Phase Distribution in Probabilistic Movement Primitives, Representing

Time Variability for the Recognition and Reproduction of Human

Movements

Vittorio Lippi and Raphael Deimel

Technische Universit

at Berlin, Fachgebiet Regelungssysteme, Sekretariat EN11, Einsteinufer 17, 10587 Berlin, Germany

Keywords:

ProMP, Human Movement, Prediction, Recognition, Time Warping, Phase.

Abstract:

Probabilistic Movement Primitives (ProMPs) are a widely used representation of movements for human-robot

interaction. They also facilitate the factorization of temporal and spatial structure of movements. In this work

we investigate a method to temporally align observations so that when learning ProMPs, information in the

spatial structure of the observed motion is maximized while maintaining a smooth phase velocity. We apply

the method on recordings of hand trajectories in a two-dimensional reaching task. A system for simultaneous

recognition of movement and phase is proposed and performance of movement recognition and movement

reproduction is discussed.

1 INTRODUCTION

Overview. Probabilistic movement primitives

(ProMP) (Paraschos et al., 2013) are a representation

of movements used in robot control and human-robot

interaction (HRI) applications (Maeda et al., 2017)

and provide several desirable properties to model

tasks for robot control and HRI(Paraschos et al.,

2018). One of these properties is temporal scaling:

the movement trajectory is not a direct function of

time but function of a phase variable, φ(t), i.e. the

temporal evolution of φ determines the the velocity of

the movement independently of the spatial structure.

A ProMP represents a sample movement y(t) as

y(t) =



˙q



= Φ(φ(t))w + ε (1)

where q is the vector of variables describing the

movement (usually joint angles or hand effector posi-

tion and orientation) and Φ is a Nx2 vector computed

with Gaussian functions and their derivatives:

(φ(t)−c

)

∑

k=1

(φ(t)−c

)

(2)

where c

represents the center and h

expresses the

spread of the bell-shaped feature. The vector w is

drawn from a multinomial distribution which is de-

ﬁned by its parameters θ:

p(w|θ) = N (µ

,Σ

) (3)

0 0.2 0.4 0.6 0.8 1

phase

0.5

0 0.5 1 1.5 2 2.5 3

time [s]

0.5

phase

0 0.5 1 1.5 2 2.5 3

time [s]

0.5

Figure 1: Features and Phase. In (a) the features Φ

(φ) are

represented as a function of the phase. In (b) a phase proﬁle,

based on the beta function as a function of time. In (c) the

features are represented as a function of the time.

In a general formulation (see for example (Colom

et al., 2014)) ε in Eq. 4 models unbiased noise as

p(ε) = N (0, Σ

). In this paper we assume that all

the variability observed on y should be accounted by

the distribution of w and that ε ≈ 0. In this way

each observed trajectory y

is represented by a single

Lippi, V. and Deimel, R.

Phase Distr ibution in Probabilistic Movement Primitives, Representing Time Variability for the Recognition and Reproduction of Human Movements.

DOI: 10.5220/0007750205710578

In Proceedings of the 16th International Conference on Informatics in Control, Automation and Robotics (ICINCO 2019), pages 571-578

ISBN: 978-989-758-380-3

571

weight vector w

. Vice versa, w can be interpreted as

a compressed representation of the movement trajec-

tory, obtained from projecting trajectory values onto a

lower dimensional subspace using the Moore-Penrose

pseudo-inverse of Φ:

w = Φ

y (4)

