arxiv: 2604.23028 · v1 · submitted 2026-04-24 · 🧮 math.OC

Recognition: unknown

Computational Method for Desensitized Optimal Guidance Using Direct Collocation

Anil Rao, Katrina Winkler

Authors on Pith no claims yet

Pith reviewed 2026-05-08 10:57 UTC · model grok-4.3

classification 🧮 math.OC

keywords desensitized optimal guidancedirect collocationsensitivity penalizationshrinking horizonZermelo navigation problematmospheric reentryMonte Carlo analysisLegendre-Gauss-Radau collocation

0 comments

The pith

Penalizing state sensitivities to uncertain parameters produces more robust optimal guidance trajectories with smaller terminal errors.

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

The paper develops a computational approach for optimal guidance that augments the objective to penalize how much the state changes with respect to uncertain model parameters. A reference trajectory computed this way then initializes repeated solves of a shrinking-horizon optimal control problem that uses Legendre-Gauss-Radau collocation on a remapped mesh at each guidance update. Monte Carlo trials on Zermelo navigation and atmospheric reentry show the resulting commands keep trajectories inside narrower envelopes and reach smaller terminal errors than commands from a standard optimal reference, while adding little extra computation. A reader would care because real guidance problems such as reentry must work when the model is imperfect, and this method supplies an explicit way to build that tolerance directly into the trajectory design.

Core claim

The paper claims that augmenting the objective functional with a penalty on the first-order sensitivities of the state to uncertain parameters produces a reference trajectory from which a shrinking-horizon collocation solver can generate guidance commands that are more robust to parameter uncertainties and external disturbances than commands generated from a standard optimal trajectory.

What carries the argument

Desensitized objective functional that penalizes state sensitivities to uncertain parameters, used to initialize remapped Legendre-Gauss-Radau collocation solves on successively shorter horizons.

If this is right

The resulting guidance commands improve robustness to external disturbances and modeling errors.
Trajectory envelopes become tighter under parameter uncertainties in the dynamic model.
Terminal state errors become smaller than those from a method that does not penalize sensitivities.
Computational cost does not increase substantially relative to the non-desensitized approach.
The method works on both simple navigation problems and complex atmospheric reentry problems, as verified by Monte Carlo analysis.

Where Pith is reading between the lines

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

The mesh-remapping technique used at each guidance update could be applied to other receding-horizon optimal control problems that require repeated solves.
If the chosen uncertain parameters miss the dominant error sources, the desensitization may provide little protection in practice.
Adding a penalty on second-order sensitivities might further reduce errors in strongly nonlinear regimes where first-order terms alone are insufficient.
Pairing the desensitized open-loop reference with an additional closed-loop feedback layer could shrink tracking errors even more.

Load-bearing premise

Penalizing first-order sensitivities to a chosen set of uncertain parameters will still produce robustness when real disturbances contain unmodeled effects, higher-order terms, or errors outside those parameters.

What would settle it

A Monte Carlo experiment on the reentry example in which the actual disturbances include unmodeled higher-order dynamics or non-parametric errors, and the desensitized method fails to produce narrower trajectory envelopes or smaller terminal errors than the non-desensitized baseline.

Figures

Figures reproduced from arXiv: 2604.23028 by Anil Rao, Katrina Winkler.

**Figure 1.** Figure 1: Partitioned mesh for guidance cycle (s − 1) [47]. of the expired horizon, T (s−1) E(s−1) , must be relocated to the point on the mesh corresponding to the terminal time of the previous guidance cycle, i.e., T (s−1) E(s−1) = T (s−1) e . Based on this notation, the first mesh point of the unexpired horizon is then denoted by T (s−1) E(s−1)+1. It is important to emphasize that T (s−1) e is not a mesh point; t… view at source ↗

**Figure 2.** Figure 2: Algorithmic flow diagram for desensitized optimal view at source ↗

**Figure 3.** Figure 3: Zermelo’s navigation problem. Determine the state, x(t), and control, u(t), that minimize the objective functional J = −x1(tf ), (32) subject to the dynamic constraints x˙ 1 = cos(u) + cx2, x˙ 2 = sin(u), (33) and the boundary conditions x1(t0) = x1(0) = 0, x2(t0) = x2(0) = 0, x2(tf ) = x2(1) = 0. (34) The nominal parameter value was chosen as c = 10. The goal of this problem is to maximize the horizontal … view at source ↗

