Equivariant Diffusion Policy
Authors: Dian Wang
Stephen Hart
David Surovik
Tarik Kelestemur
Haojie Huang
Haibo Zhao
Mark Yeatman
Jiuguang Wang
Robin Walters
Robert Platt
Venue: CoRL 2024 (2024)
Keywords: Equivariance
Diffusion Model
Robotic Manipulation
TL;DR: We propose Equivariant Diffusion Policy that leverages symmetries in diffusion policy learning to improve sample efficiency.

Abstract: Recent work has shown diffusion models are an effective approach to learning the multimodal distributions arising from demonstration data in behavior cloning. However, a drawback of this approach is the need to learn a denoising function, which is significantly more complex than learning an explicit policy. In this work, we propose Equivariant Diffusion Policy, a novel diffusion policy learning method that leverages domain symmetries to obtain better sample efficiency and generalization in the denoising function. We theoretically analyze the $\mathrm{SO}(2)$ symmetry of full 6-DoF control and characterize when a diffusion model is $\mathrm{SO}(2)$-equivariant. We furthermore evaluate the method empirically on a set of 12 simulation tasks in MimicGen, and show that it obtains a success rate that is, on average, 21.9\% higher than the baseline Diffusion Policy. We also evaluate the method on a real-world system to show that effective policies can be learned with relatively few training samples, whereas the baseline Diffusion Policy cannot.

Equivariant Diffusion Policy
Dian Wang' Stephen Hart", David Surovik', Tarik Kelestemur", Haojie Huang', Haibo Zhao', Mark Yeatman', Jiuguang Wang", Robin Walters', Robert Platt Northeastern University Boston Dynamics Al Institute https://equdiift. tit.ibb.iii
Abstract: Recent work has shown diffusion models are an effective approach to learning the multimodal distributions arising from demonstration data behavior cloning. However, a drawback of this approach is the need to learn denoising function, which is significantly more complex than learning explicit policy, In this work, we propose Equivariant Diffusion Policy, novel diffusion policy learning method that leverages domain symmetries to obtain better sample effi- ciency and generalization in the denoising function. We theoretically analyze the (0(2) symmetry of full 6-DoF control and characterize when a diffusion model is (0))))-quivariann We furthermore evaluate the method empirically on a set 12 simulation tasks MimicGen, and show that it obtains a success rate that is, on average, 21.9% higher than the baseline Diffusion Policy. We also evaluate the method on real-world system to show that effective policies can be learned with relatively few training samples, whereas the baseline Diffusion Policy cannot.
Keywords: Equivariance, Diffusion Model, Robotic Manipulation
Introduction
The recently proposed Diffusion Policy [1] for- mulates robotic manipulation action prediction as diffusion model that denoises the action con-
Denobo
A
AP
ditioned on the observation, thereby better cap-
"ifuue
turing the multimodal action distribution of the demonstration data in Behavior Cloning (BC). Although Diffusion Policy often outperforms baselines benchmarks [2, 3], key drawback is that the denoising function is more complex than a standard policy function. In particular, for a single state-action pair (x, a). the denois- ing process uses a mapping (s, a + e s k) + g for all possible i and e*, where el Gaussian noise conditioned on step k, which harder to train compared with explicit BC
Denotoo
AH
Figure 1: Equivariance in diffusion policy. Top
left: a randomly sampled trajectory. Top right: a valid trajectory after denoising. If the state and the random trajectory are both rotated (bottom left), and rotate the noise accordingly in the denoising process, we will end up with success- ful trajectory in the rotated state (bottom right).
In this paper, we leverage equivariant neural models to embed task symmetry as an inductive bias the diffusion process, making the denois- ing function easier learn. Although equivariant diffusion models have been studied by a number of prior works [4, 5, 6, 7, 8], our paper is the first study the idea in the context of visuomotor policy learning. As illustrated in Figure 1, rotation of state and noisy trajectory action about the gravity axis (i.e., rotated on the tabletop) results in corresponding rotation of the denoised trajec-
Part of the work was done as an inter at the Boston Dynamics Al Institute
8th Conference Robot Learning (CoRL. 2024), Munich, Germany.

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tory. As a result of this symmetry, our model is more data efficient and generalizes better than the non-symmetric baselines, mitigating the high data costs typically associated with diffusion.
Our contributions are as follows: 1) we propose Equivariant Diffusion Policy, a novel BC approach based on equivariant diffusion, we analyze the conditions under which the denoising function is equivariant, 3) we theoretically demonstrate the use of control for robotic manipulation, which prior methods [9, 10] leveraged in less expressive SE(2) action space, and 4) we provide a thorough demonstration of our method in both simulated and phys- ical systems. In simulation, we evaluate on 12 manipulation tasks in the MimicGen benchmark [11] and outperform the baseline Diffusion Policy by an average success rate of 21.9% when trained with 100 demos. On hardware, we show that successful policies can be learned with small number (be- tween 20 and 60) of demonstrations for six different manipulation tasks, including a long-horizon bagel baking task, while the original Diffusion Policy performs poorly in this low-data regime.
in the context of 6- 6-DoF
2 Related Work
Diffusion Models Diffusion models [12] learn distributions by modeling the reverse of diffusion process, which is Markov chain that gradually adds Gaussian noise to the data until it transitions to Gaussian distribution. Denoising diffusion models [13, 14] can be interpreted as learning the gradient field of an implicit score during training, where inference applies a sequence of score op- timization steps. This new family of generative methods has proven to be effective for capturing multimodal distributions in planning [15, 16] and policy learning [17, 18, 19, 1, 20]. However, these methods did not leverage the geometric symmetries underlying the task and the diffusion process. Xu al. [21], Hoogeboom et al. [4] show that leveraging SO(3) symmetries from the domain in the diffusion process dramatically improves sample efficiency and generalization ability in molecular generation. EDGI [6] extends diffuser [15] to equivariant diffusion planning with improved perfor- mance, but relies on the ground truth state as the input. Ryu et al. [7] propose bi-equivariant diffusion models for visual robotic manipulation, while limited to open-loop settings. By contrast, we exploit domain symmetries during the diffusion process to attain an effective closed-loop visuomotor policy.
Equivariance in manipulation policies Robots operate within a three-dimensional Fuclidean space, where manipulation tasks inherently encompass geometric symmetries such rotations. Re- cent works [9, 22, 23, 24, 25, 26, 27, 10, 28, 29, 30, 31, 32] compellingly show that improvement in sample efficiency and performance can be obtained by leveraging symmetries in policy leaning. [33, 34, 35] show the efficiency of equivariant models for on-robot learning. [36, 37, 38, 39] learn an open-loop pick and place policy with few demonstrations. While this prior work either considers symmetries in SE(3) open-loop or SE(2) closed-loop action spaces, our paper studies symmetries in an SF)) closed-loop action space, and the first one to study the symmetries diffusion policy.
Closed-loop Visuomotor control Closed-loop visuomotor policies are more robust and respon- sive but struggle with learning from diverse trajectories and predicting long-horizon actions. Pre- vious methods [40, 41, 42, 43] directly map from observations to actions. However, this type of explicit policy learning struggles to learn multimodal behavior distributions and may not be expres- sive enough to capture the full range and fidelity of trajectory data (17, 44). Several works propose implicit policies [45, 46] with meeggy-based models [47, 48]. However, training is challenging due to the necessity of substantial volume of negative samples to effectively learn an optimal energy score function for state-action pairs. Recently, [17, 1] model action generation as a conditional denoising diffusion process and demonstrate strong performance by adapting diffusion models to sequential environments. Our work builds on [1] but focuses on equivariance in the diffusion process.
3 Background
Problem Statement We study policy learning using behavior cloning. The agent is required to leam mapping from the observation to the action a that mimics expert policy. Both and can

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contain number of time steps, i.e., o o (m--(m- (:.--)))......1..000:) where m is the number history steps observed and n is the number of future action steps. The observation contains both visual information (images or voxels) and the pose vector the gripper.
{ar, a.+1...., (n-11)
Let Tr E 1x be the current SE(3) pose of the gripper in the world frame, the actions a, specify desired pose A, E R4x4 of the gripper and open-width command wy E IR. The pose can be either absolute (Tz+1 = A,, also called position control) or relative (T++1 = A,T,, also called velocity control). In order to noise and denoise via addition and subtraction as in the standard diffusion process, we vectorize the SF(() pose A, into a vector a, during diffusion and denoising, and orthogonalize the noise-free action vector after denoising.
Diffusion Policy Chi et al. [1] proposed Diffusion Policy to model the multimodal distribution in behavior cloning using Denoising Diffusion Probabilistic Models (DDPMs) [14]. Diffusion Policy lears noise prediction function (o a c* k) ex using network Eg parameterized by 8. The network is expected to predict the noise component of the input a a +** During training, transitions (o, a) are sampled from the expert dataset. Then, random noise gl (conditioned on a randomly sampled denoising step k) added to a. The loss is L = ||Ea (o, a k) 11 During inference, given an observation o, DDPM performs sequence of K denoising steps starting from random action a .N(0,)) 1) to generate action a defined inductively by a k-1
c<(a (e, a², k) +c,,
where ~ ((0,02)). 7,7 a are functions of the denoising step h (also known as the noise schedule). The action a is expected to be a sample from the expert policy π o a.
Equivariance A function f is equivariant if it commutes with the transformations of a symmetry group G. Specifically, Vg E G. (p)()) (()) where p: G - GL(n) is called the group representation that maps each group element to an invertible matrix that acts on the input and output through matrix multiplication. We sometimes leave the actions implicit and write f(xx) = (z). We mainly focus on the group SO( of planar rotations (i.e., rotation around the z-axis of the world) and its subgroup C, containing discrete rotations. There are three particular representations of SO(2) or C, that are of interest in this paper:
1) the trivial representation po defines SO(2) or C. acting on an invariant scalar x E R Ry (o(y)æ x. 2) the irreducible representation Pwr defines SO(2) or Cy acting on a vector E R² by 2 x 2 rotation matrix with frequency W, Pw = cos wg - -ain wg
sinwg cos )t. 3) the regular representation Preg that defines Cu acting on a vector x E IR_ by u u permutation matrices. Let g = T E C.
(1,r1,... ,"=1] and (.i,.... zn) E RR. Then (g) (Iu-mtl,..... Iny 31, T2. Fu-m
cyclically permutes the coordinates of L2"
A representation can also be a combination of different representations, i.., p- p= Pb 0pp pp GL(nn + 2n1 + 222). In such a case, p(g) is an (no + 22n + 2n2) x (no + 2n, + 222) block diagonal matrix that acts on x E +20+ 42n2.
4
Method
4.1
Theory of Equivariant Diffusion Policy
The main contribution of this paper is a method that incorporates equivariance in the diffusion pro- cess for policy learning. As theoretical justification, we first analyze the noise prediction function and show that it is equivariant any time the expert policy that being modeled is equivariant. This implies equivariant neural networks have the correct inductive bias model this function.
Let : o + a be the expert policy function, and let é (o, a² k) + e be the ground truth noise prediction function associated with the expert policy such that = = E(o, (o) + k.) Assume g E (2( acts upon the noise şx in the same way as acts upon the action a. Proposition 1. The noise prediction function is equivariant, i.e., E(go, gat, .t) ge a k). (2(,), when the expert policy function is
i.e., (go)) = (T(()) 9 E S0(2).
3

