On discrete symmetries of robotics systems: A group-theoretic and data-driven analysis
Authors: Daniel Ordonez-Apraez
Mario Martin
Antonio Agudo
Francesc Moreno-Noguer
Venue: Submitted to ICLR 2023 (2023)
Keywords: Morphological Symmetries
Discrete Symmetries of Dynamical Systems
Equivariant Dynamics
Equivariant Function Approximators
Geometric Deep Learning

Abstract: In this work, we study the Morphological Symmetries of dynamical systems with one or more planes of symmetry, a predominant feature in animal biology and robotic systems, characterized by the duplication and balanced distribution of body parts. These morphological symmetries imply that the system's dynamics are symmetric (or approximately symmetric), which in turn imprints symmetries in optimal control policies and in all proprioceptive and exteroceptive measurements related to the evolution of the system's dynamics. For data-driven methods, symmetry represents an inductive bias that justifies data augmentation and the construction of symmetric function approximators. To this end, we use Group Theory to present a theoretical and practical framework allowing for (1) the identification of the system's morphological symmetry Group $\G$, (2) the characterization of how the group acts upon the system state variables and any relevant measurement living in the Euclidean space, and (3) the exploitation of data symmetries through the use of $\G$-equivariant/$\G$-invariant Neural Networks, for which we present experimental results on synthetic and real-world applications, demonstrating how symmetry constraints lead to better sample efficiency and generalization while reducing the number of trainable parameters.

Under review as a conference paper at ICLR 2023
ON DISCRETE SYMMETRIES OF ROBOTICS SYSTEMS: A
AND DATA- ANALYSIS
Anonymous authors Paper under double-blind review
ABSTRACT
In this work, we study the Morphological Symmetries of dynamical systems with more planes of symmetry, predominant feature in animal biology and robotic systems, characterized by the duplication and balanced distribution of body parts. These morphological symmetries imply that the system's dynamics are symmetric (or approximately symmetric), which in turn imprints symmetries in optimal control policies and in all proprioceptive and exteroceptive measure- ments related to the evolution of the system's 's dynamics. For data-driven methods, symmetry represents an inductive bias that justifies data augmentation and the con- struction of symmetric function approximators. To this end, we use Group Theory to present a theoretical and practical framework allowing for (1) the identification of the system's 's morphological symmetry Group G, (2) the characterization of how the group acts upon the system state variables and any proprioceptive and extero- ceptive measurement, and (3) the exploitation of data symmetries through the use of gnguivariant/G-inviiian
Neural Networks, for which we present experimen- tal results on synthetic and real-worls applications, demonstrating how symmetry constraints lead to better sample efficiency and generalization while reducing the number of trainable parameters.
INTRODUCTION
Symmetries are a predominant feature in animal biology. The majority of living (and extinct) species are bilaterally or radially symmetric (i.e., having one or more planes of symmetry), a property intu- itively recognized by the patterns of balanced distribution and duplication of body parts and shapes (Hol16) 2017). Likewise, most robotic systems are symmetric, often featuring more precise symme- tries than nature due to the accurate duplication of body parts and the tendency to design mechanisms with symmetric volumes and mass distributions. These morphological symmetries animals and robots imply that the dynamics and control of body motions are also approximately symmetric, sulting in all proprioceptive and exteroceptive measurements, related to the evolution of the system's dynamics (e.g. joint torques, depth images, contact forces), to be also symmetric. This highly rel- evant inductive bias is frequently left unexploited in most data-driven applications in the fields of robotics, computer graphics, computational biology, and control.
Recent works in computer graphics (Yeh et al.] 2019 Abdolhosseini et al. 2019 Yu et al. 2018)
and robotics/dynamical systems (Van der Pol al. 20200] Ordonez-Apraez et al. 2022 Hamed & Grizzle| 2013 Finzi et al.) 202 1a) have exploited through different approaches the morphological symmetry group associated with bilateral (or sagittal) symmetry (the reflection group C2). obtaining improvements in generalization and sample efficiency of function approximators. Notably, Zinke- vich & Balch (2001) proved that Markov Decision Processes with state symmetries have symmetric optimal value and policy functions. Despite these encouraging contributions, exploiting the induc- tive bias of morphological symmetries is not widespread technique in the research community. We attribute the scarce adoption of these techniques the lack of unifying theoretical and practical framework, allowing to identify different morphological symmetries in arbitrary dynamical systems and efficiently and conveniently exploit them in data-driven applications.
This work takes a step towards this unifying framework by studying morphological symmetries through the lens of dynamical systems and group theory]] Our theoretical contributions are:
'The field of mathematics that studies symmetrics, which is broadly used in Machine leamnigg (MI.)

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Figure 1: Left: Caley diagram and top-view (see 3D animation of symmetric configurations of the quadruped robot Solo, whose morphological symmetries (described by the Klein four-group Kc4) allow it to imitate the effect of reflections (gx, 9e) and 180 rotations of space (gr). Transformations affect both proprioceptive (state space, CoM linear 1 and angular k momentum) and exteroceptive (terrain elevation, external disturbances) quantities. Right: Diagram of a Ka-equivariant NN. Each of the layer's linear maps W is constructed as a weighted average of the basis of the space of equiv- ariant linear maps B, computed from the 1C₃ symmetries of the input-output spaces (see section 5).
* A group-theoretic formalization of the concept of discrete morphological symmetry. * A characterization of how the morphological symmetry group G affects the system's 's state vari- ables and any relevant proprioceptive and exteroceptive measurements. Facilitating the identi- fication of G and the augmentation of proprioceptive and exteroceptive measurements.
Once the morphological symmetry group G is identified, our practical contributions focus on the efficient construction and versatile use of G-equivariant neural networks, for arbitrary discrete mor- phological symmetry groups G, for which we:
Derived optimal initialization for the trainable parameters of equivariant layers Demonstrate that G-equivariance reduces the trainable parameters by approximately 1/1s). & Enable the construction of large scale G-equivariant networks by mitigating the construction computational complexity and the storage memory complexity of equivariant architectures?
BACKGROUND ON SYMMETRY GROUPS
In nutshell, symmetry group in Group Theory is an abstraction of the concept of symmetries that different geometric objects might have, understanding symmetry as transformation that when ap- plied to an object conserves a relevant property of its structure. For instance, in fig. Illeft the Klein four-group XC4 describes the symmetries that vectors, pseudo-vectors rigid bodies, and quadruped robot have to 180" rotations (g,) and two perpendicular reflections (ge> ). Transformations that preserve vector magnitudes and energy. While on fig. righh the same group describes the symme- tries of vector spaces, representing the quadruped robot's s state x and legs contact state y.
Formally, symmetry group is set of invertible symmetry transformations (or actions) G (c, 91, -2.... }. containing the trivial action e (which leaves objects unchanged) and having a bi- nary operator (+) G that is associative (i.e. 91 (g2 gs) (01 92) gs), which composes group members into other group members, such as gr = Gt for K.4 (see fig. lp. Group representa- tions are characterizations of how each action g transforms specific geometrie object, say R* A representation P.. g + (k) (GL: General Linear group) is group homomorphism associ- ating each g to an invertible linear map Pa (g) *x specifying how the object æ is transformed, that is: g(x) = æ = P= (g)x. Since group actions are abstract, it is common to define different object-dependent representations for each action, as we will see throughout this work.
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A fundamental concept for this work is the notion of function Geequivariance and G-invariance. Consider the function f IR" + IR".. f is said to be G-equivariant or G G-invariant
y - fg x)) A E
f x:)
A 9 EGG.
(1)
Equivariance
Invariance
In words, an equivariant function maps symmetries of the input to symmetries of the output, while an invariant function maps symmetries the imput to an invariant output.
Being this short section undoubtedly an unsatisfactory introduction to Group Theory, refer the uninitiated and interested reader to Bronsteii et al. (2021) for a remarkable introduction to the field.
LAGRANGIAN MECHANICS AND SYMMETRIES OF DYNAMICAL. SYSTEMS
First, we provide group-theoretic perspective of symmetries in dynamical system. To this end, let us consider a dynamical system with generalized coordinates q E Q
S R" and velocities q E T R"; well Lagrangian function L Q x Q R T(q. 4) U(q. q). Being Q the constrained configuration space, Ta Q the configuration tangent space at q (i.e., the space of generalized velocities), and T(q, q). (ạ1 q) the state kinetic and potential energies, respectively.
The symmetries of a dynamical system are defined as transformations in the space of generalized coordinates that keep the energy state of the system unchanged (Ostrowski & Burdick 1996). In this work, we study time-invariant point-raansformation2] of generalized coordinates which are interpreted as actions of a symmetry group, L.e. 9 E G. Denoting Pa(g) R=x= as the action representation in Q, we define the transformed coordinates as g(q) Pa(9)q = g. q. Consequently, the velocity and acceleration of the transformed coordinates are given by g q and dq
Q Q.
g q, respectively, considering that dg(g) 0g(q) dq
0q dea = =P((g)
dq = g de
de
dt=
Formally, we say that a dynamical system has a symmetry group 9 if its Lagrangian is G-invariant:
L (qq q) Lg (.9 9.9.
A geg, g, E Q E Q.Q.
(2)
Because the Lagrangian structure differs between the original (q. q) and transformed coordinates (g 9.9 q) A E G. when we derive the Equations of Motion (EoM) of the system in the transformed coordinates, we obtain a set of EoMs describing the true system dynamics in different coordinate systems. Formally, if we derive the EoM through the Euler-Lagrange equation of the
second order @C(g.4) dc(q.é] M(q) T(q, q) 0). the distinet EoM are equivarian 8q
0g
to each other (Lanczos
2020 a property we will refer to as dynamics G-equivariance:
g- [M(g)q T(q. 4)
Mig q)g q Tg 9.9 a 0 lvgeg, E Q TETOQ.
