Frenet Vehicle Kinematics

Basic terminology and definition of the Frenet state

This article considers only vehicle kinematics: it describes how position and heading change with motion, without constructing a dynamic model of tire forces, load transfer, vehicle mass, or similar effects. The vehicle moves in a plane, and the center of its rear axle is used as the position reference point.

A state here is a set of variables that describes the vehicle’s geometric position and orientation relative to the reference path and evolves with an independent variable. An input is supplied from outside the model and determines the state derivatives. A model whose independent variable is time $t$ is called a time-domain model. A model whose independent variable is the reference-path arc length $s$ is called a reference-path-arc-length-domain model, or, more briefly, a spatial-domain model.

The reference path is a $C^2$ planar curve parameterized by arc length $s$:

$$\mathbf r:I\longrightarrow\mathbb R^2, \qquad s\longmapsto\mathbf r(s).$$

Here, $I$ is the range of the arc-length parameter $s$. Give the Cartesian plane its standard orientation and let $J$ denote counterclockwise rotation by $90^\circ$. The reference path’s unit tangent, left unit normal, signed curvature, and continuously unwrapped heading angle are denoted respectively by $\mathbf T$, $\mathbf N$, $\kappa_r$, and $\theta_r$. The following planar signed Frenet relations are used:

$$\begin{aligned} \mathbf T(s) &= \frac{\mathrm{d}\mathbf r}{\mathrm{d}s},& \mathbf N(s) &= J\mathbf T(s),\\[0.6em] \frac{\mathrm{d}\mathbf T}{\mathrm{d}s} &= \kappa_r\mathbf N,& \frac{\mathrm{d}\mathbf N}{\mathrm{d}s} &= -\kappa_r\mathbf T,\\[0.6em] \kappa_r &= \frac{\mathrm{d}\theta_r}{\mathrm{d}s}. \end{aligned}$$

The curvature sign follows the left-normal convention: when traversing the reference path in the direction of increasing $s$, a left turn has $\kappa_r>0$ and a right turn has $\kappa_r<0$. The complete construction of arc length, frames, standard orientation, and these formulas appears in Planar Frenet Frame. This article uses that construction as the geometric foundation of the vehicle kinematics.

Let $\mathbf z(t)$ denote the position vector of the rear-axle center in a fixed planar coordinate system, and let $\theta(t)$ be the vehicle heading angle. Define the corresponding heading unit vector by

$$\mathbf e_\theta(t):= \begin{bmatrix} \cos\theta(t)\\ \sin\theta(t) \end{bmatrix}.$$

Let $v(t)$ be the signed longitudinal velocity, positive in the $\mathbf e_\theta$ direction. No sideslip means that the instantaneous velocity of the rear-axle center has no component perpendicular to the vehicle’s longitudinal axis:

$$\dot{\mathbf z}=v\mathbf e_\theta.$$

Thus, $v>0$ when driving forward, $v<0$ when reversing, and $v=0$ when stationary. The vector $\mathbf e_\theta$ always describes the vehicle-body orientation; the actual direction of velocity while the vehicle is moving also depends on the sign of $v$.

Let $\ell$ be the oriented arc-length coordinate of the rear-axle-center trajectory. Given an initial value $\ell(t_0)$, define

$$\ell(t)=\ell(t_0)+\int_{t_0}^{t}v(\tau)\,\mathrm{d}\tau,$$

so that

$$\frac{\mathrm{d}\ell}{\mathrm{d}t}=v.$$

The value of $\ell$ increases while driving forward and decreases while reversing. It is not the monotonically increasing cumulative distance, whose increment would instead be computed from $|v|\,\mathrm{d}t$. On a trajectory segment where $v\ne0$, $\ell$ can serve as a local parameter. The chain rule then gives $\mathrm{d}\mathbf z/\mathrm{d}\ell=\mathbf e_\theta$, so the signed curvature of the vehicle trajectory under this orientation is defined by

$$\kappa_v := \frac{\mathrm{d}\theta}{\mathrm{d}\ell}.$$

For a given vehicle position, the closest-point projection finds the point on the reference path that minimizes the distance to the rear-axle center. This article assumes that the projection is unique and lies in the interior of the reference-path parameter interval, so the line from the projection point to the rear-axle center is perpendicular to the path. Denote the projection point’s arc-length coordinate by $s(t)$. Let $d(t)$ be the signed distance from that point to the vehicle along the left unit normal $\mathbf N$: $d>0$ places the vehicle to the left of the increasing-$s$ direction, and $d<0$ places it to the right. Define the difference between the vehicle heading and the reference heading at the projection point by