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See Appendix A for the proof. Figure 2 illustrates the equivariance property of E. If we infer & for all actions in the action space, we effectively acquire a gradient field towards the expert trajectory. The figure shows that such gradient field is equivariant when the expert policy equivariant, thus the function is also equivariant. Notice that the figure shows the average of all action time steps,
L
cia cio.r'.) 11)= >>
Gradient Field
Rot 90
eggo.ge .) /
Fott 90*
4.2 S2( Representation on 6DoF Action
A key step in defining an Equivariant Diffusion Policy is to define how actions a, transform under rotation. We
describe this transformation in terms of irreducible SO(2) representations, which allows us to build the equivariance constraint into the denoising network.
go
Gradient Field
Figure 2: Equivariance of the denois- ing function Left: In observation o, the goal for the gripper is to reach the green block while avoiding the blue obstacle. Right: The expert trajectory and the gradient field associated with the denoising function. If the policy is equivariant, both the denoising function and the entire gradient field is equivari- ant. The orange boxes show the equiv- ariance of e with a particular input ex.
Proposition 2. There exist irreducible representations SE(3) gripper action ac. In absolute pose control, let a, Vece(A.) where Vece flattens an SE(3) pose A₁ E R4xx into a vector by column, ga; = (p) @ (g)a. In relative pose control, let a = Vec, (At) where Vec, flattens Ar into a vector by row. ga, p-1 [ @ p e p2) Pa,, where a fixed change- off-sis- matrix.
that describe how S0(2) acts
We first consider absolute pose matrix corresponding to the SO(2) rota- trol, i.e., Te+1 A., Let Tg be the transformation
Absolute Control
sin 0
(g)
tion along the z-axis of the world frame, Ty sin Dos g 00
where = (pi @ (g) Vectorizing
po(g)
10
0
po(g)
(g) = ( cos g - sing
). The S0( action on A, is gA, = TyA,
sing cos
A, by column gives a (ec) [A]T A A A where A, is the äth column of A,, By the rule of matrix multiplication, we have g = (pi @ A and ga, = (p1 1 p (g)a,,
Since the gripper open width invariant, gut, Po(g) my, we can append we to a, and add an extra Po to the representation. We can also simplify the representation by removing the constants in the transformation matrix and removing the last row in the rotation part of the transformation matrix (i.e.. the 6D rotation representation [49]). The resulting action vector would be a, R 3 x R, where the first six elements are the 6D rotation, the following three elements are the translation, and the last element is the gripper open width. In such a case, we have ga, (p) @ (p1 @P)) @ (g)(g)a
Relative Control For relative gripper pose, i.c., T.+1 = A,T,, the group action on A, satisfies (gA,)T,Ti = (A,Ti) (because the rotation E SO(2) applies to both the current pose and the change of pose). Solving for gå, we get gA,
TgA,TT Let a Vec, where Vec, : _axm + (m-m)) flattens a matrix into a vector by row. Here we want to find a linear action PA that satisfies ga, = (oc,))oc,AAA
= Voc, (T, (T,A,T After solving for PA E IR 16x and calculating change-of-basé matrix such that Pps p-1 is block diagonal matrix consisting of irreducible representations, we have ga, p-1 [ @ p @ p2) Pa, (see Appendix for details). For easier implementation, we add po for the gripper action Wy, remove the constants in the transformation matrix, and decompose SE(3) SO(3) R3. See Appendix C.
4.3 Implementation of Equivariant Diffusion Policy
Now that we have the theoretical grounding of the equivariance in the noise prediction function this section will introduce the network architecture of our Equivariant Diffusion Policy. As is

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Kitchen DI
(b) Nut Assembly DO
(c) Pick Place DO
(d) Coffee Preparation DI
Figure 4: The experimental environments from MimicGen [11]. The left image in each subfigure shows the initial state of the environment; the right image shows the goal state. See Figure in the Appendix for all environments.
shown in Figure 3, our network consists of three main parts: encoding (white box), denoising (yel- low box), and decoding (gray box). We implement our network using the escnn library [50]. First, an equivariant observation encoder and an equivariant action encoder take inputs o and a², respectively, to create equivariant embeddings Co and cat. The em- beddings will tation of the subgroup C SO(2) (where number of discrete rotations in the group). The em- beddings have shape e E Ruxd, and e,² E IR Ruxd,, where each of the d, or da dimensional vectors en- codes the features for a specific group element (i.e., a rotation angle). Second, in the denoising step, let eg E Rdd and ea E dRda be pair of partial em- beddings corresponding to the same group element g. We process each pair with ID Temporal U-Net (adopted from the prior works [15, 1]) to calculate an equivariant noise embedding. Specifically, letting k be the denoising step, the U-Net, and z its out- put, we have 2 = Ueq. e k). Since the same network is applied for all g E Cu. the output an equivariant embedding of the noise the regular representation. Finally, an equivariant decoder will decode the noise et See Appendix D for details.
the form of regular represen- the
Noley Art Bequenoe
Equí Emb
Det Step
Equi
Ulee
Nokoa Prodiction
k
fanatoon
Exuil hisse Emb
Doodlig
Nocas-Fria Action Suquarcce a*
Figure 3: Overview of our Equivariant Dif- fusion Policy architecture.
Experiments
5.1 Simulation Experiment
Experimental Settings We first evaluate our Equivariant Diffusion Policy (EquiDiff) with either image (Im) or voxel (Vo) input on 12 manipulation tasks from MimicGen [11] (Figure 4). We define the rotation of the observation as a voxel grid rotation or an image rotation. Notice that in the image version of our method, there is a mismatch between the rotation of the agent view image and the rotation of the ground truth state since the agent view not orthogonally top-down. Although top- down observations could be captured,
use the observation settings in the published dataset from MimicGen [11] to demonstrate the generalizability of our method On the other hand, the voxel version eliminates this symmetry mismatch as the rotation of the voxel grid aligns with the rotation of the ground truth state. To better leverage the equivariance, we also add rotation augmentation in the voxel version of our method following our analysis in Section 4. 14.2. We compare our method with the following baselines: 1) DiffPo-C: the original diffusion policy [1] trained with the 1D UNet architecture
method, but it does not have any equivariant structure. 2) DifiPo-T: same as above, but trained with transformer. 3) DP3: the 3D diffusion policy [20] trained with a point net encoder. 4) ACT: the Action Chunking
Temporal UNet [15]. Notice that the baseline shares the
Prior work [52] demonstrate that the equivariant CNN is still able capture symmetry in such scenario.

