Performance analysis of a grade estimation method by means of simulation

(1)

Master Thesis

Performance analysis of a

grade estimation method

by means of simulation

Peter Ståhl

Stockholm, Sweden 2011

(2)

Performance analysis of a grade estimation method by

means of simulation

Peter St˚ahl

Abstract

Digital maps allow advanced driver assistance systems to adapt their behavior to the vehicle’s surroundings. Preview grade information can be used in on-board functions of a heavy duty vehicle such as adaptive lighting, predictive cruise control and gearbox control. These functions have a potential for adding safety, increasing fuel efficiency and improving driver comfort.

Grade estimation by use of existing inexpensive on-board sensors in heavy vehicles is a promising way of creating and maintaining up-to-date road grade maps with min- imal investment. This thesis examines the performance of a recently developed grade estimation method.

Simulation is used to test the grade estimation performance in a longer perspective than is possible with the experimental data currently available. Errors are introduced into the simulation model based on interviews and experience from road tests of the estimation method. Fast convergence is seen when many simulated runs are fused into a grade map.

Also, experimental data are analyzed and some systematic errors are found in the vehicle model. These errors are not explicitly accounted for in the error model for vehicle data used in the grade estimation method. By comparing grade estimated with the aid of GPS to grade calculated using only vehicle data, corrections to the vehicle model are computed. The effect of these corrections is examined and it is shown that it is feasible to take dynamic parameter errors into account in an estimation method with a potential for increased performance. For a heavy vehicle a decrease of 45 % in the residual error of fused grade estimates is observed when adding these corrections.

Candidates for a method that integrates this new knowledge of model errors into a grade estimation method are proposed. Obtained knowledge of the dynamic vehicle behavior may also be valuable in applications using the generated grade data.

(5)

1 Introduction

As computational power, data storage and communication get less expensive, functions that aid the driver of a Heavy Duty Vehicle (HDV) become more common. The general task of the driver is to adapt the way the vehicle is driven to the conditions of the road.

To assist the driver, the technical systems need knowledge of the environment in which the vehicle is driven. Some relevant data can be collected while other data may be over the sensor horizon, that is not yet available to the systems at the time when they are needed. This is why digital maps are an important component in the creation of driver support systems.

1.1 Background

One attribute of the road that is relevant to the regulation of a HDV is road grade. Even very small changes in road grade result in significant change in the driving resistance for a heavy vehicle. By planning the speed profile with access to preview grade information the amount of energy that is wasted by braking is decreased without penalty to the trip time. Grade information may also be relevant in other present and future functions in the vehicle.

Several methods for collecting the data needed for the creation of a map with grade content exist. One alternative is to survey all roads with advanced and expensive equipment built for this purpose. This is the kind of data that can be bought from digital map providers at a premium price. Another alternative is to use inexpensive sensors already present in modern HDVs to allow a large set of vehicles to collect grade data in their normal day-to-day operations. By combining the measurements from a large set of vehicles doing an even larger set of measurements, grade data of good quality may be obtained. This distributed approach to data collection has a better chance of responding to dynamic changes to road data compared to centralized measurement operations. In some parts of the world this may be a crucial point. In 2010 the length of expressways constructed reached 9000 km in China alone.

A prototype implementation of a method for data post processing and grade estimation using inexpensive on board sensors in a HDV is examined here. This method is a result of a research project conducted in collaboration between the Royal Institute of Technology (KTH) in Stockholm and the HDV manufacturer Scania CV AB. In [1] the grade estimation method is described in detail.

1.2 Thesis objective

This thesis explores the strategy of simulation as a means to overcome the difficulty of evaluating a measurement in the case of grade estimation. A simulation model can be utilized to learn how the estimation method performs on datasets that are larger than the ones currently available from experiments. Another advantage is that the properties of the data generated by simulation is well known compared to real-world experimental

(6)

data, which allows for a better understanding of the behavior of the estimation method in different circumstances.

To implement the error simulation, it is necessary to understand the errors present in the experimental data currently available. The knowledge of the structure of errors in data is an important result of this thesis.

1.3 Related work

In the literature many methods have been suggested to deal with the problem of grade estimation in different settings. Some methods are based on direct measurement of the road. Others use models of the impact road grade has on vehicle dynamics when it is driven on the road. A third approach is to measure the 3D position of a moving probe by GPS and thus gain a map of the road topography.

With modern surveying equipment, the road grade can be measured with good accuracy. Air borne laser scanning is used in a project to develop an altitude model of Sweden. Precise positioning of the surveying aircraft together with a LIDAR (light de- tection and ranging) scanner enables the generation of a 3D surface. In [2] the authors describe how this 3D data can be cross referenced with 2D road maps to obtain road grade information.

In [3] a method is investigated where the difference between longitudinal acceleration and vehicle acceleration relative to the road is used. The gravitational component of the longitudinal acceleration can be computed and a road grade estimated. In the implementation described here, its accuracy seems to be too poor for applications like look ahead cruise control.

A high precision GPS with correction services in tandem with an Inertial Navigation System (INS) yields very precise absolute altitude measurements from which the grade can be extracted by differentiation. Another alternative is to use a differential GPS capable of using two antennas. The antennas are placed along the direction of travel on a surveying vehicle. The difference in altitude of the two antennas together with their separation along the vehicle gives a direct measurement of the road grade. This method is examined in [4] and evaluated against the alternative of obtaining road grade by comparing the vertical and horizontal velocity in a single antenna GPS.

To analyze the performance of the grade estimation method, it is necessary to con- sider the nature of GPS position errors. A solid base for understanding the GPS system is given in [5]. Much research exists regarding the nature of GPS signal and pseudo- range errors, but little is found on the navigational error output by a GPS receiver.

In [6] an error model for the pseudoranges measured to optain a position is described.

This approach requires knowledge of the proprietary algorithms of the GPS receiver to obtain a positioning error. The model also does not account for the varying terrain conditions along a road that probably are the cause of sporadic but large GPS errors in the experiments.