Inertial
Mowing
Inertio!
M Moving
Denoting M(g) Q Raxx as the generalized mass matrix function and T(g, q) QTT@xx1 R" as the generalized moving forces at and q. Note that, in in/eq. (3) the original and transformed dynamics are related linearly by the Jacobian of the coordinate transformation 0g(4)/og (g (Wheeler 2014), which to preserve notation is reduced to
To ensure dynamics G-equivariance deg. (3) both the generalized inertial and moving forces need to be independently equivariant, meaning:
M(g q) gM(q)g"
g T(g.9) 9.9 - ④) IVges, q q á E TQQ. (4)
The equivariance of the generalized mass matrix provides a pathway for the identification of the symmetry group G. and the group action representations Pg(g) g E (see sectionn 4.2). While the equivariance of the generalized moving forces (which in practice usually incorporates control forces, constraint forces, and external interactions) implies that dynamics G-equivariance is held until a symmetry braking force violates the equivariance of T.
Floating robotic/dynamical systems: Let now narrow our focus floating- based dynami- cal systems. Namely, legged/flying/swimmigg robots, animals, and animated characters evolving in
'A ppint-rrassformation is a finite, invertible, continuous, and diffcrentiable function of q Lanczos 2020
"Some authors refer to this property covariance of the FoMs (Wheeler 2014
Lanezos 2020
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Euclidean space of d dimensions (with its corresponding Euclidean Lie Group Eg), whose gener- alized coordinates can be decoupled into q = X Q = E, Q Where X B E Eg represents the system's base (or center of mass (CoM)) position and orientation q Q₂ C R represents the internal Degrees of Freedom (DoF) constrained configuration. And Qu the internal configuration space commonly referred to as joint space. In this coordinate space, we can differentiate the effect of
g on E and Q2 noting that: ggg) = 9 = pa(g)q P2y(3)
X
g E G. Represent-
ing Ped (g) E Ed homogeneous matrix transformation affecting the base, and Pay(g) (g) E R a transformation on the joint-space. The differentiation becomes handy in identifying the system's continuous and discrete (section 4) symmetries.
Continuous symmetries of floating-base systems: The most commonly studied and exploited sym- metries of floating-based systems are the continuous symmetries of the Euclidean space in which the system evolves, i.e., symmetry actions J E Ea,, involving ddddimensional rotations/reflections + translations | The property of these actions g. which is of most interest to us, is the Eg-invariance of the joint-space configuration: y q = å + Pa, (g) = Las V g E Eg.
4 DISCRETE MORPHOLOGICAL SYMMETRIES (DMSs)
A Discrete Morphological symmetry (DMS) is a mathematical formalization of the intuitive property of floating- base dynamical systems that can imitate the effect of rotations, translations, and infeasible reflections of space with feasible discrete change in the system configuration. To better introduce the concept of DMS useful to first study the most simple (and most frequent) instance a DMS: the reflection symmetry, which all humans and most animals approximately possess (Holló 2017).
Reflection DMS g = C2: Despite most floating- dynamical systems being symmetric w.r.t reflections of space E Ed), in practice, it is common to ignore these reflection symmetries, since, in general, it is impossible to subject a real-world robotic/dynamical system to a true reflection of space (Selig 2005). Think of your own body your base (hip) in space but you are unable to execute a true reflection of space, which will force your heart switch sides (and certainly die). Fortunately, your body is symmetric w.r.t. the sagittal plane (we will assume perfect symmetry for now) which allows you to imitate the effect of a true reflection of space by modifying your internal configuration (your body pose), and rotating and translating your base (i.c., with a feasible discrete change in your configuration, see (supp.fig 6a). Therefore, you have a discrete morphological symmetry associated with the reflection group Cz.
floating-bas -ase system, you can move and rotate
Discrete Morphological Symmetries with higher order groups: When the finite symmetry group has group order [9] > 2, DMS can imitate both rotations/reflections and translations Ez- Therefore, the group G will be isomorphic to one of the groups in Euclidean geometry. Most fre- quently G is a Cyclic or Dihedral group. See examples for C2 on supp.figs and 66a for C3 in supp.fgg 5, and for D2 = K., in in.ig.
Definition of Discrete Morphological Symmetry: Consider a floating-bass dynamical system, with generalized coordinates q E Q = EE x QJ, evolving in d-dimensional Euclidean space. The system said to have a DMS if, for a given continuous symmetry action 7 E Ea, there exists an action g E G that is proper (1p, (g)l 1) and non-trivial in joint-space (Pa, (g) F Ia), such that:
L Qq. L q.g q) C (g 9.9 q) E Q q TTQ. g E G. g E Ed-
(5)
Where ÿ represents rotation/reellection translation in Eg, and g is the action of the DMS finite group G. forcing a transformation of the internal joint-space configuration. The difference between
TTechnically, the topology of Q = E, QJ is referred to as trivial principal fiber bundle OOtrooski & Burdick] 1996). with E, as fiber Lie group, and Q, as the base space. Note that this topology applies to larger range of dynamical systems than merely floating-base
'e deliberately abuse notation keep the homogeneous matrix representation X e of position and orien- tation. instead of the gector-uuttenion representation, common in robotics and computer graphics.
"These symmetries are commonly studied since, in conservative systems, translational and rotational sym- metries imply the conservation of linear and angular momentum, while time symmetries imply the conservation of energy (Nocther 1918).
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g and g is highlighted when reformulating eq. (5) for specific system configuration:
Pea xB
Pe, (T X = c =c ((1rn (9) pa. pa,166 (9) q Pa. (9) à 1)
(ea(G) = +1, PG)
(6)
(g) = pe, (r) (3)"
What eq. (6) highlights is that with DMS infeasible reflections (12 (7) (1)) = -1 and feasible or infeasible rotations/translationns of space are imitated by a feasible (p (g)) = 1) transformation to the system's 's base and a change in joint-space configuration. Furthermore, the structure of the proper transformation Pe, (g)x = Pe., (g X Pe, (7) along with the properties of the dynamics of symmetrical dynamical systems (eq. (4),, provide a pathway for the identification of G and the representations (g) Vg E G for any Hloatingbbas dynamical system (section 4.3).
4.1 DATA AUGMENTATION IN SYSTEMS WITH DMS
Recall from section that point-transforrmtioons E G have the same representation Pa Pg(g) for the configuration space Q its tangent space Ty and any higher order tangent spaces (e.g., the space of generalized accelerations and forces (cg. (3))). Since our floating-base systems' configuration space has the topology Q = Ed x Qu, this property passes to the representations on Eg and Qu- Meaning that the representation PEd (g) can be used to augment members of Eg (i.e., points, vectors, and orientations) and members of EE higher order tangent spaces (i.e., linear & angular veloci- ties/accelerations). Likewise the representation Pa, (g) can be used to augment members of Q, and its higher order tangent spaces joints restiionsvellottie/ecceeeratttno joint forces/torques).
In practice this means that any proprioceptive (e.g.. joint torques, contact forces) and exterocep- tive (e.g., point clouds, terrain height maps, RGBD-images) measurements relevant to the evolu- tion of the system's 's dynamics, can be augmented solely with combinations of PE and Pay (g). Since these measurements live in Q and E. and their higher order tangent spaces (see examples in supp..ig 5] and supplementary E.3.1 and E.4.I). To achieve this we need to identify the symmetry group G and its action representations (section 4.2).
4.2 DMS IN THE CASE OF RIGID-BODY DYNAMICS
Until now, we have only assumed our dynamical system is a floating base system. Now, we assume the system dynamics are ruled by ridig-body dynamics. This means that our system is a collection of n interconnected rigid bodies. This is the most frequent scenario in robotics, computer graphics, and experimental biology (see (see (uuplementary A). In rigid body dynamics the generalized mass ma- trix is given by M(q)= = Σx* nn Jr, (g)gg))))))) m r, (q) +JJR, TI RR (q), being Jr, (q) : Q + Rdxn and J JR (q) Q + _dx= the position and orientation Jacobians that are used to map generalized velocities to the linear (+x- Jr, (q)g) and angular (wx, Ra (q) velocities of the body k WWieber 2006). These Jacobians are functions of the kinematic parameters of the system. While my and Lg. the mass and inertia of body k. represent the dynamic parameters of the system dynam- ics. A DMS implies symmetries over the kinematic and dynamic parameters of the system, that in practice become useful for the identification of the DMS group G.
Symmetries of kinematic parameters (Kinematic Tree): Considering only the kinematic pa- rameters and the equivariance nature of M(q) (eq. (4)), we conclude that rigid-body system with a symmetry group must have positional and rotational Jacobians that respect J7, (g q) Jz, (q)g² A JR. (g q) = JR. (q)g AA E G. Being g continuous or discrete symmetry action. In the case of DMS. in which the discrete action g E & is designed to imitate the effect of a specific continuous symmetry action E E,, we have that the 7th body Jacobians should respect:
Jr, (g-44g Jr. (g 4) = Jr. (4)T" ^ JR, 4)g = J (g-4) = JR, (4) V(g.7)gg E G. E. (7)
Where l is the index of the body of the g transformed system (see lppendix C.3p. In practice, the symmetry in kinematic parameters described in eqq. (7) is interpreted
kinematic tree symmetry (see jecssppplemmntrr C). requiring the discrete action g to result in a kinematic tree indistinguishable from the one obtained by applying the rotation/reflection
translation g.