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Stack DI
Stack Three D1
Square 1X2
Threading D22
Method
Crrl
Obs
200
1000
200
1000
100
200
1000
100
200
1000
EquiDit (Vo)
Voxel
99 (+23)
(+3)
75 (+:37)
91 (+19)
91 (-33
(+31)
(+29)
63 (+14)
(+22)
(+18)
55 (:4)
EquiDir (Im)
RGB
93 (+17)
100 (+3)
100 (-)
55 (+17)
77 (+5)
96 (-2)
25 (+17)
41 (+22)
60 (+11)
22 (+5)
40 (+5)
59 (-)
Ditro-C [1I
DiiPo-T [1]
Abs
RGB
RGB
76
97
83
100
99
38
72
41
94
84
8
19
46
17
11
18
35
59
[20]
PCD
23
65
23
ACTISI
RGBB
73
96
37
78
18
10
21
35
Equitiir (No)
Wveel
95 (-14)
(45)
(-)
59 (+33)
76 (+24)
(-9)
(+17)
35 (+14)
(-7)
33 (+-11)
(+13)
461-1)
Equitiif (1mn)
DirPo-C [1]
Rel
RGB
RGB
75 (-6)
R1
96 (+1)
93
100 (+)
99
25 (-1)
26
63 (+11)
52
92 (-)
86
11 (45)
6
21 (-)
13
48 (-11)
37
11 (-2)
13
22 (-4)
26
49 (+2)
40
RC RNN
RGB
100
92
Coffee D2
Three Pc. Assembly D2
Hammer Cleanup DI
Mug Clesnup DI
Method
Ctrl
Obs
100
200
10001
100
200
1000
100
200
1000
100
200
1000
EquiDalt (Ni)
Voxel
66 (+18)
73
76 (:3)
37 (+33)
58 (+52)
71 (-28)
70 (+16)
66 (-5)
73 (-14)
53 (+1I))
65 (+6)
68 (+5)
Equitai (Im)
RCIR
60 (=13)
79 (413)
76 (-3)
15 (+11)
30 (433)
69 (-26)
65 (+11)
63 (-8)
77 (-10)
40 (46)
64 (4())
67 (+2)
Dillf-C C
DiIPe-T [1]
Abs
RGB
RGB
61
75
79
43
30
48
60
30
43
59
DP3 [20]
PCD
34
45
69
3
54
71
87
21
23
53
[ST
RGB
33
64
24
38
54
71
23
31
36
Equitaff (Vo)
Vonel
55 (+12)
59 (+7)
64 (-12) (+3)
5 (-)
55 (+28)
64 (+21)
62 (+8)
67 (:5)
39 (+14)
43 (144
(25)
Equitei (1m)
Rel
RCBR
41
(-10))
(c])
5 (-))
(432)
49 (46)
(-2)
(-3)
(52)
DimPo-C [1]
RGB
67
25
39
BC KNN [2]
RGBB
57
52
76
27
32
43
72
19
39
67
Kitchen D1
Nut Assembly DO
Pick Place DO
Colfee Propuration D1
Method
Ctrl
Obs
1000
1000
100
200
1000
Equitait (1e)
Vonel
85
(+4)
88 (-3)
67
(-1)
(+23)
(+3)
90 (+115
(+21)
W5 (+9)
EquiDin (Im)
RGB
67 (=)
77 (-8)
(-10)
74 (+19)
85 (+17)
94 (+10)
42 (+7)
74 (49)
(+9)
77 (+12)
(+21)
85 (-9)
DilPo-CC [1]
RGB
67
85
87
55
68
R3
35
65
83
65
62
58
DitPP-T [1]
Abs
RGB
54
75
81
31
32
46
15
37
50
38
51
76
[20]
PCD
24
58
12
22
ACT(S1]
RGB
87
46
EquiDint (Vo)
Vosel
(+27)
(+19)
89 (+1)
53 (+11)
72 (-13)
40 (+5)
58 (:1)
(+18)
(+12)
EquiDoff (Im)
Didif C [1]
Rel
RGB
RGB
61 (+19)
42
72 (+8)
64
83 (+2)
81
44 (+2)
42
65 (+:)
62
87 (+2)
75
29 (-6)
35
55 (-4)
59
91 (19)
82
49 (+7)
42
59 (16)
53
79 (+18)
51
BC OCRNN
RGB
81
58
85
21
14
32
61
Table 1: The performance of our method compared with the haselines in simulation. We experi- ment with 100, 200, and 1000 demos in each environment and report the maximum task success rate among 50 evaluations throughout training. Results averaged over three seeds. Number in parenthe- ses shows the difference between our method and the best baseline (with increment colored in blue and decrement in red). Bold performance indicates the best, bold difference is greater than 10%.
Transformer [51] trained as a conditional VAE. 5) BBBRRNNN a recurrent architecture from [2]. Notice that the voxel version of our method and DP3 utilizes the 3D inputs constructed from four cameras, while the image version of our method and the other baselines directly use the RGB images from two cameras. As our main baseline, we evaluate DifiPo-C in both absolute and relative pose control. We evaluate the other baselines in the same control mode as in the original work (absolute for DiffPo-T, DP3 and ACT, and relative for BC RNN). See Appendix E and F for the details.
Results Table shows the experimental result in terms of the maximum success rate among 50 evaluations throughout the training. First, with absolute pose con- trol, our Equivariant Diffusion Policy with voxel input achieves the best overall performance, outperforming the baselines in 11 out of the 12 environments (Ham- mer Cleanup DI being the exception). With RGB im- age inputs, our method outperforms all RGB baselines in all environments except for Kitchen D1. Second, in relative control, our method with voxel input achieves the best performance, while our method with RGB input is only marginally better than the baselines. Third, our method appears to perform particularly well in the low-
Average over 12 Envrroments
Method
Cirl
100
200
1000
EquiDen (No)
63.9(+21.9)
72.6 ()
77.9 (+6.5)
EquiDet (Im)
(411(7
(+-5(())..)
79.7 (+8.5)
DitPa- C [1]
DimPo-T [1]
Abs
29.0
42.0
57.8
71.4
DF3 [20]
23,9
35.1
56.8
ACT
21.3
38.2
63.3
Equiaat (No)
(8.5(+
(+107)
(-0.)
EquiDaif (Im)
(42.)+
50.4 (+3.1)
74.0 (+3.7)
Dift'o < 111
Rei
33.3
63.2
[2)
22.9
Table 2: The average performance over
12 tasks of Equivariant Diffusson Policy compared with baselines.

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Performance Improvement Across Environments
(a) Stuck
(b) Stack Three
(c) Square
Three Pc.
High-Equivariance Taoks
Coffee Preperation
Threading 02-
Hommer Cleanup D1
(c) Coffee
(1 Hammer Cl.
(g) Mug Cl. do Coffie Prc. (1 Threading
Coffee 02
Intermediate Equivariance Tasks
Mug Cleenup D1
Rikk Hace D0-
Kitrhen D1
2006 penoos
Assembly
① Kitchen (k) Not Assem. (1 Pick Place
Low--Eqiivariance Tasks
Partormance Difference
(a)
(b)
Figure 5: (a) The three task groups are based on the level of equivariance and their initial object distribution. Images were generated by taking the average of five random initialization states. (b) The performance improvement our Equivariant Diffusion Policy (Voxel) compared with the orig- inal diffusion policy in absolute pose control. Blue environments are high-equivariance tasks; green environments are intermediate -euuiiiatee-ivvviiinnee tasks; red environments are low-equivariance tasks.
data regime (i.e., with 100 or 200 demos). Specifically, taking the average over all environments (as shown in Table 2), our method with voxel input and absolute pose control trained with 100 demos outperforms the best baseline by 21.9%. When trained with 200 demos, it outperforms all baselines trained with 1000 demos, indicating the strong sample efficiency of our method. We also perform an ablation study in Appendix H. where we ablate the equivariant structure and the voxel input in our method. We show that though both of these items contribute to the performance improvement our method shows over the baseline, the equivariant structure is the more important factor.
5.2 Improvement with Different Levels of Equivariance
We further analyze the performance improvement of our method when the tasks have different levels of equivariance. Since equivariant models generalize automatically across different object poses, equivariance should hypothetically be more useful when there is greater variance in the distribution of initial object poses. We qualitatively group the tasks into three levels: 1) nigh-equivariance tasks where the poses of the objects are initialized randomly within the workspace; 2) intermediate- cquivariance tasks where each object initialized in a certain range, but with some randomness inside the range; 3) low-equivariance tasks where there is no randomness for the position and/or orientation of certain objects. Figure 5a shows the three task groups. We show the performance improvement of Equivariant Diffusion Policy with voxel in absolute pose control compared with the standard diffusion policy in Figure 5b. Generally, the high-equivariance tasks benefit more from injecting symmetry in the network architecture. Moreover, our method's strong performance in the intermediate and low-equivariance tasks indicates its robustness and generalizability, as the model's 'ss symmetry is helpful even when the task is partially symmetric.
5.3 Real-Robot Experiment
Experimental Settings In this section, we evaluate our method on real robot system contain- ing Franka Emika robot arm [53] equipped with pair of fin-ray [54] fingers and three Intel Realsense [55] D455 cameras. Demonstrations nexion mouse. Observations and demonstration actions were recorded at 5Hz. Similarly to prior
gathered by operator using 6DoF 3DCon-

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(a) Oven Opening
(b) Banana in Bowl
(c) Letter Alignment
(d) Trash Sweeping
(e) Hammer to Drawer
(f) Bagel Baking
Figure 6: The real- environments. The left image of each subfigure shows the initial state of the environment; the right image shows the goal state. See Appendix L for a detailed task description.
Oven Opening
Ranina in Bowl
Letter Alignment
Trash Sweeping
Hammer to Drawer
Ragel Raking
Demos
20
40
40
60
58
EquiDiiI (Vo)
(1920)
95% (19/20)
90% (18/20)
(17/20)
(16/20) 10% (2/20)
DillPo C (No)
60% (12/20)
308 (6/20)
0% (0/20)
5%2(1/20)
(12(1/2)
Table 3: Performance of Equivariant Diffusion Policy in Real- Real-World Robot Experiments.
work [1]. we use DDIM [56] in this experiment to reduce the number of denoising steps to 16. Figure shows the six tasks in this experiment. We compare our Equivariant Diffusion Policy with voxel input against a baseline Diffusion Policy, which uses the same voxel grid as the vision input and uses a non-equivariant 3D convolutional encoder with approximately the same number of train- able parameters ours. As show in the ablation study (Appendix H), this baseline works better than the original diffusion policy with image input.
Results We evaluate the trained models over 20 test trials for each task. The results are shown in Table 3. Our Equivariant Diffusion Policy can solve those tasks with only 20 to 60 demonstrations. Notably, our method achieves an 80% success rate in bagel baking, where the failures were all due to the joint limits of the robot. comparison, the baseline performs poorly in all six tasks.
Conclusion
This paper studies the leveraging of symmetries in visuomotor policy learning. We propose the novel Equivariant Diffusion Policy method and provide theoretical analysis identifying the conditions under which diffusion processes using -Suuieequive in the 6DoF control for robotic manipulation. We evaluate our method in both simulation and the real world and show in both cases that our method outperforms the baseline Diffusion Policy by a large margin.
equivariant. We also demonstrate general framework for
One limitation of this work the partial utilization of the power of equivariance due to the symmetry mismatch in the vision system. Even with the voxel input, Factors like the arm's 's occasional presence in the voxel grid and camera noise could break symmetry. Future work could address this by design- ing a vision system free of symmetry corruption. Additionally, "incorrect equivariance", as shown in prior work [57], may harm performance when the model's 's symmetry conflicts with the demonstra- tion. Another limitation is that although the theory in Section 4.2 is not limited to diffusion policies and can apply to other policy learning pipelines as well, this is not demonstrated. Specifically, given the good performance of BC RNN with the relative pose control in Table 1, experimenting with an equivariant version of BC RNN could be beneficial. Finally, extending our method to other robotic tasks like navigation, locomotion, and mobile manipulation is key future direction.

