Real-Time Generation of Near-Minimum-Energy Trajectories via Constraint-Informed Residual Learning

Basilio Lenzo; Cosimo Della Santina; Domenico Dona'; Giovanni Franzese; Paolo Boscariol

REVIEW 4 major objections 5 minor 37 references

Real-Time Generation of Near-Minimum-Energy Trajectories via Constraint-Informed Residual Learning

T0 review · 4 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read A learned residual, added to a standard cubic trajectory and forced to match boundary conditions, produces near-minimum-energy robot trajectories in real time, recovering 87.3% of the optimal planner's energy savings near the training…

desk verdict Read the abstract's headline numbers with care: the 50.8% far-from-dataset figure is a post-active-learning number on points added to the training set, the 87.3% appears in no table, and the underlying hard-boundary residual-learning construction is more solid than the headline claims. read the letter →

arxiv 2501.09450 v1 pith:IFGVM6O4 submitted 2025-01-16 cs.RO

Domenico Dona' , Giovanni Franzese , Cosimo Della Santina , Paolo Boscariol , Basilio Lenzo This is my paper · ORCID

classification cs.RO

keywords residuallearningminimum-energytrajectoryplanningreal-timeoptimalcontrolboundaryconditionembeddingGaussianprocessregressionneuralnetworkensembleactiveroboticmanipulators

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

This paper sets out to establish that a learned surrogate, trained on offline solutions of an optimal control problem, can generate near-minimum-energy trajectories for robotic manipulators fast enough for real-time use. The key move is residual learning: instead of predicting the whole trajectory, predict only the gap between a standard cubic trajectory and the optimal one, and hard-wire the boundary conditions into the predictor with a fixed scaling function. Tested on a pendulum, a SCARA arm, and a six-axis robot, the surrogate is claimed to capture 87.3% of the optimizer's energy savings near the training data, 50.8% far from it, while running two to three orders of magnitude faster. If true, this would let industrial manipulators re-plan energy-efficient motions online without invoking a numerical optimizer.

What carries the argument

The load-bearing device is the scaling function $s(\xi)=\xi^2(1-\xi)^2$. Multiplying a free-structured output by this factor forces both the value and the derivative to zero at $\xi=0$ and $\xi=1$, so the residual obeys the boundary conditions no matter what the regressor predicts; for Gaussian processes, the same multiplication is applied to the kernel, so the sampled functions obey them. The residual decomposition $q(t)=p(t)+r(t)$, with a cubic polynomial $p(t)$ as the prior, lets the regressor learn only the correction, and the probabilistic ensemble or GP sampling lets the planner choose the lowest-energy sample.

What would settle it

Take a fixed-time motion long enough that the optimal torque hits its motor limit for an interval, producing a sharp bend in the residual curve where the constraint turns on; if the learned residual, constrained to be smooth and zero at both endpoints by $\xi^2(1-\xi)^2$, cannot reproduce that bend even with abundant training data, the paper's representational premise is false.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is that the minimum-energy fixed-time point-to-point trajectory can be decomposed into a cubic prior plus a residual, and the residual can be learned while boundary conditions are enforced exactly by multiplying a free regressor's output, or its kernel, by $s(\xi)=\xi^2(1-\xi)^2$. With this architecture, both an ensemble of tanh neural networks and a sparse variational Gaussian process learned residuals from offline optimal-control data; near the training data they recover 87.3% of the optimal planner's energy savings, far from it 50.8%, while computing trajectories two to three orders of magnitude faster. The paper treats this as evidence that constraint-informed residual learning is a viable real-time surrogate for numerical optimal control in the tested regime.

Load-bearing premise

The load-bearing premise is that the gap between the simple cubic trajectory and the true minimum-energy trajectory varies smoothly with time and with the starting and ending conditions, so a smooth curve-shaping model, forced to zero at the start and end, can represent it; any sharp bend or discontinuity in that gap would defeat the method.

