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REVIEW 3 major objections 4 minor 51 references

Combining Movement Primitives with Contraction Theory

T0 review · 3 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read This paper establishes that discrete and rhythmic movement primitives can be combined in parallel and in sequence with stability guaranteed by contraction theory, while each movement remains independently scalable in space and time.

desk verdict A clean modular extension of DMPs with a sound parallel combination; the sequential combination proof cites the wrong theorem but looks fixable. read the letter →

arxiv 2501.09198 v1 pith:5I4GQWNP submitted 2025-01-15 cs.RO

Moses C. Nah , Johannes Lachner , Neville Hogan , Jean-Jacques Slotine This is my paper · ORCID
classification cs.RO
keywords movementprimitivesdynamiccontractiontheorytransverselimitcyclerobotmotionplanningmodularstabilitydiscreteandrhythmic
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper sets out to make movement primitives—modular dynamical-system building blocks for robot trajectories—combinable in the same way functions are composed, so that a programmer can mix discrete point-to-point motions with rhythmic cyclic motions without re-learning a trajectory from scratch. Its central claim is that contraction theory supplies the stability guarantee that Lyapunov-based DMP methods cannot, because those forbid limit cycles. If the claim holds, complex robot tasks such as polishing or peg-in-hole assembly can be built as parallel or sequential combinations of simpler primitives, with each primitive scaled, rotated, or timed independently. The authors prove the parallel combination by closure of contraction under convex weighting, and the rhythmic case by treating limit-cycle attractors as transverse-contracting systems.

What carries the argument

Contraction theory, a differential stability analysis: a nonlinear system is contracting if all neighboring trajectories converge exponentially in a suitable metric, and this property is preserved under parallel (convex-weighted) combination, hierarchical coupling, and one-way coupling. Rhythmic primitives are modeled as transverse-contracting systems around a stable limit cycle, so the Andronov–Hopf oscillator's phase serves as a canonical 'clock' while its amplitude is exponentially forgotten. The load-bearing identity is the parallel-combination closure: if each primitive makes the same transformation system contract with the same metric, then the weighted sum of primitive inputs contracts with a rate that is the same convex combination of the individual rates.

What would settle it

Run the sequential combination of two discrete DMPs with different goal positions, using the smooth activation function of Eq. (24) with a fast switch (e.g., transition time much shorter than the transformation system's time constant); if the output trajectory overshoots, fails to converge to the second goal, or leaves a bounded region, then the claimed stability of Eq. (23) is refuted. A simpler falsifier: check whether the conditions of the one-way coupling theorem (Section II-B.3) are satisfied by Eq. (23); they are not, since the systems are not identical and the coupling is not $u(x_1)-u(x_2)$.

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Extended reading notes

Core claim

The paper's contribution is a definition of a movement primitive input $p(x) = F(s(t)) + \alpha_z \beta_z g^{(d)}$ that, when fed into a contracting second-order transformation system, reproduces a demonstrated trajectory. Because the transformation system is contracting and the forcing terms for discrete and rhythmic movements are, respectively, contracting and transverse-contracting, any parallel combination of primitive inputs—a convex combination $\alpha_1 p_1(x_1) + \alpha_2 p_2(x_2)$—remains contracting under the same metric, giving an exponential convergence guarantee for the blended trajectory. The paper further claims that sequential combination, achieved by time-varying weights with time offsets, is stable by reference to a one-way coupling theorem for contracting systems. This extends dynamic movement primitives to rhythmic movements with stability guarantees, preserving spatial scaling, temporal scaling, and rotation of each component.

Load-bearing premise

The stability of the sequential combination (Eq. 23) is asserted rather than proven: the cited one-way-coupling theorem applies to a different architecture, so the claim that time-varying weighted sums of different primitives remain stable is an assumption that, if false, would break the sequential method.