**Figure 4.** Figure 4: The reference solution for Zermelo’s navigation p view at source ↗

**Figure 5.** Figure 5: Results computed from 1000 Monte Carlo trials for e view at source ↗

**Figure 6.** Figure 6: Trade study from 100 Monte Carlo trials for view at source ↗

**Figure 7.** Figure 7: The optimal solution for the unconstrained reusab view at source ↗

**Figure 8.** Figure 8: Penalized state sensitivities with respect to den view at source ↗

**Figure 9.** Figure 9: Results from 1000 Monte Carlo trials for terminal d view at source ↗

read the original abstract

A computational method is developed for desensitized optimal guidance using adaptive Gaussian quadrature collocation. The method computes a reference trajectory that reduces the sensitivity to uncertainties in the dynamic model by augmenting the objective functional to explicitly penalize the sensitivity of the state with respect to uncertain parameters. Using this desensitized reference trajectory as a starting point, the desensitized optimal guidance method developed in this paper computes a new optimal control on the remaining horizon at specified guidance update times. This shrinking horizon optimal control problem is solved using a Legendre-Gauss-Radau collocation method where, at each guidance update, a reduced-horizon mesh is determined by remapping the mesh to the remaining horizon and deleting the portion of the mesh associated with the expired portion of the horizon. The resulting guidance solution is found to improve robustness to external disturbances and modeling errors. The method is demonstrated on two numerical examples. The first example is Zermelo's navigation problem which illustrates the behavior of the method on a simple example. The second example is an atmospheric reentry problem that demonstrates the performance of the method on a more complex problem. For both examples, the dynamics are simulated in the presence of parameter uncertainties in the dynamic model, and Monte Carlo analysis is performed. The results show that the method developed in this paper produces tighter trajectory envelopes and smaller terminal state errors without significantly increasing the computational burden when compared with a method that does not penalize sensitivities.

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 gives a workable numerical recipe for desensitized guidance by adding first-order sensitivity penalties to the objective and solving remapped shrinking-horizon collocation problems, with Monte Carlo checks on two standard examples.

read the letter

The core contribution is the combination of an objective that penalizes state sensitivities to selected uncertain parameters with adaptive Legendre-Gauss-Radau collocation and mesh remapping at each guidance update. This produces a reference trajectory that is then re-solved on the remaining horizon without rebuilding the entire mesh from scratch. On Zermelo's navigation problem and the atmospheric reentry case, the Monte Carlo runs under parameter perturbations show visibly tighter trajectory bundles and lower terminal errors than the same collocation setup without the sensitivity term, while the extra compute stays small. That part is concrete and directly usable for anyone already running direct collocation codes in guidance loops. The approach stays within established direct-method machinery, so the implementation path is clear once the sensitivity equations and remapping rule are coded. The evidence is not circular; the Monte Carlo trials are separate forward simulations. The main limitation is scope. The reported trials only vary the parameters that were chosen for the sensitivity penalty. The abstract states the method improves robustness to external disturbances and modeling errors in general, but no additive noise, higher-order effects, or disturbances outside the parameter set are shown. If those turn out to matter, the first-order parametric penalty may not deliver the claimed benefit. A few more implementation specifics, such as how the sensitivities are integrated into the collocation defect constraints and the exact mesh adaptation tolerances, would also make the work easier to reproduce. This is the sort of paper that belongs in an optimal control or aerospace guidance journal. Readers who already use direct collocation for online trajectory updates will get practical ideas from the remapping step and the two worked examples. It is incremental rather than foundational, but the numerical results are solid enough that a serious editor should send it out for review rather than desk-reject it. The method is worth testing in other problems even if the broader robustness language needs some qualification.

Referee Report

3 major / 2 minor

Summary. The paper develops a computational method for desensitized optimal guidance via adaptive Gaussian quadrature collocation. The objective is augmented to penalize first-order state sensitivities with respect to selected uncertain parameters in the dynamics; the resulting problem is solved on a shrinking horizon with mesh remapping at guidance updates. The approach is illustrated on Zermelo's navigation problem and an atmospheric reentry example, with Monte Carlo simulations under dynamic-model parameter perturbations showing tighter trajectory envelopes and reduced terminal errors relative to a non-desensitized baseline, at comparable computational cost.

