Orientation Probabilistic Movement Primitives on Riemannian Manifolds
Authors: Leonel Rozo
Vedant Dave
Venue: CoRL2021 Poster (2021)
Keywords: learning from demonstration
movement primitives
riemannian manifolds

Abstract: Learning complex robot motions necessarily demands to have models that are able to encode and retrieve full-pose trajectories when tasks are defined in operational spaces. Probabilistic movement primitives (ProMPs) stand out as a principled approach that models trajectory distributions learned from demonstrations. ProMPs allow for trajectory modulation and blending to achieve better generalization to novel situations. However, when ProMPs are employed in operational space, their original formulation does not directly apply to full-pose movements including rotational trajectories described by quaternions. This paper proposes a Riemannian formulation of ProMPs that enables encoding and retrieving of quaternion trajectories. Our method builds on Riemannian manifold theory, and exploits multilinear geodesic regression for estimating the ProMPs parameters. This novel approach makes ProMPs a suitable model for learning complex full-pose robot motion patterns. Riemannian ProMPs are tested on toy examples to illustrate their workflow, and on real learning-from-demonstration experiments. 

Orientation Probabilistic Movement Primitives on Riemannian Manifolds
Leonel Rozo"! Vedant D, 1,2
Bossh Center for Artificial Intelligence. Renningen, Germany. University of Leoben. Leoben, Austria.
hww.:/owwwwdd.........o.ooo vedant.dave www.dddaavvnillloeenncoo
Abstract: Learning complex robot motions necessarily demands have models that are able to encode and retrieve full-pose trajectories when tasks are defined in operational spaces. Probabilistic movement primitives (ProMPs) stand out a principled approach that models trajectory distributions learned from demon- strations. ProMPs allow for trajectory modulation and blending to achieve bet- ter generalization to novel situations. However, when ProMPs are employed in operational space, their original formulation does not directly apply to full-pose movements including rotational trajectories described by quaternions. This paper proposes a Riemannian formulation of ProMPs that enables encoding and retriev- ing of quaternion trajectories. Our method builds on Riemannian manifold theory. and exploits multilinear geodesic regression for estimating the ProMPs parame- ters. This novel approach makes ProMPs a suitable model for learning complex full-pose robot motion patterns. Riemannian ProMPs are tested on toy examples illustrate their workflow, and on real learning-fromt www.--rmmmomstattiiii experiments.
Keywords:
Learning from Demonstration, Riemannian Manifolds
Introduction
For robots to perform autonomously in unstructured environments, they need to learn to interact with their surroundings. To do so, robots may rely on a library of skills to execute simple motions or perform complicated tasks as composition of several skills [1]. A well-established way to learn motion skills is via human examples, a.k.a. learning from demonstrations (LfD) [2]. This entails human expert showing once or several times specific motion to be imitated by robot. In this context, several models to encode demonstrations and synthesize motions have been proposed: Dynamical-systems approaches [3], probabilistic methods [4, 5], and lately, neural networks [6, 7].
Probabilistic movement primitives (ProMPs) [4], Stable Estimators of Dynamical Systems (SEDS) [8]. Task- Pask-Parameterized Gaussian Mixture Models (TP-GMMs) [5], Kernelized MPs (KMPs) [9] and Conditional Neural MPs (CNMPs) [6] are some of the recent probabilistic models to represent motion primitives (MPs). While some of them were originally proposed to learn joint space motions (e.g., ProMPs), others have mainly focused on MPs task space (e-g.. TP-GMM), assuming that certain tasks, such as manipulation, may be more easily represented in Cartesian coordinates. The latter approach comes with an additional challenge, namely, encoding and synthe- sizing end-effector orientation trajectories. This problem is often overlooked, but the need of robots performing complicated tasks makes imperative to extend MPs models to handle orientation data.
Orientation representation in robotics comes in different ways such as Euler angles, rotation matrices and quaternions. Euler angles are a minimal and intuitive representation, which however is not unique They are also known to be undesirable in feedback control due to singularities [10, 11]. Rotation matrices are often impractical due to their number of parameters. In contrast, quaternions are a control [10]. Despite their antipodality (i.e., each rotation represented by two antipodal points on the sphere 8 quaternions have gained interest in robot learning, control, and optimization due to their favorable properties [10, 12, 13, 14]. Pastor et [12] and Ude et al. [15] pioneered works
representation and provide strong stability guarantees in close-loop orientation
"qqual contribution.
5th Conference on Robot Learning (CoRI. 2021), London, UK.

