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arxiv: 2504.01215 · v2 · submitted 2025-04-01 · 🧮 math.OC

A New Approach to Motion Planning in 3D for a Dubins Vehicle: Special Case on a Sphere

Deepak Prakash Kumar , Swaroop Darbha , Satyanarayana Gupta Manyam , David Casbeer This is my paper

Pith reviewed 2026-05-22 21:23 UTC · model grok-4.3

classification 🧮 math.OC
keywords motion planningDubins vehiclespherecurvature constraintsoptimal pathsphase portraitCGC paths
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The pith

Optimal paths for a curvature-constrained vehicle on a sphere are CGC or concatenations of circular arcs, now proven for turning radii up to √3/2.

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

This paper reduces 3D motion planning for aerial vehicles with pitch-rate and yaw-rate limits to shortest-path problems on a sphere. It applies a phase portrait method to identify candidate paths and uses direct proofs to establish that all optimal solutions are built from fixed-radius left or right turns and great-circle arcs. The main advance is extending the range of the minimum turning radius r for which these path types are known to be optimal, from earlier limits around 1/√2 up to √3/2, with explicit lists of the admissible combinations in each interval.

Core claim

The optimal path is CGC or concatenations of C segments through simple proofs, where C = L, R denotes a turn of radius r and G denotes a great circular arc. We generalize the previous result of optimal paths being CGC and CCC paths for r ∈ (0,1/2]∪{1/√2} to r ≤ √3/2. The optimal path is CGC, CCCC for r ≤ 1/√2, and CGC, CCπC, CCCCC for r ≤ √3/2.

What carries the argument

Phase portrait approach used to enumerate and classify candidate shortest paths consisting of radius-r circular arcs and great-circle arcs on the unit sphere.

If this is right

  • All candidate optimal paths can be constructed analytically once start and end configurations are given.
  • The admissible path families change at the thresholds r = 1/√2 and r = √3/2.
  • The publicly released code implements the analytic construction for any r in the extended range.

Where Pith is reading between the lines

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

  • If the sphere model is an exact lift, the same path types give globally optimal 3D trajectories for the original pitch-yaw constrained vehicle.
  • The geometric transition points at 1/√2 and √3/2 mark where new concatenations first become shorter than pure CGC paths.
  • Phase-portrait classification may extend to other constant-curvature problems on curved manifolds.

Load-bearing premise

Solutions found on the sphere transfer exactly to the original 3D vehicle without any loss of optimality or feasibility.

What would settle it

An explicit pair of start and end configurations on the sphere together with a path shorter than every listed CGC or C-concatenation candidate when r equals √3/2 would disprove the classification.

Figures

Figures reproduced from arXiv: 2504.01215 by David Casbeer, Deepak Prakash Kumar, Satyanarayana Gupta Manyam, Swaroop Darbha.

