# Optimization Design of Parking Models Based on Complex and Random Parking Environments

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## Abstract

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## 1. Introduction

- The autonomous vehicle is a symmetrical four-wheeled passenger car with a rectangular body. It adopts front-wheel steering and rear-wheel driving;
- During the vehicle’s motion, the velocity direction at the control point aligns with the direction angle of the trajectory point. The instantaneous turning radius is equal to the curvature radius of the trajectory point;
- The autonomous vehicle travels on an ideal and flat road surface, neglecting any vertical motion of the vehicle;
- Longitudinal and lateral aerodynamic factors are ignored;
- The vehicle acceleration does not exceed 3 m/s
^{2}. During the acceleration phase, the constant jerk is assumed to be 20 m/s^{3}; - All turning maneuvers are assumed to be a constant jerk motion with a circular arc trajectory;
- The maximum centripetal acceleration during a turn is assumed to be equal to the maximum deceleration during braking;
- The time consumed by the friction between the vehicle tires and the ground is negligible;
- Some relevant data of the vehicle included the following: it was 4.9 m long, 1.8 m wide, had a maximum steering wheel angle of 470°, and a steering wheel and front wheel angle of transmission ratio of 16:1 (steering wheel rotation 16°, front wheel rotation 1°).

## 2. Model Establishment

#### 2.1. Unmanned Vehicle Turning Minimum Radius Solving Problem

#### 2.1.1. Solving the Minimum Radius of Unmanned Vehicles

- (1)
- Vehicle steering principle

_{i}and $\Phi $

_{o}, respectively, and the distance from the vehicle centerline to the instantaneous center of the rotation as R. The formula is as follows [60]:

_{i}and tan$\varphi $

_{o}in the above formula and making the difference, the Ackermann steering formula can be obtained as follows:

- (2)
- Vehicle formulas of motion

- (3)
- Vehicle kinematic model

- (4)
- Minimum turning radius solution

- R = minimum turning radius of the vehicle (minimum);
- L = vehicle length;
- W = vehicle width;
- D = vehicle minimum turning lane width;
- φ = vehicle direction maximum turning angle; (steering wheel maximum turning angle/16 = 470/16 = 29.375).

#### 2.1.2. Unmanned Vehicles Accelerate in a Straight Line

- (1)
- Modeling.

_{0}, the acceleration is ${a}_{0}$, and the acceleration and jerk are $J\left(t\right)$.

- When $t\in \left[\mathrm{0,0.15}\right]$, we assumed that $t=0.15$ and the speed is ${v}_{1}$. The route is ${s}_{1}$. The formula is as follows:$$\left.\begin{array}{c}{v}_{1}={\int}_{0}^{0.15}20tdt\\ {s}_{1}={\int}_{0}^{0.15}v\left(t\right)dt\end{array}\right\}$$The solution is: ${v}_{1}=0.225\mathrm{m}/\mathrm{s}$; ${s}_{1}=0.03375\mathrm{m}$.
- When $t>0.15$, thereafter, acceleration is ${a}_{1}=3\mathrm{m}/{\mathrm{s}}^{2}$, and the acceleration is ${J}_{1}=0\mathrm{m}/{\mathrm{s}}^{3}$. Then, the distance ${s}_{2}$ is as follows:$${s}_{2}=\frac{{v{2}_{1}}_{max}}{2{{a}_{1}}^{2}}$$The result is obtained as follows:$${s}_{all}={s}_{1}+{s}_{2}=5.16935=5.17\mathrm{m}$$
- In other cases, the distance traveled during the acceleration phase can be calculated using the following formula: ${a}_{0}$ > 0, ${v}_{0}$ > 0, $J\left(t\right)$ as the maximum value.$$\mathrm{s}\le {\mathrm{s}}_{\mathrm{a}\mathrm{l}\mathrm{l}}{\mathrm{s}}_{0},\mathrm{s}\in \left[0,{\mathrm{s}}_{\mathrm{a}\mathrm{l}\mathrm{l}}\right]$$

#### 2.1.3. Calculation of the Rate of Change of the Relative Path Curvature Length

#### 2.2. Unmanned Vehicle Parking Trajectory

#### 2.2.1. Unmanned Vehicle Reverse Parking Model

_{0}, y) and (x

_{a}, y

_{a}) represent the coordinates of the contact points a and d of the rear wheels, respectively. L denotes the wheelbase, and L

_{rw}represents the rear wheel track width.

_{f}denotes the absolute forward speed measured at the front axle center of the vehicle, where positive values indicate forward motion and negative values indicate reverse motion. θ represents the vehicle’s heading angle, with counterclockwise as positive and clockwise as negative. g represents the Ackermann steering angle, with counterclockwise as positive and clockwise as negative.

