Equivariant Motion Manifold Primitives
Authors: Byeongho Lee
Yonghyeon Lee
Seungyeon Kim
MinJun Son
Frank C. Park
Venue: CoRL 2023 Poster (2023)
Keywords: Movement primitives
Manifold
LfD
Equivariance

Abstract: Existing movement primitive models for the most part focus on representing and generating a single trajectory for a given task, limiting their adaptability to situations in which unforeseen obstacles or new constraints may arise. In this work we propose Motion Manifold Primitives (MMP), a movement primitive paradigm that encodes and generates, for a given task, a continuous manifold of trajectories each of which can achieve the given task. To address the challenge of learning each motion manifold from a limited amount of data, we exploit inherent symmetries in the robot task by constructing motion manifold primitives that are equivariant with respect to given symmetry groups. Under the assumption that each of the MMPs can be smoothly deformed into each other, an autoencoder framework is developed to encode the MMPs and also generate solution trajectories. Experiments involving synthetic and real-robot examples demonstrate that our method outperforms existing manifold primitive methods by significant margins. Code is available at https://github.com/dlsfldl/EMMP-public.

Equivariant Motion Manifold Primitives
Byeongho Lee
Yonghyeon Lee
Seungyeon Kim'
Minjun Son'
Frank C. Park'
Scoul National University (bhlee, ksy, mjson) @ www.tiic..su..c.... ylee@kias.re..... ke.
Korea Institute for Advanced Study (KIAS)
fep@ anp.aru.ckkkr
Abstract: Existing movement primitive models for the most part focus on repre- senting and generating single trajectory for given task, limiting their adaptabil- ity to situations which unforeseen obstacles or new constraints may arise. this work we propose Motion Manifold Primitives (MMP), movement primitive paradigm that encodes and generates, for given task, continuous manifold of trajectories each of which achieve the given task. To address the challenge of learning each motion manifold from a limited amount of data, we exploit in- herent symmetries in the robot task by constructing motion manifold primitives that are equivariant with respect to given symmetry groups. Under the assumption that each of the MMPs can be smoothly deformed into each other, an autoencoder framework is developed to encode the MMPs and also generate solution trajecto- ries. Experiments involving synthetic and real-robot examples demonstrate that method outperforms existing manifold primitive methods by significant mar- gins. Code is available at https: / /ttp://gtttt..t//// com/dssfld//EMma//...c p
Keywords: Movement primitives, Manifold, LID, Equivariance
Introduction
Learning basic motion skills as movement primitives has been an enduring focus of learning from demonstration (LfD) research [1, 2, 3]. A primary challenge is constructing movement primitive models adaptable diverse situations, such as when unforeseen obstacles new straints emerge. Current approaches to movement primitives encompass dynamic movement primi- tives [4. 5, 6, 7, 8, 9, 10, 11, 12, 13], stable dynamical systems [14, 15, 16, 17, 18, 19, 20, 21, 22], methods based on Gaussian processes [23, 24, 25] and Gaussian mixture models [26, 27, 28, 29], along with other methods [30, 32).
The limited adaptability existing primitive models largely stems from their design which encodes and generates single trajectory for specific task, since they have alternatives when their pri- mary trajectory becomes infeasible in new environments (e.g. when unexpected obstacle blocks the trajectory). Although dynamical system-based methods can integrate, for instance, obstacle avoidance potential function terms [33, 34, 35, 36), the resulting motions might violate other task constraints. For adaptable motion primitives, a method that encodes multiple trajectories for a single task essential.
In this paper, we propose to learn continuous manifold of motion trajectories that can perform the given task, which we refer to as Motion Manifold Primitives (MMP). As the MMP encodes multiple successful trajectories, even if some trajectories are obstructed by obstacles or violate constraints, alternative feasible trajectories remain accessible within the MMP. As such, the MMP is highly adaptable, although more diverse demonstration data are needed for training, compared to the single trajectory-baant primitives.
Given set of task-trajectory paired data where multiple demonstration trajectories are collected for single task parameter - our objective is to learn set of manifold primitives (Mr} for
The two lead co-authors contributed equally.
7th Conference Robot Learning (CoRL. 2023), Atlanta, USA.

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all т. However, given a limited amount of demonstration data, learning accurate manifolds and their boundaries is a very challenging task. In fact, the TC-VAE [37] is the first work to adopt this manifold primitives approach but shows less-than-desrrbble performance given small dataset as we show later in our experiments.
In this paper, ifold primitives learning algorithm, where we adopt the autoencoder- based manifold learning framework [38, 39, 40, 41, 42, 43, 44]. First, we propose Equivariant Motion Mani- fold Primitives (EMMP), which takes into account the inher- ent symmetry robot tasks. For example, consider water- pouring task where the initial cup and bottle's positions defined the task parameter. Given symmetry transforma- tion that preserves the relative positional relation between the cup and bottle (e.g., rotating bottle around the cup). the set of water-pouring trajectories, the motion manifold primitives Mr. should also be transformed a consistent manner, more precisely, equivariantly (see Figure 1). We show that using an invariant encoder and equivariant decoder can guarantee the equivariance of the MMP and propose a strategy construct invariant and equivariant mappings. Mcanwhile, equiv- ariance has been increasingly recognized pivotal factor in robotics tasks in general [45, 46, 47].
develop data-efficient motion man-
Trajectory manifold
Equivariantly transformed trajectories
Figure 1:
An illustration of motion manifold primitives and equivariant transformation.
Second, to further enhance data efficiency, we find a shared latent coordinate space Z, by assuming that M, for all 7 are homeomorphic (i.e., can be smoothly deformed into each other). In practice we consider the latent coordinate variable = Z to be independent T. This means that p(z/7) p(2) for T, making the support of p(2) the shared latent coordinate space 2. To enforce this condition, newly propose independence regularization ferm in autoencoder training.
Through comprehensive experiments involving synthetic and real-world robot water-pouring exper- iments, we compare our EMMP model with the existing manifold primitives method TC- VAE [37] through a systematic evaluation on (i) manifold learning, (ii) independence between and т, (ii) density learning, and (iv) success rate. We find that our method significantly outperforms TC-VAE. Further, an in-depth ablation study reveals that the use of an invariant encoder and equivariant de- coder is the primary factor driving this performance improvement.
2 MMPs: Motion Manifold Primitives
In this section, given con- figuration space Q, we con- sider motion as a dis- crete and fixed-length con- figuration trajectory æ {q. E 2) the space of all trajectories is denoted by X QT. Let com- pact space T represent a space of parameters that define specific tasks. We assume that, for each 7 E T, a set of motions that can perform the given task forms an m-dimensional differentiable manifold, which we refer to as the Motion Manifold Primitives (MMP) denoted by My. and that a submanifold lying in the ambient space X. We assume that we are given a set of task-trajectory paired data, denoted by N.j=I...,,M; D = {(T, xig) E T X"J==I,.,,N M. where M, represents the total number of trajectories for the task parameter 7; and Tay is sampled from My,, For readers unfamiliar with geometric cepts such manifold and homeomorphism, we include brief introductions of them in Appendix C.
Task parameter
Feasible trajectories
Metion Manifold Primitive
Latent Coordinete Space
M
Z
An illustration of a task parameter feasible trajectories that achieve the task, and a manifold that the trajectories form. The resulting manifold M, consists of two disjoint components.
Figure 2:

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2.1 MMP described via Manifold and Density
This section introduces how to represent the set of motion manifold primitives Mr C X for all T E T. First, we assume that each manifold primitive M, can be parametrized by a nonlinear map f R + X with a coordinate space 2, R. as MM = fz+) To represent the set of motion manifold primitives simultaneously, we consider a differentiable mapping / IR T + X' such that 72) := f+(z). Second, to represent each coordinate space Z,, employ conditional probability density function in R" given T denoted by p(zl7). Then, we consider the support of p(2|T) as 2.. As result, each motion manifold for 7 is represented as MM = f(z,, r); the motion manifold primitives Mr is said to he parametrized by the mapping f and the density p(2/7) or parametrized by f and 2, (see Figure 2).
2.2 Autoencoder-based Manifold and Density Learning
We adopt the autoencoder framework for learning My via f(2, and p(2l)). where f considered as the decoder and the latent space density in IR.. An additional component that specifies the coordinates of an input trajectory E MM needs to be introduced, called an encoder, and we denote it by : X' x T + Rn'h such that 2 = g(x, T). We use parametric models (c.g., neural networks) for encoder Ie- decoder fg. and density (=/r), where 0, Φ, and represent the model parameters. We propose a two-step approach where we first learn the coordinate systems, i.e., fo: Ie and then learn the density, i.e., p=(==))
The standard autoencoder reconstruction loss can be employed to learn fe, 9ø. that is an expectation of && (fol T.), T,), miy)) where (;, :) is a proper distance measure on X' Minimizing the reconstruction loss results in that (M, T), se MM, for the ground truth manifold M,, fo(: T) becomes a coordinate system for Mr and (M(,, becomes the coordinate space 2,- Secondly, we can learn (=|T) given a trained encoder dø via standard likelihood maximization framework, i.e., maximizing expectation of log Py (IIg,
2.3 Homeomorphic Manifold Assumption
Learning densities for all т, i.e., p(2(r) challenging given the small training dataset DD In this section, we introduce a homeomorphic manifold assumption to make the density learning problem more tractable. We assume that M, for each 7 are homcomorphic each other and there exists latent coordinate variable statistically independent of r, i.e. p(z) p(=17) for all т, which results in shared latent coordinate space Z Z, for all The training of autoencoder and density can be greatly simplified under this assumption.
First, we can restrict the encoder's input space to X, i.c., g X + R because 7 should not contribute 2. Second, p(2]T) can be replaced by a shared model p(z). Using g(x) and p(2), we may sequentially train the autoencoder and density, however, the reconstruction loss alone does guarantee the statistical independence between and hence learning the density p(z) can fail.
In this section, we introduce an independence regularization term for autoencoder training, so that and 7 become independent and = Z, for all
E, () [1())(())) go (fo ge ())))
R(0, の) =
E(,,,))D Ig. (21;))1
where 7 is randomly sampled from the uniform distribution on T. This loss enforces that g(f(2, r)) does not depend 7, and eventually, 7 does not contribute to 2, and becomes independent of z. The denominator makes the regularization term invariant to the latent value scale. This regularization term is added to the reconstruction loss with proper regularization coefficient.
EMMPs: Equivariant Motion Manifold Primitives
In this section, goal is to construct the motion manifold primitive M, that transforms equivari- antly when symmetry transformations are applied to We denote symmetry group by H where
3

