🤖 machine learning

Learning Forced Multibody Dynamics on Lie Groups

This paper proposes a geometric deep learning architecture that utilizes discrete forced Euler-Lagrange equations on Lie groups to learn mechanical system dynamics from position-only data, thereby preserving inherent geometric structures and conservation laws while accommodating multibody systems and external controls.

Original authors: Martine Dyring Hansen, Marta Ghirardelli, Elena Celledoni, David Martin de Diego, Brynjulf Owren

Published 2026-07-15

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Martine Dyring Hansen, Marta Ghirardelli, Elena Celledoni, David Martin de Diego, Brynjulf Owren

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to teach a robot how to dance. You have a video of the dance, but you only see the robot's pose (where its body parts are) every few frames. You don't have a speedometer, and you certainly don't have a map of the invisible forces pushing the robot around. Your goal is to build a computer program that can predict the next move, and then the one after that, forever, without the robot ever falling over or glitching out.

This is exactly the challenge tackled by a team of researchers who propose a new way to learn how mechanical systems move. They call their creation LieDFLNN.

The Problem: The "Gimbal Lock" Trap

Most computer models try to describe rotation using a set of three numbers, like Euler angles (think of them as "yaw, pitch, and roll"). It's like trying to describe a location on Earth using only latitude and longitude. It works fine until you get to the North Pole. Suddenly, the math breaks down. In the world of robotics, this is called gimbal lock.

If a robot spins 360 degrees, a model using Euler angles might suddenly jump from "179 degrees" to "-179 degrees" just to keep the numbers small. To the computer, the robot didn't spin smoothly; it teleported. This causes the learning model to get confused, trying to fit a smooth dance to a jagged, broken map.

The paper explicitly argues against using these standard "flat" maps (Euclidean spaces) or Euler angles for systems that spin or move in complex ways. They show that forcing rotation data into these flat coordinates introduces artificial jumps and errors that get worse the more the object rotates.

The Solution: Dancing on a Sphere

Instead of flattening the world, the authors suggest we keep the world curved, just like it is. They treat the robot's orientation not as three numbers, but as a rotation matrix—a special grid of numbers that always stays "perfectly orthogonal" (like the corners of a perfect box).

Think of it like this: If you are walking on a flat floor, you can just use a ruler to measure distance. But if you are walking on the surface of a giant beach ball, a ruler doesn't work well; you need to follow the curve of the ball. The authors' method lives directly on that "beach ball" (mathematically known as a Lie group, specifically SO(3) for rotation and SE(3) for rotation plus movement).

By doing this, the model never has to worry about "teleporting" or hitting a singularity. It stays on the surface of the sphere, just like the real robot does.

The Magic Trick: Learning Without Speed

Usually, to figure out how something moves, you need to know its speed (velocity). But in the real world, sensors often only give you positions (where things are), and calculating speed from position is messy and noisy.

The authors use a clever trick from physics called Discrete Forced Euler-Lagrange equations.

The Analogy: Imagine you are watching a movie of a ball rolling down a hill. You don't need to know the ball's exact speed at every millisecond to predict where it will be next. You just need to know where it was two frames ago and where it is now. The laws of physics (specifically, the principle of "least action") connect these dots.
The Method: Their model learns a "Discrete Lagrangian." Think of this as a secret rulebook that tells the system how to jump from one pose to the next based only on the previous two poses. It doesn't need to calculate speed explicitly. It just looks at the pattern of positions and learns the "force" that connects them.

What They Proved (and What They Didn't)

The researchers didn't just guess this would work; they tested it with simulations and real data.

The "Big Spin" Test: They simulated a rigid body spinning. When they used the old Euler angle method, the model failed miserably as the spin got larger, producing jagged, wrong paths. When they used their new Lie group method, the model stayed accurate even for huge, 360-degree spins. The error stayed low and steady, while the old method's error exploded.
The "Long Haul" Test: They tried to predict the future of a spinning object for 500 steps (much longer than the training data). The new method stayed on the "sphere" (the correct mathematical shape) the whole time. The old methods, even if they started right, slowly drifted off the sphere, becoming physically impossible (like a rotation matrix that isn't a perfect square anymore).
The "Human Cartwheel" Test: They took real data from a human doing a cartwheel (recorded at 120 Hz). This is a complex system with many moving parts (a "multibody system"). They trained the model on just two recordings of the same person doing the same move. The model learned to reproduce the motion with high fidelity. The error in their prediction was actually smaller than the natural difference between the two real recordings (meaning the model was as good as the human's own consistency).
The "Secret Controller" Test: In one experiment, they knew the physics of the robot (its weight and shape) but didn't know the "brain" (the control signal) telling it what to do. They fed the model only the position data. The model successfully figured out the hidden control signal, proving it can separate the "body" from the "brain."

The Bottom Line

The paper suggests that by respecting the curved geometry of rotation (using Lie groups) and using a discrete, step-by-step physics rulebook, we can build learning models that are:

More accurate for large rotations.
More stable over long periods (they don't drift).
Able to learn from position-only data, ignoring the noisy speed measurements.

They didn't claim this solves every problem in robotics, but they showed that for systems that spin and move like rigid bodies (drones, robots, human limbs), this approach is significantly better than the standard "flat" methods. It's a way of teaching computers to understand the world as it actually is: curved, connected, and continuous.

