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Physics-based Online Adaptive Koopman Model Predictive Attitude Control for Combined Spacecraft with Dynamic Uncertainties

Yicheng Sun, Yueyong Lyu, Yuhan Liu, Yanning Guo, Wei Pan

arXiv:2609.02534v1eess.SY

TL;DR

Combined-spacecraft attitude stabilization must handle uncertain post-capture inertia and target-induced disturbances without the computational burden of nonlinear predictive control. The paper develops a physics-based, quaternion-lifted adaptive Koopman MPC with online model updates and QP optimization. Simulations report fast computation, small steady-state attitude error, and formal feasibility and stability guarantees.

  • Problem

    Post-capture inertia uncertainty and target-induced disturbances challenge model-based combined-spacecraft attitude control, while nonlinear MPC imposes a severe onboard computational burden.

  • Method

    The paper combines a compact quaternion-based physics-based Koopman model with recursive online updates for inertial uncertainty and QP-based MPC.

  • Results

    PAKMPC achieves an average step runtime of 0.010 s and the smallest steady-state attitude error, ∥qv∥= 2.1e−3, among the compared strategies.

  • Takeaways & Limitations

    The proposed framework provides adaptive and physically interpretable predictions for LMPC while formally guaranteeing recursive feasibility and regional input-to-state stability.

Abstract

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This paper proposes a physics-based adaptive Koopman Model Predictive Control (MPC) strategy for combined spacecraft attitude stabilization under inertia uncertainties and active target maneuverability. A novel, quaternion-based Koopman model is constructed from a set of analytical lifting functions derived from the quaternion kinematics, which provides a more compact and physically interpretable linear representation of the nonlinear dynamics compared with the conventional black-box EDMD and higher-dimensional DCM-based model. Leveraging the linear structure of this nominal model, a gradient descent-based update law is employed to efficiently identify time-varying inertial uncertainties from real-time input/output data. By integrating this adaptive linear model into the MPC framework, the optimal control problem reduces to a computationally efficient Quadratic Program (QP), thereby significantly lowering the online computational burden compared to nonlinear adaptive MPC. Recursive feasibility and regional input-to-state stability are formally established through the design of terminal ingredients for the MPC. The effectiveness and superiority of the proposed strategy are validated through comparative simulations of an attitude stabilization task for combined spacecraft in a high-fidelity 3D simulator.

I. INTRODUCTION

Combined-spacecraft attitude control becomes difficult after capture because uncertain inertial properties and target-induced disturbances can undermine traditional model-based controllers. The paper proposes a physics-based adaptive Koopman MPC strategy that preserves interpretability while reducing predictive-control computation.

  • Motivation: Post-capture inertia uncertainty can make traditional model-based attitude controllers inadequate or potentially unstable during combined-spacecraft stabilization.The uncertainty concerns properties including mass, center of mass, and moments of inertia of non-cooperative targets.
  • Motivation: Nonlinear MPC captures nonlinear dynamics but imposes a severe onboard computational burden, motivating a framework that balances global dynamics representation with real-time efficiency.The introduction contrasts this burden with the limitations of locally valid feedback linearization.
  • Proposed approach: The proposed PAKMPC addresses inertia uncertainties and competitive torque exerted by the target in combined-spacecraft attitude stabilization.The overall control framework is illustrated in Fig. 1.
  • Proposed approach: Analytically constructed quaternion-based lifting functions produce a compact (4Nϕ+7)-dimensional Koopman representation instead of a generic data-driven dictionary.The model is described as more physically interpretable and less structurally redundant than data-driven EDMD and DCM-based alternatives.
  • Proposed approach: Inertia mismatch and target-induced disturbances are incorporated as online-updated parameter variations, allowing predictive control through a QP-based formulation.The strategy also includes terminal ingredients for recursive-feasibility and regional input-to-state-stability guarantees.
  • Attitude dynamics: The attitude model uses quaternion kinematics and Euler rigid-body dynamics with uncertain combined-spacecraft inertia and external disturbance torque.The nominal inertia and its deviation are separated to form the control-oriented error model.

