Orientation Probabilistic Movement Primitives on Riemannian Manifolds
Anonymous Author(s)
AfﬁliationAddress email
Abstract: Learning complex robot motions necessarily demands to have modelsthat are able to encode and retrieve full-pose trajectories when tasks are deﬁnedin operational spaces.
Probabilistic movement primitives (ProMPs) stand out asa 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 inoperational space, their original formulation does not directly apply to full-posemovements including rotational trajectories described by quaternions.
This paperproposes 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 complexfull-pose robot motion patterns.
Riemannian ProMPs are tested on toy examples toillustrate their workﬂow, and on real learning-from-demonstration experiments.
Introduction
For robots to perform autonomously in unstructured environments, they need to learn to interactwith their surroundings.
To do so, robots may rely on a library of skills to execute simple motionsor perform complicated tasks as a composition of several skills [1].
A well-established way tolearn motion skills is via human examples, a k.a. learning from demonstrations (LfD)
Thisentails a human expert showing once or several times a speciﬁc motion to be imitated by a robot.
Inthis 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-Parameterized Gaussian Mixture Models (TP-GMMs)
[5], Kernelized MPs(KMPs) [9] and Conditional Neural MPs (CNMPs) [6] are some of the recent probabilistic modelsto represent motion primitives (MPs).
While some of them were originally proposed to learn jointspace motions (e.g., ProMPs), others have mainly focused on MPs in task space (e.g., TP-GMM),assuming that certain tasks, such as manipulation, may be more easily represented in Cartesiancoordinates.
The latter approach comes with an additional challenge, namely, encoding and synthesizing end-effector orientation trajectories.
This problem is often overlooked, but the need of robotsperforming complicated tasks makes it imperative to extend MPs models to handle orientation data.
Orientation representation in robotics comes in different ways such as Euler angles, rotation matricesand quaternions.
Euler angles are a minimal and intuitive representation, which however is notunique.
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, quaternionsare a nearly-minimal representation and provide strong stability guarantees in close-loop orientationcontrol [10].
Despite their antipodality (i.e., each rotation is represented by two antipodal pointson the sphere S 3 ), quaternions have gained interest in robot learning, control, and optimization dueto their favorable properties [10, 12, 16].
Pastor et al.
[12] and Ude et al.
[13] pioneered works tolearn quaternion MPs, where the classic DMP [3] is formulated to encode quaternion trajectories for robot motion generation.
Its stability guarantees were later improved in [14], while Saverianoet al.
[15] used this new formulation into a single DMP framework to learn full-pose end-effectortrajectories.
However, DMPs do not encode the demonstrations variability or uncertainty, oftenexploited to extend robot learning capabilities.
Silv´erio et al.
[16] proposed the ﬁrst extension of TP-GMM for learning quaternion trajectories,which relies on unit-norm approximations.
This was resolved in [17], where a Riemannian manifold formulation for both motion learning and reproduction was introduced.
Despite TP-GMMoffers good extrapolation capabilities due to its task-parameterized formulation, none of the foregoing 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 TPGMM.
KMP was then extended to handle orientation data [18], via a projection onto an Euclideanspace using a ﬁxed quaternion reference, i.e. a single Euclidean tangent space.
Thus, model learning, via-point adaptation, and skill reproduction take place in Euclidean space, ignoring the intrinsicquaternions geometry.
Note that such approximations lead to data and model distortions [17].
Although ProMPs have been used to learn Cartesian movements, their formulation does not handlequaternion trajectories.
A possible solution would entail unit-norm approximations as in [16], butthis approach fully ignores the geometry of the quaternions space and may lead to inaccurate models.
An alternative and more sound solution relies on a Riemannian manifold formulation of ProMP,in the same spirit as [17].
However, two main difﬁculties arise: ( i ) learning of the model parametersdoes not accept a closed-form solution, and ( ii ) trajectory retrieval is constrained to lie on the sphere
S 3 .
We here provide solutions to these and related problems, which lead to the ﬁrst ProMP framework that makes possible encoding and reproducing full-pose end-effector trajectories.
In contrastto TP-GMM methods, our approach provides via-point modulation and blending capabilities, whichare naturally inherited from the original ProMP.
Unlike KMP, ProMP is a compact and standalonemodel, meaning that learning and reproduction do not rely on previously-learned 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 primitivesusing ProMP.
Our extension builds on the classic ProMP [4] (summarized in § 2), and considersthe space of quaternions as a Riemannian manifold.