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to learn quaternion MPs, where the classic DMP [3] is formulated to encode quaternion trajectories for robot motion generation. Its stability guarantees were later improved in [16], while Saveriano et al. [17] used this new formulation into a single DMP framework to learn full-pose end-effector trajectories. However, DMPs do not encode the demonstrations variability, which is often exploited to retrieve collision-free trajectories or to design robot controllers [4, 6].
Silvério et al. [13] proposed the first extension of TP-GMM for learning quaternion trajectories, which relies on unit-norm approximations. This was resolved in [18], where a Riemannian man- ifold formulation for both motion learning and reproduction was introduced. Despite TP-GMM offers good extrapolation capabilities due to its task paraamtterizedd formulation, none of the forego- ing works provides via-point modulation or trajectories blending. These issues were addressed by KMP [9] for position data, whose model builds on a previously-learned probabilistic MP, such as TP- GMM. KMP was then extended to handle orientation data [19], via projection onto an Fuclidean space using a fixed quaternion reference, i.e. single Euclidean tangent space. Thus, model learn- ing, via-point adaptation, and skill reproduction take place in Euclidean space, ignoring the intrinsic quaternions geometry. Note that such approximations lead to data and model distortions [18].
Although ProMPs have been used learn Cartesian movements, their formulation does not handle quaternion trajectories. A possible solution would entail unit-norm approximations as in [13], but this approach fully ignores the geometry of the quaternions space and may lead to inaccurate mod- els. An alternative and more sound solution relies on Riemannian manifold formulation of ProMP, in the same spirit as [18]. However, two main difficulties arise: (i) learning of the model parameters does not accept closed-form solution, and (ii) trajectory retrieval is constrained to lie on the sphere 53_ We here provide solutions to these and related problems, which lead to the first ProMP frame- work that makes possible encoding and reproducing full-pose end-effector trajectories. In contrast to TP-GMM methods, our approach provides via-point modulation and blending capabilities, which are naturally inherited from the original ProMP. Unlike KMP, ProMP is compact and standalone model, meaning that learning and reproduction do not rely on previously- MPs. Moreover, our approach is not prone to inaccuracies arising from geometry-unaware operations.
Specifically, we introduce a Riemannian manifold approach to learn orientation motion primitives using ProMP. Our extension builds on the classic ProMP [4] (summarized in $ 2). and considers the space of quaternions as a Riemannian manifold. We propose to estimate the ProMP parameters using multivariate geodesic regression (see $ 3), and we show how trajectory retrieval, modulation of the trajectory distributions, and MPs blending are all possible via a Riemannian probabilistic formulation. In 4, we illustrate our approach on the unit-sphere manifold S², and we learn realistic motion skills on a 7-DoF robotic manipulator featuring complex full full-pose trajectories on R x S3.
2 Background
2.1 ProMPs
Probabilistic Movement Primitives (ProMPs) [4] is a probabilistic framework for learning and syn- thesizing robot motion skills. ProMPs represent a trajectory distribution by a set of basis functions. Its probabilistic formulation enables movement modulation, parallel movement activation, and ex- ploitation of variance information in robot control. Formally, a single movement trajectory is de- noted by r { where y: is d-dimensional vector representing either a joint configuration or Cartesian position at time step t (additional time derivatives of may also be considered). Each point of the trajectory T is represented as a linear basis function model
y: V, + Ey Ply (w)) = (y N, Ey);
where is ddN dimenssiol weight vector, L is fixed d -diieensioaa block diagonal ma- trix containing No. time-dependent Gaussian basis functions di for each DoF, and E, ~ (O, Ey) is the zero mean i..dd Gaussian noise with uncertainty E, (see [4]). ProMPs employ phase vari- able E [o, 1] that decouples the demonstrations T = {yr ) from the time instances, which in turn allows for temporal modulation. A table with the relevant notation is provided in Appendix 1.
ProMPs assume that each demonstration is characterized by a different weight vector un, leading to a distribution P(w; 0) N(w pw, Ew.) Consequently, the distribution of y:, P(y 0) is
0) / N( F.w. (y))((l Hrw: Ew ddw -N V Wi V E. V. + Ey).
(2)
2