---

Acknowledgments
This work supported in part by NSF 1750649, NSF 2107256, NSF 2314182, NSF 2134 NSF 2409351, and NASA 80NSSCI9K1474, Dian Wang is supported in part by the JPMorgan Chase PhD fellowship. The authors would like to thank Dr. Osman Dogan Yirmibesoglu for the design of the fin-ray gripper fingers, Dr. Andy Park for building the teleop system for data collection, Emmanuel Panov for collecting demonstration data in the robot experiment, Dr. Thomas Weng for the proof-read of the paper, and Dr. Cheng Chi for the helpful discussion.
References
[1] C. Chi, S. Feng, Du, Z. Xu, E. Cousineau, B. Burchfiel, and S. Song. Diffusion Policy: Visuomotor Policy Learning via Action Diffusion. In Proceedings of Robotics: Science and Systems (RSS), 2023.
[2] A. Mandlekar, D. Xu, J. Wong, S. Nasiriany, C. Wang, R. Kulkarni, L. Fei-Fei, S. Savarese, Y. Zhu, and R. Martín-Martín. What Matters Learning from Offline Human Demonstrations for Robot Manipulation. In Faust, D. Hsu, and G. Neumann, editors, Proceedings of the 5th Conference Robot Learning, volume 164 of Proceedings of Machine Learning Research, pages 1678-1690. PMLR, 08-11 Nov 2022.
[3] A. Gupta, V. Kumar, C. Lynch, S. Levine, and K. Hausman. Relay Policy Learning: Solv- ing Long-Horizon Tasks via Imitation and Reinforcement Learning. In Conference on Robot Learning, pages 1025-1037. PMLR, 2020.
[4] E. Hoogeboom, V. G. Satorras, C. Vignac, and M. Welling. Equivariant Diffusion for Molecule Generation in 3D. In International Conference Machine Learning, pages 8867-8887.
PMLR, 2022.
[5] Guan, W. W. Qian, X. Peng, Y. Su, J. Peng, and J. Ma. 3D Equivariant Diffusion for Target- Aware Molecule Generation and Affinity Prediction. In The Eleventh International Conference on Learning Representations, 2023.
[6] Brehmer, J. Bose, P. De Haan, and T. Cohen. EDGI: Equivariant Diffusion for Planning with Embodied Agents. arXiv preprint PrXXiv:2303.24110 2023.
[7] H. Ryu, J. Kim, Chang, H. Ahn, J. Seo, T. Kim, J. Choi, and R. Horowitz. Diffusion- EDFs: Bi-equivariant Denoising Generative Modeling SE(3) for Visual Robotic Manipula- tion. arXiv preprint arXiv:2309.0266 02685, 2023.
[8] K. Chen, X. Chen, Yu, M. Zhu, and Yang. Equidiff: conditional equivariant diffusion model for trajectory prediction. In 2023 IEEE 26th International Conference Intelligent Transportation Systems (ITSC), pages 746-751. IEEE, 2023.
[9] D. Wang, R. Walters, and Platt. Eqqiiaaqu Reinforcement Learning. Interna- lional Conference on Learning Representations, 2022.
[10] M. Jia, D. Wang, G. Su, D. Klee, X. Zhu, R. Walters, and R. Platt. SEIL: Simullaiioonnauugmenntee Equivariant Imitation Learning. In International Conference on Robotics and Automation (ICRA), 2023.
[11] A. Mandlekar, S. Nasiriany, Wen, I. Akinola, Y. Narang, L. Fan, Y. Zhu, and D. Fox. Mim- icGen: A Data Generation System for Scalable Robot Learning using Human Demonstrations. In 7th Anmual Conference on Robot Learning, 2023.
[12] Sohl-Dickstein, Weiss, N. Maheswaranathan, and Ganguli. Deep Unsupervised Learn- ing using Nonequilibrium Thermodynamics. In Bach and Blei, editors, Proceedings of the 32nd International Conference on Machine Learning, volume 37 of Proceedings of Machine Learning Research, pages 2256 2265, Lille, France, 07-09 Jul 2015. PMLR

---

[13] Y. Song and S. Ermon. Generative Modeling by Estimating Gradients of the Data Distribution. Advances in Neural Information Processing Systems, 32, 2019.
[14] J. Ho, A. Jain, and P. Abbeel. Denoising Diffusion Probabilistic Models. Advances in Neural Information Processing Systems, 33:6840-6851, 2020.
[15] M. Janner, Y. Du, Tenenbaum, and Levine. Planning with Diffusion for Flexible Behavior Synthesis. In International Conference Machine Learning, pages 9902-9915. PMLR, 2022.
[16] Z. Liang, Y. Mu, M. Ding, F. Ni, M. Tomizuka, and P. Luo. AdaptDiffuser: Diffusion Models as Adaptive evollevol Planners. In International Conference on Machine Learning, 2023.
[17] T. Pearce, T. Rashid, A. Kanervisto, D. Bignell, M. Sun, R. Georgescu, S. V. Macua, S. Z. Tan, I. Momennejad, K. Hofmann, et al. Imitating Human Behaviour with Diffusion Models. In The Eleventh International Conference on Learning Representations, 2022.
[18] Z. Wang, J. J. Hunt, and M. Zhou. Diffusion Policies as an Expressive Policy Class for Offline Reinforcement Learning. In The Eleventh International Conference on Learning Representa- tions, 2022.
[19] 7. Xian, N. Gkanatsios, T. Gervet, T.W. Ke, and K. Fragkiadaki. ChainedDiffuser: Unify- ing Trajectory Diffusion and Keypose Prediction for Robotic Manipulation. In 7th Annual Conference on Robot Learning, 2023.
[20] Y. Ze. G. Zhang, K. Zhang, C. Hu, M. Wang, and H. Xu. 3d diffusion policy: Generalizable visuomotor policy learning via simple 3d representations. In Robotics: Science and Systems, 2024.
[21] M. Xu, L. Yu, Y. Song, C. Shi, S. Ermon, and J. Tang. GeoDiff: A Geometric Diffusion Model for Molecular Conformation Generation. In International Conference Learning Representations, 2022.
[22] D. Wang, R. Walters, X. Zhu, and R. Platt. Equivariant Q Learning in Spatial Action Spaces. In 5th Annual Conference on Robot Learning, 2021.
[23] H. Huang, D. Wang, A. Tangri, R. Walters, and R. Platt. Leveraging Symmetries in Pick and Place. The International Journal of Robotics Research, 2023.
[24] A. Simeonov, Y. Du, Y.-C. Lin, A. R. Garcia, L. P. Kaelbling, T. Lozano-Pérez, and P. Agrawal. Equivvaaan Relational Rearrangement with Neural Descriptor Fields. Conference on Robot Learning, pages 835-846. PMLR, 2023.
[25] C. Pan, B. Okorn, H. Zhang, B. Eisner, and D. Held. TAX-Pose: Task-Specific Cross-Pose Estimation for Robot Manipulation. In Conference on Robot Learning, pages 1783-1792. PMLR, 2023.
[26] H. Huang, D. Wang, X. Zhu, R. Walters, and R. Platt. Edge Grasp Network: A Graph-Based SE(3)-invariant Approach to Grasp Detection. In International Conference on Robotics and Automation (ICRA), 2023.
[27] S. Liu, M. Xu, P. Huang, X. Zhang, Y. Liu, K. Oguchi, and D. Zhao. Continual Vision-based Reinforcement Learning with Group Symmetries. In Conference on Robot Learning, pages 222- 222-240. PMLR, 2023.
[28] S. Kim, B. Lim, Y. Lee, and F. C. Park. Se (2)-equivariant pushing dynamics models for tabletop object manipulations. In Conference on Robot Learning, pages 427-436. PMLR, 2023.
[29] C. Kohler, A. S. Srikanth, E. Arora, and R. Platt. Symmetric models for visual force policy learning, arXiv preprint 2xXx:23308.16600 2023.
10

