Vehicle Dynamics Model for Estimating Maximum Light Duty Vehicle Acceleration Levels
Hesham Rakha Charles Via Department of Civil and Environmental Engineering Virginia Tech Transportation Institute 3500 Transportation Research Plaza (0536) Blacksburg. VA 24061 Phone: (540) 231-1505 Fax: (540) 231-1555 E-mail:
[email protected] Matthew Snare Charles Via Department of Civil and Environmental Engineering Virginia Tech Blacksburg, VA 24061 François Dion Department of Civil and Environmental Engineering 3568 Engineering Building Michigan State University East Lansing, MI 48824
6,474 (words) + 1000 (tables + figures) = 7,474 words
Paper submitted for Presentation and Publication at the 83rd Transportation Research Board Meeting November 14, 2003
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ABSTRACT This paper presents and validates a vehicle dynamics model for predicting maximum light-duty vehicle accelerations for use within a microscopic traffic simulation environment. The research also constructs a database of unconstrained vehicle acceleration data for 13 light-duty vehicles and trucks. Using the field data, the proposed vehicle dynamics model is validated and compared to a number of state-of-the-art vehicle acceleration models, including the Searle model and dualregime, linear decay, and polynomial, models. The advantages of the proposed model include its ability to accurately predict vehicle behavior with readily available input parameters and its flexibility in estimating acceleration rates of both large and small vehicles on varied terrain. Key words: Vehicle acceleration, vehicle dynamics models, traffic simulation, car-following behavior.
INTRODUCTION Microscopic simulation software programs use car-following models to simulate the motion of vehicles traveling in a given lane. However, a major problem with the state-of-practice carfollowing models is that the majority of them do not ensure that vehicle accelerations are realistic by imposing vehicle dynamics constraints on the vehicle acceleration behavior. To accurately model the behavior of vehicles, state-of-the art car-following models use a system of equations. The first equation typically characterizes the steady-state motion of the following vehicle (commonly denoted as vehicle “n+1”) with respect to the behavior of the lead vehicle (vehicle “n”), as was described by Rakha and Crowther (1). Examples of such car-following behavior models include the GM, Van Aerde, Greenshields, Greenberg, and Pipes models. For example, Equation 1 demonstrates the Pipes steady-state car-following model that considers a driver perception/reaction time of T seconds and a time step of ∆t seconds. For simplicity, the time lag will be ignored in the formulation. It should also be noted that in order to ensure that a vehicle converges to its steady-state behavior, the car-following model should be formulated with the speed, as opposed to vehicle acceleration, as the control variable.
[
]
⎫ ⎧1 ~ u~n +1(t ) = min⎨ ⋅ hn +1(t − T ) − h j , uf ⎬ ⎭ ⎩ c3
[1]
where: ~ (t ) u n +1 ~ hn +1(t )
T uf hj c3
= Desired speed of following vehicle at time t, = Predicted headway between following and lead vehicles at time t, = Driver perception-reaction time, = Facility free-flow speed, = Headway at jam density, and = Model constant.
A second set of equations imposes constraints on the car-following behavior to ensure that the estimated vehicle accelerations are realistic, as illustrated by Equation 2. In this case, the desired speed u~ of a following vehicle that is accelerating at an instant t is constrained by the maximum speed uˆ allowed by vehicle dynamics at instant t. This equation alone clearly points to the critical importance of correctly modeled vehicle dynamics to the accurate modeling of car-following behavior.
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⎧uˆ (t ) if (u~n +1(t ) > uˆn +1(t )) un +1(t ) = ⎨ ~n +1 ~ ⎩un +1(t ) if (un +1(t ) ≤ uˆn +1(t ))
[2]
where:
u n +1 (t ) = Predicted speed of following vehicle at time t, and
uˆ n +1 (t ) = Maximum speed of following vehicle allowed by vehicle dynamics at time t.
A third, and final, set of equations further ensures that the traffic stream is asymptotically stable by enforcing collision-avoidance rules when the lead vehicle is traveling at a lower speed than the following vehicle (un(t) < un+1(t)). Specifically, this set of constraints ensures that the following vehicle is able to decelerate to a complete stop without colliding with the lead vehicle. Equations 3 and 4 illustrate the application of such a constraint, where Equation 3 dictates the deceleration speed based on a minimum safe speed u , determined by Equation 4. ⎧u (t ) if (u~n +1 (t ) > u n +1 (t )) u n +1 (t ) = ⎨ ~n +1 ~ ⎩u n +1 (t ) if (u n +1 (t ) ≤ u n +1 (t ))
[3]
u 2 (t ) − u 2 (t − ∆t ) u n +1 (t ) = u n +1 (t − ∆t ) + n ~ n +1 ⋅ ∆t 2 hn +1 (t ) − h j
[
]
[4]
where:
u n +1 (t )
u n (t )
= Minimum safe speed of following vehicle at time t, and = Speed of leading vehicle at time t.
To address the need to accurately model vehicle-acceleration behavior, this paper presents and validates a vehicle dynamics model for predicting maximum light-duty vehicle accelerations for use within a microscopic modeling environment. The paper starts by presenting state-of-practice vehicle-acceleration models, as well as a proposed light-duty vehicle acceleration model based on vehicle dynamics. The paper then describes the research that was conducted to construct a database of field-measured vehicle accelerations for validation purposes. Comparative applications of the proposed vehicle dynamics model and existing state-of-the-art models are presented before describing the main conclusions and recommendations of the research.