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the group operation between two elements h1, h2 E H is written as h2hę E H. Then we assume symmetry transformations are defined as group actions. Two symmetry transformations in T and X' x T are considered, where we use the same symbol to denote the group action ' by T + h and that in X by (x,7) + h (x,,). r). Let [.] denote the x component, i.e., [[x,T)]= x. The ground truth motion manifold primitive is denoted by M,, while the learned motion manifold primitive is denoted by MM parametrized by a decoder f(2, 7) and p(z[r). We include preliminary knowledge on group, equivariance, and invariance in Appendix C.
3.1 Invariant Encoder and Equivariant Decoder
We begin this section with the definition of the equivariant motion manifold primitive:
Definition 3.1. Suppose M, is motion manifold primitive. Given a transformed task parameter
++ h-r, if the motion manifold M, equivariantly transformed, i.e., Ma- {[- (z, r) z Mr), then the primitive called Equivariant Motion Manifold Primitive (EMMP); see Figure 3.
We show that using an invariant encoder and an equivariant decoder can guarantee the equiv- ariance of the resulting MMP; first provide their definitions:
Equivariant Motion Manifold Primitives
Task Farameter
Transfomation
Definition 3.2. An encoder : XXX + Rm is invariant if g(h-(a, = (2, for all H and X.
Motoon manitold
Tranformation by
Definition 3.3. A decoder f R" x + +x equivariant f(z, h- = [h- for all h И and 7 T.
Figure 3: Given a symmetry transformation to the task parameter - h where the relative dis- tances between the robot, goal, and obstacle are preserved, the MMP equivariantly transformed.
Denoting the ground truth motion manifold by Mr the latent coordinate space 2, defined as the support of p(=[7) can be specified by coder g(x, T) follows: Z, (UzeM, g(z,T)) g(M,,T). If M, is equivariant, then an invariant encoder produces invariant coordinate space:
Proposition 3.1. Suppose M, an EMMP. If g is invariant, then Z, Zuu for all H.
In addition to the encoder invariance condition, the learned MMP parametrized by a decoder f(z 7) and 2,, denoted by M,. is equivariant if a decoder f is equivariant:
Proposition 3.2. Suppose My is an EMMP. If invariant and f is equivariant, then the MMP
parametrized by f and Z,, i.e., M, = f(Z,, where 2, == is equivariant.
By constructing an invariant encoder and an equivariant decoder, we can ensure the equivariance of the leamed motion manifold primitives. When the ground truth MMP equivariant, it is reasonable to expect that this equivariance guarantee would enhance the accuracy of manifold learning.
3.2 Construction of Invariant and Equivariant Mappings
In this section, propose method for converting arbitrary encoder and decoder models to that are invariant and equivariant to the symmetry group H. Let G X x T - R and Fg Im x T + X' be arbitrary parametric models for encoder and decoder, respectively. Assuming we can find an equivariant map h + H such that h(h - 7) = hā(r) for all E H, E T, an invariant encoder and equivariant decoder can be constructed as follows:
Proposition 3.3. An encoder g X T - T defined as (o(x, 7) = G2(h(r)-11 (x, (x, is invariant and a decoder f I x + X' defined as folz, == ((()) [ T)] is equivariant.

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How to construct the map is problem-specifie. As an example, let X' be the set of 3D point cloud data - point x E A' of the form x = f .x) and H be the group of translations in R In this case, one possible H -eequivariant h(x)) is the point cloud centroid: h(x)) :- & Σ Li. whose inverse would be h(z) - = -(x).. More examples are in Appendix D.2.1 and Appendix D.3.1.
When we adopt the homeomorphic manifold assumption and use a parametric model Go. R" the invariant encoder can be defined as follows: (z( 7) = Go((H(r)-1 (2,, .)), where Jo needs to take as input while Gø does not.
Experiments
In this section, we compare our EMMP framework mainly with the existing manifold primitives method, TC-VAE [37], using both synthetic and real-world robot experiments. Additionally, we compare MMP, MMP + indep, EMMP, and EMMP indep, where the EMMP uses an invariant encoder and equivariant decoder, and indep' indicates adding the independence regularization term in autoencoder training. When training TC-VAE, MMP, and MMP + indep, always apply random data augmentation on Throughout, we use Gaussian Mixture Model (GMM) to fit p(z).
We report four evaluation metrics. First, the manifold learning accuracy is measured by the Reconstruction Error (RE). Second, measure the degree independence, we report estimated Mutual Information (MI) between and T. by using MINE [48]. Third, we compute the probability density learning performance by the Negative Log-Likelihood (NLL) measured in the data space. Lastly, report the task Success Rate (SR), of which criteria is desk-dependent. Details on the computations of these measures are described in Appendix D.I.
Evaluation metrics:
4.1 Goal-Reaching Task of Planar Mobile Robot
In this section, consider goal-reaching task for planar mobile robot, given cross-shaped wall, without colliding with the wall; the robot is re- quired to enter through one of the two closest passages (see Figure 4). As shown, it is reasonable to assume a set of feasible trajectories form a con- tinuous manifold. Our goal to learn the motion manifold primitive for an arbitrarily rotated wall and robot's initial position.
The configuration space is Q = IR2, the trajectory space is X' = 2TT, The task parameter т consists of the initial position of the mobile robot, denoted by (g/- )²), and the rotated angle of the wall axis En with respect to L.,, de- noted by ww (see Figure Left). To make demonstration data, a human demonstrator has drawn feasible trajectories, 6 trajectories for randomly given task parameters. Figure illustrates exam- ple demonstration trajectories. More details data generation and data split for training, validation, and testing included in Appendix D.2.2.
Figure 4: Demo tra- jectories.
Symmetries: There exist three sym- metries that preserve the relative geo- metric relation between the wall and initial mobile robot position: flip- ping of the robot over the wall axis f, (ii) rotation of the robot around the origin by 90, 180, and 270 degrees, and (iii) rotation of the wall and mo- bile robot around the origin by the same amount (see Figure 5 Right). These symmetry transformations can be described group actions with the symmetry group Hl p4m SO(2), where p4m specific type of wallpaper group and SO(2) is the group of 2 x rotation matrices. We de- note p4m = {(i,j)l e {0.1).j E 2, where i {0, 1} represents flipping and
Flip over axis Robot / rotation
Task Parameter
Wall&rrobot Rotation
Initial
(gi-9f)
Wall
angle the
9
Figure 5: Left: The task parameter of 2D mobile robot's goal reaching task. Right: Symmetry transformations.
5

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j E {0, 1,2,3} represents n rotation. Details on the group operations in H and group actions on T and X' x T are discussed in Appendix D.2.1.
Construction of h: Figure 6 visualizes the equivariant map h(()) (hu (T), l2(TT). (T) pim is determined based on the robot's position relative to the wall state
(1(1)
(0.
p4m
(r) Ro)( S0(2)
[0.
(0,0)
(0. (0.0)
(0. 1)
shown in Figure 6 (Left). (T) E SO(2) is determined based on the wall axis an- gle wi (see Figure 6 Right). The h(T) is equivariant to p4m x SO(2); more details including the proof are in Appendix 1.2.1.
(1,2)
(L,0)
(0,2) (1,2)
Wall axis angle olw
(0.3)
3(1,3)
(0,
(1.3)
(0.3)
90" rotation flipping
L.&,
Figure 6: Illustration of the (p4m, - map h(T).
Assuming a one-dimensional latent space, we train fully connected autoencoders for MMPs and EMMPs, and TC- VAE where temporal convolutional neural networks are used as in [37]. Table shows the four evaluation metrics, where the success
Table I: Reconstruction Error (RE), Mutual Informa- tion (MI, and Negative Log-Likelihood (NLL); the lower, the better. Success Rate (SR); the higher, the better.
Method
RE(L)
ML
NLL (U)
(R(
TC-VAE [37]
0.257
0.268
1.50 x 10
53.18%
rate is measured follows: (i) sam- ple (T,2) from the uniform distribution on T and learned density p(z), (ii) generate trajectories via f(2,, T), and (iii) the gener- ated trajectory is considered successful if it is consistent with the task parameter and reaches the goal without colliding to the wall. More details are in Appendix D.2.2.
MMP
0.223
0.487
1.49 x 10
50.08%
indep
1.48 10
52.39%
EMMP
0,223
0,082
1.25 x 10"
92.40%
EMMP indep
0.229
0.077
1.24 x 10
86.66%
First of all, the EMMP methods show much higher suc- cess rates than both the MMP methods and TC-VAE, which is in large part attributed to the independence be- tween & and T and latent density fitting as shown in Ta- ble 1. As observed from the MI scores of the MMP + indep (0.329) and EMMP (0.082), using invariant and equivariant mappings is much more effective than the in- dependence regularization term making z and inde- pendent, which consequently makes learning p(z) easier (see Figure 7).
MMP+indep
EMMP
p((tt))
T1
Z
Figure 7: p(2(T) of the MMP + indep and EMMP.
Second, the success rates of MMPs and VC-VAE do not show a noticeable difference. This suggests that the net- work architecture (temporal CNN FCN) and autoen- coder method (VAE AE) do not have significant im- pact on performance. A further comparison using EMMP in this regard is provided in Appendix D.2.3. Third, while the independence regularization term slightly improves the success rate of MMP, for EMMP, it rather shows a negative effect. Given the already low MI of EMMP, fur- ther reducing it at the expense of RE appears have neg- atively impacted the success rate.
TC-VAE
EMMP indep
p(2)
90%
90%
Decoding
G
Figure& 8 (Top) visualizes learned latent space densities p(z) of TFT--VAL VAE and EMMP + indep. We select equally- spaced 6 latent points {= 1 Z for each model as shown in Figure 8 (blue points). The trajectories are then generated by the decoders as shown in Figure 8 (Bottom).
Figure 8: Top: p(z). Bottom: Generated trajectories.
The EMMP + indep shows much higher success rate than the TC-VAE the support of p(>) in our method has two connected components, each corresponding to the set of trajectories that passes through one of the two closest passages to the robot.