$$\varphi(t) := \theta(t)-\theta_r\bigl(s(t)\bigr).$$

Both $\theta$ and $\theta_r$ are continuously unwrapped along the trajectory segment under consideration, preventing an artificial jump in $\varphi$ when an angle representation crosses the $\pm\pi$ boundary. The time-domain model uses time $t$ as its independent variable and $(s,d,\varphi)$ as its state. The reference-path-arc-length-domain, or spatial-domain, model instead uses $s$ as its independent variable and therefore requires only $(d,\varphi)$ as its state. Figure 1 shows the geometric relationship among these quantities. The vectors $\mathbf T(s)$ and $\mathbf N(s)$ are the Frenet basis vectors at the projection point; the purple bidirectional dimension line is the lateral deviation $d$ from the reference path to the rear-axle center; and the arrow $\mathbf e_\theta$ at the rear-axle center is the vehicle heading unit vector.

Throughout this article, upright $\mathrm{d}$ denotes the differential symbol, distinct from the italic lateral-deviation variable $d$; $\dot d$ denotes the time derivative of that deviation.

T(s)N(s)e_θd

‖T‖ = ‖N‖ = 1

Time-domain Frenet kinematics

Position-coordinate kinematics

From the normal-coordinate representation in Figure 1, the vehicle position is

$$\mathbf z(t) = \mathbf r\bigl(s(t)\bigr) + d(t)\mathbf N\bigl(s(t)\bigr). \tag{1}$$

Differentiate Equation (1) with respect to time $t$, using $\mathrm{d}\mathbf r/\mathrm{d}s=\mathbf T$ and $\mathrm{d}\mathbf N/\mathrm{d}s=-\kappa_r\mathbf T$:

$$\begin{aligned} \dot{\mathbf z} &= \frac{\mathrm{d}\mathbf r}{\mathrm{d}s}\dot s + \dot d\,\mathbf N + d\,\frac{\mathrm{d}\mathbf N}{\mathrm{d}s}\dot s\\[0.6em] &= \mathbf T\dot s + \dot d\,\mathbf N - d\kappa_r\mathbf T\dot s\\[0.6em] &= (1-d\kappa_r)\dot s\,\mathbf T + \dot d\,\mathbf N. \end{aligned}$$

Here and below, any $\mathbf T$, $\mathbf N$, and $\kappa_r$ without an explicit argument are evaluated at the projection point $s(t)$. The no-sideslip assumption gives $\dot{\mathbf z}=v\mathbf e_\theta$. This relation remains valid for both $v>0$ while driving forward and $v<0$ while reversing. Since $(\mathbf T,\mathbf N)$ is an orthonormal basis, $\mathbf e_\theta$ can be expanded by its orthogonal projections onto the two basis directions:

$$\mathbf e_\theta = \langle\mathbf e_\theta,\mathbf T\rangle\mathbf T + \langle\mathbf e_\theta,\mathbf N\rangle\mathbf N.$$

Substituting

$$\mathbf T = \begin{bmatrix} \cos\theta_r\\ \sin\theta_r \end{bmatrix}, \qquad \mathbf N = \begin{bmatrix} -\sin\theta_r\\ \cos\theta_r \end{bmatrix}$$

into the projection coefficients gives

$$\begin{aligned} \langle\mathbf e_\theta,\mathbf T\rangle &= \cos\theta\cos\theta_r + \sin\theta\sin\theta_r\\ &= \cos(\theta-\theta_r) = \cos\varphi,\\[0.7em] \langle\mathbf e_\theta,\mathbf N\rangle &= -\cos\theta\sin\theta_r + \sin\theta\cos\theta_r\\ &= \sin(\theta-\theta_r) = \sin\varphi. \end{aligned}$$

Therefore, the vehicle heading unit vector has the following representation in the Frenet frame:

$$\mathbf e_\theta = \cos\varphi\,\mathbf T + \sin\varphi\,\mathbf N.$$

The vehicle velocity is consequently

$$\dot{\mathbf z} = v\mathbf e_\theta = v\cos\varphi\,\mathbf T + v\sin\varphi\,\mathbf N.$$

Using the same Frenet frame and the same resultant velocity vector, Figure 2 separately shows the two components obtained by differentiating the position coordinates and by expressing the no-sideslip velocity. Placing them side by side makes the subsequent coefficient comparison a direct geometric correspondence.

