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REVIEW 2 major objections 6 minor 18 references

BMP: Bridging the Gap between B-Spline and Movement Primitives

T0 review · 2 major / 6 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read By rewriting B-splines as movement primitives with control-point weights, BMPs let B-splines model trajectory distributions for imitation and reinforcement learning while exactly satisfying start and end position and velocity conditions.

desk verdict A useful but flawed twist on ProMP: the boundary-velocity equations miss a factor of 1/T, so the paper's central guarantee doesn't hold as stated. read the letter →

arxiv 2411.10336 v2 pith:RJDOAD3W submitted 2024-11-15 cs.RO

Weiran Liao , Ge Li , Hongyi Zhou , Rudolf Lioutikov , Gerhard Neumann This is my paper · ORCID
classification cs.RO
keywords B-splineMovementPrimitivesProbabilisticImitationLearningEpisodicReinforcementTrajectoryDistributionsBoundaryConditionsRobot
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 introduces B-spline Movement Primitives (BMPs), a representation that combines the boundary-condition guarantees of B-splines with the probabilistic modeling of ProMPs. The core move is to treat B-spline control points as the weights in a linear-Gaussian trajectory model, so the distribution over trajectories is Gaussian with mean and covariance given by the spline basis applied to a learned distribution over control points. Boundary conditions are enforced deterministically by fixing the first and last control points through linear equations, so every sample from the distribution respects the specified start and end positions and velocities. If correct, BMP gives imitation and episodic reinforcement learning a trajectory representation that is both expressive and constraint-satisfying, which the paper demonstrates on digit-writing, goal-reaching, and robot box-pushing tasks.

What carries the argument

The central object is the clamped uniform B-spline, defined by the Cox–de Boor recursion over a knot vector with repeated end knots so that the curve starts at the first control point and ends at the last. Its basis functions have local support, its derivative is another B-spline whose control points are differences of the original ones via $c^{(1)}_i = \frac{p}{\Delta}(c_{i+1} - c_i)$, and its convex hull property bounds the curve between its control points. Treating the interior control points as the weights of a ProMP-style linear-Gaussian model yields a trajectory distribution whose mean and covariance are computed by propagating a Gaussian over those control points through the spline basis, with boundary conditions folded in as a deterministic offset.

What would settle it

Run BMP on a demonstration set with two clearly separated movement modes, such as reaching above and below an obstacle, and evaluate the likelihood of the demonstrations under the fitted Gaussian trajectory distribution; with distinct modes the fitted Gaussian will place most probability mass between them, yielding low likelihood. A second check is a physical robot experiment that measures the achieved end velocity and compares it to the prescribed value, because the paper derives velocities from a phase derivative divided by duration $T$ and provides no hardware verification that exact end-velocity conditions are met.

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Extended reading notes

Core claim

The central claim is that B-splines, reformulated as movement primitives through their control points, inherit the probabilistic machinery of ProMPs while preserving exact boundary-condition satisfaction. Concretely, for given initial and final position and velocity conditions, the single-DoF trajectory distribution is Gaussian with mean $\Phi^T_{[2:n-3]} \mu_c + d$ and covariance $\Phi^T_{[2:n-3]} \Sigma \Phi_{[2:n-3]} + I\sigma_y^2$, where the selected control-point submatrix excludes the boundary-fixed endpoints and $d$ is the deterministic part fixed by the boundary conditions. This means the learned representation only needs to model the residual freedom of the interior control points, while the first two and last two control points are pinned down by equations such as $c_0 = y_0$ and $c_1 = \frac{\Delta}{p}\dot{y}_0 + c_0$, with the symmetric equations at the end. The paper shows that B-splines alone regress digit-writing trajectories with lower mean squared error than ProDMP and ProMP, can represent constant segments without wiggles, and, when embedded in the BBRL and TCE algorithms, reach or exceed the success rates of ProDMP while also respecting prescribed end velocities.

Load-bearing premise

The argument rests on the assumption that a Gaussian distribution over the learned control points adequately captures the trajectory variability in real imitation and reinforcement learning tasks; if the true variability is strongly non-Gaussian or multi-modal, BMP still enforces boundary conditions but its probabilistic predictions will be misleading.