_{1}and R

_{2}represent the turning radii of the right rear wheel of the car when the steering wheel is turned left and right, respectively. S

_{1}and S

_{2}represent the paths traveled by the car when the steering wheel is turned left and right, respectively. S

_{0}and h

_{0}represent the horizontal and vertical distances traveled by the car from the starting position to the destination position. θ′ represents the instantaneous direction angle of the car’s body when it reaches point f′ and begins to reverse the steering wheel.

- (1)
- Based on the collision analysis at the vertex D of the vehicle body, the width of the parking space, S
_{y}, is determined. As shown in Figure 7, the radius of the arc covered by the vertex D of the car is calculated as:$${R}_{1D}=\sqrt{{\left({R}_{1}+{L}_{rr2r}\right)}^{2}+{L}_{ra2t}^{2}}$$

_{rr}

_{2r}represents the distance from the center plane of the right rear wheel to the right side of the car; L

_{ra}

_{2t}represents the rear overhang; l

_{e}

_{2si}represents the safety clearance between the right side of the car CD and the inner side C′D′ of the parking space after the successful reverse parking and aligning of the vehicle; W represents the width of the car; δ

_{y}represents the reserved safety clearance between the car and the parking space in the y-axis direction, obtained using:

_{y_outer}represents the reserved safety clearance between the outer side of the vehicle and the outer boundary of the parking space in the y-axis direction. For now, let δ

_{y_outer}= 0.00. δ

_{y_inner}represents the reserved safety clearance between the right side of the vehicle and the inner part of the parking space in the y-axis direction.

- (2)
- To determine the length Sx of the parking space and the maximum value of R
_{1}based on the potential collision between the vehicle’s vertex C and the parking space’s endpoint B′, the distance between the turning center O_{1}of the right rear wheel when the steering wheel is turned left and the parking space’s boundary A′B′ is obtained using [62]:$$\genfrac{}{}{0pt}{}{{L}_{{O}_{1}2s}=\left({R}_{1}+{L}_{rr2r}\right)+{\delta}_{y}-{S}_{y}}{{R}_{1C}=\sqrt{{\left({R}_{1}+{L}_{rr2r}\right)}^{2}+{\left(L+{L}_{ra2t}\right)}^{2}}}$$

_{min}denotes the minimum turning radius of the vehicle; r

_{C}

_{2B′}represents the designated safety clearance between the vehicle vertex C and point B′ of the parking space; δ

_{x}denotes the reserved safety clearance between the vehicle’s rear end and the left side of the parking space, as well as between the vehicle’s front end and the right side of the parking space; L denotes the length of the vehicle.

- (3)
- The minimum value of R
_{2}, denoted as R_{2min}, is estimated based on the potential collision between the vehicle vertex B and an obstacle on the other side of the parking space, as shown in Figure 9. The radius of the curvature of the arc covered by the vehicle vertex B is obtained using:$$\genfrac{}{}{0pt}{}{{R}_{2B}=\sqrt{{\left[\left({R}_{2}-{L}_{rr2r}\right)+W\right]}^{2}+{\left(L-{L}_{ra2t}\right)}^{2}}}{{h}_{0}={\delta}_{V2S}+\left({S}_{y}-{\delta}_{y}\right)}$$

- (4)
- As shown in Figure 10, based on the analysis of potential collisions between the extended line of the wheel contact point ad and the right side of the vehicle body at point e and the parking space, the maximum value of R
_{2}, denoted as R_{2max}, can be determined according to Figure 11. The following formula can be derived:$${\left({R}_{2}-{r}_{e2{B}^{\prime}}-{L}_{rr2r}\right)}^{2}-{\left[{R}_{2}-\left({\delta}_{V2S}+{L}_{rr2r}\right)\right]}^{2}={\left[{S}_{0}-\left({S}_{x}-{\delta}_{x}-{L}_{ra2t}\right)\right]}^{2}$$

_{1}and Formulas (8)–(31), the maximum value of R

_{2}, denoted as R

_{2max}, can be determined.

_{1}and R

_{2}values. The coordinates of the starting and ending positions of the right rear wheel contact point d, as shown in Figure 12, can be expressed as follows:

_{d}

_{0}, y

_{d}

_{0}) = (x

_{a}− R

_{1}× cos(g + δ), y

_{a}− R

_{1}× sin(g + δ)).

_{d}

_{0}, y

_{d}

_{0}) = (x

_{a}− R

_{2}× cos(g − δ), y

_{a}− R

_{2}× sin(g − δ)).

