Task Generalization with Stability Guarantees via Elastic Dynamical System Motion Policies
Authors: Tianyu Li
Nadia Figueroa
Venue: CoRL 2023 Poster (2023)
Keywords: Stable Dynamical Systems
Reactive Motion Policies
Learning from Demonstrations
Task Parametrization
Task Generalization

Abstract: Dynamical System (DS) based Learning from Demonstration (LfD) allows learning of reactive motion policies with stability and convergence guarantees from a few trajectories. Yet, current DS learning techniques lack the flexibility to generalize to new task instances as they overlook explicit task parameters that inherently change the underlying demonstrated trajectories. In this work, we propose Elastic-DS, a novel DS learning and generalization approach that embeds task parameters into the Gaussian Mixture Model (GMM) based Linear Parameter Varying (LPV) DS formulation. Central to our approach is the Elastic-GMM, a GMM constrained to SE(3) task-relevant frames. Given a new task instance/context, the Elastic-GMM is transformed with Laplacian Editing and used to re-estimate the LPV-DS policy. Elastic-DS is compositional in nature and can be used to construct flexible multi-step tasks. We showcase its strength on a myriad of simulated and real-robot experiments while preserving desirable control-theoretic guarantees.

Task Generalization with Stability Guarantees via Elastic Dynamical System Motion Policies
Tianyu Li University of Pennsylvania tianyuli@seas upenn edu
Nadia Figueroa University of Pennsylvania nadiafig@seas upenn edu
Abstract: Dynamical System (DS) based Learning from Demonstration (LID) allows learning of reactive motion policies with stability and convergence guar- antees from few trajectories. Yet. current DS learning techniques lack the flexibility to generalize to new task instances as they overlook explicit task pa- rameters that inherently change the underlying demonstrated trajectories. In this work, we propose Elastic-DS, a novel DS learning and generalization approach that embeds task parameters into the Gaussian Mixture Model (GMM) based Lin- ear Parameter Varying (LPV) DS formulation. Central our approach the Elastic-GMM, GMM constrained SE(3) task-relevant frames. Given new task instance/context, the Elastic-GMM is transformed with Laplacian Editing and used to re-estimate the LPV-DS policy. Elastic-DS is compositional in nature and can be used to construct flexible multi-step tasks. We showcase its strength on a myriad of simulated and real-robot experiments while preserving desirable control-theoretic guarantees. Project website: https://sites google. com/ vies/elastic-ds.
Keywords: Stable Dynamical Systems, Reactive Motion Policies, Learning from Demonstrations, Task Parametrization Task Generalization
Introduction
With advanced development in robotics and autonomous systems in the past decades, the oppor- tunities and demands for more complex physical human-robot interaction (pHRI) in our everyday unconstrained environments are rising; thus, is critical for robots to be adaptive, compliant, re- active, safe and easy to program [1, 2, 3]. In many cases, robots will need to acquire new skills satisfy task requirements in an ever-changing environment. program robots for complex motion tasks and even tedious for experts reprogram them when task requirements change. A straightforward and intuitive approach for robots develop new skills is through Learning from Demonstration (LfD) [4, 5, 6, 7, 8). This paradigm allows robots to acquire skills, typically encoded or defined in literature as action policies, motion policies, or imita- tion policies, directly from motion examples provided by humans or even other robots, mirroring a teacher- teacher-student relationship.
usually difficult for non-experts
In recent years, significant progress has been made in using Lff to learn complex and diverse mo- tion tasks. However, many focused learning and executing tasks from static unchanged scenar- iss/enviroomment/coneextt which could lead to failures when faced with out-of-distributio cases. From the machine learning perspective, this is the covariate shift issue that exists in many super- vised learning related tasks, especially in the behavior cloning (BC) approach [7, 9]. By providing a fixed training dataset beforehand, the LfD algorithm will learn a policy that performs well for training dataset but could fail generalize to unseen input during deployment. The learned policy will become invalid due to the change of distribution. Hence, instead of memorizing human demon- strations for scenario, the robot should be able adapt and generalize novel scenarios with satisfactory performance, given the same task objective.
7th Conference Robot Learning (CoRL. 2023), Atlanta, USA.

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Drio-inn
Bodiina Soliaser-Crr Corigg
New oreet Carle
34 to 20
Executing the learned DS (h) Trajectory for the Generalize the learned (d) Trajectory for the original learned policy policy to new configuration new policy
Task generalization for bookshelf stacking with Elastic-DS. (a-b) Given single Demon- stration, the Elastic-DS can efficiently learn and reproduce and (c-d) efficiently adapt to position and orientation changes in task parameters. Our Elastic-DD DS approach generalizes seamlessly and effi- ciently to new task parameter configurations while retaining stability and convergence guarantees.
Figure 1:
The Trilemma- Generalization or adaptation abilities are particularly important to enable robots to perform effectively in dynamic environments. There are many attempts at generalization with meth- ods like BC [10]. Inverse RL (IRL) [11, 12, 13], Meta-Learning [14]. Multi-Task Learning [15], Transfer Learning [16], Multi-Task Reinforcement Learning [17], Lifelong Learning [18, 19, 20), and continual learning [21]. However, the aforementioned approaches have no emphasis on provid- guarantees on the learned policies, such stability, boundedness, and con- vergence, all of which are critical for safe pHRI. On the other hand, the Dynamical System-based (DS) motion policy approach [3] offers many advantages such as reactivity, motion- adaptation, and, most importantly, stability guarantees; and can be learned from only handful of demonstrations ensuring minimal human effort [22, 23]. However, due to the closed-form and offline learning na- ture of bS-based motion policies, they have no flexibility for generalizing novel environments as they ignore explicit task parameters that inherently change the underlying trajectories that shape the DS vector fields. This limits their generalization capability and adoption low-level policies. Hence, invoking the no free lunch theorem, we posit that the state-of-the-att currently suffers from the generalization stability vs. effort trilemma.
ing
Goal In this work, we seek to alleviate this trilemma by proposing an L.fD approach that has i) sta- bility guarantees, ii) the flexibility to generalize motions across novel scenarios, while iii) requiring minimal human effort during learning and adappattonngennraaizzaiinn as depicted in Figure 1.
Related Work The number of techniques that exist for LID/IL is vast [6, 4, 7, 8). This work fol- lows the BC approach, which learns a policy that maps states (state-action pairs, trajectories, and other contexts are also used) to control inputs [24]. DAgger [25] addressed distribution shift issues with online interactions/comrcctionss Whereas the generative adversarial learning framework [26] randomly explores for corrections that bring the policy close the demonstrated distribution. These approaches offer generalization in terms of distribution shift but require either constant human ef- fort lots of data and computation and hold no -htrool-teoorctic guarantees on the learned policy. The recently introduced TaSIL (Taylor Series IL. framework introduces a simple augmentation to the BC loss such that the trained policy is robust to distribution deviations by ensuring incremental input-to-state stability, also benefiting from reduced sample-complexin [27]. Nevertheless, it cannot generalize to novel task instances/enviromments not seen during training. A Probably Approximately Correct (PAC)-Bayer IL framework was introduced [10] that computes upper bounds on the ex- pected cost of policies in novel environments. Prior works also focus on explicit skill generalization for novel environments or task instances, such multitasking learning [28, 29, 30, 31] and meta- learning [14, 32]. While capable of generalizing learned tasks to different environments, these works require a considerable amount of offline/online training and require excessively large DNNs, which cannot be used in a reactive manner nor offer any form of control-theoretic guarantees.
A significant body of work tackles the generalization problem by emphasizing task parametriza- tion (TP) relevant task frames in SE(3) assigned to relevant objects in demonstrations, like TP- GMM [33, 34), TP-DMP [35], task invariants [36, 37] and environmental constraints [38]. Other

