Learning Articulated Rigid Body Dynamics with Lagrangian Graph Neural Network
Authors: Ravinder Bhattoo
Sayan Ranu
N M Anoop Krishnan
Venue: NeurIPS 2022 Accept (2022)
Keywords: Rigid body
dynamical systems
Lagrangian neural network
graph neural network

Abstract: Lagrangian  and Hamiltonian neural networks LNN and HNNs, respectively) encode strong inductive biases that allow them to outperform other models of physical systems significantly. However, these models have, thus far, mostly been limited to simple systems such as pendulums and springs or a single rigid body such as a gyroscope or a rigid rotor. Here, we present a Lagrangian graph neural network (LGNN) that can learn the dynamics of articulated rigid bodies by exploiting their topology. We demonstrate the performance of LGNN by learning the dynamics of ropes, chains, and trusses with the bars modeled as rigid bodies. LGNN also exhibits generalizability---LGNN trained on chains with a few segments exhibits generalizability to simulate a chain with large number of links and arbitrary link length. We also show that the LGNN can simulate unseen hybrid systems including bars and chains, on which they have not been trained on. Specifically, we show that the LGNN can be used to model the dynamics of complex real-world structures such as the stability of tensegrity structures. Finally, we discuss the non-diagonal nature of the mass matrix and its ability to generalize in complex systems.

Learning Articulated Rigid Body Dynamies with Lagrangian Graph Neural Network
Ravinder Bhattoo
Sayan Ranu
Department of Civil Engineering Indian Institute of Technology Delhi cez177518@i acc ii
Department of Computer Science and Engineering Yardi School of Artificial Intelligence
Indian Institute of Technology Delhi sayanranu@i iummiiiiii itd at..a...........................
N. M. Anoop Krishnan Department of Civil Engineering Yardi School of Artificial Intelligence Indian Institute of Technology Delhi krishnan @iitd. ac.
Abstract
Lagrangian and Hamiltonian neural networks (LNNS and HNNs, respectively) encode strong inductive biases that allow them outperform other models of physical systems significantly. However, these models have, thus far, mostly been limited to simple systems such as pendulums and springs or a single rigid body such a gyroscope
rigid rotor. Here, we present Lagrangian graph neural network (L.GNN) that can learn the dynamics articulated rigid bodies by exploiting their topology. We demonstrate the performance of L.GNN by learning the dynamics of ropes, chains, and trusses with the bars modeled as rigid bodies. LGNN also exhibits ceneralizability-ooto
trained chains with few segments exhibits generalizability to simulate a chain with large number of links and arbitrary link length. We also show that the LGNN can simulate unseen hybrid systems including bars and chains, on which they have not been trained on. Specifically, we show that the LGNN be used to model the dynamics of complex real-world structures such as the stability of tensegrity structures. Finally, we discuss the -iagonal nature of the mass matrix and its ability generalize in complex systems.
Introduction and Related Works
Movements of robotic arm, rolling ball, falling chain be characterized by rigid body motion [1, Understanding the dynamics of the motion crucial in several applications including robotics, human-robot interaction, planning, and computer graphics [3, 1]. Traditionally, the rigid body mechanics is studied in the framework of classical mechanics, which relies on either force- based or energy-based approaches [4]. Force-based approaches involve the computation of all the unknown forces based on the equations of equilibrium and hence is cumbersome for large structures. Energy-based approaches present an elegant formalism which involve the computation of scalar quantity representing the state of system. namely, Lagrangian (C = T - V). which is the difference between the kinetic (g(g. i)) and potential (V(g)) energies, or Hamiltonian (Hl T + V). which represents the total energy of the system. This scalar quantity can, in turn, be used to predict the dynamies of the system. However, the functional form governing this scalar quantity may not be
The code is available at https//gtthbb.cmm//BBB--D-D htt.e//iiiiiiddooyyyyyyymeee Sally_ddaamiss_graae
36th Conference on Neural Information Processing Systems (NcurIPS 2022).

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known a priori in many cases [5]. Thus, learning the dynamics of rigid bodies directly from the trajectory can simplify and accelerate the modeling of these systems [5. 6, 7. 8].
Learning the dynamics of particles has received much attention recently using physics-informed approaches [9]. Among these, Lagrangian neural networks (LNNS) and Hamiltonian neural networks (HNNs) are two physics-ifformed neural networks with strong inductive biases that outperform other learning paradigms of dynamical systems [10, 11, 12, 8. 6, 13, 7. 14]. In this approach, neural network is trained to learn the C (or H) of system based on its configuration (g, q). The L is then used along with the Eager-Lagrange (EL) equation to obtain the time evolution of the system. Note that the training of LNNS is performed by minimizing the error on the predicted trajectory with respect to the actual trajectory. Thus, LNNS can effectively learn the Lagrangian directly from the trajectory of multi parlti- system [6, 13].
Most of the works on LNN has focused on relatively simpler particle-based systems such as springs and pendulums [15, 16, 6, 13, 7, 10, 17J. This approach models a rigid body, for instance a ball, as a particle and predicts the dynamies. This approach thus ignores the additional rotational degrees of freedom of the body due to its finite volume. Specifically, while a particle in 3D has three degrees of freedom (translational), rigid body in 3D has six degrees of freedom (translational and rotational). Thus, the dynamics and energetics associated with these degrees of motions are lost by modeling a rigid body as a particle. To the best of authors' knowledge, thus far, only one work has attempted to learn rigid body dynamics using LNNS and HNNs, where it was demonstrated the dynamics of simple rigid bodies such as a gyroscope or rotating rotor can be learned [13]. However, the LNNS used in this work, owing to their fully connected MLP architecture, are transductive in nature. An LNN trained on a dlouble-pendulum system or 3-spring system can used only for the same system and does not generalize to a different system size such as 3- 3-pendulum or 5-spring, respectively. In realistic situations the number of particles in a system can vary arbitrarily, and accordingly, a large number of trained models might be required to model these systems.
