ww.sciencedirect.com

b i o s y s t em s e ng i n e e r i n g 1 1 6 ( 2 0 1 3 ) 3 9 9e4 0 9

Available online at w

journal homepage: www.elsev ier .com/locate/ issn/15375110

Research Paper

Using continuous-curvature paths to generatefeasible headland turn manoeuvres

Dennis Sabelhaus a,*, Frank Roben b, Lars Peter Meyer zu Helligen b,Peter Schulze Lammers a

a Institut fur Landtechnik, Nußallee 5, 53115 Bonn, GermanybClaas Agrosystems KGaA mbH & Co KG, Backerkamp 19, 33330 Gutersloh, Germany

a r t i c l e i n f o

Article history:

Received 11 June 2013

Received in revised form

16 August 2013

Accepted 20 August 2013

Published online 13 October 2013

* Corresponding author. Tel.: þ49 1746896119E-mail address: dennis.sabelhaus@claas.

1537-5110/$ e see front matter ª 2013 IAgrEhttp://dx.doi.org/10.1016/j.biosystemseng.20

Today’s agricultural engineering is characterised by automation and information tech-

nology. Automatic steering systems have become an adequate tool for guidance on a track

with accuracy in the range of centimetres. Consequently, the transition from track to track

must be planned exactly, so that the target track is achieved precisely. A method which can

generate turn trajectories e so-called headland turns - with smooth transition and a fast

computation performance is investigated. The method is based on the continuous-

curvature path planning in the field of mobile robotics and is connected to the specific

agronomic requirements. In this context the clothoid construction element constitutes the

main construction element. It enables the smooth connection from zero curvature to

maximal curvature which represents the reciprocal of the minimal turning radius. In to-

tality, a manoeuvre can be planned with modified Dubins curves, both going backwards

and forwards is feasible with modified Reeds and Shepp curves. Seven different ma-

noeuvres are useful from an agronomic point of view. It is shown that all turn manoeuvres

are feasible with this method. Also an analysis regarding the trajectory length, the head-

land width and the operation time is shown.

ª 2013 IAgrE. Published by Elsevier Ltd. All rights reserved.

1. Introduction Manufacturers are beginning to derive principles from

Climate change, resource scarcity and the growing world

population are forcing agriculture to adopt more efficient crop

farming with smaller harvesting windows. Automation and

information technology enter a new modern agricultural en-

gineering, which is facing the upcoming challenges. Auto-

matic steering systems, which can guide the vehicle within a

few centimetres by the usage of differential Global Positioning

Sysemd D-GPS with correction signals, have become the

standard on large farm machinery. Today, the machine pro-

cess parameters are monitored online and can optimise

combined fleet management.

.com (D. Sabelhaus).. Published by Elsevier Lt13.08.012

monitoring and automatically adjusting the process technol-

ogy of agricultural machinery (e.g. CLAAS KgaA Harsewinkel

Germany, CEMOS Automatic) and mission planning is the

focus of research (Bochtis, Vougioukas, Tsatsarelis, &

Ampatzidis, 2007). Bochtis and Sorensen (2012) have devel-

oped methods to control the field logistics and fleet manage-

ment in a more efficient way. Jensen and Larsen (2011)

determined the non-working distance as a major criterion for

mission planning. The turn process at the end of a field track

characterises the largest part of the non-working distance.

Engelhardt (2004) investigated the turn driving time under

agronomic and economic aspects and highlighted its major

d. All rights reserved.

Nomenclature

a,aslow down,speedup.adelay Accelerations

d Distance between centre points

l Vehicle wheelbase

lclothoid Basic clothoid length

Lengthheadland turn Length of a headland turn

LTL Lock-to-lock time. (time taken for the valves to

travel from one wheel angle maximum to the

other)

p Run parameter for clothoid length

Rbig Radius of the large CC-turn circle

Rmin Minimal turning radius

tvs,turn manoeuvre Valve switching time, turning time

v, vstart, vdirection change Velocity

w Turn width

x,y Basic clothoid parameter describing the x-

coordinate

xmin Tangential distance between Rbig and Rmin

d Steering angle

k Curvature

m Clothoid angle

s Sharpness

Ux,y,start,end Centre coordinate of CC-turn circle

_k Curvature gradient

k,kmax (Maximal) curvature

s,smax (Maximal) sharpness

q Vehicle heading

b i o s y s t em s e n g i n e e r i n g 1 1 6 ( 2 0 1 3 ) 3 9 9e4 0 9400

impact on the overall process. It implied that headland width

and driving time are values which should beminimised by the

use of computational optimisation. An automatically gener-

ated turn path, which for example minimises the path length,

represents an optimisation and minimisation of the non-

working distance.