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4.2 Water-Pouring Task of Franka Panda Robot
In this section, we consider a water- naterng task for a Franka Emika Panda robot arm; the robot is assumed to hold the bottle initially upright and is required to pour 150g of water into the cup (see Figure Left). As shown in Figure 9 Right, human demonstrator provides multi- ple water-pourrin trajectories, which are assumed to form a continuous manifold. Our goal is to learn the motion manifold primitive for an arbitrary cup position, a bot- tle's s initial pose, and the amount of water the bottle. In particular, depending on the amount of water in the bottle, the demonstration trajectories have very different characteristics (see Figure 15 Left in Appendix D.3.3).
Human demonstration
Demonstration trajectories
Figure 9: Demonstration data.
The configuration space is the space of the bottle pose, Q SE(3), the trajectory space is A' SE(3)*. The task parameter 7 consists of the cup position on the table (gè (²) E R2, the bottle's initial pose T. E SE(3), and the mass of the water mw E [0.2, 0.41] (See Figure 10 Left). We collect 5 demonstration trajectories for 35 different task parameters in total of 175 trajectories (details are included in Appendix D.3.2). The demonstration trajectories are collected by recording video of human demonstrator performing the water-pouring task and extracting the bottle's SE(3) trajectories using AprilTag [49]. Figure 9 Right shows trajectories demonstrated for given single task parameter 7. More details on the task parameter selection, data generation, and data split for training. validation, test are included in Appendix D.3.2.
Symmetries: There exist three symmetries that pre- serve the relative positional and geometric relationship between the cup and the bottle: (i) translation of the cup and the bottle by the same amount, (ii) rota- tion of the bottle around the cup, and (iii) rotation of the bottle around itself (see Figure 10 Right). The symmetry transformations can be described group actions with the symmetry group H = R² x SO(2) x SO(2). Details on the group operations in H and group actions on T and X' x T are discussed in Appendix D.3.2.
Task parameter
Translation of cup bottle
Rotation of bottle around cup axis
Rotation of battle around
mw [0.2.0.4]
(ab.92) R2
O-
T, SE(3)
Figure 10: Left: The task parameter of water-pouring task. Right:
Symmetry transformations.
Definition of H and h: We de- fine h(7) (hi(7) (T), hT)) as shown in Figure 11. (T) is deter- mined as the cup's position as shown in Figure 11 Left. h2(r) is deter- mined based on the position of the bottle relative to the cup (see Fig-
A,(z) = (2b.9)) ER? hg(t) (ax) e S0(2)
hy(t) (ci)) S0(2)
S
Rotation angle
Rotation angle around cup axis
around itself
Figure 11: (R²
module h(T))
ure 11 Middle). hz(r) is determined based on the bottle's 's orientation relative to the base frame (see Figure 11 Right).
Assuming a two-dimensional latent space dimension, we train EMMP indep with fully connected neural networks and TC-VAE. Table 2 shows the four evaluation metrics (RE, MI. NLL, SR) where the generated robot motion is considered successful even a little water can be poured into the cup without spilling and one additional metric that measures the error in the amount of poured water (150g water has been poured in demonstration data), which we refer to as the Water-Pouring Error (WPE). To generate motions given task parameter from T, we sample from the learned density p(2) and generate the bottle's 's trajectory via f(2, 3). If the generated SE(3) trajectory is out of the robot's workspace, i.c., the inverse kinematics solution does not exist, then we re-sample
7

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until we obtain a feasible trajectory. We measure the SR and WPE with task parameters each with 5 samples, where we run a total of 20 generated trajectories on the real Panda robot.
As shown in Table the EMMP indep significantly outper- forms TC-VAE. The large margin RE and MI leads to lower NLL and much higher task success rate of EMMP. Out of 20 trials of the VAE, 7 trials fail to pour water into the cup and 4 trials spill water, resulting in only 9 successful pourings. On the other hand, EMMP + indep results in 100% success rate. In addition, the WPE in EMMP + indep is also much lower than that of TA-.
Table 2: Reconstruction Error (RE), Mutual Information (MI), Negative Log-Likelihood (NLL), and Water-Pouring Error (WPE);
the lower, the better. Success Rate (SR); the higher, the better.
Method
TC-VAE
EMMP indep
RE(T)
0.183
0.129
While the EMMP's WPE (23.0g) out of 150g seems high, note that this error is not caused by the manifold primitive learning error, but rather is attributed to the error caused when processing AprilTags and smoothing the trajectories. Even when we replay the demonstration tra-
NLL
1.18 x 10
6.40 10*
SKIT)
9/20
20/20
WPE (J)
1:
23.0
jectories on the robot, the water pouring error exists and it 19.3g on average, implying 23.0g error that high.
To show the strong adaptability of our framework, we perform an obstacle avoidance task using EMMP + indep. Suppose there is an obstacle, not seen during training, that blocks some water-pouring trajectories in the learned manifold primitives (c.g.. Figure 12 Left). Since we have learned a motion manifold, not single trajectory, even if some trajectories are blocked, we can easily find alterna- tive collision-free frree trajectory from the learned manifold primitives shown in Figure 12 Right. More details the collision detection and obstacle avoidance algorithms are in Appendix D.3.2
Collision occured
Collision avoided
Figure 12: Obstacle avoid- ance.
5 Limitations
One of the key assumptions in our framework is the homeomorphic manifold assumption, that is Mr for all r E T are homeomorphic, which may not hold depending on the problem. In such cases, instcad of p(2), we need to fit p(2/T), which would require more demonstration data. Sccond, to construct an invariant encoder and equivariant decoder, we need define an equivariant map for a given symmetry group H. Although, in our case studies, relatively straightforward to construct ht this process may not be trivial even impossible depending the problem. Lastly, as experimental results show, the independence regularization term itself is not sufficient to enforce independence between and т. Finding better independence regularization method would be an important future research direction.
6 Conclusion
In this paper, we have proposed new family of highly adaptable movement primitive models, motion manifold primitives which is a set of trajectory manifolds {M,} for all task parameters r and an autoencoder-basse framework for learning them. To tackle the challenges learning M, such requiring many demonstration data, (i) under the homeomorphic manifold assumption, we develop the motion manifold primitives framework and introduce the independence regularization term where enforee independence between and that is sufficient to learn p(2) instead of p(=17)- and (ii) we propose equivariant motion manifold primitives for an arbitrary symmetry group // in the robot task and method to parameterize it by constructing an invariant encoder and equivariant decoder. Extensive experiments have confirmed the strong adaptability our framework and that the equivariant manifold modeling highly effective learning accurate M,, which leads superior performance compared to the existing method by a significant margin.