Significance. If the central numerical findings hold, the method offers a practical, direct-collocation-based route to desensitized guidance that can be recomputed online. The two-example Monte Carlo validation and the explicit shrinking-horizon remapping procedure constitute concrete, falsifiable evidence for the parametric-uncertainty case and could be useful in aerospace trajectory design where model parameters are the dominant source of uncertainty.

major comments (3)

[Abstract] Abstract: the claim that the method 'improve[s] robustness to external disturbances and modeling errors' is not supported by the reported evidence. Monte Carlo results are generated only by perturbing parameters inside the dynamic model; no additive disturbances, higher-order sensitivities, or errors outside the chosen parameter set are tested, so the broader robustness statement cannot be substantiated from the given data.
[Method (collocation and sensitivity penalization)] Method description (sensitivity augmentation and collocation): the manuscript provides no explicit description of how the sensitivity equations are adjoined to the state dynamics inside the collocation scheme, how the penalty weights on the sensitivity terms are selected, or the precise mesh-adaptation and remapping rules. These omissions are load-bearing because the central claim rests on the accuracy with which first-order sensitivities are penalized and recomputed at each guidance update.
[Numerical examples] Numerical examples: verification that the collocation discretization accurately captures the sensitivity dynamics (e.g., comparison against finite-difference or variational sensitivities, or residual checks on the sensitivity equations) is absent. Without such checks, it is unclear whether the observed Monte Carlo improvement is due to true desensitization or to numerical artifacts in the augmented problem.

minor comments (2)

[Abstract] Abstract: the phrase 'without significantly increasing the computational burden' should be accompanied by quantitative metrics (CPU time, number of collocation points, or iteration counts) rather than a qualitative statement.
[Throughout] Notation: ensure that the sensitivity variables and the penalty terms are introduced with consistent symbols and that their dimensions are stated explicitly when the augmented objective is first written.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their thorough review and constructive feedback on our manuscript. We address each of the major comments below and have made revisions to the manuscript to improve clarity and accuracy.

read point-by-point responses

Referee: [Abstract] Abstract: the claim that the method 'improve[s] robustness to external disturbances and modeling errors' is not supported by the reported evidence. Monte Carlo results are generated only by perturbing parameters inside the dynamic model; no additive disturbances, higher-order sensitivities, or errors outside the chosen parameter set are tested, so the broader robustness statement cannot be substantiated from the given data.

Authors: We concur that the Monte Carlo analysis is limited to perturbations of the dynamic model parameters. The broader phrasing in the abstract regarding external disturbances and modeling errors is not directly supported by the presented results. We will revise the abstract to more precisely state that the method improves robustness to uncertainties in the dynamic model parameters. revision: yes
Referee: [Method (collocation and sensitivity penalization)] Method description (sensitivity augmentation and collocation): the manuscript provides no explicit description of how the sensitivity equations are adjoined to the state dynamics inside the collocation scheme, how the penalty weights on the sensitivity terms are selected, or the precise mesh-adaptation and remapping rules. These omissions are load-bearing because the central claim rests on the accuracy with which first-order sensitivities are penalized and recomputed at each guidance update.

Authors: The referee correctly identifies that key implementation details were omitted. In the revised version, we will add a detailed description of the augmented dynamics, including the explicit form of the sensitivity equations adjoined to the original system. We will also specify how the penalty weights are chosen (typically through normalization or empirical tuning) and provide the exact procedure for mesh remapping on the shrinking horizon, including deletion of expired segments and interpolation if necessary. revision: yes
Referee: [Numerical examples] Numerical examples: verification that the collocation discretization accurately captures the sensitivity dynamics (e.g., comparison against finite-difference or variational sensitivities, or residual checks on the sensitivity equations) is absent. Without such checks, it is unclear whether the observed Monte Carlo improvement is due to true desensitization or to numerical artifacts in the augmented problem.

