Pith. sign in

REVIEW 3 major objections 4 minor 34 references

Locomotion on Constrained Footholds via Layered Architectures and Model Predictive Control

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read A layered architecture that separates foothold sampling from whole-body MPC lets legged robots traverse stepping-stone terrain in real time, with better cost and reliability than a conventional foot-placement heuristic.

desk verdict Solid layered MPC architecture with a genuine hardware demo; the optimality/reliability comparison is underpowered and partly circular. read the letter →

arxiv 2506.09979 v2 pith:XXIWFVF4 submitted 2025-06-11 cs.RO

Zachary Olkin , Aaron D. Ames This is my paper
classification cs.RO
keywords leggedlocomotionmodelpredictivecontrollayeredarchitectureterrainsamplingfootholdselectionquadrupedrobothumanoidreal-time
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 argues that real-time legged locomotion over constrained footholds does not need a single optimization over both discrete and continuous variables. Its claim is that a two-layer architecture—a sampling-based layer that proposes footholds, plus a full-order model predictive controller solved in parallel for each proposal—can pick terrain online while keeping control rates fast enough for hardware. On a quadruped crossing gaps and height-varying stones, the sampling layer failed once in five runs at 0.7 m/s whereas the conventional heuristic failed four times, and the successful sample-based runs had lower average cost. The same controller ran in real time on a simulated humanoid, which used its arms and waist to keep the torso upright while swinging legs over gaps. If correct, the architecture offers a general way to make hybrid locomotion decisions without hand-coded foot-selection rules.

What carries the argument

The load-bearing object is the fixed-mode MPC: a full-order nonlinear program over joint configuration, velocity, torque, and end-effector forces, in which the contact schedule is fixed and each candidate foothold enters as a polytope constraint on the stance foot, $\gamma_k^l A_k^l r^l(q_k) \le \gamma_k^l b_k^l$. The discrete layer builds a sample tree around a nominal foot position projected from the velocity command; each branch of the tree is a terrain combination, sampled in proportion to the area of each terrain piece inside the robot's kinematic reach, and each branch is evaluated by one parallel MPC solve. The MPC uses a real-time iteration scheme—one quadratic program per control step—with inverse dynamics and quaternion tangent-space coordinates, so each evaluation is cheap enough to run several in parallel and still close the loop at 100 Hz. The point of the machinery is that discrete choices are scored by the actual closed-loop cost of the full-order controller rather than by a heuristic.

What would settle it

Run the same controller on a sparse stepping-stone course while sweeping the number of parallel samples M; if success rate stops improving before worst-case terrain is handled, the remaining failures come from the fixed contact schedule rather than from insufficient terrain exploration.

Watch

Extended reading notes

Core claim

The central claim is that hybrid locomotion decisions can be split by variable type without sacrificing either optimality or real-time feasibility. Discrete choices—which foothold polytope each foot will land on—are made by a gradient-free sampling layer, fixed as constraints in a smooth nonlinear MPC, and the lowest-cost MPC solution over all samples is executed. The paper reports that on stepping stones with varying heights at 0.7 m/s, the sampling controller failed once in five runs while the nominal-foot-placement heuristic failed in four; even the heuristic's successes had a higher average cost (1.0963 versus 0.8946). The same controller steered a simulated humanoid across gaps in real time with two samples, and the whole-body formulation let the robot use its arms and waist for stabilization. This is presented as evidence that the layered approach is more optimal and reliable than heuristic terrain selection and faster than pure sampling-based full-order optimization.

Load-bearing premise

The contact schedule—which feet are in stance or swing at each instant—is fixed in advance and never sampled or optimized; only the terrain beneath those fixed foot timings is chosen.

Editorial extensions

If this is right

  • With four parallel MPC evaluations, the hardware quadruped crossed the stepping-stone course at 0.7 m/s with one failure in five runs, versus four failures for the heuristic, and with lower average cost on successes.
  • Because terrain choice is scored by the MPC cost rather than a fixed rule, the same sampling layer can be pointed at other discrete decisions the paper names as future work, such as contact schedules or arm-bracing contacts.
  • The measured solve times—about 1.2 ms preparation and 4.3 ms feedback for four parallel quadruped samples, and 1.8 ms and 6.2 ms for two humanoid samples—fit inside a 10 ms control period, leaving room for more samples or longer horizons.
  • The controller works without a task-specific foothold heuristic; the nominal position is used only to center the sampling region, not to decide where to step.