Figure 2
Figure 2. Figure 2: Visualization of segments on a sphere has been reasonably explored in the literature. X0 T0 N0 X Y Z Nf Xf Tf [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Initial and final configuration on sphere [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Turns on a sphere [31] From the identified control inputs in (9), the optimal cur￾vature κ can be observed to be dependent on the scalar function H12(s). Therefore, the candidate optimal paths can be obtained by constructing the phase portrait of H12(s). To this end, an equation relating H12(s) and dH12(s) ds is first obtained. To this end, it can first be observed that J := h 2 1 + h 2 2 + H2 12 is a cons… view at source ↗
Figure 5
Figure 5. Figure 5: Overview of cases and results. (In this figure, whenever we refer to the optimal path being a particular type, the optimal path can be a degenerate [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Phase portrait of H12 for λ = 0 Remark. A non-trivial path is defined as a path wherein all segments have a non-zero length. Remark. A C segment is said to be completely traversed if H12(s1) = 0, H12(s2) = 0, and H12(s) is greater than zero for C = R and less than zero for C = L. Here, s1 and s2 denote the arc length corresponding to the start and end, respectively, of the C segment, and s ∈ (s1, s2). Give… view at source ↗
Figure 7
Figure 7. Figure 7: A GRG path connecting the same configurations connected by an LδRπLδ path for r = 0.5, δ = 30◦ Using the previous result, it follows that the optimal path contains at most two C segments for r ≤ √ 1 2 for λ = 0. How￾ever, the maximum number of concatenations in an optimal path for r > √ 1 2 is not known. To this end, it is claimed that the maximum number of concatenations for r ≤ √ 3 2 is three through the… view at source ↗
Figure 8
Figure 8. Figure 8: An RGL path connecting the same configurations connected by an LδRπLπRδ path for r = 0.865, δ = 20◦ B. Case 2.1: λ = 1, λH12 < Umax 1+U2max Consider λ = 1, which corresponds to the case of normal controls (refer to [PITH_FULL_IMAGE:figures/full_fig_p010_8.png] view at source ↗
Figure 10
Figure 10. Figure 10: Phase portrait of H12 for λ = 1 and λH12 = Umax 1+U2max in the GCG, GCC, CCC paths, the angle of the middle C segment is 2π. The argument for the same will rely on the observation from the phase portrait given in [PITH_FULL_IMAGE:figures/full_fig_p011_10.png] view at source ↗
Figure 9
Figure 9. Figure 9: Phase portrait of H12 for λ = 1 and λH12 < Umax 1+U2max C. Case 2.2: λ = 1, λH12 = Umax 1+U2max The equation corresponding to H12(s) < 0 and H12(s) ≥ 0, given in (25), are ellipses, whose origin lies on the H12−axis. The intersection of the ellipse corresponding to H12(s) < 0 with the H12−axis are at ±λH12 − Umax 1+U2max , whereas for the ellipse corresponding to H12(s) ≥ 0, the intersection points are at … view at source ↗
Figure 11
Figure 11. Figure 11: Phase portrait of H12 for λ = 1 and λH12 > Umax 1+U2max Proposition 7. For λ = 1, λH12 > Umax 1+U2max , the optimal path is a concatenation of C segments. For the optimal path containing a concatenation of C segments, the following claim is made regarding the angle of the middle C segments. It should be noted here that while the following result has been shown in [29], a simpler proof is shown here utiliz… view at source ↗
Figure 13
Figure 13. Figure 13: A LϕRπ−βLϕ path connecting the same configurations connected by an LπRπ+βLπ path for r = 0.55, β = 40◦ 0, γ > 0. It should be noted that the considered path is non￾optimal for r ≤ √ 1 2 , since the Lπ+βRπ+βLπ+β subpath is shown to be non-optimal using Lemma 9. Hence, it is sufficient to show non-optimality for r ∈  √ 1 2 , √ 3 2 i . It is claimed that the considered path is non-optimal for r ∈  √ 1 2 , … view at source ↗
Figure 14
Figure 14. Figure 14: A LϕRπ−βLπ−βRϕ path connecting the same configurations connected by an LπRπ+βLπ+βRπ path for r = 0.72, β = 40◦ Using the results shown in this section, which are summa￾rized in [PITH_FULL_IMAGE:figures/full_fig_p014_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Instances wherein CCπC paths and CCCC paths are optimal V. CONCLUSION In this article, a new approach for motion planning in 3D was proposed, wherein the complete configuration description was considered. The motion constraints considered in this regard correspond to pitch rate and yaw rate constraints for the vehicle. As a step towards addressing this difficult problem, motion planning for a Dubins vehic… view at source ↗
read the original abstract

In this article, a new model for 3D motion planning, applicable to aerial vehicles, is proposed to connect an initial and final configuration subject to pitch rate and yaw rate constraints. The motion planning problem for a curvature-constrained vehicle over the surface of a sphere is identified as an intermediary problem to be solved, and it is the focus of this paper. In this article, the optimal path candidates for a vehicle with a minimum turning radius $r$ moving over a unit sphere are derived using a phase portrait approach. We show that the optimal path is $CGC$ or concatenations of $C$ segments through simple proofs, where $C = L, R$ denotes a turn of radius $r$ and $G$ denotes a great circular arc. We generalize the previous result of optimal paths being $CGC$ and $CCC$ paths for $r \in \left(0, \frac{1}{2} \right]\bigcup\{\frac{1}{\sqrt{2}}\}$ to $r \leq \frac{\sqrt{3}}{2}$ to account for vehicles with a larger $r$. We show that the optimal path is $CGC, CCCC,$ for $r \leq \frac{1}{\sqrt{2}},$ and $CGC, CC_\pi C, CCCCC$ for $r \leq \frac{\sqrt{3}}{2}.$ Additionally, we analytically construct all candidate paths and provide the code in a publicly accessible repository.

