DE102021112919A1 - Planning a path for a vehicle based on a Bezier curve - Google Patents

Abstract

According to a method for planning a route (7), a target position (11) for a vehicle (1) is determined and the route (7) is planned in such a way that it follows a starting position (10) of the vehicle (1) following a Bezier curve. connects to the target position (11). A minimum longitudinal distance between the starting position (10) and the target position (11) in a direction parallel to an initial longitudinal axis (14) of the vehicle (1) is determined as a function of a predetermined transverse distance between the starting position (10) and the target position (11 ) in a direction parallel to an initial transverse axis (13) of the vehicle (1) and a maximum steering angle of the vehicle (1).

Description

The present invention is directed to a computer-implemented method for planning a path for a vehicle, determining a target position for the vehicle and planning the path to connect an initial position of the vehicle to the target position. The invention is also directed to a method for at least partially automatically driving a vehicle, to an electronic vehicle guidance system and to computer program products.

Path planning, also known as motion planning, is one of the most important sub-functions for various applications in automated driving assistance systems or autonomous driving systems in both high-speed and low-speed manoeuvres. Existing approaches to path planning use straight lines, circle segments or combinations thereof. This results in a low degree of path smoothness and, in general, also in longer paths than necessary to connect an initial position with a target position of the vehicle.

An object of the present invention is to improve path planning for a vehicle such that complex paths can be planned with improved smoothness.

This object is solved by the respective subject matter of the independent claims. Further designs and preferred embodiments are the subject matter of the dependent claims.

The invention is based on the idea of using a Bezier curve for planning a path of the vehicle and deriving a longitudinal distance to travel corresponding to the path from a predetermined lateral distance to travel.

According to one aspect of the invention, a computer-implemented method for planning a path for a vehicle is provided. A target position for the vehicle is determined and the route is planned in such a way that it connects a starting position of the vehicle with the target position. A minimum longitudinal distance between the home position and the target position is determined depending on a predetermined lateral distance between the home position and the target position and a maximum steering angle of the vehicle to determine the target position. The minimum longitudinal distance corresponds to a distance between the starting position and the target position in a direction parallel to an initial longitudinal axis of the vehicle, and the predetermined lateral distance corresponds to a distance between the starting position and the target position in a direction parallel to an initial lateral axis of the vehicle, where the initial transverse axis is perpendicular to the initial longitudinal axis. The path is planned in such a way that it follows a Bezier curve from the starting position to the target position.

Here and below, all steps of a computer-implemented method according to the invention can be executed by a computing unit, in particular a computing unit of the vehicle. The processing unit can have one or more sub-units, which can be spatially distributed or not. For example, one or more electronic control units, ECUs, of the vehicle can contain the computing unit or vice versa.

The initial position of the vehicle corresponds to a predefined or predetermined position of the vehicle. For example, the home position may correspond to a current position of the vehicle at a time when the computer-implemented method for planning the path is executed. On the other hand, the target position is not predefined, but can be considered as a result or an intermediate result of the computer-implemented method in the sense that by defining the starting position and determining the path connecting the starting position to the target position, the target position is also determined. However, one or more boundary conditions may apply that restrict the target position. The boundary conditions can be predefined in some implementations.

It should be noted that the concept of path planning should not be confused with the concept of path prediction. While path planning aims at determining a desired path for the vehicle, for example to control the vehicle's actuators in such a way that the vehicle actually follows the planned path, path prediction aims at estimating which path the vehicle is likely to follow based on its current path situation or his current driving condition.

In particular, a home pose of the vehicle is defined by the home position of the vehicle in combination with a home orientation of the vehicle at the home position. Similarly a target pose of the vehicle is defined by the target position in combination with a target orientation of the vehicle at the target position. The initial orientation and the target orientation can be understood as the respective orientations of a vehicle coordinate system, which is rigidly connected to the vehicle, with respect to a reference coordinate system, which is rigidly connected to the environment of the vehicle, for example a roadway traveled by the vehicle. The choice of the orientation of the reference coordinate system is fixed but arbitrary. It can therefore be selected, for example, to be identical to the vehicle coordinate system in accordance with the initial orientation or the initial pose of the vehicle. The vehicle coordinate system can, for example, be spanned by the longitudinal axis, the transverse axis and a normal axis of the vehicle, which is perpendicular to the longitudinal axis and to the transverse axis. The initial longitudinal axis and the initial transverse axis or the target longitudinal and target transverse axis can be understood as the longitudinal axis and the transverse axis of the vehicle according to the initial orientation or the target orientation, in particular with regard to the reference coordinate system. The longitudinal axis can, for example, correspond to a direction of movement of the vehicle if the steering angle of the vehicle is equal to zero. The plane spanned by the longitudinal axis and the transverse axis can be parallel to a plane spanned by the tire contact points on the road surface.

In other words, the target position is given by the starting position which is shifted by the transverse distance along the initial transverse axis and shifted by the longitudinal distance along the initial longitudinal axis.

In general, the starting orientation may differ from the target orientation of the vehicle. The initial longitudinal axis and the initial lateral axis of the vehicle can therefore be used as reference axes to define the longitudinal distance and the lateral distance. In preferred exemplary embodiments, however, the initial orientation and the target orientation of the vehicle are identical or approximately identical, so that the initial longitudinal axis is parallel to the target longitudinal axis and the initial lateral axis is parallel to the target lateral axis of the vehicle. In this case, the longitudinal distance can also be considered as a distance between the initial transverse axis and the target transverse axis. Similarly, the lateral distance can be viewed as a distance between the initial longitudinal axis and the target longitudinal axis.

For example, the predetermined lateral distance may be predetermined by the vehicle itself, for example using an environmental sensor system to assess the vehicle's surroundings and determine the desired lateral distance for path planning. The predetermined transverse distance can also be predefined per se in such a way that it is given by a constant value. In either case, the lateral distance remains constant while the computer-implemented method of planning the path is executed.

The minimum longitudinal distance can be considered as a smallest possible longitudinal distance that can be realized by the vehicle when a predefined set of boundary conditions applies. A constraint that applies in particular implementations is that the path is planned to follow the Bezier curve from the origin position to the destination position. A further boundary condition can be established, for example, by the degree of the Bezier curve. For example, the minimum longitudinal distance in the case of using a third degree Bezier curve may be different to a situation where a second or fourth degree Bezier curve is used. Furthermore, the maximum steering angle of the vehicle also corresponds to a boundary condition. The larger the maximum steering angle of the vehicle, the shorter the minimum longitudinal distance can be. The steering angle of the vehicle may, for example, correspond to an angle of steering wheels, in other words an angle of steerable wheels of the vehicle, for example with respect to the longitudinal axis of the vehicle. A maximum rate of change of the steering angle can also impose a corresponding boundary condition. In addition, for example, a wheelbase of the vehicle can correspond to a boundary condition. The shorter the wheelbase, the shorter the minimum longitudinal distance can be. Furthermore, a predefined tolerance value can set up a boundary condition. For example, the tolerance value may be implemented by artificially reducing the maximum steering angle and/or reducing the maximum rate of change of the steering angle.

The path of the vehicle describes a planned movement of a specific predefined point on the vehicle. For example, the planned path may describe a movement of a point on an axis of the vehicle, for example at a laterally central location of the vehicle. The path can also describe the movement of a point on a front bumper of the vehicle, etc. In particular, the path corresponds to a two-dimensional curve within a plane that is perpendicular to the normal axis of the vehicle, for example. Consequently, the Bezier curve has two components ten, with each of the components describing a respective spatial coordinate. Both components of the Bezier curve can also be considered as respective Bezier curves, in particular as a function of an independent parameter t. For example, the planned path may be given by (x(t),y(t)), where t takes on values in a predefined range, for example in the range of [0,1]. It should be noted that the planned path only describes the spatial aspect of the movement of the vehicle. In particular, the actual motion of the vehicle can be described by considering t as a function of time. However, t does not correspond to time per se.

A Bezier curve can be understood as a function of the independent parameter that satisfies the respective Bezier equation according to the degree of the Bezier curve, the degree in particular being predefined. For example, the Bezier curve, in particular the components of the Bezier curve, are determined as Bezier curves of at least third degree, in particular third degree.

According to at least one embodiment, a further target position is determined for the vehicle and the route is planned in such a way that it also connects the target position with the further target position. The path is planned such that it follows the Bezier curve from the starting position to the target position and follows another Bezier curve from the target position to the further target position.

All explanations regarding the Bezier curve, the starting position and the target position can be transferred analogously to the further Bezier curve, the target position or the further target position.

The use of the Bezier curve and, if necessary, the further Bezier curve for the path planning makes it possible to use closed equations for the path, which can be planned and adapted quickly and dynamically. Furthermore, compared to a path consisting of straight and circular segments, for example, improved smoothness and fewer required steering actions and thus an improved level of comfort for a user can be achieved.

The use of the Bezier curve and optionally the further Bezier curve enables a robust and dynamic motion planning for the vehicle, especially in situations with limited space, in particular in tight parking situations or other situations in which a collision with an external object is to be avoided avoid. Furthermore, the calculations to be performed do not require extensive computing resources and can therefore also be performed by embedded systems, such as are commonly used in automobiles and other motor vehicles.

Furthermore, using the Bezier curve, simple as well as complex routes including S-shaped routes, straight routes or circular routes can be planned within the shortest possible travel distance. Using the wider Bezier curve increases the achievable path complexity even further without requiring significantly more computational resources.

According to some implementations, the path is planned to follow a third degree Bezier curve from the starting position to the target position and/or the path is planned to follow another third degree Bezier curve from the target position to the further target position.

In other words, the Bezier curve and/or the further Bezier curve are third-degree Bezier curves. This means that at least one component of the Bezier curve and/or the further Bezier curve is a tertiary component and the remaining component is a tertiary or lower component.

In such implementations, the mathematical representation of the path has a relatively low complexity, while already most relevant path types, such as straight paths, circular paths or S-shaped paths and, if the further Bezier path is included, double S-shaped paths , can be represented accurately. A particularly efficient, robust and computationally favorable solution for route planning is thus achieved.

According to some implementations, the minimum longitudinal distance is determined based on a wheelbase of the vehicle.

