Nonlinear Output Feedback for Autonomous U-turn Maneuvers of a

Robot in Orchard Headlands

E. Le Fl

echer

1,2 a

, A. Durand-Petiteville

, F. Gouaisbaut

1,2

, V. Cadenat

1,2

, T. Sentenac

and S. Vougioukas

CNRS, LAAS, 7 avenue du colonel Roche, F-31400 Toulouse, France

Univ. de Toulouse, UPS, F-31400, Toulouse, France

Departement of Biological and Agricultural Engineering, University of California, Davis, CA, 95616, U.S.A.

Keywords:

Mobile Robotics, Sensor-based Navigation, Control Theory, Agricultural Robotics, Orchards.

Abstract:

This paper is devoted to the navigation of a robot in orchard headlands using embedded sensors such as lasers,

Lidars or cameras. The main idea is to consider a differential robot model directly in polar coordinates and

not in Cartesian coordinates which makes it possible to obtain simpler expressions of the outputs. Then two

nonlinear output state feedback controllers are proposed to track two shapes based on spirals allowing to go

from one row of fruit trees to another. These controllers are based on an input to output linearization and

proved to be very efﬁcient on simulations.

1 INTRODUCTION

According to (Foley et al., 2011), agriculture will

need by 2050 to double its production to feed the in-

creasing population. Mechanization has been iden-

tiﬁed as one of the best solutions to increase signiﬁ-

cantly the food production (Reid, 2011). For mowing,

spraying, pruning, harvesting in ﬁelds or orchards,

one of the main challenge consists in navigating. In

other words, the robot has to autonomously drive from

the entrance of a row to its exit, then navigate in the

headlands to reach the entrance of the next row. This

process is repeated to cover the area of interest. This

problem has been addressed for many years as it is

shown in the review proposed in (Li et al., 2009).

Most of the presented solutions focus on open ﬁeld

navigation and rely on GPS. The presented work is in-

cluded in the orchards navigation problem where GPS

cannot be used because its signal is blocked by the

dense canopy. For this reason, orchards navigation

systems are based on sensors such as laser range ﬁnd-

ers, Lidars or cameras. For example, laser range ﬁnd-

ers and cameras are used to drive through rows (Sub-

ramanian et al., 2006), (Shariﬁ and Chen, 2015). The

headland navigation problem is addressed in works

such as (Andersen et al., 2010), (Zhang et al., 2014)

and (Bayar et al., 2015) where sensory data (proprio-

https://orcid.org/0000-0002-2683-8675

ceptive and/or exteroceptive) are used to localize the

robot in a metric map of the orchard in order to follow

a precomputed path. However, keeping an updated

metric map seems challenging due to constant varia-

tion of the environment. Indeed, over the years, trees

grow up, and are pruned; over the seasons, leaves

grow then disappear; over the days, fruits grow, bend

branches and ﬁnally fall. Moreover, metric localiza-

tion can accumulate errors and lead to the navigation

failure. To overcome these drawbacks, we have pre-

sented in (Durand-Petiteville et al., 2017) a solution

relying on a topological representation of the envi-

ronment coupled with a set of output feedback con-

trollers. Especially, the headland navigation consists

in a U-shape turn around the last tree of the row. It

is performed thanks to an output feedback controller

tracking a spiral, inspired by the work presented in

(Boyadzhiev, 1999).

However, as presented in (Bochtis and Vou-

gioukas, 2008) and in (Zhang et al., 2014), it is not

always possible to perform a simple U-shape turn be-

cause of the minimum turning radius of the robot

which might be too small to deal with narrow rows.

It is then mandatory to consider different shapes for

turns. Thus, in this paper, we address the headland

navigation by designing controllers allowing to per-

form two different shapes of turns (see ﬁgure 1). The

ﬁrst one, the classical U-shape turn, is used when

Flécher, E., Durand-Petiteville, A., Gouaisbaut, F., Cadenat, V., Sentenac, T. and Vougioukas, S.

Nonlinear Output Feedback for Autonomous U-turn Maneuvers of a Robot in Orchard Headlands.