(7)

1.4 Outline

An introduction to the vehicle model used is given in Section 2. This is followed by an introduction to the grade estimation method that is the central subject in this thesis. In Section 4 the simulation model for the vehicle is described. The GPS is also modeled, the details of the GPS error and the model chosen to represent it in the simulation are presented in Section 5. The result of the simulations and convergence of the grade estimation method is presented in Section 6.

Some new information on the errors in the grade estimate is found in the experimental data. This information is the basis for Section 7, where errors in the vehicle model are described, and Section 8 where the prospects for correcting these errors are explored.

The corrections lead to a better grade estimation result under some conditions. In Section 9 the results are summarized and suggestions are made as to how the results may be used in a real-world implementation.

2 Vehicle Model

The basis for the simulation is the longitudinal force balance model described in [7].

Engine torque is translated to an accelerating force by the powertrain model depicted in Figure 1. Longitudinal forces are related to the vehicle acceleration by means of the generalized Newton’s second law of motion.

Figure 1: A schematic drawing of the powertrain model.

The engine is modeled by a mass moment of inertia J_e with angular acceleration of

(8)

θ. The engine torque balance is a result of the engine torque T¨ o, internal friction torque T_{f ric,o} and torque from the clutch load Tc.

J_eθ = T¨ _o− T_{f ric,o}− T_c. (1) During the time a gear is engaged, the clutch is assumed stiff. Torque from the engine is propagated down the drive line by the transmission. Here itand ηtrepresent the gear ratio and efficiency of the transmission at the current gear, respectively. Replacing the torque at the clutch with torque propagated through the transmission and replacing T_o and To,f ric with the net engine torque Te the model equation becomes

itηtJeθ = i¨ tηtTe− T_f. (2) Correspondingly, the torque from the transmission is propagated through the final drive with efficiency η_f and ratio i_f to the wheels

i_tη_ti_fη_fJ_eθ = i¨ _tη_ti_fη_fT_e− T_w. (3) At the wheels, torque is converted to an accelerating force Fa,

Tw = Farw, (4)

where r_w is the effective radius of the drive wheel.

The retarding forces are air drag, rolling resistance and force related to change in potential energy due to road grade. Air drag can be modeled as

F_airdrag= 1/2c_dAaρav² (5)

where c_dis a shape-dependent drag coefficient, A_ais an effective front area of the vehicle, ρ_a is the air density and v is vehicle speed.

Rolling resistance depends on the vehicle mass mg and a rolling resistance coefficient cr

F_roll = m_ggc_r. (6)

If the road has an inclination in the direction of travel, moving along the road includes lifting or lowering the vehicle center of mass. This results in a force on the vehicle, F_grade, modeled as

F_grade= m_ggsin(α) (7)

where α is defined as in Figure 2. According to Newton’s second law of motion, the net longitudinal force results in an acceleration of the vehicle mass and angular acceleration of the wheel mass.

F_a− F_airdrag− F_roll− F_grade=

m_g+J_w r_w²

˙v (8)

When writing down the final force balance equation, angular acceleration of the engine ¨θ can be expressed as a function of vehicle acceleration ˙v. The complete model obtained by combining equations (3) - (8) and expressing ¨θ in terms of vehicle acceleration ˙v is

i_ti_fη_tη_f rw

Te= 1

2cdAaρav²+ ˙v mg+J_w+ i²_ti²_fη_tη_fJ_e r²_w

!

+ mgg(cr+ sin(α)). (9)

(9)

F _roll

F _gravity

F _airdrag

F _brake F powert rain

α

Figure 2: Forces acting in the longitudinal direction of the vehicle.

3 Road grade estimation

In this thesis, a grade estimation method based on a Kalman filter and a distributed data fusion strategy is examined. The data flow is outlined in Figure 3. From the model described in Section 2, the road grade can be calculated if all the other parameters are known. Also, the grade can be calculated by differentiating the altitude obtained by means of GPS. These sources of grade data are fused in a Kalman filter. The Kalman filter takes measurement data from the GPS, wheel speed sensors and estimated applied torque calculated from engine fueling and parameters. Signals indicating braking, gear shifts and GPS satellite loss are also taken into account. In the Kalman filter, estimates of the system noise are set in a matrix Q and measurement noise in matrix R. The values of these matrices affect the behavior of the filter and are seen as design parameters.

(10)

Kalman filter

Brake system

Shifting Braking Satellites

Q R Noise covariances

Gain K

State update Smoothing

v z

Grade map data Data fusion

gear T_e

Data bus

Engine Gearbox GPS

α

Figure 3: Block diagram illustrating the data flow in the estimation procedure.

The output of the filter is an estimate of road grade and altitude, as weel as a matrix P which contains an estimate of the error covariance in the estimated quantities. In the data fusion step, estimates from different measurements of traveled road segments are fused into a map. Here, the values of P are used as weights on the estimated altitude and grade values, causing measurements with smaller estimated errors to have a larger impact on the final combined estimate.

The vehicle model does not describe the vehicle when shifting gears or applying any braking system. At times where the model is invalid, less weight is given to this part of the input data by assuming more noise in the vehicle model. This is achieved by modifying matrices Q and R while brakes are applied or a gear shift is indicated in the input data.

3.1 Experimental data

From the research leading up to the development of the grade estimation method a limited set of experimental data is available. Experiments are carried out on the two sections of European route E4 between Södertälje and Nyköping marked in Figure 4.

(11)

Södertälje

Nyköping

Southbound

Northbound

Figure 4: Experimental data is available from runs on two sections of the E4 between Södertälje and Nyköping.

Grade data on these sections of road exist from previous measurements. These measurements were made with a high precision GPS/INS instrument. Data were collected in several runs in each direction. The measurement runs were averaged to form a reference road grade to be used in the evaluation of the estimation method. In the instrument specifications it is claimed that the output grade error has a standard deviation of 0.05 % grade. The differences between the measurements used in the creation of the reference are of approximately this size, implying that the advertised accuracy is achieved. This reference grade is used to evaluate the performance of the grade estimation method.