Symmetries of dynamic parameters (Mass and Inertia of rigid-bodiess In order for a rigid-body dynamical system to have DMS the bodies of the system must have symmetric mass distribution or the kinematic tree must be modular (subchains of the tree are symmetric to each other). To understand this morphological constraint consider the base body configuration X E E2 and the
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definition Psy (g)X8 Pe, Pe, (g) in eq. (6) Where the action the left of X is inter- preted as a Euclidean transformation of the base ın a global reference frame and the action to the right transformation in the frame attached to the base.
Recall that, for and g to be Lagrangian- eerraniian-cqivvalent (eq. (5)) the dynamics of the base body at the configuration Pe, (g) XB should be identical to the dynamics of the body at Pe, (g) X (g)xx (eqs. (3) and (5). Assuming exact kinematic parameter symmetries, both body configurations will have equivalent dynamics if their Inertia matrix I in both configurations is identical. Because in general,
Ps.(g) * pe,(g (g), the rigid-body Inertia must be invariant the right transformation X Pp, (g)) This inertia invariance implies a symmetric mass distribution of the rigid body (see geometric proof in (supplementary C.2)). And becomes a key property for the identification of the DMS group G.
As an example consider the robot Solo in ng. This robot is able to imitate two reflections of space (gt, 9.) and 180 rotation of space g.- This is possible since the base body of the robot has two symmetry planes (see seepsupp..ig 6b), making the inertia of the base I, at any arbitrary configuration, invariant under the transformation X (g)*
[T.. Fr K44-
Modular Kinematic Trees:
Theoretically, the previously described constraint of symmetric mass distribution applies to all rigid bodies in the system, limiting the applicability of DMS to diverse floating-bast systems. Conveniently, most systems of interest are modula i.e., their kinematic trees composed of subchains with identical or reflected rigid bodies (e.g., in(supp.tig 5hthe identical replication of fingers in the TriFinger robot, or the reflected arms and legs of the humanoid Atlas supp.fig 6ap. In such architectures, swapping identical/reflected bodies (and thus subchains of the tree) can satisfy the Inertia invariance required for Pe_ (g) (s.( (x) without requiring symmetric mass distributions. Refer to lappendix C.3 and supp. fg 5 for details and examples.
4.3 IDENTIFICATION OF DMS GROUP 9 IN RIGID-BODY DYNAMICS
The identification of the DMS group 9 of a floating- base dynamical system, composed of rigid- bodies, can be outlined in four steps (see simple examples in supp.fig 5
1. Identify the configuration Xg and its associated Inertia Lg. Usually the base body or the CoM.
2. Identify the symmetries in mass distribution as invariances to Euclidean transformations g
Ed of the reflected Lg- These are the candidate actions that the system could imitate.
3. Identify modularity in the kinematic tree. L.e., all pairs of identical/reffected rigid bodies.
4. From base to end-effectors use (7) determine for each g. if the action and Pa, (g) exists.
G-EQUIVARIANT AND G-INVARIANT FUNCTION CONNCTIONAPPROOOI APPROXIMATORS
Once we identified the DMS group G of our system, we know that any proprioceptive or extero- ceptive measurements have the same symmetry group 9 (section 4.1). Therefore, to improve gen- eralization and sample efficiency, we can exploit the known symmetries of the input æ and output y spaces. of any mapping we desire to approximate, by constructing Geeqiivariant or G-invariant leq. (1) NN [ax; Φ), with parameters & (Bronstein et al. 2017). This section is built on top of the Framework for the construction of G-equivariant NN of Finzi et al. (202Tb). Where our main motivation is to address the limitations that prohibit the construction of large-scale G-equivariant NN (see supplementtrry D), which are ubiquitous in real-life applications.
Consider (x; Φ) to be composed of multiple perceptron (or convolutional) layers of the form 'y := a W' x ib), where 'x E IR," 'y E RTm ' E mR and 'b are the I layer's 's linear map and bias, respectively; and : R IR is strictly monotonic nonlinearity (Ravanbakhsh et al. 2017). With this parametrization, the equivariance constraints of leq. (1) can be reduced to constraints on the linear map W (dropping the layer index / for notation clarity):
Powe (g) W = (g).(g) Egeg = (pw(g) - I)w = Vg
(8)
The RHS of (8) is reformulation of the equivariance constraints as standard set of linear equations, defining (w(g) (our(g) & p... () mmnxm as the representation of the group
*This covers most flying/swimming/legg. robots, animals and animated characters supplementary A A similar analysis can be made for the hias vector b.

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action acting the linear map as a result of a semi-direct produc 10 of the input and output group actions (& stands for the Kronecker product) and w = vec(W] (W) E R mm as a vectorized version of W (refer to Fonzi et al. (2021b) for details). Since the constraint imposed by each g is linear in W, we can stack them into a single system of linear equations Cw = 0. The nullspace of this system of equations B Rmxxr describes the basis vectors spawning the entire space of equivariant linear maps, Allowing to parameterize all G-equivariant W as:
w = EieeB
W Eics unvec(B...)
Where the basis coefficients E R" represent the free variables of the system of equations and the trainable parameters of the equivariant layer (see (seefig.. righht.
Dealing with memory complexity of equivariant layers: An equivariant layer needs to store the matrices Pw(gg and B, in addition the typical memory complexity of a perceptron or convolu- tional layer. These matrices' memory complexity quickly becomes intractable for moderate input- output dimensions (see supp.table T). Fortunately, finite symmetry groups have sparse action matrix representations, resulting in both of the aforementioned matrices being sparse. Our implementa- tio extends the Pytorch API from Finzi et al. (2021b) to process finite groups with sparse matrix definitions limiting the additional memory footprint of equívariant layers to a minimum.
Dealing with the computational complexity of determining the equivariant basis B: Computing B for layer is a process with high computational complexity. Finzi et al. (2021b) proposes a Krylov gradient-based method able to handle both finite and Lie groups' arbitrary regular representations. While Van der Pol et al. 2020) approximates B through SVD of matrix W Rxxmm (z mm). Both approaches run in polynomial time (r+2(m))) (prohibiting their use in large dimensional spaces) and approximate the space rank r numerically.
Fortunately, DMS groups G are finite and have, in general, generalized permutation matrices as regular representations. Enabling the computation of B in linear time (see supplementary B): Note that the constraints imposed by each (w(g) result in parameter sharing constraints (e.g.. W1o = W.)) In these cases, every vector of the null-space of C' (i.e., B,) simply describes the sharing scheme of a free variable of the system of equations (i.e., the trainable parameter ci), and this sharing scheme is nothing else but one of the unique r orbits of the dimensions of w when transformed by all group actions, c.g., G W1o {g W10 A g g} = {wio, -w2..... Wo
WW2
(see the parameter orbits of length 4 in fig. righh, for K,)). The orbits of all w E w are trivially computed with [w, (w(g)) ... .. Pw (g ø ) while the unique orbits can be identified in O ((m)) time. Our proposed solution can be thought of as a linear-time version of Ravanbakhsh et al. 2011
Optimal trainable parameter initialization for equivariant layers: Proper initialization of the equivariant layer's s trainable parameters C₁ (eq. (9) is required to avoid cetivations/gradients from vanishing exploding (Klambauer al., 2017). Following the same derivation of the Kaiming initialization (He et al. (015)) (see supplementary D.2). we can conclude that the parameters should be initially sampled from a distribution with Var (c) "M/Aero. ensure constant variance of activations throughout the network layers (see supp fig Where AB - Σ Σ B² jsjsk and %
a nonlinearity dependant scalar (e.g., RiLL = 1/2, TSe following Klambauer et al. (017
This initialization depends only on B. Thus, is applicable any Lie finite group.
Reduction of trainable parameters in equivariant layers: Determining analytically the number of trainable parameters (i.e. the rank r) of an G-equivariant layer is, in general, unresolved problem. However, for DMS groups, we show on (supplementary D.I that the number of trainable parameters of a G-equivariant layer can range from w1/ S r' S |w], depending on the number of dimensions of the input-output spaces left invariant by the symmetry actions. In practice, this implies that for G-equivariant layer without any input-output fixed points (e.g., all intermediate layers of a G-equivariant NN), the number of trainable parameters is reduced by 1/1g) being [9] the group order. Therefore a G-equivariant architecture with
C2 (supp.fig 6ap (or & KC4, see fig. I) will have approximately 1/2 (or 1/4) of the trainable parameters of an unconstrained NN of the same architectural size. The reduction of parameters is caused by the parameter sharing constraints leq. (9) and is visually depicted in fig. right.
10 Note that Finzi et al. 2022 considers always direct product of the input-output symmetry actions. However, for DMS the input and output groups are isomorphic, direct product over-contains the models to symmetries not present in the data.
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Solo - G A2
Atlas 6 C2
Solo - G = K2 VS. G C2
Pyper porerrs
hel20
10y
Hypae paraes
10a
Kcbb
hoze
Model
Mode
test loss
EMLP
MLP
10-1
10-2
10"
10-2
100 k
200 300
20 k
training samples
training samples
training samples
Figure 2: CoM-estimation results comparing MLP, MLP-aau, and EMLP models. Left and
Middle: Test set sample efficiency, for robot Solo and Atlas, of model variants with different ca- pacities (number of hidden layers' neurons hc). Right: Sample efficiency for robot Solo (fig. 1). of models with he = 512, when exploiting 9 Ka (sagittal and traversal symmetries) and G C2 {e, J C K4 (only sagittal symmetry). Reported values represent the average and standard deviation across 10 different seeds.