(a)

$\dot{\mathbf z}$$(1-d\kappa_r)\dot{s}\,\mathbf T$$\dot{d}\,\mathbf N$

(b)

v e_θv cosφ Tv sinφ N

Equating the coefficients of the same velocity vector in the basis $(\mathbf T,\mathbf N)$ gives

$$(1-d\kappa_r)\dot s = v\cos\varphi, \qquad \dot d = v\sin\varphi.$$

When $1-d\kappa_r\ne0$, the time-domain position kinematics are

$$\dot s = \frac{v\cos\varphi}{1-d\kappa_r}, \qquad \dot d = v\sin\varphi.$$

The normal coordinates and velocity decomposition of Equation (1) appear in [1, §2.1–2.2, Eqs. (14)–(20)], and an equivalent position representation appears in [2, §3.1, Eq. (1)]. The longitudinal and lateral state equations under the same coordinate convention appear in [3, §4.3, Eqs. (4.33a)–(4.33b)].

Heading-error kinematics

The heading error is defined by

$$\varphi(t) = \theta(t) - \theta_r\bigl(s(t)\bigr).$$

Apply the chain rule with respect to time and substitute $\kappa_r=\mathrm{d}\theta_r/\mathrm{d}s$:

$$\begin{aligned} \dot\varphi &= \dot\theta - \frac{\mathrm{d}\theta_r}{\mathrm{d}s}\dot s\\[0.6em] &= \dot\theta - \kappa_r\dot s. \end{aligned}$$

On an interval where $v\ne0$, using $\kappa_v=\mathrm{d}\theta/\mathrm{d}\ell$ and $\mathrm{d}\ell/\mathrm{d}t=v$ gives

$$\begin{aligned} \dot\theta &= \frac{\mathrm{d}\theta}{\mathrm{d}\ell}\frac{\mathrm{d}\ell}{\mathrm{d}t}\\[0.6em] &= v\kappa_v. \end{aligned}$$

When $v=0$, $\ell$ cannot locally replace $t$ as a parameter at that instant. If the curvature input $\kappa_v$ is bounded, however, the kinematic relation $\dot\theta=v\kappa_v$ extends continuously to the stationary state, where $\dot\theta=0$.

The heading-error kinematics are therefore

$$\dot\varphi = v\kappa_v - \kappa_r\dot s.$$

Three-state time-domain model

Combining the heading-error kinematics with the position-coordinate kinematics gives the three-state time-domain model for $(s,d,\varphi)$:

$$\begin{aligned} \dot s &= \frac{v\cos\varphi}{1-d\kappa_r},\\[0.65em] \dot d &= v\sin\varphi,\\[0.65em] \dot\varphi &= v\kappa_v - \dot s\,\kappa_r\\[0.65em] &= v\left( \kappa_v-\frac{\kappa_r\cos\varphi}{1-d\kappa_r} \right). \end{aligned} \tag{2}$$

Here, $\kappa_r=\kappa_r(s)$ is determined by the reference path. The signed longitudinal velocity $v$ and vehicle-trajectory curvature $\kappa_v$ are treated as inputs to the kinematic model. In a particular steering model, $\kappa_v$ is normally determined in turn by the steering geometry. Model validity requires both a unique closest-point projection and $1-d\kappa_r\ne0$. The former gives the vehicle position a unique $(s,d)$ representation, while the latter is the local regularity condition that prevents the normal-coordinate map from degenerating in the tangential direction. The region satisfying both conditions is called the Frenet-coordinate validity domain here. A standard Frenet vehicle model corresponding to Equation (2) appears in [3, §4.3, Eq. (4.41)].

Reversing in the time-domain model

Equation (2) applies to reversing without a separate derivation. The time-domain derivation already defines $v$ as the signed longitudinal velocity positive along the heading unit vector $\mathbf e_\theta$ and uses the same no-sideslip kinematics:

$$\dot{\mathbf z}=v\mathbf e_\theta, \qquad \dot\theta=v\kappa_v.$$

Neither relation requires $v>0$. When driving forward, $v>0$ and the velocity vector $\dot{\mathbf z}$ points in the same direction as $\mathbf e_\theta$. When reversing, $v<0$ and $\dot{\mathbf z}$ points in the opposite direction. Differentiating the position-coordinate relation (1) with respect to time, comparing components in the Frenet frame, and applying the chain rule to the heading error are all independent of the sign of $v$. The same three-state model therefore remains valid in reverse.