STATE-OF-PRACTICE VEHICLE ACCELERATION MODELS Several researchers have developed models that predict vehicle speed and acceleration profiles for inclusion within traffic simulation models. Essentially, these models can be classified into two categories: namely, models that predict acceleration behavior based on kinematics of motion and models that consider vehicle forces in estimating vehicle acceleration. Vehicle Kinematics Models Vehicle kinematics models take into account the mathematical relationship between acceleration, speed, and distance traveled. They are generally based on empirical mathematical relationships between acceleration and speed or acceleration and time. The most basic kinematical model is the constant acceleration model, which assumes a constant acceleration rate throughout an acceleration maneuver. Because of its simplicity, this model is used in several traffic simulation packages. However, research has shown that the assumption of constant acceleration is erroneous because vehicles tend to achieve higher acceleration rates while traveling at low speeds versus high speeds. To account for this behavior, several refined models have been
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suggested. A first model is the dual-regime model, which considers two constant acceleration rates: a high rate for low speeds and a lower rate for high speeds. Another model is the linear decay model. This model features an acceleration rate that starts at some maximum value at a zero speed and decreases linearly as a function of speed. Further research suggested that the maximum acceleration rate might not actually occur at time zero, but rather at some time shortly after the start of acceleration (2). To account for these observations, new models based on elaborate statistical distributions and mathematical functions were then designed in an attempt to duplicate the intricate pattern of observed acceleration profiles in field data. Examples of such models include a triangular model, in which acceleration increases linearly to its maximum before decreasing linearly; a Gamma model based on the gamma statistical density function; a polynomial model; and several models based on the trigonometric sine function (2, 3). Despite attempts at using elaborate statistical distributions and mathematical functions to model the complex acceleration behavior of vehicles, previous studies (2 and 3) have demonstrated that the dual-regime, linear decay and polynomial models still generally provide the best fit to field data. Based on these results, these three models were selected for analysis in this paper and are described in further detail in the following sub-sections. Dual-Regime Model As indicated, the dual-regime model uses two separate acceleration rates: one for low speeds and one for high speeds. One application of this model is presented by Bham and Benekohal (2), who recommend a transition point of 13 m/s (48.6 km/h) for making the switch from the high to the low acceleration regime, as shown in Equation 5: ⎧u(t − ∆t ) + a1∆t u(t ) = ⎨ ⎩u(t − ∆t ) + a2 ∆t
where: u(t) a1 a2 ∆t
for 0 ≤ v < 13 m/s for v > 13 m/s
[5]
= Speed at time at instant t, = Acceleration rate in first regime, = Acceleration rate in second regime, and = time increment.
In this paper, vehicle-specific optimum breakpoints were selected in order to ensure a best match between model predictions and field observations. Linear Decay Model The linear decay model assumes that the acceleration rate varies inversely with speed (4), with a maximum acceleration attained at a speed of zero and a subsequent linear decrease of the acceleration rate down to a value of 0 at the maximum speed, as illustrated in Equation 6.
a(t ) = a m −
am u (t ) ue
[6]
where: am = Maximum acceleration rate, and ue = Equilibrium speed, also known as the crawl speed. By integrating Equation 6, the two following relationships relating speed and distance traveled to time can then be determined:
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u(t ) = u e (1 − e
−
a0 t ue
5
) + u0e
−
a0 t ue
a0
[7] a0
− − t t u u u2 x (t ) = u e t − e (1 − e u e ) + 0 e (1 − e u e ) a0 a0
where: u(t) x(t) uo t
[8]
= speed of vehicle at time t, = Cumulative distance traveled since beginning of acceleration, = Initial speed of vehicle, and = Time elapsed since beginning of acceleration.
Polynomial Model The polynomial model was designed to satisfy the conditions of zero acceleration and zero jerk (first derivative of acceleration with respect to time) at the start and the end of the acceleration maneuver. This model generates a peak in the acceleration profile at time tm near the beginning of the acceleration maneuver, in an attempt to match field data. Acceleration at any instant t is computed using Equation 9 in conjunction with Equations 10 through 13 (3). a(t ) = bθ (1 − θ m )2 (m > −0.5)
[9]
with: b=
2( m + 1)( m + 2) aavg m2
[10]
m=
15 − 27 ρ + (81ρ2 − 138 ρ + 73 ) 12 ρ − 4
[11]
ρ=
uavg − u0
[12]
uf − u0
θm = (1 + 2m )
−
where: tf tm xf uf u0 uavg b aavg θ θm
1 m
[13]
= Time required for vehicle to attain desired final speed, = Time at which vehicle attains maximum acceleration, = Distance of travel required for vehicle to attain desired final speed, = Final speed of vehicle, = Initial speed of vehicle, = Average speed of vehicle (xf/tf), = Vehicle-specific calibrated parameter, = Average acceleration rate of vehicle ((uf-u0)/tf), = Relative time at instant t (t/tf), and = Relative time at which vehicle attains maximum acceleration (tm/tf).