An alternate approach model these systems would be use graph neural network (GNN) [18, 19, 5, 15, 16), which, once trained, can generalize to arbitrary system sizes. GNNS have been widely used to model physical and atomic systems extensively due to their inductive bias [20, 21, 22, 15, 16]. GNNS have also been used to model rigid bodies mainly following two approaches, namely, particle- based [19] and lumped mass [22, 23] methods. In the first approach, a rigid body is discretized into finite number of particles and the motion of the individual particles are learned to predict the dynamics of rigid body [19]. Note that this approach is philosophically similar to mess-less methods such as aoooothed-particle hydrodynamies (SPH) [24] or peridynamics (PD) [25], where the time-evolution of a continuum body is simulated by discretizing the domain using particles. This approach [19], although useful, have several limitations, namely, it does not (i) conserve physical quantities such as energy when simulated over a long duration, and (ii) generalize to a different timestep of forward simulation than the on which it is trained. In the second approach, rigid body is modeled as lumped mass [22, 26). the dynamics of which is learned by assuming this lumped mass as a particle. For instance, the dynamics of chain is modeled by discretizing the chain to smaller segments and modeling each segment as a lumped mass. As mentioned earlier, this approach leads to the loss of additional degrees of freedom that are associated with rigid body.
Here, we present a Lagrangian graph neural network (L.GNN) framework that can learn the dynamics of rigid bodies. Specifically, exploiting the topology of a physical system, we show that rigid body can be modeled as a graph. Further, the Lagrangian of the graph structure can be learned directly by minimizing the loss on the predicted trajectory with respect to the actual trajectory of the system. The major contributions of the work are as follows.
Topology aware modeling of rigid body. We present graph-basee based model for articulated rigid bodies such in-extensible ropes, chains, trusses. Further, we demonstrate using LGNN that the dynamics of these systems can be learned in the Lagrangian framework.
Generalizability to arbitrary system sizes. We show that L.GNN can generalize to arbitrary system sizes once trained.
Generalizability complex unseen topology. We demonstrate that the LGNN can generalize to unseen topology, that is, links with varying lengths, a combination of truss and chain structures, and different boundary conditions.
Altogether, we demonstrate that L.GNN can be a strong framework for simulating the dynamics of articulated rigid bodies.
2

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2 Dynamics of Rigid Bodies
The dynamics of physical system can be represented as q= F(q. q), where q.4 R is function of time (t) for system with D degrees of freedom. The future states or trajectory of the system can be predicted by integrating these equations to obtain q(t + 1) and so While there are several physics-based methods for generating the dymamics of the system such as d' Alembettt's principle, Newtonian, Lagrangian, Hamiltonian approaches, all these approaches result in the equivalent sets of equations [3].
The two broad paradigms for modeling the dynamics involve force- and energy-based approaches. Energy-based approaches is an elegant framework, which relies on the computation a single scalar quantity, for instance energy, that represents the state of system. The dynamics of the system is, in turn, computed based on this scalar quantity. Among the mnergy-based approaches, Lagrangian formulation has been widely used to predict the dynamics of particles and rigid bodies by computing the Lagrangian C of the system. The standard form of Lagrange's 's equation for a system with
O, and the Lagrangian is C(g.q.t) T(g. t) V(q, t) with T(g. q.t) and V'(g.t) t) representing the total kinetic energy of the system and the potential function from which generalized forces can be derived. Accordingly, the dynamics of
holonomic constraints is given by d
(e
de
84
the system can be represented using EL equations q.
8 30
ac 3qg
aC 0aic0..
9t
Modified Euler-Lagrange Equation. A modified version of the EL can be used in cases where some of the terms involved in the equation can be decoupled. This formulation allows explicit incorporation of constraints (holonomic and Pfaffian) and additional dissipative terms for friction or drag [3, 1J. In be crucial in applications such multi-fingerrd grasping
rigid body motion, Pfaffian constraints where, the velocity of two or more fingers are constrained so that the combined geometry formed is able to catch or hold object. A generic expression of constraints for these systems that accounts for both holonomic and Pfaffian can be A(q) = O, where, A(q) E R represents k velocity constraints. In addition, drag, friction other dissipative terms of system can be expressed as an additional forcing term in the EL equation. It is worth noting that EL equation, by nature, is energy conserving. Hence, the additional dissipative terms are crucial for modeling realistic systems with friction and drag. If these terms are not included, the system will essentially try to simulate an energy preserving trajectory, thereby resulting in huge errors in the dynamics [17].