In this paper a method suitable for generating paths for an

automatic steering system is presented. The computational

time is required to be low and the trajectory must have

continuous curvature. It is also necessary for the controller to

follow the planned path. Good trajectory tracking should

guarantee a good connection between field and headland

paths and so field operation occurs with minimal imperfec-

tions and working errors. Edwards and Brochner (2011)

demonstrated an approach to generate smooth headland

paths. Their algorithm was based on the constant average

acceleration method and they stated the time rate of change

(ROCs) as a work package for future modelling. Another tool

that can be used is Bezier splines. They are characterised by a

continuous-curvature course and are rapidly and easily

generated. Oksanen and Visala (2007) showed an application

of headland paths with piecewise Bezier splines and also

investigated the influence of angle deviation and headland

width on paths. Ferguson (2010) pursued another option by

computing turning tracks using kinematic constraints A

multi-dimensional optimisation was built-in which ran a cost

function and gave good results for speeds from 0 to 2m s�1. Jin

(2009) planned headland paths, using cost appraisal and field

valuation, by approximating the headland turn as a

conglomerate of circles and straight lines. The aim was to

provide an simple and rapid cost assessment. The approach

was sufficient for the objectives but was not suitable for

smooth path generation. Vougioukas, Blackmore, Nielsen, and

Fountas (2006) also showed another approach for headland

turn generation. Their method computed optimal turns

numerically by a two-stage motion planning algorithm for a

given cost function.

Cariou, Lenain, Berducat, and Thuilot (2010), Cariou,

Lenain, Thuilot, Humbert, and Berducat (2010, pp. 1e10)

were the first to use clothoids as transition elements for

headland turns. They presented amotion-planner and lateral-

longitudinal controllers for the autonomousmanoeuvring of a

farm vehicle. This optimisation approach built a trajectory in

two steps; the motion primitives were generated and con-

nected and then the trajectories were generated using a ki-

nematic model. Clothoids depict a spiral path with increasing

curvature and same curvature change. Although they are not

arbitrary; their application is bounded to the continuous-

curvature path generation (Fraichard & Scheuer, 2004). A ho-

listic approach is used in which segments straight lines, cir-

cles with minimal turning radius and transition clothoids are

used for a symmetric construction of headland paths. Scheuer

and Fraichard (1997) demonstrated this method for mobile

robots but after interpretation and formula connection to

steering system behaviour and calibration values. Both for-

wards and backwards manoeuvres can be planned with

feasible headland turn path.

Therefore the objective of this research was to develop a

method to generate continuous curvature headland trajectory

and to investigate the properties of different turning types.

2. Analytical method

To construct automatic steering systems, with the aim of

minimising working time, an online approach for automatic

turn generation with low computational time was developed.

For many field operations such as seeding, or for controlled

traffic farming generally, turn to track connection accuracy is

important. Parallel tracks are a significant requirement for

farmers. Most methods for headland path generation

concentrate on computational time or accuracy but the

compromise between these values is not sufficient for use in

auto-guidance systems. Also the large number of potential

headland turn manoeuvres is not compared in the current

literature. Special requests such as minimisation of the

headland width are possible with this approach and also,

realistic driving times and path length approximation are

practicable and this can result in comparisons between

different turning types. In section 2.1 we introduce a vehicle

model and the basics of path planning. It is necessary to un-

derstand the vehicle behaviour while turning and thus the

necessity of smooth paths. In later sections the continuous-

curvature path planning is explained and the mathematical

model from Scheuer and Fraichard (1997) is described. This is

required for the overall understanding of the path generation

Fig. 1 e Kinematic model of an agricultural vehicle.

Fig. 3 e General case of a CC-turn.

b i o s y s t em s e ng i n e e r i n g 1 1 6 ( 2 0 1 3 ) 3 9 9e4 0 9 401

strategy which is applied and demonstrated in this paper.

Afterwards, the model is shown for agricultural headland

generation and some conceivable turn manoeuvres are

presented.

2.1. Model of agricultural vehicle

The agricultural vehicle is an extension of the Dubins model

and is similar to the car-like vehicle explained by Boissonat,

Cerezo, and Leblond (1994). The car is considered as a four-

dimensional system with the two control units velocity v

and the sharpness s. Equation (1) describes the kinematics of

an agricultural vehicle with orientation q, curvature k, sharp-

ness s and longitudinal change _x and the lateral change _y.