EXPERIMENTS
We present two experiments of supervised learning, a regression application using synthetic data and a classification application using real-world data. Both experiments aim to illustrate the versatility of DMSs for data augmentation and training of equívariant functions, along with the impact on the model's 's sample efficiency and generalization capacity when exploiting DMSs. While we keep the presentation concise, all the technical aspects are detailed in (sppleemennrry E and B
CoM Momentum Estimation (Regression): In this experiment, we train a NN to approximate
a robot's center-of of mass momentum given the joint-space position and velocities: h = Aq) Where h = [TT ktjt are the linear and angular k momentum components And Ag is the Cen- troidal Momentum Matrix (CMM) Orin et al. (2013). This analytical function is highly non-linear and Geequivariant to the robot's 's DMS group G. Consequently, the function approximator is expected to be equivariant or to approximate equivariance.
We test two robots: (1) Atlas, n, = 30[DoF] humanoid robot with reflection DMS group G = C2 (see supp..ig 6ab. (2) Solo, r. 12 DoFF) quadruped robot with the Klein-4 group as DMS group K, (see fig. leett). the same time, we compare three variants of function approximation: standard Multi-Layer Perceptron (MLP), a version of the MLP trained using auttentgmmnttiionn (MLP-aug), and a version of the MLP with hard-cquivariance constraints (E-MLP).
On fig. middle&mmidde,, ties, the E-MLP and MLP-Aug outperform MLP on sample efficiency (better generalization with less data) and robustmess to overfitting when training data is scarce. Comparing the E-MLP and MLP-Aug model variants, we see that the lower capacity versions behave similarly, but as capacity increases, E-MLP starts to show better sample efficiency and generalization. Lastly, on (fig. right we compare, for the robot Solo, the performance of the model variants when exploiting the robots' entire symmetry group (Kc)) and a subgroup of the real symmetry group (C2 K4). The results indi- cate that sample efficiency and generalization capacity increase with the number of true symmetries of the data exploited.
compare the model variants. For both robots and all model capaci-
Static-F reectionneeggiiii Regimee Contact Detection (Classification): This experiment uses the dataset pre- sented in [in et al. (202T) for the estimation of itatic-friction-reeinn contacts in each of the four legs of the Mini-Cheetah quadruped robot. The dataset samples, collected in the real-world, consist of the history of the past 150 time-frames of proprioceptive data collected from inboard sensors of the robot during locomotion with various gaits and over several terrains, e 154xx100 and the ground truth contact state of the robot y e 18 estimated off- using a non-causa algorithm (i.c., dependant on the past and future). The objective is to train causal function approximator ff f(:; (b) for estimating the contact state.
The real-world Mini-Cheetah robot has approximate reflection DMS G Cz. Hence, both the proprioceptive data x and the contact state y share the symmetry group & (see supplementary E.4). We compare three variants of function approximators: the original Conv-NN architecture of Lin
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Mini-Cheetah C
Mini-Cheetale G
Mini-Cheelee @
COM4 ows-ag
CMN
0.90
0.05
test loss
1est legs evg-1
Metric Score
0.80
0.75
0.78
1.0
ECNN
0.75
0.65
0.73
CNN
0.00
0 100k 200 300 400
500
100k 200 300 400 SC0
1 88 State
Figure 3: training samples
samples
contact detection results comparing CNN, CNN-aug, and ECNN. Left: Sample efficiency in log-log scale. Middle: Average legs F1-score vs. training Classification metrics on test set performance of models trained with the entire training set. Selected metries are contact-state (y E 10) accuracy (Acc) and fl T1-score (FI) for each leg binary contact state. Due the sagittal symmetry of the robot the left front (LF) and right front (RF) legs are expected to be symmetric, as the left hind (LH) and right hind (RH) legs. FI-score is presented considering the dataset class imbalance (see supplementary E.4 values represent the average and standard deviation across 8 different seeds.
samples. Right:
supp.fig supp.hig 7).
Reported
ct al. (20TT) (CNN), version of CNN trained using data-augmentation (CNN-aug), and version of CNN with hard-equivariance constraints (E-CNN).
The sample efficiency of the model variants and the average legs contact state classification are illus- trated in fig. 5lleft&-miidde Where the equivariant model E-CNN presents better generalization, performance, and robustness dataset biases (see supp.fig 7 than the unconstrained models across all training set sizes, followed by CNN-aug. In fig... 3-right we evaluate test set classification metrics when using the entire training data. The E-CNN model outperforms both CNN-aug and CNN on contact state classification and average leg contact detection. Of relevant importance is the mitiga- tion of suboptimal asymmetries of the models by exploiting symmetries. Preventing the model to favor the classification of one leg above others fi. 3). Refer to tolappendix E.5 details.
7 CONCLUSIONS & DISCUSSION
In this work, we present the concept of Discrete Morphological Symmetry (DMS). These are dis- crete symmetries of dynamical systems evolving in Euclidean space, that associated with the capability of the system to imitate Euclidean transformations (rotations/reflecciios and translations) with discrete changes the system's internal state configuration. With this formalism, we can describe the bilateral and radial symmetries that are ubiquitous in robotic systems and animals in nature. By studying these symmetries with the language of Group Theory, we propose mechanism for the identification of the finite DMS group G, and of the representations of the symmetry actions in the system's state variables and relevant proprioceptive and exteroceptive measurements.
Having made the connection between Dynamical Systems and Group Theory, we show why and how these symmetries should be exploited in data-driven applications to obtain improvements in sample efficiency and generalization capacity- either by using data- aatta-aummenattion or G-equivariant Neural Networks. For the latter, we present practical contributions addressing the implementation drawbacks (intractable computational and memory complexity) of using G-equivariant architectures for real-world applications. Additionally, we release open-access code enabling the rapid prototyp- ing of G-cquivariant Neural Networks for the exploitation of DMS in applications processing data from higid-body dynamics (c.g., robotics, computer graphics, and computational biology).
Lastly, present empirical results supporting synthetic and real-world data from three different robots. In both experiments improvements in sample efficiency and generalization are obtained by exploiting the morphological symmetries bias, motivating the use of this technique in applications using data from simulation and/or the real world.
claims
data-driven applications using
L Limitations: For detail account of limitations see supplementary

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REPRODUCIBILITY STATEMENT
The experimental setup used in the experiments is described in section and suppleementary Moreover, our implementation will be open-access, where any interested party can find: (1) The original scripts used to run the experiments and generate the results, (2) the parameters the models used for comparison in the experiments, avoiding the need to retrain the models test the results, (3) the datasets used in each of our experiments, including the custom partitioning of the dataset of Lin et al. (202T), (4) the scripts used to summarize the results into the figures used in this paper, and (5) 3D interactive environments that allow for the visualization of the morphological symmetries,
one of this environments was used for capturing fig. Ieft and its 3D animation
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SUPPLEMENTARY ON DISCRETE SYMMETRIES OF ROBOTICS SYSTEMS: A GROUP-THEORETIC AND DATA-DRIVED ANALYSIS
A APPLICATIONS OF DISCRETE MORPHOLOGICAL SYMMETRIES
In this section, we provide our perspective on possible applications where DMS can be of value and the respective fields of knowledge where these applications play important role.
All applications of DMS for data-driver techniques fall within two categories (1) data augmentation of proprioceptive and exteroceptive measurements, and (2) G-Equivariant function approximation. The fields of knowledge that can benefit from the aforementioned applications are:
Biology Biomechanics, and Experimental Veterinary. Studying the biomechanics and dynamics of animals in nature is becoming a fundamental area of the fields of Biology, Biomechanics, and Experimental Veterinary (Wei & Kording 2018). Considering that around 99% of eumetazoans (most species excluding sponges and other sea species) are Bilaterian (Ferretti al., 2020) (i.e., having approximate C2 symmetry) or Radiatal (i.e., having approximate C, n 2 2 symmetry), DMS become a flexible and natural approach to study the data gathered from the study of these systems. especially of vertebrates, whose dynamics are often approximated to rigid body dynamics (Wu et al. (022).
The process of study of animal motion dynamics normally involves the use of motion capture data of animal motions using marker-based (Prankel et al. 2016) or marker-less (Mathis et al. 2018) sensor pipelines. The data of the markers then either directly pro- cessed or fitted to kinematic models (that make the assumption of exact kinematic symme- tries) and then processed for information retrieval. DMS offers a clear approach to mitigate the cost of data collection by providing simple approach for data augmentation, and for the construction of G-Equivariant NN to process the dynamics of the kinematic chains.
Computer Graphics & Vision. Computer Graphics is perhaps the de facto application field of exact DMS. this area, the kinematic structure of animated characters assumed to be symmetric, they often model the behavior of living vertebrates with C, n 2 2 symmetry. The recorded trajectories are obtained through motion capture data or expert seldomly augmented
artist animations, and to the author's knowledge, the trajectories to their symmetric equivalents. In this field, NNs are commonly used to learn projection spaces where motion matching and animation interpolation becomes easier problem than in minimal coordinate space (Holden et al.) 2015]] Starke et al. 2022) or control policies for physies-based animation (Peng al.] 20188] Ma al.) 2021 However, the exploita- tion of the DMS inductive bias is not common in the field and has been approached (with
specialized and costly algorithms) solely for the C. symmetry in Yeh al. (2019): Ordonezz
Apraez et al. 2022): Abdolhosseini et al. (2019) : Wu et al. (2022).