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Acknowledgments
B. Lee, Kim, and C. Park were supported in part by SRRC NRF grant RS-2023-00208055, IITP-MSIT grant 2021-0-02068 (SNU AT Innovation Hub), IITP-MSIT grant 2022-0-00480 (Train- ing and Inference Methods for Goal-Oriented AI Agents), KIAT grant P0020536 (HRD Program for Industrial Innovation), ATC+ MOTIE Technology Innovation Program grant 20008547, SNU-AIIS, SNU-IAMD, SNU BK21+ Program in Mechanical Engineering, and SNU Institute for Engineering Research, Lee was the beneficiary of an individual grant from CAINS supported by KIAS Indi- vidual Grant (AP092701) via the Center for and Natural Sciences Korea Institute for Advanced Study.
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A Proofs of Propositions
Proof of Group-Equivariance of (3.1). The equivariance of the ground truth EMMP and the invari- ance proves the proposition as follows:
Z = alM N r) a( U (.(ar)) = al U (x,T)) a(MriT) 2,.
(2)
MCMr
#CMr
Proof of Group-Equivariance of (3.2). The equivariance of and invariance of proves the propo- sition as follows:
M 1(2 h. U [b (((T))))) U (2, {(N (z,r)), M.).
(3)
262r
M
Proof of Group-Equivvriaacee of (3.3). Invariance Is can be seen by the equivariance h, as follows:
96(h- (2,+)) (h(hhh -) (a,, +))) Ga(((hh(T)) (a.+)) G,(h(T)) (x, go(2,))
(4)
Equivariance of fa can be seen follows:
f [h(h 7) ((b ((- (hh-)))
= [h .(r) (1))
[h. ())))
(5)
B
Related Works
In this section, we provide an overview of areas related to our work.
B.1 Movement Primitives
In this section, we consider any form of mathematical representation used to describe motions (e.g., trajectories) that perform given task-as specified by task parameter variable 7-as move- ment primitives. Dynamic movement primitives encode motion trajectories in the form of time- dependent nonlinear dynamical systems. These systems consist of mass-spring-damper systems and parametric force terms, with their task parameters defined by the initial and final configura- tions. [4, 8, 9, 10, 11, 12, 13]. Stable dynamical system-based approaches use state-dependent dynamical systems that are globally asymptotically stable [14, 15, 16, 17, 18, 19, 20, 21, 22]. In these dynamical systems- based methods, the initial and goal configurations can be considered the task parameters, and the solution trajectory that connects the two configurations is the motion described by those systems. ProMP [30] represents motions as distribution over trajectories. By conditioning the distribution with the initial configuration, motion trajectory is achieved. In ProMP, the initial and final configurations can be considered as task parameters. TP-GMM [26] tries to adapt to unseen task parameters by encoding trajectories as GMM seen from multiple frames. The task parameters of TP- TP-GMM are the frames of G.Mss. Given unseen frames, TP-GMM generates trajectories by calculating joint distributions between GMMs. [23, 24, 25] parameterize the demonstration trajec- tories using Gaussian Pross Regression, and [28, 27, 29] represents the motions using GMR (no task-parameeerizzation exists).
However, the task parameters of most movement primitives are strictly restricted to the initial and final configurations. This limits the range of tasks that can be parameterized. In the case of water pouring, the cup's position and the amount of water can be represented by the initial and final configurations. By adopting conditional variational autoencoder's structure, MMPs and EMMPs provide freedom of defining task parameters.
13

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B.2 Autoencoder-Based Manifold Learning
Autoencoders have gained prominence in recent years for identifying and generating samples from a given data distribution's underlying cow-dimensional manifold structure. The main reason autoen- coders are frequently adopted for manifold learning that they learn the latent space coordinates along with the manifolds. To learn more accurate manifolds, researchers have introduced additional regularization terms [38, 39, 40, 42, 43, 41, 50, 44]. For specific structure of conditional variational autoencoder, where the decoder gets an additional conditional parameter, the need to disentangle the conditional inputs and latent values has risen. [51] introduced a regularization term to disentangle input spaces of its decoder, by solving adding an auxiliary neural network to estimate conditional inputs from latent values, and regularizing the autoencoder by making it harder for the auxiliary network to estimate. However, unlike the independence regularization term that we introduced, the regularization term does not necessarily guarantee independence between the two spaces.
B.2.1 Autoencoder-Basee Motion Manifold Primitives
In this section, we introduce an existing motion manifold primitive framework called TC-VAE [37]. TC-VAE aims to parameterize the motion manifold given a task parameter based on autoencoder frameworks. As TC-VAE adopts the structure of [51], the decoder of it takes additional task param- eter inputs other than the latent value inputs. TC-VAE also adopts the regularization term of [51] for disentangling the task parameters and the latent values, which still shares the shortcoming of not guaranteeing independence between latent space and the conditional input space.
B.3 Equivariant Models in Roboties
Invariance and equivariance properties have played a role in deep learning models as an inductive bias to generalize well and be trained data efficiently [52]. Translation equivariance in convolu- tional neural networks (CNNs) has been effective for image recognition tasks [53]. Group equiv- ariant CNNs have expanded the equivariance in CNNs to more complex equivariance, e.g. SO(3)- equivariance achieved by Spherical CNNs [54, 55]. In robot manipulation tasks, [56] proposed an -quiivaqui object representation, and [45] introduced a SE(Da-equivariant dynamics model learning for pushing manipulation. Most of the existing equivariant models robotics are restricted to certain types of groups, whereas our work can be applied tasks with arbitrary group symmetries.
C Geometric Preliminaries
This section provides preliminary knowledge of geometric tools used the paper.
C.1 Manifold Hypothesis
Real-world observations often require large number of variables represent numerically. For ample, SE(3)-traecctory of length 500 lives in high-dimensional data space 8800 Dealing with such high-dimensional data is very challenging the amount of data needed grows exponentially with the dimensionality, known as the curse of dimensionality.
The manifold hypothesis states that high-dimensional data (e.g. trajectory) approximately lie on some sower-dimensionll manifold embedded in high-dimensional space, which suggests that the high-dimensional data can be in fact described by relatively small number of variables. For exam- plc, to describe points on two-dimensional sphere- which are represented as unit vectors in g we only need two variables, e.g., the spherical (0, cooordinatsss.
Of particular relevance to our paper, consider a set of length n trajectories in IR? which start from the robot and end the star, as shown in Figure 13 (i.e., trajectories A, B, C, D. E). We note that these five trajectories are elements the high-dimensional trajectory data space g2xn However, it is clear that they do not fill up the entire space. Rather it is suspected that they form lower-iimensional space. Each trajectory may approximately be represented with one variable that indicates how much
14

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One-dimensional manifold in trajectory data space
One- -inedimensional manifold
D
Trajectory data space
A
Figure 13: Trajectories in high-dimensional data space lie one-dimensional manifold.
bends down up compared the straight line between the robot and the star, meaning that these five trajectories approximately lie on one-dimensional manifold.
C.2 Homeomorphism
A homeomorphism is continuous, bijective function that has continuous inverse function between two topological spaces (for this paper, two manifolds). Two manifolds are said to be homeomorphic there exists homeomorphism between two manifolds. Intuitively, two manifolds are home- omorphic if one can smoothly deformed into another. For example, sphere can be smoothly deformed into an ellipsoid, hence a sphere and an ellipsoid are homeomorphic. However, there is way to smoothly deform sphere into torus, which indicates that sphere and torus not homeomorphic.
Suppose that there exist two m-dimensional manifolds M1 and M2.. Let the latent space of M1 be 2, S IR' and the coordinate space M2 be Z2 C RT" Let fi Zi - Mi and f2 Ze + M2 be invertible maps satisfying fi (Z1) M₁ and fz
.M2. If M1 and M2 are homeomorphic, iee. there exists MM M2 such that g(M1) M2. 2, (fz o (2) is satisfied. Mi and Ma can have shared latent space by replacing f2 as f((-) = (g f-)())
C.3 Group and Group Action
group H a non-empty set, combined with a group operation x
: H - H, where we simply
denote by h1 h2 = h1h2, A group must satisfy four conditions:
The group H contains an identity e.
The group contains inverses, i.e., 1-1h hh-11 for all
The group operation is associative, i.e., hi (h2ha) = h2) for all hi, Ju,, hg H.
The group is closed under operation, i.e., h1h2 E H for all h1, h2 E H.
group action H X X' function defined on the product space of group and a set satisfying two conditions:
h (h2 (ggx)) x) = (h1h2) for all h h2 e H.
where e E H is the identity.
C.4 Equivariance and Invariance
Given group H. function f X + Y and group actions defined X and Y, the function is said to be H-equivariant if satisfies
f(h x) h f(x),
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for all x E X and h E H. f is said to be H H-invariant if it satisfies
f(h x) = f(x),
for all E X and h H. Invariance is special case of equivariance where the group actions defined on Y is the identity function, i.c. h for all h E H and E Y.
D Experimental and Implementation Details
Throughout the experiments, we have used RTX 2080 Ti, RTX 3080 Ti, RTX 3090 for training the models, and each experiment takes a few hours to 10 hours depending on the model.
D.1 Evaluation Metrics
Recontruction Error: We measure reconstruction error in the test dataset using following equation:
1 N
Reconstruntion loss
dh
T.),
(6)
M T
Latent-Tass Dataset: To calculate mutual information and negative log-likelihodd, we define a dataset (2,T) r) paired dataset. To build the dataset large enough, we first randomly augment (z, r) pairs in the training dataset 100 times. Then, the every z is the encoded values from (xr, r): go(x,7) T).
Mutual Information: Mutual information between 2 and is measured using Mutual Informa-
tion Neural Estimator (MINE) [48]. MINE estimates by, which estimates mutual information by maximizing its lower bound, using the Donsker- Donsker-Varadhan representation:
DKL 2 I1T) 2 sup Ez[F] - log (Eye²), FEJ
(7)
where F is any class of functions F : Ω + R. In our case, = x T. By replacing F by parametric family Fe, the mutual information is estimated as follows:
Ie T2(2,77
sup Ep(,.r) Fo log (Ep())o(=)fF()) AE
We train MINE using the latent-task dataset for 1,500 iterations with batch size of 5,000 equally for all models.
Negative Log-Likelihood: Given (2, from the latent-task dataset, we calculate negative log- likelihood log(pM, (0o()))) in trajectory space X, using the following equation:
PM. (go(z, r)) pz, (2, (z)l det [
(9)
where Jas denotes
Oga Dz (z,
D.2 Planar Mobile Robot Experiment
D.2.1 Formulas and Proofs
Task Parameter Space: Recall that the configuration space is Q RR2 the trajectory space is X 2T The task parameter consists of the initial position of the mobile robot, denoted by (g. 9²), and the rotated angle of the wall axis f. with respect to å,, denoted by ww-
To uniformly sample from we make T compact by restricting the initial position of the motile robot on a disk whose center the origin, the inner radius is and the outer radius is 10. Also, have limited the wall rotation axis - S Ww < I make the problem easier for non-equivariant methods e.g. TC-VAE. since the wall's geometries identical every 90 degrees, the wall still can span all possible geometrical configurations.
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Trajectory Space and Distance Measure: We set the length of trajectories T = 201, which makes the trajectory space X' = IR:402, 'he distance measure is defined as: (x F2) == ||
Group Operations: Recall that the symmetry group H H = p4m SO(2), where p4m is specific type of wallpaper group and SO(2) is the group of 2 x rotation matrices. We de- note p4m = {(t,j)l E {0.1),j E (0,1,2.3)}; where i {0, represents flipping and j E {0, 1, 2) represents nnT n2 rotation. Throughout this section, we represent an SO(2) element cos a sin a
R
E SO()) simply as a.
sin a
CDs or
Given two group elements ((a,b), ((c, d) p4m SO(2), the group operation is defined as: ((a, b), (i) d), B) - ((mod(a + c, 2), mod + (-1)"d, d, 4)), a + 3),
(10)
where mod(æ, y) denotes the remainder of n
Group Actions: The procedure of the group actions of И = p4m x SO(2) can be explained as follows: (i) flip the robot over the wall axis (ii) rotate the robot around the origin nr 2 and finally, (iii) rotated the robot and the wall. Given a task parameter = (4, of. ww) and a group element
((a, b), a),, the group action h T is then defined as:
by
(Rot(a + ww flip(a) Rot( -ww (at (.);)))))))))))) a ww)
(11)
1
[oos
sin o sin a cos a
where * denotes matrix multiplication, flip(a)
0 -1 and Rot(a)
Given a trajectory { a task parameter h ((a, b), a), the group action h (x. T) ) is defined as
o2))
(gr q² uw) and group element
( ({Rot(a + ww + by 2
h (I,, T) =
(ul (T)
(12)
Group-Equivariant Map h: Given task parameter h(T) can be divided into two elements, M(T) = (T), h2()), where (T) = (T). E p4m and ha(T) E SO((). Figure 14 illustrates (u(T,, and its equivariance for h using two group actions hmpp = (1,0,0) and A,000 1,) It can be seen by the commutative diagram that Ji(hnip -) = haph(T)) and h( 7) = hov)h((). The rest cases of flipping and rotating 90, 180, 270 degrees can be shown in the same way. shown, hz(rr) is defined as (T2 = Wi. The equivariance of h2 can be shown by
hot (0,0, a) and T -at (ot ww)):
T(hmme = (4': up. ww + a.) = (0, O, (u L. uw) = h.T((T)
(13)
Equivariance for the case where h = (a, a)) then simply shown by dividing into h = aa, b,0):
b((a,bba) - = E((((o, o, (a, 6, 0))
Ac(00, 0 a) (a, 6,0)
= (0, 0, a)h(((a 6,0)
(a, 0, a)h(r)
(a. (a,
(14)
Below equation is the formal definition of h given
(g/h 9² Ww
(T(T) = [0,
kx < atan2 (g2,, . (2)) ww <
kn
+
2
2
11 otherwise
0,
if _ S atan2 (gx 4,= 0pp)
Wm
1, if x < atan2 q² 9.2) - wiw < 3x
4
4
(15)
[2((T) =
2,
if 3n < atan2 0²2 s.2) -w <
< atan2 -/ 42) Wwo
3T
3, if S atan2 ( o 97) Who
h2(T) = Whi
17

