Learning Spatially Ambiguous Trajectories from Demonstrations via Phase-Modulated Dynamical Systems

Hyunseo Jung , Jonghyeok Kim , Keehoon Kim
Pohang University of Science and Technology (POSTECH)
IEEE Robotics and Automation Letters (RA-L), 2026
Paper

Abstract

This paper presents a single-demonstration Learning from Demonstration (LfD) framework for imitating spatially ambiguous trajectories, including motions with self-intersections, reversals, and stopping behaviors. Existing approaches based on Dynamical Systems or Dynamic Movement Primitives typically either fail to reproduce such ambiguous trajectories or rely on fixed time parameterizations, making them sensitive to variations in execution speed and unsuitable for compliant human-robot interaction because task progress can become desynchronized from the executed motion. To address these limitations, we propose a phase-modulated dynamical system. The key idea is to introduce a phase-evolution law that synchronizes task progress with the executed motion, thereby making the system time-invariant. We show that the proposed system (i) exactly reproduces the demonstrated motion when initialized on the demonstrated trajectory, (ii) guarantees exponential convergence to the demonstrated trajectory from off-trajectory initial conditions, and (iii) remains robust to perturbations while preserving trajectory topology. Rigorous theoretical analysis establishes these properties, and experimental results demonstrate stable and compliant behavior under execution-speed variations and external interventions.

Introduction

Dynamical System (DS) is being a standard choice for Learning from Demonstrations (LfD). It is reactive, time-invariant, and offers stability by construction which makes itself suitable for human-robot collaboration.

However, since the velocity field depend only on the current position, it has one critical limitation: it cannot represent spatial ambiguity.

Self-intersecting trajectory with two possible branches at the crossing

(a) One position, different motion directions.

Phase selects the branch when position alone gives multiple possible velocities

(b) Phase selects the intended branch.

To solve this, we propose the Phase-Modulated Dynamical System (PMDS).

Proposed Method

Given a demonstration $X_d:[0,T]\to SE(3)$, PMDS computes the desired body twist $\text{gvf}(t)$ from the task pose $X(t)$ and phase $s(t)$:

  • Mimicking vector field $\text{mvf}(t)$: follows the demonstrated body twist $\mathcal{V}_d(s(t))$, expressed in the robot’s current frame.
  • Contracting vector field $\text{cvf}(t)$: attracts the pose toward the demonstration, orthogonally to $\text{mvf}(t)$ in the weighted inner product.
\[\text{gvf}(t) = \text{mvf}(t) + k\,\text{cvf}(t), \qquad k>0.\]

The gain $k$ controls attraction strength without directly advancing phase.

Graphical interpretation of PMDS showing the mimicking field, contracting field, and phase modulation

Mimicking follows the tangent; contraction reduces transverse error.

Phase follows executed motion

Phase evolves from the realized body twist $\mathcal{V}(t)$, using $\langle x,y\rangle_W=x^T W y$ with $W\succ0$:

\[\dot{s}(t) = \begin{cases} \dfrac{\langle \text{mvf}(t),\mathcal{V}(t)\rangle_W}{\langle \text{mvf}(t),\text{mvf}(t)\rangle_W}, & \text{mvf}(t)\ne 0,\\[4pt] 1, & \text{mvf}(t)=0. \end{cases}\]

For $\text{mvf}(t)\ne 0$, faster forward motion advances phase faster, holding freezes it, and reverse motion moves it backward. At a demonstrated stop ($\text{mvf}(t)=0$), phase advances at unit rate to complete the recorded dwell. The law has no explicit wall-clock dependence.

Theoretical Properties

Under the paper’s continuous-time modeling and regularity assumptions:

  • Exact replay with ideal velocity execution and on-trajectory initialization.
  • Exponential convergence from off-trajectory states with closest-point phase initialization.
  • Separated disturbance effects: tangential disturbances affect phase; transverse disturbances affect path error.

Experiments and Results

THREETURN
Rope winding with human intervention.
STOPGO
Preserving the demonstrated pause.
RETURN
Reversing through the same positions.
Four demonstrated trajectories named SPIRAL, THREETURN, STOPGO, and RETURN, colored by phase

Four SE(3) demonstrations, projected onto the x-y plane and colored by phase.

Benchmark Motion to reproduce
SPIRAL A smooth spiral without spatial ambiguity.
THREETURN Three turns with self-intersections, then an exit.
STOPGO A three-second dwell, then resumed motion.
RETURN Outward and return motions through the same positions.

Four complementary tests

  1. Replay (T1): reproduce the demonstration from its start.
  2. Recovery (T2): converge from three random off-trajectory poses.
  3. Perturbation (T3): execute twice the desired twist or its negative for three seconds.
  4. Time shift (T4): delay execution by five seconds.

PMDS ($k=1,3$) is compared with BCSDM, MPDS, SF-GVF, an SE(3) DMP, and GDMP.

Quantitative results

PMDS preserves task behavior in both perturbation conditions on every benchmark, at both tested gains. Each cell reports successful outcomes out of the two tested conditions (Fig. 4 of the paper).

Method SPIRAL THREETURN STOPGO RETURN
BCSDM 2/2 1/2 0/2 1/2
MPDS 1/2 1/2 0/2 1/2
SF-GVF 1/2 1/2 2/2 1/2
DMP 1/2 0/2 2/2 2/2
GDMP 1/2 1/2 2/2 1/2
PMDS, k = 1 2/2 2/2 2/2 2/2
PMDS, k = 3 2/2 2/2 2/2 2/2
  • Replay: PMDS captures both path geometry and ambiguous behaviors, including the STOPGO dwell that geometry-only metrics can miss.
  • Recovery: higher $k$ improves recovery in STOPGO and RETURN, but not every accuracy metric. In RETURN, $k=1$ gives lower tracking error and Fréchet distance than $k=3$.
  • Time invariance: PMDS shows near-zero sensitivity to the five-second execution delay.

BibTeX

@ARTICLE{11685323,
  author={Jung, Hyunseo and Kim, Jonghyeok and Kim, Keehoon},
  journal={IEEE Robotics and Automation Letters},
  title={Learning Spatially Ambiguous Trajectories From Demonstrations via Phase-Modulated Dynamical Systems},
  year={2026},
  volume={11},
  number={11},
  pages={12663-12670},
  keywords={Trajectory;Timing;Dynamical systems;Learning (artificial intelligence);Convergence;Modeling;Robots;Vectors;TV;Chromium;Learning from demonstrations;dynamical systems;path-following control},
  doi={10.1109/LRA.2026.3732892}}