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works focus on the motion policy parameter perspective. such as adapting explicit start and goal po- sitions in movement primitíves [39], conditioning on probability distributions like the Probabilistic Movement Primitives (ProMPs) for different via-points [40] and geometric descriptor [41]. While such works have demonstrated the ability for task generalization, few provide real-time reactive mo- tion and stability guarantees, and most of them rely on the availability of demonstrations in different contexts environments to extract the relevant task parameters for generalization.
Approach We propose a zero-shot approach for generalizing motion policies to novel scenarios while guaranteeing control-theoretic properties for safe deployment in pHRI. To achieve stability, we adopt the DS-based LID paradigm [3] that learns motion policies as time-invariant nonlinear DS with Lyapunov stability guarantees. To achieve generalization, we follow the task- paraaaaattrrzntoonnn perspective [33, 34, 35] and propose to embed relevant task frames directly into the DS policy. While several neural network (NN) based formulations for stable DS motion policies exist, such as neurally imprinted vector fields [42], DNN via contrastive learning [43], diffusion models [44], and euclideanizing flows [45]; given their black-box nature it is not straightforward to embed such task parameters into these formulations. Further, NN approaches need multiple demonstrations to encode the stable DS properly. To achieve minimal data, compute and human effort during learning, we adopt the Gaussian Mixture Model (GMM) based Linear Parameter Varying (LPV-DS) formulation [22, 3] which has shown to be computationally efficient and capable of learning stable vector fields of complex motions from single demonstration [23]. Finally, we take inspiration from elastic bands [46] and trajectory editing [47] and propose novel approach to generalize the GMM-based LPV-DS to novel scenarios without new demonstrations, referred to as Elastic-DS.
Contributions We introduce the Elastic-DS formulation as a solution to the LID trilemma (See Fig- ure 2). The Elastic-DS is constrained set of task parameters described as geometric descriptors representing the invariant features of task (e.g., object, via-point, target configurations). It is ca- pable of efficiently generating novel DS policies upon task parameter changes without requiring new data or human input for single, multi-step tasks and the composition of new tasks via DS stitching.
2 Problem Statement
Let D == ffEr. I 1 be set of N demonstration trajectories collected from kinesthetic teaching for a task, where & E Rd, Er.m E IRd represent the kinematic robot state and velocity vectors at time t, respectively, for the n-th trajectory with length TN, In this work, we consider & E Rd to be the end-effector Cartesian position. Let g f({) be a first-order DS that describes a motion policy in Th" state space. Given D, the goal is infer f(()) I IR" such that any point in the state space leads to stable attractor E*, with f(E) described by set of parameters θ and g*
É f(E (,E) + lim ||C --1l =.
(I)
Usually, such DS motion policies are learned in an offline manner and fixed for the execution phase [23]. The main contribution of this work is to introduce an update stage that parametrizes the system further, allowing the motion policies to adapt and generalize to new tasks, as shown in Figure
Task Parameters A task is defined as a combination of multiple trajectories where each of them is grounded on geometric constraint descriptor set O {Oenter; Gezit] E S3) describing the two endpoint poses. Let B, E (X++++ be generalization parameters in the state space conditioned on the geometric descriptors O,, As O changes, 3, will change accordingly to generate a new set of DS to reach O2 with the correct poses. In this work, the geometric descriptors O is assumed to be known or given during demonstrations. However, it could come from various upstream sources such as human specifications [23] or generative segmentation algorithms [48].
Motion Policy We propose the following motion policy for task paramearaeteriiee generalization:
E Σ (.,)),((E,,,,
where 5(g, O,) is activation function determining the sequence of execution for M DSs describing multi-step sequential tasks. Hence, given D with the same behavior and M geometric descriptor
3

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Train
Update
Execution
Demrmshraiios
LPV-88
Passive 05 Cuntolllr
Recoredras Velxatty
Elstic-OMM
(a) Original
(b) Adapt
Av Rip P
PC GMM
Fla f
Transtor Gauss
Caussian
Desorptors
Adapt
(d) Adapt
Elastic-DS Learning and Control Flowchart
Figure 2: (a) Elastic-DS reproducing stable LPV-DS vector field on the originally demonstrated data. (b-d) Elastic-DS generating stable LPV-DS motion policies from peg attractor changes (trans- lation and rotation) without new demonstration. (e) Flowchart of Elastic-DSS approach.
O, configurations, approach finds 33 that will generate DS motion policies with i) stability guarantees with respect to their corresponding attractors E7 and ii) the flexibility to achieve the same task with new geometric constraint descriptor configurations.
3 Preliminaries: PC-GMM and LPV-DS Motion Policy [22]
The GMM-based LPV-DS [22] motion policy has the following formulation.
{ P = 2+, 20, (Q+)) - (3)
=() (E) (A +
s.t.
6² A
where (E) is the state-dependent mixing function that quantifies the weight of each linear time- invariant (LTI) system (A&E + b"). (N(((02) denotes the probability of observation from the k-th Gaussian component parametrized by 0 = {eu Σx}, }, and represents the prior probability of an observation from this particular component and the a posteriori probability
(E(00) (E) Σ5= (E10,)
from (EH 0+))
Σ TWW(EE Ex).
Intuitively, Eq. fits mixture of linear DS to complex non-linear trajectory, with 7x(E)) ensuring the smoothness of the reproduced trajectories. Hence, each Gaussian component must be placed on quasi-linear segments of D. With PC-GMM, the optimal number of linear DS 4 for given D can be automatically inferred.
Stability To guarantee global asymptotic stability of Eq. 3, a Lyapunov function V(E)
<*)T P(g (*) with P pT 0, is used derive the stability constraints in Eq. 3. Minimizing the fitting error of Eq. 3 with respect to demonstrations subject to constraints in Eq. 3 yields
non-linear DS with a stability guarantee via a Semi-Defing Program (SDP) [22]. Implementation details are provided in Appendix A.
ELastic-DS
4.1 Elastic-GMM
To adapt generalization parameters 3, for the corresponding spatial change in the geometric de- scriptors O,, we introduce Elastic-GMM as the core component of augmenting LPV-DS (Eq. 3) into Elastic-DS (Eq. 2). Figure 2e shows the pipeline Elastic-DS, During the training stage, we use PC-GMM [22] obtain set of initial Gaussian parameters (m,, 00) as well as the initial pg The update stage will produce the updated Gaussian parameters { 0x) as well the updated 3,, which are key way-points in the state space. A trajectory will be generated based on the key points to specify the velocity for the DS after the update. With the new Gaussian parameters and

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the new velocity information, an updated LPV-DS can be learned. If further updates happen in the environment, we can directly update LPV-DS using the transform without re-estimating the GMM, which avoids a time-consuming stage. During the execution, a passive-DS controller [3] will take newest LPV-DS output velocity & to generate the corresponding joint torque т for the robots. The upcoming section will discuss each key component in detail. The algorithm Appendix D.
(a) GMM from trajectories
(b) GMM Chain
(c) Gaussian joints illustration 2D
Figure 3: Illustrations of PC-GMM fitted trajectory data (a) and the Gaussian chain transform from Section 4.1.1, (c) a closer look at the Gaussian joints introduced in Section 4.1.3.
4.1.1 GMM Chain
Following the LPV-DS pipeline, the demonstration trajectory is encoded into GMM { 0%) using PC-GMM [22]. As shown in Figure 3a, a trajectory extracted and simplified into chain of Gaussians links. In the update stage (Figure 2e), transform the spatial relationships among the Gaussians to achieve task adaptation. Imagine analogy of the Gaussian chain being robot arm, which could be rotated around each joint to achieve a specific geometric configuration as shown in Figure 3b. Note the robot arm analogy is on the end-effector trajectory instead of the actual robot arm. The generalization parameters 3; are the joints between each pair of the neighboring Gaussians, which describes the spatial relationship between the neighbors. After the PC-GMM step, we can obtain the initial 3 and later update the 8 & achieve transform. The joint between two Gaussians, which is the 8g, is the mean of the product between them as described by the picture the left in Figure 3c, which could be obtained by,
Σ, = +
122 (22
(5)
where y and E are the mean and covariance of the Gaussians { 0 and { 09). To complete the robot analogy, also need determine the links position and orientation TEMM E SE(3)K with respect to the joints. Figure 3e depicts the Gaussian mean position Hí and orientation (described by the eigenvectors éÿ of the covariance matrix Σj) with respect to the frame at the last joint with the x-axis pointing towards the next joint (in the direction of the demonstration). All of the above are constructed as the initial condition, which no update is involved yet. Later when the states of the generalization parameter 33 change, we will recover the same transformation of the mean and covariance with respect to the corresponding 3.. Before introducing the approach obtain the new joint positions 3; we provide brief summary of the Laplacian editing approach [49, 47].
4.1.2 Laplacian Editing Primer
Laplacian Editing allows directly modifying an existing trajectory defined by m waypoints r E Rdxm while capturing local properties. First, we convert the waypoints r in cartesian space into Laplacian coordinates A with the graph Laplacian matrix mmx [47],
if if
Wij
Lij
Σ
EjEN, Wi
(6)
otherwise.
where N; are set of neighbor points ry for waypoint r;, and Way is weight set for this work. One can obtain A Lr. where A is a concatenation of the Laplacian coordinate for each waypoint 5, EjEN. ΣjeN, myy (r: rý). The matrix I. can be singular, the system Lr = A when solving for new waypoints to achieve editing [47].
can impose constraints
5

