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Evaluation of fractal terrain model for vehicle dynamic simulations
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Journal of Terramechanics Journal of Terramechanics 49 (2012) 299–307 www.elsevier.com/locate/jterra
Evaluation of fractal terrain model for vehicle dynamic simulations Jeremy J. Dawkins ⇑, David M. Bevly 1, Robert L. Jackson 2 1418 Wiggins Hall, Department of Mechanical Engineering, Auburn University, AL 36849, USA Received 2 March 2012; received in revised form 22 August 2012; accepted 5 October 2012
Abstract Fractals are a popular method for modeling terrains that include various scales. This paper investigates the effectiveness of using fractals for generating artificial terrains which can be used for vehicle simulations. The 3-D Weierstrass–Mandelbrot function was used to generate surfaces based on experimentally measured terrains. There is an exponential relationship between the root means squared elevation of the surfaces and the fractal scaling parameter. This relationship was used to determine the required fractal parameters to generate a surface with a desired roughness. A light detection and ranging (LiDAR) sensor coupled with a global positioning system (GPS) and inertial navigation system (INS) was used to measure two off road surfaces. The experimental terrain was then compared to the simulated terrain. Based on the comparison, the fractal model can capture the general roughness of the experimentally measured terrains as determined by the dynamic response of a suspension model. However, the fractal model fails to capture some of the nuances and nonperiodic events observed in experimental terrains. Ó 2012 ISTVS. Published by Elsevier Ltd. All rights reserved. Keywords: Fractal terrain models; Terrain vehicle interaction; Root mean squared elevation; Off-road vehicle dynamics
1. Introduction The roughness of the terrain will have a significant impact on the dynamic behavior of a vehicle driving off-road. It is desirable to conduct experiments on off-road terrains to better understand the vehicle dynamics under varying conditions. Unfortunately, experimental testing of off road vehicle dynamic maneuvers can prove to be problematic for various reasons. It is difficult to find testing locations which accurately represent the terrain on which the vehicle will be required to operate. Additionally, off-road vehicle dynamic experiments can be more dangerous than their on-road counterparts. Developing these methodologies in simulation allows one to cover a wider range of scenarios while avoiding the danger and expense of running numerous
⇑ Corresponding author. Tel.: +1 334 844 3267.
E-mail addresses: [email protected] (J.J. Dawkins), bevlydm@au burn.edu (D.M. Bevly), [email protected] (R.L. Jackson). 1 Tel.: +1 334 844 3446. 2 Tel.: +1 334 844 3340.
vehicle tests. Thus, it is desirable to develop methods for generating artificial terrains for the purpose of vehicle dynamic simulation. There have been several works which have generated terrain profiles for the purpose of vehicle modeling. Li and Sandu have developed methods of modeling profiles using polynomial chaos methods [1] and using ARMA models [2], both of which are effective in generating profiles which can be used for simulation. A work by Lee and Sandu used stochastic partial differential equations to model profiles [3]. The ARIMA models used by Kern and Ferris can also be used to generate rough terrain profiles accounting for the non-stationary nature of the terrain profiles [4]. Another common method for modeling terrain roughness is by using fractals. A fractal profile is one which has the property of self-similarity, which is to say that smaller scales are a reduced size copy of the whole [5]. The most popular uses for fractal terrain models have been for large scale elevation data. However, fractals have also been used for modeling smaller scales of terrain roughness relevant to the vehicle dynamic response. Howe et al. have
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generated 2-D profiles using fractal Brownian motion, implemented with the midpoint displacement method [6]. A previous work has demonstrated how fractal terrain profiles of varying roughness can be generated with the 2-D Weierstrass–Mandelbrot function [7]. There are several methods for using 2-D terrain profiles for vehicle simulations. Depending on the desired fidelity of the model, a quarter car model, a planar heave and pitch model (half car), or a full suspension model assuming the left and right wheel paths have the same terrain input. It is also possible to generate two separate profiles for the left and right wheel paths of the vehicle for the full car simulation. However, for higher fidelity vehicle models it is beneficial for the vehicle simulation to use fully 3-D surface models. Although it can be assumed that there is an effective 2-D profile for any trajectory the vehicle may travel, this effective 2-D profile may be difficult to determine. The 3-D surface also allows the vehicle response to be more accurately determined with regards to lateral dynamics. Several of the 2-D methods can be extended quite nicely to 3-D. Li and Sandu extended the ARMA model to create 3-D terrain surfaces [2]. They simulated quarter car suspension