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

1 major / 2 minor

Summary. The manuscript reduces 3D Dubins motion planning with pitch- and yaw-rate bounds to a curvature-constrained shortest-path problem on the unit sphere. Via phase-portrait analysis it classifies candidate optimal paths as CGC or concatenations of C arcs (C = L or R of radius r, G = great-circle arc) and generalizes earlier results: for r ≤ 1/√2 the families are CGC and CCCC; for r ≤ √3/2 the families are CGC, CC_πC and CCCCC. Analytic constructions of all candidates are supplied together with publicly accessible code.

Significance. If the phase-portrait classification and completeness arguments hold, the work extends the known catalog of shortest-path families for the spherical Dubins problem to a larger interval of turning radii. The provision of reproducible code for candidate construction is a concrete strength that supports verification and downstream use.

major comments (1)
  1. [phase-portrait analysis] The central generalization (abstract and § on phase-portrait analysis) asserts that all singular cases are covered by the listed families up to r = √3/2, yet the manuscript supplies only a high-level description of the phase portrait without the explicit switching curves, boundary conditions, or exhaustive case enumeration needed to confirm that no additional concatenations arise for r > 1/√2.
minor comments (2)
  1. Notation: the symbol CC_πC is introduced without an explicit definition of the subscript π (presumably a fixed angular length); a short clarifying sentence or equation would remove ambiguity.
  2. The publicly accessible repository is mentioned but no URL or commit hash appears in the text; adding the link would improve reproducibility.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their thorough review and constructive feedback. The recommendation for major revision is noted, and we address the major comment point by point below with a commitment to strengthen the presentation where appropriate.

read point-by-point responses

  1. Referee: [phase-portrait analysis] The central generalization (abstract and § on phase-portrait analysis) asserts that all singular cases are covered by the listed families up to r = √3/2, yet the manuscript supplies only a high-level description of the phase portrait without the explicit switching curves, boundary conditions, or exhaustive case enumeration needed to confirm that no additional concatenations arise for r > 1/√2.

    Authors: We appreciate this observation and agree that greater explicitness would improve clarity. The phase-portrait analysis in the manuscript proceeds from the necessary conditions of the Pontryagin Maximum Principle applied to the spherical curvature-constrained problem. Geometric properties of the unit sphere and the fixed turning radius r are used to identify admissible switching loci, leading to the conclusion that only CGC paths and the indicated concatenations of C arcs (L or R) satisfy optimality for the stated ranges of r. The families CGC and CCCC for r ≤ 1/√2, and CGC, CC_πC, CCCCC for r ≤ √3/2, follow directly from the reachable-set geometry and the prohibition on certain switch sequences that would violate the curvature bound or increase length. To address the request for explicit details, the revised manuscript will include the explicit equations of the switching curves, the boundary conditions separating the families, and a tabulated case enumeration for the interval (1/√2, √3/2]. These additions will be cross-referenced with the already-provided analytic constructions and the publicly available code, which generates and evaluates all candidate paths for any admissible r. We maintain that no additional families arise, but the expanded exposition will make the completeness argument fully self-contained. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper derives candidate optimal paths (CGC, CCCC, CCπC, CCCCC families) for the curvature-constrained problem on the unit sphere via phase-portrait analysis of the associated optimal-control problem. It generalizes earlier results on admissible path concatenations for specified ranges of the turning radius r through direct analytical arguments and explicit construction of candidates. No parameters are fitted to data and then relabeled as predictions, no result is defined in terms of itself, and no load-bearing step reduces to a self-citation chain or an imported ansatz. The sphere problem is treated as the primary object of study, with publicly released code enabling external reproduction; the derivation therefore rests on standard spherical geometry and Pontryagin’s principle rather than any internal circular reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The derivation relies on standard facts of spherical geometry and the phase-portrait method for time-optimal control; no free parameters, ad-hoc axioms, or new entities are introduced.

axioms (2)
  • domain assumption The vehicle is constrained to the unit sphere and obeys constant minimum turning radius r.
    Stated in the abstract as the intermediary problem.
  • domain assumption Phase-portrait analysis enumerates all candidate extremals.
    Invoked to conclude that only CGC and C-concatenations need be considered.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. A Novel Model for 3D Motion Planning for a Generalized Dubins Vehicle with Pitch and Yaw Rate Constraints

    cs.RO 2025-09 unverdicted novelty 6.0

    New 3D Dubins-style motion planner for vehicles with bounded pitch and yaw rates that uses rotation-minimizing frames and concatenates optimal paths on spherical, cylindrical, and planar surfaces.

