Nonlinear Guidance for Arrival Time and Arrival Angle Control Using Trajectory Shaping

Andr\'es Infante Adri\'an; Kun Wang

arxiv: 2606.21643 · v1 · pith:UDYBBT2Jnew · submitted 2026-06-19 · 🧮 math.OC

Nonlinear Guidance for Arrival Time and Arrival Angle Control Using Trajectory Shaping

Kun Wang , Andr\'es Infante Adri\'an This is my paper

Pith reviewed 2026-06-26 13:20 UTC · model grok-4.3

classification 🧮 math.OC

keywords nonlinear guidancetrajectory shapingarrival time controlarrival angle controlUAV guidancepolynomial parameterizationnonlinear integral equations

0 comments

The pith

A two-stage solver with polynomial look-angle parameterization achieves near-optimal UAV arrival time and angle control.

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

The paper develops a nonlinear guidance law for constant-speed fixed-wing UAVs that must meet both a prescribed arrival time and a prescribed arrival angle. The look angle is expressed as a fourth-order polynomial, converting the guidance problem into two coupled nonlinear integral equations in two unknown parameters. A two-stage procedure first builds an analytical warm start from a linear approximation between the parameters that reduces to a scalar quadratic equation, then refines the guess by solving a one-dimensional nonlinear equation before addressing the original two-dimensional system. Simulations indicate that the resulting trajectories remain close to the open-loop optimum in regimes where prior methods produce no feasible solution.

Core claim

The nonlinear guidance problem for simultaneous arrival-time and arrival-angle control is transformed by a fourth-order polynomial parameterization of the look angle into two coupled nonlinear integral equations; these equations are solved reliably by a two-stage procedure that first obtains an analytical initial guess via linear approximations yielding a quadratic equation, then refines that guess through a one-dimensional nonlinear solve before tackling the full two-dimensional system, producing solutions numerically indistinguishable from the open-loop optimum even in strongly nonlinear test cases.

What carries the argument

The two-stage solution procedure for the pair of coupled nonlinear integral equations that result from the fourth-order polynomial parameterization of the look angle.

If this is right

The procedure supplies feasible trajectories in highly nonlinear regimes where other guidance methods fail.
The obtained trajectories lie close to the open-loop time-and-angle optimal solution for the tested constant-speed cases.
The method simultaneously enforces both arrival time and arrival angle without requiring iterative re-planning.

Where Pith is reading between the lines

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

The same polynomial parameterization and staged solver structure could be tested on vehicles whose speed varies slowly rather than remaining strictly constant.
Real-time onboard implementation would require checking whether the one-dimensional refinement step can be replaced by a closed-form correction without loss of accuracy.
The approach might extend to three-dimensional guidance by adding a second look-angle polynomial for the vertical plane.

Load-bearing premise

The fourth-order polynomial look-angle parameterization and the linear approximations used to relate the two guidance parameters remain accurate enough to yield near-optimal feasible trajectories over the full range of arrival-time and arrival-angle constraints.

What would settle it

A new simulation test case in which the linear approximation between the two guidance parameters deviates measurably from the true nonlinear coupling, yet the two-stage procedure still returns a trajectory whose cost differs from the true open-loop optimum by more than the margin reported for existing methods.

read the original abstract

This paper proposes a nonlinear trajectory shaping guidance strategy for arrival time and angle control of a constant-speed fixed-wing unmanned aerial vehicle. The look angle is parameterized by a fourth-order polynomial. The nonlinear guidance problem is transformed into solving two coupled nonlinear integral equations with respect to two unknown guidance parameters. Directly solving these equations is challenging due to the strong coupling between the parameters. To address this, a two-stage solution procedure is developed. In the first stage, an analytical warm start is constructed. Specifically, by applying approximations, a linear relationship between the two guidance parameters is obtained. With this relationship, a scalar quadratic equation in the first guidance parameter is derived, leading to a good initial guess for the first guidance parameter. In the second stage, this initial guess is refined by solving a one-dimensional nonlinear equation. The obtained solution is finally used as the initial guess for solving the original two-dimensional nonlinear system of equations. Numerical simulations demonstrate that for the test cases, the obtained solution is very close to the open-loop optimal solution, even in highly nonlinear scenarios where some existing methods fail to generate a feasible solution.