---

Mpp
hot99
Wall axis
Wall axis
Wall axis
hppp
Rrot90
h
h
11.1)
(1,1)
(0.0)
(0. 1)
(1,1)
(0.0)
(0. 1)
(1,1)
(0,0)
hplpp
fott0
(1.2)
O1,0)
(1.2)
(1,0) 0)
(1,2)
(0,
(0.3)
(0.2)
(1, 3)
(0.3)
(0. 2)
(1, 3) (0.3)
i [hhmp r) = Anip ((t)
h(r)= = (0, 1, 1,w)
h(hror90 r) h()
(1,3, whw)
(0, 2, w)w)
Illustration of h and its equivariance. hap = (1,0,0) is the flipping motion of the mobile robot over the wall axis, and 010990 == 1,0) the rotation of the mobile robot 90 degrees around the origin. It can be seen that hmph(r)) = (1,0,0 w.) = (1, mod - 1, 4), ww) = (1, (,,) = (haipp T) and (()) = (0, 1, (0, w.w) = (0, mod + 1, ww)) = (0, 2, ww = h(hm019077 T).
Figure 14:
where k E 1-2, -1,,-1,0,13 and (atan2 (2) ww) is assumed to be satisfying - S atan2 (22) Wwi < I.
D.2.2 Experimental Details
For dataset generation, we first uniformly sample from the smallest space that can span T by symmetry transformations, in which the robot's initial position ς. satisfies 5 < ||4~|| 10 and 0 S atan2(g; vf) < T. T/4, and the wall axis angle is O. We collect trajectory data by generating B-splines given via points labeled by humans. The B-splines time length of the splines becomes seconds, where the splines accelerate for the first second and decelerate for the last second. We finally sample 201 points from the splines.
Datasets:
then reparameterized
that the
For the training dataset, we have gathered 6 trajectories for 75 randomly given task parameters, a total of 300 trajectories for training. For the validation dataset, have gathered trajectory for 80 randomly given task parameters and randomly augmented them using symmetry transformations 100 times. For the test dataset, have gathered trajectory for 40 randomly given task parameters and randomly augmented them using symmetry transformations 1000 times. The number of validation and test datasets are then 8,000 and 40,000 repectively.
Network Architectures and Training Details: A task parameter J is represented as (q. 92 ww). where (at q²) is the mobile robot's initial position and ww is the wall's axis angle. In practical implementation, we use (4/- q,2, cos ww, sin wy) E IR as an input parameter vector. Since T = 201, the output space is 402
We use two-layer fully connected neural networks of 512 nodes for MMPs and EMMPs with elu as their activation function. TA-VAE's encoder includes a fully connected network and temporal convolutional network, and the decoder includes two fully connected networks for z and т, a tem- poral convolutional network, and a fully connected network. All four fully connected networks used TC-VAL are of two layers with size 434. The output sizes of fully connected networks for z and in the decoder are 36 and 72 respectively. The two temporal convolutional layers in TC- VAE are
18

---

both with channel sizes (18, 36, 72) and kerne size 3. More details on the structure of VC- are in [37]. All models in the experiments have similar number of parameters (=e 9.4 x 10%).
We consider a trajectory successful if it consistent with the task parameter and reaches the goal without colliding with the wall. More specifically, check (i) collision avoidance, (ii) the robot's s initial position, and the robot's final position. We consider the trajectory satisfies (ii) and (iii) if the initial configuration and the final configuration are within radius of 0.3 at the initial position specified in the task parameter and origin, respectively. The number of sample (2, we use for success rate calculation is 50,000.
Success Criterion:
D.2.3 Additional results
Architecture Comparison: We compare MMPs and EMMPs of fully connected autoencoders (de- noted as AE), fully connected variational autoencoders (denoted as VAE), and variational autoen- coders of the same structure with TC-VAE (denoted as TC-VAE). Table 3 shows the four evaluation metrics. Overall, as shown in the success rate scores, regardless of network architecture and autoen- coder method, EMMPs without regularization perform the best, and MMPs without regularization perform the worst. Although EMMP (TC-VAD) excels in most measures (MI and NLL) its success rate (91.2%) is still lower than EMMP (AE)'s (92.40%) and EMMP (VAE)'s 's (95.72%), which is caused by the tendency of (TC-VAE) that it violates the initial and final condition in about 6% of trials, whereas EMMP (AE) only violates them and EMMP (VAE) almost never violate them (0% 0.01%)
Equivariance Comparison: Here, we qualitatively compare the equivariance performance of ran- dom data augmentation and equivariant learning method by comparing MMP (AE) and EMMP (AE). Figure 15 shows trajectories generated by T and h T, with same z. If the decoder f is equiv- ariant, [=, (- (blue lines in the figure) and [z-(f(z,T), r), +)]m (grey lines in the figure) must overlap. However, as shown in the Figure 15 Left, trajectories of the MMP do not, whereas trajectories of the EMMP perfectly overlap shwon in Figure 15 Right.
D.3 Water-Pouring Experiment
D.3.1 Formulas and Proofs
Task Parameter Space: The input space Q = R2 x SB(3) [0.2, 0.41]. A task parameter can be represented as ((g2, d²), (gh oh oh, Ro)), mw), where (g- q.)) is the cup's position, (g), qh qi Ra)) the bottle's s initial position and orientation, and m the weight of water in the bottle. To construct compact T, we limit (g., q to be inside a square at the origin with edge length of 0.5, i.e., -.25 < 9 c S 0.25, and limit the distance between (q.,g²) 9²) and (g. g ) to satisfy 0.3 < l1 n: o z 2 I 0.78. Since the bottle on the table upright, og is constant.
Trajectory Space and Distance Measure: We set the length of trajectories T to be 480, which makes trajectory space X SE(3) (50 Given æ {-1 1,400 and 22 {xt ).80 where each 2/ can be represented by (Ryg E =SO(3), Paj RR the distance measure on X' is defined as:
dx(21,
(| R2 11 + plppa -2221
(16)
Table 3: Reconstruction Error (RE), Mutual Information (MI), and Negative Log-Likelihood (NLL); the lower, the better. Success Rate (SR); the higher, the better.
Method
(E)
ML
NLL (L)
SR (?)
MMP (AE)
0.223
0.487
1.49
50.08%
MMP (VAE)
0.233
0.687
1.57
1
42.98%
MMP (AF) indep
0.225
0.329
1.48
104
52.39%
MMP (VAB) + indep
0.229
0.652
1.56 x 10
44.28%
EMMP (AE)
0.223
0.082
1.25 10
92.40%
(AE) + indep
0.229
1.21 10
EMMP (VAE)
0.231
0.066
1.23 10
95.72%
EMMP (VAE) indep
0.225
0.167
1.28 10
82.02%
EMMP (TC-VAF)
0.227
0.065
1.00 x 10
91.20%
EMMP (TC-VAB) + indep
0.247
0.071
1.15 1
88.22%
19

