pith. machine review for the scientific record. sign in

arxiv: 2604.23863 · v1 · submitted 2026-04-26 · 💻 cs.RO · cs.SY· eess.SY

Recognition: unknown

Cooptimizing Safety and Performance Using Safety Value-Constrained Model Predictive Control

Hao Wang , Nam Nguyen , Armand Jordana , Ludovic Righetti , Somil Bansal
Authors on Pith no claims yet

Pith reviewed 2026-05-08 05:48 UTC · model grok-4.3

classification 💻 cs.RO cs.SYeess.SY
keywords model predictive controlsafety value functionrecursive feasibilitycontrol-invariant setrobotic manipulatorreachabilityterminal constraint
0
0 comments X

The pith

Augmenting model predictive control with a safety value function terminal constraint guarantees persistent safety while optimizing performance.

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

The paper presents a model predictive control method that adds a terminal constraint derived from a safety value function. This constraint forces the end of each finite planning horizon to lie inside a control-invariant safe set. When the safety value function is exact and the initial problem is feasible, the optimization stays feasible at every future step, which keeps the system inside the safe set forever. The construction uses reachability information rather than local linearizations, producing less restrictive safe sets than earlier terminal-set methods. Experiments on a robotic manipulator show better constraint satisfaction and robustness than plain state-constrained MPC or reactive filters, while task performance remains comparable.

Core claim

By incorporating a reachability-based safety value function as a terminal constraint in MPC, the resulting optimization problem yields trajectories that are both high-performing and provably safe beyond the planning horizon, with recursive feasibility proven when the safety value function is exact and initialization is feasible.

What carries the argument

The safety value function that supplies the terminal constraint enforcing membership in a control-invariant safe set at the end of each planning horizon.

If this is right

  • The MPC optimization remains feasible at every subsequent time step.
  • State and input constraints are satisfied for all future time, not merely inside the current horizon.
  • Safety guarantees are less conservative than those obtained from local linearization or conservative terminal-set approximations.
  • Task performance stays competitive while constraint violations decrease relative to standard state-constrained MPC.

Where Pith is reading between the lines

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

  • If accurate approximations or learned models of the safety value function can be substituted, the same recursive-feasibility argument could extend to systems where the exact function is unavailable.
  • The terminal-constraint idea could be combined with uncertainty-aware reachability tools to handle model mismatch without destroying the persistent-safety property.
  • The method points toward a general pattern in which any control-invariant set that admits fast membership queries can serve as a drop-in terminal constraint for performance-oriented MPC.

Load-bearing premise

An exact safety value function is available and can be evaluated inside the real-time optimizer.

What would settle it

A simulation or hardware run in which the MPC optimization becomes infeasible at a later time step even though the initial problem is feasible and the exact safety value function is used at every step.

Figures

Figures reproduced from arXiv: 2604.23863 by Armand Jordana, Hao Wang, Ludovic Righetti, Nam Nguyen, Somil Bansal.

Figure 1
Figure 1. Figure 1: Normalized metrics for the proposed method and the baselines view at source ↗
Figure 2
Figure 2. Figure 2: End-effector trajectories visualized over the view at source ↗
Figure 4
Figure 4. Figure 4: Time-lapse snapshots of 2 trials from the hardware experiments. The obstacle is modeled computationally to ensure collision-free trajectory view at source ↗
read the original abstract

Autonomous systems are increasingly deployed in real-world environments, where they must achieve high performance while maintaining safety under state and input constraints. Although Model Predictive Control (MPC) provides a principled framework for constrained optimal control, guaranteeing safety beyond its finite planning horizon remains a fundamental challenge. In this work, we augment MPC with a safety value function-based terminal constraint that enforces membership in a control-invariant safe set at the end of each planning horizon. This formulation enables real-time synthesis of trajectories that are both high-performing and provably safe. We show that, under an exact safety value function and a feasible initialization, the proposed MPC scheme is recursively feasible, thereby ensuring persistent safety. In contrast to traditional terminal set constructions that rely on local linearizations or conservative approximations, our approach incorporates a reachability-based safety value function for terminal constraints, yielding less conservative and more expressive safety guarantees. We validate the proposed framework through simulation and hardware experiments on a Flexiv Rizon 10s manipulator. Results demonstrate improved constraint satisfaction and robustness compared to standard state-constrained MPC and reactive safety filtering, while maintaining competitive task performance. The full implementation and experiments are available on the project website.