Robotics. NN function approximators are becoming valuable tool in robotics applica- tions of perception, control (Miki et al. 2022) and statc-estimation (Lin et al. (202]). In all applications, NNs are used to approximate functions processing proprioceptive or ex- teroceptive measurements related to the evolution of the dynamics of the robot. Despite the majority of legged platforms (Radford al. [015] (215))0Giiiiii Grimminger et al.) 22200) Miki al.) [2022) and manipulators having C. 2 2 symmetries, the use of data augmentation (or G- equivariant NN) to mitigate the high cost and risks of collecting data with robotic systems in the real-world (or simulation) is not widespread technique. We believe the framework of DMS and our accn-sccess code can contribute to the adoption of G-equivariant function approximators in the field.
Control In the case of model-based control, it is common to exploit the symmetries of the Euclidean Lie Group in which the robot evolves (section Wu & Sreenath, 2015) and to inherently exploit the equivariance of inertial forces (eq. (4)) by assuming approxi- mate exact symmetries in the kinematic and dynamic parameters of the dynamics model (Mastalli et al. 2022). However, to the authors' knowledge, no approach exploits DMSs (and especially the discrete symmetries of joint space Q,)) in applications of exploration
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of space, planning. and trajectory optimization, where DMS offers technique to avoid the computation of symmetric trajectories.
In model-free control, specifically in reinforcement learning (RL), the exploitation of sym- metries in the dynamics implies mitigation of the sample inefficiency and sensibility to local optima that these learning algorithms have. Previous works have shown the impact of symmetry data- augmeetatinnnnn (Weissenbacher et al.] [022] Ordonez-Apraez et al. 2022) and of G-equivariant function approximators (Van der Pol et al. 2020 Ordonez-Apraez et al. 2022) on model-free RI.
B LIMITATIONS
Our work makes two main assumptions:
1. Symmetries exact: By assuming that dynamical system has exact and not approx- imate symmetries we are departing from the real-world nature of DMSs, since for any robotic system in the real-world the manufacturing and assembly process introduces rors/tolerances in the kinematic and dynamic parameters of each of the robot's bodies. Likewise, the dynamics of animals in nature not perfectly equivariant since morpho- logical symmetries are only approximate symmetries. Although exact symmetries seem to be a strong assumption, in practice, the reality is that it is a common assumption in the fields of robotics and control theory, in which idealized models of the dynamics are often assumed, implying exact DMSs through the exact symmetries in kinematic and dynamic parameters (which are responsible for the equivariant nature of the generalized mass matrix M(g 9) = gM)g)) deq. (4) and therefore, for the equivariance of inertial, centripetal, gravitational, and Coriolis generalized forces).
On Onssection 6) ww show that the exact symmetry bias is justifiable and beneficial for learning function approximators processing the dynamics of approximately symmetrical systems in the real world. However, the authors highlight the necessity to properly address the case of approximate equivariance, which we leave to future work. To address this case, system identification techniques (Simpkins] (2012) have been wildly used to approximate the deviation of the kinematic and dynamic parameters from the assumed values. While in the case of Geqqiivariant NN Wang et al. (2022); Finzi et al. (2021a) provide clear and valuable approaches to learn approximate G G-equivariant NN.
It is relevant to highlight that, in physics-based simulation, the most common practice is to work with the idealized model of dynamics. Thus, the assumption of exact symme- tries is justifiable and encouraged in applications where simulation is relevant tool (see supplementary A).
2. Linear time computation of the basis B of equivariant linear maps is restricted to group actions with regular matrix representations that are generalized permutation matrices: The algorithm for computing B in ((m)) time and determining analytically the rank of this space (see section 5) is restricted to the scenario where Pa (g) is a generalized permutation matrix (which occurs when both (g) and po.)(g) are also generalized permutation matrices). Although this assumption always holds true for g = C2 (the most common DMS) and for all action representations described in |section 6 and supp.fig 5 and fig. 1 in general, it might not hold for |9| 2. In cases where either P... P.)((g and Po [g)) are not generalized permutation matrices, the computation of can be approached using the Krylov subspace method proposed by Finzi et al. (2021b) with complexity (r² (mm)?) and numerical approximation of r = rank
Pw(g) E R
Although this seems like strong assumption, consider that in the case of DMS groups G:
All action representations (.) acting on the joint-space manifold Q, and its tan- gent space, are generalized permutation matrices. This property holds true when using the common convention of minimum coordinates for q and q. in which each vector of the orthogonal basis of Q corresponds with degree of freedom of the system. With this convention any DMS action g acting on a single degree of freedom can be defined as a function of a single degree of freedom g q = g(ū;) s.t. q 4 E q.
All action representations (g..(g), P... (g)) of the latent vector spaces of internal layers of equívariant neural network (e.g.. in fig. 1) can be arbitrarily parameterized
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(while respecting the group axioms). Singe G is by definition finite symmetry group, we can parameterize (..(g) and Powe (g) to be generalized permutation matrices.
C PROPERTIES OF ROBOTIC SYSTEMS WITH MORPHOLOGICAL SYMMETRIES
For clarity of the explanation, let us imagine two different Euclidean spaces and two versions of the robot: the original space (with reference frame o) and robot with coordinates q and q. and the virtual rotated/reflected space (with a reference frame J, with configuration "X² RS rg and virtual robot with coordinates I and 4 referenced to Noting that in the case of a reflection, the virtual robot has reflected versions of each rigid body.
Foreqs. (5) hold, there must exist action y E 9 transforming the real robot configuration 9 9.9 resulting in the same kinetic energy as the virtual robot's 's kinetic energy:
nn T( 9.9 2.9.00 9) >
nn >
fgg g.s + T 9. m
L.
+ W = T 9.9-9))
(10)
where gg.i Wg.i m, and I, are the linear and angular velocity, mass, and inertia matrix of the transformed body i (referenced to o). Likewise, F,, Wú, mi and I, are the equivalent quantities for the virtual robot and body (referenced σ).
C.I KINEMATIC SYMMETRIES:
Ignore momentarily the influence of the mass and inertia terms of the real and virtual bodies. We can assert that for sorpsppe.c to hold, the transformed configuration should result in a kinematic tree indistinguishable from the virtual robot's. Thus, for everybody i in the real robot kinematic tree, there should exist an equivalent virtual body l (as seen in suspp.figg 5] not always k = i). By equating the linear and angular velocities of the real and virtual bodies, referenced to o, and expressing the velocities as functions of the generalized coordinates we obtain:
gg i Rg i&s Jn (g 9)9 Rg Jr. (q)4 Jr. q)g Rg (q)
Wg wy Rg Ry wk (g g)g
R2 Rg JR. (q)4.
(11)
Jn. q)g
Ry R Jn, (q).
where Jr, (q), JR. (q) E 3 are the position and orientation analytical Jacobians (describing the instantancous velocity vectors contributed by each DoF to body i) of the real robot at configuration (Wieber 2006). Formulating supp.eq 11 for each of the n. bodies of the robot we obtain at best ng x 3 X non-linear equations that can be used assert if exists. In practice, the action representation (g( and especially its component acting on the joint space Pa, (g) can be trivially determined by solving supp.eq II (or equivalently eq. (7)) for each body from top to bottom of the kinematic tree (i.e., base first, end-effectors last), if g exists.
C.2 REFLECTIONS/ROTATION REQUIRE MODULARITY OR SYMMETRIC RIGID BODIES
Let us assume kinematic symmetry and direct our attention now to the influence of the mass and inertia terms on the kinetic energy of single rigid body when it is transformed with the action g. which imitates a true reflection of space g. Focus on the first two columns of supp.fig 4 Because of the kinematic symmetry the CoM of the reflected and transformed bodies coincide, both bodies have equivalent linear components of kinetic energy. However, for arbitrary rigid bodies, the reflected and transform bodies will have different angular components of kinetic energy. Note that in the general case, the transformed and reflected bodies' inertia will differ, thus even if both bodies have the same angular velocities, their kinetic energy will differ.
Let p, Py and Py be frames located at the CoM of the original, reflected and transformed bodies, aligned with the principal axes of inertia of each of the bodies. Similarly, denote 'I and II as the original and transformed bodies inertias referenced to o, and ", as the reflected body inertia referenced to the reflected Fuclidean space o. In order to comply with ssuppeq 11 must ensure
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2
e
PW
9
Wiy
Pid.
10gg
Supplementary Figure 4: Properties of bodies capable of imitating a true reflection of space (w.r.t the yz-plane in this case), with proper transformation g involving only a rotation and translation. The first row shows the original bodies with their respective angular velocities w, subjected to trivial symmetry transformation (dashed lines represent the principle axes of inertia of the bodies), and the second and third rows display the effect of and on the bodies and angular velocities, respectively. The first column displays rigid body with symmetric mass distribution, for which g exists, as the reflected and rotated bodies share an equivalent angular kinetic energy. The second column shows a rigid body with asymmetrical mass distribution, for which the rotation y, that produces a kinematic symmetry, results in the reflected and rotated bodies having different angular kinetic energies leg. (2).. The third column shows two bodies with asymmetrical mass distributions, each reflected version of the other, in this case, the action swaps bodies configurations to imitate the configuration and energy state of the reflected bodies transformed with g. Angular velocity is pseudo-vector (or axcal-vector), for which reflection transformation is computed wa RR Ry w (see Quigley (1973).
that:
w T II o w
-w.
I
w
g g
R w T I, RR ww) "w 1 w
R7 Rr ww
"I Rg I R
R Rgg I, b R. " RT
IR*PPo TT Rg RR' P
RR
Rg
II=
R Po RR Rg R
P
P R
T R TRR.