Reading between the lines

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

  • Editorial inference: the reliability advantage of sampling over the heuristic should grow with speed and terrain sparsity, because the nominal foot position is exactly the predictor that degrades in those regimes; the 0.7 m/s experiment is one point on that curve.
  • Editorial inference: the fixed contact schedule, not the sampling distribution, is the operative limit; a terrain that demands an atypical gait phase cannot be handled by any amount of foothold sampling, so generalizing the layer to contact timing is the natural next test.
  • Editorial inference: the same split of discrete sampling plus parallel gradient-based evaluation could transfer to manipulation, where the discrete options are grasp points or wall contacts and the continuous problem is a whole-body MPC.
Share X Bluesky LinkedIn Reddit HN

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 proposes a layered controller for legged locomotion over constrained footholds. A gradient-free sampling layer selects which terrain polytopes to use, and a fixed-mode whole-body MPC is solved in parallel for each sampled option; the lowest-cost MPC solution is then executed. The fixed-mode MPC uses a quaternion tangent-space parameterization, inverse-dynamics constraints, polytope foot-placement constraints, and a Raibert-style end-effector cost term. Results include hardware experiments on a Unitree Go2 quadruped and asynchronous MuJoCo simulations of a Unitree G1 humanoid. The authors claim that the layered approach is more optimal and reliable than heuristic terrain selection and faster to compute than pure sampling methods.

Significance. If fully supported, the paper would be a useful practical contribution: the two-layer decomposition is clean, the hardware demonstration on a real quadruped is encouraging, and the real-time humanoid simulation suggests the architecture scales to higher-dimensional systems. The technical formulation is largely standard but competently presented, and the detailed timing breakdown is helpful. The paper does not provide code or formal proofs, and the central comparative claims currently rest on weak evidence; the feasibility result is the strongest part of the paper, while the optimality and reliability claims are not yet established.

major comments (3)
  1. [Sec. IV, reliability experiment] The reliability comparison is based on only five simulation runs per method, with the Raibert heuristic failing 80% of the time and the sampling method failing once. With n=5, this difference is not statistically meaningful; a Fisher exact test would not reach conventional significance. The statement that the sampling method 'outperforms the Raibert heuristic by a significant margin' is therefore unsupported. The authors should run more trials, report confidence intervals or a statistical test, or weaken the claim to an anecdotal observation.
  2. [Sec. IV and Eq. (8)] The optimality comparison is largely circular. The sampling layer selects the lowest-cost MPC solve, and the reported cost is exactly the MPC cost in Eq. (8), so lower average cost for the sampling method is expected by construction. In addition, Sec. III states that the cost function includes the Raibert end-effector placement term, which is the very heuristic used by the baseline; with only one heuristic success versus four sampling successes, averaging the cost over successes introduces further selection bias. An external performance metric (for example, success rate with a larger number of trials, distance traveled, or time to completion) or a cost ablation without the Raibert term is needed.
  3. [Sec. IV and Table I] The claim that the method is 'faster to compute than pure sampling methods' is not supported by direct comparison. Table I reports only the proposed algorithm's own timings on a single CPU, while the comparisons to CIMPC and to sampling-based methods rely on cited timings from different implementations and different hardware. A head-to-head benchmark under the same timing protocol, or at least a same-hardware reimplementation of a baseline, is required before this claim can be evaluated.
minor comments (4)
  1. [Sec. II-A] The sentence 'This will effect how the constraints are formulated' should read 'affect'.
  2. [Sec. I] The phrase 'Simulation results shown the efficacy' should read 'show the efficacy'.
  3. [Sec. II-B] The kinematic-limit circle radius and the polytope shrink amount are not quantified; these are additional free parameters that should be listed explicitly in the experimental setup.
  4. [Sec. III] The description of the humanoid actuation (19 of 27 actuators optimized, with ankle and wrist joints fixed) appears only in the implementation section; stating the resulting degrees of freedom and how the fixed joints are held would improve reproducibility.