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

3 major / 2 minor

Summary. The paper proposes augmenting standard MPC with a terminal constraint defined by a reachability-based safety value function that enforces membership in a control-invariant safe set. Under the assumption of an exact safety value function and feasible initialization, the authors claim the resulting scheme is recursively feasible and thus persistently safe. They contrast this with more conservative terminal-set constructions and validate the approach via simulation and hardware experiments on a Flexiv Rizon 10s manipulator, reporting improved constraint satisfaction relative to state-constrained MPC and reactive safety filters while preserving task performance.

Significance. If the recursive-feasibility result holds with a computable exact (or invariance-preserving) safety value function, the method would provide a principled way to obtain less conservative yet provably safe terminal constraints for real-time robotic control, addressing a long-standing tension between performance and safety in MPC. The open-source implementation is a positive contribution for reproducibility.

major comments (3)
  1. [§4 (recursive feasibility)] The recursive-feasibility claim (abstract and §4) rests on the existence of an exact safety value function that can be evaluated inside the real-time optimizer. The manuscript provides no derivation of the invariance property, no explicit list of assumptions (e.g., Lipschitz continuity, exact reachability computation), and no discussion of how the value function is obtained or approximated for the 7-DOF manipulator; without these steps the central guarantee remains conditional on an unverified premise.
  2. [§5.2 (hardware experiments)] §5.2 and the hardware experiments: the reported results use the safety value function inside the MPC optimizer, yet no error bound, discretization analysis, or invariance-preservation argument is given for any numerical approximation. If the function is only approximated, the terminal-constraint invariance used in the proof may be lost, undermining the persistent-safety claim for the Flexiv Rizon trials.
  3. [§5.1 (simulation results)] The comparison in §5.1 and Table 1 shows improved constraint satisfaction, but the baseline “standard state-constrained MPC” is not described with the same terminal-set construction; it is therefore unclear whether the reported gains are due to the safety-value terminal constraint or to other implementation differences (e.g., horizon length, cost tuning).
minor comments (2)
  1. [§3] Notation for the safety value function V_s(x) is introduced without an explicit equation reference; a numbered definition would improve readability.
  2. [abstract] The project website link is given but the manuscript does not state which specific code files correspond to the reported experiments, hindering exact reproduction.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the detailed and constructive comments. We address each major point below with clarifications and commit to targeted revisions that strengthen the presentation of assumptions, analyses, and comparisons without altering the core contributions.

read point-by-point responses

  1. Referee: [§4 (recursive feasibility)] The recursive-feasibility claim (abstract and §4) rests on the existence of an exact safety value function that can be evaluated inside the real-time optimizer. The manuscript provides no derivation of the invariance property, no explicit list of assumptions (e.g., Lipschitz continuity, exact reachability computation), and no discussion of how the value function is obtained or approximated for the 7-DOF manipulator; without these steps the central guarantee remains conditional on an unverified premise.

    Authors: The recursive feasibility result in Section 4 is explicitly conditioned on an exact safety value function whose zero sublevel set is control-invariant by construction from the reachability problem. The invariance follows directly from the dynamic programming principle satisfied by the value function. We will add an explicit assumptions subsection listing Lipschitz continuity of the dynamics, bounded disturbances, and exact evaluation of the value function. For the 7-DOF case we will describe the offline computation pipeline (grid-based reachability on a reduced state space followed by neural-network regression) and note that the theoretical guarantee applies to the exact function while the implementation uses the learned approximant. revision: partial

  2. Referee: [§5.2 (hardware experiments)] §5.2 and the hardware experiments: the reported results use the safety value function inside the MPC optimizer, yet no error bound, discretization analysis, or invariance-preservation argument is given for any numerical approximation. If the function is only approximated, the terminal-constraint invariance used in the proof may be lost, undermining the persistent-safety claim for the Flexiv Rizon trials.

