous manifold dynamics ẋ = f(x), allowing discontinuous vector fields whose solutions are interpreted in the Filippov sense.
- Definitions: Almost-global finite-time stability requires a full-measure subset containing the desired equilibrium whose trajectories reach that equilibrium in finite time.
- Definitions: Almost-global fixed-time stability additionally requires a uniform upper bound Tmax on all settling times in the full-measure subset.
- Definitions: Almost-global fixed-time attractivity requires eventual entry into a forward-invariant target set within a common fixed-time bound for almost all initial conditions.
- Lemmas: Lemma 1 uses a nonnegative Lyapunov function and a differential inequality to establish fixed-time attraction to a closed target set containing the desired equilibrium in its interior.
- Lemmas: When the target set is the singleton equilibrium, the lemma reduces to the standard almost-global fixed-time stability criterion.
C. Spacecraft Attitude Kinematics and Dynamics
The spacecraft attitude is represented intrinsically by a rotation matrix on SO(3), with angular velocity and tracking errors defined relative to a desired attitude and frame.
- Attitude Representation: The spacecraft attitude R relative to the inertial frame belongs to SO(3), and its kinematics satisfy Ṙ = Rω×.
- Reference Frames: The desired frame FD provides the reference configuration that the body-fixed frame FB must track.
- Tracking Errors: The attitude error is represented by Re = RdᵀR, while the angular-velocity error compares spacecraft and desired angular velocities in compatible frames.
- Dynamics: Euler’s equation yields the attitude error dynamics using the spacecraft inertia matrix and control and disturbance torques.
- Assumptions: Desired angular-velocity signals are locally bounded, and the matched disturbance torque has a known uniform bound smaller than the prescribed constant γ.
D. Attitude Pointing Constraints and Forbidden Regions
The paper models forbidden pointing directions as regions on SO(3) that the spacecraft attitude must avoid. The admissible set excludes these regions and a measure-zero singular set under assumptions ensuring interior initial and desired attitudes and disjoint forbidden regions.
- Forbidden regions: Each pointing constraint specifies a forbidden inertial direction that the body-fixed sensitive-axis direction Rg_d must not approach.The sensitive axis is represented by g_d in the body frame, while the forbidden direction is represented in the inertial frame.
- Forbidden regions: The forbidden region O_i is determined by a minimum allowable angle θ_i between Rg_d and the forbidden inertial direction y′_i.The constraint requires the line of sight to remain outside a cone of half-angle θ_i centered on y′_i.
- Admissible set: The admissible attitude set excludes every forbidden region O_i and the set L of relative attitudes satisfying tr(R_d^T R) ≤ −1.The admissible set is defined through the interior of the resulting manifold with boundary.
- Geometric assumptions: The forbidden regions are assumed pairwise disjoint, so their boundaries do not intersect and the admissible set has no corner singularities.Under this assumption, the admissible sets M and M_e are smooth 3-dimensional manifolds with boundary.
- Geometric assumptions: The initial and desired attitudes are assumed to remain in the interior of the admissible attitude manifold.This assumption is imposed for the initial attitude and for the desired trajectory at all nonnegative times.
E. Problem Statement
The control objective is to track a desired attitude while satisfying all pointing constraints for every admissible initial attitude. It also requires fixed-time attraction of the error state to a sufficiently small compact neighborhood while preserving forward invariance.
- Control objective: The controller must keep R(t) in int(M) for all t ≥ 0, thereby satisfying every attitude pointing constraint throughout the maneuver.This requirement applies for every initial attitude R(0) in int(M).
III. A NOVEL ATTITUDE POTENTIAL FUNCTION ON SO(3)
The paper constructs an intrinsic attitude potential on SO(3) after characterizing the admissible attitude and error sets geometrically. Under suitable tuning, the potential has a unique nondegenerate minimum at the identity and is locally strongly convex there.
- Geometric characterization: The diffeomorphism φ_t(R_e)=R_d(t)R_e maps int(M_e) onto int(M) and preserves the attitude flow.Thus, trajectories in the error coordinates correspond uniquely to trajectories in the actual attitude coordinates.
- Geometric characterization: The admissible sets M and M_e are smooth 3-dimensional manifolds with boundaries formed by the constraint hypersurfaces and the excluded trace sets.Pairwise disjoint constraints ensure the boundaries remain geometrically regular.
