Development of a dynamic stability simulator for articulated and conventional Tractors useful for...

Development of a Dynamic Stability Simulator for Articulated and Conventional Tractors Useful for Real-time Safety Devices

Fabrizio Mazzetto1,a, Marco Bietresato1,b and Renato Vidoni1,c 1Free University of Bozen-Bolzano, Faculty of Science and Technology – Fa.S.T.,

piazza Università 5, P.O. Box 276, I-39100 Bolzano (BZ), Italy

[email protected], [email protected] (corresp. author), [email protected]

Keywords: Agricultural Tractors, Overturning, Dynamic stability, Safety index, Matlab® simulation.

Abstract. The safety of agricultural tractors’ drivers is a very actual topic, especially when tractors

operate on side slopes, such as in terraced vineyards. This work approaches the problem of

articulated tractors’ stability by modelling, simulating and quantifying the safety of the driver with

respect to both roll and pitch overturns. First of all, an articulated tractor has been modelled and

simplified; after that, a stability index has been defined and calculated in several simulated slope

conditions when the tractor travels along a circular trajectory. Then, the obtained results have been

compared with respect to a conventional tractor. This work is a preliminary study for a tilting test

platform for real vehicles, capable to reproduce real field conditions (slope, obstacles, roughness).

Finally, some directives on how exploiting the obtained results for real-time safety devices have

been formulated.

Introduction

General considerations and problem description. The study of the dynamic behaviour of off-

road vehicles, in particular of agricultural tractors, is a very actual research topic in the field of

engineering, since it allows having a greater awareness of how design choices influence the

stability, the handling and the operational limits of a vehicle, and hence the safety level of the

occupants [1].

One of the most interesting topics in studying the rollover stability of agricultural (or “farm”)

tractors operating on steep hillsides is the capability for the analyst to predict, prevent or limit the

damage caused by a possible overturning. From a general point of view, these situations can be

described through analytical equations relating the vehicle attitude to the incipient overturning

condition. This problem can be addressed in two different ways: (i) energetic or (ii) Newtonian,

more widespread than the former.

In [2], for example, the energetic approach allowed to analyse the different initial conditions for

rollover of tractors and to evaluate the energy available at rollover start. Thanks to this study it has

been possible to realize that the available energy may not be a linear function of tractor mass, as

assumed by the international testing procedure, and to define the application range of the energy

formula used in Code 6 [3].

State of the art. Most of the works, however, use a Newtonian approach and some of them

consider also a three-dimensional tire–terrain interaction model [4] or the effects on the tractor

stability of the rear track width and additional weight placed on the wheels when driving on side

slopes [5]. Trailers or agricultural implements attached to the tractor change substantially the

behaviour of the whole vehicle and could easily lead it to critical conditions, thus they deserve

deeper investigations. For example, in [6] a linear dynamics model of tractor/full trailer with six

degrees-of-freedom is presented and used to evidence critical situations occurring when avoiding an

obstacle (rearward amplification ratio phenomenon).

The model of the lateral dynamics of a tractor with a single-axle grain cart was studied in [7]; a

sensitivity analysis allowed to identify the effect of uncertainty/variation of some parameters on

system responses. In [8] the longitudinal stability of the tractor-front-end loader system and of

tractor-forklift system was studied in the most difficult work situations: braking and moving the

load on the forks while descending on a slope. The same analytical approach is used in [9], where a

Applied Mechanics and Materials Vol. 394 (2013) pp 546-553Online available since 2013/Sep/03 at www.scientific.net© (2013) Trans Tech Publications, Switzerlanddoi:10.4028/www.scientific.net/AMM.394.546

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geometrical model for predicting under quasi-static conditions the rollover initiation angle of

conventional farm tractors fitted with front-axle pivot is presented. The model uses a kineto-static

approach based on two rigid bodies: the front axle and wheels (anterior body) and the remaining

machine and rear wheels (posterior body). Conventional tractors have been studied also in [10]

through a dynamic model capable of investigating the effects of forward speed, ground slope and

wheel–ground friction coefficient on lateral stability at the presence of position disturbances.

The present work uses an approach similar to [9] but deals with articulated farm tractors, i.e.

wheeled tractors having a central joint used for steering [11], [12]. The rollover angle is calculated

in a quasi-static condition and the attitude of the tractor in every slope condition is quantified by a

stability index, in a way similar to [13] and [14]. This index synthetizes possible overturn conditions

(e.g., roll and pitch angles) in a single number and it could be very useful if used as input signal for

many real-time active safety devices acting on several systems, e.g.: braking system, limited-slip

differentials [15], variable-geometry roll-over structures [16], self-levelling cab system [17].

