DE102022101054A1 - Method for planning a path for an at least partially automated motor vehicle, computer program product, computer-readable storage medium and assistance system - Google Patents

Abstract

The invention relates to a method for planning a path (7) for a motor vehicle (1) from an initial position (10) to a target position (11), with the steps of receiving data representing the surroundings (21) of the motor vehicle (1), determining an end point (15) of an obstacle (16) using the data, determining the target position (11) of the motor vehicle (1) as a function of the detected end point (15) and as a function of the initial position (1 0) of the motor vehicle (1), and planning the path (7) of the motor vehicle (1) from the starting position (10) to the target position (11) depending on the starting position (10), the target position (11) and a maximum turning radius (R) of the motor vehicle (1), the target position (11) being the closest position between the end point (15) and the path (7), and wherein an orientation (θ1) of the vehicle (1) in the target position (11) by means of the maximum turning radius (R) and the end point (15). The invention also relates to a computer program product, a computer-readable storage medium and an assistance system (2).

Description

The invention relates to a method for planning a path for an at least partially automated motor vehicle from a starting position of the motor vehicle to a target position in an area surrounding the motor vehicle using an assistance system of the motor vehicle. Furthermore, the invention relates to a computer program product, a computer-readable storage medium and an assistance system.

Path planning, also known as motion planning, is one of the most important sub-functions for various applications for automated driver assistance systems or automated driving systems in high-speed as well as low-speed maneuvers. Existing path planning methods use straight lines, circular segments, or combinations thereof. This results in a low degree of smoothness of the path and also generally in longer paths than required to connect an initial position to a target location of the vehicle.

It is an object of the invention to provide a method, a computer program product, a computer-readable storage medium and an assistance system, through which path planning for a motor vehicle can be presented in an improved manner.

This object is achieved by a method, a computer program product, a computer-readable storage medium and an assistance system according to the independent claims. Advantageous forms of combinations are presented in the dependent claims.

One aspect of the invention relates to a method for planning a path for an at least partially automated motor vehicle from a starting position of the motor vehicle to a target position in an area surrounding the motor vehicle using an assistance system of the motor vehicle. Data of the surroundings are recorded, in particular by means of surroundings sensors of the motor vehicle. The sensor can contain ultrasonic sensors and/or a camera and/or radar and/or lidar, in particular flash lidar, which is set up to provide the data. The data can be sent to an electronic computing device, for example a control unit, in particular a central control unit, which is connected to all environment sensors. An end point of an obstacle in the environment is determined. The target position of the motor vehicle is determined depending on the detected end point and depending on the starting position of the motor vehicle. The planning of the path of the motor vehicle from the starting position to the target position is carried out depending on the starting position, the target position and a maximum turning radius of the motor vehicle, the target position being the closest position between the end point and the path and with an orientation of the vehicle in the target position is constructed using the maximum turning radius and the end point.

"Maximum Turn Radius" means a radius that can be achieved when an actuator turns the steered wheels to a limit determined by software and/or mechanical limitations of a steering. For example, the limit for the actuator may be set to protect the actuator or steering from damage before the mechanical limit is reached.

This has the benefit of being able to provide higher robust motion planning that can cover various tangential maneuvers, particularly tight ones, while being simple as well as being expandable to run on inexpensive targets and paths with multiple steps in the same special processing timing constraints. This is in addition to optimizing the number of steps associated with driving actions, since particularly tangential maneuver generation could take the best advantage of the available space.

In order to have a robust dynamic motion planning, especially in tight parking situations, a simplified Bezier path is generated. The main idea of the invention is to provide simple Bezier path calculation steps that can run on inexpensive targets. This is in addition to the ability to provide general functionality by providing paths that are steerable and tangent to a detected object/obstacle, referred to as an end point. In such a way, the proposed simplified calculations can be viewed as a general library to be used for robust motion planning in different situations, where planning paths tangent to the end point could result in taking best advantage of the available space and maneuvers are performed in a safe manner with fewer driving actions.

In other words, the invention introduces a method to generate navigable Bezier paths tangent to the end point with a tailored maximum wheel angle using the closed form of the Bezier curve equations.

Here and below, all steps of the method, which can be a computer-implemented method in particular, can be executed by an electronic computing device, in particular an electronic computing device of the motor vehicle. The electronic computing device may include one or more sub-units, which may or may not be specially distributed. For example, one or more electronic control units, ECUs, of the vehicle may include the electronic computing device, or vice versa. The initial position of the motor vehicle corresponds to a predefined or predetermined position of the motor vehicle. For example, it may correspond to a current position of the motor vehicle at a time when the path planning method is executed. On the other hand, the target position is not predefined but can be considered as the result or intermediate result of the method in that it is determined by defining the initial position and determining the path connecting the initial position to the target position. However, one or more constraints may apply that restrict the target position. The boundary conditions may be predefined in some implementations.

It is noted that the concept of path planning should not be confused with the concept of path prediction. While path planning aims at finding a desired path for the motor vehicle, for example to control actuators of the vehicle such that the vehicle actually follows the planned path, path prediction aims at estimating which path the vehicle is likely to follow on the basis his actual situation or his actual driving condition.

In particular, an initial position of the motor vehicle is defined by the initial position of the motor vehicle in combination with an initial orientation of the motor vehicle in the initial position. Analogously, a target position of the motor vehicle is defined by the target position in combination with a target orientation of the motor vehicle in a target position. The initial orientation and the target orientation can be understood as respective orientations of a motor vehicle coordinate system, which is originally connected to the vehicle, with reference to a reference coordinate system, which is rigidly connected to the environment of the vehicle, for example to a road on which the motor vehicle is driving become. The choice of the orientation of the reference coordinate system is fixed, but arbitrary. Therefore, for example, it can be chosen to be identical to the motor vehicle coordinate system according to the initial orientation or position of the motor vehicle. The vehicle coordinate system can be generated, for example, by the longitudinal axis, the transverse axis and a normal axis of the motor vehicle, which is perpendicular to the longitudinal axis and to the transverse axis. The initial longitudinal and initial transverse axis or the target longitudinal and target transverse axis can be understood as the longitudinal axis and transverse axis of the motor vehicle according to the initial orientation or the target orientation, in particular with reference to the reference coordinate system. The longitudinal axis can, for example, correspond to a direction of movement of the motor vehicle if the steering angle of the motor vehicle is equal to zero. The plane created by the longitudinal axis and the lateral axis may be parallel to a plane created by the tire contact points of the road surface.

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

The predetermined lateral distance may be predetermined by the motor vehicle itself, for example using a sensing device for sensing the surroundings of the motor vehicle in order to assess the surroundings of the motor vehicle and to determine the desired lateral distance for path planning. The predetermined transverse distance can also be predefined per se, so that it is given by a constant value. In any case, while the process of planning the path is being carried out, the transverse distance in particular remains constant.

