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arxiv: 2605.03246 · v1 · submitted 2026-05-05 · 📡 eess.SY · cs.SY

Model Predictive Static Programming for Discrete-Time Optimal Control on Lie Groups

Akhil B Krishna , Mangal Kothari This is my paper

Pith reviewed 2026-05-07 14:31 UTC · model grok-4.3

classification 📡 eess.SY cs.SY
keywords model predictive static programmingLie groupsoptimal controldiscrete-time systemsquadrotorhelicoptersensitivity matricesquadratic programming
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The pith

The Lie-group MPSP framework converts finite-horizon optimal control on curved spaces into sequences of static quadratic programs with closed-form solutions.

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

This paper develops an extension of model predictive static programming to handle optimal control of mechanical systems whose states live on Lie groups rather than flat Euclidean spaces. By relying on left-trivialized variations and intrinsic linearizations native to these groups, along with recursive terminal sensitivity matrices, it turns the usual hard dynamic optimization into a series of easier static quadratic programs. A sympathetic reader would care because traditional methods based on Pontryagin's principle require solving nonlinear two-point boundary value problems that are computationally heavy for real-time use in vehicles like quadrotors. The result is a method that generates control updates in closed form, demonstrated on flipping maneuvers that require negative thrust.

Core claim

The Lie-group MPSP method reformulates the finite-horizon optimal control problem for systems on Lie groups as a sequence of static quadratic programs that admit closed-form control updates. This is achieved through left-trivialized variations, intrinsic linearization on Lie groups, and recursive computation of terminal sensitivity matrices. The approach is validated by deriving continuous-time necessary and sufficient optimality conditions and comparing the MPSP trajectories against TPBVP solutions and iLQR in simulations of variable-pitch quadrotor and single-main-rotor helicopter flips.

What carries the argument

Left-trivialized variations and intrinsic linearization on Lie groups with recursive terminal sensitivity matrices that enable reformulation as static quadratic programs yielding closed-form control updates.

Load-bearing premise

Left-trivialized variations and intrinsic linearization on Lie groups together with the recursive terminal sensitivity matrices remain accurate for the discrete-time approximation to align with continuous-time optimality conditions.

What would settle it

Comparing the control trajectories produced by the MPSP algorithm for the variable-pitch quadrotor flipping maneuver against the solutions of the derived continuous-time necessary and sufficient optimality conditions from Pontryagin's principle.

read the original abstract

This paper extends the Model Predictive Static Programming (MPSP) framework for nonlinear systems evolving on Euclidean spaces to simple mechanical systems evolving on Lie groups. Classical optimal control approaches based on Pontryagin's Maximum Principle (PMP) lead to nonlinear two-point boundary value problems (TPBVPs), whose numerical solution becomes particularly challenging on nonlinear configuration spaces. To overcome this difficulty, the proposed Lie-group MPSP framework reformulates the finite-horizon optimal control problem as a sequence of static quadratic programs that admit closed-form control updates, thereby avoiding the need to solve TPBVPs directly. The development relies on left-trivialized variations, intrinsic linearization on Lie groups, and a recursive computation of terminal sensitivity matrices, which together enable computationally efficient real-time implementation. The proposed method is demonstrated through optimal flipping maneuvers of a variable-pitch quadrotor (VPQ) and a single-main-rotor helicopter (SMRH), both of which are capable of generating negative thrust. For validation, continuous-time necessary and sufficient optimality conditions are derived, and the corresponding TPBVP solutions are compared against the trajectories generated by the proposed MPSP method in numerical simulations. In addition, the proposed algorithm is systematically compared with the iterative Linear Quadratic Regulator (iLQR) method, and a detailed numerical study is presented to highlight the relative performance and computational features of the two approaches.

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

2 major / 2 minor

Summary. The paper extends the Model Predictive Static Programming (MPSP) framework from Euclidean spaces to discrete-time optimal control of simple mechanical systems on Lie groups. It reformulates the finite-horizon problem as a sequence of static quadratic programs that yield closed-form control updates via left-trivialized variations, intrinsic linearization, and recursive terminal sensitivity matrices, thereby avoiding direct solution of nonlinear TPBVPs from the Pontryagin Maximum Principle. The method is demonstrated on large-angle flipping maneuvers for a variable-pitch quadrotor and single-main-rotor helicopter (both with negative thrust capability), with trajectories compared to TPBVP solutions and to iLQR, plus a numerical performance study.

