Relaxed Reeds–Shepp Metric Overview

• Relaxed Reeds–Shepp metric is a continuous anisotropic Riemannian/Finsler approximation of sub-Riemannian geometry that replaces hard nonholonomic constraints with steep penalty terms.
• It uses a relaxation parameter to penalize deviations from permissible directions, thereby transforming singular costs into a smooth metric structure amenable to viscosity solutions and fast marching methods.
• This regularized metric underpins dynamic curve regularization in imaging and robot motion planning, converging to classical Reeds–Shepp geometry as the penalty parameter approaches zero.

Searching arXiv for recent and foundational papers on relaxed Reeds–Shepp metrics and closely related formulations. The relaxed Reeds–Shepp metric is a continuous, highly anisotropic Riemannian or Finsler approximation of the classical Reeds–Shepp sub-Riemannian geometry on position–orientation spaces such as $\mathbb{R}^2\times \mathbb{S}^1$ or $\mathbb{R}^d\times \mathbb{S}^{d-1}$. Its defining purpose is to replace hard nonholonomic constraints—most notably the prohibition of sideways motion, and in some variants backward motion—by finite but very large penalties governed by a relaxation parameter $\varepsilon$. In the limit $\varepsilon\to 0$, the relaxed metric converges to the singular Reeds–Shepp geometry; for fixed $\varepsilon>0$, it yields a regular metric structure compatible with viscosity-solution theory, fast marching discretizations, and variational formulations in imaging and inverse problems (Duits et al., 2016, Laville et al., 14 Jul 2025).

1. Classical Reeds–Shepp geometry and the motivation for relaxation

The classical Reeds–Shepp car is posed on the position–orientation manifold

$$\mathbb{M}:=\mathbb{R}^d\times \mathbb{S}^{d-1}, \qquad d\in\{2,3\},$$

with state $p=(x,n)$, where $x\in\mathbb{R}^d$ is position and $n\in\mathbb{S}^{d-1}$ is orientation. Its kinematics enforce

$$\dot x(t)=u_1(t)\,n(t), \qquad \dot n(t)=u_2(t), \qquad u_2(t)\perp n(t),$$

equivalently

$\mathbb{R}^d\times \mathbb{S}^{d-1}$0

Thus, the vehicle can translate only along its current orientation and can change orientation, but cannot move sideways. This is encoded by the singular metric

$\mathbb{R}^d\times \mathbb{S}^{d-1}$1

Its forward-only counterpart additionally requires $\mathbb{R}^d\times \mathbb{S}^{d-1}$2, producing an asymmetric distance (Duits et al., 2016).

The associated path length is

$\mathbb{R}^d\times \mathbb{S}^{d-1}$3

and the induced distance is the infimum of this length over Lipschitz curves joining two configurations. Because forbidden directions have infinite cost, the resulting geometry is sub-Riemannian or sub-Finsler rather than globally Riemannian; the metric tensor exists only on the horizontal distribution $\mathbb{R}^d\times \mathbb{S}^{d-1}$4. Analytically, the eikonal equation is singular; numerically, infinite costs and degeneracies obstruct the construction of causal stencils for fast marching. The relaxed metric is introduced precisely to remove these obstructions while preserving the preferred directions of motion.

2. Canonical relaxed metrics on position–orientation manifolds

For $\mathbb{R}^d\times \mathbb{S}^{d-1}$5, the standard relaxed Reeds–Shepp metrics soften the nonholonomic constraints by penalizing, rather than forbidding, undesirable directions. Writing

$\mathbb{R}^d\times \mathbb{S}^{d-1}$6

and

$\mathbb{R}^d\times \mathbb{S}^{d-1}$7

the symmetric relaxed metric is

$\mathbb{R}^d\times \mathbb{S}^{d-1}$8

and the forward-only relaxed metric is

$\mathbb{R}^d\times \mathbb{S}^{d-1}$9

The orthogonal spatial component $\varepsilon$0 is therefore allowed but penalized by $\varepsilon$1. In the forward-only model, negative longitudinal motion is also penalized by $\varepsilon$2, whereas positive motion retains unit weight. Pointwise on $\varepsilon$3,

$\varepsilon$4

so the original Reeds–Shepp geometry is recovered in the singular limit (Duits et al., 2016).