The wheelbase corresponds to a distance along the longitudinal axis of the vehicle between a front axle and a rear axle of the vehicle, with two on opposite sides of the front wheels of the vehicle mounted on the axle are steerable wheels and two wheels mounted on opposite ends of the rear axle are non-steerable wheels.

bestimmt, wobei Δx den minimalen longitudinalen Abstand bezeichnet, Δy den Querabstand bezeichnet, 1 den Radstand bezeichnet, b einen Parameter innerhalb des Intervalls ]0, 0.5] bezeichnet, φ_max den maximalen Lenkwinkel bezeichnet und φ_tol einen vordefinierten Toleranzwert in dem Intervall [0, φ_max[, zum Beispiel in dem Intervall [0, φ_max/10], bezeichnet.According to some implementations, the minimum longitudinal distance is given according to the relationship: $${\left(\mathrm{\Delta }\text{x}\right)}^2=\left(\mathrm{\Delta }\text{y/}2\right)\left[-\mathrm{\Delta }\text{y}+\sqrt{{\left(\mathrm{\Delta }\text{y}\right)}^2+432{\left(\text{blue/tan}\left({\phi }_{\text{Max}}-{\phi }_{\text{great}}\right)\right)}^2}\right]$$

determined, where Δx denotes the minimum longitudinal distance, Δy denotes the lateral distance, 1 denotes the wheelbase, b denotes a parameter within the interval ]0, 0.5], φ _max denotes the maximum steering angle and φ _tol a predefined tolerance value in the interval [0 , φ _max [, for example in the interval [0, φ _max /10].

$${\text{x}}_{\text{P}}\left(\text{t}=1\right)=\text{M},$$

$${\text{y}}_{\text{P}}\left(\text{t}=0\right)=\mathrm{0,}$$

$${\text{y}}_{\text{P}}\left(\text{t}=1\right)=0.$$

Said relationship is obtained by assuming symmetrical control points of the Bezier curve at bM or (1 - b)M, with the following boundary conditions applying: $${\text{x}}_{\text{P}}\left(\text{t}=0\right)=\mathrm{0,}$$

$${\text{x}}_{\text{P}}\left(\text{t}=1\right)=\text{M},$$

$${\text{y}}_{\text{P}}\left(\text{t}=0\right)=\mathrm{0,}$$

$${\text{y}}_{\text{P}}\left(\text{t}=1\right)=0$$

(x _P ,y _P )(t) represent the spatial coordinates of the components of the path in a local coordinate system that is defined by the said boundary conditions. The starting position of the vehicle then corresponds to (x _p ,y _p )(t=0)=(0,0) and the target position corresponds to (x _p ,y _p )(t=1)=(M,0). In other words, M is the Euclidean distance between the starting position and the target position, such that $\text{M}=\sqrt{{\left(\mathrm{\Delta }\text{x}\right)}^2+{\left(\mathrm{\Delta }\text{y}\right)}^2}.$

und $${\text{y}}_{\text{p}}\left(\text{t}\right)=\left(3{\text{bMt}}_0\right)\text{t}-9{\text{bMt}}_0{\text{t}}^2+6{\text{bMt}}_0{\text{t}}^3$$

geplant.According to some statements, the path becomes according to the relationships $${\text{x}}_{\text{p}}\left(\text{t}\right)=\left(3\text{bM}\right)\text{t}+3\text{M}\left(1-3\text{b}\right){\text{<figure-callout id="2" label="t" filenames="DE102021112919A1\_0097.png" state="{{state}}">t</figure-callout>}}^2-2\text{M}\left(1-3\text{b}\right){\text{<figure-callout id="3" label="t" filenames="DE102021112919A1\_0097.png" state="{{state}}">t</figure-callout>}}^3$$

and $${\text{y}}_{\text{p}}\left(\text{t}\right)=\left(3{\text{bMt}}_0\right)\text{t}-9{\text{bMt}}_0{\text{<figure-callout id="2" label="t" filenames="DE102021112919A1\_0097.png" state="{{state}}">t</figure-callout>}}^2+6{\text{bMt}}_0{\text{<figure-callout id="3" label="t" filenames="DE102021112919A1\_0097.png" state="{{state}}">t</figure-callout>}}^3$$

planned.

Here, t ₀ is a real constant that describes the initial and target orientation of the vehicle. In other words, the initial and target orientations are identical in such implementations. In particular applies $\frac{{\dot{\text{y}}}_{\text{P}}\left(\text{t}=0\right)}{{\dot{\text{x}}}_{\text{P}}\left(\text{t}=0\right)}={\text{<figure-callout id="0" label="t" filenames="DE102021112919A1\_0098.png" state="{{state}}">t</figure-callout>}}_0=\frac{{\dot{\text{y}}}_{\text{P}}\left(\text{t}=1\right)}{{\dot{\text{x}}}_{\text{P}}\left(\text{t}=1\right)}.$

According to some embodiments, to determine the further target position, a further minimum longitudinal distance between the target position and the further target position in the direction parallel to the initial longitudinal axis of the vehicle is dependent on a predetermined further lateral distance between the target position and the further target position in the direction parallel to the initial lateral axis of the vehicle and is determined as a function of the maximum steering angle and in particular the wheelbase of the vehicle.

In particular, in such embodiments, the initial orientation of the vehicle at the initial position is identical to the target orientation of the vehicle at the target position. The explanations regarding the minimum longitudinal distance and corresponding explanations of the method according to the invention can be transferred analogously to the further minimum longitudinal distance.

According to a further aspect of the invention, a method for at least partially automatically driving a vehicle is provided. In this case, sensor data representing an environment of the vehicle is generated by an environment sensor system of the vehicle. A path for the vehicle is planned by a computing unit of the vehicle using a computer-implemented method according to the invention as a function of the sensor data and/or as a function of the predetermined transverse distance. The vehicle is guided at least partially automatically according to the planned route, in particular by the computing unit or by an electronic vehicle guidance system of the vehicle which contains the computing unit.

In particular, the computing unit or a control unit of the electronic vehicle guidance system is set up to generate control signals for at least partially automatically guiding the vehicle along the planned route.

An environmental sensor system can be understood as a sensor system capable of generating sensor data or sensor signals that reflect, represent or map an environment of the environmental sensor system. In particular, the ability to sense or detect electromagnetic or other signals from the environment may be considered a sufficient condition for a sensor system to be an environment sensor system. For example, cameras, lidar systems, radar systems, or ultrasonic sensor systems can be considered environmental sensor systems.

For example, the lateral distance for the route to be planned for the vehicle is determined by the computing unit based on the sensor data.

According to some embodiments of the method for driving a vehicle according to the invention, a position of an obstacle in the area is determined by the computing unit based on the sensor data and the transverse distance is determined as a function of the position of the obstacle.

In particular, the lateral distance can be determined in such a way that a collision of the vehicle with the obstacle is avoided according to the planned route.

To determine the obstacle and the position of the obstacle based on the sensor data, in particular camera images or lidar point clouds and/or radar data et cetera, the processing unit can use one or more algorithms for computer vision or conventional image processing or image analysis.

According to some embodiments, a target lane, in particular a position of the target lane, for example a transverse position of the target lane with respect to the vehicle, in the vicinity of the vehicle is determined by the computing unit based on the sensor data. The transverse distance is determined depending on the target lane or the position of the target lane.

In other words, the lateral spacing may be chosen such that the path takes the vehicle from a current lane of the roadway to an adjacent lane of the roadway or a lane closest to the adjacent lane of the roadway and, in some implementations, back to the current lane.

According to some implementations, a position of a parking space in the area is determined by the computing unit based on the sensor data and the lateral distance is determined depending on the position of the parking space. In particular, the route is planned in order to automatically carry out a parking maneuver of the vehicle.

According to a further aspect of the invention, an electronic vehicle guidance system, which has a computing unit and an environment sensor system for a vehicle, is provided. The environment sensor system is set up to generate sensor data that represent an environment of the vehicle. The processing unit is set up to plan a route for the vehicle using a computer-implemented method according to the invention depending on the sensor data and/or depending on the lateral distance for the route and to guide the vehicle at least partially automatically according to the planned route.

According to some implementations, the computing unit is set up to determine the lateral distance for the path based on the sensor data.

An electronic vehicle guidance system can be understood to mean an electronic system which is set up to guide a vehicle fully automatically or fully autonomously and in particular without the need for manual intervention or control by a driver or user of the vehicle. The vehicle automatically carries out all necessary functions, such as steering manoeuvres, braking manoeuvres, and/or acceleration manoeuvres, as well as monitoring and registering road traffic and responding accordingly. In particular, the electronic vehicle guidance system can implement a fully automated or fully autonomous driving mode according to level 5 according to the SAE J3016 classification. An electronic vehicle guidance system can also be implemented as a driver assistance system, ADAS, to support a driver in semi-automatic or semi-autonomous driving. In particular, the electronic vehicle guidance system can implement a semi-automatic or semi-autonomous driving mode according to levels 1 to 4 according to the SAE J3016 classification. Here and in the following, "SAE J3016" refers to the respective standard in the June 2018 version.

Further versions of the electronic vehicle guidance system result directly from the different versions of the computer-implemented method according to the invention for planning a route and from the different versions of the method according to the invention for at least partially automatically guiding a vehicle and vice versa. In particular, the electronic vehicle guidance system is set up to execute a computer-implemented method according to the invention or a method according to the invention, or the electronic vehicle guidance system according to the invention executes such a method or computer-implemented method.

According to a further aspect of the invention, a first computer program containing first instructions is provided. If the first commands or the first computer program are executed by a computer system, in particular by an electronic vehicle guidance system according to the invention, for example by the processing unit of the electronic vehicle guidance system, the first commands cause the computer system to execute a computer-implemented method according to the invention for planning a route for a vehicle .

According to a further aspect of the invention, a second computer program containing second instructions is provided. If the second computer program or the second commands are executed by an electronic vehicle guidance system according to the invention, in particular by the computing unit of the electronic vehicle guidance system, the second commands cause the electronic vehicle guidance system to execute a method according to the invention for at least partially automatically driving a vehicle.

According to another aspect of the invention, a computer-readable storage medium is provided. The computer-readable storage medium stores a first computer program and/or a second computer program according to the invention.

The first computer program, the second computer program and the computer-readable storage medium can be viewed as respective computer program products that contain first instructions and second instructions, respectively.

Further features of the invention result from the claims, the figures and the description of the figures. The features and combinations of features mentioned above in the description and the features and combinations of features mentioned below in the description of the figures and/or shown in the figures can be included in the invention not only in the respective combination mentioned, but also in other combinations. In particular, the invention also encompasses configurations and combinations of features that do not have all the features of an originally formulated claim. The invention also encompasses configurations and combinations of features that go beyond or deviate from the combinations of features set out in the listed claims.

• 1 eine schematische Darstellung eines Fahrzeugs mit einer beispielhaften Ausführung eines erfindungsgemäßen elektronischen Fahrzeugführungssystems;
• 2 ein Beispiel einer Bezier-Kurve;
• 3 ein Beispiel von Grenzpunkten und Kontrollpunkten einer Bezier-Kurve in globalen Koordinaten;
• 4 ein Beispiel von Grenzpunkten und Kontrollpunkten einer Bezier-Kurve in lokalen Koordinaten;
• 5 ein Beispiel einer S-förmigen Bezier-Kurve in lokalen Koordinaten;
• 6 einen gemäß einer beispielhaften Ausführung der Erfindung geplanten Weg für ein Fahrzeug; und
• 7 einen gemäß einer weiteren beispielhaften Ausführung der Erfindung geplanten Weg für ein Fahrzeug.