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 two-stage analytical warm-start plus 1D refinement is a solid practical addition for this guidance problem, though the missing error bounds on the key approximation limit how far the claims can be trusted without more checks.

read the letter

The paper's main contribution is a two-stage procedure to solve the coupled nonlinear integral equations that arise when you parameterize the look angle with a fourth-order polynomial for arrival time and angle control in constant-speed UAVs. They derive an approximate linear relationship between the two guidance parameters to get an analytical warm start, solve a quadratic for an initial guess, refine it with a 1D nonlinear solve, and then use that to initialize the full 2D system.

This approach is new in the way it combines the analytical warm-start with the 1D refinement for this specific setup. It builds on existing polynomial ideas but gives a concrete way to handle the strong coupling without jumping straight into a hard multidimensional solve. The simulations indicate it produces solutions close to the open-loop optimum and finds feasible points in highly nonlinear cases where other methods fail, which is a practical plus for fixed-wing guidance.

The weak point is the lack of any error analysis on those approximations used to get the linear relationship. Without bounds or a sensitivity study, it's not clear how much the warm start deviates from the true relation as nonlinearity grows, which could affect how close the final solution gets to the real optimum. The abstract also gives no quantitative error metrics from the simulations and doesn't explain how the open-loop reference was computed, so the claim of being "very close" is hard to evaluate precisely. The assumption that the fourth-order polynomial plus the specific approximations stay accurate enough across the constraint range is the load-bearing one, and it isn't stress-tested in the reported work.

This paper is for engineers and researchers working on trajectory shaping guidance laws for UAVs with simultaneous time and angle constraints. A reader in that niche would find a usable method to implement and test against their own problems.

It deserves a serious referee because the technique is specific and the problem is well-posed, even though the current validation leaves some questions open on the approximation quality.

Referee Report

2 major / 2 minor

Summary. The paper proposes a nonlinear trajectory shaping guidance law for simultaneous arrival time and arrival angle control of a constant-speed fixed-wing UAV. The look angle is parameterized as a fourth-order polynomial, converting the problem into two coupled nonlinear integral equations in two unknown guidance parameters. A two-stage solver is introduced: stage one uses unspecified approximations to derive a linear relationship between the parameters, solves a scalar quadratic for an initial guess, and refines it via a one-dimensional nonlinear equation; stage two uses this as a warm start to solve the original two-dimensional system. Numerical simulations are reported to yield solutions very close to open-loop optima, including in highly nonlinear regimes where competing methods fail to produce feasible trajectories.

Significance. If the approximation error can be bounded and the closeness to optimality quantified, the two-stage procedure would provide a practical, low-dimensional numerical method for real-time nonlinear guidance with terminal constraints. The polynomial parameterization and staged warm-start construction are systematic and could generalize to other coupled integral-equation guidance problems.

major comments (2)

[Abstract] Abstract and the description of the two-stage procedure: the central claim that 'the obtained solution is very close to the open-loop optimal solution' lacks any quantitative error metrics (e.g., terminal time/angle errors, cost-function differences) or a description of how the open-loop reference trajectories were computed; without these, the simulation evidence cannot be assessed for the claimed accuracy in highly nonlinear cases.
[Two-stage solution procedure] Stage-one derivation (the approximations yielding the linear relationship between the two guidance parameters): no error analysis, sensitivity study, or comparison of the approximated versus exact relationship is provided, yet this linear relation is load-bearing for generating the initial guess that is asserted to remain reliable across the full range of arrival-time and arrival-angle constraints.

minor comments (2)

Clarify the exact form of the two coupled nonlinear integral equations and the fourth-order polynomial coefficients before the approximation step.
Add a brief statement on the numerical solver used for the final two-dimensional system and any convergence tolerances.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback on our manuscript. We address the two major comments below and will incorporate revisions to provide the requested quantitative metrics and analysis of the stage-one approximations.

read point-by-point responses

Referee: [Abstract] Abstract and the description of the two-stage procedure: the central claim that 'the obtained solution is very close to the open-loop optimal solution' lacks any quantitative error metrics (e.g., terminal time/angle errors, cost-function differences) or a description of how the open-loop reference trajectories were computed; without these, the simulation evidence cannot be assessed for the claimed accuracy in highly nonlinear cases.

