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Stable Gaussian Process based Tracking Control of Euler-Lagrange Systems
Thomas Beckers, Dana Kulić, Sandra Hirche
TL;DR
Unknown Euler-Lagrange dynamics make low-gain, high-performance tracking difficult. The paper uses Gaussian Process regression to compensate residual dynamics and adapts feedback gains using model confidence; simulations report lower tracking error than classic computed-torque control, with variable gains improving performance outside the training region.
Problem
Accurate models of unknown dynamics are difficult to obtain, limiting stable low-gain tracking control for real-world Euler-Lagrange systems.
Method
The CTC-GPR controller uses the GP mean to compensate residual dynamics and GP confidence to adapt feedback gains.
Results
Both CTC-GPR approaches achieve lower tracking error than classic computed-torque control, while variable gains keep errors low and bounded outside the training area.
Takeaways & Limitations
The method links tracking-error bounds to model uncertainty and feedback gains while enabling lower gains in regions of high model confidence.
Takeaways & Limitations
The approach requires computationally demanding GP predictive-mean and marginal-variance calculations, and gain-function design remains future work.
Abstract
from arXiv · showhide
Perfect tracking control for real-world Euler-Lagrange systems is challenging due to uncertainties in the system model and external disturbances. The magnitude of the tracking error can be reduced either by increasing the feedback gains or improving the model of the system. The latter is clearly preferable as it allows to maintain good tracking performance at low feedback gains. However, accurate models are often difficult to obtain. In this article, we address the problem of stable high-performance tracking control for unknown Euler-Lagrange systems. In particular, we employ Gaussian Process regression to obtain a data-driven model that is used for the feed-forward compensation of unknown dynamics of the system. The model fidelity is used to adapt the feedback gains allowing low feedback gains in state space regions of high model confidence. The proposed control law guarantees a globally bounded tracking error with a specific probability. Simulation studies demonstrate the superiority over state of the art tracking control approaches.
1 Introduction
Euler-Lagrange tracking control relies on accurate feed-forward models, but unknown dynamics and external forces make precise modeling difficult. The paper addresses stable tracking for unknown systems using Gaussian Process regression and confidence-adaptive feedback gains.
- 1 Introduction: Accurate models are needed for low-gain computed-torque control, but friction, payload, contact forces, and unstructured environments make unknown dynamics difficult to model precisely.Increasing feedback gains can compensate uncertainty, but high-gain control is undesirable.
- 1 Introduction: Gaussian Process regression is used as a data-driven approach for stable tracking control of Euler-Lagrange systems with unknown dynamics.The method requires little prior knowledge and can represent complex functions while generalizing from small training sets.
- 1 Introduction: The CTC-GPR law uses the GP mean for feed-forward compensation and model confidence to adapt feedback gains.This enables lower gains where the model is reliable while retaining stability-oriented gain adaptation.
- 1 Introduction: The proposed method explicitly computes tracking-error bounds and guarantees ultimate boundedness within a specified radius and probability.Compared with earlier work, it removes the need for a diagonal feedback-gain matrix and relaxes restrictions on the generalized inertia matrix.
2 Preliminaries and Definitions
The paper formulates fully actuated Euler-Lagrange systems with known structural dynamics and unknown external dynamics, then introduces noisy-data Gaussian Process regression for nonlinear function modeling.
- 2 Preliminaries and Definitions: The system uses generalized coordinates, kinetic and potential energy, control input, and unknown dynamics within the Euler-Lagrange formulation.Unknown dynamics include generalized external forces in addition to the control input.
- 2 Preliminaries and Definitions: Unknown dynamics are modeled as a continuous function of state-related variables and are assumed not to depend directly on time.This includes common robotic effects such as Coulomb and viscous friction.
- 2 Preliminaries and Definitions: Gaussian Process regression models noisy nonlinear function observations using training inputs and outputs, with predictive means and variances obtained from conditional Gaussian distributions.Separate covariance functions and hyperparameters characterize component-wise correlations and are optimized through likelihood maximization.
- 2 Preliminaries and Definitions: The GP prediction combines component-wise outputs into a multivariate Gaussian distribution with diagonal predictive covariance.Marginalization permits variance computation with respect to subsets of the test input.
3 Gaussian Process Model
The Gaussian Process model learns residual dynamics relative to an estimated Euler-Lagrange model from finite system data. Under covariance-function assumptions, the residual-model error is bounded, although misspecified kernels or hyperparameters can loosen that bound.
- 3 Gaussian Process Model: A hybrid model combines estimated parametric Euler-Lagrange dynamics with a GP trained on residual dynamics between the real system and the estimated model.Training data can be collected under an arbitrary controller, provided a finite sequence is available; stability during collection is not required.
- 3 Gaussian Process Model: The estimated inertia and Coriolis terms must satisfy structural bounds, while no prior system knowledge permits identity inertia and zero Coriolis and gravity estimates.These properties are required for the estimates rather than the true system.
- 3 Gaussian Process Model: The covariance function is assumed to place each residual-dynamics component in a bounded reproducing kernel Hilbert space on compact sets.Universal covariance functions can approximate any continuous function arbitrarily precisely on compact sets, so smooth residual dynamics can be covered.
- 3 Gaussian Process Model: The GP model error is bounded with a stochastic bound because only finitely many noisy training points are available.The information capacity has sub-linear dependence on training-set size for many common covariance functions, supporting increasingly accurate learning.