The minimum longitudinal distance can be viewed as the minimum possible longitudinal distance that can be realized by the motor 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 initial position to the target position. Another constraint can be imposed by the degree of the Bezier curve, for example. The minimum longitudinal distance can be may be different if a third degree Bezier curve is used compared to a situation where a second or fourth degree Bezier curve is used. Furthermore, the maximum steering angle of the motor vehicle also corresponds to a boundary condition. The larger the maximum steering angle of the motor vehicle, the shorter the minimum longitudinal distance can be. The steering angle of the motor vehicle can correspond, for example, to a steering wheel angle or, in other words, to an angle of steerable wheels of the motor vehicle, for example in relation to the longitudinal axis of the motor vehicle. A maximum rate of change of steering angle may also impose a respective constraint. In addition, for example, a wheelbase of the motor vehicle can correspond to a boundary condition. The shorter the wheelbase, the shorter the minimum longitudinal spacing can be. Furthermore, a predefined tolerance value can also impose a constraint. 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 in the motor vehicle. The planned path can, for example, describe a movement of a point on an axis of the motor vehicle, for example at the lateral center of the motor vehicle. The path can also describe the movement of a point on a front bumper of the motor vehicle etc. In particular, the path corresponds to a two-dimensional curve within a plane that is perpendicular to the axis of the motor vehicle, for example. Consequently, the Bezier curve has two components, 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 can be given by (x(t),y(t)), where t takes on values in the predefined range, for example in the range of [0,1]. It is noted that the planned path describes the spatial aspect of the movement of the vehicle. In particular, the actual movement of the motor vehicle can be described by considering t as a function of time. However, t does not correspond to time itself. 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. The Bezier curve, in particular the components of the Bezier curve, are determined, for example, as Bezier curves of at least the third degree, in particular the third degree.

All explanations relating to the Bezier curve, the starting position and the target position are transferred analogously to a further Bezier curve, the target position or the further target position.

Using the Bezier curve and the other Bezier curve for path planning allows the use of closed form equations for the path that can be dynamically planned and adjusted in a fast manner. Furthermore, compared to paths consisting of straight and circular segments, for example, improved smoothness and less required steering actions and therefore an improved level of comfort for a user can be achieved.

The use of the Bezier curve and the further Bezier curve, which will be explained later, enables a robust and dynamic movement planning for the vehicle, especially in situations with limited space, especially in tight parking situations or other situations where contact with an external object is to be avoided. Furthermore, the calculations required to do this do not require extensive computational resources and can therefore also be performed by embedded systems as commonly used in automobiles and other motor vehicles.

Furthermore, the use of the Bezier curve enables planning of simple as well as complex paths including S-shaped paths, straight paths or circular paths within a minimum possible driving distance. The use of the further Bezier curve further increases the achievable complexity of the path without increasing the computational resources required.

According to one embodiment, an assist point is determined, where a distance from the assist point to the start position is equal to the maximum turning radius, and where the target position is on a line constructed by the end point and the assist point.

This can mean that the line intersects the end point and the auxiliary point.

According to a further embodiment, the orientation of the vehicle in the target position is perpendicular to the line constructed by the end point and the auxiliary point.

Therefore, when the vehicle passes through the target position, the vehicle has a tangential orientation with respect to the end point and the line.

According to another embodiment, an extension of a rear axle of the vehicle intersects the auxiliary point when the vehicle is in the initial position.

This means that the help point is determined based on the position and orientation of the rear axle in combination with the maximum steering radius and the end point of the obstacle.

According to another embodiment, the end point is a corner point that represents a corner of the obstacle.

According to a further specific embodiment, a map of the surroundings of the motor vehicle is provided, with sensor data from at least two, three or four different types of surroundings sensors being provided for the map.

Sensor data from ultrasonic sensors and/or at least one camera and/or a lidar and/or radar can, for example, be combined and made available on the map.

According to a further embodiment, the path is planned in such a way that a safe distance from the motor vehicle to the end point is maintained. In particular, the safety distance from a side wall of the motor vehicle and from the end point is measured. The safety distance can be predetermined by the assistance system or can be adjusted by a user of the motor vehicle. Therefore, a collision between the motor vehicle and the obstacle, particularly a collision between the end point and the motor vehicle, is prevented.

In another embodiment, the path is planned using a third degree Bezier curve. In particular, the path is planned to follow a third degree Bezier curve from the initial position to the target position and/or the path is planned to follow another third degree Bezier curve from the target position to another 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 of the third degree and the remaining component is of the third or lower degree. In such implementations, the mathematical representations of the path still have a relatively low complexity, while already the most relevant types of paths, such as straight paths, circular paths or an S-shaped path, and in the wider Bezier curve double S-shaped paths , can be represented accurately. A particularly efficient, robust and computationally favorable solution for path planning is therefore achieved.

According to another embodiment, a width of the motor vehicle is taken into account when planning the path. For example, the width can be stored in a memory device of the motor vehicle's electronic computing device. In particular, when the longitudinal axis of the motor vehicle is given in the center of the motor vehicle, half the width of the motor vehicle is taken into account. In particular, by considering the width, a collision between the end point and the motor vehicle is prevented. Furthermore, a reliable path to the target position and/or the further target position can be implemented by taking into account the width of the motor vehicle.

In another embodiment, a path into a parking space, in particular into a vertical parking space, is planned by the assistance system. In particular, the end point can be a corner point of the parking space. In particular, the motor vehicle can plan a path into the parking space by reversing, which is in particular perpendicular to the direction of travel of the motor vehicle. Therefore, a robust path planning into the parking space can be realized using the Bezier curve. In particular, when the parking space is very tight, the method can also plan a robust route into the tight parking space.

In another advantageous embodiment, a further target position is planned from the target position to a parking position in the parking space using a further Bezier curve. In particular, after arriving at the target position, the detection device can detect the parking space as a whole, in particular without detecting restrictions caused by the end point or the obstacle, wherein another target position can be determined as a parking position. The planned path from the target position to the further target position/to the parking position can be referred to as a further Bezier curve. Therefore, a smooth parking maneuver can be realized by the motor vehicle.

Furthermore, it has proven to be advantageous if a safety value is subtracted from the maximum turning radius of the motor vehicle in order to obtain a safety turning radius and the path is planned using the safety turning radius. Therefore, the path is not directly planned using the maximum turning radius, instead the path is planned using the safety turning radius, and a more comfortable way of driving along the planned one can be realized. Furthermore, when an unexpected situation happens, the motor vehicle can use the maximum turning radius to avoid collision with an object in the vicinity.

According to another specific embodiment, an initial position of the motor vehicle in the initial position and/or a target position in the target position are taken into account for path planning. In particular, the motor vehicle cannot be oriented directly perpendicular to the parking space. In the initial position, therefore, a different attitude from the vertical position can be used to plan the path. Furthermore, the target position cannot be perpendicular or parallel to the parking position. Therefore, the target location can also be taken into account to provide a robust way of planning the path to the target position.