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4.1.3 Transform Gaussians with Constraints
The initial joints o are converted into the Laplacian coordinate as in Section 4.1.2 and used to construct a least-square objective Eq. Then, we align the first link (formed by 80 and (3) and the last link (formed by fn-1 and B,)) with the geometric descriptor O, which forms the constraints for the least-square formulation. When solving for this optimization, the other Br will adjust based on the Laplacian objective, softly preserving local position properties,
1o Br,1) Ostart T, (3 Bs.n) Oend
min J (3.) = 1|L3; - A subject to
where L E _ and A are the same in Section 4.1.2. To, (3.. 31) represents the frame trans- formation from Bi,0 to 8r,1- The solution of this optimization will produce new joints positions Bi. After that, the link position and orientation (which are the Gaussians' means and covariances), as well as the scale, are recovered using the TGMM recorded from the previous section. The orientation of the Gaussian determined by the eigenvector shown in red and green in Figure 3c. Each orien- tation will remain fixed with respect the last joint frame (312 Figure 3c). If rotation happens to the last joint frame, the Gaussian frame associated with the last joint will be moved together in the global frame but remain the same in the last joint frame. The scale is determined by the change from the original distance between each pair of neighboring joints, which scales the eigenvalues of the Gaussians' covariances. Referring back to the flowchart in Figure 2e, this section outputs the updated joint positions 8; and updated GMM parameters 0₂ (T stay unchanged).
Figure 4: After obtaining the joints between neighboring Gaussians, piecewise linear trajectory formed. The green polygon is the geometric descriptor at the exit of the trajectory. We set a constraint the last segment to align with the pose of the geometric descriptor. After solving (7), we achieve the final transformation (depicted in the rightmost image). Note: This illustration only shows the constraint the exit, while in general, the constraints could be the entry and the exit.
4.2 Create Velocity Profile
Depending on the new task constraints, the velocity requirement could be different from the original demonstration. Therefore, this approach offers the opportunity modify the velocity by regenerat- ing trajectory along the Gaussians joints the waypoints. There are many ways to achieve this with known waypoints, such as splines minimum trajectory. We provide simple example of using Laplacian Editing to generate new trajectory. First, we connect 8., and 8t.x to form linear trajectory E Rdxx, p is the number of points on this trajectory. We then force this trajectory to pass through 3, the Gaussian joints, with Laplacian editing.
min (C) [T - Acll2 subject to
C = B
where j E (O....,P- 1} is the index in Ç for matching the corresponding 3.. More details can be found in Appendix B. The velocity will be determined by the finite difference between the edited trajectory neighboring data points divided by the dt collected from the demonstrations. After this section, the velocity information and the updated GMM will become the input for learning new DS motion policy following Section 3, which by nature preserves the stability guarantees.
4.3 Multiple Segments
Multiple Elastic-DS could be stitched together achieve via-point trajectory and even long- horizon multi-segment tasks. The index in the task parameters & represents multiple segments,

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consequently multiple DSs in (2) when M > 1. To allow adding spatial constraints along the trajectory, one can split the task into multiple segments and process them divide-and-conquer
manner. There are two possible task-specific cases for stitching the segments: (i) The activation function ó(S, Oi) is in charge the switch for multiple DSs. The next DS will activated by when the last DS reaches the attractor. (ii) To create smooth movement, this case will first connect all the Elastic-GMMs from different segments and then learn single DS. The function is not in use for this case. As mentioned in Section 2, the interesting separation points described by the geo- metric descriptors are specified by some upstream sources. For more information about the split and stitching process, please refer to Appendix C. The flexibility of composing and regrouping different transformed DS with the new constraint poses allows the possibility for multi-task scenarios.
Experimental Results
5.1 2D Experiments
This section shows 2D examples of using Elastic-DS. The learned DS is plotted as steamplots de- scribing a velocity vector field (in blue) with GMM (in orange) geometric descriptors (in green) and rollout trajectory (in black) overlaying on top. The 2D simulation in Figure shows the atomic case of one segment trajectory conditioned on a geometric descriptor with constraints at the endpoints. The two ends of the geometric descriptor (green polygons) are shifted and rotated to show different configurations and the changes in the DS vector field. Figure 6 shows the example of using via- point to modify the policy in the middle of single DS corresponding case (ii) in section 4.3. The DS motion policy is able to adapt to the changes. For more details about stitching the trajectory, please refer to Appendix C. Figure 7 and Table display a comparison to TP-GMM-DS [50, 51], TP-GPR-DS [52], and TP-proMP [52, 40J. Appendix shows the failure cases TP-GMM with fewer demonstrations. With single demonstration, Elastic-DS is able to generalize well, while other methods fail. For more comparison details, please refer to Appendix F.
(a) Demonstration
(b) Reverse
(c) Shape
(d) Checkmark
Figure 5: Elastic-DS motion policy generalizes based on changes of start and goal poses
(a) Demonstration (b) Set via-point
(c) Shifted
(d) Rotated
Figure 6: Elastic-DS motion policy generalizes based changes of the via-point
demonstration new situation
demonstration new stuation
demonstration new situstion
demonstration sttoonttttion
new situation
ELastic-DS (Ours)
TP-GPR-DS
TP-GMM-DS
TP-proMP
Figure 7: Elastic-DS is able to handle the situation while the other methods fail with single demonstration. Appendix E shows that TP approaches require more demonstrations to succeed.
7

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Metric
ELastic-DS (Ours)
TP-GPR-DS
TP-GMM-DS
TP-ProMP
Start Cosine Similarity t
0.9843
-0.3981
0.9405
0.4777
Goal Cosine Similarity T
0.9998
0.5453
0.6324
0.8965
Endpoints Distance
0.0008
0.8360
0.07642
0.1788
The metrics include the orientation alignment of the start and goal as well as the position distance with the geometric descriptors in the new instance. The resulting data corresponds to the orange trajectories in Figure 7. With both ends moving, Elastic-DS remains in good performance on all three metrics. More details about the metrics are in Appendix F.
Table I:
5.2 Robot Experiments
We demonstrate the validation result through four different real robot experiments on the Franka Emika Panda robot: Bookshelf, Pick and Place, Tunnel, and Combination, see Figure 2, and Appendix G. In these experiments, the geometric descriptors are detected from a motion capture system. By attaching motion capture markers objects, specify geometric descriptors anchored on the objects of interest. Three experiments will start with a human performing kinesthetic teaching by moving the end-effector. Then, the execution of the original learned DS from the demonstration will be shown. The objects interest will then be shifted and rotated with different configurations. The last experiment shows the ability to compose new tasks, Without any new demonstration, the robot can still achieve the required tasks. Please refer to the video and Appendix G.
Comeratadon
(a) Exccuting the learned DS (b) Original trajectory
(c) Generalize to a rotation
(d) Adapted trajectory
Figure 8: Example of Elastic-DS adaptation in tunnel passing task
6 Conclusions and Limitations
We propose ELastic-DS in this work, which allows modifying DS with task parameters conditioned on geometric features to achieve task generalization. As the core component, we introduced Elastic- GMM to augment the original LPV-DS to create more flexible task-specific motions with as low as one demonstration. By showing both 2D simulation and different 31) robot experiments, we validate the ability of Elastic-DS perform task generalization as well as the potential for multi-task and hong-horizon motion policies. Following discussed the limitations of our approach.
Limitations First, this work only considers end-effector motions in the Cartesian position space. The task constraints are only for translational motion as well. To achieve more variety of tasks and extend to more possible poses, the orientation space has to be considered. Further directions could adopt works like [53] and [54] produce DS motion policies and meet task constraints in the full pose space. To go even further, the full pose could also contain the gripper state, which could be achieved with coupled DS approach such as [55] and [3]. Motion policy in the joint space should also be considered [56]. Second, the geometric descriptors are assumed to be given by human specification in this work. To address this limitation, we could utilize rigid object tracking methods like BundleTrack [57] to identify the geometric interactions between the robot demonstration and objects. Finally. the task adaptation this work is fast yet not not real-time time (*s 1s for 2D and 3D data on a typical laptop with Intel 17-12700H and 16GB memory, depending on the complexity of the task). Computation time analysis is provided in Appendix H. With the dynamic nature of physical human- robot interaction, it is important to provide continuous adaptation the fly to create seamless experience. Hence, our immediate next step is to accelerate adaptation time to ms scale.