Editorial extensions

If this is right

  • BMP can be plugged into episodic reinforcement learning algorithms such as BBRL and TCE, and its trajectory distribution enables segment-level advantage-weighted policy updates that step-based algorithms cannot perform.
  • In the box-pushing task, BMP combined with TCE improves convergence speed and final success rate compared to BMP with BBRL, while enforcing zero end-velocity conditions that ProDMP fails to meet.
  • B-splines represent constant trajectory segments without oscillation, which ProDMP cannot do, so BMP is better suited for tasks with holding phases.
  • Imposing initial position and velocity and end position and/or velocity requires only fixing two control points at each end, and the same derivative recursion extends the construction to acceleration-level boundary conditions.

Reading between the lines

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

  • The derivative recursion means BMP could impose acceleration or jerk boundary conditions as well, by fixing additional control points, which would matter for kinodynamic planning and physical feasibility.
  • The convex hull property of B-splines, which the authors mention as future work, could be used to certify velocity or torque bounds without sampling, giving hard safety guarantees in RL.
  • The linear-Gaussian assumption limits BMP to unimodal trajectory distributions; multi-modal demonstrations would need mixtures or latent variables, and the boundary-condition construction itself would survive but the probabilistic inference would not.
  • B-spline basis locality means changing one control point affects only a local segment of the trajectory, which may make BMP's optimization landscape smoother for black-box RL than the global-basis ProDMP.
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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

2 major / 6 minor

Summary. This paper introduces B-spline Movement Primitives (BMPs), a representation that casts clamped uniform B-splines as probabilistic movement primitives in the style of ProMP. Trajectories are written as linear combinations of B-spline basis functions with control-point weights, and a Gaussian distribution over the interior control points yields a Gaussian trajectory distribution (Eq. 13). Boundary conditions on position and velocity are enforced by fixing the first and last one or two control points via linear equations (Eqs. 11-12). The paper reports three sets of experiments: a regression comparison of B-spline versus ProMP/ProDMP on digit-writing trajectories, a toy planning task, an IL experiment with a conditional encoder-decoder for digit images, and an episodic RL experiment on a box-pushing task comparing BMP with ProDMP under BBRL and TCE.

Significance. The core idea is attractive and the mathematical development in Section 3 is mostly sound: the Cox-de Boor recursion, derivative relation (Eqs. 9-10), and clamped-endpoint construction are correctly stated, and the formulation as a linear-Gaussian trajectory distribution is a natural and potentially useful bridge between B-spline planning and probabilistic MP learning. The regression and RL results are suggestive of practical benefits. However, the central claim of exact boundary-velocity satisfaction is undermined by a missing phase-to-time conversion factor in the boundary-condition equations, and the IL experiment does not include any baseline MP. These issues need to be resolved before the paper's claims are fully supported.

major comments (2)
  1. [Section 3, Eqs. (11)-(12)] The boundary-condition equations are off by a factor of 1/T. The text after Eq. (9) explicitly states that the phase velocity c^(1) must be divided by the duration T to obtain real velocity. Yet Eq. (11) sets c^(1)_0 = ˙y0 and Eq. (12) sets c^(1)_{n−2} = ˙ye, where ˙y0 and ˙ye are described as desired initial and end velocities. If these are physical velocities, the actual real velocity at the start is c^(1)_0 / T = ˙y0 / T, so the condition is satisfied only for T = 1; if they are phase velocities, the paper never provides a mechanism to impose a real velocity boundary condition. This directly affects the central claim that BMPs retain their ability to satisfy boundary conditions and the velocity demonstration in Fig. 5(b). The equations should use T ˙y0 and T ˙ye (or equivalently define c^(1) as time-derivative control points), and the authors should verify that the experiments used the corrected formula.
  2. [Section 4.2, Imitation Learning] The digit-writing IL experiment reports only the BMP model and does not include ProMP or ProDMP baselines trained with the same encoder-decoder architecture and the same pairwise log-likelihood loss. As a result, this experiment cannot support the abstract's claim that BMPs offer greater expressiveness compared to existing MP variants in an IL setting. Adding at least a ProMP baseline (and ideally ProDMP) is necessary to substantiate the comparative claim, or the claim should be explicitly restricted to the regression and RL experiments.
minor comments (6)
  1. [Section 2, Eq. (2)] The noise term is written as "ϵy ∼ (0, σy)" without the normal-distribution symbol; it should be "ϵy ∼ N(0, σy)".
  2. [Section 3, Eq. (6)] The definition of N_{i,0} is garbled: "if ui ≤ u < ui+1, if ui ≤ u ≤ ui+1 for i = n − 1" should be "if ui ≤ u < ui+1, and if ui ≤ u ≤ ui+1 for i = m − 1" (or similar).
  3. [Section 3, after Eq. (8)] The notation "u0, u1, ..., up = 0" is imprecise; it should be "u0 = u1 = ... = up = 0", and similarly for the right end. Also, "i ∈ [p, ..., m-p-1]" is nonstandard notation for an index range.
  4. [Section 3, Eq. (13)] The index set [2:n-3] is ambiguous; with n control points and four boundary constraints, the free control points are c_2, ..., c_{n-3}, and the notation should be defined explicitly as inclusive of both endpoints or as a set.
  5. [Section 3, Fig. 1(a) and text] The basis-function notation N_{1,0} and N_{2,0} used for higher-degree basis functions is inconsistent with the standard B-spline notation; the figure and text should use N_{0,1} and N_{0,2} (or clearly redefine the indexing).
  6. [Acknowledgments] The acknowledgments section contains placeholder boilerplate ("If a paper is accepted, the final camera-ready version will (and probably should) include acknowledgments...") that should be removed or replaced with actual acknowledgments.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: BMP is an explicit combination of standard B-spline parameterization and the ProMP linear-Gaussian model, with boundary conditions solved from linear equations.