---

[30] H. H. Nguyen, A. Baisero, D. Klee, D. Wang, R. Platt, and C. Amato. Equivariant reinforce- ment learning under partial observability. In Conference on Robot Learning, pages 3309-3320. PMLR, 2023.
[31] H. Nguyen, T. Kozuno, C. C. Beltran- Hernandez, and M. Hamaya. Symmetry- aware reinforce- ment learning for robotic assembly under partial observability with soft wrist, arXiv preprint arXiv:2402. 18002, 2024.
[32] B. Eisner, Yang. Davchev, M. Vecerik, J. Scholz, and D. Held. Deep -quivvaquiaa geometric reasoning for precise placement tasks. In The Twelfth International Confer- Learning Representations, 2024. URL http::///pereeiee.neee 2inBuwTyL2.
[33] X. Zhu, D. Wang, O. Biza, G. Su, R. Walters, and R. Platt. Sample Efficient Grasp Learning Using Equivariant Models. Robotics: Science and Systems, 2022.
[34] D. Wang, M. Jia, X. Zhu, R. Walters, and R. Platt. On-Robot Learning With Equivariant Models. In 6th Annual Conference on Robot Learning, 2022.
[35] X. Zhu, D. Wang. G. Su, O. Biza, R. Walters, and R. Platt. On robot grasp learning using equivariant models. Autonomous Robots, 47(8):1175-1199, 2023.
[36] H. Huang, D. Wang, R. Walters, and R. Platt. Equivariant Transporter Network. In Robotics: Science and Systems, 2022.
[37] A. Simeonov, Y. Du, A. Tagliasacchi, B. Tenenbaum, A. Rodriguez, P. Agrawal, and V. Sitz- mann. Neural Descriptor Fields: SE(3)-Equivariant Object Representations for Manipula- tion. In 2022 International Conference on Robotics and Automation (ICRA), pages 6394-6400. IEEE, 2022.
[38] H. Ryu, H. in Lee, J.-H. Lee, and Choi. Equivariant Descriptor Fields: SEQiv--quivariant Energy Based Models for End-to-End Visual Robotic Manipulation Learning. In The Eleventh International Conference on Learning Representations, 2023.
[39] H. Huang, Howell, X. Zhu, Wang, R. Walters, and R. Platt. Fourier Transporter: Bi- Equivariant Robotic Manipulation 3D. arXiv preprint arXiv:2401.12046 2024.
[40] T. Zhang, Z. McCarthy, O. Jow, D. Lee, X. Chen, K. Goldberg, and P. Abbeel.. Deep Imitation Learning for Complex Manipulation Tasks from Virtual Reality Teleoperation. In 2018 IEEE International Conference on Robotics and Automation (ICRA), pages 5628-5635 IEEE, 2018.
[41] S. Toyer, R. Shah, A. Critch, and S. Russell. The Magical Benchmark for Robust Imitation. Advances in Neural Information Processing Systems, 33:1 13:18284-18295 2020.
[42] Florence, Manuelli, and R. Tedrake. Self-Supervised Correspondence in Visuomotor Policy Learning. IEEE Robotics and Automation Letters, 5(2): 499, 2019.
[43] R. Rahmatizadeh, P. Abolghasemi, L.. Bölöni, and S. Levine. Vision-Based Multi-Taa Task Ma- nipulation for Enexpensive Robots Using End-to-End Learning from Demonstration. In 2018 IEEE International Conference on Robotics and Automation (ICRA), pages 3758-3765. IEEE, 2018.
[44] M. Orsini, A. Raichuk, L. Hussenot, D. Vincent, R. Dadashi, S. Girgin, M. Geist, O. Bachem, O. Pietquin, and M. Andrychowicz. What Matters for Adversarial Imitation Learning? Ad- vances in Neural Information Processing Systems, 34:14656-14668, 2021.
[45] D. Jarrett, I. Bica, and M. van der Schaar. Strietly Batch Imitation Learning by Energy-Based Distribution Matching. Advances in Neural Information Processing Systems, 33:7354-7366 2020.

---

[46] P. Florence, C. Lynch, A. Zeng, O. Ramirez, A. Wahid, L. Downs, A. Wong. J. Lee, I. Mor- datch, and Tompson. Implicit Behavioral Cloning. Conference on Robot Learning (CoRL., 2021.
[47] Y. Du and L. Mordatch. Implicit generation and generalization in energy-based models. In Advances in Neural Information Processing Systems, 2019.
[48] W. Grathwohl, K.-C. Wang, J.-H. Jacobsen, D. Duvenaud, and R. Zemel. Learning the Stein Discrepancy for Training and Evaluating Energy-Based Models Without Sampling. In Inter- national Conference on Machine Learning, pages 3732-3747. PMLR, 2020.
[49] Y. Zhou, C. Barnes, J. Lu, J. Yang, and Li. On the Continuity of Rotation Representations in Neural Networks. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 5745-5753, 2019.
[50] G. Cesa, L. Lang, and M. Weiler. A Program to Build Eqn)--quivaaiint Steerable CNNs. In International Conference on Learning Representations, 2021.
[51] T. Z. Zhao, V. Kumar, S. Levine, and C. Finn. Learning Fine-Grained Bimanual Manipulation with Low-Cost Hardware. In Proceedings of Robotics: Science and Systems (RSS), 2023.
[52] D. Wang, J. Park, N. Sortur, L. L. Wong, R. Walters, and R. Platt. The Surprising Effective- ness of Equivariant Models in Domains with Latent Symmetry. In International Conference on Learning Representations, 2023.
[53] Haddadin, S. Parusel, Johannsmeier, S. Golz, Gabl, Walch, M. Sabaghian, C. Jähne,
L. Hausperger, and S. Haddadin. The Franka Emika robot: Reference Platform for Robotics Research and Education. IEEE Robotics & Automation Magazine, 29(2):46 64, 2022.
[54] W. Crooks, G. Vukasin, M. O'Sullivan, W. Messner, and C. Rogers. Fin ray@ effect inspired soft robotic gripper: From the robosoft grand challenge toward optimization. Frontiers in Robotics and AI, 3:70, 2016.
[55] L. Keselman, J. Iselin Woodfill, A. Grunnet-Jepsen, and A. Bhowmik. Intel Realsense Stereo- scopie Depth Cameras. In Proceedings of the IEEE Conference on Computer Vision and Pat- tern Recognition Workshops, pages 1-10, 2017.
[56] J. Song, C. Meng, and S. Ermon. Denoising diffusion implicit models. In International Con- ference on Learning Representations, 2020.
[57] D. Wang, X. Zhu, Y. Park, M. Jia, G. Su, Platt, and R. Walters. general theory of correct, incorrect, and extrinsic equivariance. Advances in Neural Information Processing Systems, 36, 2024.
[58] R. Wang, R. Walters, and Yu. Incorporating symmetry into deep dynamics models for improved generalization. In International Conference on Learning Representations (ICLR), 2021.
[59] K. He, X. Zhang, S. Ren, and Sun. Deep Residual Learning for Image Recognition. In Proceedings of the IEEE Conference on Computer Vision and Puttern recognition, pages 770- 778. 2016.
(60] I. Loshchilov and F. Hutter. Decoupled Weight Decay Regularization. In International Con- ference on Learning Representations, 2018
12

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A Proof of Proposition
Proof. Consider the observation o and the action a = T()), let a = a + et. By the definition of noise prediction function E, we have
Ex =(0 +E*,),
(2)
Applying E SO(2) to we have
Ek == + +
(3)
Since is equivariant,
ck E(go, ga + t k).
(4)
Since the noise prediction function predicts the noise as long as e is the same on both sides of the equation, we can substitute ex with ge
gE Ego, ga gE k).
(5)
By linearity, ga ge g(a + et = ga and thus
ge = E(go, ga k).
(6)
Replacing * with e(o, at k) gives (o a k:) E(go, ga k) as desired.
L
B Decomposing Group Representation in Relative Pose Control into Irreducible Representations
In Section 4.2, we want to find a linear action PA that satisfies ga, = (g(g) Vec,(A)) = Vec, (Ty (T solving for PA
R16x16 we have the group action of SO(2) on Vcc,(Ar)
2
0
s2
0
0
8
2
0
0
0
0
0
0
0
0
3
0
0
22
3
0
1000
0
0
PA
00
00
0
Pi(g)
Ig
(T(g)
T2
where cos g, 8
sin g, 02
2g, 82
sin 2g.
Define P
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0 0 1000
0
0
0
0
0
O
0
0
0
0
0
0
1
0
0000
0
0
0
0
0
0
0
1
P
1
1
0
0
0
0
0
0
0
0000
0
0
0
0
0
0
13

---

We then have
16
pi(0)
pi(g)
PpAP-1
(9)
pi
pi(g)
p2(0)
C Simplifying Group Action in Relative Pose Control
In Section 4.2, we want to find linear action PA that satisfies
PA (e (A,) Vec, (TgAiIg
(10)
To simplify the problem, we decompose A, [ar, y, 2| is the translation. Define Rg the rotation matrix in Tg.
where R, the SO(3) rotation and D,
cos g sin g Rg = sin
cos g 0 = (g(()
po(o)
(11)
Since the conjugate does not apply to translation, we can write gD, = (p ( For rotation, similar as before, we need to find the representation PR that satisfies
PR (R) Vec, Vec, (Rg
(12)
Solving for pR(9) E R99 we have
C8
C8
32
0
0
0
C8
2
08
0
0
0
0
0
0
0
0
0
0
C8
c22
68
0
0
0
0
PR
s
C8
Cs
c2
(13)
0
0
0
0
0
0
0
0
C
8
0
0
where c = cos 9, 8 = sin g.
To decompose into the irreducible representations of S0(2), we
define
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
1
0
1
0
0
0
P
0
0
0
0
0
0
(14)
0
0
0
1
0
100
0
0
0
0
1
1
0
0
0
Such that PPRP-1 is block diagonal matrix consisting of irreducible representations
[Pu(g)
Po(g)
pa(g)
PPR
(15)
(i(
(i(g)
P2(g)
14