Circularity Check

1 steps flagged · score 6.0 of 10

Optimality comparison is circular: the reported cost metric is the exact quantity the sampling layer minimizes; reliability and timing evidence are independent.

  1. self definitional [Section II-B (sampling) and Section IV (Results), with cost defined in Eq. (8)]
    "To select terrain for locomotion, we use a sampling based method where we evaluate the cost of stepping on each piece of terrain throughout the MPC horizon and choose the lowest cost set of terrain. [...] For the sampling, the average cost of the successes was 0.8946 and for the heuristic it was 1.0963, showing an improvement in the optimality even across successes."

    The optimality comparison in Section IV uses the MPC cost J (Eq. 8) as the metric. But Section II-B defines the sampling layer precisely as minimizing that same cost: each terrain sample is an MPC solve, and the lowest-cost solve is selected. Hence the sampling controller's reported average cost is, by construction, a minimum over the sample set, while the Raibert-baseline cost is not minimized. Lower average cost for the sampling method is therefore baked into the selection rule, not an independent measure of optimality. The reliability comparison (failure counts) is separate and non-circular, but the abstract's 'more optimal' claim rests on this self-defined metric.

full rationale

The paper's layered architecture and MPC derivation are self-contained; the sampling layer is described and implemented in full, and the controller's real-time feasibility is demonstrated by hardware and asynchronous simulation. No load-bearing self-citation is present: the citation to the authors' prior work [6] is a general reference for layered architectures, not a uniqueness or derivation step. The central circularity is confined to the optimality comparison. Section II-B states that terrain is selected by evaluating the MPC cost for each sample and choosing the lowest-cost solve, and Section IV then reports the average MPC cost of the successful runs as evidence of 'improvement in the optimality' over the Raibert heuristic. Since the sampling controller's output is defined as the argmin over that same cost (Eq. 8), while the heuristic baseline does not minimize it, the sampling method's lower cost is a direct consequence of the selection rule, not an independent finding. The reliability comparison (1/5 vs 4/5 failures) is external and non-circular, but is statistically weak; the timing comparison is also external. Additional confounds in the same comparison include the Raibert end-effector term in the cost and the fact that average cost is computed only over successes, with a different number of successes per method. These issues affect the strength of the abstract's 'more optimal and reliable' claim, but the reliability and real-time claims have independent content, so the overall circularity is partial rather than total.

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

The central claim depends on tuned cost weights, a fixed contact schedule, known terrain polytopes, and a solver approximation (single SQP iteration). These are either free parameters chosen by hand or domain assumptions stated in the formulation.

free parameters (5)
  • MPC cost weights (Wx, Wu, Wr, WxN, eta)
    The weights for state tracking, input regularization, end-effector tracking, and terminal cost are not reported numerically; they are tuned to make the controller work and the paper's experiments depend on them.
  • Friction coefficient mu
    Used in the linearized friction cone constraints (Sec. II-A.2); no value is given.
  • Desired swing height z_k = 8 cm for quadruped
    The swing height for the foot during swing phase is user-specified; 8 cm is mentioned in the hardware experiment caption.
  • Number of MPC samples M = 4 (quadruped), 2 (humanoid)
    The number of parallel MPC solves is chosen by the user; no sensitivity analysis is provided.
  • Polytope shrink amount and kinematic-limit circle radius
    The terrain polytopes are shrunk to avoid edges, and a circle approximates kinematic limits during sampling; exact values are not specified.
assumptions (5)
  • domain assumption Rigid-body inverse dynamics model f(q,v,a,F) is accurate for the robot
    The MPC uses inverse dynamics (Eq. 2) with finite-differenced accelerations; the hardware model mismatch is acknowledged but the controller robustly handles small errors.
  • domain assumption Contact schedule (which feet are in stance/swing at each node) is fixed and known a priori
    Sec. II-A states gamma_l^k is not an optimization variable; the controller only samples terrain, so it cannot adapt gait timing to terrain.
  • domain assumption Terrain geometry is known and representable as polytopes
    The foothold constraint (Eq. 6) requires polytope descriptions; the paper does not address perception or mapping uncertainties.
  • ad hoc to paper A single SQP iteration with a capped number of QP iterations produces a usable control action
    The MPC is computed with Real Time Iterations, fixing the QP iteration count (10 for quadruped, 8 for humanoid); this is a practical approximation with no convergence or stability guarantee.
  • standard math Quaternion tangent-space linearization adequately captures orientation dynamics over the horizon
    The paper uses the manifold tangent space for quaternion updates (Eq. 5) to avoid Euler singularities; this is a standard Lie-group technique.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Locomotion on Constrained Footholds via Layered Architectures and Model Predictive Control." pith.science (2026). https://pith.science/paper/XXIWFVF4