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12

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Appendix
A LPV-DS Parameter Optimization
GMM-based LPV-DS formulation It described in Section 3 Preliminaries: PC-GMM and LPV-
DS Motion Policy.
s.t. (A4) T ( =Q = (Q+)) <0 6t = -A
{ = f() = Ex (E) (A*E + b*)
(9)
DS Estimation The set of DS parameters AD (* for f(E) is estimated with LPV-DS by minimizing the Mean Square Error (MSE) against the demonstrations [22] subject stability constraints in Equation 9.
Nect TN
min J (0ps) Bps
Aref St,
f (C1.1) ||²
(10)
(A*) P+ Q+, = (Qt)² 00 6 AA =k= L...,..
s. t.
which is constrained non-convex semi-defmite program (SDP). Further, when P is known (or estimated beforehand as in [22]) the problem becomes a convex SDP that can be solved highly efficiently with off-the-shelf QP solvers [22, 3].
13

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B Creating Velocity Profile with Laplacian Editing
Figure 9: Connect endpoints and set constraints along the trajectory progress at the Gaussian joints
The main goal is to force linear trajectory E dRdxp to pass through 3, the Gaussian joints, with Laplacian editing, min J (C) = 1| - A 21l subject to
G Bi.g
(II)
We first calculate the total Euclidean distance along the piecewise connected line between neigh- boring joints with total distance D Σ || B:ll. Then we have the percentage Ay of the progress that the joints positions in 3g make along the total distance D. By using Ag. the correspond- ing 33 is mapped to the index (p 1)J. In practice, by also enforcing constraints between the joints with linearly interpolation will help the trajectory become more aligned with the GMM link and the geometric descriptors. The velocity will be determined by the finite difference between the edited trajectory neighboring data points divided by the dt collected from the demonstrations. One can specify the velocity by controlling the spacing between the edited trajectory neighboring data points and the number of points p.
14

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C Stitching from Multiple Segments
As mentioned in the main paper, there are two design trade-off approaches for stitching multiple segments. Depending on the task, one can choose either to learn a DS for each Flastic-GMM (Sequential DS) or learn a single DS for the stitched Elastic-GMM (Combined DS). Figure 10a shows the flow describing the former case, and Figure 10b shows the flow for the latter case.
Derronathatiors
Demonsstoooon
M Geameenc Deseriptons
GMM
ElasteGGMM
GMM
IPWOS
LPVIDS
LPVOS
LIVeSS
Sequence
(a) This flow ensures that the position geometric (b) This flow produces smooth motion with a sin- constraints will be reached for the task with mul- gle DS. One adapt [23] achieve task satis- tiple DS
faction.
Figure 10: Different task split flows for design trade-off.
Here is an example of the two cases in which they are used to modify a DS with via-point. By specifying an interesting point in the demonstration (usually by human specification or upstream computer vision method), the trajectory will be split into separate components. Each individual segment will be processed with Elastic-GMM to meet the new interesting via-point geometric con- straints as depicted in green polygons. By performing such steps, can pose constraints not just on the endpoints of the demonstration but also on the intermediate points of the demonstration. This via-point experiment shows that with changes in the via-point constraints, Elastic-DS can adapt to them.
C.1 Sequential Elastic-DS
Figure 11: An example of the pipeline for sequential DS. In this case, the single demonstration (in red) is separated at the blue spot. Both segments perform Elastic GMM to meet the geometric descriptor constraints and learn DS individually. The end results are two DS which will run quentially, After the first DS reaches its attractor, the robot will start to follow the second DS. This corresponds to the flowchart in Figure 10a
15

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(a) I= first DS
(b) 1= DS
(c) t= in second
(d) 1=2 in second
Figure 12: Modified DS based the via point multi-segment DS without demonstration. The plots here show the rollout trajectory the switch DS. It uses the same GMM as in Figure 14c.
C.2 Combined ELastic-DS
Figure 13: An example of the pipeline for building single DS. In this case, the single demonstration (in red) is separated the blue spot. After obtaining the GMM for each segment, they are stitched together and generate single DS; such an approach will generate smooth velocity but could result trajectories missing geometric constraints with perturbation. One could adapt LTL-DS [23] to alleviate this and achieve task satisfaction. This corresponds to the flowchart in Figure 10b
(a) The original DS Figure 14: Modified DS based on the via point as a single DS without new demonstrations
(b) Original with a via- (c) Shift the via-point point constraint
(d) Shift and rotate the via-point
16

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D Transforming Elastic-GMM
Algorithm 1: Transform Elastic-GMM for Generalization
Input: (1k, )=
(14: Eg 1= St=1, Çt=T, Ostart, Oend
Output:
kk]==1
f-
k - 1;
while K do
In (Σ- + +22 1-1)-1; (Eq (5) in the main text]
An = En E W1 bn p2); {Eq (5) the main text)
λ; eigl Exg(
Mn-1 create a frame at 8n-1 using 3n - Bn-1 as the x-axis;
(= creaee frame with é,, p4g
I'k---1 =th the transformation from Mn-1 to C;
=
1;
end
31 = S1, AN Er:
Construct L via Eq (6) in the main text;
A LB;
To. Create a frame with 3p and B1 at 39:
Tn In Create a frame with Bn-1 and B, at Bn;
arg ming (B) = LB - Al subject to constraints in Eq (7) in the main text;
- 1;
while K do
Recover Mk: Gii with F W.r.t 31 Scale X Maz according the new distance between neighboring 3; Reconstruct Eg with X Cbe
1;
end +
Return (p 1 1K
17

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E Failure Cases for TP-GMM Task-Parameterized Learning
This section shows the performance mask-parametrized policy learning under a different number of demonstrations. Specifically, the method shown here uses lask-Parameterized Gaussian Mixture Model as the encoding strategy and Gaussian mixture regression as the trajectory reproduction [52, 51]. There two examples in total. Each example will start with four demonstrations (samples) in blue (but faded), moving from the bottom geometric descriptor (frame) to four various other geometric descriptors top. For the new situation, the two geometric descriptors will be placed at new positions (in deep green). The orange trajectory shows the reproduction in the new situation, with the orange ellipses being the GMM encoding. The number of demonstrations will decrease in each example to show the generalization ability to the new situation with less training data as well as the sensitivity to the coverage of the training data.
E.1
Case Example
atinss nus stuution
domersbatiios now sitsaion
demorstrttion
new situstion
demoratrations stustion
4 Samples
3 Samples
2 Samples
Samples
Figure 15: With four demonstrations and the frames being placed among the samples, the motion policy performs well. As the number of demonstrations decreases to three and two, the new motion policies still reach the goal with reasonable behaviors though the initial movements are in the opposite direction. It clearly does not perform well with a single demonstration.
E.2 Case Example 2
atteratratiooss near abuation
demeratrations near whastion
demorataatona new sitsaion
demorstrations new situaion
5
42
Samples
3 Samples
Samples
Samples
The new frames are placed further away from the demonstration's 's coverage area, but TP- GMM can still generate a correct reproduction with four demonstrations. However, as the number of demonstrations decreases, its performance decays and eventually fails with the single demonstration case.
Figure 16:
In conclusion, TP-GMM does not generalize well when reducing the number of demonstrations in these two examples. It requires more effort to determine the appropriate placement the demon- strations to generalize well. With a single demonstration, it tends to overfit that trajectory. The next section will show the performance of Elastic-DS being able to generalize well with a single demonstration compared other TP approaches.
18