- Potential construction: The potential Ψ combines an attractive component toward the desired attitude with a repulsive component penalizing proximity to constraint boundaries.It is constructed intrinsically on the Riemannian manifold SO(3).
- Potential properties: The potential satisfies Ψ(R_e)=0 only at I_3×3 and has no other critical point in int(M_e) under the stated tuning condition.The excluded antipodal singular rotations do not belong to the admissible set.
- Potential properties: For α > max{α∗_1(δ), α∗_2(δ)}, Ψ admits a unique nondegenerate minimum and is locally strongly convex around I_3×3.The Riemannian Hessian is uniformly positive definite on an open neighborhood of the identity.
IV. DESIGN OF CONSTRAINED GEOMETRIC FIXED-TIME SLIDING MODE CONTROLLER
The controller uses the intrinsic potential to define a nonsingular fixed-time sliding variable and a geometric sliding-mode torque law. The resulting closed loop reaches the sliding manifold in fixed time, preserves the admissible attitude set, and converges to a small neighborhood of the desired equilibrium.
- Controller design: The proposed sliding variable is built from the attitude error vector and the potential-based geometric quantities, with exponents satisfying p < 1 and q > 1.The construction is designed to support fixed-time convergence without singularity.
- Sliding-phase convergence: On the sliding manifold, the error state converges within fixed time T_s to a sufficiently small compact neighborhood of (I_3×3, 0).The neighborhood size depends on the design parameters and the stated Lyapunov bounds.
- Closed-loop guarantees: The proposed torque law reaches S=0 within a prescribed fixed time T_r despite matched bounded disturbances.The discontinuous term is interpreted in the Filippov sense.
- Closed-loop guarantees: The closed-loop system admits a Filippov solution for all nonnegative time and keeps trajectories in int(M) × R^3.Existence follows from the regularity properties of the associated differential inclusion and bounded disturbance assumption.
- Closed-loop guarantees: The admissible set int(M) is the largest positively invariant subset of SO(3) under the closed-loop attitude dynamics.The result is transferred from the error-coordinate set int(M_e) through the flow-preserving diffeomorphism.
V. RESULTS AND DISCUSSIONS
Numerical examples show that the proposed controller avoids single and multiple forbidden pointing regions while preserving final attitude tracking. Constraint enforcement reshapes transient trajectories and increases transient angular-velocity and torque responses.
- Example 1: The constrained trajectory remains outside a single forbidden cone by steering around the restricted region before reaching the desired attitude.The unconstrained trajectory follows the shortest rotation path and intersects the exclusion cone.
- Example 1: Both constrained and unconstrained cases converge to the same terminal attitude, with tr(Re) reaching 3 and configuration errors reaching zero.The constrained response includes a transient plateau and slower decay near the constraint boundary.
- Example 1: Constraint enforcement produces larger transient angular-velocity deviations, secondary oscillations, higher peak torques, and slightly increased settling time.These effects arise as the controller modifies the rotation trajectory to avoid restricted orientations.
- Example 2: With three forbidden zones and 20° minimum allowable angles, the constrained path becomes longer and more curved while still reaching the desired final attitude.The unconstrained motion passes through or near exclusion cones, whereas the constrained motion remains outside them.
- Example 2: In Example 2, both constrained and unconstrained cases achieve tr(Re) = 3 and zero error measures, while constraints introduce transient plateaus and larger angular-velocity overshoots.The constrained trajectory detours around forbidden regions, preserving final tracking performance.
VI. CONCLUSIONS
The paper develops and validates an intrinsic geometric fixed-time control framework for constrained spacecraft attitude tracking on SO(3). The analysis establishes admissible-space invariance and prescribed-time convergence to a small neighborhood of the desired equilibrium.
- VI. CONCLUSIONS: The framework constructs an SO(3) attitude potential and establishes its unique nondegenerate minimum and local strong convexity using Riemannian analysis.A nonsingular fixed-time geometric sliding manifold and constrained control law are built from the potential’s Riemannian gradient.
- VI. CONCLUSIONS: Closed-loop analysis guarantees admissible-space invariance and convergence to a sufficiently small neighborhood of the desired equilibrium within a prescribed fixed time.The result holds with multiple pointing constraints and matched external disturbances.
- VI. CONCLUSIONS: Numerical simulations validate the theoretical results and demonstrate the effectiveness of the proposed approach.