Considering the additional forces mentioned above, the modified EL equation can be written as:
dt V
(1)
where A forms non-normalized basis for the constraint forces, E IR* known as the Lagrange multipliers, gives the relative magnitudes of these force constraints, Y represents the non-conservative forces, such as friction or drag, which are not directly derivable from a potential, and F represents any external forces acting on the system. This equation can be modified to obtain as:
q M- +1+++1++++ A (Q)A++)
where M 8 ac represents the mass matrix, C = a ac represents Coriolis-like forces, and aq 0j 3c represents the conservative forces derivable from a potential. Differentiating the constraint By equation gives A(q)ă + Á(q)4 = 0. Solving À (see A.2) and substituting in Eq, 2, we obtain q as
II
q ÿ= 8= (1- -I- CQ
(AAM-1AA)- (I CQ+ F) + Ag) + +F)
(3)
For system subjected to these forces, the dynamics can be learned using LNN by minimizing the loss on the predicted and observed trajectory, where the predicted acceleration / is obtained using the Equation 3. It is worth noting that in this equation, M, C, and II can be directly derived from the L. Constraints on the systems are generally known as they generally form part of the topology. worth noting that there are some recent works that focus on learning constraints as well [8].
3

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3 Lagrangian Mechanics for Articulated Rigid Bodies
In the case of particle systems such as spring or pendulum systems, the approach mentioned in Sec.2 can be directly used in conjunction with an LNN to learn the dynamics. In this case, the mass matrix (q) remains constant with only diagonal entries mas in Cartesian coordinates. Inducing this as a prior knowledge, wherein the masses are parameterized as a diagonal matrix is shown to simplify the learning process [13]. However, in the case of an articulated rigid body, the mass matrix is non-diagonal in the Cartesian coordinates. Further, the kinetic energy term 7 becomes function of both position and velocity. In other words, the kinetic energy also becomes a function of the topology. This makes learning the dynamics a complex problem especially in real-world complex structures such as trusses or tensegrities, which are a combination of bars, ropes, and chains.
To this extent, we briefly review the mechanics of falling rope or chain as an example. Note that simple rigid bodies such as a gyroscope or rotating rotor has already been studied using LNNs [13]. Of our special interest are articulated rigid bodies that can be arbitrarily large such as chains, ropes or trusses, that can be divided into smaller constituent members. This is because, it generally assumed that extending LNNS to large structures a challenging problem [17]. Traditionally, the mechanics of chains or ropes are modeled using discrete models [2]. Figure shows discrete model of rope of mass M and length L. The rope is discretized into n cylindrical rods or segments each having mass m, .M/ /n and length 4 L/n. These segments are considered to be rigid, and with a finite uniform cross sectionltiinna area and volume. In order to replicate realistic dynamics of rope, the 4 should be significantly smaller than L. Note that in the case of chain or truss, such artificial discretization is not required and the bars associated with each segment can be directly considered as a rigid body.
(h. Ju
(4., yd
mg
mg
D6. Vn)
+mg
Di. ya
+mg
Figure Chain considering rigid body dynamics. Spheres repre- sent the connections and bars rep- resent the links of the chain.
To formulate the C, the generalized coordinates with orientation of each link represented by Φ; tan-1
y/- W can be considered. Placing the origin at the beginning of first segment (see Figure x x--1-
1), the center of mass of 7th segment (x.Tm, y)") can be written in terms of generalized coordinates as
1 cos di, 23 Σ 1 cos oj 5
1 2 sisin sin di
yi
Σ
1 sin Φs
(4)
j=1
Accordingly, the kinetic energy of the system is given by [2]
1 2
T
(mi(sf,cm + (...o))) + T:db?)
(5)
where 1; mlll represents the moment of inertia of the rigid segment i, Similarly, the potential energy of the system can be expressed as:
V
magy
cm
where g represents the acceleration due to gravity. Finally, the Lagrangian of the system can be obtained as L T V. which can be substituted in the EL equation to obtain the dynamics of the rigid body.
To learn the dynamics of an articulated rigid body, we employ the approach shown in Figure Specifically, we model a physical system as graph. Further, the Lagrangian of system is learned by decoupling the potential and kinetic energy, each which are learned by two GNNS, namely, Gy and GT. Finally, the Lagrangian is computed as L T - V. This framework is trained end-to-end based by minimizing the loss on the acceleration predicted by the L.GNN using EL equation with respect to the ground truth. In this section, we describe the L.NN architecture for rigid bodies in detail (See

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Node unneson
Massagee pass ng GMN
Summation of potertisihimetic
(g.Ap-A)
%
Lagranglan (3 9))
Node feadures
Embedsing
Emandding
TM
2 5
18.0%
Inttal Edge
oquction to updet (4-9)
Eddge fundtiin
potentiahineticc energy
Figure 2: (a) Learning articulated rigid body dynamics with L.GNN (b) Architecture of LGNNN for rigid bodies.
Figure 2 for overview). We empirically show that the dynamics of rigid body be learned by LGNN. In addition, due to the inductive nature of the graph architecture, once trained on a small system, LGNN can generalize to arbitrary system sizes and topology.
Graph structure. Figure shows a chain. The (undirected) graph of the physical system is constructed by considering the bars/segments of the chain the edges and the connections nodes. Here, edges represent the rigid bodies and nodes represent the connection between these rigid bodies. This is in contrast earlier approaches used for particle-based systems, where node represented the particle position and edge represented the connections between them. Hereon, we use the notation g(u, E) to to represent the graph representation of a rigid body with U and its node and edge sets.