Show in the figures

• 1 a schematic representation of a vehicle with an exemplary embodiment of an electronic vehicle guidance system according to the invention;
• 2 an example of a Bezier curve;
• 3 an example of boundary points and control points of a Bezier curve in global coordinates;
• 4 an example of boundary points and control points of a Bezier curve in local coordinates;
• 5 an example of an S-shaped Bezier curve in local coordinates;
• 6 a planned path for a vehicle according to an exemplary embodiment of the invention; and
• 7 a planned path for a vehicle according to another exemplary embodiment of the invention.

1 shows a schematic representation of a vehicle 1 with an exemplary embodiment of an electronic vehicle guidance system 2 according to the invention.

The electronic vehicle guidance system 2 has a computing unit 3 that can plan a route 7 for the vehicle 1 that has a starting position 10 of the vehicle 1 with a target position 11 (see FIG 2 ) connects. In this case, the path 7 is determined as a Bezier curve, in particular as a third-degree Bezier curve.

For example, the electronic vehicle guidance system 2 can have an environment sensor system 5 . The environment sensor system 5 can have a camera and/or a radar system and/or a lidar system and/or an ultrasonic sensor system, for example. The computing unit 3 receives input data containing environmental sensor data generated by the environmental sensor system 5 and can plan the route 7 depending on the input data.

The electronic vehicle guidance system 2 can optionally have a status sensor system 4 and/or an input device 6 . The state sensor system 4 can have a speed sensor, a steering angle sensor and/or a yaw rate sensor, for example. The input device 6 can, for example, have direction indicators of the vehicle 1 which are to be actuated by a driver of the vehicle 1 . The condition sensor system 4 may generate respective condition sensor data related to a current condition of the vehicle and/or an input device 6 may generate a corresponding user input signal. The user input signal and/or the state sensor data can also be part of the input data and the computing unit 3 can take them into account for the planning of the path 7 in some implementations.

The functionality of the electronic vehicle guidance system 2 and respective versions of computer-implemented methods according to the invention for planning the route 7 and methods according to the invention for at least partially automatically guiding the vehicle 1 are described below with reference to FIG 2 until 7 explained in more detail.

2 shows an exemplary Bezier path 7 planned according to the invention as well as a path 7' with the same starting position 10 and target position 11, but which consists of rectilinear segments and circular segments. The path 7 planned according to the invention has a significantly improved regularity and requires fewer steering actions in comparison to the other path 7'.

In the following it is explained how an asymptotic optimal path 7, which connects the starting position 10 with the target position 11 by means of a Bezier curve, can be obtained.

The starting position 10 is denoted by (x ₀ ,y ₀ ,θ ₀ ) and the target position 11 by (x _F ,y _F ,θ _F ) as in FIG 3 shown. The circles between the starting position 10 and the target position 11 represent the control points of a cubic, third-order, or third-degree Bezier curve.

The cubic Bezier equations are particularly appropriate for representing path 7, since they can describe all of the most relevant path types, ranging from a straight line to sinusoidal maneuvers. New coordinates (x _P ,y _P ) can be used to simplify the functional equations instead of non-functional in (x,y) coordinates. To generate the asymptotic optimal Bezier curve in the new coordinates, note that input angles in the range of [0°, 360°[ are normalized to the range of [-180°,180°[.

$$\begin{array}{l}wenn\left(x_F-x_0\right)<0\\ \text{ wenn }\mathrm{\alpha }>0\\ \mathrm{\alpha }=\mathrm{\alpha }-\mathrm{180,}\\ \text{ sonst}\\ \mathrm{\alpha }=\mathrm{\alpha }+\mathrm{180,}\end{array}$$

$${\mathrm{\theta }}_{{\text{r}}_0}={\mathrm{\theta }}_0-\mathrm{\alpha }\left(\text{normiert auf den Bereich}[-180°\mathrm{,180}°[\right),$$

$${\mathrm{\theta }}_{{\text{r}}_{\text{F}}}={\mathrm{\theta }}_{\text{F}}-\mathrm{\alpha }\left(\text{normiert auf den Bereich}[-180°\mathrm{,180}°[\right),$$

$$\text{M}=\sqrt{{\left({\text{x}}_{\text{F}}-{\text{x}}_0\right)}^2+{\left({\text{y}}_{\text{F}}-{\text{y}}_0\right)}^2},$$

$${\text{t}}_0=\text{tan}\left({\mathrm{\theta }}_{{\text{r}}_0}\right),$$

$${\text{t}}_{\text{F}}=\text{tan}\left({\mathrm{\theta }}_{{\text{r}}_{\text{F}}}\right).$$

Regarding 3 , is applicable $$\mathrm{a}={\text{tan}}^{-1}\left(\frac{{\text{y}}_{\text{f}}-{\text{y}}_0}{{\text{x}}_{\text{f}}-{\text{x}}_0}\right),$$

$$\begin{array}{l}wenn\left(x_f-x_0\right)<0\\ \text{if}\mathrm{a}>0\\ \mathrm{a}=\mathrm{a}-\mathrm{180,}\\ \text{otherwise}\\ \mathrm{a}=\mathrm{a}+\mathrm{180,}\end{array}$$

$${\mathrm{\theta }}_{{\text{right}}_0}={\mathrm{\theta }}_0-\mathrm{a}\left(\text{normalized to the area}[-180°\mathrm{,180}°[\right),$$

$${\mathrm{\theta }}_{{\text{right}}_{\text{f}}}={\mathrm{\theta }}_{\text{f}}-\mathrm{a}\left(\text{normalized to the area}[-180°\mathrm{,180}°[\right),$$

$$\text{M}=\sqrt{{\left({\text{x}}_{\text{f}}-{\text{x}}_0\right)}^2+{\left({\text{y}}_{\text{f}}-{\text{y}}_0\right)}^2},$$

$${\text{t}}_0=\text{tan}\left({\mathrm{\theta }}_{{\text{right}}_0}\right),$$

$${\text{t}}_{\text{f}}=\text{tan}\left({\mathrm{\theta }}_{{\text{right}}_{\text{f}}}\right).$$

$$\begin{array}{l}\text{ wenn }\left({\mathrm{\theta }}_{{\text{r}}_0}>90°\&\&{\mathrm{\theta }}_{{\text{r}}_{\text{F}}}>90°\right)\\ {\mathrm{\theta }}_{{\text{r}}_0}={\mathrm{\theta }}_{{\text{r}}_0}-180°,\\ {\mathrm{\theta }}_{{\text{r}}_{\text{F}}}={\mathrm{\theta }}_{{\text{r}}_{\text{F}}}-180°,\\ Fahrtrichtung=\text{'}R\ddot{u}ckw\ddot{a}rts\text{'}\end{array}$$

$$\begin{array}{l}\text{ wenn }\left({\mathrm{\theta }}_{{\text{r}}_0}<-90°\&\&{\mathrm{\theta }}_{{\text{r}}_{\text{F}}}>90°\right)\\ {\mathrm{\theta }}_{{\text{r}}_0}={\mathrm{\theta }}_{{\text{r}}_0}+180°,\\ {\mathrm{\theta }}_{{\text{r}}_{\text{F}}}={\mathrm{\theta }}_{{\text{r}}_{\text{F}}}-180°,\\ Fahrtrichtung=\text{'}R\ddot{u}ckw\ddot{a}rts\text{'}\end{array}$$

$$$$

$$\begin{array}{l}\text{ wenn }\left({\mathrm{\theta }}_{{\text{r}}_0}>90°\&\&{\mathrm{\theta }}_{{\text{r}}_{\text{F}}}<-90°\right)\\ {\mathrm{\theta }}_{{\text{r}}_0}={\mathrm{\theta }}_{{\text{r}}_0}-180°,\\ {\mathrm{\theta }}_{{\text{r}}_{\text{F}}}={\mathrm{\theta }}_{{\text{r}}_{\text{F}}}+180°,\\ Fahrtrichtung=\text{'}R\ddot{u}ckw\ddot{a}rts\text{'}\end{array}$$

$$\begin{array}{l}\text{wenn }\left(\left|{\mathrm{\theta }}_{{\text{r}}_0}\right|\ge 90°\Vert \left|{\mathrm{\theta }}_{{\text{r}}_{\text{F}}}\right|\ge 90°\right)\\ \to \text{kein entsprechender Weg existiert}.\end{array}$$

Taking into account the orientation 14, 14' of the vehicle 1, which is given by t ₀ or t _F , the direction of travel can be estimated, see also 4 . By using a pseudocode notation as follows $$\begin{array}{l}faH\mathrm{right}t\mathrm{right}icHt\mathrm{and}nG=\text{'}VO\mathrm{right}w\ddot{a}\mathrm{right}ts\text{'}\\ \text{if}\left({\mathrm{\theta }}_{{\text{right}}_0}<-90°\&\&{\mathrm{\theta }}_{{\text{right}}_{\text{f}}}<-90°\right)\\ {\mathrm{\theta }}_{{\text{right}}_0}={\mathrm{\theta }}_{{\text{right}}_0}+180°,\\ {\mathrm{\theta }}_{{\text{right}}_{\text{f}}}={\mathrm{\theta }}_{{\text{right}}_{\text{f}}}+180°,\\ faH\mathrm{right}t\mathrm{right}icHt\mathrm{and}nG=\text{'}R\ddot{\mathrm{and}}ckw\ddot{a}\mathrm{right}ts\text{'}\end{array}$$

$$\begin{array}{l}\text{if}\left({\mathrm{\theta }}_{{\text{right}}_0}>90°\&\&{\mathrm{\theta }}_{{\text{right}}_{\text{f}}}>90°\right)\\ {\mathrm{\theta }}_{{\text{right}}_0}={\mathrm{\theta }}_{{\text{right}}_0}-180°,\\ {\mathrm{\theta }}_{{\text{right}}_{\text{f}}}={\mathrm{\theta }}_{{\text{right}}_{\text{f}}}-180°,\\ faH\mathrm{right}t\mathrm{right}icHt\mathrm{and}nG=\text{'}R\ddot{\mathrm{and}}ckw\ddot{a}\mathrm{right}ts\text{'}\end{array}$$

$$\begin{array}{l}\text{if}\left({\mathrm{\theta }}_{{\text{right}}_0}<-90°\&\&{\mathrm{\theta }}_{{\text{right}}_{\text{f}}}>90°\right)\\ {\mathrm{\theta }}_{{\text{right}}_0}={\mathrm{\theta }}_{{\text{right}}_0}+180°,\\ {\mathrm{\theta }}_{{\text{right}}_{\text{f}}}={\mathrm{\theta }}_{{\text{right}}_{\text{f}}}-180°,\\ faH\mathrm{right}t\mathrm{right}icHt\mathrm{and}nG=\text{'}R\ddot{\mathrm{and}}ckw\ddot{a}\mathrm{right}ts\text{'}\end{array}$$

$$$$

$$\begin{array}{l}\text{if}\left({\mathrm{\theta }}_{{\text{right}}_0}>90°\&\&{\mathrm{\theta }}_{{\text{right}}_{\text{f}}}<-90°\right)\\ {\mathrm{\theta }}_{{\text{right}}_0}={\mathrm{\theta }}_{{\text{right}}_0}-180°,\\ {\mathrm{\theta }}_{{\text{right}}_{\text{f}}}={\mathrm{\theta }}_{{\text{right}}_{\text{f}}}+180°,\\ faH\mathrm{right}t\mathrm{right}icHt\mathrm{and}nG=\text{'}R\ddot{\mathrm{and}}ckw\ddot{a}\mathrm{right}ts\text{'}\end{array}$$

$$\begin{array}{l}\text{if}\left(\left|{\mathrm{\theta }}_{{\text{right}}_0}\right|\ge 90°\Vert \left|{\mathrm{\theta }}_{{\text{right}}_{\text{f}}}\right|\ge 90°\right)\\ \to \text{no corresponding way exists}.\end{array}$$