We propose to estimate the ProMP parametersusing multivariate geodesic regression (see § 3), and we show how trajectory retrieval, modulationof the trajectory distributions, and MPs blending are all possible via a Riemannian probabilisticformulation.
In § 4, we illustrate our approach on the unit-sphere manifold S 2 , and we learn realisticmotion skills on a 7 -DoF robotic manipulator featuring complex full-pose trajectories on R 3 × S 3 .
Background2.1 ProMPs
Probabilistic Movement Primitives (ProMPs) [4] is a probabilistic framework for learning and synthesizing robot motion skills.
ProMPs represent a trajectory distribution by a set of basis functions.
Its probabilistic formulation enables movement modulation, parallel movement activation, and exploitation of variance information in control.
Formally, a single movement trajectory is denotedby τ = { y t } Tt =1 , where y t is a d -dimensional vector representing either a joint conﬁguration or a
Cartesian position at time step t (additional time derivatives of y may also be considered).
Eachpoint of the trajectory τ is represented as a linear basis function model
[Equation]
where w is a dN φ -dimensional weight vector, Ψ t is a ﬁxed d × dN φ -dimensional blockdiagonal matrix containing N φ time-dependent Gaussian basis functions φ t for each DoF, and(cid:15)
y ∼ N ( 0 , Σ y ) is the zero mean i.i.d.
Gaussian noise with uncertainty Σ y
(see [4]).
ProMPsemploy a phase variable z ∈ [0 , 1] that decouples the demonstrations τ = { y t } z T t =
z 0 from the timeinstances, which in turn allows for temporal modulation.
ProMPs assume that each demonstration is characterized by a different weight vector w , leading toa distribution P ( w ; θ )
=
N ( w | µ w , Σ w ) .
Consequently, the distribution of y t , P ( y t ; θ ) is
P
[Equation]
Learning from demonstrations: The learning process of ProMPs mainly consists of estimatingthe weight distribution P ( w ; θ ) .
To do so, a weight vector w i , representing the i -th demonstrationas in (1), is estimated by maximum likelihood, leading to solution of the form
[Equation]
where Y i = (cid:2) y T i, 1 . .
y T i,T (cid:3) T concatenates all the trajectory points, and Ψ consists of all the timeinstances for the basis-functions matrix Ψ t .
Given a set of N demonstrations, the weight distributionparameters θ = { µ w , Σ w } are estimated by maximum likelihood (see Algorithm 1 in Appendix 1 ).
Trajectory modulation: To adapt to new situations, ProMPs allow for trajectory modulation to via-points or target positions by conditioning the motion to reach a desiredpoint y ∗ t with associated covariance Σ ∗ y .
This results into the conditional probability
P
[Equation]
|
[Equation]
N
[Equation]
w |
[Equation]
w , Σ w ) , whose parameters can be computed as follows
[Equation]
Blending: By computing a product of trajectory distributions, ProMPs blend different movement primitives into a single motion.
The blended trajectory follows a distribution P ( y + t )
= 103 (cid:81)
Ss =1 P s ( y t )
α t,s , for a set of S ProMPs P s ( y t ) =
N ( y t | µ t,s , Σ t,s ) , with associated blendingweight α t,s ∈ [0 , 1] .
The parameters of P ( y + t ) =
N ( y + t | µ + t , Σ + t ) are estimated as follows
[Equation]
+ t
[Equation]
Task parametrization: ProMPs also exploit task parameters to adapt the robot motion to, forexample, target objects for reaching tasks.
Formally, ProMPs consider an external state ˆ s andlearn an afﬁne mapping from ˆ s to the mean weight vector µ w , leading to the joint distribution
P ( w , ˆ s ) =
N ( w | O ˆ s + o , Σ w ) N
(ˆ s | µ ˆ s , Σ ˆ s ) , 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 a vector space, thusthe use of traditional Euclidean methods for operating these variables is inadequate.
We exploit
Riemannian geometry to formulate ProMPs on quaternion space as presented in § 3.
Formally, a
Riemannian manifold M is a m -dimensional topological space with a globally deﬁned differentialstructure, where each point locally resembles an Euclidean space R m .
For each point x ∈M , thereexists a tangent space T x M that is a vector space consisting of the tangent vectors of all the possiblesmooth curves passing through x .
A Riemannian manifold is equipped with a smoothly-varyingpositive deﬁnite inner product called a Riemannian metric, which permits to deﬁne curve lengths in
M .