Reference graph

Works this paper leans on

36 extracted references · 36 canonical work pages · cited by 1 Pith paper

  1. [1]

    On curves of minimal length with a constraint on average curvature, and with prescribed initial and terminal positions and tangents,

    L. E. Dubins, “On curves of minimal length with a constraint on average curvature, and with prescribed initial and terminal positions and tangents,” American Journal of Mathematics , vol. 79, 1957

    work page 1957

  2. [2]

    R. W. Beard and T. W. McLain, Small Unmanned Aircraft: Theory and Practice. Princeton University Press, 2012

    work page 2012

  3. [3]

    Dubins path generation for a fixed wing uav,

    I. Lugo-C ´ardenas, G. Flores, S. Salazar, and R. Lozano, “Dubins path generation for a fixed wing uav,” in 2014 International Conference on Unmanned Aircraft Systems (ICUAS) , 2014, pp. 339–346

    work page 2014

  4. [4]

    Optimal paths for a car that goes both forwards and backwards,

    J. A. Reeds and L. A. Shepp, “Optimal paths for a car that goes both forwards and backwards,” Pacific Journal of Mathematics , vol. 145, 1990

    work page 1990

  5. [5]

    L. S. Pontryagin, V . G. Boltyanskii, R. V . Gamkrelidze, and E. F. Mishchenko, The mathematical theory of optimal processes . Inter- science Publishers, 1962, ch. 1, pp. 1–71

    work page 1962

  6. [6]

    Shortest path synthesis for dubins non-holonomic robot,

    X.-N. Bui, J.-D. Boissonnat, P. Soueres, and J.-P. Laumond, “Shortest path synthesis for dubins non-holonomic robot,” in Proc. 1994 IEEE Int. Conf. Robot. Automat. , 1994, pp. 2–7

    work page 1994

  7. [7]

    Shortest paths for the reeds-shepp car: a worked out example of the use of geometric techniques in nonlinear optimal control,

    H. J. Sussmann and G. Tang, “Shortest paths for the reeds-shepp car: a worked out example of the use of geometric techniques in nonlinear optimal control,” Rutgers University, Tech. Rep., 1991

    work page 1991

  8. [8]

    Non-euclidean dubins’ problem,

    F. Monroy-P ´erez, “Non-euclidean dubins’ problem,”Journal of Dynam- ical and Control Systems , vol. 4, pp. 249–272, 1998

    work page 1998

  9. [9]

    The weighted markov-dubins problem,

    D. P. Kumar, S. Darbha, S. G. Manyam, and D. Casbeer, “The weighted markov-dubins problem,” IEEE Robotics and Automation Letters, vol. 8, no. 3, pp. 1563–1570, 2023

    work page 2023

  10. [10]

    The asymmetric sinistral/dextral markov- dubins problem,

    E. Bakolas and P. Tsiotras, “The asymmetric sinistral/dextral markov- dubins problem,” in Proceedings of the 48h IEEE Conference on De- cision and Control (CDC) held jointly with 2009 28th Chinese Control Conference, 2009, pp. 5649–5654

    work page 2009

  11. [11]

    Shortest dubins paths to intercept a target moving on a circle,

    S. G. Manyam, D. W. Casbeer, A. V . Moll, and Z. Fuchs, “Shortest dubins paths to intercept a target moving on a circle,” Journal of Guidance, Control, and Dynamics , vol. 45, pp. 2107–2120, 2022

    work page 2022

  12. [12]

    3-dimensional path planning for autonomous underwater vehicle,

    Y . Wang and Y . R. Zheng, “3-dimensional path planning for autonomous underwater vehicle,” in OCEANS 2018 MTS/IEEE Charleston, 2018, pp. 1–6

    work page 2018

  13. [13]

    Motion planning for two 3D-Dubins vehicles with distance constraint,

    H. Marino, M. Bonizzato, R. Bartalucci, P. Salaris, and L. Pallottino, “Motion planning for two 3D-Dubins vehicles with distance constraint,” in 2012 IEEE/RSJ International Conference on Intelligent Robots and Systems, 2012, pp. 4702–4707

    work page 2012

  14. [14]