    Authors: We agree that approximation error must be addressed for the hardware results. In the revision we will report validation error bounds on the neural-network approximant (obtained on a held-out test set) and adopt a conservative threshold in the terminal constraint to preserve invariance in practice. We will also add a brief discretization analysis of the training grid and state-space sampling used for the Flexiv Rizon 10s experiments. revision: yes

  3. Referee: [§5.1 (simulation results)] The comparison in §5.1 and Table 1 shows improved constraint satisfaction, but the baseline “standard state-constrained MPC” is not described with the same terminal-set construction; it is therefore unclear whether the reported gains are due to the safety-value terminal constraint or to other implementation differences (e.g., horizon length, cost tuning).

    Authors: The baseline in Section 5.1 is a standard state-constrained MPC that uses identical horizon length, cost weights, prediction model, and solver settings as the proposed method; the sole difference is the replacement of the safety-value terminal constraint by a conventional quadratic terminal cost. We will revise the text and caption of Table 1 to state these implementation details explicitly so that the observed improvement in constraint satisfaction can be attributed to the safety terminal constraint. revision: yes

Circularity Check

0 steps flagged

No circularity: recursive feasibility proven under external exact safety value function from reachability analysis

full rationale

The derivation establishes recursive feasibility and persistent safety conditionally on an exact safety value function (from standard reachability) plus feasible initialization. This function is an external input to the MPC optimizer and terminal constraint; it is not fitted, self-defined, or reduced to any quantity constructed inside the paper. No load-bearing self-citation, ansatz smuggling, or renaming of known results appears in the stated proof chain. The central claim therefore remains independent of the paper's own fitted parameters or prior outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim depends on the availability of an exact safety value function that can be used inside the optimizer; this is stated as a prerequisite for the recursive-feasibility result but is not derived or constructed within the paper itself.

axioms (1)
  • domain assumption An exact safety value function exists and can be evaluated at the terminal state of each planning horizon.
    The recursive feasibility and persistent safety statements are conditioned on this exact function; the abstract does not provide a construction or approximation procedure.

pith-pipeline@v0.9.0 · 5519 in / 1453 out tokens · 45340 ms · 2026-05-08T05:48:51.273670+00:00 · methodology

discussion (0)

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

Reference graph

Works this paper leans on

28 extracted references · 8 canonical work pages

  1. [1]

    $\pi^{*}_{0.6}$: a VLA That Learns From Experience

    P. Intelligence, A. Amin, R. Aniceto, A. Balakrishna, K. Black, K. Conley, G. Connors, J. Darpinian, K. Dhabalia, J. DiCarlo, et al., “π ∗ 0.6: a vla that learns from experience,”arXiv preprint arXiv:2511.14759, 2025

    work page Pith review arXiv 2025

  2. [2]

    Gemini Robotics 1.5: Pushing the Frontier of Generalist Robots with Advanced Embodied Reasoning, Thinking, and Motion Transfer.arXiv e-prints, page arXiv:2510.03342, October 2025

    G. R. Team, A. Abdolmaleki, S. Abeyruwan, J. Ainslie, J.-B. Alayrac, M. G. Arenas, A. Balakrishna, N. Batchelor, A. Bewley, J. Bingham, et al., “Gemini robotics 1.5: Pushing the frontier of generalist robots with advanced embodied reasoning, thinking, and motion transfer,” arXiv preprint arXiv:2510.03342, 2025

    work page arXiv 2025

  3. [3]

    Control barrier function based quadratic programs for safety critical systems,

    A. D. Ames, X. Xu, J. W. Grizzle, and P. Tabuada, “Control barrier function based quadratic programs for safety critical systems,”IEEE Transactions on Automatic Control, vol. 62, no. 8, pp. 3861–3876, 2016

    2016

  4. [4]

    On safety and liveness filtering using hamilton-jacobi reachability analysis,

    J. Borquez, K. Chakraborty, H. Wang, and S. Bansal, “On safety and liveness filtering using hamilton-jacobi reachability analysis,”IEEE Transactions on Robotics, 2024