Editorial extensions

If this is right

Fixed-time point-to-point energy-optimal replanning for a six-axis manipulator can run within a real-time budget, here defined as one-tenth of the task time.
Because boundary conditions are enforced exactly, the generated trajectories are endpoint-feasible and can be handed directly to a controller without post-correction.
The same residual-plus-scaling template should transfer to any planning problem where a known feasible prior exists, not just cubic trajectories.
Active learning with uncertainty sampling sharply improves out-of-distribution performance, moving a GP from 2.98% to 50.93% energy savings, close to the 53.84% reference.
The speed gain of two to three orders of magnitude opens the door to online trajectory updates in adaptive industrial settings.

Reading between the lines

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

The paper leaves implicit that the sharp performance drop from 87.3% to 50.8% outside the training region makes the epistemic uncertainty a natural trigger: a controller could use high uncertainty to fall back to the cubic prior or summon the optimizer.
A testable extension is to apply the same hard-constraint residual scheme to other optimal-control tasks, such as time-optimal or minimum-jerk planning, where a simple baseline trajectory is already known; the method is a template, not an energy-specific trick.
Adding torque or velocity limits, deferred to future work, would likely create non-smooth optimal residuals; if so, the smooth function class induced by $s(\xi)$ would need richer bases or a constraint-aware prior, and the current architecture gives no hard guarantee there.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Read the abstract's headline numbers with care. The 50.8% far-from-dataset figure is the GP's post-active-learning value on four high-uncertainty outside cases that were added to the training set and then re-evaluated. That is not a held-out extrapolation result. The 87.3% near-dataset figure does not appear in any table I can find: Table III gives inside-dataset savings of 25.8% (NN) and 21.8% (GP) against an OCP reference of 28.5%, which would be 90.5% and 76.5% of OCP performance, not 87.3%. The method underneath is better than the abstract suggests.

What is actually new: the residual-learning formulation for fixed-time point-to-point minimum-energy trajectory planning, with boundary conditions imposed as hard constraints through the scaling s(ξ)=ξ^2(1−ξ)^2 applied both to NN outputs and to GP kernels. That construction is clean, the appendix proof is coherent, and the ablation shows vanilla regressors violate the boundary conditions by roughly 0.2–0.3 rad. The measured two-to-three orders-of-magnitude speedup over the OCP solver is credible. The related work is honest that residual learning and surrogate trajectory optimization exist; the specific combination here is new.

Soft spots, in proportion: the untraceable abstract numbers are the main problem, not a minor one. Table IV is explicitly an active-learning table, and the four high-uncertainty cases became training data before the post-AL savings were computed. So 50.8% measures fit to newly added training points, not generalization far from the dataset. The pre-AL outside-dataset numbers are also weak: 6.8% (NN) and 2.8% (GP) against a 31.6% OCP reference, which undercuts the "near-minimum-energy far from dataset" claim. There are no confidence intervals, no code or data release, and the only baseline is the naive ablation rather than a comparison with another surrogate method. That said, the stress-test's circularity concern is not a load-bearing flaw: this is a standard supervised surrogate trained on OCP solver outputs, and the evaluation is relative to that solver. The flaw is overclaiming what the protocol shows, not circular reasoning.

This paper will be useful to people building real-time energy-conscious planners and to anyone working with GP/NN surrogates that need hard boundary-condition satisfaction. It deserves a serious referee: a competent reviewer can fix the traceability problem with a clean held-out evaluation and a revised abstract while keeping the hard-constraint construction. The central method is worth engaging with; the headline numbers need to be either traced or withdrawn.

Referee Report

4 major / 5 minor

Summary. This paper proposes a residual-learning paradigm for real-time fixed-time point-to-point minimum-energy trajectory planning. An offline optimal control solver generates a dataset of residuals between the OCP solution and a cubic polynomial prior; the regressor learns only the residual. Boundary conditions are enforced exactly by multiplying the NN output or the GP kernel by s(ξ)=ξ^2(1−ξ)^2. Uncertainty from NN ensembles or SVGPs is used both to select the best sampled trajectory and to drive active learning. Experiments on a pendulum, a SCARA robot, and a UR5e report energy savings relative to the standard planner and computation times, with the abstract claiming 87.3% of OCP performance near the training data, 50.8% far from it, and two-to-three-orders-of-magnitude speedups.