Let $\operatorname{sgn}(x)$ denote the sign function, whose value is respectively $1$, $0$, or $-1$ when $x>0$, $x=0$, or $x<0$. From Equation (2), the direction of change of the reference-path arc-length coordinate generally satisfies

$$\operatorname{sgn}(\dot s)= \operatorname{sgn}(v) \operatorname{sgn}\!\left(\frac{\cos\varphi}{1-d\kappa_r}\right).$$

In the usual Frenet tracking region where $1-d\kappa_r>0$ and $|\varphi|<\pi/2$, forward motion has $\dot s>0$ and reversing has $\dot s<0$. More generally, the sign must be determined from the equation above; “reversing” alone does not determine the sign of $\dot s$ independently of $\varphi$ and $1-d\kappa_r$.

For a strict comparison, fix the same state $(s,d,\varphi)$, the same vehicle curvature $\kappa_v$, and the same speed magnitude $u>0$, changing only the longitudinal velocity from forward $v=u$ to reverse $v=-u$:

Quantity	Forward: $v=u$	Reverse: $v=-u$	Definition or model changed?
Vehicle velocity $\dot{\mathbf z}$	$u\mathbf e_\theta$	$-u\mathbf e_\theta$	No-sideslip relation unchanged
Oriented arc-length rate $\mathrm{d}\ell/\mathrm{d}t$	$u$	$-u$	Always $\mathrm{d}\ell/\mathrm{d}t=v$
Reference-path coordinate rate $\dot s$	$\dfrac{u\cos\varphi}{1-d\kappa_r}$	$-\dfrac{u\cos\varphi}{1-d\kappa_r}$	Determined by the same state equation
Lateral-deviation rate $\dot d$	$u\sin\varphi$	$-u\sin\varphi$	Definition of $d$ unchanged
Heading rate $\dot\theta$	$u\kappa_v$	$-u\kappa_v$	$\kappa_v$ is not artificially negated
Heading-error rate $\dot\varphi$	$u\left(\kappa_v-\dfrac{\kappa_r\cos\varphi}{1-d\kappa_r}\right)$	$-u\left(\kappa_v-\dfrac{\kappa_r\cos\varphi}{1-d\kappa_r}\right)$	Definition of $\varphi$ unchanged
Reference curvature $\kappa_r$ and vehicle curvature $\kappa_v$	Existing sign conventions	Same sign conventions	Neither is modified
Equation (2)	Original equations	Original equations	Completely unchanged

Under these comparison conditions, changing only the sign of $v$ reverses all three state derivatives in Equation (2). Time $t$ continues to advance in its positive direction. Unlike reversing in the reference-path-arc-length-domain model, time-domain reversing is represented by $v<0$, not by integrating time backward. When stationary, $v=0$, and Equation (2) gives $\dot s=\dot d=\dot\varphi=0$.

Reference-path-arc-length-domain Frenet kinematics

This section constructs the model directly from the arc-length-parameterized position-coordinate relation and heading-error relation. It neither invokes the time-domain state equations nor uses time derivatives.

Oriented arc-length coordinate of the vehicle trajectory

The variable $s$ is the arc-length coordinate of the projection point on the reference path. The variable $\ell$ is the oriented arc-length coordinate of the rear-axle-center trajectory, positive along the vehicle heading unit vector $\mathbf e_\theta$. On a nonstationary trajectory segment where the Frenet coordinates are valid, $s$ and $d$ can both be treated as functions of $\ell$, so Equation (1) becomes

$$\mathbf z(\ell) = \mathbf r\bigl(s(\ell)\bigr) + d(\ell)\mathbf N\bigl(s(\ell)\bigr).$$

Differentiate directly with respect to $\ell$:

$$\begin{aligned} \frac{\mathrm{d}\mathbf z}{\mathrm{d}\ell} &= \frac{\mathrm{d}\mathbf r}{\mathrm{d}s}\frac{\mathrm{d}s}{\mathrm{d}\ell} + \frac{\mathrm{d}\,d}{\mathrm{d}\ell}\mathbf N + d\frac{\mathrm{d}\mathbf N}{\mathrm{d}s}\frac{\mathrm{d}s}{\mathrm{d}\ell}\\[0.8em] &= \mathbf T\frac{\mathrm{d}s}{\mathrm{d}\ell} + \frac{\mathrm{d}\,d}{\mathrm{d}\ell}\mathbf N + d\frac{\mathrm{d}\mathbf N}{\mathrm{d}s}\frac{\mathrm{d}s}{\mathrm{d}\ell}. \end{aligned}$$

