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REVIEW 3 major objections 5 minor 29 references

Hierarchical Reduced-Order Model Predictive Control for Robust Locomotion on Humanoid Robots

T0 review · 3 major / 5 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read The paper claims that a hierarchical ROM-MPC with adaptive step timing and arm/torso-aware mid-level control raises push-recovery success by 36% and cuts pelvis yaw drift under torso twists.

desk verdict Solid, useful humanoid locomotion controller; the headline robustness number is real but oversold by simulation-only results and a confounded baseline comparison. read the letter →

arxiv 2509.04722 v1 pith:Q6YRVILV submitted 2025-09-05 cs.RO

Adrian B. Ghansah , Sergio A. Esteban , Aaron D. Ames This is my paper · ORCID
classification cs.RO
keywords humanoidlocomotionmodelpredictivecontrolreduced-ordermodelsangularmomentumlinearinvertedpendulum(ALIP)adaptivesteptimingpushrecoveryupper-bodyyawdisturbancerejection
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 is trying to establish that a humanoid robot can be made more robust for real-world walking by splitting locomotion control into two predictive layers built on simplified models: a slower nonlinear planner that chooses step lengths, step durations, and ankle torques using the angular-momentum ALIP model, and a fast linear MPC that tracks the resulting plan while also using arm accelerations and torso rotation to resist perturbations. If true, it would mean humanoid locomotion controllers can keep the tractability of reduced-order models while recovering some of the stabilizing behavior usually reserved for whole-body control. The paper reports a 36% increase in push-recovery success rate with adaptive step timing over a fixed step period, and reduced pelvis yaw deviations when the robot is twisted about the vertical axis, validated in simulation and on the Unitree G1 humanoid. The upper-body yaw-rejection result is shown in simulation; the hardware terrain experiments run with upper-body control disabled.

What carries the argument

The load-bearing object is the step-to-step (S2S) ALIP dynamics in (11), which maps pre-impact angular-momentum states across steps under step length, step period, and ankle torque. It carries the high-level NMPC cost and constraints, letting the planner vary step timing and placement. The second mechanism is the Decomposed SRB (DSRB) dynamics in (28), an extension of single-rigid-body dynamics with torso yaw and two arm point masses whose x-axis acceleration produces equal-and-opposite forces and moments; the linearity comes from fixing the arm y/z positions and forces to zero, so the moment from each arm is a constant cross product times a scalar force. This keeps the mid-level MPC a quadr

What would settle it

Run the robot at 0.3 m/s on hardware, apply a 30–150 Nm yaw moment to the torso for 0.05 s, and compare pelvis yaw recovery with DSRB-MPC versus SRB-MPC; if the yaw deviation curves overlap, the upper-body contribution claim fails. Equally, log commanded versus measured arm force during recovery: if the arm joints cannot deliver the commanded x-axis forces within their torque limits, the mechanism is not realized.

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Extended reading notes

Core claim

The central claim is that an ALIP-based step-to-step nonlinear MPC can solve for foot placement, step timing, and ankle torque together, and that its plan can be converted into references for a linear SRB MPC whose model is decomposed to include arms and torso without losing linearity. Concretely, the paper shows the dynamics in (28): arms are modeled as point masses constrained to move along the x-axis, generating a force and coupled moments, and a torso yaw actuator generates an opposite moment on the lower body. With this Decomposed SRB model, the mid-level MPC can command arm forces and torso torques to reject yaw moments. The authors report that adaptive step periods in the range 0.25–0

Load-bearing premise

The mid-level DSRB model assumes each arm is a point mass that can only be accelerated along the x-axis with fixed y/z lever arms, and that the whole upper-body effect on the pelvis is a single z-axis torque; if real arm inertia, joint limits, or off-axis forces violate this, the predicted yaw rejection may not appear on hardware.

