research article feasible path planning for autonomous ...path planning is one of the most important...
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Research ArticleFeasible Path Planning for Autonomous Vehicles
Vu Trieu Minh1 and John Pumwa2
1 Department of Mechatronics, Tallinn University of Technology, Ehitajate tee 5, 19086Tallinn, Estonia2Department of Mechanical Engineering, Papua New Guinea University of Technology, Lae 411, Papua New Guinea
Correspondence should be addressed to Vu Trieu Minh; [email protected]
Received 8 January 2014; Revised 10 February 2014; Accepted 11 February 2014; Published 18 March 2014
Academic Editor: Sivaguru Ravindran
Copyright Β© 2014 V. T. Minh and J. Pumwa. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.
Theobjective of this paper is to find feasible path planning algorithms for nonholonomic vehicles including flatness, polynomial, andsymmetric polynomial trajectories subject to the real vehicle dynamical constraints. Performances of these path planning methodsare simulated and compared to evaluate the more realistic and smoother generated trajectories. Results show that the symmetricpolynomial algorithm provides the smoothest trajectory. Therefore, this algorithm is recommended for the development of anautomatic control for autonomous vehicles.
1. Introduction
Path planning is one of the most important parts of anautonomous vehicle. It deals with, searching a feasible path,taking into consideration the geometry of the vehicle andits surroundings, the kinematic constraints and others thatmay affect the feasible paths. Study of this paper can beused to develop a real-time control system for autonomousground vehicles which can track exactly on any feasible pathsfrom the global positioning system (GPS) maps or/and fromunmanned aerial vehicle (UAV) images. This system can bealso applied to autonomous unmanned ground vehicles (onroad or off road) and autoparking and autodriving systems.
The main idea of this paper is a system which canautomatically generate an optimal feasible trajectory and thencontrol the vehicle to track exactly on this path fromany givenstarting points to any given destination points from a mapsubject to the vehicle physical constraints on obstacles, speed,steering angle, and so forth.
From the existed pieces of literature on this researchtopic, there are still few research papers dealing with optimaltrajectory generation subject to real automotive engineeringconstraints. Basic studies on the flatness, nonholonomic, andnonlinear systems can be read from the recent book of LeΜvinein [1] where the fundamental motion planning of a vehicleis presented. The flatness and controllability conditions forthis nonlinear system are also well investigated and defined.
However, in this research, parking simulation of a two-trailervehicle is also demonstrated but without the constraint ofsteering angle and steering angular velocity.
The problem of trajectory generation for nonholonomicsystem is also presented by Dong and Guo in [2] wheretwo trajectory generation methods are proposed.The controlinputs are the second-order polynomial equations. By inte-grating those control inputs, coefficients for those second-order polynomial equations are found. However this paperis lacking constraint analysis on the vehicle velocity and thesteering angle.
The cell mapping techniques and reinforcement learningmethods to obtain the optimal motion planning by Gomezet al. in [3] are tested for vehicles considering kinematics,dynamics, and obstacle constraints. This paper shows asimulation of a wheeled mobile robot moving on a pathgenerated by the bang-bang trajectory generation. And thepaper does not mention the algorithms used for generatingthe vehicle trajectory. Several other research papers onoptimal trajectories and control of autonomous vehicles canbe read in [4β7]; however, most of those studies are based onthe real traffic flow roads and the control algorithms are toperform the maneuver tasks such as lane changing, merging,distance keeping, velocity keeping, stopping, and collisionavoidance.
This paper, therefore, focuses on the applicable algo-rithms to generate feasible trajectories from a starting point
Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014, Article ID 317494, 12 pageshttp://dx.doi.org/10.1155/2014/317494
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2 Mathematical Problems in Engineering
to any destination point subject to vehicle constraints. Thispaper refers to the vehicle sideslip model and estimation byMinh in [8, chapter 8].The nonlinear computational schemesfor the nonlinear systems are referred to by Minh andAfzulpurkur in [9]. Although the holonomic dynamical tra-jectories generation is not new, recent research articles havetried to develop those trajectories in different applications.Trajectory for autonomous ground vehicles based on Beziercurves operating under waypoints and corridor constraintscan be referred to by Choi et al. in [10] and a real-timetrajectory generation for car-like vehicles navigating dynamicenvironments can be read by Delsart et al. in [11]. A newpaper of trajectory generation model-based IMM trackingfor safe driving in intersection scenario can be read by Zhouet al. in [12] to reduce traffic accidents at intersections.Another paper of dynamic trajectory generation for wheeledrobots is referred to by Missura and Behnke in [13] wherea dynamic motion of wheeled robots can be determined inreal-time onboard.The real vehiclemodels and conditions forstabilizability of dynamic systems are referred to in [14, 15].
