The two-dimensional simulation model of an ATV-sized compact wheeled electric agricultural robot pulling a tine harrow was developed in MATLAB/Simulink software version R2022b. As was previously mentioned, RWD, FWD, and AWD versions of the model were created. The RWD and FWD versions feature a single electric motor (EM), while the AWD version has a motor on each axle. Each version utilizes a fixed gear ratio. In order to examine the effect of longitudinal weight distribution, two versions were created of the RWD and FWD models. For the RWD model, weight distributions of 40/60 and 30/70 (front/rear) were used. Conversely, the two FWD models featured weight distributions of 60/40 and 70/30. The AWD model had a weight distribution of 50/50.
The main components of the simulation model are the powertrain model, the tire–soil interaction model, the equations of motion of the robot, the harrow dynamics, and the control algorithm. Each of these components will be described in the upcoming subsections. The robot and soil parameters are presented in
Section 3.4.
Figure 1 and
Figure 2 show diagrams of the RWD and AWD models where the different subsystems and the flow of information between them can be seen. A diagram of the FWD version would be identical to the RWD version except the front and rear axles would be switched around. The numbers in the brackets represent the equations utilized by the subsystems. In the figures,
v and
d represent speed and depth, and the variable
x passed from the robot chassis to the harrow subsystem represents a vector of states of the robot, including its longitudinal and vertical positions and speeds as well as the pitch angle and rate. These states are then used by the harrow subsystem to calculate the current states of the harrow.
3.2. Tire–Soil Interaction Model
The tire model used in the simulation model assumes the tire to be rigid. This assumption was considered to be acceptable due to the relatively small tire size in the simulated robot. The diameter of the robot wheels is 0.635 m, and the harrow wheels have a diameter of 0.584 m. The rigid wheel model used is based on the classic widely used model developed by Wong and Reece [
32,
33]. A similar tire modeling approach was adopted by Senatore and Sandu in their research focusing on off-road tire modeling and the multipass effect [
38]. The tire model is based on calculating the normal and shear stress distributions beneath the tire. The shear stress distribution is calculated utilizing the equation originally defined by Janosi and Hanamoto [
84]:
where
is the maximum shear stress,
j is the terrain shear displacement,
k is the shear deformation modulus, and
is the angle that describes the angular position of the currently examined tire element. The value of
is zero at the bottom of the tire, and the value increases when rotating counterclockwise, as demonstrated in
Figure 3.
The maximum shear stress is calculated according to the Mohr–Coulomb equation:
where
c is the soil cohesion,
is the normal stress, and
is the angle of internal shear resistance of the soil. The definition of the shear displacement depends on the sign of the slip ratio, which is conventionally defined as follows [
85,
86]:
where
is the rotational speed of the wheel,
R is the radius of the wheel, and
is the longitudinal speed of the wheel. The wheel is slipping when the slip ratio is positive and skidding when it is negative. However, it can be seen in Equation (
10) that the slip ratio cannot be calculated for a wheel that has zero longitudinal speed, and at near-zero speeds it can be unstable. Thus, a different formulation was developed to be used at very low speeds. Below a specified threshold speed
, the slip ratio is instead calculated as
The model progressively switches to the formulation presented in Equation (
10) as the vehicle speed increases. Thus, the full formulation of the slip ratio can be expressed as
where
v is the longitudinal speed of the robot and
is a speed threshold above which the conventional slip ratio definition given in Equation (
10) is used. The threshold speeds were experimentally determined utilizing different soils to find the lowest speeds at which the tire model was found to behave in a stable manner in all conditions. For
, the speed was determined as 0.1 km/h, and the value for
was set to 2.0 km/h.
For slip (i.e., the wheel has a positive slip ratio), the shear displacement is calculated as follows [
32]:
where
is the entry angle of the wheel, as seen in
Figure 3. As described by Wong and Reece [
33], when a wheel is skidding (i.e., it has a negative slip ratio) on a loose surface, there exists a point where the tangential stress changes its direction from opposite to wheel rotation (defined as positive) to the same direction as the wheel rotation (defined as negative). The model accomplishes simulating this phenomenon by accounting for the change in the direction of the shear displacement in the rear region of the tire. An example of the difference in the tangential stress distribution between slipping and skidding is shown in
Figure 4. The shear displacement for a skidding wheel is calculated as follows [
33]:
where
is the angle of maximum stress, and
is a supporting coefficient, which is defined as follows [
33]:
The definition of the angle of maximum stress also depends on whether the wheel is slipping or skidding. For slip, it is defined as follows [
32]:
where
and
are empirically estimated constants. For skidding, the angle of maximum stress is defined as follows [
33]:
where
and
are defined as
The normal stress in the contact patch is defined as a piece-wise function. The stress between the leading edge of the contact patch at the angle
and the angle at which maximum normal stress occurs
) is defined as follows [
38]:
where
and
are cohesion-dependent and frictional-dependent coefficients specific to the soil material,
is the soil bulk density (in N/m
3),
n is the sinkage index, and
b is a parameter related to the contact geometry. The normal stress between the angle
and the trailing edge of the contact patch at angle
is calculated as follows [
38]:
where
is the angle of the trailing edge of the contact patch. The sinkage index depends on the slip ratio and is defined as follows [
87]:
where
and
are empirically approximated constants. The geometry-related parameter
b is calculated as follows [
38]:
where
is the projected contact patch length and
w is the width of the tire.