---

Ced Rellee-1
Agen Vicw
Doul
Reahes10
Eyc-ns fand Image
Skop
Poo P) Ae
Tenpooa
At
Realo:
127-200
2,1-44
Orippor Nosoo
how
Liner
Figure 7: The detailed network architecture of our Equivariant Diffusion Policy in the simulation experiments.
We can then use p(g) PPRP-1- pā(g) @ pi(g)
p2(9) E R the group representation
of the output of the equivariant network, then construct the rotation matrix R, using P. Specifically, let V R be the output of the network associated with the representation p(g) (i.c., acts on V through V). Define
Vec, (R.) P-VV.
(16)
Applying p(g) on will lead to
p P(gVV
(17)
= -P- -P-tPpn--Ty
(18)
=PR Vec,
(19)
which is the desired property in the equivariant network.
In the end, adding the group action for the translation (pi @ Po) and gripper open width (po), we have Pa p + p + P2.
D Network Architecture Detail
In the image version, implement the equivariant observation encoder with equivariant ResNet [58] for the agent view image, a standard ResNet [59] for the eye-in-hand image, and an cquivariant MLP for the robot states. We implement the equivariant layers in the group Cg. Figure shows the detailed network architecture of our Equivariant Diffusion Policy in the simulation exper- iments. The network is defined under group Cg. First, in the encoding phase, the agent view image processed with an equivariant ResNet-18, whose output is 128 8-dimensional regular repre- sentation vector of group. non-equivariant ResNet-18 with a spatial maxpool at the end processes the eye-in-hand image and outputs 128- 228diimensional representation vector that uses the trivial in- variance representation. Those two vectors are concatenated with the gripper position (represented
15

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(a) Stack D1
(b) Stack Three DI
(c) Square D2
(d) Threading D2
(e) Coffee D2
(f) Three Pc, Assembly D2 (g) Hammer Cleanup D1
(h) Mug Cleanup DI
(i) Kitchen DI
(j) Nut Assembly DO
(k) Pick Place DO
(I) Coffee Preparation DI
Figure 8: The experimental environments from MimicGen [11]. The left image in each sub figure shows an initial state of the environment; the right image shows the goal state.
Kitchen DI
(b) Coffee Preparation D1
Figure 9: Illustration an episode of Kitchen DI task and Coffee Preparation DI task.
using Pi + Po). gripper orientation (in the format of 6D rotation, represented using pi), and the gripper finger position (represented using pQ). The concatenated mixxedreprresentation vector is sent to equivariant linear layer, whose output is 128 x 8-dimensional regular representation obser- vation embedding. The noisy action is also encoded using equivariant linear layer, whose output is a 64 x 8-dimensional regular representation action embedding. Second, in the denoising phase, we process each part of the observation embedding and the action embedding that corresponds to the same group element with Temporal UNet with hidden dimensions [512, 1024, 2048) to get a 64-dimensional vector. Doing so for each pair, we will recover a 64 x 8-dimensional regular representation noise embedding. the end, an equivariant linear layer will decode the noise.
In the voxel version, the agent view image is replaced with voxel grid, and we replace the equiv- ariant ResNet with an 8-layer 3D equivariant convolutional encoder. The ID Temporal UNet has hidden dimensions of [256, 512, 1024]. The other part of the network stays the same. In the real- world, we remove the eye-in-hand image and only use the voxel grid as vision input (the gripper state vector stays the same).
E Simulation Environments
Figure shows the initial and goal states of each tasks. Figure shows an example trajectory for finishing the Kitchen and Coffee Preparation tasks. The RGB observation is agent-view image and an eye-in- eyn-in-hand image with size of 3 x 84 x 84. The voxel grid observation has size of
16

---

Task
Max Steps
Max Out of Plane Rot in Demo
Stack DI
400
11.2
Stack Three D1
400
13.2
Square D2
400
14.7
Threading D2
400
13.4
Coffee D2
400
Three Piece Assembly D2
500
16.2
Hammer Cleanup DI
500
16.4
Mug Cleanup DI
500
13.0
Kitchen D1
800
16.2
Nut Assembly
500
15.5
Pick Place DO
1000
18.0
Coffee Preparation
59.0
Table 4: The maximum number of time steps and the maximum out of plane rotation (in degrees) in the demo for each simulation environments. The maximum out of plane rotation in the demo is the maximum angular difference between the SO(3) rotation and the SO(2) rotation (i.e., only rotating around the axis) over all demonstration steps, averaged over 1000 demonstration episodes.
Method
Obs
Ohs Step
Action Pred. Step
Action Exec. Step
EquiDiff (Vo)
Voxel Grid, Eye-In Hand Imuge, Gripper State
1
16
8
EquiDiff (Im)
Agent Image, Eye- In Hand Image, Gripper State
16
DittPo-C
Agent View Image, Eye-In-Hon Hand Image, Gripper State
2
16
8
DifiPo-T
Agent View Image, Eye-ln-Hand Image, Gripper State
2
10
DP3
Point Cloud, Gripper State
2
16
ACT
Agent View Image, Eye-In-Hand Image, Gripper State Agent Image, Eye-In-Haare Image, Gripper State
1
10
10
BC-RNN
Table 5: The observation format, observation step, action prediction step, and action execution step
for all methods. The gripper state is a vector including a dimensional position vector, a rotation vector in the format of 6D rotation representation or 4D quaternion, and 2 dimensional finger position.
4 x 64 64 64 where the first channel is binary occupancy and the remaining three channels RGB. The point cloud observation has size of 1024 x (i.e, xyzrgb). The point cloud only contains points above the table, as suggested in [20]. All tasks have full 6 DoF SE(3) action space. Table 4 shows the maximum number of time steps (following [1 IJ) and the maximum out of plane rotation in the demo, calculated by taking the maximum angular difference between the SO(3) rotation and the SO(2) rotation (i.e., only rotating around the z axis) across the entire demonstration episode. Results averaged for 1000 demonstrations.
F Training Detail
In the simulation experiments, we follow the hyper-parameters of the prior work [1] for the image version of our method, where the only change is that increase the batch size to 128 for faster training. Specifically, the observation contains two steps of history observation, and the output of the denoising process is a sequence of 16 action steps. We use all 16 steps for training but only execute eight steps in evaluation. We train our models with the AdamW [60] optimizer (with a learning rate of 10 -4 and weight decay of 10 66 and Exponential Moving Average (EMA). We use a cosine learning rate scheduler with 500 warm-up steps. We use DDPM [14] with 100 denoising steps for both training and evaluation. For each different number of demos (100, 200, 1000), we maintain roughly the same number of training steps by the training for 50000/n epochs where is the number of demos. Evaluations are conducted every 1000/n epochs (50 evaluations in total), In the voxel version, we use only one step of history observation, and keep the other hyper-parameters the same.
The hyper-parameters for the diffusion policy and BC RNN baselines exactly follow [1]. We follow the original work [20] for the hyper-parameters of DP3, except that we use the same action sequence length (16 for training and 8 for evaluation) as [1] and our method. For the ACT baseline, follow the parameees provided in the prior work [51], except that
chunk size of 10, KL
17