full rationale

The derivation chain is self-contained in the relevant sense. Equation (8) writes a clamped uniform B-spline in ProMP-like basis form, Eqs. (9)-(10) use the standard B-spline derivative identity, and Eqs. (11)-(12) solve linear equations to express the first two or last two control points in terms of imposed position and velocity conditions. This is inversion of a linear relationship, not fitting a parameter to the quantity later reported as a prediction. Equation (13) explicitly transfers the ProMP linear-Gaussian model to the interior control-point weights; the Gaussian assumption is inherited from ProMP rather than derived from B-splines, so there is no hidden equivalence between the model's output and its input. The paper's self-citations ([4], [9], [10]) appear as baselines and as algorithm components, not as evidence that BMP is valid or as imported uniqueness theorems. The empirical claims are checked against external data and tasks: digit-trajectory regression, MNIST image-conditioned generation, and a box-pushing benchmark. The paper even admits that the pairwise log-likelihood loss in Eq. (14) is biased relative to the exact trajectory log-likelihood, which is a stated limitation rather than a circular step. No fitted parameter is renamed as a prediction, and no self-cited result is load-bearing for the central representation. Any concern about the missing 1/T factor in phase-velocity boundary conditions is a correctness issue, not a circularity issue. Therefore the appropriate finding is no significant circularity, score 0.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The derivation uses standard B-spline facts and the ProMP linear-Gaussian model. The only hand-chosen quantities are basis count, degree, and noise variance, none of which are fitted to make the central claim. No new physical entities are introduced.

free parameters (3)
  • Number of B-spline basis functions / control points n = 9 for RL experiments; swept in Fig. 2a
    Chosen by hand for the comparisons; expressiveness depends on this count and no sensitivity analysis is given.
  • B-spline degree p = not reported for experiments
    Degree controls smoothness and local support; the experimental sections do not state the degree used.
  • Noise variance sigma_y^2 = not reported
    Additive observation variance in Eq. (13); no fitting or tuning procedure is stated.
assumptions (5)
  • standard math Cox-de Boor recursion defines the B-spline basis functions.
    Used in Eqs. (6) and (7), taken as background from Prautzsch et al. [14].
  • standard math Clamped uniform knot vectors make the curve pass through the first and last control points.
    Repeated end knots make the first and last basis functions evaluate to 1 at the endpoints, used in Section 3 and Fig. 1c to impose boundary conditions.
  • standard math The derivative of a B-spline is a lower-degree B-spline whose control points are differences of the original control points.
    Used in Eqs. (9) and (10) to prescribe velocity boundary conditions.
  • domain assumption A Gaussian distribution over control points induces a Gaussian trajectory distribution.
    Eq. (13) assumes the linear-Gaussian model inherited from ProMP; the paper does not validate this for non-Gaussian trajectory distributions.
  • standard math The convex hull property of B-splines can be used to impose soft velocity bounds via control point constraints.
    Used in the goal-reaching toy task in Section 4.1 and Fig. 3.

