pith. sign in

arxiv: 2511.14311 · v2 · submitted 2025-11-18 · 📡 eess.SY · cs.RO· cs.SY

Multi-Timescale Model Predictive Control for Slow-Fast Systems

Lukas Schroth , Daniel Morton , Amon Lahr , Daniele Gammelli , Andrea Carron , Marco Pavone This is my paper

Pith reviewed 2026-05-17 20:54 UTC · model grok-4.3

classification 📡 eess.SY cs.ROcs.SY
keywords model predictive controlmulti-timescale controlslow-fast systemsexponential decay of sensitivitiesrobotic controlcomputational efficiencyreal-time optimizationconstrained control
0
0 comments X

The pith

A multi-timescale MPC scheme for slow-fast systems switches to reduced slow models and uses exponentially increasing integration steps to cut computation time by up to an order of magnitude while preserving closed-loop performance.

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

This paper sets out to solve the real-time computation bottleneck in model predictive control when systems have both fast and slow dynamics and long prediction horizons are needed. The key idea is to exploit the exponential decay of sensitivities, which means that model inaccuracies far ahead in the horizon have little effect on the current decision. The authors therefore replace the full model with a reduced slow-only model for later parts of the horizon and let the integration step size grow exponentially, lowering the detail of the predictions progressively. They demonstrate the approach on three simulated robotic control tasks and report speed-ups of up to ten times compared with standard MPC. If the approach works as claimed, real-time constrained control becomes feasible for a wider class of robotic and autonomous systems that mix fast actuation with slow task-level behavior.

Core claim

The central claim is that, for systems with fast and slow dynamics, a multi-timescale MPC scheme improves computational efficiency by switching to a reduced model that captures only the slow dominant dynamics and by exponentially increasing integration step sizes to progressively reduce model detail along the horizon, while closed-loop performance remains acceptable because of the exponential decay of sensitivities property.

What carries the argument

The multi-timescale MPC scheme that progressively reduces model fidelity along the prediction horizon by switching to a slow-only reduced model and exponentially increasing integration step sizes, grounded in the exponential decay of sensitivities.

If this is right

  • Real-time MPC becomes practical for robotic systems that require long horizons yet must run on limited onboard compute.
  • Longer prediction horizons can be used without exceeding real-time budgets in slow-fast dynamics.
  • The same reduction strategy applies directly to other constrained optimization-based controllers that face similar timescale separation.
  • Resource-constrained platforms can now host higher-fidelity models earlier in the horizon while still meeting timing constraints.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Hardware experiments on the same robotic platforms would reveal whether sensor noise or model mismatch breaks the exponential decay assumption observed in simulation.
  • The scheme could be combined with learned reduced-order models to further improve the slow-dynamics approximation without manual derivation.
  • Explicit stability or performance certificates that quantify the degradation introduced by the multi-timescale approximations would make the method easier to certify for safety-critical use.

Load-bearing premise

The exponential decay of sensitivities property continues to hold for the target robotic systems, so that the reduced slow model and coarser integration steps do not push closed-loop performance outside acceptable limits.

What would settle it

If the closed-loop state trajectories or accumulated costs produced by the multi-timescale controller differ substantially from those of the full-fidelity MPC on the same three robotic simulation tasks, the performance-preservation claim would be falsified.

Figures

Figures reproduced from arXiv: 2511.14311 by Amon Lahr, Andrea Carron, Daniele Gammelli, Daniel Morton, Lukas Schroth, Marco Pavone.

Figure 1
Figure 1. Figure 1: Multi-Timescale MPC for slow-fast systems. (a) The [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Mean closed-loop cost increase (relative to the full [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: System schematics and Pareto frontiers. MTS-MPC (orange) consistently outperforms all baselines in the trade-off [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Closed-loop trajectories: MTS-MPC achieves the same performance as the full-resolution baseline while being 16×, [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Frobenius norm of the sensitivity matrix [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Step size schedule versus stage k. The plotted schedules are those used in the experiments. B. Experiments: further implementation details As mentioned, we implement all predictive controllers using the multi-phase interface of the open-source SQP framework acados [35], [34], which allows for conve￾nient formulation and solution of optimal control problems with multiple phases that may involve different co… view at source ↗
read the original abstract