Editorial extensions

If this is right

  • A programmer can independently scale, rotate, or time-shift each primitive in a combination without destabilizing the overall motion.
  • Rhythmic movements (limit cycles) can now be included in DMP-based planning with exponential stability guarantees, which Lyapunov-based DMP methods exclude.
  • Sequential combination of discrete and rhythmic primitives enables composed tasks such as reach-then-polish or peg-in-hole sequences with smooth transitions.
  • Obstacle-avoidance coupling terms and temporal coupling to the canonical system can be added reactively while preserving contraction-based stability.
  • The divide-and-conquer programming strategy lets a complex motion be built by reusing and refining existing primitives rather than re-learning from demonstration.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Beyond the paper: the sequential-combination stability proof is the fragile link—Eq. (23) uses time-varying weights over different primitives, whereas the cited theorem concerns one-way coupling of two identical systems; verifying, or repairing, this step is a natural next problem.
  • Beyond the paper: the same convex-weighting closure should extend to other primitive families (e.g., orientation DMPs on SO(3)) as long as they share a contraction metric, which would generalize the framework to manifolds.
  • Beyond the paper: the framework's reactive obstacle-avoidance claim suggests a concrete test: add a repulsive coupling term to a rhythmic parallel combination in simulation and measure whether the limit cycle's transverse contraction rate is preserved.
  • Beyond the paper: because the weights $\alpha_i(t)$ are user-defined, one could adapt them online via sensory feedback (e.g., switching rhythmic gaits) and still retain stability if the switching is slow relative to the contraction rate—a testable extension not stated in the paper.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper presents a modular motion-planning framework that combines discrete and rhythmic Dynamic Movement Primitives (DMPs) using contraction theory. The authors define a movement primitive input, show how parallel combinations can be formed by weighted sums of primitive inputs, and propose sequential combinations via time-varying weights with smoothstep activations. They claim stability for both combination modes by invoking contraction-theoretic results, and they illustrate the approach with simulations of letter-like and circular trajectories. The central contribution is the claimed ability to compose discrete and rhythmic primitives while preserving independent spatial, temporal, and rotational modulation, with stability guarantees.

Significance. If the stability claims were fully established, the framework would be a useful practical contribution: it offers a modular, divide-and-conquer approach to motion planning, preserves the invariance properties of DMPs, and includes simulation code on GitHub. The parallel-combination analysis is consistent with a linear-filter view of the transformation system, and the examples are illustrative. However, the sequential-combination stability proof is currently missing or mis-cited, and that gap is load-bearing for the abstract's promise that 'Stability is proven by Contraction Theory.' The contribution is therefore significant but conditional on a rigorous proof or a suitably weakened claim.

major comments (3)
  1. [Section IV-B.2, Eq. (23)] The stability of the sequential combination is asserted by referring to Section II-B.3, but the theorem in Eq. (8) is a partial-contraction result for two identical systems coupled one-way through u(x1)-u(x2), and it requires f-u to be contracting. Equation (23), by contrast, is a single transformation system driven by a time-varying weighted sum of different primitive inputs, which is structurally a parallel combination with time-dependent weights rather than a one-way coupling. The paper does not verify the hypotheses needed for the parallel theorem (a common contraction metric, and in Remark 1 the condition that the weights sum to one), nor does it provide a separate proof for Eq. (23). Since the smoothstep activation in Eq. (24) does not guarantee that the weights sum to one during overlaps, the cited theorem does not cover the sequential method as stated.
  2. [Section IV-B.2 and Eq. (21)] Because the transformation system in Eq. (21) is linear, its output is a filtered version of the input; it is not equal to the weighted sum of the individual primitive outputs, y(t) = sum_i alpha_i(t) y_i(t - t_off), even when each primitive individually tracks its demonstration. Switching between primitives with time-varying weights introduces transients that depend on derivatives of alpha_i, and the paper supplies no bound on these transients and no proof that the trajectory converges to the intended primitive during each active interval. The simulations in Figure 3 demonstrate particular parameter choices, but they do not establish the general stability or convergence property promised in the abstract and contributions.
  3. [Section IV-B.1, Eq. (22)] The parallel-combination theorem in Section II-B.1 (Remark 1) requires the weights to satisfy alpha_i(t) >= 0 and sum_i alpha_i(t) = 1. Equation (22) is written for arbitrary nonnegative alpha_1, alpha_2, and the examples use the specific choice alpha_2 = 1 - alpha_1 without stating that this sum-to-one condition is part of the method. If arbitrary weights are allowed, the stability guarantee does not follow from the cited theorem; if the weights must sum to one, that constraint should be stated explicitly because it has implications for the claimed independent modulation of each primitive.
minor comments (4)
  1. [Section II-A.2, Eq. (3)] The orthogonality condition is written as 'delta x^T M_T(x) f(x)' without the required equality to zero; it should be 'delta x^T M_T(x) f(x) = 0'.
  2. [Section III-A.2 and Remark 3] There is an inconsistency in the definition of the phase variable: the text defines s_r(t) = arctan(x_2/x_1), while Remark 3 states theta = arctan(x_1/x_2). Please correct the typo in Remark 3.
  3. [Section III-D.2, Eq. (18)] The notation s_r^(d)(t) is used for the demonstration phase, but s_r is also the canonical phase variable of the rhythmic system; please distinguish these or add a clarifying sentence.
  4. [Section IV-B.2, Eq. (24)] The activation function alpha_i(t) in Eq. (24) uses the same parameters t_1, t_2, t_3, t_4 for every primitive i, but Eq. (23) gives each primitive its own time offset; please clarify that these parameters are per-primitive and chosen by the user, and state the constraints needed to ensure the intended smooth transition.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the contraction-theory results are independent theorems, and the parallel combination is a direct application; the sequential-combination gap is an unsupported assertion rather than a by-construction reduction.

