2021

These authors contributed equally to this work.

These authors contributed equally to this work.

These authors contributed equally to this work.

These authors contributed equally to this work.

These authors contributed equally to this work.

These authors contributed equally to this work.

These authors contributed equally to this work.

These authors contributed equally to this work.

1]Department of Mechanical Engineering, IIT Kharagpur, WB, India

2]Department of Industrial & Systems Engineering, IIT Kharagpur, WB, India

3]Department of Aerospace Engineering, IIT Kharagpur, WB, India

# Design of an All-Purpose Terrace Farming Robot

Vibhakar Mohta  Adarsh Patnaik  Shivam Kumar Panda  Siva Vignesh Krishnan
Abhinav Gupta  Abhay Shukla  Gauri Wadhwa  Shrey Verma  Aditya Bandopadhyay  [
[  [

###### Abstract

Automation in farming processes is a growing field of research in both
academia and industries. A considerable amount of work has been put into this
field to develop systems robust enough for farming. Terrace farming, in
particular, provides a varying set of challenges, including robust stair
climbing methods and stable navigation in unstructured terrains. We propose
the design of a novel autonomous terrace farming robot, ‘Aarohi’, that can
effectively climb steep terraces of considerable heights and execute several
farming operations. The design optimisation strategy for the overall
mechanical structure is elucidated. Further, the embedded and software
architecture along with fail-safe strategies are presented for a working
prototype. Algorithms for autonomous traversal over the terrace steps using
the scissor lift mechanism and performing various farming operations have also
been discussed. The adaptability of the design to specific operational
requirements and modular farm tools allow Aarohi to be customised for a wide
variety of use cases.

###### keywords:

Robotics, Terrace Farming, Stair Climbing, Agriculture Automation

##  1 Introduction

According to the economic survey 2020-21 GDP , the share of agriculture in the
gross domestic product (GDP) is only about 20.19% even though agriculture is
the primary source of livelihood for 58% of India’s population. This can
highly be regarded to the labour-intensive practices being followed in Indian
agriculture due to the lack of resources. On steep slopes, terrace farming is
one of the most efficient ways of conserving soil and water engdawork2014long
. It helps in reducing runoff by over 41.9% and sediment loss by over 52%
deng2021advantages . By 2050, it is projected that the food demand of India
will almost double hamshere2014india . To meet such demands, the need for
revolutionising the field of agriculture arises, which can be achieved by
automating certain agricultural practices. The agriculture sector has a 57%
potential for automation and was ranked 4th in a McKinsey report that assessed
the automation potential of major global sectors mckinsey . Thereby this
sector presents a great opportunity for reducing the cost of production and
increasing productivity. However, there are multiple challenges in automating
the process of terrace farming. The use of petrol or diesel tractors is non-
viable due to the narrow size of steps spugnoli2013environmental . As a
result, most of the farmers rely on the use of traditional hand tools by
abundant household labortiwari2004implications  varisco1991future . Moreover,
socio-economic challenges, such as poverty and illiteracy, further exacerbate
the problem of automating terrace farming chapagain2017agronomic .

Over the last few decades, significant advances in sensor technology and
computer capacity have paved the path for autonomous systems to be deployed in
a variety of sectors 10.1145/3387304.3387321 ; 9265493 ; 9476722 ;
singhal2019real . Likewise, the challenges faced in the agricultural sphere
have been addressed in the literature using drones, robotic arms, and
autonomous vehicles. Ladybird Ladybird , an autonomous field robot was
designed at the Australian Centre for Field Robotics to collect data in an
agricultural field. An autonomous mobile plant irrigation robot, using
wireless communication to communicate with a moisture sensor, was developed
and tested xiong2020autonomous . Conventional methods of seed sowing, using
animals or tractors, are time and energy-consuming. To tackle this problem at
a low cost, a swarm of robots with minimal individual intelligence was
developed for sowing tasks blender2016managing . A centralised coordinator,
OptiVisor is used here for path planning and optimisation. Another autonomous
robot prototype developed particularly for the task of seed sowing was
developed in naik2016precision . Autonomous robots for sweet pepper harvesting
in greenhouses arad2020development , apple harvesterssilwal2017design  have
been developed using fruit detection algorithms, path planning, localisation
and motion control. Conventionally, pesticide spraying is done by the means of
backpack sprays or tractor sprayers. A human-robot collaborative method of
spraying specific targets berenstein2017human  was developed using different
image based targeting algorithms. These studies have been done to automate a
particular process in the process of agricultural production, but there is a
need to integrate such automated processes into one. Numerous studies have
developed multipurpose agricultural robots sowjanya2017multipurpose ;
gollakota2011agribot  which can perform all agricultural tasks but are capable
of traversing plain fields only. To overcome the challenge of uneven terrains
and steps, drones have been used in terraces for pesticide spraying and crop
health monitoring kong2018autonomous . Studies to understand the application
of machine learning, image processing and pattern recognition in drones for
agricultural purposes have also been conductedkulbacki2018survey . However,
there are multiple challenges posed by the usage of drones in agriculture such
as high cost, and the inability to perform tasks like ploughing and
harvesting.

Full-scale development and commercialisation of a terrace farming robot is
inherently a challenging task. Firstly, it requires the robot to be able to
traverse long and curved individual terraces on a hilly slope safely and
efficiently. The terraces in India are typically 60-80m long with varying
widths along the length. This necessitates the farm robot to be as compact as
possible to safely pass through these converging terraces. Active mapping of
the environment is a primary requirement for all autonomous field robots, more
so in the case of a terrace farming robot where a minor misjudgement in step
dimensions can cause the robot to fall from huge heights. Secondly, for the
commercial success of the robot, both the initial cost and the running cost of
the system should be kept as low as possible keeping in mind the low income of
farming households farmer_income . The third and the most important
consideration while designing such a terrace farming robot is to develop an
effective mechanism to safely climb up the terraces whose step heights can
easily reach up to 40 cm.

To this end, we reviewed several stair-climbing mechanisms proposed by
researchers and engineers to date. Broadly, various stair-climbing robots can
be put into either of these categories: (a) Wheeled robots, (b) Tracked or
rail-based robots and (c) Legged robots.

Wheeled stair-climbing robots typically use multiple wheels in combination to
roll, crawl and finally climb the stairs Siegwart1998DesignAI . MSRox proposed
by Mohsen et al. is one such wheeled robot Dalvand2006StairCS . The authors
introduced a unique star-wheel design which consisted of an assembly of three
wheels powered by electric motors on a freely rotating base. This
configuration with a star-wheel showed good adaptability on rugged terrain and
stairs, however, it suffers from a major flaw. The radius of the star wheel
should be significantly greater than the step height for the robot to climb
the stairs. This constraint makes the entire system bulky and complex. While
it may work well for small stair heights (10-17 cm), it can not be used on
terrace farms that have small terrace widths and steep heights. Moreover, the
chassis of the robot is also very unstable for supporting huge weights
required for terrace farming.

Tracked robots are also a popular choice for stair-climbing as they have been
found to work well for rough terrains Lee2009ASR ; Michaud2003AZIMUTAL ;
Kim2010WheelH . Suyang et al. designed a wheelchair robot with variable
geometry tracked mechanism for assistance in stair climbing
Yu2010ConfigurationAT . The robot could adapt to convex terrain by active
control of tension in the track. Yoon-Gu et al. also developed a terrain
adaptive wheel track hybrid robot that used both wheel and track-based systems
Kim2012AutonomousTA . However, similar to wheeled robots, both of them suffer
from large size and complexity which makes them impractical to use in narrow,
steep terraces.

Humanoids, such as HONDA ASIMO 1389593 , NAO 6094533 , and HRP-2 4399104 ,
have demonstrated capability for stair climbing but they are extremely
expensive and difficult to put into farming use. On the other hand, Legged
robots such as RHex Saranl2001RHexAS  could be augmented with farming tools
easily but the unstable chassis motion discourages their use for farming
applications that require high precision and accuracy.

Figure 1: Aarohi’s Robot Model

(a) Prototype with extended scissors

(b) Prototype with closed scissors

Figure 2: Images of the robot prototype in a test arena

Apart from the aforementioned categories, there are other robots proposed by
researchers which can be used for stair climbing. Zhang et al. proposed the
design for a stair-cleaning robot that combined lift or drop motion with the
telescopic linear motion to achieve stair climbing Zhang2016ANC . However, the
long telescopic mechanism accounts for large weight and size which not only
restricted its quick turning capability but also imbalanced the entire robot
in one of the stair descent configurations. Later, the design was modified to
a small leg-wheel hybrid robot which showed promising results for cleaning
applications on small stairs Zhang2016ANC . Unfortunately, the entire system’s
load is supported by a single actuated joint in one of the configurations,
which would make the robot highly unstable if this mechanism were replicated
for terrace farming with heavy payloads.

From our discussion so far, it can be ascertained that there is a lack of
agriculture robots that can provide complete automation of common farming
processes such as ploughing, sowing, watering, etc. Moreover, existing
mechanical designs of stair climbing robots can not be directly put to use for
building a terrace-climbing farming robot. Therefore, to overcome this issue,
we propose the novel design of ‘Aarohi’, an autonomous terrace farming robot.
The key innovations associated with this work are summarised as follows:

  1. 1.

Novel scissor lift based mechanism for stair climbing to handle steep terrace
steps relevant to terrace farming scenarios

  2. 2.

Re-configurable design for attaching and detaching different modules for
ploughing, sowing, pesticide spraying, watering etc.