@misc{pith2026250609979,
  author       = {Pith},
  title        = {Pith review of: Locomotion on Constrained Footholds via Layered Architectures and Model Predictive Control},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XXIWFVF4}},
  note         = {Machine review of arXiv:2506.09979}
}
read the original abstract

Computing stabilizing and optimal control actions for legged locomotion in real time is difficult due to the nonlinear, hybrid, and high dimensional nature of these robots. The hybrid nature of the system introduces a combination of discrete and continuous variables which causes issues for numerical optimal control. To address these challenges, we propose a layered architecture that separates the choice of discrete variables and a smooth Model Predictive Controller (MPC). The layered formulation allows for online flexibility and optimality without sacrificing real-time performance through a combination of gradient-free and gradient-based methods. The architecture leverages a sampling-based method for determining discrete variables, and a classical smooth MPC formulation using these fixed discrete variables. We demonstrate the results on a quadrupedal robot stepping over gaps and onto terrain with varying heights. In simulation, we demonstrate the controller on a humanoid robot for gap traversal. The layered approach is shown to be more optimal and reliable than common heuristic-based approaches and faster to compute than pure sampling methods.

Figures

Figures reproduced from arXiv: 2506.09979 by the authors.

Figure 1
Figure 1. Demonstration of the layered controller dynamically navigating over [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Diagram of the control architecture. Velocity commands from the user are used for sampling and go into the MPC cost. A set of contacts (terrain [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. Example result from the sampling. At sampling time, all feet are [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: Comparison between using our sampling based approach and a [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]
Figure 5
Figure 5. Figure 5: Hardware experiment on the Unitree Go2 quadruped where the terrain is composed of two large tiles separated by three smaller blocks. The [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]
Figure 6
Figure 6. Figure 6: Simulation demonstration of the humanoid traversing stepping [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

34 extracted references · 20 canonical work pages

  1. [1]

    Cafe-Mpc: A Cascaded-Fidelity Model Pre- dictive Control Framework with Tuning-Free Whole-Body Control,

    H. Li and P. M. Wensing, “Cafe-Mpc: A Cascaded-Fidelity Model Pre- dictive Control Framework with Tuning-Free Whole-Body Control,” Mar. 2024, arXiv:2403.03995 [cs]

    arXiv 2024

  2. [2]

    Tailoring Solution Accuracy for Fast Whole-body Model Predictive Control of Legged Robots,

    C. Khazoom, S. Hong, M. Chignoli, E. Stanger-Jones, and S. Kim, “Tailoring Solution Accuracy for Fast Whole-body Model Predictive Control of Legged Robots,” Jul. 2024, arXiv:2407.10789 [cs]. [Online]. Available: http://arxiv.org/abs/2407.10789 TABLE I MPC T IMINGS Robot Optimization DoF Nodes # Parallel MPC Prep Time (ms) Feedback Time (ms) Average / Std....

    arXiv 2024

  3. [3]

    A Unified MPC Framework for Whole-Body Dynamic Locomotion and Manipulation,

    J.-P. Sleiman, F. Farshidian, M. V . Minniti, and M. Hutter, “A Unified MPC Framework for Whole-Body Dynamic Locomotion and Manipulation,” Mar. 2021, arXiv:2103.00946 [cs]

    arXiv 2021

  4. [4]

    Momentum-Aware Trajectory Optimization and Control for Agile Quadrupedal Locomotion