The third-degree Bezier equation, in its generic form, can be written as follows: $$\text{right}\left(\text{t}\right)={\left(1-\text{t}\right)}^3{\text{<figure-callout id="0" label="R" filenames="DE102021112919A1\_0098.png" state="{{state}}">R</figure-callout>}}_0+3{\left(1-\text{t}\right)}^2{\text{<figure-callout id="1" label="t R" filenames="DE102021112919A1\_0099.png" state="{{state}}">t R</figure-callout>}}_1+3\left(1-\text{t}\right){\text{t}}^2{\text{<figure-callout id="2" label="R" filenames="DE102021112919A1\_0097.png" state="{{state}}">R</figure-callout>}}_2+{\text{t}}^3{\text{<figure-callout id="3" label="R" filenames="DE102021112919A1\_0097.png" state="{{state}}">R</figure-callout>}}_3$$

beziehungsweise $\left[\text{r}\left(\text{t}=1\right)-\frac{\dot{\text{r}}\left(\text{t}=1\right)}{3}\right].$

Unter Anwendung der Randbedingungen $${\text{x}}_{\text{P}}\left(\text{t}=0\right)=\mathrm{0,}$$

$${\text{x}}_{\text{P}}\left(\text{t}=1\right)=\text{M},$$

$${\text{y}}_{\text{P}}\left(\text{t}=0\right)=\mathrm{0,}$$

$${\text{y}}_{\text{P}}\left(\text{t}=1\right)=0.$$

erhält man $${\text{x}}_{\text{P}}\left(\text{t}\right)={\left(1-\text{t}\right)}^2{\text{t }\dot{\text{x}}}_{\text{P}}\left(\text{t}=0\right)+3\left(1-\text{t}\right){\text{t}}^2\text{M}-\left(1-\text{t}\right){\text{t}}^2{\dot{\text{x}}}_{\text{P}}\left(\text{t}=1\right)+{\text{t}}^3\text{M},$$

$${\text{y}}_{\text{P}}\left(\text{t}\right)={\left(1-\text{t}\right)}^2{\text{t }\dot{\text{y}}}_{\text{P}}\left(\text{t}=0\right)-\left(1-\text{t}\right){\text{t}}^2{\dot{\text{y}}}_{\text{P}}\left(\text{t}=1\right),$$

wobei $$\begin{array}{cc}\frac{{\dot{\text{y}}}_{\text{P}}\left(\text{t}=0\right)}{{\dot{\text{x}}}_{\text{P}}\left(\text{t}=0\right)}={\text{t}}_0& \frac{{\dot{\text{y}}}_{\text{P}}\left(\text{t}=1\right)}{{\dot{\text{x}}}_{\text{P}}\left(\text{t}=1\right)}={\text{t}}_{\text{F}}\end{array}.$$

where t is an independent variable in the range [0, 1]. R ₀ and R ₃ are the values of r(t=0) and r(t=1) respectively. R ₁ and R ₂ are the control points and the values of $\left[\frac{\dot{\text{right}}\left(\text{t}=0\right)}{3}+\text{right}\left(\text{t}=0\right)\right]$

respectively $\left[\text{right}\left(\text{t}=1\right)-\frac{\dot{\text{right}}\left(\text{t}=1\right)}{3}\right].$

Applying the boundary conditions $${\text{x}}_{\text{P}}\left(\text{t}=0\right)=\mathrm{0,}$$

$${\text{x}}_{\text{P}}\left(\text{t}=1\right)=\text{M},$$

$${\text{y}}_{\text{P}}\left(\text{t}=0\right)=\mathrm{0,}$$

$${\text{y}}_{\text{P}}\left(\text{t}=1\right)=0$$

you get $${\text{x}}_{\text{P}}\left(\text{t}\right)={\left(1-\text{t}\right)}^2{\text{t}\dot{\text{x}}}_{\text{P}}\left(\text{t}=0\right)+3\left(1-\text{t}\right){\text{t}}^2\text{M}-\left(1-\text{t}\right){\text{t}}^2{\dot{\text{x}}}_{\text{P}}\left(\text{t}=1\right)+{\text{t}}^3\text{M},$$

$${\text{y}}_{\text{P}}\left(\text{t}\right)={\left(1-\text{t}\right)}^2{\text{t}\dot{\text{y}}}_{\text{P}}\left(\text{t}=0\right)-\left(1-\text{t}\right){\text{t}}^2{\dot{\text{y}}}_{\text{P}}\left(\text{t}=1\right),$$

whereby $$\begin{array}{cc}\frac{{\dot{\text{y}}}_{\text{P}}\left(\text{t}=0\right)}{{\dot{\text{x}}}_{\text{P}}\left(\text{t}=0\right)}={\text{t}}_0& \frac{{\dot{\text{y}}}_{\text{P}}\left(\text{t}=1\right)}{{\dot{\text{x}}}_{\text{P}}\left(\text{t}=1\right)}={\text{t}}_{\text{f}}\end{array}.$$

und ${\text{x}}_{\text{P}}\left(\text{t}=\frac{2}{3}\right)=\left(1-\text{b}\right)\text{M}$

und $${\dot{\text{x}}}_{\text{P}}\left(\text{t}=0\right)=3\text{bM},$$

$${\dot{\text{x}}}_{\text{P}}\left(\text{t}=1\right)=3\text{bM},$$

$${\dot{\text{y}}}_{\text{P}}\left(\text{t}=0\right)={\text{t}}_0{\dot{\text{x}}}_{\text{P}}\left(\text{t}=0\right)=3{\text{bMt}}_0,$$

$${\dot{\text{y}}}_{\text{P}}\left(\text{t}=1\right)={\text{t}}_{\text{f}}{\dot{\text{x}}}_{\text{P}}\left(\text{t}=1\right)=3{\text{bMt}}_{\text{f}}.$$

Assuming symmetric control points for x _P (t) at {bM, (1 - b)M} leads to control points ${\text{x}}_{\text{P}}\left(\text{t}=\frac{1}{3}\right)=\text{bM}$

and ${\text{x}}_{\text{P}}\left(\text{t}=\frac{2}{3}\right)=\left(1-\text{b}\right)\text{M}$

and $${\dot{\text{x}}}_{\text{P}}\left(\text{t}=0\right)=3\text{bM},$$

$${\dot{\text{x}}}_{\text{P}}\left(\text{t}=1\right)=3\text{bM},$$

$${\dot{\text{y}}}_{\text{P}}\left(\text{t}=0\right)={\text{t}}_0{\dot{\text{x}}}_{\text{P}}\left(\text{t}=0\right)=3{\text{bMt}}_0,$$

$${\dot{\text{y}}}_{\text{P}}\left(\text{t}=1\right)={\text{t}}_{\text{f}}{\dot{\text{x}}}_{\text{P}}\left(\text{t}=1\right)=3{\text{bMt}}_{\text{f}}.$$

$${\text{y}}_{\text{P}}\left(\text{t}\right)=3{\text{bMt}}_0{\left(1-\text{t}\right)}^2\text{t}-3{\text{bMt}}_{\text{f}}\left(1-\text{t}\right){\text{t}}^2.$$

Substituting into [2] and [3] we get $${\text{x}}_{\text{P}}\left(\text{t}\right)=3\text{bM}{\left(1-\text{t}\right)}^2\text{t}+3\left(1-\text{t}\right){\text{t}}^2\text{M}-3\text{bM}\left(1-\text{t}\right){\text{t}}^2+{\text{t}}^3\text{M},$$

$${\text{y}}_{\text{P}}\left(\text{t}\right)=3{\text{bMt}}_0{\left(1-\text{t}\right)}^2\text{t}-3{\text{bMt}}_{\text{f}}\left(1-\text{t}\right){\text{<figure-callout id="2" label="t" filenames="DE102021112919A1\_0097.png" state="{{state}}">t</figure-callout>}}^2.$$

$${\text{y}}_{\text{P}}\left(\text{t}\right)=\left[3{\text{bMt}}_0\right]\text{t}-\left[3\text{bM}\left(2{\text{t}}_0+{\text{t}}_{\text{f}}\right)\right]{\text{t}}^2+\left[3\text{bM}\left({\text{t}}_0+{\text{t}}_{\text{f}}\right)\right]{\text{t}}^3,$$

This results in formulas for the third-order polynomial Bezier curve with control points at bM and (1 - b)M as follows: $${\text{x}}_{\text{P}}\left(\text{t}\right)=\left[3\text{bM}\right]\text{t}+\left[3\text{M}\left(1-3\text{b}\right)\right]{\text{t}}^2-\left[2\text{M}\left(1-3\text{b}\right)\right]{\text{t}}^3,$$

$${\text{y}}_{\text{P}}\left(\text{t}\right)=\left[3{\text{bMt}}_0\right]\text{t}-\left[3\text{bM}\left(2{\text{t}}_0+{\text{t}}_{\text{f}}\right)\right]{\text{t}}^2+\left[3\text{bM}\left({\text{t}}_0+{\text{t}}_{\text{f}}\right)\right]{\text{t}}^3,$$

$$\text{y}={\text{y}}_0+{\text{x}}_{\text{P}}\text{ sin}\left(\mathrm{\alpha }\right)+{\text{y}}_{\text{P}}\text{ cos}\left(\mathrm{\alpha }\right).$$

