Orbitally Stable Motion Primitives (OSMPs)

• Orbitally Stable Motion Primitives (OSMPs) are dynamical policies that learn periodic behaviors via convergence to stable limit cycles, differing from fixed-point methods.
• They use techniques like diffeomorphic encoders and supercritical Hopf oscillators to map robot states into latent spaces with guaranteed transverse contraction.
• Alternative approaches such as CLF-CBF-NODE and distance-field methods demonstrate OSMPs’ practical success in cyclic tasks like manipulation, handwriting, and locomotion.

Orbitally Stable Motion Primitives (OSMPs) are a class of dynamical motion policies for learning complex periodic or rhythmic behaviors from demonstration while maintaining formal convergence guarantees to a periodic orbit. In the formulation explicitly named OSMPs, a task-conditioned diffeomorphic encoder maps robot states into a latent space governed by a supercritical Hopf bifurcation, yielding global orbital stability and transverse contraction; adjacent formulations realize the same underlying objective through Neural ODEs combined with control Lyapunov and control barrier functions, or through autonomous dynamical systems derived from distance fields of demonstrated curves (Stölzle et al., 12 Jul 2025). Across these variants, the common objective is not convergence to a single fixed point, but convergence to a closed orbit that remains attractive under perturbation and can reproduce cyclic manipulation, wiping, stirring, sketch-following, or locomotion-like behaviors (Nawaz et al., 2023).

1. Definition and stability notion

The stability notion underlying OSMPs is orbital asymptotic stability (O.A.S.). For a dynamical system

$\dot x = f(x),\quad x\in\mathbb R^n,$

O.A.S. means that there exists a closed, non-self-intersecting periodic trajectory $\bar x(\tau)$, $\bar x(\tau+T)=\bar x(\tau)$, and a basin of attraction $\mathcal X\subset\mathbb R^n$ such that, for every initial condition $x_0\in\mathcal X$,

$$\lim_{t\to\infty}\min_{0\le\tau<T}\|x(t;x_0)-\bar x(\tau)\|=0.$$

Equivalent certification routes include Lyapunov-type functions that decrease off the cycle, or a Poincaré section whose return-map linearization has Floquet multipliers strictly inside the unit circle (Zhi et al., 2023).

This notion differs fundamentally from pointwise asymptotic stability. Pointwise-stable motion primitive frameworks such as SEDS require a single fixed-point attractor $x\to x_{\rm goal}$, whereas OSMPs enforce convergence to a limit cycle $x^*(t)$. In the CLF-CBF-NODE construction, the tracking error $e(t)=x(t)-x^*(t)$ is driven to zero so that $\|x(t)-x^*(t)\|\to0$ as $\bar x(\tau)$0; the resulting trajectory converges onto the nominal periodic orbit rather than to a stationary goal (Nawaz et al., 2023).

Within the formulation explicitly titled OSMPs, the class is defined as learning complex periodic behaviors from demonstration while providing formal convergence guarantees in the form of global orbital stability and transverse contraction. The motivation is to address the limitations of classical Dynamic Movement Primitives for sharp corners, discontinuous velocity profiles, curved orbits, and zero-shot task interpolation (Stölzle et al., 12 Jul 2025).

2. Canonical latent-space OSMP formulation

A canonical OSMP policy is specified in an “oracle” space, such as joint space or task space, by

$\bar x(\tau)$1

where $\bar x(\tau)$2 is a smooth bijection parameterized as a task-conditioned diffeomorphic encoder, $\bar x(\tau)$3 is its Jacobian, $\bar x(\tau)$4 is an optional velocity-scaling factor, and $\bar x(\tau)$5 is a continuous task-conditioning variable (Stölzle et al., 12 Jul 2025). Pulling back the latent velocity through $\bar x(\tau)$6 yields a velocity field in $\bar x(\tau)$7 that inherits the stability of the latent dynamics by construction.