Which weight vector is most likely depends on the

optimization criterium, e.g. jerk minimization as reg-

ularization principle (Paraschos et al., 2018). Alter-

native features for movement representation can be

designed to guarantee other properties such as min-

imal error at the ﬁnal position and speed (Lippi et al.,

2012). Due to the probabilistic nature of the ProMP

framework constrains on generated trajectories can

be imposed by conditioning the probability distribu-

tion(Paraschos et al., 2018). Figure 1 illustrates the

features as either a function of time or phase. The

time modulation performed by the phase function also

modulates the resolution of the low dimensional rep-

resentation during movement execution. In Figure

1 (c) features in the middle of the movement are more

dense than at the beginning or at the end, due to the

choice of a sigmoidal phase proﬁle. We assume that

part of the variability observed among the repetitions

of a movement is caused by shifted and stretched

phase signals. Time scaling can be performed by

means of a linear constant, i.e. φ = αt (Maeda et al.,

2017; Ewerton et al., 2015). In this work we esti-

mate phase proﬁle parameters for each observation

that minimize the variances over all observed w (or,

equivalently of the response in phase domain), under

the constraint of an assumed structure for the phase

proﬁle. In particular we propose to restrict phase sig-

nals to the beta function:

∆

,∆

(t) =

2,2



τ − ∆

∆

− ∆



dτ (5)

where β

2,2

is the incomplete beta function with pa-

rameters a = b = 2. This sigmoidal function, shown

in 1 (b), is differentiable, monotonic and saturates at

0 and 1. The parameters ∆

, ∆

associated with each

trajectory depend on the distribution of the whole

sample set. Notice that weights w are in general spe-

ciﬁc w.r.t. to the chosen phase proﬁle (see Eq. 4). The

average of the sample set distribution is deﬁned as:

y = argmin

∑

i=1

(φ) −

y(φ))

dφ (6)

For the i

sample the phase is deﬁned as:

= argmin



φ(τ(φ))



− y(φ)



dφ (7)

The term τ(φ) represents alignment between the av-

erage y and the y

sample, it express the fact that as

φ and φ are both monotonically dependent on t, the

former can be expressed as a function of the latter, in

particular:

τ(φ) = arg



φ(t) =

φ(t)



(8)

As y depends on all phase proﬁles, Eq.6 and Eq.7

need to be optimized together to ﬁnd the optimal

phase proﬁles. The problem is solved by iteratively

optimizing φ(t) and φ

, which yields phase proﬁle pa-

rameters for each sample movement. The phase pro-

ﬁle parameters of the average trajectory is ﬁxed a pri-

ori to ∆

= ∆

= 0, as the time scaling when com-

paring two or more movements (e.g. when comput-

ing the average) depends on the relationship between

the respective phase proﬁles and hence there is a de-

gree of redundancy. The described time scaling can

also be performed on time domain values. In order to

take into account the movement primitive representa-

tion the y in Eq. 6 and Eq. 7 can be projected on the

MP representation y

= Φw. Once optimized time

scaling parameters of the sample set are obtained,

we can obtain a probability distribution describing

the movements using empirical estimators from lit-

erature (Maeda et al., 2017; Paraschos et al., 2018).

In the following paragraphs we demonstrate (a) how

to identify a model for movements, (b) how to recog-

nize an observed movement given a set of movement

primitives, (c) how to estimate the current phase, (d)

how to integrate perception in the phase recognition

process and (e) how to generate a movement.

Model Identiﬁcation. The estimation of ProMP pa-

rameters from of a sample set can be based on dif-

ferent principles, e.g. linear regression of each ob-

servation individually (Paraschos et al., 2013) or

maximizing the likelihood of the complete observed

data(Paraschos et al., 2018). As introduced in the

overview in this work we assume ε ≈ 0 in Eq. 4 and

that the phase proﬁle can be chosen to minimize the

variability on w. This leads to a two step procedure:

First, each observed trajectory is assigned a set of pa-

rameters (w

,∆

). This is done on the basis of Eq.

6 and Eq. 7 to obtain the phase proﬁle, and Eq. 4 to

obtain w, then the distributions for w (Eq. 2) and φ(t)

is obtained with an empirical estimator. An example

of model identiﬁcation is shown in Fig.2. The train-

ing set is generated artiﬁcially and consists of planar

motion with random time proﬁle (with the form spec-

iﬁed in Eq.5) and non-intersecting parabolic trajecto-

ries. In Fig.2 (b) movements generated with the iden-

tiﬁed distribution are shown. Notice that assuming

a normal distribution of w produces a distribution of

trajectories that is different from the more “uniform”

training set.

