Extrapolation of Physics-Inspired Deep Networks in Learning Robot Inverse Dynamics

Abstract

Accurate robot dynamics models are crucial for safe and stable control as well as for generalization to new conditions. Data-driven methods are increasingly used in robotics dynamics modeling for their superior approximation, with extrapolation performance being a critical efficacy indicator. While deep learning is widely used, it often overlooks essential physical principles, leading to weaker extrapolation capabilities. Recent innovations have introduced physics-inspired deep networks that integrate deep learning with physics, leading to improved extrapolation due to their informed structure, but potentially to underfitting in real-world scenarios due to the presence of unmodeled phenomena. This paper presents an experimental framework to assess the extrapolation capabilities of data-driven methods. Using this framework, physics-inspired deep networks are applied to learn the inverse dynamics models of a simulated robotic manipulator and two real physical systems. The results show that under ideal observation conditions physics-inspired models can learn the system’s underlying structure and demonstrate strong extrapolation capabilities, indicating a promising direction in robotics by offering more accurate and interpretable models. However, in real systems their extrapolation often falls short because the physical priors do not capture all dynamic phenomena, indicating room for improvement in practical applications.

1. Introduction

Accurately mapping inverse dynamics to connect joint movements with applied torques and forces is crucial for controlling robotic arms. Multi-degree robotic arms pose modeling challenges due to their nonlinearity and strong coupling [1]. Traditional methods relying on manually derived dynamics equations are labor-intensive and only accurate under ideal assumptions. Parameter uncertainties, omission of dynamics such as complex friction, and structural changes all reduce their effectiveness [2].

Black box data-driven approaches such as Gaussian process regression, recurrent neural networks, and feed-forward neural networks (FF-NN) are popular in robotic arm dynamics modeling for their approximation capabilities. These models derive the inverse dynamics directly from experimental data, bypassing the need for extensive physical system knowledge. For data-driven approaches, a reliable dynamics modeling method should not only possess precise predictive capabilities within the training domain but also exhibit good predictive capabilities outside the training domain (i.e., extrapolation) [3], ensuring that the robot maintains sufficient precision even in novel operational conditions. However, Black box data-driven approaches struggle to learn kinematic constraints and energy conservation laws from the data, leading to weak extrapolation, low data utilization, and a lack of physical plausibility [4,5].

To address the drawbacks of black box models, physically inspired data-driven learning methods are increasingly used in robotic dynamics modeling [6]. These approaches integrate existing knowledge with deep networks, leveraging deep learning to extract hidden structures and abstract features while preserving traditional knowledge-based benefits. These methods are known as physics-inspired deep networks [7] or physics-informed neural networks [8,9]; in this paper, these models are referred to as physics-inspired deep networks. Classic examples include deep Lagrangian networks (DeLaN) [10] and Hamiltonian neural networks [11], which integrate Lagrangian and Hamiltonian mechanics into deep learning frameworks. Lagrangian-based methods [1,4,7,8,10,12,13,14,15] design neural network models for physical systems using the Euler–Lagrange differential equations of motion, expressed in terms of generalized coordinates q, their velocity $\dot{q}$, and a Lagrangian function $\mathcal{L}(q,\dot{q})$. The Lagrangian function, representing the difference between kinetic and potential energies, is modeled by neural networks. These energy components can be modeled either separately [7,10,15] or together [12]. This enhances physical plausibility and interpretability, allowing for more precise dynamics models and better extrapolation in rigid systems under ideal conditions [7,8,10,11,12,13]. However, the use of physical priors can lead to less accuracy compared to black box methods [14], as they often fail to capture complex nonlinear phenomena such as friction, hysteresis, and contact [16]. Thus, while physics-inspired networks outperform black box models for rigid-body dynamics, their modeling and extrapolation capabilities are limited in real robotic systems with uncertain forces due to the constraints of physical priors [1,8,14,17].

In addressing the limitations of physics-inspired networks in modeling friction due to conservative dynamics, research has focused on integrating key friction effects from robotic manipulators’ actuators into model learning [14,15]. To handle uncertainties such as nonlinear friction and flexibility, the augmented deep Lagrangian network, DeLaN-FFNN [1], was developed. By incorporating a feed-forward neural network, this enhanced model can represent uncertainties beyond rigid body dynamics, improving its effectiveness for real systems.

Existing research has not systematically experimented with or analyzed the extrapolation capabilities of physics-inspired networks. This paper introduces four different structures of DeLaN, examines their characteristics, and conducts a comprehensive evaluation of their extrapolation capabilities across various scenarios, including simulations of real systems, different control policies, and varying end-effector loads. The contributions of this paper are as follows: (i) an experimental framework for assessing data-driven methods is introduced and used to systematically analyze how physics-inspired networks can improve performance in real systems; (ii) it is shown that under ideal conditions, physics-inspired networks can learn underlying system structures and outperform standard deep learning methods in extrapolation tasks; (iii) we demonstrate that for real systems, adding incomplete physical priors can paradoxically reduce the extrapolation abilities of physics-inspired networks compared to black box models.

In the following sections, the mathematical formulation for inverse dynamics modeling is introduced (Section 2.1), followed by a summary of the theory behind Lagrangian mechanics (Section 2.2). The incorporation of rigid-body dynamics in deep learning is then explained (Section 2.3.1), along with the methods for introducing friction (Section 2.3.2) or uncertain forces (Section 2.3.3) into the learning of models. Section 3 introduces an experimental framework for assessing extrapolation. The extrapolation capabilities of physics-inspired networks and black box models are compared in both simulated (Section 4.2) and real (Section 4.3) robotic systems, with an analysis of the results presented in Section 4.4.

2. Preliminaries

The modeling of inverse dynamics and the derivation of rigid body dynamics equations through Lagrangian mechanics are outlined in this section, and four different structures of DeLaN are introduced.

2.1. Inverse Dynamics Modeling

The goal of inverse dynamics modeling is to find the function f mapping from system change to control input, i.e.,

$$f(q,\dot{q},\ddot{q})=\tau ,$$

(1)

where q, $\dot{q}$, and $\ddot{q}$ represent the vectors for joint positions, joint velocities, and joint accelerations, respectively. For a robot with n degree of freedom (DoF), these vectors are of the dimension $R^{n\times 1}$, with $\tau \in R^{n\times 1}$ denoting the unknown joint torques that need to be learned. Coupling of the control input $\tau$ and system state q is essential for model-based control approaches in order to ensure precise control and responsiveness to changes in the system state. Using these definitions, the general inverse dynamics model of a robot is provided by

$$\begin{array}{cc}\hfill H\left(q\right)\ddot{q}& \hfill + c(q,\dot{q}) + g\left(q\right) + \epsilon (q,\dot{q},\ddot{q})=\tau ,\hfill \end{array}$$

(2)

where $H\left(q\right)\in R^{n\times n}$ is the symmetric and positive definite mass matrix, $c(q,\dot{q})\in R^{n\times 1}$ represents the matrix of the centrifugal force and Coriolis force, $g\left(q\right)\in R^{n\times 1}$ represents the gravity vector, and $\epsilon (q,\dot{q},\ddot{q})\in R^{n\times 1}$ stands for the uncertain torque/force effects of those robot elements that are not considered elsewhere in the dynamics model (e.g., elasticities in the mechanical designs, model parameter inaccuracies in the masses or inertiae, vibrational effects, Stribeck friction, couplings, sensor noise) [1,2,18]. Nonlinear friction factors are the main factors affecting the robot’s high-precision motion [19]. If only friction forces are considered in $\epsilon (q,\dot{q},\ddot{q})$, then the expression for the inverse dynamics model can be performed as follows [20]:

$$H\left(q\right)\ddot{q}+c(q,\dot{q})+g\left(q\right)+f_r\left(\dot{q}\right)=\tau$$

(3)

where $f_r\left(\dot{q}\right)\in R^{n\times 1}$ represents the component of joint friction forces. There exist various methods for deriving an inverse dynamics model from the above equation of motion when completely neglecting $\epsilon (q,\dot{q},\ddot{q})$ [21]. The most commonly used methods for modeling robot dynamics are based on Lagrangian mechanics derived from the principles of rigid body dynamics. Robot rigid-body dynamics (${\tau }_{RBD}$) can be defined by the specific mathematical formulation

$$H\left(q\right)\ddot{q}+c(q,\dot{q})+g\left(q\right)={\tau }_{\mathrm{RBD}}.$$

(4)

Therefore, because Lagrangian mechanics are based on rigid body dynamics, they tend to overlook certain aspects such as $\epsilon (q,\dot{q},\ddot{q})$ in robot inverse dynamics. As a result, they fail to accurately describe complex nonlinear phenomena such as friction, hysteresis, contact, and forces generated by flexibility and other uncertainties [16]. For ideal rigid-body dynamics data, as described by Equation (4), the data can be collected in a simulated environment. However, in a real-world environment, dynamic phenomena are not limited to rigid-body dynamics, as real-world data include the uncertain forces described in Equation (2). For a more detailed analysis of these uncertain forces, please refer to [1].

2.2. Lagrangian Mechanics

The Lagrangian $\mathcal{L}$ in Lagrangian mechanics is a function of the generalized coordinates q that describe the dynamics of a system. The Lagrangian is generally chosen to be

$$\mathcal{L}(q,\dot{q})=T(q,\dot{q})-V\left(q\right)={\displaystyle \frac{1}{2}}{\dot{q}}^{T}H\left(q\right)\dot{q}-V\left(q\right),$$

(5)

where $T(q,\dot{q})$ is the kinetic energy and $V\left(q\right)$ is the potential energy. Utilizing the calculus of variations, the Euler–Lagrange equation with non-conservative forces is derived by

$${\displaystyle \frac{d}{dt}}{\displaystyle \frac{𝜕\mathcal{L}(q,\dot{q})}{𝜕\dot{q}}}-{\displaystyle \frac{𝜕\mathcal{L}(q,\dot{q})}{𝜕q}}=\tau ,$$

(6)

$${\displaystyle \frac{{𝜕}^2\mathcal{L}(q,\dot{q})}{{𝜕}^2\dot{q}}}\ddot{q}+{\displaystyle \frac{{𝜕}^{ }\mathcal{L}(q,\dot{q})}{𝜕q𝜕\dot{q}}}\dot{q}-{\displaystyle \frac{{𝜕}^{ }\mathcal{L}(q,\dot{q})}{𝜕q}}=\tau ,$$

(7)

where $\tau$ stands for the generalized forces. Substituting $\mathcal{L}$ for the kinetic and potential energy in Equation (5) yields the following second-order ordinary differential equation (ODE):

$$H\left(q\right)\ddot{q}+\underset{=c(q,\dot{q})}{\underbrace{\dot{H}\left(q\right)\dot{q}-{\displaystyle \frac{1}{2}}{\left({\dot{q}}^T{\displaystyle \frac{𝜕H\left(q\right)}{𝜕q}}\dot{q}\right)}^T}}+\underset{=g\left(q\right)}{\underbrace{{\displaystyle \frac{𝜕V\left(q\right)}{𝜕q}}}}=\tau .$$

(8)

Simplifying this expression yields Equation (4). The Lagrangian method is a commonly used approach for inverse dynamics analysis, as it allows for the simplest form of inverse dynamics models to be established when ignoring the uncertain forces $\epsilon (q,\dot{q},\ddot{q})$.