In another advantageous embodiment, a tangential path from the motor vehicle in the starting position to the end point is planned as the path of the motor vehicle. Therefore, the Bezier curve is planned tangential to the endpoint for a perpendicular parking scenario. The path is planned in such a form that a tangential path is planned for the motor vehicle when the motor vehicle reverses into the parking space. The first reverse stroke must cause the motor vehicle to dive deeply and safely into the parking space, which requires the motor vehicle to be as close to the end point as possible. The end point considered is that of the obstacle/gap object last detected by the detection device, which is called the object/obstacle. Consequently, the reverse path is planned to be as close as possible to the circle tangent to the endpoint while maintaining a safety margin. The planning of the tangential first backward step in a perpendicular parking scenario must cause the motor vehicle to be better able to delve deeper into the parking space, allowing for better detection of the space structure. Furthermore, the motor vehicle may be better able to park in a reduced number of steps and within a limited driving range, particularly when the final parking position of the parking space is shifted due to space remeasurements after the first reverse plunge stroke. These advantages have a great impact in improving automatic parking, especially for tight parking situations.

Furthermore, a speed of the motor vehicle can be taken into account when planning the path. In particular, the steering angle of the motor vehicle depends on the current speed of the motor vehicle. When the motor vehicle drives very slowly, a high driving angle can be realized. When the motor vehicle drives faster, the steering angle is limited. For example, path planning for driving into a parking space uses a maximum speed of 20 km/h for planning the path.

In another embodiment, a direction of movement of the motor vehicle while following the path is taken into account when planning the path. In particular, when the motor vehicle is planning the path into a parking space, a reversing direction of the motor vehicle is used to plan the path. If the motor vehicle is planning an overtaking maneuver, for example, a forward direction of the motor vehicle is taken into account. In particular, the direction of travel has an effect on the steering angle of the motor vehicle, in particular on the orientation of the motor vehicle as a function of the direction of travel.

In another advantageous embodiment, control points of the Bezier curve are assumed to be symmetrical. Therefore, a closed form of the Bezier curve can be realized by adopting the symmetrical control points. In particular, by assuming the control points to be symmetric, a computationally efficient method for planning the path across the Bezier curve is provided.

As already mentioned, the method is in particular a computer-implemented method. Therefore, a further aspect of the invention relates to a computer program product with program code means, which are stored in a computer-readable medium, in order to carry out the method for determining a state of a tailgate according to the preceding aspect, when the computer program product is executed on a processor of an electronic computing device.

Yet another aspect of the invention relates to an electronic computing device and/or a computer-readable storage medium including a computer program product according to the preceding aspect.

Another aspect of the invention relates to an assistance system for planning a path for an at least partially automated motor vehicle from a starting position of the motor vehicle to a target position in an area surrounding the motor vehicle, which includes at least one detection device and at least one electronic computing device, the assistance system for performing a method according to the preceding aspect is set up. In particular, the method is carried out by the assistance system.

Yet another aspect of the invention relates to a motor vehicle with the assistance system according to the preceding aspect. In particular, the motor vehicle is at least partially automated or automated.

Advantageous forms of the method are to be regarded as advantageous forms of the computer program product, the computer-readable storage medium, the assistance system and the motor vehicle. The assistance system and the motor vehicle therefore contain means for carrying out the method.

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 alone in the figures can be used not only in the combination specified in each case, but also in other combinations without departing from the scope of the invention . Embodiments of the invention are therefore also to be regarded as included and disclosed which are not explicitly shown and explained in the figures, but which result from the explained embodiments and can be generated by means of separate combinations of features. Versions and combinations of features are also to be regarded as disclosed which therefore do not have all the features of an originally formulated independent claim. Furthermore, embodiments and combinations of features, in particular through the embodiments presented above, are to be regarded as disclosed which go beyond or deviate from the combinations of features presented in the back references of the claims.

The invention will now be explained in more detail on the basis of preferred embodiments and with reference to the accompanying drawings.

• 1 zeigt eine schematische Darstellung eines Kraftfahrzeugs mit einer beispielhaften Implementierung eines Assistenzsystems gemäß der Erfindung;
• 2 zeigt ein Beispiel der Bezierkurve;
• 3 zeigt ein Beispiel von Randpunkten und Kontrollpunkten einer Bezierkurve in globalen Koordinaten;
• 4 zeigt ein Beispiel von Randpunkten und Kontrollpunkten einer Bezierkurve in lokalen Koordinaten;
• 5 zeigt ein Beispiel einer Ausführungsform gemäß der Erfindung; und
• 6 zeigt eine andere Ausführungsform gemäß dem Verfahren der Erfindung.

These show in:

• 1 shows a schematic representation of a motor vehicle with an exemplary implementation of an assistance system according to the invention;
• 2 shows an example of the Bezier curve;
• 3 shows an example of edge points and control points of a Bezier curve in global coordinates;
• 4 shows an example of edge points and control points of a Bezier curve in local coordinates;
• 5 shows an example of an embodiment according to the invention; and
• 6 shows another embodiment according to the method of the invention.

In the figures, the same elements are marked with the same reference numbers.

1 shows a schematic representation of a vehicle 1 with an exemplary implementation of an assistance system 2 according to the invention.

The assistance system 2 has an electronic computing device 3 that can plan a path 7 for the vehicle 1 that connects an initial position 10 of the vehicle 1 with a target position 11 and the target position 11 with a further target position 12 (see FIG 2 ). The path 7 is designed as a combination of two Bezier curves 17, 20 (see 4 ), in particular as two third-order Bezier curves 17, 20, one of them connecting the starting position 10 to the target position 11 and the other connecting the target position 11 to the further target position 12.

The assistance system 2 can have a detection device 5, for example. The detection device 5 can contain, for example, a camera and/or a radar system and/or a lidar system and/or an ultrasonic sensor system. The electronic computing device 3 receives input data with environment sensor data generated by the detection device 5 and can plan the path 7 depending on the input data.

The assistance system 2 optionally contains a state sensor system 4 and/or an input device 6. The state sensor system 4 can contain, for example, a speed sensor, a steering angle sensor and/or a yaw rate sensor. The input device 6 can contain, for example, turn signals of the vehicle 1 that are intended to be actuated by a driver of the vehicle 1 . The condition sensor system 4 can generate respective condition sensor data regarding a current condition of the vehicle and/or an input device 6 can generate a respective user input signal. The user input signal and/or the state sensor data may also be part of the input data and the electronic computing device 3 may consider them for planning the path 7 in some implementations.

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

2 Figure 1 shows an exemplary Bezier path 7 planned according to the invention and a path 7' with the same starting position 10 and the same target positions 11, 12, but consisting of rectilinear segments and circular segments. The path 7 planned according to the invention has a significantly increased smoothness and requires less steering action compared to the other path 7'.

In the following it is explained how an asymptotic optimal part of the path 7, which connects the initial position 10 with the target position 11 by means of a Bezier curve, can be obtained. All explanations and calculations apply accordingly to the portion of path 7 that connects target position 11 to further target position 12 . In this case, the starting position 10 is replaced by the target position 11 in the following explanations and the target position is replaced by the further target position 12 in the following explanations.