Vehicle Type Configuration Weight (t) Experiment

A P310LA4x2MNA Tractor & semi-trailer 39 1,2,3

B R420LA4x2MNA Tractor 12 4,5

Table 1: Vehicles used in experiments

Five northbound and five southbound experiments are considered in this thesis. They are named according to the direction traveled and the experiment number. Two different vehicle are used. They differ in configuration and mass as described in Table 1. The vehicles are driven either on cruise control or under manual control. A data logging

(12)

system is connected to the vehicle Controller area network (CAN) and a GPS receiver as illustrated in Figure 5. The vehicle signals used in the estimation method are: wheel speed, applied torque, brake usage, and gear shift. These are logged from the vehicle CAN. The GPS receiver provides longitude, latitude and altitude information.

CAN

Vehicle

card

CAN bus GPS can

bus

Figure 5: Measurement setup for experiments.

The collected data are logged in the time domain. To simplify the processing and comparison of data, a distance domain dataset is used. This dataset is created by linear interpolation of the time domain data onto the reference road. Thus, a map is created where the signals are stored as functions of discretized road distance. This method may destroy some information but analysis is greatly simplified. It also facilitates fusion of several measurements without further processing.

The raw vehicle data is very noisy. Figure 6 illustrates the nature of three types of errors present. If grade is calculated using the vehicle model and only vehicle data, the speed sensor noise is very evident as shown in the top graph. If a low-pass filter is applied, much of the noise is removed. This leaves an error that is dominated by bias, as seen in the middle graph. The GPS introduces errors with different characteristics.

This section of the experiment was chosen to illustrate the effect of a GPS error on the estimated grade. The lower graph is the error when grade is estimated by the grade estimation method where GPS data is used. Considering the quality of the available input, the output grade error is very small in the experiments but some errors remain.

(13)

Figure 6: Road grade error in a section of the third northbound experiment. Note the different scales on the error axis.

Top: Error in the pointwise calculated grade from measured data without filtering.

Middle: Error in the pointwise calculated grade from measured data with a low-pass filter applied to the speed signal.

Bottom: Error in the grade estimated using the Kalman filter grade estimation method.

4 Simulation model

For the simulation model, the vehicle model described in Section 2 is used. A PI regulator emulates cruise control, attempting to keep the vehicle at a set speed whenever possible.

4.1 Sensitivity analysis

Figure 7 illustrates the impact of the most significant parameter errors in the vehicles used in experiments. Errors in output grade are calculated using the vehicle model.

(14)

Figure 7: Error inflicted on the pointwise calculated grade as a function of true road grade. The top graph shows the sensitivity in a vehicle using the parameters from vehicle A, the bottom graph corresponds to a lighter vehicle using the parameters from vehicle B.

(15)

Driving at a steady speed of 80 km/h whenever possible was chosen as the working point for this analysis. The parameters are varied one at a time to 120 % of the nominal parameter values, each parameter giving rise to one line in the plot. Road grade is varied under the circumstances mentioned.

In both the lighter and the heavier vehicle errors in vehicle mass and the torque signal are large contributors to the total estimation error. The effect of an error in mass and a proportional error in applied torque only differs by vertical translation and scaling.

Vehicle speed only enters the model in the drag term. When driving in windy conditions, the speed of the air flowing past the vehicle is not equal to the vehicle speed.

In the plots, v describes the effects of this difference. In the lighter vehicle, the effect of wind is significant, even at moderate wind speed. Here, the yellow dash-dotted line corresponds to a longitudinal wind of roughly 5 m/s.

4.2 Distribution fitting

Figure 8: A typical error distribution in an experiment run. The blue histogram shows the relative frequency of errors, the solid line is a normal distribution fitted to experimental data. To the left is the pointwise grade error using only the vehicle model. To the right is the error distribution after grade estimation.

Run µ σ RMSE

1 0.0627 1.6817 1.7140 2 0.0152 1.6369 1.6679 3 -0.1389 1.5805 1.6229 4 -0.3028 1.2234 1.2962 5 -0.6932 1.1624 1.3834

Table 2: Southbound pointwise grade calculation (% grade)

(16)

Run µ σ RMSE 1 -0.2901 1.6973 1.7569 2 -0.3278 1.6426 1.7078 3 -0.3443 1.6420 1.7130 4 -0.6760 1.1692 1.3824 5 -0.6616 1.1604 1.3674

Table 3: Northbound pointwise grade calculation (% grade)

Errors in experimental data are analyzed to form the basis for simulated errors.

The grade is calculated pointwise from the vehicle data and model, using a forward difference approximation to the acceleration. Also, the grade is estimated using the grade estimation method. For each run the grade error is formed as the difference of computed grade and the reference grade. A histogram of grade errors after pointwise calculation and grade estimation in a typical run is shown in Figure 8. Also, a normal distribution is fitted to the data, the fitted distribution is plotted as a solid line. It is clear that the errors are not normally distributed, but the distribution is an adequate description of the shape and is used to measure of the size of the error distribution.

Parameters µ and σ for the distributions fitted to the pointwise errors of the runs can be are found in Table 2 and Table 3 together with the Root Mean Square Error (RMSE).

4.3 Simulated vehicle

In the simulation, the vehicle model as described in Section 2 is used. A vehicle to be simulated is defined by setting vehicle parameters. Simulation is done in the spatial domain. An initial state is chosen and the model equation is used to determine vehicle speed at each point along a chosen road. The applied torque is determined by a PI controller in each step, attempting to keep a set speed of 80 km/h whenever possible. If the speed exceeds 89 km/h, brakes are applied.

Input for road grade and altitude is the same reference data that the experiment measurements are compared to. This data is collected with a high precision GPS unit coupled to an INS. In the specifications, precision of this instrument is claimed to be 0.05 % grade (1σ).

A preliminary interview study was conducted, establishing that very little is known about how the parameters and quantities measured in the model vary from their assumed or measured value. At best, a number can be guessed based on experience. This leaves the alternative of examining experimental data and try to obtain a measure of the error distribution. Then the simulation model can be disturbed within the the approximate range obtained in the interview study together with some knowledge about the physical nature of underlying system. If the distribution of error in the simulation of a disturbed system matches the distribution from experimental data, the simulation model could be used to investigate the behavior of the estimation on a larger and more controlled set of data than is available from actual experiments.