full rationale

The paper's central derivation applies standard contraction theory to dynamic movement primitives. The transformation system (Eq. 15 and Eq. 21) is a linear stable system driven by a primitive input p(x) (Eq. 19). Reproducing a demonstrated trajectory after least-squares imitation learning (Eq. 16) is a fitting procedure, not a prediction, and the spatial/temporal scaling and rotation properties follow by linearity of the transformation system rather than by circular definition. The parallel combination (Eq. 22) is a weighted sum of inputs into one linear system; the cited parallel-contraction result (Eq. 4 and Eq. 6) is an established, parameter-free theorem whose assumptions are at least stated (identical dimensions, common contraction metric, nonnegative weights summing to one). The contraction-theory references [31], [32], [39], and [40] are independent mathematical results, not fitted values or conclusions derived from the present paper's own output. The sequential combination (Eq. 23) is essentially the same weighted-sum form with time-varying weights, but the paper cites Section II-B.3 for stability even though the theorem there (Eq. 8) concerns one-way coupling of two identical systems via u(x1)-u(x2), not a weighted superposition of different primitive inputs. This is a genuine gap in the stability argument for the sequential case, but it is a missing or inapplicable proof, not a circular reduction: the claim does not become true by construction, and no fitted parameter is renamed as a prediction. Therefore no circular step meets the evidentiary threshold; the appropriate finding is no significant circularity.

Assumptions & free parameters 5 free parameters · 4 assumptions · 0 invented entities

The framework rests on standard contraction theory, on the assumption of exact imitation learning, and on the transverse contraction property of the oscillator. The sequential combination stability is not derived from the cited theorems, and the metric proof in Remark 3 is under-specified.

free parameters (5)
  • m_theta_theta(r,theta) = unspecified; chosen to satisfy an inequality in Remark 3
    Introduced to make the transverse contraction metric positive definite; no explicit form is given.
  • gamma (oscillator radius) = 1 (common choice)
    Sets the limit cycle radius of the Andronov-Hopf oscillator; arbitrary positive value.
  • epsilon (transverse contraction region) = arbitrary positive constant
    Defines the region r >= epsilon for the transverse contraction proof; lambda_T = 4*epsilon^2 depends on it.
  • activation function times t1..t4 = hand-picked per simulation (e.g., Figure 3)
    Used in Eq. (24) for sequential combination; chosen to achieve smooth transitions.
  • time offsets t_off = e.g., 0.0, 0.969, 1.939 s in Figure 3A
    Introduced in Eq. (23) for sequential combination; hand-tuned for the examples.
assumptions (4)
  • standard math The contraction and transverse contraction criteria (Eqs. 2, 3) and the combination theorems (Eqs. 4, 7, 8) are valid and applicable to the DMP systems.
    Section II states these as known results from [31], [32], [35]-[39] and omits proofs.
  • domain assumption Imitation learning (Eq. 16) yields weights W such that the primitive input p(x) in Eq. (19) exactly reproduces the demonstrated trajectory y_d(t).
    Section III-D and III-E use this to claim y(t) -> y_d(t); in practice there is approximation error.
  • domain assumption The Andronov-Hopf oscillator (Eq. 10) is transverse contracting on R^2 without the origin, with metric in Eq. (12).
    Remark 3 sketches a metric but leaves m_theta_theta unspecified and asserts lambda_T = 4*epsilon^2 without a complete verification.
  • domain assumption All primitives combined in parallel must be learned under identical DMP parameters so they share a common contraction metric.
    Section V acknowledges this constraint on the method; it limits modularity across differently trained primitives.