By integrating Equation 9, the speed and distance traveled at any instant t can then be computed as demonstrated in Equations 14 and 15: ⎡ θ 2m ⎤ 2θm u(t ) = u0 + t f bθ 2 ⎢0.5 − + ⎥ m + 2 2m + 2 ⎦ ⎣
[14]
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⎡1 ⎤ θ 2m 2θ m x (t ) = u0t + t f2bθ ⎢ − + ⎥ ⎣ 6 (m + 2)(m + 3) (2m + 2)(2m + 3) ⎦
6
[15]
Vehicle Dynamics Model A problem with vehicle-kinematics models is that by empirically developing mathematical expressions that describe the acceleration patterns of the vehicle, the actual components that affect the motion of the vehicle – the tractive force provided by the engine and the resistance forces opposing the vehicle’s motion–are not modeled explicitly. Therefore, these models are difficult to calibrated and do not generally provide a good fit to field data for each of the acceleration, speed, distance, and time domains. They also do not account for different vehicle types, roadway grades, and other factors that affect the vehicle acceleration patterns. To better account for an accelerating vehicle’s actual physics of motion, a number of acceleration models have been developed based on vehicle dynamics. These models characterize the acceleration of a vehicle based on the power output from the vehicle’s engine and power losses resulting from internal mechanical friction losses and external resistance forces, such as air resistance, rolling resistance, grade resistance forces. The following paragraphs describe in more detail two vehicle dynamics models that were considered in the study: the Searle model (5), a standard vehicle dynamics model (6), and an enhancement to the standard vehicle-dynamics model proposed by Rakha and Lucic (7). Searle Model Searle used vehicle dynamics to generate equations of motion that compute the vehicle speed and distance traveled for vehicles accelerating at maximum acceleration rates to help with the investigation of road accidents (5). The model does not account for the specific effects of the resistance forces on the vehicle but incorporates them in the estimation of a power constant k that links vehicle performance to the ratio of engine output to the weight of the vehicle, according to Equation 16. k=
7.9 ηa Pmax M
where: k
ηa
Pmax M
[16]
= Power constant (khp/ton or kilowatt/ton), = Acceleration efficiency, = Maximum engine power (bhp or kilowatts), and = Mass of vehicle (tons).
In the model, the parameter η accounts for losses in the transmission as well as external resistances to motion, such as air, grade, rolling, and friction resistances. Once determined, the parameter k is then used to predict the speed of the vehicle and the distance traveled, as follows:
[u(t )]2 = u02 + 2kt
[17]
[u( x )]3
[18]
= u 03 + 3kx
(
)
⎡ u 2 + 2kt 1.5 − u 3 ⎤ 0⎥ ⎢ 0 ⎦ x (t ) = ⎣ 3k
[19]
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The Searle model provides a reasonable approximation of speeds within a specific time frame or over a certain distance for the purpose of recreating accidents for investigations. However, the model is not as useful in describing the overall acceleration behavior of vehicles, as is required in simulation software packages, since it does not accurately describe the motion of a vehicle at the beginning of an acceleration from a stop or at high speeds. Rakha and Lucic Vehicle Dynamics Model Rakha et al. described and applied a state-of-practice vehicle dynamics model for the modeling of truck acceleration behavior (6), while Rakha and Lucic (7) enhanced the state-of-practice model by accounting for losses in vehicle power during gear shifts. The resulting vehicle dynamics model is based on the basic principle that force equals the product of the vehicle mass and vehicle acceleration. If the net force on the vehicle and the vehicle mass are known, the acceleration of the vehicle can be determined by the relationship of Equation 20. a( t ) =
F (t ) − R (t ) M
[20]
where: F(t) = Residual force at instant t (N), R(t) = Total resistance force at instant t (N), and M = Vehicle mass (kg). Note that the net force on the vehicle is the difference between the force applied by the vehicle and the various resistance forces the vehicle encounters. The mass of the vehicle is constant, but the magnitude of the applied force and the resistance forces change as a function of vehicle speed and distance traveled. Therefore, given that acceleration is the second derivative of distance with respect to time, Equation 20 resolves to a second-order Ordinary Differential Equation (ODE) of the form indicated in Equation 21. dx d 2x = f ( , x) 2 dt dt
[21]
Because the tractive effort includes a minimum operand, the derivative of acceleration becomes a non-continuous function. However, Equation 20 can be recast as two first-order ODE's, and a numerical solution can then be reached using a first-order Euler approximation, as shown by Equations 22 to 25: In Equations 24 and 25, the smaller the value of the time-step ∆t, the more accurate are the model predictions. For purposes of this study, a time step of 0.1 seconds was considered. F (t ) − R (t ) M
[22]
⎧u (t )⎫ ⎧a(t )⎫ ⎬ ⎨ ⎬ = ⎨ ⎩ x (t )⎭ ⎩u(t )⎭
[23]
u(t ) = u(t − ∆t ) + a(t )∆t
[24]
x(t ) = x(t − ∆t ) + u(t )∆t
[25]
a(t ) =
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The vehicle dynamics model developed by Rakha and Lucic computes the effective tractive force as the minimum of the engine tractive force and the maximum force that can be sustained between the vehicle tires and the roadway pavement, as demonstrated in Equation 26. While the proposed model does not include gear shifting, it does account for the major behavioral characteristics that result from gear shifting: namely, the reductions of power as gearshifts are being engaged. Specifically, the model uses a variable power efficiency factor β that is dependent on the vehicle speed, as opposed to the constant factor that is currently utilized in state-of-practice vehicle dynamics models. The factor is a linear relation of vehicle speed with an intercept of 1/up and a maximum value of 1.0 at a speed up (i.e, at the speed at which maximum power occurs), as demonstrated in Equation 27. The intercept guarantees that the vehicle has enough power to accelerate from a complete stop. The adjustment factor is then multiplied by the vehicle power to compute the tractive force, as demonstrated in Equation 26. ⎡ ⎤ P Fn +1 (t ) = min⎢3600 ⋅ η ⋅ β ⋅ , 9.8066 ⋅ M ta ⋅ µ ⎥ u n +1 (t − ∆t ) ⎣ ⎦
β=
1 up
⎡ ⎛ 1 ⎢1 + min(u n +1 (t − ∆t ),u p )⎜1 − ⎜ ⎢⎣ ⎝ up
⎞⎤ ⎟⎥ ⎟⎥ ⎠⎦
[26]
[27]
where:
β η
P Mta
µ
up
= Variable power factor, = Transmissions efficiency, = Engine power (kW), = Mass of vehicle on tractive axle (kg), = Coefficient of friction between tire and pavement, and = Speed at which vehicle attains maximum power (km/h).