0B@

_x_y_q_k

1CA ¼

0B@

cos qsin q

k0

1CAvþ

0B@

0001

1CAs (1)

The restrictions required for planning manoeuvres are

curvature and sharpness. The velocity is constant.

kmaxzdmax

l; smax ¼ k$max ¼

_dmax

l cos2 dmax(2)

All listed variables are shown in Fig. 1, l is the wheelbase of

the tractor. The steering angle d and the steering angle velocity_d are limited to an upper boundary. Detailed information is

given by Fraichard and Scheuer (2004). Rear steered vehicles

Fig. 2 e Comparison of curvature progress of left-right-left-stee

planning (right).

such as combine harvesters can also be modelled. In this case

the vehicle rotates in opposite direction to the vehicle shown

in Fig. 1, around the front axle. The restrictions kmax(¼1/Rmin)

and smax are determined in the same way and they are further

limitations for turn manoeuvres.

2.2. Continuous-curvature path planning e

Continuous-curvature (CC) path planning can be divided into

CC-turns and CC-paths. CC-turn circles represent a path to

drive from straight to the minimum turning radius and to get

back on straight. CC-paths are composed of individual CC-

turn circles so that each start and end configuration is

reached and the path is feasible. The generation of CC-turns in

agricultural is similar to that in mobile robotics. However, the

CC-paths are customised for use in headland turns and

adjusted by farmer requests, operation and machine

parameters.

2.2.1. CC-turnIn most cases, a CC-turn consists of three parts. The pre-

liminary segment is a clothoid arc with maximum curvature

change, also known as sharpness, s ¼ þsmax. The curvature

increases from k ¼ 0 to k ¼ kmax. The second segment is a

ring combination of Dubins curves (left) and CC-path

Fig. 4 e Construction of an elementary clothoid.

b i o s y s t em s e n g i n e e r i n g 1 1 6 ( 2 0 1 3 ) 3 9 9e4 0 9402

circular arc with k ¼ kmax. The third, and closing segment, is a

clothoid arc with the opposite properties than the preliminary

clothoid segment. The curvature decreases from k ¼ kmax to

k ¼ 0 with the minimal sharpness s ¼ �smax. The closing clo-

thoid has the same properties and is the reverse of the pre-

liminary clothoid (Fraichard & Scheuer, 2004). Figure 3 shows

the explained construction of a typical CC-turn. The associ-

ated courses of curvature and sharpness are shown in Fig. 2.

The first part that is computed is the clothoid arc. The

characteristic of a clothoid is that the curvature changes lin-

early with its arc length. Kimia, Frankel, and Popescu (2003)

described the equations for a clothoid as the Fresnel in-

tegralsCf and Sf. For path computation, only one basic clothoid

with sharpness smax is computed. This means the clothoid

increases with maximal curvature change from k ¼ 0 to

k ¼ kmax. The clothoids are shown in Fig. 3 by the red line.

Based on the clothoid, the centre point of the CC-turn circle U

can be computed as shown by Scheuer and Fraichard (1997)

with the run parameter p as pmax ¼ kmax2 /psmax and the basic

clothoid coordinates x,y:

Ux ¼ xClothoid

�k2max

psmax

�� k�1

max � sinðqÞ (3)

Uy ¼ yClothoid

�k2max

psmax

�þ k�1

max � cosðqÞ (4)

Fig. 5 e Sequence diagram of autom

The radius Rbig is defined as the distance between centre

point and start position or end position of the CC-turns

(Fraichard & Scheuer, 2004).

Rbig ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�Ux � yClothoidð0Þ2

�þ�Uy � yClothoidð0Þ2

�r(5)

The angle m and the angle d are supplementary, m is defined

as:

m ¼ p� d ¼ atan

�Ux � yClothoidð0ÞUy � yClothoidð0Þ

�(6)

The angle d depicts the change of orientation from the start

position to the end position of the CC-turn. d is the deflection

of the CC-turn. The minimal length of deflection is defined as

dmin ¼ k2max � s�1max. In case of small deflections (jdj � dmin), the CC-

turn makes a loop or the clothoid arcs of the CC-turn self-

intersect. To avoid this case, Fraichard and Scheuer (2004)

suggested an elementary path. This elementary path is a

clothoid arc with sharpness s � smax. Self-intersections

happen by deflections of 0 < d < dmin. The required sharpness

is defined as:

s ¼p�cos

�d2

�Cf

� ffiffidp

q �þ sin

�d2

�Sf

� ffiffidp

q ��2

r2 sin2�d2 þ m

� (7)

This sharpness can be used to compute the two mirrored

clothoids using the previously listed equations. An elemen-

tary path is presented in Fig. 4.