This paper introduces three (flatness, polynomial, andsymmetric) conventional holonomic trajectories generationsubject to steering angle constraint for autonomous vehicles.Performances of these three algorithms are compared. Datataken from simulation on the vehicle longitudinal velocityand its steering angular velocity will be used to develop thecontrol algorithms for the vehicle tracking on those trajecto-ries in the next part of this paper. The outline of this paper isas follows. Section 2 describes the kinematicmodel; Section 3presents the flatness method; Section 4 produces the trajec-tory generation subject steering angle constrained violation;Section 5 presents the polynomial method; Section 6 devel-ops the symmetric polynomial method; Section 7 analysesthe performance of the three methods; finally conclusion isdrawn in Section 8. Simulations in this paper are conductedin the MATLAB Simulink R2009a.
2. Kinematic Model of a Vehicle
The constraints for this vehicle model are based on theassumption that the wheels are rolling without slipping andthe steering angle is simplified as a single wheel in themidpoint of the two frond wheels. Then, a kinematic modelof a vehicle can be drawn in Figure 1.
The kinematic model of a forward rear-wheel drivingvehicle can be written as
[[[[
οΏ½ΜοΏ½Μπ¦Μπ
οΏ½ΜοΏ½
]]]]=[[[[[
cos πsin πtanππ0
]]]]]
πV1+ [[[[
0001
]]]]V2, (1)
where π = [π₯, π¦, π, π] is the system state variables, (π₯, π¦)are the Cartesian coordinates of the middle point of the rearwheel axis, π is the angle of the vehicle body to the π₯-axis,π is the steering angle, π is the vehicle wheel base, π is thewheel radius, V
1is the angular velocity of the rear wheel,
and V2is the angular steering velocity. Given the initial state
π(0) = [π₯0, π¦0, π0, π0] at time π‘ = 0 and the final state
y
y
xx
π
π
l
r
Figure 1: A simplified vehicle model.
π(π) = [π₯π, π¦π, ππ, ππ] at time π‘ = π, the paper generates
a feasible trajectory for this vehicle.Similarly, the model for a forward front-wheel driving
vehicle is presented as
[[[[
οΏ½ΜοΏ½Μπ¦Μπ
οΏ½ΜοΏ½
]]]]=[[[[[
cos π cosπsin π cosπ
tanππ0
]]]]]
πV1+ [[[[
0001
]]]]V2. (2)
The model for a forward rear-wheel driving vehicle in (1)will be used for all calculations and simulations in this paper.From this kinematic model, flatness equations for the vehicletrajectory generation are presented in the next section.
3. Vehicle Flatness Trajectory Generation
The flatness system is the system of nonlinear differentialequations in (1) whose movement curves are in smooth andflat space. From Figure 1, the vehicle angular velocity can becalculated as
V1=βοΏ½ΜοΏ½2 + Μπ¦2
π .(3)
Transforming from (1), the vehicle body angle is
π = arctan( Μπ¦οΏ½ΜοΏ½) . (4)
From the derivative of the above trigonometric, π, the bodyangular velocity, Μπ, is achieved as follows:
Μπ = Μπ¦οΏ½ΜοΏ½ β οΏ½ΜοΏ½ Μπ¦οΏ½ΜοΏ½21
( Μπ¦/οΏ½ΜοΏ½)2 + 1= Μπ¦οΏ½ΜοΏ½ β οΏ½ΜοΏ½ Μπ¦οΏ½ΜοΏ½2 + Μπ¦2 =
tanππ πV1. (5)
Therefore, π and π can be directly calculated from variables:οΏ½ΜοΏ½, οΏ½ΜοΏ½, and Μπ¦, Μπ¦. And it means that the above system is flat [1].Thus, all state and input variables can be presented by the flatoutputs π₯ and π¦.