Significance. If the claims held, the paper would provide a practical method for embedding hard boundary conditions in learned trajectory planners and for reducing offline data requirements via residual learning. The hard-constraint construction in Section III-B.2 is simple, provably correct (Appendix Theorem 5.1), and likely to be of independent use; the speed advantage is plausible and supported by Fig. 9. However, the headline performance figures are not traceable to the reported tables, and the far-from-dataset claim is conflated with the active-learning update. These issues must be resolved before the main contribution can be fully assessed.

major comments (4)

[Abstract; Tables III and IV] The abstract's 87.3% and 50.8% performance claims are not supported by the reported data. Table III gives inside-dataset savings of 25.8% (NN) and 21.8% (GP) against an OCP reference of 28.5%, which correspond to 90.5% and 76.5% of OCP performance, not 87.3%; Table IV reports the 50.8% GP figure only after active learning. Please state exactly how 87.3% is computed or remove it from the abstract, and clearly label the 50.8% as a post-active-learning value on cases that were added to the training set.
[Section IV-C.5; Table IV] The 'far from dataset' 50.8% result is not a held-out generalization result. The four outside-dataset cases with highest uncertainty were solved by the OCP oracle and added to the training set before re-evaluation; the post-AL saving therefore measures interpolation to newly incorporated points, while the pre-AL savings are 2.98% (GP) and 17.63% (NN). A clean evaluation of generalization requires an outside dataset that is never used for model updates, with the active-learning protocol reported separately.
[Tables III and IV; Figs. 7-8] All energy-savings numbers are point estimates without confidence intervals or sample sizes, and the reported savings are obtained by selecting the best of multiple sampled trajectories (e.g., 100 samples for the pendulum). The difference between the NN and GP results may therefore not be statistically meaningful. Please report the distribution over test cases (box plots or mean ± std) and, where possible, a paired comparison between NN and GP.
[Section III-B.2, Eqs. (10)-(12)] The hard-constraint construction rests on the premise that the OCP residual lies in the smooth function class induced by tanh networks or RBF kernels multiplied by s(ξ)=ξ^2(1−ξ)^2. The paper gives no evidence that residuals from the OCP solver are smooth in this sense, and the outside-dataset degradation shown in Fig. 10 suggests the prior/residual parameterization may not capture the target structure. Please add a diagnostic of residual approximation error or an ablation with alternative scaling functions, and discuss the resulting domain of validity.

minor comments (5)

[Section II] There is a typo: 'the seconds formulate the problem' should be 'the second formulates the problem'; please proofread the text.
[Table I] The range formatting for the SCARA and UR5e datasets is difficult to parse (e.g., '0.0 × −0.5'); please present each joint range as an explicit interval.
[Section III-B.2 and Appendix] The text says 'respects propriety (9)' but should say 'property (9)'.
[Figure 6] The legend 'Boundary & Model' is unclear; please specify which visual element corresponds to which boundary condition (q0 or qF) and which model.
[Figure 7] The caption mentions 'best sample' and 'mean results', but the violin plot does not distinguish these clearly; please add explicit markers for the best sample and the mean.

Circularity Check

1 steps flagged · score 6.0 of 10

The abstract's 50.8% far-from-dataset number is the post-active-learning value on the four uncertainty-selected cases after adding them to the training set, so it is a fitted evaluation, not a held-out prediction; the residual-learning derivation itself is self-contained.

fitted input called prediction [Abstract; Section IV-C.5; Table IV]
"For this reason, the 4 cases with the higher uncertainty of the outside dataset were used to update the models, for the pendulum system. The savings before and after the AL phase are reported in Tab. IV. ... GPs can easily incorporate new data by updating their training dataset. On the other hand, NNs do not possess the same flexibility when it comes to incorporating new information. One common strategy for NNs is fine-tuning, which involves training the model for additional epochs while enriching the original batches with new data. ..."