To connect these values, described by Fraichard and

Scheuer (2004) in detail, the valve switching time tvs and the

velocity v need to be linked to the parameter clothoid length

lclothoid. This fact enables the CC-path planning for auto-

guidance systems. The following formula shows the

correlation:

lclothoid ¼ tvs2v (8)

2.3. CC-path planning for headland turn manoeuvres

CC-path planning is a good method to compute paths with

continuous-curvature limited to the minimal turning radius

and adjustable curvature change. However for agricultural

path generation not only is the shortest path searched, but

also the headland width and driving time are major criteria.

Therefore another, strict planning algorithm must be

atic headland path generation.

Fig. 7 e U-turn.

b i o s y s t em s e ng i n e e r i n g 1 1 6 ( 2 0 1 3 ) 3 9 9e4 0 9 403

developed in contrast to the CC-path planning for mobile ro-

botics where all possible paths are generated and only the

shortest possible path is selected. To give an overview, a

sequence diagram is shown in Fig. 5. The exemplary compu-

tation of an omega-turn is shown. The following listing shows

agricultural manoeuvres, and their geometry restrictions for

the utilisation of CC-turns. Firstly, the three typical turns

without self-intersections are computed. These turns are

restricted to the distance between the start and end positions.

The turn manoeuvres have special validity ranges which are

described as distances between the start and end positions.

The detailed distance formulae are listed in the following turn

descriptions. The fourth turn mode shows the slope turn,

which is an alternative to the omega-turn and is also applied

in practical applications. The next twomanoeuvres are able to

go both forward and backward in what is called a fishtail-turn.

All of these turn modes are typical for agricultural applica-

tions but also some other turn modes can be planned. Lastly,

some new ideas for turnings are presented and analysed. A

seventh headland turnmanoeuvre is considered with the idea

of minimising the headland width and restricting only driving

forward. Also, two last turns are computed to cover all com-

binations of feasible headland turn manoeuvres. Because of

safety aspects, auto-guidance systems have to switch off

during stops so that these turn paths are not considered.

2.3.1. Omega-turnThe omega-turn is the usual turn manoeuvre for skipping one

row. In the environment of CC-paths, it is defined as a left-

right-left/right-left-right turn. That means that the curvature

changes twice and the whole manoeuvre consists of three

circles. The CC-turn circles for the start and end positions are

computed first. The middle CC-turn circle affects these two

circles tangential. The tangency is the entry/exit point out of

the middle CC-turn circle so that the CC-path is an assembly

of the single CC-turn circles. If the distance between the

centre points of the start and end CC-turn circle is

d � 4*Rbig*cos(m), the an omega-turn is not feasible since the

Fig. 6 e Omega-turn (legend applies for all manoeuvres).

middle CC-circle would not provide any connecting points

with the start and end CC-circles. Thus one of the following

turn manoeuvres must be planned. Figure 6 shows the con-

struction of an omega-turn. The specified legend is valid for all

listed manoeuvre plots.

2.3.2. U-TurnFigure 7 shows the construction of a U-turn. Similar to the

planning of an omega-turn, the CC-turns for start and end

positions are planned first. A line segment connects the exit

point of the first CC-turn circle and the entry point of the

second CC-turn circle. The line segment is parallel to the

connection line between the two centre points and has the

length lline segment ¼ �Ux,start � Rbig*sin(m)þUx,end � Rbig*sin(m).

The U-turn is the only option for turning in case that the turn

distance between start and end position is d �2 � Rbig � cosðmÞ þ 2 �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR2big � ðRbig � cosðmÞÞ2

q.

2.3.3. Gap-turn manoeuvreA special case is the gap-turn manoeuvre. If an omega-turn

and aU-turn are not feasible, because of the condition 2 � Rbig �cosðmÞ � d � 2 � Rbig � cosðmÞ þ 2 �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR2big � ðRbig � cosðmÞÞ2

q, then a

gap-turn manoeuvre is planned as shown in Fig. 8.

Fig. 8 e Gap-turn.

Fig. 9 e Derivation of xmin.

Fig. 11 e Fishtail-turn.

b i o s y s t em s e n g i n e e r i n g 1 1 6 ( 2 0 1 3 ) 3 9 9e4 0 9404

It also consists of three CC-turn circles. The middle circle

centre point has the distance xmin from the start circle and end

circle centre point. Figure 9 shows how the distance xmin is

established.