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Mathematical Problems in Engineering 3
From (5), the boundary conditions for the outputs π₯ andπ¦ are
π2π¦ππ₯2 =
tanππcos3π .
(6)
The initial state at time, π‘ = 0, to the final state at time, π‘ = π,for π₯(π‘), is
π₯ (0) = π₯0, π₯ (π) = π₯
π. (7)
The initial state for π¦(π‘) = π¦(0) is
π¦ (0) = π¦0= tan π
0β π
2π¦ππ₯2
π‘=0= tanπ0πcos3π
0
. (8)
The final state for π¦(π‘) = π¦(π) is
π¦ (π) = π¦π= tan π
πβ π
2π¦ππ₯2
π‘=π= tanπππcos3π
π
. (9)
From the initial state (π₯0, π¦0, π0, π0) at the time π‘ = 0 to the
final state (π₯π, π¦π, ππ, ππ), under a real condition that |οΏ½ΜοΏ½(π‘)| β₯
π > 0, if it is assumed that = |π₯πβ π₯0|/2π > 0, the trajectory
of π₯(π‘) can be written freely as
π₯ (π‘) = (π β π‘π ) π₯0 +π‘ππ₯π +
π₯π β π₯0π‘ (π‘ β π)2π2 . (10)
And the trajectory of π¦(π‘) can be selected as
π¦ (π‘) = π¦0+ π‘πΌ1tan π0+ π‘2 πΌ2 tanπ02πcos3π
0
+ π‘3π1+ π‘4π2+ π‘5π3,(11)
where πΌ1= (2(π₯
πβπ₯0)β|π₯πβπ₯0|)/2π, πΌ
2= |π₯πβπ₯0|/π2, and
πΌ3= (2(π₯
πβ π₯0) + |π₯πβ π₯0|)/2π; and π = [π
1, π2, π3] = π΄β1π
with
π΄ = [[
π3 π4 π53π2 4π3 5π46π 12π2 20π3
]],
π =
[[[[[[[[[
π¦πβ π¦0β ππΌ1tan π0β π2πΌ2 tanπ02πcos3π
0
πΌ3tan ππβ πΌ1tan π0β ππΌ2 tanπ0πcos3π
0
πΌ2tanππ
πcos3ππ
β πΌ2 tanπ0πcos3π0
]]]]]]]]]
.
(12)
From (10),
π = arctan(2π2 (πΌ1tan π0+ πΌ2 tanπ0πcos3π
0
π‘
+3π1π‘2 + 4π
2π‘3 + 5π
3π‘4)
Γ (2π (π₯πβ π₯0)
β π π₯π β π₯0 + 2 π₯π β π₯0 π‘)β1)
(13)
And, from (5),
π = arctan((πΌ2 tanπ0πcos3π0
+ 6π1π‘ + 12π
2π‘2 + 20π
3π‘3)
Γ (2π2)2πcos3π
Γ ( (2π (π₯πβ π₯0)
βπ π₯π β π₯0 + 2 π₯π β π₯0 π‘)2)β1) .
(14)
The angular velocity of the vehicle in (1) can be calculatedfrom (13) and (14).
For derivative of π¦,
Μπ¦ (π‘) = (πΌ1tan π0+ πΌ2 tanπ0πcos3π
0
π‘ + 3π1π‘2 + 4π
2π‘3 + 5π
3π‘4)(15)
and, then,
Μπ¦ (π‘) = (πΌ2 tanπ0πcos3π0
+ 6π1π‘ + 12π
2π‘2 + 20π
3π‘3) (16)
and derivative of π₯ is
οΏ½ΜοΏ½ (π‘) = ((2π (π₯π β π₯0) β ππ₯π β π₯0 + 2 π₯π β π₯0 π‘)2π2 )
(17)
And, then,
οΏ½ΜοΏ½ (π‘) = (π₯π β π₯0
π2 ) . (18)
The absolute vehicle velocity can be calculated from (15) to(18) with
V1(π‘) =
βοΏ½ΜοΏ½2 + Μπ¦2π
(19)
or with another formula to calculate V1(π‘) is V
1(π‘) =
(οΏ½ΜοΏ½(π‘) cos π/π) + ( Μπ¦(π‘) sin π/π)); then
V1(π‘) = ((2π (π₯π β π₯0) β π
π₯π β π₯0 + 2 π₯π β π₯0 π‘)2ππ2 )
Γ cos π + ((πΌ1tan π0+ πΌ2 tanπ0πcos3π
0
+ 3π1π‘2
+ 4π2π‘3 + 5π
3π‘4) sin πΓ (π)β1) .