2.3. Physics-Inspired Deep Networks

This subsection provides an introduction on incorporating rigid-body dynamics into deep learning, based on which methods for introducing friction or uncertain forces into model learning are discussed.

2.3.1. Deep Lagrangian Networks

The DeLaN structure can be seen in Figure 1a. A DeLaN parameterizes the mass matrix H and potential energy V as two separate deep networks. Therefore, the approximate Lagrangian $\mathcal{L}$ is described by

$$\widehat{\mathcal{L}}(q,\dot{q};\theta ,\varphi )={\displaystyle \frac{1}{2}}{\dot{q}}^T\widehat{H}(q;\theta )\dot{q}-\widehat{V}(q;\varphi ),$$

(9)

where $\theta$ and $\varphi$ are the respective network parameters. Then, the inverse model $\tau =f(q,\dot{q},\ddot{q};\theta ,\varphi )$ is approximated by

$$\begin{array}{cc}\hfill \widehat{f}(q,\dot{q}& \hfill ,\ddot{q};\theta ,\varphi )=\widehat{H}(q;\theta )\ddot{q}+\widehat{\dot{H}}(q;\theta )\dot{q}-{\displaystyle \frac{1}{2}}{\left({\dot{q}}^T{\displaystyle \frac{𝜕\widehat{H}(q;\theta )}{𝜕q}}\dot{q}\right)}^T+{\displaystyle \frac{𝜕\widehat{V}(q;\varphi )}{𝜕q}}.\hfill \end{array}$$

(10)

The input of the network consists of position-related information on the generalized position, velocity, and acceleration. The output of the network is the approximated torque $\tau$, presented in Equation (10). The optimization problem is described as follows:

$$({\theta }^{*},{\varphi }^{*})=\underset{\theta ,\varphi }{argmin}{{∥\widehat{f}(q,\dot{q},\ddot{q};\theta ,\varphi )-{\tau }_R∥}^2}_{W_{{\tau }_R}}$$

(11)

where ${\tau }_R$ represents the real torque, which is used to compute the loss function with the corresponding torque $\tau$ output by the network, ${∥\cdot ∥}_W$ represents the Mahalanobis norm, and $W_{{\tau }_R}$ represents the diagonal covariance matrix of the generalized forces. It is beneficial to normalize the loss using the covariance matrix, as the magnitude of the residual might vary between different joints.

Previously, it was noted that the Lagrangian $\mathcal{L}$ can be parameterized using two networks for the mass matrix $H\left(q\right)$ and potential energy $V\left(q\right)$. This is referred to as ‘structured’, and is shown in Figure 1a. An alternative approach employs a single network for both tasks, bypassing the need to learn the mass matrix $H\left(q\right)$, kinetic energy $T(q,\dot{q})$, and potential energy $V\left(q\right)$, resulting in what is termed the black-box Lagrangian, shown in Figure 1b.

2.3.2. Introducing Friction to Model Learning

DeLaN cannot model friction directly, as the learned dynamics are conservative. Referencing [15], we utilized a model assuming that the motor friction depends solely on the joint velocity ${\dot{q}}_i$ of the i-th joint, and is independent of other joints. Depending on the model’s complexity, a combination of static, viscous, or Stribeck friction is assumed as a model prior, with their superposition described by

$${\tau }_{f_i}=-\left({\tau }_{C_v}+{\tau }_{C_s}exp\left(-{\displaystyle \frac{{\dot{q}}_i^2}{v}}\right)\right)\mathrm{sign}\left({\dot{q}}_i\right)-d{\dot{q}}_i,$$

(12)

where ${\tau }_{C_v}$ is the coefficient of static friction, d is the coefficient of viscous friction, and ${\tau }_{C_s}$ and v are the coefficients of Stribeck friction. In the following, the friction coefficients are abbreviated as $\phi =\left\{{\tau }_{C_v},{\tau }_{C_s},v,d\right\}$. Because the friction model is a function of the generalized positions, the friction force ${\tau }_{f_i}$ is a nonconservative generalized force that can be added to the dynamics model derived using Lagrangian mechanics. Equation (8) is described by

$$H\left(q\right)\ddot{q}+\underset{=c(q,\dot{q})}{\underbrace{\dot{H}\left(q\right)\dot{q}-{\displaystyle \frac{1}{2}}{\left({\dot{q}}^T{\displaystyle \frac{𝜕H\left(q\right)}{𝜕q}}\dot{q}\right)}^T}}+\underset{=g\left(q\right)}{\underbrace{{\displaystyle \frac{𝜕V\left(q\right)}{𝜕q}}}}=\tau +{\tau }_f.$$

(13)

The simplified form of this expression is Equation (3). Given the model prior of Equation (12), the friction coefficients $\phi$ can be learned by treating the coefficients as network weights. In this paper, friction is introduced into model learning within DeLaN, as depicted in Figure 1c. This approach is named DeLaN-Friction.

2.3.3. Introducing Uncertain Forces to Model Learning

To address the limitation of DeLaN, which encompasses only rigid-body dynamics priors, DeLaN was combined with a standard deep network to ensure that the model can incorporate all of the force decomposition structures outlined in Equation (2). The structure of DeLaN-FFNN is shown in Figure 1d. The inverse model $\tau =\widehat{f}(q,\dot{q},\ddot{q};\theta ,\varphi ,\psi )$ can be described by

$$\widehat{f}(q,\dot{q},\ddot{q};\theta ,\varphi ,\psi )=\widehat{H}(q;\theta )\ddot{q}+\widehat{c}(q,\dot{q};\theta )+\widehat{g}(q;\varphi )+\widehat{\epsilon }(q,\dot{q},\ddot{q};\psi ),$$

(14)

where $\psi$ are the network parameters corresponding to torque $\epsilon (q,\dot{q},\ddot{q})$. To prevent scenarios where potential and kinetic energy are overlooked, with the black box model primarily influencing the predicted dynamics, the optimization objective is defined as follows:

$$\begin{array}{cc}\hfill ({\theta }^{*},{\varphi }^{*},{\psi }^{*})=\underset{\theta ,\varphi ,\psi }{argmin}(& \hfill {∥\widehat{f}(q,\dot{q},\ddot{q};\theta ,\varphi ,\psi )-{\tau }_R∥}_{W_{{\tau }_R}}^2+\lambda \epsilon {(q,\dot{q},\ddot{q};\psi )}^2)\hfill \end{array}$$

(15)

where $\lambda$ is a positive value used to adjust the proportion of the second loss term in the overall loss value.

3. Extrapolation

The inverse dynamics modeling problem can be addressed using data-driven methods, avoiding the tedious derivation of mathematical formulas [22]. The extrapolation capability of deep learning is crucial, and is being actively investigated [23]. For data-driven methods, models that are only applicable under the specific conditions of their training data collection are of limited use; it is crucial that these models, learned under a specific source distribution, maintain their effectiveness across different target distributions [24]. Evaluating a data-driven model’s stability and reliability hinges on its ability to predict inverse dynamics in unexplored configuration spaces [3]. The iid test error (accuracy under the training data distribution) is often a poor indicator of the extrapolation error (accuracy under a different distribution) [25].

Therefore, for a physical system, training trajectories are collected using certain initial conditions $\pi$. Based on these data, a dynamics model $\widehat{f}$ can be learned. This model is then evaluated under a new condition ${\pi }^{*}$, with this new condition used to collect a dataset that has a distribution different from the training set. For a robotic manipulator, two different conditions $\pi$ and ${\pi }^{*}$ can represent two different control policies, end effector holding different loads, or any other two distinct conditions; hence, it is not clear that a model $\widehat{f}$ learned under $\pi$ will also be accurate under ${\pi }^{*}$.

Inspired by the experimental setup in [25] (illustrated in Figure 2), where dynamics models were learned using trajectories sampled from a condition $\pi$ and their extrapolation performance was assessed on trajectories sampled from some unseen conditions ${\pi }^{*}$, our inverse dynamics model $\widehat{f}$ learned on the training set is then tested on two types of test sets in subsequent experiments. First, the accuracy of the model’s predictions on an iid test set collected under the same conditions $\pi$ as the training set is assessed. Second, the extrapolation capabilities of the data-driven model on non-iid test sets collected under different conditions ${\pi }^{*}$ from the training set are evaluated.

4. Experiment

In the experiments, physics-inspired networks are compared to black box methods for learning the nonlinear dynamics of both simulated and real systems, and their modeling effectiveness and extrapolation capabilities evaluated. For simulations, a 3-DoF robotic manipulator is constructed to follow rigid-body dynamics and provide ideal observation data for model learning. In experiments with a real system, these models are applied to a 3-DoF robot arm and KUKA LWR, where the dynamics exceed the rigid-body priors due to uncertain forces. These experiments aim to answer the following questions:

• Q1: Under ideal observation data, can physics-inspired networks learn the underlying structure of physical systems?
• Q2: When applied to real physical systems, will the extrapolation capabilities of physics-inspired networks be limited by physical priors?
• Q3: What are the main factors that affect the extrapolation performance of physics-inspired networks in real systems?

4.1. Experiment Setup

In our experiments, a deep Lagrangian network using a single network to represent the Lagrangian (Figure 1b) is defined as black-box DeLaN. When two separate networks represent the mass matrix and potential energy (Figure 1a), this is referred to as structured DeLaN. Various physics-inspired networks were used to assess the inverse dynamics modeling, including structured/black-box DeLaN, DeLaN-Friction, DeLaN-FFNN, and a black-box feedforward network (FF-NN) without priors.