The tire forces are then calculated as resultant forces caused by the normal and tangential stress distributions. The longitudinal tire force produced by the normal stress is calculated as follows [
32]:
and the vertical force produced by the normal stress is defined as follows [
32]:
The longitudinal force produced by the tangential stress can be calculated as follows [
32]:
and the vertical force produced by the tangential stress is defined as follows [
32]:
The total longitudinal force produced by the tire is the sum of
and
:
while the total vertical force is the sum of
and
. In addition, vertical damping was added to the tire model in order to improve stability. Thus, the total vertical tire force would be calculated as
where
is the damping coefficient and
is the vertical speed of the wheel. The driving torque is given as follows [
32]:
As the robot was modeled featuring beam axles, the speed of the wheels on an axle can be calculated as
where
is the torque provided to the axle,
is the total inertia at the axle, and
is the initial rotation speed of the wheels, which in the simulations conducted in this research was always set to 0.
As was shown in Equations (
13), (
14), (
16), and (
17), the shear displacement
j and the angle of maximum stress
depend on the sign of the slip ratio. Some previous studies utilized the slip version of the equations for both slipping and skidding [
38]. However, as can be observed in
Figure 5, such an approach causes the tires to produce excessive force at high negative slip values. Thus, in the developed model, the skid versions of the equations were utilized for skidding calculations. In order to prevent instability caused by discontinuity at zero slip when the model switches between the slip and skid versions, smoothing was introduced into the calculations. The smoothing progressively switches the model from the slip to the skid model at low skid values according to the following equations:
where
is the slip ratio threshold value below which the skid model is utilized. In this model, the threshold value was set to −0.1.
Figure 5 shows the difference between using the combined model and only the slip model. It can be seen in the figure that utilizing the slip model for negative slip ratio values would result in excessively negative tire force. The figure also shows that according to the model, the longitudinal tire force can be nonzero when the slip ratio is zero. Thus, in order to enable the vehicle to stay stationary, an additional rule was added to the model that if the wheel is not rotating and the vehicle speed is zero, the longitudinal tire force must then also be zero. Additionally, wheel rotational speed was not allowed to be less than zero.
For modeling the multipass effect, the approach developed by Senatore and Sandu was utilized [
38]. As described by Holm, the changes in terrain properties following each pass are primarily a function of slip [
88]. The more slip the tire has, the stronger its effect on the soil is. The multipass effect influences the density
, cohesion
, and shear deformation modulus
utilizing the following equations:
where
,
, and
are constants,
is the slip ratio during the previous pass, and
is the number of passes. The values for
,
, and
were acquired from [
38]. In addition to the parameter changes, the soil is also compacted by the tires. Thus, due to the robot front and rear axles having the same track width, the rear tires traverse in the ruts created by the front tires. Furthermore, some of the tines on the harrow traverse in the ruts left by the tires of the robot, which in turn affects the tine forces. In the developed simulation model, the ground was given a resolution of 1 cm. The values for soil density, cohesion, shear deformation modulus, and ground height used for calculating the tire forces of the rear tires are taken as the average of each of the points currently in contact with the tire.
3.3. Robot Equations of Motion and Harrow Dynamics
The dynamics of the robot is described by Equations (
38)–(
40). The longitudinal (
x), lateral (
y), and vertical (
z) axes and coordinates referred to in the following sections refer to the world-fixed coordinate system. It should be noted that the points at which the resultant forces of the normal stress and tangential stress affect the tires are not the same. Hence, the longitudinal and vertical components of the normal and shear stress resultant forces are separated in Equation (
40). In previous tire–soil interaction models, it has been common to make the assumption that all tire forces always apply at the bottom of the tire [
38,
39]. Aerodynamic drag was omitted from the model due to the low speeds at which the robot would operate. The longitudinal dynamics of the robot in the developed model is described as
where
and
represent the longitudinal force produced by the front and rear tires;
is the longitudinal force, also known as draft force, exerted on the hook at the back of the robot by the harrow; and
is the longitudinal acceleration of the center of gravity (CoG) of the robot. The vertical dynamics is described as
where
and
are the vertical forces produced by the front and rear tires,
is the vertical force exerted on the hook by the harrow, and
is the vertical acceleration of the CoG of the robot. The rotational dynamics of the robot with respect to the lateral axis is described as
where
,
,
,
,
,
,
, and
represent the longitudinal and vertical forces at the front and rear tires caused by the shear (
) and normal (
) stresses;
,
,
,
,
,
,
, and
are the longitudinal and vertical coordinates of the shear and normal stress resultant forces;
and
are the longitudinal and vertical coordinates of the CoG;
and
are the vertical speeds of the front and rear wheels; and
and
are the inertia and rotational acceleration of the robot.