The latent dynamics are based on a supercritical Hopf oscillator. Writing $\bar x(\tau)$8, with radius $\bar x(\tau)$9 and gains $\bar x(\tau+T)=\bar x(\tau)$0,

$\bar x(\tau+T)=\bar x(\tau)$1

$\bar x(\tau+T)=\bar x(\tau)$2

$\bar x(\tau+T)=\bar x(\tau)$3

In polar coordinates,

$\bar x(\tau+T)=\bar x(\tau)$4

The set $\bar x(\tau+T)=\bar x(\tau)$5 is the unique attracting orbit. The radial term has supercritical Hopf form, while $\bar x(\tau+T)=\bar x(\tau)$6 advances with either constant or learned angular speed to reproduce nonuniform phase-speed profiles (Stölzle et al., 12 Jul 2025).

The encoder $\bar x(\tau+T)=\bar x(\tau)$7 is implemented as a Real NVP / Euclideanizing-flow style normalizing flow with coupling blocks. Conditioning is introduced by embedding $\bar x(\tau+T)=\bar x(\tau)$8 via a small MLP or random-Fourier embedding and concatenating it to each coupling-layer MLP. This produces a single model that can represent multiple motion objectives and support zero-shot interpolation within the training distribution (Stölzle et al., 12 Jul 2025).

The formal guarantees are established in two complementary ways. First, a Lyapunov candidate in latent space,

$\bar x(\tau+T)=\bar x(\tau)$9

vanishes on the cycle and is strictly positive elsewhere; its derivative is strictly negative transverse to the orbit. Pulling back through $\mathcal X\subset\mathbb R^n$0 yields $\mathcal X\subset\mathbb R^n$1, proving global asymptotic orbital stability. Second, a transverse-contraction metric can be constructed in polar coordinates and mapped back by congruence transformations involving $\mathcal X\subset\mathbb R^n$2, giving exponential convergence to the limit cycle at rate $\mathcal X\subset\mathbb R^n$3 (Stölzle et al., 12 Jul 2025).

3. Alternative realizations of orbitally stable motion primitives

Several neighboring constructions instantiate the same core idea—an attracting orbit rather than a fixed-point goal—through different mathematical mechanisms.

Formulation	Core mechanism	Stability / safety mechanism
CLF-CBF-NODE	Neural ODE nominal field $\mathcal X\subset\mathbb R^n$4 plus online virtual control $\mathcal X\subset\mathbb R^n$5	CLF for orbital tracking, CBF for forward-invariant safety (Nawaz et al., 2023)
OS-MP via distance fields	Demonstration encoded as quadratic Bézier spline and distance field $\mathcal X\subset\mathbb R^n$6	Lyapunov decrease of $\mathcal X\subset\mathbb R^n$7 and LaSalle invariance (Li et al., 13 Apr 2025)
SDDT	Diffeomorphic morphing of a known O.A.S. seed system	Topological conjugacy preserves O.A.S. under the diffeomorphism (Zhi et al., 2023)

In the CLF-CBF-NODE framework, a Neural ODE learns a nominal vector field

$\mathcal X\subset\mathbb R^n$8

from demonstrations by minimizing trajectory reconstruction error after integrating the learned field. The learned field is then augmented online as

$\mathcal X\subset\mathbb R^n$9

where $x_0\in\mathcal X$0 is obtained from a quadratic program. Orbital stability is encoded through a control Lyapunov function on the tracking error $x_0\in\mathcal X$1, using $x_0\in\mathcal X$2, while safety is encoded through control barrier functions $x_0\in\mathcal X$3 defining a collision-free set $x_0\in\mathcal X$4. The unified QP minimizes $x_0\in\mathcal X$5 subject to an affine CBF inequality and a softened CLF decrease constraint, producing a unique minimizer $x_0\in\mathcal X$6 used as the correction $x_0\in\mathcal X$7. This realizes practical orbital stability in the presence of obstacle-avoidance requirements (Nawaz et al., 2023).