4.1.1. Evaluation Metrics

In the experiments, the mean square error (MSE), normalized mean square error (nMSE), and coefficient of determination ($R^2$) are used to evaluate the performance of all the learned models. The MSE and nMSE metrics are described by

$$MSE={\displaystyle \frac{1}{N}}\sum _{i=1}^N{\left({\tau }_i-{\widehat{\tau }}_i\right)}^2,$$

$$nMSE={\displaystyle \frac{{\sum }_{i=1}^N{\left({\tau }_i-{\widehat{\tau }}_i\right)}^2}{{\sum }_{i=1}^N{\left({\tau }_i+\delta \right)}^2}},$$

where ${\tau }_i$ and ${\widehat{\tau }}_i$ are the actual torque value and predicted torque value, respectively, N is the total amount of data, and $\delta$ is a small constant to ensure numerical stability. Lower the MSE and nMSE values indicate the better prediction performance. Additionally, the coefficient of determination $R^2$ is utilized, defined as follows:

$$R^2=1-{\displaystyle \frac{{\sum }_{i=1}^N{({\tau }_i-{\widehat{\tau }}_i)}^2}{{\sum }_{i=1}^N{({\tau }_i-\stackrel{-}{\tau })}^2}}$$

where $\stackrel{-}{\tau }$ is the mean torque value of the datasets and takes values generally ranging from 0 to 1. This metric is used to evaluate and compare the overall performance of torque predictions for all joints of the robot, with higher values indicating better predictions. A value of 1 represents perfect accuracy, while values outside the 0–1 range suggest poor torque prediction performance [2].

4.1.2. Neural Network Training Details

In each experiment, the dynamics models were trained to convergence using a fixed dataset. The evaluations focused on two types of test datasets: an iid test set, which shows model performance under familiar conditions, and several non-iid test sets designed to challenge extrapolation capabilities (Figure 2). To reduce randomness from single random seed training, the results were averaged over five seeds. Neural networks were constructed using the JAX deep learning framework, which employs automatic differentiation to calculate partial derivatives within the dynamics model.

The ADAM optimizer was employed for the simulated 3-DoF manipulator ass well as for the real 3-DoF robot arm and KUKA LWR physical systems. The learning rate was set to ${10}^{-4}$ and the weight decay to ${10}^{-5}$. The distinct hyperparameters of the dynamics models for each system are outlined in

Table 1. Hyperparameters used for the dynamics models across different dynamical systems.

	3-DoF Manipulator	3-DoF Robot Arm	KUKA LWR
Batch Size	512	1024	512
Activation	Tanh	Sigmoid	Tanh
Network Dimension	[2 × 64]	[2 × 128]	[2 × 128]

. The selection of these parameters was based on experience and experimentation, representing the optimal identified values.

4.2. Simulated Robot Experiments

For the simulation experiments, a 3-DoF robotic manipulator was used to collect ideal observation data under rigid-body dynamics conditions. The goal was to assess whether physics-inspired networks could learn the underlying system dynamics and demonstrate superior extrapolation capabilities compared to black box methods.

4.2.1. Dataset Construction

The 3-DoF robotic manipulator features three continuous revolute joints and operates in the vertical x-z plane under the influence of gravity. In order to collect ideal data, no non-conservative forces were present during motion. The manipulator was simulated using the Bullet physics engine.

To smoothly change velocity and direction without causing abrupt reactions in the robotic manipulator, the trajectory planning was based on sinusoidal functions. This ensures a smooth transition from the initial angle ($q_0$) to the final angle ($q_f$). For the i-th joint, the trajectory is defined by

$$q_t^i=q_0^i+{\displaystyle \frac{q_f^i-q_0^i}{T}}\left(t-{\displaystyle \frac{T}{2\pi }}sin\left({\displaystyle \frac{2\pi }{T}}t\right)\right).$$

In this definition, T represents the total movement time and t denotes the current time. The angle of each joint of the robotic manipulator is constrained within $[-3,3]$, and 200 sets of initial and final angles for the three joints are randomly generated. A total of 200 trajectories were collected, each comprising 400 data points, with a sampling period T of 8 s and a frequency of 50 Hz. This process generated a total of 80,000 datasets, which served as the training dataset, encompassing joint positions q, joint velocities $\dot{q}$, joint accelerations $\ddot{q}$, and corresponding joint torques $\tau$. Calculation of the inverse dynamics based on the Newton–Euler formulation method was conducted in Pybullet.

Following the same method used to collect the training set, data from three additional trajectories were collected as the iid test set (Figure 3a). The manipulator then executed the same trajectories, except with the total time T reduced to one-2.5th of its original duration, effectively increasing velocity values by 2.5 times. This resulted in non-iid test set 0, which was used for evaluating velocity extrapolation. Additionally, with the total time unchanged, the initial and final angle constraints were expanded to $[-10,10]$ and three trajectories were collected to form non-iid test set 1, (Figure 3b), in which each joint experiences unseen positions and higher velocities.

4.2.2. Modeling Experiments

The outcomes of the simulated robot inverse model experiments are summarized in

Table 2. The nMSE for torque predictions and torque decomposition predictions across the three joints, MSE for each joint’s torque predictions, and coefficient of determination ($R^2$) for overall modeling effectiveness. The results are averaged over five random seeds, with corresponding confidence intervals.