(17)

Since no data is available on the distribution of errors in production vehicles an ad- hoc strategy of applying Gaussian noise to the vehicle parameters was chosen. Noise applied to the mass, drag and torque is randomized every run.

4.3.1 Engine

The engine mass moment of inertia is small relative to the total vehicle mass, even when translated through transmission ratios and efficiencies at a reasonable working point of the HDV. Since it would require a large error in engine mass and engine friction to influence the grade estimation result and it seems reasonable that these quantities could be measured with good precision in a lab, no errors are modeled here.

4.3.2 Transmission

In the gear box the ratio of rotational speeds on the inbound and outbound axis is determined by the number of cogs in the gears. This ratio is well known. The gear box and final gear have efficiencies measured in a caloric chamber, these numbers are also considered well known and no errors are modeled.

4.3.3 Wheels

The wheel radius is a well known parameter compared to the modeled errors, and it is possible to calibrate this parameter by comparing integrated wheel speed over a length of road with the distance traveled measured by GPS. Like in the case of the engine, the rotational inertia here is small compared to the total mass of the vehicle. No errors are modeled.

4.3.4 Vehicle mass

To estimate the vehicle mass is a difficult problem. According to interviews errors of 10−15 % are common. A large part of the total grade error is attributed to error in vehicle mass. In the simulation, a random number drawn from a normal distribution with µ = 1 and σ = 0.2 is multiplied by the true vehicle weight to determine the weight to be reported for later grade estimation. This is most likely an overestimation of the error, but reasonable in an attempt to establish an upper bound for the grade error.

4.3.5 Drag

The drag term depends on two parameters related to the shape and size of the vehicle:

the front area A and the drag coefficient c_d. It also depends on the air density ρ. Lacking information about the variation in these parameters, an error of the total drag term is modeled by a normal distribution in the same way as for the mass.

Since the drag force depends on the velocity of air flowing past the vehicle, wind affects the total drag. In the model used for grade estimation no wind is assumed. Wind

(18)

is not explicitly modeled in the simulation, but is included as a constant error under the assumption that changes in wind speed and direction are slow.

Total drag is multiplied by a random number drawn from a normal distribution with µ = 1 and σ = 0.2. This assumed error is probably also an overestimation.

4.3.6 Rolling resistance

Rolling resistance is a parameter that varies with different road surfaces. This factor is also temperature dependent. When the tire is at operational steady state temperature less energy is lost due to hysteresis than when the tire is cold. However, these effects are neglected in this simulation assuming steady state temperature has been reached.

To account for the fact that the rolling resistance varies from the assumed value, an error of the same nature as above is induced. c_r is multiplied by a random number drawn from a normal distribution with µ = 1, σ = 0.2.

4.3.7 Vehicle speed

The wheel speed signal in experimental data contains noise. The high frequency component of this noise generates a large error in estimated grade when the wheel speed signal is differentiated to obtain the vehicle acceleration. This noise is modeled as Gaussian white noise in the simulation with µ = 0 and sigma chosen to yield a pointwise grade error distribution of similar width as the distributions seen in experimental data.

4.3.8 Torque estimate

For the torque estimate, an error in the maximum available torque is set. Since the torque signal is given as a percentage of maximum torque this scales the entire range of applied torque. Maximum torque is scaled by a random number drawn from a Gaussian distribution with µ = 1, σ = 0.2.

5 GPS

One important component of the estimation method is a GPS receiver. From the receiver the vehicle position can be obtained, which is essential to the mapping mission. Also, the GPS receiver reports the vehicle altitude. This measurement is unbiased when taken over a sufficiently long time. This property is important in the grade estimation method since the grade estimation by vehicle model only is prone to bias errors. By including altitude measured by GPS this bias can be compensated for.

The GPS system is comprised of three segments. The space segment consists of positioning satellites broadcasting a time signal, their position and some diagnostic information. On earth the control segment monitors the satellites and transmits time and position corrections to the satellites. The user segment is the receivers that measure the time it takes for radio signals to travel from satellites to receiver antenna. From

(19)

the combination of position and time information gathered distance to each satellite the location and velocity of the receiver is determined.

5.1 GPS error

There are many components that lead to errors in the GPS system. Actual satellite position compared to the broadcast position, atmospheric conditions, radio signals bouncing on terrain and objects causing multipath errors and a receiver that tries to interpret a very low power signal in an environment full of radio noise.

The errors accumulate throughout this system to give a positioning error in the user end, each error component with its own characteristics. A large part of the error comes from atmospheric delays. These errors vary slowly over the course of time. The GPS position signal typically has a small relative error over times that are short compared to the timescale of the atmospheric changes. A typical time for the change of atmospheric conditions that affect GPS navigation is 24 hours. Since the experiments for grade estimation are much shorter, a large part of the absolute error in altitude manifests in the measurements as a bias.

The primary task of the estimation method is to find the road grade. A near-constant error in absolute altitude is not of much concern since the altitude is differentiated to find a road grade. However, the higher frequency errors influence the estimation result and need to be modeled. In [8] it is suggested that the GPS error may be modeled as a Markov chain. A discrete-time Markov chain with state transition probabilities inferred from experimental data was deemed to be a sufficiently accurate model of this error component.

5.2 GPS model

Experimental data used to create the model consists of a time series of altitude measurements taken while driving. This data is processed to form a set of evenly spaced points in space with interpolated values of the measured quantities. The sampling rate of the measurements used to train the Markov chain is therefore not constant in time. Since the speed is kept in a narrow range most of the time and similar in all experiments, this violation of the stationarity criterion for a Markov process is acceptable.

A state model with 61 states representing vertical errors of −15 to +15 meters in half meter increments is chosen. The number of states is denoted by m. The reference altitude obtained in the high precision GPS/INS measurements is subtracted from the altitude data in the grade estimation experiments, resulting in a series of GPS errors.