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Stack DI
Stack Three D1
Square D22
Threading D2
Method
Ctrl
100
200
1000
100
200
1000
100
200
1000
100
200
1000
EquiDifi (Vo3
10004:00
100:0:000
14.74-44
91.440077
9074-13
48.0
6334:13
38.740.77
52.74229
54.74-299
EquiDif (Im)
93.3=00.7
10001:00
10000:000
54.7::22
77331
960011
25.3:8.77
41.3299
500142
22.0:112
10.0:112
1.
DiaPo-T [1]
DidPo-C 111
Abs
1.
76.0=4,0
97.4407
K2.14.0.
1000=00
98.74007
16.71077
380+00
413429
120+20
9404-12
840111
4.21
8.+12
11
19.3+5.3
64.74:477
460+72
10.7407
133118
35.3+1.3
18.0::..
58.7+0.77
00.7::0.7
[20]
69.3=3.7
96.71:4.7
99.3:007
0.300.7
22.71:37
65.31:18
674007
6.000.0
1931333
12.0:311
23.3:13.
10.0::2.0
ACT 1511
34.7=007
7274-73
96.0=1122
6.0=2.3
36.74-277
78.0-112
6.0:0000
1800:12
4934443
10.0+1122
20.7+2299
35.3+22,4
Equitit (Way
94.7=13
1000400
1000:000
59.3487
760000
82.7-07
347418
347252
520423
33,111
18.7429
46/0112
Equitit (tm)
74.7258
900000
100000000
2..
62.7535
92.0:12
11.31:3
20.744.1
480440
11.34:13
22.0:112
49.3+224
DimPo-C [1]
Rel
80.7=24
93.34-0.7
99.3207
260044
52.0220
860112
6.0
13
367448
13.1118
36.0411
40,02233
59.3270
3.
10000000
12.0225
480453
92.023
8.0:1.2
20.7277
587215
73407
13.3224
46.72007
Coffee D2
Three Pc. Assembly D2
Hammer Cleaaup D1
Mug Clemuu D1
Method
Ctl
100
200
1000
100
200
1000
100
200
1000
100
200
1000
EquiDir (Ve)
64.7:007
7333118
76.0:20.0
37.32277
58.05..00
71.34333
7..0::0.0
6..0223
72.71:077
52.11:33
64.7124
68.0
EquiDiir (Im)
60.0=2.0
1.
760422
153311
99.3
69.3435
65.
63.320.7
63.32444
7674477
403107
6.002:2
66.72007
DiiD.-C [1]
44.0=12
66.01..3
78.7:007
0.0000
60112
30.0:112
52.0:1.2
1.
7334124
42.71077
58.7113
65.3:224
DimPoTT [1]
4..3=0.7
60
7477227
0.7
4.0200
42.7::13
48.022.2
600212
760212
300112
42.7:29
63.3::0.7
DP3 [20]
34,0=440
45.3-14.11
68.7-42.4
0,0-0.0
0.74-077
3.34:077
54,0:3..1
70.75:4.1
86.74-0.7
21.3+-277
32.7
$2.7
ACT [S1]
19,3:244
33.34224
61.012.3
0.0:0.0
331.07
24.0:3.1
38.01442
14.212
70.113
23.34077
31,3113
56.01.200
Equitii (Va)
55.3=0.7
5934407
64.0+0.0
4.7-077
53+07
547433
640412
620+12
673+13
393407
43.3+18
62.0+:12
EquiDir (tm)
40.7=0.7
58
66021.
133007
47407
5931448
48.7227
320025
687424
20.1429
16.02:2
65.3224
Dirfo-C [1]
42.7=18
507418
66.722
20-12
20+00
200+12
43.3-13
540=12
653118
25,3407
39,3-18
54.72077
BC RNN [3]
37.0-10
52042.
76.0:23
0.000.0
534
27.04-10
32.0-00.0
42.75007
720022
19:34077
39.0 66.74-0.7
Kitchen
Noi Assembly DO
Pick Place DO
Coffee Prepuntion DI
Method
200
1000
100
EquiDit (V))
85.3=0.7
893340-7
88.022.3
67.3209
770000
8333007
57,7418
685206
82:240
100412
83
EquiDin (Im)
67.3=0.7
767433
81.31077
14.2:12
850115
9171008
41.7433
742232
920111
767407
82.7107
85.3:007
DimPs-C [1]
66.7224
84.7200
867218
54.7223
680224
13
35.3222
6502228
82.7104
65.1107
62.0:442
3..02..1
DirPP-T [1]
Abs
54.0=2.3
75334-0.7
81.3:224
30.7550
5233552
45.7559
14771.
16
500260
38.0220
51
76.0660
14.7=1.
71.3124
91.3222
157713
23.733
57.7419
11,7109
150
340100
100:223
32.0:333
63.31:..1
ACT 151)
37.323.5
6071:5
873333
42.3229
63.7235
843309
722299
1722111
500229
33.0230
46.023.
64.7222
EquiDiff (Vo2
6..3
62.74:13
893:18
53.01100
650020
720120
40.31:16
58.2109
788108
48.01112
70.7112
13.3:--88
DidPo-C []]
Equitbif (im)
Rel
60.7=13
42.0=23
7204331
64.45..0
827727
$13++13
41.7+22
44.3112
683115
620115
573300
7534-12
29.343.1
34.7+111
58.7-10
147415
913+12
822
487413
42.04-111
52.7-43.3
39.3422
51,3-1188
79.3---.77
BC RNN [2]
29
46.116.7
80.7113
3533107
5k0121
850112
2122107
41,0290
773:25
10011
32.0:112
60.71.4.1
Table 6: The performance of our Equivariant Diffusion Policy compared with the baselines in simu- lation. We experiment with 100, 200, and 1000 demos in each environment and report the maximum task success rate among 50 evaluations throughout training. Results averaged over three seeds. + indicates standard error.
weight of 10, batch size of 64 with learning rate of 5 x 10-5, and no temporal aggregation. following the tuning tips provided by the authors. See Table 5 for the observation format, observation step, action prediction step, and action execution step for all methods.
In the real-world experiments, we use a batch size of 64, one step of observation, and disable the EMA. We use DDIM [56] with 100 denoising steps for training and 16 denoising steps for evalua- tion. The other hyper-parameters stay the same in simulation.
G Simulation Experiment Result with Standard Error
Table 6 shows the same result in Table with the standard error.
H Ablation Study
We perform an ablation study regarding the equivariant structure and the voxel input in our method. We consider the following four candidates: 1) Ours: our Equivariant Diffusion Policy with voxel input: 2) Ours no Voxel: our Equivariant Diffusion Policy with RGB input; 3) Ours no Equi.: the baseline Diffusion Policy with voxel input; 4) Ours no Voxel no Equi.: the baseline Diffusion Policy with RGB input, same as [1]. Table shows the result and Table shows the average over all 12 en- vironments. Though both the equivariant structure and the voxel input contribute the performance improvement of our method, the equivariant structure plays a more important rule, as removing
18

---

Stack
Stock Three
Square D2
Threading D2
Ablation
Method
Ctrl
Obs
100
200
1000
100
200
1000
100
200
1000
(Va)
Vonel
75
No Voxel
EquiDii (Im)
RGB
100
55
77
60
22
40
No Equi.
DdfPo C (Ne)
Abs
Vonel
87
99
100
33
79
94
10
24
60
19
43
54
No Youel No
DiiPo
RGB
76
97
38
72
17
25
Coffee 122
Three Pc. Assp TD2
Hammer Cleamup TJ
Mug Cleamup D1
Ablation
Method
Ctrl
Obs
100
200
1000
100
200
1000
100
200
1000
100
200
1000
Equitiiff (Vo)
Voxel
73
37
5M
71
73
53
65
68
No Voxel
Equi.
DeMPP
EquiDiti (Im)
Abs
Vosel
RGB
60
79
72
76
15
39
69
65
03
77
47
49
64
07
Voxel Equi.
De1Po-C111
RGB
Kiichen
Nut Assemblly
Coffee Prep DI
Ablation
Method
Ctrl
Obs
100
200
100
200
1000
100
100
200
1000
EquiDit (Wo)
Voxell
85
89
88
67
77
83
58
60)
82
80
83
85
Vonel
EquiDitf (lm)
RGB
77
74
85
77
83
Equi..
DiTTP--C
Woxel
87
77
No Vonel No Equi.
DATPP-C
RGB
85
87 55 68
83 65
62
58
Table 7: The ablation study that ablates the voxel input and the equívariant structure in our method. We experiment with 100, 200, and 1000 demos in each environment and report the maximum task success rate among 50 evaluations throughout training. Results averaged over three seeds.
Average Emirunments
Ablation
Method
Cull
100
200
1030
EquiDit (Vo)
63.9
72.6
77.9
No Vosel
DinPo
EquiDitt (Im)
Abs
53.7 (-10.3)
(215
(-.11
79.7
(+1.8)
(:2.5)
Equi.
DeNPo-C[1]
(-21922
(-.8
(1.4()6.5
Table 8: The average performance over 12 tasks of the ablation study. Number in parenthesis shows the performance difference after removing different components in our Equivariant Diffusion Policy with voxel input.
(No Equi.) lead to a more significant performance drop compared with removing the voxel input (No Voxel). Note that by using the voxel input, Diffpo-C (Vo) is marginally better than the original Diffusion Policy (DiffPo-C), thus we use Diffpo-C (Vo) as the baseline in our robot experiment in Section 5.3.
I Implementing Equivariance via Data Augmentation
In this section, evaluate implementing cquivariance through data augmentation instead of using equivariant networks. Specifically, we applied random rotation data augmentation based on our anal- ysis in Sections 4.1 and 4.2 to standard, unconstrained CNN. We then compared this CNN + Aug baseline against using equivariant neural networks (Equi. Net) and not implementing equivariance at all (CNN).
As is shown in Table 9, CNN Aug achieve good performance, occasionally even outperforming equivariant networks in simpler tasks like Stack and Stack Three. However, it performs poorly in more challenging tasks. When averaged across 12 environments, CNN + Aug performs better than CNN but still underperforms compared to Equi. Net by a significant margin.
J SE(2) Action Space Variation
In this section, we evaluate a variation of our Equivariant Diffusion Policy in an SE(2) (with trans- lation) action space to demonstrate the necessity of leveraging an SF(33 action space. Specifically, the SEE(2) agent only learns the top-down rotation and the out-of-plane rotations will be constantly set to O. As is shown in Table 10, the SE(2) variation achieves similar performance the SE(3) version in Stack Three, as the demonstration data in this task has the least amount of out-of-plane
19

---

Method
Clbs
Stack DI
Stack Three
Square 132
Threading
Equi Net (Mo)
Vouel
74:07
74.7:244
38.72:133
38.70077
CNN Aug (Vo)
Vosel
99.3:10077
84,0 1.2
36,0:1: 1.2
20.7:4:1.8
CNN (No)
Vowel
86.74:1.3
33.32 L.
10.0:22.0
19.3:12.4
Equi Net tim)
RGB
92.3+0077
54,7=t5.2
25.5++8.7
22.0++1.22
CNN
98 08.77077
68.0 1.2
26.7211. 1.8
22.0:::222
CNN (ho)
76.0s::00
Method
Cleanup
Cleanup
Net
Voxel
64.7-20.77
31.3-2:277
70.06:2.0
52.74:133
CNN Aug (Wo)
56.7429
7.3-00.7
70.7=11.8
$2.0+1.22
CNN (No)
Voxel
50.0 3.1
2.0:20.0
54.0g23.1
46.750077
Equi Net (Im)
RGB
60.0:2:00
13.3:118
65.3-20.7
49.3
CNN Aug (Im)
RGB
58.0:t-122
5.3
61.3::2999
50.0:1-1.2
CNN (lm)
RGB
4..0:1:122
4.0:1..0
52.0 1.2
42.7:1077
Method
Olbs
Kitchen D1
Nut Assemblly DO
Pick Place DO
Coffee Prep. DI
Equi Net (Mi)
Vosel
83 15.3::00.7
67.3:20.9
$7.72:188
80.0:2:122
CNN Aug (Vi)
Voxel
62.02220
$1.721.5
19.522.8
48.7422.4
CNN (No)
82.0:1:.33
Equi Net
67.34-077
74.0=:122
41.74:3.2 35.3
76.74-0.7
CNN Aug
24.7-4-2.9
CNN (1m)
RGB
66.7
5..7=2.3
65.35b0.7
Method
Avenge 12 Environmnnts
Equi, Net (No)
Vouel
63.9
CNN
Vonel
CNN (Vo)
Voxel
46.3
Equi. Net
53.7
CNN a Aug (lm)
RGB
46.2
CNN (Im)
RGI
42.0
Table Comparing implementing equivariance via equivariant network or data augmentation. We experiment with 100 demos in each environment and report the maximum task success rate among 50 evaluations throughout training. Results averaged over three seeds.
Three
Threading
Coffee Preparation
EquiDiff (1m)) SE(3) Action
77.3
85.3
EquiDiff (Jm). SE(2) Action
75.3
12.7
0.0
Table 10: Performance of Equivariant Diffusion Policy in SE(2) action space compared with SE(3)
action space. 200 demos are used this experiment.
rotation (as shown in Table 4). On the other hand, the SE(2) variation significantly underperforms in Threading, since the ability of wiggling the out-of-plane rotation helps the agent to precisely insert the tool. In the end, the SE(2) agent cannot solve Coffee Preparation at all, because the task requires a significant amount of out-of-plane rotation (as shown in Figure 9b).
K Robomimic Experiment
In this section, we compare our Equivariant Diffusion Policy with the original Diffusion Policy across four Robomimic tasks (Figure 10). Both methods trained with 100 Proficient-Human (PH) demonstrations 100 Multi-Human (MH) demonstrations. Other hyperparameters mirror those used in our MimicGen experiment.
Table 11 shows the result. Our method achieves similar or slightly better performance compared to the baseline Diffusion Policy. The improvements in Robomimie tasks are smaller than Mimic- Gen tasks. This because the Robomimic tasks can be classified as Low-Equivariance Tasks (as illustrated in Figure 5, bottom), with minimal randomness in the initial distribution (except for Lift), making the symmetry in our method less advantageous.
20