Choosing the control points at these positions leads to a simplification in the equations and the consequent acceptance checks. Also, it can be considered as an asymptotic optimal path. The equation of the generated Bezier path can be represented in global coordinates as follows $$\text{x}={\text{x}}_0+{\text{x}}_{\text{P}}\text{cos}\left(\mathrm{a}\right)-{\text{y}}_{\text{P}}\text{sin}\left(\mathrm{a}\right),$$

$$\text{y}={\text{<figure-callout id="0" label="y" filenames="DE102021112919A1\_0098.png" state="{{state}}">y</figure-callout>}}_0+{\text{x}}_{\text{P}}\text{sin}\left(\mathrm{a}\right)+{\text{y}}_{\text{P}}\text{cos}\left(\mathrm{a}\right).$$

The vehicle orientation (θ) and the front wheel angle (φ) of the vehicle 1, which may also be referred to as the steering wheel angle or steering angle, can be represented by the following relationship $$\text{tan}\left(\mathrm{\theta }\right)=\frac{\text{dy}}{\text{dx}}=\frac{\frac{\text{dy}}{\text{German}}}{\frac{\text{dx}}{\text{German}}}\approx \frac{\text{y}\left(\text{i}\right)-\text{y}\left(\text{i}-1\right)}{\text{x}\left(\text{i}\right)-\text{x}\left(\text{i}-1\right)}.$$

$$\dot{\phi }=\left[\frac{\left({\dot{\text{x}}}_{\text{p}}{}^2+{\dot{\text{y}}}_{\text{p}}{}^2\right)\left({\dot{\text{x}}}_{\text{p}}{\stackrel{⃛}{\text{y}}}_{\text{p}}-{\dot{\text{y}}}_{\text{p}}{\stackrel{⃛}{\text{x}}}_{\text{p}}\right)-3\left({\dot{\text{x}}}_{\text{p}}{\ddot{\text{y}}}_{\text{p}}-{\dot{\text{y}}}_{\text{p}}{\ddot{\text{x}}}_{\text{p}}\right)\left({\dot{\text{x}}}_{\text{p}}{\ddot{\text{x}}}_{\text{p}}+{\dot{\text{y}}}_{\text{p}}{\ddot{\text{y}}}_{\text{p}}\right)}{{\left({\dot{\text{x}}}_{\text{p}}{}^2+{\dot{\text{y}}}_{\text{p}}{}^2\right)}^3+{\mathcal{l}}^2{\left({\dot{\text{x}}}_{\text{p}}{\ddot{\text{y}}}_{\text{p}}-{\dot{\text{y}}}_{\text{p}}{\ddot{\text{x}}}_{\text{p}}\right)}^2}\right]\times \mathcal{l}\text{v}\le {\dot{\phi }}_{\text{max}},$$

wobei ℓ den Radstand des Fahrzeugs 1 bezeichnet und v = (ẋ_P ² + ẏ_P ²)^1/2.A front steering wheel angle acceptability check (φ) can be performed as follows, where φ _max and φ _max are the specified maximum value and maximum rate of front wheel angle to be achieved, respectively, $$\text{tan}\left(\mathrm{\phi }\right)=\frac{\mathcal{l}\dot{\theta }}{\text{v}}=\frac{\mathcal{l}\left({\dot{\text{x}}}_{\text{P}}{\ddot{\text{y}}}_{\text{P}}-{\dot{\text{y}}}_{\text{P}}{\ddot{\text{x}}}_{\text{P}}\right)}{{\left({\dot{\text{x}}}_{\text{<figure-callout id="2" label="P" filenames="DE102021112919A1\_0097.png" state="{{state}}">P</figure-callout>}}{}^2+{\dot{\text{y}}}_{\text{P}}{}^2\right)}^{3\text{/}2}}\le \text{tan}\left({\mathrm{\phi }}_{\text{Max}}\right)$$

$$\dot{\phi }=\left[\frac{\left({\dot{\text{x}}}_{\text{p}}{}^2+{\dot{\text{y}}}_{\text{p}}{}^2\right)\left({\dot{\text{x}}}_{\text{p}}{\stackrel{⃛}{\text{y}}}_{\text{p}}-{\dot{\text{y}}}_{\text{p}}{\stackrel{⃛}{\text{x}}}_{\text{p}}\right)-3\left({\dot{\text{x}}}_{\text{p}}{\ddot{\text{y}}}_{\text{p}}-{\dot{\text{y}}}_{\text{p}}{\ddot{\text{x}}}_{\text{p}}\right)\left({\dot{\text{x}}}_{\text{p}}{\ddot{\text{x}}}_{\text{p}}+{\dot{\text{y}}}_{\text{p}}{\ddot{\text{y}}}_{\text{p}}\right)}{{\left({\dot{\text{x}}}_{\text{p}}{}^2+{\dot{\text{y}}}_{\text{p}}{}^2\right)}^3+{\mathcal{l}}^2{\left({\dot{\text{x}}}_{\text{p}}{\ddot{\text{y}}}_{\text{p}}-{\dot{\text{y}}}_{\text{p}}{\ddot{\text{x}}}_{\text{p}}\right)}^2}\right]\times \mathcal{l}\text{v}\le {\dot{\phi }}_{\text{Max}},$$

where ℓ denotes the wheelbase of the vehicle 1 and v = (ẋ _P ² + ẏ _P ² ) ^1/2 .

To obtain an expression for the minimum distance between two points that results in a third-order feasible S-shaped Bezier curve connecting the points, the Bezier equation for this inverse S-shaped path is analyzed. The third-order Bezier equation [5] in local coordinates can be analyzed as follows to arrive at an expression for the minimum distance between source and destination frames, see also 5 .

und Einsetzen in die Gleichung [5] für y_P(t) liefert: $${\text{y}}_{\text{P}}\left(\text{t}\right)=\left[3{\text{bMt}}_0\right]\text{t}-\left[3{\text{bMt}}_0\left(2+\text{K}\right)\right]{\text{t}}^2+\left[3{\text{bMt}}_0\left(1+\text{K}\right)\right]{\text{t}}^3,$$

$${\dot{\text{x}}}_{\text{P}}\left(\text{t}\right)=3\text{bM}+\left[6\text{M}\left(1-3\text{b}\right)\right]\text{t}-\left[6\text{M}\left(1-3\text{b}\right)\right]{\text{t}}^2,$$

$${\dot{\text{y}}}_{\text{P}}\left(\text{t}\right)=3{\text{bMt}}_0-\left[6{\text{bMt}}_0\left(2+\text{K}\right)\right]\text{t}+\left[9{\text{bMt}}_0\left(1+\text{K}\right)\right]{\text{t}}^2,$$

$${\ddot{\text{x}}}_{\text{P}}\left(\text{t}\right)=6\text{M}\left(1-3\text{b}\right)-\left[12\text{M}\left(1-3\text{b}\right)\right]\text{t},$$

$${\ddot{\text{y}}}_{\text{P}}\left(\text{t}\right)=-6{\text{bMt}}_0\left(2+\text{K}\right)+\left[18{\text{bMt}}_0\left(1+\text{K}\right)\right]\text{t}\text{.}$$

Denoting the ratio between the tangents of the initial and final angles by (K), $${\text{t}}_{\text{f}}=\text{K}\times {\text{<figure-callout id="0" label="t" filenames="DE102021112919A1\_0098.png" state="{{state}}">t</figure-callout>}}_0$$

and substituting into equation [5] for y _P (t) yields: $${\text{y}}_{\text{P}}\left(\text{t}\right)=\left[3{\text{bMt}}_0\right]\text{t}-\left[3{\text{bMt}}_0\left(2+\text{K}\right)\right]{\text{t}}^2+\left[3{\text{bMt}}_0\left(1+\text{K}\right)\right]{\text{t}}^3,$$

$${\dot{\text{x}}}_{\text{P}}\left(\text{t}\right)=3\text{bM}+\left[6\text{M}\left(1-3\text{b}\right)\right]\text{t}-\left[6\text{M}\left(1-3\text{b}\right)\right]{\text{t}}^2,$$

$${\dot{\text{y}}}_{\text{P}}\left(\text{t}\right)=3{\text{bMt}}_0-\left[6{\text{bMt}}_0\left(2+\text{K}\right)\right]\text{t}+\left[9{\text{bMt}}_0\left(1+\text{K}\right)\right]{\text{t}}^2,$$

$${\ddot{\text{x}}}_{\text{P}}\left(\text{t}\right)=6\text{M}\left(1-3\text{b}\right)-\left[12\text{M}\left(1-3\text{b}\right)\right]\text{t},$$

$${\ddot{\text{y}}}_{\text{P}}\left(\text{t}\right)=-6{\text{bMt}}_0\left(2+\text{K}\right)+\left[18{\text{bMt}}_0\left(1+\text{K}\right)\right]\text{t}\text{.}$$

According to equation [9] follows $$\text{tan}\left(\mathrm{\phi }\right)=\frac{\mathcal{l}\dot{\theta }}{\text{v}}=\frac{\mathcal{l}\left(\dot{\text{x}}\ddot{\text{y}}-\dot{\text{y}}\ddot{\text{x}}\right)}{{\left({\dot{\text{x}}}^2+{\dot{\text{y}}}^2\right)}^{3\text{/}2}}=\frac{\mathcal{l}\left({\dot{\text{x}}}_{\text{P}}{\ddot{\text{y}}}_{\text{P}}-{\dot{\text{y}}}_{\text{P}}{\ddot{\text{x}}}_{\text{P}}\right)}{{\left({\dot{\text{x}}}_{\text{<figure-callout id="2" label="P" filenames="DE102021112919A1\_0097.png" state="{{state}}">P</figure-callout>}}{}^2+{\dot{\text{y}}}_{\text{P}}{}^2\right)}^{3\text{/}2}}\le \text{tan}\left({\mathrm{\phi }}_{\text{Max}}\right).$$

$$\dot{\theta }=\left(\frac{1}{1+{\left(\frac{\dot{\text{y}}}{\dot{\text{x}}}\right)}^2}\right)\left(\frac{\dot{\text{x}}\ddot{\text{y}}-\dot{\text{y}}\ddot{\text{x}}}{{\dot{\text{x}}}^2}\right)=\frac{\dot{\text{x}}\ddot{\text{y}}-\dot{\text{y}}\ddot{\text{x}}}{{\dot{\text{x}}}^2+{\dot{\text{y}}}^2},$$

$$\text{v}=\sqrt{{\dot{\text{x}}}^2+{\dot{\text{y}}}^2},$$

$$\text{tan}\left(\mathrm{\phi }\right)=\frac{\mathcal{l}\dot{\theta }}{\text{v}}=\frac{\mathcal{l}\left(\dot{\text{x}}\ddot{\text{y}}-\dot{\text{y}}\ddot{\text{x}}\right)}{{\left({\dot{\text{x}}}^2+{\dot{\text{y}}}^2\right)}^{3\text{/}2}}.$$