DOI: 10.5220/0007918803550362

In Proceedings of the 16th International Conference on Informatics in Control, Automation and Robotics (ICINCO 2019), pages 355-362

ISBN: 978-989-758-380-3

355

the robot turning radius is large enough to allow the

vehicle directly reach the middle of the next row.

When it is not the case, a more complex shape for the

turn known as Ω-shape turn is chosen as proposed in

(Bochtis and Vougioukas, 2008). This shape is clas-

sically used when the robot can only drive forward.

The lack of sensors to detect hazards while driving

backward usually motivates this choice.

Figure 1: Dashed red: U-shape k Solid blue: Ω-shape.

The problem considered in this paper is centered

around control only (the planning level is not consid-

ered here). Thus, this work aims i) at creating the

paths of reference to perform U-turn maneuvers (ei-

ther Ω or U, built from spirals, following the ideas

proposed in (Boyadzhiev, 1999)) and ii) at design-

ing nonlinear controllers able to track the predeﬁned

shapes. The whole problem is stated in terms of po-

lar coordinates instead of Cartesian ones as presented

in (Siegwart et al., 2011). This formulation allows

to design output feedback controllers which depend

on the measurements and not only on the robot ab-

solute position for instance as it has been done in

(d’Andrea Novel et al., 1992; d’Andr

ea Novel et al.,

1995). Therefore, unlike the works (Asif et al., 2016;

Yang et al., 2016; Shi et al., 2016), there is no need to

consider the design of observers to recover the whole

state. In this way, metric localization and cumulative

errors can be avoided. From a technical point of view,

the proposed controllers rely on an input to output lin-

earization based on the expression of the errors dy-

namics. A nonlinear diffeomorphism based only on

the outputs allows to transform the nonlinear model

into a linear system. This latter is then controlled

using classical linear control laws. Finally, notice

that both controllers induce a zero dynamic which is

proved to be stable only.

This paper is organized as follows. The next sec-

tion is devoted to the problem statement. An exoge-

nous model of the spiral shape is ﬁrst derived using

polar information. Then, based on this reference, a

nonlinear model of the error dynamics is established.

The nonlinear output feedbacks controllers, based on

input to state linearization techniques (Isidori, 2013),

allowing to vanish these errors are designed in section

3. Finally, simulation results allowing to highlight the

interest and the efﬁciency of the approach end the ar-

ticle.

2 ROBOT AND U-TURNS

MODELING

In this section, the mathematical models used in this

paper are introduced. First the model of a differen-

tial robot is presented. Next, the spiral model from

(Boyadzhiev, 1999) is recalled. Finally, based on the

geometry of the spiral, the two shapes of turns are

modeled.

2.1 System Modeling

We consider a differential robot driving in the head-

land by turning around the last tree of a row. Firstly,

a global frame F

= (O

,~x

,~y

,~z

) represents the

position of the tree of interest (see ﬁgure 2(a)). Next,

the frame F

= (O

,~x

,~y

,~z

) is attached to the dif-

ferential robot. The robot states in F

are deﬁned by

χ(t) = [d(t) β(t) α(t)]

, where d(t) is the norm of

the vector

d connecting O

and 0

, β(t) the angle be-

tween ~x

and

d , and α(t) the angle between ~x

and

d. d(t) and β(t) are the polar coordinates of the robot

in F

, and α(t) is its orientation. This choice for the

state representation is motivated by the fact that both

d(t) and α(t) can be directly measured by embedded

sensors such as laser range ﬁnders, stereo visions sys-

tems or Lidars. Thus, it is assumed that the measure

is given by:

y(t) =



1 0 0

0 0 1



χ(t) (1)

Finally, the robot control input is deﬁned by

[v(t) ω(t)]

, with v(t) the linear velocity along~x

and

ω(t) the angular velocity around ~z

. Thus, the kine-

matic equations are given by:





d(t)

β(t)

α(t)











−cos(α(t)) 0

−sin(α(t))

d(t)

sin(α(t))

d(t)

−1









v(t)

ω(t)



(2)