B. Control Objectives

The control objectives are robust attitude stabilization and satisfaction of actuator torque limits. A Koopman transformation is used to obtain a linear prediction model suited to predictive control.

  • Control objectives: The controller must drive the attitude and angular-velocity error state to an arbitrarily small neighborhood of the origin despite inertia uncertainties and target-induced disturbances.The objective is stated as uniform ultimate boundedness of the error state.
  • Control objectives: The control torque must satisfy actuator saturation limits at all times.The maximum available torque for each axis is denoted by τmax.
  • Prediction model: Koopman operator theory transforms the nonlinear dynamics into a linear system for designing a predictive controller whose prediction model supports computationally efficient control.The practical model uses a finite-dimensional subspace spanned by selected observables and represented by a constant matrix A.

B. Physics-based Nominal Koopman Model

The paper constructs a finite-dimensional Koopman model tailored to combined-spacecraft attitude dynamics using quaternion-based analytical lifting functions. The resulting representation preserves key properties needed for control while reducing dimensionality relative to DCM-based and generic EDMD liftings.

  • Physics-based construction: The physics-based model exploits Euler’s equation and quaternion kinematics to construct lifting functions recursively for combined-spacecraft attitude dynamics.Euler’s equation is made linear through a redefined input, while quaternion bilinearity enables recursive observable construction.
  • Interpretability and efficiency: The quaternion-based lifting provides a compact, physically interpretable, and computationally efficient alternative to high-dimensional black-box EDMD liftings.The paper contrasts generic polynomial or radial-basis dictionaries with lifting functions designed from spacecraft dynamics.
  • Finite-dimensional model: The infinite-dimensional Koopman representation is truncated by retaining the original states and the first Nϕ observables, yielding an approximate (4Nϕ + 7)-dimensional model.The truncation makes the model suitable for controller design while retaining a three-dimensional control input.
  • Theoretical properties: The Koopman model preserves Lyapunov stability of the original equilibrium and is controllable under the stated assumptions.Controllability follows from the full-column-rank controllability matrix and the surjective mapping from transformed input v to original input u.
  • Discrete-time representation: Under ideal zero-order hold, the continuous-time Koopman system admits a unique exact discrete-time map for MPC design.The result is supported by existence and uniqueness of solutions over each sampling interval.
  • Compactness: The proposed model is 5Nϕ + 2 dimensions more compact than DCM-based lifting for the same number of dynamic terms.The comparison is made against approaches based on DCM dynamics.

C. Recursive Online Update Law

The online update law adapts the Koopman model by estimating input-channel uncertainty from recent prediction errors. It uses a regularized, exponentially weighted objective and Adam-based gradient descent to improve robustness under streaming data, while explicitly targeting local control accuracy rather than exact parameter convergence.

  • Adaptive identification: Online adaptation compensates for model mismatch caused by unknown inertia and external noise by recursively updating the uncertainty matrix ΔB.The update minimizes prediction error between lifted true-state transitions and nominal-model estimates.
  • Data window: The identification objective uses recent input and state-increment data in a sliding window, retaining the most recent Nw time steps.The window forms the dataset Dk from Uk and ΔZk.
  • Regularized objective: An ℓ1-regularized exponentially weighted objective discounts older data and promotes sparsity to mitigate overfitting with noisy or limited data.The forgetting factor satisfies 0 < λ < 1, while γ > 0 induces sparsity.
  • Gradient update: Adam is used for gradient descent to improve numerical stability and convergence speed during online estimation.The update uses first- and second-moment estimates with bias correction and has complexity linear in the size of ΔB.
  • Adaptation objective: Persistent excitation is not explicitly enforced because the adaptation targets local model accuracy for closed-loop control rather than exact parameter convergence.Excitation mainly arises during transient operation, while regularization and forgetting mitigate drift near steady state.
  • Uncertainty scope: The adaptive matrix ΔB represents input-dependent uncertainty, primarily covering inertia-induced errors and bounded target torques that can be locally input-matched.It is not a physical decomposition of every uncertainty source.

A. Model Predictive Controller Design

The controller embeds the Koopman linear model and an interpolated initial state into an MPC formulation. By expressing predicted trajectories and the cost in terms of control inputs and an interpolation parameter, the optimization becomes a standard quadratic program solvable online.