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F Compare to Existing Methods
To show the advancement of our method, we create both qualitative and quantitative comparisons against various benchmark methods in the task-parametrized (TP) approach [52, 50J. Specifically, the benchmarks include lisk-Parameterized Gaussian Process Regression with DS-GMR for mo- tion reproduction (TP-GPR-DS) [52], Lask-Parameterized Gaussian Mixture Model with DS-GMR for motion reproduction (TP-GMM-DSS) [51, 50, 52], Pasamearrmtterizee Probabilistic Movement Primitives (TP-proMPP) [52, 40J. We use different quantitative metrics to show satisfaction in gener- alizing tasks:
Start Cosine Similarity: It describes the starting direction of the trajectory and how it aligns with the entry/starting geometric descriptor. We take the first two data points to cre- ate vector Vs and compare it against the pointing direction of the entry/starting geometric descriptor wOs- The closer this value is to one, the better.
cos(O,) = 0x VOs
(12)
lee uo
Goal Cosine Similarity: It describes the goal reaching direction the trajectory and how it aligns with the goal/exit geometric descriptor. We take the last two data points of the trajectory to create vector Un and compare against the pointing direction the goal/exit geometric descriptor tog. The closer this value is to one, the better.
2g 20g
cos(O,) (g))
(13)
| ||ol||
Endpoints Distance; Besides the pointing direction, it important that the trajectory starts from the center the starting geometric descriptor Po. and reach the center of the goal geometric descriptor Pog. Let &o be the start of the trajectory and &r be the end of the trajectory. The metric will be the sum of the two Euclidean distances. The smaller this value, the better.
D = d(Eo, Pos) + d(Er, Pog)
(14)
We compare four different trials with the same training data (a single demonstration): Close, Far, Both Ends Shifted, and Both Ends Shifted Far with increasing difficulty levels. Each subsection below describes trial with a plot showing with red arrows how the geometric descriptors (in green) are being changed. Then it will be followed by four plots showing our method compared to three other methods. Each subsection will include a table showing the quantitative comparison. The single demonstration data taken from the attached library code in [52]. The parameters for the benchmark methods remain in default as in the code from [52]. There are no required tuning parameters for Elastic-DS.
F.1
Close
Metric
Elastic-DS (Ours)
TP-GPR-DS
TP-GMM-DS
TP-proMP
Start Cosine Similarity t
0.9857
-0.8222
0.9794
0.2062
Goal Cosine Similarity T
0.9999
0.7397
0.5422
0.9889
Endpoints Distance I
0.0008
0.4298
0.7564
0.3062
Flastic-DS outperforms other methods for meeting the new geometric descriptor con- straints. Its cosine similarities are the closest 1, and the endpoints distance the closest O. Both TP-GMM-DS and TP-proMP have large endpoints distances. TP-GPR-DS starts its movement in the opposite direction.
Table 2:
19

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demonstration
E
Figure 17: The single demonstration (in blue) goes from the bottom to the top, constrained by the two geometric descriptors. In the new scenario, the goal/exit geometric descriptor is shifted to closed position, as indicated by the red arrow. The start/entry geometric descriptor remains at the same pose. We will need generate new motion policy that adapts to such change.
demonstration
demonstration
demonstration
demonstration
situation
Elastic-DS (Ours)
TP-GPR-DS
TP-GMM-DS
TP-proMP
Figure 18: The demonstration and its probabilistic encoding are indicated as blue. The new motion
policy and its probabilistic encoding for the new situation are in orange. While the other benchmark methods fail to meet the constraints given only one demonstration, Elastic-DS can generalize to the geometric descriptors constraints both ends
F.2 Far
demonstration
f
Figure 19: In the new scenario, the goal/exit geometric descriptor is shifted to a further position with rotation, indicated by the red arrow. The start/entry geometric descriptor remains at the same pose.
20

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demonstration new situation
demonstration new situation
demonstration
new
situation
demonstration situation
Elastic-DS (Ours)
TP-GPR-DS
TP-GMM-DS
TP-proMP
Figure 20: Only Elastic-DS generates motion policy that meets the geometrie descriptor constraints
Metric
EL (Ours)
TP-GPR-DS
TP-GMM-DS
TP-proMP
Start Cosine Similarity 1
0.9971
-0,9999
0.7872
0.3141
Goal Cosine Similarity f
0.9997
0.5451
0.6724
0.9611
Endpoints Distance 4.
0.0009
0.8459
1.552
0.6480
Table 3: As the goal geometric descriptor is moved further away, the performances of the other
methods start to decay. The Start Cosine Similarities for TP-GMR-DS and TP-proMP decrease.
Elastic-DS remains in good performance the three metrics.
F.3
Both Ends Shifted
demonstration
f
Figure 21: The new situation includes translation and rotation for both the starting and goal geomet- ric descriptors, as indicated by the red arrows.
demonstration new situation
demonstration new situation
demonstration new situation
demonstration new situation
TP-proMP Figure 22: Elastic-DS is able to handle the new situation while the other benchmark methods fail.
Elastic-DS (Ours)
TP-GPR-DS
TP-GMM-DS
21

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Metric
Elastic-DS (Ours)
TP-GPR-DS
TP-GMM-DS
TP-proMP
Start Cosine Similarity 1
0.9843
-0.3981
0.9405
0.4777
Goal Cosine Similarity 1
0.9998
0.5453
0.6324
0.8965
Endpoints Distance
0.0008
0.8360
0.07642
0.1788
Table With both geometric descriptors moving, Flastic-DS remains in good performance on all three metrics.
F,4 Both Ends Shifted Far
demonstration
f
Figure 23: In the new scenario, the goal/exit geometric descriptor is shifted to even further position with rotation, as indicated by the red arrow
demonstration new situstion
demonstration new situation
demonstration new situation
demonstration new situation
W
Elastic-DS (Ours)
TP-GPR-DS
TP-GMM-DS
TP-proMP
Figure 24: Only Elastic-DS generates new motion policy that meets the new geometric descriptor constraints
Metric
ELastic-DS (Ours)
TP-GPR-DS
TP-GMM-DS
TP- TP-proMP
Start Cosine Similarity 1
0.9869
-0.3827
0.8540
0.1295
Goal Cosine Similarity 1
0.9997
0.9378
0.7151
0.9766
Endpoints Distance
0.0012
1.265
1.143
0.6468
With the challenging new scenario, the performances for the benchmark methods decay even more. The endpoint distances increase for all the TP approaches. Elastic-DS remains in good performance on all three metries.
Table 5:
22