ICINCO 2019 - 16th International Conference on Informatics in Control, Automation and Robotics

572

Movement Recognition. An observed movement y

can be classiﬁed by comparing likelihoods between

ProMPs. The likelihood

p(y(t

)|µ

,Σ

)



y(t

)|Φ(φ)µ

,Φ(φ)Σ

Φ(φ)



p(φ|t

)dφ

(9)

of observing the movement given model k is com-

puted and the most likely model is selected, i.e. L =

argmax

p(y|θ

). Notice that the probability of y(t

)

is considered independent of the previous observa-

tions, i.e. p(y(t

)|θ

,y(t

i−1

)) = p(y(t

)|θ

). The prob-

ability of an observed sequence can be obtained by

multiplying the probabilities of the observed samples,

p(y|θ

) =

∏

p(y

|θ

). In the described examples

we will include all the past observations of y

, in the

general case not all points of the trajectory may be

available, due to occlusion for example. Applying the

ProMP framework to such cases is demonstrated in

(Maeda et al., 2017).

Phase Recognition. Using the Bayes rule we can

estimate the most likely phase of an observed sample:

p(φ|y(t

)) =

p(y(t

)|Φ(φ)µ

,Φ(φ)Σ

Φ(φ)

)p(φ|t

)

∑

p(θ

)p(φ|y(t), θ

)

(10)

Notice that p(φ|t

) is assumed to be normally dis-

tributed and hence parameterized by mean µ

(t) and

standard deviation σ

(t) as functions over time. ∆

and ∆

are not normally distributed due to the nonlin-

ear relationship in Eq. 5. In most prior work on the

topic, φ is assumed to be a linear function over time

and that therefore φ(t) can be described by a normal

distribution, e.g. in (Ewerton et al., 2015). Dynamic

time warping (DTW) is another, nonlinear, approach

proposed recently (Ewerton et al., 2018).

Perception. Eq. 10 provides an estimate given a

time-dependent but otherwise ﬁxed p(φ|t

) derived

from the training set. Alternatively, we can use an

explicit phase estimate provided by some perceptual

system. Examples of fusing ProMP and perception

have been proposed, e.g. in (Dermy et al., 2019)

ProMPs are used in the context of predicting human

intentions. In general the sensor input will have many

more dimensions than y. In this work we will estimate

the phase from recent observations of the motion. We

will consider a phase estimator based on previous val-

ues of y over a sliding time window to yield an esti-

mate φ

∗

) of the current phase. Such estimator is

used to compute the probability of the observed y(t

Instead of integrating over the distribution of proba-

bility of φ as shown in Eq. 9, only the estimated value

is used to compute y(t

p(y(t

)|µ

,Σ

)

= p



y(t

)|Φ(φ

∗

))µ

,Φ(φ

∗

))Σ

Φ(φ

∗

))



(11)

Movement Generation. The model can be used

to generate movements. Depending on the task the

movements can be generated deterministically using

Eq. 1 or stochastically by sampling w from the dis-

tribution. In certain cases the term ε in Eq. 1 may be

nonzero due to actuation and external noise, but is not

the case in the presented example. Generating move-

ments from ProMP can be used both for robot con-

trol and to predict human movements in HRI, see, for

example, (Oguz et al., 2017). Samples can obey con-

straints such as a speciﬁc target position or velocity,

by conditioning the distribution to the constraint prior

to sampling (Paraschos et al., 2013), which can be

done parameterically (and therefore efﬁciently) with

normal distributions. A constraint can be expressed

as a target point y

∗

(φ

) to be reached at a given phase.