In the distance-field construction, a demonstrated trajectory $x_0\in\mathcal X$8 is approximated as a concatenation of quadratic Bézier curves $x_0\in\mathcal X$9. For a state $$\lim_{t\to\infty}\min_{0\le\tau<T}\|x(t;x_0)-\bar x(\tau)\|=0.$$0, the unsigned distance field is

$$\lim_{t\to\infty}\min_{0\le\tau<T}\|x(t;x_0)-\bar x(\tau)\|=0.$$1

Its gradient is

$$\lim_{t\to\infty}\min_{0\le\tau<T}\|x(t;x_0)-\bar x(\tau)\|=0.$$2

where segment $$\lim_{t\to\infty}\min_{0\le\tau<T}\|x(t;x_0)-\bar x(\tau)\|=0.$$3 attains the minimum distance. The autonomous system is then defined as

$$\lim_{t\to\infty}\min_{0\le\tau<T}\|x(t;x_0)-\bar x(\tau)\|=0.$$4

with $$\lim_{t\to\infty}\min_{0\le\tau<T}\|x(t;x_0)-\bar x(\tau)\|=0.$$5, $$\lim_{t\to\infty}\min_{0\le\tau<T}\|x(t;x_0)-\bar x(\tau)\|=0.$$6, and $$\lim_{t\to\infty}\min_{0\le\tau<T}\|x(t;x_0)-\bar x(\tau)\|=0.$$7. Using $$\lim_{t\to\infty}\min_{0\le\tau<T}\|x(t;x_0)-\bar x(\tau)\|=0.$$8, one obtains $$\lim_{t\to\infty}\min_{0\le\tau<T}\|x(t;x_0)-\bar x(\tau)\|=0.$$9, and LaSalle’s invariance principle gives global asymptotic attraction to the demonstrated curve, after which trajectories slide along it (Li et al., 13 Apr 2025).

In Stable Diffeomorphic Diagrammatic Teaching (SDDT), a simple O.A.S. seed system is chosen in $x\to x_{\rm goal}$0, comprising a planar oscillator in $x\to x_{\rm goal}$1 and a vertical attractor in $x\to x_{\rm goal}$2: $x\to x_{\rm goal}$3 A smooth invertible map $x\to x_{\rm goal}$4 then morphs the seed limit cycle to match a user-provided sketch. The $x\to x_{\rm goal}$5-submap is a coupling-based Real-NVP style network, and the optimization objective is the Hausdorff distance between the sketch-derived point set and a discretization of the transformed limit cycle, regularized to discourage extreme distortions. Because the transformed dynamics are topologically conjugate to the seed system, O.A.S. is preserved exactly (Zhi et al., 2023).

4. Learning objectives and computational procedures

The training objective in the canonical OSMP formulation is a weighted sum

$x\to x_{\rm goal}$6

The constituent terms are velocity imitation $x\to x_{\rm goal}$7, limit-cycle matching $x\to x_{\rm goal}$8, time-guidance $x\to x_{\rm goal}$9, encoder regularization $x^*(t)$0, velocity regularization $x^*(t)$1, and smooth conditioning interpolation $x^*(t)$2. Optimization uses AdamW with $x^*(t)$3, small weight decay, linear warm-up for 10 epochs, constant learning rate, then cosine annealing; all demonstrations are processed in a single batch, and the encoder is initialized near identity (Stölzle et al., 12 Jul 2025).

The CLF-CBF-NODE procedure separates offline model learning from online control. Offline, one learns $x^*(t)$4 by integrating

$x^*(t)$5

from initial conditions $x^*(t)$6 using a 5th-order Runge–Kutta solver and minimizing

$x^*(t)$7

Back-propagation is performed through the ODE integrator using a binomial-checkpointing scheme. Online, a target trajectory $x^*(t)$8 is precomputed by integrating $x^*(t)$9, and at each control step the closest index $e(t)=x(t)-x^*(t)$0 is found. A short look-ahead window is searched, and the candidate requiring the smallest QP correction norm $e(t)=x(t)-x^*(t)$1 is selected as the moving goal $e(t)=x(t)-x^*(t)$2. Because the CLF and CBF constraints are affine in $e(t)=x(t)-x^*(t)$3 and the cost is strictly convex, the QP is convex and can be solved in $e(t)=x(t)-x^*(t)$4 time or faster using operator-splitting such as OSQP (Nawaz et al., 2023).