Model Predictive Control (MPC) has established itself as the primary methodology for constrained control, enabling autonomy across diverse applications. While model fidelity is crucial in MPC, solving the corresponding optimization problem in real time remains challenging when combining long horizons with high-fidelity models that capture both short-term dynamics and long-term behavior. Motivated by results on the Exponential Decay of Sensitivities (EDS), which imply that, under certain conditions, the influence of modeling inaccuracies decreases exponentially along the prediction horizon, this paper proposes a multi-timescale MPC scheme for fast-sampled control. Tailored to systems with both fast and slow dynamics, the proposed approach improves computational efficiency by i) switching to a reduced model that captures only the slow, dominant dynamics and ii) exponentially increasing integration step sizes to progressively reduce model detail along the horizon. We evaluate the method on three practically motivated robotic control problems in simulation and observe speed-ups of up to an order of magnitude.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The paper proposes a multi-timescale MPC scheme for slow-fast systems. Motivated by the Exponential Decay of Sensitivities (EDS) property, it improves efficiency by switching to a reduced model capturing only slow dominant dynamics and by exponentially increasing integration step sizes to reduce model detail along the horizon. The approach is evaluated in simulation on three robotic control problems, with reported speed-ups of up to an order of magnitude.

Significance. If the EDS assumption holds with adequate decay rate for the target systems and closed-loop performance remains acceptable, the method could meaningfully extend the applicability of long-horizon MPC to high-fidelity models in real-time robotic settings. The simulation results provide preliminary evidence of computational gains, though stronger quantitative validation of performance preservation would increase the impact.

major comments (1)
  1. [Abstract and Evaluation] The central efficiency claim depends on the EDS property holding under the operating conditions of the three robotic systems so that the reduced slow model and exponentially increasing step sizes do not cause unacceptable closed-loop degradation. The manuscript provides no theorem, proof, or numerical verification that EDS decays at a sufficient rate for these specific fast-slow dynamics (see Abstract and the evaluation description). This verification is load-bearing; without it, early-horizon modeling errors could propagate and undermine the reported speed-ups.
minor comments (1)
  1. [Abstract] The abstract states that a reduced model is used but does not specify how it is obtained from the full model or the criteria for selecting the slow dynamics; adding this detail would aid reproducibility.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their constructive comments on our manuscript. We address the major comment below and outline the revisions we will make to strengthen the presentation of the EDS-based efficiency claims.

read point-by-point responses

  1. Referee: [Abstract and Evaluation] The central efficiency claim depends on the EDS property holding under the operating conditions of the three robotic systems so that the reduced slow model and exponentially increasing step sizes do not cause unacceptable closed-loop degradation. The manuscript provides no theorem, proof, or numerical verification that EDS decays at a sufficient rate for these specific fast-slow dynamics (see Abstract and the evaluation description). This verification is load-bearing; without it, early-horizon modeling errors could propagate and undermine the reported speed-ups.

    Authors: We agree that explicit verification of the EDS decay rate for the specific robotic systems is important to support the central efficiency claims. The manuscript is motivated by and cites established EDS results from the literature that apply under the stated conditions for slow-fast systems; however, we acknowledge that the current evaluation section does not include dedicated numerical checks of the decay rate or sensitivity propagation for the three example dynamics. In the revised manuscript we will add a new subsection to the evaluation that numerically verifies the EDS property for each robotic system. This will include computed sensitivity decay curves (or equivalent metrics) under the operating conditions used in the closed-loop simulations, together with a direct comparison of closed-loop performance between the full multi-timescale scheme and a baseline that retains full model fidelity throughout the horizon. These additions will confirm that modeling inaccuracies remain tolerable and do not undermine the reported speed-ups. revision: yes