    Path generation and tracking in 3-d for uavs,

    G. Ambrosino, M. Ariola, U. Ciniglio, F. Corraro, E. De Lellis, and A. Pironti, “Path generation and tracking in 3-d for uavs,” IEEE Transactions on Control Systems Technology , vol. 17, no. 4, pp. 980– 988, 2009

    work page 2009

  15. [15]

    A mixed local-global solution to motion planning within 3-d environments,

    R. Hurley, R. Lind, and J. Kehoe, “A mixed local-global solution to motion planning within 3-d environments,” in AIAA Guidance, Navigation, and Control Conference , 2009. [Online]. Available: https://arc.aiaa.org/doi/abs/10.2514/6.2009-6297

    work page doi:10.2514/6.2009-6297 2009

  16. [16]

    Accessibility region for a car that only moves forwards along optimal paths,

    X.-N. Bui and J.-D. Boissonnat, “Accessibility region for a car that only moves forwards along optimal paths,” INRIA, Tech. Rep., 1994

    work page 1994

  17. [17]

    Path planning using 3D Dubins curve for un- manned aerial vehicles,

    Y . Lin and S. Saripalli, “Path planning using 3D Dubins curve for un- manned aerial vehicles,” in2014 International Conference on Unmanned Aircraft Systems (ICUAS), 2014, pp. 296–304

    work page 2014

  18. [18]

    Shortest 3-dimensional paths with a prescribed curvature bound,

    H. Sussmann, “Shortest 3-dimensional paths with a prescribed curvature bound,” in Proc. 1995 34th IEEE Conf. Decision Control , 1995, pp. 3306–3312. IEEE TRANSACTIONS ON ROBOTICS. SUBMISSION VERSION. 16

    work page 1995

  19. [19]

    Optimal geometrical path in 3D with curvature constraint,

    S. Hota and D. Ghose, “Optimal geometrical path in 3D with curvature constraint,” in 2010 IEEE/RSJ International Conference on Intelligent Robots and Systems , 2010, pp. 113–118

    work page 2010

  20. [20]

    Optimal path planning for an aerial vehicle in 3D space,

    ——, “Optimal path planning for an aerial vehicle in 3D space,” in 49th IEEE Conference on Decision and Control (CDC), 2010, pp. 4902–4907

    work page 2010

  21. [21]

    Reparametrization of 3D csc Du- bins paths enabling 2d search,

    L. Xu, Y . Baryshnikov, and C. Sung, “Reparametrization of 3D csc Du- bins paths enabling 2d search,” in Algorithmic Foundations of Robotics XVI, 2024

    work page 2024

  22. [22]

    An analytic solution to the 3D csc Dubins path problem,

    V . M. Baez, N. Navkar, and A. T. Becker, “An analytic solution to the 3D csc Dubins path problem,” in 2024 IEEE International Conference on Robotics and Automation (ICRA) , 2024, pp. 7157–7163

    work page 2024

  23. [23]

    Time-optimal paths for a Dubins airplane,

    H. Chitsaz and S. M. LaValle, “Time-optimal paths for a Dubins airplane,” in2007 46th IEEE Conference on Decision and Control, 2007, pp. 2379–2384

    work page 2007

  24. [24]

    M. Owen, R. W. Beard, and T. W. McLain, Implementing Dubins Airplane Paths on Fixed-Wing UAVs* . Dordrecht: Springer Netherlands, 2015, pp. 1677–1701. [Online]. Available: https://doi.org/ 10.1007/978-90-481-9707-1 120

    work page doi:10.1007/978-90-481-9707-1 2015

  25. [25]

    Vector fields for robot navigation along time-varying curves in n -dimensions,

    V . M. Goncalves, L. C. A. Pimenta, C. A. Maia, B. C. O. Dutra, and G. A. S. Pereira, “Vector fields for robot navigation along time-varying curves in n -dimensions,” IEEE Transactions on Robotics, vol. 26, no. 4, pp. 647–659, 2010

    work page 2010

  26. [26]

    Minimal 3D Dubins path with bounded curvature and pitch angle,

    P. V ´aˇna, A. Alves Neto, J. Faigl, and D. G. Macharet, “Minimal 3D Dubins path with bounded curvature and pitch angle,” in 2020 IEEE International Conference on Robotics and Automation (ICRA) , 2020, pp. 8497–8503

    work page 2020

  27. [27]