The distance-field pipeline begins with least-squares fitting of a quadratic spline in Bernstein basis. With reparameterized phase $e(t)=x(t)-x^*(t)$5,

$e(t)=x(t)-x^*(t)$6

At run time, each segment solves the cubic orthogonality condition

$e(t)=x(t)-x^*(t)$7

for $e(t)=x(t)-x^*(t)$8, then selects the closest segment and computes $e(t)=x(t)-x^*(t)$9, $\|x(t)-x^*(t)\|\to0$0, $\|x(t)-x^*(t)\|\to0$1, and finally

$\|x(t)-x^*(t)\|\to0$2

The system can be integrated with standard ODE solvers such as Euler or RK4 (Li et al., 13 Apr 2025).

The SDDT optimization procedure starts from a user-drawn closed 2-D sketch over an image. Using depth $\|x(t)-x^*(t)\|\to0$3 and camera intrinsics, each sketch point is ray-traced to a world point $\|x(t)-x^*(t)\|\to0$4 on the target cycle in the plane $\|x(t)-x^*(t)\|\to0$5. The transformed limit cycle is discretized, and the loss

$\|x(t)-x^*(t)\|\to0$6

is minimized with ADAM at learning rate $\|x(t)-x^*(t)\|\to0$7, back-propagating through both the INN and an Euler-style ODE integration of the base system. The reported implementation uses PyTorch and FrEIA for coupling-based INNs, PyBullet for simulation, and a custom Euler integrator in Python/CUDA (Zhi et al., 2023).

5. Empirical evaluation and application domains

The reported empirical record spans handwriting, periodic planar curves, 3-D motions, task-space and joint-space manipulation, and multi-platform robotic systems. On the LASA handwriting dataset with 30 2-D patterns, NODE alone achieves roughly half the DTW error of SEDS/LPV-DS on both train and test, with mean DTW approximately $\|x(t)-x^*(t)\|\to0$8 versus $\|x(t)-x^*(t)\|\to0$9. CLF-NODE recovers from large spatial perturbations, and CLF-CBF-NODE safely avoids circular and non-convex obstacles. For periodic 2-D letters $\bar x(\tau)$00 and 3-D spirals and wiping motions, the NODE model is reported as faster to execute than Imitation-Flow and GP-based learning from demonstration by a factor of $\bar x(\tau)$01 to $\bar x(\tau)$02, with DTW lower by $\bar x(\tau)$03 to $\bar x(\tau)$04. On a Franka Emika robot arm, wiping a mannequin and wiping a whiteboard were demonstrated under unforeseen human perturbations and moving obstacles, with the online QP running at $\bar x(\tau)$05 kHz on a laptop; the robot remained compliant, avoided collisions, and rejoined the periodic orbit (Nawaz et al., 2023).

The distance-field OS-MP formulation was validated numerically on 2-D LASA handwriting patterns such as S-, J-, and G-shapes and compared favorably in reconstruction error to other basis sets. GPU/CPU benchmarks indicate sub-10 ms update rates for $\bar x(\tau)$06 and $\bar x(\tau)$07. Hardware experiments on a Franka 7-DoF arm include task-space S- and L-shapes, joint-space pick-and-place, and a human-to-robot handover in a 10-D experiment consisting of 3-D hand motion plus 7-D robot state; in each case the same distance-gradient dynamical system returned to the learned curve after human disturbances (Li et al., 13 Apr 2025).