		Inverse Model (nMSE)				Inverse Model (MSE)
Iid Test Set		Torque $\tau$	Inertial Torque ${\tau }_I$	Coriolis Torque ${\tau }_c$	Gravitational Torque ${\tau }_g$	Joint 0	Joint 1	Joint 2	$R^2$
DeLaN	Structured Lagrangian	$\mathbf{8}.\mathbf{1}\times {\mathbf{10}}^{-\mathbf{6}}\pm \mathbf{4}.\mathbf{7}\times {\mathbf{10}}^{-\mathbf{6}}$	$\mathbf{1}.\mathbf{6}\times {\mathbf{10}}^{-\mathbf{4}}\pm \mathbf{1}.\mathbf{3}\times {\mathbf{10}}^{-\mathbf{4}}$	$\mathbf{1}.\mathbf{1}\times {\mathbf{10}}^{-\mathbf{4}}\pm \mathbf{5}.\mathbf{8}\times {\mathbf{10}}^{-\mathbf{5}}$	$\mathbf{7}.\mathbf{8}\times {\mathbf{10}}^{-\mathbf{6}}\pm \mathbf{5}.\mathbf{0}\times {\mathbf{10}}^{-\mathbf{6}}$	$\mathbf{2}.\mathbf{2}\times {\mathbf{10}}^{-\mathbf{3}}\pm \mathbf{9}.\mathbf{7}\times {\mathbf{10}}^{-\mathbf{4}}$	$\mathbf{1}.\mathbf{3}\times {\mathbf{10}}^{-\mathbf{3}}\pm \mathbf{1}.\mathbf{5}\times {\mathbf{10}}^{-\mathbf{3}}$	$\mathbf{8}.\mathbf{3}\times {\mathbf{10}}^{-\mathbf{4}}\pm \mathbf{6}.\mathbf{2}\times {\mathbf{10}}^{-\mathbf{4}}$	$\mathbf{1}.\mathbf{0}\times {\mathbf{10}}^{\mathbf{0}}\pm \mathbf{4}.\mathbf{8}\times {\mathbf{10}}^{-\mathbf{6}}$
DeLaN	Black-Box Lagrangian	$3.2\times {10}^{-4}\pm 2.8\times {10}^{-4}$	$3.5\times {10}^{-3}\pm 8.7\times {10}^{-4}$	$1.4\times {10}^{-2}\pm 1.2\times {10}^{-2}$	$\mathbf{1}.\mathbf{1}\times {\mathbf{10}}^{-\mathbf{5}}\pm \mathbf{1}.\mathbf{0}\times {\mathbf{10}}^{-\mathbf{5}}$	$1.4\times {10}^{-1}\pm 1.5\times {10}^{-1}$	$2.5\times {10}^{-2}\pm 1.6\times {10}^{-2}$	$9.4\times {10}^{-3}\pm 3.2\times {10}^{-3}$	$\mathbf{1}.\mathbf{0}\times {\mathbf{10}}^{\mathbf{0}}\pm \mathbf{2}.\mathbf{8}\times {\mathbf{10}}^{-\mathbf{4}}$
FF-NN	Feed-Forward Network	$3.7\times {10}^{-3}\pm 1.9\times {10}^{-3}$	$3.7\times {10}^{-2}\pm 2.1\times {10}^{-2}$	$2.3\times {10}^{-1}\pm 9.7\times {10}^{-2}$	$\mathbf{9}.\mathbf{8}\times {\mathbf{10}}^{-\mathbf{5}}\pm \mathbf{6}.\mathbf{7}\times {\mathbf{10}}^{-\mathbf{5}}$	$1.8\times {10}^0\pm 9.6\times {10}^{-1}$	$1.2\times {10}^{-1}\pm 6.6\times {10}^{-2}$	$6.6\times {10}^{-2}\pm 5.4\times {10}^{-2}$	$\mathbf{1}.\mathbf{0}\times {\mathbf{10}}^{\mathbf{0}}\pm \mathbf{1}.\mathbf{9}\times {\mathbf{10}}^{-\mathbf{3}}$
DeLaN-Friction	Introducing Friction	$\mathbf{8}.\mathbf{3}\times {\mathbf{10}}^{-\mathbf{6}}\pm \mathbf{6}.\mathbf{2}\times {\mathbf{10}}^{-\mathbf{6}}$	$\mathbf{3}.\mathbf{6}\times {\mathbf{10}}^{-\mathbf{4}}\pm \mathbf{3}.\mathbf{9}\times {\mathbf{10}}^{-\mathbf{4}}$	$\mathbf{1}.\mathbf{6}\times {\mathbf{10}}^{-\mathbf{4}}\pm \mathbf{1}.\mathbf{4}\times {\mathbf{10}}^{-\mathbf{4}}$	$\mathbf{6}.\mathbf{6}\times {\mathbf{10}}^{-\mathbf{6}}\pm \mathbf{5}.\mathbf{0}\times {\mathbf{10}}^{-\mathbf{6}}$	$\mathbf{2}.\mathbf{0}\times {\mathbf{10}}^{-\mathbf{3}}\pm \mathbf{1}.\mathbf{8}\times {\mathbf{10}}^{-\mathbf{3}}$	$\mathbf{1}.\mathbf{5}\times {\mathbf{10}}^{-\mathbf{3}}\pm \mathbf{8}.\mathbf{5}\times {\mathbf{10}}^{-\mathbf{4}}$	$\mathbf{9}.\mathbf{5}\times {\mathbf{10}}^{-\mathbf{4}}\pm \mathbf{1}.\mathbf{2}\times {\mathbf{10}}^{-\mathbf{3}}$	$\mathbf{1}.\mathbf{0}\times {\mathbf{10}}^{\mathbf{0}}\pm \mathbf{6}.\mathbf{3}\times {\mathbf{10}}^{-\mathbf{6}}$
DeLaN-FFNN	An Augmented DeLaN	$\mathbf{6}.\mathbf{2}\times {\mathbf{10}}^{-\mathbf{6}}\pm \mathbf{3}.\mathbf{0}\times {\mathbf{10}}^{-\mathbf{6}}$	$\mathbf{1}.\mathbf{6}\times {\mathbf{10}}^{-\mathbf{4}}\pm \mathbf{1}.\mathbf{2}\times {\mathbf{10}}^{-\mathbf{4}}$	$\mathbf{1}.\mathbf{1}\times {\mathbf{10}}^{-\mathbf{4}}\pm \mathbf{5}.\mathbf{3}\times {\mathbf{10}}^{-\mathbf{5}}$	$\mathbf{5}.\mathbf{4}\times {\mathbf{10}}^{-\mathbf{6}}\pm \mathbf{4}.\mathbf{7}\times {\mathbf{10}}^{-\mathbf{6}}$	$\mathbf{1}.\mathbf{6}\times {\mathbf{10}}^{-\mathbf{3}}\pm \mathbf{1}.\mathbf{7}\times {\mathbf{10}}^{-\mathbf{3}}$	$\mathbf{1}.\mathbf{0}\times {\mathbf{10}}^{-\mathbf{3}}\pm \mathbf{6}.\mathbf{8}\times {\mathbf{10}}^{-\mathbf{4}}$	$\mathbf{6}.\mathbf{5}\times {\mathbf{10}}^{-\mathbf{4}}\pm \mathbf{4}.\mathbf{1}\times {\mathbf{10}}^{-\mathbf{4}}$	$\mathbf{1}.\mathbf{0}\times {\mathbf{10}}^{\mathbf{0}}\pm \mathbf{3}.\mathbf{0}\times {\mathbf{10}}^{-\mathbf{6}}$
Non-iid Test Set 0		Torque $\tau$	Inertial Torque ${\tau }_I$	Coriolis Torque ${\tau }_c$	Gravitational Torque ${\tau }_g$	Joint 0	Joint 1	Joint 2	$R^2$
DeLaN	Structured Lagrangian	$\mathbf{2}.\mathbf{7}\times {\mathbf{10}}^{-\mathbf{5}}\pm \mathbf{1}.\mathbf{5}\times {\mathbf{10}}^{-\mathbf{5}}$	$\mathbf{1}.\mathbf{7}\times {\mathbf{10}}^{-\mathbf{4}}\pm \mathbf{1}.\mathbf{3}\times {\mathbf{10}}^{-\mathbf{4}}$	$\mathbf{1}.\mathbf{1}\times {\mathbf{10}}^{-\mathbf{4}}\pm \mathbf{5}.\mathbf{8}\times {\mathbf{10}}^{-\mathbf{5}}$	$\mathbf{7}.\mathbf{7}\times {\mathbf{10}}^{-\mathbf{6}}\pm \mathbf{4}.\mathbf{9}\times {\mathbf{10}}^{-\mathbf{6}}$	$\mathbf{2}.\mathbf{2}\times {\mathbf{10}}^{-\mathbf{2}}\pm \mathbf{1}.\mathbf{4}\times {\mathbf{10}}^{-\mathbf{2}}$	$\mathbf{1}.\mathbf{2}\times {\mathbf{10}}^{-\mathbf{2}}\pm \mathbf{9}.\mathbf{5}\times {\mathbf{10}}^{-\mathbf{3}}$	$\mathbf{8}.\mathbf{2}\times {\mathbf{10}}^{-\mathbf{3}}\pm \mathbf{1}.\mathbf{8}\times {\mathbf{10}}^{-\mathbf{3}}$	$\mathbf{1}.\mathbf{0}\times {\mathbf{10}}^{\mathbf{0}}\pm \mathbf{1}.\mathbf{5}\times {\mathbf{10}}^{-\mathbf{5}}$
DeLaN	Black-Box Lagrangian	$2.4\times {10}^{-1}\pm 4.9\times {10}^{-2}$	$3.7\times {10}^{-3}\pm 8.6\times {10}^{-4}$	$4.9\times {10}^{-1}\pm 1.5\times {10}^{-1}$	$\mathbf{1}.\mathbf{1}\times {\mathbf{10}}^{-\mathbf{5}}\pm \mathbf{1}.\mathbf{0}\times {\mathbf{10}}^{-\mathbf{5}}$	$3.4\times {10}^2\pm 6.9\times {10}^1$	$3.4\times {10}^1\pm 6.5\times {10}^0$	$2.0\times {10}^0\pm 4.9\times {10}^{-1}$	$7.6\times {10}^{-1}\pm 4.9\times {10}^{-2}$
FF-NN	Feed-Forward Network	$2.2\times {10}^{-1}\pm 3.7\times {10}^{-2}$	$3.6\times {10}^{-1}\pm 7.9\times {10}^{-2}$	$7.9\times {10}^{-1}\pm 6.9\times {10}^{-2}$	$\mathbf{9}.\mathbf{6}\times {\mathbf{10}}^{-\mathbf{5}}\pm \mathbf{6}.\mathbf{5}\times {\mathbf{10}}^{-\mathbf{5}}$	$3.2\times {10}^2\pm 5.6\times {10}^1$	$1.7\times {10}^1\pm 3.6\times {10}^0$	$9.2\times {10}^0\pm 2.1\times {10}^0$	$7.8\times {10}^{-1}\pm 3.7\times {10}^{-2}$
DeLaN-Friction	Introducing Friction	$\mathbf{3}.\mathbf{7}\times {\mathbf{10}}^{-\mathbf{5}}\pm \mathbf{2}.\mathbf{7}\times {\mathbf{10}}^{-\mathbf{5}}$	$\mathbf{3}.\mathbf{8}\times {\mathbf{10}}^{-\mathbf{4}}\pm \mathbf{4}.\mathbf{2}\times {\mathbf{10}}^{-\mathbf{4}}$	$\mathbf{1}.\mathbf{6}\times {\mathbf{10}}^{-\mathbf{4}}\pm \mathbf{1}.\mathbf{4}\times {\mathbf{10}}^{-\mathbf{4}}$	$\mathbf{6}.\mathbf{5}\times {\mathbf{10}}^{-\mathbf{6}}\pm \mathbf{5}.\mathbf{0}\times {\mathbf{10}}^{-\mathbf{6}}$	$\mathbf{2}.\mathbf{7}\times {\mathbf{10}}^{-\mathbf{2}}\pm \mathbf{2}.\mathbf{9}\times {\mathbf{10}}^{-\mathbf{2}}$	$\mathbf{1}.\mathbf{7}\times {\mathbf{10}}^{-\mathbf{2}}\pm \mathbf{9}.\mathbf{1}\times {\mathbf{10}}^{-\mathbf{3}}$	$\mathbf{1}.\mathbf{4}\times {\mathbf{10}}^{-\mathbf{2}}\pm \mathbf{1}.\mathbf{2}\times {\mathbf{10}}^{-\mathbf{2}}$	$\mathbf{1}.\mathbf{0}\times {\mathbf{10}}^{\mathbf{0}}\pm \mathbf{2}.\mathbf{7}\times {\mathbf{10}}^{-\mathbf{5}}$
DeLaN-FFNN	An Augmented DeLaN	$\mathbf{2}.\mathbf{7}\times {\mathbf{10}}^{-\mathbf{5}}\pm \mathbf{9}.\mathbf{9}\times {\mathbf{10}}^{-\mathbf{6}}$	$\mathbf{1}.\mathbf{7}\times {\mathbf{10}}^{-\mathbf{4}}\pm \mathbf{1}.\mathbf{3}\times {\mathbf{10}}^{-\mathbf{4}}$	$\mathbf{1}.\mathbf{1}\times {\mathbf{10}}^{-\mathbf{4}}\pm \mathbf{5}.\mathbf{4}\times {\mathbf{10}}^{-\mathbf{5}}$	$\mathbf{5}.\mathbf{3}\times {\mathbf{10}}^{-\mathbf{6}}\pm \mathbf{4}.\mathbf{6}\times {\mathbf{10}}^{-\mathbf{6}}$	$\mathbf{2}.\mathbf{0}\times {\mathbf{10}}^{-\mathbf{2}}\pm \mathbf{7}.\mathbf{5}\times {\mathbf{10}}^{-\mathbf{3}}$	$\mathbf{1}.\mathbf{3}\times {\mathbf{10}}^{-\mathbf{2}}\pm \mathbf{6}.\mathbf{1}\times {\mathbf{10}}^{-\mathbf{3}}$	$\mathbf{8}.\mathbf{9}\times {\mathbf{10}}^{-\mathbf{3}}\pm \mathbf{3}.\mathbf{6}\times {\mathbf{10}}^{-\mathbf{3}}$	$\mathbf{1}.\mathbf{0}\times {\mathbf{10}}^{\mathbf{0}}\pm \mathbf{9}.\mathbf{9}\times {\mathbf{10}}^{-\mathbf{6}}$
Non-iid Test Set 1		Torque $\tau$	Inertial Torque ${\tau }_I$	Coriolis Torque ${\tau }_c$	Gravitational Torque ${\tau }_g$	Joint 0	Joint 1	Joint 2	$R^2$
DeLaN	Structured Lagrangian	$\mathbf{5}.\mathbf{9}\times {\mathbf{10}}^{-\mathbf{4}}\pm \mathbf{3}.\mathbf{1}\times {\mathbf{10}}^{-\mathbf{4}}$	$\mathbf{1}.\mathbf{0}\times {\mathbf{10}}^{-\mathbf{4}}\pm \mathbf{2}.\mathbf{9}\times {\mathbf{10}}^{-\mathbf{5}}$	$\mathbf{1}.\mathbf{1}\times {\mathbf{10}}^{-\mathbf{3}}\pm \mathbf{5}.\mathbf{8}\times {\mathbf{10}}^{-\mathbf{4}}$	$\mathbf{3}.\mathbf{3}\times {\mathbf{10}}^{-\mathbf{6}}\pm \mathbf{1}.\mathbf{6}\times {\mathbf{10}}^{-\mathbf{6}}$	$\mathbf{3}.\mathbf{9}\times {\mathbf{10}}^{-\mathbf{1}}\pm \mathbf{3}.\mathbf{1}\times {\mathbf{10}}^{-\mathbf{1}}$	$\mathbf{3}.\mathbf{7}\times {\mathbf{10}}^{-\mathbf{1}}\pm \mathbf{3}.\mathbf{1}\times {\mathbf{10}}^{-\mathbf{1}}$	$\mathbf{2}.\mathbf{0}\times {\mathbf{10}}^{-\mathbf{1}}\pm \mathbf{5}.\mathbf{7}\times {\mathbf{10}}^{-\mathbf{2}}$	$\mathbf{1}.\mathbf{0}\times {\mathbf{10}}^{\mathbf{0}}\pm \mathbf{3}.\mathbf{1}\times {\mathbf{10}}^{-\mathbf{4}}$
DeLaN	Black-Box Lagrangian	$4.6\times {10}^{-1}\pm 4.2\times {10}^{-2}$	$2.7\times {10}^{-3}\pm 1.2\times {10}^{-3}$	$7.9\times {10}^{-1}\pm 7.9\times {10}^{-2}$	$\mathbf{8}.\mathbf{3}\times {\mathbf{10}}^{-\mathbf{6}}\pm \mathbf{5}.\mathbf{1}\times {\mathbf{10}}^{-\mathbf{6}}$	$6.8\times {10}^2\pm 5.9\times {10}^1$	$4.9\times {10}^1\pm 9.1\times {10}^0$	$2.7\times {10}^1\pm 5.7\times {10}^0$	$5.4\times {10}^{-1}\pm 4.2\times {10}^{-2}$
FF-NN	Feed-Forward Network	$5.3\times {10}^{-1}\pm 4.4\times {10}^{-2}$	$3.4\times {10}^{-1}\pm 1.5\times {10}^{-1}$	$9.0\times {10}^{-1}\pm 6.2\times {10}^{-2}$	$\mathbf{6}.\mathbf{9}\times {\mathbf{10}}^{-\mathbf{5}}\pm \mathbf{4}.\mathbf{2}\times {\mathbf{10}}^{-\mathbf{5}}$	$7.6\times {10}^2\pm 7.0\times {10}^1$	$4.3\times {10}^1\pm 1.9\times {10}^1$	$5.5\times {10}^1\pm 8.9\times {10}^0$	$4.7\times {10}^{-1}\pm 4.5\times {10}^{-2}$
DeLaN-Friction	Introducing Friction	$\mathbf{7}.\mathbf{5}\times {\mathbf{10}}^{-\mathbf{4}}\pm \mathbf{4}.\mathbf{3}\times {\mathbf{10}}^{-\mathbf{4}}$	$\mathbf{1}.\mathbf{2}\times {\mathbf{10}}^{-\mathbf{4}}\pm \mathbf{7}.\mathbf{5}\times {\mathbf{10}}^{-\mathbf{5}}$	$\mathbf{1}.\mathbf{4}\times {\mathbf{10}}^{-\mathbf{3}}\pm \mathbf{7}.\mathbf{7}\times {\mathbf{10}}^{-\mathbf{4}}$	$\mathbf{3}.\mathbf{2}\times {\mathbf{10}}^{-\mathbf{6}}\pm \mathbf{1}.\mathbf{9}\times {\mathbf{10}}^{-\mathbf{6}}$	$\mathbf{5}.\mathbf{9}\times {\mathbf{10}}^{-\mathbf{1}}\pm \mathbf{5}.\mathbf{8}\times {\mathbf{10}}^{-\mathbf{1}}$	$\mathbf{3}.\mathbf{4}\times {\mathbf{10}}^{-\mathbf{1}}\pm \mathbf{1}.\mathbf{0}\times {\mathbf{10}}^{-\mathbf{1}}$	$\mathbf{2}.\mathbf{9}\times {\mathbf{10}}^{-\mathbf{1}}\pm \mathbf{1}.\mathbf{7}\times {\mathbf{10}}^{-\mathbf{1}}$	$\mathbf{1}.\mathbf{0}\times {\mathbf{10}}^{\mathbf{0}}\pm \mathbf{4}.\mathbf{3}\times {\mathbf{10}}^{-\mathbf{4}}$
DeLaN-FFNN	An Augmented DeLaN	$\mathbf{4}.\mathbf{8}\times {\mathbf{10}}^{-\mathbf{4}}\pm \mathbf{1}.\mathbf{5}\times {\mathbf{10}}^{-\mathbf{4}}$	$\mathbf{8}.\mathbf{8}\times {\mathbf{10}}^{-\mathbf{5}}\pm \mathbf{3}.\mathbf{4}\times {\mathbf{10}}^{-\mathbf{5}}$	$\mathbf{8}.\mathbf{8}\times {\mathbf{10}}^{-\mathbf{4}}\pm \mathbf{2}.\mathbf{9}\times {\mathbf{10}}^{-\mathbf{4}}$	$\mathbf{2}.\mathbf{8}\times {\mathbf{10}}^{-\mathbf{6}}\pm \mathbf{1}.\mathbf{6}\times {\mathbf{10}}^{-\mathbf{6}}$	$\mathbf{3}.\mathbf{5}\times {\mathbf{10}}^{-\mathbf{1}}\pm \mathbf{2}.\mathbf{7}\times {\mathbf{10}}^{-\mathbf{1}}$	$\mathbf{2}.\mathbf{5}\times {\mathbf{10}}^{-\mathbf{1}}\pm \mathbf{1}.\mathbf{3}\times {\mathbf{10}}^{-\mathbf{1}}$	$\mathbf{1}.\mathbf{8}\times {\mathbf{10}}^{-\mathbf{1}}\pm \mathbf{6}.\mathbf{8}\times {\mathbf{10}}^{-\mathbf{2}}$	$\mathbf{1}.\mathbf{0}\times {\mathbf{10}}^{\mathbf{0}}\pm \mathbf{1}.\mathbf{5}\times {\mathbf{10}}^{-\mathbf{4}}$