R = R R RT
P Po
P9
(12a)
What lqq. (12a) states that in order for the reflected and transformed bodies to have equivalent angular kinetic energy, both bodies should have co-linear (or aligned) principal axes of inertia. This allows us to describe 'R's as a function of the original body configuration "R" and two reflection matrices: the true reflection of space R. and body specific diagonal reflection matrix " "R"a, which exists only if the rigid body has a symmetric mass distribution. A visual example for symmetric and asymmetrical rigid bodies is presented in supp.fig leeft and middle columns.
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C.3 MODULARITY KINEMATIC TREES
As previously described, body i of the real robot should have reflected equivalent body k with the same CoM position and aligned principle axes of inertia, up to reflection of any of these axes. If body is unique in the kinematic tree, then & and the body must have at least symmetric mass distribution. When body is not unique (i.e. there exists another body h * i that is its true reflected version), then constraint of symmetric mass distribution is imposed i, only the alignment of the principal axes of inertia is required. A didactic example of this scenario is presented in supp.fig 4rright, for Bolt's legs in supp.fig and for Solo's legs in fig.
C.4 SYMMETRIC POSITION AND VELOCITY CONSTRAINT CONFIGURATION SPACES
Although it is implicitly implied on eq. (2) that the constrained position Q and velocity T con- figuration vector spaces should also be symmetric or equivariant, this property might be casily over- looked. As mentioned in section 4 the relevance of morphological symmetries relies on the equívari- ant nature of the system dynamics (egs. and
(3) which imprints symmetry constraints on optimal control policies and proprioceptive and exteroceptive measurements. However, with non-symmetric constrained configuration spaces, eq. (2) will not hold for every system state q E Q. á E TqQ, and any uncontrolled controlled trajectory of the system dynamics shall not have a symmetric equivalent trajectory, as this has the potential to violate the constraints of the configuration space.
D EFFICIENT CONSTRUCTION OF G-EQUIVARIANT NNs FOR DISCRETE MORPHOLOGICAL SYMMETRY (DMS) GROUPS G
As mentioned in section our work builds upon the framework for the construction of G-equivariant NN of Finzi et al. (b) The core limitation of this framework is the inability to handle large dimensional spaces, due to the computational and memory complexities. For instance, for an equiv- ariant layer with input dimension and output dimension m, the computational complexity of find- ing the equivariant linear map basis B (which is quadratic ((mn) (r2m))22)) through the Krylov sub- space method) and the memory complexity of B gmaxr S mn, become easily intractable for moderate and m dimensions. This limitation is openly discussed in the EMLP repository README. but regretfully not in the original paper.
In practice, we found these limitations when trying to construct the equivariant version of the Contact CNN (Lin al. 202T) p in our second experiment. This architecture its internal layers has n, > 2000, for which: (i) the Krylov subspace method complexity renders the operation intractable with standard hardware and (ii) the matrices B of internal layers required storage of [ >> for moderate input output dimensions (m, n ~ 250) and 1 [Pb) > for m, n > 2000). See ssupp.table 1 for comparison between dense and sparse matrix representations.
D.1 TRAINABLE PARAMETER REDUCTION OF G-EQUIVARIANT LAYERS (FOR DMS GROUP)
Determining analytically the number of trainable parameters (i.e. the rank r) of an G-equivariant layer is, in general, an unresolved problem. However, for DMS groups, r can be computed once the input-output action representations are known. The requirement to compute r is that actions affecting the linear maps are a semi-direct product of the input-output groups, and the input- output representations are generalized permutation matrices. These conditions are met for most DMS groups (see supplementary B).
The equivariance constraints eq. (9) linear maps of perceptron (or convolutional) layers imply a reduction of trainable parameters from w = mn to |c] = r S mn. For DMS groups, r is associated with the number of unique orbits of the elements of w. Thus we can compute this value using the orbit-counting theorem (also known as Burnside's ss Lemma), which states that the number of orbits the average number of fix-points of G, that is r Р ΣnES /w0), where w = {w w. g w = w represents the set of elements of w that are invariant to (i.c. fix-points). Those fix- points be identified by the elements the diagonal of (w(g) that are equal to one. Therefore,
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Tri-Finger Robot C3
Bolt Bipedal Robot G C2
122
Bolt a bipedal robot with sagittal plane reflec- tion symmetry (g = C2). This morphological sym- metry allows it to imitate the effect of arbitrary re- flections of space (g Es) by re-configurnng its base and legs. To identify this symmetry group we apply the procedure section 4.3)
This fixed-based robot is symmetric space by 120 the vertical axis. There- fore, its symmetry group is the cyclic group of or- der three (g = Ca). To identify this symmetry group apply the procedure in insscciion 4.3
rotations
Identify XB and Tg: As fix-based robot, define Xa to the mounting structure support- ing each finger, and the gray disk delimiting the workspace (see image).
1. Identify X and Ta: X is the robot hase (hips) body, with its corresponding inertia 1 Identify symmetries of Iæ: The base body has symmetrical mass distribution w.r. sagittal plane. Thus, Ig invariant to the transforma- tion X (7) E {e, (e, Ca (eq. (6)).
2. Identify symmetries of Is: The inertia of this virtual base L invariant to rotations by 120"
in the vertical axis. L.c. Ig is invariant to Xap:, (7) E (e, Do. 9ol C3 (cq. (6).
the kinematic tree: There are two symmetric kinematic subchains. The left leg subchain and bodics are reflected versions of the right leg subchain and bodics.
Identify modularity
3. Identify modularity in the kinematic tree:
There three symmetric kinematic subchains. Each finger composed of replicated versions of the same bodies.
Identify the DMS group G:
Since reflection w.r.t to the sagittal plane would imply a true reflection of the rigid bodies of the legs, we need to permate each body in the kine- matic tree with each reflected version. Denote the joint-space á = [4] 4 composed the left L and right R legs' DoF (qL/R E RR'), Denote the sign- relation between the DoF of the Left and right legs' degrees freedom as $LJR E R.
4. Identify the DMS group G:
Consider that the transformation Pe, XB)
Peg (8) P2, (7)* lcq. (6) can be interpreted as rotation of the virtual base 8 followed by rotation -0t in the z axis. Thus respecting the fix-base constraint the system. Denote joint- space = q = [4)1 9/2- Ffl be composed of each finger's s DoF (qfz E R².).
Pg)(g) & C2. Being p-)(-) the unique rep- resentation the actions of C2 2-dimensional space (2 subchains). the group this is (7.) = 19
Then we can define Pa. (g) (s))
Then define pa, (9) = Pao @ Is 1 Cg. Being (-) the unique representation
the actions of C3 on a 3-dimensional space (3 sub-
the non-trivial action
chains). For the generator action of the group this [ 000
is Pas (g)
1100
Lastly, recalling the definition of Peg (g) in eq. (6)
Lastly, verify if C. by testing all tentative group actions for DMSs cq. (5)
C2 by testing all tentative group actions for DMSs oq. (5)
we verify
Augmentation of data samples: Say collect dataset of robot states (q, q)) and cube states Xe every time step f, to train the manipulation policy (Funk et al. 2021). ). To obtain the symmetric states, at every I, we need to understand that since we are imitating the effect of true rotation of space g. the symmetric states obtained by (g q. I 9) and g X Pzg (T) Xc).
Augmentation of data samples: Say collect a dataset of robot states (q, q) and ground reaction space generalized forces as (T Tfx). at every timestep I. Aiming to train reactive locomotion policy (Ordonez-Apracz
forces (fr, fR). that we transform
all 2022). The symmetric states, every are then defined as: 19.9.9 ②) and (g T ) = (pa(g)T Pa(g)T/
Supplementary Figure 5: Example morphological symmetries of the Tri-Finger (Funk et al., 2021) and Bolt robots, allowing to imitate a rotation space (left) and a reflection space (right).
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Supplementary Table Comparison of memory complexity individual layers of the equiv- ariant version of Contact-CNN Lin et al. (221)) (ECNN). This example compares the sparse and dense representations of matrices Pu (g) E Rm mmn for the symmetry group G C2 of the Mini-Cheetah robot, with r m"/2 (see supp.eq 13). Here, Nt m represents the input and output dimensions each layer. The dense memory complexity of all action representations increases with the group order [9] while the mem- ory complexity for decreases with larger group orders (since r S mn becomes smaller). We assume floating point representations with 32 bits.
Faxx x and the [g] group action representations
Dense Memory [Bytes]
Sparse Memory [Bytes]
Layer Type
771
(g)
B
P.. (g)
ID-Conv
54
64
764.41M
191.10M
221.18k
110.59k
ID-Conv
64
64
1.07G
268.43M
262.14A:
131.07k
ID-Conv
64
128
4.29G
1.07G
524.28k:
262.14k
ID-Conv
128
128
17.18G
4.29G
1.04M
524.28k
Percept
4736
2048
6.02P
1.50P
620.75M
310.37M
Percept
2048
512
70.367
17.597
67.10M
33.55M
Percept
512
16
4.29G
1.07G
524.28k
262.14k
for G-equivariant layer, the number of trainable parameters is determined by:
7 3 Egeo (g)= E Xpmn((")) Xpous (g),
(13)
denoting (g) : G N the number of fix-points of the action representation p(g). Therefore, the number of trainable parameters can range from /wl/ja| I S |w), depending the fix-points of the layers' input and output spaces.
D.2 PARAMETER INITIALIZATION OF EQUIVARIANT LAYERS FOR DMS
Consider a Equivariant Neural Network architecture composed of multiple layers of equivariant linear (or convolutional) layers of the form 'y == a ('W'a x + 'b), being / the layer index, 'æ E IR" and " E R the layer's input and output vector spaces, ' = >ix lc 'B.,. mg the layer's linear map. 'B mmxxxx the layer's basis vectors spawning the space of equivariant linear maps, cc E R' the layer's trainable parameters, and E R" the layer's bias vector.