---

MMP (AE)
EMMP (AE)
T(ZT))
((2,T)
((2,z)
h(2
Dh (l)).
flzh- [h r),
Figure 15: Equivariance comparison between MMP (AB) and EMMP (AE). If the decoder f is equivariant, f(2, h T) (blue lines in the figure) and h (f(2, 7), T)]2 (grey lines in the figure) must overlap. It can be that the decoder of MMP is not equivariant, whereas the EMMP' decoder is equivariant.
where 7 5 is constant.
The symmetry group H IR2 SO(2) x SO(2), each is for translation of the cup and the bottle, rotation of the bottle around the cup, and rotation of the bottle around itself. Given two group elements (a, b, Rai, Ra1). ) (c, d, Ra2, Ra2) E R22 x SO(2) x SO(2), the group operation defined as follows:
Group Operations:
b, Rg),, d, R.y,, Rs) (a+ c, R. Ry Rg Rs).
(17)
Group Actions: The procedure of group actions of h E H can be explained as follows: (i) translate the cup and the bottle, (ii) rotate the bottle around the cup, and (iii) rotate the bottle around itself. Given a task parameter т = ((g. d²), (4b L. oh Rb)), mw), and a group element (a, b, Rai Rg), the group action h is defined as:
T (l12 + a. d + b). ((a) + a. o b. 0) + (R., - (o) 4. o a d)),, Bu Ba - Rg); mw).
(18)
Group-Equivariane Map h: Given task parameter = (lub v2), (ab, oh ui Ro)), m.), h(T) (fi(T), ha()), h(())) is defined as follows:
(r) (g. d.) E R2
(19)
h2(T) Rot(2, 01),
(20)
ha(T) Rot(2, (02-01))
(21)
where 0, := atan2( g2: 9h g.)) 02 = atan2(s?,, and 2. denotes the first column of Rg.
Given an arbitrary h = (a, b, R., Rz), where Rt = Rot(2, a) and R3 Rot(2, B3), the equivariance of h is shown by the following equation:
h(h 7) = hi((n + a, o b). ((at +a. 0) 0)+ (R., + (ab d of 2..021 R.. Rs Rg). m.)
= ((a_ + q² b), Ra(2, 01), Rox(O2 0, + 8)
((g) a, o b),
RRo((, 0, Rox(2, 0s 0,) R₂))
((q) + a, d + b), R., Roi(E, 0;), Rg Ra(E2, 0g 0,))
= (a, b, R.. Re)l(a. (.), Rot(a, 6,), Rod(s, (0s
1))
= -h(r).
(22)
D.3.2 Experimental Details
Datasets:
The water-pouring demonstration trajectories are collected by recording videos of water- pouring motions of a human demonstrator for seconds (intended for 3.5 seconds of reaching mo- tion and 4.5 seconds of pouring motion) at 60fps, with three AprilTags,, resulting in 480 frames [49].
20

---

Distance from the cup
Weight of the water
30cm
8cm 8cm 8cm 8cm 8cm 8cm
26.5cm
12cm
6.7cm
0
12.5cm
14.5cm
2
0 8,
200g
270g
305g
340g
410g
Figure 16: Task parameters for demonstration. The cup is always at the origin, and the bot- lle is at (0,r 0.3, 0.38, 0.54, 0.62, 0.78 The mass of water in the bottle is me 0.2, 0.27, 0.305, 0.34, 0.41.
Then we extract the SE(3) trajectories of the bottle, whose length T 480. We perform trajectory smoothing and transform task parameters and the trajectories using group actions of H for the cup's position to be the origin, and the bottle be initially the ă. &.-direction from the bottle, and 2, to be aligned with ă.. The resulting task parameters are in the form of (),,)) (r. O, h, R. 0)), mw))
Assuming the bottle to be initially on the table upright, h 0.145 is constant, and R = T. We gather 5 trajectories for 7 different r and different m. in a total of 175 trajectories. As shown in Figure 16 Left, we choose r every 8em, from 30cm 78cm, i.e., E {0.3, 0.38, 0.46, 0.54, 0.62, 0.7, 0.78). and as shown in Figure 16 Righ/ we choose mhw E (0.2, 0.27, 0.305, 0.34, 0.413. The dimensions of illustrated in Figure 16 right. The
the cup and the bottle, and the position of the bottle frame five demonstrations of each task parameter are intended to pour water gradually from the left side of the cup and the right side of the cup.
We use 125 trajectories of E 0.3, 0.38, 0.54, 0.62, 0.78 as the training dataset, and we randomly split the other 50 trajectories into half for validation and test dataset. We randomly augment the dataset 100 and 1,000 times for validation and test dataset respectively, resulting 2,500 validation trajectories and 25,000 test trajectories.
Network Architectures and Training Details: The space of the task parameters R² x SE(3) x [0.2, 0.41). where task parameter can be represented in the form of ((g.. 9²), (gb, c. db. Ro)), mw). Assuming that the bottle is initially on the table upright, since E is a constant variable and Rg can be represented as Roté, 0z. in practical implementation, we use (gt g2, q) 4h mmy, cos 0g, sin 0₁) IR? for input of the decoder.
The output of the model is an element in SE(3)480 which is not vector space. A naive parame- terization or SE(() element (e.g. as 12-dimensional vector) does enforce the model outputs to satisfy SE(3) constraints. To constraint the model output space to be SE(3) we first set all model output sizes be 480 x 2880, and add an additional layer Vec2SE3 the end of every decoder.
Given vector y = (v' (v'....,ef) ) E IR, Vec2SE3 is defined as:
0
1/2
Vec2SE3 6
exp(
U3
0
11
)
26
5
E SE(3).
122
We finally vectorize the first three rows of the SE(3) matrix, since the last row is constant at (0, 0,
We use two-layer fully connected neural networks of 168 nodes for the EMMP with elu as its activa- tion function. VA-V s encoder includes a fully connected network and a temporal convolutional network, and the decoder includes two fully connected networks for 2 and т. a temporal convolu- tional network, and a fully connected network. All four fully connected networks used in TC-VAE are of two layers with size 512. The output sizes of fully connected networks for and in the decoder are 40 and 80 respectively. The two temporal convolutional layers TC-VAE are both with channel sizes (36,7 72, 144) and kernel size 3. More details on the structure of TC-VAE are in [37].
21

---

Demonstration
EMMP indep
TC-VAE
Wb
410g
200g
-1.59
time
time
time
Figure 17: Graphes of bottle angle time. The pouring angle wh is the angle between the bottle axis & and the ry-plane. The orange lines are pouring angles for the my = 0.41 case, and the blue lines are pouring angles for the mh 0.20 case. It can be observed that the pouring angle decreases as the mass of water increases.
All models in the experiments have similar number of parameters, where EMMP contains (1.51 x 10%) parameters and TC-VAE contains (1.56 x 10%) parameters.
Task Parameters for Success Rate Measure: We sample five feasible trajectories for four task parameters. Throughout the four task parameters, the cup's position (g., 9.) = (-0.2, 0), the bottle is initially in the y-direction from the cup, i.e., of -.22, the bottle initially aligned with the base frame, i.c., Rs I. The rest parts, (g mw) for the four task parameters (0.35, 0.25), (0.45, 0.275), (0.40, 0.35), (0.55, 0.400). These task parameters are picked within the robot's workspace.
Obstacle Avoidance Algorithm: Given a task parameter 7 and an obstacle, the obstacle avoidance task is performed as follows: (i) we sample 2 from p(z), (ii) generate the bottle's 's trajectories via f2, (iii) check the collision between the bottle and the obstacle and pick collision-free trajecto- ries, and (iv) solve the inverse kinematics problem of the robot and choose one that is feasible and also collision-free.
We check collisions between the bottle and the obstacle and between the robot and the obstacle by converting the meshes of the bottle and robot point clouds, and parameterizing the obstacle superquadric, which represents objects as sign distance function [57]. As a sign distance function, superquadrics have benefits in checking if point inside or outside them. We consider trajectory of point cloud and a superquadric to be collision-free none of the points in the point cloud gets inside the superquadric at every timestep, and consider they collide otherwise.
D.3.3 Additional results
Water-Pouring Performance Comparison: The motions of the bottle pouring water near the cup are highly dependent on the amount of water in the bottle. A bottle of small water needs to tilted more than a bottle that is almost full to pour the same amount of water into the cup. The amount of tilting the bottle can captured in the angle between its axis and the table, which we denote as the bottle angle.
TC-VAE
EMMP + indep
hla.h-3 x)
1
f(zT)
((a
Figure 18: f(2,T) T) (blue) and f(z, h r) (orange), and Ih (), (apricot). [h- (f(z, 7), r) and f(z, h 7) should be overlapped if the trajectories are generated equivariantly with the task parameters.
22