---

(a) Lift
(b) Can
(c) Square
(d) Tool Hang
Figure 10: The experimental environments from Robomimic. The left image in each subfigure shows the initial state of the environment; the right image shows the goal state.
Square
tool hang 100 PH
Average
100 PH
100 MH
100 PH
100 MH
100 PH
100 MH
EquiDiff
100.0:10.0
100.0:10.00
9.73
96. 0.7
84.
1.
76.0.....
90.4
DifiPo
100.0:0..0
100.0=0.00
100.0:00.0
95.3=0.7
85.3 0.77
70.7
64.0
87,9
Table 11: The performance of our Equivariant Diffusion Policy compared with the Diffusion Policy baseline in Robomimic. We experiment with 100 Humaann (PH) or Multi-Human (MH) de- mos in each environment and report the maximum task success rate among 50 evaluations through- out training. Results averaged over three seeds. + indicates standard error.
L Real-Robot Environment Details
Figure 11 shows our real-world experimental platform containing Franka Emika [53] and three Intel Realsense[55] D455 cameras. Compared with simulation, we use pair of fin-ray [54] gripper fingers instead of the original Franka fingers. Figure 6 shows the five tasks in this experiment. In Oven Opening, the oven is randomly initialized at one the four borders of the workspace. In Banana in Bowl, the initial poses of the banana and the bowl are both randomly sampled. In Trash Sweeping, the robot needs to use tool brush to sweep two pieces of crumpled paper out of its workspace. The initial poses of the objects randomly sampled. In Letter Alignment, the robot needs to align the letters to form "Al". The letter A is randomly initialized at one of the four corners of the workspace, and the pose of the is randomly sampled. In Hammer to Drawer, the robot needs to open a drawer, pick up a hammer, place it inside the drawer, and close the drawer. The drawer initialized at one of the four borders of the workspace, and the hammer is randomly initialized at the opposite side of the drawer. Lastly, we also evaluate a Bagel Baking task with an extremely long time horizon, where the robot needs to open the oven, pull out the tray inside the oven, pick up the bagel, place it inside the tray, close the tray, and close the oven. In this task, the oven is randomly initialized at one of the three borders of the workspace (where from the robot to avoid joint limits of the robot), and the bagel is randomly initialized along the opposite side of the oven. The observation is voxel grid with a resolution of 64 x 64 x 64 and the gripper pose and open width. The voxel grid covers the (0.4m)3 workspace. During training, we apply a random crop augmentation to crop the voxel grid 58 x 58. In Banana in Bowl and Trash Sweeping, we train the model with an additional random rotation augmentation. The baseline trained with the same data augmentation our method.
climinate the side that is furthest
M Generalization Experiment
In this experiment, we evaluate the generalizability of Equivariant Diffusion Policy to unseen object poses. We conduct this evaluation in the Bagel Baking experiment in the real world, where the oven is initialized in three different poses during training (Fig 12a). At test time, we rotate the oven to eight different, unseen poses (Fig 12b). We found that the learned policy can zero-shot generalize to these unseen rotations, with the exception of the scenario where the oven is rotated to the bottom- right corner. this case, the policy constrained by the robot's ss joint limits. Specifically, the policy was able to open the oven and pull out the tray, but when picking up the bagel, although the policy could generate good gripper poses, the actions were infeasible for the robot due joint limits. This generalization demonstrates the power of the equivariant structure in our policy.
21

---

Figure 11: Our real-robot platform contains Franka Emika robot equipped with pair of fin- ray fingers, and three Intel Realsense D455 cameras.
(a) Oven Poses in Training Set
(b) Oven Poses in The Generalization Experiment
Figure 12: (a) The initial oven poses in the training set. (b) The oven poses in the generalization experiment. Those poses are unseen during training. both training and testing, the pose of the bagel is random.
22


Reviews:
[{"review_id": "Dj2vTWNXCm", "reviewer": "Reviewer_T42w", "summary": "", "questions": "", "limitations": "", "rating": 4, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": 4, "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": "The authors propose Equivariant Diffusion Policy, an imitation learning policy based on diffusion policy that is SE(3) equivariant, i.e., if the observation inputs undergo an homogeneous transformation H, the output actions also expected to undergo the same transformation.  Importantly, the authors consider visual inputs and not state based ones, which shows the higher relevance of this work for real world applications.", "summary_of_recommendation": "This paper is a great contribution to the field of structured policies for imitation learning. The theory is sound and clearly explained. The experiments, both in simulation and real world, showcase the benefit of using equivariant representations over the baselines. Most importantly, the authors show that using equivariant networks improves greatly on sample efficiency, which is a bottleneck for learning from demonstrations.", "technical_quality": 5, "clarity_of_presentation": 5, "potential_impact": 4, "robotics_focus": 4, "quality_of_limitations": 3, "questions_for_rebuttal": "One of the benefits of diffusion models is that they model highly multimodal distributions quite well. \nCompared to the diffusion policy, do you notice any drop in multimodality due to the equivariant networks? \n\nSince your observation encoder is equivariant, do you still need to do data augmentation for the voxel and image-based observations?\nIf yes, did you also use data augmentation when training the (unstructured) diffusion policy?\n\nIn the introduction, you state: “…we theoretically demonstrate the use of SO(2)-equivariance in the context of 6-DoF control for robotic manipulation, which prior methods [9, 10] leveraged in a less expressive SE(2) action space…” I’d expect the SE(2) space to be “more expressive” than SO2 since it includes translations. Could you clarify why you say SE(2) is less expressive?", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "983uRLqxM5", "reviewer": "Reviewer_1QLT", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": 3, "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": "This paper introduces equivariant diffusion policy, which improves the performance of the original diffusion policy, especially in the low-data regime. To this end, the major contribution is action equivariance in SO(2) space. Finally, extensive experimental results in both simulation and the real world are included to show the performance improvement.", "summary_of_recommendation": "It has good quality and could be potentially improved.", "technical_quality": 3, "clarity_of_presentation": 3, "potential_impact": 3, "robotics_focus": 4, "quality_of_limitations": 2, "questions_for_rebuttal": "Listed in the \"Review\" section.", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "qRK7tAnxk9", "reviewer": "Reviewer_TQso", "summary": "", "questions": "", "limitations": "", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": 4, "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": "This paper aims to leverage domain symmetries in robot arm manipulation tasks to improve the performance and sample efficiency of Diffusion Policies. Specifically, the proposed framework uses the SO(2)-equivariant of planar rotations, i.e., rotation around the z-axis of the world), employing an equivariant observation encoder, an equivariant action encoder, and an equivariant action decoder. The experiments in both simulation (12 manipulation tasks from MimicGen) and a real-world system (Franka Emika robot arm) show that the proposed framework outperforms existing methods, including Diffusion Policy, 3D Diffusion Policy, Action Chunking Transformer, and BC-RNN, in terms of the performance of learned policies as well as data efficiency, i.e., the amount of demonstration required. This work is well-motivated, tackles an important problem, and presents a reasonable framework with sufficient evaluation. I believe this work will interest a wide range of audiences at CoRL and therefore recommend accepting this work.", "summary_of_recommendation": "This work is well-motivated, tackles an important problem, and presents a reasonable framework with sufficient evaluation. I believe this work will interest a wide range of audiences at CoRL and therefore recommend accepting this work.", "technical_quality": 5, "clarity_of_presentation": 4, "potential_impact": 3, "robotics_focus": 4, "quality_of_limitations": 2, "questions_for_rebuttal": "See above", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: # Strength 
- All reviewers are mostly aligned on the technical strength of the paper

# Weakness 
- No major weakness highlighted by the reviewers 

# Recommendation 
Reviewers have made excellent recommendations to convert a good to a great paper. 
- Why MimicGen instead of Robomimic?
- Demonstration on locomotion tasks on uneven terrains. Additionally validates its effectiveness on nonflat surfaces. 
- Extension to continuous rotations.

Authors are highly encouraged to work with the reviewers to accommodate these suggestions to further strengthen the submission.

# Rebuttal 
All reviewers are aligned on their recommendations post rebuttal.
Decision: Accept