Circularity Check

0 steps flagged

No significant circularity; proposal builds independently on external EDS results

full rationale

The paper motivates its multi-timescale MPC approach by citing prior results on Exponential Decay of Sensitivities (EDS) from the literature and then presents an independent algorithmic construction: switching to a reduced slow-dynamics model and using exponentially increasing integration steps. This construction is evaluated empirically on three robotic simulation problems rather than being derived from fitted parameters or self-referential definitions within the paper. No load-bearing step reduces by construction to its own inputs, and the central efficiency claims rest on the external EDS motivation plus simulation outcomes, making the derivation self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the domain assumption that Exponential Decay of Sensitivities applies to the robotic systems of interest and that a reduced slow-dynamics model remains adequate for the tail of the horizon; no free parameters or new invented entities are mentioned.

axioms (1)
  • domain assumption Exponential Decay of Sensitivities (EDS) implies that the influence of modeling inaccuracies decreases exponentially along the prediction horizon under certain conditions.
    Cited as the key motivation that justifies switching to reduced models and coarser steps farther along the horizon.

pith-pipeline@v0.9.0 · 5477 in / 1401 out tokens · 42449 ms · 2026-05-17T20:54:51.621531+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

81 extracted references · 81 canonical work pages

  1. [1]

    An overview of industrial model predictive control technology,

    S. J. Qin and T. A. Badgwell, “An overview of industrial model predictive control technology,” inAIche symposium series, vol. 93, no. 316. New York, NY: American Institute of Chemical Engineers, 1971-c2002., 1997, pp. 232–256

    work page 1971

  2. [2]

    Geyer,Model predictive control of high power converters and industrial drives

    T. Geyer,Model predictive control of high power converters and industrial drives. John Wiley & Sons, 2016

    work page 2016

  3. [3]

    Nonlinear model predictive control of wind turbines using lidar,

    D. Schlipf, D. J. Schlipf, and M. K ¨uhn, “Nonlinear model predictive control of wind turbines using lidar,”Wind energy, vol. 16, no. 7, pp. 1107–1129, 2013

    work page 2013

  4. [4]

    Real-time model predictive control for quadrotors,

    M. Bangura and R. Mahony, “Real-time model predictive control for quadrotors,”IFAC Proceedings Volumes, vol. 47, no. 3, pp. 11 773– 11 780, 2014

    work page 2014

  5. [5]

    Offline motion li- braries and online mpc for advanced mobility skills,

    M. Bjelonic, R. Grandia, M. Geilinger, O. Harley, V . S. Medeiros, V . Pajovic, E. Jelavic, S. Coros, and M. Hutter, “Offline motion li- braries and online mpc for advanced mobility skills,”The International Journal of Robotics Research, vol. 41, no. 9-10, pp. 903–924, 2022

    work page 2022

  6. [6]

    Robots with flexible elements,

    A. De Luca and W. J. Book, “Robots with flexible elements,”Springer handbook of robotics, pp. 243–282, 2016

    work page 2016

  7. [7]

    A stiffness-oriented model order reduction method for low-inertia power systems,

    S. Muntwiler, O. Stanojev, A. Zanelli, G. Hug, and M. N. Zeilinger, “A stiffness-oriented model order reduction method for low-inertia power systems,”Electric Power Systems Research, vol. 235, p. 110630, 2024

    work page 2024

  8. [8]

    Hierarchical predictive control of integrated wastewater treatment systems,

    M. Brdys, M. Grochowski, T. Gminski, K. Konarczak, and M. Drewa, “Hierarchical predictive control of integrated wastewater treatment systems,”Control Engineering Practice, vol. 16, no. 6, pp. 751–767, 2008

    work page 2008

  9. [9]

    Control of integrated process net- works—a multi-time scale perspective,

    M. Baldea and P. Daoutidis, “Control of integrated process net- works—a multi-time scale perspective,”Computers & chemical en- gineering, vol. 31, no. 5-6, pp. 426–444, 2007

    work page 2007

  10. [10]

    Exponential decay in the sensitivity analysis of nonlinear dynamic programming,

    S. Na and M. Anitescu, “Exponential decay in the sensitivity analysis of nonlinear dynamic programming,”SIAM Journal on Optimization, vol. 30, no. 2, pp. 1527–1554, 2020

    work page 2020

  11. [11]

    On the convergence of overlapping schwarz decomposition for nonlinear optimal control,