The runs have a different altitude bias due to the slow atmospheric changes between experiments. In order to preserve stationarity, the mean altitude error is subtracted from the GPS error series to obtain an error series with mean 0. The errors are binned into the model states by rounding and errors of more than 15 m are put into the −15 m or +15 m bin.

The m × m frequency matrix F is computed by counting the number of transitions in the discretized error series. F_ij is the number of transitions from state i to state j in

(20)

all the runs used.

The m × m transition probability matrix G is computed by scaling the rows of F .

G_ij =











Fij m

P

i=1

Fij

if

m

P

i=1

Fij > 0

0 if

m

P

i=1

F_ij = 0

5.2.1 GPS error generation

To generate a new N point error series for simulation, an initial state in ε⁽¹⁾, the first element in the series, is chosen. For each point ε^(k), k = 2 . . . N , the state is chosen at random according to the probability distribution given in row ε^(k−1) of G. A low-pass filter is applied to the generated error signal to obtain an error that varies smoothly.

5.2.2 GPS model verification

The effect of the simulated GPS error on the grade estimation method is examined by substituting altitude data in the experimental datasets with reference altitude plus simulated vertical error. Using the simulated data, the estimation result is typically worse than in the original experimental dataset. The worst runs with simulated data, however, are similar in total error to the worst runs with experimental data, but runs with large RMSE are more common. Since the model does not underestimate the error it was deemed to be satisfactory in this aspect.

6 Simulation output

The simulation script outputs data that is analyzed with the pointwise grade calculation and the Kalman filter grade estimation method. In some aspects the simulated data differs from experimental data.

6.1 Raw error

To the left in Figure 9 is a histogram of the error when grade is calculated pointwise from a typical simulated run. The raw data pointwise error has a distribution that is closer to the Gaussian distribution than the error distribution after grade estimation.

The simulations do not include errors present in experimental data such as varying wind that may not be normally distributed within one run.

6.2 Estimation error

Although the raw data distribution is similar to the experimental data in both size and position, the error in the estimate from simulated data has a flatter error distribution.

This indicates that the errors introduced in the simulation cannot be compensated for

(21)

Figure 9: A typical error distribution in a simulated run. The blue histogram shows the relative frequency of errors, the line is a normal distribution fitted to experimental data.

To the left is the pointwise grade error using only the vehicle model. To the right is the error distribution after grade estimation.

quite as well as the errors in the experimental data. This is due to influence of GPS errors. Simulated GPS data are calibrated to be pessimistic in the sense that their influence on grade estimation is ment to be at least as severe as measurement data.

6.3 Convergence

In an attempt to test the performance over many runs, 800 simulated runs on the northbound reference grade were divided into 20 sets of 40 runs. In each set the runs were merged to form a map of the road grade. At first, the map consists of the estimated grade in one run. Then the other runs in the set are added one by one. For each run added to the map, the RMSE of the map is calculated. In Figure 10 the RMSE at each step in the creation of maps from the 20 sets are shown.

In Figure 11 the mean RMSE for each number of runs added in the sets is plotted in blue. For the first ten runs the RMSE decreases as N^−c where N is the number of runs used and c in this case is close to the theoretical 0.5. The dashed red line is fitted to the first ten experiments.

After ten experiments the rate of convergence decreases. The reference road grade has high frequency components that are not recreated in the grade estimation method due to its low-pass filtering property. This causes the error to converge to some value larger than 0.

(22)

Figure 10: Accumulated map error as a function of the number of simulated runs used in the creation of the map. The 20 solid blue lines are sets of 40 runs each. The dashed red line is a function fitted to first ten runs of an average of the sets.

Figure 11: Log-Log plot of the data in Figure 10. The blue line is the average of 20 sets of 40 runs each. The dashed red line is the fitted function.

(23)

7 Vehicle model errors

The vehicle model and its parameters differ from the vehicle whose behavior it is meant to capture. These differences cause errors in the grade estimation based on the model.

Some of the differences are examined here by investigation of the experimental data.

7.1 Mean error

A large mean grade error can be seen in eight out of ten experiments when grade is computed pointwise, see Table 4. The four runs made with the lighter vehicle exhibit the largest bias, 0.69 % grade. This bias is as large as the entire modeled rolling resistance.

To compensate for this error, GPS altitude data is used in the estimation method.

Vehicle data is assumed to be so noisy that the bias is small compared to the error component modeled as white noise. In the resulting estimate the bias component of the error is well compensated for by relying more on the GPS. By setting the elements of Q and R the vehicle data is treated as a signal with large amounts of white noise, thus giving more weight to GPS measurements and compensating for the vehicle model grade bias. This introduces more errors from the GPS in a frequency range where the vehicle model has a smaller error. Since the GPS data introduces larger errors in the estimate than the velocity and torque signals when bias is disregarded, it is in some cases beneficial to compensate for vehicle model bias error before processing and decrease the weight on GPS data in the Kalman filter.

Run Southbound Northbound

1 0.0628 -0.2931

2 0.0164 -0.3304

3 -0.1424 -0.3434

4 -0.3028 -0.6784

5 -0.6941 -0.6620

Table 4: Mean grade error (% grade)

7.2 Warm-up and slowly varying errors

In some runs there is not only a static bias. There is also a component of the error that varies slowly over the length of the experiment. The theory that some of the non- stationarity can be explained as a warm-up phenomenon was checked against time stamp information in the original data files from the experiments. Runs where the negative bias drift is most evident were made in the morning, implying that the truck was cold from standing still overnight.

In southbound run four the warm-up effect is very clear, as can be seen in Figure 12.

The upper plot shows the error in the estimate using only vehicle data. The vehicle model grade bias drifts towards the negative over the course of the experiment. The

(24)

lower plot is the error in the estimate of this run when using both vehicle and GPS data.

Here, the bias is well corrected for.

Figure 12: Road grade estimated from one run.

Top: Error using only the vehicle data, exhibiting a drifting bias.

Bottom: Error in estimation using both vehicle and GPS data.