    Z. Zhou, B. Wingo, N. Boyd, S. Hutchinson, and Y . Zhao, “Momentum-Aware Trajectory Optimization and Control for Agile Quadrupedal Locomotion,” Jun. 2022, arXiv:2203.01548 [cs]

    work page Pith review arXiv 2022

  5. [5]

    Planar Bipedal Locomotion with Nonlinear Model Predictive Control: Online Gait Generation using Whole-Body Dynamics

    M. Y . Galliker, N. Csomay-Shanklin, R. Grandia, A. J. Taylor, F. Farshidian, M. Hutter, and A. D. Ames, “Planar Bipedal Locomotion with Nonlinear Model Predictive Control: Online Gait Generation using Whole-Body Dynamics,” Nov. 2022, arXiv:2203.07429 [cs]

    work page Pith review arXiv 2022

  6. [6]

    Layered Nonlinear Model Predictive Control for Robust Stabilization of Hybrid Systems

    Z. Olkin and A. D. Ames, “Layered Nonlinear Model Predictive Control for Robust Stabilization of Hybrid Systems,” Mar. 2025, arXiv:2503.12810 [eess]. [Online]. Available: http://arxiv.org/abs/ 2503.12810

    work page Pith review arXiv 2025

  7. [7]

    Simultaneous Contact, Gait and Motion Planning for Robust Multi-Legged Locomotion via Mixed-Integer Convex Optimization

    B. Aceituno-Cabezas, C. Mastalli, H. Dai, M. Focchi, A. Radulescu, D. G. Caldwell, J. Cappelletto, J. C. Grieco, G. Fernandez- Lopez, and C. Semini, “Simultaneous Contact, Gait and Motion Planning for Robust Multi-Legged Locomotion via Mixed-Integer Convex Optimization,” IEEE Robotics and Automation Letters , pp. 1–1, 2017, arXiv:1904.04595 [cs]. [Online]...

    work page Pith review arXiv 2017

  8. [8]

    Footstep planning on uneven terrain with mixed-integer convex optimization,

    R. Deits and R. Tedrake, “Footstep planning on uneven terrain with mixed-integer convex optimization,” in 2014 IEEE-RAS International Conference on Humanoid Robots . Madrid, Spain: IEEE, Nov. 2014, pp. 279–286. [Online]. Available: http://ieeexplore.ieee.org/document/ 7041373/

    work page 2014

Show all 34 references
  1. [9]

    Multi- Layered Safety for Legged Robots via Control Barrier Functions and Model Predictive Control,

    R. Grandia, A. J. Taylor, A. D. Ames, and M. Hutter, “Multi- Layered Safety for Legged Robots via Control Barrier Functions and Model Predictive Control,” in 2021 IEEE International Conference on Robotics and Automation (ICRA) . Xi’an, China: IEEE, May 2021, pp. 8352–8358

    2021

  2. [10]

    Perceptive Locomotion through Nonlinear Model Predictive Control,

    R. Grandia, F. Jenelten, S. Yang, F. Farshidian, and M. Hutter, “Perceptive Locomotion through Nonlinear Model Predictive Control,” Aug. 2022, arXiv:2208.08373 [cs]. [Online]. Available: http: //arxiv.org/abs/2208.08373

    2022 arXiv

  3. [11]

    Inverse Dynamics Tra- jectory Optimization for Contact-Implicit Model Predictive Control,

    V . Kurtz, A. Castro, A. O. Onol, and H. Lin, “Inverse Dynamics Tra- jectory Optimization for Contact-Implicit Model Predictive Control,” Sep. 2023, arXiv:2309.01813 [cs]

    2023 arXiv

  4. [12]

    Whole-Body Nonlinear Model Predictive Control Through Contacts for Quadrupeds,

    M. Neunert, M. St ¨auble, M. Giftthaler, C. D. Bellicoso, J. Carius, C. Gehring, M. Hutter, and J. Buchli, “Whole-Body Nonlinear Model Predictive Control Through Contacts for Quadrupeds,” IEEE Robotics and Automation Letters , vol. 3, no. 3, pp. 1458–1465, Jul. 2018, arXiv:171...