    S. Na, S. Shin, M. Anitescu, and V . M. Zavala, “On the convergence of overlapping schwarz decomposition for nonlinear optimal control,” IEEE Transactions on Automatic Control, vol. 67, no. 11, pp. 5996– 6011, 2022

    work page 2022

  12. [12]

    Exponential decay of sensitivity in graph-structured nonlinear programs,

    S. Shin, M. Anitescu, and V . M. Zavala, “Exponential decay of sensitivity in graph-structured nonlinear programs,”SIAM Journal on Optimization, vol. 32, no. 2, pp. 1156–1183, 2022

    work page 2022

  13. [13]

    Model hierarchy predictive control of robotic systems,

    H. Li, R. J. Frei, and P. M. Wensing, “Model hierarchy predictive control of robotic systems,”IEEE Robotics and Automation Letters, vol. 6, no. 2, pp. 3373–3380, 2021

    work page 2021

  14. [14]

    Cafe-mpc: A cascaded-fidelity model predictive control framework with tuning-free whole-body control,

    H. Li and P. M. Wensing, “Cafe-mpc: A cascaded-fidelity model predictive control framework with tuning-free whole-body control,” IEEE Transactions on Robotics, 2024

    work page 2024

  15. [15]

    Adaptive complexity model predictive control,

    J. Norby, A. Tajbakhsh, Y . Yang, and A. M. Johnson, “Adaptive complexity model predictive control,”IEEE Transactions on Robotics, 2024

    work page 2024

  16. [16]

    Continuous dynamic bipedal jumping via adaptive-model optimization,

    J. Li, O. Kolt, and Q. Nguyen, “Continuous dynamic bipedal jumping via adaptive-model optimization,”arXiv e-prints, pp. arXiv–2404, 2024

    work page 2024

  17. [17]

    Multi-fidelity receding horizon planning for multi-contact locomotion,

    J. Wang, S. Kim, S. Vijayakumar, and S. Tonneau, “Multi-fidelity receding horizon planning for multi-contact locomotion,” in2020 IEEE-RAS 20th International Conference on Humanoid Robots (Hu- manoids). IEEE, 2021, pp. 53–60

    work page 2021

  18. [18]

    Multi-fidelity models for model predictive control,

    S. Kameswaran and N. Subrahmanya, “Multi-fidelity models for model predictive control,” inComputer Aided Chemical Engineering. Elsevier, 2012, vol. 31, pp. 1627–1631

    work page 2012

  19. [19]

    Long-horizon vehicle motion planning and control through serially cascaded model complexity,

    V . A. Laurense and J. C. Gerdes, “Long-horizon vehicle motion planning and control through serially cascaded model complexity,” IEEE Transactions on Control Systems Technology, vol. 30, no. 1, pp. 166–179, 2021

    work page 2021

  20. [20]

    Model predictive control with non-uniformly spaced optimization horizon for multi-timescale processes,

    C. K. Tan, M. J. Tippett, and J. Bao, “Model predictive control with non-uniformly spaced optimization horizon for multi-timescale processes,”Computers & Chemical Engineering, vol. 84, pp. 162– 170, 2016

    work page 2016

  21. [21]

    Model predictive control method for autonomous vehicles using time-varying and non- uniformly spaced horizon,

    M. Kim, D. Lee, J. Ahn, M. Kim, and J. Park, “Model predictive control method for autonomous vehicles using time-varying and non- uniformly spaced horizon,”IEEE Access, vol. 9, pp. 86 475–86 487, 2021

    work page 2021

  22. [22]

    Exploiting models of different granularity in robust predictive control,

    T. B ¨athge, S. Lucia, and R. Findeisen, “Exploiting models of different granularity in robust predictive control,” in2016 IEEE 55th Confer- ence on Decision and Control (CDC). IEEE, 2016, pp. 2763–2768

    work page 2016

  23. [23]

    Model predic- tive control with models of different granularity and a non-uniformly spaced prediction horizon,