Editorial extensions

If this is right

  • A humanoid can absorb pushes by shortening or lengthening steps on the fly rather than keeping a fixed cadence, improving recovery without heavier whole-body optimization.
  • The DSRB formulation gives a computationally cheap way for upper-body motion to actively cancel yaw moments, reducing reliance on stance-foot friction for torsion disturbances.
  • The 40 Hz high-level / 500 Hz mid-level split suggests the full stack can run on low-power onboard compute while keeping real-time quadratic-program solves.
  • The same S2S ALIP plan can feed different mid-level models, making the high-level planner a reusable interface for step placement and timing.
  • The approach is robust across uneven outdoor and indoor terrains despite the ROM assumptions, pointing toward deployment in varied real environments.

Reading between the lines

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

  • An implication the authors leave implicit is that the yaw-rejection benefit should transfer best to disturbances whose moment-arm geometry matches the assumed fixed r_LA,y and r_LA,z; a testable extension is to add asymmetric torso loads or off-axis pushes and observe when arm joint torque limits saturate.
  • The 36% push-recovery gain in simulation may understate or overstate hardware performance depending on push phase; a hardware experiment sweeping push timing uniformly over the step cycle would give a tighter estimate of the mean recovery probability.
  • Because the high-level ALIP planner ignores arm and torso dynamics, feeding DSRB state limits and upper-body constraints into the high-level planner is a natural next step; one concrete test is whether adding arm-state bounds preserves yaw rejection under larger twists.
  • The arm yaw-rejection mechanism depends on the controller being able to command x-axis arm forces fast enough; logging commanded versus measured arm force during a recovery would reveal whether hardware joints can actually realize the simulated benefit.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper proposes a hierarchical model predictive control framework for humanoid locomotion. A high-level NMPC optimizes step lengths, step periods, and ankle torques using the step-to-step dynamics of the ALIP model at 40 Hz; the resulting plan is converted into references for a 500 Hz linear MPC based on a decomposition of the single rigid body dynamics into lower body, torso, and arms (DSRB). The manuscript reports simulation studies showing that the DSRB model reduces pelvis yaw deviation after yaw moment disturbances and that adaptive step periods increase push-recovery success by 36%, plus hardware push-recovery and terrain-walking experiments on the Unitree G1.

Significance. If validated, the proposed hierarchical architecture is practically important: it demonstrates that nonlinear step-timing and step-length optimization can run on an onboard mini-PC at 40 Hz while a convex MPC runs at 500 Hz, and the DSRB model is an elegant way to incorporate simplified upper-body actuation into a linear MPC. The hardware demonstrations on indoor/outdoor terrain support feasibility and robustness. However, the two headline quantitative claims rest on modest simulation evidence, and the upper-body benefit is not demonstrated on hardware. The paper would be strengthened by more rigorous experimental reporting and by qualifying simulation-only claims.