The initial 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 22, 23 of a cubic or third degree Bezier curve.

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

$$\begin{array}{l}\text{falls}\left(x_{ƒ}-x_0\right)<0\\ \text{ falls}\alpha >\text{0}\\ \alpha =\alpha -\text{180}\\ \text{ ansonsten}\\ \alpha =\alpha +\text{180}\\ {\theta }_{r_0}={\theta }_0-\alpha ,normiertaufdenBereich[-\mathrm{180,180}[\\ {\theta }_{r_f}={\theta }_f-\alpha ,normiertaufdenBereich[-\mathrm{180,180}[\end{array}$$

$$M=\sqrt{{\left(x_{ƒ}-x_0\right)}^2+{\left(y_{ƒ}-y_0\right)}^2}$$

$$t_0=tan\left({\theta }_{r_0}\right)$$

$$t_f=tan\left({\theta }_{r_f}\right)$$

Regarding 3 one has $$a={\text{tan}}^{-1}\left(\frac{y_{ƒ}-y_0}{x_{ƒ}-x_0}\right)$$

$$\begin{array}{l}\text{if}\left(x_{ƒ}-x_0\right)<0\\ \text{if}a>\text{0}\\ a=a-\text{180}\\ \text{otherwise}\\ a=a+\text{180}\\ {\theta }_{{\mathrm{right}}_0}={\mathrm{<figure-callout\; id="0"\; label="\theta "\; filenames="DE102022101054A1\_0065.png"\; state="\{\{state\}\}">\theta </figure-callout>}}_0-a,nO\mathrm{right}mie\mathrm{right}ta\mathrm{and}f\mathrm{i.e}enBe\mathrm{right}eicH[-\mathrm{180,180}[\\ {\theta }_{{\mathrm{right}}_f}={\theta }_f-a,nO\mathrm{right}mie\mathrm{right}ta\mathrm{and}f\mathrm{i.e}enBe\mathrm{right}eicH[-\mathrm{180,180}[\end{array}$$

$$M=\sqrt{{\left(x_{ƒ}-x_0\right)}^2+{\left(y_{ƒ}-y_0\right)}^2}$$

$${\mathrm{<figure-callout\; id="0"\; label="t"\; filenames="DE102022101054A1\_0065.png"\; state="\{\{state\}\}">t</figure-callout>}}_0=tan\left({\theta }_{{\mathrm{right}}_0}\right)$$

$$t_f=tan\left({\theta }_{{\mathrm{right}}_f}\right)$$

With regard to 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 . Using a pseudocode notation, as follows

The third degree Bezier equation can be written in its general form as follows: $$\mathrm{right}\left(t\right)={\left(1-t\right)}^3{\mathrm{<figure-callout\; id="0"\; label="R"\; filenames="DE102022101054A1\_0065.png"\; state="\{\{state\}\}">R</figure-callout>}}_0+3{\left(1-t\right)}^2\mathrm{<figure-callout\; id="1"\; label="t\; R"\; filenames="DE102022101054A1\_0063.png,DE102022101054A1\_0066.png"\; state="\{\{state\}\}">t</figure-callout>}{R}_{}{}_1+3\left(1-t\right)t^2{\mathrm{<figure-callout\; id="2"\; label="R"\; filenames="DE102022101054A1\_0063.png"\; state="\{\{state\}\}">R</figure-callout>}}_2+t^3{R}_3$$

beziehungsweise $\left[r\left(t=0\right)\frac{\dot{r}\left(t=0\right)}{3}\right].$

Durch Ersetzen mit den Randbedingungen in den kubischen Beziergleichungen für sowohl x_P(t) als auch y_P(t) können die folgenden Gleichungen erhalten werden. $$x_P\left(t\right)={\left(1-t\right)}^{2}t{\dot{x}}_P\left(t=0\right)+3\left(1-t\right)t^{2}M-\left(1-t\right)t^2{\dot{x}}_P\left(t=1\right)+t^{3}M$$

$$y_P\left(t\right)={\left(1-t\right)}^{2}t{\dot{y}}_P\left(t=0\right)-\left(1-t\right)t^2{\dot{y}}_P\left(t=1\right)$$

wobei $$\frac{{\dot{y}}_P\left(t=0\right)}{{\dot{x}}_P\left(t=0\right)}=t_0\frac{{\dot{y}}_P\left(t=1\right)}{{\dot{x}}_P\left(t=1\right)}=t_{ƒ}$$

where t is an independent variable in the range [0, 1]. R0 and R3 are the values of r(t=0) and r(t=1) respectively. R1 and R2 are the control points 22, 23 and the values of $\left[\frac{\dot{\mathrm{right}}\left(t=0\right)}{3}+\mathrm{right}\left(t=0\right)\right]$

respectively $\left[\mathrm{right}\left(t=0\right)\frac{\dot{\mathrm{right}}\left(t=0\right)}{3}\right].$

Substituting with the boundary conditions in the cubic Bezier equations for both x _P (t) and y _P (t), the following equations can be obtained. $$x_P\left(t\right)={\left(1-t\right)}^{2}t{\dot{x}}_P\left(t=0\right)+3\left(1-t\right)t^{2}M-\left(1-t\right)t^2{\dot{x}}_P\left(t=1\right)+t^{3}M$$

$$y_P\left(t\right)={\left(1-t\right)}^{2}t{\dot{y}}_P\left(t=0\right)-\left(1-t\right)t^2{\dot{y}}_P\left(t=1\right)$$

whereby $$\frac{{\dot{y}}_P\left(t=0\right)}{{\dot{x}}_P\left(t=0\right)}=t_0\frac{{\dot{y}}_P\left(t=1\right)}{{\dot{x}}_P\left(t=1\right)}=t_{ƒ}$$

und $X_2=x_P\left(t=1\right)-\frac{{\dot{x}}_P\left(t=1\right)}{3}=\left(1-b\right)M.$

$$

Dann wird der asymptotische optimale Bezierpfad in Anbetracht der folgenden Randgleichungen erhalten. $$x_P\left(t=0\right)=0$$

$$x_P\left(t=1\right)=M$$

$$y_P\left(t=0\right)=0$$

$$y_P\left(t=1\right)=0$$

$${\dot{x}}_P\left(t=0\right)=3bM$$

$${\dot{x}}_P\left(t=1\right)=3bM$$

$${\dot{y}}_P\left(t=0\right)=t_0{\dot{x}}_P\left(t=0\right)=3bM{t}_0$$

$${\dot{y}}_P\left(t=1\right)=t_{ƒ}{\dot{x}}_P\left(t=1\right)=3bM{t}_{ƒ}$$

Assuming symmetric control points 22,23 for x _p (t) at {bM,(1-b)M} leads to control points 22,23 $X_1=\frac{{\dot{x}}_P\left(t=0\right)}{3}+x_P\left(t=0\right)=bM$

and ${\mathrm{<figure-callout\; id="2"\; label="X"\; filenames="DE102022101054A1\_0063.png"\; state="\{\{state\}\}">X</figure-callout>}}_2=x_P\left(t=1\right)-\frac{{\dot{x}}_P\left(t=1\right)}{3}=\left(1-b\right)M.$