Exploiting the linear dependency between w and y the

updated parameters for p(w|y

= y

∗

(φ

)) are

∗

= µ

+Σ

(φ)



Φ(φ)Σ

(φ)



−1

∗

− Φ(φ)µ

)

(12)

∗

= Σ

+ Σ

(φ)



Φ(φ)Σ

(φ)



−1

Φ(φ)Σ

(13)

An example of generated movements is shown in

Fig.2 (b,c,d). The conditioned distributions in 2 (c)

exhibit very small Σ

(i.e. all the trajectories start-

ing from a given point are indistinguishable in shape),

capturing the regularity of the training set. In Fig.2

(d) the distribution is forced to pass through two via-

points that are unlikely to be observed in the same

movement, i.e. it is conditioned on an outlier. As

a consequence, the resulting sampled trajectories do

not resemble the observations presented in the train-

ing set.

2 DATASET AND BENCHMARK

PROBLEM

The Reaching data set consists in a reaching task:

Participants have been asked to move the mouse cur-

sor from a starting point to targets appearing in 4

different positions. Each target identiﬁes a different

movement, associated to a different ProMP model.

The samples are shown in Fig. 3. Given the na-

ture of the task, consisting in reaching movements,

it comes natural to perform the segmentation on the

Phase Distribution in Probabilistic Movement Primitives, Representing Time Variability for the Recognition and Reproduction of Human

Movements

573

0 0.5 1 1.5

0.5

1.5

Training set

0.1

0.2

0.3

0.4

0.5

0.6

0.7

time[s]

0 0.5 1 1.5

0.5

1.5

Movements Generated from distribution

0 0.5 1 1.5

0.5

1.5

Conditioned distributions

0 0.5 1 1.5

0.5

1.5

Unlikely trajectory

Figure 2: Parabolic lines synthetic data set. In (a) an artiﬁcial data-set of trajectories with a parabolic shape and random

duration is shown, with only 10 of the 100 produced samples shown but covering more intermediate positions. The color

gradient indicates the temporal evolution of the trajectory, the black line the average trajectory in time. The resulting shape

does not match the presented examples well, especially in the end when sample count drops. The data-set has been used as a

training set for a proMP model. In (b) trajectories generated with the proMP distribution are shown, the black line represent

the average (computed respect to phase). Notice that while the model captured very accurately the shape of the movements,

the gaussian distribution of parameters results in a distribution of trajectory that is different from the one presented in the

training set. In (c) conditioned trajectories with imposed via-points a time t = 0 are shown, indicated by black stars. The

conditioned Σ

is so small that the trajectories generated for each via-point are indistinguishable. In (d) the trajectory is

conditioned to pass through two via-points unlikely to appear in the same movement. The resulting shape is dissimilar to

those presented in the training set. For reference, trajectories from the training set are shown in gray.

basis of events, in particular: (a) a movement starts

when the mouse cursor is in the starting position and

the target is shown on the screen and (b) it stops

when a target is reached. In general different heuris-

tics are possible for segmentation, notably the ProMP

framework provides the possibility to segment the

signal exploiting an expectation-maximization algo-

rithm, treating segmentation as a latent variable to be

optimized together with the movement models param-

eters(Lioutikov et al., 2017). The dataset includes 100

movement samples used as training-set for the model

identiﬁcation and 100 samples used as test set for the

movement recognition and phase recognition. Move-

ments were sampled at 100 Hz. The number of fea-

tures was set to N = 9, with parameters, in Eq. 2,

= (k − 1)/(N − 1) and h

= 0.15.

