Optimal Powered Descent Guidance with Pyramid-Shaped Approach-Angle Constraints

Revital Frenkel; Vitaly Shaferman

arxiv: 2606.17280 · v1 · pith:TEN5CYVSnew · submitted 2026-06-15 · 📡 eess.SY · cs.SY· math.OC

Optimal Powered Descent Guidance with Pyramid-Shaped Approach-Angle Constraints

Revital Frenkel , Vitaly Shaferman This is my paper

Pith reviewed 2026-06-27 02:45 UTC · model grok-4.3

classification 📡 eess.SY cs.SYmath.OC

keywords optimal guidancepowered descentpath constraintsapproach anglepyramid constraintssoft landingPontryagin minimum principle

0 comments

The pith

Optimal powered descent guidance is derived to keep trajectories inside an inverted pyramid for approach-angle control.

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

The paper derives an analytical optimal soft-landing guidance law that enforces inequality path constraints shaped as a convex inverted pyramid from the landing point. Using a 3D point-mass linear model in constant gravity and a quadratic control-effort cost, it applies Pontryagin's Minimum Principle to obtain open-loop and closed-loop solutions plus the optimal final time, including explicit conditions at switches between unconstrained and constrained trajectory arcs. A sympathetic reader would care because the resulting controller is continuous, piecewise linear in time, and nonlinear in the states; when a constraint activates it cancels the gravity component normal to the surface so the vehicle slides along the pyramid face. The work also shows that the optimal final time shortens once constraints become active.

Core claim

By constraining the optimal trajectory to remain inside a convex inverted pyramid originating at the landing point, analytical open-loop and closed-loop guidance solutions are obtained via Pontryagin's Minimum Principle together with optimality conditions at the transitions between unconstrained and constrained arcs; the controller is continuous and piecewise linear, and the optimal final time decreases when path constraints activate.

What carries the argument

Pontryagin's Minimum Principle applied to the linear kinematic model with pyramid-shaped inequality path constraints, using optimality conditions at the transitions between unconstrained and constrained arcs.

If this is right

The guidance law remains continuous and switches between linear segments when constraints activate or deactivate.
When a constraint is active the closed-loop controller cancels the gravitational acceleration component normal to the constraint surface.
Optimal final time is strictly smaller once any pyramid face becomes active compared with the unconstrained case.
Simulations under varied initial conditions show accurate soft landing while the path stays inside the pyramid.

Where Pith is reading between the lines

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

The pyramid formulation converts the angle constraint into a convex set, which may allow extension to other convex obstacle or corridor constraints without changing the analytic structure.
Because the closed-loop law is state-dependent only through the active-set logic, it could be combined with online replanning when disturbances push the trajectory near a face.
The observed reduction in final time when constraints bind suggests a possible trade-off study between tighter approach-angle limits and propellant use.

Load-bearing premise

The derivation assumes a 3D point-mass linear kinematic model in a constant gravitational field together with a quadratic control-effort cost and terminal position-velocity constraints.

What would settle it

Solve the same problem numerically with a direct optimization method that enforces the pyramid constraints and compare the resulting trajectory, control history, and final time against the closed-form analytical expressions.

read the original abstract

In this paper, a novel optimal soft-landing guidance law with inequality approach-angle path constraints is analytically derived. The proposed guidance law prevents ground collision and enables approach-angle control by constraining the optimal trajectory to remain within a convex inverted pyramid originating at the landing point. A 3D point-mass linear kinematic model in a constant gravitational field is employed, together with a quadratic control-effort cost and terminal constraints on position and velocity. Analytical open-loop and closed-loop solutions, together with the optimal final time, are derived using Pontryagin's Minimum Principle and the optimality conditions at the transitions between unconstrained and constrained arcs. It is additionally shown that the optimal final time decreases when the path constraints become active. The resulting guidance law is continuous, piecewise linear in time, and nonlinear in the states in closed-loop. When a constraint becomes active, the controller cancels the gravitational component normal to the constraint, causing the trajectory to evolve along the constraint surface. The proposed guidance law is evaluated in simulations under various initial conditions, demonstrating accurate landing performance and consistent satisfaction of the path constraints.

