Multi-Modal Reeds-Shepp Curves

• Multi-Modal Reeds-Shepp Curves are trajectory models that integrate multiple motion modes with explicit mode-switch costs and tailored Reeds-Shepp connectors.
• They extend classical Reeds-Shepp families by incorporating discrete steering regimes, enabling varied kinematic constraints for Ackermann and four-wheel systems.
• These approaches support applications in hybrid A* planning, multi-robot coordination, and non-Euclidean geometries while preserving analytic tractability.

Multi-modal Reeds-Shepp curves are Reeds-Shepp-type trajectories whose segments are selected from more than one motion mode. In the most explicit recent usage, the state is extended from $(x,y,\theta)$ to $(x,y,\theta,m)$, where $m$ is a discrete steering mode, and path planning combines per-mode Reeds-Shepp connectors with explicit mode-switch costs. Closely related work treats the same idea more broadly: as concatenations of forward/backward, left/right, straight, wait, or greedy Reeds-Shepp primitives for Ackermann robots; as a partition of classical Reeds-Shepp path families into analytically selected modes; as joint geodesics for fleets of Reeds-Shepp vehicles under an $L^p$-type coupling; and as generalized Reeds-Shepp geometries on spheres or lifted position-orientation spaces (Bao et al., 7 Sep 2025, Guo, 11 Jun 2025, Ibrahim et al., 8 Apr 2025, Lukyanenko et al., 2021).

1. Terminological scope and principal meanings

The classical Reeds-Shepp car is posed on $\mathrm{SE}(2)$ with six primitive motions: $s^{+}$, $s^{-}$, $l^{+}$, $l^{-}$, $r^{+}$, and $(x,y,\theta,m)$0, corresponding to forward/backward straight motion and forward/backward maximal left/right turns. Reeds-Shepp paths are concatenations of these primitive segments, and cusps are the points where the sign of velocity changes (Ibrahim et al., 8 Apr 2025). In that classical setting, “mode” may denote a path family such as $(x,y,\theta,m)$1, $(x,y,\theta,m)$2, $(x,y,\theta,m)$3, $(x,y,\theta,m)$4, or $(x,y,\theta,m)$5, or a specific pattern such as $(x,y,\theta,m)$6 (Ibrahim et al., 8 Apr 2025).

In multi-robot Ackermann planning, the same notion is instantiated at the primitive level. The primitive set $(x,y,\theta,m)$7 contains up to eight actions: Forward max-left, Forward straight, Forward max-right, Backward max-left, Backward straight, Backward max-right, Wait, and a Greedy motion primitive derived from the first segment of a shortest single-robot path, which is the optimal Reeds-Shepp path in obstacle-free space and a hybrid A* path in clutter (Guo, 11 Jun 2025). In four-wheel independent steering planning, the mode variable is explicit and discrete: Ackermann steering, lateral steering, and parallel movement, each with distinct kinematics and a dedicated Reeds-Shepp-like connector (Bao et al., 7 Sep 2025).

Formulation	Mode set	Role
Classical planar Reeds-Shepp	46 patterns, or 20 canonical types $(x,y,\theta,m)$8 after partitioning	Analytic path-family selection
Ackermann MRPP	FL, FS, FR, BL, BS, BR, Wait, GM	Discrete primitive library in configuration-time
4WIS Hybrid A*	Ackermann, lateral, parallel	Explicit hybrid state $(x,y,\theta,m)$9
Fleet planning in product spaces	$m$0	Coupling mode for joint cost
Spherical convexified Reeds-Shepp	$m$1, $m$2, $m$3	Time-optimal multimode concatenation
Lifted data-driven Reeds-Shepp	Modality-dependent costs	Geometry adapted to data modalities

This suggests that “multi-modal Reeds-Shepp curve” is not a single standardized object. In current literature it can denote a path assembled from distinct steering regimes, a selector over analytic Reeds-Shepp families, a joint cost mode for multiple vehicles, or a Reeds-Shepp geometry adapted to data or manifold structure (Bao et al., 7 Sep 2025, Ibrahim et al., 8 Apr 2025, Lukyanenko et al., 2021).