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G Robot Experiments Details
G.1 Software and Hardware Details
For all of these experiments, we used the 7DOF Franka Emika Panda robotic arm controlled via ROS and the libfranka C++ interface. The computer for the experiments ran Ubuntu 20.04 with Intel 17-11700K 3.6GHz CPU and 32GB memory. To track the geometric descriptors O2 for each task, we use the Optitrack Motion Capture system, which provides us 6DoF frames of the rigid bodies at 100Hz. We first attached a set of motion capture markers to the base of the Franka Panda robot arm, which served as the fixed frame. For each experiment, we attached motion capture markers to the task-relevant objects.
To record demonstrations, published the robot end-effector position to ROSbag recording. Dur- ing the demonstration, the orientation R and position of the task-relevant objects will be recorded with the Motion Capture system. We used the finite difference of the collected position data to calculate the trajectory velocity, which forms the training data D == {Er. & Jim
In the training phase, we first use Elastic-GMM implemented in both MATLAB and Python to learn GMM encoding of the training data. Then, during the testing phase, use Elastic-GMM implemented in Python modify the encoded data with the geometric descriptors O, The Elastic- GMM output will then become the input to the MATLAB code for learning the motion policy. The execution of the Elastic-DS motion policy is implemented in a C++ ROS node, which takes the current state of the end-effector of the robot & E R as the input. The output of this ROS node is the desired end-effector velocity & E RR,, which is sent to low-level cartesian velocity impedance controller implemented in C++ and running at 1kHz. To achieve the required velocity the end- effector, it performs torque control at the joints. The stiffnnsss parameter of the controller was set to 180.0. The orientation of the end-effector is fixed for every task the Elastic-DS DS motion policy Ç = (() is only learned in 3D space.
G.2 Bookshelf Experiment
The goal of this experiment is to teach the robot how to insert book into desktop bookshelf. With the bookshelf being moved to different locations and orientations on the table, the robot should be able to generalize and reproduce new motion policies for inserting the book into the bookshelf.
Prepare for the experiment:
1. We attached a motion capture marker object the side of the bookshelf. The goal geomet- ric descriptor O was an offset from the marker object that it was inside one of the slots in the bookshelf. The orientation R SO(3) of the geometric descriptor Og was the same as the opening of the bookshelf. Both the position p R and the orientation R were with respect to the fixed frame.
2. Closed the gripper to hold the center of the book vertically.
3. Calibrated the weight of the book so that the robot would not move in gravity compensation.
Collect data:
Recorded the robot end-effector position & E R a ROSbag.
2. At the same moment, the motion capture system recorded the bookshelf (geometric de- scriptor) position and orientation Og,
3. As shown in the figure below, person used hands directly in touch with the robot to perform kinesthetic teaching demonstration. The ROSbag recording ended as the end effector position & reached the bookshelf slot p. which is the position in Og
4. The single demonstration (end effector position and dime-derivative computed numeri- cally with timestamp data {), was then used as the training data D (f 1Tn
23

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The training data D was encoded as Elastic-GMM Hi. Eg tion 4.1. There was no required tuning parameter.
} =====> as described in Sec-
Execution:
We moved the robot end-effector (with the book) and the bookshelf Og to different config- urations, as shown different figures below and the video. The end-effector orientation with the book was always aligned with the bookshelf opening that a translational move- ment could insert the book into the bookshelf.
2. The motion capture system recorded the new configuration of the bookshelf.
3. For the updated bookshelf configuration Ogneww we updated the Elastic-GMM {TKs M2g - 1K to the new situation and learned the Elastic-DS as described in Section 4.1.
4. The robot then executed the DS motion policy & = f(E) in the task space with velocity- based impedance controller [58].
5. The gripper released the book once it reached the attractor Eourr p.
6. Repeated with different configurations.
Figure 25: Demonstration for inserting book into desktop bookshelf
Figure 26: The execution for the learned DS the original configuration
Figure 27: The bookshelf was shifted closer to the robot (The shifting direction is indicated by the red arrow). Without any new demonstration, the learned DS was able to adapt to the new configura- tion.
24

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Figure 28: The bookshelf was shifted up (The shifting direction is indicated by the red arrow).
Without any new demonstration, the Elastic-DS was able to adapt to the new configuration.
Figure 29: The bookshelf was shifted up (The shifting direction is indicated by the red arrow). Without any new demonstration, the Flastic-DS was able to adapt to the new configuration, The robot was still able to reach the new bookshelf position with human disturbances
Figure 30: The bookshelf is rotated and shifted to the left side of the robot. Without any new demonstration, the Elastic-DS able to adapt to the new configuration. The robot was still able reach the new bookshelf configuration. We rotate the end-effector to be parallel with the bookshelf beforehand.
25

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G.3 Pick and Place Experiment
In this task, we will show the robot how to pick and place a cube in a bin. The cube position p E R3 can be changed (labeled by the motion capture marker on the box as the geometric descriptors OD while the bin position is fixed). Based on the nature of this task, we manually set two motion segments. However, the cutoff location of the two motion segments is determined automatically by the motion capture data. The first segment is the picking motion with a geometric descriptor O the end of the trajectory (at the cube). The second segment is the placing motion with a geometric descriptor O2 at the beginning to ensure the robot with the cube will move upward first to reach enough height to approach the bin from the top. So there are two geometric descriptors O and O₂ at the cube to serve as a via-point. During the demonstration, the gripper open/close done through voice commands (with the microphone at the bottom right of the snapshots). The gripper state is memorized and associated with each segment. At the end of each segment (reaching the attractor), the robot will open/close the gripper depending on commands during the demonstration.
Prepare for the experiment:
1. We attached a motion capture marker set to a base box for placing the cube.
Collect data:
1. Record robot end-effector position & E 3 to ROSbag.
2. At the same moment, the motion capture system recorded the box (geometric descriptors ) position P1 and p2 from O and O2. This became the cutoff the two motion segments. As shown in the figure below, person used hands directly touch with the robot to perform a kinesthetic teaching pick and place demonstration
single trajectory.
3. The human used voice command (with the mic at the bottom of the figures) to control the gripper state (open/close). The ROSbag recorded the gripper state.
4. The single demonstration (end-effector position, timestamp data gripper state) was then separated into two parts D == {f ) N= and used for training. The two seg- ments of training data were encoded as two Elastic-GMM {{ Ma D 1K 1N=2 as described Section 4.1. There was no required tuning parameter.
Execution:
1. We moved the robot end-effector and the box to different positions, as shown in the different figures below. The gripper always pointed downward all time.
2. The motion capture system recorded the new positions of the box. It served the new geometric descriptors' positions P1 and p2 from O₂ and O₂ well the switch position of the two segments. The first geometric descriptor orientation R1 SO(3) was always pointing down to make the gripper approach the cube from the top. The second geometric descriptor orientation Rt E SO(3) was always pointing to allow the gripper to reach enough height before placing the cube in the bin.
3. For the updated box (geometric descriptors) configurations O and Oz, we updated the Elastic-GMMs {{TE, 1 N to the new situation and learned the Elastic- DSs for the two segments as described in Section 4.1.
4. The multiple DS motion policies & = 8(fx(E) with one-hot activation were executed in order separated by via-point at the cube, switching of them automatically happens the via-point as described in Appendix C.I.
5. The gripper released the cube once reached the attractor in f2(E).
26

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Figure 31: Demonstration for a pick and place task
Figure 32: The execution for the learned DS in the original configuration
Figure 33: The cube was shifted further away from the robot (The shifting direction is indicated by the red arrow).
Figure 34: The cube was shifted to the right side of the robot (The shifting direction is indicated by the red arrow).
Figure 35: The cube was shifted slightly to the right side of the robot (The shifting direction is indicated by the red arrow). During the execution, the human held the robot and switched to another cube. After that, the robot finished the task
27

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G.4 Tunnel Experiment
In this experiment, we will show the robot how to pass through a tunnel, mimicking a scan- ning/inspection task. Two different motion capture marker objects label the entrance and the exit of the tunnel. Separating by the two markers, there are total of three segments in this task. It is task with via points.
Prepare for the experiment:
1. We attached two motion capture objects to two sides of the tunnel (Each object has three markers).
Collect data:
Recorded robot end-effector position & E R$ ROSbag.
2. At the same moment, the motion capture system recorded the entry and exit positions 1B=4 [pt) from {O+)_.t. They became the two cutoffs of the three motion segments. A person then used hands directly touch with the robot perform kinesthetic teaching tunnel demonstration in single trajectory.
3. The single demonstration (end-effector position and timestamp data) was separated into three segments D [(Er,n: ) \ for individual training. The three segments of training data were encoded as three Elastic-GMMs (TK,isM,s - scribed Section 4.1 There was no required tuning parameter.
N==3 as de-
Execution:
1. We moved the robot end-effector and changed the tunnel position and orientation. The gripper always pointed downward.
2. The motion capture system recorded the new positions {pb) of the tunnel entry and exit. They served the new geometric descriptor position in {O+) as well as the switch position of the three segments. The first segment had geometric descriptor O at the end (at the tunnel entry). The second segment was within the tunnel, so it had two geometric descriptors O2 and O3 at two ends. The third segment had geometric descriptor O4 at the beginning (at the tunnel exit). All of the geometric descriptors {O, were predefined to point along the tunnel movement direction. They will change based on the relative position of the entry and exit.
3. We updated the three Flastic-GMMs {( Mk,,0 E Ni the new tunnel con- figurations and learned single Elastic-DS & { f(E) in Appendix C.2. Except for the k=] to learn sequential Elastic-DS
flip tunnel case, where we combined (mk, M4: Σ 1
{ ff((E) as in Appendix C.I.
4. Execution of the motion policy (E) via the cartesian veloeity impedance controller
Figure 36: The human guides the end-effector for an inspection task, starting from the left side, passing through tunnel, and stopping on the right side
28