_{a}, y

_{a}) represents the coordinates of the rear wheel contact point a, R

_{1}and R

_{2}are the turning radii of the right rear wheel, g is the Ackermann steering angle, and δ is the predetermined safety gap.

_{1}and R

_{2}values, the corresponding ranges of the starting and ending position coordinates can be calculated, thereby determining the preliminary reverse area of the autonomous vehicle.

_{d}

_{0}, y

_{d}

_{0}) and (x

_{ds}, y

_{ds}) represent the coordinates of the right rear wheel contact point d when the vehicle is located at the preliminary reverse position and the endpoint position, respectively. The determined preliminary reverse area has a fixed vertical coordinate. However, as shown in Figure 11, to increase the size of the preliminary reverse area and facilitate the search for a reverse position, the value of the gap δ

_{v}

_{2s}between the vehicle’s body and the parking space at the preliminary reverse position can be adjusted (increased or decreased). This adjustment will change the initial vertical coordinate of the vehicle’s parking position. Then, based on the previously described method, the values of R

_{1}and R

_{2}can be re-determined, thereby expanding the preliminary reverse area of the vehicle.

#### 2.2.2. Unmanned Vehicle Vertical Parking Space Parking Model

_{3}, is determined. At the same time, the nearest starting position for parking is obtained. The definition of the initial parking area is the same as in parallel parking. Considering the reverse motion of the vehicle coming out of the garage, after avoiding collision points A and B, a larger radius of curvature than R

_{min}is used to achieve a new starting position for parking. It is also possible to use a radius of curvature larger than R

_{min}for both maneuvers. However, in this case, as shown in Figure 12, the starting parking position will be farther away from the horizontal obstacle in the vertical direction, and the vehicle’s entry into the parking space will be shallower. This requires a larger space on the left side of the vehicle.

_{1}. The subsequent stages include uniform acceleration, constant velocity, and deceleration until reaching a complete stop, denoted as t

_{2}. The reverse parking stage can be calculated to obtain the time duration, denoted as t

_{3}. Finally, the total time for this parking process can be obtained by summing up the durations of each stage:

- (1)
- Steering system$$s=\frac{\pi}{2}R=\frac{3.14}{2}\cdot 4.99=7.83\phantom{\rule{0ex}{0ex}}{t}_{1}=\frac{s}{v}=\frac{7.83}{2.78}=2.82s\phantom{\rule{0ex}{0ex}}{t}_{2}=0.93s$$
- (2)
- Uniform acceleration and uniform linear travelUniform acceleration: $t\approx 0.93s$Uniform linear: $t=3.67;x=3.86;t=7.55$
- (3)
- Slow down and reverse parking$$\mathit{tan}\alpha =\frac{4.99}{5.3}=0.942\alpha ={43.3}^{\xb0}L=\frac{43.3\times \pi \times 4.99}{180}=3.77\phantom{\rule{0ex}{0ex}}v={v}_{0}+at\phantom{\rule{0ex}{0ex}}x=\frac{1}{2}a{t}^{2}\phantom{\rule{0ex}{0ex}}10{t}^{2}=3.77\phantom{\rule{0ex}{0ex}}{t}_{4}=0.61st=2.82+0.93+4.84+0.61+2.71=11.91s$$

#### 2.2.3. Unmanned Vehicle Parking Path Model

- (1)
- Parking path tracking control law

_{1}and k

_{2}in Formula (44) represent the feedback coefficients for the position deviation and heading deviation of the vehicle, respectively.

_{1}> 0 and k

_{2}> 0, according to the first stability theorem of Lyapunov, it can be concluded that the origin is the unique equilibrium point of Formula (45), i.e., the point where the system reaches a stable state [65]:

- (2)
- Parallel parking space parking model

_{1}. Then, the vehicle undergoes uniform acceleration, followed by a period of constant velocity until reaching the second turning point. The time for this stage is calculated as t

_{2}. The second turning maneuver takes time t

_{3}, and the third turning maneuver takes time t

_{4}. Afterward, the vehicle accelerates uniformly and maintains a constant velocity during straight-line driving until finally decelerating to a stop. The time for this straight-line segment is denoted as t

_{5}. Lastly, the time for the reverse parking maneuver is calculated as t

_{6}. By summing up the individual times, the total time for the parallel parking process can be determined.