Note: Bold indicates better results.

. The table includes the nMSE for the torques learned through supervised learning and for the torque decompositions learned through unsupervised learning across the three joints for each dynamics model, with the torque decomposition encompassing the inertial torque ${\tau }_I$, Coriolis torque ${\tau }_c$, and gravitational torque ${\tau }_g$. The MSE serves as the evaluation metric for the torque predictions of each joint. Additionally, the coefficient of determination $R^2$ is presented on the far right as the overall modeling accuracy for inverse dynamics modeling.

Using nMSE as the evaluation metric, the dynamics models for structured DeLaN, DeLaN-Friction, and DeLaN-FFNN achieve consistent results across all test sets, accurately predicting the inertial, Coriolis, and gravitational forces, indicating their ability to learn the system’s underlying structure. Their prediction accuracy surpasses that of black-box DeLaN and FF-NN. Visualization results for the iid test set are shown in Figure 4, with non-iid test set results included in the appendix (Appendix B). For the non-iid test sets, all models experience varying degrees of accuracy deterioration, with FF-NN and black-box DeLaN showing significant declines compared to the slight deterioration in structured DeLaN, DeLaN-Friction, and DeLaN-FFNN. Notably, all models effectively learn gravitational decomposition without deterioration, even in extrapolation tests, demonstrating their ability to extract gravitational features from the data.

For each joint’s prediction error, structured DeLaN, DeLaN-Friction, and DeLaN-FFNN consistently achieve superior accuracy compared to black-box DeLaN and FF-NN on both iid and non-iid test sets. The coefficient of determination ($R^2$) offers an intuitive comparison of each network’s extrapolation capabilities. On the iid test set, the $R^2$ for all networks is close to 1, indicating near-perfect predictive performance. For non-iid test sets, the $R^2$ for structured DeLaN, DeLaN-Friction, and DeLaN-FFNN remains very close to 1, showing continued effectiveness, while the $R^2$ for black-box DeLaN and FF-NN drops significantly, reaching around 0.5 on non-iid test set 1. Based on this analysis, structured DeLaN, DeLaN-Friction, and DeLaN-FFNN have superior extrapolation capabilities within physics-inspired networks. Notably, DeLaN-Friction and DeLaN-FFNN perform well even with data containing only conservative forces. This is because in DeLaN-Friction the friction coefficients are trained as network weights, while in DeLaN-FFNN the loss function (Equation (15)) reduces the output of the uncertainty force network when there are no uncertainties to learn.

Figure 5 illustrates each model’s performance under increased velocities to assess their ability to handle new conditions. All models are initially trained on trajectories at a velocity scale of $1\times$, then tested at increased velocities. Data collected at $2.5\times$ the velocity correspond to iid test set 0. Structured DeLaN, DeLaN-Friction, and DeLaN-FFNN show slight linear deterioration with increasing velocity. In contrast, black-box DeLaN and FF-NN experience severe deterioration, especially before reaching $2\times$ velocity. At $3\times$ velocity, structured DeLaN, DeLaN-Friction, and DeLaN-FFNN outperform black-box DeLaN and FF-NN at $1\times$ velocity. This advantage arises because structured DeLaN’s input domain consists solely of q, making it unaffected by changes in velocities or accelerations. In contrast, FF-NN’s input domain includes $(q,\dot{q},\ddot{q})$ and black-box DeLaN’s includes $(q,\dot{q})$. DeLaN-Friction and DeLaN-FFNN, which build on DeLaN, do not show improved extrapolation because complex friction and uncertain forces were not incorporated into the simulated robotic manipulator’s setup.

The experiments show that structured DeLaN, DeLaN-Friction, and DeLaN-FFNN, which incorporate the mass matrix $H\left(q\right)$, potential energy $V\left(q\right)$, and kinetic energy $T(q,\dot{q})$ as physical priors, are able to effectively learn the system’s structure from ideal observation data. These models demonstrate superior modeling accuracy and more reliable extrapolation capabilities compared to FF-NN, which lacks any priors. In contrast, black-box DeLaN, which skips learning the mass matrix, kinetic energy, and potential energy by directly training the Lagrangian $\mathcal{L}(q,\dot{q})$ through a single network, performs similarly to FF-NN.

4.3. Real Robot Experiments

In the real robot experiments, a 3-DoF robotic arm was used to evaluate the extrapolation capabilities of each dynamics model with data collected under different control policies. Additionally, data collected using a KUKA LWR robot pushing flasks of different volumes were utilized to assess the extrapolation capabilities of the dynamics models with data collected under varying end-effector loads.

4.3.1. Dataset Construction

The datasets for the 3-DoF robot arm and KUKA LWR were sourced from publicly available real-system datasets. Detailed descriptions of these datasets are provided below.

3-DoF Robot arm Dataset. (Click to enter the 3-DoF Robot dataset website: https://edmond.mpg.de/dataset.xhtml?persistentId=doi:10.17617/3.ZT6K7P, accessed on 15 April 2024). Data used to evaluate extrapolation ability to different policies was recorded on a real 3-DoF robotic system (shown in Figure 6a). Dynamics data from various distributions were collected by tracking trajectories with different frequency settings and joint ranges. The sine trajectories $X_t$ in joint space were generated according to

$$X_t=C+A\odot \left(\sum _{i=1}^k{B}_i\odot sin({\omega }_{i}t+{\phi }_i)\right),$$

where C, A, $B_i$, ${\omega }_i$, ${\phi }_i$ are all three-dimensional vectors, k denotes the number of sine components that are superimposed, and ⊙ denotes the element-wise product. By changing the angular frequency ${\omega }_i$, different frequency settings are controlled, while the joint range reachable by the manipulator is altered by adjusting C and A. The frequency is categorized into high or low, while the joint constraints are divided into the end-effector moving only in the left part of the task space or in the entire range of safe movements.