The canonical OSMP formulation was evaluated on a UR5 manipulator, a Helix continuum soft robot, the Crush rigid-soft turtle robot, a KUKA LBR iiwa cobot, and kinesthetic demonstrations on a UR5. Tasks included tracking 2-D image contours such as ellipse, square, star, logos, Dolphin, Bat, and Eagle; bio-inspired swimming oracles defined either in 3-D flipper-tip space or in 3-joint space; and whiteboard erasing and brooming. Baselines included MLPs, RNNs, LSTMs, Neural ODEs, Diffusion Policies, ImitationFlow, and SPDT. Reported metrics included Traj RMSE, normalized Traj DTW, Vel RMSE, directed Hausdorff distance, ICP-MED, and runtime in microseconds per step on CPU/GPU. Across dozens of tasks and seeds, OSMPs match or exceed the imitation fidelity of baselines, with global convergence errors approximately $\bar x(\tau)$08 units Hausdorff versus greater than $\bar x(\tau)$09 for other stable-DMP methods, and runtimes around $\bar x(\tau)$10 kHz eager and $\bar x(\tau)$11 kHz compiled on modern CPU, $\bar x(\tau)$12 to $\bar x(\tau)$13 faster than diffusion policies. Additional demonstrations include compliant recovery under time-phase perturbations, phase synchronization of two to six independent OSMPs, and online shaping through affine translation/scaling, speed modulation, and convergence-tube gating (Stölzle et al., 12 Jul 2025).

SDDT was evaluated using simulation silhouettes such as whale, dog, flower, eagle, star, arrow, and knight, as well as freehand sketches on a live RGB image. Hausdorff distance was the principal cycle-accuracy metric. Mean performance over three random initializations was reported as $\bar x(\tau)$14 m for star, $\bar x(\tau)$15 m for knight, and $\bar x(\tau)$16 m for arrow, compared with $\bar x(\tau)$17 m, $\bar x(\tau)$18 m, and $\bar x(\tau)$19 m for a Neural ODE-based free-form model without stability constraint, and $\bar x(\tau)$20 to $\bar x(\tau)$21 m for a tuned base system that adjusted only circle radius and center. On a Unitree Aliengo quadruped with Z1 arm, a pentagon and a star sketched on the egocentric camera feed were traced continuously by a pen-equipped end effector while maintaining near-constant normal contact force (Zhi et al., 2023).

6. Conceptual distinctions, limitations, and open problems

A recurring misconception is that stable motion primitives must converge to a fixed point. The OSMP literature explicitly separates orbital convergence from pointwise convergence: the objective is attraction to a periodic orbit, not collapse to a stationary state. Another misconception is that rhythmic stability requires a globally enforced time clock. The CLF-CBF-NODE construction replaces explicit clocking with a horizon-based moving goal $\bar x(\tau)$22, selected from a look-ahead window on the target trajectory, and the distance-field construction is autonomous from the outset because the demonstrated motion is converted from a time-dependent curve into a spatial distance field (Nawaz et al., 2023).

The principal limitations are also explicit. In the canonical OSMP formulation, pure periodic segments must be extracted or segmented from demonstrations, and non-cyclic or multiple-stable tasks require extension. Intersecting or crossing trajectories are ill-posed for single-valued flows; suggested remedies include second-order augmentation or phase lifting. Large encoder deformations can push pullback Jacobians near singularity, creating numerical issues, with mitigation through encoder regularization, velocity gating, Jacobian regularization, or CBF constraints (Stölzle et al., 12 Jul 2025).

In SDDT, the current setting is restricted to flat planar surfaces at known depth. Extending to curved surfaces would require replacing the global diffeomorphism with a surface-intrinsic embedding, for example by using a local chart and a Riemannian metric. Higher-dimensional cycles in full $\bar x(\tau)$23 would require diffeomorphisms on $\bar x(\tau)$24 and careful treatment of orientation; the paper suggests coupling translational cycles with an independent contraction in rotation space, for example using quaternions. Force and torque embedding are omitted, with port-Hamiltonian formulations or hybrid dynamical systems identified as possible extensions (Zhi et al., 2023).

The relation among these strands suggests a broader interpretation of OSMPs as a design space rather than a single algorithmic template. Some realizations derive attraction from a latent Hopf prior and diffeomorphic pullback, others from Lyapunov decrease in a distance field, and others from online CLF-CBF correction of a learned Neural ODE. A plausible implication is that future systems may combine these ingredients: expressive diffeomorphic orbit encoding, transverse contraction or Lyapunov certificates, and real-time safety filtering through barrier constraints.