major comments (3)
  1. [§IV-C, Fig. 5] The 36% improvement in push-recovery success is not supported by the reported data. Each force combination was tested only 5 times, with a binary success/failure outcome; with n=5, a single trial changes the success rate by 20 percentage points, so no statistically meaningful comparison can be made. The figure also appears to conflate two factors: the fixed-step condition uses T=0.35 s while the adaptive condition uses a desired period of 0.4 s with bounds 0.25–0.5 s. The improvement may be due to the lower nominal step frequency rather than to adaptation. Please report per-cell success counts, confidence intervals (or a formal test), and run control experiments that vary the fixed step period and the desired adaptive period independently.
  2. [§IV-B and §IV-D.3] The central claim that upper-body control improves yaw disturbance rejection is supported only by simulation. The hardware terrain experiments in Sec. IV-D.3 explicitly state that upper-body control was disabled, so no hardware evidence validates the DSRB model. Moreover, the DSRB model in Sec. III-C assumes each arm is a point mass moving only along the x-axis, with F_LA,y=F_LA,z=0 and fixed moment arms r_LA,y/r_LA,z, and that the torso–lower-body interaction is a single z-axis torque. There is no demonstration that these assumptions are realizable on the G1 hardware, e.g., under joint limits, arm inertia, and torque limits. The abstract and conclusion currently state the benefit without this simulation-only caveat. Please either add hardware yaw-disturbance experiments with upper-body control enabled or, at minimum, substantially rephrase the claims and add a sensitivity/discussion sec
  3. [§IV-C, general experimental methodology] The push-recovery simulation experiments are not described in enough detail to be reproducible or to assess the effect's magnitude. The text reports 'external force (F_x: –600 – 600 N, F_y: 0 – 400 N) was applied for 0.1 s' and 'the robot was walking in place,' but does not state how the force was applied in simulation (e.g., direction relative to the walking frame, the exact timing distribution, or the robot's state distribution at push). The success rate depends strongly on push phase; with only 5 trials, the phase coverage is likely poor. Please provide a detailed experimental protocol, including the number of trials per condition, the phase sampling method, and the variance across trials.
minor comments (5)
  1. [§II-D, Eq. (13)] The state x_SRB in Eq. (2) has 15 components (p, Θ, v, ω, g), while Eq. (13) uses an identity matrix I_13×13. This is a dimension mismatch that should be corrected or clarified.
  2. [§IV-B, Fig. 4] The figure shows raw data and binned means/standard deviations, but no sample size per bin is given. The binning in 10 Nm steps is coarse; please report the number of trials per bin and consider plotting individual trajectories or medians with interquartile ranges.
  3. [§III-B.1, Eq. (23)] The notation for the stance foot position p_STF,xy,k and the use of k_n in the following section is not fully defined; please clarify the indexing and the relationship between step index k and MPC node n.
  4. [Abstract] The phrase 'the upper body control improved the yaw disturbance rejection' should be qualified as a simulation result, since hardware experiments did not exercise upper-body control.
  5. [General] There are several minor typographical issues in equations, e.g., the multiplication dots in Eq. (16) and the layout of Eq. (28). A careful proofread would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the controller claims are empirical comparisons, not derivations from fitted inputs or self-cited uniqueness theorems.

full rationale

The paper's central claims are experimental: the 36% push-recovery improvement is a measured success-rate comparison between fixed and adaptive step periods (Sec. IV-C), and the yaw-rejection benefit is a simulated comparison between SRB-MPC and DSRB-MPC (Sec. IV-B). Neither quantity is fitted from the data it is compared against, nor is any prediction equivalent to an input by construction. The DSRB model in Sec. III-C is an explicit modeling choice (arm point masses constrained to x-motion, torso z torque) and the yaw-rejection mechanism follows from the Newton-Euler equations (28)-(30); however, the claimed result is the closed-loop behavioral outcome of the MPC, not an equation that reduces to the model definition. Self-citations [9], [23], [6], [4] provide standard ROMs and trajectory tools; they are externally published and not used as a uniqueness or existence theorem to force the paper's conclusions. The main limitation noted in Sec. IV-D.3 is that hardware terrain experiments disabled upper-body control, so the yaw-rejection result is simulation-only; this is a generalization/correctness caveat, not circularity.

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

The central claims rest on the ALIP/SRB/DSRB model assumptions and several hand-tuned controller parameters. The main support is empirical: simulation and hardware experiments. No code or data are released, and the cost weights and some model parameter values are not reported.

free parameters (4)
  • Step period bounds T_lb, T_ub = 0.25 s, 0.5 s
    Chosen by hand for the adaptive step timing experiments; directly bounds the optimization search space and affects the reported 36% improvement.
  • Desired step period T_des = 0.4 s (adaptive), 0.35 s (fixed)
    Baseline step frequency; the fixed vs adaptive comparison uses different nominal periods, potentially confounding the success-rate gain.
  • Desired step width ell_offset_y = not specified
    Used in equation (18) to compute desired lateral step length; a nominal constant chosen by hand.
  • MPC cost weights = not reported
    Weights Qx, R_l, R_T, R_tau, Q_SRB, R_SRB in equations (16) and (15) are tuned but values are not given, limiting reproducibility.
assumptions (4)
  • domain assumption ALIP model: constant COM height p_z and conserved angular momentum about the stance foot during foot switching (Section II-C).
    The S2S dynamics (11) and the NMPC (16) rest on these assumptions; they are standard for LIP/ALIP but violated during aggressive push recovery.
  • domain assumption Step-to-step impact model: pre- and post-impact states are related by B_d * step length (Eq. 9), which assumes an instantaneous, perfectly plastic impact with zero double-support time.
    This is implicit in the S2S dynamics derivation and affects the validity of the planner's predictions.
  • ad hoc to paper DSRB decomposition: arm masses move only along the x-axis and F_LA,y=F_LA,z=0, so (29) reduces to (30); torso-lower body interaction is a single z-axis torque (Section III-C).
    This linearity-forcing constraint is introduced solely to keep the mid-level MPC linear; it is a modeling assumption not derived from the robot's physical actuation capabilities.
  • domain assumption Current ALIP state estimate (21)-(22) uses diagonal inertia and COM velocity to approximate angular momentum; this is an estimate rather than a direct measurement.
    The planner initializes x_ALIP^0 from this approximation; errors could corrupt the optimized step plan.

