Post on 24-Feb-2023
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
afabrizio.mazzetto@unibz.it, bmarco.bietresato@unibz.it (corresp. author), crenato.vidoni@unibz.it
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
All rights reserved. No part of contents of this paper may be reproduced or transmitted in any form or by any means without the written permission of TTP,www.ttp.net. (ID: 79.20.191.57-13/09/13,14:20:25)
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.
Applied Mechanics and Materials Vol. 394 549
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.
550 Mechatronics, Applied Mechanics and Energy Engineering
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.
Applied Mechanics and Materials Vol. 394 551
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.
552 Mechatronics, Applied Mechanics and Energy Engineering
References
[1] A. Guarnieri and A. Fabbri, in: Convegno Nazionale AIIA - La sicurezza delle macchine
agricole e degli impianti agro-industriali, 2002, pp. 1–21.
[2] A. L. Guzzomi, V. Rondelli, A. Guarnieri, G. Molari, and P. G. Molari: Biosystems
Engineering, vol. 104, no. 3, pp. 318–323, Nov. 2009.
[3] OECD, “Code 6 - OECD Standard Code for the Official Testing of Front Mounted Rollover
Protective Structures on Narrow Track Wheeled Agricultural and Forestry Tractors.”
Organisation for the Economic Co-operation and Development, Paris, France, 1990.
[4] A. Pazooki, S. Rakheja, and D. Cao: Mechanical Systems and Signal Processing, vol. 28, pp.
679–695, Apr. 2012.
[5] I. Gravalos, T. Gialamas, S. Loutridis, D. Moshou, D. Kateris, P. Xyradakis, and Z.
Tsiropoulos: Journal of Terramechanics, vol. 48, no. 4, pp. 319–323, Aug. 2011.
[6] B. Ji-hua, L. Jin-liang, and Y. Yan, in: 2011 International Conference on Electric Information
and Control Engineering, 2011, pp. 2294–2298.
[7] M. Karkee and B. L. Steward: Biosystems Engineering, vol. 106, no. 4, pp. 352–366, Aug.
2010.
[8] S. Popescu and N. Sutru, in: Engineering for Rural Development - International Scientific Con,
2009, pp. 165–170.
[9] A. L. Guzzomi: Biosystems Engineering, vol. 113, no. 1, pp. 65–75, Sep. 2012.
[10] I. Ahmadi: Journal of Terramechanics, vol. 48, no. 5, pp. 339–346, Oct. 2011.
[11] F. Mazzetto, R. Gallo, R. Vidoni, and C. Bisaglia, in: International Conference of Agricultural
Engineering CIGR-Ageng, 2012.
[12] F. Mazzetto, R. Gallo, R. Vidoni, C. Bisaglia, and A. Calcante, in: International Conference
RAGUSA SHWA 2012 - “Safety Health and Welfare in Agriculture and in Agro-food Systems”,
2012, pp. 243–250.
[13] M. G. Yisa, H. Terao, N. Noguchi, and M. Kubota: Journal of Terramechanics, vol. 35, no. 1,
pp. 1–19, Jan. 1998.
[14] J. Liu and P. D. Ayers, in: 1999 Summer Conference of the NATIONAL INSTITUTE FOR
FARM SAFETY, 1999, no. 99, p. Paper No. 99–11.
[15] R. Huang, J. Zhan, and J. Wu, in: Proceedings of the FISITA 2012 World Automotive Congress,
2012, vol. 198, pp. 441–448.
Applied Mechanics and Materials Vol. 394 553
[16] H. Silleli, M. a. Dayıoğlu, A. Gültekin, K. Ekmekçi, M. a. Yıldız, E. Akay, and G. Saranlı:
Biosystems Engineering, vol. 97, no. 2, pp. 153–161, Jun. 2007.
[17] B. Mashadi and H. Nasrolahi: Journal of Terramechanics, vol. 46, no. 6, pp. 299–311, Dec.
2009.
[18] G. Genta, Meccanica dell’autoveicolo - Collana di Progettazione e Costruzione di Macchine.
Torino, Italy: Levrotto & Bella, 2000, p. 568.
554 Mechatronics, Applied Mechanics and Energy Engineering
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
Real-Time Safety Devices 10.4028/www.scientific.net/AMM.394.546