  3. 3.

Autonomous solution to traverse the terrace and perform a multitude of
agricultural activities.

The following aspects are needed to be emphasised before proceeding further:

  1. 1.

Only the generalised design, design process, control and software architecture
has been proposed in this work. The specifications of hardware mentioned in
this paper are for the prototype developed by us and are in no way exclusive
to the design.

  2. 2.

Steps and stairs mentioned are the steps in the terrace farming region and are
interchangeably used throughout the paper. It is also assumed that the steps
are flat and not ramped.

  3. 3.

It is assumed that the robot is not operated in a heavy windy environment.

  4. 4.

The number of pairs of powered wheels and the number of scissor levels used
for climbing is prototype specific. It could be extended to $n$ number of
wheels and $m$ scissor levels depending on the size of the robot, where $n\geq
2$ and $m\geq 2$. Fig. 3(a) and fig. 3(b) shows the design of the chassis and
stair climbing module with two and three pairs of wheels, respectively.

(a) Design with two pairs of scissors

(b) Design with three pairs of scissors

Figure 3: Design with varying wheel pairs

The rest of the paper is organised as follows. Section 2 introduces the
overall design and discusses the mechanical structure of the entire assembly
and the design optimisation process in detail. Finally, section 3 describes
the software architecture of the robot in-depth, including navigation
strategies, stair climbing strategy and control of various modules. All the
results are finally concluded in section 4.

##  2 Mechanical Design

The novel modular design enables the robot to traverse the terrain of terrace
farming steps, climb up and down the terrace steps, and perform various
farming operations such as ploughing, sowing, watering, pesticide spraying,
etc. The scissor lift mechanism allows the robot to climb up and down the
steps. Independent detachable modules have been included in the robot to
perform basic farming activities. The advantage of having separate modules is
that these can be attached or detached to the robot depending upon the work at
hand. Also, since each module works independently of the others, it makes it
easier to repair or replace any module in case of damage.

During prototyping, V-slotted aluminium extrusions were used to build the
chassis of the robot. The locomotion is carried out by a four-wheel
differential drive. This drive mechanism allows the robot to turn about its
position, change its orientation during a task and proceed in the desired
direction. Pneumatic wheels are employed for the drive as they provide higher
load capacity and better traction while traversing on different kinds of soil.
To prevent any damage to the mechanisms inside the prototype due to
vibrations, an independent 3-revolute-1-prismatic (3R1P) suspension mechanism
with a coaxial spring cum damper system has been attached to each wheel. Dummy
wheels that are not powered are attached to the chassis in order to help the
robot climb the steps.

Table 1: Parameters in scissor-lift mechanism design Parameter |  Description
---|---
$n$ |  Number of scissor levels
$\theta$ |  Angle made by a positive sloping link with x-axis
$D$ |  Length of the scissor-link
$h$ |  Height of the scissor lift mechanism in its current configuration
$H$ |  Height of the stairs that the robot needs to climb
$l$ |  Length of the actuator AB
$S$ |  Stroke length of the linear actuator
$i$ |  Number of scissors below the point B
$a$ |  Fractional number between $(0,0.5]$. $aD$ denotes the length PA
$b$ |  Fractional number between $(0,0.5]$. $bD$ denotes the length OB
$F$ |  Linear actuator force required to actuate the scissor-lift at current configuration
$L$ |  Load lifted by the scissor-lift, incorporating both robot and scissor-lift weight
$t$ |  Thickness of the scissor-link

###  2.1 Stair Climbing Mechanism

The scissor lift mechanism allows the robot to climb up and down the step. It
also facilitates variable height adjustment during watering, pesticide
spraying, and harvesting. Only two pairs of scissors are used in the prototype
developed. However, multiple pairs of scissors could be used for climbing
depending upon the dimensions and weight of the robot. For example, Fig. 3(b)
shows a modified design with three pairs of scissors.

The pair of front wheels and rear wheels are attached to one scissor lift
mechanism each. These two scissor lift mechanisms are differentially actuated
to climb up and down the stairs. There are four pairs of dummy wheels – one
ahead of the front wheel, two between the front and rear wheels and one behind
the rear wheels – to support the climbing motion. The number of dummy wheels
can be changed according to the dimension of the robot and the number of
scissor lift pairs used in the robot. Scissors connecting to the front and
rear pair of wheels are attached with one linear actuator each to support the
lifting of the robot. Guide rails are attached to the scissors to ensure
proper vertical lift motion. The stair climbing and descending algorithms are
discussed in detail in section 18.

The scissor-lift mechanism, which is considered as one of the most stable
mechanisms for lifting actions in industries, has been used instead of fork-
lift or lead-screw for lifting the robot because of the relative easiness in
manufacturing and assembling.

####  2.1.1 Design of scissor-lift mechanism

When designing a scissor-lift mechanism, the location of the linear actuator’s
attachment is critical because it directly impacts the potency of the system,
that is, the mechanical advantage and the velocity ratio, and thereby
influencing the choice of the linear actuator to actuate the system, and
hence, the overall cost. In short, the position variables of the linear
actuator attachment affect the force output and the stroke length of the
actuator. There are other design parameters such as the link lengths, link
thickness, number of scissors, etc. that directly influence the force output
and the stroke length of the actuator.

Before picking an ideal set of position variables, it is necessary to
establish equations for force output and the stroke length of a linear
actuator in terms of the scissor-lift position variables. Without loss of
generalisation, we can consider a $n$ stage scissor lift mechanism as shown in
Fig. 4. The linear actuator has one end static and the other translating. The
translating end is attached to a point on a positively sloping link in the
scissors and the static end is attached to a fixed support. When the
translating end of the actuator extends, it will cause the scissor-lift to
extend and vice versa.

As shown in Fig. 4, A is the point where the translating end of the linear
actuator is attached to the scissor and hence, it is the point where force is
applied. B is the point where the static end of the linear actuator is
attached. O is the origin and XY is the reference co-ordinate axes for the
analysis. Various symbols used and their corresponding descriptions are
tabulated in Table 1.

Here, $a$, $b$, $\theta$ and $n$ are the only independent variables. $D$ and
$i$ are assumed to be constant in our analysis. $i$ is set equal to $n-1$. The
expressions $h$, $l$ and $F$, derived in scissor-lift , in terms of the
independent variables, are shown in (1), (2), and (3), respectively. The
stroke length of the linear actuator can be obtained by subtracting the length
of the actuator in the maximum extended state and the maximum compressed state
as expressed in (5). It is also required to find out the maximum force
required to actuate the scissor-lift to choose the linear actuator. The force
required to actuate the scissor-lift will be maximum when the scissors are in
the most compressed state, i.e., when $\theta=\theta_{min}$. The maximum
force, $F_{max}$, can be expressed as in (4). The variation of $F_{max}$ and
$S$ with respect to $a$, $b$, and $n$ is clearly illustrated in Fig. 5(a) and
Fig. 5(b), respectively. With an increase in both $a$ and $b$, $F_{max}$
decreases and $S$ increases. Also, with a increase in $n$, $F_{max}$ increases
and $S$ decreases.

$\displaystyle h=nD\sin\theta$ (1) $\displaystyle
l=D\left[\lambda\cos^{2}{\theta}-2b\overline{a}\cos{\theta}+b^{2}+(i+a)^{2}\right]^{\frac{1}{2}}$
(2) $\displaystyle
F=nL\left[\frac{\lambda\cos^{2}{\theta}-2b\overline{a}\cos{\theta}+b^{2}+(i+a)^{2}}{\left(b\overline{a}\tan\theta-\lambda\sin\theta\right)^{2}}\right]^{\frac{1}{2}}$
(3) $\displaystyle F_{max}=F|_{\theta=\theta_{min}}$ (4) $\displaystyle
S=l|_{\theta=\theta_{max}}-l|_{\theta=\theta_{min}}$ (5)

Figure 4: Representation of an n-level scissor lift mechanism with the
variables used.

(a) Variation of maximum Force output with position variables $a$, $b$ and
number of scissors $n$

(b) Variation of the maximum stroke length with position variables $a$, $b$
and number of scissors $n$

Where $\lambda=\left(\overline{a}^{2}-(i+a)^{2}\right)$ and
$\overline{a}=1-a$. Clearly, $F_{max}$ and $S$ are conflicting functions with
respect to $a$ and $b$. Hence, we formulate this problem as a multi-objective
optimization problem, where the conflicting objectives, $F_{max}$ and $S$, are
to minimized. The problem is mathematically formulated as shown in (6).

Figure 6: Pareto Front showing various values of $F_{max}$ and $S$ for
different values of $n$

$\displaystyle\min_{a,\;b,\;\theta_{min},\;n,\;D,\;i}$ $\displaystyle
f=(F_{max},S)^{T}$ (6) S.T $\displaystyle 0<a,b\leq 0.5$ $\displaystyle
5^{\circ}\leq\theta_{min}\leq 10^{\circ}$ $\displaystyle D_{min}\leq D\leq H$
$\displaystyle i=n-1$

Here, $D_{min}$ is the minimum link length required to climb the step of
height, $H$, calculated as shown in (7).

$D_{min}=\frac{H}{n(1-\sin(min(\theta_{min})))}$ (7)

Since a single solution cannot optimise all the conflicting objectives
simultaneously, we seek to optimise the objective functions and present the
Pareto Front. The Decision Maker, then, can choose a linear actuator from the
Pareto Front solution set based on the use case of the robot.