- First turn:$${t}_{1}=2.82s$$
- The first straight-line travel:Uniform straight line:$${t}_{2}=10.43s$$Deceleration:$${t}_{3}=1.93s$$
- Second turn:$${t}_{4}=2.82s$$
- Third turn:$${t}_{5}=2.82s$$
- Second straight sectionUniform straight line:$${v}^{2}-{{v}_{0}}^{2}=2ax\phantom{\rule{0ex}{0ex}}x=3.86\phantom{\rule{0ex}{0ex}}v={v}_{0}+at\phantom{\rule{0ex}{0ex}}{t}_{6}=0.927$$
- Slow down and reverse parking:$${t}_{7}=\frac{6.24+9\times 5.3-3.86}{5.56}=9s\phantom{\rule{0ex}{0ex}}{t}_{8}=0.657s$$
- Find the total time:22.404 + 1.1 = 23.504 s

- (3)
- Unmanned vehicle 45° inclined parking space parking model

_{1}. Then, the vehicle undergoes uniform acceleration, followed by a period of constant velocity until reaching the second turning point. The time for this stage is calculated as t

_{2}. After the second turning maneuver, the vehicle maintains a constant velocity and travels a certain distance forward. The time for this segment is denoted as t

_{3}. Finally, the time for the reverse parking maneuver is calculated as t

_{4}. By summing up the individual times, the total time for the parking process in an inclined parking space can be determined.

## 3. Problem Solving

#### 3.1. Optimal Parking Space Parking Model

- (1)
- Parking-oriented analysis

_{i}, y

_{i}, θ

_{i}, φ

_{i}), where:

_{i}represents the x-coordinate in the direction of the vehicle’s entrance.

_{i}represents the y-coordinate perpendicular to the entrance.

_{i}represents the orientation of the autonomous vehicle at the moment before reversing.

_{i}represents the direction of rotation needed during the reverse parking process, with 0 indicating the positive x-axis direction and values ranging from 0 to 2π for a complete counterclockwise revolution.

- (2)
- Curve Analysis

_{i}, Δθ

_{i}= θ − θ

_{i}

_{1i}= 2πRΔθ

_{i}, where R is the radius of the curve and Δθ

_{i}is the angular displacement.

_{1}, we can use the formula:

_{1i}= S

_{1i}/v

_{1}

- (3)
- Straight-line path analysis

_{i}= x − x

_{i}and Δy

_{i}= y − y

_{i}; then, the distance traveled in a straight line is:

_{1i}for each curved path segment, we can determine the time required to traverse them. Let us consider the following speed values:

_{2}: Speed within 5 m before the deceleration zone.

_{3}: Speed on other straight sections of the road.

_{1}is the maximum speed for curved paths, V

_{2}is the speed within the 5-m zone, and V

_{3}is the speed on straight sections of the road.

- (A)
- Δθ
_{i}= 0, time required t_{i}:$$\begin{array}{l}\u2460b-x5,\Delta x0:{t}_{2i}=(b-x-5)/{v}_{3}+(-x)/{v}_{2}\\ \u2461b-x\le 5,\Delta x0:{t}_{2i}=(b-x)/{v}_{2}\\ \u2462\Delta x0:{t}_{2i}=(-\Delta x)/{v}_{2}\end{array}$$ - (B)
- Δθ
_{i}= π/2, time required t_{i}:$$\begin{array}{l}\u2460b-x5,\Delta x0:{t}_{2i}=(b-x-5)/{v}_{3}+(b-x+\Delta {y}_{i})/{v}_{2}\\ \u2461b-x\le 5,\Delta x0:{t}_{2i}=(b-x+\Delta {y}_{i})/{v}_{2}\end{array}$$ - (C)
- Δθ
_{i}= π, time required t_{i}:$$\begin{array}{l}\u2460b-{x}_{i}\ge 5,b-x5:{t}_{2i}=(2b-x-{x}_{i}-10)/{v}_{3}+(b-x+\Delta {y}_{i}+5)/{v}_{2}\\ \u2461b-{x}_{i}\le 5,b-x5:{t}_{2i}=(b-x-5)/{v}_{3}+(2b+\Delta {y}_{i}-x-{x}_{i})/{v}_{2}\\ \u2462b-{x}_{i}\ge 5,b-x\le 5:{t}_{2i}=(2b+\Delta {y}_{i}+x+5)/{v}_{2}+(b-5-{x}_{i})/{v}_{3}\\ \u2463b-{x}_{i}\le 5,b-x\le 5:{t}_{2i}=(2b+\Delta {y}_{i}+x-{x}_{i}+5)/{v}_{2}\end{array}$$

- (4)
- Reversing path analysis

_{3i}= 2πR·ϕi

- reversing process vehicle position in the speed bump or reverse driving: t
_{3i}= S_{3i}/V_{2}. - reversing process of vehicle position in a straight line in addition to speed bumps: t
_{3i}= S_{3i}/V_{3}.