The abstract's 50.8% far-from-dataset figure matches Table IV's post-AL GP saving of 50.93%, which is reported after the four highest-uncertainty outside cases were 'used to update the models,' i.e., incorporated into the training set by GP dataset updates or NN fine-tuning. Measuring savings on those same four cases after updating the models evaluates fit to newly added training data, not generalization to unseen far-from-dataset inputs. Reporting this number as 'far from the dataset' thus presents a fitted evaluation as a held-out prediction; the result is forced by the protocol rather than derived from the model's generalization capacity. The 87.3% near-dataset figure is not traceable to any reported table, which is a separate evidence gap rather than this circular step.

full rationale

The residual-learning derivation itself is self-contained and not circular. The residual is defined as q(t)=p(t)+r(t) (Eq. 7) with p a cubic prior satisfying the boundary conditions; since the OCP solution also satisfies them, the residual necessarily has r(0)=r(1)=r'(0)=r'(1)=0, so the hard-constrained parameterizations in Eqs. (11) and (12) are faithful rather than assumption-laden. Training on OCP-generated residuals and evaluating against the same OCP is standard supervised surrogate modeling: the regressor does not define its own target, and no fitted parameter is fed back into the cost. There is no load-bearing self-citation: the cited prior work supplies data or parameters, not the correctness of the core claim. The genuine circularity is confined to the headline far-from-dataset number: the 50.8% equals the post-active-learning GP saving in Table IV, obtained after the four highest-uncertainty outside cases were added to the training set and re-evaluated, so it is a fit to newly added training data presented as a generalization prediction. The 87.3% near-dataset figure cannot be reproduced from Table III (NN 25.8/28.5 is about 90.5%; GP 21.8/28.5 is about 76.5%) and no other table contains it; this is an unmet-evidence problem rather than a circularity. Similarly, Table III does not separate the 80/20 held-out split mentioned in Section IV-B, so even the inside-dataset phrasing may overstate generalization. On balance, the architecture, boundary-condition embedding, and speed advantage are independent contributions, but one of the two central quantitative claims reduces to a fitted evaluation, warranting a partial-circularity score of 6.

Assumptions & free parameters 5 free parameters · 6 assumptions · 0 invented entities

The central claim rests on the OCP solver's output being the true optimum, the dynamics and cost models being accurate, and the representational assumption that the optimal residual lies in the smooth class induced by the scaling and kernel. There are no new physical entities. The free parameters are the hand-chosen design choices and the fitted model hyperparameters, all of which shape the reported performance.

free parameters (5)

NN/GP hyperparameters (learning rate, weight decay, hidden layers, width, kernel lengthscale, noise variance, inducing…
From Table II and Section IV-B; these are tuned by hand or by standard training, and the central results depend on them.
Scaling function s(ξ)=ξ^2(1-ξ)^2 = functional form chosen by hand
Defined in Eq. (10); it enforces boundary conditions but also restricts the shape of the achievable residual.
Dataset ranges for tf, q0, qf (pendulum, SCARA, UR5e) = Table I ranges
The training domain determines what 'inside' versus 'outside' dataset means; results degrade outside these ranges.
Number of samples for best-sample selection = 100 (pendulum), 10 (SCARA, UR5e)
Section IV-B; the best-sample selection relies on this hand-chosen number.
Number of active-learning points = 4
Section IV-C.5; only four outside-dataset points are used to update the models.

assumptions (6)

domain assumption The numerical solutions produced by IPOPT for the OCP (4) are the true energy-optimal trajectories.
The training labels and reference savings come from Rockit/IPOPT; no verification of global optimality is reported (Section IV-A).
domain assumption The DC motor power model P = u^T R u + u^T N x (Eq. 1) accurately represents the robot's energy consumption.
The cost function is the basis for all optimality claims.
domain assumption The robot dynamics models (CasADi for pendulum, Pinocchio for manipulators) and the friction parameters from [30] are accurate.
The OCP solutions and all energy evaluations use these models.
standard math The residual representation r̃(ξ,X)=s(ξ)⊙o(ξ,X) and the GP kernel (12) guarantee that boundary conditions are satisfied.
The scaling function has zeros of order 2 at ξ=0,1; the appendix proves the GP mean and variance vanish at the boundaries.
domain assumption The prior cubic polynomial is a valid and useful baseline for all systems.
The residual learning performance is measured relative to the cubic prior; a poor prior could bias the results.
domain assumption Torque and velocity limits are not binding for the tested trajectories.
The paper explicitly defers torque/speed constraints to future work (Section V), so the 'minimum-energy' solution ignores them.