Multi-objective evolutionary algorithms (MOEAs) have grown in popularity in
recent years as a result of their ability to find many Pareto-optimal
solutions in a single simulation run. Algorithms such as Non-Dominated Sorting
Genetic Algorithms (NSGA-II) 996017 , Multi-Objective Grey Wolf Optimizer
(MOGWO) mirjalili2016multi , Multi-Objective Particle Swarm Optimizer (MOPSO)
reyes2006multi , Multi-Objective Bonobo Optimizer (MOBO) das2020multi  are
widely used to obtain Pareto-front in real-world multi-objective problems.

Table 2: Linear actuator specifications and position variables for the developed prototype. Parameter | Datasheet Values | Final values used
---|---|---
$F_{max}$ | 576.840 N | 1000 N
S | 0.242 m | 0.25 m
a | 0.499 m | 0.5 m
b | 0.158 m | 0.16 m
$\theta_{min}$ | $10^{\circ}$ | $10^{\circ}$
D | 0.409 m | 0.4 m
n | 2 | 2
t | N/A | 3 mm Steel

Here, we are employing gamultiobj, a MATLAB package that uses a controlled,
elitist genetic algorithm, to solve the constrained multi-objective
optimisation problem (6). We set the population size to be $200$ and the
Pareto fraction as $0.7$. We designed a robot for climbing stairs of height up
to $40$ cm. In our case, choosing $n>4$ seemed sub-optimal and hence, we added
an additional constraint on the number of scissors, $n=\\{2,3,4\\}$. The load
$L$ moved by the scissor-lift and the height $H$ of the stair is fixed as 250
N and 40 cm, respectively. Solving the optimisation problem separately for
each value of $n$, three different Pareto optimal fronts were obtained as
shown in Fig. 6.

Figure 7: Simplified Model of the scissor-lift mechanism both in the
undeformed and deformed position. Forces and torque are applied at ends marked
as 1, 2, 5, and 6. Joints marked as 3 and 4 are fixed during the simulation.
Figure 8: (a) Plot showing the variation of the factor of safety against
various thicknesses considered for steel and aluminium. It can be observed
that the values are similar for both the metals at a particular thickness; (b)
Plot showing the variation of maximum deformation ($\delta_{max}$) and its
components along X- ($\delta_{max}|_{X}$) and Y- ($\delta_{max}|_{Y}$)
direction against various thickness considered for steel and aluminium.

We have added additional constraints on the maximum force ($F_{max}\leq 1000$
N) and stroke length ($S\leq H$) to filter out the non-feasible values. The
filtered values of $F_{max}$ and $S$ along with the corresponding independent
variables have been tabulated: (LINK). The Decision Maker can use the table as
a datasheet to compare them to other linear actuators on the market and select
one based on price and availability. In our case, we chose a linear actuator
with $25$ cm stroke length and $1000$ N maximum load. This choice of linear
actuator introduced a factor of safety of approximately $1.74$ in the design.
The final scissor-lift design parameters along with the linear actuator
specifications chosen for the robot prototype are tabulated in Table 2.

Further, the thickness of the scissor links needs to be optimised between
maximum stress, deformation due to bending and buckling. Also, while
optimising the thickness, the space for the placement of the linear actuators
between the scissors has to be maximised. Given the problem, a simplified
model of the scissor lift mechanism with two crosses used in the prototype was
imported to ANSYS 17.1 for structural simulation analysis as shown in fig. 7.
This model refers to the maximum extended position of the scissor lift
mechanism. In the simulation, considering 40 kg as the weight of the robot,
respective forces and torques were applied to the ANSYS model at the ends
marked as 1,2,5 and 6 in fig. 7. The joints indicated as 3 and 4 in fig. 7 are
fixed during the simulation. Two commonly used materials – Steel and
Aluminium, each with four commercially available and relevant thicknesses – 2
mm, 3 mm, 5 mm and 7 mm – were selected for the simulation study.

The factor of safety due to the maximum stress ($N_{S}$), and the absolute
maximum total deformation ($\delta_{max}$) and its components in X -
($\delta_{max}|_{X}$) and Y-direction ($\delta_{max}|_{Y}$) were analysed
during the ANSYS simulation study. Figure 8 shows the values of parameters
analysed during the study. It is to be noted that $\delta_{max}|_{X}$
physically signifies the deformation due to bending and buckling of the
scissor links. It was found that the safety factors ($N_{S}$) for all the four
thicknesses considered were similar for both metals. However, aluminium showed
a higher maximum deformation than steel with the same thickness both in X- and
Y-direction. Selecting a thicker aluminium link would lead to reduced space
for the linear actuator. Even so, choosing steel would mean a heavier
mechanism. Hence, optimising all the requirements for the prototype developed
by us, steel links of 3 mm thickness were used to provide sufficient strength,
lower maximum deformation and more space for accommodating the linear
actuator.

The above-mentioned procedure shall be followed to choose various design
parameters of the scissor lift mechanism optimally, depending on the size,
weight and use case of the robot.

###  2.2 Watering and Pesticide Spraying

Figure 9: CAD model of Watering and Pesticide Spraying Mechanism Figure 10:
CAD Models of isometric and side views of the ploughing sowing rolling (PSR)
mechanism

Irrigation and pest control are the two most important farming operations as
they directly affect the quality of the crops and hence, the yield. The
prevalent methods for irrigation and pesticide spraying have a lot of
drawbacks. Installing systems for drip irrigation, sprinkler irrigation, etc.
consumes a significant amount of space and is time-consuming, costly and
labour-intensive. There have been multiple attempts to robotise the irrigation
process through mobile robots in order to address the drawbacks posed by the
traditional irrigation systems in indoor and outdoor environments
nagaraja2012plant ; chen2020design ; bodunde2019architectural ;
adeodu2019development ; rafi2016design . Traditional manual pesticide spraying
is highly ineffective and sometimes dangerous. Improper use of pesticides can
contaminate groundwater and affect soil fertility yallappa2017development . In
order to address these issues, several Unmanned Aerial Vehicles (UAVs) have
been developed to automate the pesticide spraying process xue2016develop ;
huang2015development . Although it quickens the pesticide spraying, it does
not optimise pesticide usage. To solve this problem, mobile robots were then
employed to minimise the use of pesticides through controlled spraying. The
robot was developed by Geng et al. geng2012assessment  minimized the use of
pesticides up to 60% when compared to the traditional spraying methods.
Similar works related to pesticide spraying were proposed in
sammons2005autonomous ; oberti2016selective ; bechar2016agricultural .

All the above-mentioned works provide separate solutions for irrigation and
pesticide spraying. We propose a design that provides a combined solution for
irrigation and pesticide spraying. The design has been inspired
bhattacharyya2020design  for accommodating more than one type of liquid. When
moving through the field, this module allows the robot to spray water or
pesticide.

The subsystem consists of storage tanks, pipes, proportional solenoid valves,
nozzles, and DC motor-driven pump as shown in Fig. 9. There are two storage
tanks - one for water and one for pesticide - fixed inside the chassis of the
prototype robot. Multiple tanks can be attached to store different chemicals
such as insecticides, herbicides, fungicides, etc. depending on the size of
the final robot. The solenoid valves are appended below each of the tanks to
ensure flow from only one tank at a time. The valves from all tanks are
normally kept closed in order to allow flow only when they are actuated. Pipes
from these valves extend to the DC motor-driven pump. The pump pressurises the
liquid that needs to be sprayed, and also controls the mass flow rate from the
nozzles. We use an array of five nozzles, extending from the pump, and placed
below the chassis to spray the liquid.

###  2.3 Ploughing, Sowing and Rolling (PSR) Mechanism

A novel mechanism to perform farming operations such as ploughing, sowing, and
rolling in series has been developed. The mechanism consists of sub-modules,
namely disc plough, a primary roller, furrower, seed metering module and a
secondary roller. They are concatenated in series and placed vertically at
different heights based upon their respective functions as shown in Fig. 10.
All these tools are mounted on the base plate which is connected to the
chassis through a linear actuator Fig. 10. Since this mechanism performs the
farming operations sequentially, it reduces the overall work time, total power
consumed and cost of operation when performed separately. Moreover, each sub-
module is detachable enabling the robot to be used for different types of
crops.

####  2.3.1 Disc Plough

(a) CAD model of the Disc Plough

(b) Disc Angle

(c) Tilt Angle

Figure 11: Illustration of the CAD model and disc and tilt angles of the disk
plough

The disc plough attached to the robot performs secondary tillage that helps in
seedbed preparations and intercultural operations. It creates refined soil
conditions following primary tillage and provides breaking of clods and mixing
of crop residues smith1976farm . Disc plough is chosen because it is long-
lasting, has more mixing action than inversion, and works well in sticky soil
conditions CHANDRAMOULI201823702 . Smooth edged discs have been used as they
have low maintenance costs and work well even if substantial parts of the disc
are worn out in abrasive soil.

The disc angle is defined as the angle at which the plane of cutting edge of
the disc is inclined to the direction of travel, as shown in Fig. 11(b). The
tilt angle is defined as the angle at which the plane of cutting edge of the
disc is inclined to vertical, as illustrated in Fig. 11(c). In
kepner1990principle , it was found out that these angles influence the depth
and width of cut of the soil. It was concluded in osman2011effects  and
bukhari1992effect  that, with an increase in disc and tilt angles, the field
capacity and the power required to plough the field increased. Hence, it
becomes important to choose suitable values for these angles while designing
the mechanism. From article_el_naim , the optimum values of disc and tilt
angles are found to be between $42^{\circ}$ to $45^{\circ}$ and $15^{\circ}$
to $20^{\circ}$.