Authors: We agree that the current presentation lacks sufficient quantitative support for the closeness claim. In the revised manuscript, we will add explicit error metrics (terminal time and angle deviations, cost-function differences) in the numerical results section, along with a description of the open-loop reference computation (numerical solution of the underlying optimal control problem via direct collocation or equivalent method). revision: yes
Referee: [Two-stage solution procedure] Stage-one derivation (the approximations yielding the linear relationship between the two guidance parameters): no error analysis, sensitivity study, or comparison of the approximated versus exact relationship is provided, yet this linear relation is load-bearing for generating the initial guess that is asserted to remain reliable across the full range of arrival-time and arrival-angle constraints.

Authors: We acknowledge that the manuscript provides no quantitative assessment of the approximation error in deriving the linear relationship. The revision will include a dedicated subsection comparing the approximated linear relation to the exact nonlinear coupling (via numerical sampling over the constraint space) and a sensitivity study showing the effect on the quality of the initial guess and overall solver reliability. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation uses explicit approximations on problem equations without self-definition or fitted predictions.

full rationale

The paper parameterizes the look angle as a fourth-order polynomial, yielding two coupled nonlinear integral equations in the guidance parameters. It then applies (unspecified) approximations to obtain a linear relationship, solves a quadratic for an initial guess, and refines via one-dimensional nonlinear solve before the full two-dimensional system. This chain is constructed directly from the stated problem equations and standard numerical techniques. No self-citations, uniqueness theorems, or fitted inputs renamed as predictions appear in the derivation. The central numerical claim rests on simulation comparison to open-loop optima rather than any quantity defined by the authors' own prior results or by construction from the inputs.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim rests on the validity of the fourth-order polynomial parameterization and on the accuracy of the unspecified approximations that produce the linear relationship between the two guidance parameters; no new physical entities are introduced.

free parameters (1)

two guidance parameters
Unknown scalars solved for by the integral equations; their values are not fitted in advance but obtained numerically for each boundary condition.

axioms (2)

domain assumption The look angle can be represented exactly by a fourth-order polynomial without loss of feasibility or significant optimality gap for the arrival constraints considered.
Invoked when the nonlinear guidance problem is transformed into the two integral equations.
ad hoc to paper The approximations used to obtain the linear relationship between the two guidance parameters preserve enough accuracy to yield a useful initial guess.
Stated in the description of the first-stage analytical warm start.

pith-pipeline@v0.9.1-grok · 5721 in / 1537 out tokens · 20010 ms · 2026-06-26T13:20:33.664878+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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Nicola, M

Shanmugavel, M., Tsourdos, A., White, B., and Żbikowski, R., “Co-operative path planning of multiple UA V s using Dubins paths with clothoid arcs,” Control Engineering Practice , Vol. 18, No. 9, 2010, pp. 1084–1092. https://doi.org/10.1016/j. conengprac.2009.02.010

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Elongation of curvature-bounded path,

Chen, Z., Wang, K., and Shi, H., “Elongation of curvature-bounded path,” Automatica, Vol. 151, 2023, p. 110936. https: //doi.org/10.1016/j.automatica.2023.110936

work page doi:10.1016/j.automatica.2023.110936 2023

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Linear quadratic guidance laws for imposing a terminal intercept angle,

Shaferman, V ., and Shima, T., “Linear quadratic guidance laws for imposing a terminal intercept angle,” Journal of Guidance, Control, and Dynamics, Vol. 31, No. 5, 2008, pp. 1400–1412. https://doi.org/10.2514/1.32836

work page doi:10.2514/1.32836 2008

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Guidance law to control impact time and angle,