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Cite this review

Pith. "Pith review of Real-Time Generation of Near-Minimum-Energy Trajectories via Constraint-Informed Residual Learning." pith.science (2026). https://pith.science/paper/IFGVM6O4

@misc{pith2026250109450,
  author       = {Pith},
  title        = {Pith review of: Real-Time Generation of Near-Minimum-Energy Trajectories via Constraint-Informed Residual Learning},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IFGVM6O4}},
  note         = {Machine review of arXiv:2501.09450}
}

read the original abstract

Industrial robotics demands significant energy to operate, making energy-reduction methodologies increasingly important. Strategies for planning minimum-energy trajectories typically involve solving nonlinear optimal control problems (OCPs), which rarely cope with real-time requirements. In this paper, we propose a paradigm for generating near minimum-energy trajectories for manipulators by learning from optimal solutions. Our paradigm leverages a residual learning approach, which embeds boundary conditions while focusing on learning only the adjustments needed to steer a standard solution to an optimal one. Compared to a computationally expensive OCP-based planner, our paradigm achieves 87.3% of the performance near the training dataset and 50.8% far from the dataset, while being two to three orders of magnitude faster.

Figures

Figures reproduced from arXiv: 2501.09450 by the authors.

**Figure 1.** Schematic of the proposed residual learning paradigm. Offline, a dataset of residuals is generated and used to train a specifically designed probabilistic regressor that embeds the boundary conditions. remains is the (residual) component required for achieving optimality. By proper regressor design, the proposed paradigm (i) allows the embedding of Boundary Conditions (BCs) as hard constraints, (ii) requires less tr… view at source ↗

**Figure 2.** The proposed planner: the trained probabilistic reg [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗

**Figure 3.** Proposed neural network scheme. The scaling functio [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗

Figures from the paper (7 more)

**Figure 4.** Figure 4: Active learning approach: the uncertainty of the sol [PITH_FULL_IMAGE:figures/full_fig_p004_4.png]

**Figure 5.** Figure 5: Example trajectories of the NN and GP naive models. Th [PITH_FULL_IMAGE:figures/full_fig_p005_5.png]

**Figure 7.** Figure 7: Inside and Outside savings with respect to the standar [PITH_FULL_IMAGE:figures/full_fig_p005_7.png]

**Figure 8.** Figure 8: Energy savings with respect to the standard solution [PITH_FULL_IMAGE:figures/full_fig_p005_8.png]

**Figure 9.** Figure 9: Computation times for each model in seconds. The piec [PITH_FULL_IMAGE:figures/full_fig_p006_9.png]

**Figure 10.** Figure 10: Comparison of example trajectories for the pendulu [PITH_FULL_IMAGE:figures/full_fig_p007_10.png]

**Figure 11.** Figure 11: Comparison of example trajectories for the robotic [PITH_FULL_IMAGE:figures/full_fig_p007_11.png]

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Works this paper leans on

37 extracted references · 33 canonical work pages

[1]

A review on energy- saving optimization methods for robotic and automatic systems,

G. Carabin, E. Wehrle, and R. Vidoni, “A review on energy- saving optimization methods for robotic and automatic systems,” Robotics, vol. 6, no. 4, p. 39, 2017

work page 2017
[2]

A real-time capable method for planning minimum energy trajectories for one deg ree- of-freedom mechatronic systems,

D. Dona’, B. Lenzo, P . Boscariol, G. Rosati et al., “A real-time capable method for planning minimum energy trajectories for one deg ree- of-freedom mechatronic systems,” Control Engineering Practice , vol. 142, p. 105766, 2024

work page 2024
[3]

Practical and accurate gene ration of energy-optimal trajectories for a planar quadrotor,

F. Morbidi and D. Pisarski, “Practical and accurate gene ration of energy-optimal trajectories for a planar quadrotor,” in 2021 IEEE International Conference on Robotics and Automation (ICRA ). IEEE, 2021, pp. 355–361

work page 2021
[4]