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Figure 37: The execution for the learned DS in the original configuration from the demonstration
Figure 38: The tunnel rotated. We rotate the end-effector to be parallel with the tunnel before the exccution.
Figure 39: The tunnel is shifted further away from the robot, indicated by the red arrow.
Figure 40: The tunnel is flipped, indicated by the arrow in the tunnel (as opposed to the direction from the demonstration). The robot needs to move to the right side enter the tunnel and exit the left side before reaching the end pose.
29

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G.5 Combined Experiment (Tunnel + Pick and Place)
There is no demonstration or training in this task. We reuse the Elastic-GMMs (each as {[TR, My = ()= learned from the previous experiments to compose new sequences which perform new tasks. We manually defined the sequence of the task with the one-hot encoding activation 8(C, oz). However, in the future, plan to develop high-level planning algorithms to determine the sequence automatically.
Prepare for the experiment:
1. We used all the previous components except for the bookshelf. The motion capture markers were placed in the same way as in the previous experiments. This time, we also added markers to the bin for the cube placing. There were total of 7 geometric descriptors {O,)
2. We defined the sequence of the execution PPik-SSaaning-PPacee in ō(S, There were total of four motion segments in this task, corresponds to (E)(E))
Execution:
1. We moved the robot end-effector and changed the object positions. The gripper always pointed downward.
2. The motion capture system recorded the new positions of the objects. The geometric de- scriptors [OT were updated.
3. We updated the four Elastic-GMMs ((TT, Pi, 2 -= 1= to the new geometric de- scriptors' configurations BOT based on the motion capture data and learned four Elastic-DSs (E) as described in Section 4.1.
4. Execution of the Sequential Elastic-DS motion policy & = (5) (Appendix C.I) via the cartesian velocity impedance controller
Figure 41: Composing the learned DSs with task transfer parameters from the "pick and place" and "tunnel" tasks. The robot is able to pick up block, pass through the tunnel for scanning. and place the block in the bin. The entire motion does not require extra demonstration. Note the positions of the objects are not the same as in the original demonstrations.
Figure 42: We shift the cube starting platform, the tunnel, and the bin. By reusing and composing the previously learned DS with task transfer, the robot is able to finish the tasks of picking, scanning, and placing without new demonstration in this new environment configuration. Also, there is human disturbance involved during the task execution.
30

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G.6 TP-GMM + LPV-DS vs. Elastic-DS
The inputs for LPV-DS include a GMM (TTR Mb: = and velocity reference É., which the TP-GMM approach is able to provide. Therefore, one could imagine using TP-GMM with LPV-DS create generalizable motion policies. This section showcases robot experiment in the bookshelf case using TP-GMM LPV-DS (TP-LPV-DS). We reused the single demonstration DD from the bookshelf case in the previous section. Then, the bookshelf was moved and rotated to new instance different from the previous section. After encoding the demonstration in TP-GMM and calculating the product GMM at the new instance O, we used GMR to produce the velocity reference and then generated a motion policy with LPV-DS. With the single demonstration D, TP-LPV-DS was not able to complete the book insertion task as shown in Figure 43. Therefore, more demonstrations were provided in different contexts for TP-LPV-DS to help generate a successful new policy for the new instance, as shown Figure 44. On the other hand, using the single demonstration, Elastic-DS was able to complete the book insertion task the instance, shown in Figure 46. The preparation and data collection process was the same G.2. A single demonstration was collected.
Itlanded shon
Figure 43: The bookshelf was shifted and rotated to a configuration {Or). The execution steps the same G.2 except that at step the products of TP-GMM { Hig E k was calculated for LPV-DS. The robot then executed the DS motion policy & f({) in the task space with a velocity- based impedance controller [58]. After training with one demonstration, when the bookshelf was set a new configuration, TP-LPV-DS landed short and hit the table.
We collected 5 more demonstrations with kinesthetic teaching + original
0.8
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Figure 44: From the original single demonstration bookshelf configuration, rotated the bookshelf to more different configurations and collected new demonstrations. However, these were not suf- ficient for generating the successful motion policy for O,. We then collected 3 more demonstrations with different end poses closer to O,, The figure shows all the new demonstrations trajectory and all endpoints geometric deseriptors {O ej. Ogi 1/=6
31

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Figure 45: After training with multiple demonstrations, it used the same execution steps the single demonstration case. When the bookshelf was set to a new configuration, TP-LPV-DS successfully completed the task.
New
New Inscance with position orientalion
New with position orientinie
with poaition orientation
Figure 46: After training with a single demonstration, The execution steps were identical to G.2 with bookshelf configuration O, Elastic-DS successfully completed the task.
32

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H
Training and Adaptation Computation Times
Training and adaptation of the Elastic-DS are performed on laptop with Intel i7-12700H and 16GB memory. Initial training time a single demonstration with roughly T, 200 datapoints
Original PC-GMM implementation Matlab [22] takes around 2-4 seconds.
An improved parallelized PC-GMM implementation C++ takes around 100-200ms.
For parameter adaptation, the recorded computation times are below (considering 3-4 Gaussians):
Elastic-GMM parameter transfer takes around 30ms-80ms in Python.
DS parameter learning (SDP optimization) takes around = 800ms in Matlab.
Hence, for single demonstrations with T, = 200 datapoints initial training is 1s whereas generating a new policy from task parameter changes takes around 1-2s. The robot experiments presented in this section contain between Tù [500, 1000) with & TR3. For such datasets the average computation time generate computation times function of increasing Tn.
policy 4s. In Fig. 47 plot the trend of such
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Figure 47: This plot shows the trend of DS computation time for generating new policy under different lengths of demonstration in 2D and 3D. Each data point is collected from an average of runs.
33