2. Classical Reeds-Shepp families as a modal partition

Classical shortest Reeds-Shepp paths lie within a finite family of sequences of at most five segments, built from constant-curvature arcs $m$4 and straight segments $m$5. After reductions, the minimal family has 46 distinct patterns grouped as $m$6, $m$7, $m$8, $m$9, and $L^p$0, each with closed-form formulas for segment lengths and feasibility conditions (Ibrahim et al., 8 Apr 2025). From a modal viewpoint, these 46 patterns are already a discrete mode library.

The state-space partitioning in the accelerated Reeds-Shepp algorithm makes this modal structure explicit. After transforming the terminal pose into a mirrored first-quadrant local frame, the method computes distances between left-hand and right-hand circle centers and applies a Boolean test based on $L^p$1. This separates the query into Set A, containing $L^p$2, $L^p$3, and related variants, and Set B, containing $L^p$4, $L^p$5, and $L^p$6 variants. Within those two sets, additional geometric predicates reduce the effective candidate library from 46 patterns to 20 canonical path types $L^p$7, with mutually exclusive and exhaustive regions in $L^p$8-space (Ibrahim et al., 8 Apr 2025).

Algorithmically, the method performs one analytic mode decision per query rather than evaluating every candidate family. OMPL evaluates 46 types per query with average runtime about $L^p$9; the Desaulniers reimplementation runs in about $\mathrm{SE}(2)$0; the proposed partition-based method runs in about $\mathrm{SE}(2)$1, with path-length differences at machine precision relative to OMPL (Ibrahim et al., 8 Apr 2025). The same paper also treats the under-specified terminal-orientation problem by showing that the minimizing orientation lies on boundaries between path-type regions, yielding three geometric regions $\mathrm{SE}(2)$2 and corresponding closed-form solutions $\mathrm{SE}(2)$3 (Ibrahim et al., 8 Apr 2025).

This suggests that multimodality need not begin with heterogeneous vehicle hardware. Even the standard planar Reeds-Shepp problem already has a modal interpretation: the optimal trajectory is selected from a finite set of analytically characterized motion families, and mode transitions correspond to region boundaries in configuration space (Ibrahim et al., 8 Apr 2025).

3. Hybrid and steering-mode constructions

For Ackermann robots in continuous state space, multi-modal Reeds-Shepp behavior is realized by discretizing the kinematics

$\mathrm{SE}(2)$4

with $\mathrm{SE}(2)$5, $\mathrm{SE}(2)$6, and minimal turning radius $\mathrm{SE}(2)$7. The primitive library contains six core Reeds-Shepp actions—FL, FS, FR, BL, BS, BR—plus Wait and Greedy motion primitive. Each primitive is one time step of constant control: a constant-curvature arc of radius $\mathrm{SE}(2)$8 or a straight segment of length $\mathrm{SE}(2)$9. The Greedy motion primitive is obtained by truncating the first segment of length $s^{+}$0 from the shortest single-robot path to the goal, which is the optimal Reeds-Shepp path in obstacle-free space and a hybrid A* path in clutter. Low-level search uses hybrid A* over discretized $s^{+}$1 with a heuristic equal to the maximum of holonomic cost with obstacles, shortest Reeds-Shepp path length, and Euclidean distance; this heuristic is stated to be admissible (Guo, 11 Jun 2025).

The dissertation does not introduce a separate curvature-optimization smoothing stage. Instead, smoothness is induced by continuous-time kinematics for each primitive, yielding $s^{+}$2 continuity with piecewise constant curvature; each online step remains curvature-bounded and nonholonomic by construction (Guo, 11 Jun 2025). In that setting, a robot trajectory is a concatenation of forward/backward arcs, straight segments, waits, and Greedy motion segments. The dissertation explicitly states that it does not coin the term “multi-modal Reeds-Shepp curves,” but that it effectively constructs them (Guo, 11 Jun 2025).

The 4WIS formulation makes the hybrid interpretation explicit. The state is $s^{+}$3 with $s^{+}$4 for Ackermann steering, lateral steering, and parallel movement. For Ackermann mode, the effective curvature bound is

$s^{+}$5

while in lateral mode the same construction is carried out in a rotated coordinate frame with

$s^{+}$6

Parallel mode keeps the body orientation constant and uses

$s^{+}$7

The planner does not define a single closed-form Reeds-Shepp solution in the full hybrid space. Instead, it uses per-mode Reeds-Shepp connectors, discrete zero-length mode-switch edges, and a switch penalty

$s^{+}$8

Its heuristic is

$s^{+}$9

and near the goal it attempts terminal connections in every mode, sorts them by cost, and accepts the first collision-free candidate (Bao et al., 7 Sep 2025).