Iterative dynamic programm ing: an approach to minimum energy trajectory planning for robotic manipu- lators,

G. Field and Y . Stepanenko, “Iterative dynamic programm ing: an approach to minimum energy trajectory planning for robotic manipu- lators,” in Proceedings of IEEE international conference on robotics and automation , vol. 3. IEEE, 1996, pp. 2755–2760

work page 1996
[5]

A. E. Bryson, Applied optimal control: optimization, estimation and control. Routledge, 2018

2018
[6]

The explicit solution of model predictive control via multipar ametric quadratic programming,

A. Bemporad, M. Morari, V . Dua, and E. N. Pistikopoulos, “ The explicit solution of model predictive control via multipar ametric quadratic programming,” in Proceedings of the 2000 American control conference. ACC (IEEE Cat. No. 00CH36334) , vol. 2. IEEE, 2000, pp. 872–876

work page 2000
[7]

Learning the problem-optimum map: Analysis and ap- plication to global optimization in robotics,

K. Hauser, “Learning the problem-optimum map: Analysis and ap- plication to global optimization in robotics,” IEEE Transactions on Robotics, vol. 33, no. 1, pp. 141–152, 2016

work page 2016
[8]

End-to- end learning to warm-start for real-time quadratic optimization,

R. Sambharya, G. Hall, B. Amos, and B. Stellato, “End-to- end learning to warm-start for real-time quadratic optimization,” in Learning for Dynamics and Control Conference . PMLR, 2023, pp. 220–234

work page 2023

Show all 37 references

[9]

Real-time optim al control for spacecraft orbit transfer via multiscale deep neural ne tworks,

L. Cheng, Z. Wang, F. Jiang, and C. Zhou, “Real-time optim al control for spacecraft orbit transfer via multiscale deep neural ne tworks,” IEEE Transactions on Aerospace and Electronic Systems , vol. 55, no. 5, pp. 2436–2450, 2018

2018
[10]

A machine learning enhanced algorithm for the optimal landing proble m,

Y . Zang, J. Long, X. Zhang, W. Hu, E. Weinan, and J. Han, “A machine learning enhanced algorithm for the optimal landing proble m,” in Mathematical and Scientiﬁc Machine Learning . PMLR, 2022, pp. 319–334

2022
[11]

A data-driven indirect method fo r nonlinear optimal control,

G. Tang and K. Hauser, “A data-driven indirect method fo r nonlinear optimal control,” Astrodynamics, vol. 3, no. 4, pp. 345–359, 2019

2019
[12]

Real-time optimal c ontrol via deep neural networks: study on landing problems,

C. S´ anchez-S´ anchez and D. Izzo, “Real-time optimal c ontrol via deep neural networks: study on landing problems,” Journal of Guidance, Control, and Dynamics , vol. 41, no. 5, pp. 1122–1135, 2018

2018
[13]

0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 1 . 2 Time (s) − 0. 2
[14]

6 Position (rad) Inside dataset Cubic OCP NN GP
[15]

0 0 . 5 1 . 0 1 . 5 2 . 0 2 . 5 Time (s) − 0. 5
[16]

0 Outside dataset Cubic OCP NN GP Fig. 10. Comparison of example trajectories for the pendulu m, both within and outside the dataset. Inside the dataset, t he regressor accurately captures the solution, while outside the dataset, the complexity of t he solution is less represe...
[17]

Deep neural networks based re al- time optimal control for lunar landing,

L. Zhu, J. Ma, and S. Wang, “Deep neural networks based re al- time optimal control for lunar landing,” in IOP Conference Series: Materials Science and Engineering , vol. 608, no. 1. IOP Publishing, 2019, p. 012045

2019
[18]

Learning the optimal state-feed back via supervised imitation learning,

D. Tailor and D. Izzo, “Learning the optimal state-feed back via supervised imitation learning,” Astrodynamics, vol. 3, no. 4, pp. 361– 374, 2019

2019
[19]

End-to -end neural network based optimal quadcopter control,

R. Ferede, G. de Croon, C. De Wagter, and D. Izzo, “End-to -end neural network based optimal quadcopter control,” Robotics and Autonomous Systems, vol. 172, p. 104588, 2024

