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Methodology for determining real time safety margin in a road vehicle
Transportation Research Procedia 00 (2018) 000–000 Available online at www.sciencedirect.com www.elsevier.com/locate/procedia
ScienceDirect
Available online at www.sciencedirect.com
Transportation Research Procedia 00 (2018) 000–000
www.elsevier.com/locate/procedia ScienceDirect XIII Conference on Transport Engineering, CIT2018 Transportation Research Procedia 33 (2018) 331–338
www.elsevier.com/locate/procedia Methodology for determining real time safety margin in a road vehicleEngineering, CIT2018 XIII Conference on Transport
a
b*
c
Marta Alonso , Daniel A. Mántaras , Pablo Luque Methodology for determining real time safety margin in a road Edificio Departamental Oeste - módulo 5. Escuela Politécnica de Ingeniería de Gijón , Gijón, Asturias, España. vehicle a,b,c
Marta Alonsoa, Daniel A. Mántarasb*, Pablo Luquec
Abstract a,b,c
Edificio Departamental Oeste - módulo 5. Escuela de Ingeniería Gijón, Asturias, España. There are multiple factors that can compromise safetyPolitécnica during driving suchdeasGijón the , tire-road friction coefficient, an inadequate speed, or an aggressive driving by the driver, leading to dangerous situations for all users of the way.
It is difficult to measure quantitatively the level of safety during driving and the change in it, depending, at each Abstract instant of time, of the situations that occur, changes in the adherence by a puddle of oil, soil wet from rain, sudden There are multiple factors that canchange compromise safety duringavailable. driving such as the tire-road friction coefficient, an accelerations and brakings, which the margin of safety inadequate speed, or an aggressive driving by the driver, leading to dangerous situations for all users of the way. In this paper presents a methodology that allows to determine the level of safety with which it is circulating through It isuse difficult measure regions quantitatively thethe level of safety duringofdriving andThe the validation change in of it, the depending, at each the of 𝑟𝑟̇- to β stability used for predictive control vehicles. methodology is instant of of thevirtual situations occur, the adherence a puddledynamic of oil, soil wet fromsoftware rain, sudden carried outtime, through tests that carried outchanges with theinMSC Adams® by multibody simulation that accelerations and brakings, which change the margin safety available. allows reproducing the behaviour of the vehicle and of knowing the values of the parameters that are necessary, such as longitudinal and lateral accelerations, yaw rate, the slips angles of the wheels, among others. In this paper presents a methodology that allows to determine the level of safety with which it is circulating through the use of 𝑟𝑟̇- β stability regions used for the predictive control of vehicles. The validation of the methodology is carried out through virtual tests carried out with the MSC Adams® multibody dynamic simulation software that © 2018 The Authors. Published by Elsevier Ltd. allows reproducing the behaviour of the vehicle and knowing the values of the parameters that are necessary, such as longitudinal lateral accelerations, yaw rate, the slips angles of the wheels, among others. ) This is an open and access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/ © 2018 Theand Authors. Publishedunder by Elsevier Ltd. Selection peer-review responsibility of the scientific committee of the XIII Conference on Transport This is an openCIT2018. access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) Engineering, © 2018 The Published by Elsevier of Ltd. Selection and Authors. peer-review under responsibility the scientific committee of the XIII Conference on Transport Engineering, CIT2018. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) Keywords: road safety, safety levels, vehicles, stability boundaries
Selection and peer-review under responsibility of the scientific committee of the XIII Conference on Transport Engineering, CIT2018. *Corresponding author. Tel.: +34 985181910 E-mail address: [email protected] Keywords: road safety, safety levels, vehicles, stability boundaries *Corresponding author. Tel.: +34 985181910 2352-1465 © [email protected] The Authors. Published by Elsevier Ltd. E-mail address: This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) Selection and peer-review under responsibility of the scientific committee of the XIII Conference on Transport Engineering, CIT2018. 2352-1465 © 2018 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) Selection and peer-review under responsibility of the scientific committee of the XIII Conference on Transport Engineering, CIT2018. 2352-1465 2018 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) Selection and peer-review under responsibility of the scientific committee of the XIII Conference on Transport Engineering, CIT2018. 10.1016/j.trpro.2018.10.110
Marta Alonso et al. / Transportation Research Procedia 33 (2018) 331–338 Author name / Transportation Research Procedia 00 (2018) 000–000
332 2
1.