Figure 6 shows a schematic of the forces affecting the robot and their application points. As can be deduced from Equations (
24)–(
27), the application points of the tangential (
) and normal
) tire forces can vary within the contact patch. The case shown in
Figure 6 represents the AWD robot where both the front and rear axles provide tractive force. The height of the ground behind the tires represents the height of the ruts left by the tires.
The harrow model was designed based on the approach originally developed by Söhne [
81]. In the model, the longitudinal and vertical tine forces depend on the tine dimensions, speed, depth, and angle in addition to the cohesion, density, and internal friction of the soil. The relevant geometry of the soil deformation used in the calculations is presented in
Figure 7, where
represents the shear angle of the soil and
represents the angle of the tine.
The longitudinal and vertical forces (
and
) produced by a tine are calculated using the following Equations [
89]:
where
b and
are the width and speed of the tine,
and
are the cohesion and acceleration forces the soil exerts on the tine,
is the coefficient of friction between the tine and the soil, and
Z is a support variable. For the harrow dynamics calculations, it is assumed that the forces
and
apply to the center point of the tine. The longitudinal dynamics of the harrow is described as
where
represents the longitudinal force produced by a harrow tire,
is the longitudinal force exerted on the harrow by the hook,
and
represent the total longitudinal force produced by the front and rear row of tines,
is the mass of the harrow, and
is the longitudinal acceleration of the CoG of the harrow. The value of
is equal and opposite to
. The vertical dynamics of the harrow is described as
where
represents the vertical force produced by a harrow tire,
is the vertical force exerted on the harrow by the hook,
and
represent the total vertical force produced by the front and back row of tines, and
is the vertical acceleration of the CoG of the harrow. The value of
is equal and opposite to
. The rotational dynamics of the harrow is described as
where
,
,
, and
represent the longitudinal and vertical forces of a harrow tire caused by the shear (
) and normal (
) stresses;
,
,
, and
are the longitudinal and vertical coordinates of the shear and normal stress resultant forces;
,
,
, and
are the longitudinal and vertical coordinates of the tines in the front and back rows;
and
are the longitudinal and vertical coordinates of the CoG of the harrow;
is the vertical speed of the harrow wheels; and
and
are the inertia and rotational acceleration of the harrow.
Figure 8 shows a schematic of the forces affecting the harrow and their application points. The lighter ground color behind the tires represents the height of the ruts left by the harrow tires.
3.4. Simulation Parameters
The parameters of the robot are presented in
Table 1. As was mentioned earlier, the robot was modeled to be the size of a typical ATV. The dimensions of the chassis and tires were made to match those of an ATV the Faculty of Agriculture and Forestry at the University of Helsinki is in possession of, the FBH version of the Blade ATV by Taiwan Golden Bee Co., Ltd. [
90]. The mass of the chassis without the engine and drivetrain was measured to be approximately 286 kg. The height of the center of gravity was defined as 0.6 m. The weight distributions of the five different models are shown in
Table 2.
The efficiency map for a permanent magnet motor, shown in
Figure 9, was acquired from the Autonomie software, which contains a vast library of powertrain components [
91]. The map combines the efficiency of both the electric motor and the inverter. The map shown in the figure is for the AWD robot, which uses two motors powering each axle independently. For the RWD and FWD models, the motor torque was doubled so that the RWD, FWD, and AWD versions would all feature the same total maximum power and torque. The gear ratio of the final drive was selected such that the theoretical maximum speed of the robot would be 20 km/h. With the chosen gear ratio, the robot would operate at the region of maximum motor efficiency when driving at typical operation speeds of 5 to 10 km/h.
For the battery pack, data for the dependence of the internal resistance and the open-circuit voltage on the SOC were acquired from a previous study examining nickel manganese cobalt (NMC) battery cells, more specifically NMC111 [
92]. The battery pack was configured to operate at 48 V and have an energy capacity of approximately 17 kWh.