For the optimal flow of information throughout the network, it's relevant initialize the trainable parameters such that the variance of activations (during inference/forward wttp:n///fooaaadprpaaaaiooo and gradi- ents (during back- sock-propaggtion) is kept constant, avoiding activationssgradients from vanishing or exploiting (Glorot & Bengio 2010
The derivation based on the equivalent process for unconstrained layers presented He et al. (2015). Let the layer's activations before the non-linearity be denoted by 'z '''x + "b such that y = ('z), and note that 'r: = I-ly. Furthermore, we will assume the elements of L and " mutually independent and sampled from two independent distributions, denoting the random variables of the two distributions as lc and xx
See Pierre Ouannes blog: httpp///wmmooggooooooooooooooooa
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The core difference in the initialization of unconstrained and equivariant layers lies in the way the linear map is parameterized. For equivariant layers we have:
Var( W x b))
Var ('Wi, xx))
Var( (b) =0
Σ
Var
ck B Bmnnn
'xj
Var ( ('c'x)
B
m,n,k
Var
Σ
So const
Σ
2 Var
Aig
(14)
In the forward dewaar-proppgation scenario, we are interested in conserving the variance of the activations throughout layers, that is we must ensure Var( 'z) = Var( zz). Using supp.eq. 14 we obtain:
Var( 22) = Var( W''e + '6)
Var('v) '2) XuB Var ('c'x)
AIB Var( 'z)
E =((=)) ( (x)
m
Var(te)
Var( 'z) ABB Var( c)L (1-1y2)
1 l-ly o o((-1z)
Var( 'z) = >ιgÀo Var( Var( (-1
E 1- (1=1y2) to Var (1
(15)
Var( 'z) = Var( (-1z)
Var (*c) Au
m
(16)
where Aa in supp.eq eq 15 is a non-linearity dependent scalar computed analytically or empirically (see He al. (2015)) In supp.eq 16 we conclude that if we sample the equivariant layer trainable parameters 'c from a distribution ensuring Var('c) c) m A. the variance the activations across
equivariant layers remain constant. similar procedure can be applied to the backward propagation case, concluding that in order to maintain a constant variance of the gradients across the network layers we should sample the trainable parameters ensuring Var(fc) Ar As remarked He
al. (2015) both variance values for the forward and backward propagation cases lead the proper flow of information in the network. On figp.fig& 8 it can be appreciated that our method achieves equivalent results for equivariant architectures as He al. (2015) does for standard linear and con- volutional architectures.
E IMPLEMENTATION DETAILS & CODE
Additional to this section, provide open-access code with the scripts for reproducing the periments of this work, the parameters of the models used for comparison, along with additional interactive examples visualizing morphological symmetries of both robotic systems and data.
E. EFFICIENT DATA AUGMENTATION
Since any input æ and output y spaces of equivariant architectures have matrix symmetry action representations, Pa (g), Pv(g). it is possible to perform batched data augmentation, reducing the computational complexity of augmenting batch of N, samples from N, matrix-vecte multiplica- tions to a single matrix-mate matrix multiplication, preferably performed after data loaded to GPU for optimal performance.
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(h)
Supplementary Figure 6: (a) Atlas robot C2 symmetry affecting proprioceptive (g. á. q. T) and exteroceptive measurements (ground reaction forces, terrain surface and height estimations) (see 3D-animation). (b) Robot Solo sagittal (blue) and transversal (red) symmetry planes of the base rigid body, allowing the system to possess K4 symmetry
E.2 HYPERPARAMETER TUNNING
The only hyper-parameter tunned for each model and model variant was the learning rate. For all model variants presented in this work (except the original Contact-CNN model from Lin et al. 22221), which we retrained using the same hyperparameters reported by the authors) we ran grid- search in log-scale among 20 different learning rates. In this scenario, always used the entire training dataset and optimized w.r.t computed loss in the entire validation partition. The learning rate values used for each model are depicted ssupp.table
E.3 EXPERIMENT: COM MOMENTUM ESTIMATION
The dataset for the CoM estimation experiment was generated using Pinocchio (Carpentier et al., 2019), which in turn uses the URDF models of the robots Solo and Atlas, to extract the kinematic and dynamic parameters required to compute the Centroidal Momentum Matrix Ac(g) matrix OOrn al. 2013), with which computing the CoM momentum reduces to:
g h Ag(g 4)g-6
Bg E G. 4g E EG.
(17)
g h / (g a. (
(18)
Where supp.eq 17 expresses the analytical G-equivariant function to compute the CoM momentum. While supp.eq is the approximation this function by an G-equivariant NN, with parameters Φ.
E.3.1 DETERMINATION OF THE INPUT AND OUTPUT REPRESENTATIONS Pa (g), (.(g) I E G
Both robots Solo and Atlas evolve in the Fuclidean space of 3-dimensions. Therefore their configu- ration space can be decoupled into Q = E, x Q., After identifying their symmetry groups and their corresponding E3 and QJ representations (Pe, (g), (g)g) V g 9), identifying the representations of the input and output spaces of the NN function approximator (supp.eq 18 becomes a trivial task considering that:
Pru f
RJ /
((Pooo
(g)
(g. F E EG E3 (19)
Pe (
k
Pa (o)
Po
Pu (g)
Defining æ
[ E R and y h E 2d RC. Note that by definition any improper transformation applied to a pseudo-vector (e.g. angular velocity/momentum, torques) computed R R k.
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E.3.2 PRACTICAL DETAILS OF THE DATASET GENERATION
The URDF files of the robots Solo and Atlas generated using XACRO scripts, which repli- cate the structure of limbs to their symmetric counterparts, making the dynamics of the robots in simulation exactly G-equivariant. However, the algorithm for computing the CoM momen- tum from Pinocchio is numerically sensitive, resulting in the orbits of the momentum g h de- viating slightly from the theoretical orbits. Therefore to reduce numerical errors and ensure theoretical equivariance of the data, we replace every target variable by the average of its orbit | EEG.
h
. E g 1(AG Pa
E.4 EXPERIMENT: STATIC-FRICTION REGIME CONTACT DETECTION
The dataset presented in (Lin al., 2021 composed of output samples y E R where each dimension of y represents logit of a specific contact state, among the 16 different combinations of each of the legs possible binary contact states. The input samples y {2, {2,);=0 150 R54x are history 150 samples 2 = (g. q, a, w p. v| E R244, Where q E R"", E R aR w E R2, . E R E IR are the MIT-Mini- MIT-Mini-Cheetah robot joint-space positions, velocities, base linear accel- eration, base angular velocity, and each of the four legs feet's position and velocities, respectively, referenced to the robots base frame B.
The function approximator to learn is expected to be approximately equivariant to the reflection group C2, considering the sagittal symmetry of the robot morphology. Therefore:
g y f (g x; Φ)
g g C2
(20)
E.4.1 DETERMINATION OF THE INPUT AND OUTPUT REPRESENTATIONS pa Pz( g
The MiniCheetah robot evolves in the Euclidean space of 3-dimensions. Therefore its configuration space can be decoupled into Q = E3 x QJ. After identifying their symmetry groups and their corre- sponding Ez and Q representations (Pe, (g), Pa (g) V 9), we can identify the representations of the input and output spaces of the NN function approximator (supp.eq 20), considering that:
Pqy
0
0
0
PQ2
0
0
0
Pv(g)
PEg
0
0
(g. F)lg e &gg g e Es
0
08g
(
0
Pp(0)
0
0
Pp(a)
(g)
Where the representation Pr(g) acting p E I2 and R is determined understanding that each of the feet positions (PRE. PLF: PRH. PLH) and velocities (URF, ULF, URH (LH) are simply vectors living in Ez. Thus, must apply the euclidean action Pr. (g) while at the same time permuting the feets (similar to the permutation of the kinematic subchains described by Pa,(g)) (g)):
= Pat (g) 89 Prg (g)p.
(g) 8 Pr. (g)
V4g. g)lg E G,9 E Eg
Pp(9)
Po(a)
(21)
D 1
PRP PLF DRH pi.
10100 0
URF e PRH DLH
0
8 Pcg (g)
1000
X P (g)
( g)[C2 = {e,g}
0 0
000
10010
(22)
Being (g) Vg E G the unique representations of the finite group G in a 4-dimensional space, representing the 4 kinematic tree's subchains. This representation, for the non-trivial action of the MiniChectah DMS group G x C2 is expressed in in/supp.eq 22 The nature of this representation might be better understood if you consider that Pa, (g) Pa+ (g) I,,, being n, the number of degrees of freedom of the kinematic subchain (which for the case of MiniCheetah 3 DoF). See simpler examples in supp..f.g 5
Lastly, the representation for the contact state (g) is given by the permutation matrix relating y and g y described in supp.ttable
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Supplementary Table 2: Robot and Model parameters for CoM and Contact state estimation.
Robot
Model
Ir
Data Samples
Solo
MLP/EMLP
Ca/K3
2.4
10-3
100k
Atlas
MLP/EMLP
C2
1.5 x 10-3
500k
MIT-MiniCheetah
CNN
C2
1.0 x 10-4
730k
MrT-MiniCheetah
E-CNN
C2
1.0 x 10-5
E.4.2 DETAILS ON DATASET PARTITIONING
We modified the original dataset partitioning properly evaluate the generalization capacity of the models. The original dataset was composed of 15 different recordings varying ground type and gait type used during data collection (most recordings were performed on a trot gait, which heavily biased the dataset to contact states 0, 6, and 9 of supp.table 3 See supp..ig 7).
