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arxiv: 2504.00966 · v2 · submitted 2025-04-01 · 💻 cs.RO · cs.SY· eess.SY

Time-optimal Convexified Reeds-Shepp Paths on a Sphere

Sixu Li , Deepak Prakash Kumar , Swaroop Darbha , Yang Zhou This is my paper

Pith reviewed 2026-05-22 21:43 UTC · model grok-4.3

classification 💻 cs.RO cs.SYeess.SY
keywords time-optimal path planningReeds-Shepp pathsspherical geometryPontryagin maximum principlemotion primitivessatellite attitude controlrolling robots
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The pith

Time-optimal paths for a convexified Reeds-Shepp vehicle on a sphere consist of at most six segments from three motion primitives.

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

The paper establishes that for vehicles moving on a sphere with bounded speed and turning rate, the shortest-time paths between two configurations follow a simple structure when the turning bound is sufficiently large. Using optimal control theory, it classifies all such paths into 23 types, each with explicit formulas for the lengths of the segments. This structure allows direct computation of optimal trajectories instead of relying on numerical optimization methods. A sympathetic reader would care because this applies directly to controlling satellites or spherical robots where quick path computation matters for real-time operation.

Core claim

Using Pontryagin's Maximum Principle and a phase-portrait analysis, the optimal path connecting a given initial configuration to a desired terminal configuration consists of at most six segments drawn from three motion primitives: tight turns, great circular arcs, and turn-in-place motions. A complete classification yields a finite sufficient list of 23 optimal path types with closed-form segment angles derived.

What carries the argument

Pontryagin's Maximum Principle applied to derive extremal trajectories, followed by phase-portrait analysis to classify switches between motion primitives.

If this is right

  • The global optimum can always be found by checking only the 23 candidate path types.
  • Each of the 23 types has closed-form expressions for its segment angles.
  • The same structure applies to underactuated satellite attitude control and spherical rolling robots.
  • Turning-rate bounds below 1 are handled by an equivalent reformulation of the problem.

Where Pith is reading between the lines

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

  • The finite list of 23 types could support a discrete graph search for real-time planning on spheres.
  • Similar phase-portrait methods might classify optimal paths on other constant-curvature surfaces.
  • The closed-form solutions provide exact benchmarks for testing numerical optimal-control solvers.

Load-bearing premise

The phase-portrait analysis and PMP application fully capture all extremal trajectories without additional singular cases or missed switches for the spherical geometry when the turning-rate bound is at least 1.

What would settle it

A pair of initial and terminal configurations whose time-optimal path requires more than six segments or a motion primitive outside the three listed would show the classification is incomplete.

Figures

Figures reproduced from arXiv: 2504.00966 by Deepak Prakash Kumar, Sixu Li, Swaroop Darbha, Yang Zhou.