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Pith. "Pith review of Combining Movement Primitives with Contraction Theory." pith.science (2026). https://pith.science/paper/5I4GQWNP

@misc{pith2026250109198,
  author       = {Pith},
  title        = {Pith review of: Combining Movement Primitives with Contraction Theory},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5I4GQWNP}},
  note         = {Machine review of arXiv:2501.09198}
}
read the original abstract

This paper presents a modular framework for motion planning using movement primitives. Central to the approach is Contraction Theory, a modular stability tool for nonlinear dynamical systems. The approach extends prior methods by achieving parallel and sequential combinations of both discrete and rhythmic movements, while enabling independent modulation of each movement. This modular framework enables a divide-and-conquer strategy to simplify the programming of complex robot motion planning. Simulation examples illustrate the flexibility and versatility of the framework, highlighting its potential to address diverse challenges in robot motion planning.

Figures

Figures reproduced from arXiv: 2501.09198 by the authors.

Figure 1
Figure 1. Dynamic movement primitives (DMP) for (A-D) discrete and (E-F) rhythmic movements and their (B, F) scaling, (C, G) rotational, and [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Parallel combination of both discrete and rhythmic DMPs. (A) Parallel combination of discrete DMPs. For the combination, primitive inputs to [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Sequential combination of both discrete and rhythmic DMPs. (A) Sequential combination of three discrete DMPs, which generate Alphabets “R”, [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: Parallel and sequential combinations of both discrete and rhythmic [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]

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Reference graph

Works this paper leans on

51 extracted references · 43 canonical work pages

  1. [1]

    Robot arm control exploiting natural dynamics,

    M. M. Williamson, “Robot arm control exploiting natural dynamics,” Ph.D. dissertation, Massachusetts Institute of Technology, 1999

    work page 1999

  2. [2]

    Learning attractor landscapes for learning motor primitives,

    A. Ijspeert, J. Nakanishi, and S. Schaal, “Learning attractor landscapes for learning motor primitives,” Advances in neural information pro- cessing systems, vol. 15, 2002

    work page 2002

  3. [3]

    Dynamics systems vs. optimal control—a unifying view,

    S. Schaal, P. Mohajerian, and A. Ijspeert, “Dynamics systems vs. optimal control—a unifying view,” Progress in brain research , vol. 165, pp. 425–445, 2007

    work page 2007

  4. [4]

    Pattern generators with sensory feedback for the control of quadruped locomotion,

    L. Righetti and A. J. Ijspeert, “Pattern generators with sensory feedback for the control of quadruped locomotion,” in 2008 IEEE International Conference on Robotics and Automation . IEEE, 2008, pp. 819–824

    work page 2008

  5. [5]

    Learning and generalization of motor skills by learning from demonstration,

    P. Pastor, H. Hoffmann, T. Asfour, and S. Schaal, “Learning and generalization of motor skills by learning from demonstration,” in2009 IEEE International Conference on Robotics and Automation . IEEE, 2009, pp. 763–768

    work page 2009

  6. [6]

    Dynamic primitives of motor behavior,

    N. Hogan and D. Sternad, “Dynamic primitives of motor behavior,” Biological Cybernetics, vol. 106, no. 11-12, pp. 727–739, 2012

    work page 2012

  7. [7]

    Dynamical movement primitives: learning attractor models for motor behaviors,

    A. J. Ijspeert, J. Nakanishi, H. Hoffmann, P. Pastor, and S. Schaal, “Dynamical movement primitives: learning attractor models for motor behaviors,” Neural Computation, vol. 25, no. 2, pp. 328–373, 2013

    work page 2013

  8. [8]

    A correct formulation for the orientation dynamic movement primitives for robot control in the cartesian space,