In the two-dimensional roto-translation space

$\varepsilon$5

the same construction appears as a Riemannian metric tensor

$\varepsilon$6

where $\varepsilon$7 and $\varepsilon$8 is a curvature scaling parameter. Motion along $\varepsilon$9 is cheap, motion orthogonal to $\varepsilon\to 0$0 is expensive, and orientation change is penalized through $\varepsilon\to 0$1. The induced norm is

$\varepsilon\to 0$2

and this quantity is used directly as a curve regularizer in variational formulations (Laville et al., 14 Jul 2025).

Geometrically, these constructions retain the Reeds–Shepp coupling between translation and orientation while replacing infinite costs by stiff anisotropy. In the ideal regime of no lateral motion, the term $\varepsilon\to 0$3 acts as curvature penalization, since for tangent-aligned planar curves one has $\varepsilon\to 0$4.

3. Dual metrics, eikonal equations, geodesics, and convergence

Given a continuous Finsler metric $\varepsilon\to 0$5, its dual metric on the cotangent bundle is

$\varepsilon\to 0$6

For a source $\varepsilon\to 0$7, the distance map $\varepsilon\to 0$8 is the unique viscosity solution of

$\varepsilon\to 0$9

For the relaxed metrics, the duals are explicit: $\varepsilon>0$0 and

$\varepsilon>0$1

The relaxed Hamiltonians are therefore finite and strongly anisotropic rather than degenerate, which makes them amenable to eikonal solvers (Duits et al., 2016).

Fast marching implementations rely on stencils adapted to anisotropy and on a generalized acuteness property ensuring causality of local Hopf–Lax updates. For the symmetric Riemannian approximation $\varepsilon>0$2, lattice-basis–reduction-based stencils due to Mirebeau are used in 2D and 3D. For the asymmetric forward-only metric $\varepsilon>0$3 in 2D, stencils are constructed via the Stern–Brocot tree; in 3D, the Hamiltonian may be discretized directly using upwind finite differences aligned with certain vectors. These constructions permit distance-map computation on $\varepsilon>0$4 despite very large anisotropy.

Once the distance map is known, geodesics are recovered by characteristic backtracking. In the general Finsler setting,

$\varepsilon>0$5

where $\varepsilon>0$6. In the symmetric relaxed case this becomes an intrinsic gradient descent with respect to the Riemannian tensor $\varepsilon>0$7: $\varepsilon>0$8 For the forward-only model, backtracking is piecewise Riemannian: on the region $\varepsilon>0$9 it uses $$\mathbb{M}:=\mathbb{R}^d\times \mathbb{S}^{d-1}, \qquad d\in\{2,3\},$$0, and on $$\mathbb{M}:=\mathbb{R}^d\times \mathbb{S}^{d-1}, \qquad d\in\{2,3\},$$1 a more isotropic metric associated with in-place rotations. In the limit $$\mathbb{M}:=\mathbb{R}^d\times \mathbb{S}^{d-1}, \qquad d\in\{2,3\},$$2, these rotational segments become the “keypoints” that replace the spatial cusps of the reversible model.

The justification for calling these objects relaxed Reeds–Shepp metrics is their convergence behavior. For any $$\mathbb{M}:=\mathbb{R}^d\times \mathbb{S}^{d-1}, \qquad d\in\{2,3\},$$3,

$$\mathbb{M}:=\mathbb{R}^d\times \mathbb{S}^{d-1}, \qquad d\in\{2,3\},$$4

and, under uniqueness assumptions, minimizing geodesics converge uniformly to minimizing sub-Riemannian geodesics. In the forward-only case the limiting distance can be discontinuous at some points, so convergence is only pointwise; in the symmetric case it is locally uniform. This distinction is central: relaxation regularizes the PDE and the numerics, but it does not erase the asymmetry and discontinuity phenomena intrinsic to the no-reverse limit (Duits et al., 2016).

4. Roto-translation regularization in variational and inverse problems

A prominent use of the relaxed Reeds–Shepp metric is as a dynamic regularizer on the space of curves in $$\mathbb{M}:=\mathbb{R}^d\times \mathbb{S}^{d-1}, \qquad d\in\{2,3\},$$5. In dynamic off-the-grid reconstruction, a planar trajectory $$\mathbb{M}:=\mathbb{R}^d\times \mathbb{S}^{d-1}, \qquad d\in\{2,3\},$$6 is lifted to $$\mathbb{M}:=\mathbb{R}^d\times \mathbb{S}^{d-1}, \qquad d\in\{2,3\},$$7, where $$\mathbb{M}:=\mathbb{R}^d\times \mathbb{S}^{d-1}, \qquad d\in\{2,3\},$$8 represents its tangent orientation. The regularization term is then built from the relaxed Reeds–Shepp norm: $$\mathbb{M}:=\mathbb{R}^d\times \mathbb{S}^{d-1}, \qquad d\in\{2,3\},$$9 with $p=(x,n)$0 a Radon measure on curve space. The full variational problem takes the form

$p=(x,n)$1

Here $p=(x,n)$2 controls the overall regularization strength, $p=(x,n)$3 enforces planarity or alignment with orientation, and $p=(x,n)$4 penalizes local curvature (Laville et al., 14 Jul 2025).