To illustrate the vehicle model bias, the difference of mean pointwise calculated grade and mean reference grade in 1250 m segments is plotted as red dots in Figure 13. In an attempt to capture the model bias using only experimental data, this error is estimated by averaging the difference of the pointwise calculated grade based on the vehicle model and a grade estimate made using the estimation method. The difference is averaged over segments of 500 data points, equivalent to 1250 m of road. For this purpose, the design parameters of the estimation method are adjusted to give greater weight to the GPS data, resulting in an estimate that has a large RMSE but very little bias.

(25)

Figure 13: The stars mark estimated error in 1250 m segments calculated by comparing the pointwise grade from the vehicle model to the grade in an estimation with large weight given to GPS data. Red dots mark the error in vehicle model grade compared to the reference. Blue solid and red dashed lines are fitted to estimated and true error respectively.

7.3 Mass

A correlation between estimation grade error and reference road grade is found in the experimental data. This correlation is how an error in vehicle mass or the maximum available torque would manifest, according to Figure 7. When the vehicle is driving uphill the cruise control or driver will apply more torque attempting to keep speed constant.

Therefore, applied torque and road grade is correlated in the experiments and it is not possible to say with any certainty whether the observed error is due to an error in vehicle mass or in the estimation of applied torque.

In Figure 14 the red dots relate the mean reference road grade in each 1250 m segment to the mean vehicle model grade error in that segment. The red dashed line is fitted to the red reference dots. It is desirable to find this relation in experimental data to be able to correct for this error in the model. When the slope of this line is known, the input data can be corrected by changing either the vehicle mass or maximum available torque. In this case vehicle mass was chosen as the parameter to add the correction to, but the correction is considered to be a model fitting parameter rather than an actual mass.

The blue stars represent an estimation of the relation between vehicle model error and road grade using only experimental data. As in Section 7.2, a GPS-heavy grade estimate is used as a replacement for the grade reference. The blue solid line is a robust fit to the blue stars, representing the relation found in experimental data.

To give an indication of the certainty of this estimate the relation of estimated line slope k and the sample standard error σ_k is examined. In Figure 14 the dash dotted green lines show the slopes k ± 2σ_k.

(26)

Figure 14: The stars mark estimated mean error in 1250 m segments calculated by comparing the pointwise grade from the vehicle model to the grade in an estimation with large weight given to GPS data. Red dots mark the error in vehicle model grade compared to the reference. Solid blue and dashed red lines are linear functions fitted to estimated and true error respectively. Dash-dotted green lines delimit the ±2σkinterval used to determine the significance in the mass correction.

8 Error impact

A naive method for correcting errors detailed in Section 7 is described here. This experiment with a modified estimation method is only meant to verify the structure of the errors, give an approximate figure of their impact on the estimation error and provide an argument for including a model of these errors in a grade estimation method.

The general idea is to add a structure for bias and mass errors in the vehicle model.

Each run is processed separately. In the first step, the grade estimation method is applied to experimental data. In this estimation the tuning parameters in the Q and R matrices of the Kalman filter are modified to rely more on GPS data. Then, the difference of the GPS-heavy estimate and pointwise grade calculation is used to determine error parameters. The bias, warm up and mass errors are corrected for in the original set of vehicle data and vehicle parameters. When the estimation is run on this modified data less weight can be given to GPS altitude in the Kalman filter by, again, changing the Q and R matrices. The result is a grade estimate less sensitive to GPS errors without excessive bias errors being introduced from the vehicle model.

8.1 Correction procedure

The estimation method is run on the experimental data with more weight on the GPS altitude data and less on the vehicle model, resulting in an estimated grade α_G. It is expected that αG will be a poor grade estimate in terms of RMSE. When averaged over

(27)

longer distances this estimate is closer to the reference grade.

Grade is also computed directly from vehicle data by means of the vehicle model and a forward difference on the velocity signal to obtain the acceleration. This grade is denoted by αp. The difference of this pointwise grade and the estimated grade is used as an approximation to the error in the vehicle model,

em= αp− α_G. (10)

This error is modeled as a linear function of road grade, em= k ˜α + c, where ˜α is the true road grade. To calculate parameters k and c, ˜α is approximated by α_G. Averaging e_m and α_G over segments of 3000 m, much of the high frequency noise from the speed sensor and GPS cancel out. A line is fitted to the averaged values, thus k and c are obtained. The quality of the fit in k is gauged by comparing k to the standard error in k, σ_k. If 2σ_k > |k| no correction is made. If the correlation passes the significance test, a new estimated mass mc is calculated from the assumed mass m.

m_c= m(1 + k) (11)

With this new value for vehicle mass, the pointwise grade α⁰_p is recalculated. A new approximation to the vehicle model error e is formed in the same manner as in Equation (10):

e = α⁰_p− α_G. (12)

The error is modeled as e = ax+b where x is the distance traveled. The same averaging and fitting procedure is carried out. To insert this correction into the measurement data, the torque signal is manipulated. Each discretization point is associated with a distance x and the amount of applied torque at that point. The equivalent amount of torque required to compensate for the grade ax + b is computed and added to the torque signal.

8.2 Correction results

Table 5 and Table 6 give the performance figures on the modified method. These can be compared to the results of the original estimation method in Table 7 and Table 8.

RMSE(v) and RMSE(a) is the root mean square error of the run. The first is the RMSE of all data points where the vehicle model is valid, i.e. when no braking or gearshift occurs. The latter is the RMSE of all data points in the run except the first and last 200 points. At the beginning and end of the dataset, the grade estimation method has not yet found an equilibrium. Excluding the some of the first and last data is a way of avoiding to include such effects into the result. RMSE(m) is the merged measurement of all previous runs using the data fusion method mentioned in Section 3. Here the first and last 200 points are also excluded for the same reasons as before. µ and σ are calculated as described in Section 4.2.