Lee, J.-I., Jeon, I.-S., and Tahk, M.-J., “Guidance law to control impact time and angle,” IEEE Transactions on Aerospace and Electronic Systems, Vol. 43, No. 1, 2007, pp. 301–310. https://doi.org/10.1109/TAES.2007.357135

work page doi:10.1109/taes.2007.357135 2007

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Supertwisting Control-Based Cooperative Salvo Guidance Using Leader–Follower Approach,

Kim, T.-H., Lee, C.-H., Jeon, I.-S., and Tahk, M.-J., “Augmented polynomial guidance with impact time and angle constraints,” IEEE Transactions on Aerospace and Electronic Systems, Vol. 49, No. 4, 2013, pp. 2806–2817. https://doi.org/10.1109/TAES. 2013.6621856. 9 X (m) -12000 -10000 -8000 -6000 -4000 -2000 0 2000 Y (m) -1500 -1000 -500 0 500 1000 1500 UA ...

work page doi:10.1109/taes 2013

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Guidance law with impact time and impact angle constraints,

Zhang, Y ., Ma, G., and Liu, A., “Guidance law with impact time and impact angle constraints,”Chinese Journal of Aeronautics, Vol. 26, No. 4, 2013, pp. 960–966. https://doi.org/10.1016/j.cja.2013.04.037

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Chen, X., and Wang, J., “Optimal control based guidance law to control both impact time and impact angle,”Aerospace Science and Technology, Vol. 84, 2019, pp. 454–463. https://doi.org/10.1016/j.ast.2018.10.036

work page doi:10.1016/j.ast.2018.10.036 2019

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Wang, K., Chen, Z., Wang, H., Li, J., and Shao, X., “Nonlinear optimal guidance for intercepting stationary targets with impact- time constraints,” Journal of Guidance, Control, and Dynamics, Vol. 45, No. 9, 2022, pp. 1614–1626. https://doi.org/10.2514/ 1.G006666

2022

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Three-dimensional nonlinear impact time guidance considering field-of-view constraints,

Majumder, K., and Kumar, S. R., “Three-dimensional nonlinear impact time guidance considering field-of-view constraints,” IEEE Control Systems Letters, Vol. 7, 2023, pp. 3848–3853. https://doi.org/10.1109/LCSYS.2023.3341447

work page doi:10.1109/lcsys.2023.3341447 2023

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Terminal Time Constrained Guidance Against Stationary Target with Bounded Input and Rate,

Singh, S., Kumar, S. R., and Mukherjee, D., “Terminal Time Constrained Guidance Against Stationary Target with Bounded Input and Rate,” IEEE Control Systems Letters, 2025. https://doi.org/10.1109/LCSYS.2025.3578569

work page doi:10.1109/lcsys.2025.3578569 2025

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Sliding mode guidance for impact time and angle constraints,

Kumar, S. R., and Ghose, D., “Sliding mode guidance for impact time and angle constraints,” Proceedings of the Institution of Mechanical Engineers, Part G: Journal of Aerospace Engineering , Vol. 232, No. 16, 2018, pp. 2961–2977. https://doi.org/ 10.1177/0954410017719304

work page doi:10.1177/0954410017719304 2018

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Nonlinear optimal guidance with constraints on impact time and impact angle,

Wu, F., Chen, Z., Shao, X., and Wang, K., “Nonlinear optimal guidance with constraints on impact time and impact angle,” Automatica, Vol. 181, 2025, p. 112500. https://doi.org/10.1016/j.automatica.2025.112500. 10

work page doi:10.1016/j.automatica.2025.112500 2025

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Generalized Impact Time and Angle Control Via Look-Angle Shaping,

Kang, S., Tekin, R., and Holzapfel, F., “Generalized impact time and angle control via look-angle shaping,” Journal of Guid- ance, Control, and Dynamics , Vol. 42, No. 3, 2019, pp. 695–702. https://doi.org/10.2514/1.G003765

work page doi:10.2514/1.g003765 2019

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New Impact Time and Angle Guidance Strategy via Virtual Target Approach,