Overview of the architecture. As shown in Figure 2, we use two GNNs; one to predict the potential energies and the other to predict kinetic energies. From these predictions the Lagrangian is computed. The error the Lagrangian is minimized through an RMSE loss function jointly train both both the GNNS, shown in Figure 2, are identical. Note that the specific graph architecture used in the present work is inspired from previous works L. LGNNNS for particle-base based systems [15, 16].
the GNNS. The architecture
Input features. Each node u, E U is characterized by position yi = (xi, Mrs =:), and velocity (.)) Each edge eij is characterized by its type lij. and the relative differences in the positions (Aqiz = q: 9j) its connecting nodes, and wij = Agi x Aijj- The type lij a discrete variable and is useful in distinguishing edges of different characteristies within system (Ex. moment inertia or area of cross section of the edge). Note that the velocity of a rigid body represented by an edge is a function of the velocities of its end points in two and three dimensional spaces. Hence, do not explicitly track edge velocities.
Pre-Processing. the pre-processing layer, we construct dense vector representation for each node E ll and edge eij & using MLPs (multi-layer perceptrons). The exact operation for potential energy is provided below in Eqs.7-8. For kinetic energy, we input ÿ in Eq 7 instead of gi and wiy in Eq. 8 instead of Agj--
h = squareplus
0
squareplus (MLpppllusuMMPPPMMMP (MLP (MLP (one- hot,t),, Aq;))
(8)
squareplus is an activation function.
Message passing. To infuse structural information in the edge and node embeddings, perform L layers of message passing, wherein the embedding each layer / [1, L] computed as follows:
I+1 h
squareplus (MLP (hl, + w (b'llh())
(9)
Here, W! is layer-specific learnable weight vector and ll represents concatenation operation. The node embeddings in a given layer I are learned as follows:
W b/y ))
h ++ squareplus
MLP
h +
Σ
(10)
JEN:
5

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Figure 3: (Topologies of the n-segment chain/rope structures with n = (a) 4, (b) 8, and (c) 16. Topologies of complex tensegrity structures (d) T1, (e) T2, and (f) T3 made of chains and rods. Note that the 4-link system in (a) is used train the LGNN and the trained model is used for inferring the dynamics on all other systems.
Here, .M. = {uj (us, uy) E } denotes the edges incident on node U₁. Similar to W W is layer-specific learnable weight vector, which performs linear transformation the embedding of each incident edge. Following L layers of message passing, the final node and edge representations the th layer denoted by z; h and Zij h respectively.
Potential and kinetic energy prediction. The predicted potential energy of each edge (rigid body) is computed by passing its final layer embedding through MLP, i.e., Vij MLP(Z.,)) The global predicted potential energy of the rigid body system is therefore the sum of the individual energies, i..., V- = Eve, EE Fiy. For kinetic energy, the computation is identical except that it occurs in the other GNN with parameters optimized for kinetic energy.
Loss function. The predicted Lagrangian is simply the difference between the predicted kinetic energy and the potential energy. Using Fuler-Lagrange equations, we obtain the predicted acceleration (t) for each node u, The ground truth acceleration is computed directly from the ground truth trajectory using the Verlet algorithm as:
1
(t)
(it) (q:(t At) + (t - At) - (t)
(11)
The parameters the GNNS are trained to minimize the RMSE loss over the entire trajectory T:
1 U
Σ
a.(0) ()))))
(12)
VuLEM
Since the integration of the equations of motion for the predicted trajectory is also performed using the same algorithm as: q(t At) = 2q(() - q(t At) + ((At), this method is equivalent to training from trajectory/positions..
4 Empirical Evaluation
In this section, we evaluate the ability of LGNN to learn rigid body dynamics. In addition, we evaluate the ability of LGNN generalize larger unseen system sizes, complex topology, and realistic structures such as tensegrity.
4.1 Experimental setup
Simulation environment. All the training and forward simulations are carried out in the JAX environment [21]. The graph architecture is implemented using the jraph package [27]. All
the codes related to dataset generation and training are available in https///gitubb.oo//MBGGG IITD/rigid sol__dynmmiccs.
Software packages: numpy-1.20.3, jax-0.2.24, max-mm..01.20. jaxlib-O.1.73, jraph-O.0.1.dewv Hardware: Memory: 16GiB System memory, Processor: Intek(R) Core(TM) 17-10750H CPU @ 2.60GHz
Baselines. As outlined earlier, there are very few works on rigid body simulations using graph-based approaches, where the graph is used to model the topology of the rigid body. To compare the

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performance of LGNN, we employ three baselines, namely, (i) graph network simulator GNS, (ii) Lagrangian graph network (LGN), and (iii) constrained Lagrangian neural network (CLNN). GNS employs full graph network architecture [5, 12, 19] to predict the update in the position and velocity of node based on the present position and velocity. GNS has been shown to be versatile model with the capability to simulate wide range of physical systems [19]. LGN and CLNN employs the exact same equations as LGNN for computing the acceleration and trajectory and hence has the same inductive biases LGNN in terms of the training and inference. However, while LGN employs a full graph network, CLNN employs a feed-forward multilayer perceptron. Details of the architectures and the hyperparameters of the baselines are provided in the Appendix A.5 and Appendix A.6, respectively.
Datasets and systems. To evaluate the performance LGNN, we selected n-chain/rope systems, where n = (4, 8, 16). All the graph based models are trained only on 4-segment chain system, which are then evaluated on other system sizes. Further, to evaluate the zero-shot generalizability of L.GNN to large-scale unseen systems, we simulate 8-, and 16-segment chain systems. Further, to push the limits of LGNN, we evaluate the model trained 4-segment chain on 100-link system, and to complex shaped topologies involving truss members (long rigid members) and chains (short rigid members), which have more than 40 segments (see Figure 3). The mass m, and moment of inertia I, of all the members are maintained to be the same for all the segments irrespective of their length. a 4-li
To evaluate the generalizability to realistic systems, we also evaluate the performance system with different link properties and also with an external drag. The details of the experimental systems are given in Appendix A. Further, the detailed data- tata-generation procedure is given in the Appendix A.4.