Introduction
From the road safety point of view, the driving habits or the type of driving increases or decreases the likelihood of suffer an accident, and the response you have to a sudden and compromised situation will determine the magnitude of the accident. Hence, the importance of monitoring and knowing the safety limits at every moment. These limits are constantly changing, depending on the road conditions (the pavement, external agents such as an oil stain or the weather itself), the tire (its own wear) and the type of driving that takes place (Luque and Mántaras, 2007), Figure 1.
Figure 1. Evolution of the driving safety demand and representation of the safety margins available at each instant of time.
Several authors (Bobier, 2012, Beal, 2013, Bobier and Gerdes, 2013, Erlien, 2015) identify the safety limits of vehicles according to the so-called Stability Boundaries. These are obtained from the parameters 𝑟𝑟̇ − 𝛽𝛽 (yaw rateslip angle) and are applied for the development of stability control systems ESP, Figure 2. These stability boundaries depend on the load configuration, the speed of movement and the tire-road friction coefficient. So, with its use, there are several parameters that are taken into account, and that significantly affect the dynamic behaviour of the vehicle. In the present work the definition of the safety limits is developed and, based on the simulation with virtual models, the reliability of said region is analysed as a reference for the estimation of the security in circulation.
Figure 2. Stability boundaries defined from the limit values of the yaw rate and the vehicle slip angle.
2.
Methodology
2.1 Stability boundaries calculation Driving through a curve, considering the absence of traction and braking, the tire develops forces perpendicular to its central plane Figure 3.
Martaname Alonso et al. / Transportation 33000–000 (2018) 331–338 Author / Transportation ResearchResearch ProcediaProcedia 00 (2018)
3333
Figure 3. Representation of the forces due to the turn. Fy is the lateral force per tire and r the yaw rate at the center of gravity
Analysing the cornering in a quasi-static situation (Wong, 1978), the angular acceleration (yaw acceleration) must satisfy the following equation: .
r=
L1·( Fyfr + Fyfl ) − L2 ·( Fyrr + Fyrl ) I zz
(1)
Where 𝑟𝑟̇ yaw acceleration, Fyf front axle lateral force, Fyr rear axle lateral force, L1 distance between front axle and center of gravity, L2 distance between rear axle and center of gravity and Izz inertia moment in vertical axle. From a balance of forces in the vertical axis, it is arrive at the following equation:
Fyf =
L2 ·Fyr L1
(2)
One of the limits of the stability area is obtained based on the maximum lateral force in one of the axes. The maximum lateral force on the axle depends on the maximum lateral force on each wheel; this is function of the vertical load on the tire, the tire-road friction coefficient and the load transfer. Below, it is shown the lateral forces in the tires, obtained from a Tire Test, performed with the MSC Adams® software for each load configuration and an adherence of 0.6, Figure 4. The process is repeated for each value of adherence considered.
Figure 4. Tire Test performed with MSC Adams® for the two load configurations and an adhesion of 0.6
When expression (3) is met, saturation of the front axle has been reached, so it is this axle that defines the limits of the stability area.
Fyf max <
L2 ·Fyr max L1
(3)
Otherwise, it is the rear axle that reaches the saturation in the first place and therefore, which defines the limits of
Marta Alonso et al. / Transportation Research Procedia 33 (2018) 331–338 Author name / Transportation Research Procedia 00 (2018) 000–000
334 4
the stability area. In that case, the following equation (4) is fulfilled:
Fyf max ≥
L2 ·Fyr max L1
(4)
Considering the above, the yaw rate limit is calculated according to the following equations:
Fyr max ·(1 +
L2 ) L1
m·vx
Fyf max ≥
L2 ·Fyr max L1
rb =
(5)
Fyf max ·(1 + m·vx
L1 ) L2
Fyf max <
L2 ·Fyr max L1
The other limit of the stability area, limit due to the maximum slip angle, uses a simplified vehicle model, known as bicycle model with two degrees of freedom (dof), Figure 5.