The estimation of the variable power factor β requires the calibration of a single parameter: namely, the speed at optimum power up. Rakha and Lucic calibrated this parameter for heavy-duty vehicles using four trucks with vehicle rated powers ranging from 260 to 375 kW (350 to 500 hp), each involving ten weight configurations. The calibration demonstrated that higher weight-to-power ratios required a lower optimum speed and a higher minimum power, as demonstrated in Equation 28. The research presented in this paper extends the research presented by Rakha and Lucic by quantifying typical values of the variable power factor for use in modeling light-duty vehicle accelerations. u p = 1164w −0.75
[28]
where: w = Vehicle weight-to-power ratio (kg/kW) The model also computes the various resistance forces acting on the vehicle: namely, the aerodynamic, rolling, and grade resistance forces, as demonstrated in Equation 29. The maximum feasible acceleration of the following vehicle within a simulation software is then computed using the acceleration upper bound, as demonstrated in Equation 30. Consequently, the upper bound for the speed at instant t can be computed by solving Equation 31, as demonstrated in Equation 32. Rn +1(t ) = c1Cd Ch Af ⋅ u n2+1 (t − ∆t ) + 9.8066MCr [c 2 u n +1 (t − ∆t ) + c 3 ] + 9.8066MG
[29]
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F (t ) − R n +1 (t ) a n +1 (t ) = n +1 M u (t ) − u n +1 (t − ∆t ) Fn +1 (t ) − R n +1 (t ) = an +1 (t ) = n +1 ∆t M ⎡ F (t ) − R n +1 (t ) ⎤ u n +1 (t ) = u n +1 (t − ∆t ) + ⎢ n +1 ⎥ ∆t M ⎣ ⎦
9
[30] [31] [32]
where: R(t) = Total resistance force; sum of the aerodynamic, rolling, and grade resistance forces (N), c1 = Constant accounting for density of air at sea level (0.047285), c2,c3 = Rolling resistance coefficients, Cd = Vehicle drag coefficient, Ch = Altidute coefficient, Cr = Rolling coefficient, Af = Vehicle frontal area (m2), and G = Percent grade (m/100m). It should be noted that Equation 31 assumes that vehicles accelerate at the maximum rate when they are constrained by their vehicle dynamics. However, drivers typically do not utilize the full acceleration capabilities of a vehicle, and thus a typical acceleration reduction factor γ is introduced to the model, as demonstrated in Equation 33. This model is incorporated within the INTEGRATION software (8). ⎡ F (t ) − R n +1 (t ) ⎤ u n +1 (t ) = u n +1 (t − ∆t ) + γ ⎢ n +1 ⎥ ∆t M ⎣ ⎦
[33]
where: γ = Typical acceleration reduction factor.
CONSTRUCTION OF VEHICLE ACCELERATION DATASET The validation of vehicle acceleration models requires comprehensive field data. Unfortunately, the available datasets are outdated. For example, in their comparative study of existing models, Bham and Benekohal were forced to use datasets that were collected in 1968 and 1983 (2) since more recent datasets were not extensive enough and did not capture the acceleration behavior of vehicles for a sufficient amount of time. Recognizing that vehicle acceleration capabilities have changed dramatically since the early 1980s, Bham and Benekohal tried to extrapolate vehicle acceleration trends from the data compiled from 1968 and 1983 to account for modern vehicle capabilities. Another important limitation of existing datasets is the limited information that is provided with regards to the vehicle types, roadway characteristics, and drivers involved in the data-collection effort. Specifically, information on the vehicle characteristics is typically not provided. Another important issue is that the existing datasets were often collected from a traffic stream, where acceleration is affected by vehicle-to-vehicle interaction and might not, therefore, truly reflect the maximum envelope of a vehicle operation. Because of the limitations in existing acceleration data, it was determined that it was essential to collect new acceleration data. In particular, in addition to documenting all available information with regards to the vehicle and roadway characteristics, the data were gathered under conditions
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in which vehicle accelerations were not constrained by surrounding traffic. Specifically, the datacollection effort was designed so that vehicle accelerations could be measured in a controlled environment in which accelerations were not constrained by external factors. The modeling of vehicle acceleration in a platoon of vehicles is an area of research that is beyond the scope of this paper but that requires further investigation.
Test Facility Testing of vehicles was performed during the summer of 2001 on a 1.6-km (1-mile) section of the Smart Road test facility at the Virginia Tech Transportation Institute in Blacksburg, Virginia. The selected test section featured a relatively straight horizontal layout with a minor horizontal curvature that had no effect on vehicle speeds, a good asphalt roadway surface, as well as a substantial upgrade that ranged from 6% at one end to 2.8% at the other end. Since no flat sections of significant length were available, vehicle accelerations were measured by driving vehicles uphill. An equation characterizing the grade of the test section was derived from the elevations of 15 stations along the test section. The vertical profile of the test section was then generated by interpolating between station elevations using a cubic spline interpolation procedure at 1-m (3.28ft) increments. The qubic spline interporlation ensured that the elevations, slopes, and slopes’ rates of change were identical at the boundary conditions (in this case every meter). The grade was then computed for each 1-m section (3.28-ft), and a polynomial regression model was fit to the grade data (R2 of 0.951) to ensure a smooth transition in the roadway grade while maintaining the same vertical profile, as demonstrated in Equation 34. The regression equation also facilitated the solution of the ODE by ensuring that the grade function was continuous. G = 0 . 059628 + 3 .32 × 10 − 6 x − 3 .79 × 10 − 8 x 2 + 1 .42 × 10 − 11 x 3
[34]
where: x = Distance from beginning of test section (m). G = Grade magnitude (m/100 m).