Evaluation Metric. Following the work of [13), we evaluate performance by computing the relative error in (1) the trajectory, known as the rollout error, given by RE(t) = l|ā(t) - (()
l1((t)l2) and (2) energy violation error given by 1|72 - H/l2/ [17/21 + ||H(|2). In addition, we also compute the geometric mean of rollout and energy error to compare the performance of different models [13]. Note that all the variables with a hat, for example å; represent the predicted values based on the trained model and the variables without hat, that is x. represent the ground truth.
Model architecture and training setup. For the graph architectures, namely, LGNN and GNS, all the neural networks are modeled one hidden layer MLPs with varying number of hidden units. For all the MLPs, a square-plus activation function is used due to its double differentiabilityy In contrast to the earlier approaches, here, the training is not performed on trajectories. Rather, it is performed on 10000 data points generated from 100 trajectories for all the models. This dataset is divided randomly in 75:25 ratio training and validation set. The model performance is evaluated on a forward trajectory, a task it was not explicitly trained for, of 1s. Note that this trajectory is 2-- orders of magnitude larger than the training trajectories from which the training data has been sampled. The dynamics of n-body system is known to be chaotic for n 2 2. Hence, all the results are averaged over trajectories generated from 100 different initial conditions. Detailed model architecture associated with each of the models and the hyperparameters used in the training are provided in the Appendices A.5 and A.6, respectively.
4.2 Comparison with baselines
Model performance. To compare the performance of LGNN with baselines, GNS, LGN [12, 6] and CLNN [13], evaluate the evolution of energy violation and rollout error. It worth noting that GNs and LGN have been demonstrated only particle-based systems and not on rigid bodies. Hence, to make a fair comparison, we give the same node and edge input features as provided for the LGNN for both GNS and LGN, while training. All the models are trained on a 4-link system and evaluated on all other systems. In the case of CLNN, due the fully connected architecture, the model is no inductive in nature. Hence, the model is trained and tested on the same system only. that is, the 4-li system. Detailed architecture of each of these systems are provided in Appendix A.5. Figure 4 shows the error in energy and rollout for LGNN, GNS, LGN, and CLNN. We observe that GNS, LGN and CLNN have a larger error in comparison to LGNN as shown in Figure 4 for both energy and rollout error, establishing the superiority of LGNN. To test the ability of LGGNN to learn more complex systems, we consider two additional experiments. Specifically, two similar 4-link systems, one with varying masses and moment of inertia, and the other subjected to a linear drag are evaluated in the
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links
0 Inks
16 links
Enongy GroP
10
0.90
0.2
0.04
9.88
0.10
0.09
632
024
6.10
0.00
0.02
6:84
0.03
6.10
Rollout GITOP
10
0.30
0.22
0.04
0.10
0.03
0:32
0.24
0.10
0.00
0.02
6.54
0.03
6.10
Time
LGNN
GNS
LGN
CLNN
Figure 4: Energy and rollout егrоr of n n-linnk link chains predicted using LGNN, GNS, LGN and CLNN comparison with ground truth for n = 4, 8, 16. Note that LGNN, GnS and LGN is trained only on the 4-link chain and predicted all the systems while CLNN was trained and predicted only on 4-link system. Shaded region shows the 95% confidence interval based on 100 forward simulations with random initial conditions.
Appendix A.7. Figures 8 and 14 show that LGNN is able to infer the dynamics in both these systems, respectively.
Generalizability to different system sizes. Now, we analyze the performance of L.GNN, trained on link segment, on 8- and 16-link segments. We observe that LGNN exhibits comparable performances with respect to the 4-segment model, in terms of both energy violation error and rollout error, on systems with 8-, and 16-segments that are unseen by the model. In contrast, GNS exhibits relatively increased error in energy violation error and rollout error, although the error LGN remains comparable for all systems. This suggests that the inductive bias in terms of the EL equations prevent the accumulation of error and allow improved generalization. However, the error in LGN is still orders magnitude higher than LGNN. This suggests that the architecture employed in LGNN is leading improved learning of the dynamics of the system. This confirms that LGNN can generalize to larger unseen system sizes when trained
significantly smaller system size. Note that the plots for CINN are not shown for 8 and 16-links as the architecture cannot exhibit generalizability to larger system sizes. Finally, push the limits, infer the dynamics of 100-link chain (see Fig. 15). We observe that the LGNN trained on 4-link can scale to a 100-link chain with comparable errors, confirming its ability to model large-scale structures. The trajectories of actual and trained models for some of these systems are provided as videos in the supplementary material (see Appendix A.3 for details).
Generalizability to systems with different edge properties and external drag. Although the framework presented here is generie, the results were limited to systems with similar edge properties. Further, dissipative forces such as drag were not considered in these systems. order to evaluate the model to incorporate these effect, consider 4-link system with different edge properties (see Appendix A.J)and also a system with drag. We observe that the LGNN presented can model systems with varying link properties and drag with comparable errors (see Figures and 14). These results confirm that the LGNN framework can be used for realistic systems with arbitrary link properties and external dissipative forces.