    L. Koutras and Z. Doulgeri, “A correct formulation for the orientation dynamic movement primitives for robot control in the cartesian space,” in Conference on robot learning . PMLR, 2020, pp. 293–302

    work page 2020

Show all 51 references
  1. [9]

    Learning dynamical systems with bifurcations,

    F. Khadivar, I. Lauzana, and A. Billard, “Learning dynamical systems with bifurcations,” Robotics and Autonomous Systems , vol. 136, p. 103700, 2021

    2021

  2. [10]

    Locally active globally stable dynam- ical systems: Theory, learning, and experiments,

    N. Figueroa and A. Billard, “Locally active globally stable dynam- ical systems: Theory, learning, and experiments,” The International Journal of Robotics Research , vol. 41, no. 3, pp. 312–347, 2022

    2022

  3. [11]

    Billard, S

    A. Billard, S. Mirrazavi, and N. Figueroa, Learning for adaptive and reactive robot control: a dynamical systems approach . Mit Press, 2022

    2022

  4. [12]

    Dynamic movement primitives in robotics: A tutorial survey,

    M. Saveriano, F. J. Abu-Dakka, A. Kramberger, and L. Peternel, “Dynamic movement primitives in robotics: A tutorial survey,” The International Journal of Robotics Research, vol. 42, no. 13, pp. 1133– 1184, 2023

    2023

  5. [13]

    Robot control based on motor primitives: A comparison of two approaches,

    M. C. Nah, J. Lachner, and N. Hogan, “Robot control based on motor primitives: A comparison of two approaches,” The International Journal of Robotics Research , p. 02783649241258782, 2024

    2024

  6. [14]

    A unified formulation of geometry-aware discrete dynamic movement primitives,

    F. J. Abu-Dakka, M. Saveriano, and V . Kyrki, “A unified formulation of geometry-aware discrete dynamic movement primitives,” Neuro- computing, p. 128056, 2024

    2024

  7. [15]

    Learning stable nonlinear dynamical systems with gaussian mixture models,

    S. M. Khansari-Zadeh and A. Billard, “Learning stable nonlinear dynamical systems with gaussian mixture models,” IEEE Transactions on Robotics, vol. 27, no. 5, pp. 943–957, 2011

    2011

  8. [16]

    A novel dmp formulation for global and frame independent spatial scaling in the task space,

    L. Koutras and Z. Doulgeri, “A novel dmp formulation for global and frame independent spatial scaling in the task space,” in 2020 29th IEEE International Conference on Robot and Human Interactive Communication (RO-MAN). IEEE, 2020, pp. 727–732

    2020

  9. [17]

    Learning periodic skills for robotic manipulation: Insights on orientation and impedance,

    F. Abu-Dakka, M. Saveriano, and L. Peternel, “Learning periodic skills for robotic manipulation: Insights on orientation and impedance,” Robotics and Autonomous Systems , p. 104763, 2024

    2024

  10. [18]

    Biologically- inspired dynamical systems for movement generation: Automatic real-time goal adaptation and obstacle avoidance,

    H. Hoffmann, P. Pastor, D.-H. Park, and S. Schaal, “Biologically- inspired dynamical systems for movement generation: Automatic real-time goal adaptation and obstacle avoidance,” in 2009 IEEE international conference on robotics and automation . IEEE, 2009, pp. 2587–2592

    2009

  11. [19]

    A dynamical system approach to realtime obstacle avoidance,

    S. M. Khansari-Zadeh and A. Billard, “A dynamical system approach to realtime obstacle avoidance,” Autonomous Robots, vol. 32, pp. 433– 454, 2012

    2012

  12. [20]

    Natural actor-critic,

    J. Peters and S. Schaal, “Natural actor-critic,” Neurocomputing, vol. 71, no. 7-9, pp. 1180–1190, 2008

    2008

  13. [21]

    Learning motor primitives for robotics,

    J. Kober and J. Peters, “Learning motor primitives for robotics,” in 2009 IEEE International Conference on Robotics and Automation . IEEE, 2009, pp. 2112–2118

    2009

  14. [22]

    Task-oriented generalization of dynamic movement primitive,

    Y . Zhou and T. Asfour, “Task-oriented generalization of dynamic movement primitive,” in 2017 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS) . IEEE, 2017, pp. 3202–3209

    2017

  15. [23]