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Figure 1: Left: Points on the surface of the sphere S² such as x and y belong to the manifold. The short- path between and y is the geodesic (-). which differs from
TxM
X
U
u
Euclidean path (-). The vector lies on the tangent space of x such that u = Lygx(y), Right: xx (u) and I (x) are the parallel trans- ported vectors and from TxM to M. The inner product between vectors conserved by this operation.
M
M
Learning from demonstrations: The learning process of ProMPs mainly consists of estimating the weight distribution P(w; 0). To do so, weight vector wn, representing the n-th demonstration as in (1), is estimated by maximum likelihood, leading to the solution of the form
W ( AI) TY
where Y, [yT.1 yI.rI concatenates all the trajectory points, and consists of all the time matrix r Given a set of N demonstrations, the weight distribution
instances for the parameters 0 Hw. Ew} are estimated by maximum likelihood (see Algorithm 1 in Appendix 1).
Trajectory modulation: To adapt to new situations, ProMPs allow for trajectory mod- ulation to via-points or target positions by conditioning the motion to reach a desired point yi with associated covariance Σn This results into the conditional probability P(wlyr) & N L,w, (w))(WW fm: Ew) whose parameters can be computed as follows
F. = E ( V Vi Ew' DN = 3(2-² V. v.
(4)
Blending: By computing a product of trajectory distributions, ProMPs blend different movement primitives into single motion. The blended trajectory follows a distribution P(y)) IT. P)(y)) for a set of ProMPs, where P,(y.) NN pesss Er,s), with associated blending weight dr,s E [0, 1]. The parameters of P(y,) = N(y, lut Et) are esti- mated follows
-1
S and mt Σt Σ
Σ
Ot.s I-1
Ort DT. H4.
(5)
Task parametrization: ProMPs also exploit task parameters to adapt the robot motion to, for example, target objects for reaching tasks. Formally, ProMPs consider an external state & and learn affine mapping from á to the mean weight vector fhus leading the joint distribution P(w, 8) = N(w Os + o W Ez), where (O, o) are learned using linear regression.
2.2 Riemannian manifolds
Since unit quaternions must satisfy a unit-norm constraint, they do not lie on vector space, thus the use of traditional Euclidean methods for operating these variables is inadequate. We exploit Riemannian geometry to formulate ProMPs on quaternion space as presented Riemannian manifold M a m-dimensional topological space with a globally defined differential structure, where each point locally resembles an Euclidean space IR. For each point x M. there exists tangent space M that is vector space consisting of the tangent vectors of all the possible smooth curves passing through x. A Riemannian manifold is equipped with a smoothly-varyin positive definite inner product called a Riemannian metric, which permits to define curve lengths in M. These curves, called geodesics, are the generalization straight lines on the Euclidean space to Riemannian manifolds, they are minimum-lente curves between two points in M (see Fig. 1).
Formally,
We exploit the Euclidean tangent spaces to operate with Riemannian data. To do so, we need map- pings back and forth between T&M and M, which are the exponential and logarithmic maps. The exponential map Expx T&M - M maps a point u in the tangent space of x point y on the manifold, so that it lies on the geodesic starting at x in the direction u, and such that the

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geodesic distance d between x and y equals the distance between x and u. The inverse oper- ation is the logarithmic map Logx . Mx Another useful operation is the parallel transport Tx T&M - Ty.M, which allows us to operate with manifold elements lying on different tangent spaces. The parallel transport moves elements between tangent spaces such that the inner product between two elements in the tangent space remains constant (see [20, 21] for further de- tails). Finally, let us introduce a Riemannian Gaussian distribution of a random variable x E M as
(x Σ)
(x) (X),
(6)
V(2T)44I)
with mean E M,, and covariance E E T...M This Riemannian Gaussian corresponds to an approximated maximum-entropp distribution for Riemannian manifolds, introduced by Pennec et al. [22]. Table 2 in Appendix 1 provides the different expressions for the Riemannian distance, exponential and logarithmic maps, and parallel transport operation for the sphere manifold S
2.3 Geodesic regression
geodesic regression model generalizes linear regression to Riemannian manifolds. The regression is defined as y Expy with y Expp(ru), (ru). where y M and x E IR are respectively the output and input variables, p .M is a base point on the manifold, u E TnM is a tangent vector at p. and the error is a random variable taking values in the tangent space y. As an analogy to linear regression, one can interpret (p, u) as an intercept p and slope u (see [23] for details). Given a set of points (y1..... Yr) .M and {z1..... FT E R. geodesic regression finds a geodesic curve y E M that best models the relationship between all the T pairs (x y,). To do so, minimize the sum-of --uurredddd Riemannian distances (i.e., eπrors) between the model estimates and the observations, that is, E(p, u) 1 E (F, where &, Expp (2.u) is the model estimate on M. df (. y,) = [[Logy, (y;))l ll the Riemannian error, and the pair (p. u) E TM is an element of the tangent bundle T.M. We can formulate a least-squares estimator of the geodesic model as minimizer of such sum-of-squared Riemannian distances, i.c.,
(6. û) argmin 1 Σ dn((oN))?
(P.W)TTM 2
The problem in (7) does not yield an analytical solution like (3). As explained by Fletcher [23], a so- lution can be obtained via gradient descent on Riemannian manifolds. Note that this geodesic model considers only scalar independent variable x E R, meaning that the derivatives are obtained along a single geodesic curve parametrized by a single tangent vector u. The extension to multivariate cases proposed by Kim al. [24], where æ E RM M requires a slightly different approach identify multiple geodesic curves (viewed as "basis" vectors in Euclidean space). Multivariate general linear models on Riemannian manifolds (MGLM) [24] provides a solution to this problem. MLGM uses a geodesic basis U = [un... Uss) formed by multiple tangent vectors umm E M of dimensionality d = dim(Tp (p.M), one for each dimension x. Then, the problem (7) can be reformulated as
(p, ()
argmin (p. u. JET Vm
Σ dd with y. Expp Exp,(Ux;).
(8)
This multivariate framework allows us to compute the weight vector, analogous (3), for a demon- stration lying on a Riemannian manifold M, for example M = $3.
3 Orientation ProMPs
When human demonstrations involve Cartesian motion patterns (via kinesthetic teaching or tele- operation), it is necessary to have a learning model that encapsulates both translation and rotation movements of the robot end-effector. This means that a demonstration trajectory T = [ is E R x S3, representing the full Cartesian pose of the end-effector
now composed of datapoints at time step I. this case, the challenge learning a ProMP in the orientation space, as the Eu- clidean case in 3 follows the classic ProMP introduced in & 2.1. Therefore, we focus on how to extend ProMP to learn trajectories on M S3, First of all, us introduce an equivalent ex- pression for y, in the MGLM framework, such that it resembles the linear basis-function model