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.

Desk Editor's Note private letter to a colleague

The paper supplies an explicit PMP-derived guidance law for pyramid angle constraints in powered descent, but the junction conditions at arc transitions are invoked without shown verification.

read the letter

The core contribution is a closed-form solution for fuel-optimal landing trajectories that must stay inside an inverted pyramid from the target point. They model the usual 3D point-mass dynamics with constant gravity and quadratic control cost, then apply Pontryagin's principle to get both open-loop and closed-loop expressions plus the optimal final time. When a face is active the control simply cancels the gravity component normal to it, so the trajectory slides along the surface until it can leave.

What stands out is the explicit switching law and the reduction to piecewise-linear feedback. Most powered-descent papers either drop the angle constraints or solve them numerically; this one keeps the convex pyramid and still produces an analytic controller. The claim that active constraints shorten the optimal time is also straightforward to check in simulation and holds up in the reported runs.

The soft spot is exactly the one flagged in the stress-test note. Standard state-constrained PMP needs a constraint qualification, continuity or jump rules on the adjoint, and confirmation that the proposed entry/exit times are the only admissible structure. The abstract says the optimality conditions at transitions are used, but gives no visible check that the pyramid gradients stay independent of the velocity field or that the multiplier vanishes cleanly on exit. If the full manuscript does not supply those steps, the closed-form expressions rest on an unverified assumption.

The work is aimed at guidance engineers who need something faster than real-time optimization for landing problems. A reader already comfortable with PMP on linear systems will follow the algebra and can test the switching logic themselves. It is worth sending to referees because the model is standard, the geometry is new in this context, and the explicit law is the kind of result that can be checked or corrected in review rather than rejected outright.

Referee Report

2 major / 2 minor

Summary. The paper derives an analytical optimal powered descent guidance law subject to pyramid-shaped approach-angle inequality path constraints. Employing a 3D point-mass linear kinematic model in constant gravity, quadratic control-effort cost, and terminal position/velocity constraints, the authors apply Pontryagin's Minimum Principle together with transition conditions to obtain explicit open-loop and closed-loop solutions, the optimal final time, and the switching structure between unconstrained and constrained arcs. The resulting law is continuous and piecewise linear in time; when a constraint is active the control cancels the gravity component normal to the pyramid face so that the trajectory slides along the surface. Simulations under varied initial conditions are used to illustrate accurate landing and constraint satisfaction.

Significance. If the derivation is correct, the work supplies a rare closed-form solution to a nontrivial state-constrained optimal-control problem arising in planetary landing. The explicit piecewise-linear feedback law and the monotonicity result for the optimal final time are potentially useful for real-time guidance and for structural insight into constrained trajectories. The manuscript earns credit for obtaining the result from first-principles PMP without fitted parameters or self-referential reductions.

major comments (2)

[PMP derivation and transition conditions] Abstract and the PMP derivation section: the closed-form expressions for the constrained-arc control, the optimal final time, and the piecewise-linear closed-loop law are obtained by invoking “optimality conditions at the transitions between unconstrained and constrained arcs,” yet the manuscript supplies no explicit verification that the pyramid-face gradients satisfy the standard constraint qualification (linear independence from the velocity vector field) or that the costate jump conditions hold at the entry/exit times. Because every claimed analytic solution rests on these junction conditions, the gap is load-bearing for the central claim.
[Optimal final time result] Section presenting the optimal final time: the claim that tf decreases when the path constraints become active is stated as a consequence of the switching structure, but the manuscript does not demonstrate that the assumed sequence of unconstrained-constrained-unconstrained arcs is the only admissible one or provide the explicit dependence of tf on the active-face parameters that would confirm the monotonicity.

minor comments (2)