3 RESULTS

Model Identiﬁcation, Movement Recognition and

Phase Recognition. The distribution of phase pro-

ﬁles obtained with Eq. 6, Eq. 7 is shown in Fig 4

(b). Each sample is transformed in an MP, i.e. the pa-

rameters (w,φ) are extracted, 4 ProMP models repre-

senting the 4 movements are estimated. The observed

movement is identiﬁed as belonging to the most likely

model to produce it according to Eq. 9. The move-

ment recognition is performed at each time step. The

classiﬁcation accuracy increases with time as shown

in Fig. 5 (a), and reaches 100% at the end of the

movement when movements are unambiguously dif-

ferentiated by their position. Phase recognition is also

performed for each time step using Eq. 10. The phase

estimate is obtained sample-by-sample and need not

ICINCO 2019 - 16th International Conference on Informatics in Control, Automation and Robotics

574

0 0.2 0.4 0.6 0.8 1

Horizontal Position [screen ratio]

0.2

0.4

0.6

0.8

Vertical Position [screen ratio]

mov 1

mov 2

mov 3

mov 4

target

Figure 3: Reaching data set. A two-dimensional reaching

task is performed by a test participant on a computer screen

using a mouse. The participant was asked to perform the

movement as fast as possible to reach circular targets that

were appearing in a position randomly decided among 4.

The resulting set is composed by 25 repetitions of 4 distinct

targets.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Time [s]

0.5

position [screen ratio]

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Time [s]

0.5

Phase

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Phase

0.5

position [screen ratio]

Average Horizontal Trajectory Average Vertical Trajectory Samples

Figure 4: Dataset ”mov 1” (see Fig. 3). (a) shows vertical

and horizontal coordinates over time, gray lines represent

training data, black line the average. In (b) the correspond-

ing phase proﬁles are shown, in gray the ones associated

with the observations, in black the average one. In (c) the

resulting movements over phase are shown.

be monotonic as shown in 5 (b).

Movement Generation. In order to generate con-

crete movements the parameters w are sampled from

the distribution. Fig. 6 shows movements generated

with and without constraints. The constraint is ex-

pressed as a via-point to be reached at a given phase.

The constrained movements are generated from the

conditioned distribution with parameters from Eq. 12,

Eq. 13. In Fig. 6(a) the imposed via-point is within

the space of the presented examples and hence trajec-

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Time [s]

100

Classification Error [%]

100

Sample Trajectories [%]

Error # Samples

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Time [s]

0.2

0.4

0.6

0.8

Phase

Average Repetitons

Figure 5: Movement and phase recognition accuracy eval-

uated on a test set of 100 samples (25 for each of the 4

recognized movements). In (a) the relative error is shown

as a function of time (blue). As the different test samples

have different durations. The number of available move-

ment samples compared at a given time is shown (orange).

The classiﬁcation error decreases from random (75%) as

more of the sample movement is observed. In (b) the es-

timated phases are shown (gray). Notice that the likelihood

of every phase value is computed at each point in time using

Eq. 10 and the most likely phase is selected. The resulting

phase signal differs from the ones assumed to generate the

training set (Eq. 5), and in general is not guaranteed to be

monotonic. Nevertheless, the average phase (black) recov-

ers the typical sigmoidal proﬁle.

tories resemble those presented in the training set. In

Fig. 6 (b) the via-point is more distant from the posi-

tions presented in the training set and hence the condi-

tioned trajectories are less representative of the orig-

inal distribution, e.g. forming loops and missing the

ﬁnal target. In (c) the conditioned distribution is used

to predict a continuation of the current motion. The

prediction is also affected by the uncertainty about the

current phase, which is estimated according to Eq. 10.

Perception. Perceptual estimation of phase was im-

plemented with a feedforward neural network with

three layers with 40, 20 and 10 neurons respectively,

taking as input the current time and a vector of 20

previous samples. The network has been trained

with the Levenberg-Marquardt backpropagation algo-

rithm(Levenberg, 1944; Marquardt, 1963). The re-

sults are shown in Fig. 7 where the classiﬁcation task

(shown in Fig. 7) is repeated by exploiting the es-

timated phase, according to Eq. 11. Classiﬁcation

performance using the neural network is comparable

to that obtained by integrating the phase distribution

(Eq. 9). In Fig. 6 (d) the phase estimated by the neural

Phase Distribution in Probabilistic Movement Primitives, Representing Time Variability for the Recognition and Reproduction of Human