models on the terrains for validation with the international roughness index. Additionally Wang et al. developed methods of generating 3-D fractal terrains using Brownian motion for vehicle testing in simulation, but the implications on the vehicle simulation aspect was not thoroughly discussed [8]. It has also been shown that the Weierstrass–Mandelbrot function can be extended to 3D which can be used for vehicle simulations [9]. Durst et al. [10] introduced a related work which developed a method to predict the small scale RMS roughness of a 2-D terrain profile based on a lower resolution scan by using the fractal dimension. Using a linear regression between the fractal parameters, the RMS roughness, and the PSD fit parameters. The predicted RMS roughness could then be related to maximum vehicle speeds based on empirical data. While fractals have been used to model the roughness of a terrain in various works, few efforts have discussed the fidelity of the fractal model for vehicle dynamic simulations, especially with respect to experimental terrain data. When predicting or generating the roughness of a terrain surface using a fractal model, one inherently assumes the surface is self-similar. While this may be valid for certain scales of a terrain surface it may break down for others. The primary purpose of this paper is to determine the effectiveness of using the fractal model for capturing the vehicle dynamic behavior of an experimental vehicle driving on a rough terrain. The approach here differs from that presented by Durst et al. since here the roughness is compared by generating surfaces for comparison. Additionally, here the similarity of the specific vehicle dynamic response model is desired. 2. Methodology One implementation of creating a fractal surface is to use the 3-D Weierstrass–Mandelbrot (W–M) function as
developed by Ausloos and Berman [11]. In this method, the surface is built from stacking sinusoids of increasing frequencies up to a maximum frequency level. Fig. 1 shows a simplified graphical representation of four iterations of this process for a 2-D profile. Note this specific example is not intended to be representative any particular terrain profile but simply to demonstrate the concept of the W–M function. This surface elevation is a function of the position in two perpendicular directions. The 3-D W–M function uses superimposed ridges to build the surface. The surface height (z(x, y)) can be described by the following, nmax M X X zðx;yÞ ¼ A cðD3Þn cosð/m;n Þ m¼1 n¼0
( )# 1=2 2pcn ðx2 þ y 2 Þ Y pm þ /m;n cos cos atan X M L
where D2 1=2 G lnc A¼L L M
ð1Þ
ð2Þ
The roughness of the surface is governed by the parameters D and G, the fractal dimension and scaling coefficient respectively. G is a measure of the amplitude roughness which effectively scales the surface, and L is the transverse width of the profile. Since the vehicle will typically travel along a trajectory, the roughness of the terrain to the sides of the vehicle will not generally affect the dynamics. Thus, it is assumed the path is much longer than it is wide. In order to not improperly scale the surface, the transverse width of the profile is used for the parameter L rather than the longitudinal length. The parameter c controls the distribution of the frequency levels in the surface. Depending on the desired surface roughness, c can range between 1.2 and 1.7. M is the number of superimposed ridges used to construct the surface profile and the parameter n is a frequency index with nmax being some upper cutoff frequency. /m,n are random phases among the sinusoids which create the surface. For a 3-D surface 2 < D < 3, and in general the surface will appear more flat when D is close to 2 and become jagged as D approaches 3. Typically the values of these parameters will be in the middle of the range, not near the extremes. The overall roughness of the surface is defined by the combination of G and D. Additionally by randomly selecting the phase shift, unique surfaces can be created using the same fractal parameters. This can be beneficial for simulating a vehicle on multiple scenarios with the same statistical roughness. Note as the frequencies in the surface increase, the amplitude decreases. Thus, at a certain frequency level the changes in the surface will no longer affect the dynamics of the vehicle. Fig. 2 shows four surfaces which were generated using this method, two surfaces with D = 2.25 and two with D = 2.5, each surface has a different G value. It can be seen that surfaces with a large range of roughness can be gener-
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fractal dimension D can be determined from the power spectral density of an experimental surface. However, the accuracy of D determined from this method will be dependent on the appropriateness of the fractal assumption for the surface. The root mean squared elevation (RMSE) or RMS roughness is a common method of describing the roughness of any type of surface, thus it is a popular way to characterize terrain profiles as well. The RMSE can be calculated with the following expression, rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 XN 2 RMSE ¼ Z ð3Þ i¼0 i N Fig. 1. Simplified 2-D graphical representation of generating a fractal rough surface profile using Weierstrass–Mandelbrot function.