A recurring structural point is that multi-modal Reeds-Shepp planning typically preserves the classical within-mode Reeds-Shepp grammar while placing discrete switching logic above it. In explicit hybrid formulations, mode changes are graph edges; in implicit lattice formulations, they are primitive choices in configuration-time (Bao et al., 7 Sep 2025, Guo, 11 Jun 2025).

4. Joint-space and multi-robot generalizations

For multiple Reeds-Shepp vehicles, the most rigorous generalization is the product-space metric construction. A single Reeds-Shepp vehicle is modeled as a sub-Finsler metric space $s^{-}$0 with $s^{-}$1, dynamics

$s^{-}$2

$s^{-}$3, and $s^{-}$4. The induced distance is the infimum of the corresponding trajectory length and is interpreted as the shortest possible transition time between poses. Small balls satisfy $s^{-}$5-Ahlfors regularity with $s^{-}$6, and this dimension controls sampling-based convergence rates (Lukyanenko et al., 2021).

For a fleet of $s^{-}$7 vehicles, the joint space is $s^{-}$8, and the joint metric is the $s^{-}$9 combination

$l^{+}$0

The parameter $l^{+}$1 interpolates between a preference for minimizing total energy and a preference for minimizing travel time. For Reeds-Shepp fleets, $l^{+}$2 gives an additive cost over vehicles, while $l^{+}$3 gives the completion time of the fleet motion. The joint extend operator is built by time-rescaling the single-vehicle geodesics to a common horizon $l^{+}$4, producing a joint unit-speed geodesic in the product metric. The product space remains $l^{+}$5-Ahlfors regular with $l^{+}$6, so PRM*, RRT, and RRT* retain probabilistic completeness or asymptotic optimality under the paper’s radius conditions (Lukyanenko et al., 2021).

In multi-robot Ackermann planning, joint-space coupling appears in configuration-time rather than as a closed-form product metric. Collision avoidance is enforced at each discrete step by non-overlap of robot footprints, and multi-robot interaction constrains which primitive or mode sequence remains admissible for each robot. The centralized methods use Conflict-Based Search with hybrid A* at the low level, while the decentralized method resolves conflicts by priority inheritance and backtracking. The dissertation states that, from a “multi-modal Reeds-Shepp curve” perspective, this becomes a constrained mode-switching problem in joint configuration-time space. Empirically, the count-based CL-PIBT variant attains success rate above $l^{+}$7 for 60 car-like robots in a $l^{+}$8 environment with obstacles, within 4 seconds per instance; ECCR improves path quality but is less scalable (Guo, 11 Jun 2025).

This suggests two distinct but compatible meanings of multimodality in fleet settings: one at the level of individual vehicle primitives or steering regimes, and another at the level of the fleet coupling norm or conflict structure that determines how those per-vehicle modes can be synchronized (Lukyanenko et al., 2021, Guo, 11 Jun 2025).

5. Continuous-curvature, spherical, and lifted variants

A major extension replaces discontinuous-curvature Reeds-Shepp arcs with continuous-curvature surrogates while preserving the same driving patterns. In continuous-curvature steering, the planning state is $l^{+}$9 with controls $l^{-}$0, where $l^{-}$1 is sharpness. The paper derives a special class of continuous-curvature paths that admit the same driving patterns as Reeds-Shepp paths but consist of cusp-free clothoid turns. Its geometric machinery is organized around clothoid turns, CC circles, and $l^{-}$2-tangency, leading to explicit existence conditions for classes such as $l^{-}$3 and $l^{-}$4. When $l^{-}$5, the shortest continuous-curvature steering problem reduces to Reeds-Shepp steering. In closed-loop parking experiments, using exactly the same NMPC, continuous-curvature references substantially outperform Reeds-Shepp references (Dai et al., 2024).

A different non-Euclidean generalization appears on the unit sphere. There the configuration is a rotation matrix in $l^{-}$6, the cost is minimum time, and the primitive modes are $l^{-}$7 for tight turning, $l^{-}$8 for great-circle motion, and $l^{-}$9 for turn-in-place. For $r^{+}$0, every time-optimal path belongs to a sufficient list of 23 path types, each with at most six segments from $r^{+}$1. The resulting spherical convexified Reeds-Shepp problem is an explicit three-mode optimal-control system in which switching between motion primitives is part of the optimal structure (Li et al., 1 Apr 2025).