  • MPC formulation: The MPC formulation uses the Koopman linear model together with an interpolated initial state at each sampling step.The interpolation parameter is included among the optimization variables.
  • Cost design: The stage cost, terminal cost, and regularization term penalize tracking, control effort, terminal deviation, and mismatch between interpolated and lifted initial states.Positive-definite Q, R, and P matrices and a positive regularization coefficient are adopted.
  • Trajectory prediction: Predicted state trajectories and control sequences are rewritten as functions of a stacked optimization variable using the lifted system dynamics.This matrix formulation supports conversion of the MPC problem into a standard QP.
  • Online solution: The resulting QP is solved at each sampling time with standard numerical solvers to obtain the optimal control sequence and interpolation parameter.Only the first optimal control input is applied in the receding-horizon implementation.

B. Terminal ingredients design

The MPC terminal design combines a terminal cost, terminal set, and terminal feedback gain to support recursive feasibility and asymptotic convergence. Fixed terminal ingredients provide tractable guarantees under online model adaptation but introduce conservatism.

  • Terminal ingredients: Terminal ingredients consist of a terminal cost, terminal set, and terminal control gain designed to ensure convergence after the state enters the terminal region.The stated objective is asymptotic convergence toward the equilibrium point while preserving recursive feasibility.
  • Uncertain model: The adaptive Koopman model is decomposed into a constant matrix B0, a bounded state-dependent term B1, and an extended perturbation.The perturbation is defined as B1uk + ΔBuk, and the decrease condition is modified accordingly.
  • Lyapunov condition: The terminal weighting matrix is constrained through a Lyapunov decrease inequality involving the nominal closed-loop matrix and stage-cost weights.The nominal inequality is (A + BF)⊤P(A + BF) − P + Q + F⊤RF ≤ 0.
  • LMI design: A Schur-complement transformation converts the terminal-gain and weighting design into an LMI-based optimization problem.The substitutions O = σP^-1 and Y = FO recover P and F after solving for O and Y.
  • Robustness margin: Fixed terminal ingredients retain tractable recursive-feasibility and stability guarantees under online model adaptation but may introduce conservatism.The paper notes that less conservative alternative terminal constructions exist.
  • Implementation: Algorithm 1 initializes the nominal model and terminal matrix, updates ΔB after the learning window, solves the MPC problem, and applies the first control input.Before sufficient data are available, the online uncertainty update is set to zero.

C. Recursive Feasibility and Stability Analysis

The analysis establishes recursive feasibility and ISS properties for the Koopman-based MPC, including the actual lifted state under bounded prediction uncertainty.

  • Recursive Feasibility: The Koopman-based MPC problem is recursively feasible.A shifted candidate control sequence preserves the terminal-set condition at the next time step.
  • Stability Analysis: The optimal predicted initial state is ISS under Assumption 1 for trajectories in compact set X.The result characterizes robustness with respect to bounded uncertainties in the prediction model.
  • Stability Analysis: The actual lifted Koopman state Φ(x_k) is also ISS under the same compact-set assumptions.The proof uses feasibility, boundedness, continuity of the lifting map, and local Lipschitz continuity of the finite-horizon QP value function.
  • Stability Analysis: The established regional ISS property targets robust practical stabilization under bounded uncertainty rather than global asymptotic regulation.This scope matches the combined-spacecraft control objective.

V. SIMULATION RESULTS

Numerical simulations assess the proposed control strategy for stabilizing the combined spacecraft attitude in the presence of dynamic uncertainty.

  • Simulation Objective: The simulations evaluate attitude stabilization of a combined spacecraft under dynamic uncertainty.The section is explicitly framed as a numerical efficacy assessment.
  • Simulation Objective: The assessment focuses on the efficacy of the proposed control strategy.The supplied passage does not report quantitative outcomes or comparative results.
  • Simulation Objective: The simulation task concerns the combined spacecraft rather than an isolated servicer or target.

A. Simulation Platform and Parameters

The simulations use a MATLAB/Simscape Multibody model of a servicer-target pair with specified inertial, initial-condition, constraint, weighting, and timing parameters.