_{i}; then, for the total interval t, there is an objective function:

- the driver parks at point i, which means there is no parking space in front of him, otherwise he may consider the space in front of him;
- If i is the optimal parking space, it means that there is no more parking space after i. Summing up the above two points, the probability of successful parking can be found as follows:

#### 3.2. Unmanned Vehicle Parking Modeling

- In order to facilitate the study so that only one vehicle can enter at a time and adjacent vehicles will not have an impact on each other.
- The vehicles leave the parking space so that the neighboring vehicles enter the parking lot between t
_{1}(x_{1}).t_{2}(x_{2}), respectively; then, Δt = t_{2}(x_{2}) − t_{1}(x_{1}). - The number of vehicles leaving this parking lot in the Δt interval is randomly generated, and the corresponding parking space number is also randomly generated.
- In the process of vehicles entering the parking lot until the completion of the parking behavior, no other vehicles leave the parking space.
- The above algorithm was implemented and simulated using MATLAB 2021a.

#### 3.3. Optimal Parking Space Driving Trajectory Simulation

## 4. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 7.**Schematic diagram of the collision analysis between point D of the car body and C′D′ of the parking space.

**Figure 8.**Schematic diagram of the collision analysis between point C of the car body and point B′ of the parking space.

**Figure 9.**Analysis diagram of the collision between point B of the car and the obstacle on the other side of the parking space.

**Figure 10.**Schematic diagram of the collision analysis between point e of the car body and point B′ of the parking space.

**Figure 15.**Collision point studies: (

**a**). Left front-end collision; (

**b**). Right-side collision; (

**c**). Left rear-end collision; (

**d**). Right front-end collision 1; (

**e**). Right rear-end collision 1; (

**f**). Right front-end collision 2; (

**g**). Right rear-end collision 2; (

**h**). Front-end collision; (

**i**). Not fully seated.

**Figure 16.**Simplified parking lot (the black square represents the car, and the circle represents the parking space).

**Figure 17.**Schematic diagram of the three strategies: (

**a**) car discovery target; (

**b**) car selection of parking space; (

**c**) car final parking point.

**Figure 18.**Schematic diagram of the inter cost between Strategy B and Strategy C: (

**a**) internal costs of Strategy B; (

**b**)internal costs of Strategy C.

Symbol | Meaning |
---|---|

L | Unmanned vehicle captain (m) |

W | The width of the unmanned vehicle (m) |

r | Maximum steering wheel turning angle (°) |

φ | Maximum front wheel turning angle (°) |

R_{min} | Minimum turning radius for unmanned vehicles (m) |

v_{f} | The maximum speed of the steering wheel of the unmanned vehicle (m/s) |

J | Maximum acceleration and jerk (m/s^{3}) |

a_{m}_{1} | Maximum throttle acceleration (m/s^{2}) |

a_{m}_{2} | Ultimate brake acceleration (m/s^{2}) |

a_{1} | The magnitude of the acceleration of uniformly accelerated motion during the acceleration phase (m/s^{2}) |

a_{2} | The magnitude of the acceleration of uniformly decelerating motion during the acceleration phase (m/s^{2}) |

t_{1} | Time of acceleration phase (s) |

v_{1} | Speed after acceleration t_{1} time (m/s) |

t_{2} | Duration of deceleration phase (s) |

v_{2} | Speed after acceleration t_{2} time (m/s) |

v_{3} | Unmanned vehicle turnaround speed when turning (m/s) |

t_{3} | The time of unmanned vehicles traveling at a constant speed during a turn (s) |

θ | Unmanned vehicle reversing process of the corner (°) |

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## Share and Cite

**MDPI and ACS Style**

Liu, X.; Zhu, S.; Fang, Y.; Wang, Y.; Fu, L.; Lei, W.; Zhou, Z.
Optimization Design of Parking Models Based on Complex and Random Parking Environments. *World Electr. Veh. J.* **2023**, *14*, 344.
https://doi.org/10.3390/wevj14120344

**AMA Style**

Liu X, Zhu S, Fang Y, Wang Y, Fu L, Lei W, Zhou Z.
Optimization Design of Parking Models Based on Complex and Random Parking Environments. *World Electric Vehicle Journal*. 2023; 14(12):344.
https://doi.org/10.3390/wevj14120344

**Chicago/Turabian Style**

Liu, Xunchen, Siqi Zhu, Yuan Fang, Yutong Wang, Lijuan Fu, Wenjing Lei, and Zijian Zhou.
2023. "Optimization Design of Parking Models Based on Complex and Random Parking Environments" *World Electric Vehicle Journal* 14, no. 12: 344.
https://doi.org/10.3390/wevj14120344