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T MM
Illustration of multivariate general lincar regression on the sphere manifold 8 used to learn the weights of orientation ProMPs. Given the trajectory y, the origin p of the tangent space TpM, and the langent weight vectors wm are estimated via (12).
Figure 2:
M
W
in (I). Specifically, the estimate 8, = Expp
Expp(Ux:) (Ux;)) Expp (X,u) with U E gdxad 2, E RM E Raxx ard and u = [u] ...u, uM E IR.dd This equivalence proves useful when establishing analogies between the classic ProMPs formulation and approach for orientation trajectories.
X; = blockdiag( [a: ........................... TMM)
Similarly to (I), point y, E of T can be represented as geodesic basis-function model ((y,/w) = NM Expp v w Ey), with My = Exp, (.a))) E M,
(9)
where p fixed base point on M, w T. w N, is large weight vector concatenating N₂ weight vectors wn, E TpM, V is the same matrix of time-dependent basis functions as in (1), and Σy is a covariance matrix encoding the uncertainty on M. Two specific aspects about this formulation deserve special attention: (i) the mean My of the Riemannian Gaussian distribution (9), exploits the aforementioned equivalent formulation of MGLM; and (ii) the weight vectors forming w in (9) correspond the vector composing the geodesic basis of MGLM.
As a demonstration is characterized by different weight vector w. again we can compute a dis- tribution Pw 0) = =N(w| )w). Therefore, the marginal distribution of y, can be defined as
0) = f /N Exp, (w)), (y Pwu: Ew, with My = Exp, (So)) (Wr)) E M, (10)
which depends two probability distributions that lie different manifolds. Using the Rieman- nian operations described in $ 2, we obtain the final marginal (see Appendix 2.1)
0) -N Ey), with ny = Exp, ( pw)) Ey = p EWW V + Ey), (1I) where Σy and Σy are parallel- transpottr covariances of the geodesic model (9) and the final marginal, respectively.
Learning from demonstrations via MGLM: For each demonstration n, we estimate a weight vector w ut ... v. using MGLM (illustrated in Fig. 2). Firstly, we resort to the equivalent expression for y, introduced previously, where Exp, W [w.... WN.] and No being the number of basis functions. Secondly, we consider a demon- strated quaternion trajectory Tn {yr)²= with E SA, Then, analogous to (3) in Euclidean space, w estimated by leveraging (8), leading
Expp( with
T Σ t=1
(p, (m)) argmin (p.w... JEM Vm
wn,,w,), with E(p, w.)) 2 (Exp v.))).
(12)
where Φi is the vector of Gaussian basis functions at time I, and W contains the set of estimated tangent weight vectors um.. Tp.M (i.e., N₂ tangent vectors emerging out from the point p M).
To solve (12), we need to compute the gradients of E(p, w.)) with respect to and each wn- As explained in Appendix 2.2, these gradients depend the so-called adjoint operators, which broadly speaking, bring each error term Logy, (yx) from M M to TpM, with ý, = Exp,(Wes (W Φ,). Therefore, these adjoint operators can be approximated as parallel transport operations as proposed in [24]. This leads to the following reformulation of the error function of (12)
T
E(p, Wm.)
[Ty -p (Logy, (x.))l?.
(13)
5