From the Frenet relation $\mathrm{d}\mathbf N/\mathrm{d}s=-\kappa_r\mathbf T$ and $\langle\mathbf T,\mathbf N\rangle=0$,

$$\left\langle \frac{\mathrm{d}\mathbf N}{\mathrm{d}s}, \mathbf N \right\rangle =-\kappa_r\langle\mathbf T,\mathbf N\rangle=0.$$

Project the position derivative onto $\mathbf N$ and use $\langle\mathbf T,\mathbf N\rangle=0$ and $\langle\mathbf N,\mathbf N\rangle=1$:

$$\left\langle \frac{\mathrm{d}\mathbf z}{\mathrm{d}\ell}, \mathbf N \right\rangle = \frac{\mathrm{d}\,d}{\mathrm{d}\ell}.$$

By the orientation of $\ell$ and the no-sideslip constraint, the unit tangent of the vehicle trajectory with respect to $\ell$ is the heading unit vector:

$$\frac{\mathrm{d}\mathbf z}{\mathrm{d}\ell} = \mathbf e_\theta = \cos\varphi\,\mathbf T + \sin\varphi\,\mathbf N.$$

Its normal component is

$$\left\langle \frac{\mathrm{d}\mathbf z}{\mathrm{d}\ell}, \mathbf N \right\rangle = \sin\varphi.$$

Equating the same normal component gives

$$\frac{\mathrm{d}\,d}{\mathrm{d}\ell} = \sin\varphi = \sin(\theta-\theta_r).$$

This result follows directly from the position relation parameterized by $\ell$ and is not inferred backward from time derivatives. Reference-path curvature does not appear explicitly because the projection of $\mathrm{d}\mathbf N/\mathrm{d}s=-\kappa_r\mathbf T$ onto the $\mathbf N$ direction is zero. General regularity of the closest-point map within a tubular neighborhood is discussed in [4, §2.1, Eq. (4), Proposition 2 and Theorem 3].

Local relationship between reference-path and vehicle-trajectory arc lengths

Substituting $\mathrm{d}\mathbf N/\mathrm{d}s=-\kappa_r\mathbf T$ into the position derivative from the previous subsection gives

$$\frac{\mathrm{d}\mathbf z}{\mathrm{d}\ell} = (1-d\kappa_r)\frac{\mathrm{d}s}{\mathrm{d}\ell}\mathbf T + \frac{\mathrm{d}\,d}{\mathrm{d}\ell}\mathbf N.$$

At the same time, the vehicle-trajectory unit tangent with respect to oriented arc length $\ell$ is

$$\frac{\mathrm{d}\mathbf z}{\mathrm{d}\ell} = \cos\varphi\,\mathbf T + \sin\varphi\,\mathbf N.$$

Equating tangential components gives

$$(1-d\kappa_r)\frac{\mathrm{d}s}{\mathrm{d}\ell} = \cos\varphi.$$

When $1-d\kappa_r\ne0$ and $\cos\varphi\ne0$, this equation gives $\mathrm{d}s/\mathrm{d}\ell\ne0$, so $s$ and $\ell$ are locally interchangeable parameters. “Locally interchangeable” means that, near the current point, the same vehicle-trajectory segment can be described using either $s$ or $\ell$, with a differentiable local inverse transformation between them. Their local arc-length relationship is

$$\frac{\mathrm{d}\ell}{\mathrm{d}s} = \frac{1-d\kappa_r}{\cos\varphi}. \tag{3}$$

The inverse relation is

$$\frac{\mathrm{d}s}{\mathrm{d}\ell} = \frac{\cos\varphi}{1-d\kappa_r}.$$

For an interactive examination of how $d$, $\kappa_r$, and $\varphi$ jointly affect the local arc-length relationship in Equation (3), together with the Frenet geometric configurations corresponding to degeneration of the coordinate map and failure of the parameterization, see Frenet Arc-Length Conversion.