Aim of the research. The aim of this work was studying a dynamic stability model to simulate

the operation of narrow-track wheeled articulated farm tractors. Since tractors of this type are

mainly used in vineyards or orchards and since these cultivations can also extend on sloping

ground, the simulator is intended to lay the basis for setting up and preliminary testing real-time

safety devices for agricultural tractors. This work can be also configured as a preliminary study of

an innovative test platform for real vehicles. The dimension of this platform will allow a tractor to

travel on it along a complete circular trajectory; moreover, the experimenter will have the chance to

vary the overall slope and the surface conditions (different roughness) of the platform but also to

insert in the tractor’s trajectory some small obstacles/rapid changes of the local slope, thus

simulating the critical points a tractor can encounter during field operations.

Materials and Methods

Kinematic models of a tractor. In order to evaluate the stability and the behavior of an

articulated farm tractor in different working and slope conditions, a simplified model has been

developed. The vehicle-system has been evaluated in its main geometrical and mechanical

parameters (Fig. 1a and Table 1) and the steering articulation has been considered as a revolute

kinematic pair. In this first work, no pivot on the front axis or independent suspensions have been

considered, thus allowing to consider the tractor footprint given by the four wheels in contact with

the surface. The revolute joint, which is in charge to simulate the steering system, is the active joint

of the tractor and it is limited by the maximum steering angle, max_θ.

Moreover, a conventional configuration, i.e. with four wheels and rigid chassis system (Fig. 1b),

has also been considered to allow a suitable stability comparison and evaluation.

a) b)

Fig. 1 – Tractor geometrical parameters. a) Articulated Tractor; b) Conventional Tractor.

Applied Mechanics and Materials Vol. 394 547

Table 1 – Nomenclature.

wb_f Forward wheelbase width

wb_b Backward wheelbase width

wb_b_d Distance joint-backward wheelbase

wb_f_d Distance joint-forward wheelbase

p_b Backward tread width

p_f Forward tread width

COG_b Backward Centre Of Gravity position

COG_f Forward Centre Of Gravity position

COG Centre Of Gravity

l Distance between the wheelbases

max_θ Maximum steering angle

As regards the steering, the overall model has been developed by means of the classical steering

kinematics [18], thus neglecting the friction contribute and the aerodynamic friction. In such this

way, it is possible to evaluate the position of the wheels in any steering condition, i.e. along circles

with different radii, fundamental elements for the stability evaluation.

Wheeled system stability. The conventional agricultural tractor has been considered as a rigid

central body with Centre Of Gravity (COG), having a mass m and four contact points with the

ground surface. The stability of a tractor can be classified at least into a longitudinal (pitch) and a

lateral (roll) turning stability, both related to the tractor stability baseline, i.e. the polygonal line that

connects all the wheels of the tractor as the wheels set on level ground. Generally speaking, the

system can be considered stable if the projection of the COG of the system on the supporting plane

surface is inside the stability baseline. Thanks to this assumption, both the longitudinal and the

lateral stability can be defined.

In the case of the articulated tractor, the stability baseline can be defined in the same way as for

the conventional tractor; the system COG has instead to be computed at each step/configuration

since it is directly related to the steering angle. If the Centres Of Gravity of the forward and

backward parts are considered (COG_f, COG_b), the system COG can be computed.

As the tractor usually moves at reasonably low speeds during its normal operations (lower than

1.5 m s-1

), the dynamic stability can be treated with a quasi-static approach, i.e. the inertial terms

can be considered negligible, without the risk of invalidating the results. This means that the only

force to be considered is the weight (Fp), which is distributed on the four wheels (Fig. 2a) in

correspondence to the four ground-tractor points of contact.

a) b)

Fig. 2 – Tractor: a) baseline (ABCD) and weight force; b) stability index computation.

548 Mechatronics, Applied Mechanics and Energy Engineering

The weight force is applied on the system COG, it is Fp = -mgz, directed along the absolute

vertical axis. Hence, with the aim of evaluating the stability of the system, it has been chosen to

simulate the tractor while travelling along a circumference with radius r and each time in different

slope conditions, i.e. turning on a plane with different values of the slope (referred to as α).

Stability Indexes. Following the approach in [14], roll and pitch stability indexes have been

defined and implemented. As previously stated, the main idea is to design a simple method for

evaluating and monitoring the stability of articulated tractors operating in slope conditions, in order

to design real-time active safety systems.

The generic (percentage) stability index (SI) is:

�� = �1 − �� ∙ 100 (1)

where X is the state variable for evaluating the stability and Xcri its critical value.

In order to compute the SI, the following procedure has been followed.

First of all the following data have been computed:

• current position of the four wheels;

• projection of the COG on the inclined plane (P in Fig. 2b);

• position of the geometric centre of the footprint (G) with respect to the absolute reference

system;

• spatial definition of the tangential (u2) and orthogonal (u1) unit vectors with respect to the

system path.

With this data available, the indexes are calculated by evaluating the distance of the projection of

the tractor COG on the running plane to the symmetry line (X), with respect to the zero stability

condition (Xcri), i.e. when the COG projection falls on the baseline border. In Fig. 2b these distances

are shown.