4.3 Zero-shot generalizability
In the conventional LNNS employing feed forward MIPs, the training and test system have the same number of particles and degrees freedom. other words, an LNN trained for an n-particle system cannot be used to perform inference on m-particle system. In contrast, we show here that

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10
Energy oroer
0.0
0.1
01
Rollout eirer
Time
LONN
Figure 5: Energy and rollout error of systems predicted using LGNN in comparison with ground truth for T1, T2, and T3. Note that LGNN is trained only on the 4-segment chain and predicted on all the systems. Since the starting configuration of the simulation fixed (perfect structure shown in Figure 3), there are no eπror bars generated for this system.
LGNN trained on a small 4-link system can be used to perform forward simulations on other unseen complex systems such as 100-link system, and tensegrity structures. This ability infer different unseen system sizes and topology is referred to as zero-shot generalizability. In order to analyze the zero-shot generalizability of the trained LGNN simulate complex real-world geometries and structures, we evaluate the ability LGNN to model the dynamics tensegrity and lattice-like truss-like structures comprising of both
structures (see Fig. 3). Note that tensegrity structures The topology of a tensegrity structure is designed so that the
tension and compression-memble compression members always bars and the tension members always ropes. Here, analyse the ability LGNN to model the equilibrium dynamics of two complex tensegrity structures and the lattice-like structure shown in Figure 3.
To this extent, we use the LGNN trained on the 4-segment structure. We convert the rigid body structure to equivalent graph and use the trained LGNN predict the dynamics of the structure when released from the original configuration under gravity. Figure 5 shows the energy error and rollout for both the complex structures and the lattice-like like structure shown in Figure 3. We note that the L.GNN is able to generalize to a complex structure with varying bar lengths and topology with high accuracy. Specifically, the energy violation and rollout error exhibits very low values for L.GNN (~ 10-4). Further, it saturates after a few initial timestep suggesting an equilibrium dynamics. In contrast, we observe that the eтror in GNS very high and continues to increase until it reaches 1, which is the maximum can take. This confirms the superior nature of LGNN to generalize to arbitrary topology, boundary conditions, and bar lengths, after training on simple 4-segment chain with constant length segments. Visualization of the dynamics of the system T1, predicted by LGNN and the ground truth, shown in Fig. 6. We observe that the deformed shapes predicted by LGNN are in excellent agreement with the ground truth. Note that since the initial configuration for the forward simulation is fixed, not possible to generate error bars for the trajectory.
4.4 Nature the learned mass matrix
Finally, we investigate the nature of the mass matrix of LGNNN for different systems. Note that in earlier approaches either the mass matrix was learned directly for given system based on the EL equations [6], or it was assumed to be diagonal in the Cartesian coordinates [13], or the functional
form of kinetic energy was assumed [7]. In the present approach, we do not make any assumptions on the nature of the mass matrix. In fact, for rigid body, the mass matrix need not be diagonal in nature and depends on the actual topology of the structure. This raises an interesting question about the nature of the mass matrix learned by the LGNN and how it generalizes to arbitrary topologies.

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Accual Posterte
Auul
Actal
Tirs 8D
TirawAT
The
Actal
Asaud
Aroul
Figure 6: Snapshots of system D during simulation.
8
Figure 7: (a) Mass matrix and (b) binary mass matrix showing non-zer values for chain of 16 nodes. (c) Mass matrix and (d) binary mass matrix showing non-zero values for the complex structure T1 made of chains and rods.
In order to investigate the nature of the mass matrix, we plot the mass matrix of the LGNN in Figure 7. Note that the mass matrix computed directly from the Lagrangian as M 8 L/aq², where is obtained from the LGNN. First, we analyze the mass matrix of the 16-segment structure. We observe that the mass matrix is banded with a penta-diagonal band as expected for chain structure. Now, we analyze the mass matrix for complex structure T1. Interestingly, we observe that the mass matrix learned is non-diagonal in nature and is congruent with the complex topology of the structure (see Figure 7). This confirms that the mass matrix of LGNN is learned on-the-fly during the forward simulation that provides the versatility for LGNN to simulation complex structures.
5 Conclusions
In this work, we present a LGNN-bbased framework that can be used to simulate the dynamics of articulated rigid bodies. Specifically, we present the graph architecture, which allows the decoupling of kinetic and potential energies, that can used to compute the Lagrangian of the system, which when applied with EL equations can infer the dynamics. We show that LGNN can learn the dynamics from a small 4-segment chain and then generalize to larger system sizes. We also demonstrate the zero- shot generalizability of LGNN to arbitrary topology including tensegrity structures. Interestingly, we show that LGNN can provide insights into the learned mass matrix, which can exhibit non- trivial structures in complex systems. This suggests the ability of LGNN learn and infer the dynamics of complex real-life structures directly from the observables such as their trajectory.
Limitations and future works. From the mechanics perspective, the LGNN assumes the knowledge of constraints. Learning constraints directly from the trajectory useful. Similarly, extending LGNN to model contacts, collisions. and deformations allows more comprehesive learning of realistic systems. From the modeling perspective, our message passing LGNN, messages are provided equal important Attention heads in sassaage-pssiigg neural networks have been shown to improve performance remarkably in several domains [28]. We plan to study the impact attention in L.GNN in our future works.
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Acknowledgments and Disclosure of Funding
The authors thank the ITT Delhi HPC facility for providing the computational and storage resources.