A brief proof of equation [9] can be as follows $$\mathrm{\theta }={\text{tan}}^{-1}\left(\frac{\dot{\text{y}}}{\dot{\text{x}}}\right),$$

$$\dot{\theta }=\left(\frac{1}{1+{\left(\frac{\dot{\text{y}}}{\dot{\text{x}}}\right)}^2}\right)\left(\frac{\dot{\text{x}}\ddot{\text{y}}-\dot{\text{y}}\ddot{\text{x}}}{{\dot{\text{x}}}^2}\right)=\frac{\dot{\text{x}}\ddot{\text{y}}-\dot{\text{y}}\ddot{\text{x}}}{{\dot{\text{x}}}^2+{\dot{\text{y}}}^2},$$

$$\text{v}=\sqrt{{\dot{\text{x}}}^2+{\dot{\text{y}}}^2},$$

$$\text{tan}\left(\mathrm{\phi }\right)=\frac{\mathcal{l}\dot{\theta }}{\text{v}}=\frac{\mathcal{l}\left(\dot{\text{x}}\ddot{\text{y}}-\dot{\text{y}}\ddot{\text{x}}\right)}{{\left({\dot{\text{x}}}^2+{\dot{\text{y}}}^2\right)}^{3\text{/}2}}.$$

Since the representations of tan(φ), θ̇̇̇, and v are the same in local and global coordinates, one can obtain the following equation for the steering $$\text{tan}\left(\mathrm{\phi }\right)=\frac{\mathcal{l}\dot{\theta }}{\text{v}}=\frac{\mathcal{l}\left({\dot{\text{x}}}_{\text{p}}{\ddot{\text{y}}}_{\text{p}}-{\dot{\text{y}}}_{\text{p}}{\ddot{\text{x}}}_{\text{p}}\right)}{{\left({\dot{\text{x}}}_{\text{<figure-callout id="2" label="p" filenames="DE102021112919A1\_0097.png" state="{{state}}">p</figure-callout>}}{}^2+{\dot{\text{y}}}_{\text{p}}{}^2\right)}^{3\text{/}2}}.$$

In order to obtain the optimal value t where the absolute steering is maximum, the following equation can be solved $$\frac{\text{i.e}}{\text{German}}\left(\text{tan}\left(\mathrm{\phi }\right)\right)=\frac{\text{i.e}}{\text{German}}\left(\frac{\mathcal{l}\left({\dot{\text{x}}}_{\text{p}}{\ddot{\text{y}}}_{\text{p}}-{\dot{\text{y}}}_{\text{p}}{\ddot{\text{x}}}_{\text{p}}\right)}{{\left({\dot{\text{x}}}_{\text{<figure-callout id="2" label="p" filenames="DE102021112919A1\_0097.png" state="{{state}}">p</figure-callout>}}{}^2+{\dot{\text{y}}}_{\text{p}}{}^2\right)}^{3\text{/}2}}\right)=0$$

For simplicity, the absolute value is approximated as reaching the maximum with some tolerance when the local transverse position (y _p ) reaches its local maximum. Thus, maximum steering minus tolerance is approximated at ẏ _p = 0. It is clear from the equations above that steering increases as tan(φ) increases, which corresponds to a decrease in ẏ _p (ẏ _p ≈ 0). As a result, the corresponding steering equation can be simplified as follows $$\text{tan}\left(\mathrm{\phi }\right)=\frac{\mathcal{l}{\ddot{\text{y}}}_{\text{p}}}{{\stackrel{<figure-callout\; id="2"\; label="\; ̇\; p"\; filenames="DE102021112919A1\_0097.png"\; state="\{\{state\}\}">\dot{}</figure-callout>}{\text{x}}}_{\text{p}}{}^2}.$$

$$\left[3\left(1+\text{K}\right)\right]{\text{t}}^2-\left[2\left(2+\text{K}\right)\right]\text{t}+1=\mathrm{0,}$$

$$\text{t}|{}_{{\dot{\text{y}}}_{\text{p}}=0}=\frac{\left(2+\text{K}\right)\pm \sqrt{{\left(2+\text{K}\right)}^2-3\left(1+\text{K}\right)}}{3\left(1+\text{K}\right)},$$

$${\text{t}}_1|{}_{{\dot{\text{y}}}_{\text{p}}=0}=\frac{\left(2+\text{K}\right)+\sqrt{{\left(2+\text{K}\right)}^2-3\left(1+\text{K}\right)}}{3\left(1+\text{K}\right)},$$

$${\text{t}}_2|{}_{{\dot{\text{y}}}_{\text{p}}=0}=\frac{\left(2+\text{K}\right)-\sqrt{{\left(2+\text{K}\right)}^2-3\left(1+\text{K}\right)}}{3\left(1+\text{K}\right)},$$

$${\dot{\text{x}}}_{\text{P}}\left({\text{t}}_1|{}_{{\dot{\text{y}}}_{\text{p}}=0}\right)=3\text{bM}+\left[6\text{M}\left(1-3\text{b}\right)\right]\left[{\text{t}}_1|{}_{{\dot{\text{y}}}_{\text{p}}=0}\right]-\left[6\text{M}\left(1-3\text{b}\right)\right]{\left[{\text{t}}_1|{}_{{\dot{\text{y}}}_{\text{p}}=0}\right]}^2,$$

$${\dot{\text{x}}}_{\text{P}}\left({\text{t}}_2|{}_{{\dot{\text{y}}}_{\text{p}}=0}\right)=3\text{bM}+\left[6\text{M}\left(1-3\text{b}\right)\right]\left[{\text{t}}_2|{}_{{\dot{\text{y}}}_{\text{p}}=0}\right]-\left[6\text{M}\left(1-3\text{b}\right)\right]{\left[{\text{t}}_2|{}_{{\dot{\text{y}}}_{\text{p}}=0}\right]}^2,$$

$${\ddot{\text{y}}}_{\text{P}}\left(\text{t}|{}_{{\dot{\text{y}}}_{\text{p}}=0}\right)=-6{\text{bMt}}_0\left(2+\text{K}\right)+\left[18{\text{bMt}}_0\left(1+\text{K}\right)\right]\left[\frac{2+\text{K}\pm \sqrt{1+\text{K}+{\text{K}}^2}}{3\left(1+\text{K}\right)}\right],$$

$${\ddot{\text{y}}}_{\text{P}}\left(\text{t}|{}_{{\dot{\text{y}}}_{\text{p}}=0}\right)=\pm 6{\text{bMt}}_0\sqrt{1+\text{K}+{\text{K}}^2},$$

$$\left|\text{tan}\left(\mathrm{\phi }\right)|{}_{{\dot{\text{y}}}_{\text{p}}=0}\right|=\left|\frac{\mathcal{l}{\ddot{\text{y}}}_{\text{p}}}{{\dot{\text{x}}}_{\text{P}}{}^2}\right|=\left(\frac{2\mathcal{l}{\text{bt}}_0}{3\text{M}}\right)\text{Max}\left[\frac{\sqrt{1+\text{K}+{\text{K}}^2}}{{\left[\text{b}+\left[2\left(1-3\text{b}\right)\right]\left[{\text{t}}_{\mathrm{1,2}}|{}_{{\dot{\text{y}}}_{\text{p}}=0}\right]-\left[2\left(1-3\text{b}\right)\right]{\left[{\text{t}}_{\mathrm{1,2}}|{}_{{\dot{\text{y}}}_{\text{p}}=0}\right]}^2\right]}^2}\right],$$

$$\left|\text{tan}\left(\mathrm{\phi }\right)|{}_{\text{t}|{}_{{\dot{\text{y}}}_{\text{p}}=0}}\right|=\left(\frac{2\mathcal{l}{\text{bt}}_0}{3\text{M}}\right)\text{Max}\left[{\text{Term}}_1,{\text{Term}}_2\right]\le \text{tan}\left({\mathrm{\phi }}_{\text{max}}-{\mathrm{\phi }}_{\text{tol}}\right)=\text{T},$$

$${\text{Term}}_1=\frac{\sqrt{1+\text{K}+{\text{K}}^2}}{{\left[\text{b}+\left[2\left(1-3\text{b}\right)\right]\left[{\text{t}}_1|{}_{{\dot{\text{y}}}_{\text{p}}=0}\right]-\left[2\left(1-3\text{b}\right)\right]{\left[{\text{t}}_1|{}_{{\dot{\text{y}}}_{\text{p}}=0}\right]}^2\right]}^2},$$

$${\text{Term}}_2=\frac{\sqrt{1+\text{K}+{\text{K}}^2}}{{\left[\text{b}+\left[2\left(1-3\text{b}\right)\right]\left[{\text{t}}_2|{}_{{\dot{\text{y}}}_{\text{p}}=0}\right]-\left[2\left(1-3\text{b}\right)\right]{\left[{\text{t}}_2|{}_{{\dot{\text{y}}}_{\text{p}}=0}\right]}^2\right]}^2}.$$

Solving for t, where ẏ _p = 0, gives the following: $${\dot{\text{y}}}_{\text{p}}\left(\text{t}\right)=3{\text{bMt}}_0-\left[6{\text{bMt}}_0\left(2+\text{K}\right)\right]\text{t}+\left[9{\text{bMt}}_0\left(1+\text{K}\right)\right]{\text{t}}^2=\mathrm{0,}$$

$$\left[3\left(1+\text{K}\right)\right]{\text{t}}^2-\left[2\left(2+\text{K}\right)\right]\text{t}+1=\mathrm{0,}$$

$$\text{t}|{}_{{\dot{\text{y}}}_{\text{p}}=0}=\frac{\left(2+\text{K}\right)\pm \sqrt{{\left(2+\text{K}\right)}^2-3\left(1+\text{K}\right)}}{3\left(1+\text{K}\right)},$$

$${\text{t}}_1|{}_{{\dot{\text{y}}}_{\text{p}}=0}=\frac{\left(2+\text{K}\right)+\sqrt{{\left(2+\text{K}\right)}^2-3\left(1+\text{K}\right)}}{3\left(1+\text{K}\right)},$$

$${\text{t}}_2|{}_{{\dot{\text{y}}}_{\text{p}}=0}=\frac{\left(2+\text{K}\right)-\sqrt{{\left(2+\text{K}\right)}^2-3\left(1+\text{K}\right)}}{3\left(1+\text{K}\right)},$$