The training set included 532,000 samples of low-frequency data with joints constrained to the left side. The iid test set retained the same settings. Non-iid test set 0 included high-frequency trajectories with left-side joint constraints, non-iid test set 1 featured low-frequency trajectories spanning the entire joint space, and non-iid test set 2 contained high-frequency trajectories across the entire joint space. Each test set contained 126,000 samples. The trajectory settings are summarized in Figure 6b; for more details, please refer to [25].

This dataset provides joint positions q, velocities $\dot{q}$, and corresponding torques $\tau$ for each sample, but does not include acceleration data $\ddot{q}$. In real-world systems, $\ddot{q}$ is often unobserved and is estimated using finite differences [7,14,26], which can amplify high-frequency noise and cause inaccuracies. To mitigate this, low-pass filters are used for more accurate acceleration estimates. Following the approach in [7,26], the accelerations were computed using finite differences and low-pass filters.

KUKA LWR Dataset. (https://cloud.cps.unileoben.ac.at/index.php/s/6aG9gynxDYMmW8M (accessed on 15 April 2024)). In the scenarios involving different end-effector loads, the task involved the KUKA LWR robot pushing flasks filled with varying volumes of liquid to a designated goal location along a curved trajectory. The dataset included two sets. The first set consisted of data collected by the robot while pushing flasks filled with either 200 mL or 400 mL of water, including 112,761 data samples. The second set consisted of data collected by the robot while pushing flasks filled with 300 mL of water, including 16,940 data samples. Each sample included joint positions q, joint velocities $\dot{q}$, joint accelerations $\ddot{q}$, and corresponding torques $\tau$ for the first five joints of the robotic manipulator. For more information on the data, please refer to [18,27].

The first set was divided into two parts, randomly extracting 12,761 data samples as the iid test set and using the remaining 100,000 samples as the training set. The second set of data was used as the non-iid test set to evaluate the model’s extrapolation capabilities for different loads.

4.3.2. Extrapolation across Different Policies

The purpose of modeling robots is to use these models to synthesize new control policies. Therefore, it is crucial to assess whether dynamics models can effectively extrapolate to changes in data distribution caused by different control policies.

The performance of each model in predicting joint torques for each joint is summarized in

Table 3. The MSE for torque predictions of each joint and coefficient of determination $R^2$ for overall modeling effectiveness. The results are averaged over five random seeds with corresponding confidence intervals.

		Inverse Model (MSE)
Iid Test Set		Joint 0	Joint 1	Joint 2	${\mathit{R}}^{\mathbf{2}}$
DeLaN	Structured Lagrangian	$1.2\times {10}^{-1}$ ± $5.9\times {10}^{-4}$	$3.0\times {10}^{-2}$ ± $4.4\times {10}^{-4}$	$1.2\times {10}^{-1}$ ± $9.9\times {10}^{-4}$	−
DeLaN	Black-Box Lagrangian	$1.8\times {10}^{-2}$ ± $6.2\times {10}^{-5}$	$2.6\times {10}^{-3}$ ± $3.1\times {10}^{-5}$	$2.6\times {10}^{-3}$ ± $2.8\times {10}^{-5}$	$8.159\times {10}^{-1}$ ± $7.200\times {10}^{-4}$
FF-NN	Feed-Forward Network	$\mathbf{1}.\mathbf{0}\times {\mathbf{10}}^{-\mathbf{2}}\pm \mathbf{3}.\mathbf{4}\times {\mathbf{10}}^{-\mathbf{4}}$	$\mathbf{7}.\mathbf{7}\times {\mathbf{10}}^{-\mathbf{4}}\pm \mathbf{6}.\mathbf{3}\times {\mathbf{10}}^{-\mathbf{5}}$	$\mathbf{6}.\mathbf{6}\times {\mathbf{10}}^{-\mathbf{4}}\pm \mathbf{2}.\mathbf{0}\times {\mathbf{10}}^{-\mathbf{5}}$	$\mathbf{9}.\mathbf{092}\times {\mathbf{10}}^{-\mathbf{1}}\pm \mathbf{2}.\mathbf{783}\times {\mathbf{10}}^{-\mathbf{3}}$
DeLaN-Friction	Introducing Friction	$1.1\times {10}^{-1}$ ± $8.2\times {10}^{-4}$	$2.9\times {10}^{-2}$ ± $3.9\times {10}^{-4}$	$1.1\times {10}^{-1}$ ± $9.6\times {10}^{-4}$	−
DeLaN-FFNN	An Augmented DeLaN	$\mathbf{1}.\mathbf{0}\times {\mathbf{10}}^{-\mathbf{2}}\pm \mathbf{5}.\mathbf{3}\times {\mathbf{10}}^{-\mathbf{4}}$	$\mathbf{7}.\mathbf{6}\times {\mathbf{10}}^{-\mathbf{4}}\pm \mathbf{1}.\mathbf{3}\times {\mathbf{10}}^{-\mathbf{4}}$	$\mathbf{7}.\mathbf{1}\times {\mathbf{10}}^{-\mathbf{4}}\pm \mathbf{3}.\mathbf{1}\times {\mathbf{10}}^{-\mathbf{5}}$	$\mathbf{9}.\mathbf{107}\times {\mathbf{10}}^{-\mathbf{1}}\pm \mathbf{4}.\mathbf{870}\times {\mathbf{10}}^{-\mathbf{3}}$
Non-iid Test Set 0		Joint 0	Joint 1	Joint 2	${\mathit{R}}^{\mathbf{2}}$
DeLaN	Structured Lagrangian	$1.1\times {10}^{-1}$ ± $3.8\times {10}^{-2}$	$5.0\times {10}^{-2}$ ± $7.8\times {10}^{-3}$	$8.1\times {10}^{-1}$ ± $3.9\times {10}^{-2}$	−
DeLaN	Black-Box Lagrangian	$\mathbf{4}.\mathbf{0}\times {\mathbf{10}}^{-\mathbf{2}}\pm \mathbf{4}.\mathbf{2}\times {\mathbf{10}}^{-\mathbf{4}}$	$8.8\times {10}^{-3}$ ± $2.7\times {10}^{-3}$	$3.0\times {10}^{-2}$ ± $4.7\times {10}^{-3}$	$5.170\times {10}^{-1}$ ± $3.568\times {10}^{-2}$
FF-NN	Feed-Forward Network	$\mathbf{4}.\mathbf{6}\times {\mathbf{10}}^{-\mathbf{2}}\pm \mathbf{1}.\mathbf{8}\times {\mathbf{10}}^{-\mathbf{3}}$	$\mathbf{4}.\mathbf{7}\times {\mathbf{10}}^{-\mathbf{3}}\pm \mathbf{3}.\mathbf{9}\times {\mathbf{10}}^{-\mathbf{4}}$	$\mathbf{1}.\mathbf{4}\times {\mathbf{10}}^{-\mathbf{2}}\pm \mathbf{5}.\mathbf{5}\times {\mathbf{10}}^{-\mathbf{4}}$	$\mathbf{6}.\mathbf{070}\times {\mathbf{10}}^{-\mathbf{1}}\pm \mathbf{1}.\mathbf{444}\times {\mathbf{10}}^{-\mathbf{2}}$
DeLaN-Friction	Introducing Friction	$1.2\times {10}^{-1}$ ± $3.6\times {10}^{-2}$	$5.0\times {10}^{-2}$ ± $3.4\times {10}^{-3}$	$8.1\times {10}^{-1}$ ± $4.0\times {10}^{-2}$	−
DeLaN-FFNN	An Augmented DeLaN	$\mathbf{4}.\mathbf{8}\times {\mathbf{10}}^{-\mathbf{2}}\pm \mathbf{1}.\mathbf{9}\times {\mathbf{10}}^{-\mathbf{3}}$	$\mathbf{5}.\mathbf{0}\times {\mathbf{10}}^{-\mathbf{3}}\pm \mathbf{8}.\mathbf{4}\times {\mathbf{10}}^{-\mathbf{4}}$	$\mathbf{1}.\mathbf{2}\times {\mathbf{10}}^{-\mathbf{2}}\pm \mathbf{1}.\mathbf{3}\times {\mathbf{10}}^{-\mathbf{3}}$	$\mathbf{6}.\mathbf{020}\times {\mathbf{10}}^{-\mathbf{1}}\pm \mathbf{9}.\mathbf{690}\times {\mathbf{10}}^{-\mathbf{3}}$
Non-iid Test Set 1		Joint 0	Joint 1	Joint 2	${\mathit{R}}^{\mathbf{2}}$
DeLaN	Structured Lagrangian	$7.1\times {10}^{-1}$ ± $1.7\times {10}^{-1}$	$3.8\times {10}^{-1}$ ± $1.5\times {10}^{-1}$	$3.0\times {10}^0$ ± $1.1\times {10}^0$	−
DeLaN	Black-Box Lagrangian	$3.5\times {10}^{-1}$ ± $7.3\times {10}^{-2}$	$4.6\times {10}^{-2}$ ± $1.7\times {10}^{-2}$	$4.9\times {10}^{-2}$ ± $1.3\times {10}^{-2}$	−
FF-NN	Feed-Forward Network	$\mathbf{1}.\mathbf{1}\times {\mathbf{10}}^{-\mathbf{1}}\pm \mathbf{1}.\mathbf{8}\times {\mathbf{10}}^{-\mathbf{2}}$	$\mathbf{7}.\mathbf{3}\times {\mathbf{10}}^{-\mathbf{3}}\pm \mathbf{1}.\mathbf{4}\times {\mathbf{10}}^{-\mathbf{3}}$	$\mathbf{1}.\mathbf{7}\times {\mathbf{10}}^{-\mathbf{3}}\pm \mathbf{1}.\mathbf{1}\times {\mathbf{10}}^{-\mathbf{4}}$	$\mathbf{5}.\mathbf{914}\times {\mathbf{10}}^{-\mathbf{1}}\pm \mathbf{6}.\mathbf{576}\times {\mathbf{10}}^{-\mathbf{2}}$
DeLaN-Friction	Introducing Friction	$8.1\times {10}^{-1}$ ± $2.9\times {10}^{-1}$	$4.6\times {10}^{-1}$ ± $2.4\times {10}^{-1}$	$3.9\times {10}^0$ ± $2.2\times {10}^0$	−
DeLaN-FFNN	An Augmented DeLaN	$\mathbf{1}.\mathbf{0}\times {\mathbf{10}}^{-\mathbf{1}}\pm \mathbf{2}.\mathbf{8}\times {\mathbf{10}}^{-\mathbf{2}}$	$2.4\times {10}^{-2}$ ± $1.9\times {10}^{-2}$	$6.3\times {10}^{-2}$ ± $7.8\times {10}^{-2}$	$3.519\times {10}^{-1}$ ± $2.594\times {10}^{-1}$
Non-iid Test Set 2		Joint 0	Joint 1	Joint 2	${\mathit{R}}^{\mathbf{2}}$
DeLaN	Structured Lagrangian	$1.4\times {10}^0$ ± $6.5\times {10}^{-1}$	$3.8\times {10}^{-1}$ ± $2.8\times {10}^{-1}$	$4.0\times {10}^0$ ± $1.5\times {10}^0$	−
DeLaN	Black-Box Lagrangian	$\mathbf{1}.\mathbf{8}\times {\mathbf{10}}^{-\mathbf{1}}\pm \mathbf{2}.\mathbf{5}\times {\mathbf{10}}^{-\mathbf{2}}$	$6.4\times {10}^{-2}$ ± $2.5\times {10}^{-2}$	$1.1\times {10}^{-1}$ ± $1.8\times {10}^{-2}$	$2.965\times {10}^{-1}$ ± $1.245\times {10}^{-1}$
FF-NN	Feed-Forward Network	$\mathbf{2}.\mathbf{4}\times {\mathbf{10}}^{-\mathbf{1}}\pm \mathbf{4}.\mathbf{7}\times {\mathbf{10}}^{-\mathbf{2}}$	$\mathbf{2}.\mathbf{2}\times {\mathbf{10}}^{-\mathbf{2}}\pm \mathbf{2}.\mathbf{5}\times {\mathbf{10}}^{-\mathbf{3}}$	$\mathbf{6}.\mathbf{4}\times {\mathbf{10}}^{-\mathbf{2}}\pm \mathbf{2}.\mathbf{5}\times {\mathbf{10}}^{-\mathbf{3}}$	$\mathbf{3}.\mathbf{501}\times {\mathbf{10}}^{-\mathbf{1}}\pm \mathbf{1}.\mathbf{002}\times {\mathbf{10}}^{-\mathbf{1}}$
DeLaN-Friction	Introducing Friction	$1.4\times {10}^0$ ± $7.0\times {10}^{-1}$	$5.0\times {10}^{-1}$ ± $5.3\times {10}^{-1}$	$4.5\times {10}^0$ ± $2.2\times {10}^0$	−
DeLaN-FFNN	An Augmented DeLaN	$\mathbf{2}.\mathbf{8}\times {\mathbf{10}}^{-\mathbf{1}}\pm \mathbf{5}.\mathbf{3}\times {\mathbf{10}}^{-\mathbf{2}}$	$\mathbf{3}.\mathbf{5}\times {\mathbf{10}}^{-\mathbf{2}}\pm \mathbf{1}.\mathbf{7}\times {\mathbf{10}}^{-\mathbf{2}}$	$1.2\times {10}^{-1}$ ± $4.3\times {10}^{-2}$	$1.453\times {10}^{-1}$ ± $7.404\times {10}^{-2}$