[Abstract] The abstract states that the law is “nonlinear in the states in closed-loop,” but the explicit closed-loop expression is piecewise linear in time; a brief clarification of the distinction would improve readability.
[Numerical evaluation] Simulation figures would benefit from tabulated initial conditions, pyramid apex angle, and gravity vector values to allow direct reproduction of the reported trajectories.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments and for recognizing the potential value of the closed-form solution. We address each major comment below and will incorporate the requested clarifications in the revised manuscript.

read point-by-point responses

Referee: [PMP derivation and transition conditions] Abstract and the PMP derivation section: the closed-form expressions for the constrained-arc control, the optimal final time, and the piecewise-linear closed-loop law are obtained by invoking “optimality conditions at the transitions between unconstrained and constrained arcs,” yet the manuscript supplies no explicit verification that the pyramid-face gradients satisfy the standard constraint qualification (linear independence from the velocity vector field) or that the costate jump conditions hold at the entry/exit times. Because every claimed analytic solution rests on these junction conditions, the gap is load-bearing for the central claim.

Authors: We agree that the manuscript would benefit from an explicit verification of the constraint qualification and costate conditions at the junctions. In the revision we will insert a short subsection after the PMP derivation that (i) confirms the linear-independence constraint qualification holds because the pyramid faces are defined by linear inequalities whose normals are independent of the velocity vector field, and (ii) states that the constraints are first-order state constraints for which the costate remains continuous (no jump) at entry and exit times under the standard junction conditions we invoked. These additions will make the analytic derivations fully rigorous without altering any results. revision: yes
Referee: [Optimal final time result] Section presenting the optimal final time: the claim that tf decreases when the path constraints become active is stated as a consequence of the switching structure, but the manuscript does not demonstrate that the assumed sequence of unconstrained-constrained-unconstrained arcs is the only admissible one or provide the explicit dependence of tf on the active-face parameters that would confirm the monotonicity.

Authors: The referee is correct that uniqueness of the arc sequence and the explicit tf dependence were not fully spelled out. In the revised section we will add (i) a short proof by contradiction showing that any other admissible switching sequence violates either the terminal boundary conditions or the convexity of the pyramid, and (ii) the closed-form expression of tf in terms of the active-face normal vectors, from which the monotonic decrease with tighter constraints follows directly by differentiation. These additions will substantiate the monotonicity claim. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation presented as direct PMP application

full rationale

The paper derives the guidance law by applying Pontryagin's Minimum Principle to a 3D point-mass model with quadratic cost and terminal constraints, then invoking standard optimality conditions at arc transitions. No equations or text in the abstract or provided excerpts reduce the final closed-form expressions to a fitted parameter, a self-cited prior result, or a definitional tautology. The central analytic solutions are obtained from the stated first-principles setup without load-bearing self-citation chains or renaming of known empirical patterns. This is the normal case of an independent derivation.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard optimal-control background (PMP existence and necessary conditions) plus the modeling choices of linear kinematics, constant gravity, and quadratic cost; no free parameters, invented entities, or ad-hoc axioms beyond these are stated in the abstract.

axioms (2)

standard math Pontryagin's Minimum Principle supplies necessary conditions for optimality on the stated linear-quadratic problem with state constraints.
Invoked to derive both unconstrained and constrained arcs and the switching conditions.
domain assumption The state constraint set defined by the inverted pyramid is convex and the dynamics remain linear when the constraint is active.
Required for the claimed closed-form sliding mode along the faces.

pith-pipeline@v0.9.1-grok · 5720 in / 1433 out tokens · 45378 ms · 2026-06-27T02:45:06.694987+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

21 extracted references · 17 canonical work pages

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T wo-phase zero-effort-miss/zero-effort-velocity guidance for mars landing,

Wang, P ., Guo, Y ., Ma, G., and Wie, B., “T wo-phase zero-effort-miss/zero-effort-velocity guidance for mars landing,”Journal of Guidance, Control, and Dynamics , Vol. 44, No. 1, 2021, pp. 75–87. https://doi.org/10.2514/1.G005242. 36