    T. Br ¨udigam, D. Prader, D. Wollherr, and M. Leibold, “Model predic- tive control with models of different granularity and a non-uniformly spaced prediction horizon,” in2021 American Control Conference (ACC). IEEE, 2021, pp. 3876–3881

    work page 2021

  24. [24]

    Time-scale decomposition of an optimal control problem in greenhouse climate management,

    E. Van Henten and J. Bontsema, “Time-scale decomposition of an optimal control problem in greenhouse climate management,”Control Engineering Practice, vol. 17, no. 1, pp. 88–96, 2009

    work page 2009

  25. [25]

    Architectures for distributed and hierarchical model predictive control–a review,

    R. Scattolini, “Architectures for distributed and hierarchical model predictive control–a review,”Journal of process control, vol. 19, no. 5, pp. 723–731, 2009

    work page 2009

  26. [26]

    An mpc approach to the design of two-layer hierarchical control systems,

    B. Picasso, D. De Vito, R. Scattolini, and P. Colaneri, “An mpc approach to the design of two-layer hierarchical control systems,” Automatica, vol. 46, no. 5, pp. 823–831, 2010

    work page 2010

  27. [27]

    Data-driven spectral submanifold reduction for nonlinear optimal control of high-dimensional robots,

    J. I. Alora, M. Cenedese, E. Schmerling, G. Haller, and M. Pavone, “Data-driven spectral submanifold reduction for nonlinear optimal control of high-dimensional robots,” in2023 IEEE international conference on robotics and automation (ICRA). IEEE, 2023, pp. 2627–2633

    work page 2023

  28. [28]

    A parallel decomposition scheme for solving long-horizon optimal control prob- lems,

    S. Shin, T. Faulwasser, M. Zanon, and V . M. Zavala, “A parallel decomposition scheme for solving long-horizon optimal control prob- lems,” in2019 IEEE 58th conference on decision and control (CDC). IEEE, 2019, pp. 5264–5271

    work page 2019

  29. [29]

    Accelerated gradient methods and dual decomposition in distributed model predictive control,

    P. Giselsson, M. D. Doan, T. Keviczky, B. De Schutter, and A. Rantzer, “Accelerated gradient methods and dual decomposition in distributed model predictive control,”Automatica, vol. 49, no. 3, pp. 829–833, 2013

    work page 2013

  30. [30]

    Benchmarking large-scale distributed convex quadratic programming algorithms,

    A. Kozma, C. Conte, and M. Diehl, “Benchmarking large-scale distributed convex quadratic programming algorithms,”Optimization methods and software, vol. 30, no. 1, pp. 191–214, 2015

    work page 2015

  31. [31]

    Diffusing-horizon model predictive con- trol,

    S. Shin and V . M. Zavala, “Diffusing-horizon model predictive con- trol,”IEEE Transactions on Automatic Control, vol. 68, no. 1, pp. 188–201, 2021

    work page 2021

  32. [32]

    Move blocking strategies in receding horizon control,

    R. Cagienard, P. Grieder, E. C. Kerrigan, and M. Morari, “Move blocking strategies in receding horizon control,”Journal of Process Control, vol. 17, no. 6, pp. 563–570, 2007

    work page 2007

  33. [33]

    A multiple shooting algorithm for direct solution of optimal control problems,

    H. G. Bock and K.-J. Plitt, “A multiple shooting algorithm for direct solution of optimal control problems,”IFAC Proceedings Volumes, vol. 17, no. 2, pp. 1603–1608, 1984

    work page 1984

  34. [34]

    aca- dos—a modular open-source framework for fast embedded optimal control,

    R. Verschueren, G. Frison, D. Kouzoupis, J. Frey, N. v. Duijkeren, A. Zanelli, B. Novoselnik, T. Albin, R. Quirynen, and M. Diehl, “aca- dos—a modular open-source framework for fast embedded optimal control,”Mathematical Programming Computation, vol. 14, no. 1, pp. 147–183, 2022

    work page 2022

  35. [35]

    Multi-phase optimal control problems for efficient nonlinear model predictive control with acados,

    J. Frey, K. Baumg ¨artner, G. Frison, and M. Diehl, “Multi-phase optimal control problems for efficient nonlinear model predictive control with acados,”Optimal Control Applications and Methods, 2024

    work page 2024

  36. [36]