The harrow parameters are presented in
Table 3. The harrow was modeled as a typical compact tine harrow with a total of ten tines. The back row of tines is mounted at the back of the harrow and the front row is mounted 0.8 m ahead of the back row. The tires are mounted 0.5 m in front of the back end of the harrow.
The parameters for the two soils used in the simulations are presented in
Table 4. The parameters were acquired from previous works by Wong as well as Senatore and Sandu [
38,
93]. The parameters for slip sinkage and multipass calculations, shown in
Table 5, were acquired from the aforementioned research by Senatore and Sandu [
38].
3.5. Control
The control system of the robot model utilizes a proportional speed controller. The aim with the choice of controller type was to facilitate a fair comparison between the different models. Thus, a simple proportional controller was chosen. As is described later, a traction control (TC) system was also added to ensure that none of the models would be disadvantaged by excess wheelspin causing significant energy losses. The proportional controller outputs a value from −1 to 1. The control value is then multiplied by the maximum torque of the motor at the current rotational speed to acquire the torque command for the electric motor based on the difference between the reference speed and the actual speed of the robot. During deceleration maneuvers, the torque demand may be negative, in which case the motor slows the robot down with regenerative braking. As has been discussed in previous works, there is a distinct lack of well-established reference operation cycles for wheeled agricultural machines [
94,
95]. Thus, a basic reference operation cycle was defined for the simulations to facilitate a fair comparison between the different robot models. The reference cycle describes the target speed of the robot as a function of distance traveled. Before the actual cycle begins, there is first a precycle portion where the robot moves for three meters. The robot initially spawns and sinks into the ground, and thus there are no ruts yet formed by the wheels. The purpose of the precycle maneuver was to ensure that the rear wheels of the robot would be in the ruts left by the front wheels when the actual cycle begins. The reference cycle starts with an acceleration maneuver at 0.1 g, which continues until the cycle reaches the defined maximum speed. After the robot has traversed 100 m on the operation cycle, a deceleration maneuver at 0.15 g is then carried out until the robot has stopped.
Three different tine depths were simulated with each of the five models on both of the soils: 5, 10, and 15 cm. The tine vertical positions were defined such that the actual average tine depths during the simulations would be approximately at the target depth when the harrow tires would sink into the soil, thus lowering the tines from their initial depth. In the context of this research, the term “operation cycle” is considered to encompass both the speed of the robot as well as the tine configuration.
Four different target speeds were defined: 4, 6, 8, and 10 km/h. Thus, simulation results from a total of 120 different combinations of five robot models, two soil types, three tine depths, and four target speeds were acquired. Examples of the operation cycles are shown in
Figure 10, which shows cycles completed by the AWD model with four different target speeds on clayey loam. For each combination of robot model, soil type, tine depth, and target speed, the gain value of the proportional controller was iterated on to ensure the robot would provide sufficient torque to the wheels to reach the target speed. However, with some combinations, the robot was unable to reach the target speed even with full throttle applied, as is shown in the Results and Discussion Section.
The control system features a simple TC algorithm that prevents the driven wheels from slipping excessively. The TC starts to limit the torque demand provided by the controller when the slip ratio of the driven wheels gets near the ratio at which maximum longitudinal tire force is achieved. The behavior of the longitudinal tire force as a function of slip is heavily dependent on the soil properties. Thus, the threshold values at which the TC would start to intervene were defined separately for each soil type. Utilizing the tire–soil model described in
Section 3.2, the threshold values were determined for each soil and model combination by observing the optimal slip with an approximated 10% vertical load transfer to the rear axle. The determination of the threshold values revealed the optimal slip to vary significantly between sandy loam and clayey loam. On sandy loam, the optimal slip ratio was found to be between 0.25 and 0.35 depending on the robot model, while on clayey loam, the range was from 0.10 to 0.18.
In addition, an algorithm controlling the distribution of motor power between front and rear axles was devised for the AWD robot. At lower torque levels, the algorithm provides more power for the rear axle than the front axle to make use of the greater traction available at the rear. As the control value output by the proportional controller increases, the power difference between the axles reduces. At maximum throttle (control value of 1), both motors provide maximum torque. This type of power distribution control was found to reduce the energy consumption of the AWD robot. The power distribution is performed with the following equations:
where
and
are the new weighted control signals for the rear and front axles,
is the original control value output by the proportional controller, and
is the power distribution factor. With the weighted control values for the axles formulated in this way, the average of the rear and front control values is still equal to the original control value. As can be seen in the equations, a higher power distribution factor will shift the power distribution further to the rear axle. By iterating through different power distribution factors in varying conditions, it was determined that a factor of 1.4 would provide the overall lowest energy consumption.