(20)
To calculate Μπ from (5),
Μπ = Μπ¦οΏ½ΜοΏ½ β οΏ½ΜοΏ½ Μπ¦οΏ½ΜοΏ½2 + Μπ¦2 =tanππ πV1. (21)
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4 Mathematical Problems in Engineering
0 1 2 3 4 5 6 7 8 9 100
5
10
Posit
ion y
Position x
0 10 20 30 40 50 60 70 80 90 1000
1
2
3
4
Time t
Vehi
cle v
eloc
ityοΏ½1
(km
)
Figure 2: Trajectory and velocity.
To calculate V2(π‘) = οΏ½ΜοΏ½, from (14), if π = arctan(π), then,
οΏ½ΜοΏ½ = οΏ½ΜοΏ½ ( 11 + π2 ) with
π = ((πΌ2 tanπ0πcos3π0
+ 6π1π‘ + 12π
2π‘2 + 20π
3π‘3) (2π2)2πcos3π
Γ ( (2π (π₯πβ π₯0)
β π π₯π β π₯0 + 2 π₯π β π₯0 π‘)2)β1) .
(22)
Another formula to calculate V2(π‘) = οΏ½ΜοΏ½which is derived from
(5) is
οΏ½ΜοΏ½ = π(π Μπ¦οΏ½ΜοΏ½ β οΏ½ΜοΏ½ Μπ¦π(οΏ½ΜοΏ½2 + Μπ¦2)3/2
)
Γ( 11 + (π ( Μπ¦οΏ½ΜοΏ½ β οΏ½ΜοΏ½ Μπ¦/π(οΏ½ΜοΏ½2 + Μπ¦2)3/2))2
).(23)
Simulation for this flatness technique and all other simu-lations for different path planning algorithms in this paperare conducted with the vehicleβs parameters of π = 2m andπ = 0.25m; the vehicle initial state at time π‘ = 0 is π₯(0) =[π₯0, π¦0, π0, π0] = [0, 0, 0, 0]; the vehicle final state at time
π‘ = π = 100 is π₯(π) = [π₯π, π¦π, ππ, ππ] = [10, 10, 0, π/6].
Figure 2 shows the trajectory (π₯, π¦) for the vehicle fromthe initial position to the final position and its velocity.Figure 3 shows the vehicle body angle, π, and the steeringangle, π, corresponding to their angular velocities, Μπ and οΏ½ΜοΏ½.
Themaximum steering angle for this movement is πmax =64β; it exceeds the physical constraint of steering angle withβ45β β€ πmax β€ 45β. Therefore, the above trajectory is notfeasible. The problem is that the distance from the startingpoint to the destination point is too short subject to the realvehicle physical constraints. A solution for this problem is tolengthen the travelling distance until it meets the constrainton steering angle. This issue is discussed in the next section.
4. Trajectory Subject toSteering Angle Constraint
The maximum steering angle of a vehicle depends on its realstructure space anddesign. In this paper, it is assumed that thereal structure constraint for the vehicle is (βπ/4) β€ π β€ π/4or the maximum steering angle for this vehicle is in a rangeof β45β β€ π β€ 45β:
βπ4 β€ π β€π4 . (24)
Simulation in the previous part shows the maximum steeringangle, πmax = 64β, exceeding the limitation in (24).Therefore,a new technique for the vehicle trajectory generation subjectthe constraint in (25) is proposed.
For time π‘ from initial state, π‘ = 0, to the final state, π‘ = π,the feasible trajectories π₯(π‘) and π¦(π‘) subject to conditions in(10) and (11) can be generated. During the progress of timing,π‘, recalculate the steering angle, π, in (14) and then checkfor the steering constraint in (24). If the constraint in (24) isviolated, the distance from the initial position (π₯
0, π¦0) to the
final position (π₯π, π¦π) is too short for satisfying the steering
angle limit.In this case, it is proposed that to lengthen the distance
π0= β(π₯
πβ π₯0)2 + (π¦
πβ π¦0)2 to a new distance,
πππ= πππ0
with π > 1 (25)for π = 1, 2, 3, . . . π until the constraint in (24) is satisfied andwith π being an amplification coefficient, π > 1.