The geometric role of the lifting is decisive. Two planar trajectories that cross at the same spatial point but with different directions are separated in $p=(x,n)$5 because their orientations differ. The relaxed metric then promotes trajectories that are short and smooth in position–orientation space, with limited lateral drift and controlled orientation change. This prevents the regularizer from preferring nearly touching non-crossing paths when the data contain genuine crossings. In the reported localization experiments, the orientation-aware model successfully untangles crossings that Euclidean Benamou–Brenier or WFR-type dynamic regularizers fail to recover.

For each fixed $p=(x,n)$6, the relaxed metric is smooth and Riemannian, so geodesics satisfy the standard geodesic ODE with Christoffel symbols computed from the metric tensor. This allows piecewise geodesic discretizations in addition to polygonal and Bézier approximations. The corresponding constrained energies

$p=(x,n)$7

$p=(x,n)$8-converge to the continuous energy $p=(x,n)$9 when the discrete families are dense and the approximation maps do not increase the curve weight. Consequently, polygonal, Bézier, and piecewise geodesic discretizations are all justified as consistent approximations of the continuous relaxed Reeds–Shepp functional.

Algorithmically, the minimization is handled by an adaptation of the Sliding Frank–Wolfe procedure, called the Unravelling Frank–Wolfe algorithm. The relaxed metric enters simultaneously through the energy, the admissible set $x\in\mathbb{R}^d$0, and the Riemannian geometry used for nonconvex refinement via exponential maps and geodesic ODE integration. The framework retains the sparse representer structure: the feasible set remains compact and convex in the weak-* topology, and extreme points are scaled Dirac masses on single curves.

The limitations are equally explicit. Performance depends sensitively on $x\in\mathbb{R}^d$1, $x\in\mathbb{R}^d$2, and $x\in\mathbb{R}^d$3; piecewise geodesic discretization can be up to two orders of magnitude more expensive than Bézier discretization; and strong curvature penalization can hinder untangling in highly curved regions. These are not incidental implementation issues but structural trade-offs induced by the relaxed geometry itself (Laville et al., 14 Jul 2025).

5. Related relaxations and generalizations

The phrase “relaxed Reeds–Shepp metric” does not denote a single construction across the literature. Several nearby notions relax different parts of the original model.

A first variant relaxes the specification of the terminal state rather than the sideways-motion constraint. In the under-specified Reeds–Shepp problem, the final orientation is free, and the induced cost is

$x\in\mathbb{R}^d$4

This is a position-only cost-to-come derived from the classical Reeds–Shepp metric on $x\in\mathbb{R}^d$5. The optimal terminal heading is characterized by boundary cases of the standard Reeds–Shepp synthesis, typically where one segment collapses to zero length, and the resulting distance can be precomputed as a grid-based transform (Ibrahim et al., 8 Apr 2025).

A second variant imposes continuous curvature. The augmented state is $x\in\mathbb{R}^d$6, with dynamics

$x\in\mathbb{R}^d$7

subject to $x\in\mathbb{R}^d$8 and $x\in\mathbb{R}^d$9. When $n\in\mathbb{S}^{d-1}$0, the shortest continuous-curvature steering problem reduces to Reeds–Shepp steering. The admissible paths are built from clothoid turns, circles, lines, and cusps, and the special class developed in the parking-planning setting admits the same driving patterns as Reeds–Shepp paths but consists of cusp-free clothoid turns. This is a relaxation of curvature discontinuities rather than of sub-Riemannian singularity (Dai et al., 2024).

A third variant is convexified and spherical. On $n\in\mathbb{S}^{d-1}$1, representing a vehicle moving on the unit sphere with forward/backward capability and bounded turning rate, the speed control is relaxed from $n\in\mathbb{S}^{d-1}$2 to $n\in\mathbb{S}^{d-1}$3. The cost is pure travel time,

$n\in\mathbb{S}^{d-1}$4

and the induced distance

$n\in\mathbb{S}^{d-1}$5

defines a convexified or relaxed Reeds–Shepp metric on $n\in\mathbb{S}^{d-1}$6. Convexification introduces turn-in-place motions $n\in\mathbb{S}^{d-1}$7 and yields optimal paths composed of segments from $n\in\mathbb{S}^{d-1}$8. For $n\in\mathbb{S}^{d-1}$9, the time-optimal paths belong to a sufficient list of 23 path types, each with at most 6 segments (Li et al., 1 Apr 2025).