Figure 2
Figure 2. Figure 2: Illustration of ug and v two control inputs suffice to control the pose of a satellite. The spherical CRS model is equivalent to the satellite model featuring two reaction wheels [24], [25]. This equivalence will be shown in more detail in Remark 2 in Section II. This work distinguishes from earlier work in the following two aspects: (1) the objective is to minimize the time to change pose with limited con… view at source ↗
Figure 1
Figure 1. Figure 1: Configurations on a sphere The spherical CRS problem is motivated by three primary types of real-world applications: 1) Optimal attitude control of underactuated satellites: In scenarios where actuators fail, a satellite could become un￾deractuated [20]; research from [21]–[23] shows that only (a) fixed v, different constant ug (b) fixed ug, different con￾stant v [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: Schematic plot of the spherical rolling robot [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Repositioning of the shutter negative Gaussian curvature. Such terrain characteristics are seen in “gently rolling terrain”, as illustrated, for example, in [PITH_FULL_IMAGE:figures/full_fig_p003_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Approximating local uneven terrain with a spherical patch [PITH_FULL_IMAGE:figures/full_fig_p003_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Relation between [Xv(s), Tv(s), Nv(s)] and [X(s), T(s), N(s)] Analogous to [8], we derive the CRS model by expanding the admissible set of v to [−1, 1], resulting in the formal formulation of the time-optimal spherical CRS problem: J = min Z T 0 1 dt (2) subject to dXv dt = v(t)Tv(t), (3) dTv dt = −v(t)Xv(t) + ug(t)Nv(t), (4) dNv dt = −ug(t)Tv(t), (5) R(0) = I3, R(T) = Rf , (6) where v ∈ [−1,1] and ug ∈ [−… view at source ↗
Figure 8
Figure 8. Figure 8: Possible segment types on an extremal path [PITH_FULL_IMAGE:figures/full_fig_p006_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Phase portrait of A − C Remark 5. (a) By Lemma 2 and equation (29) (shown in [PITH_FULL_IMAGE:figures/full_fig_p006_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Connections between proofs Remark 7. The starting and terminal segments of any can￾didate optimal path with at least n ≥ 3 segments will sometimes be referred to as boundary segments, and the other n − 2 segments will be referred to as middle segments. For example, in the path CCβ|C, the middle segment is Cβ and in C|CβGCβ|C, the middle segments are Cβ, G. Lemma 7. For g = 1 and Umax ≥ 1 (or r ≤ √ 1 2 ), … view at source ↗
Figure 12
Figure 12. Figure 12: Illustration of a middle L− segment with switched evolution directions [PITH_FULL_IMAGE:figures/full_fig_p010_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: A middle L− segment with switched evolution directions on a sphere, for Umax = 1.1 (or r = √ 1 2.21 ) case (2) does not enter the G− segment in [PITH_FULL_IMAGE:figures/full_fig_p010_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Illustrative evolution of A − C for g = 1 Using similar proofs for the middle sub-path entered at (− 1 Umax , 0) counter-clockwise, and the sub-paths entered at ( 1 Umax , 0) in both clockwise and counter-clockwise directions, it can be proved that the middle sub-paths are of type |CβGCβ| or |CβCβ|. For starting/terminal sub-paths that do not start/terminate at cusps, the starting sub-paths may be truncat… view at source ↗
Figure 15
Figure 15. Figure 15: Non-optimality of L − β R − β |R + β Using similar proofs for paths L + β R + β |R − β , R − β L − β |L + β , and R + β L + β |L − β , the lemma is proved. The following lemma states that with any non-negative σ, GσCβ|Cβ and Cβ|CβGσ paths can replace each other. Lemma 10. A path of type GσCβ|Cβ can be replaced by a path of type Cβ|CβGσ, and vice versa. Proof. Suppose a path is of type G− σ L − β |L + β wi… view at source ↗
Figure 16
Figure 16. Figure 16: Non-optimality of R − β G − σ L − β |L + β are redundant. They may be replaced respectively by paths of type C|C, C|CβG, CGCβ|C, C|CβG, C|CβG, and CGCβ|C. Proof. The replacement of CCβ|CβC, CCβ|CβG, and CGCβ|CβC paths follows directly from Lemma 8, and the replacement of GCβ|CβG, CGCβ|CβG, and CGCβ|CβGC paths follows directly from Lemmas 8 and 10. B. Optimal Paths Containing T Segments As shown in the pha… view at source ↗
Figure 17
Figure 17. Figure 17: Non-optimality of L0 ρL − π Using similar proofs for paths L 0 ρL + π , R0 ρR− π , and R0 ρR+ π , the lemma is proved. Since for g = 1 and Umax ≥ 1, an optimal path may include T segments solely when Umax = 1 (by Remark 9), and a Cπ segment corresponds to a middle C segment when Umax = 1, the following corollary directly follows from Lemma 13. Corollary 1. For g = 1 and Umax ≥ 1 (or r ≤ √ 1 2 ), a path co… view at source ↗
Figure 18
Figure 18. Figure 18: , it is clear that the optimal path is a concatenation of |CC| sub-paths; for start/terminal sub-paths that do not start/terminate at cusps, they are truncated but the evolution directions remain the same. Furthermore, it is clear that all middle C segments are completely traversed once, either clockwise or counter-clockwise; hence, they all have an angle of µ < arctan(√ 1 U4max−1 ) + π 2 according to Lem… view at source ↗
Figure 19
Figure 19. Figure 19: Replacing a L + µ R + ϵ L + µ path with a G + θ path With similar proofs for R+ µ L + ϵ R+ µ , L − µ R− ϵ L − µ , and R− µ L − ϵ R− µ paths, the lemma is proved. Now, utilizing Lemma 17, the key result Lemma 18 can be proved. Lemma 18. For Umax ≥ 1 (or r ≤ √ 1 2 ), a path of type CµCµ|CµCµ is not optimal. Proof. Consider a R+ µ L + µ |L − µ R− µ path where 0 < µ < arctan(√ 1 U4max−1 ) + π 2 . It is claime… view at source ↗
Figure 21
Figure 21. Figure 21: Phase portrait of C￾dC dt for g < 1 The key result Lemma 19 can be obtained by directly utilizing the phase portrait of C(t). Lemma 19. For g < 1 and Umax ≥ 1 (or r ≤ √ 1 2 ), the optimal path is of type C, T, T C (or CT), or a concatenation of C segments joined by cusps or joined by T segments, and any path containing a middle C segment is not optimal. Proof. From the phase portrait shown by [PITH_FULL_… view at source ↗
Figure 20
Figure 20. Figure 20: Non-optimality of R + µ L + µ |L − µ R − µ With similar proofs for R− µ L − µ |L + µ R+ µ , L + µ R+ µ |R− µ L − µ , and L − µ R− µ |R+ µ L + µ paths, the lemma is proved. With the key results, Lemmas 16 and 18, the finite sufficient list for g > 1 listed in Proposition 2 can be obtained. VI. OPTIMAL PATHS FOR g < 1 Referring to [PITH_FULL_IMAGE:figures/full_fig_p015_20.png] view at source ↗
Figure 22
Figure 22. Figure 22: Optimal and feasible paths connecting the initial and desired terminal [PITH_FULL_IMAGE:figures/full_fig_p017_22.png] view at source ↗
read the original abstract