Then the new coordinates are
π₯ππ
= ππ (π₯πβ π₯0) , π¦
ππ= ππ (π¦
πβ π¦0) . (26)
The next simulation is done with the same parameters in theprevious part and subject to the constraint in (24) with anamplification coefficient π = 1.1. The constraint in (24) willbe satisfied after π = 4 iterations.Thenew coordinate isπ₯
ππ=
π¦ππ
= 23.579.Figure 4 shows the new trajectory and new velocity of the
vehicle with the new lengthened distance. The velocity of thevehicle is increased since the final time π is not changed.
Figure 5 shows the new vehicle body angle, π, velocity, Μπ,steering angle, π, and velocity, οΏ½ΜοΏ½, for the new trajectory. Thenew steering angle has now satisfied the constraint π = 40β.
Due to the size of this paper, the sideslip of the vehiclemodel is ignored. In reality, the sideslip of a vehicle dependson the tire stiffness and the cornering velocity. Then thetrajectory generation in this study does not depend on thevehicle velocity. In the next part, a new vehicle trajectorygeneration based on polynomial equations is investigated.
5. Polynomial Trajectory Generation
For faster generation of a feasible vehicle tracking, a second-order polynomial for trajectory generation is presented.Equation (1) is separated into the following forms:
π§1= π₯, π§
2= tanππcos3π , π§3 = tan π, π§4 = π¦. (27)
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Mathematical Problems in Engineering 5
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Time t Time t
0 50 100
0
50
100
0 50 100
0
0.5
1
0 50 100
0
50
100
0 50 100
0
1
2
3
β0.5
β50 β50
β1
Body
angl
eΜ
π(r
pm)
Stee
r vel
ocity
π(r
pm)
Body
angl
eπβ
Stee
r ang
leπβ
Figure 3: Body and steering angle.
0 5 10 15 2005
10152025
0 10 20 30 40 50 60 70 80 90 1000
2
4
6
8
Posit
ion y
Position x
Time t
Vehi
cle v
eloci
tyοΏ½1
(km
)
Figure 4: Trajectory and velocity.
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6 Mathematical Problems in Engineering
0 50 100
0
50
100
0 50 100
0
20
40
0 50 100
0
0.5
1
β50 β20
β0.50 50 100
0
0.5
1
β0.5
Time t Time t
Time t Time t
Body
angl
eΜ
π(r
pm)
Stee
r vel
ocity
π(r
pm)
Body
angl
eπβ
Stee
r ang
leπβ
Figure 5: Body and steering angle.
Then, οΏ½ΜοΏ½1= οΏ½ΜοΏ½, (28)
οΏ½ΜοΏ½2= V2π cos
2π + 3πV1cos π sin πsin2π
π2cos5πcos2π ,
οΏ½ΜοΏ½3= tanππcos2ππV1,
οΏ½ΜοΏ½4= Μπ¦ = sin ππV
1.
(29)
The vehicle will move from the initial state (π₯0, π¦0, π0, π0) at
time π‘ = 0 to the final state (π₯π, π¦π, ππ, ππ) at time π‘ = π
corresponding from the initial state at (π§1.0, π§2.0, π§3.0, π§4.0) to
the final state at (π§1.π, π§2.π, π§3.π, π§4.π).
For 0 β€ π‘ β€ π, the calculation of [π§1(π‘), π§2(π‘), π§3(π‘), π§4(π‘)]
will be
π§1(π‘) = π§
1.0+ ππ‘,
π§2(π‘) = π§
2.0+ β1π‘ + 12β2π‘
2 + 13β3π‘3,
π§3(π‘) = π§
3.0+ ππ§2.0π‘ + 12πβ1π‘
2 + 16πβ2π‘3 + 112πβ3π‘
4,
π§4(π‘) = π§
4.0+ ππ§3.0π‘ + 12π
2π§2.0π‘2
+ 16π2β1π‘3 + 124π
2β2π‘4 + 160π
2β3π‘5
(30)
with
π = π§1.π β π§1.0π ,[β1, β2, β3] = π·β1π,
π· =[[[[[[[[
π 12π2 1
3π3
12ππ2 1
6ππ3 1
12ππ4
16π2π3 124π
2π4 160π2π5
]]]]]]]]
,
π =[[[[[
π§2.π
β π§2.0
π§3.π
β π§3.0β ππ§2.0π
π§4.π
β π§4.0β ππ§3.0π β 12π
2π§2.0π2]]]]]
.