Figure 5. The bicycle model parameters representation
The most critical situation for the lateral stability of the vehicle occurs when the slip limit value on the rear axle is reached, since the vehicle is directionally unstable. This occurs when the outer wheels reach the maximum lateral force, the slip angle corresponding to this situation is as follows:
= b b α r lim +
L2 ·rb vx
(6)
Under the simplification of linear model, the equations that delimit the stability areas are the following:
m·L1 1− ·vx2 2· L · L · C αr 2 b ss = m L1·Cα f − L2 ·Cα r 2 ·vx 1 − 2 · Cα f ·Cα r 2·L
·L2 ·δ L
(7)
1 rss = m L1·Cα f − L2 ·Cα r 2 ·vx 1 − 2 · Cα f ·Cα r 2·L
·vx ·δ L
(8)
Marta Alonso et al. / Transportation Research Procedia 33 (2018) 331–338 Author name / Transportation Research Procedia 00 (2018) 000–000
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2.2. Virtual model To evaluate the safety area, a multibody bus model shown in Figure 6 is used and the MSC Adams® multibody dynamic simulation software (Prado, 2001), (Drugge and Magnus, 2014). The virtual model, with 29 degrees of freedom, reproduces a 45-seat bus (plus 3 crew) with pneumatic suspension of the front parallelogram type and a rigid multi-link rear axle. The tire used is a complex PAC2002 model, based on the Pacejka Magic Formula (Pacejka, 2002).
Figure 6. Virtual bus model used for Ramp Steer tests under full load
The inputs of the system are the speed and the steering wheel angle. A PID control is implemented to regulate the torque on the wheels of the rear axle according to the reference speed (Rajamani, 2012). In this way the driver's actions on the vehicle are reproduced in the same way as in the real system. Two variants are developed on this model that represent two load cases: running order and maximum laden (full load) masses, these being the limit cases.
Figure 7. Trajectory carried out by the vehicles, making turns both to the right and to the left (Case of full load (CC), speed of 75 km / h and tire adhesion / road 1) Table 1. Summary table with the tests carried out Load Case
Mass (kg)
Load Ratio (%)
Centre of gravity height
Running order
13542
33%-67%
1.08
Maximum laden mass
17887
35%-65%
1.3
Speed (km/h) 50 75 100 50 75 100
Friction coefficient 0.3 0.6 1 0.3 0.6 1
Marta Alonso et al. / Transportation Research Procedia 33 (2018) 331–338 Author name / Transportation Research Procedia 00 (2018) 000–000
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2.3. Training Test Ramp Steer tests are carried out, which consist of introducing to ramp-shaped address input, progressively without producing sudden changes in it. The speed of rotation will be the same in all cases, varying only the speed and the tire-road friction coefficient. Maneuvers will be carried out to the left and to the right, in order to take into account the possible asymmetry in the behavior of the vehicle. The load cases are simulated for the speeds of 50 km / h, 75 km / h and 100km / h and three tire-road friction values 1, 0, .6 and 0.3 are considered. In the following table, Table 1, a summary of the tests carried out is shown. The simulations have been carried out in accordance with ISO standards for heavy vehicles and buses, ISO 15037 - ISO 11026 - ISO 14793, until the imminent rollover condition is reached, this being the driving situation. 3.
Results
With the virtual tests carried out, the reliability of the stability region defined in point 2.1 is analyzed. For this, it is analyzed if the vehicle is dynamically stable when the relationship between the yaw rate and the slip angle of the vehicle reach one of the limits defined by the mentioned region and if the safety demand in those limits is similar for the different operating conditions.
Figure 8. Representation of the lateral forces against the slip angle of the tires obtained from the simulation for full load (CC), 50 km / h and an adhesion of 0.6
The evaluation of the stability of the vehicle in the limit situation is carried out by analyzing how the outer tires are demanded at that point. This analysis is based on comparing the slope of the lateral force-slip angle curve at that instant with the slope in the linear zone (Figure 8). In the first place, the so-called stability regions are obtained for each of the load, speed and adherence configurations. Next, the simulations are carried out until the limit situation defined by the stability region 𝑟𝑟̇ − 𝛽𝛽 is reached (Figure 9).