$$

Then the asymptotic optimal Bezier path is obtained considering the following boundary equations. $$x_P\left(t=0\right)=0$$

$$x_P\left(t=1\right)=M$$

$$y_P\left(t=0\right)=0$$

$$y_P\left(t=1\right)=0$$

$${\dot{x}}_P\left(t=0\right)=3bM$$

$${\dot{x}}_P\left(t=1\right)=3bM$$

$${\dot{y}}_P\left(t=0\right)=t_0{\dot{x}}_P\left(t=0\right)=3\mathrm{<figure-callout\; id="0"\; label="b\; M\; t"\; filenames="DE102022101054A1\_0065.png"\; state="\{\{state\}\}">b</figure-callout>}M{t}_{}{}_0$$

$${\dot{y}}_P\left(t=1\right)=t_{ƒ}{\dot{x}}_P\left(t=1\right)=3bM{t}_{ƒ}$$

$$\therefore y_P\left(t\right)=3bM{t}_0{\left(1-t\right)}^{2}t-3bM{t}_{ƒ}\left(1-t\right)t^2$$

Substituting into (2) and (3) we get $$\therefore x_P\left(t\right)=3bM{\left(1-t\right)}^{2}t+3\left(1-t\right)t^{2}M-3bM\left(1-t\right){\mathrm{<figure-callout\; id="2"\; label="t"\; filenames="DE102022101054A1\_0063.png"\; state="\{\{state\}\}">t</figure-callout>}}^2+t^{3}M$$

$$\therefore y_P\left(t\right)=3bM{t}_0{\left(1-t\right)}^{2}t-3bM{t}_{ƒ}\left(1-t\right){\mathrm{<figure-callout\; id="2"\; label="t"\; filenames="DE102022101054A1\_0063.png"\; state="\{\{state\}\}">t</figure-callout>}}^2$$

$$y_P\left(t\right)=\left[3bM{t}_0\right]t-\left[3bM\left(2t_0+t_{ƒ}\right)\right]t^2+\left[3bM\left(t_0-t_{ƒ}\right)\right]t^3$$

Consequently, formulas for the third-order polynomial Bezier curve with control points 22, 23 at bM and (1 - b)M are obtained as follows: $$x_P\left(t\right)=\left[3bM\right]t+\left[3M\left(1-3b\right)\right]{\mathrm{<figure-callout\; id="2"\; label="t"\; filenames="DE102022101054A1\_0063.png"\; state="\{\{state\}\}">t</figure-callout>}}^2-\left[2M\left(1-3b\right)\right]t^3$$

$$y_P\left(t\right)=\left[3bM{t}_0\right]t-\left[3bM\left(2{\mathrm{<figure-callout\; id="0"\; label="t"\; filenames="DE102022101054A1\_0065.png"\; state="\{\{state\}\}">t</figure-callout>}}_0+t_{ƒ}\right)\right]{\mathrm{<figure-callout\; id="2"\; label="t"\; filenames="DE102022101054A1\_0063.png"\; state="\{\{state\}\}">t</figure-callout>}}^2+\left[3bM\left({\mathrm{<figure-callout\; id="0"\; label="t"\; filenames="DE102022101054A1\_0065.png"\; state="\{\{state\}\}">t</figure-callout>}}_0-t_{ƒ}\right)\right]t^3$$

$$y=y_0+x_P\text{ sin}\left(\alpha \right)+y_P\text{ cos}\left(\alpha \right)$$

Choosing the control points 22, 23 in these positions leads to a simplification in the equations and the consequent acceptance checks. Moreover, it can be considered as an asymptotic optimal path. The equation of the generated Bezier path can be represented in global coordinates as follows $$x=x_0+x_P\text{cos}\left(a\right)-y_P\text{sin}\left(a\right)$$

$$y={\mathrm{<figure-callout\; id="0"\; label="y"\; filenames="DE102022101054A1\_0065.png"\; state="\{\{state\}\}">y</figure-callout>}}_0+x_P\text{sin}\left(a\right)+y_P\text{cos}\left(a\right)$$

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

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

A front wheel steering angle (φ) acceptability check 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 reached, respectively. $$\text{tan}\left(\phi \right)=\frac{\mathcal{l}\dot{\theta }}{v}=\frac{\mathcal{l}\left({\dot{x}}_P{\ddot{y}}_P-{\dot{y}}_P{\ddot{x}}_P\right)}{{\left({\dot{x}}_{\mathrm{<figure-callout\; id="2"\; label="P"\; filenames="DE102022101054A1\_0063.png"\; state="\{\{state\}\}">P</figure-callout>}}{}^2+{\dot{y}}_P{}^2\right)}^{3/2}}\le \text{tan}\left({\phi }_{max}\right)$$

$$\dot{\phi }=\left[\frac{\left({\dot{x}}_{\mathrm{<figure-callout\; id="2"\; label="p"\; filenames="DE102022101054A1\_0063.png"\; state="\{\{state\}\}">p</figure-callout>}}{}^2+{\dot{y}}_p{}^2\right)\left({\dot{x}}_p{\stackrel{⃛}{y}}_p-{\dot{y}}_p{\stackrel{⃛}{x}}_p\right)-3\left({\dot{x}}_p{\ddot{y}}_p-{\dot{y}}_p{\ddot{x}}_p\right)\left({\dot{x}}_p{\ddot{x}}_p+{\dot{y}}_p{\ddot{y}}_p\right)}{{\left({\dot{x}}_{\mathrm{<figure-callout\; id="2"\; label="p"\; filenames="DE102022101054A1\_0063.png"\; state="\{\{state\}\}">p</figure-callout>}}{}^2+{\dot{y}}_p{}^2\right)}^3+{\mathcal{l}}^2\left({\dot{x}}_p{\ddot{y}}_p-{\dot{y}}_p{\ddot{x}}_p{}^2\right)}\right]\times \mathcal{l}v\le {\dot{\phi }}_{max}$$

5 shows a schematic view according to an embodiment of the method according to the invention. In particular shows 5 the method for planning the path 7 for the motor vehicle 1 from the initial position 10 to the target position 11 in the vicinity of the motor vehicle 1 by the assistance system 2.

The environment 21 is captured by the motor vehicle 1 . An end point 15 of an obstacle 16 in the area 21 is detected. The target position 11 of the motor vehicle 1 is determined depending on the detected end position 15 and depending on the starting position 10 of the motor vehicle 1 . The path 7 of the motor vehicle 1 is planned from the starting position 10 to the target position 11 depending on the starting position 10, the target position 11 and a maximum turning radius R of the motor vehicle 1, the path 7 being planned using a Bezier curve 17.

In particular, the path 7 is planned in such a way that a safety distance 18 from the motor vehicle 1 to the end point 15 is maintained. Furthermore, the path 7 is planned using a third-degree Bezier curve 17 . Furthermore, the width W, in particular half the width W/2, of the motor vehicle 1 is taken into account when planning the path 7 .