These curves, called geodesics, are the generalization of straight lines on the Euclidean space to
Riemannian manifolds, as they are minimum-length curves between two points in M (see Fig. 1).
We exploit the Euclidean tangent spaces to operate with Riemannian data.
To do so, we need mappings back and forth between T x M and M , which are the exponential and logarithmic maps.
Theexponential map Exp x : T x M → M maps a point u in the tangent space of x to a point y onthe manifold, so that it lies on the geodesic starting at x in the direction u , and such that thegeodesic distance d M between x and y equals the distance between x and u .
The inverse operation is the logarithmic map Log x : M → T x M .
Another useful operation is the parallel transport
Γ x → y : T x M → T y M , which allows us to operate with manifold elements lying on differenttangent spaces.
The parallel transport moves elements between tangent spaces such that the innerproduct between two elements in the tangent space remains constant (see [19, 20] for further details).
Finally, let us introduce a Riemannian Gaussian distribution of a random variable p ∈ M as
[Equation]
with mean µ ∈ M , and covariance Σ ∈ T µ M .
This Riemannian Gaussian corresponds to anapproximated maximum-entropy distribution for Riemannian manifolds, as introduced by Pennecet al.
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 m .
2.3 Geodesic regression
A geodesic regression model generalizes linear regression to Riemannian manifolds.
The regressionis deﬁned as y =
Exp ˜ y
( (cid:15) ) , with ˜ y
=
Exp p ( x u ) , where y ∈ M and x ∈ R are respectivelythe output and input variables, p ∈ M is a base point on the manifold, u ∈ T p M is a tangentvector at p , and the error (cid:15) is a random variable taking values in the tangent space at ˜ y .
As ananalogy to linear regression, one can interpret ( p , u ) as an intercept p and a slope u (see [22] fordetails).
Given a set of points { y 1 , . . .
, y T } ∈ M and { x 1 , . . .
, x T } ∈ R , geodesic regression ﬁndsa geodesic curve γ ∈ M that best models the relationship between all the T pairs ( x i , y i ) .
To do so,we minimize the sum-of-squared Riemannian distances (i.e., errors) between the model estimatesand the observations, that is, E ( p , u ) = 12 (cid:80)
Ti =1 d M (ˆ y i , y i ) 2 , where ˆ y i
=
Exp p ( x i u ) is the modelestimate on M , d M (ˆ y i , y i )
= (cid:107) Log ˆ
y i ( y i ) (cid:107) is the Riemannian error, and the pair ( p , u ) ∈ T M isan element of the tangent bundle T M .
We can formulate a least-squares estimator of the geodesicmodel as a minimizer of such sum-of-squared Riemannian distances, i.e.,
[Equation]
The problem in (7) does not yield an analytical solution like (3).
As explained by Fletcher [22],a solution can be obtained via gradient descent on Riemannian manifolds.
Note that this geodesicmodel considers only a scalar independent variable x ∈ R , meaning that the derivatives are obtainedalong a single geodesic curve parametrized by a single tangent vector u .
The extension to multivariate cases proposed by Kim et al.
[23], where x ∈ R n , identiﬁes multiple geodesic curves (viewed as“basis” vectors in Euclidean space).
Multivariate general linear models on Riemannian manifolds(MGLM) [23] provides a solution to this problem.
MLGM uses a geodesic basis U =
[ u 1 . . .
u n ]formed by multiple tangent vectors u j ∈ T p M of dimensionality d = dim( T p M ) , one for eachdimension of x .
Then, the problem (7) can be reformulated as
[Equation]
Exp [Equation] This multivariate framework allows us to compute the weight vector, analogous to (3), for a demonstration lying on a Riemannian manifold M , for example M ≡ S 3 .
Orientation ProMPs
When human demonstrations involve Cartesian motion patterns (via kinesthetic teaching or teleoperation), it is necessary to have a learning model that encapsulates both translation and rotationmovements of the robot end-effector.
This means that a demonstration trajectory τ = { y t } Tt =1 isnow composed of datapoints y t ∈ R 3 × S 3 , representing the full Cartesian pose of the end-effectorat time step t .
In this case, the challenge is learning a ProMP in the orientation space, as the Euclidean case in R 3 follows the classic ProMP introduced in § 2.1.
Therefore, we focus on how toextend ProMP to learn trajectories on M = S 3 .
First of all, let us introduce an equivalent expression for ˆ y i , in the MGLM framework, such that it resembles the linear basis-function modelin (1).