In Aarohi, six circular smooth edge discs attached to a shaft have been used.
They are oriented with disc angle and tilt angle as $45^{\circ}$ and
$15^{\circ}$, respectively, in order to reduce the radial force. Alternative
discs are oriented in opposite directions to eliminate side thrust force on
the robot. This novelty in design has allowed removing the additional rear
furrow wheel which is used in the reduction of side thrust in conventional
mechanisms. The shaft that connects the discs is powered by a motor to control
the rotational speed of the discs. The shaft is rotated such that the friction
acts on the discs in the direction that supports the motion of the robot
reducing the load on the motors that power the wheels.

####  2.3.2 Primary Roller

It is placed immediately after the disc plough with sufficient clearance. The
primary roller levels the ploughed soil, thereby flattening the soil.
Torsional springs are mounted at the ends of the roller to give compliance and
reduce additional loads on the wheels of the robot and ensure proper levelling
of the soil.

####  2.3.3 Furrower

V-shaped furrowers are used to make grooves of appropriate depth on the
flattened land. In the robot, four V-shaped metal plates are connected to the
PSR mechanism’s base plate through a bracket. Although the V-shaped metal
plates are not the most optimal design for furrowers smith1976farm , they have
been considered due to their relative simplicity in manufacturing and
assembly. The linear actuator connected to the base plate provides the
downward force required for furrowing.

####  2.3.4 Seed Metering Module

Figure 12: CAD Models of the seeder wheel and the seeder box of the seed
metering module Figure 13: Sensing and Interfacing

Sowing seeds precisely is the next and a very crucial step in precision
agriculture. Recently, a lot of researchers have been focussing on precision
seeding techniques as a driving direction for modern agriculture technology
singh2005optimisation ; yasir2012design . It can save a lot of seeds and
effectively control the inter-seed distance as well. Broadly, two kinds of
seed metering devices for grains have been proposed in literature based on the
mode of operation: (a) Mechanical seeder, and (b) Pneumatic seeder. While
pneumatic seed metering devices have proven to have better performance in
terms of a more uniform distribution and quality of feed index, several
researchers have advocated the use of mechanical seeding for its low cost and
reliability especially for the sowing of crops like wheat minfeng2018optimal ;
yu2021design  . In conventional mechanical feeders, a ground wheel is
typically used to drive the seed metering device article_ground_wheel .
However, occasional wheel slips and resistance in an uneven terrain results in
poor performance of this system in controlling the seed rate. Zhai et al.
jianbo2014design  improved upon this system by designing a control system that
could keep the rotational speed of the seed-metering device consistent with
the seeder’s working speed and achieved good performance. Jin et al.
minfeng2018optimal  also solved the problem of poor consistency management by
adjusting the opening of fluted rollers by a novel mechanism. Our seeder
system is a low-cost yet efficient solution to the automated seeding process
and drives inspiration from these designs.

As shown in Fig. 12, the seed metering module in Aarohi consists of four boxes
from where a seed feeder wheel puts the seeds in the funnel, which is then
connected to a gravity drop seed tube. The other end of the tube is placed
just behind the V-shaped metal plates of the liner, to ensure the falling of
seeds inside the furrow. The seeder boxes are slotted and attached to the
chassis. The seed feeder wheels are connected to a shaft which is in turn
connected to a stepper motor. This stepper motor is attached to the chassis
through L-brackets and it is responsible for the prevention of interlocking
among the seed granules and also controlling the longitudinal inter-seed
distance during sowing operation.

####  2.3.5 Secondary Roller

A secondary roller is used to fill the furrows and level the ground. This
rolling operation creates a smooth and firm seedbed, presses the seeds into
the soil thereby eliminating air pockets, and helps in retaining the soil
moisture for a longer period, thereby allowing for faster germination
smith1976farm .

Table 3: Description of the count and locations of 1D LiDAR sensors. Here $N$ is the number of wheel pairs in the robot, and since the current prototype has two-wheel pairs $N=2$ Location | Count | Symbols
---|---|---
At far ends of all sides | 2 per side x 4 sides = 8 | $(L_{fl},L_{fr}),(L_{bl},L_{br}),(L_{lf},L_{lb}),(L_{rf},L_{rb})$
Downward Facing |  $N$($=2$ scissors) + 2(chassis) = 4 |  $(L_{d1},\ldots,L_{dN})$, $(L_{df},L_{db})$
Width Estimation (at $45^{\circ}$) | 2 (one for each side) | $(W_{l},W_{r})$
Failsafe (at $25^{\circ}$) | 1 per side x 4 sides = 4 | $(F_{f},F_{b},F_{l},F_{r})$
Figure 14: Software Architecture Overview

##  3 Software Architecture

A modular Robot Operating System (ROS) based software stack is developed to
achieve fully autonomous terrace navigation and farming, as shown in Fig. 14.
We first discuss the localisation module, which uses a combination of 1D
lidars, RGB camera, GPS, IMU and wheel encoders (as shown in Fig. 13) to
localize the robot within its environment and estimate terrace edge widths.
Next, we discuss the planning module that outputs the desired high-level plan
to traverse the terrain and perform the given farming operations. Finally, we
discuss the controls module that calculates actuator inputs based on the
generated high-level plan, while accounting for any external disturbances and
uncertainties.

###  3.1 Perception and Localisation

Figure 15: Illustration of various robot lengths and respective notation
Figure 16: Illustration for width estimation

####  3.1.1 Localisation

The estimation of the pose and orientation of the robot is done using ground
odometry data from wheel encoders and inertial measurements from IMU. As
ground odometry tends to be inaccurate in the presence of wheel slipping, it
is often fused with other forms of inputs like lidars and GPS for accurate
localisation. We use an Extended Kalman Filter (EKF) method to fuse different
localisation estimates. While GPS sensors provide global position data, they
were not found to be of much use in the robot due to large errors (1̃m). The
EKF method effectively weights the different estimates with the uncertainty
associated with them and outputs the robot position. Based on these low-level
inputs, the overall localisation module as shown in Fig. 14, outputs high-
level location information that includes the step number (from the top), crop
row number and distance from sides of the terrace.

We use the width estimation lidars $W_{l}$ and $W_{r}$ to estimate the minimum
terrace width at the current step. These lidars are pointed downwards at an
angle of $45^{\circ}$ from the horizontal. For the purpose of describing the
method for terrace width estimation, assume that the robot is moving such that
the terrace wall is towards its left, as shown in Fig. 16. A new width
measurement is taken when there is a sudden change in the readings of lidars
$W_{l}$ and $W_{r}$. Using simple geometry, the terrace step width can be
estimated as

$w_{\rm
terrace}=\left[\frac{L_{lf}+L_{lb}}{2}+d_{W}+L_{db}\tan{45^{\circ}}\right]\cos\phi$
(8)

Where $\phi$ is the angle between robot heading and the terrace wall, given by

$\phi=\tan^{-1}\left({\frac{L_{lf}-L_{lb}}{d_{L}}}\right)$ (9)

Here $d_{W}$ and $d_{L}$ are the robot width and lengths, as illustrated in
Fig. 15. This width estimation system allows for the robot to handle terrace
steps with variable widths along the length, and allows the planning module to
account for the variations accordingly. Since the system depends on sudden
changes in the width estimation lidar readings, it can estimate width only
when the nearest distance between robot midline and the terrace step end
crosses the value $L_{db}\tan 45^{\circ}$.

To measure the distance from the robot’s body-fixed frame and the terrace
edge, an image processing based edge detection algorithm is deployed. It uses
a modified urban road stop-line detection algorithm as proposed in
seo2014vision . A vertical gradient filter is applied on the input image which
highlights horizontal features. Next, the probabilistic hough transform is
applied on sufficiently large clusters with a horizontal spread to detect
horizontal lines, and relatively close horizontal lines are merged into super-
clusters, and the bottom-most super-cluster is detected as the terrace edge,
as shown in Fig. 17.

Figure 17: Terrace Edge Detection

Inverse homography and static frame transforms are used to convert the
distance pixels to meters and calculate the distance between the terrace edge
and the robot’s body-fixed frame. We test and validate the algorithm multiple
times on even and uneven edges to ensure robust detection.

####  3.1.2 Fail-safes for fall avoidance

The fail-safe system for fall avoidance comprises of four 1D lidars mounted at
all the edge centres of the robot, looking downwards making an angle
$25^{\circ}$ with the horizontal Fig. 16. These lidars form a safe zone around
the robot, and if a sudden increase in values of any of these lidars is
detected, an emergency stop is triggered and all autonomous operations are
stopped. The safe zone is an imaginary rectangle with its edges located at an
inner distance $L_{db}\tan 45^{\circ}$ from the robot edges, which, when the
linear actuators are fully contracted, is about 12 centimetres. This system
stops the robot from falling off terrace edges, which might happen due to
planning errors or system uncertainties.

Figure 18: Traversal along Terrain Figure 19: Stair Climbing Algorithm
Illustration

###  3.2 Planner

####  3.2.1 Row Traversal

Based on the number of rows of crops present in the terrace, the planner
generates the primary path that the robot has to traverse on the terrace.
Within the current method, the primary path is a straight line along the
terrain at a constant distance from the wall of the terrace. The planner takes
as input the number of rows of crops to devise an optimal path that traverses
through each plant row while minimising time and actuation cost. As
exemplified in Fig. 18, if $n_{\rm croprow}=3$, the path generated ensures
that the robot returns in the same orientation as it entered in order to save
actuation costs of zero degrees turn. The planner also takes into account the
spatial constraints of the robot and the terrace width estimate to maintain
safe margins from both sides of the terrace. The planner prevents the robot
from entering into any unsafe zones along the terrace.