5 shows that a path 7 into a parking space 19, in particular into a vertical parking space 19, is planned by the assistance system 2. Furthermore, the further target position 12 is planned from the target position 11 into a parking position in the parking space 19 using a further Bezier curve 20 .

Furthermore shows 1 that a safety value is subtracted from the maximum turning radius R of the motor vehicle 1 to obtain a safety turning radius and the path 7 is planned using the safety turning radius.

According to another embodiment described in 5 is shown, a tangential path 7 from the motor vehicle 1 to the end point 15 is planned as the path 7 of the motor vehicle 1 . Furthermore, a speed of the motor vehicle 1 is taken into account when planning the path 7 . In another specific embodiment, a direction of movement of motor vehicle 1 while tracking path 7 is taken into account when planning path 7 .

Given the geometry of 5 the target frame representing the construction point tangent to end point 15 can be calculated from the initial (current) frame, the intended turning radius and the designed safety tolerance as follows. It should be noted that the intended turning radius of the motor vehicle 1 may be the designed minimum turning radius corresponding to the peak rearward steering angle of the front wheels (φ_max). To ensure the passability of the preserved path , a special defined steering tolerance (φ_tolerance) is subtracted from the given maximum allowable steering.

Regarding 5 all positions are shown with respect to specific gap coordinates (x,y). Such coordinates are considered to be global coordinates, and the generated Bezier curve 17 is shown. The vehicle frame is represented with a particular position and orientation (vehicle heading angle) in these global coordinates.

The main challenge is to derive a closed-form mathematical equation to represent the tangential Bezier curve 17 to be generated, given the inputs: {start and target frames, gap vertex and safety tolerance distance to keep from them}. In addition to the motor vehicle design with its dimensions and its maximum permitted wheel angle, this is certain.

$$\because x_0-x_s=Rsin\left(-{\theta }_0\right)+\left[R-\frac{W}{2}-SicherheitsToleranz\right]sin\left({\theta }_1\right)$$

$$\therefore {\theta }_1=si{n}^{-1}\left[\frac{x_0-x_s+Rsin\left({\theta }_0\right)}{R-\frac{W}{2}-SicherheitsToleranz}\right]$$

$$\therefore x_1=x_s-\left[\frac{W}{2}+SicherheitsToleranz\right]sin\left({\theta }_1\right)$$

$$\therefore y_1=y_s+\left[\frac{W}{2}+SicherheitsToleranz\right]cos\left({\theta }_1\right)$$

Regarding 5 the following equations are used to plan path 7: $$R=\frac{Ra\mathrm{i.e}stan\mathrm{i.e}}{tan\left({\phi }_{max}-{\phi }_{tOle\mathrm{right}ance}\right)}=R_{min}+TOle\mathrm{right}an\mathrm{e.g}$$

$$\because x_0-x_s=Rsin\left(-{\theta }_0\right)+\left[R-\frac{\mathrm{<figure-callout\; id="2"\; label="W"\; filenames="DE102022101054A1\_0063.png"\; state="\{\{state\}\}">W</figure-callout>}}{2}-SicHe\mathrm{right}HeitsTOle\mathrm{right}an\mathrm{e.g}\right]sin\left({\theta }_1\right)$$

$$<figure-callout\; id="1"\; label="\therefore \; \theta "\; filenames="DE102022101054A1\_0063.png,DE102022101054A1\_0066.png"\; state="\{\{state\}\}">\therefore </figure-callout>{\theta }_{}{}_1=si{n}^{-1}\left[\frac{x_0-x_s+Rsin\left({\theta }_0\right)}{R-\frac{\mathrm{<figure-callout\; id="2"\; label="W"\; filenames="DE102022101054A1\_0063.png"\; state="\{\{state\}\}">W</figure-callout>}}{2}-SicHe\mathrm{right}HeitsTOle\mathrm{right}an\mathrm{e.g}}\right]$$

$$\therefore x_1=x_s-\left[\frac{\mathrm{<figure-callout\; id="2"\; label="W"\; filenames="DE102022101054A1\_0063.png"\; state="\{\{state\}\}">W</figure-callout>}}{2}+SicHe\mathrm{right}HeitsTOle\mathrm{right}an\mathrm{e.g}\right]sin\left({\theta }_1\right)$$

$$<figure-callout\; id="1"\; label="\therefore \; y"\; filenames="DE102022101054A1\_0063.png,DE102022101054A1\_0066.png"\; state="\{\{state\}\}">\therefore </figure-callout>y_{}{}_1=y_s+\left[\frac{\mathrm{<figure-callout\; id="2"\; label="W"\; filenames="DE102022101054A1\_0063.png"\; state="\{\{state\}\}">W</figure-callout>}}{2}+SicHe\mathrm{right}HeitsTOle\mathrm{right}an\mathrm{e.g}\right]cOs\left({\theta }_1\right)$$

$$b=\frac{2}{3}\left[\frac{cos\left(\frac{\Delta \theta }{2}\right)-co{s}^2\left(\frac{\Delta \theta }{2}\right)}{si{n}^2\left(\frac{\Delta \theta }{2}\right)}\right]$$

$$M=\sqrt{{\left(x_1-x_0\right)}^2+{\left(y_1-y_0\right)}^2}$$

Based on the calculated intermediate frame (x _{1 ,} y ₁ ,θ ₁ ), the corresponding Bezier curve 17 connecting the initial frame (x ₀ ,y ₀ ,θ ₀ ) and the intermediate frame (x ₁ ,y ₁ ,θ ₁ ) can be be calculated as follows. {x _P (t),y _P (t)} are the parametric Bezier equations in the local coordinates, while {x(t) ,y(t)} are the final Bezier curve equations in the global coordinates, where t is the parametric independent variable of is 0 to 1. $$\Delta \theta =\left|{\mathrm{<figure-callout\; id="1"\; label="\theta "\; filenames="DE102022101054A1\_0063.png,DE102022101054A1\_0066.png"\; state="\{\{state\}\}">\theta </figure-callout>}}_1-{\mathrm{<figure-callout\; id="0"\; label="\theta "\; filenames="DE102022101054A1\_0065.png"\; state="\{\{state\}\}">\theta </figure-callout>}}_0\right|$$

$$b=\frac{2}{3}\left[\frac{cOs\left(\frac{\Delta \theta }{2}\right)-cO{s}^2\left(\frac{\Delta \theta }{2}\right)}{si{n}^2\left(\frac{\Delta \theta }{2}\right)}\right]$$

$$M=\sqrt{{\left(x_1-x_0\right)}^2+{\left({\mathrm{<figure-callout\; id="1"\; label="y"\; filenames="DE102022101054A1\_0063.png,DE102022101054A1\_0066.png"\; state="\{\{state\}\}">y</figure-callout>}}_1-y_0\right)}^2}$$