$${\dot{\text{x}}}_{\text{P}}\left({\text{t}}_1|{}_{{\dot{\text{y}}}_{\text{p}}=0}\right)=3\text{bM}+\left[6\text{M}\left(1-3\text{b}\right)\right]\left[{\text{t}}_1|{}_{{\dot{\text{y}}}_{\text{p}}=0}\right]-\left[6\text{M}\left(1-3\text{b}\right)\right]{\left[{\text{t}}_1|{}_{{\dot{\text{y}}}_{\text{p}}=0}\right]}^2,$$

$${\dot{\text{x}}}_{\text{P}}\left({\text{t}}_2|{}_{{\dot{\text{y}}}_{\text{p}}=0}\right)=3\text{bM}+\left[6\text{M}\left(1-3\text{b}\right)\right]\left[{\text{t}}_2|{}_{{\dot{\text{y}}}_{\text{p}}=0}\right]-\left[6\text{M}\left(1-3\text{b}\right)\right]{\left[{\text{t}}_2|{}_{{\dot{\text{y}}}_{\text{p}}=0}\right]}^2,$$

$${\ddot{\text{y}}}_{\text{P}}\left(\text{t}|{}_{{\dot{\text{y}}}_{\text{p}}=0}\right)=-6{\text{bMt}}_0\left(2+\text{K}\right)+\left[18{\text{bMt}}_0\left(1+\text{K}\right)\right]\left[\frac{2+\text{K}\pm \sqrt{1+\text{K}+{\text{K}}^2}}{3\left(1+\text{K}\right)}\right],$$

$${\ddot{\text{y}}}_{\text{P}}\left(\text{t}|{}_{{\dot{\text{y}}}_{\text{p}}=0}\right)=\pm 6{\text{bMt}}_0\sqrt{1+\text{K}+{\text{K}}^2},$$

$$\left|\text{tan}\left(\mathrm{\phi }\right)|{}_{{\dot{\text{y}}}_{\text{p}}=0}\right|=\left|\frac{\mathcal{l}{\ddot{\text{y}}}_{\text{p}}}{{\dot{\text{x}}}_{\text{P}}{}^2}\right|=\left(\frac{2\mathcal{l}{\text{bt}}_0}{3\text{M}}\right)\text{Max}\left[\frac{\sqrt{1+\text{K}+{\text{K}}^2}}{{\left[\text{b}+\left[2\left(1-3\text{b}\right)\right]\left[{\text{t}}_{1.2}|{}_{{\dot{\text{y}}}_{\text{p}}=0}\right]-\left[2\left(1-3\text{b}\right)\right]{\left[{\text{t}}_{1.2}|{}_{{\dot{\text{y}}}_{\text{p}}=0}\right]}^2\right]}^2}\right],$$

$$\left|\text{tan}\left(\mathrm{\phi }\right)|{}_{\text{t}|{}_{{\dot{\text{y}}}_{\text{p}}=0}}\right|=\left(\frac{2\mathcal{l}{\text{bt}}_0}{3\text{M}}\right)\text{Max}\left[{\text{term}}_1,{\text{term}}_2\right]\le \text{tan}\left({\mathrm{\phi }}_{\text{Max}}-{\mathrm{\phi }}_{\text{great}}\right)=\text{T},$$

$${\text{term}}_1=\frac{\sqrt{1+\text{K}+{\text{K}}^2}}{{\left[\text{b}+\left[2\left(1-3\text{b}\right)\right]\left[{\text{t}}_1|{}_{{\dot{\text{y}}}_{\text{p}}=0}\right]-\left[2\left(1-3\text{b}\right)\right]{\left[{\text{t}}_1|{}_{{\dot{\text{y}}}_{\text{p}}=0}\right]}^2\right]}^2},$$

$${\text{<figure-callout id="2" label="term" filenames="DE102021112919A1\_0097.png" state="{{state}}">term</figure-callout>}}_2=\frac{\sqrt{1+\text{K}+{\text{K}}^2}}{{\left[\text{b}+\left[2\left(1-3\text{b}\right)\right]\left[{\text{<figure-callout id="2" label="t" filenames="DE102021112919A1\_0097.png" state="{{state}}">t</figure-callout>}}_2|{}_{{\dot{\text{y}}}_{\text{p}}=0}\right]-\left[2\left(1-3\text{b}\right)\right]{\left[{\text{<figure-callout id="2" label="t" filenames="DE102021112919A1\_0097.png" state="{{state}}">t</figure-callout>}}_2|{}_{{\dot{\text{y}}}_{\text{p}}=0}\right]}^2\right]}^2}.$$

Assuming the term T as the tangent of the maximum steering angle minus a predefined tolerance (φ _tol ), the following formula results for the minimum distance between start and end frames to produce an allowable third order Bezier path $${\text{M}}_{\text{at least}}=\left(\frac{2\mathcal{<figure-callout\; id="0"\; label="l\; bt"\; filenames="DE102021112919A1\_0098.png"\; state="\{\{state\}\}">l</figure-callout>}{\text{bt}}_{}{}_0}{3\text{T}}\right)\text{Max}\left[{\text{<figure-callout id="1" label="term" filenames="DE102021112919A1\_0099.png" state="{{state}}">term</figure-callout>}}_1,{\text{term}}_2\right].$$

Since the tolerance (φ _tol ) could reach relatively large values, for example 10° to 15°, especially for tight situations where the maximum steering is reached much faster than the peak point (ẏ _p = 0) and with b = 1/ 3, a factor of 2 is introduced to compensate for the inaccuracy in the maximum steering assumption at (ẏ _p = 0) and to achieve limited tolerances (φ _tol ≈ 2°). Note that the steering at b = 1/3 is the same at the two peaks, in other words term ₁ = term ₂ and $${\text{M}}_{\text{at least}}|{}_{\text{b}=1\text{/}3}=\left(\frac{4\mathcal{<figure-callout\; id="0"\; label="l\; t"\; filenames="DE102021112919A1\_0098.png"\; state="\{\{state\}\}">l</figure-callout>}{\text{t}}_{}{}_0}{3\text{T}}\right)\sqrt{1+\text{K}+{\text{<figure-callout id="2" label="K" filenames="DE102021112919A1\_0097.png" state="{{state}}">K</figure-callout>}}^2}.$$

To formulate a generic approach to generate a steerable S-shaped Bezier path connecting two parallel frames (x ₁ ,y ₁ ,θ) and (x ₂ ,y ₂ ,θ), Equation [11] is used to obtain to get the path with the smallest Euclidean distance between the two parallel frames. This results in a path with a minimal travel distance.

$${\mathrm{\theta }}_{{\text{r}}_0}={\mathrm{\theta }}_{{\text{r}}_{\text{f}}},$$

$$\text{K}=\mathrm{1,}$$

$${\text{t}}_1|{}_{{\dot{\text{y}}}_{\text{p}}=0}=\frac{1}{2}+\frac{1}{2\sqrt{3}},$$

$${\text{t}}_2|{}_{{\dot{\text{y}}}_{\text{p}}=0}=\frac{1}{2}-\frac{1}{2\sqrt{3}},$$

$${\text{Term}}_1={\text{Term}}_2=\frac{\sqrt{3}}{{\left[\text{b}+\left[2\left(1-3\text{b}\right)\right]\left[\frac{1}{2}+\frac{1}{2\sqrt{3}}\right]-\left[2\left(1-3\text{b}\right)\right]{\left[\frac{1}{2}+\frac{1}{2\sqrt{3}}\right]}^2\right]}^2},$$

$${\text{Term}}_1={\text{Term}}_2=9\sqrt{3},$$

$$\left|\text{tan}\left(\mathrm{\beta }\right)\right|=\frac{\mathrm{\Delta }\text{y}}{\mathrm{\Delta }\text{x}},$$

$${\text{M}}_{\text{min}}=\left(\frac{2\mathcal{l}\text{b}}{3\text{T}}\right)\left(9\sqrt{3}\right)\left(\frac{\mathrm{\Delta }\text{y}}{\mathrm{\Delta }\text{x}}\right),$$

$${\text{M}}_{\text{min}}=\sqrt{\mathrm{\Delta }{\text{x}}^2+\mathrm{\Delta }{\text{y}}^2},$$

$$\sqrt{{\left(\mathrm{\Delta }\text{x}\right)}^2+{\left(\mathrm{\Delta }\text{y}\right)}^2}=\frac{6\text{b}\mathcal{l}}{\text{T}}\frac{\mathrm{\Delta }\text{y}}{\mathrm{\Delta }\text{x}}\sqrt{3},$$

$${\left(\mathrm{\Delta }\text{x}\right)}^2+{\left(\mathrm{\Delta }\text{y}\right)}^2={\left(\frac{6\text{b}\sqrt{3}\mathcal{l}}{\text{T}}\right)}^2{\left(\frac{\mathrm{\Delta }\text{y}}{\mathrm{\Delta }\text{x}}\right)}^2,$$

$${\left({\left(\mathrm{\Delta }\text{x}\right)}^2\right)}^2+{\left(\mathrm{\Delta }\text{y}\right)}^2{\left(\mathrm{\Delta }\text{x}\right)}^2-108{\left(\frac{\text{b}\mathcal{l}}{\text{T}}\right)}^2{\left(\mathrm{\Delta }\text{y}\right)}^2=\mathrm{0,}$$

$${\left(\mathrm{\Delta }\text{x}\right)}^2=\frac{-{\left(\mathrm{\Delta }\text{y}\right)}^2+\sqrt{{\left(\mathrm{\Delta }\text{y}\right)}^4+4\times 108{\left(\frac{\text{b}\mathcal{l}}{\text{T}}\right)}^2{\left(\mathrm{\Delta }\text{y}\right)}^2}}{2},$$

$${\left(\mathrm{\Delta }\text{x}\right)}^2=\left(\frac{\mathrm{\Delta }\text{y}}{2}\right)\left[-\mathrm{\Delta }\text{y}+\sqrt{{\left(\mathrm{\Delta }\text{y}\right)}^2+432{\left(\frac{\text{b}\mathcal{l}}{\text{T}}\right)}^2}\right].$$