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Cite this review

Pith. "Pith review of Hierarchical Reduced-Order Model Predictive Control for Robust Locomotion on Humanoid Robots." pith.science (2026). https://pith.science/paper/Q6YRVILV

@misc{pith2026250904722,
  author       = {Pith},
  title        = {Pith review of: Hierarchical Reduced-Order Model Predictive Control for Robust Locomotion on Humanoid Robots},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Q6YRVILV}},
  note         = {Machine review of arXiv:2509.04722}
}
read the original abstract

As humanoid robots enter real-world environments, ensuring robust locomotion across diverse environments is crucial. This paper presents a computationally efficient hierarchical control framework for humanoid robot locomotion based on reduced-order models -- enabling versatile step planning and incorporating arm and torso dynamics to better stabilize the walking. At the high level, we use the step-to-step dynamics of the ALIP model to simultaneously optimize over step periods, step lengths, and ankle torques via nonlinear MPC. The ALIP trajectories are used as references to a linear MPC framework that extends the standard SRB-MPC to also include simplified arm and torso dynamics. We validate the performance of our approach through simulation and hardware experiments on the Unitree G1 humanoid robot. In the proposed framework the high-level step planner runs at 40 Hz and the mid-level MPC at 500 Hz using the onboard mini-PC. Adaptive step timing increased the push recovery success rate by 36%, and the upper body control improved the yaw disturbance rejection. We also demonstrate robust locomotion across diverse indoor and outdoor terrains, including grass, stone pavement, and uneven gym mats.

Figures

Figures reproduced from arXiv: 2509.04722 by the authors.

Figure 1
Figure 1. The G1 humanoid robot traversing a variety of terrains [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. In the high-level planner, NMPC is used on the S2S dynamics of the ALIP model to generate a step plan. The step [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. (a) The full humanoid robot model and (b) the [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: Pelvis yaw after recovery from torso yaw disturbances [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]
Figure 5
Figure 5. Figure 5: Success rates for push recovery under external dis [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]
Figure 6
Figure 6. Figure 6: The robot is exposed to external disturbances while walking in place. The plots shows how the stepping periods and [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

29 extracted references · 22 canonical work pages

  1. [1]

    Hybrid zero dynamics of planar biped walkers,

    E. R. Westervelt, J. W. Grizzle, and D. E. Koditschek, “Hybrid zero dynamics of planar biped walkers,”IEEE transactions on automatic control, vol. 48, no. 1, pp. 42–56, 2003

    2003

  2. [2]

    Realizing dynamic and efficient bipedal locomotion on the humanoid robot durus,

    J. Reher, E. A. Cousineau, A. Hereid, C. M. Hubicki, and A. D. Ames, “Realizing dynamic and efficient bipedal locomotion on the humanoid robot durus,” in2016 IEEE International Conference on Robotics and Automation (ICRA). IEEE, 2016, pp. 1794–1801

    work page 2016

  3. [3]

    Dynamic walking with compliance on a cassie bipedal robot,

    J. Reher, W.-L. Ma, and A. D. Ames, “Dynamic walking with compliance on a cassie bipedal robot,” in2019 18th European Control Conference (ECC). IEEE, 2019, pp. 2589–2595

    work page 2019

  4. [4]