    Hpipm: a high-performance quadratic programming framework for model predictive control,

    G. Frison and M. Diehl, “Hpipm: a high-performance quadratic programming framework for model predictive control,”IFAC- PapersOnLine, vol. 53, no. 2, pp. 6563–6569, 2020

    work page 2020

  37. [37]

    Dynamic modelling of differential- drive mobile robots using lagrange and newton-euler methodologies: A unified framework,

    R. Dhaouadi and A. A. Hatab, “Dynamic modelling of differential- drive mobile robots using lagrange and newton-euler methodologies: A unified framework,”Advances in robotics & automation, vol. 2, no. 2, pp. 1–7, 2013

    work page 2013

  38. [38]

    Enabling bidirectional thrust for aggressive and inverted quadrotor flight,

    W. Jothiraj, C. Miles, E. Bulka, I. Sharf, and M. Nahon, “Enabling bidirectional thrust for aggressive and inverted quadrotor flight,” in2019 International Conference on Unmanned Aircraft Systems (ICUAS). IEEE, 2019, pp. 534–541

    work page 2019

  39. [39]

    Taming high-dimensional dynamics: Learning optimal projections onto spectral submanifolds,

    H. Buurmeijer, L. A. Pabon, J. I. Alora, R. S. Kaundinya, G. Haller, and M. Pavone, “Taming high-dimensional dynamics: Learning optimal projections onto spectral submanifolds,”arXiv preprint arXiv:2504.03157, 2025

    work page arXiv 2025

  40. [40]

    Nocedal and S

    J. Nocedal and S. J. Wright,Numerical optimization. Springer, 1999. APPENDIX A. Exponential Decay of Sensitivities

    work page 1999

  41. [41]

    Simple example and visualization:To build intuition for EDS, we consider a simple setting with quadratic costs, linear dynamics affected by additive disturbancesp k, and no inequality constraints: min xk,uk xT N SxN + N−1X k=0 xT k Qxk +u T k Ruk (27a) s.t.x k+1 =Ax k +Bu k +p k, k∈[N−1](27b) x0 =x(k),(27c) where we use the compact notation[N−1] :={0, . ....

    work page

  42. [42]

    [12], who establish EDS for general graph-structured nonlinear programs, and present them in the context of MPC

    Previous Work on EDS:In this subsection, we briefly summarize the results by Shin et al. [12], who establish EDS for general graph-structured nonlinear programs, and present them in the context of MPC. For simplicity of notation, we neglect the structure of optimal control problems and instead consider the standard form in the following: NLP(P) := min X L...

    work page

  43. [43]

    Oscillation-aware drone control:For completeness, we report the equations of motion used in the drone-spring- pendulum experiments. The full model is given by ¨py = −κ(w1 +w 2) sinφ−k Sl0 sinϑ+k Srsinϑ md ,(41a) ¨pz = −mdg+κ(w 1 +w 2) cosφ+k Sl0 cosϑ−k Srcosϑ md , (41b) ¨φ=Lrot κ(−w 1 +w 2) Ixx ,(41c) ¨r= kS(l0 −r) ml +r ˙ϑ2 + κ(w1 +w 2) cos(φ−ϑ) md + kS(...

    work page

  44. [44]

    Baseline4705.448 (±12928.892) 0.237 (±0.357) 0.017 (±0.016)

    work page

  45. [45]

    Shorter Horizon4955.323 (±12642.376) 0.029 (±0.040) 0.004 (±0.004)

    work page

  46. [46]

    Larger Step Size4736.789 (±12929.419) 0.068 (±0.105) 0.004 (±0.004)

    work page

  47. [47]

    Increasing Step Sizes4705.674 (±12934.705) 0.020 (±0.031) 0.001 (±0.001)

    work page

  48. [48]

    Model Switching4705.710 (±12925.564) 0.132 (±0.203) 0.009 (±0.009)

    work page

  49. [49]