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Cite this review

Pith. "Pith review of BMP: Bridging the Gap between B-Spline and Movement Primitives." pith.science (2026). https://pith.science/paper/RJDOAD3W

@misc{pith2026241110336,
  author       = {Pith},
  title        = {Pith review of: BMP: Bridging the Gap between B-Spline and Movement Primitives},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RJDOAD3W}},
  note         = {Machine review of arXiv:2411.10336}
}
read the original abstract

This work introduces B-spline Movement Primitives (BMPs), a new Movement Primitive (MP) variant that leverages B-splines for motion representation. B-splines are a well-known concept in motion planning due to their ability to generate complex, smooth trajectories with only a few control points while satisfying boundary conditions, i.e., passing through a specified desired position with desired velocity. However, current usages of B-splines tend to ignore the higher-order statistics in trajectory distributions, which limits their usage in imitation learning (IL) and reinforcement learning (RL), where modeling trajectory distribution is essential. In contrast, MPs are commonly used in IL and RL for their capacity to capture trajectory likelihoods and correlations. However, MPs are constrained by their abilities to satisfy boundary conditions and usually need extra terms in learning objectives to satisfy velocity constraints. By reformulating B-splines as MPs, represented through basis functions and weight parameters, BMPs combine the strengths of both approaches, allowing B-splines to capture higher-order statistics while retaining their ability to satisfy boundary conditions. Empirical results in IL and RL demonstrate that BMPs broaden the applicability of B-splines in robot learning and offer greater expressiveness compared to existing MP variants.

Figures

Figures reproduced from arXiv: 2411.10336 by the authors.

Figure 1
Figure 1. (a) The first basis for B-spline of degree 0, 1, 2, 3. (b) Basis functions for a degree [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. (a) The log-scaled averaged regression MSE loss on 20000 3-second digit-writing tra [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Using CMA-ES [15] algorithm for 50 iterations to generate trajectories from given initial states to reach goal states while avoiding obstacles. (a) Generated best trajectory path within 50 iterations. (b),(c) The velocity profile of the B-spline and ProDMP trajectory, where the red dash lines are the velocity bounds. Cholesky decomposition Lw of the covariance Σw of the weights distribution of the BMP. Then BMP maps… view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: (a) A batch of digit images and reconstructed trajectories using BMP. (b) Sampling trajec [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]
Figure 5
Figure 5. Figure 5: (a) The averaged success rate of 4 random seeds. Both BMP (B-spline) and ProDMP use [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

18 extracted references · 11 canonical work pages

  1. [1]

    A. J. Ijspeert, J. Nakanishi, H. Hoffmann, P. Pastor, and S. Schaal. Dynamical movement primitives: Learning attractor models for motor behaviors. Neural Computation, 25(2):328– 373, 2013. doi:10.1162/NECO a 00393

    doi:10.1162/neco 2013

  2. [2]

    Paraschos, C

    A. Paraschos, C. Daniel, J. R. Peters, and G. Neumann. Probabilistic movement prim- itives. In C. Burges, L. Bottou, M. Welling, Z. Ghahramani, and K. Weinberger, ed- itors, Advances in Neural Information Processing Systems , volume 26. Curran Asso- ciates, Inc., 2013. URL https://proceedings.neurips.cc/paper_files/paper/ 2013/file/e53a0a2978c28872a4505bd...

    work page 2013

  3. [3]

    Y . Zhou, J. Gao, and T. Asfour. Learning via-point movement primitives with inter-and ex- trapolation capabilities. In 2019 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), pages 4301–4308. IEEE, 2019

    work page 2019

  4. [4]

    G. Li, Z. Jin, M. V olpp, F. Otto, R. Lioutikov, and G. Neumann. Prodmp: A unified perspective on dynamic and probabilistic movement primitives. IEEE Robotics and automation letters , page 1–8, 2023. ISSN 2377-3766. doi:10.1109/LRA.2023.3248443

    arXiv 2023

  5. [5]

    Usenko, L

    V . Usenko, L. V on Stumberg, A. Pangercic, and D. Cremers. Real-time trajectory replanning for MA Vs using uniform B-splines and a 3D circular buffer. In 2017 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS) , pages 215–222, Vancouver, BC, Sept

    work page 2017

  6. [6]