Speciﬁcally, the estimate ˆ y i
=
Exp p ( Ux i ) ≡ Exp p
( X i u ) with U ∈ R d × n , x i ∈ R n , X i =
blockdiag ( x T 1 , . .
, x T n ) ∈ R d × nd and u =
[ u T 1 . . .
u T n ] T ∈ R nd .
This equivalence provesuseful when establishing analogies between the classic ProMPs formulation and our approach fororientation trajectories.
Similarly to (1), a point y t ∈ M of τ can be represented as a geodesic basis-function model
[Equation]
where p is a ﬁxed base point on M , w = (cid:104) w T 1 . .
w T N φ (cid:105) T is a large weight vector concatenating
N φ weight vectors w n ∈ T p M , Ψ t is the same matrix of time-dependent basis functions as in (1),and Σ
y is a covariance matrix encoding the uncertainty on T µ y M .
Two speciﬁc aspects aboutthis formulation deserve special attention: ( i ) the mean µ y of the Riemannian Gaussian distributionin (9), exploits the aforementioned equivalent formulation of MGLM; and ( ii ) the weight vectorsforming w in (9) correspond to the vector composing the geodesic basis of MGLM.
As a demonstration is characterized by a different weight vector w , again we can compute a distribution P ( w ; θ ) =
N ( w | µ w , Σ w ) .
Therefore, the marginal distribution of y t can be deﬁned as
P ( y ; θ ) = (cid:90) N M ( y | Exp p ( Ψ w ) , Σ y ) N ( w | µ w , Σ w ) d w , with µ y =
Exp p ( Ψ w ) ∈ M , (10) which depends on two probability distributions that lie on different manifolds.
Using the Riemannian operations described in § 2, we obtain the ﬁnal marginal (see Appendix 2 . 1 )
[Equation]
where ˜ Σ y and ˆ Σ y are parallel-transported covariances of the geodesic model 9 and the ﬁnalmarginal, respectively.
Learning from demonstrations via MGLM:
For each demonstration i , we estimate a weightvector ˆ w i =
(cid:104)
ˆ w T 1 . . .
ˆ w T N φ (cid:105) T using MGLM (illustrated in Fig. 2).
Firstly, we resort to theequivalent expression for y t introduced previously, where Exp p ( W φ t ) ≡ Exp p ( Ψ t w ) , with
W = (cid:2) w 1 . . .
w N φ (cid:3) and N φ being the number of basis functions.
Secondly, we consider a demonstrated quaternion trajectory τ
i = { y t } Tt =1 with y t ∈ S 3 .
Then, analogous to (3) in Euclideanspace, ˆ w i is estimated by leveraging (8), leading to(ˆ p , ˆ [Equation] argmin [Equation]
p , w
[Equation]
p [Equation] with E ( p , w n
[Equation]
T (cid:88) [Equation] 12) where φ t is the vector of Gaussian basis functions at time t , and W contains the set of estimatedtangent weight vectors ˆ w n ∈ T ˆ p M (i.e., N φ tangent vectors emerging out from the point p ∈ M ).
To solve (12), we need to compute the gradients of E ( p , w n ) with respect to p and each w n .
As explained in Appendix 2 . 2 , these gradients depend on the so-called adjoint operators, whichbroadly speaking, bring each error term Log ˆ y t ( y t ) from T ˆ y t M to T p M , with ˆ y t
=
Exp p ( W φ t ) .
Therefore, these adjoint operators can be approximated as parallel transport operations as proposedin [23].
This leads to the following reformulation of the error function of (12)
[Equation]
With the gradients of (13) (given in Appendix 2 ), we can now estimate both the vector p i andthe weight matrix W i , for each demonstration i .
Note that each demonstration may lead to different estimates of p , which deﬁnes the base point on M used to estimate w n ∈ T p M .
Thismay produce different tangent spaces across demonstrations, and therefore diverse tangent weightvectors.
An effective way to overcome this is to assume that all demonstrations share the same tangent space base p , which is the same assumption made when deﬁning the geodesic basis-functionmodel (9).
So, we only need to estimate p for a single demonstration, and use it to estimate alltangent weight vectors for the whole set of demonstrations.
Then, given a set of N demonstrations,the weight distribution parameters θ = { µ w , Σ w } are estimated by standard maximum likelihoodas w n ∈ T p M = R 3 ⊂ R 4 .