Note: The bolded part refers to better results, and − representing a coefficient of determination $R^2$ range exceeding 0–1.

. Across all test sets for each joint, better torque prediction results are concentrated among black-box DeLaN, FF-NN, and DeLaN-FFNN, with a particular focus on FF-NN and DeLaN-FFNN. For evaluation using the coefficient of determination $R^2$, it is noteworthy that DeLaN and DeLaN-Friction score beyond the range of 0–1 on every test set, indicating poor prediction performance across all groups of test sets. For the iid test set and non-iid test set 0, the highest torque prediction scores are achieved by FF-NN and DeLaN-FFNN. For non-iid test sets 1 and 2, the highest scores are achieved by FF-NN. Notably, black-box DeLaN scores beyond the range of 0–1 on non-iid test set 1, indicating poor extrapolation capabilities.

Figure 7 presents the overall torque prediction performance, allowing for a direct comparison. Structured DeLaN and DeLaN-Friction exhibit the poorest accuracy across all test sets, struggling even on the iid test set. Better accuracy is achieved by FF-NN, black-box DeLaN, and DeLaN-FFNN. For the iid test set and non-iid test set 0, FF-NN and DeLaN-FFNN provide the best torque predictions. However, for the more complex non-iid test sets 1 and 2, DeLaN-FFNN’s accuracy is slightly inferior to FF-NN.

From this analysis, structured DeLaN and DeLaN-Friction lack accuracy and extrapolation capabilities, with the former limited by rigid-body dynamics priors and the latter adding actuator friction. Black-box DeLaN performs better due to fewer physical priors. FF-NN, without physical priors, achieves the best results, indicating that inadequate priors compromise performance in physics-inspired networks. DeLaN-FFNN matches the accuracy of FF-NN on the iid tests by capturing all of the dynamics in Equation (2). However, DeLaN-FFNN has inferior extrapolation in non-iid tests due to introducing uncertain forces in a black-box manner, reducing transparency and the benefits of physical principles.

4.3.3. Extrapolation across Different Loads

A robotic manipulator often needs to handle different end-effector loads, making it essential to evaluate a dynamics model’s extrapolation capabilities for varying loads.

Table 4. The MSE for each joint and coefficient of determination $R^2$ for each dynamics model on each set of test data, along with the average and confidence interval calculated over five random seeds.

		Inverse Model (MSE)
Iid Test Set		Joint 0	Joint 1	Joint 2	Joint 3	Joint 4	${\mathit{R}}^{\mathbf{2}}$
DeLaN	Structured Lagrangian	$6.703\times {10}^{-4}$ ± $6.984\times {10}^{-5}$	$2.579\times {10}^{-4}$ ± $7.915\times {10}^{-5}$	$1.939\times {10}^{-4}$ ± $4.009\times {10}^{-5}$	$1.008\times {10}^{-3}$ ± $4.744\times {10}^{-4}$	$4.378\times {10}^{-4}$ ± $1.355\times {10}^{-4}$	$\mathbf{9}.\mathbf{787}\times {\mathbf{10}}^{-\mathbf{1}}\pm \mathbf{6}.\mathbf{316}\times {\mathbf{10}}^{-\mathbf{3}}$
DeLaN	Black-Box Lagrangian	$\mathbf{2}.\mathbf{337}\times {\mathbf{10}}^{-\mathbf{4}}\pm \mathbf{1}.\mathbf{312}\times {\mathbf{10}}^{-\mathbf{5}}$	$\mathbf{6}.\mathbf{853}\times {\mathbf{10}}^{-\mathbf{5}}\pm \mathbf{4}.\mathbf{182}\times {\mathbf{10}}^{-\mathbf{6}}$	$\mathbf{5}.\mathbf{890}\times {\mathbf{10}}^{-\mathbf{5}}\pm \mathbf{6}.\mathbf{225}\times {\mathbf{10}}^{-\mathbf{6}}$	$\mathbf{1}.\mathbf{689}\times {\mathbf{10}}^{-\mathbf{4}}\pm \mathbf{4}.\mathbf{673}\times {\mathbf{10}}^{-\mathbf{6}}$	$\mathbf{1}.\mathbf{390}\times {\mathbf{10}}^{-\mathbf{4}}\pm \mathbf{1}.\mathbf{238}\times {\mathbf{10}}^{-\mathbf{5}}$	$\mathbf{9}.\mathbf{945}\times {\mathbf{10}}^{-\mathbf{1}}\pm \mathbf{2}.\mathbf{411}\times {\mathbf{10}}^{-\mathbf{4}}$
FF-NN	Feed-Forward Network	$\mathbf{2}.\mathbf{735}\times {\mathbf{10}}^{-\mathbf{4}}\pm \mathbf{3}.\mathbf{426}\times {\mathbf{10}}^{-\mathbf{5}}$	$\mathbf{8}.\mathbf{273}\times {\mathbf{10}}^{-\mathbf{5}}\pm \mathbf{1}.\mathbf{074}\times {\mathbf{10}}^{-\mathbf{5}}$	$\mathbf{6}.\mathbf{578}\times {\mathbf{10}}^{-\mathbf{5}}\pm \mathbf{9}.\mathbf{738}\times {\mathbf{10}}^{-\mathbf{6}}$	$\mathbf{1}.\mathbf{915}\times {\mathbf{10}}^{-\mathbf{4}}\pm \mathbf{1}.\mathbf{823}\times {\mathbf{10}}^{-\mathbf{5}}$	$\mathbf{1}.\mathbf{611}\times {\mathbf{10}}^{-\mathbf{4}}\pm \mathbf{1}.\mathbf{659}\times {\mathbf{10}}^{-\mathbf{5}}$	$\mathbf{9}.\mathbf{936}\times {\mathbf{10}}^{-\mathbf{1}}\pm \mathbf{6}.\mathbf{993}\times {\mathbf{10}}^{-\mathbf{4}}$
DeLaN-Friction	Introducing Friction	$6.685\times {10}^{-4}$ ± $1.044\times {10}^{-4}$	$2.568\times {10}^{-4}$ ± $7.258\times {10}^{-5}$	$2.021\times {10}^{-4}$ ± $5.867\times {10}^{-5}$	$9.732\times {10}^{-4}$ ± $4.707\times {10}^{-4}$	$4.413\times {10}^{-4}$ ± $8.665\times {10}^{-5}$	$\mathbf{9}.\mathbf{789}\times {\mathbf{10}}^{-\mathbf{1}}\pm \mathbf{6}.\mathbf{276}\times {\mathbf{10}}^{-\mathbf{3}}$
DeLaN-FFNN	An Augmented DeLaN	$\mathbf{2}.\mathbf{362}\times {\mathbf{10}}^{-\mathbf{4}}\pm \mathbf{4}.\mathbf{264}\times {\mathbf{10}}^{-\mathbf{5}}$	$\mathbf{8}.\mathbf{470}\times {\mathbf{10}}^{-\mathbf{5}}\pm \mathbf{1}.\mathbf{571}\times {\mathbf{10}}^{-\mathbf{5}}$	$\mathbf{6}.\mathbf{816}\times {\mathbf{10}}^{-\mathbf{5}}\pm \mathbf{6}.\mathbf{304}\times {\mathbf{10}}^{-\mathbf{6}}$	$\mathbf{1}.\mathbf{861}\times {\mathbf{10}}^{-\mathbf{4}}\pm \mathbf{3}.\mathbf{278}\times {\mathbf{10}}^{-\mathbf{5}}$	$\mathbf{1}.\mathbf{660}\times {\mathbf{10}}^{-\mathbf{4}}\pm \mathbf{2}.\mathbf{211}\times {\mathbf{10}}^{-\mathbf{5}}$	$\mathbf{9}.\mathbf{939}\times {\mathbf{10}}^{-\mathbf{1}}\pm \mathbf{8}.\mathbf{587}\times {\mathbf{10}}^{-\mathbf{4}}$
Non-iid Test Set		Joint 0	Joint 1	Joint 2	Joint 3	Joint 4	${\mathit{R}}^{\mathbf{2}}$
DeLaN	Structured Lagrangian	$2.728\times {10}^2$ ± $2.981\times {10}^2$	$1.948\times {10}^3$ ± $2.788\times {10}^3$	$3.054\times {10}^3$ ± $3.370\times {10}^3$	$1.779\times {10}^3$ ± $2.983\times {10}^3$	$2.212\times {10}^3$ ± $1.770\times {10}^3$	−
DeLaN	Black-Box Lagrangian	$\mathbf{1}.\mathbf{172}\times {\mathbf{10}}^{-\mathbf{1}}\pm \mathbf{4}.\mathbf{657}\times {\mathbf{10}}^{-\mathbf{2}}$	$\mathbf{1}.\mathbf{006}\times {\mathbf{10}}^{-\mathbf{1}}\pm \mathbf{9}.\mathbf{268}\times {\mathbf{10}}^{-\mathbf{2}}$	$\mathbf{9}.\mathbf{031}\times {\mathbf{10}}^{-\mathbf{2}}\pm \mathbf{7}.\mathbf{375}\times {\mathbf{10}}^{-\mathbf{2}}$	$\mathbf{1}.\mathbf{169}\times {\mathbf{10}}^{-\mathbf{1}}\pm \mathbf{2}.\mathbf{503}\times {\mathbf{10}}^{-\mathbf{2}}$	$\mathbf{1}.\mathbf{105}\times {\mathbf{10}}^{-\mathbf{1}}\pm \mathbf{1}.\mathbf{928}\times {\mathbf{10}}^{-\mathbf{2}}$	−
FF-NN	Feed-Forward Network	$\mathbf{2}.\mathbf{319}\times {\mathbf{10}}^{-\mathbf{1}}\pm \mathbf{2}.\mathbf{948}\times {\mathbf{10}}^{-\mathbf{2}}$	$\mathbf{1}.\mathbf{220}\times {\mathbf{10}}^{-\mathbf{1}}\pm \mathbf{4}.\mathbf{778}\times {\mathbf{10}}^{-\mathbf{2}}$	$\mathbf{7}.\mathbf{723}\times {\mathbf{10}}^{-\mathbf{2}}\pm \mathbf{3}.\mathbf{886}\times {\mathbf{10}}^{-\mathbf{2}}$	$\mathbf{1}.\mathbf{996}\times {\mathbf{10}}^{-\mathbf{1}}\pm \mathbf{5}.\mathbf{962}\times {\mathbf{10}}^{-\mathbf{2}}$	$\mathbf{2}.\mathbf{661}\times {\mathbf{10}}^{-\mathbf{1}}\pm \mathbf{1}.\mathbf{213}\times {\mathbf{10}}^{-\mathbf{1}}$	−
DeLaN-Friction	Introducing Friction	$3.139\times {10}^2$ ± $3.369\times {10}^2$	$1.019\times {10}^3$ ± $1.540\times {10}^3$	$2.755\times {10}^3$ ± $1.507\times {10}^3$	$5.593\times {10}^2$ ± $6.002\times {10}^2$	$3.693\times {10}^3$ ± $4.022\times {10}^3$	−
DeLaN-FFNN	An Augmented DeLaN	$\mathbf{4}.\mathbf{361}\times {\mathbf{10}}^{-\mathbf{1}}\pm \mathbf{6}.\mathbf{790}\times {\mathbf{10}}^{-\mathbf{1}}$	$\mathbf{1}.\mathbf{373}\times {\mathbf{10}}^{-\mathbf{1}}\pm \mathbf{8}.\mathbf{175}\times {\mathbf{10}}^{-\mathbf{2}}$	$1.060\times {10}^{-1}$ ± $7.969\times {10}^{-2}$	$\mathbf{2}.\mathbf{760}\times {\mathbf{10}}^{-\mathbf{1}}\pm \mathbf{2}.\mathbf{681}\times {\mathbf{10}}^{-\mathbf{1}}$	$\mathbf{3}.\mathbf{950}\times {\mathbf{10}}^{-\mathbf{1}}\pm \mathbf{3}.\mathbf{146}\times {\mathbf{10}}^{-\mathbf{1}}$	−

