INISTA 2010 International Symposium on INnovations in Intelligent SysTems and Applications, 21-24 June 2010, Kayseri & Cappadocia, TURKEY
Direct-Motion Parallel Parking for a Vehicle with and without a Trailer Necip Gürler , Fırat Yılmaz Cevher , Alpaslan Yufka , and Osman Parlaktuna Department of Electrical and Electronics Engineering, Eskisehir Osmangazi University, Eskisehir, Turkey { necipgrlr, fycevher, ayufka } @gmail.com,
[email protected] directly even if the vehicle can park by using iterative motions [1]. The paper is organized as follows: Section 2 introduces kinematic models of a vehicle with and without a trailer. Section 3 shows how these two types of vehicles park in direct-motion parallel parking. In Section 4, simulations of the proposed approach in MATLAB environment are given, and Section 5 concludes the study.
Abstract In this study, the method of direct-motion parallel parking is introduced both in theory and in simulations for the vehicle with and without a trailer. The results show that not only a vehicle with a trailer park in parallel using direct motion, but also a vehicle with a trailer park in parallel in direct motion after the final value of the orientation angle is satisfied. Simulation results are given to verify the effectiveness of the proposed method.
2. Kinematic model of the Vehicle
1. Introduction
It is well known that the wheeled vehicles are a non-holonomic system because there are differential constraints that cannot be completely integrated [4]. To generate a feasible path for these vehicles, first of all the kinematic models of the vehicle with and without a trailer need to be obtained.
In daily life, everyone begins to have a car, and automotive companies introduce cars with new features. It’s believed that new intelligent and autonomous vehicles will replace old-style cars in the future. These future-tech cars will make our lives easier by using automatic parking, routing to destination, auto-driving, etc.. Present study addresses the autonomous direct-motion parallel parking for the new intelligent vehicles. In literature, there are several methods in parking. In [1], an iterative algorithm of parallel parking maneuver for a car-like vehicle is introduced and sinusoidal reference functions are used to control the steering angle and translational speed during parking process. In [2], a direct-motion parallel parking process, instead of the iterative approach, for both a car-like vehicle and a tractor-trailer is addressed. Moreover, in [3], it is shown that trailer of the tractor can track the desired trajectory using its globally asymptotically stable tracking control. There are different types of parking according to the geometric considerations of the environment. These are parallel, perpendicular and diagonal parking [2]. This paper focuses on the maneuvering of direct-motion parallel parking for a vehicle with and without a trailer. The advantage of direct-motion parallel parking is that the vehicle with and without a trailer can park without any iterative motions. However there is a restriction about the parking area in this method. Unless the parking area is enough to park for direct-motion, the vehicle cannot park
2.1. A vehicle without a trailer system A In many autonomous vehicle applications, ‘Carts’ and ‘Cars’, as shown in Figure 1, are widely used. The only difference between these systems is the number of the front wheels.
Figure 1. a) Cart-like robot
b) Car-like robot
Kinematic models of these types of vehicles can be expressed in [9]: ‘ cos ’ = sin 0 ′
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0 ( ) 0 ( ) ( ) 1
(1)
INISTA 2010 International Symposium on INnovations in Intelligent SysTems and Applications, 21-24 June 2010, Kayseri & Cappadocia, TURKEY
where C(x,y) is the midpoint of the rear wheels of the car-like vehicle, θ is the vehicle’s heading, v(t) is the linear velocity of point C, and ρ(t) is the curvature of the path described by point C. In addition to this kinematic model, the nonholonomic constraint can be written as: x ′ (t) sin θ − y ′ (t) cos θ = 0 (2)
The kinematic model of these types of system can be expressed as:
Furthermore, the radius for any curvature path can be expressed as:
where C(x,y) represents the midpoint of the rear wheels of the tractor vehicle, T(xt,yt) represents the midpoint of the two wheels of the trailer and L is the distance between point C and T. θ is the orientation angle of the tractor , β is the orientation angle of the trailer and α is the relative orientation of the trailer with respect to the vehicle. Moreover, nonholonomic constraints of this system can be written as: ′ sin() − ′ cos() = 0 (7) $ ′ sin(%) − $ ′ cos(%) = 0
1
ℓ
R(t) = ρ(t) = tan ϕ
′ ′ ′ = ′
(3)
where ℓ is the distance between the midpoint of the front wheels and the rear wheels, and ϕ which is the steering angle and is defined as the angle between the main axis of the vehicle and the velocity vector of the front wheels. In the end, to model the motion of the vehicle, equations in [5] are used. For ϕ(t) = 0̊ the model of the vehicle is given in (Eq. 4) (4)
(6)
( ) = (0) + & $'*,(-)
and for the case ϕ(t) ≠ 0̊ the model becomes as follows:
2(-) $
. /(0) +
&
(0)3 − (0)4
( ) = (0) − & $'*,(-)
( ) = (0) +
sin (t) − (0) (0) ( ) = (0) −
cos (t) − cos (0) (0)
0 0 ⎤ ( ) ⎥ 1 ( ) ( ) ⎥ 1⎦
In this study, to model the motion of the vehicle with a trailer, equations in [6] are used as given in (Eq. 8 and Eq. 9).