Figure 9. Example of stability region with r -̇ β simulation data. The represented case is for the configuration of load, order of march (OM) and a speed of 100km / h
Marta Alonso et al. / Transportation Research Procedia 33 (2018) 331–338 Author name / Transportation Research Procedia 00 (2018) 000–000
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These limit values are represented on the curve 𝐹𝐹𝑦𝑦 − 𝛼𝛼 of the tire and the value of the slope around that point is analyzed. In Figure 10, the 𝐹𝐹𝑦𝑦 − 𝛼𝛼 curves for the load configuration OM and a speed of 100 km / h are represented, on the curves the limit values of the region and the slope have been represented.
Figure 10. Location of the limits of the stability region in the tire curve. Slope m for adhesion 0.6
The value of the slope around that point (m) is compared with the theoretical slope in the linear zone (m0), the following equation is applied. %𝑀𝑀 =
𝑚𝑚·100
(9)
𝑚𝑚0
This calculated percentage allows to determine if the vehicle has already reached the saturation of the tire, or, on the contrary, it is still in a safe area and the degree to which the tire is demanded. In Table 2, the results obtained for the cases considered are shown. It can be seen that the percentage of the slope that you have is above 70%, regardless of the traffic conditions. This means that the stability region allows establishing limits as a reference for safety in circulation. These limits are stable, within each condition, and weighted to obtain a single reference. Table 2. Results obtained for each case Adherence
Load Configuration CC
0.3 OM CC 0.6 OM CC 1 OM
Velocity
m0
m
%M
50.0 75.0 100.0 50.0 75.0 100.0 50.0 75.0 100.0 50.0 75.0 100.0 50.0 75.0 100.0 50.0 75.0 100.0
3262.7 3240.8 3228.7 2521.3 2521.3 2521.3 3584.3 3626.3 3583.6 2828.2 2828.2 2828.2 3408.2 3408.2 3408.2 2875.0 2875.0 2875.0
2749.7 2683.5 1987.0 1985.1 1970.7 2728.9 2712.2 2647.3 2095.8 2054.7 2027.2 2524.0 2509.0 × 2477.5 2342.1
84.3 83.1 78.8 78.7 78.2 76.1 74.8 73.9 74.1 72.7 71.7 74.1 73.6 × 86.2 81.5
Marta Alonso et al. / Transportation Research Procedia 33 (2018) 331–338 Author name / Transportation Research Procedia 00 (2018) 000–000
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4.
Conclusions
The definition of an objective metric for the evaluation of safety in road traffic is an issue not resolved in the current state of the art. In the present work, the validity of the stability region, defined based on the relationship 𝑟𝑟̇ − 𝛽𝛽, is evaluated as a reference to establish the limits of safe driving in any operating condition. The evaluation is done through the simulation of a virtual vehicle model, analyzing the demand of the tire in the limits of the region. It has been found that the stability regions define limits that guarantee that the vehicle, if it does not exceed them, maintains a safe circulation. The level of tire demand is below 30% for any operating condition, which means that the regions are conservative. Finally, it is noted that the limits are stable, within each condition, and can be weighted to obtain a single reference.