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Reviews:
[{"review_id": "mOxoaEzc0Oa", "reviewer": "Reviewer_EADF", "summary": "The paper presents a GNN-based Lagrangian formulation for learning the dynamics of chains and ropes, and achieve better accuracy and energy behavior than an unconstrained model (GNS).", "questions": "- the terms gamma, a(q) etc. need to be properly defined in the methods section\n- why do you even need a full GNN with L layers of message passing? kinetic/potential energy are local, so it at most would need 1 round of collecting neighborhood information\n- it sounds like the paper makes a distinction between bars, chains and ropes. In a rigid-body approximation, aren't those exactly the same (the only difference being that ropes having shorter segment lengths than bars) ?\n- in the appendix, why is LGN (which I think refers to a _Hamiltonian_ GNN) much worse? The core method is very similar, and I'd expect it to preserve energy very well.", "limitations": "The method is quite limited in what it can be applied to, which is pointed out briefly in the conclusion section.", "rating": 5, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Most papers on HNN, LNN approaches only demonstrate results on mass/spring and n-body system, and the premise of using these methods on more complex, realistic systems is alluring. However, while the paper is motivated by the challenges of rigid-body simulation, I'd argue the method doesn't actually tackle any issue that make rigid-body simulation a hard problem, such as collision between complex object shapes, frictional contact, or even 3D rotation for objects with nontrivial inertial tensor. \n\nContrary to what the introduction states, 2D ropes can be solved pretty well with simple particle-based methods (see e.g. the 2016 Interaction Nets paper). The non-diagonal mass matrix and inertial tensor is (a) only a problem for method which have an explicit formulation for it (e.g. LNN/HNN), end-to-end learned models (such as plain GNNs) can learn the effect from data and neighborhood relations and (b) the effect disappears in the limit of small edge lengths.\n\nThe main contribution of this paper hence is demonstrating that graph-based LNN approaches work for bars/chains/ropes, and make use of the energy-stability of those methods. However, it's worth noting that properties such as length preservation, generalization that the paper shows stem from the fact that very little is learned here; all the constraints (segment length constraint, friction, drag etc.) are manually encoded, only inertial dynamics are learned, and those equations are both universal, and quite straightforward to infer from geometry. Normally you'd want a learned method to do the opposite, i.e. learn complicated interactions/friction models that may be hard to measure for real systems, and manually encode the known priors. The only case I could think of where the proposed approach would be useful is for inferring non-trivial mass distributions which may not be visible (say, on a drawbridge made from composite anisotropic material); but those use cases would need to be demonstrated.\n\nSo while I do think there's value in expanding the type of systems that can be tackled with LNN/HNN methods, the paper in its current form is overclaiming its contributions (rigid dynamics) and quite limited in what it adds. It could be extended into a much stronger paper by showing how any of the harder aspects of rigid simulation can be tackled, or how the tricky bits (e.g. constraints, friction, ...) can be learned.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "Ro_Ji2s6AYn", "reviewer": "Reviewer_VUdP", "summary": "In this paper, the authors propose a Lagrangian graph neural network (LGNN) that can learn the dynamics of rigid bodies which can be modelled as a graph. The LGNN consists of two GNNs, which learn the potential energy and kinetic energy of the system Lagrangian respectively. The performance of the model is verified on 2D articulate body simulations based on in-extensible chains and rods. The experiments show that the proposed method can outperform baseline method Graph Neural Simulator (GNS). The generalizability of the model is evaluated on system with larger sizes and unseen topology. ", "questions": "- The simulations shown can be solved by traditional numerical methods fairly accurately and effectively. I think experiments on larger scale or more complex scenarios where the traditional numerical methods are slow or struggling, will greatly enhance the statements made in the paper.\n- The term `rigid body dynamics` is somehow too general I think. Actually the simulations learned here are limited to articulate-body dynamics which can be modelled as a graph. \n- The term `zero-shot generalizability` is mentioned in the paper multiple times but without explanations or references. It will be helpful to explain what the `zero-shot` here indicates.  \n- Videos shown in the supplemental materials demonstrate a simulation of 1s physical time. However, in the paper, the Figure 5 and Figure 6 only show the errors  for 0.1s and 0.3s physical time. It will be more convincing to show how the errors evolving during the whole 1s simulation.\n- There are no visual results of the simulation shown in the paper. Putting some key snapshots of the videos will help the readers to easily understand the problems to solve here.\n- In the appendix A.6 training details, the author mentioned the baseline method Lagrangian Graph Network (LGN), I am wondering what is the difference between the LGN and proposed LGNN?\n\nMinors:\n- Mixed usages of `colon` and `period`, e.g., line 221 `loss function` uses colon while other paragraph titles in the same section uses period; the paragraph titles 5.1 use colon but that of 5.2 use period.\n\n", "limitations": "The authors have discussed the limitations in the conclusion to some extent.", "rating": 4, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths \n- Learning the Lagrangian mechanics using GNN is interesting and the experiments results and the videos shown in the supplemental materials are neat.\n- The paper is written clearly and well organized.\n- The introduction to the background knowledge helps understand the proposed idea.