6 shows how to derive a generic relationship between Δy and Δx to obtain a drivable (steerable) S-shaped path. Based on this relationship for a given Δy, the minimum value of Δx to obtain a drivable path is found. As in 6 shown, the angle β represents both θ _{r 0} as well as _{θr f} for the two parallel frames. Thus, according to [11], the following equations can be derived $${\text{M}}_{\text{at least}}=\left(\frac{2\mathcal{l}{\text{bt}}_0}{3\text{T}}\right)\text{Max}\left[{\text{term}}_1,{\text{term}}_2\right],$$

$${\mathrm{\theta }}_{{\text{right}}_0}={\mathrm{\theta }}_{{\text{right}}_{\text{f}}},$$

$$\text{K}=\mathrm{1,}$$

$${\text{t}}_1|{}_{{\dot{\text{y}}}_{\text{p}}=0}=\frac{1}{2}+\frac{1}{2\sqrt{3}},$$

$${\text{t}}_2|{}_{{\dot{\text{y}}}_{\text{p}}=0}=\frac{1}{2}-\frac{1}{2\sqrt{3}},$$

$${\text{term}}_1={\text{term}}_2=\frac{\sqrt{3}}{{\left[\text{b}+\left[2\left(1-3\text{b}\right)\right]\left[\frac{1}{2}+\frac{1}{2\sqrt{3}}\right]-\left[2\left(1-3\text{b}\right)\right]{\left[\frac{1}{2}+\frac{1}{2\sqrt{3}}\right]}^2\right]}^2},$$

$${\text{term}}_1={\text{term}}_2=9\sqrt{3},$$

$$\left|\text{tan}\left(\mathrm{\beta }\right)\right|=\frac{\mathrm{\Delta }\text{y}}{\mathrm{\Delta }\text{x}},$$

$${\text{M}}_{\text{at least}}=\left(\frac{2\mathcal{l}\text{b}}{3\text{T}}\right)\left(9\sqrt{3}\right)\left(\frac{\mathrm{\Delta }\text{y}}{\mathrm{\Delta }\text{x}}\right),$$

$${\text{M}}_{\text{at least}}=\sqrt{\mathrm{\Delta }{\text{x}}^2+\mathrm{\Delta }{\text{y}}^2},$$

$$\sqrt{{\left(\mathrm{\Delta }\text{x}\right)}^2+{\left(\mathrm{\Delta }\text{y}\right)}^2}=\frac{6\text{b}\mathcal{l}}{\text{T}}\frac{\mathrm{\Delta }\text{y}}{\mathrm{\Delta }\text{x}}\sqrt{3},$$

$${\left(\mathrm{\Delta }\text{x}\right)}^2+{\left(\mathrm{\Delta }\text{y}\right)}^2={\left(\frac{6\text{b}\sqrt{3}\mathcal{l}}{\text{T}}\right)}^2{\left(\frac{\mathrm{\Delta }\text{y}}{\mathrm{\Delta }\text{x}}\right)}^2,$$

$${\left({\left(\mathrm{\Delta }\text{x}\right)}^2\right)}^2+{\left(\mathrm{\Delta }\text{y}\right)}^2{\left(\mathrm{\Delta }\text{x}\right)}^2-108{\left(\frac{\text{b}\mathcal{l}}{\text{T}}\right)}^2{\left(\mathrm{\Delta }\text{y}\right)}^2=\mathrm{0,}$$

$${\left(\mathrm{\Delta }\text{x}\right)}^2=\frac{-{\left(\mathrm{\Delta }\text{y}\right)}^2+\sqrt{{\left(\mathrm{\Delta }\text{y}\right)}^4+4\times 108{\left(\frac{\text{b}\mathcal{l}}{\text{T}}\right)}^2{\left(\mathrm{\Delta }\text{y}\right)}^2}}{2},$$

$${\left(\mathrm{\Delta }\text{x}\right)}^2=\left(\frac{\mathrm{\Delta }\text{y}}{2}\right)\left[-\mathrm{\Delta }\text{y}+\sqrt{{\left(\mathrm{\Delta }\text{y}\right)}^2+432{\left(\frac{\text{b}\mathcal{l}}{\text{T}}\right)}^2}\right].$$

Equation [13] provides a generic expression relating Δx and Δy. Given an initial frame (x ₁ ,y ₁ ,θ) and a specified Δy to a second parallel target frame with the same orientation, Δx can be calculated to give a steerable S-shaped path 7 connecting the frames. The generated Bezier path 7 is steerable in terms of the steering angle and its rate of change. In addition, the path 7 generated has a smooth progression and is easy to calculate, for example compared to a combination of circular and straight paths. This leads to a robust and inexpensive solution for planning complex paths 7 where there could be a combination of S-shaped manoeuvres. For b=0.4, which is particularly suitable for S-shapes, the following equation is obtained $${\left(\mathrm{\Delta }\text{x}\right)}^2=\left(\frac{\mathrm{\Delta }\text{y}}{2}\right)\left[-\mathrm{\Delta }\text{y}+\sqrt{{\left(\mathrm{\Delta }\text{y}\right)}^2+69.12{\left(\frac{\mathcal{l}}{\text{T}}\right)}^2}\right].$$

An exemplary application in which the invention proves to be particularly advantageous is when planning an S-shaped path to avoid an obstacle 8a, 8b, 8c in the vicinity of the vehicle 1 or when planning a path to a parking space 9, as in 7 shown. Such an obstacle 8a, 8b, 8c requires a minimum transverse distance Δy along the transverse axis 13 of the vehicle 1 in order to maneuver it. Based on this minimum transverse distance Δy, equation [14] can be used to calculate the corresponding minimum longitudinal distance Δx in order to obtain a drivable S-shaped Bezier path 7 that avoids the obstacle 8a, 8b, 8c with the minimum possible route can be bypassed. This is a great advantage when planning the route, for example if the vehicle 1 still has enough space after driving around the obstacle 8a, 8b, 8c to carry out further maneuvers, for example corresponding to a further S-shaped path from the target position 11 to a further target position 12, or perform an equivalent parking maneuver.

Claims (15)

Computer-implemented method for planning a route (7) for a vehicle (1), wherein a target position (11) for the vehicle (1) is determined and the route (7) is planned in such a way that it has a starting position (10) of the vehicle ( 1) connects to the target position (11), characterized in that - to determine the target position (11), a minimum longitudinal distance between the starting position (10) and the target position (11) in a direction parallel to an initial longitudinal axis (14) of the vehicle (1) depending on a predetermined transverse distance between the starting position (10) and the target position (11) in a direction parallel to an initial transverse axis (13) of the vehicle (1) and a maximum steering angle of the vehicle (1); and - the path (7) is planned in such a way that it follows a Bezier curve from the starting position (10) to the target position (11). Computer-implemented method claim 1 , characterized in that the path (7) is planned in such a way that it follows a third degree Bezier curve from the starting position (10) to the target position (11). Computer-implemented method according to one of the present claims, characterized in that the minimum longitudinal distance is determined as a function of a wheelbase of the vehicle (1). Computer-implemented method claim 3 , characterized in that the minimum longitudinal distance according to the relationship $${\left(\mathrm{\Delta }\text{x}\right)}^2=\left(\mathrm{\Delta }\text{y/}2\right)\left[-\mathrm{\Delta }\text{y}+\sqrt{{\left(\mathrm{\Delta }\text{y}\right)}^2+432{\left(\text{blue/tan}\left({\mathrm{\phi }}_{\text{Max}}-{\mathrm{\phi }}_{\text{great}}\right)\right)}^2}\right]$$

is determined, where Δx denotes the minimum longitudinal distance, Δy denotes the lateral distance, 1 denotes the wheelbase, b denotes a parameter within the interval ]0, 0.5], φ _max denotes the maximum steering angle and φ _tol a predefined tolerance value in the interval [ 0, φ _max [ denotes. Computer-implemented method according to one of claims 3 or 4 , characterized in that the path (7) according to the relationships $${\text{x}}_{\text{p}}\left(\text{t}\right)=\left(3\text{bM}\right)\text{t}+3\text{M}\left(1-3\text{b}\right){\text{t}}^2-2\text{M}\left(1-3\text{b}\right){\text{t}}^3$$

and $${\text{y}}_{\text{p}}\left(\text{t}\right)=\left(3{\text{bMt}}_0\right)\text{t}-9{\text{bMt}}_0{\text{t}}^2+6{\text{bMt}}_0{\text{t}}^3$$

is planned, where x _p (t) describes a first spatial coordinate of the path (7) as a function of an independent parameter t within the interval [0, 1], y _p (t) describes a second spatial coordinate of the path (7) as a function of the independent Parameters t describes, t ₀ is a real constant and $\text{M}=\sqrt{\mathrm{\Delta }{\text{x}}^2+\mathrm{\Delta }{\text{y}}^2}.$

Computer-implemented method according to one of the preceding claims, characterized in that - a further target position (12) is determined for the vehicle (1) and the route (7) is planned in such a way that it further connects the target position (11) to the further target position; and - the path (7) is planned in such a way that it follows a further Bezier curve from the target position (11) to the further target position (12). Computer-implemented method claim 6 , characterized in that to determine the further target position (12) a further minimum longitudinal distance between the target position (11) and the further target position (12) in the direction parallel to the initial longitudinal axis (14) of the vehicle (1) depending on a predetermined further transverse distance between the target position (11) and the further target position (12) in the direction parallel to the initial transverse axis (13) of the vehicle (1) and the maximum steering angle and in particular the wheelbase of the vehicle (1) is determined. Computer-implemented method according to one of Claims 6 or 7 , characterized in that the path (7) is planned in such a way that it follows a further third degree Bezier curve from the target position (11) to the further target position (12). Method for at least partially automatically driving a vehicle (1), wherein sensor data representing an environment of the vehicle (1) is generated by an environment sensor system (5) of the vehicle (1), characterized in that - a lateral distance for a for the Vehicle (1) to be planned path (7) is determined by a computing unit (3) of the vehicle (1) based on the sensor data; - the path (7) is planned by the computing unit (3) using a computer-implemented method according to one of the preceding claims depending on the transverse distance; and - the vehicle (1) is guided at least partially automatically according to the planned route (7). procedure after claim 9 , characterized in that - a position of an obstacle (8a, 8b, 8c) in the vicinity of the computing unit (3) is determined based on the sensor data; and - the transverse distance is determined as a function of the position of the obstacle (8a, 8b, 8c). procedure after claim 9 , characterized in that - a target lane in the area is determined by the computing unit (3) based on the sensor data; and - the lateral distance is determined as a function of the target lane. procedure after claim 9 , characterized in that - a position of a parking space (9) in the vicinity of the computing unit (3) is determined based on the sensor data; and - the transverse distance is determined as a function of the position of a parking space (9). Electronic vehicle guidance system (2), which has a computing unit (3) and an environment sensor system (5) for a vehicle (1), the environment sensor system (5) being set up to generate sensor data which represent an environment of the vehicle (1), characterized in that the computing unit (3) is set up to - determine a lateral distance for a route (7) to be planned for the vehicle (1) on the basis of the sensor data; - The way (7) using a computer-implemented method according to claims 1 until 8th to plan depending on the transverse distance; and - to guide the vehicle (1) at least partially automatically according to the planned route (7). Computer program product including instructions which, when executed by a computer system, cause the computer system to perform a computer-implemented method according to any one of Claims 1 until 8th to execute. Computer program product, which includes commands that, if they are from an electronic vehicle guidance system (2). Claim 13 be executed, the electronic vehicle guidance system (2) cause a method according to one of claims 9 until 12 to execute.

Images