The distance xmin represents the minimum distance at

which the middle CC-turn circle can be entered without sur-

roundings. The path provides a smooth path, loops and cur-

vature sign changes are avoided. If the farmer wants the

restriction, that the headland turn manoeuvre does not self-

intersect, then the slope-turn is not feasible. The centre

point of the middle CC-turn circle is defined as:

xCC�Turn;middle ¼ xCenter;start þ xCenter;end � xCenter;end

2(9)

yCC�Turn;middle¼yCenter;start

þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR2big�

�Rbig�cosðmÞ

�2q 2

�ðxCenter;end�xCenter;startÞ2r

(10)

Fig. 10 e Slope-turn.

2.3.4. Slope-turn1

The last Dubins planned CC-turns is the slope-turn. Slope

means that the planned trajectory self-intersects as shown in

Fig. 10, a right-left-right steering combination is applied. It is

the opposite of the omega-turn, which would be a left-right-

left-combination. This turn manoeuvre can be the shortest

drivable trajectory.

2.3.5. Fishtail-turnThe first four turn options can be used if the vehicle only goes

forward. They are planned with Dubins curves. In case of ve-

hicles going forwards and backwards, more options are

feasible. The so-called fishtail-turn, with or without a slope, is

drivable. It is planned using Reeds and Shepp’s curves. The

CC-turn circles for a fishtail-turn without a slope are planned

like the CC-turn circles of an omega-turn, But in the middle

circle, the direction changes and the vehicle goes backwards

Fig. 12 e Fishtail-turn with slope.

Fig. 13 e Headland width minimised turn.

Fig. 15 e Reversal pinhole turn.

b i o s y s t em s e ng i n e e r i n g 1 1 6 ( 2 0 1 3 ) 3 9 9e4 0 9 405

using theminimal turning radius. Figure 11 shows the fishtail-

turn for typical agricultural usage.

2.3.6. Fishtail-turn with slopeThe last of the standard headland turns manoeuvre is the

fishtail-turn with a slope. It is planned like the slope-turn, but

like the normal fishtail-turn, the middle part of the trajectory

goes backwards with the minimal turning radius. Figure 12

shows the fishtail-turn with slope.

2.3.7. Minimal longitudinal widthThe first six headland turn options are sufficient to cover all

start-/endpoint configurations. However, it can be useful to

generate aheadland turnwith aminimal longitudinalwidth, so

that the estimated headland width can be minimised. There-

fore, this turnmode is presented for feasible headland turnings

with this condition ofminimal longitudinalwidth, but the total

headland turn lengthwill be increased. This turnmode has the

restriction of only going forward and is not self-intersecting. It

is a useful manoeuvre for combine harvesters. As shown in

Fig. 14 e Pinhole turn.

Fig. 13, a third CC-turn circle is inserted with the restriction of

having the longitudinal distance to start point d¼Rbig. Pictori-

ally, it is a left-right-straight-right steering-combination. The

fact that the centre of U2 has the distance Rmin to the headland

boundary ensures that the width is minimised.

As shown in Fig. 13, the second circle has the minimal

feasible distance to the headland boundary. The important

correlation between the CC circles is given and all important

values can be computed. To get an overall overview, twomore

feasible turn manoeuvres are presented.

2.3.8. Pinhole turnThe eighth turnmanoeuvre is the pinhole turn. Pictorially, it is

a right-left-straight steering-combination. The two CC-circles

intersect at one point which is also the transition point. All

geometrical constraints are shown in Fig. 14. This manoeuvre

has no big advantages. The only visible benefit is the mini-

misation of computational time. The usage of gap turns can be

circumvented and no elementary paths are embedded. This is

Fig. 16 e Simulation results of curvature and curvature

change.

-35 -30 -25 -20 -15 -10 -5 0 5 10

0

5

10

15

20

25

30

Width [m]

Leng

th [m

]

Omega-Turn

Fig. 17 e Comparison of omega-turn and headland width

minimised turn.

b i o s y s t em s e n g i n e e r i n g 1 1 6 ( 2 0 1 3 ) 3 9 9e4 0 9406

the only construction element which needs to be computed

individual for each turn.

2.3.9. Reversal pinhole turnA ninth feasible turn is the reversal pinhole turn. The con-

struction is similar to the pinhole turn. The centres of the CC

circles are the same, but the major difference is that the

vehicle changes the direction twice. This turnmanoeuvre also

does not provide a length or driving time benefit. The advan-

tage is the less computational time. Figure 15 visualises the

reverse pinhole turn.