Authors: We agree that independent verification of the sensitivity propagation within the collocation scheme would strengthen the results. We will include additional analysis in the numerical examples section, such as a comparison of the collocated sensitivities with those obtained via finite differences for selected cases, along with checks on the satisfaction of the sensitivity differential equations. revision: yes

Circularity Check

0 steps flagged

No significant circularity; method is a direct numerical procedure validated externally

full rationale

The paper describes a computational algorithm: augment the optimal control objective with first-order state sensitivities to selected uncertain parameters, solve the resulting problem via adaptive Gaussian quadrature collocation, then apply shrinking-horizon remapping at guidance updates. No equations are shown that reduce a claimed result to the inputs by construction. Monte Carlo validation is performed on separate simulation runs with injected parameter perturbations; the robustness claim is therefore an empirical outcome, not a tautological renaming or self-referential fit. No self-citation load-bearing steps, uniqueness theorems, or ansatzes imported from prior work by the same authors appear in the derivation chain. The approach is self-contained as a direct collocation-based optimizer.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The approach rests on standard optimal control assumptions plus one tunable penalty weight for sensitivities; no new entities are postulated.

free parameters (1)

sensitivity penalty weight
The scalar multiplier on the sensitivity term in the objective is introduced to trade off nominal performance against robustness and must be selected for each problem.

axioms (2)

domain assumption The uncertain parameters enter the dynamics in a differentiable manner that allows first-order sensitivity equations to be integrated alongside the state.
Required for the augmented objective to be well-defined in the collocation transcription.
domain assumption The nominal dynamic model remains a sufficiently accurate base for the sensitivity calculation even when actual disturbances occur.
Implicit in the claim that penalizing nominal sensitivities improves real-world robustness.

pith-pipeline@v0.9.0 · 5546 in / 1522 out tokens · 49802 ms · 2026-05-08T10:57:11.256312+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

49 extracted references · 39 canonical work pages

[1]

A. E. Bryson, Applied Optimal Control: Optimization, Es timation and Control, Rout- ledge, 2018

2018
[2]

L. W. Neustadt, Optimization: A Theory of Necessary Cond itions, Princeton University Press, 2015

2015
[3]

Kwakernaak, R

H. Kwakernaak, R. Sivan, Linear Optimal Control Systems , Vol. 1, Wiley-Interscience New York, 1972

1972
[4]

G. F. Franklin, J. D. Powell, A. Emami-Naeini, J. D. Powel l, Feedback Control of Dynamic Systems, Vol. 4, Prentice Hall Upper Saddle River, 2 002
[5]

B. C. Kuo, Automatic Control Systems, Prentice Hall PTR, 1987

1987
[6]

Seywald, R

H. Seywald, R. Kumar, Desensitized optimal trajectorie s, Analytical Mechanics Asso- ciates Rept (2003) 03–16

2003
[7]

H. Shen, H. Seywald, R. W. Powell, Desensitizing the mini mum-fuel powered descent for mars pinpoint landing, Journal of Guidance, Control, an d Dynamics 33 (1) (2010) 108–115. doi:10.2514/1.44649

work page doi:10.2514/1.44649 2010
[8]

S. Li, Y. Peng, Mars entry trajectory optimization using DOC and DCNLP, Advances in Space Research 47 (3) (2011) 440–452. doi:10.1016/j.asr.2010.09.005. 21

work page doi:10.1016/j.asr.2010.09.005 2011
[9]

Seywald, K

H. Seywald, K. L. Seywald, Desensitized optimal control , Journal of Guidance, Control, and Dynamics 47 (8) (2024) 1542–1555. doi:10.2514/1.G008284

work page doi:10.2514/1.g008284 2024
[10]

Jawaharlal Ayyanathan, E

P. Jawaharlal Ayyanathan, E. Taheri, Reduced desensit ization formulation for op- timal control problems, The Journal of the Astronautical Sc iences 71 (21) (2024). doi:10.1007/s40295-024-00435-w

work page doi:10.1007/s40295-024-00435-w 2024
[11]

Zames, B

G. Zames, B. Francis, Feedback, minimax sensitivity, a nd optimal robust- ness, IEEE Transactions on Automatic Control 28 (1) (2003) 5 85–601. doi:10.1109/TAC.1984.1103357

work page doi:10.1109/tac.1984.1103357 2003
[12]