    2018 arXiv

  5. [13]

    Hybrid iLQR Model Predictive Control for Contact Implicit Stabilization on Legged Robots,

    N. J. Kong, C. Li, and A. M. Johnson, “Hybrid iLQR Model Predictive Control for Contact Implicit Stabilization on Legged Robots,” Jul. 2022, arXiv:2207.04591 [cs]

    2022 arXiv

  6. [14]

    iLQR for Piecewise- Smooth Hybrid Dynamical Systems,

    N. J. Kong, G. Council, and A. M. Johnson, “iLQR for Piecewise- Smooth Hybrid Dynamical Systems,” Sep. 2021, arXiv:2103.14584 [cs]. [Online]. Available: http://arxiv.org/abs/2103.14584

    2021 arXiv

  7. [15]

    Contact- Implicit MPC: Controlling Diverse Quadruped Motions Without Pre- Planned Contact Modes or Trajectories,

    G. Kim, D. Kang, J.-H. Kim, S. Hong, and H.-W. Park, “Contact- Implicit MPC: Controlling Diverse Quadruped Motions Without Pre- Planned Contact Modes or Trajectories,” Dec. 2023, arXiv:2312.08961 [cs]

    2023 arXiv

  8. [16]

    Direct Trajectory Optimization of Rigid Body Dynamical Systems through Contact,

    M. Posa and R. Tedrake, “Direct Trajectory Optimization of Rigid Body Dynamical Systems through Contact,” in Algorithmic F oundations of Robotics X . Berlin, Heidelberg: Springer Berlin Heidelberg, 2013, vol. 86, pp. 527–542, series Title: Springer Tracts in Advanced Robotics....

    2013 doi

  9. [17]

    Fast Contact-Implicit Model- Predictive Control,

    S. L. Cleac’h, T. Howell, S. Yang, C.-Y . Lee, J. Zhang, A. Bishop, M. Schwager, and Z. Manchester, “Fast Contact-Implicit Model- Predictive Control,” Jan. 2023, arXiv:2107.05616 [cs, eess]. [Online]. Available: http://arxiv.org/abs/2107.05616

    2023 arXiv

  10. [18]

    Optimization-Based Control for Dynamic Legged Robots,

    P. M. Wensing, M. Posa, Y . Hu, A. Escande, N. Mansard, and A. Del Prete, “Optimization-Based Control for Dynamic Legged Robots,” Nov. 2022, arXiv:2211.11644 [cs]

    2022 arXiv

  11. [19]

    Predictive Sampling: Real-time Behaviour Synthesis with MuJoCo,

    T. Howell, N. Gileadi, S. Tunyasuvunakool, K. Zakka, T. Erez, and Y . Tassa, “Predictive Sampling: Real-time Behaviour Synthesis with MuJoCo,” Dec. 2022, arXiv:2212.00541 [cs, eess]. [Online]. Available: http://arxiv.org/abs/2212.00541

    2022 arXiv

  12. [20]

    Full-Order Sampling- Based MPC for Torque-Level Locomotion Control via Diffusion-Style Annealing,

    H. Xue, C. Pan, Z. Yi, G. Qu, and G. Shi, “Full-Order Sampling- Based MPC for Torque-Level Locomotion Control via Diffusion-Style Annealing,” Sep. 2024, arXiv:2409.15610 [cs]. [Online]. Available: http://arxiv.org/abs/2409.15610

    2024 arXiv

  13. [21]

    Real-Time Whole-Body Control of Legged Robots with Model-Predictive Path Integral Control,

    J. Alvarez-Padilla, J. Z. Zhang, S. Kwok, J. M. Dolan, and Z. Manchester, “Real-Time Whole-Body Control of Legged Robots with Model-Predictive Path Integral Control,” Sep. 2024, arXiv:2409.10469 [cs]. [Online]. Available: http: //arxiv.org/abs/2409.10469

    2024 arXiv

  14. [22]

    Model-Based Diffusion for Trajectory Optimization,

    C. Pan, Z. Yi, G. Shi, and G. Qu, “Model-Based Diffusion for Trajectory Optimization,” May 2024, arXiv:2407.01573 [cs]. [Online]. Available: http://arxiv.org/abs/2407.01573