    B. Zhou, F. Gao, L. Wang, C. Liu, and S. Shen. Robust and Efficient Quadrotor Trajec- tory Generation for Fast Autonomous Flight. IEEE Robotics and Automation Letters , 4(4): 3529–3536, Oct. 2019. ISSN 2377-3766. doi:10.1109/LRA.2019.2927938. URL https: //ieeexplore.ieee.org/document/8758904. Conference Name: IEEE Robotics and Au- tomation Letters

    arXiv 2019

  7. [7]

    Kicki, P

    P. Kicki, P. Liu, D. Tateo, H. Bou-Ammar, K. Walas, P. Skrzypczy ´nski, and J. Peters. Fast Kinodynamic Planning on the Constraint Manifold with Deep Neural Networks, Jan. 2023. URL http://arxiv.org/abs/2301.04330. arXiv:2301.04330 [cs]

    arXiv 2023

  8. [8]

    Bridging the gap between Learning-to-plan, Motion Primitives and Safe Reinforcement Learning

    P. Kicki, D. Tateo, P. Liu, J. Guenster, J. Peters, and K. Walas. Bridging the gap between learning-to-plan, motion primitives and safe reinforcement learning, 2024. URL https:// arxiv.org/abs/2408.14063

    work page Pith review arXiv 2024

Show all 18 references
  1. [9]

    G. Li, H. Zhou, D. Roth, S. Thilges, F. Otto, R. Lioutikov, and G. Neumann. Open the black box: Step-based policy updates for temporally-correlated episodic reinforcement learning. In ICLR 2024 : The Twelfth International Conference on Learning Representations, Vienna, 7th- 11...

    2024

  2. [10]

    F. Otto, O. Celik, H. Zhou, H. Ziesche, V . A. Ngo, and G. Neumann. Deep black-box rein- forcement learning with movement primitives. In 6th Annual Conference on Robot Learning (CoRL 2022), volume 205 of Proceedings of Machine Learning Research , pages 1244–1265. Machine Learn...

    2022

  3. [11]

    F. Otto, H. Zhou, O. Celik, G. Li, R. Lioutikov, and G. Neumann. Mp3: Movement primitive- based (re-) planning policy. arXiv preprint arXiv:2306.12729, 2023. 9

    2023 arXiv

  4. [12]

    G. Li, D. Tian, H. Zhou, X. Jiang, R. Lioutikov, and G. Neumann. Top-erl: Transformer-based off-policy episodic reinforcement learning. arXiv preprint arXiv:2410.09536, 2024

    2024 arXiv

  5. [13]

    Saveriano, F

    M. Saveriano, F. J. Abu-Dakka, A. Kramberger, and L. Peternel. Dynamic movement primi- tives in robotics: A tutorial survey. The International Journal of Robotics Research , 42(13): 1133–1184, 2023. doi:10.1177/02783649231201196. URL https://doi.org/10.1177/ 02783649231201196

    2023 doi

  6. [14]

    Prautzsch, W

    H. Prautzsch, W. Boehm, and M. Paluszny. B´ezier and B-Spline Techniques. Springer, Berlin, Heidelberg, 2002. ISBN 978-3-642-05240-8. doi:10.1007/978-3-662-04947-3

    2002 doi

  7. [15]

    N. Hansen. The CMA Evolution Strategy: A Tutorial, Mar. 2023. URL http://arxiv.org/ abs/1604.00772. arXiv:1604.00772 [cs, stat]

    2023 arXiv

  8. [16]

    Pahi ˇc, B

    R. Pahi ˇc, B. Ridge, A. Gams, J. Morimoto, and A. Ude. Training of deep neural net- works for the generation of dynamic movement primitives. Neural Networks, 127:121–131,

  9. [2017]

    ISBN 978-1-5386-2682-5

    IEEE. ISBN 978-1-5386-2682-5. doi:10.1109/IROS.2017.8202160. URL http:// ieeexplore.ieee.org/document/8202160/

    2017

  10. [2020]

    doi:https://doi.org/10.1016/j.neunet.2020.04.010

    ISSN 0893-6080. doi:https://doi.org/10.1016/j.neunet.2020.04.010. URL https: //www.sciencedirect.com/science/article/pii/S0893608020301301. 10

    2020 doi

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