3. Results and discussion

Becauseof theabundanceofpotentialmanoeuvres available to

be investigated; only the results from one exemplary

manoeuvre were investigated for drivability. The headland

Table 1 e Headland width comparison for increasing valve tim

V ¼ 10 km h�1, R ¼ 6 m LTL ¼ 2 s LTL ¼ 4 s

Length 1 [m] 73.7866 81.8686

Length 2 [m] 42.9585 51.1286

Difference [m] 30.8281 30.74

Headland Width 1 [m] 12.3493 13.3339

Headland Width 2 [m] 17.3195 21.6375

Difference [m] 4.9702 8.3036

Table 2 e Comparison of different turn modes.

V ¼ 6 km h�1, R ¼ 6 m, 3 m distance Omega-turn S

Length [m] 43.5145

Driving Time [s] 26.1035

Headland Width [m] 17.07

turn manoeuvre into one neighbouring track was chosen. The

most important variables for the evaluation of a trajectory

are the curvature and the curvature change. They can be

computed without a kinematic vehicle model, it only depends

on the track. For analysis, a cubic spline was chosen and the

common curvature computation for this method is chosen.

The curvature with respect to time represents the curvature

change. The number of points represents the trajectory

coordinates.

The curvature and curvature change behaved as expected.

Curvature did not pass the maximal curvature of 0.1 m. It was

the reciprocal of the minimal turning radius of 10 m. The

curvature change was limited at smax z 0.4 � 10�10 m-2. The

curvature and the curvature changes eare shown in Fig. 16

with no discontinuities.

It was shown that the generated turn paths are feasible

trajectories. In the next steps, the comparable turn modes are

evaluated for the constraints track length and headland

width. Two types need to be compared. First, the headland

minimisation with restriction of only going forward is ana-

lysed. Figure 17 shows the turn modes omega-turn and

headland width minimized turn. Table 1 displays the mea-

surements for different values: velocity, minimal turning

radius and valve switching time (Tables 2 and 3).

Table 1 shows the generated headland path results for

different valve switching times. The velocity is 6 km h�1. The

minimal turning radius is 6m.The index 1 depends on the blue

headland width minimised trajectory, the red line is the

omega-turn with index 2. The headland width minimised tra-

jectoryhas a significant higher lengthvarying from38% to 72%.

However, the headland width can be minimised from 40% to

88%. The driver needs to choose the best mode based on his

requirements for his field conditions and field operation.

For distance � 2 � R � cosðmÞ þ 2 � R � cosðmÞ þ 2�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR2big � ðRbig � cosðmÞÞ2

q, only the U-turn track needs to be

planned. However, for travelling into a row with lower dis-

tance, four turn options are possible and they need to be

validated for the requested turn mode (special turns gap-turn

and headland width minimised turn are not considered). The

es (LTL [ Lock-To-Lock-Time).

LTL ¼ 6 s LTL ¼ 8 s LTL ¼ 10 s

94.9163 105.6048 117.3384

63.7368 74.1744 84.8675

31.1795 31.4304 32.4709

15.8367 17.9649 20.3639

27.9321 33.1133 38.3781

12.0954 15.1484 18.0142

lope-turn Fishtail-turn with slope Fishtail-turn

50.2719 33.51 33.5093

30.1571 22.5692 22.5688

18.77 8.913 8.643

Table 3 e Comparison of different turn modes.

V ¼ 6 km h�1, R ¼ 6 m, 10 m distance Omega-turn Slope-turn Fishtail-turn with slope Fishtail-turn

Length [m] 32.6717 57.3816 33.5097 33.5178

Driving Time [s] 19.5991 34.3816 20.1018 20.1067

Headland Width [m] 12.58 19.58 8.56 7.031

-5 0 5 100

5

10

15

Width [m]

Leng

th [m

]

Omega-Turn

-10 -5 0 5 100

5

10

15

Width [m]

Leng

th [m

]

Slope-Turn

-4 -2 0 2 4 60

2

4

6

8

Width [m]

Leng

th [m

]

Fishtail-Turn

-4 -2 0 2 4 60

2

4

6

8

Width [m]

Leng

th [m

]

Fishtail-Turn with Slope

Fig. 18 e Comparison of headland turns with

distance [ 3 m.

b i o s y s t em s e ng i n e e r i n g 1 1 6 ( 2 0 1 3 ) 3 9 9e4 0 9 407

three most important values are the driving time, turn length,

headland width.

To compute driving time, some values need to be approx-

imated. The fishtail-turn manoeuvre contains two

0 5 10

0

2

4

6

8

Width [m]

Leng

th [m

]

Fishtail-Turn

0 5 100

5

10

Width [m]

Leng

th [m

]

Omega-Turn

Fig. 19 e Comparison of headland

decelerations from 6 to 0 km h�1 and two accelerations from

0 to 6 km h�1. The speeding up and braking distances need to

be computed and subtracted from the total distance. The

remaining distances are assumed to be travelled at 6 km h�1.