Movements

575

0 0.2 0.4 0.6 0.8 1

Horizontal Position [screen ratio]

0.2

0.4

0.6

0.8

Vertical Position [screen ratio]

Movement from MU e sigma

Target

Average

Repetitions

Conditioned Average

Conditioned repetitions

Conditioned Waypoint

0 0.2 0.4 0.6 0.8 1

Horizontal Position [screen ratio]

0.2

0.4

0.6

0.8

Vertical Position [screen ratio]

0 0.2 0.4 0.6 0.8 1

Horizontal Position [screen ratio]

0.2

0.4

0.6

0.8

Vertical Position [screen ratio]

Prediction

Target

Observed Movement

Predicted

Position at Prediction Time

Conditioned repetitions

0 0.2 0.4 0.6 0.8 1

Horizontal Position [screen ratio]

0.2

0.4

0.6

0.8

Vertical Position [screen ratio]

Figure 6: Movements generated using the ProMP model for the observations set ”mov 1” (see Fig. 3). Trajectories in red

are samples from the distribution over w. Trajectories in blue are samples constrained to pass through a via-point at phase

∗

= 0.5, indicated by a black star. Thick lines indicate the average trajectories of the respective distributions. The red circle

marks the ﬁnal target. In (a) the via-point is within the space of the presented examples and hence the trajectories resemble

thoses presented in the training set. In (b) the via-point is an outlier to the positions presented in the training set and hence the

conditioned trajectories are not representative of the observed movements. In (c) the conditioned distribution is used together

with classiﬁcation and phase identiﬁcation to produce a prediction of the motion in progress (black dotted line). The most

likely trajectory is plotted with the blue thick line, and samples as thin blue lines. In (d) the prediction is performed using a

neural network to estimate the phase.

network is used to perform a prediction on the basis

of the observed trajectory. The result is similar to that

which is shown in Fig. 6 (c).

A 7-DoF Robot Arm Example. In additon to previ-

ous low-dimensional examples, model identiﬁcation

is applied to movements of a 7-DoF robotic arm.

The task consists in picking up an object using a

hook and handing it to an user in a single continuous

movement. The setup is shown in Fig. 8. The ob-

ject’s loop represents a spatial constraint for the tra-

jectory, since the object is always picked up in the

same position. Trajectory examples were recorded by

manually moving the robot, producing a total of 11

examples. The task was represented in seven dimen-

sional joint space. The number of features was set to

9, hence the ProMP distribution has 63 dimensions.

Fig. 9. shows the covariance matrices of the ProMP’s

distributions learned on data without and with opti-

mized phase proﬁles. The use of phase modulation

decreases variation in adjacent feature weights in the

covariance matrix elements as temporal noise is ex-

ternalized to the phase signal. A more compact de-

scription of the concept is shown in Fig. 10, where

the eigenvalues of the two covariance matrices, i.e.

the principal components, are compared. All the non-

zero eigenvalues are smaller with phase modulation.

4 DISCUSSION AND FUTURE

WORK

In this work we used ProMPs to model human move-

ments and demonstrations to robots in order to per-

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576

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Time [s]

100

Classification Error [%]

100

Sample Trajectories [%]

Error # Samples

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Time [s]

0.2

0.4

0.6

0.8

Phase

Average Repetitons

Figure 7: Classiﬁcation and phase evaluation accuracy us-

ing a neural network for the identiﬁcation of phase. The

training set and the test set are the same as used in Fig. 5.

In (a) the relative error is shown as a function of time (blue).

As the movements have different durations, the number of

available movement samples at a given time is indicated

in orange. The classiﬁcation accuracy increases as more

of the test movements are observed. In (b) the identiﬁed

phase proﬁles are shown in gray, where phase is computed

at each time step. The average phase (black) shows the typ-

ical sigmoidal proﬁle. Although in some cases the phase

appears less accurate using the neural network estimator,

overall classiﬁcation performance is comparable to that of

Fig. 5.