####  3.2.2 Agriculture Strategy

The software stack provides the provision for the farmers to input a desired
agricultural strategy which acts as an input to the planner. Before starting
the work on terrain, the planner sends a query to the agriculture strategy
module as shown in Fig. 14. Based on the user inputs provided, the planner
module accordingly plans the traversal path as well as the activation required
for different farming tools to perform agricultural activities. Hence a
flexible strategy can be maintained from terrain to terrain and crop to crop.
Different combinations of farming activities can be fed into the strategy
module in order to achieve different goals under varying circumstances.

###  3.3 Stair Climbing and Descending Module

On completion of all the tasks on the respective step, the planner activates
the stair climbing/descending module. As given in Fig. 14, the planner sends a
high-level command for performing the climb up or climb down activity. Before
the actual stair climbing or descending operation, the robot is aligned
parallel to the stairs. Using the pair of lidars facing the wall, the robot
aligns itself parallel to the wall by using a yaw control algorithm that
provides actuation inputs according to the differential value of distances as
measured by both the lidar. Once the robot is aligned, the stair climb/descend
algorithm takes over.

1 $N=\text{Number of wheel pairs}$

2 $yawAlignment()$

3 while  _$!DetectSuddenChange(L_{fl,fr}.distance)$_ do

4 ExtendAllScissors()

5$stepHeight=L_{df}.distance+d_{hb}$

6 while  _$L_{df}.distance\leq StepHeight$_ do

7 ExtendAllScissors()

8while  _$L_{df}.distance\geq StepHeight$_ do

9 MoveForward()

10for  _$i\leftarrow 1$ to $N$ _ do

11 while  _$L_{di}.distance\leq StepHeight$_ do

12 CompressScissor($i^{th}$ pair)

13 $targetDistance=L_{fl,fr}.distance-(d_{of}+d_{ob})$

14 while  _$L_{fl,fr}.distance\geq targetDistance$_ do

15 MoveForward()

16

Algorithm 1 Climb Up

As seen in Fig. 19, while climbing, initially the robot is at its bottom-most
position. In the first stage, the scissor mechanism lifts the robot to the
height of the step. In the second stage, the moves forward such that the front
dummy wheels are in contact with the horizontal surface of the next step. Here
we use LiDAR $L_{df}$ to detect the edge. In the next stage, the front pair of
wheels are lifted up by the scissor mechanism while the rear wheels are still
in contact with the present step. In the fourth stage, the robot moves forward
such that the front wheels and the third pair of dummy wheels come in contact
with the next step. Here we use LiDAR $L_{fr}$ and $L_{fl}$ to ensure precise
movement. In the fifth stage, the rear wheels are lifted up and in the last
stage, the robot moves forward such that the whole robot finally moves on to
the next step. Fig. 19 illustrates the above-explained stages. In every stage,
it is ensured that the centre of mass of the whole robot stays inside the
polygon made by the contact points of the wheels and the surface to ensure its
stability. For climbing down, the same stages would be followed in the reverse
direction.

1 $N=\text{Number of wheel pairs}$

2 $yawAlignment()$

3 for  _$i\leftarrow 1$ to $N$ _ do

4 $targetDistance=L_{bl,br}.distance+d_{of}+d_{ob}$ while
_$L_{bl,br}.distance\leq targetDistance$_ do

5 MoveForward()

6 while  _$L_{di}.distance\geq d_{hb}$_ do

7 ExtendScissor($i^{th}$ pair)

8

9$targetDistance=L_{bl,br}.distance+d_{of}$

10 while  _$L_{bl,br}.distance\leq targetDistance$_ do

11 MoveForward()

12while  _$L_{db}.distance\geq d_{hb}$_ do

13 CompressAllScissors()

Algorithm 2 Climb Down

The scissor mechanism is actuated using a linear actuator fixed at orientation
with optimised stroke length and force during the process. The mechanism is
stable in lifting up to 50 kgs in our robot. It also enables intermediate
height control for the robot within 40 cm which is useful for variable height
steps in real-life situations. It’s advantageous during the watering and
harvesting of crops of different heights. Lastly, the mechanism dimensions can
be modified easily for higher steps.

###  3.4 Controls Module

Using the task strategy and the trajectories from the high-level planner, the
control module provides the required actuator inputs for realising the plans
as shown in Fig. 14. The overall control module for the robot is solves two
broad sub-problems. First is controlling the height of the robot and scissor
mechanism actuated with linear actuators and the second is to accurately track
a path on the uneven terrain as a differential drive robot. Considering that
the control strategy for each of these sub-problems are not correlated, we
treat both problems independently and show the controller performance and
system stability for each of the sub-problem in this section.

####  3.4.1 Pitch Control

Figure 20: Illustration for pitch control of the robot

Due to the difference in the mechanical response of both scissor lift
mechanisms, the pitch of the chassis while lifting up may not be zero when
both the actuators are given the same input causing instability. We use a PID
controller for maintaining zero pitch that ensures the robot remains stable at
all times. We estimate the error in pitch and provide corrective feedback to
the linear actuators to keep the pitch zero at all times.

$\displaystyle A_{f}-A_{b}=$ $\displaystyle k_{p}\theta+k_{i}\int\theta
dt+k_{d}\frac{d\theta}{dt}$ (10a) $\displaystyle A_{f}+A_{b}=$ $\displaystyle
c_{1}-c_{2}\Delta v$ (10b)

Here $A_{f}$ and $A_{b}$ actuator inputs for the front and back linear
actuators. $\theta$ denotes the pitch of the robot, as shown in Fig. 20. This
method allows us to achieve two different objectives while controlling the
pitch of the robot,

  1. 1.

Difference in input is controlled based on the pitch of the chassis.

  2. 2.

Average actuation input is controlled based on the change in velocity of the
base to ensure a constant velocity of ascent/descent.

Figure 21: Performance of standard PID controller

We observe that a standard PID performs well during normal operation but the
controller performance depletes on addition of a payload and in presence of
external forces. As observed in Fig. 21, a standard PID in presence of an
external weight leads to a change in the asymptotic value of the pitch which
can cause instability. In order to address this we use an adaptive PID
algorithm based on (Alonso2013SelftuningPC, ). The algorithm uses a modified
version of the MIT rule for altering PID gains given as,

$\displaystyle k_{p}(t+1)=k_{p}(t)+\gamma_{P}\cdot[e(t)-e_{m}(t)]$ (11a)
$\displaystyle k_{i}(t+1)=k_{i}(t)+\gamma_{I}\cdot e_{m}(t)$ (11b)
$\displaystyle k_{d}(t+1)=k_{d}(t)+\gamma_{D}\cdot[\dot{e}(t)-\dot{e}_{m}(t)]$
(11c)

where $\gamma_{p},\gamma_{I},\gamma_{D}$ are tunable parameters and
$e(t),\dot{e}$ are standard error and error rate measurements whereas
$e_{m}(t),\dot{e}_{m}(t)$ are filtered values through a low pass filter. This
algorithm helps to handle noises in the system effectively while preventing
extremely high values for the PID gains.

Figure 22: Performance of Adaptive PID controller

As shown in Fig.22. We observe improved performance with adaptive PID in
comparison to standard PID controller in maintaining zero pitch with external
disturbances.

####  3.4.2 Path Tracking Control

Figure 23: Pure Pursuit Geometry

Given an optimal path from the planning module, we need to ensure accurate
tracking of the path considering external forces and environment
uncertainties. We use a modified pure-pursuit based controller for this
purpose which takes into consideration the kinematic constraints of the robot
and external disturbances due to uneven terrains. Pure pursuit is a goal-
oriented approach where the wheel velocities are calculated to drive the robot
along a circular path from the current position to a final goal position on
the optimal path which is at a particular look-ahead distance from the robot.
As shown in Fig. 23, the robot considers the point on the path at a distance
of $l_{d}$ as the goal. For smooth trajectories, the robot is treated as an
ackermann vehicle, and based on Ackermann geometry, $\delta$ is the steering
angle, $v_{x}$ is the longitudinal velocity and $R$ is the turning radius.
$\alpha$ is the central angle for the turn and its value is calculated from
the differential value of the distance measured by the pair of lidars facing
the adjacent wall. The look-ahead distance is the parameter that needs to be
tuned. We use an adaptive look-ahead distance proportional to the velocity of
the robot to ensure smooth control inputs. Hence, we obtain the required
steering angle using

$\displaystyle\tan(\delta)$ $\displaystyle=\frac{2Lsin(\alpha)}{l_{d}}$ (12a)
$\displaystyle l_{d}$ $\displaystyle=l_{0}+\beta\lvert v\rvert$ (12b)

Using the steering angle, we can calculate the angular velocity using

$\omega=\frac{v}{R}=\frac{v}{L}\tan(\delta)=\frac{2vsin(\alpha)}{l_{d}}$ (13)

Once we have the required velocity $v$ and angular velocity $\omega$, we can
estimate the left and right wheel velocities using

$\displaystyle v_{l}$ $\displaystyle=v-\frac{\omega L}{2}$ (14a)
$\displaystyle v_{r}$ $\displaystyle=v+\frac{\omega L}{2}$ (14b)

The standard pure pursuit algorithm only considers kinematic constraints and
ignores dynamic forces on the tires and external disturbances which might
cause instability especially in uneven terrains like agricultural fields. In
order to address this issue we implement an adaptive Pure Pursuit algorithm
similar to (Hoffmann2007AutonomousAT, ) accounting for ground reactions
causing side slips. An added gain term is added to the nominal input which
acts as an active damper against the ground reaction forces causing slip. The
modified steering control input is calculated as,

$\delta=\tan^{-1}\Big{\\{}\frac{2Lsin(\alpha)}{l_{d}}\Big{\\}}+k_{d,yaw}(r_{ref}-r)$
(15)

where $r_{ref}$ is the rate of change of yaw for the reference trajectory and
$r$ is the change of yaw for the robot. The added gain of $k_{d,yaw}$ requires
to be tuned based on the range of operating velocity of the robot.