References Beal, C.E. and Gerdes, J.C. Model Predictive Control for Vehicle Stabilization at the Limits of Handling. IEEE Transactions on Control Systems Technology. 2013; 21:1258–1269. doi:10.1109/TCST.2012.2200826 Bobier C.G. A phase portrait approach to vehicle stabilization and envelope control. Stanford University; 2012. Bobier, C.G. and Gerdes J.C. Staying within the nullcline boundary for vehicle envelope control using a sliding surface. Vehicle System Dynamics. 2013; 51:199–217. doi:10.1080/00423114.2012.720377 Drugge and Magnus J. (2010) Aerodynamic loads on buses due to crosswind gusts: extended analysis. Vehicle System Dynamics: International Journal of Vehicle Mechanics and Mobility, 48:S1, 287-297, DOI: 10.1080/00423111003739814 Erlien S.M. Shared vehicle control using safe driving envelopes for obstacle avoidance and stability. Stanford University; 2015. ISO 11026:2010. Heavy commercial vehicles and buses -- Test method for roll stability -- Closing-curve test ISO 14793:2011. Road vehicles -- Heavy commercial vehicles and buses -- Lateral transient response test methods ISO 15037-2. Road vehicles - Vehicle dynamics test methods - Part 2: General conditions for heavy vehicles and buses Luque, P. and Mántaras, D. (2007). Investigación de Accidentes de Tráfico. Manual de reconstrucción. Netbiblo. La Coruña. Pacejka, B.H. (2002). Tyre and Vehicle Dynamics. Butterworth-Heinemann, Oxford. Prado, M., Cunha, R., Neto and Á., Costa, A. et al. (2001). Bus Handling Analysis and Experimental Validation Using the Multibody System Technique. SAE Technical Paper 2001-01-3966, 2001, doi:10.4271/2001-01-3966 Rajamani, R. Vehicle Dynamics and Control. Springer Science & Business Media; 2012. ISBN 978-1-4614-1432-2 Wong J.Y. Theory of ground vehicles. John Wiley & Sons, Inc. Canada. 1978.
ScienceDirect
Available online at www.sciencedirect.com
Transportation Research Procedia 00 (2018) 000–000
www.elsevier.com/locate/procedia ScienceDirect XIII Conference on Transport Engineering, CIT2018 Transportation Research Procedia 33 (2018) 331–338
www.elsevier.com/locate/procedia Methodology for determining real time safety margin in a road vehicleEngineering, CIT2018 XIII Conference on Transport
a
b*
c
Marta Alonso , Daniel A. Mántaras , Pablo Luque Methodology for determining real time safety margin in a road Edificio Departamental Oeste - módulo 5. Escuela Politécnica de Ingeniería de Gijón , Gijón, Asturias, España. vehicle a,b,c
Marta Alonsoa, Daniel A. Mántarasb*, Pablo Luquec
Abstract a,b,c
Edificio Departamental Oeste - módulo 5. Escuela de Ingeniería Gijón, Asturias, España. There are multiple factors that can compromise safetyPolitécnica during driving suchdeasGijón the , tire-road friction coefficient, an inadequate speed, or an aggressive driving by the driver, leading to dangerous situations for all users of the way.
It is difficult to measure quantitatively the level of safety during driving and the change in it, depending, at each Abstract instant of time, of the situations that occur, changes in the adherence by a puddle of oil, soil wet from rain, sudden There are multiple factors that canchange compromise safety duringavailable. driving such as the tire-road friction coefficient, an accelerations and brakings, which the margin of safety inadequate speed, or an aggressive driving by the driver, leading to dangerous situations for all users of the way. In this paper presents a methodology that allows to determine the level of safety with which it is circulating through It isuse difficult measure regions quantitatively thethe level of safety duringofdriving andThe the validation change in of it, the depending, at each the of 𝑟𝑟̇- to β stability used for predictive control vehicles. methodology is instant of of thevirtual situations occur, the adherence a puddledynamic of oil, soil wet fromsoftware rain, sudden carried outtime, through tests that carried outchanges with theinMSC Adams® by multibody simulation that accelerations and brakings, which change the margin safety available. allows reproducing the behaviour of the vehicle and of knowing the values of the parameters that are necessary, such as longitudinal and lateral accelerations, yaw rate, the slips angles of the wheels, among others. In this paper presents a methodology that allows to determine the level of safety with which it is circulating through the use of 𝑟𝑟̇- β stability regions used for the predictive control of vehicles. The validation of the methodology is carried out through virtual tests carried out with the MSC Adams® multibody dynamic simulation software that © 2018 The Authors. Published by Elsevier Ltd. allows reproducing the behaviour of the vehicle and knowing the values of the parameters that are necessary, such as longitudinal lateral accelerations, yaw rate, the slips angles of the wheels, among others. ) This is an open and access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/ © 2018 Theand Authors. Publishedunder by Elsevier Ltd. Selection peer-review responsibility of the scientific committee of the XIII Conference on Transport This is an openCIT2018. access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) Engineering, © 2018 The Published by Elsevier of Ltd. Selection and Authors. peer-review under responsibility the scientific committee of the XIII Conference on Transport Engineering, CIT2018. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) Keywords: road safety, safety levels, vehicles, stability boundaries