Test Vehicles Table 1 identifies the thirteen test vehicles that were selected for the study. These vehicles were selected to cover a wide range of light-duty vehicles. All test vehicles were personal cars driven by various faculty and students working and/or studying at the Virginia Tech Transportation Institute. As indicated in the table, the selected vehicles represent a wide range of sizes and a variety of EPA vehicle classes. Table 1 presents the main characteristics of each vehicle and related parameters for use in the acceleration models described earlier. The numerous input parameters shown in the table emphasize the complexity of vehicle acceleration behavior. Below is a description of each of the parameters listed in the table and how the values used in the study were obtained: •
Vehicle Engine Power: The engine power can be easily obtained from the vehicle specifications and is usually given in horsepower. To convert from horsepower to kilowatts, multiply the value by 0.746.
•
Engine Efficiency: Power losses in the engine due to internal friction and other factors generally account for between 20-35% of the engine losses for light-duty vehicles. Therefore, typical efficiency values range between 0.65-0.80. The actual values were computed by minimizing the sum of squared error between field-observed power estimates and calibrated power estimates for different efficiency factors, as demonstrated in Figure 1.
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•
Vehicle Mass: Vehicle mass is an important parameter in the model as it determines the force required to accelerate a vehicle. The curb weight is usually available within the vehicle specifications. However, care should be exercised to include the weight of the passengers in the vehicle during the tests.
•
Percentage of Vehicle Mass on the Tractive Axle: For a four-wheel drive vehicle, this value is 100%. However, most light-duty vehicles are two-wheel drive. Typical values for front-wheel drive vehicles are in the range of 50-65%, reflective of the high weight of the engine sitting on top of the axle. For rear-wheel drive vehicles, the mass on the tractive axle typically ranges between 35-50% of the total mass. For each vehicle listed in the table, axles were weighed separately to obtain the correct percentage of mass on the tractive axle using a scale of accuracy ±10 percent.
•
Frontal Area: The frontal area of the vehicle can be approximated as 85% of the height times the width of the vehicle if it was not given directly in the vehicle specifications.
•
Air Drag Coefficient: The air drag coefficient is given in the vehicle specifications. Typical values for light-duty vehicles range from 0.30 to 0.35, depending on the aerodynamic features of the vehicle.
Data Collection Procedures Each of the 13 test vehicles was subjected to the same set of tests. The test runs involved accelerating the vehicles from a complete stop at the maximum acceleration rate over the entire length of the 1.6-km test section. Three speed ranges were tested: accelerations from 0 to 56 km/h (35 mph), 0 to 88 km/h (55 mph), and 0 km/h to the maximum attainable speed within the test section. The latter set of runs was utilized for the analysis that is presented in this paper because it covered a wide range of the envelop of operation of each vehicle. Depending on the type of vehicle, maximum speeds attained by the end of the test section varied between 128 and 160 km/h (80 and 100 mph). In conducting the study, a minimum of five repetitions were executed for each test set in order to provide a sufficient sample size for the validation analysis. This generated a total of 15 test runs for each vehicle. In each test run, the speed and position of the vehicle was recorded using a portable Global Positioning System (GPS) receiver connected to a laptop. Outputs from GPS receiver included latitude, longitude, altitude, speed, heading, and time stamp once every second. Nominal position accuracy was specified with a 25-m (82-ft) spherical error probability, while nominal velocity accuracy was specified within 0.1 m/s (0.31 ft/s) error probability.
ROADWAY PARAMETERS AND MODEL ASSUMPTIONS As mentioned earlier, the research effort involved applying the Rakha and Lucic model that was originally developed for trucks to a variety of passenger vehicles and comparing the model predictions against field data and other state-of-the-art models. To do so, a number of roadway, tire, and vehicle-specific parameters were input to the model. The determination of these parameters is described in the following sections.
Roadway Characteristics To apply the Rakha model, five parameters linked to roadway characteristics must be determined: pavement type, pavement coefficient of friction, roadway grade, rolling coefficients, and altitude of roadway.
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•
Pavement: The pavement type and condition are required to determine several parameters. As indicated earlier, the selected test section on the Smart Road facility had a Pavement Serviceability Index greater than 3.0 and thus was classified as “good.” The pavement condition affects the coefficient of friction and rolling coefficients, as described in detail by Rakha et al. (6). Consequently, a coefficient of friction of 0.6 and values of 1.25, 0.0328 and 4.575 were selected for the coefficients Cr, C2 and C1, respectively.
•
Grade: The roadway grade was computed using Equation 33 at each vehicle position.
•
Altitude: This is the altitude above sea level for the testing location, in meters. Since the Smart Road sits at an altitude of 600m, this lead to determination of an altitude coefficient of 0.95, as described by Rakha et al. (6).