Hu, Q., Han, T., and Xin, M., “New impact time and angle guidance strategy via virtual target approach,” Journal of Guidance, Control, and Dynamics, Vol. 41, No. 8, 2018, pp. 1755–1765. https://doi.org/10.2514/1.G003436

work page doi:10.2514/1.g003436 2018

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Virtual target approach-based optimal guidance law with both impact time and terminal angle constraints,

Zhang, Z., Ma, K., Zhang, G., and Y an, L., “Virtual target approach-based optimal guidance law with both impact time and terminal angle constraints,” Nonlinear Dynamics, Vol. 107, No. 4, 2022, pp. 3521–3541. https://doi.org/10.1007/s11071-021- 07142-3

work page doi:10.1007/s11071-021- 2022

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A physics-informed indirect method for trajectory optimization,

Wang, K., Lu, F., Chen, Z., and Li, J., “A physics-informed indirect method for trajectory optimization,” IEEE Transactions on Aerospace and Electronic Systems, Vol. 60, No. 6, 2024, pp. 9179–9192. https://doi.org/10.1109/TAES.2024.3438687

work page doi:10.1109/taes.2024.3438687 2024

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Nonlinear Optimal Impact Angle Control Guidance With Acceleration Constraints,

Wang, K., Lu, F., and Chen, Z., “Nonlinear Optimal Impact Angle Control Guidance With Acceleration Constraints,” IEEE Transactions on Aerospace and Electronic Systems, Vol. 61, No. 4, 2025, pp. 8907–8921. https://doi.org/10.1109/TAES.2025. 3551283

work page doi:10.1109/taes.2025 2025

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Polynomial shaping of the look angle for impact-time control,

Tekin, R., Erer, K. S., and Holzapfel, F., “Polynomial shaping of the look angle for impact-time control,” Journal of Guidance, Control, and Dynamics, Vol. 40, No. 10, 2017, pp. 2668–2673. https://doi.org/10.2514/1.G002751

work page doi:10.2514/1.g002751 2017

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Look-Angle-Shaped Impact Time Control Guidance with Field-of-View Constraints,

Wang, K., Wei, Z., and Wang, P ., “Look-Angle-Shaped Impact Time Control Guidance with Field-of-View Constraints,” Aerospace Science and Technology, 2026, p. 112131. https://doi.org/10.1016/j.ast.2026.112131

work page doi:10.1016/j.ast.2026.112131 2026

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Trajectory Shaping Guidance for Field-of-View Constrained Impact Angle Control,

Wang, K., “Trajectory Shaping Guidance for Field-of-View Constrained Impact Angle Control,” The CEAS Conference on Guidance, Navigation and Control, EuroGNC 2026, Council of European Aerospace Societies (CEAS), 2026. https://doi.org/ 10.82124/CEAS-GNC-2026-027

work page doi:10.82124/ceas-gnc-2026-027 2026

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Guaranteeing Convergence in Trajectory Shaping Guidance for Impact Time Control,

Wang, K., Ding, C., Wei, Z., Wang, P ., and Chen, Z., “Guaranteeing Convergence in Trajectory Shaping Guidance for Impact Time Control,” Automatica, Vol. 191, 2026, p. 113107. https://doi.org/10.1016/j.automatica.2026.113107

work page doi:10.1016/j.automatica.2026.113107 2026

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GPOPS-II: A MATLAB software for solving multiple-phase optimal control problems using hp-adaptive Gaussian quadrature collocation methods and sparse nonlinear programming,

Patterson, M. A., and Rao, A. V ., “GPOPS-II: A MATLAB software for solving multiple-phase optimal control problems using hp-adaptive Gaussian quadrature collocation methods and sparse nonlinear programming,” ACM Transactions on Math- ematical Software (TOMS), Vol. 41, No. 1, 2014, pp. 1–37. https://doi.org/10.1145/2558904. 11

work page doi:10.1145/2558904 2014