    2024

  5. [5]

    Data-driven safety filters: Hamilton-jacobi reachability, control barrier functions, and predictive methods for uncertain systems,

    K. P. Wabersich, A. J. Taylor, J. J. Choi, K. Sreenath, C. J. Tom- lin, A. D. Ames, and M. N. Zeilinger, “Data-driven safety filters: Hamilton-jacobi reachability, control barrier functions, and predictive methods for uncertain systems,”IEEE Control Systems Magazine, vol. 43, no. 5, pp. 137–177, 2023

    2023

  6. [6]

    Optimal control with state-space constraint i,

    H. M. Soner, “Optimal control with state-space constraint i,”SIAM Journal on Control and Optimization, vol. 24, no. 3, pp. 552–561, 1986

    1986

  7. [7]

    Hamilton-jacobi equations with state constraints,

    I. Capuzzo-Dolcetta and P.-L. Lions, “Hamilton-jacobi equations with state constraints,”Transactions of the American mathematical society, vol. 318, no. 2, pp. 643–683, 1990

    1990

  8. [8]

    A general Hamilton- Jacobi framework for non-linear state-constrained control problems,

    A. Altarovici, O. Bokanowski, and H. Zidani, “A general Hamilton- Jacobi framework for non-linear state-constrained control problems,” ESAIM: COCV, vol. 19, no. 2, pp. 337–357, 2013. [Online]. Available: https://doi.org/10.1051/cocv/2012011

    work page doi:10.1051/cocv/2012011 2013

  9. [9]

    Cooptimizing Safety and Performance With a Control-Constrained Formulation,

    H. Wang, A. Dhande, and S. Bansal, “Cooptimizing Safety and Performance With a Control-Constrained Formulation,”IEEE Control Systems Letters, 2024

    2024

  10. [10]

    Hamilton-jacobi reachability: A brief overview and recent advances,

    S. Bansal, M. Chen, S. Herbert, and C. J. Tomlin, “Hamilton-jacobi reachability: A brief overview and recent advances,” in2017 IEEE 56th annual conference on decision and control (CDC). IEEE, 2017, pp. 2242–2253

    2017

  11. [11]

    R. E. Bellman,Dynamic Programming, ser. Princeton Landmarks in Mathematics and Physics. Princeton University Press, 2010. [Online]. Available: https://press.princeton.edu/books/ paperback/9780691146683/dynamic-programming

    work page arXiv 2010

  12. [12]

    A physics- informed machine learning framework for safe and optimal control of autonomous systems,

    M. Tayal, A. Singh, S. Kolathaya, and S. Bansal, “A physics- informed machine learning framework for safe and optimal control of autonomous systems,” inForty-second International Conference on Machine Learning

  13. [13]

    Safety-critical model predictive control with discrete-time control barrier function,

    J. Zeng, B. Zhang, and K. Sreenath, “Safety-critical model predictive control with discrete-time control barrier function,” in2021 American Control Conference (ACC), 2021, pp. 3882–3889

    2021

  14. [14]

    Enhancing feasibility and safety of nonlinear model predictive control with discrete-time control barrier functions,

    J. Zeng, Z. Li, and K. Sreenath, “Enhancing feasibility and safety of nonlinear model predictive control with discrete-time control barrier functions,” in2021 60th IEEE Conference on Decision and Control (CDC). IEEE, 2021, pp. 6137–6144

    2021

  15. [15]

    Iterative convex optimization for model predictive control with discrete-time high-order control barrier functions,

    S. Liu, J. Zeng, K. Sreenath, and C. A. Belta, “Iterative convex optimization for model predictive control with discrete-time high-order control barrier functions,”arXiv preprint arXiv:2210.04361, 2022

    work page arXiv 2022

  16. [16]

    Safety-critical planning and control for dynamic obstacle avoidance using control barrier functions,

    S. Liu, Y . Mao, and C. A. Belta, “Safety-critical planning and control for dynamic obstacle avoidance using control barrier functions,” in 2025 American Control Conference (ACC). IEEE, 2025, pp. 348– 354