(31)
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Mathematical Problems in Engineering 7
0 1 2 3 4 5 6 7 8 9 1002468
10
0 10 20 30 40 50 60 70 80 90 1000
2
4
6
8
Posit
ion y
Position x
Time t
Vehi
cle v
eloc
ityοΏ½1
(km
)
Figure 6: Trajectory and velocity.
Simulation of this method is conducted with the sameparameters in Section 2 and is shown in Figures 6 and 7 usingthe following equations:
π = arctan (π§3) ,
π = arctan (π§2πcos3π) ,
οΏ½ΜοΏ½ (π‘) = οΏ½ΜοΏ½1(π‘) = π,
Μπ¦ (π‘) = οΏ½ΜοΏ½4(π‘) = ππ§
3.0+ π2π§
2.0π‘
+ 12π2β1π‘2 + 18π
2β2π‘3 + 112π
2β3π‘4,
V1(π‘) =
βοΏ½ΜοΏ½2 (π‘) + Μπ¦2 (π‘)π ,
Μπ = οΏ½ΜοΏ½3cos2π = (ππ§
2.0+ πβ1π‘ + 12πβ2π‘
2 + 13πβ3π‘3) cos2π,
(32)
And, from (28),
οΏ½ΜοΏ½ = οΏ½ΜοΏ½2(π‘) πcos3π cos2π β 3 Μπ tan π sinπ cosπ. (33)
Figure 6 shows the trajectory (π₯,π¦) for the vehicle from initialpoint to the final point and the velocity of the vehicle.
Figure 7 shows the vehicle body angle, π, and the steeringangle,π, corresponding to the angular velocity, Μπ and οΏ½ΜοΏ½, of thevehicle.
As shown in Figure 7, the maximum steering angle forthis trajectory generation is π = 41.5736β and satisfiedthe constraint β45β β€ π β€ 45β. Therefore this trajectorygeneration is better than the flatness method in the previouspart.
From Figure 6, it is really not realistic when the vehiclespeed is increasing exponentially. It is expected that, when avehicle is moving from one point to another point, the speedwill increase gradually at the starting point and decreasegradually to the destination point. Therefore, in the nextpart, new symmetric polynomial equations in third order areinvestigated.
6. Symmetric PolynomialTrajectory Generation
Since the system is flat and each flat output can be parame-terized by a sufficiently smooth polynomial, symmetric third-order polynomial equations are tried for this trajectory gen-eration. Because the sideslip is ignored, the vehicle trajectorydoes not depend on its speed, V
1, and on the travelling time,
π. A new time variable, therefore, for this system is appliedwith a ratio of π‘/π for π‘ = 0 Γ· π:
π₯ (π‘) = β( π‘π β 1)3
π₯0+ ( π‘π)
3
π₯π
+ ππ₯( π‘π)2
( π‘π β 1) + ππ₯π‘π(
π‘π β 1)
2(34)
and the trajectory of π¦ is
π¦ (π‘) = β( π‘π β 1)3
π¦0+ ( π‘π)
3
π₯π
+ ππ¦( π‘π)2
( π‘π β 1) + ππ¦π‘π(
π‘π β 1)
2
.(35)
Derivation of π₯(π‘) is
οΏ½ΜοΏ½ (π‘) = β3( π‘π β 1)2
π₯0+ 3( π‘π)
2
π₯π+ ππ₯2 π‘π (
π‘π β 1)
+ ππ₯( π‘π)2
+ ππ₯( π‘π β 1)
2
+ ππ₯2 π‘π (
π‘π β 1)
(36)
and derivation of π¦(π‘) is
Μπ¦ (π‘) = β3( π‘π β 1)2
π¦0+ 3( π‘π)
2
π¦π+ ππ¦2 π‘π (
π‘π β 1)
+ ππ¦( π‘π)2
+ ππ¦( π‘π β 1)
2
+ ππ¦2 π‘π (
π‘π β 1) .