Lateral-deviation equation

On an interval where $\mathrm{d}s/\mathrm{d}\ell\ne0$, the chain rule gives

$$\frac{\mathrm{d}\,d}{\mathrm{d}\ell} = \frac{\mathrm{d}\,d}{\mathrm{d}s}\frac{\mathrm{d}s}{\mathrm{d}\ell}.$$

Substituting $\mathrm{d}\,d/\mathrm{d}\ell=\sin\varphi$ and the inverse of Equation (3) yields

$$\begin{aligned} \frac{\mathrm{d}\,d}{\mathrm{d}s} &= \frac{\sin\varphi} {\cos\varphi/(1-d\kappa_r)}\\[0.6em] &= (1-d\kappa_r)\tan\varphi. \end{aligned}$$

The relation $\mathrm{d}\,d/\mathrm{d}\ell=\sin\varphi$ contains no explicit reference-path curvature because it describes normal displacement per unit oriented vehicle-trajectory arc length. By contrast, $\mathrm{d}\,d/\mathrm{d}s$ explicitly contains $\kappa_r$, because changing the independent variable to the reference-path arc-length coordinate requires the inverse of Equation (3) for the scale conversion.

Heading-error equation

Under the conditions for Equation (3), $s$ and $\ell$ are locally interchangeable parameters. Along the same vehicle trajectory, write the heading error directly as

$$\varphi(s)=\theta\bigl(\ell(s)\bigr)-\theta_r(s).$$

Differentiate with respect to reference-path arc length $s$, using $\kappa_v=\mathrm{d}\theta/\mathrm{d}\ell$, $\kappa_r=\mathrm{d}\theta_r/\mathrm{d}s$, and Equation (3):

$$\begin{aligned} \frac{\mathrm{d}\varphi}{\mathrm{d}s} &= \frac{\mathrm{d}\theta}{\mathrm{d}\ell}\frac{\mathrm{d}\ell}{\mathrm{d}s}-\frac{\mathrm{d}\theta_r}{\mathrm{d}s}\\[0.65em] &= \kappa_v\frac{1-d\kappa_r}{\cos\varphi}-\kappa_r. \end{aligned}$$

The first term is the rate of change of vehicle heading per unit reference-path arc length, while the second term, $\kappa_r=\mathrm{d}\theta_r/\mathrm{d}s$, is the rate of change of reference heading under the same independent variable.

Two-state reference-path-arc-length-domain model

Combining the lateral-deviation and heading-error equations gives the spatial-domain model with independent variable $s$ and state $(d,\varphi)$:

$$\begin{aligned} \frac{\mathrm{d}\,d}{\mathrm{d}s} &= (1-d\kappa_r)\tan\varphi,\\[0.7em] \frac{\mathrm{d}\varphi}{\mathrm{d}s} &= \frac{(1-d\kappa_r)\kappa_v}{\cos\varphi}-\kappa_r. \end{aligned} \tag{4}$$

Here, $\kappa_r=\kappa_r(s)$ is determined by the reference path, and $\kappa_v$ is the vehicle-trajectory curvature input. Equation (4) follows entirely from arc-length-parameterized geometric relations and does not depend on the time-domain model. In addition to a unique closest-point projection and $1-d\kappa_r\ne0$, it requires $\cos\varphi\ne0$. Otherwise, $\mathrm{d}s/\mathrm{d}\ell=0$, and the reference-path arc-length coordinate $s$ cannot serve as a local independent variable of the vehicle trajectory at that point.

Reversing in the arc-length-domain model

Equation (4) also applies to reversing without a separate derivation. The reason is that the preceding derivation uses the oriented arc-length coordinate $\ell$, positive along the vehicle heading unit vector, and relies only on the no-sideslip geometric differential relations

$$\mathrm{d}\mathbf z=\mathbf e_\theta\,\mathrm{d}\ell, \qquad \mathrm{d}\theta=\kappa_v\,\mathrm{d}\ell.$$

Neither relation requires $\mathrm{d}\ell>0$. For forward motion, $\mathrm{d}\ell>0$ and the displacement $\mathrm{d}\mathbf z$ points with $\mathbf e_\theta$. For reverse motion, $\mathrm{d}\ell<0$ and $\mathrm{d}\mathbf z$ points against $\mathbf e_\theta$, while $\mathrm{d}\mathbf z/\mathrm{d}\ell=\mathbf e_\theta$ remains true. The component comparison of the position differential, arc-length conversion, and chain rule are therefore unchanged in reverse, yielding the same two-state model.