2.2 Spiral Modeling

In this work, it is proposed to perform different shapes

of turns by tracking paths created thanks to spirals,

whose a model is given in (Boyadzhiev, 1999). Let

deﬁne a point O

moving on a plane with respect to

a ﬁxed point O

(see ﬁgure 2(b)). From now on, O

will be considered as the center of the spiral. ~v

is the

velocity vector applied to O

and its norm is denoted

ICINCO 2019 - 16th International Conference on Informatics in Control, Automation and Robotics

356

(a) System model (b) Spiral model

Figure 2: Models.

by v

(t). Moreover

is the vector connecting O

whose norm is d

(t). Finally α

(t) is deﬁned as

the oriented angle between~v

and

. In (Boyadzhiev,

1999) it is shown that if both v

(t) and α

(t) are con-

stant then O

describes a spiral whose center is O

For this reason they are respectively denoted v

and

from now on. Moreover the author shows that the

dynamics of the distance d

is deﬁned as follows:

= −v

cos(α

) (3)

As it can be seen in this equation, the type of per-

formed spiral depends on the sole parameter α

. First

if 0 < α

< π, O

turns counter-clockwise with re-

spect to O

otherwise if 0 < α

< −π it turns clock-

wise. Then if 0 ≤ α

< π/2 or 0 ≤ α

< −π/2,

(t) decreases with time. In other words, O

describing an inward spiral around O

. If π/2 <

≤ π or −π/2 < α

≤ −π then d

(t) increases with

time which means O

is describing an outward spi-

ral around O

. Finally, if α

= π/2 or α

= −π/2,

(t) = d

(0). O

then describes a circle of radius

(0) around O

Equation (3) and its analysis highlight our interest

in the spirals. Indeed, adapting the spiral model to our

system, i.e., the center of the spiral is the position of

the tree of interest with O

= O

, deﬁnes a reference

path, whose distance dynamics is known, solely based

on one point. The reference path is thus deﬁned in the

robot sensor space.

2.3 U-shape Turn Modeling

2.3.1 U-shape Turn in the Navigation Problem

In the context of an orchard navigation, the robot has

to drive from p

, end of the current row, to p

, begin-

ning of the next row (see ﬁgure 3(a)). In this spe-

ciﬁc application, the width of each row is roughly

known (e.g., farmers data or Google map) and saved

in the robot database. Thus, when the end of the

row is detected by a dedicated data processing algo-

rithm, the robot measures d

and extracts d

from

its database. It is then possible to compute the spiral

linking those two points, i.e., to compute α

for a

given v

. Moreover, for a given spiral, one can calcu-

late the required turning radius. If this latter is smaller

than the robot one, then a U-shape turn is tracked to

enter the next row.

2.3.2 U-shape Turn Error

As previously explained, U-shape turns are performed

by tracking a spiral. To do so, it is mandatory to re-

spectively make α(t) and d(t) converge towards α

and d

(t). Thus, the problem of tracking a U-shape

turn can be seen as the problem of ﬁnding a controller

such that the error (4) converges to zero.



(t) = d(t) − d

(t)

(t) = α(t) − α

(4)

(a) U-shape and Ω-shape

turns

0 10 20 30 40 50

(t) (deg)

d ( (t)) (m)

(b) Dotted: d

& d

Solid: d

(β(t))

Figure 3: U-shape, Ω-shape and distance proﬁle d

(β(t)).

2.4 Ω-shape Turn

2.4.1 Ω-shape Turn in the Navigation Problem

In case the spiral S

computed in 2.3.1 requires a too

large turning radius, the robot has to perform an Ω-

shape turn. One recalls that this choice is motivated

by the inability of the robot to go backward. An Ω-

shape turn is made of three parts: i) moving from p

to a spiral S

(ϕ

), ii) tracking S

(ϕ

), and iii) mov-

ing from S

to p

(ϕ

). For each part, the robot turn-

ing radius has to be large enough to track the refer-

ence. Moreover, one has to guarantee the Ω shape,

and thus the success of the U-turn. Let deﬁne, β

the angular distance to switch from one row to the

next one. Moreover, β

, β

and β

are respec-

tively the angular distances of parts ϕ

, ϕ

and ϕ

Because β

+ β

= β

, with β

≥ 0, β

≥ 0

and β

≥ 0, it is mandatory that:

+ β

≤ β

(5)

Next, let deﬁne S

min

, the spiral centered on O

with

the smallest radius d

min

such that the robot turning ra-

dius is large enough to track it. Thus, parts ϕ

and ϕ

Nonlinear Output Feedback for Autonomous U-turn Maneuvers of a Robot in Orchard Headlands

357

have to make the robot move from/to spiral S

to/from

a spiral S

centered on O

with a radius d

≥ d

min

while respecting the constraint introduced by equa-

tion (5). For this reason, it is proposed to track a

distance proﬁle d

(β(t)), guaranteeing that the robot

reaches the spiral for a given angular distance.

2.4.2 Distance Proﬁle Design

To design the distance proﬁle d

(β(t)), we ﬁrst deﬁne

normalized angle

β(t), with 0 ≤

β(t) ≤ 1, as:

β(t) =

β(t) − β(t

)

β(t

) − β(t

)

(6)

where t

and t

are the initial and ﬁnal time, and β(t)

is based on the encoders measurements. β(t

) and

β(t

) represent therefore the starting angle and the ﬁ-

nal angle. In addition,

(

β(t)) is deﬁned as the nor-

malized distance to track, with 0 ≤

(

β(t)) ≤ 1. In

order to obtain a distance proﬁle dealing with the ini-

tial and ﬁnal robot orientations, the path described by

the proﬁle is tangent to the spirals. Thus, it is pro-

posed to deﬁne

(

β(t)) such as:

(

β(t)) = k

tan

−1

(

β(t) + k

)) + k

(7)

with k

and k

, scalar terms used to design

the shape of the normalized function. Moreover the

choice of those parameters has to guarantee that the

distance proﬁle to track does not require a too large

turning radius. Finally, using equations (6) and (7),

we obtain:

(β(t)) = d(t

) +

(

β(t))[d(t

) − d(t

)] (8)

An example of this distance proﬁle is given in ﬁgure

3(b) with k

= 0.365, k

= 10, k

= −0.5, k

= 0.5,

β(t

) = 0 rad, β(t

) = π/4 rad, d(t

) = 5 m and

d(t

) = 10 m.

2.4.3 Ω-shape Turn Error

As previously explained, Ω-shape turns are made of

three parts. ϕ

consists in a spiral tracking and thus

requires to ﬁnd a controller vanishing the error (4).

Regarding parts ϕ

and ϕ

, it is mandatory to make

d(t) converge towards d

(β(t)). Thus, the problem of

making the robot reach a spiral turns into the prob-

lem of ﬁnding a controller such that the error (9) con-

verges to zero.

(t) = d(t) − d

(β(t)) (9)

U-shape and Ω-shape turns can thus be performed

by following a spiral and moving from one spiral to

the other. At this stage, as the different turns have

been expressed by introducing a set of errors, the next

section will be dedicated to the design of controllers

allowing to make them vanish.

3 CONTROLLERS DESIGN

This section proposes to design an output feedback

control law which makes the errors (4) and (9) con-

verge toward zero asymptotically. As the system

is nonlinear, the main idea is to use an exact input

to state linearization method proposed by (Isidori,

2013). We ﬁrst present the controller allowing the

tracking of a spiral. In a second subsection, an adap-

tation of this controller is presented to move from one

spiral to another one. In this section, it is assumed that

v(t) is a constant input set at v

. The only remaining

input is therefore ω(t).

3.1 Spiral Tracking

The errors (4) dynamics are deﬁned by:



˙e

(t) = v

[cos(α

) − cos(α(t))]

˙e

(t) = −ω(t) +

d(t)

sin(α(t))

(10)

Which can be written as follows:



˙e

(t) = v

[cos(α

) − cos(e

(t) + α

)]

˙e

(t) = −ω(t) +

(t)+d

(t)

sin(e

(t) + α

)

(11)

The main idea of this section is to linearize the error

system. To this end, consider the transformation:

z =







(t)

[cos(α

) − cos(e

(t) + α

)]



= T (e)

(12)

Notice that T(0) = 0 and in the domain D deﬁned by:

D =



) ∈ ℜ

∈ ℜ,e

∈] − α

,−α

+ π[}

(13)

T deﬁnes a diffeomorphism. Applying this transfor-

mation to the error system leads to:



˙z

= z

˙z

= v

sin(e

(t) + α

) ˙e

(t)

(14)

and



˙z

= z

˙z

= v

sin(e

(t) +α

)(−ω(t) +

(t)+d

(t)

sin(e

(t) +α

))

(15)

Taking:

ω(t) =

sin(e

(t) +α

)

(t) +

(t) +d

(t)

sin(e

(t) +d

)

(16)

we obtain:



˙z

= z

˙z

= ω

(t)

(17)

where ω

(t) is a new control law, which has to be

designed. At this stage, as the system (17) is linear, a

classical linear control law:

ICINCO 2019 - 16th International Conference on Informatics in Control, Automation and Robotics

358

(t) = −λ

(t) − λ

(t) (18)

with λ

,λ

> 0, allows to stabilize asymptotically sys-

tem (17).

Therefore, we propose the following theorem:

Theorem 1. Consider two positive scalars, λ

,λ

the error system (10) in closed loop with the control

law (16), (17), (18), where z = T (e) in (12) is locally

asymptotically stable.

Proof 1. It is sufﬁcient to notice that z converge

asymptotically to zero and z = T (e) deﬁnes a local

diffeomorphism with T(0) = 0.

Remark 1. λ

and λ

are two gains used to tune the

speed of convergence of the system in closed loop.

Remark 2. The relative degree of the chosen output

used for the linearization is two while the original

state space dimension is three. Therefore, there exists

a zero dynamic. Deﬁning β

the reference trajectory

for β, the β error dynamics is driven by

˙e

(t) = (−

sinα

d(t)

sinα

Following (Isidori, 2013), it is straightforward to see

that the zero dynamic deﬁned by

˙e

(t) = 0

is stable only. As a consequence, in steady state, we

may expect a static error between β(t) and its ex-

pected value β

(t).

3.2 Distance Proﬁle Tracking

From now on, a path leading to the convergence to-

ward a speciﬁc spiral is described and thus a function

of d

(β(t)) is proposed. Therefore, the error to be

minimized can be rewritten as:

(t) = d(t) − [d(

β(t

)) +

(

β(t))d

gap

] (19)

The controller designed to ensure the convergence

of the error will use the same input to output feed-

back linearization method than the previous one. Let

introduce the following new states







(t)

˙z



and therefore the transformation T





(t)

−v

cos(α) + d

(β(t))

sin(α)

= T

(χ(t))

(20)

where d

(β(t)) stands for the derivative of d

with

respect to β. It can be proved that the transformation

deﬁnes a local diffeomorphism and applying this

transformation leads to:

˙e

(t) = −v

cos(α(t)) − k

β(t)d

gap

(

β(t) + k

)

+ 1

(21)

Calculating the derivative of z

along the trajectories

of the original system gives:

¨e

(t) =v

α(t)sin(α(t)) −

β(t)d

gap

(

β(t) + k

)

+ 1)

gap

(

β(t) + k

)

(

β(t) + k

)

+ 1)

(22)

where, thanks to the Eq.6,

β(t) and

β(t) are computed

as:

β(t) =

sin(α(t))

d(t)β

gap

(23)

β(t) = −

sin(α(t))

d(t)

(t) β

gap

α(t)cos(α(t))

d(t)β

gap

(24)

Hence, the z

system can be rewritten as :



˙z

= z

˙z

= f (α(t),β(t),d(t)) + g(α(t),β(t), d(t))ω

(25)

where the function f is deﬁned accordingly to

(22),(23),(24).

Choosing the control law for ω such that

ω(t) =

g(α,β,d)

( f (α,β,d) + ω

(t)) (26)

where ω

(t) is a new control allowing to obtain the

following system deﬁned by a simple double integra-

tor:



˙z

= z

˙z

= ω

(t)

(27)

We propose therefore the following control law for

(t)

(t) = −λ

˙e

(t) − λ

(t), (28)

where λ

and λ

are two positive scalars which en-

sure the asymptotic stability of the closed loop system

(25) with the control (26), (28).