(28)

Run µ σ RMSE(v) RMSE(a) RMSE(m) 1 0.0210 0.1098 0.1117 0.1109 0.1109 2 -0.0148 0.1235 0.1244 0.1873 0.0862 3 0.0011 0.1355 0.1355 0.1878 0.0846 4 0.0599 0.2282 0.2359 0.2372 0.0973 5 0.0423 0.2352 0.2390 0.2467 0.1091 Table 5: Southbound modified grade estimation error (% grade)

Run µ σ RMSE(v) RMSE(a) RMSE(m)

1 -0.0030 0.1250 0.1251 0.1315 0.1315 2 -0.0126 0.0840 0.0849 0.1138 0.1071 3 -0.0148 0.1011 0.1022 0.1391 0.0930 4 0.0686 0.2286 0.2386 0.2308 0.0922 5 0.0982 0.2087 0.2306 0.2251 0.1048 Table 6: Northbound modified grade estimation error (% grade)

1 0.0108 0.1738 0.1741 0.1744 0.1744 2 0.0185 0.1878 0.1887 0.2022 0.1421 3 -0.0126 0.2248 0.2252 0.2948 0.1464 4 -0.0311 0.1891 0.1916 0.1871 0.1269 5 -0.0999 0.2527 0.2717 0.2760 0.1322 Table 7: Southbound original grade estimation error (% grade).

1 -0.0496 0.2104 0.2161 0.2165 0.2165 2 -0.0627 0.2258 0.2344 0.2488 0.2039 3 -0.0525 0.2403 0.2459 0.2780 0.1731 4 -0.0763 0.4078 0.4149 0.4098 0.1747 5 -0.0717 0.2021 0.2144 0.2029 0.1615 Table 8: Northbound original grade estimation error (% grade).

One effect of the modified method seen from the perspective of these tables is that it harmonizes the experiments. There, northbound and southbound runs display approximately the same error in the total grade estimate. The variation in error between runs in the same direction with the same vehicle is also small.

For the heavier vehicle the modified method results in both better estimates for each run and a better fused estimate. It does not improve the performance when the vehicle

(29)

model is invalid, leaving a significant difference in RMSE when including or excluding points where braking or gearshifts occur. The total bias over the run is not affected, µ is still small in all runs. When only considering points where the vehicle model is valid, the modified method yields a 46 % decrease in RMSE(v) on average compared to the original method. The smallest relative improvement is seen in southbound run 2 with an RMSE(v) decrease of 34 %. Northbound run 2 exhibits the largest improvement.

Here, the RMSE(v) is decreased by 64 %. The merged grade from the three runs in each direction is improved by 46 % (southbound) and 44 % (northbound).

Runs where the lighter vehicle is used do not show much improvement. On average, runs with this vehicle show a small deterioration, the modified method yields an increase in RMSE(v) of 7 %. The worst deterioration is seen in southbound run 4, where the modified method yields a RMSE(v) 23 % larger than the original method. In northbound run 4 the largest improvement is found, a decrease of the RMSE(v) of 45 %. When only the runs with this vehicle in each direction are fused the modified method increases the fused RMSE(m) by 4 % in the southbound direction and decreases it by 17 % in the northbound direction.

Figure 15: The upper plot shows the error in the estimate with the original method as a function of distance. The lower is the error with bias correction, mass correction and lower weight on GPS data. Braking occurs in segments with gray background.

Gearshifts are indicated by vertical black lines over the distance axis.

Some runs exhibit a drastic error reduction when the modified estimation method is

(30)

used. The effect is most significant in northbound runs 1-3 where vehicle A, the heavier vehicle, is used. Figure 15 shows the difference in grade error when the grade estimation methods are run on northbound run 3. It is very clear in the lower plot when brake systems are used or gearshifts occur. Between these times the error is kept at very low levels. This is typical of the northbound runs with vehicle A.

In Figure 16 the same relation is shown for northbound run 4. This is a run where vehicle B, the lighter vehicle, is used. The RMSE for this run decreased by using the modified method, but not much improvement can be seen in the plot. The entire improvement in performance is obtained by mitigating the impact of a large GPS error at 42 km. If this section of the experiment is excluded the performance of the modified estimation method is equivalent to the performance of the original method. This is in line with what is observed in the other runs made with vehicle B. It seems that the errors in the corrected vehicle model are of approximately the same severity as the errors in the GPS. The effect of using the modified method is that the grade estimation error becomes more predictable and less sensitive to the some times erratic behavior of the GPS.

Figure 16: Error in northbound run 4.

Top: Using original estimation method.

Bottom: Using modified estimation method.

(31)

Figure 17: Accumulated map after merging the three northbound runs with vehicle A.

In the top plot the original estimation method is used. In the lower plot the modified method is used. Braking occurs in segments with gray background. Gearshifts are indicated by vertical black lines over the distance axis.

Just as in the individual runs the merged grade estimate from the three runs made with the lighter vehicle is improved by using the modified grade estimation method.

This is illustrated in Figure 17. The gray areas mark sections of road where a vehicle used a braking system in any of the runs used in the creation of the merged estimate.

The vertical black bars indicate that a gearshift occurred at that point in any of the runs used. A bad estimate in a portion of one run has a significant effect in the merged grade estimate, even when it is known that the quality of the data from that segment of that run is poor.

(32)

Figure 18: Accumulated map after merging all five northbound runs. In the top plot the original estimation method is used. In the lower plot the modified method is used.

Braking occurs in segments with gray background. Gearshifts are indicated by vertical black lines over the distance axis.

When all five northbound runs have been merged, illustrated in Figure 18, the difference in results between the methods is smaller. The fourth run with a large GPS error leaves a mark in the merged grade when using the original method.

8.3 Discussion

The parameters used in the vehicle model are chosen according to best knowledge of the vehicles used in experiments. The mass of the vehicles is measured by weighing them on a scale. The vehicles used in the experiments are lab vehicles that are not in use as much as would be expected for a vehicle used in customer operations. Even so, systematic errors are present in the vehicle model. Some appear to be static, some vary in time while the vehicle is in operation. It is plausible that vehicles in heavy use would exhibit even larger variations in vehicle parameters.