Selection and peer-review under responsibility of the scientific committee of the XIII Conference on Transport Engineering, CIT2018. *Corresponding author. Tel.: +34 985181910 E-mail address: [email protected] Keywords: road safety, safety levels, vehicles, stability boundaries *Corresponding author. Tel.: +34 985181910 2352-1465 © [email protected] The Authors. Published by Elsevier Ltd. E-mail address: This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) Selection and peer-review under responsibility of the scientific committee of the XIII Conference on Transport Engineering, CIT2018. 2352-1465 © 2018 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) Selection and peer-review under responsibility of the scientific committee of the XIII Conference on Transport Engineering, CIT2018. 2352-1465 2018 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) Selection and peer-review under responsibility of the scientific committee of the XIII Conference on Transport Engineering, CIT2018. 10.1016/j.trpro.2018.10.110
Marta Alonso et al. / Transportation Research Procedia 33 (2018) 331–338 Author name / Transportation Research Procedia 00 (2018) 000–000
332 2
1.
Introduction
From the road safety point of view, the driving habits or the type of driving increases or decreases the likelihood of suffer an accident, and the response you have to a sudden and compromised situation will determine the magnitude of the accident. Hence, the importance of monitoring and knowing the safety limits at every moment. These limits are constantly changing, depending on the road conditions (the pavement, external agents such as an oil stain or the weather itself), the tire (its own wear) and the type of driving that takes place (Luque and Mántaras, 2007), Figure 1.
Figure 1. Evolution of the driving safety demand and representation of the safety margins available at each instant of time.
Several authors (Bobier, 2012, Beal, 2013, Bobier and Gerdes, 2013, Erlien, 2015) identify the safety limits of vehicles according to the so-called Stability Boundaries. These are obtained from the parameters 𝑟𝑟̇ − 𝛽𝛽 (yaw rateslip angle) and are applied for the development of stability control systems ESP, Figure 2. These stability boundaries depend on the load configuration, the speed of movement and the tire-road friction coefficient. So, with its use, there are several parameters that are taken into account, and that significantly affect the dynamic behaviour of the vehicle. In the present work the definition of the safety limits is developed and, based on the simulation with virtual models, the reliability of said region is analysed as a reference for the estimation of the security in circulation.
Figure 2. Stability boundaries defined from the limit values of the yaw rate and the vehicle slip angle.
2.
Methodology
2.1 Stability boundaries calculation Driving through a curve, considering the absence of traction and braking, the tire develops forces perpendicular to its central plane Figure 3.
Martaname Alonso et al. / Transportation 33000–000 (2018) 331–338 Author / Transportation ResearchResearch ProcediaProcedia 00 (2018)
3333
Figure 3. Representation of the forces due to the turn. Fy is the lateral force per tire and r the yaw rate at the center of gravity
Analysing the cornering in a quasi-static situation (Wong, 1978), the angular acceleration (yaw acceleration) must satisfy the following equation: .
r=
L1·( Fyfr + Fyfl ) − L2 ·( Fyrr + Fyrl ) I zz
(1)
Where 𝑟𝑟̇ yaw acceleration, Fyf front axle lateral force, Fyr rear axle lateral force, L1 distance between front axle and center of gravity, L2 distance between rear axle and center of gravity and Izz inertia moment in vertical axle. From a balance of forces in the vertical axis, it is arrive at the following equation:
Fyf =
L2 ·Fyr L1
(2)
One of the limits of the stability area is obtained based on the maximum lateral force in one of the axes. The maximum lateral force on the axle depends on the maximum lateral force on each wheel; this is function of the vertical load on the tire, the tire-road friction coefficient and the load transfer. Below, it is shown the lateral forces in the tires, obtained from a Tire Test, performed with the MSC Adams® software for each load configuration and an adherence of 0.6, Figure 4. The process is repeated for each value of adherence considered.