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With the gradients of (13) (given in Appendix 2), we can now estimate both the vector Pn. and the weight matrix n.. for each demonstration n. Note that each demonstration may lead dif- ferent estimates of p. which defines the base point on M used to estimate Wmm. E Tp.M. This may produce different tangent spaces across demonstrations, and therefore diverse tangent weight vectors. An effective way to overcome this is to assume that all demonstrations share the same tan- gent space base p. which is the same assumption made when defining the geodesic basis-function model (9). So, we only need to estimate p for single demonstration, and use it to estimate all tangent weight vectors for the whole set of demonstrations. Then, given a set of N demonstrations, the weight distribution parameters O = {Mw: Ew are estimated by standard maximum likelihood as Wn. E TpMM = R c IR.. The complete learning process for orientation ProMPs is given in Algorithm 2 of the supplementary material.
Trajectory modulation: We formulate a trajectory modulation technique (i.e., to adapt to new situations) by conditioning the motion to reach a desired trajectory point yi E M with associated covariance Σy E My This results into the conditional probability P(wlly)) x (y, [Eppp (w)) Ey)W(wll Ew), which depends on two probability dis- tributions that lie on different manifolds, similarly to (10). We leverage the fact that the mean My depends p E .M, which is the basis of M where the weight distribution lies on. Thus, we rewrite the conditional distribution as follows
P(w|Log, & (V(Log ( L,w, Σÿ Wy) pw: Ew) = NNWw Ew,,,
(14)
where Σ; (Ep)) and {m En) are the parameters to estimate for the resulting condi- tional distribution. Since both distributions now lie on Tp which is embedded in the Euclidean space, we can estimate (P Σn} similarly to the classic ProMP conditioning procedure, with spe- cial care of
the covariance matrices. So, the updated parameters are
m' - Ei (w! ! Logp(yi) + (w'mw) and Σn Ww + +
W. W:)² (15)
From the new weight distribution, we can obtain a new marginal distribution P(y; 0*) via (11).
Blending: When it comes to blend motion primitives in M, one needs to consider that each of them is parametrized by set of weight vectors lying different tangent spaces Tp.M. There- fore, the weighted product of Gaussian distributions needs to be reformulated. To do so, we resort to the Gaussian product formulation on Riemannian manifolds introduced by Zeestraten [18], where the log-likelihood of the product is iteratively maximized using a gradient- based ap- proach as proposed in [25]. We here provide the iterative updates of the blended distribution P(y+)= = Ny lut,, )+, while the full solution is given in Appendix 2.3. The itcrative es- timation of f is
Aut Σ Ag,Ays
Σ Ag (Myy)) and Mt+1 Exp (A (16)
for a set of S skills, where Ay,a = r the skill After convergence iteration K, obtain the final parameters follows
My.. - (2 and a, is the blending weight associated to
8 Σ
n²
1 and Σ
(17)
Task parametrization: Classic ProMP allows for adapting the weight distribution P(w; 0) N(w (wwWHw Ew) as a function of an external task parameter 8, as explained in Section 2.1. This task parametrization straightforwardly applies our method as the weight vectors wn, E TpM C R4, long as the task parameter š is Euclidean. However, if & E M,, we can lear a joint probability distri- bution P(w, 8) using a Gaussian mixture model on Riemannian manifolds as proposed in [18, 26). Subsequently, we can employ Gaussian mixture regression to compute P P((w(=*) during reproduc- tion when a new task parameter 8* provided. We refer the reader to the works of Zeestraten [18] and Jaquier et al. [26) for details on how to compute the distributions P (w, 8) and P ((w)8*).

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SSGG
Figure 3: Demonstrated trajectories (-) in S marginal distribution Ply: 0) (mean -, covariance
and via-point adaptation (mean covariance and via-point +) for models trained over S and G datasets.
O
Figure 4: Trajectory distribution blending via orien- tation ProMP for datasets (S and (G, The re- sulting (blended) distribution trajectory (mean and covariance -)) starts from the first letter in the dataset (+) and then smoothly joins the trajectory of the second letter (+).
4 Experiments
Synthetic data on S2: To illustrate how model learning, trajectory reproduction, via-point adap- tation, and skills blending work in approach, original trajectories were generated in I and later projected to S by a unit-norm mapping. Each letter in the dataset demonstrated N 8 times, and a simple smoothing filter was applied for vi- sualization purposes. We trained 4 ProMPs, one for each letter in the dataset I, J, SJ. All models were trained following Algorithm 2 with hyperparameters given in Appendix 3.1. Figure shows the demonstration data, marginal distribution Ply; 0) computed via (10), and via-point adaptation obtained from (14) and (15), corresponding the G and S models. The mean of P(y; 0) follows the demonstrations pattern, and the covariance profile captures the demonstrations variability in S². Note that the trajectories and display very elaborated "motion" patterns that might be more complex than those observed in realistic robotic settings. Concerning the via-point adaptation, we chose a random point y" E 2 with associated Σy 10 I (i.e., high precision while passing through y*). As shown in Fig. 3, our approach smoothly adapts the trajectory and its covariance while accurately passing through
used dataset of hand-written letters. The
To test the blending process, we used the following subsets of our original dataset: {G, l} and {S, J}. The goal was to generate a trajectory starting from the first letter the set, and then smoothly switching midway to the second letter. Figure 4 shows the resulting blended trajectories for the two aforementioned cases, where our approach smoothly blends the two given trajectory distributions by following the procedure introduced in $ 3. Note that the blending behavior strongly depends on the temporal evolution of the weights a, E [0, 1] associated each skill s. We used sigmoid -ike
function for the weights a² (I) and as. while az (G) 1 a and a
(J) The foregoing
results show that our approach successfully learns and reproduces trajectory distributions on S². and provides full via-point adaptation and blending capabilities. We now turn our attention to robotic experiments where we learn and synthesize full-pose movement primitives.
Manipulation skills on R33 83 To test our approach in robotic setting, we consider re- re-orient skill from [27], which involves lifting a previously-grappe object, rotating the end- effector, and placing the object back on its original location with modified orientation (see Fig. 5). This skill features significant position and orientation changes, and is therefore suit- able to showcase the functionalities of our Riemannian ProMP. We collected 4 demonstrations of the re-orient skill via kinesthetic teaching on a Franka Emika Panda robot, where full-pose end- effector trajectories (p. were recorded, with p. R3 x S being the end-effector pose at time step f. The data was used to train a ProMP on R x S with position and orientation models learned using respectively classic ProMP and our approach with hyperparameters given in Appendix 3.2.
Figure 6 shows the demonstrations (gray solid lines) and the mean of the marginal distribution P(p: 0) depicted as black trajectory. Our model properly captures the motion pattern on IR3 x S3
7