    Trends and challenges in robot manipula- tion,

    A. Billard and D. Kragic, “Trends and challenges in robot manipula- tion,” Science, vol. 364, no. 6446, p. eaat8414, 2019

    2019

  16. [24]

    Fast obstacle avoidance based on real-time sensing,

    L. Huber, J.-J. Slotine, and A. Billard, “Fast obstacle avoidance based on real-time sensing,” IEEE Robotics and Automation Letters , vol. 8, no. 3, pp. 1375–1382, 2022

    2022

  17. [25]

    Divide et impera: Learning impedance families for peg-in-hole as- sembly,

    J. Lachner, F. Tessari, A. M. West Jr, M. C. Nah, and N. Hogan, “Divide et impera: Learning impedance families for peg-in-hole as- sembly,” arXiv preprint arXiv:2410.01054 , 2024

    2024 arXiv

  18. [26]

    S. H. Strogatz, Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering . CRC press, 2018

    2018

  19. [27]

    Sequential com- position of dynamically dexterous robot behaviors,

    R. R. Burridge, A. A. Rizzi, and D. E. Koditschek, “Sequential com- position of dynamically dexterous robot behaviors,” The International Journal of Robotics Research , vol. 18, no. 6, pp. 534–555, 1999

    1999

  20. [28]

    Funnel libraries for real-time robust feedback motion planning,

    A. Majumdar and R. Tedrake, “Funnel libraries for real-time robust feedback motion planning,” The International Journal of Robotics Research, vol. 36, no. 8, pp. 947–982, 2017

    2017

  21. [29]

    Movement generation using dynamical systems: a humanoid robot performing a drumming task,

    S. Degallier, C. P. Santos, L. Righetti, and A. Ijspeert, “Movement generation using dynamical systems: a humanoid robot performing a drumming task,” in 2006 6th IEEE-RAS International Conference on Humanoid Robots. IEEE, 2006, pp. 512–517

    2006

  22. [30]

    Encoding of periodic and their transient motions by a single dynamic move- ment primitive,

    J. Ernesti, L. Righetti, M. Do, T. Asfour, and S. Schaal, “Encoding of periodic and their transient motions by a single dynamic move- ment primitive,” in 2012 12th IEEE-RAS international conference on humanoid robots (humanoids 2012) . IEEE, 2012, pp. 57–64

    2012

  23. [31]

    On contraction analysis for non- linear systems,

    W. Lohmiller and J.-J. E. Slotine, “On contraction analysis for non- linear systems,” Automatica, vol. 34, no. 6, pp. 683–696, 1998

    1998

  24. [32]

    Modular stability tools for distributed computation and control,

    J.-J. E. Slotine, “Modular stability tools for distributed computation and control,” International Journal of Adaptive Control and Signal Processing, vol. 17, no. 6, pp. 397–416, 2003

    2003

  25. [33]

    Motion primitives for robotic flight control,

    B. E. Perk and J.-J. E. Slotine, “Motion primitives for robotic flight control,” arXiv preprint cs/0609140 , 2006

    2006 arXiv

  26. [34]

    Sparse control for dynamic move- ment primitives,

    P. M. Wensing and J.-J. Slotine, “Sparse control for dynamic move- ment primitives,” IFAC-PapersOnLine, vol. 50, no. 1, pp. 10 114– 10 121, 2017

    2017

  27. [35]

    Modularity, evolution, and the binding problem: a view from stability theory,

    J.-J. Slotine and W. Lohmiller, “Modularity, evolution, and the binding problem: a view from stability theory,”Neural networks, vol. 14, no. 2, pp. 137–145, 2001

    2001

  28. [36]

    Contraction theory for nonlinear stability analysis and learning-based control: A tutorial overview,

    H. Tsukamoto, S.-J. Chung, and J.-J. E. Slotine, “Contraction theory for nonlinear stability analysis and learning-based control: A tutorial overview,” Annual Reviews in Control , vol. 52, pp. 135–169, 2021

    2021

  29. [37]

    Bullo, Contraction theory for dynamical systems

    F. Bullo, Contraction theory for dynamical systems. Francesco Bullo, 2022

    2022

  30. [38]