Therefore we propose the following theorem:

Theorem 2. Consider two positive scalars, λ

,λ

the error system (25) in closed loop with the con-

trol law (26), (28), where z

= T

(χ) in (20) is locally

asymptotically stable.

Proof 2. Omitted.

Nonlinear Output Feedback for Autonomous U-turn Maneuvers of a Robot in Orchard Headlands

359

Remark 3. The controllers (16), (26), shows two sin-

gularities, when α(t) = π and α(t) = 0. These singu-

larities are due to the polar coordinates model and the

construction of the transformation T . Furthermore,

these ones occurs when the robot is facing the last tree

or let it on its back. Dealing with these conﬁgurations

are then easily avoidable.

As previously noted in the last subsection, 3.1, the

system in closed loop exibits also a stable zero dy-

namic.

In this section, we have proposed two output state

feedbacks controlling the convergence of the states to-

ward its references. A ﬁrst one follows a speciﬁc spi-

ral based on d

(t) and α

(t). It is used for the part

of our shapes of turn. The second controller al-

lows performing the convergence toward a spiral with

a given angle β

through a arctan proﬁle. It will be

used during the part ϕ

and ϕ

of our turns.

4 SIMULATIONS

In this section, simulations realized with the

MATLAB



software are presented. First, we fo-

cus on the previously designed controllers, then we

couple them in order to simulate Ω-shape turns. For

all the simulations: the sampling time is setup as

= 0.1 s, the spiral center is deﬁned as [0,0] and

the linear velocity is given as v = 0.15 m.s

−1

. More-

over, the robot is represented by a set of green and red

lines, respectively ~x

and ~y

4.1 Spiral Tracking

The ﬁrst set of simulations shows the performance of

the spiral tracking controller given by equation (16)

setup with λ

= 0.1 and λ

= 0.5. Here, two different

situations are presented. The ﬁrst column (i.e., ﬁgures

4(a), 4(c), 4(e) and 4(g)) presents the behavior of the

robot when the initial robot pose is on the spiral to

be tracked. The robot starts at χ(t) = [7,0, π/2]

and

tracks an inward spiral deﬁned by α

(t) = 15π/32 rad

and d

) = 7 m. The second column (i.e., ﬁgures

4(b), 4(d), 4(f) and 4(h)) presents the behavior of the

robot when its initial pose is not on the spiral. The

robot starts at χ(t) = [6, 0,π/2]

and tracks the same

spiral. In ﬁgure 4(a) it can be seen that the robot tracks

accurately the spiral when its initial position belongs

to it. In ﬁgure 4(b), the robot ﬁrst converges toward

the spiral and then follows it. However, the tracking

is not fully accurate and there is an error between the

current position and the desired one. As mentioned in

section 3.1, it can be explained by a static error on β(t)

in steady state. Indeed, as it can be seen in ﬁgures 4(d)

(a) Dotted: Reference spiral

- Solid: Robot trajectory

(b) Dotted: Reference spiral

- Solid: Robot trajectory

β(t) robot

(d) solid: β(t) spiral - Solid:

β(t) robot

(e) Dotted: errors - Solid:

(t) - Dashed: e

(t)

(f) Dotted: errors - Solid:

(t) - Dashed: e

(t)

(g) ω(t) (h) ω(t)

Figure 4: Simulation of spiral tracking.

and 4(c) displaying the evolution of β(t) and β

(t),

a static error exists on steady state when the initial

robot state does not belong to the spiral. However,

the static error on β does not modify the controller

performances regarding e

(t) and e

(t) which both

converge towards zero (see ﬁgures 4(f), 4(e)). Finally,

ﬁgures 4(h) and 4(g) show the evolution of ω(t) used

to track the spiral.

