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A Black-Box Physics-Informed Estimator based on Gaussian Process Regression for Robot Inverse Dynamics Identification

Giulio Giacomuzzos, Ruggero Carli, Diego Romeres, Alberto Dalla Libera

TL;DR

This article proposes a black-box model based on Gaussian process (GP) regression for the identification of the inverse dynamics of robotic manipulators that outperforms state-of-the-art black-box estimators based both on Gaussian processes and neural networks in terms of accuracy, generality, and data efficiency.

Abstract

Learning the inverse dynamics of robots directly from data, adopting a black-box approach, is interesting for several real-world scenarios where limited knowledge about the system is available. In this paper, we propose a black-box model based on Gaussian Process (GP) Regression for the identification of the inverse dynamics of robotic manipulators. The proposed model relies on a novel multidimensional kernel, called \textit{Lagrangian Inspired Polynomial} (\kernelInitials{}) kernel. The \kernelInitials{} kernel is based on two main ideas. First, instead of directly modeling the inverse dynamics components, we model as GPs the kinetic and potential energy of the system. The GP prior on the inverse dynamics components is derived from those on the energies by applying the properties of GPs under linear operators. Second, as regards the energy prior definition, we prove a polynomial structure of the kinetic and potential energy, and we derive a polynomial kernel that encodes this property. As a consequence, the proposed model allows also to estimate the kinetic and potential energy without requiring any label on these quantities. Results on simulation and on two real robotic manipulators, namely a 7 DOF Franka Emika Panda, and a 6 DOF MELFA RV4FL, show that the proposed model outperforms state-of-the-art black-box estimators based both on Gaussian Processes and Neural Networks in terms of accuracy, generality and data efficiency. The experiments on the MELFA robot also demonstrate that our approach achieves performance comparable to fine-tuned model-based estimators, despite requiring less prior information.

A Black-Box Physics-Informed Estimator based on Gaussian Process Regression for Robot Inverse Dynamics Identification

TL;DR

This article proposes a black-box model based on Gaussian process (GP) regression for the identification of the inverse dynamics of robotic manipulators that outperforms state-of-the-art black-box estimators based both on Gaussian processes and neural networks in terms of accuracy, generality, and data efficiency.

Abstract

Learning the inverse dynamics of robots directly from data, adopting a black-box approach, is interesting for several real-world scenarios where limited knowledge about the system is available. In this paper, we propose a black-box model based on Gaussian Process (GP) Regression for the identification of the inverse dynamics of robotic manipulators. The proposed model relies on a novel multidimensional kernel, called \textit{Lagrangian Inspired Polynomial} (\kernelInitials{}) kernel. The \kernelInitials{} kernel is based on two main ideas. First, instead of directly modeling the inverse dynamics components, we model as GPs the kinetic and potential energy of the system. The GP prior on the inverse dynamics components is derived from those on the energies by applying the properties of GPs under linear operators. Second, as regards the energy prior definition, we prove a polynomial structure of the kinetic and potential energy, and we derive a polynomial kernel that encodes this property. As a consequence, the proposed model allows also to estimate the kinetic and potential energy without requiring any label on these quantities. Results on simulation and on two real robotic manipulators, namely a 7 DOF Franka Emika Panda, and a 6 DOF MELFA RV4FL, show that the proposed model outperforms state-of-the-art black-box estimators based both on Gaussian Processes and Neural Networks in terms of accuracy, generality and data efficiency. The experiments on the MELFA robot also demonstrate that our approach achieves performance comparable to fine-tuned model-based estimators, despite requiring less prior information.
Paper Structure (27 sections, 2 theorems, 56 equations, 9 figures, 3 tables)

This paper contains 27 sections, 2 theorems, 56 equations, 9 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Background
  3. Inverse dynamics
  4. Gaussian Process Regression for multi-output models
  5. Polynomial kernel
  6. Combination of kernels
  7. Linear operators and GPs
  8. GPR for Inverse Dynamics Identification
  9. Lagrangian Inspired Polynomial Model
  10. From energies to torques GP models
  11. Kinetic and potential energy polynomial priors
  12. Potential energy
  13. Kinetic energy
  14. Energy estimation
  15. Discussion on related works
  16. ...and 12 more sections

Key Result

Proposition 1

Consider a manipulator with $n+1$ links and $n$ joints. The total potential energy $\mathcal{V}(\boldsymbol{q})$ belongs to the space $\mathbb{P}_{(n)}(\boldsymbol{q}_{c_{(1)}}, \boldsymbol{q}_{s_{(1)}}, \boldsymbol{q}_{p_{(1)}})$, namely it is a polynomial function in $(\boldsymbol{q}_c, \boldsymbo

Figures (9)

  • Figure 1: Box plots of the average torque nMSE obtained with the simulations described in Section \ref{['sec:gen']}.
  • Figure 2: Box plots of the torque nMSE (a) and enregy nMSE (b) obtained with the simulations described in Section \ref{['sec:gen']}. The tables below the figures report the numerical values of the median nMSE percentages.
  • Figure 3: Kinetic energy (T), potential energy (U), and Lagrangian (L) estimated with the LIP and LSE models on one of the trajectories of the MC experiment described in Section \ref{['sec:exp_sim']}. While the LSE model correctly reconstructs the Lagrangian, it fails to estimate the potential and kinetic energy. In contrast, the LIP model correctly reconstructs all the energy components.
  • Figure 4: Results of the data efficiency test described in section \ref{['sec:data_eff']}. The plot shows the evolution of the Global MSE on the test set, as a function of the number of training samples.
  • Figure 5: Box plot of the torque nMSE obtained with the experiment on the PANDA robot described in section \ref{['sec:real_panda']}. The table below reports the mean nMSE percentage for each joint and each estimator. J1J2J3J4J5J6J7SE3.870.582.491.0229.7212.817.39GIP1.130.240.780.2921.935.3415.29LSE0.890.100.530.2418.388.0515.77LIP0.490.050.280.1011.084.105.03
  • ...and 4 more figures

Theorems & Definitions (3)

  • Proposition 1
  • Proposition 2
  • Remark 1