This article studies the time-optimal path planning problem for a convexified Reeds-Shepp (CRS) vehicle on a unit sphere, capable of both forward and backward motion, with speed bounded in magnitude by 1 and turning rate bounded in magnitude by a given constant. For the case in which the turning-rate bound is at least 1, using Pontryagin's Maximum Principle and a phase-portrait analysis, we show that the optimal path connecting a given initial configuration to a desired terminal configuration consists of at most six segments drawn from three motion primitives: tight turns, great circular arcs, and turn-in-place motions. A complete classification yields a finite sufficient list of 23 optimal path types with closed-form segment angles derived. The complementary case in which the turning-rate bound is less than 1 is addressed via an equivalent reformulation. The proposed formulation is applicable to underactuated satellite attitude control, spherical rolling robots, and mobile robots operating on spherical or gently curved surfaces. The source code for solving the time-optimal path problem and visualization is publicly available at https://github.com/sixuli97/Optimal-Spherical-Convexified-Reeds-Shepp-Paths.

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

0 major / 1 minor

Summary. This paper studies the time-optimal path planning problem for a convexified Reeds-Shepp (CRS) vehicle on a unit sphere, with speed bounded in magnitude by 1 and turning rate bounded by a constant κ. For the case κ ≥ 1, Pontryagin's Maximum Principle combined with phase-portrait analysis is used to prove that any optimal path consists of at most six segments drawn from three primitives (tight turns, great circular arcs, turn-in-place motions). A complete classification produces a finite list of 23 admissible path types together with closed-form expressions for the segment angles. The case κ < 1 is handled by an equivalent reformulation. The formulation targets applications in underactuated satellite attitude control and spherical rolling robots; public code is provided.

Significance. If the classification and closed-form derivations hold, the work supplies a complete, finite catalogue of time-optimal trajectories for this nonholonomic system on the sphere, directly extending the classical Reeds-Shepp result to spherical geometry. The explicit list of 23 word types and the public repository constitute a practical, reproducible contribution that can be used for real-time planning in satellite and mobile-robot applications.

minor comments (1)
  1. The abstract states that the source code is publicly available; the manuscript would benefit from an explicit citation of the repository in the main text (e.g., in the introduction or conclusion) so that readers can locate the implementation without searching the abstract.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, including the detailed summary of our contributions and the recommendation to accept. No major comments requiring response or revision were raised.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper's central derivation applies Pontryagin's Maximum Principle (a standard external theorem) to the spherical control system and performs a phase-portrait analysis to enumerate extremals, yielding the 23 path types and closed-form angles. These steps rely on established optimal-control machinery and geometric properties of the sphere rather than any self-definition, fitted inputs renamed as predictions, or load-bearing self-citations. The restriction to turning-rate bound ≥1 is an explicit modeling choice, and the complementary case is handled by reformulation; no equation reduces to its own input by construction. The public code further supports independent verification outside the paper.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard mathematical tools from optimal control; no free parameters or new entities are introduced based on the abstract. Review is abstract-only so ledger is necessarily incomplete.

axioms (2)
  • standard math Pontryagin's Maximum Principle can be applied to characterize the time-optimal trajectories
    Invoked to find the structure of optimal paths.
  • domain assumption The phase-portrait analysis on the sphere covers all possible switching sequences without missed cases
    Used to limit paths to at most 6 segments and derive the 23 types.

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