In the original estimation method, the errors in the vehicle model are assumed to be Gaussian white noise. Some of the noise in the vehicle is of this type, but in the experiments a significant portion of the vehicle model error is not. If these errors are addressed by modeling and estimation there is a potential for improvement, as described by the numbers in Section 8.2, at least for individual runs and merged maps of a smaller

(33)

number of runs.

For the heavy vehicle, the improvement in the modified method is substantial. The residual error of the merged grade is reduced by 45 % on average on the two roads where experiments have been conducted. The RMSE of 0.08 % grade and 0.09 % grade approaches the accuracy of the reference grade used. The difference between the reference grade and grade data from the digital map provider Navteq is 0.07 % grade in the root mean square sense.

Grade estimation using a lighter vehicle is more sensitive to wind and other distur- bances. The grade error is larger for every Newton of error in the driving resistance when a light vehicle is used. This is why the modified method does not improve grade estimation performance in vehicle B. The corrected vehicle model does not perform much better than the GPS in vehicle B, while the difference is significant in vehicle A. The modified method does, however, move the vulnerability to input errors away from the GPS, and consequently increases the vulnerability to errors in the vehicle model. In the GPS altitude signal, large local errors are common and in the current method no method of detecting them is implemented. The vehicle model exhibits large local errors where brakes are applied or gear shifts occur. During normal driving conditions large errors are rare in experimental data. Braking and shifting events are indicated in vehicle data and easily detected. Some of the results indicate that more corrections in the time varying components of the grade error can be made for the lighter vehicle, this might yield some grade estimation improvement in vehicle B.

In the original method the Q and R matrices of the Kalman filter are seen as tuning parameters. They are tuned on a training set of experiment data. In the modified method, these have been chosen manually to yield ”good” results on a subset of runs.

Also, an additional tuning parameter is introduced: the number of points of each segment in the bias and mass correction steps. While care has been taken to avoid overfitting, there is a risk that some of the improvement is due to such effects. There may also exist a potential for further improvement if a more careful tuning procedure is carried out.

If a more general method is developed where vehicle model errors are modeled, then analysis and tuning will be a major task.

When the vehicle model is relied on too much in the estimation method, a risk exists that systematic errors are introduced in the grade map. The original estimation method introduces GPS errors to a larger extent but corrects very well for most of the vehicle model errors. In some situations, the effects of GPS errors can be very large. It takes many runs to cancel out these errors, but it may be possible to find these errors in the data and reduce their impact. Over long periods of time and many runs the GPS errors can be expected to be unbiased. Whether this is true for the vehicle model is difficult to determine, even if many different vehicles are used in many runs over a long period of time.

(34)

9 Conclusions and Future work

The error distribution of the simulated runs resemble the ones found in experimental data. The GPS error is modeled in a somewhat pessimistic way, but the overall behavior seems to be captured in the model. The simulations show that under the assumptions made the grade estimation method will exhibit very fast convergence. Some caution must be used when interpreting these results. The modeled errors in the convergence test have very little in common with actual vehicles. It is the best test available to examine the long term behavior of the grade estimation method without extensive and expensive road- and/or lab tests, but the limitations of this approach must be kept in mind.

It was found that the bias, bias drift and mass/torque errors mentioned in Section 7 can be partially corrected for with a modified estimation procedure, thus improving the total estimation result. Since there are only a few short experiments made with two vehicles that are not subjected to heavy use, there are limits to what conclusions can be drawn. The observations and corrections presented in Section 8 make it plausible that the estimation method would benefit from an error model that includes some errors that are not currently taken into account explicitly.

The modified grade estimation method presented here is not a suitable choice of method for a real world implementation. A more general method is needed, capable of dealing with runs of varying lengths and possibly different noise characteristics. Looking at Figure 13, it seems there is potential for estimating a bias with wavelengths of a few kilometers. It is also clear that when large GPS errors occur, like the ones at around 25 km in the same plot, the bias estimation may yield incorrect results if care is not taken.

The findings of correctable bias in the experimental data open an avenue for future work. In [9] two methods of dealing with bias in a Kalman filter are reviewed. Either the bias is explicitly modeled in the filter, adding a bias or bias drift state, or a separate bias estimation Kalman filter is used. The estimated bias is later used in the primary grade estimation Kalman filter. The method used to show feasibility of correction in this thesis is similar to the separate bias estimation approach. If a proper implementation of the corrections is to be made, this seems like a smooth way of introducing it into the current grade estimation framework. Compared to the situation explored in [9], there is a considerable advantage in the grade estimation case. In the grade estimation setting two separate sources of road grade data are available: GPS measurements and vehicle model derived grade. Of these, only the vehicle model has a long term bias.

(35)

References

[1] Per Sahlholm. Distributed Road Grade Estimation for Heavy Duty Vehicles. PhD thesis, Royal Institute of Technology (KTH), March 2011. TRITA-EE 2011:008.

[2] Carsten Hatger and Claus Brenner. Extraction of road geometry parameters from laser scanning and existing databases. In International Archives of Photogrammetry, Remote Sensing and Spatial Information Sciences, pages 225–230, 2003.

[3] Hiroshi Ohnishi. A study on road slope estimation for automatic transmission control.

JSAE Review, 21:235–240(6), April 2000.

[4] Hong S. Bae. Road grade and vehicle parameter estimation for longitudinal control using gps. In Proceedings of IEEE Conference on Intelligent Transportation Systems, 2001.

[5] Pratap Misra and Per Enge. Global positioning system : signals, measurements and performance. Ganga-Jamuna Press, 2006.

[6] J. Rankin. An error model for sensor simulation gps and differential gps. In Position Location and Navigation Symposium, 1994., IEEE, pages 260 –266, apr 1994.

[7] U. Kiencke and L. Nielsen. Automotive Control Systems. Springer Verlag, Berlin, 2003.

[8] David Allerton. Principles of Flight Simulation. John Wiley & Sons, 2009.

[9] Jean-Philippe Dr´ecourt, Henrik Madsen, and Dan Rosbjerg. Bias aware kalman filters: Comparison and improvements. Advances in Water Resources, 29(5):707 – 718, 2006.