Piprek, H

P. Piprek, H. Hong, F. Holzapfel, Optimal trajectory de sign accounting for robust stability of path-following controller, Journal of Guidance, Control, and Dynamics 45 (8) (2022) 1385–1398. doi:10.2514/1.G006383

work page doi:10.2514/1.g006383 2022
[13]

Pilipovsky, P

J. Pilipovsky, P. Tsiotras, J. Hart, B. van Bloemen Waan ders, F-DOC: feedback de- sensitized optimal control, Journal of Guidance, Control, and Dynamics 48 (6) (2025) 1397–1406. doi:10.2514/1.G008468

work page doi:10.2514/1.g008468 2025
[14]

Robbiani, M

T. Robbiani, M. Sagliano, F. Topputo, H. Seywald, Fast d esensitized optimal control for rocket-powered descent and landing, Journal of Guidanc e, Control, and Dynamics 48 (11) (2025) 2480–2494. doi:10.2514/1.G009058

work page doi:10.2514/1.g009058 2025
[15]

V. R. Makkapati, M. Dor, P. Tsiotras, Trajectory desens itization in optimal control problems, in: 2018 IEEE Conference on Decision and Control ( CDC), IEEE, 2018, pp. 2478–2483. doi:10.1109/CDC.2018.8619577

work page doi:10.1109/cdc.2018.8619577 2018
[16]

J. T. Betts, Practical Methods for Optimal Control and E stimation Using Nonlinear Programming, SIAM, 2010

2010
[17]

D. E. Kirk, Optimal Control Theory: An Introduction, Co urier Corporation, 2004

2004
[18]

L. T. Biegler, V. M. Zavala, Large-scale nonlinear prog ramming using IPOPT: An in- tegrating framework for enterprise-wide dynamic optimiza tion, Computers & Chemical Engineering 33 (3) (2009) 575–582. doi:10.1016/j.compchemeng.2008.08.006

work page doi:10.1016/j.compchemeng.2008.08.006 2009
[19]

P. E. Gill, W. Murray, M. A. Saunders, SNOPT: An SQP algor ithm for large-scale constrained optimization, SIAM review 47 ( 1) (2005) 99–131. doi:10.1137/S0036144504446096

work page doi:10.1137/s0036144504446096 2005
[20]

R. H. Byrd, J. Nocedal, R. A. Waltz, KNITRO: An integrate d package for nonlinear optimization, in: Large-scale Nonlinear Optimization, Sp ringer, 2006, pp. 35–59

2006
[21]

Kraft, On converting optimal control problems into n onlinear programming prob- lems, in: Computational Mathematical Programming, Spring er, 1985, pp

D. Kraft, On converting optimal control problems into n onlinear programming prob- lems, in: Computational Mathematical Programming, Spring er, 1985, pp. 261–280. doi:10.1007/978-3-642-82450-0_9

work page doi:10.1007/978-3-642-82450-0_9 1985
[22]

C. R. Hargraves, S. W. Paris, Direct trajectory optimiz ation using nonlinear program- ming and collocation, Journal of Guidance, Control, and Dyn amics 10 (4) (1987) 338–

1987
[23]

Von Stryk, R

O. Von Stryk, R. Bulirsch, Direct and indirect methods f or trajectory optimization, Annals of Operations Research 37 (1992) 357–373. doi:10.1007/BF02071065. 22

work page doi:10.1007/bf02071065 1992
[24]

P. J. Enright, B. A. Conway, Discrete approximations to optimal trajectories using direct transcription and nonlinear programming, Journal o f Guidance, Control, and Dynamics 15 (4) (1992) 994–1002. doi:10.2514/3.20934

work page doi:10.2514/3.20934 1992
[25]

Elnagar, M

G. Elnagar, M. A. Kazemi, M. Razzaghi, The pseudospectr al legendre method for discretizing optimal control problems, IEEE Transactions on Automatic Control 40 (10) (1995) 1793–1796. doi:10.1109/9.467672

work page doi:10.1109/9.467672 1995
[26]