Significance. If the discrete approximation is shown to be sufficiently accurate, the framework would provide an efficient, real-time-capable alternative to TPBVP solvers for Lie-group optimal control problems arising in robotics and aerospace. The explicit comparisons to both TPBVP and iLQR, together with the emphasis on computationally cheap closed-form updates, would be a useful contribution to the literature on geometric control and model-predictive methods.

major comments (2)
  1. [Numerical validation / simulation results] Numerical validation section: the trajectories generated by the Lie-group MPSP method are stated to be compared against continuous-time TPBVP solutions, yet no quantitative error metrics (e.g., integrated state deviation norms, terminal constraint violation, or cost-function differences), step-size convergence study, or discretization-error bounds are reported. This omission leaves the central claim that the discrete closed-form updates remain consistent with the continuous necessary and sufficient optimality conditions without verifiable support, especially for the large-angle maneuvers shown.
  2. [Lie-group MPSP derivation / sensitivity recursion] Derivation of recursive sensitivity matrices (around the left-trivialized linearization): the paper relies on first-order variations and recursive terminal sensitivities to obtain the static QP, but provides neither an a-priori error bound on the neglected curvature terms nor a proof that the discrete necessary conditions converge to the continuous PMP conditions as the time step tends to zero. For the demonstrated negative-thrust flips, accumulation of linearization error could undermine the claimed fidelity.
minor comments (2)
  1. [Preliminaries / Lie-group linearization] Notation for left-trivialized quantities and the precise definition of the discrete dynamics map should be clarified with an explicit example for SE(3) or SO(3) to aid readers unfamiliar with geometric mechanics.
  2. [Numerical study] The manuscript would benefit from a short table summarizing wall-clock times, iteration counts, and final costs for MPSP versus iLQR across the two vehicle examples.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below and indicate the revisions we plan to incorporate.

read point-by-point responses

  1. Referee: [Numerical validation / simulation results] Numerical validation section: the trajectories generated by the Lie-group MPSP method are stated to be compared against continuous-time TPBVP solutions, yet no quantitative error metrics (e.g., integrated state deviation norms, terminal constraint violation, or cost-function differences), step-size convergence study, or discretization-error bounds are reported. This omission leaves the central claim that the discrete closed-form updates remain consistent with the continuous necessary and sufficient optimality conditions without verifiable support, especially for the large-angle maneuvers shown.

    Authors: We agree that quantitative metrics would strengthen the validation. Although the manuscript presents visual trajectory comparisons and states consistency with the continuous TPBVP solutions, explicit error norms, terminal violations, cost differences, and a step-size study are not included. In the revised manuscript we will add these quantitative results, including integrated state deviation norms, terminal constraint violations, cost-function differences, and discretization convergence data for the demonstrated maneuvers. revision: yes

  2. Referee: [Lie-group MPSP derivation / sensitivity recursion] Derivation of recursive sensitivity matrices (around the left-trivialized linearization): the paper relies on first-order variations and recursive terminal sensitivities to obtain the static QP, but provides neither an a-priori error bound on the neglected curvature terms nor a proof that the discrete necessary conditions converge to the continuous PMP conditions as the time step tends to zero. For the demonstrated negative-thrust flips, accumulation of linearization error could undermine the claimed fidelity.

    Authors: The framework is formulated directly in discrete time using first-order left-trivialized variations and intrinsic linearization, which is standard for geometric control. We do not provide a priori curvature-error bounds or a formal convergence proof to the continuous PMP because the contribution centers on the closed-form static-programming reformulation rather than discretization analysis. The numerical results for the large-angle negative-thrust flips show close agreement with the continuous TPBVP solutions, indicating that linearization errors remain manageable in practice. We will add a concise discussion of the first-order approximation and its practical implications in the revised manuscript. revision: partial

Circularity Check

0 steps flagged

No significant circularity: Lie-group MPSP closed-form updates derived from intrinsic linearization and sensitivity recursion without self-referential reduction

full rationale

The derivation begins from left-trivialized variations and intrinsic linearization on Lie groups to obtain discrete dynamics and cost approximations, then applies recursive terminal sensitivity matrices to convert the finite-horizon problem into a sequence of static quadratic programs whose solutions yield explicit control updates. These steps are constructed directly from the Lie-group geometry and the standard MPSP static-programming idea without fitting parameters to the target trajectories or defining any quantity in terms of itself. Validation proceeds by independently deriving continuous-time PMP necessary conditions, solving the resulting TPBVPs numerically, and comparing trajectories, which constitutes an external benchmark rather than a tautology. No load-bearing premise reduces to a self-citation chain or to an ansatz smuggled from prior work by the same authors; the Lie-group extension is self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard Lie-group differential geometry and prior MPSP results; no new free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption Lie groups admit left-trivialized variations and intrinsic linearization that preserve the manifold structure
    Invoked to convert the nonlinear optimal control problem into quadratic programs on the tangent space.

pith-pipeline@v0.9.0 · 5543 in / 1207 out tokens · 76589 ms · 2026-05-07T14:31:07.810110+00:00 · methodology

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