The following formulae generally show how the driving time

is calculated. The approximated values are: acceleration

aslow down, speed up¼1.5 m s�2, velocity¼6 km h�1.

Dv ¼ vstart � vdirectionchange ¼ 6km h�1 (11)

Dt ¼ Dvadelay

¼ 6 kmh

3;61; 5ms2¼ 0:74 s (12)

lengthdelay¼12adelayDt

2 ¼ 0;4107 m (13)

lengthfishtale ¼ lengthfishtale � 4ldelay (14)

tfishtale ¼lengthfishtale

vþ 4 Dt (15)

tlrl;rlr ¼ lengthlrl;rlr

v(16)

The headland width and the turn manoeuvre length can be

computed directly from the turn manoeuvre computation.

The results are shown in Figs. 18 and 19. The turn ma-

noeuvreswere computed for a turn distance d of 3m and 10m.

0 5 100

2

4

6

8

Width [m]

Leng

th [m

]

Fishtail-Turn with Slope

-5 0 5 10 150

5

10

15

Width [m]

Leng

th [m

]

Slope-Turn

turns with distance [ 10 m.

-20 -15 -10 -5 0 5 10 15 20 250

5

10

15

20

25

30

35

Comparison of turn manoeuvre with different end orientation

Length [m]

Wid

th [m

]

1 2 3 4 5 6 7 8 9 10 110

10

20

30

40

50

60

70

80

90

100Histogram of Turn Manoeuvre Length

Number of Investigation

Leng

th [m

]

Fig. 20 e Path variations outcome of angle deviation with corresponding path length.

b i o s y s t em s e n g i n e e r i n g 1 1 6 ( 2 0 1 3 ) 3 9 9e4 0 9408

For the lower turn distance of 3 m, the fishtail-turn had the

shortest turn length, the lowest driving time and the lowest

headland width. These important values changed by

enlarging the turn distance. For a turn distance of 10 m, the

omega-turn had the shortest turn length and driving time.

Only in case of minimising the headland width, was the

fishtail-turn still the best choice.

The influence of angle deviation and offset of the start and

end points was investigated. The tractor settings were

selected with usual sizes. The velocity is 15 km h�1, the valve

switching time was 4 s and the minimal turning radius was

6 m. The angle deviation varied from �45� to 45� which are

usual deviations for e.g. in contour farming. Figure 20 shows

the turn manoeuvres and also the histogram with the track

lengths. In this comparison, only track length need to be

investigated because driving time can be derived by this value.

It is clearly shown that a deviation towards the start orienta-

tion extends the path length. This result is easy to interpret

because the vehicle has to change the direction. The same

effect that caused an angle deviation opposed to start orien-

tation reduced the total path length.

-20 -15 -10 -5 0 5 10 15 20 25 30

-5

0

5

10

15

20

25

30

35Comparison of turn manoeuvre with different endpoints

Length [m]

Wid

th [m

]

Fig. 21 e Path variations outcome of longitudin

The longitudinal offset of the end point was investigated.

The histogram in Fig. 21 shows offsets from�10m toþ10m in

2m steps. The importance of this offset is much less than that

of angular deviation. It was shown that an offset (positive or

negative) causes an extension of the path length. The offset

sign has only a minor influence on the extension; the offset

distance determines the path length extension.

The lateral offset from start and end point has been

investiagted. This offset implies that the turn manoeuvre

changes. The three typical agricultural turn types omega

turn, gap turn and U-turn were chosen and the offset

covered the range of 1 me35 m with a step size of 1 m. It was

shown that the U-turn has the shortest path length. In case

of offset extension, the U-turn only enlarged to the order of

this offset extension. The other two turn types behaved

differently. The gap turn had approximately the same length

in all shown cases but this path length was in a range be-

tween U-turn length and omega turn length. The omega-turn

produces the longest turns and a higher offset reduces the

total path length. Figure 22 visualises the influence of the

lateral offset.

1 2 3 4 5 6 7 8 9 10 110

10

20

30

40

50

60

70

80

90Histogram of Turn Manoeuvre Length

Number of Investigation

Leng

th [m

]

al offset with corresponding path length.