Constant Power Assumption As was mentioned earlier, Rakha and Lucic observed a reduction in truck power at low speeds caused by the build-up of power that occurs as a truck accelerates through the various gears. However, it was unclear whether this behavior would also occur for light-duty vehicles because these vehicles have fewer gears, carry less mass, and generally use automatic transmissions. The first step was to investigate whether the assumption of constant power was reasonable. Specifically, Figure 1 illustrates the field-estimated power for the Acura Integra vehicle (power was computed as the resultant force multiplied by the speed at any instant t). The figure clearly demonstrates an initial linear increase in power followed by a regime with fairly constant power after the maximum power is attained. Similar observations were made for other test vehicles. Consequently, it was concluded that the assumption of a constant power at speeds higher than 40 km/h was reasonable for modeling purposes. The next step was to characterize the slope of the power buildup and to test its consistency with the Fmax assumption. The fact that the power buildup appears to be linear suggests that the resultant force is constant for the initial stage of acceleration (given that power is equal to the product of force and speed). The dashed line reflects the estimated power based on the computed Fmax multiplied by a power adjustment factor β (β=0.95). The relatively good fit between the fieldobserved power estimates and the predicted power (R2 of 0.93) suggests that a gear-shift reduction factor (β) that is within 10 percent of 1.0 is sufficient for modeling purposes. Similar results were observed for the other test vehicles. For example, the Dodge Neon and BMW 740i also exhibited reasonable correlation with field data (R2 values of 0.86 and 0.88, respectively). However, the Geo Metro had the lowest coefficient of determination (R2 = 0.74), demonstrating the need to incorporate a lower power reduction factor, which could be partly attributed to the fact that the vehicle had a manual transmission and also had a low engine power. Variations in the power adjustment factor for the 13 test vehicles suggested minimum losses in vehicle power during gear shifts in general (β ranged from 0.9 to 1.0). Consequently, the use of a power reduction factor of 1.0 appears to be sufficient for modeling purposes.
COMPARISON OF VEHICLE DYNAMICS MODEL PREDICTIONS TO STATE-OFPRACTICE MODELS To assess the validity of the vehicle dynamics model, the model predictions were compared to state-of-the-art acceleration model predictions. Four existing models were chosen for comparison purposes, namely: the dual-regime, linear decay, polynomial, and Searle models. The dual-regime model was chosen because it was recommended by Bham and Benekohal following a comprehensive comparison of fourteen models. The linear decay model was recommended for use in a paper by Long (9) and also appears in several textbooks, including one by Drew (4). The
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Polynomial model was further recommended in a comparison study performed by Akçelik (3). Finally, the Searle model was chosen for comparison as it exemplifies a different vehicle-dynamics model. It should be noted that the polynomial model was calibrated by computing the maximum vehicle speed that the vehicle can attain based on its dynamics, commonly known as the equilibrium speed (speed of vehicle when tractive force equals summation of resistance forces). Specifically, the equilibrium speed (uf), the distance traveled to attain the equilibrium speed (xf), and the duration of time spent accelerating to the equilibrium speed (tf) were computed using the vehicle dynamics model. Subsequently, the uf, xf, and tf parameters served as input to the polynomial model, as was described earlier using Equations 10 through 13. For example, the Dodge Intrepid parameters (uf, xf, and tf ) were set at 169 km/h, 5994 m, and 152 s, respectively. It should be noted that the calibration of the model utilizing a final speed equal to the equilibrium speed ensures that the vehicle acceleration and jerk are zero at the conclusion of the acceleration cycle, as recommended in the literature (3). Furthermore, this calibration ensures that the vehicle is only constrained by its dynamics while accelerating and that the acceleration covers the entire envelope of operation of the vehicle.
Vehicle Dynamics Model Application Results Application of the Rakha and Lucic vehicle dynamics model to the 13 test vehicles in conjunction with a simple exponential smoothing procedure, as demonstrated in Equation 34, demonstrated a strong correlation to field measurements. It should be noted that a smoothing factor of 0.08 appeared produced an optimum fit to field data (minimum error in the speed-acceleration domain). A sample of the evaluation results is shown in Figure 2, which illustrates the acceleration predictions for the Chevy S-10 truck. As observed in the figure, the Rakha model demonstrates good fits in the acceleration versus speed, acceleration versus time, acceleration versus distance, speed versus time, and speed versus distance domains. While not shown, similar results were also obtained for other test vehicles. Considering that the test vehicles covered various vehicle types, including small cars, large cars, SUVs, and pickup trucks, the evaluation results clearly demonstrate the flexibility and validity of the proposed model in predicting maximum vehicle acceleration levels for a wide range of vehicle types. While the proposed model did offer accurate modeling of vehicle acceleration behavior, the model slightly over-estimated vehicle speeds (less than 5 km/h) for two of the test vehicles, namely: the Dodge Neon and the Blazer. These two vehicles seemed to demonstrate a slight drop in vehicle acceleration towards the end of the test runs that was not captured by the vehicle dynamics model. However, these drops were attributed to the fact that the test vehicle did not properly shift into the final gear, and thus accelerated below its maximum potential. a~ (t ) = α ⋅ a (t ) + (1 − α ) ⋅ a~ (t − ∆t ) [35] n +1
n +1
n +1
where:
a~n +1 (t ) = Exponentially smoothed acceleration of vehicle “n+1” at instant “t”,
a n+1 (t ) = Instantaneous acceleration of vehicle “n+1” at instant “t”,
α
= Smoothing constant (ranges from 0.0 to 1.0).
State-of-the-Art Model Application Results Comparative predictions with the Rakha model using existing models were made for 5 of the 13 test vehicles. The vehicles selected were the Mazda Protégé, Dodge Intrepid, Chevy Blazer, Ford Windstar, and Chevy S-10. These vehicles were selected because they cover a variety of vehicle types. Only the results from the Dodge Intrepid are shown in Figure 3; however, similar results were observed for other vehicles.