D. A. Benson, G. T. Huntington, T. P. Thorvaldsen, A. V. R ao, Direct trajectory optimization and costate estimation via an orthogonal coll ocation method, Journal of Guidance, Control, and Dynamics 29 (6) (2006) 1435–1440. doi:10.2514/1.20478

work page doi:10.2514/1.20478 2006
[27]

A. V. Rao, D. A. Benson, C. Darby, M. A. Patterson, C. Fran colin, I. Sanders, G. T. Huntington, Algorithm 902: GPOPS, a matlab software for solving multiple-phase optimal control problems using the gauss pseudospectral me thod, ACM Transactions on Mathematical Software (TOMS) 37 (2) (2010) 1–39. doi:10.1145/1731022.1731032

work page doi:10.1145/1731022.1731032 2010
[28]

Kameswaran, L

S. Kameswaran, L. T. Biegler, Convergence rates for dir ect transcription of optimal control problems using collocation at radau points, Comput ational Optimization and Applications 41 (2008) 81–126. doi:10.1007/s10589-007-9098-9

work page doi:10.1007/s10589-007-9098-9 2008
[29]

D. Garg, M. Patterson, W. W. Hager, A. V. Rao, D. A. Benson , G. T. Huntington, A uniﬁed framework for the numerical solution o f optimal control problems using pseudospectral methods, Automatica 46 (11) (2010) 1843–1851. doi:10.1016/j.automatica.2010.06.048

work page doi:10.1016/j.automatica.2010.06.048 2010
[30]

D. Garg, W. W. Hager, A. V. Rao, Pseudospectral methods f or solving inﬁnite-horizon optimal control problems, Automatica 47 ( 4) (2011) 829–837. doi:10.1016/j.automatica.2011.01.085

work page doi:10.1016/j.automatica.2011.01.085 2011
[31]

D. Garg, M. A. Patterson, C. Francolin, C. L. Darby, G. T. Huntington, W. W. Hager, A. V. Rao, Direct trajectory optimization and costat e estimation of ﬁnite- horizon and inﬁnite-horizon optimal control problems usin g a radau pseudospec- tral method, Computational Optimization and Applications 49 (2) (2011) 335–358. doi:10.1007/s10589-009-9291-0

work page doi:10.1007/s10589-009-9291-0 2011
[32]

W. W. Hager, H. Hou, A. V. Rao, Convergence rate for a gaus s collocation method applied to unconstrained optimal control, Journal of Optim ization Theory and Appli- cations 169 (3) (2016) 801–824. doi:10.1007/s10957-016-0929-7

work page doi:10.1007/s10957-016-0929-7 2016
[33]

W. W. Hager, H. Hou, A. V. Rao, Lebesgue constants arisin g in a class of col- location methods, IMA Journal of Numerical Analysis 37 (4) ( 2017) 1884–1901. doi:10.1093/imanum/drw060

work page doi:10.1093/imanum/drw060 2017
[34]

W. W. Hager, J. Liu, S. Mohapatra, A. V. Rao, X.-S. Wang, C onvergence rate for a gauss collocation method applied to constrained optimal c ontrol, SIAM Journal on Control and Optimization 56 (2) (2018) 1386–1411. doi:10.1137/16M1096761

work page doi:10.1137/16m1096761 2018
[35]

W. Chen, W. Du, W. W. Hager, L. Yang, Bounds for integrati on matrices that arise in gauss and radau collocation, Computational Optimizatio n and Applications 74 (1) (2019) 259–273. doi:10.1007/s10589-019-00099-5 . 23

work page doi:10.1007/s10589-019-00099-5 2019
[36]

W. W. Hager, H. Hou, S. Mohapatra, A. V. Rao, X.-S. Wang, C onver- gence rate for a radau hp collocation method applied to const rained optimal control, Computational Optimization and Applications 74 ( 1) (2019) 275–314. doi:10.1007/s10589-019-00100-1

work page doi:10.1007/s10589-019-00100-1 2019
[37]

Lu, Introducing computational guidance and control , Journal of Guidance, Control, and Dynamics 40 (2) (2017) 193–193

P. Lu, Introducing computational guidance and control , Journal of Guidance, Control, and Dynamics 40 (2) (2017) 193–193. doi:10.2514/1.G002745

work page doi:10.2514/1.g002745 2017
[38]