( ) = (0) + (0) cos (0) ( ) = (0) + (0) sin (0) ( ) = (0)
cos ⎡ sin ⎢ 0 ⎢ ⎣−
.56 /(0) +
( ) = (0) +
2(-) 7 &
2(-) $ &
(0)3 − 56 (0)4
(8)
(0)
( ) = 2 95 ( (0):);)
(5)
where
(0) ( ) = (0) + (0)
δ = tan ?
−(0) @A ( + B (0) (0)) D @A − 2 C :
where t is the elapsed time since the beginning of the motion.
A = C (0) :C − C
2.2. A vehicle with a trailer system
B = 2
A vehicle with a trailer system consists of a tractor vehicle (similar to a vehicle without a trailer) towing a two-wheeled trailer as illustrated in Figure 2.
3. Parallel parking for a non-holonomic vehicle with and without a trailer
(9)
'EF$'*/G HI⁄J(; 7K (-.M (-) 7K ,(-) ;&)3 & J @G 2(-)J 7K ,(-)
In this section, direct-motion parallel parking method for both a vehicle with and without a trailer is discussed. 3.1. Parallel parking for a non-holonomic vehicle without a trailer Several approaches have been proposed for direct-motion parallel parking for non-holonomic vehicles with and without a trailer. In [5], a path-
Figure 2. A vehicle with a trailer system
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INISTA 2010 International Symposium on INnovations in Intelligent SysTems and Applications, 21-24 June 2010, Kayseri & Cappadocia, TURKEY
planning based on piecewise path generation is addressed. To change the value of the dependent variables, (x, y, θ), the technique of a restricted maneuver in [7] is used to construct closed paths for some independent variables. The restricted maneuver used in this paper consists of a canonical path for changing only one dependent variable, y, as shown in Figure 3.. In this method, collision-free maneuvers are directly obtained by means of geometric considerations.
Figure 4. The parking area There are three possible collisions during the virtual parking process. Firstly, when the virtual vehicle goes from P0 to P1, the first possible collision may occur when point F collides with the obstacle. To avoid this collision, R1, which is the radius of the curvature from P0 to P1, is chosen in [7] between lower and upper limits of as given in following equations R1min = R1max =
ℓ
(10)
$'* ,N'O PO J Q PSJ ; ℓJ
(11)
CPS
The second possible collision may occur when the virtual vehicle goes from P1 to P2. To avoid this collision, R2, which defines the curvature path from P1 to P2, is chosen in [7] between lower and upper limits of R2 as indicated in (Eq. 12 to Eq. 15). R2min =
U J Q ℓJ ; (V;SW) J
(12)
C (V ; SW;U)
R2Tmax = | R2T (Rmin)|
Figure 3. A restricted maneuver to change y.
(13) SW
As shown in Figure 3, position of the car in y direction can be changed by following curves of two equal-radius circles and a straight-line motion. Depending on the order of straight-line motion and curve following, change in y could be in (+) or (-) direction. In this study, a restricted maneuver based on direct-motion parallel parking for a vehicle without a trailer is evaluated by following steps. First, to park the vehicle as given in Figure 3.(b), virtual parking process is applied as shown in Figure 3.(a). In virtual parking process, the virtual vehicle plans a collisionfree path to leave the park place from P0 to P2. Thus, a final maneuver is obtained from this design in the reverse order. Since the trajectory, which should be followed by the vehicle, and kinematic equations of the vehicle are known, a collision-free path is generated using the geometric constraints of the environment. In Figure 4, both the real car and the virtual car are indicated. The path generated for the virtual car to leave the parking place is as follows: from P0 to P1, P1 to P2. The vehicle and the obstacles are assumed as rectangular objects and a right-hand side parking is used.
R2 = R2T (R1 ) = R1 −