-5 0 5 10 15 20 25 30 35

-5

0

5

10

15

20

25

Width [m]

Leng

th [m

]

Comparison of Different Lateral Distances

0 5 10 15 20 25 30 35 400

5

10

15

20

25

30

35

40

45

50Histogram of lateral offset

Number of investigation

Pat

h Le

ngth

[m]

Fig. 22 e Path variations outcome of lateral offset with corresponding path length.

b i o s y s t em s e ng i n e e r i n g 1 1 6 ( 2 0 1 3 ) 3 9 9e4 0 9 409

4. Conclusions

The construction of CC-circles with its basic components

simplifies the calculation and guarantees suitable paths with

acceptable computational time, and without time-consuming

approximations or interpolations. The initial computed seg-

ments circle, straight and basic clothoid only need to be shifted

and rotated to the requested position. The Dubins curve path

planning with CC-turns can be extended, so that the gap-turn

or the headland width minimised turns can enable vehicles

to travel between each start and end point configuration.

This planning strategy is well suited for straight connec-

tions from track to track. Extensions for application in contour

farming need to be carried out in furtherwork. TheU-turnwas

established as the best turn mode. By choosing a track

sequence with skipping rows so that U-turns are feasible, the

total non-working distance, and as a consequence the total

route length can be minimised.

Manoeuvres with start or end curvature ks0 are not

feasible with CC turns. There is no known method for

computing an elementary path with this special condition.

Mathematically some more clothoid constructions are

feasible, but too many input values are missing for this situ-

ation so that only an iterative solution can solve this problem.

Completely different concepts such as spline interpolation

could be used for this special parts of the turn paths because

this case can be detected very clearly. Other track sections

could be generated with the presented method. Headland

turns for contour tracks cannot be planned. The gap-turn

enables CC-path planning for headland turn manoeuvres for

all start and end configurations but it is feasible that a shorter

path could be found by applying so called composite curves.

r e f e r e n c e s

Bochtis, D. D., & Sorensen, C. G. (2012). Conceptual model of fleetmanagement in agriculture. Biosystems Engineering, AarhusEngineering, 105, 41e50.

Bochtis, D., Vougioukas, S., Tsatsarelis, C., & Ampatzidis, Y.(2007). Optimal dynamic motion sequence generation for

multiple harvesters. Agricultural Engineering International: TheCIGREjournal, 1e9.

Boissonat, J., Cerezo, A., & Leblond, J. (1994). A note on shortestpaths in the plane subject to a constraint on the derivative of thecurvature. INRIAResearch Report 2160.

Cariou, C., Lenain, R., Berducat, M., & Thuilot, B. (2010).Autonomous maneuvers of a farm vehicle with a trailedimplement in headland. In Proceedings of the 7th internationalconference on informatics in control, automation and robotics, 2; pp.109e115).

Cariou, C., Lenain, R., Thuilot, B., Humbert, T., & Berducat, M.(2010). Manoeuvres automation for agricultural vehicle in headland.AgEng 2010, International Conference on AgriculturalEngineering, Clermont Ferrand.

Edwards, G., & Brochner, T. (2011). A method for smoothedheadland paths generation for agricultural vehicles. CIGR/NJFseminar 441-Automation and system technology in plantproduction.

Engelhardt, H. (2004). Auswirkungen von Flachengroße undFlachenform auf Wendezeiten, Arbeitserledigung undverfahrenstechnische Maßnahmen im Ackerbau (pp. 54e62).Dissertatioon Universitat Giessen: Institut fur Landtechnik.

Ferguson, D. (2010). Path generation and control for end of row turningin an orchard environment. Master Thesis. Pittsburgh PA, USA:Carnegie-Mellon Robotics Institute http://www.dhferguson.com/DavidFergusonThesis.pdf.

Fraichard, T., & Scheuer, A. (2004). From Reeds and Shepp’s tocontinuous-curvature paths. IEEE Transactions on Robotics, 6,1025e1035.

Jensen, M. F., & Larsen, J. J. (2011). Analytical derived headlandturning costs. In Conference poster on NJF seminar 441automation and system technology in plant production.

Jin, J. (2009). Optimal field coverage path planning on 2D and 3Dsurfaces. Ames, IA, USA: Iowa State University. GraduatedTheses and Dissertations. Paper 11054.

Kimia, B., Frankel, I., & Popescu, A.-M. (2003). Euler spiral for ShapeCompletion. International Journal of Computer Vision, 54, 159e182.

Oksanen, T., & Visala, A. (2007). Path planning algorithms foragricultural machines. Agricultural Engineering International: TheCIGREjournal, 2009, 1e19.

Scheuer, A., & Fraichard, T. (1997). Continuous-curvature pathplanning for car-like vehicles. In International conference onintelligent robots and systems (pp. 1e7).

Vougioukas, S., Blackmore, S., Nielsen, J., & Fountas, S.(2006). A two-staged optimal motion planner forautonomous agricultural vehicles. Precision Agriculture,7(5), 361e377.