    Steps4707.832 (±12923.114) 0.015 (±0.023) 0.001 (±0.001) Drone

    Model Switching + Inc. Steps4707.832 (±12923.114) 0.015 (±0.023) 0.001 (±0.001) Drone

    work page

  50. [50]

    Baseline25.070 (±40.842) 0.627 (±1.217) 0.016 (±0.027)

    work page

  51. [51]

    Shorter Horizon26.661 (±43.050) 0.424 (±1.250) 0.010 (±0.017)

    work page

  52. [52]

    Larger Step Size26.043 (±41.446) 0.279 (±0.374) 0.005 (±0.006)

    work page

  53. [53]

    Model29.428 (±40.149) 0.324 (±0.524) 0.009 (±0.009)

    Approx. Model29.428 (±40.149) 0.324 (±0.524) 0.009 (±0.009)

    work page

  54. [54]

    Increasing Step Sizes25.132 (±41.088) 0.204 (±0.262) 0.005 (±0.005)

    work page

  55. [55]

    Model Switching25.143 (±41.124) 0.346 (±0.459) 0.009 (±0.009)

    work page

  56. [56]

    Steps25.142 (±41.052) 0.189 (±0.301) 0.004 (±0.005) Trunk-like

    Model Switching + Inc. Steps25.142 (±41.052) 0.189 (±0.301) 0.004 (±0.005) Trunk-like

    work page

  57. [57]

    Baseline0.00972 (±0.01283) 0.089 (±0.012) 0.031 (±0.002)

    work page

  58. [58]

    Shorter Horizon0.00993 (±0.01356) 0.052 (±0.008) 0.019 (±0.001)

    work page

  59. [59]

    Larger Step Size0.00985 (±0.01271) 0.048 (±0.008) 0.014 (±0.001)

    work page

  60. [60]

    Increasing Step Sizes0.00975 (±0.01270) 0.054 (±0.010) 0.018 (±0.001)

    work page

  61. [61]

    Model Switching0.00973 (±0.01301) 0.021 (±0.003) 0.008 (±0.001)

    work page

  62. [62]

    Model Switching + Inc. Steps0.00973 (±0.01299) 0.017 (±0.002) 0.006 (±0.0005) TABLE II: Mean values (±standard deviation) of stage cost, total solve time, and solve time per SQP iteration for each example system and MPC variant. System Approach∆t 0 [s]N ¯k T[s]f[Hz] Differential Drive

    work page

  63. [63]

    Baseline 0.01 1000 – 10.0 100

    work page

  64. [64]

    Shorter Horizon 0.01 250 – 2.5 100

    work page

  65. [65]

    Larger Step Size 0.04 250 – 10.0 25

    work page

  66. [66]

    Increasing Step Sizes 0.01 80 – 10.0 100

    work page

  67. [67]

    Model Switching 0.01 1000 36 10.0 100

    work page

  68. [68]

    Steps 0.01 80 21 10.0 100 Drone

    Model Switching + Inc. Steps 0.01 80 21 10.0 100 Drone

    work page

  69. [69]

    Baseline 0.01 150 – 1.50 100

    work page

  70. [70]

    Shorter Horizon 0.01 100 – 1.00 100

    work page

  71. [71]

    Larger Step Size 0.02 75 – 1.50 50

    work page

  72. [72]

    Model 0.01 150 – 1.50 100

    Approx. Model 0.01 150 – 1.50 100

    work page

  73. [73]

    Increasing Step Sizes 0.01 75 – 1.50 100

    work page

  74. [74]

    Model Switching 0.01 150 45 1.50 100

    work page

  75. [75]

    Steps 0.01 75 33 1.50 100 Trunk-like

    Model Switching + Inc. Steps 0.01 75 33 1.50 100 Trunk-like

    work page

  76. [76]

    Baseline 0.005 40 – 0.200 200

    work page

  77. [77]

    Shorter Horizon 0.005 25 – 0.125 200

    work page

  78. [78]

    Larger Step Size 0.010 20 – 0.200 100

    work page

  79. [79]

    Increasing Step Sizes 0.005 25 – 0.200 200

    work page

  80. [80]

    Model Switching 0.005 40 10 0.200 200

    work page

Showing first 80 references.