2024
[20]

Learning trajectories f or real- time optimal control of quadrotors,

G. Tang, W. Sun, and K. Hauser, “Learning trajectories f or real- time optimal control of quadrotors,” in 2018 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS) . IEEE, 2018, pp. 3620–3625

2018
[21]

Learning-based warm-starting for fast sequential convex programming and trajectory optimizatio n,

S. Banerjee, T. Lew, R. Bonalli, A. Alfaadhel, I. A. Alom ar, H. M. Shageer, and M. Pavone, “Learning-based warm-starting for fast sequential convex programming and trajectory optimizatio n,” in 2020 IEEE Aerospace Conference . IEEE, 2020, pp. 1–8

2020
[22]

Deep learnin g for optimization of trajectories for quadrotors,

Y . Wu, X. Sun, I. Spasojevic, and V . Kumar, “Deep learnin g for optimization of trajectories for quadrotors,” IEEE Robotics and Au- tomation Letters , 2024

2024
[23]

Deep networks for motor c ontrol functions,

M. Berniker and K. P . Kording, “Deep networks for motor c ontrol functions,” Frontiers in computational neuroscience , vol. 9, p. 32, 2015

2015
[24]

Neural path planning: Fixed time, near-optimal path generation via ora cle imitation,

M. J. Bency, A. H. Qureshi, and M. C. Yip, “Neural path planning: Fixed time, near-optimal path generation via ora cle imitation,” CoRR, vol. abs/1904.11102, 2019. [Online]. Available: http://arxiv.org/abs/1904.11102

1904 arXiv
[25]

Long short-term memory,

S. Hochreiter, “Long short-term memory,” Neural Computation MIT- Press, 1997

1997
[26]

Attention is all you need,

A. V aswani, “Attention is all you need,” Advances in Neural Informa- tion Processing Systems , 2017

2017
[27]

Tr ansformers for trajectory optimization with application to spacecraf t rendezvous,

T. Guffanti, D. Gammelli, S. D’Amico, and M. Pavone, “Tr ansformers for trajectory optimization with application to spacecraf t rendezvous,” in 2024 IEEE Aerospace Conference . IEEE, 2024, pp. 1–13

2024
[28]

Transformer-based model predictive control: T rajectory optimization via sequence modeling,

D. Celestini, D. Gammelli, T. Guffanti, S. D’Amico, E. C apello, and M. Pavone, “Transformer-based model predictive control: T rajectory optimization via sequence modeling,” IEEE Robotics and Automation Letters, 2024

2024
[29]

C onstraint- informed learning for warm starting trajectory optimizati on,

J. Briden, C. Choi, K. Y un, R. Linares, and A. Cauligi, “C onstraint- informed learning for warm starting trajectory optimizati on,” arXiv preprint arXiv:2312.14336, 2023

2023 arXiv
[30]

E ffortless modeling of optimal control problems with rockit,

J. Gillis, B. V andewal, G. Pipeleers, and J. Swevers, “E ffortless modeling of optimal control problems with rockit,” in 39th Benelux Meeting on Systems and Control , vol. 138. Elspeet, The Netherlands, 2020

2020
[31]

Large-scale nonlinear p rogramming using ipopt: An integrating framework for enterprise-wide dynamic optimization,

L. T. Biegler and V . M. Zavala, “Large-scale nonlinear p rogramming using ipopt: An integrating framework for enterprise-wide dynamic optimization,” Computers & Chemical Engineering , vol. 33, no. 3, pp. 575–582, 2009

2009
[32]

CasADi – A software framework for nonlinear optimization a nd optimal control,

J. A. E. Andersson, J. Gillis, G. Horn, J. B. Rawlings, an d M. Diehl, “CasADi – A software framework for nonlinear optimization a nd optimal control,” Mathematical Programming Computation , vol. 11, no. 1, pp. 1–36, 2019

2019
[33]