4.2 Distance Proﬁle Tracking

The second set of simulations presents the perfor-

mances of controller given by equation (26), allow-

ing to move toward a speciﬁc spiral. The controller is

set up with λ

= 0.05, λ

= 0.1 and d

(β(t)) is de-

ﬁned with k

= 0.365, k

= 10, k

= −0.5, k

= 0.5,

β(t

) = 0 and β(t

) = 3π/8. The robot starts at

χ(t) = [5,0,π/2] and has to reach a spiral deﬁned with

ICINCO 2019 - 16th International Conference on Informatics in Control, Automation and Robotics

360

) = 10 m and α

= π/2 rad. As it can be seen

in ﬁgure 5(c), the robot reaches the desired spiral at

the given angle β(t

). Moreover, the distance proﬁle

is accurately tracked (5(b)) with a smooth command

ω(t) (see ﬁgure 5(a)).

(a) ω(t) (b) e

(t)

Dotted: desired trajectory - Solid:

Robot trajectory

Figure 5: Distance proﬁle tracking.

4.3 Ω-shape Turn

The third set of simulations presents an example of

Ω-turn in the context of a headland navigation. First,

a row following task (see (Durand-Petiteville et al.,

2017) for more details) is performed, followed by

an Ω-turn driving the robot to the next row. Fi-

nally, a new row following task is started. For this

simulation, the turn is performed with ω

MIN−MAX

±0.08 rad/s

−1

and it requires both controllers given

by equations (16), (18), (26), and (28). The parame-

ters λ

= 0.01, λ

= 0.05 and λ

= 0.01, λ

= 0.05

are used for their respective controller. A proﬁle

is created for ϕ

and ϕ

with k

= 0.365, k

= 10,

= −0.5 and k

= 0.5. The robot starts at χ(t) =

[3,0,π/2] and needs to reach a spiral deﬁned at d

8m and α

= π/2 rad with a respective angle of refer-

ence of β(ϕ

) = 3π/8 rad. Finally, d

= 2.5m with

an angle of reference of β(ϕ

) = 3π/8 rad. In addi-

tion, to assess the sensitivity of the controllers to er-

rors, a centered Gaussian noise has been added to d(t)

and α(t), with a respective amplitude of 0.05m and

◦

. As shown in 6(c), the robot successfully performs

a Ω-shape turn by using alternatively both controllers

presented above. Indeed the designed distance pro-

ﬁles, as well as the spiral, are accurately tracked by

the robot (see ﬁgure 6(b)). Moreover, it should be no-

ticed that the static error on steady state during the

spiral tracking does not affect the turn. Indeed, the

distance proﬁle for ϕ

is designed to allow to start the

spiral tracking when the robot state belongs to the spi-

ral. Thus, despite this drawback, the controller given

by (16) provides suitable performances for our spe-

ciﬁc navigation problem. It can also be noticed that

the noise added to the measures does not disturb the

behavior of the robot. Finally, an appropriate tuning

of parameters deﬁning the distance proﬁles and spiral

allows to perform an Ω-shape turn while dealing with

a maximal robot turning radius.

(a) Dashed: ω

MIN−MAX

Solid: ω(t)

(b) Dashed: d

(t) -

Dashed: d

(β(t)) - Solid:

d(β(t))

Robot trajectory - Pink crosses: controller

switches - Black crosses: trees

Figure 6: Ω-shape turn.

5 CONCLUSION

In this work, the problem of driving in headlands dur-

ing an orchard navigation has been addressed. First,

U-shape and Ω-shape turns have been modeled using

polar coordinates in order to generate the paths to fol-

low when driving in the headland. Then, two nonlin-

ear output state feedbacks controllers have been de-

signed to track the designed paths. Their design was

based on input to state linearization techniques.

Simulations have highlighted the suitable perfor-

mances of the controllers and their usefulness to efﬁ-

ciently drive the robot in orchards headlands. The ob-

tained results are encouraging and must be integrated

on our testbed. To do so, it is planned to couple the

controllers with a perception system allowing to de-

Nonlinear Output Feedback for Autonomous U-turn Maneuvers of a Robot in Orchard Headlands

361

tect the trees in the surroundings of the robot. Thus, it

will be possible to evaluate the efﬁciency and robust-

ness of the proposed approach.