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 tonew situations) by conditioning the motion to reach a desired trajectory point y ∗ t ∈ Mwith associated covariance Σ ∗ y ∈ T y ∗ t M .
This results into the conditional probability
P ( w
[Equation]
y ∗ t ) ∝ N M ( y
[Equation]
t | Exp p ( Ψ t w ) , Σ ∗ y ) N ( w | µ w , Σ w ) , which depends on two probability distributions that lie on different manifolds, similarly to (10).
We leverage the fact that the mean µ ydepends on a single and ﬁxed p ∈ M , which is the basis of T p M where the weight distribution lieson.
Thus, we rewrite the conditional distribution as follows
[Equation]
where ˜ Σ ∗ y =
Γ y ∗ t → p ( Σ ∗ y ) , and { µ ∗ w , Σ ∗ w } are the parameters to estimate for the resulting conditional distribution.
Since both distributions now lie on T p M , which is embedded in the Euclideanspace, we can estimate { µ ∗ w , Σ ∗ w } similarly to the classic ProMP conditioning procedure, with special care of parallel-transporting the covariance matrices.
So, the updated parameters are
[Equation]
From the new weight distribution, we can obtain a new marginal distribution P ( y ; θ ∗ ) via (11).
Blending: When it comes to blend motion primitives in M , one needs to consider that each ofthem is parametrized by a set of weight vectors lying on different tangent spaces T p M .
Therefore, the weighted product of Gaussian distributions needs to be reformulated.
To do so, weresort to the Gaussian product formulation on Riemannian manifolds introduced by Zeestraten[17], where the log-likelihood of the product is iteratively maximized using a gradient-based approach as proposed in [24].
We here provide the iterative updates of the blended distribution
P ( y + ) =
N M ( y + | µ + , Σ + ) , while the full solution is given in Appendix 2 . 3 .
The iterative estimation of µ + is
[Equation]
[Equation]
for a set of S skills, where Λ y ,s
[Equation] ,s ) , and α s is the blending weight associated tothe skill s .
After convergence at iteration K , we obtain the ﬁnal parameters as follows
[Equation]
Task parametrization: Classic ProMP allows for adapting the weight distribution P ( w ; θ ) =
N ( w | µ w , Σ w ) as a function of an external task parameter ˆ s , as explained in Section 2.1.
This taskparametrization straightforwardly applies to our method as the weight vectors w n ∈ T p M ⊂ R 4 , aslong as the task parameter ˆ s is Euclidean.
However, if ˆ s ∈ M , we can learn a joint probability distribution P ( w , ˆ s ) using a Gaussian mixture model on Riemannian manifolds as proposed in [17, 25].
Subsequently, we can employ Gaussian mixture regression to compute P ( w | ˆ s ∗ ) during reproduction when a new task parameter ˆ s ∗ is provided.
We refer the reader to the works of Zeestraten [17]and Jaquier et al.
[25] for details on how to compute the distributions P ( w , ˆ s ) and P ( w | ˆ s ∗ ) .
Experiments
Synthetic data on S 2 : To illustrate how model learning, trajectory reproduction, via-point adaptation, and skills blending work in our approach, we used a dataset of hand-written letters.
Theoriginal trajectories were generated in R 2 and later projected to S 2 by a unit-norm mapping.
Eachletter in the dataset was demonstrated N = 8 times, and a simple smoothing ﬁlter was appliedfor visualization purposes.
We trained 4 ProMPs, one for each letter in { G , I , J , S } .
All modelswere trained following Algorithm 2 with hyperparameters given in Appendix 3 . 1 .
Figure 3 showsthe demonstration data, marginal distribution P ( y ; θ ) computed via (10), and via-point adaptationobtained from (14) and (15), corresponding to the G and S models.
The mean of P ( y ; θ ) followsthe demonstrations pattern, and the covariance proﬁle captures the demonstrations variability in S 2 .
Note that the trajectories G and S display very elaborated “motion” patterns that might be morecomplex than those observed in realistic robotic settings.
Concerning the via-point adaptation, wechose a random point y ∗ ∈ S 2 with associated Σ ∗ y = 10 − 3 I (i.e., high precision while passingthrough y ∗ ).
As shown in Fig.
3, our approach smoothly adapts the trajectory and its covariancewhile accurately passing through y ∗ .
To test the blending process, we used the following subsets of our original dataset: { G , I } and{ S , J } .
The goal was to generate a trajectory starting from ﬁrst letter in the set, and then smoothlyswitching midway to the second letter.
