Sample Efﬁcient Dynamics Learning for Symmetrical Legged Robots Leveraging Physics Invariance and Geometric Symmetries Jee-eun Lee1and Jaemin Lee3and Tirthankar Bandyopadhyay2and Luis Sentis1

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Sample Efﬁcient Dynamics Learning for Symmetrical Legged Robots:

Leveraging Physics Invariance and Geometric Symmetries

Jee-eun Lee1and Jaemin Lee3and Tirthankar Bandyopadhyay2and Luis Sentis1

Abstract— Model generalization of the underlying dynamics

is critical for achieving data efﬁciency when learning for robot

control. This paper proposes a novel approach for learning

dynamics leveraging the symmetry in the underlying robotic

system, which allows for robust extrapolation from fewer sam-

ples. Existing frameworks that represent all data in vector space

fail to consider the structured information of the robot, such

as leg symmetry, rotational symmetry, and physics invariance.

As a result, these schemes require vast amounts of training

data to learn the system’s redundant elements because they

are learned independently. Instead, we propose considering the

geometric prior by representing the system in symmetrical

object groups and designing neural network architecture to

assess invariance and equivariance between the objects.Finally,

we demonstrate the effectiveness of our approach by comparing

the generalization to unseen data of the proposed model and the

existing models. We also implement a controller of a climbing

robot based on learned inverse dynamics models. The results

show that our method generates accurate control inputs that

help the robot reach the desired state while requiring less

training data than existing methods.

Index Terms— Model Learning for Control; Representation

Learning; Group-equivalent Neural Networks

I. INTRODUCTION

Various types of legged systems have been developed

with the ability to traverse extreme terrains to increase the

versatility of autonomous robots. Model-based approaches

have been extensively used for controlling these robots [1],

[2], [3], [4]. However, such strategies often fail due to

complex and unpredictable effects generated from dynamic

contacts and other environmental interactions. Also, model-

based control requires good dynamic models for accurate tra-

jectory tracking. Alternatively, model-free approaches have

shown remarkable success in this ﬁeld recently. Using the

latest deep reinforcement learning techniques, robots can

learn complex tasks [5], [6], and locomotion [7], [8], [9].

However, the lack of physical plausibility in learned mod-

els limits their applicability to the vicinity of the training

data. Hence, the efﬁcacy of the model depends heavily on

the sample complexity. To mitigate this limitation, exhibiting

1Jee-eun Lee is with Human Centered Robotics Laboratory, Department

of Aerospace Engineering and Engineering Mechanics, The University of

Texas at Austin, USA, jelee@utexas.edu

3Jaemin Lee is with the Department of Mechanical and Civil En-

gineering, California Institute of Technology, Pasadena, CA, USA,

jaemin87@caltech.edu

1Luis Sentis is with Faculty of Department of Aerospace Engineering

and Engineering Mechanics, The University of Texas at Austin, USA,

lsentis@austin.utexas.edu

2Tirthankar Bandyopadhyay is with Robotics and Autonomous

Systems Group, Data61, CSIRO, QLD 4069, Australia

tirtha.bandy@csiro.au

{w}

{b}



translational

invariance

rotational

symmetry

gravity-axis

rotational invariance

Fig. 1: The ﬁgure shows examples of physical invariance

found in a ﬂoating-based robot where the legs are symmet-

rical. Robots in different conﬁgurations in the ﬁgure above

have the same physical properties. e.g., the torque required

at each joint to track the given motion is the same.

combinatorial generalization is also suggested as a key aspect

for modern AI to resemble human intelligence, which can

generalize beyond one’s experience [10]. There are several

network architectures proposed to incorporate prior knowl-

edge. One example is utilizing a differential equation form

to encode a physics prior in the Euler-Lagrange equation

[11], [12]. By constraining the model to satisfy a differential

equation, it extrapolates more accurately to unseen samples

than Feed Forward Neural Networks (FFNNs). Using a

mixture of experts (MOE) composed of a system equation

(white-box) and a deep neural network (black-box) to learn

complex dynamics model is proposed in [13]. It leverages

model plausibility from the white-box model while the black-

box eliminates the residual error.

Every ﬂoating base robot has translation invariance and

gravity-axis rotational invariance in dynamics as shown in

Fig. 1.Speciﬁcally, multi-legged robots with rotational sym-

metry like Magneto [14] have a set of conﬁgurations that

works under the same physics. However, current schemes

that represent the states and actions of the robot as a stacked

vector hardly capture these properties, meaning that they only

learn these symmetric properties through data, resulting in

sample inefﬁciency and a lack of generalization capability.

In this paper, we suggest a network topology that works on

structured data–a graph or grouped-by-leg data– reﬂecting

the robot symmetry.

One popular neural network for structured data learning

is graph neural networks (GNNs) [15],[16] that is known

for its ability to consider the relations between objects by

using graph structure for learning. In robotics, there are some

previous works that use GNNs to learn dynamics [17], [18],

[19]. However, most of these researches focus on the model

transfer capability of GNNs over various types of robots.

arXiv:2210.07329v1 [cs.RO] 13 Oct 2022

Reference [17] emphasizes that the model can predict the

dynamics of a new robot(i.e., where its data has not been

used for training) by learning the physical relations between

links and joints. Notably, GNNs can even be applied to zero-

shot transfer learning for modular robots [18], [19].

However, asking models to learn all types of robot physics

is unrealistic. It would require tremendous training data; it

is not applicable to robots with different types of sensors

and actuators; and sensors produced by different manufac-

turers can have different performances. Instead, we focus

on reducing required sample complexity for system identi-

ﬁcation using physical priors. In this paper, we introduce

new data representations for symmetrical-legged robots and

how GNNs and other group-equivariant neural networks

can improve sample efﬁciency based on the permutation

equivariance in legged-robot dynamics.