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For the dual-regime model, the calibrated input parameters include the vehicle’s final speed, the total acceleration time, the speed breakpoint between acceleration regimes, and the time to reach the second regime. These parameters were calibrated for each vehicle by minimizing the sum of squared error in the speed-time domain. As illustrated in Figure 3, the dual-regime model appears to provide a reasonable fit to the observed speed profiles with respect to distance and time. However, the results demonstrate that the model tends to over-estimate speeds in the middle of the distance profile, towards the end of the first regime. In the case of the Blazer and Windstar, the model further tends to under-estimate vehicle speeds at the beginning of the time profiles. Finally, the acceleration plots demonstrate a poor fit to the field data for all test vehicles. An analysis of the above results further leads to the conclusion that the major disadvantage of the model is the discontinuity in the acceleration profile. However, the simplicity of the model and its ability to generate reasonable speed profiles makes it useful in limited applications. In the case of the linear decay model, the maximum acceleration and speed are calibrated to obtain an optimum fit (minimum sum of squared error) between the model estimates and field data in the acceleration-speed domain. As illustrated in Figure 3, the linear decay model enhances the dual-regime model by providing a continuous acceleration function that decreases with vehicle speed. The model also accounts for the apparent curvature in the data in the acceleration versus time and distance domains. However, the data in the acceleration-versus-speed plot exhibits curvature, which is not captured in the linear function, creating a tendency to underestimate the acceleration rate of the vehicle when the vehicle is traveling at low speeds. Nevertheless, this model appears to be the best kinematics model, even though it requires calibration to field data before it can be applied. Calibration of the polynomial model involved estimating four parameters, including the vehicle’s initial and final speeds, acceleration time, and acceleration distance. As was described earlier, these parameters were estimated when the vehicle attains equilibrium using the Rakha vehicle dynamics model, thus ensuring a zero speed and acceleration at the conclusion of the acceleration cycle. The sample results of Figure 3 demonstrate that the polynomial model provides a good fit to the observed speed profiles. However, the model tends to estimate a higher maximum acceleration rate than is observed in the field data and the maximum acceleration tends to occur earlier in time and at lower speeds than is observed in the field data. In applying the Searle model, the power constant k was calibrated for each of the test vehicles. In Figure 3, the Searle model appears to break down at very low speeds and speeds outside the normal driving range. The model predicts acceleration rates approaching infinity for low speeds because it does not account for the potential of wheel spinning at low speeds. Alternatively, the vehicle dynamics model presented by Rakha et al. (6) introduces this constraint by ensuring that the maximum force value does not exceed the frictional force that can be sustained between the vehicle tires and the pavement surface, as indicated by Equation 26. Application of the model to field data further demonstrates that the Searle model overestimates speeds beyond speeds of 120 km/h (75 mph). Because of these limitations, the Searle model is therefore deemed useful only in limited applications.
Advantages of Proposed Vehicle Dynamics Model Based on the above comparison, three primary advantages of the proposed vehicle dynamics model can be identified: 1. Good Fit in all Domains: The model offers a good fit to field data in each of the five domains: namely speed versus distance, speed versus time, acceleration versus speed, acceleration versus time, and acceleration versus distance. The other models, especially the
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empirical models, are designed to fit a single domain and, therefore, do not fit well over all domains. 2. Model Flexibility: The model is able to adapt to different vehicle types, different pavement conditions, and travel over different terrain (e.g. effect of grades). 3. Simple Calibration: Another advantage of the model is that it does not require field datasets to calibrate input parameters. Instead, input parameters can be obtained from vehicle manuals, car magazines or the Internet. The other models require the collection of field data for the calibration of their respective input parameters.
CONCLUSIONS AND RECOMMENDATIONS The main objective of the paper was to validate a vehicle dynamics model for the modeling of light-duty vehicle acceleration. Following the development of a new database of field-observed accelerations, accelerations predictions from the proposed model were compared to predictions from four other state-of-the-art models. The advantages of the vehicle dynamics model included its ability to accurately predict vehicle behavior with readily available input parameters and its flexibility in estimating acceleration rates of both large and small vehicles on varied terrain. Following the efforts described in this paper, it is recommended that further research be conducted in a number of areas. First, there is a need to collect field data on typical lead-vehicle acceleration (not interacting with other preceding vehicles) as well as on vehicle acceleration within a platoon (interacting with other vehicles). Second, data are required on typical vehicle acceleration during critical events, including merging with freeway traffic at an on-ramp and accelerating to overtake a vehicle. For all these scenarios vehicle acceleration models should also be developed to capture driver behavior in the non-steady state car-following mode of operation.
ACKNOWLEDGEMENTS This research was funded by the Mid-Atlantic University Transportation Center (MAUTC) and the Virginia Department of Transportation (VDOT).