    Humanoid robot co-design: Coupling hardware design with gait generation via hybrid zero dynamics,

    A. B. Ghansah, J. Kim, M. Tucker, and A. D. Ames, “Humanoid robot co-design: Coupling hardware design with gait generation via hybrid zero dynamics,” in2023 62nd IEEE Conference on Decision and Control (CDC). IEEE, 2023, pp. 1879–1885

    work page 2023

  5. [5]

    Whole-body model predictive control for biped locomotion on a torque-controlled humanoid robot,

    E. Dantec, M. Naveau, P. Fernbach, N. Villa, G. Saurel, O. Stasse, M. Taix, and N. Mansard, “Whole-body model predictive control for biped locomotion on a torque-controlled humanoid robot,” in 2022 IEEE-RAS 21st International Conference on Humanoid Robots (Humanoids). IEEE, 2022, pp. 638–644

    work page 2022

  6. [6]

    Reduced-Order Model Guided Contact-Implicit Model Predictive Control for Humanoid Locomotion

    S. A. Esteban, V . Kurtz, A. B. Ghansah, and A. D. Ames, “Reduced- order model guided contact-implicit model predictive control for humanoid locomotion,”arXiv preprint arXiv:2502.15630, 2025

    work page Pith review arXiv 2025

  7. [7]

    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,”IEEE Robot. Autom. Lett., 2024

    work page 2024

  8. [8]

    The 3d linear inverted pendulum mode: A simple modeling for a biped walking pattern generation,

    S. Kajita, F. Kanehiro, K. Kaneko, K. Yokoi, and H. Hirukawa, “The 3d linear inverted pendulum mode: A simple modeling for a biped walking pattern generation,” inin Proc. IEEE/RSJ Int. Conf. Intell. Robots Syst, vol. 1. IEEE, 2001, pp. 239–246

    work page 2001

Show all 29 references
  1. [9]

    3-d underactuated bipedal walking via h-lip based gait synthesis and stepping stabilization,

    X. Xiong and A. Ames, “3-d underactuated bipedal walking via h-lip based gait synthesis and stepping stabilization,”IEEE Transactions on Robotics, vol. 38, no. 4, pp. 2405–2425, 2022

    2022

  2. [10]

    Angular momentum about the contact point for control of bipedal locomotion: Validation in a lip-based controller,

    Y . Gong and J. Grizzle, “Angular momentum about the contact point for control of bipedal locomotion: Validation in a lip-based controller,” arXiv preprint arXiv:2008.10763, 2020

    2008 arXiv

  3. [11]

    The spring-mass model for running and hopping,

    R. Blickhan, “The spring-mass model for running and hopping,” Journal of biomechanics, vol. 22, no. 11-12, pp. 1217–1227, 1989

    1989

  4. [12]

    Global position control on underactuated bipedal robots: Step-to-step dynamics approximation for step planning,

    X. Xiong, J. Reher, and A. D. Ames, “Global position control on underactuated bipedal robots: Step-to-step dynamics approximation for step planning,” in2021 IEEE International Conference on Robotics and Automation (ICRA). IEEE, 2021, pp. 2825–2831

    2021

  5. [13]

    Terrain- adaptive, alip-based bipedal locomotion controller via model predictive control and virtual constraints,

    G. Gibson, O. Dosunmu-Ogunbi, Y . Gong, and J. Grizzle, “Terrain- adaptive, alip-based bipedal locomotion controller via model predictive control and virtual constraints,” in2022 IEEE/RSJ International Con- ference on Intelligent Robots and Systems (IROS). IEEE, 2022, pp. 6724–6731

    2022

  6. [14]

    Stair climbing using the angular momentum linear inverted pendulum model and model predictive control,

    O. Dosunmu-Ogunbi, A. Shrivastava, G. Gibson, and J. W. Grizzle, “Stair climbing using the angular momentum linear inverted pendulum model and model predictive control,” in2023 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS). IEEE, 2023, pp. 8558–8565

    2023

  7. [15]

    A model predictive capture point control framework for robust humanoid balancing via ankle, hip, and stepping strategies,