Dueri, B

D. Dueri, B. Açıkmeşe, D. P. Scharf, M. W. Harris, Custom ized real-time interior-point methods for onboard powered-descent guidance, Journal of G uidance, Control, and Dynamics 40 (2) (2017) 197–212. doi:10.2514/1.G001480

work page doi:10.2514/1.g001480 2017
[39]

D. P. Scharf, B. Açıkmeşe, D. Dueri, J. Benito, J. Casoli va, Implementation and ex- perimental demonstration of onboard powered-descent guid ance, Journal of Guidance, Control, and Dynamics 40 (2) (2017) 213–229. doi:10.2514/1.G000399

work page doi:10.2514/1.g000399 2017
[40]

P. Lu, C. W. Brunner, S. J. Stachowiak, G. F. Mendeck, M. A . Tigges, C. J. Cerimele, Veriﬁcation of a fully numerical entry guidance algorithm, Journal of Guidance, Control, and Dynamics 40 (2) (2017) 230–247. doi:10.2514/1.G000327

work page doi:10.2514/1.g000327 2017
[41]

Ferranti, T

L. Ferranti, T. Keviczky, Operator-splitting and grad ient methods for real-time predic- tive ﬂight control design, Journal of Guidance, Control, an d Dynamics 40 (2) (2017) 265–277. doi:10.2514/1.G000288

work page doi:10.2514/1.g000288 2017
[42]

C. L. Darby, W. W. Hager, A. V. Rao, An hp-adaptive pseudo spectral method for solving optimal control problems, Optimal Control Applica tions and Methods 32 (4) (2011) 476–502. doi:10.1002/oca.957

work page doi:10.1002/oca.957 2011
[43]

C. L. Darby, W. W. Hager, A. V. Rao, Direct trajectory opt imization using a variable low-order adaptive pseudospectral method, Journal of Spac ecraft and Rockets 48 (3) (2011) 433–445. doi:10.2514/1.52136

work page doi:10.2514/1.52136 2011
[44]

M. A. Patterson, W. W. Hager, A. V. Rao, A ph mesh reﬁnemen t method for op- timal control, Optimal Control Applications and Methods 36 (4) (2015) 398–421. doi:10.1002/oca.2114

work page doi:10.1002/oca.2114 2015
[45]

F. Liu, W. W. Hager, A. V. Rao, Adaptive mesh reﬁnement me thod for optimal con- trol using nonsmoothness detection and mesh size reduction , Journal of the Franklin Institute 352 (10) (2015) 4081–4106. doi:10.1016/j.jfranklin.2015.05.028

work page doi:10.1016/j.jfranklin.2015.05.028 2015
[46]

F. Liu, W. W. Hager, A. V. Rao, Adaptive mesh reﬁnement me thod for optimal control using decay rates of legendre polynomial coeﬃcients, IEEE T ransactions on Control Systems Technology 26 (4) (2017) 1475–1483. doi:10.1109/TCST.2017.2702122

work page doi:10.1109/tcst.2017.2702122 2017
[47]

M. E. Dennis, W. W. Hager, A. V. Rao, Computational metho d for optimal guid- ance and control using adaptive gaussian quadrature colloc ation, Journal of Guidance, Control, and Dynamics 42 (9) (2019) 2026–2041. doi:10.2514/1.G003943

work page doi:10.2514/1.g003943 2019
[48]

M. A. Patterson, A. V. Rao, GPOPS − II: A matlab software for solving multiple-phase optimal control problems using hp-adaptive gaussian quadr ature collocation meth- ods and sparse nonlinear programming, ACM Transactions on M athematical Software (TOMS) 41 (1) (2014) 1–37. doi:10.1145/2558904. 24

work page doi:10.1145/2558904 2014
[49]

M. J. Weinstein, A. V. Rao, Algorithm 984: ADiGator, a to olbox for the algorithmic diﬀerentiation of mathematical functions in MATLAB using s ource transformation via operator overloading, ACM Transactions on Mathematical So ftware (TOMS) 44 (2) (2017) 1–25. doi:10.1145/3104990. 25

work page doi:10.1145/3104990 2017