Note: Bold indicates better results, while − represents a coefficient of determination $R^2$ range exceeding 0–1.

summarizes the torque prediction errors for each joint and the overall $R^2$ for the KUKA LWR achieved by each dynamics model. On the iid test set, black-box DeLaN, FF-NN, and DeLaN-FFNN achieve the best accuracy, while structured DeLaN and DeLaN-Friction perform worse. All models score close to 1 for $R^2$, indicating nearly perfect predictions. For non-iid test sets, torque errors for black-box DeLaN, FF-NN, and DeLaN-FFNN range from ${10}^{-1}$ to ${10}^{-2}$, while the errors for structured DeLaN and DeLaN-Friction range from ${10}^2$ to ${10}^3$, demonstrating superior extrapolation for the former models. However, all models score poorly on $R^2$ for the non-iid test sets.

Figure 8 visualizes the overall prediction performance of each model for the two test sets. For the iid test set, structured DeLaN and DeLaN-Friction have slightly higher prediction errors than black-box DeLaN, FF-NN, and DeLaN-FFNN, though all achieve nearly ideal predictions. In the non-iid test sets, all models show some deterioration, with structured DeLaN and DeLaN-Friction degrading significantly more than the others.

Despite the varying degrees of physical constraints, all physics-inspired networks and FF-NN achieve ideal predictions on the iid test set. However, structured DeLaN and DeLaN-Friction are constrained by physical priors, and lack extrapolation ability for different loads. In contrast, DeLaN-FFNN and black-box DeLaN show extrapolation capabilities similar to FF-NN.

4.4. Results Analysis and Discussion

Returning to the initial questions (Section 4), the experimental results show the following:

(1)

Under ideal observation data, structured DeLaN, DeLaN-Friction, and DeLaN-FFNN are capable of learning the underlying structure of physical systems and possess strong extrapolation capabilities. However, black-box DeLaN, which is also a physics-inspired network, cannot learn the underlying structure of the system from the data due to its lack of learning ability for the mass matrix $H\left(q\right)$, kinetic energy $T(q,\dot{q})$, and potential energy $V\left(q\right)$.

(2)

When applied to real systems, structured DeLaN and DeLaN-Friction show weaker extrapolation than black-box DeLaN, with DeLaN-FFNN slightly behind the black-box model. The failure of physics-inspired networks to capture all dynamics phenomena indicates that excessive physical constraints can limit extrapolation abilities. Improved extrapolation in physics-inspired networks requires more comprehensive capture of the real system dynamics.

(3)

In order for physics-inspired networks to achieve better extrapolation on real systems, they rely on two key factors: high-fidelity dynamics data that accurately reflects system phenomena, and the integration of comprehensive dynamics priors. These priors should transparently capture the real system dynamics instead of using black-box methods such as the uncertain forces in DeLaN-FFNN.

For Result 1, it is evident that physics-inspired deep networks are a promising direction in robotics, offering more accurate and interpretable models. These networks enhance physical plausibility and interpretability, resulting in more precise dynamics models and better extrapolation in rigid systems under ideal conditions. Similar experimental results can be found in the following papers: [1,7,8,10,11,12,13].

For Result 2, although the black-box networks achieved better results in real-world scenarios with uncertainties due to their lack of prior constraints, they can approximate any mapping relationship without restrictions, even if the relationship is spurious. Therefore, this result does not prove that they have good extrapolation capabilities; rather, it merely indicates that when the physical priors integrated into physics-inspired networks fail to capture all the dynamics phenomena of real systems, this hinders their extrapolation performance.

For Result 3, two key factors are highlighted for better extrapolation in physics-inspired networks: high-fidelity dynamics data, and comprehensive dynamics priors. For high-fidelity dynamics data, the challenge in real systems stems from often unobserved or unknown generalized coordinates q, $\dot{q}$, compounded by the difficulty of inferring complete system states from partial observations, making high-fidelity data acquisition difficult. This issue does not affect black-box models that do not depend on specific observations [7]. Additionally, in experiments, inverse dynamics modeling requires the state derivative (angular accelerations), which cannot be directly observed. This extra computation step, combined with measurement noise, introduces noisy spikes in the computed accelerations, limiting the learning capabilities of physics-inspired networks [28]. For incorporating more comprehensive dynamics priors, in structured DeLaN theory, non-conservative forces are assumed to be fully known and equal to the actuation forces directly acting on the Lagrangian coordinates q. However, this approach lacks an energy dissipation mechanism [8]. For DeLaN-Friction, the findings reveal that merely adding friction is inadequate. For DeLaN-FFNN, introducing uncertain forces through a black-box approach improves performance but sacrifices interpretability. Current attempts to embed contact forces into physics-inspired networks often target specific scenarios with restrictive assumptions, and consequently lack widespread applicability [13,29]. Several recent works have introduced an energy dissipation mechanism by incorporating the learning of a dissipation matrix in physics-inspired networks, resulting in improved modeling performance. However, their extrapolation capabilities remain to be demonstrated [8,30]. Therefore, improving the extrapolation of physics-inspired networks in real systems continues to pose significant challenges.

5. Conclusions

This work introduces four physics-inspired networks and an experimental design to evaluate the extrapolation capabilities of data-driven models in robotics inverse dynamics modeling. Experiments were conducted to assess these networks’ extrapolation ability from simulated to real systems. The results indicate that physics-inspired networks can recover the system’s underlying structure from ideal simulated data, demonstrating strong extrapolation capabilities. It is evident that physics-inspired deep networks are a promising direction in robotics, offering more accurate and interpretable models. However, in real systems these networks often fall short compared to black-box models due to insufficient physical priors and their inability to capture all dynamics phenomena, highlighting the need for further improvements. To improve extrapolation in real systems, high-fidelity data and more comprehensive physical priors are needed. Future studies should focus on these areas in order to enhance the performance of physics-inspired networks.