Learning collaborative manipulation tasks by ... - LASA - EPFL

Velocity target ˆẋVelocitytarget at t=0Position target ˆxInitial positionfor reproductionẋ20x2Initial velocityfor reproduction0ẋ 1Influence of ẍ VPosition target at t=0x 1Influence of ẍ Phkx2x2x 1Final reproductionx 1x2Fig. 2. Schematic of the Gaussian Mixture Regression (GMR) process.Top: By considering a single Gaussian distribution. Bottom: By consideringa GMM composed of two Gaussian distributions.distribution of each state is represented by a Gaussiandistribution representing locally the correlations between thedifferent variables. The parameters of the model {Π,a,µ,Σ}are learned through Baum-Welch algorithm, a variant ofExpectation-Maximization (EM) for HMM [16]. Π i is theinitial probability of being in state i, a ij is the probability totransit from state i to state j. µ i and Σ i represent the centerand covariance matrix of the i-th Gaussian distribution ofthe HMM with K states. The different variables of thedataset and associated model are labelled separately as[[µIiµ O iµ O′iµ I′i]=]=[ξIξ O ]=[ µxiµ F iµẋi[ µxiµ F iµẋi[ xFẋ],[] [, ΣIi ΣIO i] [,Σ OIiΣ O′iΣ O iΣ OI′iΣ IO′i Σ I′i]ξ O′=ξ I′]=]=[ xFẋ[Σxi Σ xFi],Σ xẋiΣ FxiΣ F iΣ FẋiΣẋx iΣẋF i[Σxi Σ xFiΣ FxΣẋiΣ xẋiiΣ F iΣ FẋiΣẋx iΣẋF iwhere the uppercase indices x, F and ẋ refer respectivelyto position, force and velocity components. 33 Note that this process can similarly be used to encode trajectories definedby position and velocity recordings (i.e. without force), where we simplyhave ξ I = x and ξ O = ẋ.Σẋi],],Fig. 3. Example of a dynamical system used to reproduce a demonstratedmotion by starting from a different initial position. A 2D motion isconsidered here as an illustrative example. The first row shows the HMM invelocity and position space encoding the {ẋ, x} relationships. In the secondand third row, the initial positions are represented by points and the retrievedtrajectories are represented in bold line.x2x20.150.10.050−0.05Low variability acrossthe 10 demonstrations−0.1−0.15 −0.1 −0.05 0 0.05 0.1x 10.150.10.050Initial positions−0.05for reproductionReproductions forlow variability dataReproductions forhigh variability data−0.1−0.15 −0.1 −0.05 0 0.05 0.1x 1x 1x2κ P0.150.10.050−0.05High variability acrossthe 10 demonstrations−0.1−0.15 −0.1 −0.05 0 0.05 0.1x 10.080.060.040.020GMM encoding datawith low variability.GMM encoding datawith high variability.0 2 4 6 8Fig. 4. Influence of the variability observed during demonstrations for thereproduction of the skill. To illustrate the influence of consistency acrossthe different observations, two datasets have first been generated from onereference 2-dimensional trajectory (in dashed line), where an HMM of 5states is trained on each dataset. The first dataset consists of 10 trajectorieswith strong consistency among the different demonstrations (first graph).The second dataset presents more variability (second graph). For each ofthe two models, two reproductions are then computed by starting from newinitial positions (third graph). The two trajectories in black line are retrievedwith the HMM represented in first graph, while the two trajectories in greyline are retrieved by the HMM represented in second graph.t