Figure 24: Convergence of Tracking Controller

The controller shows good performance in terrace farming tasks and converges
to zero cross track error even in rough terrains. The Fig.24 shows the
convergence data of the controller implemented in a real-world testing
environment.

####  3.4.3 Farm Tool Control

##### Pesticide Spraying and Watering:

The robot possesses two different containers for pesticides and water which
are connected to a DC motor-driven pump. On the basis of the agricultural
strategy, the control system needs to provide the required actuation signal to
the DC motor. Taking inspiration from the prior work in controlled pesticide
spraying in Deshmukh2020DesignAD , we employ solenoid valves to switch between
the two tanks. The inlet of the DC motor pump is connected to the junction of
the two pipes coming from the respective tanks through a solenoid valve, and
the outlet of the pump is connected to the series nozzle system which can
spray the pesticide or water based on the agriculture strategy. The solenoid
valves have to be in a closed condition in order to allow flow only when they
are actuated.

##### Ploughing, Sowing and Rolling:

The distance between two seeds for sowing is controlled by a stepper. Let
$n_{t}$, $r$, $\omega$ be the number of teeth, the inner radius of the seed
feeding wheel, and the RPM of the stepper motor respectively. We denote the
forward velocity of the robot at time $i\Delta t$ as $v_{i}$ (m/s), where
$\Delta t$ is the sampling time of the control module. Given a desired seed
separation of $d$ meters, and assuming one seed is being carried per tooth and
a constant robot velocity for the next $\Delta t$ seconds, the seeder stepper
motor RPM $\omega_{i}$ for this time step can be given by the Eq. 16.
Therefore, the desired motor RPM is achieved by controlling the pulse speed
$p$ [Hz] to the stepper motor to attain this desired angular velocity, given
by Eq. 17.

$\displaystyle\omega_{i}$ $\displaystyle=\frac{v_{i}}{n_{t}d}$ (16)
$\displaystyle p_{i}$ $\displaystyle={\omega_{i}}{\alpha}$ (17)

Where $\alpha$ is the step angle of the stepper motor, with units rad/pulse.

##  4 Conclusion

This work describes Aarohi – a fully automated solution to terrace farming,
which is widely used for agriculture in hilly areas. The robot incorporates a
novel stair climbing mechanism design, which facilitates climbing terrace
steps of large heights. An optimisation-based design technique has been
presented to determine optimal design parameters for the stair climbing
mechanism based on the given use case. The agricultural module has been
developed to perform farming activities like ploughing, sowing, rolling,
watering and pesticide spraying efficiently and autonomously. A robust
software stack has been developed, which uses a combination of 1D lidars, GPS
and IMU sensors, wheel encoders and RGB camera images for robot localisation,
task planning and execution. The proposed robot design is modular, and
parameters like the number of wheel pairs, maximum stair climbing distance and
maximum load-carrying capacity can be customised according to specific use
cases or needs of the farmer. The prototype of this robot secured the First
Position in the DIC’s Autonomous Terrace Farming event at the 8th Inter IIT
Tech Meet, IIT Roorkee. We would like to conclude that from a commercial point
of view, ‘Aarohi’ could bridge the current gap in extending automated farming
to hilly regions, and prove to be a great solution to all terrace farming
needs at an affordable price.

Acknowledgments We would like to thank the MN Faruqui Innovations Center
(MNFIC), IIT Kharagpur for the continuous support in developing and testing
the prototype. We would also like to extend a special thanks to Mr. Utkarsh
Agrawal, Mr. Himanshu Raj Khandelwal, Mr. Yash Agarwal, Mr. Aditya Rathore,
Mr. Het Shah, Mr. Jaydeep Godbole and Mr. Utkarsh Sharma for their extensive
efforts and contribution during the prototype development stage.

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# Charged resonances and MDM bound states at a multi-TeV muon collider

Natascia Vignaroli  Physics Department, University of Naples and INFN Naples.

###### Abstract

A multi-TeV muon collider proves to be very efficient not only for the search
for new heavy neutral particles, but also for the discovery of charged bosons
of the $W^{\prime}$ type. We find that, by analyzing the associated production
with a Standard Model W, charged resonances can be probed directly up to
multi-TeV mass values close to the collision energy, and for very small
couplings with the SM fermions, of the order of $10^{-3\div 4}$ times smaller
than the SM weak coupling. Additionally, charged bound states of WIMP Minimal
Dark Matter can be discovered with low statistics by running above the
kinematic threshold, at a center-of-mass energy just slightly above the mass
of the MDM bound state. This opens up a very interesting possibility for the
discovery of WIMPs, complementary to the search for the resonant production of
the neutral MDM bound state component, which relies on an on-peak search, and,
for 5-plet MDM, more efficient than the WIMP searches based on mono-X,
missing-mass and disappearing tracks signatures.

##  I Introduction

The design of a future $\mu^{+}\mu^{-}$ collider (MuCol) with multi-TeV energy
has recently been proposed, showing outstanding possibilities to discover and
test different aspects of high energy physics [1, 2, 3]. Among these, very
interesting prospects have been highlighted for the discovery of Weakly
Interacting Massive particles (WIMPs) and for new heavy neutral bosons.

For example, as shown in [4], $Z^{\prime}$ type of resonances from gauged
$L_{\mu}-L_{\tau}$ models, produced through s-channel radiative return, can be
probed directly up to very small couplings, of the order of $10^{-3}$, for a 1
TeV $Z^{\prime}$, by a 3 TeV MuCol with 1 ab-1. A powerful test, and a higher
reach on the mass of the resonance, can also be obtained indirectly, via
precision measurements. In this case, for example, a $Y$-universal
$Z^{\prime}$ model can be ruled out by a 10 TeV MuCol, for $Z^{\prime}$ masses
of the order of 100 TeV and couplings of the order of $10^{-1}$ [5].

WIMP dark matter can be realistically discovered by a multi-TeV MuCol up to
the thermal target of the electroweak (wino-like) triplet, considering the
channel where the WIMPs are produced in pairs, generating mono-X signals with
large missing energy, or by the decays of heavier charged states of the WIMP
EW multiplet, which give raise to disappearing tracks [6, 7]. A powerful
alternative strategy to test the WIMP scenario in its minimal realization, the
minimal dark matter (MDM) hypothesis, is to consider the detection of bound
states formed by two MDM fermionic weak multiplets [8]. For the Majorana
5-plet, MDM bound states can be produced resonantly with large cross section
at a muon collider, provided the MuCol runs “on-peak” at a center-of-mass
energy close to the mass of the bound state, $\sqrt{s}\approx 2M$. This would
permit to discover MDM 5-plets with low statistics, few fb-1, in the first
phase of the collider operation [8]. Conversely, mono-X signals show lower
sensitivities and would hardly be able to test the 5-plet MDM thermal target
[9]. A better coverage on WIMPs could be obtained by considering disappearing
tracks [6] and missing-mass searches, even if the 5-plet target would be
reached only for $\sqrt{s}\gtrsim 30\div 50$ TeV and a large amount of
integrated luminosity, $L\gtrsim 2\div 100$ ab-1 [7].

In this letter we propose a strategy that would allow to detect directly
charged resonances of the $W^{\prime}$ type at a muon collider, and a new
efficient way to test MDM bound states. This opportunity is offered by the
analysis of the channel where the new charged state, either a $W^{\prime}$ or
a MDM bound state, is produced in association with the $W$ boson of the
Standard Model (SM). Besides the unique opportunity to directly test a
$W^{\prime}$ new boson, with a very high reach, this analysis offers a very
efficient way to test the WIMP scenario through the detection of MDM bound
states, which is complementary to the search for the resonant production of
the neutral bound state. Furthermore, it presents the advantage of not needing
an ”on-peak” focus of the MuCol beam energy near the bound state mass, but
just requiring the experiment to operate slightly above the kinematic
threshold, $\sqrt{s}\gtrsim 2M$.

In the following, after having introduced our theoretical framework for $X$
generic charged resonances of the $W^{\prime}$ type (Sec. II) and for the MDM
bound states (Sec. III), we will present our search strategy in Sec. IV and
will offer our conclusions in Sec. V. Details on the analysis will be reported
in the Appendices.