A fourth use of the terminology appears in terrain-aware path planning. There the cost of a primitive $$\dot x(t)=u_1(t)\,n(t), \qquad \dot n(t)=u_2(t), \qquad u_2(t)\perp n(t),$$0 is

$$\dot x(t)=u_1(t)\,n(t), \qquad \dot n(t)=u_2(t), \qquad u_2(t)\perp n(t),$$1

evaluated along exact Dubins, Reeds–Shepp, or bicycle primitives on a lattice. The paper explicitly states that the authors use “metric” in the broad sense of a state–cost function, not in the strict mathematical sense of a metric on configuration space: non-negativity and identity hold, but symmetry is not guaranteed, and triangle inequality is not analyzed. This is best viewed as a generalized Reeds–Shepp cost functional rather than a strict distance (Naik et al., 14 Oct 2025).

6. Analytical interpretations, misconceptions, and broader significance

A recurring misconception is to identify the relaxed Reeds–Shepp metric with the exact Reeds–Shepp distance. The exact model is singular and sub-Riemannian or sub-Finsler; the relaxed model is finite, continuous, and numerically tractable. The two coincide only in the limit $$\dot x(t)=u_1(t)\,n(t), \qquad \dot n(t)=u_2(t), \qquad u_2(t)\perp n(t),$$2, and even then the forward-only limit may be discontinuous. Another misconception is that “metric” always means a symmetric distance on configuration space. In forward-only models the cost is asymmetric, and in terrain-weighted formulations the term may refer only to a state–cost function (Duits et al., 2016, Naik et al., 14 Oct 2025).

A complementary analytical perspective arises from planar norm decompositions. Any norm on $$\dot x(t)=u_1(t)\,n(t), \qquad \dot n(t)=u_2(t), \qquad u_2(t)\perp n(t),$$3 admits an explicit additive decomposition

$$\dot x(t)=u_1(t)\,n(t), \qquad \dot n(t)=u_2(t), \qquad u_2(t)\perp n(t),$$4

equivalently an explicit isometric embedding into $$\dot x(t)=u_1(t)\,n(t), \qquad \dot n(t)=u_2(t), \qquad u_2(t)\perp n(t),$$5: $$\dot x(t)=u_1(t)\,n(t), \qquad \dot n(t)=u_2(t), \qquad u_2(t)\perp n(t),$$6 Using this representation, the Buja–Logan–Reeds–Shepp inequality extends from the Euclidean norm to any norm on a two-dimensional space: $$\dot x(t)=u_1(t)\,n(t), \qquad \dot n(t)=u_2(t), \qquad u_2(t)\perp n(t),$$7 for iid random vectors $$\dot x(t)=u_1(t)\,n(t), \qquad \dot n(t)=u_2(t), \qquad u_2(t)\perp n(t),$$8. The paper itself does not define a relaxed Reeds–Shepp metric, but it provides a way to represent any planar norm as an $$\dot x(t)=u_1(t)\,n(t), \qquad \dot n(t)=u_2(t), \qquad u_2(t)\perp n(t),$$9-integral of absolute linear functionals, which is naturally applicable when a relaxed model uses a planar norm as an effective motion cost (Pinelis, 2015).

This suggests that “relaxed Reeds–Shepp metric” is best understood as a family of geometric regularizations that preserve the coupling of translation, orientation, and curvature while replacing exact nonholonomic constraints by analytically manageable surrogates. In image analysis and sub-Riemannian numerics, the canonical surrogate is the $\mathbb{R}^d\times \mathbb{S}^{d-1}$00-anisotropic Riemannian/Finsler metric on $\mathbb{R}^d\times \mathbb{S}^{d-1}$01. In inverse problems, it is a curve-space regularizer on $\mathbb{R}^d\times \mathbb{S}^{d-1}$02. In robot motion planning, it may instead mean convexification of speed, continuous-curvature refinement, or relaxation of terminal orientation. Across these settings, the unifying feature is not a single formula but a common strategy: preserve Reeds–Shepp kinematics at the structural level, relax the singular or overly rigid part of the model, and recover the original geometry as a limiting case.