Reasoning about the physical world requires models that are endowed with the right inductive biases to learn the underlying dynamics. Recent works improve generalization for predicting trajectories by learning the Hamiltonian or Lagrangian of a system rather than the differential equations directly. While these methods encode the constraints of the systems using generalized coordinates, we show that embedding the system into Cartesian coordinates and enforcing the constraints explicitly with Lagrange multipliers dramatically simplifies the learning problem. We introduce a series of challenging chaotic and extended-body systems, including systems with $N$-pendulums, spring coupling, magnetic fields, rigid rotors, and gyroscopes, to push the limits of current approaches. Our experiments show that Cartesian coordinates with explicit constraints lead to a 100x improvement in accuracy and data efficiency.

This work proposes PH-based metrics for benchmarking impedance control. A causality-consistent PH model is introduced for mass-spring-damper impedance in Cartesian space. Based on this model, a differentiable, force-torque sensing-independent, n-DoF passivity condition is derived, valid for time-varying references. An impedance fidelity metric is also defined from step-response power in free motion, capturing dynamic decoupling. The proposed metrics are validated in Gazebo simulations with a six-DoF manipulator and a quadruped leg. Results demonstrate the suitability of the PH framework for standardized impedance control benchmarking.

The introduction is divided into two parts. Section 1.1 attempts to succinctly describe ways in

For the first argument, we use induction. For the second part, we it is essentially a Coupon Collector's problem. The colors represent the target environment. The environment is shown in Figure 6. The results are shown in Figure 5. Forward} to reach the target grid (green).

Reasoning about the physical world requires models that are endowed with the right inductive biases to learn the underlying dynamics. Recent works improve generalization for predicting trajectories by learning the Hamiltonian or Lagrangian of a system rather than the differential equations directly. While these methods encode the constraints of the systems using generalized coordinates, we show that embedding the system into Cartesian coordinates and enforcing the constraints explicitly with Lagrange multipliers dramatically simplifies the learning problem. We introduce a series of challenging chaotic and extended-body systems, including systems with N -pendulums, spring coupling, magnetic fields, rigid rotors, and gyroscopes, to push the limits of current approaches. Our experiments show that Cartesian coordinates with explicit constraints lead to a 100x improvement in accuracy and data efficiency. Figure 1: By using Cartesian coordinates with explicit constraints, we simplify the Hamiltonians and La-grangians that our models learn, resulting in better long term predictions and data-efficiency than Neural ODEs and Hamiltonian Neural Networks (HNNs).

SymODEN, SRNN and LieConv, only DeLaN was tested on real world mechanical data.

Furthermore, this method can be applied to any network by replacing an existing layer.

We briefly explain the theoretical formulation highlighting the similarities and differences in the inductive biases and graph architecture of these systems.