  • Simulation Platform: The simulation models a rigid cube-shaped servicer with two three-link robotic arms and a rigid cubesat target with solar panels and a docking ring.
  • Physical Parameters: The servicer and target masses are 1080 kg and 75 kg, with specified diagonal inertia matrices.The servicer inertia is diag(405, 405, 405) kg·m^2; the target inertia is diag(36.8, 37.5, 36.8) kg·m^2.
  • Initial Conditions: The initial quaternion is [0.5, 0.5, 0.5, 0.5]^T, and the initial angular velocity is [0.02, −0.01, −0.01]^T rad/s.
  • Constraints: The control-torque constraint is |u_max| ≤ 20 N·m.
  • MPC Parameters: The simulation runs for 200 s with a 0.1 s step, using Q = 10^2I_7, R = 10^−2I_3, and terminal-set bound σ = 208.68.The terminal weighting matrix is solved using MOSEK, and the regularization coefficient is γ_e = 1e6.
  • Parameter Selection: N_Φ is selected from an offline accuracy-complexity comparison, while other parameters trade off adaptation speed, noise attenuation, and computational burden.

B. Comparison Study of Nominal Koopman Model

The comparison study evaluates the quaternion-based nominal Koopman model against DCM-based and RBF-EDMD alternatives in prediction accuracy, stabilization behavior, robustness, and computational performance. The proposed model provides higher-fidelity prediction while supporting efficient adaptive control under dynamic uncertainty.

  • Nominal model accuracy: Increasing Nϕ reduces approximation error for both models, while the quaternion-based model achieves significantly smaller errors than the DCM-based model at the same Nϕ.The error is defined as the 2-norm between Koopman-predicted and nonlinear-dynamics states.
  • Closed-loop stabilization: All three nominal models stabilize the attitude, but the DCM-based and quaternion-based models produce faster and smoother responses than the RBF-EDMD model.The RBF-EDMD response is slower and more oscillatory in the closed-loop comparison.
  • Computational performance: The proposed quaternion-based model is significantly more efficient than the DCM-based and RBF-EDMD models in simulation runtime, average step runtime, and memory usage.The computational comparison is summarized using the metrics reported in Table II.
  • Adaptive attitude stabilization: Under dynamic uncertainty, PAKMPC stabilizes the combined spacecraft faster than KMAE-AKMPC, achieves the lowest state error, and avoids the severe overshoot of non-adaptive MPC.Non-adaptive MPC converges most rapidly but uses larger saturated control torques; PAKMPC achieves comparable accuracy with lower control effort.
  • Online adaptation: The adaptive weighting variable µk starts at its limit during large-error transients and decreases as model predictive confidence increases, while the uncertainty estimate eventually converges.The uncertainty norm rises and oscillates after the first complete data window before converging.

D. Structured Robustness Evaluation

A structured sweep varying target inertia, mass, and disturbance levels shows that PAKMPC maintains the most favorable balance of attitude accuracy, control effort, robustness, and runtime among the compared controllers.

  • Robustness: PAKMPC is consistently more robust than the three baseline methods across the structured operating-point sweep.The sweep jointly varies target inertia and mass from 50% to 150% and disturbance torque from 0% to 200%.
  • Runtime-performance trade-off: PAKMPC provides the most favorable overall balance between computational cost and control performance over the operating-point sweep.The evaluation keeps the initial condition, reference, prediction horizon, and controller hyperparameters fixed.
  • Control response: The non-adaptive MPC experiences severe control saturation because it does not compensate for inertial uncertainty.The saturation behavior is shown in the control-torque responses.
  • Robustness: Across the scanned uncertainty-disturbance range, PAKMPC achieves low steady-state quaternion-vector error with moderate total control effort.The comparison includes PAKMPC, non-adaptive MPC, KMAE-AKMPC, and INMPC.
  • Runtime-performance trade-off: PAKMPC lies closest to the lower-left region of runtime-performance scatter plots, indicating the best overall trade-off among the compared controllers.The plots compare runtime with both total control effort and steady-state quaternion error.
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