##  II Charged resonances

We focus on a heavy spin-1 state transforming as a triplet under the SM
electroweak group. We consider the case of an effective $W^{\prime}$ boson,
which we indicate as $X$, which interacts with the SM particles analogously to
the SM $W$. The relevant Lagrangian reads:

$\displaystyle\begin{split}\mathcal{L}^{W^{\prime}}_{eff}=\frac{g_{X}}{\sqrt{2}}\left[\right.&V^{CKM}_{ij}\bar{u}_{i}\gamma^{\mu}P_{L}d_{j}\\\
&+V^{PMNS}_{ij}\bar{\nu}_{i}\gamma^{\mu}P_{L}\ell_{j}\left.\right]X_{\mu}+\text{H.c.}\,,\end{split}$
(1)

with $V^{CKM}$ and $V^{PMNS}$ denoting the CKM and PMNS matrices. In the case
$g_{X}=g_{2}$, where $g_{2}=e/\sin\theta_{w}$ is the SM weak coupling, the $X$
interactions are identical to those of the SM $W$. This reproduces the
Sequential SM scenario (SSM) of Ref. [10]. In our analysis we will leave
$g_{X}$ as a free parameter. The $X$ decay rates read

$\displaystyle\begin{split}\Gamma(X^{\pm}\to\ell^{\pm}\nu)\simeq\frac{g^{2}_{X}}{48\pi}m_{X}\quad\Gamma(X^{\pm}\to\bar{q}q^{\prime})\simeq\frac{g^{2}_{X}}{16\pi}m_{X}\,.\end{split}$
(2)

The decays to Higgs and SM gauge bosons are suppressed. Charged $W^{\prime}$’s
also appear in composite Higgs theories [11, 12, 13], generated by a new
strong dynamics as composite spin-1 weak triplet resonances. The composite
$W^{\prime}$ couplings to Higgs and gauge bosons, differently from the SSM
case, are typically relevant.

##  III Minimal Dark Matter Bound States

A compelling and Minimal solution to the Dark Matter puzzle is to consider, in
addition to the SM, a new fermionic multiplet under the SM weak gauge group
[14], whose neutral and, as such, lightest component constitutes a good DM
candidate. The cosmological DM abundance can be then reproduced thermally, via
freeze-out, for TeV-scale values of the DM mass, $M$ [14, 15]. For $M\gtrsim
M_{W,Z}/\alpha_{2}$, pairs of MDM multiplets can form Coulombian-like
electroweak bound states. This is verified for larger representations, and in
particular in the case of 5-plets under $SU(2)_{L}$ of Majorana fermions with
zero hypercharge. After taking into account Sommerfeld and bound-state
corrections [15], the 5-plet thermal abundance matches the DM density for a
mass $M\approx 14$ TeV.

Bound states with the same quantum numbers as the weak vectors inherit, via
mixing, couplings to SM fermions. We are especially interested in such bound
states, as they can thereby be directly produced in $\mu^{+}\mu^{-}$
collisions. In particular, the charged components of the bound state can be
produced in the associated $W$ channel we are studying. For the 5-plet MDM,
these special bound states exist: the spin-1 ${}^{n}s_{3}$ vector triplets
[8]. The ground state ${}^{1}s_{3}$, which will be the focus of our analysis,
decays into fermions with a rate
$\Gamma_{\text{a}nn}=15625\,\alpha_{2}^{5}M/48\approx 0.17$ GeV [8]. Its
collider phenomenology can be reproduced by the effective $W^{\prime}$
description of the previous section with an effective coupling:

$g_{X}=g_{{}^{1}s_{3}}\simeq 0.014\,g_{2}\,,$ (3)

while its mass is $m_{X}=m_{{}^{1}s_{3}}\approx 2M\approx 28$ TeV.

##  IV $\bf X^{\pm}W^{\mp}$ associated production

We focus on the possibility to directly search for a charged spin-1 resonance
$X$ at a muon collider, in the SM $W$ associated production channel
$\mu^{+}\mu^{-}\to X^{\pm}W^{\mp}$. The leading Feynman diagram is shown in
Fig. (1).111 In composite $W^{\prime}$ scenarios [11, 12, 13] further
contributions to the $XW$ associated production, which we will not include in
our analysis, since they are suppressed for our SSM effective description,
come from the exchange of an EW gauge boson in the $s$-channel and from
vector-boson-fusion (VBF). We will leave the analysis of these signal
topologies to a future study (cfr. [16] for a study on the efficiency of the
VBF channel at a multi-TeV MuCol). Therefore, the results we will show in this
letter represent only a conservative estimate of the reach on a composite
$W^{\prime}$ resonance.

Figure 1: Leading Feynman diagram for the SM $W$ associated production of a
charged spin-1 resonance $X$ at a muon collider.

The cross section can be expressed in the analytic form:

$\displaystyle\begin{split}&\sigma(\mu^{+}\mu^{-}\to
X^{+}W^{-})=\sigma(\mu^{+}\mu^{-}\to X^{-}W^{+})\simeq\\\
&\frac{g^{2}_{2}\,g^{2}_{X}}{1536\,\pi\,s^{2}\,m^{2}_{X}\,m^{2}_{W}}\big{[}s^{2}+10\,m^{2}_{X}s+m^{4}_{X}+m^{4}_{W}\\\
&+10\,m^{2}_{W}(s-5m^{2}_{X})\big{]}\sqrt{(s-m^{2}_{X})^{2}-2m^{2}_{W}(s+m^{2}_{X})+m^{4}_{W}}\\\
\end{split}$ (4)

We find good agreement between the analytic evaluation of the cross section
and the numeric calculation with MadGraph5 [17]. The cross section is shown in
Fig. 2 for different MuCol $\sqrt{s}$, as a function of $m_{X}$, and in Fig.
3, as a function of $\sqrt{s}$, in the specific case of the $W$ associated
production of the ${}^{1}s_{3}$ MDM bound state.

Figure 2: Cross section for the associated production of a charged spin-1
resonance $X$ with a SM $W$ (Fig. 1) at a muon collider with different beam
energies, as function of the charged resonance mass. The plotted cross
sections are calculated in the SSM case, with $g_{X}=g_{2}$. The cross section
depends quadratically on $g_{X}$.  Figure 3: Cross section for the associated
production of a 5-plet MDM bound state with a SM $W$ at a muon collider, as
function of the center-of-mass energy.

###  IV.1 Selection strategy and reach

We consider the fully hadronic final state $\mu^{+}\mu^{-}\to(W\to jj)(X\to
jj)$. The SM $W$ decay products, as a consequence of the large Lorentz boost,
are emitted very collimated, and will be mostly collected in a single jet.

We thus require at least 3 hard jets in the central region, with sufficient
separation from each other:

$p_{T}\,j>30\;\text{GeV}\,,\quad|\eta_{j}|<2.5\,,\quad\Delta R_{jj}>0.4\;,$
(5)

where $\Delta R_{jj}=\sqrt{\Delta\phi_{jj}^{2}+\Delta\eta_{jj}^{2}}$ denotes
the angular separation between two jets. We also assume a detection efficiency
of 70$\%$ for each jet in this acceptance region.

Signal and background events are simulated with MadGraph5 [17]. Events are
then passed to Pythia8 [18] for showering. Jets are clustered with Fastjet
[19] by using an anti-kt algorithm with cone size $R=0.4$. We also apply a
smearing to the jet 4-momenta, following the Delphes [20] default card, in
order to minimally take into account detector effects. 222According to studies
on the expected performance of a future muon collider, even better detector
performances, than those considered in this analysis, can be envisaged [21].
However, we prefer to remain conservative in our predictions.

The SM background, which we find to be of the order of a few fb in the
acceptance region, is given mainly by events with jets emitted by the
radiation of an s-channel photon, where the third jet is produced by a gluon
radiated from a quark. A subdominant background component consists of jets
from $WW$ production, while the background contribution from $t\bar{t}$ events
is almost negligible.

We apply a simple strategy to identify the jets coming from the $X$ decays,
and consequently to reconstruct the heavy charged resonance. We observe (cfr.
Fig. 6) that in the case of heavier $X$ resonances, the jets from the $X$ are
mostly emitted back-to-back and have a large $\Delta R$ separation; for
lighter $X$, instead, they tend to be collimated and closer. We thus consider
the three $\Delta R$ separations among the three $p_{T}$-leading jets and, for
$m_{X}\leq\sqrt{s}/2$, we identify the two jets from the $X$ by assuming that
they are those with the smallest separation (consequently the third remaining
jet is identified with the $W$ jet), while, for $m_{X}>\sqrt{s}/2$, the two
$X$ jets are identified with those with the largest separation (the third
remaining jet constitutes the hadronically decayed $W$). In order to reduce
the background and to obtain a clean reconstruction of the $X$ resonance, we
then consider a cut on the invariant mass of the reconstructed $W$ boson
($M_{W}$):

$50\,\text{GeV}<M_{W}<110\,\text{GeV}\,.$ (6)

The acceptances to this $W$ and $X$ reconstruction strategy range from 98% for
the lightest $X$ values to about 40% for $m_{X}\simeq\sqrt{s}/2$ in the lower
$X$ mass case, and from about 67% to 90% in the case of heavier $X$. The
reconstruction strategy based on the $\Delta R$ separation we apply is
conservative. More refined techniques, indeed, could be also applied, which we
leave for future investigation. For example, one could identify the different
jets origin by analyzing the invariant mass and structures of the final state
jets or of their combinations. We prefer, however, to rely on the simple
strategy described above, since it shows already good efficiencies and,
moreover, it is much less dependent on still-unknown detector performance
details and on yet-to-be-tuned modelling of jet showering effects.