$$t_0=\text{tan}\left(\frac{\Delta \theta }{2}\right)$$

According to equation [4]: $$x_P\left(t\right)=\left[3bM\right]t+\left[3M\left(1-3b\right)\right]{\mathrm{<figure-callout\; id="2"\; label="t"\; filenames="DE102022101054A1\_0063.png"\; state="\{\{state\}\}">t</figure-callout>}}^2-\left[2M\left(1-3b\right)\right]t^3$$

$${\mathrm{<figure-callout\; id="0"\; label="t"\; filenames="DE102022101054A1\_0065.png"\; state="\{\{state\}\}">t</figure-callout>}}_0=\text{tan}\left(\frac{\Delta \theta }{2}\right)$$

$$\alpha =\left(\frac{{\theta }_1+{\theta }_0}{2}\right)$$

According to equation [5] with t ₀ = -t _f : $$y_P\left(t\right)=\left[3bM{t}_0\right]\left(t-t^2\right)$$

$$a=\left(\frac{{\mathrm{<figure-callout\; id="1"\; label="\theta "\; filenames="DE102022101054A1\_0063.png,DE102022101054A1\_0066.png"\; state="\{\{state\}\}">\theta </figure-callout>}}_1+{\mathrm{<figure-callout\; id="0"\; label="\theta "\; filenames="DE102022101054A1\_0065.png"\; state="\{\{state\}\}">\theta </figure-callout>}}_0}{2}\right)$$

ansonsten $$\alpha =\alpha +180°$$

Since the direction of travel is backwards, then the orientation of the Bezier coordinates with respect to the global coordinates (α) must be added or subtracted by 180° based on their sign.
if a ≥ 0 $$a=a-180°$$

otherwise $$a=a+180°$$

$$y\left(t\right)=y_0+x_P\left(t\right)sin\left(\alpha \right)+y_P\left(t\right)cos\left(\alpha \right)$$

According to equations [6] and [7]: $$x\left(t\right)=x_0+x_P\left(t\right)cOs\left(a\right)-y_P\left(t\right)sin\left(a\right)$$

$$y\left(t\right)={\mathrm{<figure-callout\; id="0"\; label="y"\; filenames="DE102022101054A1\_0065.png"\; state="\{\{state\}\}">y</figure-callout>}}_0+x_P\left(t\right)sin\left(a\right)+y_P\left(t\right)cOs\left(a\right)$$

It should be noted that other Bezier types ({1/3-2/3} and {7/24-17/24}) can also be tried (if necessary). It could be possible that steerable paths exist based on these according to the given frame positions.

It should also be clear that as soon as there is not enough space to have a navigable path, the calculation of θ ₁ automatically fails since sin(θ ₁ ) is unrealistic (> 1).

6 shows another embodiment of the method according to the invention. In 6 shows the generation of the path 7 using the Bezier curve 17 tangential to the end point 15 for parallel parking and/or overtaking scenarios.

Similar to the one in the previous one 5 Given the geometry, the following figure shows the case of a tangential path planning to the end point 15 of a front detected object 16. The main idea here of planning the tangential path 7 is to provide the ability to move the front object 16 with less possible steering according to the space available maneuver. In such a way, the motor vehicle 1 can avoid a collision with the front object 16 with less possible steering operation, and can exit the parallel parking smoothly with less lateral clearance from the road. Moreover, for the overtaking scenario, it can provide a smooth path 7 with less required steering, which could be passable in a wide range of speeds, starting with low speeds on city streets (traffic jam assistant) and ending with higher speeds on highways (lane change assistant). Consequently, the turning radius used in the following mathematical derivation is based on a tailored target wheel angle (φ_target) rather than the maximum wheel angle considered in the previous section. Such a target wheel angle can be set based on the current vehicle speed at the initial frame. For example, it may be defined as the maximum allowed wheel angle that the vehicle motion controller can support at a particular vehicle speed.

$$\because x_0-x_s=Rsin\left(-{\theta }_0\right)+\left[R+\frac{W}{2}+SicherheitsToleranz\right]sin\left({\theta }_1\right)$$

$$\therefore {\theta }_1=si{n}^{-1}\left[\frac{x_0-x_s+Rsin\left({\theta }_0\right)}{R+\frac{W}{2}+SicherheitsToleranze}\right]$$

$$\therefore x_1=x_s-\left[\frac{W}{2}+SicherheitsToleranz\right]sin\left({\theta }_1\right)$$

$$\therefore y_1=y_s+\left[\frac{W}{2}+SicherheitsToleranz\right]cos\left({\theta }_1\right)$$

Regarding 6 the following equations can be used: $$R=\frac{Ra\mathrm{i.e}stan\mathrm{i.e}}{tan\left({\phi }_{ta\mathrm{right}Get}\right)}$$

$$\because x_0-x_s=Rsin\left(-{\theta }_0\right)+\left[R+\frac{\mathrm{<figure-callout\; id="2"\; label="W"\; filenames="DE102022101054A1\_0063.png"\; state="\{\{state\}\}">W</figure-callout>}}{2}+SicHe\mathrm{right}HeitsTOle\mathrm{right}an\mathrm{e.g}\right]sin\left({\theta }_1\right)$$

$$<figure-callout\; id="1"\; label="\therefore \; \theta "\; filenames="DE102022101054A1\_0063.png,DE102022101054A1\_0066.png"\; state="\{\{state\}\}">\therefore </figure-callout>{\theta }_{}{}_1=si{n}^{-1}\left[\frac{x_0-x_s+Rsin\left({\theta }_0\right)}{R+\frac{\mathrm{<figure-callout\; id="2"\; label="W"\; filenames="DE102022101054A1\_0063.png"\; state="\{\{state\}\}">W</figure-callout>}}{2}+SicHe\mathrm{right}HeitsTOle\mathrm{right}an\mathrm{e.g}e}\right]$$

$$\therefore x_1=x_s-\left[\frac{\mathrm{<figure-callout\; id="2"\; label="W"\; filenames="DE102022101054A1\_0063.png"\; state="\{\{state\}\}">W</figure-callout>}}{2}+SicHe\mathrm{right}HeitsTOle\mathrm{right}an\mathrm{e.g}\right]sin\left({\theta }_1\right)$$

$$<figure-callout\; id="1"\; label="\therefore \; y"\; filenames="DE102022101054A1\_0063.png,DE102022101054A1\_0066.png"\; state="\{\{state\}\}">\therefore </figure-callout>y_{}{}_1=y_s+\left[\frac{\mathrm{<figure-callout\; id="2"\; label="W"\; filenames="DE102022101054A1\_0063.png"\; state="\{\{state\}\}">W</figure-callout>}}{2}+SicHe\mathrm{right}HeitsTOle\mathrm{right}an\mathrm{e.g}\right]cOs\left({\theta }_1\right)$$

$$b=\frac{2}{3}\left[\frac{cos\left(\frac{\Delta \theta }{2}\right)-co{s}^2\left(\frac{\Delta \theta }{2}\right)}{si{n}^2\left(\frac{\Delta \theta }{2}\right)}\right]$$