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Figure 5: Snapshots of the human demonstrations of the ze-orient skill [27]
0.05
0.8
1.0
0.55
qu
00.5-
0.25
-0.35
20 0.2
0.1
-0.05
10
20
0
10
20
10
f.
t
Figure 6: Time-series plot of the re-orient skill demonstrations (-), original mean trajectory (-) of the marginal distribution P(p; 9). and resulting mean trajectory (-) of the new marginal distribution P (p: 0*) passing through a given via-point (+) p" E IR x S End-effector orientation represented as quaternion (9w, 9v, qu: qz]. Time axis given in sec.
for this skill. We then evaluated how this learned skill may adapt to via point p" R S representing a new position and orientation of the end-effector at t = 8.5 sec. By using the approach described in $ 3, computed new P(p: 0*), where the updated mean is required to pass through p". Figure 6 displays the updated mean (light blue lines), which successfully adapts to pass through the given via-point. Note that the adapted trajectory exploits the variability of the demonstrations (i.e. the associated covariance) to adapt the trajectory smoothly. We also learned two classic ProMPs using Euler-angle representation and unit-norm approximation, using the same hyperparameters set. While both models retrieved a distribution P(p: 0) similar to our approach, their performance severely compromised in the via-point case, they retrieve jerky trajectories with lower accuracy tracking w.r.t p' (see Appendix 3.2 and supplemental simulation videos using PyRoboL [28]). These results confirm the importance of our Riemannian formulation for ProMP when learning and adapting full-poose end-effecte skills. Appendix 3.2 reports learning and reproduction of two addi- tional skills featuring motions of diverse complexity. demonstrating the versatility of our approach.
Discussion
Two main issues were identified in our approach: The effect of the hyperparameters on the tra- jectory distributions, and the increased computational cost of the weights estimation and blending process. The former problem is a limitation inherited from the classic ProMP, as the more complex the trajectory, the more basis functions we need. The smoothness of this encoding also relies on the basis functions width, which is often the same for all of them. We hypothesize that considering the trajectory curvature to distribute the basis functions accordingly and define different widths may alleviate this hyperparameter tuning issue. The second problem arises as our weights estimation has no closed-form solution, and relies on Riemannian optimization methods [29, 21] for solving the geodesic regression and for estimating mean vectors for distributions lying on M. However, note that the weights are estimated only once and offline, meaning that the computations of P(p: 0) or P(p; 0*) are not compromised. Finally, the gradient-based approach to blend several ProMPs often converges in less than 10 iterations, making a fairly fast approach for online blending settings.
As mentioned in Section 1, several methods overlook the problem of encoding orientation trajecto- ries, or propose approximated solutions that produce distortion and inaccuracy issues (as shown in Appendix 3.2). Our Riemannian approach explicitly considers the geometric constraints of quater- nion data, and therefore it represents a potential alternative to the Riemannian TP-GMM [18]. Nev- ertheless, the benefits of ProMP when compared to TP-GMM lie on the trajectory modulation and blending features, which endow a robot with larger set of adaptation capabilities. It may be worth investigating how these formal methods dealing with full-pose trajectories compare to each other in terms of accuracy, adaptation, and extrapolation capabilities. ilar to [30] may bring important insights to the field. On a different note, the classic ProMP [4] also includes time derivatives of the considered variable, which was not covered in this paper. However, this extension is straightforward: This involves to include linear and angular velocities of the end- effector into the trajectory variables. Given that these are Fuclidean variables, the main challenge arises when quaternions are part of the trajectory data.
this context, benchmark works sim-
8