Figure 4. Tire Test performed with MSC Adams® for the two load configurations and an adhesion of 0.6
When expression (3) is met, saturation of the front axle has been reached, so it is this axle that defines the limits of the stability area.
Fyf max <
L2 ·Fyr max L1
(3)
Otherwise, it is the rear axle that reaches the saturation in the first place and therefore, which defines the limits of
Marta Alonso et al. / Transportation Research Procedia 33 (2018) 331–338 Author name / Transportation Research Procedia 00 (2018) 000–000
334 4
the stability area. In that case, the following equation (4) is fulfilled:
Fyf max ≥
L2 ·Fyr max L1
(4)
Considering the above, the yaw rate limit is calculated according to the following equations:
Fyr max ·(1 +
L2 ) L1
m·vx
Fyf max ≥
L2 ·Fyr max L1
rb =
(5)
Fyf max ·(1 + m·vx
L1 ) L2
Fyf max <
L2 ·Fyr max L1
The other limit of the stability area, limit due to the maximum slip angle, uses a simplified vehicle model, known as bicycle model with two degrees of freedom (dof), Figure 5.
Figure 5. The bicycle model parameters representation
The most critical situation for the lateral stability of the vehicle occurs when the slip limit value on the rear axle is reached, since the vehicle is directionally unstable. This occurs when the outer wheels reach the maximum lateral force, the slip angle corresponding to this situation is as follows:
= b b α r lim +
L2 ·rb vx
(6)
Under the simplification of linear model, the equations that delimit the stability areas are the following:
m·L1 1− ·vx2 2· L · L · C αr 2 b ss = m L1·Cα f − L2 ·Cα r 2 ·vx 1 − 2 · Cα f ·Cα r 2·L
·L2 ·δ L
(7)
1 rss = m L1·Cα f − L2 ·Cα r 2 ·vx 1 − 2 · Cα f ·Cα r 2·L
·vx ·δ L
(8)
Marta Alonso et al. / Transportation Research Procedia 33 (2018) 331–338 Author name / Transportation Research Procedia 00 (2018) 000–000
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2.2. Virtual model To evaluate the safety area, a multibody bus model shown in Figure 6 is used and the MSC Adams® multibody dynamic simulation software (Prado, 2001), (Drugge and Magnus, 2014). The virtual model, with 29 degrees of freedom, reproduces a 45-seat bus (plus 3 crew) with pneumatic suspension of the front parallelogram type and a rigid multi-link rear axle. The tire used is a complex PAC2002 model, based on the Pacejka Magic Formula (Pacejka, 2002).
Figure 6. Virtual bus model used for Ramp Steer tests under full load
The inputs of the system are the speed and the steering wheel angle. A PID control is implemented to regulate the torque on the wheels of the rear axle according to the reference speed (Rajamani, 2012). In this way the driver's actions on the vehicle are reproduced in the same way as in the real system. Two variants are developed on this model that represent two load cases: running order and maximum laden (full load) masses, these being the limit cases.
Figure 7. Trajectory carried out by the vehicles, making turns both to the right and to the left (Case of full load (CC), speed of 75 km / h and tire adhesion / road 1) Table 1. Summary table with the tests carried out Load Case
Mass (kg)
Load Ratio (%)
Centre of gravity height
Running order
13542
33%-67%
1.08
Maximum laden mass
17887
35%-65%
1.3
Speed (km/h) 50 75 100 50 75 100
Friction coefficient 0.3 0.6 1 0.3 0.6 1
Marta Alonso et al. / Transportation Research Procedia 33 (2018) 331–338 Author name / Transportation Research Procedia 00 (2018) 000–000
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2.3. Training Test Ramp Steer tests are carried out, which consist of introducing to ramp-shaped address input, progressively without producing sudden changes in it. The speed of rotation will be the same in all cases, varying only the speed and the tire-road friction coefficient. Maneuvers will be carried out to the left and to the right, in order to take into account the possible asymmetry in the behavior of the vehicle. The load cases are simulated for the speeds of 50 km / h, 75 km / h and 100km / h and three tire-road friction values 1, 0, .6 and 0.3 are considered. In the following table, Table 1, a summary of the tests carried out is shown. The simulations have been carried out in accordance with ISO standards for heavy vehicles and buses, ISO 15037 - ISO 11026 - ISO 14793, until the imminent rollover condition is reached, this being the driving situation. 3.