    M.-J. Kim, D. Lim, G. Park, and J. Park, “A model predictive capture point control framework for robust humanoid balancing via ankle, hip, and stepping strategies,”arXiv preprint arXiv:2307.13243, 2023

    2023 arXiv

  8. [16]

    Seamless reaction strategy for bipedal locomotion exploiting real-time nonlinear model predictive control,

    J. Choe, J.-H. Kim, S. Hong, J. Lee, and H.-W. Park, “Seamless reaction strategy for bipedal locomotion exploiting real-time nonlinear model predictive control,”IEEE Robotics and Automation Letters, vol. 8, no. 8, pp. 5031–5038, 2023

    2023

  9. [17]

    Adapting gait frequency for posture-regulating humanoid push-recovery via hierarchical model predictive control,

    J. Li, Z. Le, J. Ma, and Q. Nguyen, “Adapting gait frequency for posture-regulating humanoid push-recovery via hierarchical model predictive control,”arXiv preprint arXiv:2409.14342, 2024

    2024 arXiv

  10. [18]

    Whole-body motion planning with centroidal dynamics and full kinematics,

    H. Dai, A. Valenzuela, and R. Tedrake, “Whole-body motion planning with centroidal dynamics and full kinematics,” in2014 IEEE-RAS International Conference on Humanoid Robots. IEEE, 2014, pp. 295–302

    2014

  11. [19]

    Humanoid arm motion planning for im- proved disturbance recovery using model hierarchy predictive control,

    C. Khazoom and S. Kim, “Humanoid arm motion planning for im- proved disturbance recovery using model hierarchy predictive control,” in2022 International Conference on Robotics and Automation (ICRA). IEEE, 2022, pp. 6607–6613

    2022

  12. [20]

    Arm/trunk motion generation for humanoid robot,

    D. Xing and J. Su, “Arm/trunk motion generation for humanoid robot,” Science China Information Sciences, vol. 53, pp. 1603–1612, 2010

    2010

  13. [21]

    Force-and-moment-based model predictive control for achieving highly dynamic locomotion on bipedal robots,

    J. Li and Q. Nguyen, “Force-and-moment-based model predictive control for achieving highly dynamic locomotion on bipedal robots,” in 2021 60th IEEE Conference on Decision and Control (CDC). IEEE, 2021, pp. 1024–1030

    2021

  14. [22]

    Dynamic locomotion in the mit cheetah 3 through convex model-predictive control,

    J. Di Carlo, P. M. Wensing, B. Katz, G. Bledt, and S. Kim, “Dynamic locomotion in the mit cheetah 3 through convex model-predictive control,” in2018 IEEE/RSJ international conference on intelligent robots and systems (IROS). IEEE, 2018, pp. 1–9

    2018

  15. [23]

    Dynamic walking on highly underactuated point foot humanoids: Closing the loop between hzd and hlip,

    A. B. Ghansah, J. Kim, K. Li, and A. D. Ames, “Dynamic walking on highly underactuated point foot humanoids: Closing the loop between hzd and hlip,” in2024 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS). IEEE, 2024, pp. 12 686– 12 693

    2024

  16. [24]

    CasADi – A software framework for nonlinear optimization and optimal control,

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

    2019

  17. [25]

    Wachter,An interior point algorithm for large-scale nonlinear optimization with applications in process engineering

    A. Wachter,An interior point algorithm for large-scale nonlinear optimization with applications in process engineering. Carnegie Mellon University, 2002

    2002

  18. [26]

    Eigen v3,

    G. Guennebaud, B. Jacobet al., “Eigen v3,” http://eigen.tuxfamily.org, 2010

    2010

  19. [27]

    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,” in2019 IEEE/SICE International Symposium on System In...

    2019

  20. [28]

    Mujoco: A physics engine for model-based control,

    E. Todorov, T. Erez, and Y . Tassa, “Mujoco: A physics engine for model-based control,” in2012 IEEE/RSJ international conference on intelligent robots and systems. IEEE, 2012, pp. 5026–5033

    2012

  21. [29]

    T. I. Fossen,Guidance and control of ocean vehicles, 1999

    1999

Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.