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Acknowledgments
The authors thank Noémie Jaquier (KIT) and Andras Kupesik (BCAT) for their support and useful feedback on this work. We would also like to thank the reviewers and area chair for their highly useful recommendations.
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Reviews:
[{"review_id": "szIZ-3P8_iM", "reviewer": "Reviewer_RYej", "summary": "This paper presents a principled extension of \"probabilistic movement\nprimitives\" or ProMPs from linear Gaussian models to data defined on\nRiemannian manifolds. This is important for learning trajectories, since often\nthe trajectory of interest is specified in terms of orientations and poses,\nwhich can be elegantly represented in terms of Riemannian manifolds. Consequently, the\nextension of ProMPs to Riemannian manifolds is well-motivated, but involves a\nnontrivial application of some fairly advanced machinery from Riemannian\ngeometry. The resulting approach is demonstrated on synthetic data and using a\nreal robot.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "## Strengths\n\nOverall, I felt this was an excellent paper.\n\n- The paper is well-written and organized. \n\n- The contribution is clear and relevant; the extension of probabilistic\nmovement primitives to Riemannian manifolds is well-motivated.\n\n- The formulation is concise and easy to follow (not always easy to achieve when\n  explaining Riemannian geometry)\n  \n- The method is demonstrated on synthetic data (which were really helpful for\n  \"visualizing\" what was going on) and with real robot experiments.\n  \n- The animations in the supplemental video (e.g. adding via points) were very\n  well done.\n\n## Weaknesses\n\nI have only minor comments for possible improvement of the paper, see Issues\nbelow.\n\n- The only \"weakness\" that stands out is the demonstration of only a single\nskill with the real robot. It would be really neat to show more trajectories. As\nmentioned below, though, the magnitude of the \"theoretical\" contribution makes\nthis a bit minor.\n\n- I'd also suggest discussing a bit more about the timing / computational\nefficiency. I'm particularly interested in the second issue raised in the\nDiscussion (regarding geodesic regression). How much more costly (e.g. in units\nof time or relative order of magnitude) is this vs. the corresponding linear\nsystem for ProMP?", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "I am recommending strong acceptance. This is a really nice paper. The proposed\nwork is well-motivated and clearly explained. Beyond that, results with\nsynthetic data and real robot experiments are provided. My only comments on\npossible areas for improvement are minor.", "technical_quality": "Excellent: Very convincing and thorough.", "clarity_of_presentation": "Excellent: Enjoyable to read.", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"impact": "4: The work contains interesting new ideas that will potentially have major impact in robotics or machine learning.", "reviewer_expertise": "Good: General knowledge of the area"}, "extra_text": {"issues": "## \"Orientation\" ProMPs\n\n- The paper refers to \"Orientation\" ProMPs, but I don't see anything in the\n  derivation in Section 3 that restricts this particular approach to\n  orientations specifically - rather, this seems to be a general extension of\n  ProMPs to arbitrary Riemannian manifolds (though admittedly, I don't have an\n  immediate application to trajectory learning employing other manifolds that\n  comes to mind). Might be worth a mention (maybe the authors have thoughts on\n  this?)\n\n## Discussion\n\n- In the discussion, the issue of computational cost is raised, but I didn't see\n  any quantitative mention about this (e.g. is there a noticeable increase in\n  computation time). It's possible I missed something, but clarification here\n  would be helpful.\n\n## Minor Comments\n\n- I recommend the excellent reference [1] for more detail on manifolds,\n  especially for the concept of parallel transport. This might help a reader\n  needing help with these concepts (it is a good supplement to the Absil,\n  Mahony, and Sepulchre reference).\n\n- Maybe more results could be provided (e.g. learning other skills than\n  \"re-orient\"), but given the theoretical contribution this is minor.\n\n[1] Boumal, Nicolas. \"An introduction to optimization on smooth manifolds.\"\n Available online, Aug (2020). http://sma.epfl.ch/~nboumal/book/index.html"}}, {"review_id": "Vo2YG-0Ozhk", "reviewer": "Reviewer_MyGv", "summary": "The paper presents Riemannian formulation of Probabilistic Movement Primitive (ProMP) that enables the representation of quaternion trajectories which adheres to Riemannian geometry. In doing so, this paper extends ProMP which previously can only represent trajectories with Euclidean geometry, to those with Riemannian geometry, unlocking the possibility of encoding the full 6D pose trajectories and its uncertainty/probabilistic representation. The paper also explains how to perform Learning from Demonstration, trajectory modulation, blending, and task parametrization with this Riemannian ProMP.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\nThe paper explains the background on ProMP, Riemannian Manifold (tangent spaces, geodesic, exponential and logarithmic maps, and parallel transport), and geodesic regression very well.\n\nWeaknesses:\nThe explanation of Orientation ProMPs is sometimes unclear, or difficult to grasp/follow; sometimes I lose track during some algebraic manipulations.\n\nMoreover, the Log() and Exp() mapping in the supplementary materials are different from those in the referenced paper: A. Ude, B. Nemec, T. Petric, and J. Morimoto. Orientation in Cartesian space dynamic movement primitives.\nNo clarification is provided yet.\n\nFurthermore, naturally the dimension of the tangent space of 3D orientations (regardless of representation: quaternion, axis-times-angle, or rotation matrices) are 3; which makes sense because this tangent space is the same space as the space of angular velocities (omega vector in R^3). However, it is stated in line 205 that wn in tangent space TpM is a subset of R^4. No clarification is provided yet.\n\nSee `Issues` for suggestions of improvements.