    Contraction theory on riemannian manifolds,

    J. W. Simpson-Porco and F. Bullo, “Contraction theory on riemannian manifolds,” Systems & Control Letters , vol. 65, pp. 74–80, 2014

    2014

  31. [39]

    Transverse contraction criteria for existence, stability, and robustness of a limit cycle,

    I. R. Manchester and J.-J. E. Slotine, “Transverse contraction criteria for existence, stability, and robustness of a limit cycle,” Systems & Control Letters, vol. 63, pp. 32–38, 2014

    2014

  32. [40]

    On partial contraction analysis for coupled nonlinear oscillators,

    W. Wang and J.-J. E. Slotine, “On partial contraction analysis for coupled nonlinear oscillators,” Biological cybernetics, vol. 92, no. 1, pp. 38–53, 2005

    2005

  33. [41]

    Movement imitation with nonlinear dynamical systems in humanoid robots,

    A. J. Ijspeert, J. Nakanishi, and S. Schaal, “Movement imitation with nonlinear dynamical systems in humanoid robots,” in Proceedings 2002 IEEE International Conference on Robotics and Automation (Cat. No. 02CH37292) , vol. 2. IEEE, 2002, pp. 1398–1403

    2002

  34. [42]

    Control, plan- ning, learning, and imitation with dynamic movement primitives,

    S. Schaal, J. Peters, J. Nakanishi, and A. Ijspeert, “Control, plan- ning, learning, and imitation with dynamic movement primitives,” in Workshop on Bilateral Paradigms on Humans and Humanoids, IEEE International Conference on Intelligent Robots and Systems , 2003, pp. 1–21

    2003

  35. [43]

    Ren, Synthesis of Linear Distributed Control and Coupled Os- cillators with Multiple Limit Cycles

    K. Ren, Synthesis of Linear Distributed Control and Coupled Os- cillators with Multiple Limit Cycles . University of California, Los Angeles, 2021

    2021

  36. [44]

    A contraction theory approach to stochastic incremental stability,

    Q.-C. Pham, N. Tabareau, and J.-J. Slotine, “A contraction theory approach to stochastic incremental stability,” IEEE Transactions on Automatic Control, vol. 54, no. 4, pp. 816–820, 2009

    2009

  37. [45]

    A generalized path integral control approach to reinforcement learning,

    E. Theodorou, J. Buchli, and S. Schaal, “A generalized path integral control approach to reinforcement learning,” The Journal of Machine Learning Research, vol. 11, pp. 3137–3181, 2010

    2010

  38. [46]

    Adaptive control of exoskeleton robots for periodic assistive be- haviours based on emg feedback minimisation,

    L. Peternel, T. Noda, T. Petri ˇc, A. Ude, J. Morimoto, and J. Babi ˇc, “Adaptive control of exoskeleton robots for periodic assistive be- haviours based on emg feedback minimisation,” PloS one , vol. 11, no. 2, p. e0148942, 2016

    2016

  39. [47]

    Reinforcement learning in robotics: A survey,

    J. Kober, J. A. Bagnell, and J. Peters, “Reinforcement learning in robotics: A survey,” The International Journal of Robotics Research , vol. 32, no. 11, pp. 1238–1274, 2013

    2013

  40. [48]

    Is imitation learning the route to humanoid robots?

    S. Schaal, “Is imitation learning the route to humanoid robots?” Trends in cognitive sciences , vol. 3, no. 6, pp. 233–242, 1999

    1999

  41. [49]

    On the modularity of elementary dynamic actions,

    M. C. Nah, J. Lachner, F. Tessari, and N. Hogan, “On the modularity of elementary dynamic actions,” in 2024 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS) . IEEE, 2024

    2024

  42. [50]

    Merging position and orienta- tion motion primitives,

    M. Saveriano, F. Franzel, and D. Lee, “Merging position and orienta- tion motion primitives,” in 2019 International Conference on Robotics and Automation (ICRA) . IEEE, 2019, pp. 7041–7047

    2019

  43. [51]

    Dynamic movement primitives for moving goals with temporal scaling adaptation,

    L. Koutras and Z. Doulgeri, “Dynamic movement primitives for moving goals with temporal scaling adaptation,” in 2020 IEEE In- ternational Conference on Robotics and Automation (ICRA) . IEEE, 2020, pp. 144–150

    2020

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.