For the roll stability index, first of all the COG projection point P is evaluated in its u1

coordinate to select the nearer stability baseline side, e.g. BC in the figure. After that, the line

orthogonal to BC and passing through the point S, intersection between this line and the line

defined by the unit vector u2, is computed. Thus, given P(xp,yp) and ax+by+c=0, i.e. the equation

for BC, the parameters a´, b´ and c´ of the orthogonal line become:

� �´ = ��´ = −��´ = −�� + �� (2)

while the coordinates of S result:

� �� = 0�� = − �´�´ (3)

By computing the coefficients and the coordinates of S, it is possible to evaluate the SP and SQ

distances, i.e. the X and Xcri quantities for the roll stability index:

�� = �1 − �� ∙ 100 (4)

The SIpitch is defined and computed in a similar way.


Simulator. The kinematics of the tractors together with the computation of the COG and of the

stability indexes have been implemented in a simulator developed in Matlab®

(The Mathworks,

Natick, Massachusetts, USA), in order to evaluate and compare the two different tractor

architectures and to lay the basis for future safety applications.

In Table 2 the main parameters for the two chosen architectures are reported. As can be appreciated,

the total weight of the two tractors is the same, as well the maximum distance between the two

wheelbases.

Table 2 – Main geometrical and dynamic parameters. Conventional Tractor Articulated Tractor

COG coordinates [0, 0.80, 0.60] m COG coordinates (f and b) [0, 0, 0.35] m

l 1.20 m l (distance wb vs. joint) 0.60 m

wb_f 0.60 m wb_f 0.70 m

wb_b 0.80 m wb_b 0.70 m

p_b 0.20 m p_b 0.20 m

p_f 0.26 m p_f 0.20 m

mass 950 kg mass fw 570 kg

mass bw 380 kg

Fig. 3 represents a visual output of the Matlab®

simulator and shows the different positions of the

stability baseline of the conventional tractor along a circular trajectory lying on a sloped plane. The

φ angle is the angular coordinate representing the current position of the vehicle along the

circumference. The tractor travels anticlockwise along the circumference, starting from the point

with maximum x coordinate.

Fig. 3 – Visual output of the simulator.

Results

The simulations have been carried out for different circle radii and slopes.

Fig. 4 and Fig. 5 show the SIroll for the two tractors when describing a circle radius of 5 m, i.e.

the φ angle that spans the entire circumference, and for slope values ranging from 0° to 40°.

Looking at the results (Table 4), it can be easily appreciated how, in a quasi-static configuration, the

roll stability index for an articulated tractor has better values than a conventional tractor.

Moreover, the adopted approach allows a fast assessment of the system stability and, if proper

inertial measurement devices are installed on a tractor, it can be directly exploited for (i) a real-time

computation of the system stability degree and (ii) the control and actuation of some active safety

devices.


Table 4 – Values of the SIroll for some slopes and in the most critical angular positions along

the circumference.

αααα φ=90° φ=270°

Conventional tr. Articulated tr. Absolute diff. Conventional tr. Articulated tr. Absolute diff.

10° 83% 89% +6 79% 83% +4

20° 62% 74% +12 59% 67% +8

30° 39% 57% +18 34% 51% +17

40° 11% 36% +25 6% 30% +24

a)

b)

Fig.4 – Roll stability index as a function of the slope and the tractor’s position on the

circumference.

a) Conventional Tractor; b) Articulated Tractor.


a) b)

Fig.5 – Roll stability index for different slope values. a) Conventional Tractor; b) Articulated

Tractor.

Conclusions and future work

In this work, the stability and human safety of narrow-track wheeled articulated tractors have

been evaluated. After having modeled both a conventional tractor and an articulated tractor from a

kinematic point of view, the vehicle dynamics has been modeled through a quasi-static approach.

Then, a stability index both for the roll and pitch axis has been defined and the overall algorithm

implemented in a Matlab®

simulator. The results show the goodness of the articulated architecture

for tractors operating on hillsides (articulated tractors’ SIroll is greater than conventional tractors’

SIroll of up to 25 percentage points) and encourage the exploitation, in a future work, of this

simplified model for its implementation in real-time safety control systems related to the vehicle

stability.

Moreover, future activities will interest some upgrades of the simulator: (i) the inclusion in the

stability index of a speed-dependent term, to considerate also the inertial phenomena occurring

when transporting fluids in a tank connected to the tractor (i.e., the equipment used for pesticide

treatments), (ii) the study of a possible additional passive degree-of-freedom on the steering joint of

the articulated tractor and its effect on the overall stability of the vehicle. Afterwards, the simulator

will be used to orient the design of an articulated tractor prototype with the aim of reaching better

stability performances on sloping lands than conventional tractors.

Acknowledgements

The authors wish to thanks eng. Pietro Alfier for his help in the simulator implementation.


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Mechatronics, Applied Mechanics and Energy Engineering 10.4028/www.scientific.net/AMM.394 Development of a Dynamic Stability Simulator for Articulated and Conventionaltractors Useful for

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