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "The paper contains novelty and deserves a publication, but can be improved by following the recommendations in the `Strengths and Weaknesses` or `Issues`.", "technical_quality": "Very Good:  Convincing.", "clarity_of_presentation": "Very Good:  Well organized and clear.", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"impact": "4: The work contains interesting new ideas that will potentially have major impact in robotics or machine learning.", "reviewer_expertise": "Very good: Comprehensive knowledge of the area"}, "extra_text": {"issues": "1. The explanation of Orientation ProMPs is sometimes unclear, or difficult to grasp/follow; sometimes I lose track during some algebraic manipulations. For example, on line 167, it would be helpful to more explicitly state what are the dimensions of the matrices U, xi, Xi, and u. Also, it would be very helpful (in maximizing the impact of the paper) to upload the source code implementation of the paper into the supplementary materials.\n\n2. Moreover, the Log() and Exp() mapping in the supplementary materials are different from those in a referenced paper A. Ude, B. Nemec, T. Petric, and J. Morimoto: Orientation in Cartesian space dynamic movement primitives, as well as in a non-referenced paper J. Solà, J. Deray, and D. Atchuthan: A micro Lie theory for state estimation in robotics (https://arxiv.org/pdf/1812.01537.pdf).\nMaybe it's useful to comment on what caused the differences.\n\n3. Furthermore, naturally the dimension of the tangent space of 3D orientations (regardless of representation: quaternion, axis-times-angle, or rotation matrices) are 3; which makes sense because this tangent space is the same space as the space of angular velocities (omega vector in R^3). Can the author comment on why it is stated in line 205 that wn in tangent space TpM is a subset of R^4?"}}, {"review_id": "Ktbk8xQBp-o", "reviewer": "Reviewer_qZSt", "summary": "This paper generalizes \"Probabilistic Movement Primitives\" from Euclidean spaces to Riemannian manifold.\nThe basic concepts in PMP (learning from demonstration, trajectory modulation, blending, task parametrization) \nare mapped to the corresponding concept on the manifold. The mapping is done via geodesic regression, utilizing  \nthe properties of smooth manifold. the method is demonstrated both on a synthetic data, and on a physical platform.\nthe simulations confirm the main claims of the paper. ", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n- the paper is well written, provides the required background\n- the analogy between the PMP and the proposed method and the advantage is clear.  \n- the experiments are convincing. \n- the videos in the supplementary are helpful and insightful. \n- the limitations are reasonable. they are discussed, and the future steps are proposed.\n\nWeakness:\n- it would be worth to elaborate on the complexity of different complement of the method. \n- adding more diverse experiment on the physical platform will be useful\n", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "I like this paper, because it has all the required: motivation, problem definition, rigour solution, validation. \n\n\n\n", "technical_quality": "Excellent: Very convincing and thorough.", "clarity_of_presentation": "Very Good:  Well organized and clear.", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"impact": "4: The work contains interesting new ideas that will potentially have major impact in robotics or machine learning.", "issues": "--", "reviewer_expertise": "Very good: Comprehensive knowledge of the area"}, "extra_text": {}}, {"review_id": "CnitfUAtV6M", "reviewer": "Reviewer_G9zn", "summary": "This paper presents an extension of the probabilistic movements primitives framework for Riemannian Manifolds.\nThe authors propose a full ProMP framework adapted to the Riemannian Manifolds, which includes imitation learning, conditioning, blending, task parametrization.\n", "questions": "", "limitations": "", "rating": 2, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "The theory is well presented and well exposed. The formulation seems solid and quite easy to use.\nHowever, the contribution is really incremental, as it consists mainly of combining existing techniques to enhance the ProMP framework.\nThe experimental evaluation is good, but a bit too simplistic, and it's not showing in a sufficiently convincing manner the advantage of considering the quaternions instead of euclidean variables to control robotic manipulator. While the authors prove that their method is better than plain ones, I would like to see more complex robotics experiments where orientation is fundamental e.g. peg-in-a-hole or other similar robotics tasks.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "I think this is a borderline paper. The theoretical framework is solid, and the solved problem is really interesting. However the contribution is a bit incremental and the experimental evaluation, particularly on real robots is quite weak.\nIf I had the possibility, I would vote for borderline (nor weak accept, nor weak reject). \nHowever, as I think that this paper could be improved, also in the rebuttal, I selected weak reject.", "technical_quality": "Excellent: Very convincing and thorough.", "clarity_of_presentation": "Excellent: Enjoyable to read.", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"impact": "3: The work is incremental and unlikely to have much impact, even though it may be technically correct and well executed.", "issues": "- Add a more complex robotic experiment to justify your claims and to show the capabilities of the robot.\n", "reviewer_expertise": "Very good: Comprehensive knowledge of the area"}, "extra_text": {}}]

Author rebuttal:

Meta-review: The reviewers agree that this is a very well written and thought through paper.
Most minor concerns regard some additional clarifications that should be included in the revised version.
The only major complaint is about the complexity of the provided experiments.
If possible some additional more complex real robot experiments would benefit the paper and make a stronger case for the approach.

## Post-Rebuttal Update:
The authors successfully addressed and clarified most of the raised issues and further improved the paper through the rebuttal phase.
Even if I would agree that the work is incremental, this would not a be a major concerns as long as the work is not trivial, which is not the case here.
I would ask the authors, however, to reference and discuss

[Gaussians on Riemannian Manifolds: Applications for Robot Learning and Adaptive Control](http://ras.papercept.net/images/temp/IROS/files/3131.pdf)  by Sylvain Calinon

in the camera ready version.
Decision: Accept (Poster)