$$M=\sqrt{{\left(x_1-x_0\right)}^2+{\left(y_1-y_0\right)}^2}$$

Based on the calculated intermediate frame (x ₁ , y ₁ , θ ₁ ), the corresponding circular Bezier path that includes the initial frame (x ₀ , y ₀ , θ ₀ ) and the intermediate frame ({x _P (t) ,y _P (t )}) connects can be calculated as follows. are the parametric Bezier equations in the local coordinates, while {x(t) ,y(t)} are the final Bezier path equations in the global coordinates, where t is the parametric independent variable from 0 to 1. $$\Delta \theta =\left|{\mathrm{<figure-callout\; id="1"\; label="\theta "\; filenames="DE102022101054A1\_0063.png,DE102022101054A1\_0066.png"\; state="\{\{state\}\}">\theta </figure-callout>}}_1-{\mathrm{<figure-callout\; id="0"\; label="\theta "\; filenames="DE102022101054A1\_0065.png"\; state="\{\{state\}\}">\theta </figure-callout>}}_0\right|$$

$$b=\frac{2}{3}\left[\frac{cOs\left(\frac{\Delta \theta }{2}\right)-cO{s}^2\left(\frac{\Delta \theta }{2}\right)}{si{n}^2\left(\frac{\Delta \theta }{2}\right)}\right]$$

$$M=\sqrt{{\left(x_1-x_0\right)}^2+{\left({\mathrm{<figure-callout\; id="1"\; label="y"\; filenames="DE102022101054A1\_0063.png,DE102022101054A1\_0066.png"\; state="\{\{state\}\}">y</figure-callout>}}_1-y_0\right)}^2}$$

$$t_0=tan\left(\frac{\Delta \theta }{2}\right)$$

According to equation [4]: $$x_p\left(t\right)=\left[3bM\right]t+\left[3M\left(1-3b\right)\right]{\mathrm{<figure-callout\; id="2"\; label="t"\; filenames="DE102022101054A1\_0063.png"\; state="\{\{state\}\}">t</figure-callout>}}^2-\left[2M\left(1-3b\right)\right]t^3$$

$${\mathrm{<figure-callout\; id="0"\; label="t"\; filenames="DE102022101054A1\_0065.png"\; state="\{\{state\}\}">t</figure-callout>}}_0=tan\left(\frac{\Delta \theta }{2}\right)$$

$$\alpha =\left(\frac{{\theta }_1+{\theta }_0}{2}\right)$$

According to equation [5] with t ₀ = -t _f : $$y_p\left(t\right)=\left[3bM{t}_0\right]\left(t-t^2\right)$$

$$a=\left(\frac{{\mathrm{<figure-callout\; id="1"\; label="\theta "\; filenames="DE102022101054A1\_0063.png,DE102022101054A1\_0066.png"\; state="\{\{state\}\}">\theta </figure-callout>}}_1+{\mathrm{<figure-callout\; id="0"\; label="\theta "\; filenames="DE102022101054A1\_0065.png"\; state="\{\{state\}\}">\theta </figure-callout>}}_0}{2}\right)$$

Since the direction of travel is forward, there is no need to add or subtract 180° to/from the orientation of the Bezier coordinates with respect to the global coordinates (α).

$$y\left(t\right)=y_0+x_P\left(t\right)sin\left(\alpha \right)+y_P\left(t\right)cos\left(\alpha \right)$$

According to equations [6] and [7]: $$x\left(t\right)=x_0+x_P\left(t\right)cOs\left(a\right)-y_P\left(t\right)sin\left(a\right)$$

$$y\left(t\right)={\mathrm{<figure-callout\; id="0"\; label="y"\; filenames="DE102022101054A1\_0065.png"\; state="\{\{state\}\}">y</figure-callout>}}_0+x_P\left(t\right)sin\left(a\right)+y_P\left(t\right)cOs\left(a\right)$$

Therefore, the Bezier curve 17 can be planned in a robust way.

Claims (15)

Method for planning a path (7) for an at least partially automated motor vehicle (1) from a starting position (10) of the motor vehicle (1) to a target position (11) in an environment (21) of the motor vehicle (1) using an assistance system (2 ) of the motor vehicle (1), with the steps: - Receiving data representing the environment (21) of the motor vehicle (1); - Determining an end point (15) of an obstacle (16) by means of the data; - Determining the target position (11) of the motor vehicle (1) as a function of the detected end point (15) and as a function of the initial position (10) of the motor vehicle (1); and - planning the path (7) of the motor vehicle (1) from the initial position (10) to the target position (11) depending on the initial position (10), the target position (11) and a maximum turning radius (R) of the force vehicle (1), wherein the target position (11) is the closest position between the end point (15) and the path (7) and wherein an orientation (θ ₁ ) of the vehicle (1) in the target position (11) by means of the maximum turning radius (R) and the end point (15). procedure after claim 1 , characterized in that an auxiliary point (24) is determined, wherein a distance from the auxiliary point (24) to the starting position (10) is equal to the maximum turning radius (R), and wherein the target position (11) lies on a line passing through the End point (15) and the auxiliary point (24) is constructed. procedure after claim 2 , characterized in that the orientation (θ ₁ ) of the vehicle (1) in the target position (11) is perpendicular to the line constructed by the end point (15) and the auxiliary point (24). procedure after claim 2 or 3 , characterized in that an extension of a rear axle of the vehicle (1) intersects the auxiliary point (24) when the vehicle is in the initial position (10). Method according to one of the preceding claims, characterized in that the path (7) is planned using a Bezier curve (17), in particular of the third degree. Method according to one of the preceding claims, characterized in that a path (7) into a parking space (19), in particular into a vertical parking space (19), is planned by the assistance system (2). procedure after claim 6 , characterized in that from the target position (11) a further target position (12) to a parking position in the parking space (19) is planned using a further Bezier curve (20). Procedure according to one of Claims 1 until 5 , characterized in that the path (7) is provided for an overtaking maneuver. Method according to one of the preceding claims, characterized in that the end point (15) is a corner point which represents a corner of the obstacle (16). Method according to one of the preceding claims, characterized in that a speed of the motor vehicle (1) is taken into account when the path (7) is planned. Method according to one of the preceding claims, characterized in that a direction of movement of the motor vehicle (1) during the tracing of the path (7) is taken into account when the path (7) is planned. Method according to one of the preceding claims, characterized in that a map of the surroundings (21) of the motor vehicle (1) is provided, sensor data from at least two, three or four different sensor types being provided for the map. Computer program product with program means, which are stored in a computer-readable medium, for the method according to one of the preceding Claims 1 until 12 to be carried out when the computer program product is executed on a processor of an electronic computing device (3). Computer-readable storage medium containing a computer program product Claim 13 . Assistance system (2) for planning a path (7) for an at least partially automated motor vehicle (1) from a starting position (10) of the motor vehicle (1) to a target position (11) in an environment (21) of the motor vehicle (1), the includes at least one detection device (5) and at least one electronic computing device (3), wherein the assistance system (2) for performing a method according to one of Claims 1 until 12 is set up.

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