Reviews:
[{"review_id": "yVGOuI1SeO5", "reviewer": "Reviewer_4LLc", "summary": "", "questions": "", "limitations": "", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "The paper proposes a dynamical system (DS) based task parameterized policy using  geometric descriptors for generalization. To generate the task parameters for the corresponding spatial change in the geometric descriptors, the paper proposes an augmented LPV-DS [22] in a developed dynamical system addressed as Elastic-DS. The deformation of this DS for new situation utilizes Laplacian Trajectory Editing [47].  By showing both 2D simulation and different 3D robot experiments, the paper validates the ability of Elastic-DS to perform task generalization as well as the potential for multi-task and long-horizon motion policies.", "summary_of_recommendation": "The paper is interesting and proposes guaranteed policy learning. However, The significance of the proposed approach with respect to the state-of-the-art is not highlighted clearly in terms of task performance. Therefore it is difficult to judge the contribution at the current stage.\n\n-----------\nUpdate: I think the reviewers have done a great job during rebuttal. My concerns are well addressed. I recommend acceptance.", "technical_quality": "Good: Generally convincing, but some additional experiments or analyses needed.", "clarity_of_presentation": "Good: Minor revision needed to improve organization or clarity.", "potential_impact": "", "robotics_focus": "Sufficient demonstration on hardware", "quality_of_limitations": "Additional details required", "questions_for_rebuttal": "The paper should compare and establish some quantitative metric for comparison against established task-parameterized motion policy learning in robotics e.g. [33], [36]. Consider also the time constraint on finding the motion in a new situation.", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "The paper has a good introduction and related works.\nThe motivation is convincing. However, it can work on clarifying the significance and experimental evaluations with respect to the state-of-the-art.\n\nStrengths: \nThe paper addresses a very important problem in robot learning from demonstration. The generalization of learnt policy in new situations is formulated as a dynamical system with stability and convergence guarantees.\nThe experiments on real robot demonstrate the validity of the approach\n\nWeakness:\nThe significance of the proposed is not so well placed with respect to the state-of-the-art.\nThe experimental test cases are relatively simple (no orientation are considered)", "extra_scores": {"impact": "3: The work is incremental and unlikely to have much impact, even though it may be technically correct and well executed."}, "extra_text": {}}, {"review_id": "u3grqIfe8xF", "reviewer": "Reviewer_KgCv", "summary": "", "questions": "", "limitations": "", "rating": 2, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "The authors propose Elastic-Dynamical System (DS), a novel DS learning and transfer approach that embeds task parameters into the Gaussian Mixture Model (GMM) based Linear Parameter Varying (LPV) DS formulation. Their motivation for proposing this is to address the shortcomings of current methods in LFD i.e. generalization vs. stability vs. effort. Their method promises i) stability guarantees, ii) the flexibility to transfer skills across novel scenarios, while iii) requiring minimal human effort during learning and transfer. They demonstrate their work in 2D experiments and also with the real robot (the Franka Emika 7 DoF manipulator in the following scenarios: Bookshelf, Pick and Place, Tunnel, and Combination). ", "summary_of_recommendation": "I think the paper will need further details before I can recommend an acceptance.", "technical_quality": "Good: Generally convincing, but some additional experiments or analyses needed.", "clarity_of_presentation": "Good: Minor revision needed to improve organization or clarity.", "potential_impact": "", "robotics_focus": "Sufficient demonstration on hardware", "quality_of_limitations": "Limitations are addressed clearly", "questions_for_rebuttal": "(1) How do you demonstrate stability guarantees in your real experiments?\n(2) Could you give more detail in the appendix on your experiments s.t. your results could be more reproducible?\n\n", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "The main strengths of the paper are that while it lacks experimental detail, the idea promises to push forward research efforts in LfDs with a particular focus on generalization, stability and effort. This is addressed by demonstrating particularly a zero-shot method. Overall, the paper is well-written and is of a good quality. The main drawbacks are the lack of experimental details and lack of stability verification.", "extra_scores": {"impact": "2: Unsure, difficult to assess."}, "extra_text": {}}, {"review_id": "qlgoYC3KlX", "reviewer": "Reviewer_uBxH", "summary": "", "questions": "", "limitations": "", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "This paper proposes to endow the linear parameter varying dynamical system (LVP-DS) formulation with generalization capabilities. To do so, it presents an elastic Gaussian mixture model (GMM) to update the underlying GMM using task parameters (start-, end-, and eventually via-points). The updated GMM is then used as a basis for an updated LVP-DS that generalizes to shifted and rotated instances of the initially-demonstrated task.", "summary_of_recommendation": "The paper presents interesting ideas and is generally convincing, but requires several additional explanations and some polishing before being ready for publication. Therefore, I now suggest to weakly reject the paper, but would be glad to update my score if the rebuttal answers my main concerns.\n\n------------\nPost-rebuttal:\n\nAs my main concerned have been answered by the rebuttal, I updated my score from \"weak reject\" to \"strong accept\".", "technical_quality": "Very Good:  Convincing.", "clarity_of_presentation": "Very Good:  Well organized and clear.", "potential_impact": "", "robotics_focus": "Sufficient demonstration on hardware", "quality_of_limitations": "Limitations are addressed clearly", "questions_for_rebuttal": "- Clarifications concerning the Gaussian transformations presented in Section 4.1.3;\n- Clarifications concerning the sequential aspect of the presented approach;\n- Extension of the results presented in the main paper;\n- Potential comparisons with respect to the generalization capabilities of state-of-the-art LfD approaches;\n- Clarify the focus and contributions of the paper in the title, abstract, and introduction.", "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": "In general, the ideas and methods presented in the paper are interesting and aim at solving the relevant problem of generalization when learning robot skills from demonstrations. The combination between DS --- which are learned from few demonstrations while offering stability guarantees --- with generalization capabilities is generally elegant. I also appreciate that the proposed approach is validated both in 2D examples and with various real-world robotic tasks. \n\nThe idea of elastic GMMs is generally elegant and seems efficient to generalize GMMs to new start- and end-points. However, I missed a clear connection between the Laplacian editing primer (Section 4.2.2) and the transformations of Gaussians described in Section 4.1.3 (beyond the fact that both involve waypoints and a least square formulation). More precisely, how are the graph Laplacian matrix $L$ and the Laplace coordinate matrix $\\Delta$ computed? How do they relate to the Laplacian editing primer? \n\nMoreover, the solution to Eq. (7) produces new joint positions $\\beta_i$. However, it doesn't seem to produce new covariances $\\Sigma_t$ although these covariances seem to control the orientation of the next Gaussian component in the chain. Instead, the covariances seem to simply be scaled as a function of the distance between neighboring joints. Therefore, how is the orientation of the GMM component modified? \n\nAlthough the motion policy proposed in Eq. (2) describes multi-step sequential task, this aspect is only briefly discussed (Section 4.3) and experimented (in Appendices C, D.1.3) later on. The paper assumes that the sequence in created manually by appending several tasks together. In this case, how are the activation functions designed to cope with the fact that the velocity is 0 at the end of each DS? How depend is the final execution of the design of these activation functions? Which system (Fig. 5b and 5c) is used in Appendix D.1.3 and in the video? \n\nOverall, the experimental results are very briefly shown in the paper and the reader must refer to Appendix to get an overall idea of the performance of the proposed approach. Given that the fact that the proposed DS can deal with via-points constraints and with sequences is given a high importance by being part of the general motion policy given by Eq. 2, I would have liked to see the related results in the main paper. In general, a comparison of the generalization capabilities of the proposed approach compared to the generalization capabilities of movement-primitives-based approaches (based on via points or task parameters) would strengthen the paper. Similarly, the proposed elastic GMM presents some similarities with TP-GMM as both adapt the Gaussian components of the GMM based on task parameters. One could therefore ask how would a TP-GMM compare to the proposed elastic GMM if used within LVP-DS? Although the main difference may be the fact that TP-GMM requires demonstrations in different context, the two approaches may also display different generalization capabilities.\n\nAlthough the contributions of the paper become clear in Section 2, I find the title and introduction misleading in that regard. While the paper is clearly concerned with generalization over instances of the same task, the title, abstract, and introduction suggest that the proposed approach instead aims a performing task transfer, which is generally understood as transferring knowledge between **different** tasks in the context of transfer learning. The sentences \"lack the flexibility to transfer to new tasks\" (line 4), \"learning from single tasks\" (line 37), \"humans tend to borrow knowledge from prior experience of skill acquisition and reuse them for acquiring and executing new tasks\" (lines 38-9), \"to achieve transfer\" (line 94) also imply a transfer between different tasks, which does not correspond to the focus of the paper. I would suggest to rephrase the title, abstract, and introduction with a focus on generalization instead of on task transfer / transfer to novel tasks to better reflect the frame and contributions of the paper. \nAlong a similar line, the sentence \"the robot should intelligently understand the objective, which means [...]\" (line 40) is not reflected in the paper, as the important task parameters are provided by hand. \n\n\nMinor:\n- Line 27-8: \"LfD, also known as imitation learning\": LfD is a type of imitation learning instead of being a synonym of it.\n- End of line 63: while ii) requiring -> iii) instead \n- Missing spaces after some section title (e.g., section 2), and above/below equations (e.g., Eq. (1) ) \n\n\n------------\nPost-rebuttal comments:\n\nThe rebuttal answered my initial concerns about the paper. Several theoretical aspects were clarified, and I believe that the contributions and scope of the paper --- in relation to generalizations over instances of the same task --- are properly reflected in the title, abstract, and introduction. The sequential aspect of the approach is now better covered in the main text. Finally, the added comparisons with state-of-the-art successfully demonstrate the advantages of the proposed approach.\n\nFollowing the comment of Reviewer 4LLc, extensions of the proposed approach to orientation data would be valuable as future work.\n\n", "extra_scores": {"impact": "3: The work is incremental and unlikely to have much impact, even though it may be technically correct and well executed."}, "extra_text": {}}]

Author rebuttal:

Meta-review: This paper proposes a LfD approach that is able to adapt to new tasks without requiring new demonstrations. Overall the reviewers found that the work addresses an important problem and is a clear contribution to the field. As such, I am happy to recommend acceptance.
Decision: Accept (Poster)