---

Figure 17 illustrates bottle angle mean and standard deviation graphs of demonstration trajectories of the training dataset (Left), generated trajectories of EMMP indep (Middle) and generated tra- jectories of TC-VAE (Right) with The = 200g (blue) and mhe 410g (orange). We randomly augment 50 task parameters of validation and test datasets times, and pick 1,000 task parameters for m = 200g and 1,000 task parameters for m. = 410g. We generate 1,000 trajectories for both cases using sampled from p(z).
Figure 17 Left shows that as the mass of water increases, the pouring angle increases, which means the bottle is tilted less. It can be seen that the minimum mean angles of EMMP for m 200g and m = 410g (-1.6 degrees and -9.5 degrees) are very much alike that of the demonstration trajectories (-1.5 degrees and -8.5 degrees). On the other hand, the minimum mean angles of TC- VAE (5.6 degrees and -3.1 degrees) are very much distant from the demonstration trajectories'.
Equivariance Comparison: For motion manifold primitive framework to be equivariant, decoded trajectories must equivariantly transform as task parameters undergo a symmetry transformation. We qualitatively compare the equivariance performance of random data augmentation method and cquivariant learning method by comparing TC-VAE and EMMP + indep.
Figure 15 Left visualizes two trajectories generated from and h where h is the rotation of the bottle around the cup and itself, without translation. the model is equivariant, the orange-colored bottle and the apricot-colored bottle in the left upper corner should overlap. However, the condition not satisfied for TC- VAE, whereas the orange trajectory of EMMP is equivariantly transformed with
23