Results
With the virtual tests carried out, the reliability of the stability region defined in point 2.1 is analyzed. For this, it is analyzed if the vehicle is dynamically stable when the relationship between the yaw rate and the slip angle of the vehicle reach one of the limits defined by the mentioned region and if the safety demand in those limits is similar for the different operating conditions.
Figure 8. Representation of the lateral forces against the slip angle of the tires obtained from the simulation for full load (CC), 50 km / h and an adhesion of 0.6
The evaluation of the stability of the vehicle in the limit situation is carried out by analyzing how the outer tires are demanded at that point. This analysis is based on comparing the slope of the lateral force-slip angle curve at that instant with the slope in the linear zone (Figure 8). In the first place, the so-called stability regions are obtained for each of the load, speed and adherence configurations. Next, the simulations are carried out until the limit situation defined by the stability region 𝑟𝑟̇ − 𝛽𝛽 is reached (Figure 9).
Figure 9. Example of stability region with r -̇ β simulation data. The represented case is for the configuration of load, order of march (OM) and a speed of 100km / h
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These limit values are represented on the curve 𝐹𝐹𝑦𝑦 − 𝛼𝛼 of the tire and the value of the slope around that point is analyzed. In Figure 10, the 𝐹𝐹𝑦𝑦 − 𝛼𝛼 curves for the load configuration OM and a speed of 100 km / h are represented, on the curves the limit values of the region and the slope have been represented.
Figure 10. Location of the limits of the stability region in the tire curve. Slope m for adhesion 0.6
The value of the slope around that point (m) is compared with the theoretical slope in the linear zone (m0), the following equation is applied. %𝑀𝑀 =
𝑚𝑚·100
(9)
𝑚𝑚0
This calculated percentage allows to determine if the vehicle has already reached the saturation of the tire, or, on the contrary, it is still in a safe area and the degree to which the tire is demanded. In Table 2, the results obtained for the cases considered are shown. It can be seen that the percentage of the slope that you have is above 70%, regardless of the traffic conditions. This means that the stability region allows establishing limits as a reference for safety in circulation. These limits are stable, within each condition, and weighted to obtain a single reference. Table 2. Results obtained for each case Adherence
Load Configuration CC
0.3 OM CC 0.6 OM CC 1 OM
Velocity
m0
m
%M
50.0 75.0 100.0 50.0 75.0 100.0 50.0 75.0 100.0 50.0 75.0 100.0 50.0 75.0 100.0 50.0 75.0 100.0
3262.7 3240.8 3228.7 2521.3 2521.3 2521.3 3584.3 3626.3 3583.6 2828.2 2828.2 2828.2 3408.2 3408.2 3408.2 2875.0 2875.0 2875.0
2749.7 2683.5 1987.0 1985.1 1970.7 2728.9 2712.2 2647.3 2095.8 2054.7 2027.2 2524.0 2509.0 × 2477.5 2342.1
84.3 83.1 78.8 78.7 78.2 76.1 74.8 73.9 74.1 72.7 71.7 74.1 73.6 × 86.2 81.5
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4.
Conclusions
The definition of an objective metric for the evaluation of safety in road traffic is an issue not resolved in the current state of the art. In the present work, the validity of the stability region, defined based on the relationship 𝑟𝑟̇ − 𝛽𝛽, is evaluated as a reference to establish the limits of safe driving in any operating condition. The evaluation is done through the simulation of a virtual vehicle model, analyzing the demand of the tire in the limits of the region. It has been found that the stability regions define limits that guarantee that the vehicle, if it does not exceed them, maintains a safe circulation. The level of tire demand is below 30% for any operating condition, which means that the regions are conservative. Finally, it is noted that the limits are stable, within each condition, and can be weighted to obtain a single reference.
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