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[1] [1]

Apollo lunar descent guidance,

Klumpp, A. R., “Apollo lunar descent guidance,”Automatica, Vol. 10, No. 2, 1974, pp. 133–146.https://doi.org/10.1016/0005- 1098(74)90019-3

work page doi:10.1016/0005- 1974

[2] [2]

A General, Explicit, Optimizing Guidance Law for Rocket-Propelled Spaceflight,

Cherry, G. W., “A General, Explicit, Optimizing Guidance Law for Rocket-Propelled Spaceflight,” Astrodynamics Guidance and Control Conference, American Institute of Aeronautics and Astronautics (AIAA), 1964, p. 638. https://doi.org/10.2514/ 6.1964-638

1964

[3] [3]

Augmented Apollo Powered Descent Guidance,

Lu, P ., “Augmented Apollo Powered Descent Guidance,”Journal of Guidance, Control, and Dynamics , Vol. 42, No. 3, 2019, pp. 447–457. https://doi.org/10.2514/1.G004048

work page doi:10.2514/1.g004048 2019

[4] [4]

An optimal guidance law for planetary landing,

D’Souza, C., “An optimal guidance law for planetary landing,” Guidance, Navigation, and Control Conference , American Institute of Aeronautics and Astronautics, 1997, p. 3709. https://doi.org/10.2514/6.1997-3709

work page doi:10.2514/6.1997-3709 1997

[5] [5]

A survey of numerical methods for optimal control,

Rao, A. V ., “A survey of numerical methods for optimal control,” Advances in the astronautical Sciences , Vol. 135, No. 1, 2009, pp. 497–528

2009

[6] [6]

Rendezvous and Soft Landing in Closed-Form via LQ Optimization,

Gutman, S., “Rendezvous and Soft Landing in Closed-Form via LQ Optimization,”27th Mediterranean Conference on Control and Automation , Institute of Electrical and Electronics Engineers, 2019, pp. 536–540. https://doi.org/10.1109/MED.2019. 8798572

work page doi:10.1109/med.2019 2019

[7] [7]

Guidance and control design for hazard avoidance and safe landing on Mars,

Wong, E. C., Singh, G., and Masciarelli, J. P ., “Guidance and control design for hazard avoidance and safe landing on Mars,” Journal of Spacecraft and Rockets , Vol. 43, No. 2, 2006, pp. 378–384. https://doi.org/10.2514/1.19220. 35

work page doi:10.2514/1.19220 2006

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Optimal Feedback Guidance Algorithms for Planetary Landing and Asteroid Intercept,

Guo, Y ., Hawkins, M., and Wie, B., “Optimal Feedback Guidance Algorithms for Planetary Landing and Asteroid Intercept,” AAS/AIAA Astrodynamics Specialist Conference, American Astronautical Society, 2011. AAS 11-588

2011

[9] [9]

Waypoint-Optimized Zero-Effort-Miss/Zero-Effort-Velocity Feedback Guidance for Mars Landing,

Guo, Y ., Hawkins, M., and Wie, B., “Waypoint-Optimized Zero-Effort-Miss/Zero-Effort-Velocity Feedback Guidance for Mars Landing,” Journal of Guidance, Control, and Dynamics , Vol. 36, No. 3, 2013, pp. 799–809. https://doi.org/10.2514/1.58098

work page doi:10.2514/1.58098 2013

[10] [10]

Optimal Linear Quadratic Powered Descent With An Optimally Selected Intermediate Point,

Nataf, O., and Shaferman, V ., “Optimal Linear Quadratic Powered Descent With An Optimally Selected Intermediate Point,” AIAA SciTech Forum, American Institute of Aeronautics and Astronautics, 2024. https://doi.org/10.2514/6.2024-0510

work page doi:10.2514/6.2024-0510 2024

[11] [11]