Reviews:
[{"review_id": "X7qt1a1aXp", "reviewer": "Reviewer_3SRv", "summary": "", "questions": "", "limitations": "", "rating": 4, "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": "In this work, the authors address the task of learning sets of valid configuration-space trajectories for arbitrary control tasks. The authors propose Motion Manifold Primitives, which is a mapping from a simpler low-dimensional space (i.e. R^m) to the configuration space. The core idea is that one can learn a density function over the latent space which can then be sampled to recover samples from the manifold in the configuration space. The authors’ second contribution is noticing that if the task parameter space has symmetries, and those symmetries can be cleanly and precisely described, and an equivariant group action can be defined, then one can formulate a mapping from the latent space to configuration space which is equivariant under all enumerated symmetries. The authors provide convincing empirical evidence that, in two scenarios with straightforward symmetries, their method demonstrates superior performance compared to other manifold-learning baselines.\n", "summary_of_recommendation": "This paper offers a compelling formal perspective on handling symmetry in robot learning tasks. While I'm not entirely sure about the applicability of this method to a broader range of more-complicated robot problems because it is difficult to exactly satisfy the assumptions, I certainly learned something by reading this paper and believe that the CoRL community would benefit from considering the implications of the theoretical treatments of symmetry/equivariance.", "technical_quality": "Very Good:  Convincing.", "clarity_of_presentation": "Good: Minor revision needed to improve organization or clarity.", "potential_impact": "", "robotics_focus": "Sufficient demonstration on hardware", "quality_of_limitations": "Limitations are addressed clearly", "questions_for_rebuttal": "How do the inductive biases of symmetry+homeomorphism affect sample complexity compared to other approaches?\n\nIs the independence regularization term from prior work? Or designed for this work?\n\nHow would the authors extend their work to classes of tasks where each instance of hte task may have different symmetries?\n\n\n(Not really asking for this experiment in rebuttal, but I am interested in some discussion): How would this method compare to diffusion-based approaches, such as Diffusion Policy, which have been shown empirically to represent configuration-space manifolds quite accurately?\n", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "# Originality\n\nManifold learning in and of itself is not novel - mapping a density function on a lower-dimensional manifold which is easy to sample from to a high-dimensional manifold is the backbone of most autoencoding methods. So learning a mapping between these manifolds for robotics tasks is not novel.\n\nHowever, this work attempts to drill down into the structure of these manifolds in many real robotics tasks, and make some more precise guarantees on performance by exploiting this structure. Broadly speaking, this work is attempting to formally nail down some of the properties which are reasonably common in robotics tasks: \nProblem symmetries: the solutions set to motion planning problems, for instance, is often equivariant under some certain transformations of initial conditions. We can often enumerate these symmetries intuitively, but the community frequently struggles to deal with symmetries systematically (often offloading them to multimodal generative modeling frameworks).\nHomeomorphism: If you change the task parameters slightly, you’ll probably be able to deform the solution manifolds smoothly as well. Intuitively, if you move an object back a little bit during a pick-and-place task, your manifold of pick-and-place motions will probably deform accordingly, and not change topologically. Some methods exhibit these properties (for instance, DMPs), but not necessarily when learning manifolds.\n\nThese observations are not in and of themselves novel - lots of work has noticed these things. But very little work has tried to tackle them explicitly and from first principles, and design solutions which exploit the mathematical structure of these properties. And I haven’t really seen much / any work which tries to rope in group theory / formal symmetry properties into trajectory learning. So to me, this is quite original for robot learning - I learned something by reading the paper, and I think the community would benefit from thinking about these properties with a more rigorous lens.\n\n# Quality\n\n## Theoretical\n\nAs far as I can tell, the theoretical contributions of this work are sound. \n\nOne question: is the independence regularization term from prior work? Or designed for this work?\n\n## Empirical\n\nThe two examples are reasonably well-executed. Both the toy task and the real-world robot task have rigorously-defined symmetries, and\n\nHowever, I guess I have the following issue with the baselines: if a task is simple enough to write down all the precise symmetries for, then sufficiently rich neural network models and a sufficiently large set of demonstrations should be able to represent these symmetries as well. More recent work than TC-VAE has shown impressive ability to represent multimodal manifolds (i.e. [1]), so I’d be curious how much of a benefit leveraging these simple explicit symmetries actually has over more recent techniques.\n\n[1] Diffusion Policy: Visuomotor Policy Learning via Action Diffusion\n\n# Clarity\n\nMy biggest issue with the paper is that the mathematical treatment of symmetry was a bit rushed + inaccessible. For this community, it’s unlikely that people will be familiar with the differences between group operations and group actions (and may conflate the two), when in fact that distinction is actually quite important to understanding what the symmetry operators are doing. The symmetry operators should at the very least have a larger diagram which relates the notation to an actual example. Figure 2 should be improved, made larger, and related to the operators better. Something with transition arrows like found on [1]. Also, section 3.2 doesn’t have much intuition, and it’s really not clear intuitively how h^bar can be constructed or is invertible. It would be really, really useful to incorporate the toy example here while explaining it.\n\nThere isn’t really enough of a related work section. If it doesn’t fit in the main paper, it should be put in the appendix and referenced in the main paper.\n\n[1] https://en.wikipedia.org/wiki/Symmetry_group\n\n# Significance\n\nAnother issue I have is in the significance of the work given the assumptions. I’m not as concerned with the homeomorphism assumption as I am with the assumption that mathematically enumerating the symmetries in arbitrary robot learning problems is simple or even possible. In most real-world problems, there are only rough/fuzzy/approximate symmetries that can be exploited, and it would very difficult to write down these symmetries, let alone define an h^bar which can be used to construct an equivariant function under those symmetries. Furthermore, if you’re trying to solve a CLASS of tasks with different symmetries, it’s unclear how you’de represent that in this framework without explicitly enumerating each element in that class. For instance, consider hanging a mug on a rack - there are discrete and rotational symmetries which may be different for different mugs and racks, particularly if the geometries are rather different.\n\n# Relevance\n\nThe work is undeniably relevant to robot learning literature - we deal with symmetries very frequently, and many works fail to address them at all, let alone with mathematical rigor. I think many people will find this treatment very interesting + inspiring in this setting.\n\n# Limitations\n\nThe authors do an adequate job addressing the two assumptions (homeomorphism, ease of defining h^bar). However, they don’t really mention how the method might be used in a situation where you have a set of symmetry groups you might want one method to solve. This puts a pretty hard cap on the direct utility of this method in real-world learning scenarios.\n\nNot sure if this is a limitation per se, but the homeomorphic assumption is not particularly elegant when it comes to predicting trajectories which are time-parametereized. For instance, stretching a short trajectory to a long trajectory with the (presumably) same set of time-indexed waypoints doesn’t really make so much sense for control. Instead, would be interesting to see if the method could be adapted to some kind of time-invariant trajectory shape through configuration space.\n\nFinally, the method doesn’t really make any discussion of manifold boundaries, which are quite relevant for many real robot tasks.\n", "extra_scores": {"impact": "3: The work is incremental and unlikely to have much impact, even though it may be technically correct and well executed."}, "extra_text": {}}, {"review_id": "P9k1Qu7M-3", "reviewer": "Reviewer_ZGAZ", "summary": "", "questions": "", "limitations": "", "rating": 3, "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": "This paper proposes to use a manifold learning technique to learn manifolds representing the set of all feasible trajectories that successfully execute a task (given a parameter task $\\tau$).\n\nLearning the task for all parameters $\\tau$ can be very data-intensive. Still, the authors propose some techniques to increase data efficiency and show that their technique is feasible whenever the manifolds (parametrized by $\\tau$) are homeomorphic to each other. \n\nTo learn such manifolds, the paper proposes an auto-encoder architecture that possesses an invariant encoder and an equivariant decoder. Those take into account the symmetries of a task, and this leads to improved generalization and efficiency in learning tasks.\n", "summary_of_recommendation": "The reviewer finds the paper interesting, with clear presentation and high impact in robotics and machine learning. ", "technical_quality": "Excellent: Very convincing and thorough.", "clarity_of_presentation": "Very Good:  Well organized and clear.", "potential_impact": "", "robotics_focus": "Sufficient demonstration on hardware", "quality_of_limitations": "Limitations are addressed clearly", "questions_for_rebuttal": "- In the first paragraph of p. 4, I think it makes more sense to define the symmetry transformation on a generic manifold rather than on $\\mathcal{T}$,  since the same notation (i.e. $\\cdot$) is used to denote the group action of $H$ on $\\mathcal{T}$ and on $\\mathcal{X} \\times \\mathcal{T}$.\n\n- As the paper incorporates abstract algebraic definitions, the term \"module\" used for $\\bar{h}$ might cause confusion initially, as there exists an algebraic structure referred to as a module. To avoid potential confusion, it would be advisable to consider replacing the term \"module\" with a different term to ensure clarity and prevent any misunderstanding.\n\n- Some titles of the referenced papers should be correctly capitalized, e.g. \"Spherical cnns\" should be \"Spherical CNNs\".\n\n- Typos:\n    - \"equivaraint\" at p. 4\n    - In B.1. the text \"one standard way @ @\" seems to be forgotten to be erased\n", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "This paper makes an interesting contribution to the field of Learning-by-Demonstration. It introduces a manifold learning technique that enables the learning of the entire space of trajectories required to successfully execute a task.\n\nThe paper is commendable for its clear illustrations and explanations. It also includes comprehensive experiments that effectively showcase the technique and demonstrate how to leverage the task symmetries using the proposed framework.\n\nHowever, to ensure acceptance, a few minor issues need to be addressed.\n", "extra_scores": {"impact": "4: The work contains interesting new ideas that will potentially have major impact in robotics or machine learning."}, "extra_text": {}}, {"review_id": "5ARdgH4vOy", "reviewer": "Reviewer_K1vn", "summary": "", "questions": "", "limitations": "", "rating": 4, "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": "This paper proposes learning a continuous manifold of trajectories, referred to as Motion Manifold Primitives (MMP), to achieve a specified task. Compared to single trajectories, MMP is able to address unforeseen constraints and obstacles. The authors further propose EMMP, which additionally accounts for equivariant transformations through using an invariant encoder and an equivariant decoder. The proposed methods were applied to synthetic and real-world robot experiments and compared against exiting manifold primitive methods, with EMMP showing superior results. The authors demonstrated that the invariant encoder and equivariant decoder are key factors for performance improvement through ablations and swapping different network architectures.", "summary_of_recommendation": "This paper presents a method that is rigorously justified, the results are favorable, and the comparisons are comprehensive. I recommend accepting this paper to CoRL 2023.", "technical_quality": "Excellent: Very convincing and thorough.", "clarity_of_presentation": "Excellent: Enjoyable to read.", "potential_impact": "", "robotics_focus": "Sufficient demonstration on hardware", "quality_of_limitations": "Limitations are addressed clearly", "questions_for_rebuttal": "1.\tThe trajectories illustrated in figure 1 do not belong to the same homotopy class. Doesn’t this break the assumption that $\\mathcal{M}_{\\tau}$ is homeomorphic?\n\n2.\tI think the architecture comparison in C.2.3 are quite interesting and should be included in the main submission if possible.\n\n3.\tThe speculation in section 4.2 on “slipping between the two-finger gripper and the bottle” seems to be something easily fixable by attaching the bottle rigidly to the end effector. I would be very interested in the performance of the method once the slippage is fixed.\n\n4.\tTypo: the definition of “SR” is omitted in Table 2. Is it “success rate”?", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "The paper is well written, the derivations are thorough, and the proposed method is well supported with simulation and real-world results. Ample comparison and ablation of the methods are provided.\n\nThe appendix and supplementary video provide detailed information about the submission and are both helpful for understanding the paper.\n\nPlease refer to the “questions for rebuttal” section for more details.", "extra_scores": {"impact": "4: The work contains interesting new ideas that will potentially have major impact in robotics or machine learning."}, "extra_text": {}}, {"review_id": "-fbcTAbxo9", "reviewer": "Reviewer_rfrA", "summary": "", "questions": "", "limitations": "", "rating": 4, "confidence": 4, "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": "This paper focues on modeling multiple solution trajectories for one task, using movement pritimives and latent manifold.\n\n\n$\\textbf{Updated review after rebuttal}$:\n\nThe paper proposed a novel method to learn the trajectory manifold by using the equivariant nature of movements. By constructing the invariant encoder and equivariant decoder strucures, this proposed method is able to capture the equivariant motion manifold primitives. During the rebuttal, the authors have significantly improved the readbility of the paper, makeing it smooth to read and technically sound.", "summary_of_recommendation": "I regret to express my reservation about recommending this paper as I encountered considerable difficulty comprehending the technical details. The abundance of assumptions presented throughout the paper lacks adequate and reasonable explanations, contributing to the overall confusion in the writing.\n\n\n$\\textbf{Updated review after rebuttal}$:\n\nI recommend this paper now, as the revised version has improved the readability a lot. After breaking through the bottleneck of the technical part, I found the idea of the paper is very interesting and should be capable to generalize to a lot of similar tasks.", "technical_quality": "Very Good:  Convincing.", "clarity_of_presentation": "Very Good:  Well organized and clear.", "potential_impact": "", "robotics_focus": "Sufficient demonstration on hardware", "quality_of_limitations": "Limitations are addressed clearly", "questions_for_rebuttal": "I would like to offer some suggestions to the author for improving the paper:\n\nReevaluate the assumptions: It would be beneficial to reconsider the number of assumptions made in the paper. Aim to retain only the most essential ones while ensuring that they are reasonable and comprehensible to the readers. Over-reliance on assumptions without proper justification can hinder the paper's credibility.\n\nEnhance technical and mathematical descriptions: Provide clearer and more intuitive explanations of the technical and mathematical aspects of the paper. Consider the knowledge level of the readers and avoid assuming that they have the same level of familiarity with the subject matter as the authors. A well-explained and accessible presentation will make it easier for readers to follow and understand the content.\n\nEstablish logical connections between sentences: Ensure that the sentences and sections in the paper are logically connected. Show how certain conclusions can be reached based on specific conditions. This will strengthen the coherence and flow of the paper, making it easier for readers to grasp the ideas and arguments presented.\n\nSeek feedback from non-experts: Prior to submitting the manuscript, consider involving individuals with less knowledge in the field to conduct a proofreading and provide feedback. This will help identify areas that may be unclear or challenging for readers who are not experts in the subject matter. Incorporating their feedback can greatly enhance the paper's overall clarity and accessibility.\n\nBy considering these improvements, the author can create a more polished and reader-friendly paper that effectively communicates the research findings to a broader audience.\n\n\n$\\textbf{Updated review after rebuttal}$:\nMy major concern has been addressed and the paper is very friendly to the readers in a broad research area.", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "Unfortunately, I found it challenging to comprehend the technical aspects of the paper, even though I believe I possess adequate knowledge in robot learning and movement primitives. My reading stopped at Line 93 on page 3 because I became completely lost. The primary reasons for my inability to continue are as follows:\n\nThe paper heavily relies on numerous assumptions without providing compelling justifications for their reasonability. Throughout the paper, the phrases \"We assume\" and \"XXX is assumed to be\" appear approximately 20 times, particularly on page 2. However, it remains unclear to me why these assumptions are necessary, making it difficult to assess the correctness of the proposed techniques. For instance, I came across an example from Line 53 to Line 56 that discusses the symmetry transformation, where the reasoning behind such an assumption was unclear.\n\nThe mathematical descriptions appear disconnected and disjointed, resembling a collection of random sentences. In particular, the lack of coherence between sentences from Line 88 to Line 93 makes it challenging to establish the causality between assumptions and the corresponding mathematical inferences.\n\nTo enhance the paper's clarity and credibility, I suggest addressing these issues and providing clearer justifications for the underlying assumptions, as well as improving the flow and coherence of the mathematical descriptions.\n\n\n$\\textbf{Updated review after rebuttal}$:\n\nI appreciate the effort of the authors to improve the readability of the paper. After reading the revised version, I see that my main concern has been addressed. I'm in favor of this paper now.\n\nIn the first version, I struggled to understand the theory because of missing explanations and assumed prior knowledge. But in this revised paper:\n\n- The authors added an introduction to the Geometric Preliminaries, which helped me grasp the main theory.\n- They've rewritten the methods section and updated the figures, offering a clearer explanation of the method.\n- They removed some unnecessary assumptions, making the theory much sounder.\n\nWith these changes, I can now see the paper's novelty and contribution. The theory is well-presented now, even for readers without much knowledge in the trajectory manifold. I think the method can be applied to many robot learning tasks. Despite my strong rejection earlier, I'm now leaning towards a strong accept.", "extra_scores": {"impact": "4: The work contains interesting new ideas that will potentially have major impact in robotics or machine learning."}, "extra_text": {}}]

Author rebuttal:

Meta-review: The paper presents an imitation learning framework to model a continuous manifold of demonstrated trajectories. To address the issue of the limited data, the proposed method leverages symmetries in robot tasks.
While issues of the presentation in the paper were raised in the initial review, the presentation was significantly improved after the rebuttal. 
As the reviewers agreed to accept the paper, AE recommends the acceptance of the paper.
Decision: Accept (Poster)