Fig. 7 in the Appendix shows the reconstructed $W$ and $X$ invariant mass
distributions for the signals of an effective $W^{\prime}$ resonance in two
cases of a lighter and a heavier $X$ scenario, and for the background. The
analysis described above can be applied as well to the case of the charged
component of the MDM bound state ${}^{1}s_{3}$. The corresponding $M_{X}$
invariant mass distribution, at a 30 TeV muon collider, is shown in Fig. 4
together with the background distribution, before and after the cut on the
reconstructed $W$ invariant mass, Eq. (6). It is evident how, after the
complete signal selection strategy, the ${}^{1}s_{3}$ resonance is clearly
distinguishable from the background.

Figure 4: Invariant mass distribution of the reconstructed $X$ resonance for
the signal of a ${}^{1}s_{3}$ MDM bound state and for the background, before
(in dashed light-red) and after (continuous thick line) the cut on the
invariant mass of the reconstructed $W$ boson, Eq. (6). The plot shows the
event distribution at a 30 TeV muon collider with an integrated luminosity of
100 fb-1.

The final reach of our analysis on a generic $X$ charged resonance is shown in
Fig. 5. We have estimated the statistical significance as $S/\sqrt{S+B}$, with
$S$ ($B$) denoting the signal (background) number of events. We indicate the
5$\sigma$ discovery and the 2$\sigma$ exclusion reach for a 10, 30 and a 50
TeV muon collider with different collected integrated luminosities, up to the
maximum achievable value, $L=10\,(\frac{\sqrt{s}}{10\,\text{TeV}})^{2}$ ab-1
[9]. The MuCol can probe with this analysis in the associated $WX$ channel,
charged $X$ resonances up to mass values close to the center-of-mass energy,
and for couplings as small as $10^{-2}g_{2}$, $10^{-3}g_{2}$ and $10^{-(3\div
4)}g_{2}$ for $\sqrt{s}=$10, 30 and 50 TeV respectively.
Furthermore, a $W^{\prime}$ in the SSM scenario, corresponding to the case
$g_{X}/g_{2}=1$, can be discovered at the very early stage of running by a
muon collider. We find that with just 50 pb-1 of integrated luminosity, a SSM
$W^{\prime}$ with a mass up to 9, 28 and 46 TeV can be discovered by the MuCol
with $\sqrt{s}=$ 10, 30 and 50 TeV respectively. In the case of the MDM bound
state, we find the following expected reach for a 5-plet MDM ${}^{1}s_{3}$
charged bound state of 28 TeV:  it can be excluded with about 34 fb-1 and
discovered with 210 fb-1 by a 30 TeV muon collider. This muon collider reach
is even higher for larger collision energies and/or mass values lower than 28
TeV. That is, for $\sqrt{s}/m_{{}^{1}s_{3}}\gtrsim 1.07$, the values we
indicate represents a conservative estimate of the MuCol reach on a 5-plet MDM
bound state of mass $m_{{}^{1}s_{3}}$.

##  V Conclusions

In this letter we have proposed a new channel and strategy to probe directly
heavy charged resonances at a future multi-TeV muon collider: The associated
production of the charged new state with a SM W. The projected sensitivities
of the MuCol in the channel are shown in Fig. 5 and indicate that a charged
resonance of the SSM $W^{\prime}$ type333As explained, the reach of this
analysis is expected to be conservative in the case of $W^{\prime}$ resonances
from composite Higgs theories. can be discovered up to multi-TeV mass values
close to the beam-colliding energy, and for very small couplings with the SM
fermions, of the order of $10^{-3\div 4}$ times smaller than the SM weak
coupling. Furthermore, the channel offers a very efficient and alternative way
to probe the WIMP scenario for MDM in the 5-plet EW representation, by
allowing the direct detection of the charged component of the MDM bound state.
A MDM Majorana 5-plet bound state can be excluded with about 34 fb-1 and
discovered with 210 fb-1 by a 30 TeV MuCol. This reach on the WIMP 5-plet
thermal target is much higher than those of mono-X, missing-mass and
disappearing tracks signatures.

Figure 5: 5$\sigma$ discovery (left plot) and 2$\sigma$ exclusion (right plot)
reach on the plane mass of the $X$ versus coupling, expressed in terms of the
SM weak coupling, for a 10 TeV (blue curve), 30 TeV (black) and 50 TeV (green)
MuCol with a collected integrated luminosity of 1 fb-1 (dotted curves), 100
fb-1 (dashed) and for the maximum achievable luminosity (continuous curves).

###### Acknowledgements.

The author thanks Salvatore Bottaro and Alessandro Strumia for previous
collaboration on related topics, and Roberto Franceschini for discussions
leading to this work.

We report in the following appendixes details on the signal selection strategy
and analysis.

##  Appendix A Distributions

As shown in Fig. 6, in the case of heavy $X$ resonances, $m_{X}>\sqrt{s}/2$,
the two jets emitted by the $X$ decays have a large $\Delta R$ separation. For
the majority of the events, this separation is the largest one among the
$\Delta R$’s of the three $p_{T}$-leading jets. For lighter $X$, instead, the
two emitted jets are collimated and closer. In this case, for most of the
events, their separation is the smallest one.

Figure 6: From Monte Carlo truth: Signal event distributions (normalized to
unit area) in the $\Delta R$ separation between two jets: between the two jets
coming from the $X$ decays (in red), between the single jet from the $W$ decay
and one of the two jets from the $X$ decay (the two histograms in blue). We
consider $\sqrt{s}=30$ TeV. The plot on the left refers to a lighter $X$
scenario, with $m_{X}=10$ TeV, the plot on the right to a heavier case with
$m_{X}=25$ TeV.

Fig. 7 shows the $W$ and the $X$ invariant mass distributions, at a 30 TeV
muon collider, for the signals of an effective $W^{\prime}$ resonance in two
cases corresponding to a lighter ($m_{X}=10$ TeV) and a heavier ($m_{X}=25$
TeV) $X$ scenario, and for the background.

Figure 7: Event distributions, at a 30 TeV muon collider with an integrated
luminosity of 100 fb-1, for the signals of an effective $W^{\prime}$ of 10 and
of 25 TeV, with a coupling $g_{X}=0.01g_{2}$, and for the background,
resulting from the two cases of a lighter and a heavier $X$ selection
strategy. Plot on the left: Invariant mass distribution of the reconstructed
$W$ boson. Plot on the right: Invariant mass distribution of the reconstructed
$X$ resonance. The background distributions are shown before (in dashed light-
red) and after (continuous thick lines) the cut on the invariant mass of the
reconstructed $W$ boson, Eq. (6).

##  Appendix B Cut flow of the cross sections

Table 1 reports the cross section values for the different signal points
considered in the analysis and for the relevant backgrounds in each of the
main steps of the selection: after the acceptance cuts in Eq. (5), after the
$W$ and $X$ reconstruction, with the subsequent constraint on $M_{W}$ in Eq.
(6).

$\sqrt{s}=$ 10 TeV | accept. |  $W$ reco.
---|---|---
$m_{X}$ (TeV) |  |
5 | 4.66 | 1.94
6 | 3.63 | 1.19
7 | 2.69 | 2.14
8 | 1.82 | 1.67
9 | 0.939 | 0.897
9.9 | 0.177 | 0.152
$Z/\gamma^{*}\to jets$ |  1.95 $\cdot 10^{-3}$ |  1.49 [1.26] $\cdot 10^{-4}$
$VV\to jets$ |  0.077 $\cdot 10^{-3}$ |  0.27 [0.58] $\cdot 10^{-4}$
Total |  |
background |  2.03 $\cdot 10^{-3}$ |  1.76 [1.84] $\cdot 10^{-4}$

$\sqrt{s}=$ 30 TeV | accept. |  $W$ reco.
---|---|---
$m_{X}$ (TeV) |  |
5 | 20.3 | 19.9
10 | 7.53 | 6.23
15 | 4.65 | 1.93
20 | 2.97 | 1.97
25 | 1.52 | 1.40
28 | 0.640 | 0.603
29.9 | 0.151 | 0.137
$Z/\gamma^{*}\to jets$ |  0.246 $\cdot 10^{-3}$ |  1.84 [1.17] $\cdot 10^{-5}$
$VV\to jets$ |  3.2 $\cdot 10^{-7}$ |  0.68 [1.6] $\cdot 10^{-7}$
Total |  |
background |  0.246 $\cdot 10^{-3}$ |  1.84 [1.18] $\cdot 10^{-5}$

$\sqrt{s}=$ 50 TeV | accept. |  $W$ reco.
---|---|---
$m_{X}$ (TeV) |  |
5 | 49.8 | 49.6
10 | 15.2 | 14.7
20 | 6.11 | 4.21
30 | 3.62 | 1.15
35 | 2.70 | 2.12
40 | 1.80 | 1.64
45 | 0.937 | 0.874
49.9 | 0.146 | 0.132
$Z/\gamma^{*}\to jets$ |  9.10 $\cdot 10^{-5}$ |  6.60 [3.82] $\cdot 10^{-6}$
$VV\to jets$ |  $<$ 1 $\cdot 10^{-7}$ |  $<$ 1 $\cdot 10^{-8}$
Total |  |
background |  9.10 $\cdot 10^{-5}$ |  6.60 [3.82] $\cdot 10^{-6}$

Table 1:  Cross section values (in pb) for signal (assuming $g_{X}=g_{2}$) and
background, in the acceptance region, and after the reconstruction of the $W$
boson, including the cut in Eq.(6). $V$ denotes a SM weak gauge boson,
$V\equiv Z,W,\gamma^{*}$. For the background, the values on the third column
refer to those following the reconstruction strategy for heavier [lighter] $X$
resonances.

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