The authors of (in et al. 2021 partitioned all 15 recordings into (70%, 15%, 15%) training, vali- dation and testing. This partition was made such that the first 70% time-samples of each recording assigned for training, the following 15% validation, and the rest for testing.
Because we are interested studying the generalization capacity of the models and the out-of- caaniingdistributiin performance, modified this partitioning such that among the 15 different recordings we selected randomly 5 recordings for testing, and the remaining 10 recordings were used for training splitting these recordings into (85,, 15%) training and validation splits as in Lin et al. (2021), that is, for each recording, the first 85% data-samples go for training and the remaining for validation.
The selected training-validation recordings were: air walking gait, concrete .iffretulifficllt.cmmm difficcllt.sipprrry concrete left. circle, biiddle-pebble roak asphalt concrete pnecrtt-gaaloping.. grass, old_asphalt_road, sidewalk. While the selected testing recordings were: air.jumping eircle, forest, small pebble.
concrete concrete_pronkingg pronking, concrete rightteircle,
E.5 MITIGATION OF SUBOPTIMAL ASYMMETRIES IN MODEL PERFORMANCE
When comparing individual leg classification a similar performance for each symmetric pair of legs, while the unconstrained models converge to an asymmetrical suboptimal state favoring the contact detection of leg the expense of reduced performance for the symmetric leg (see LF and RF fl-scores). This asymmetrical performance is attributed the CNN and CNN-aug models learning extract temporal features for both symmetric legs separately, increasing the likelihood of converging to asymmetrical local minima. On the con- trary, the equivariant model E-CNN can thought of as learning extract single set of symmetric temporal features for each symmetric pair of states (a consequence of the model's 's equivariance and parameter sharing). This implies that the temporal features used for determining the contact state of, say the left frontal leg, would also be used to determine the contact state of the symmetric leg, the right frontal leg, and vice-versa.
that the equivariant model converges to having
E.6 EQUIVARIANT CONV ID LAYERS
For details the construction of the Equivariant ID Convolutional layers reefer P Note that the symmetry of a single time-sample z; is shared across all time-samples = 120
150
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0.5
2300C0
2000C0
150300
R01
U1
06-
100000
04
62-
0.1
-25 0.0
05
-25
0.0 0.5 1.0 15
Supplementary Figure 7: Sample distribution of the training set of the contact dataset presented by
Lin et al. (202T). The left figure shows the distribution of the binary contact state for each of the mini-cheetah legs, clearly displaying similar distribution among legs favoring states of no contact (approximately for each leg 60% of data samples represent states no contact). The right figure shows the distribution of contact states (y E RR1),, clearly showing the dataset imbalance caused by the majority of the data being collected by the robot performing a trot gait.
Supplementary Table 3: Symmetric contact state for Mini-Cheetah quadruped robot, considering its sagittal symmetry, and morphological symmetry group Cz. Each leg binary contact state (LF: Left Front, RF: Right Front, LH: Left Hind, RH: Right Hind) displayed with its corresponding robot contact state
RF
LF
RH
LH
y
g
y
LF
RF
LH
RH
0
0
0 0
0
0
0
0
0
2
1
0
0
2
0
1
0
O
3
0
COTO
0
OOTTOO
4
0
5
0
6
7
OOTO
8
4
OOTTOO
9
10
OO
VOII
TOOOO
11
I
12
12
O
I
I
13
14
14
13
0
15
15
1
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salo_Act ReLU Activations Distribuse iuutions
solo_Act- tant Activations Distributions
Intiddicaaion Mode lan
nitielized kon Mide - fan
emlp
enlp
2,5
Activatton
6.0
1
-5.0
-7.5
Intiiaization Meda Fan,
Intiiaization Mode Tan Lout
15-
Adtvssion
-15
Intielization Mode no0mall.05
Intielization Mode normato
0.5
Altnelan
Acteasion
6.0
-0.5
005-
-1.0
168
Actvaton
Activaton
-1
5
Layer
Layer
Supplementary Figure 8: Comparison of the initialization method of unconstrained layers of al. (2015) with our initialization method for equivariant layers. Left and right columns correspond to MLP and EMLP architectures with a ReLu and Tanh lincaarriiiiss respectively. Each row shows different initialization methods differing in the variance of the initialization distribution of the layer's trainable parameters. First and second rows show the forward and backward propagation cases of He et al. (215)) for MLP and of supplementary D.2 for EMLP, with Var('c) - m//(Are>.))) and Var('c) = r/(AtgAa), Ar), respectively. In these cases, the variance of activations through the net- work depth remains nearly constant, as desired. The last two rows show the initialization of layer parameters with a constant variance of 0.052 and 0.82. illustrating scenarios of activations vanish- ings and exploiting. All architectures are composed of 10-layers with 256 neurons on intermediate layers. In the equivariant case, the network Ka-equivariant.
25


Reviews:
[{"review_id": "xdluLq8QYF", "reviewer": "Reviewer_Yn9H", "summary": "", "questions": "", "limitations": "", "rating": 5, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "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": "this paper proposes to mimic physically unrealizable symmetries such as reflections, by reconfiguration of dynamics such as left/right legs shift in bipedal. ", "summary_of_the_review": "as a whole it is a good paper. the idea is interesting. the novelty is limited. the applicability is limited.  \n", "strength_and_weaknesses": "Strength:\n1. DMS is an interesting idea\n2. fabulous representation of the background\n\nWeaknesses:\nAs a whole this paper is good. It does contribute to the field and combines different advanced approaches. My main concerns are \n1. that the DBS itself is a tiny part of the paper, which is built on and plugged in to the existing methods.\n2. DBS can not be applicable to many dynamics, which is mentioned also in \"Imitating reflections of space requires symmetric rigid bodies and/or modularity\"\n3. DBS requires significant domain knowledge (when and how to mimic the original symmetry). there is no general principle how to calculate it in an arbitrary dynamics. it is not unique, as there can be multiple reconfigurations mimicking the original symmetry. \n4. the experiments are limited to supervised learning, even though the author method in the text uncontrollable and consolable trajectories, which alluded there would be application to control problems. \n\n\n\n\n\nsmall comments:\ntypos: \"Kinematic three\",\" an angle $\\theta$ unfeasible\".  ", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "4: All of the claims and statements are well-supported and correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}, {"review_id": "t4fiQSNB-r", "reviewer": "Reviewer_vevv", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "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": "This paper studies the discrete symmetries of robotics systems. The authors present a set of results: (1) the identification of the system’s morphological symmetry Group G, (2) the characterization of how the group acts upon the system state variables and any relevant measurement living in the Euclidean space, and (3) the exploitation of data symmetries through the use of G-equivariant/G-invariant Neural Networks. However, the motivations and insights are not clear to me.", "summary_of_the_review": "I find the results are novel and interesting but I am concerned about the motivation and impact of the results.", "strength_and_weaknesses": "Pros: \n+ The paper is well-written and easy to read.\n+ The literature review is comprehensive. The presented results are well-positioned in the existing works.\n+ The presented theoretical foundations for robotics systems are novel and make sense. The authors also present detailed proofs.\n\nCons:\n- The presented results lack a clear motivation. The authors are encouraged to discuss what the presented results can contribute to. The paper says “addressing the memory\nand computational complexity drawbacks of using this type of architecture.” However, no theory or experiments are give to support this. Overall, I am concerned about the impact of this paper.\n", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "4: All of the claims and statements are well-supported and correct.", "technical_novelty_and_significance": "4: The contributions are significant, and do not exist in prior works.", "empirical_novelty_and_significance": "Not applicable"}, "extra_text": {}}, {"review_id": "1BmMN1GFc3", "reviewer": "Reviewer_tbqU", "summary": "", "questions": "", "limitations": "", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "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": "This study focuses on the morphological symmetries in dynamical systems; for example, a bipedal robot is symmetric to the reflection. When applying data-driven method to such systems, the function is desired to have corresponding symmetries. To this end, this study uses the method proposed in ICML,2021 to build symmetric neural networks.\n", "summary_of_the_review": "While the problem statement and the proposed method are clear, they seem combinations of existing studies. It is better to emphasize the novelty.\n", "strength_and_weaknesses": "**Strengths**\n\nThe problem setting is clear, and the proposal is a nice application of Finzi et al., ICML, 2021.\n\n**Weakness**\n\nNovelty and significance appear to be limited. Several studies have already investigated the symmetries in dynamical systems.\nEspecially, Finzi et al., ICML, 2021, on which the proposed method is based, already examined dynamical systems symmetric to continuous groups such as $O(2)$. Discrete symmetries have also examined in the references in Finzi et al., ICML, 2021. It is not clear what main difficulties were and how the proposed method resolved.\n\nThe morphological discrete symmetries in robots have also examined in Finzi et al., \"Residual Pathway Priors for Soft Equivariance Constraints\", NeurIPS, 2021, which is not cited in the present manuscript.\n\nAlso, the differences from previous studies (especially, Weissenbacher et al., 2022) are still unclear. Please introduce Limitations in the Appendix in more detail.\n\n\nIn Section3, the definition and expression about the velocity contain several abuses of notions. $TQ$ is often used to represent a tangent bundle (the space of the pair of position and velocity). $T_qQ$ is the notion for a tangent space (the space of the velocity) at point $q\\in Q$. The map $\\dot g$ accepts $q$ as an argument, but if so, its domain should be $Q$, not $TQ$. Nothing can map a tangent vector $\\dot q$ independently from the position $q$. The map should be the pushforward by $g$ at the position $q$ and can be denoted by (but not limited to) $g^*_q$.\n\n", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}]

Author rebuttal:

Decision: Reject