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REFERENCES 1. Rakha H. and Crowther B, (2003), Comparison and Calibration of FRESIM and INTEGRATION Steady-state Car-following Behavior, Transportation Research, 37A, pp. 127. 2. Bham, G.H., and Benekohal, R.F. (2002). “Development, Evaluation, and Comparison of Acceleration Models.” Pre-print CD-ROM, 81st Annual Meeting of the Transportation Research Board, Washington, D.C. 3. Akcelik, R., and Biggs, D.C. (1987). “Acceleration Profile Models for Vehicles in Road Traffic.” Transportation Science, Vol. 21, No. 1. 4. Drew, D.R. (1968). Traffic Flow Theory and Control. McGraw-Hill. 5. Searle, J. (1999). “Equations for Speed, Time and Distance for Vehicles Under Maximum Acceleration.” Advances in Safety Technology, Special Publication No. 1433, Society of Automotive Engineers, Inc., Warrendale, PA. 6. Rakha, H., Lucic, I., Demarchi, S., Setti, J., and Van Aerde, M. (2001). “Vehicle Dynamics Model for Predicting Maximum Truck Acceleration Levels.” Journal of Transportation Engineering, Vol. 127, No. 5. 7. Rakha H. and Lucic I., (2002), Variable Power Vehicle Dynamics Model for Estimating Maximum Truck Acceleration Levels, Journal of Transportation Engineering, Vol. 128(5), Sept./Oct., pp. 412-419. 8. Rakha H. and Ahn K. (2004), INTEGRATION Modeling Framework for Estimating Mobile Source Emissions. Journal of Transportation Engineering, Vol. 130(2), March/April, pp. 183193. 9. Long, G. (2000). “Acceleration Characteristics of Starting Vehicles.” Transportation Research Record 1737, TRB, National Research Council, Washington, D.C.
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NOTATIONS A(t) aavg am A1 A2 Af C3 Cd Ch Cr C1 C2,c3 F(t) G hj
~ hn +1(t )
K M Mta P Pmax R(t) T ∆t T u(t) uavg ue uf uo up u(t)
u n (t ) u n +1 (t ) ~ (t ) u n +1
uˆ n +1 (t ) u n +1 (t )
x(t) W
β η ηa θ µ γ
= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
Vehicle acceleration at instant t Average acceleration rate Maximum acceleration rate First regime acceleration rate in dual-regime model Second regime acceleration rate in dual-regime model Vehicle frontal area (m2) Pipes Model constant Vehicle drag coefficient Altitude coefficient Rolling coefficient Constant accounting for density of air at sea level (0.047285) Rolling resistance coefficients Total resistance force at instant t Percent grade (m/100 m) Headway at jam density Desired headway of following vehicle at time t Searle model power constant Vehicle mass (kg) Vehicle mass on tractive axle (kg) Engine power (kW) Maximum engine power in Searle model Total resistance force, which is the sum of the aerodynamic, rolling, and grade resistance forces (N) Time elapsed since beginning of acceleration Duration of time interval used for solving the ODE (in this case 1-second duration) Driver perception-reaction time Vehicle speed at instant t Average speed of vehicle Equilibrium speed, also known as the crawl speed Facility free-flow speed Initial speed of vehicle Speed at which vehicle attains maximum power Speed at instant t Speed of leading vehicle at time t Speed of following vehicle at time t Desired speed of following vehicle at time t Maximum speed of following vehicle allowed by vehicle dynamics at time t Minimum safe speed of following vehicle at time t Cumulative distance traveled at time t since beginning of acceleration Vehicle weight-to-power ratio Variable power factor Transmission efficiency Acceleration efficiency in Searle model Calibrated constant in polynomial model Coefficient of friction between tires and pavement Maximum acceleration factor
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LIST OF TABLES Table 1: Characteristics of Test Vehicles
LIST OF FIGURES Figure 1: Model versus Field Measured Power versus Speed Relationship (Acura Integra) Figure 2: Example Application of the Rakha Model to the Chevy S-10 Vehicle Figure 3: Models Predictions versus Field Data (Dodge Intrepid)
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Table 1: Characteristics of Test Vehicles Vehicle
EPA Class
1996 Geo Metro Hatchback 1995 Acura Integra SE 1995 Saturn SL 2001 Mazda Protégé LX 2.0 2001 Plymouth Neon 1998 Ford Taurus 1998 Honda Accord 1995 BMW 740I 1995 Dodge Intrepid 1999 Ford Crown Victoria 1998 Ford Windstar LX 1995 Chevy S-10 1995 Chevy Blazer
Subcompact Compact
Midsize Large Minivan Pickup SUV
Engine Power (hp) 55 142 124 130 132 145 150 282 161 200 200 155 195
Engine Power (kW) 41.0 105.9 92.5 97.0 98.5 108.2 111.9 210.4 120.1 149.2 149.2 145.47 145.47
Engine Efficiency 0.65 0.68 0.72 0.70 0.75 0.80 0.75 0.70 0.68 0.70 0.65 0.72 0.65
Vehicle Mass (kg) 1130 1670 1240 1610 1650 1970 1770 2370 2040 2300 2270 1930 2310
Mass on Tractive Axle (%) 0.380 0.515 0.560 0.525 0.495 0.575 0.610 0.515 0.535 0.590 0.550 0.605 0.560
Frontal Area (m2) 1.88 1.94 1.95 2.04 2.07 2.26 2.12 2.27 2.30 2.44 2.73 2.31 2.49
Air Drag Coefficient 0.34 0.32 0.33 0.34 0.36 0.30 0.34 0.32 0.31 0.34 0.40 0.45 0.45
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140 120
Power (kW)
100 80 60 40
R2 = 0.995
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Figure 1: Model versus Field Measured Power versus Speed Relationship (Acura Integra)
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Figure a: Speed versus distance Figure b: Speed vesus time Figure c: Acceleration versus distance Figure d: Acceleration versus time Figure e: Acceleration versus speed
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Figure 2: Example Application of the Rakha Model to the Chevy S-10 Vehicle
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Figure 3: Models Predictions versus Field Data (Dodge Intrepid)