(37)
Then,
οΏ½ΜοΏ½ (π‘) = β6 ( π‘π β 1) π₯0 + 6π‘ππ₯π + ππ₯2 (2
π‘π β 1)
+ ππ₯2 π‘π + ππ₯2 (
π‘π β 1) + ππ₯2 (2
π‘π β 1) ,
(38)
Μπ¦ (π‘) = β6 ( π‘π β 1)π¦0 + 6π‘ππ¦π + ππ¦2 (2
π‘π β 1)
+ ππ¦2 π‘π + ππ¦2 (
π‘π β 1) + ππ¦2 (2
π‘π β 1) .
(39)
The constraint on speed is
πV1= οΏ½ΜοΏ½cos π =
Μπ¦sin π . (40)
The constraint at starting point π‘ = 0 is
οΏ½ΜοΏ½ (0) = π0cos π0, Μπ¦ (0) = π
0sin π0. (41)
The constraint at destination point π‘ = π is
οΏ½ΜοΏ½ (π) = ππcos ππ, Μπ¦ (π) = π
πsin ππ. (42)
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8 Mathematical Problems in Engineering
0 50 100
0
50
100
0 50 100
0
0.5
0 50 100
0
50
0 50 100
0
0.5
1
β50 β50
β0.5β0.5
Time t Time t
Time t Time t
Body
angl
eΜ
π(r
pm)
Stee
r vel
ocity
π(r
pm)
Body
angl
eπβ
Stee
r ang
leπβ
Figure 7: Body and steering angle.
0 1 2 3 4 5 6 7 8 9 100
5
10
0 10 20 30 40 50 60 70 80 90 1000
1
2
3
Posit
ion y
Position x
Time t
Vehi
cle v
eloci
tyοΏ½1
(km
)
Figure 8: Trajectory and velocity.
-
Mathematical Problems in Engineering 9
0 50 1000
20
40
60
0 50 100
0
20
40
60
0 50 100
0
20
40
60
0 50 100
0
1
2
β20
β20
β1
Time t Time t
Time t Time t
Body
angl
eΜ
π(r
pm)
Stee
r vel
ocity
π(r
pm)
Body
angl
eπβ
Stee
r ang
leπβ
Figure 9: Body and steering angle.
0 1 2 3 4 5 6 7 8 9 100
5
10
FlatnessPolynomialSymmetric polynomial
0 10 20 30 40 50 60 70 80 90 1000
2
4
6
8
Posit
iony
Position x
Time t
Velo
cityοΏ½1
Figure 10: Comparison of trajectory and velocity.
-
10 Mathematical Problems in Engineering
0 50
50
100
0
20
40
60
FlatnessPolynomialSymmetric polynomial Symmetric polynomial
FlatnessPolynomial
0 100
0
50
0 50 100
0
50
0 50 100
0
0.5
1
1.5
β50
β50
Time t Time t
Time t Time t
Body
angl
eΜ
π(r
pm)
Stee
r vel
ocity
π(r
pm)
Body
angl
eπβ
Stee
r ang
leπβ
Figure 11: Comparison of body and steering angle.
0 2 4 6 8 100
5
10
0 10 20 30 40 50 60 70 80 90 100
0
2
4
β2
Posit
ion y
Position x
Time t
Vehi
cle v
eloci
tyοΏ½1
(km
)
Figure 12: Trajectory and velocity in reverse speed.
-
Mathematical Problems in Engineering 11
0 50 100
0
50
100
0 50 100
0
50
0 50 100
0
20
40
0 50 100
0
1
2
β1
β2
β20
β50
β50
β100
Time t Time t
Time t Time t
Body
angl
eΜ
π(r
pm)
Stee
r vel
ocity
π(r
pm)
Body
angl
eπβ
Stee
r ang
leπβ
Figure 13: Body and steering angle in reverse speed.