    2024 arXiv

  15. [23]

    Model Predictive Path Integral Control: From Theory to Parallel Computation,

    G. Williams, A. Aldrich, and E. A. Theodorou, “Model Predictive Path Integral Control: From Theory to Parallel Computation,” Journal of Guidance, Control, and Dynamics , vol. 40, no. 2, pp. 344–357, Feb. 2017

    2017

  16. [24]

    Raibert, Legged Robots That Balance

    M. Raibert, Legged Robots That Balance . The MIT Press, Mar. 2000

    2000

  17. [25]

    A Real-Time Iteration Scheme for Nonlinear Optimization in Optimal Feedback Control,

    M. Diehl, H. G. Bock, and J. P. Schl ¨oder, “A Real-Time Iteration Scheme for Nonlinear Optimization in Optimal Feedback Control,” SIAM Journal on Control and Optimization , vol. 43, no. 5, pp. 1714– 1736, Jan. 2005

    2005

  18. [26]

    Planning With Attitude,

    B. E. Jackson, K. Tracy, and Z. Manchester, “Planning With Attitude,” IEEE Robotics and Automation Letters , vol. 6, no. 3, pp. 5658–5664, Jul. 2021

    2021

  19. [27]

    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,” Jun. 2020, arXiv:2003.02547 [cs, eess, math]. [Online]. Available: http: //arxiv.org/abs/2003.02547

    2020 arXiv

  20. [28]

    A micro Lie theory for state estimation in robotics,

    J. Sol `a, J. Deray, and D. Atchuthan, “A micro Lie theory for state estimation in robotics,” Dec. 2021, arXiv:1812.01537 [cs]. [Online]. Available: http://arxiv.org/abs/1812.01537

    2021 arXiv

  21. [29]

    An auto-generated real- time iteration algorithm for nonlinear MPC in the microsecond range,

    B. Houska, H. J. Ferreau, and M. Diehl, “An auto-generated real- time iteration algorithm for nonlinear MPC in the microsecond range,” Automatica, vol. 47, no. 10, pp. 2279–2285, Oct. 2011

    2011

  22. [30]

    acados: a modular open-source framework for fast embedded optimal control,

    R. Verschueren, G. Frison, D. Kouzoupis, J. Frey, N. van Duijkeren, A. Zanelli, B. Novoselnik, T. Albin, R. Quirynen, and M. Diehl, “acados: a modular open-source framework for fast embedded optimal control,” Nov. 2020, arXiv:1910.13753 [math]

    2020 arXiv

  23. [31]

    The Pinocchio C++ library : A fast and flexible implementation of rigid body dynamics algorithms and their analytical derivatives,

    J. Carpentier, G. Saurel, G. Buondonno, J. Mirabel, F. Lamiraux, O. Stasse, and N. Mansard, “The Pinocchio C++ library : A fast and flexible implementation of rigid body dynamics algorithms and their analytical derivatives,” in 2019 IEEE/SICE International Symposium on System ...

    2019

  24. [32]

    Real-time optimization and non- linear model predictive control of processes governed by differential- algebraic equations,

    M. Diehl, H. G. Bock, and Z. Nagy, “Real-time optimization and non- linear model predictive control of processes governed by differential- algebraic equations,” Journal of Process Control , 2002

    2002

  25. [33]

    Implementing Regularized Predictive Control for Simultaneous Real-Time Footstep and Ground Reaction Force Optimization,

    G. Bledt and S. Kim, “Implementing Regularized Predictive Control for Simultaneous Real-Time Footstep and Ground Reaction Force Optimization,” in 2019 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS) . Macau, China: IEEE, Nov. 2019, pp. 6316–6323. [On...

    2019

  26. [34]

    Hydrax: Sampling-based model predictive control on GPU with JAX and MuJoCo MJX,

    V . Kurtz, “Hydrax: Sampling-based model predictive control on GPU with JAX and MuJoCo MJX,” 2024. [Online]. Available: https://github.com/vincekurtz/hydrax

    2024

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.