A. Contribution

This paper proposes sample efﬁcient dynamics learning for

symmetrical-legged robots. By using grouped-by-leg struc-

tured data, we can deﬁne group invariance and equivariance

followed by geometric symmetries of a robot based on the

data representation. Then, by training a model based on

the networks designed to preserve the group equivariance,

we can learn the dynamics that incorporate physics priors,

providing sample efﬁciency and generalization capability.

II. PRELIMINARIES

Using a priori representational and computational assump-

tions to achieve generalization is actively studied in the

deep learning. This can be realized by representing prior

knowledge as equivariance or invariance under group actions.

Below we introduce the relevant concepts and prior works.

A. Invariance, Equivariance, and Symmetries

In geometry, we say that an object has symmetry if the

object has an invariance under the group action. In group

theory, group action, invariance, and equivariance can be

deﬁned as follows:

Deﬁnition 1 (Group Action) A group G is said to act

on a set Xif there is a map ψ:G×X→Xcalled

the Group Action such that ψ(e, x) = xfor all x∈X,

and ψ(g, ψ(h, x)) = ψ(gh, x)for all g, h ∈G, x ∈X.

For compactness we follow the practice of denoting a group

action by a ‘◦’ e.g. g◦x.

Deﬁnition 2 (Invariance) Let Gbe a group which acts

on the sets X. We say that the function f:X→Yis

G-invariant if f(g◦x) = f(x)for all x∈X, g ∈G.

Deﬁnition 3 (Equivariance) Let Gbe a group which acts

on the sets Xand Y. We say that the function f:X→Yis

G-equivariant if f(g◦x) = g◦(f(x)) for all x∈X, g ∈G.

B. Equivariance and Invariance in Neural Networks

Many works in deep learning leverage equivariance and

invariance for constructing the neural network architecture.

Data augmentation is a straightforward way to learn group-

equivariant representation [20]; however, it is computation-

ally inefﬁcient due to the increased data volume to train the

model.

Convolutional Neural Networks One well-known example

to consider equivariance and invariance is convolutional neu-

ral networks (CNNs) [21], [22], which utilize convolutional

layer for preserving object identity over the translation group.

Group-Equivariant Neural Networks A group convolu-

tional layer is proposed to consider other symmetries like

rotation and reﬂection [23],[24]. For more general structures,

deep symmetry networks [25] capture a wide variety of

invariances deﬁned over arbitrary symmetry groups using

kernel-based interpolation and symmetric space pooling.

For a set structure like point clouds, simple permutation-

equivariant layers [26] are shown to be beneﬁcial.

Graph Neural Networks Graph neural networks [15], [27]

is a type of neural networks directly operating on graph struc-

tures where a system of objects and relations is represented

through nodes and edges. GNNs can capture all permutation

invariance, and equivariance [28] in nodes and edges. Graph

Networks (GNs), proposed by [10], provides a general form

of graph structure for learning, G= (u,{ni},{ej, sj, rj})

that includes a graph embedding vector called global features

u, a set of node features {ni}i=1,..,Nnand a set of edges

{ej, sj, rj}j=1,..,Ne, where ejis a vector representing edge

features and sj,rjare the indices of the sender and receiver

nodes ranging from 1to Nn. This data structure allows

GNNs to capture the permutation equivariance of nodes and

edges. Then graph networks are deﬁned as a ”graph2graph”

module, which takes an input graph and returns an output

graph with the updated features.

III. STRUCTURED REPRESENTATION FOR SYMMETRICAL

LEGGED ROBOT DYNAMICS

This section describes the equivariance and invariance in

inverse dynamics of symmetrical-legged robots. Then, we

suggest data representation that can capture those properties

in neural networks.

A. Dynamics of Floating-base Legged Robots

The dynamics of legged robots are commonly described as

a function of the generalized coordinates q= [q>

b,q>

j]>∈

Rnq, where qb∈R6,qj∈Rnjrepresent the conﬁguration of

the ﬂoating base and joints, respectively. As such, the rigid-

body dynamics of a ﬂoating base robot can be obtained using

Euler-Lagrange formalism yielding:

M(q)¨

q+b(q,˙

q) = S>

aτa+Jc(q)>Fc(1)

where M(q),b(q,˙

q)represent the mass/inertia matrix and

the Coriolis/centrifugal force plus the gravitational force.

Sa∈Rna×nqdenotes the selection matrix corresponding to

the index set of actuated joints, which maps τa(the actuated

joint torques) into the generalized forces and, Jc(q)denotes

the stacked contact Jacobian matrix that maps contact wrench

vector Fcinto the generalized forces. Finally, from the

dynamics equation, we can derive the forward model ffwd

and inverse model finv as:

q=ffwd(q,˙

q,τ),τ=finv(q,˙

q,¨

q)(2)

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SampleEfcientDynamicsLearningforSymmetricalLeggedRobots:LeveragingPhysicsInvarianceandGeometricSymmetriesJee-eunLee1andJaeminLee3andTirthankarBandyopadhyay2andLuisSentis1AbstractModelgeneralizationoftheunderlyingdynamicsiscriticalforachievingdataefciencywhenlearningforrobotcontrol.Thispaperpropos...

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Sample Efﬁcient Dynamics Learning for Symmetrical Legged Robots Leveraging Physics Invariance and Geometric Symmetries Jee-eun Lee1and Jaemin Lee3and Tirthankar Bandyopadhyay2and Luis Sentis1

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