Figure 8: The robot task conﬁguration (A). The robot’s

gripper is holding a hook (B) which it uses to pick up the

object (C).

form movement recognition and movement genera-

tion, with particular emphasis on leveraging a para-

-0.3

-0.2

-0.1

0.1

0.2

-0.2

0.2

Figure 9: Color representation of the covariance matrix for

w, for the robot arm movements set without phase modula-

tion (left) and with phase scaling (right). Most of the values

in the covariance matrix are smaller.

2 4 6 8 10 12

Eigenvalue #

0.5

1.5

2.5

3.5

Var

With Phase scaling

Without Phase scaling

Figure 10: Non-zero eigenvalues of the covariance matrices

shown in Fig. 9. With the phase modulation all eigenvalues

are smaller, which indicates a more precise reproduction of

movements.

metric phase proﬁle to modulate movement execution

speed. The approach showed to properly classify and

generalize the movements in a data-set of reaching

movements. The proposed parametric phase proﬁle

has a sigmoidal shape that is particularly suitable to

reaching tasks such as the one presented into the ex-

ample and other ”real world” tasks, such as pick and

place operations where it is relevant to the control of

initial and ﬁnal position. However, a sigmoidal phase

proﬁle enforces zero velocity at the beginning and at

the end of the movement. Therefore, in tasks where

ﬁnal velocity must be controlled, other proﬁles (e.g.

a beta function with a = 2, b = 1) may be favorable.

In general it should be considered that time scaling of

human movements can work differently depending on

the task and the working conditions(Zhang and Chaf-

ﬁn, 1999). The example reaching task is represented

as a library of 4 ProMPs describing movements to 4

possible targets. In general, the diversity of move-

ments in a task can be represented with a manifold in

the space of hyper-parameters and mixture distribu-

tions (Rueckert et al., 2015), or speciﬁcally with gaus-

sian mixtures describing the distribution of w (Ewer-

ton et al., 2015). The parameters can be constrained

with linear relationships for dimensionality reduction,

as recently shown in (Colom

e and Torras, 2018). With

human movements, dependencies between target po-

sitions and the moving parameters can be represented

by a linear model(Avizzano and Lippi, 2011; Lippi

et al., 2012). In all those cases phase can be estimated

online while observing the movement by modifying

Phase Distribution in Probabilistic Movement Primitives, Representing Time Variability for the Recognition and Reproduction of Human

Movements

577

Eq. 10 to take into account the chosen distribution.

Further, a speciﬁc perceptual estimator dedicated to

the identiﬁcation of phase using a feed-forward neu-

ral network was described and implemented. The es-

timator led to a performance in terms of classiﬁcation

and prediction accuracy that was comparable to that

of the system exploiting the empirical probability dis-

tribution of the phase directly. An advantage of us-

ing a neural network is, that it avoids the potentially

costly computation of the integral in Eq.9 at each inte-

gration step. Future work on phase recognition will be

focus on the identiﬁcation of state of dynamic system

models tasked with generating phases, e.g. phase-

state machine models (Deimel, 2019). Such mod-

els could provide phase proﬁles and could be used

in conjunction with a library of ProMPs to control a

task. Furthermore, current research on ProMPs in-

cludes the design of feedback control systems consid-

ering physical interaction with the environment and

the users (Paraschos et al., 2018; Paraschos et al.,

2013). Time scaling does not apply to the control of

arm kinematics and the external contact forces in the

same way as for movements (this holds for each kind

of rescaling, also the linear one). A general frame-

work that can scale kinematics and contact instants in

time should be integrated with a consistent control of

applied forces and torques.

ACKNOWLEDGEMENTS

We gratefully acknowledge ﬁnancial support for the

project MTI-engAge (16SV7109) by BMBF.

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