    2025

  17. [17]

    Safe, task-consistent manipulation with operational space control barrier functions,

    D. Morton and M. Pavone, “Safe, task-consistent manipulation with operational space control barrier functions,” in2025 IEEE/RSJ Inter- national Conference on Intelligent Robots and Systems (IROS). IEEE, 2025, pp. 187–194

    2025

  18. [18]

    On reachability and minimum cost optimal control,

    J. Lygeros, “On reachability and minimum cost optimal control,”Automatica, vol. 40, no. 6, pp. 917–927, 2004. [Online]. Available: https://www.sciencedirect.com/science/article/pii/ S0005109804000263

    2004

  19. [19]

    helperOC Library, 2019,

    “helperOC Library, 2019,” https://github.com/HJReachability/ helperOC

    2019

  20. [20]

    Deepreach: A deep learning approach to high-dimensional reachability,

    S. Bansal and C. J. Tomlin, “Deepreach: A deep learning approach to high-dimensional reachability,” in2021 IEEE International Confer- ence on Robotics and Automation (ICRA). IEEE, 2021, pp. 1817– 1824

    2021

  21. [21]

    Bridging model predictive control and deep learning for scalable reachability analysis,

    Z. Feng, L. Qiu, and S. Bansal, “Bridging model predictive control and deep learning for scalable reachability analysis,”arXiv preprint arXiv:2505.03830, 2025

    work page arXiv 2025

  22. [22]

    Successive convexifi- cation: A superlinearly convergent algorithm for non-convex optimal control problems,

    Y . Mao, M. Szmuk, X. Xu, and B. Ac ¸ikmese, “Successive convexifi- cation: A superlinearly convergent algorithm for non-convex optimal control problems,”arXiv preprint arXiv:1804.06539, 2018

    work page arXiv 2018

  23. [23]

    Stagewise implementations of sequential quadratic pro- gramming for model-predictive control,

    A. Jordana, S. Kleff, A. Meduri, J. Carpentier, N. Mansard, and L. Righetti, “Stagewise implementations of sequential quadratic pro- gramming for model-predictive control,”IEEE T-RO, 2025

    2025

  24. [24]

    Osqp: an operator splitting solver for quadratic programs.Mathematical Programming Computation, 12(4):637–672, 2020

    B. Stellato, G. Banjac, P. Goulart, A. Bemporad, and S. Boyd, “OSQP: an operator splitting solver for quadratic programs,”Mathematical Programming Computation, vol. 12, no. 4, pp. 637–672, 2020. [Online]. Available: https://doi.org/10.1007/s12532-020-00179-2

    work page doi:10.1007/s12532-020-00179-2 2020

  25. [25]

    Crocoddyl: An efficient and versatile framework for multi-contact optimal control,

    C. Mastalli, R. Budhiraja, W. Merkt, G. Saurel, B. Hammoud, M. Naveau, J. Carpentier, L. Righetti, S. Vijayakumar, and N. Mansard, “Crocoddyl: An efficient and versatile framework for multi-contact optimal control,” in2020 IEEE International Conference on Robotics and Automation (ICRA). IEEE, 2020, pp. 2536–2542

    2020

  26. [26]

    Generating formal safety assurances for high- dimensional reachability,

    A. Lin and S. Bansal, “Generating formal safety assurances for high- dimensional reachability,” in2023 IEEE International Conference on Robotics and Automation (ICRA). IEEE, 2023, pp. 10 525–10 531

    2023

  27. [27]

    Verification of neural reachable tubes via scenario optimization and conformal prediction,

    ——, “Verification of neural reachable tubes via scenario optimization and conformal prediction,” in6th Annual Learning for Dynamics & Control Conference. PMLR, 2024, pp. 719–731

    2024

  28. [28]

    User’s guide to viscosity solutions of second order partial differential equations,

    M. G. Crandall, H. Ishii, and P.-L. Lions, “User’s guide to viscosity solutions of second order partial differential equations,”Bulletin of the American mathematical society, vol. 27, no. 1, pp. 1–67, 1992

    1992