Proximal and Sparse Resolution of Constrained Dynamic Equations,

J. Carpentier, R. Budhiraja, and N. Mansard, “Proximal and Sparse Resolution of Constrained Dynamic Equations,” in Robotics: Science and Systems 2021 , Austin / Virtual, United States, Jul. 2021. [Online]. Available: https://hal.inria.fr/hal-03271811

2021
[34]

Electro- mechanical modeling and identiﬁcation of the UR5 e-series r obot,

E. Clochiatti, L. Scalera, P . Boscariol, and A. Gaspare tto, “Electro- mechanical modeling and identiﬁcation of the UR5 e-series r obot,” Robotica, pp. 1–23, 2024

2024
[35]

Decoupled weight decay re gularization,

I. Loshchilov and F. Hutter, “Decoupled weight decay re gularization,” arXiv preprint arXiv:1711.05101 , 2017

2017 arXiv
[36]

Pytorch: An imperative style, high-performance deep learning library ,

A. Paszke, S. Gross, F. Massa, A. Lerer, J. Bradbury, G. C hanan, T. Killeen, Z. Lin, N. Gimelshein, L. Antiga et al. , “Pytorch: An imperative style, high-performance deep learning library ,” Advances in neural information processing systems , vol. 32, 2019

2019
[37]

Gpytorch: Blackbox matrix-matrix gaussian process infer ence with gpu acceleration,

J. Gardner, G. Pleiss, K. Q. Weinberger, D. Bindel, and A . G. Wilson, “Gpytorch: Blackbox matrix-matrix gaussian process infer ence with gpu acceleration,” Advances in neural information processing systems , vol. 31, 2018. APPENDIX This section demonstrates that a GP with a k...

2018

Pith tools

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

[1] [1]

A review on energy- saving optimization methods for robotic and automatic systems,

G. Carabin, E. Wehrle, and R. Vidoni, “A review on energy- saving optimization methods for robotic and automatic systems,” Robotics, vol. 6, no. 4, p. 39, 2017

work page 2017

[2] [2]

A real-time capable method for planning minimum energy trajectories for one deg ree- of-freedom mechatronic systems,

D. Dona’, B. Lenzo, P . Boscariol, G. Rosati et al., “A real-time capable method for planning minimum energy trajectories for one deg ree- of-freedom mechatronic systems,” Control Engineering Practice , vol. 142, p. 105766, 2024

work page 2024

[3] [3]

Practical and accurate gene ration of energy-optimal trajectories for a planar quadrotor,

F. Morbidi and D. Pisarski, “Practical and accurate gene ration of energy-optimal trajectories for a planar quadrotor,” in 2021 IEEE International Conference on Robotics and Automation (ICRA ). IEEE, 2021, pp. 355–361

work page 2021

[4] [4]

Iterative dynamic programm ing: an approach to minimum energy trajectory planning for robotic manipu- lators,

G. Field and Y . Stepanenko, “Iterative dynamic programm ing: an approach to minimum energy trajectory planning for robotic manipu- lators,” in Proceedings of IEEE international conference on robotics and automation , vol. 3. IEEE, 1996, pp. 2755–2760

work page 1996

[5] [5]

A. E. Bryson, Applied optimal control: optimization, estimation and control. Routledge, 2018

2018

[6] [6]

The explicit solution of model predictive control via multipar ametric quadratic programming,

A. Bemporad, M. Morari, V . Dua, and E. N. Pistikopoulos, “ The explicit solution of model predictive control via multipar ametric quadratic programming,” in Proceedings of the 2000 American control conference. ACC (IEEE Cat. No. 00CH36334) , vol. 2. IEEE, 2000, pp. 872–876

work page 2000

[7] [7]

Learning the problem-optimum map: Analysis and ap- plication to global optimization in robotics,

K. Hauser, “Learning the problem-optimum map: Analysis and ap- plication to global optimization in robotics,” IEEE Transactions on Robotics, vol. 33, no. 1, pp. 141–152, 2016

work page 2016

[8] [8]

End-to- end learning to warm-start for real-time quadratic optimization,

R. Sambharya, G. Hall, B. Amos, and B. Stellato, “End-to- end learning to warm-start for real-time quadratic optimization,” in Learning for Dynamics and Control Conference . PMLR, 2023, pp. 220–234

work page 2023