Technical Summary: Learning Forced Multibody Dynamics on Lie Groups

Problem Statement
Learning dynamical systems from observational data often involves states evolving on nonlinear configuration spaces, such as the rotation group $SO(3)$ or the special Euclidean group $SE(3)$. Existing approaches frequently model these systems in Euclidean embeddings (e.g., using Euler angles or generalized coordinates), which introduces coordinate singularities (such as gimbal lock), discontinuities, and representation ambiguities. Furthermore, standard learning methods often require velocity data, which may be unavailable, noisy, or difficult to obtain from discrete position measurements. Additionally, models that do not inherently respect the manifold structure of the configuration space can suffer from geometric drift, where predicted trajectories violate physical constraints (e.g., orthogonality of rotation matrices) over time.

Methodology
The paper proposes LieDFLNN (Lie Group Discrete Forced Euler-Lagrange Neural Network), an architecture that learns the dynamics of mechanical systems directly on Lie groups using only position data. The methodology is grounded in discrete variational mechanics:

Discrete Forced Euler-Lagrange Equations: The core of the approach utilizes the discrete forced Euler-Lagrange equations on Lie groups. Instead of approximating continuous-time derivatives, the method formulates dynamics using discrete Lagrangians $L_d: G \times G \to \mathbb{R}$ and discrete external forces. This formulation relies solely on pairs of consecutive configurations $(g_k, g_{k+1})$ and their group increments $W_k = g_k^{-1}g_{k+1}$ , eliminating the need for explicit velocity measurements.
Trivialization: The equations are expressed using left-trivialization, mapping the tangent bundle $TG$ to $G \times \mathfrak{g}$ (where $\mathfrak{g}$ is the Lie algebra). This allows the dynamics to be computed entirely in terms of group elements and algebra elements, preserving the geometric structure.
Neural Parameterization: The discrete Lagrangian and external forces are parameterized by neural networks ( $L_\theta$ $L_{θ}$ and $F_\theta$ $F_{θ}$ ). The architecture supports:
- Rigid Bodies: Specific formulations for $SO(3)$ and $SE(3)$, including inertia matrices and potential energies.
- Multibody Systems: Extension to product manifolds (e.g., $SE(3) \times SO(3)^{N}$ ) to model articulated systems like human skeletons.
- Control Inputs: The framework can incorporate unknown control laws (e.g., feedback controllers) as part of the external force term.
Regularization: To prevent degenerate solutions (e.g., constant Lagrangians), a regularization term is added to the loss function. This term enforces the regularity of the discrete Lagrangian by penalizing Hessians that approach singularity, ensuring the system remains well-posed.
Inference: Prediction is performed by iteratively solving the discrete forced Euler-Lagrange equations for the group increments, ensuring that the generated trajectory remains strictly on the configuration manifold without ad-hoc projection steps.

Key Contributions

Manifold-Consistent Learning: The method operates directly on group-valued observations, guaranteeing that learned states remain on the configuration manifold (e.g., $SO(n)$, $SE(n)$), thereby avoiding geometric drift and the need for re-orthogonalization.
Velocity-Free Dynamics: By utilizing the discrete variational formulation, the approach recovers physical structure from position measurements alone, avoiding the noise and complexity associated with numerical differentiation.
Modularity and Flexibility: The framework is modular, capable of learning full dynamics (Lagrangian and forces) jointly or isolating specific components (e.g., identifying unknown control laws when the mechanical properties are known).
Generalization to Multibody Systems: The approach naturally extends to complex multibody systems with coupled interactions, handling product manifolds of $SO(3)$ and $SE(3)$.

Experimental Results
The authors validate the approach through five experimental settings:

Coordinate Representation: On $SO(3)$, LieDFLNN significantly outperforms Euler-angle-based baselines, particularly for large rotations where Euler angles suffer from singularities and discontinuities. The Lie group representation maintains smooth, low-error trajectories across all rotation magnitudes.
Discrete vs. Continuous: The discrete variational formulation demonstrates superior data efficiency and accuracy compared to continuous-time baselines (including those using Lie group integrators). The discrete model achieves high accuracy even with very short training windows (3 consecutive poses), whereas continuous models require longer trajectories to improve and still lag in performance.
Manifold Constraints: When applied to $SE(3)$, the proposed model maintains geometric consistency (orthogonality and unit determinant) by construction. In contrast, baselines that treat matrix entries as independent Euclidean coordinates (even those with variational structures) accumulate drift and violate physical constraints over long rollouts.
Real-World Multibody Systems: Applied to human cartwheel motion capture data (a product manifold of $SE(3)$ and $SO(3)$), the model successfully reproduces complex, high-dimensional coupled dynamics with large rotations. The prediction errors remain below the inherent variability between repeated ground-truth recordings.
Control Identification: In a setting where the Lagrangian is known, the framework successfully identifies an unknown PD control torque from a single trajectory of position data alone, accurately recovering the control signal and stabilizing the system.

Significance and Claims
The paper claims that embedding the geometric structure of Lie groups directly into the variational integrator eliminates the coordinate singularities and geometric drift common in Euclidean baselines. The authors assert that the discrete variational structure is the key to learning effectively from discrete observations, as it allows the model to extract maximal dynamical information from minimal temporal context. The framework is presented as a robust tool for system identification in settings where velocity data is unavailable or unreliable, offering improved physical consistency, generalization, and long-term predictive accuracy, particularly for systems involving large rotations or complex multibody interactions. The work aligns with the broader effort in geometric deep learning to incorporate group and manifold structures into learning algorithms, specifically focusing on mechanical systems via discrete variational formulations.