- 3 Gaussian Process Model: If the covariance function or hyperparameters violate the RKHS assumption, the model error may remain bounded on compact sets but the resulting upper bound can be looser.Tighter bounds may require alternative methods.
4 Tracking control with GPR
The paper develops a Gaussian-process-based computed-torque controller whose gains adapt to model confidence. Under bounded-reference assumptions, its Lyapunov analysis guarantees probabilistic ultimate boundedness and exponential convergence to a tracking-error ball.
- Tracking control with GPR: The controller uses GP marginal variances to adapt proportional and derivative feedback gains according to model confidence.The gain functions depend on position variance for Kp and position-velocity variance for Kd.
- Tracking control with GPR: Theorem 1 guarantees a compact state region and bounded model error for the CTC-GPR control law, yielding a probabilistic ultimate bound on tracking error.The guarantee depends on the stated assumptions and holds with probability δ.
- Tracking control with GPR: The Lyapunov candidate is positive definite and radially unbounded when ε satisfies 0 < ε < min {kp1/h2, h1/h2}.The proof establishes positivity through lower and upper bounds involving the gain matrix eigenvalues.
- Tracking control with GPR: Lemma 3 upper-bounds the Lyapunov-function drift with probability δ under bounded desired trajectories and bounded model-error variance.The drift bound is obtained from the closed-loop dynamics and matrix bounds.
- Tracking control with GPR: The closed-loop system is uniformly ultimately bounded and exponentially convergent to a ball with probability at least δ.The bounded-state result also permits defining a compact region and maximum model error for the analysis.
- Tracking control with GPR: The tracking-error radius shrinks quadratically with the upper model-error bound and can decrease with lower GP variance or larger derivative-gain lower bounds.The paper connects lower model variance and gain choices to smaller ultimate bounds and lower feasible feedback gains.
5 Numerical Illustration
Numerical studies compare CTC-GPR with classical CTC and static-gain CTC-GPR in one-dimensional and two-link Euler-Lagrange systems. The results show lower tracking error, improved noise attenuation, and reduced control action, while variable gains preserve performance outside the training region.
- 5.1 Noise attenuation and saturation: 61.6% median reduction in maximal tracking error is achieved by CTC-GPR across 30 randomly generated systems.The comparison also reports higher signal-to-noise ratio and reduced maximal control action for CTC-GPR.
- 5.1 Noise attenuation and saturation: CTC-GPR reduces tracking error and control action relative to classical CTC by using learned dynamics compensation with lower feedback gains.The lower gains can prevent actuator saturation, while the mean prediction compensates unknown dynamics.
- 5.2 Case study: The two-link case study uses Gaussian-process training data, noisy measurements, and variance-dependent proportional and derivative gains.The variable gains are Kp(Σp) = 7I+400 Σp(q) and Kd(Σd) = 6I + 400 Σd( ˙q, q), with classical CTC as the comparison.
- 5.3 Discussion: Variable-gain CTC-GPR outperforms static-gain CTC-GPR for position and velocity error when trajectories leave the training area.The gains increase outside regions covered by training data, keeping tracking error low and bounded there.
- 5.3 Discussion: The improved tracking performance requires computationally demanding predictive-mean and marginal-variance calculations.The design of the variable-gain functions and their closed-loop effects remain topics for future work.
Conclusion
The paper proposes data-driven tracking control that combines Gaussian-process mean predictions for residual-dynamics compensation with variance-based feedback-gain adaptation. It establishes probabilistic ultimate boundedness and exponential convergence of the tracking error, linking the bound to model uncertainty and gains.
- Conclusion: The proposed controller adapts feedback gains using model fidelity while compensating residual dynamics with the Gaussian-process mean prediction.The approach is framed as a computed-torque control law for high-performance tracking.
- Conclusion: The tracking error is uniformly ultimately bounded and exponentially convergent to a ball with a given probability.The bound correlates with model uncertainty and feedback gains.
A Proof of Lemma 1
The proof establishes a probabilistic upper bound on the multidimensional Gaussian-process model error by combining one-dimensional bounds across system dimensions. The bound depends on finite noisy training data and the covariance structure used by the GP.
- A Proof of Lemma 1: The proof applies a one-dimensional Gaussian-process result to each dimension of the unknown dynamics.Each dimension uses a separate GP, with covariance information entering the multidimensional variance calculation.
- A Proof of Lemma 1: The underlying GP observations model noisy function values as y = f(x) + η with η drawn from a Gaussian distribution.The proof uses this noisy-training setup to derive the probabilistic model-error bound.
- A Proof of Lemma 1: The multidimensional model-error probability is lower-bounded by δ = ˜δ^n under the assumed uncorrelated residual dynamics.Because ΠA ⊆ ΠB, the resulting probability bound yields an upper bound on the model-error norm.
B Negative definiteness of M in (23)
The proof establishes that matrix M is negative definite by separately ensuring negative definiteness of its upper-left block and Schur complement through a sufficiently small ε.
- B Negative definiteness of M in (23): A sufficiently small ε makes both M11 and the Schur complement S negative definite, thereby establishing that M is negative definite.M11 is controlled by the bounded positive-definite matrices Kd, Ĥ, and Kp; S combines a negative linear term and a positive quadratic term in ε.
- B Negative definiteness of M in (23): M11 becomes negative definite because Kd, Ĥ, and Kp are positive definite and bounded, allowing ε to be chosen sufficiently small.
- B Negative definiteness of M in (23): The Schur complement S is negative definite because its negative linear ε term dominates its positive quadratic ε term for an appropriate ε.