Moreover, a couple of challenges have to be ad-

dressed to obtain a fully autonomous system. The

ﬁrst one consists in automatically generating the spi-

ral and distance proﬁle parameters based on the a pri-

ori known and on-line acquired data related to the or-

chard structure. Moreover, the controller sensitivity

to robot state has to be investigated. Thus, it might

be required to design a recursive estimation process

based on the acquired data to improve the accuracy

of the state knowledge. Finally, it seems relevant to

guarantee the continuity of the control law when the

robot switches from one controller to the other one.

REFERENCES

Andersen, J. C., Ravn, O., and Andersen, N. A. (2010).

Autonomous rule-based robot navigation in orchards.

IFAC Proceedings Volumes, 43(16):43–48.

Asif, M., Memon, A. Y., and Junaid Khan, M. (2016). Out-

put feedback control for trajectory tracking of wheeled

mobile robot. Intelligent Automation & Soft Comput-

ing, 22(1):75–87.

Bayar, G., Bergerman, M., Koku, A. B., and ilhan Konuk-

seven, E. (2015). Localization and control of an au-

tonomous orchard vehicle. Computers and Electron-

ics in Agriculture, 115:118–128.

Bochtis, D. and Vougioukas, S. (2008). Minimising the

non-working distance travelled by machines operating

in a headland ﬁeld pattern. Biosystems engineering,

101(1):1–12.

Boyadzhiev, K. N. (1999). Spirals and conchospirals in the

ﬂight of insects. The college mathematics Journal,

30(1):23.

d’Andrea Novel, B., Bastin, G., and Campion, G. (1992).

Dynamic feedback linearization of nonholonomic

wheeled mobile robots. In Robotics and automation,

1992., pages 2527–2532. IEEE.

d’Andr

ea Novel, B., Campion, G., and Bastin, G. (1995).

Control of nonholonomic wheeled mobile robots by

state feedback linearization. The International journal

of robotics research, 14(6):543–559.

Durand-Petiteville, A., Le Flecher, E., Cadenat, V., Sen-

tenac, T., and Vougioukas, S. (2017). Design of a

sensor-based controller performing u-turn to navigate

in orchards. International Conference on Informatics

in Control, Automation and Robotics, 2:172–181.

Foley, J. A., Ramankutty, N., Brauman, K. A., Cas-

sidy, E. S., Gerber, J. S., Johnston, M., Mueller,

N. D., O’Connell, C., Ray, D. K., West, P. C., et al.

(2011). Solutions for a cultivated planet. Nature,

478(7369):337.

Isidori, A. (2013). Nonlinear control systems. Springer

Science & Business Media.

Li, M., Imou, K., Wakabayashi, K., and Yokoyama, S.

(2009). Review of research on agricultural vehicle au-

tonomous guidance. International Journal of Agricul-

tural and Biological Engineering, 2(3):1–16.

Reid, J. F. (2011). The impact of mechanization on agricul-

ture. Bridge, 41(3):22–29.

Shariﬁ, M. and Chen, X. (2015). A novel vision based row

guidance approach for navigation of agricultural mo-

bile robots in orchards. In Automation, Robotics and

Applications (ICARA), 2015 6th International Confer-

ence on, pages 251–255. IEEE.

Shi, S., Yu, X., and Khoo, S. (2016). Robust ﬁnite-time

tracking control of nonholonomic mobile robots with-

out velocity measurements. International Journal of

Control, 89(2):411–423.

Siegwart, R., Nourbakhsh, I. R., and Scaramuzza, D.

(2011). Introduction to autonomous mobile robots.

MIT press.

Subramanian, V., Burks, T. F., and Arroyo, A. (2006). De-

velopment of machine vision and laser radar based au-

tonomous vehicle guidance systems for citrus grove

navigation. Computers and electronics in agriculture,

53(2):130–143.

Yang, H., Fan, X., Xia, Y., and Hua, C. (2016). Ro-

bust tracking control for wheeled mobile robot based

on extended state observer. Advanced Robotics,

30(1):68–78.

Zhang, J., Maeta, S., Bergerman, M., and Singh, S. (2014).

Mapping orchards for autonomous navigation. In

2014 Montreal, Quebec Canada July 13–July 16,

2014, page 1. American Society of Agricultural and

Biological Engineers.

ICINCO 2019 - 16th International Conference on Informatics in Control, Automation and Robotics

362