From (38), (39), (41), and (42), for the calculation simplicity,it is assumed that the speed coefficients at the starting anddestination points, π
0= ππ= π; then
ππ₯= π cos π
πβ 3π₯π, π
π₯= π cos π
0β 3π₯0. (43)
Similarly,
ππ¦= π sin π
πβ 3π¦π, π
π¦= π sin π
0β 3π¦0. (44)
Simulation is conducted with the same parameters in theprevious parts and is shown in Figures 8 and 9. The speedcoefficients are π
0= ππ= π = 1. Other parameters are
calculated in the following equations:
V1=βοΏ½ΜοΏ½2 + Μπ¦2
π ,
π = arctan( Μπ¦οΏ½ΜοΏ½) ,
π = arctan(πcos3π Μπ¦
(οΏ½ΜοΏ½)2 )
or π = arctan(π Μπ¦οΏ½ΜοΏ½ β οΏ½ΜοΏ½ Μπ¦π(οΏ½ΜοΏ½2 + Μπ¦2)3/2
) ,
Μπ = Μπ¦οΏ½ΜοΏ½ β οΏ½ΜοΏ½ Μπ¦οΏ½ΜοΏ½21
( Μπ¦/οΏ½ΜοΏ½ )2 + 1= Μπ¦οΏ½ΜοΏ½ β οΏ½ΜοΏ½ Μπ¦οΏ½ΜοΏ½2 + Μπ¦2 =
tanππ πV1,
οΏ½ΜοΏ½ =π (arctan(π (( Μπ¦οΏ½ΜοΏ½ β οΏ½ΜοΏ½ Μπ¦) /π(οΏ½ΜοΏ½2 + Μπ¦2)3/2)))
ππ‘ .(45)
It can be seen from Figure 8 that the trajectory of thissymmetric polynomial is more realistic because the velocityincreases from the starting point and decreases in the desti-nation point as per the expectation.
The maximum steering angle for the symmetric polyno-mial method is π = 41.1622β and satisfied the constraintβ45β β€ π β€ 45β. This steering angle is smaller than thesteering angle in the second-order polynomial method. Inthe next part, a comparison of the previous three methodsis presented.
-
12 Mathematical Problems in Engineering
7. Performance Analysis
Performances of the three methods on the trajectory genera-tion and the vehicle velocity are shown in Figures 10 and 11.
It can be seen that the symmetric polynomial generationcan produce a more realistic speed and a smoother trajectorysince it allows the vehicle gradually to increase the speed atthe starting point and reduce the speed at the destinationpoint.
Figure 11 shows the symmetric polynomial method pro-viding the lowest body angle, π(π‘), the body angle velocity,Μπ(π‘), the steering angle, π(π‘), and the steering angle velocity,οΏ½ΜοΏ½(π‘). Therefore, this method is recommended for the devel-opment of an automatic control of tracking vehicles.
For the vehicle moving in reverse speeds, as mentionedin (1), the vehicle velocity, V
1, will change the sign. The
speed coefficients in (43) and (44) will be minus values.Simulations for the reverse speeds are conducted with thespeed coefficients π
0= ππ= π = β1. Results of the simulation
are shown in Figures 12 and 13.Since the vehicle is forced to reverse at the starting point
with π0= β1 and at the destination pointwith π
πβ1, Figure 12
shows the vehicle reversing out at the starting point, goingforward to the destination, and then again reversing to theparking space. The vehicle velocity is changing the directionin three times.
Figure 13 shows the body angle, π, switching 180β in twotimes corresponding to each reverse speed. The maximumsteering angle in this reverse speed is πmax = 33.7097β. Thisnumber ismuch lower than the numbers of the forward speedmovements. However the travelling distance is also longer inthe reverse speed trajectory.
8. Conclusion
The paper has presented three methods of trajectory genera-tion for autonomous vehicles subject to constraints. Regard-ing the real vehicle speed development, the third-ordersymmetric polynomial trajectory is recommended. Simula-tions and analyses are also conducted for vehicle moving inforward and in reverse speeds. Results from this study canbe used to develop a real-time control system for autodrivingand autoparking vehicles. The limitation of this study is theignorance of the vehicle sideslip due to the cornering velocity.However this error can be eliminated with the feedbackcontrol loop and some offset margins of the steering angleconstraint.
Conflict of Interests
The authors declare that there is no conflict of interests regar-ding the publication of this paper.
References
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