Lifted and relaxed Reeds-Shepp geometries provide another axis of generalization. In image analysis, optimal paths are computed on $r^{+}$2 with data-driven Finsler metrics that penalize both longitudinal motion and orientation change, optionally without reverse gear. The paper explicitly frames such constructions as a natural setting for “multi-modal Reeds-Shepp curves,” because the costs $r^{+}$3 can be adapted to 2D retina, 3D diffusion MRI, and, in principle, several modalities simultaneously (Duits et al., 2016). In dynamic off-the-grid tracking, the problem is lifted to the roto-translation space $r^{+}$4, regularized by a relaxed Reeds-Shepp metric, and used to untangle crossing trajectories that are ambiguous in $r^{+}$5. The work derives properties of polygonal, Bézier, and piecewise-geodesic discretizations and proves a $r^{+}$6-convergence result for the discretized energies (Laville et al., 14 Jul 2025).

These variants show that multimodality can arise from steering hardware, from continuity constraints on curvature, from non-Euclidean manifold geometry, or from modality-dependent costs on lifted state spaces (Dai et al., 2024, Li et al., 1 Apr 2025, Duits et al., 2016, Laville et al., 14 Jul 2025).

6. Interpretation, limitations, and recurrent design principles

A common misconception is that multi-modal Reeds-Shepp curves refer to a single universally accepted model. The literature does not support that interpretation. The term is explicit in four-wheel independent steering, where it denotes a hybrid state $r^{+}$7 and per-mode Reeds-Shepp planning. In other settings, the same phrase is interpretive: it may denote a primitive library for Ackermann robots, a selector over classical path families, a fleet coupling parameter, or a geometry adapted to sensing modalities. This suggests that “multimodal” is best understood as a structural property of the path model rather than a fixed canonical formalism (Bao et al., 7 Sep 2025, Guo, 11 Jun 2025, Ibrahim et al., 8 Apr 2025, Duits et al., 2016).

A second misconception is that multimodality requires abandoning Reeds-Shepp structure. Recent work generally does the opposite. Per-mode or per-family Reeds-Shepp solutions are retained as local connectors or analytic primitives, while the higher-level planner adds a small number of switching mechanisms: mode-switch edges, focal search constraints, priority inheritance, or $r^{+}$8 joint metrics. In the 4WIS planner, no single closed-form distance is written for the full hybrid space; in the Ackermann multi-robot dissertation, the implicit lattice uses motion primitives and hybrid A* rather than a fully explicit augmented-state geodesic. Multimodality is therefore usually layered on top of classical Reeds-Shepp geometry rather than replacing it (Bao et al., 7 Sep 2025, Guo, 11 Jun 2025).

The main limitations are likewise mode-specific. The 4WIS formulation does not give a full Reeds-Shepp family for parallel mode, ignores obstacles during Reeds-Shepp computation, and is not globally optimal in continuous $r^{+}$9 (Bao et al., 7 Sep 2025). The multi-robot Ackermann methods guarantee kinematic feasibility by construction and collision avoidance at discrete time steps, but the dissertation does not provide a closed-form bound for the full multi-robot problem under Ackermann kinematics (Guo, 11 Jun 2025). The non-Euclidean sampling theory assumes metric-based cost, geodesic Extend, and static free space (Lukyanenko et al., 2021). The continuous-curvature construction is restricted to a special class of valid CC paths and zero-curvature boundary conditions (Dai et al., 2024). The spherical classification is stated for $(x,y,\theta,m)$00 (Li et al., 1 Apr 2025). The lifted imaging and trajectory-untangling formulations are relaxed or approximating geometries rather than exact classical Reeds-Shepp boundary-value solvers (Duits et al., 2016, Laville et al., 14 Jul 2025).

Across these formulations, three design principles recur. First, define a finite mode library whose elements retain clear kinematic meaning. Second, attach an analytic or efficiently computable local connector to each mode or path family. Third, make switching explicit, either as a geometric partition, a graph edge with penalty, or a joint metric coupling. That pattern underlies classical path-family partitioning, hybrid A* with steering modes, multi-robot primitive planning, fleet product metrics, continuous-curvature surrogates, and non-Euclidean generalizations alike (Ibrahim et al., 8 Apr 2025, Bao et al., 7 Sep 2025, Lukyanenko et al., 2021).