Figure 4 shows the resulting blended trajectories for the twoaforementioned cases, where our approach smoothly blends the two given trajectory distributionsby following the procedure introduced in § 3.
Note that the blending behavior strongly depends onthe temporal evolution of the weights α s ∈ [0 , 1] associated to each skill s .
We used a sigmoid-likefunction for the weights α ( I ) s and α ( J ) s , while α ( G ) s = 1 − α
( I ) s and α ( S ) s
= 1 − α
( J ) s .
The foregoingresults show that our approach successfully learns and reproduces trajectory distributions on S 2 , andprovides full via-point adaptation and blending capabilities.
We now turn our attention to roboticexperiments where we learn and synthesize full-pose movement primitives.
Manipulation skills on R 3 × S 3 :
To test our approach in a robotic setting, we consider are-orient skill, which involves lifting a previously-grasped object, rotating the end-effector,and placing the object back on its original location with a modiﬁed orientation (see Fig. 5).
Thisskill features signiﬁcant position and orientation changes, and it is therefore suitable to showcase thefunctionalities of our Riemannian ProMP.
We collected 4 demonstrations of the re-orient skillvia kinesthetic teaching on a Franka Emika Panda robot, where full-pose end-effector trajectories{ p t } Tt =1 were recorded, with p t ∈ R 3 × S 3 being the end-effector pose at time step t .
The data wasused to train a ProMP on R 3 × S 3 , with position and orientation models learned using respectivelya 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 ; θ ) depicted as a black trajectory.
Our model properly captures the motion pattern on R 3 × S 3 for this skill.
We then evaluated how this learned skill may adapt to a via point p ∗ ∈ R 3 × S 3 ,representing a new position and orientation of the end-effector at t = 10 sec .
By using the approachdescribed in § 3, we computed a new P ( p ; θ ∗ ) , where the updated mean is required to pass throughp ∗ .
Figure 6 displays the updated mean (light blue lines), which successfully adapts to pass throughthe 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 ProMPsusing an Euler-angle representation and a unit-norm approximation, using the same hyperparameters.
While both models retrieved a distribution P ( p ; θ ) similar to our approach, their performanceis severely compromised in the via-point case, as they retrieve jerky trajectories with lower accuracytracking w.r.t p ∗ (see details in Appendix 3 . 2 ).
These results conﬁrm the importance of our Riemannian formulation for ProMP when learning and adapting full-pose end-effector skills.
Appendix 3 .
2reports learning and reproduction results for two additional manipulation skills featuring motions ofdiverse complexity, demonstrating the versatility of our approach.
Discussion
Two main issues were identiﬁed in our approach: The effect of the hyper-parameters on the trajectory distributions, and the increased computational cost of the weights estimation and blendingprocess.
The former problem is a limitation inherited from the classic ProMP, as the more complexthe trajectory, the more basis functions we need.
The smoothness of this encoding also relies on thebasis functions width, which is often the same for all of them.
We hypothesize that considering thetrajectory curvature to distribute the basis functions accordingly and to deﬁne different widths mayalleviate this hyper-parameter tuning issue.
The second problem arises as our weights estimationhas no closed-form solution, and it relies on Riemannian optimization methods [26] for solving thegeodesic regression and for estimating mean vectors for distributions lying on M .
However, notethat the weights are estimated only once, meaning that the computation of P ( p ; θ ) or P ( p ; θ ∗ ) arenot compromised.
Finally, the gradient-based approach to blend several ProMPs often converges inless than 10 iterations, making it a fairly fast approach for online blending settings.
As mentioned in Section 1, several methods overlook the problem of encoding orientation trajectories, or propose approximated solutions arising distortion and inaccuracy issues (as shown in Appendix 3 . 2 ).
Our Riemannian approach explicitly considers the geometric constraints of quaterniondata, and therefore it represents a potential alternative to the Riemannian TP-GMM [17].
Nevertheless, the beneﬁts of ProMP when compared to TP-GMM lie on the trajectory modulation andblending features, which endow a robot with a larger set of adaptation capabilities.
It may be worthinvestigating how these formal methods dealing with full-pose trajectories compare to each other interms of accuracy, adaptation, and extrapolation capabilities.
In this context, benchmark works similar to [27] may bring important insights to the ﬁeld.
On a different note, the classic ProMP [4] alsoincludes 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 endeffector into the trajectory variables.
Given that these are Euclidean variables, the main challengearises when quaternions are part of the trajectory data.