\n\nWeakness\n- The method is validated on a simple (less than 20 links) rigid body simulation scenario, where the traditional numerical methods can fairly accurately and effectively solve the problem. Therefore, the key point I am concerning is what the advantage of the learning based method proposed comparing to traditional method. The advantages can be either more efficient than traditional simulations or exploring dynamics the traditional simulations do not easy to capture. However, the comparisons on such cases are missing which makes the contribution somehow hard to evaluate. I think experiments on more large scale or complex scenarios will make the proposed method more convincing.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "KxyJS9CzguH", "reviewer": "Reviewer_aiXF", "summary": "The authors introduce a Lagrangian graph neural network to predict the trajectory of a rigid body. The rigid body is represented as a graph where edges represent rigid bodies – e.g., links in a chain – and nodes represent connections between those rigid bodies. The position and velocity of each node is assumed to be given. From the input graph, the authors construct two graph neural networks which are trained simultaneously. One learns to predict the kinetic energy of the rigid body and the other learns to predict the potential energy of the rigid body and the Lagrangian is formed by computing the difference of these quantities. The networks are trained by minimizing the difference between the predicted acceleration (determined by the EL equations) and the ground truth acceleration of the system. Against previous work, the proposed approach shows strong predictive performance and an ability to generalizability to unseen structures. The mass and moment of inertia of all members in the graph are the same irrespective of their length.", "questions": "- Questions:\n  - [55] should it be \\psi = sin^(-1)((y_i - y_{i-1})/x_i - x_{i-1}))?\n  - Eqn 5: it looks like the kinetic energy is mass times position, shouldn’t it be mass times velocity\n  - [208] what is || in eqn 9?\n  - [309] How does the learned mass matrix compare to the gt mass matrix?\n- Comments:\n  - [47] Possibly relevant citations [1].\n  - The notation in eq 5 is a bit challenging (e.g., x_i^cm2)\n  - Consider using \\mathcal{l}(\\ddot{q}, \\hat{\\ddot{q}}) for the loss since \\mathcal{L} is used for Lagrangian\n- Possible typos:\n  - [37] abbreviation EL is used before it is defined\n  - [143] “This is because…”\n  - [152] “significantly smaller that” → “significantly smaller than”\n  - [184, 185] node → nodes\n  - [211] N_i = {u_j | (u_i, u_j) \\in E}\n  - [215] “denoted as” → “denoted by”\n\n[1] Duong, Thai, and Nikolay Atanasov. \"Hamiltonian-based Neural ODE Networks on the SE (3) Manifold For Dynamics Learning and Control.\" Robotics: Science and Systems (RSS). 2021.\n", "limitations": "The authors note that the proposed approach assumes knowledge of system constraints.", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "- Originality: The method appears to be new. The related work is well organized and the proposed approach is adequately situated in the context of existing methods.\n- Quality: The submission appears to be technically sound, with the claims well supported by the empirical analysis.\n- Clarity: The paper is well written and organized\n- Significance: The paper addresses the issue of predicting physically plausible trajectories of a rigid body system, a challenging problem which has the potential to impact scientific discovery. However, there are several limitations of the approach, i.e., The mass and moment of inertia of all members in the graph are the same, and the constraints of the system are assumed to be known.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "Iu1XG8k9uX", "reviewer": "Reviewer_6uQz", "summary": "A novel framework is introduced for Lagrangian Neural Networks using graph networks to model rigid body dynamics. Comparisons are made with classical Lagrangian and Hamiltonian Neural Networks, showing the generalizability to arbitrary topologies and system sizes.", "questions": "1) Same from before: line 100 the $\\nabla_{\\dot{q}_i \\dot{q}_i}$ operator, would that be a second-order derivative?\n2) How did the choice of Section 4.1 come about, where the edges represent rigid bodies and the nodes represent connections between the rigid bodies? What benefit/drawback does this have?\n3) Did you perform any experiments on the Hamiltonian extensions? \n4) Figure 5 shows a quasi constant error between the two methods, is there a reason for this? Is this not the initial error propagating throughout the rollout trajectory? \n5) Why is the GNS result in Figure 6 so smooth, while the LGNN is more chaotic?", "limitations": "The limitations are clearly addressed, though focusing more on that contact modeling is a limitation would be preferred, since the main focus is rigid body mechanics, where contact and collision is essential. Adding some initial ideas towards addressing contact would be appreciated as well.", "rating": 8, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 4, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "The contributions are very clear, and the authors did a great job explaining their framework for rigid bodies. The benchmarks are clear, and the non-diagonal mass matrices show the capabilities nicely of the model. Some steps could, however, involve a few more explanations for readers that are not familiar with the topic, such as in line 100 the $\\nabla_{\\dot{q}_i \\dot{q}_i}$ operator, would that be a second-order derivative? It is not a notation I have seen before. Also, in line 121 you mention solving and substituting the lambda, it would be great if this derivation were present in the Appendix.\n\nThe information presented in Figure 2 and 3 were a bit overlapping, and it could perhaps be combined to make the paper visuals more concise. Figure 4 also has text \"ABC\" which are not really explained (can be inferred from caption), and ideally the purple lines can be explained in the caption as well.\n\nThe extension to the Hamiltonian feels oversimplified, though I'm not sure about the topic. It would absolutely be worth investigating. If no experiments were performed, section 4.2 could be left out.\n\nIn line 276 the authors refer to an appendix, but no appendix was present in the file. \n\nAll in all, more results with exciting articulated bodies could strengthen the paper a lot, and especially scenarios such as Pfaff et al. 2020 (Learning mesh-based simulation with Graph Networks) with deforming plate and waving flag can be interesting to try.\n\n", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}]

Author rebuttal:

Decision: Accept