The inverse of Equation (3) gives

$$\frac{\mathrm{d}s}{\mathrm{d}\ell}=\frac{\cos\varphi}{1-d\kappa_r}.$$

In general, the signs of $\mathrm{d}s$ and $\mathrm{d}\ell$ are related by

$$\operatorname{sgn}(\mathrm{d}s)= \operatorname{sgn}(\mathrm{d}\ell) \operatorname{sgn}\!\left(\frac{\cos\varphi}{1-d\kappa_r}\right).$$

In the usual Frenet tracking region where $1-d\kappa_r>0$ and $|\varphi|<\pi/2$, $\mathrm{d}s/\mathrm{d}\ell>0$, so forward motion has $\mathrm{d}s>0$ and reverse motion has $\mathrm{d}s<0$. Given the same vehicle-curvature function $\kappa_v(s)$ and the same initial state, the same spatial-domain equations determine the same $(d,\varphi)$ state curve for forward and reverse motion; the curve is simply traversed in opposite directions of $s$. The right-hand side of Equation (4) requires no modification.

Within this usual tracking region, while keeping the same state, initial condition, and curvature input, the quantities correspond as follows:

Quantity	Forward	Reverse	Definition or model changed?
Oriented vehicle-trajectory arc-length increment $\mathrm{d}\ell$	$\mathrm{d}\ell>0$	$\mathrm{d}\ell<0$	Definition unchanged
Vehicle displacement $\mathrm{d}\mathbf z=\mathbf e_\theta \mathrm{d}\ell$	Same direction as $\mathbf e_\theta$	Opposite direction to $\mathbf e_\theta$	Geometric relation unchanged
Reference-path arc-length increment $\mathrm{d}s$	$\mathrm{d}s>0$	$\mathrm{d}s<0$	Determined by Equation (3)
Heading increment $\mathrm{d}\theta=\kappa_v \mathrm{d}\ell$	Accumulates along positive arc length	Increment direction reverses for the same $\kappa_v$	$\kappa_v$ is not artificially negated
States $d$, $\varphi$	Evolve forward in $s$	Evolve backward in $s$	State definitions unchanged and not pre-negated
Reference curvature $\kappa_r$ and vehicle curvature $\kappa_v$	Existing sign conventions	Same sign conventions	Neither is modified
Two right-hand sides of Equation (4)	Original form	Original form	Completely unchanged

Accordingly, when solving directly in $s$ within this usual tracking region, forward motion integrates from a smaller $s$ to a larger $s$, while reverse motion integrates from a larger $s$ to a smaller $s$. On a reversing segment beginning at $s_{\mathrm{start}}$ with monotonically decreasing $s$, a numerical implementation may define an increasing progress variable $\sigma=s_{\mathrm{start}}-s$. Since $\mathrm{d}/\mathrm{d}\sigma=-\mathrm{d}/\mathrm{d}s$, both state-equation right-hand sides then acquire an overall factor of $-1$. This is a consequence of changing the independent variable, not a modification of Equation (4) caused by reversing.

References

1. Jan Eilbrecht and Olaf Stursberg,

   Challenges of Trajectory Planning with Integrator Models on Curved Roads

   , IFAC-PapersOnLine, 53(2), 2020, doi: 10.1016/j.ifacol.2020.12.2492, §2.1–2.2, Eqs. (14)–(20). This reference is used to verify the Frenet position coordinates, velocity decomposition, and time-domain state equations.

2. Keisuke Yoneda et al.,

   Trajectory Optimization and State Selection for Urban Automated Driving

   , Artificial Life and Robotics, 23, 474–480, 2018, §3.1, Eq. (1). This reference is used to verify the normal-coordinate position representation $\mathbf z=\mathbf r(s)+d\mathbf N(s)$.

3. Oskar Ljungqvist,

   Motion Planning and Feedback Control Techniques with Applications to Long Tractor-Trailer Vehicles

   , Linköping Studies in Science and Technology, Dissertation No. 2070, Linköping University, 2020, doi: 10.3384/diss.diva-165246, §4.3. This reference is used to verify the regular domain of Frenet coordinates and the three-state vehicle-kinematics model.

4. Catherine Kublik and Richard Tsai,

   Integration over Curves and Surfaces Defined by the Closest Point Mapping

   , Research in the Mathematical Sciences, 3, Article 3, 2016, doi: 10.1186/s40687-016-0053-1, §2.1, Eq. (4), Proposition 2 and Theorem 3. This reference supplies the general regularity background for the closest-point map and its tubular neighborhood.