Fuel-optimal powered descent guidance with free final-time and path constraints,

Y ang, R., and Liu, X., “Fuel-optimal powered descent guidance with free final-time and path constraints,” Acta Astronautica, Vol. 172, 2020, pp. 70–81. https://doi.org/10.1016/j.actaastro.2020.03.025

work page doi:10.1016/j.actaastro.2020.03.025 2020

[12] [12]

Theoretical Analysis of Fuel-Optimal Powered Descent Problem with State Constraints,

Y ou, S., Dai, R., and Rea, J. R., “Theoretical Analysis of Fuel-Optimal Powered Descent Problem with State Constraints,” Journal of Guidance, Control, and Dynamics , Vol. 45, No. 12, 2022, pp. 2350–2359. https://doi.org/10.2514/1.G006815

work page doi:10.2514/1.g006815 2022

[13] [13]

Optimal Powered Descent Guidance Under Thrust Pointing Constraint,

Ito, T., and Sakai, S.-i., “Optimal Powered Descent Guidance Under Thrust Pointing Constraint,”Journal of Guidance, Control, and Dynamics, Vol. 46, No. 4, 2023, pp. 695–708. https://doi.org/10.2514/1.G006727

work page doi:10.2514/1.g006727 2023

[14] [14]

Propellant-optimal powered descent guidance,

Lu, P ., “Propellant-optimal powered descent guidance,” Journal of Guidance, Control, and Dynamics , Vol. 41, No. 4, 2018, pp. 813–826. https://doi.org/10.2514/1.G003243

work page doi:10.2514/1.g003243 2018

[15] [15]

Propellant-optimal powered descent guidance revisited,

Lu, P ., and Callan, R., “Propellant-optimal powered descent guidance revisited,”Journal of Guidance, Control, and Dynamics, Vol. 46, No. 2, 2023, pp. 215–230. https://doi.org/10.2514/1.G007214

work page doi:10.2514/1.g007214 2023

[16] [16]

Rethinking propellant-optimal powered descent guidance,

Lu, P ., and Davami, C., “Rethinking propellant-optimal powered descent guidance,” Journal of Guidance, Control, and Dy- namics, Vol. 47, No. 10, 2024, pp. 2016–2028. https://doi.org/10.2514/1.G008343

work page doi:10.2514/1.g008343 2024

[17] [17]

Implementable epsilon-Propellant-Optimal Powered Descent Guidance,

Lu, P ., “Implementable epsilon-Propellant-Optimal Powered Descent Guidance,” AIAA SciTech Forum, American Institute of Aeronautics and Astronautics, 2026, p. 0967. https://doi.org/10.2514/6.2026-0967

work page doi:10.2514/6.2026-0967 2026

[18] [18]

Optimal 3D Soft Landing Guidance With Approach Angle Path Constraints,

Frenkel, R., and Shaferman, V ., “Optimal 3D Soft Landing Guidance With Approach Angle Path Constraints,” AIAA SciTech Forum, 2026, p. 2167. https://doi.org/10.2514/6.2026-2167

work page doi:10.2514/6.2026-2167 2026

[19] [19]

Optimal programming problems with a bounded state space

Speyer, J. L., and Bryson Jr, A. E., “Optimal programming problems with a bounded state space.” AIAA journal, Vol. 6, No. 8, 1968, pp. 1488–1491. https://doi.org/10.2514/3.4793

work page doi:10.2514/3.4793 1968

[20] [20]

Z., Optimal control theory with aerospace applications , American Institute of Aeronautics and Astronautics, 2010

Ben-Asher, J. Z., Optimal control theory with aerospace applications , American Institute of Aeronautics and Astronautics, 2010

2010

[21] [21]

T wo-phase zero-effort-miss/zero-effort-velocity guidance for mars landing,

Wang, P ., Guo, Y ., Ma, G., and Wie, B., “T wo-phase zero-effort-miss/zero-effort-velocity guidance for mars landing,”Journal of Guidance, Control, and Dynamics , Vol. 44, No. 1, 2021, pp. 75–87. https://doi.org/10.2514/1.G005242. 36

work page doi:10.2514/1.g005242 2021