    Finding 3D Dubins paths with pitch angle constraint using non-linear optimization,

    J. Herynek, P. V ´aˇna, and J. Faigl, “Finding 3D Dubins paths with pitch angle constraint using non-linear optimization,” in 2021 European Conference on Mobile Robots (ECMR) , 2021, pp. 1–6

    work page 2021

  28. [28]

    Towards finding the shortest-paths for 3D rigid bodies,

    W. Wang and P. Li, “Towards finding the shortest-paths for 3D rigid bodies,” in Robotics: Science and Systems 2021 , 2021

    work page 2021

  29. [29]

    Optimal geodesic curvature constrained dubins’ paths on a sphere,

    S. Darbha, A. Pavan, R. Kumbakonam, S. Rathinam, D. W. Casbeer, and S. G. Manyam, “Optimal geodesic curvature constrained dubins’ paths on a sphere,” Journal of Optimization Theory and Applications , vol. 197, pp. 966–992, 2023

    work page 2023

  30. [30]

    Optimal geodesic curvature constrained dubins’ path on sphere with free terminal orientation,

    D. P. Kumar, S. Darbha, S. G. Manyam, D. Tran, and D. W. Casbeer, “Optimal geodesic curvature constrained dubins’ path on sphere with free terminal orientation,” in 2023 Ninth Indian Control Conference (ICC), 2023, pp. 377–382

    work page 2023

  31. [31]

    Generalization of optimal geodesic curvature constrained dubins’ path on sphere with free terminal orientation,

    D. P. Kumar, S. Darbha, S. G. Manyam, and D. Casbeer, “Generalization of optimal geodesic curvature constrained dubins’ path on sphere with free terminal orientation,” IEEE Control Systems Letters , vol. 8, pp. 2991–2996, 2024

    work page 2024

  32. [32]

    Time-optimal synthesis for left-invariant control systems on so(3),

    U. Boscain and Y . Chitour, “Time-optimal synthesis for left-invariant control systems on so(3),” SIAM Journal on Control and Optimization , vol. 44, no. 1, pp. 111–139, 2005

    work page 2005

  33. [33]

    Time optimal control of a satellite with two rotors,

    U. Boscain, Y . Chitour, and B. Piccoli, “Time optimal control of a satellite with two rotors,” in 2001 European Control Conference (ECC), 2001, pp. 1721–1725

    work page 2001

  34. [34]

    Equivalence of dubins path on sphere with geographic coordinates and moving frames,

    D. P. Kumar, S. Darbha, S. G. Manyam, D. W. Casbeer, and M. Pachter, “Equivalence of dubins path on sphere with geographic coordinates and moving frames,” Journal of Guidance, Control, and Dynamics , vol. 48, no. 2, pp. 437–442, 2025. APPENDIX A. Computations for non-optimality of RLRLR path for r ≤ 1√ 2 in Lemma 9 The equation to be satisfied by the alte...

    work page 2025

  35. [35]

    First, the expression for A2 + B2 can be obtained as A2 + B2 = 4 1 − 2r2 1 − r2 (1 + cβ) 2

    α33 = β33 To obtain the expression for sϕ and cϕ, the definition of sθ and cθ in (29) in terms of A and B, which are defined in the same equation, is desired to be used. First, the expression for A2 + B2 can be obtained as A2 + B2 = 4 1 − 2r2 1 − r2 (1 + cβ) 2 . Since β ∈ (0, π), 1 − 2r2 1 − r2 (1 + cβ) > 1 − 4r2(1 − r2) = (2r2 −1)2 ≥ 0. Hence, the expres...

    work page

  36. [36]

    As the solution for ϕ from (33) is given by ϕ = δ−π, it follows that sϕ = −sδ, cϕ = −cδ

    η33 = ¯β33 To show the above claims, the expressions for sϕ and cϕ are required. As the solution for ϕ from (33) is given by ϕ = δ−π, it follows that sϕ = −sδ, cϕ = −cδ. First, the expression for C2 + D2 is obtained as C2 + D2 = 5 − 10r2 + 6r4 + 4 2r2 − 1 r2 − 1 cos β − 2r2 1 − r2 cos 2β 2 := (g(r, β))2 . To obtain the expression for the positive square r...

    work page