ated using this method. In general as G is increased the surface will become rougher for a given D. However, the particular magnitude of G required to yield a desired roughness can change significantly based on the current D. This relationship can be seen by examining Eq. (2) which describes the overall scaling of the surface. When D is closer to 2 the exponent is closer to zero making the quantity less sensitive to variations in G. As D approaches 3 the scaling factor becomes more dependent on G. The effect of this on generating a desired terrain surface is that it requires the correct combination of D and G to be selected in order to yield the desired roughness. It should be noted that the roughness of the surfaces created using the W–M function are not strictly isotropic, although any variations in roughness along different directions are negligible from a vehicle dynamics standpoint. In general the
where Z is the profile height in meters and N is the number of samples. One of the drawbacks to this method is that in order for it to accurately represent the profile’s roughness the profile must be stationary. That is to say, the statistics do not change over the length of the profile. If one considers a profile for which the first half is smooth and the last half is rough, the RMSE for the entire profile will be an average of the two, which does not accurately represent the true profile. Since the profiles being tested here are non-stationary the RMSE can vary for subsections of the profiles. It has also been shown that the measured RMSE is dependent on the length of the profile [12,13]. If the roughness (RMSE) of the generated terrains can be related to the fractal parameters based on the W–M function, the appropriate fractal parameters can be determined to yield the desired surface roughness. To determine the relationship between the RMSE and the fractal parameters several surfaces were generated with various fractal parameters. Fig. 3 shows the RMSE as a function of scaling coefficient G for each of the terrain surfaces generated. The data points are the calculated RMSE values for each
Fig. 2. Examples of surfaces generated using Weierstrass–Mandelbrot function with various values of D and G.
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Fig. 3. Exponential relationship between RMSE and Scaling parameter, for various fractal dimensions D.
of the surfaces. Since there is an exponentially increasing trend in RMSE as G is increased, an exponential equation was fit to each set of data. The regression carried out here is similar to that presented by Durst et al. [10] except for it is applied to simulated fractal surfaces rather than experimental surfaces. It is interesting to observe that there are several combinations of G and D that will yield a surface with the same RMSE. This figure also shows how the range of G values can change based on D. It should be noted that the RMSE will vary based on the length of the profile and the parameter c. The equations shown here were determined assuming the length of the profile is 100 m and c = 1.2. These equations can be used to determine the appropriate G value to generate a fractal terrain surface with the measured surface roughness. The equation determined for the relationship when D = 2.25 can be solved to yield an expression for G as a function of the profile RMSE as seen in 1 Z rms 0:244 G¼ ð4Þ 2:62
beam to measure ranges to objects at which the sensor is pointed. The LiDAR was mounted to the front of the experimental vehicle, shown in Fig. 4, scanning the ground in front of the vehicle. Since the LiDAR is mounted to the vehicle, the motion of the vehicle will affect the measurements taken by the sensor. One of the key challenges in accurately mapping the terrain is to estimate the states of the vehicle in order to account for the change in the LiDAR position and orientation. The experimental vehicle was equipped with a Novatel Propak v3 and a Crossbow 440 six degree of freedom inertial measurement unit for the global positioning system (GPS) and inertial navigation system (INS) respectively. The measurements from these sensors were used in a loosely coupled GPS/INS Extended Kalman filter to estimate vehicle position, velocity and attitudes. The estimated current states from the GPS/INS filter are used to correct the position and orientation of the LiDAR. The result of the blended solution of the LiDAR and GPS/INS system is a 3-D point cloud of the terrain. However, in order to compare the experimental terrain with the simulated terrain it is beneficial to do so using 2-D profiles extracted from the terrain. To determine the longitudinal profile from the terrain map the values which are within a narrow range ±0.05 m of a desired lateral position are determined. If the value for this range is too large, the profiles will have increased noise from the laterally neighboring points. If the value is too small the profile will have gaps where no points exist within the tolerance. Three profiles were extracted from each of terrain maps. Two of the profiles are from the left and right wheel paths and one is from center of the vehicle. It should be noted that the further away from the center of the vehicle the profiles are taken the more error which will be present from lack of resolution and the increased effects of roll dynamics. Longitudinally the profiles have a resolution of approximately 5 cm although this will change based on vehicle velocity. Fig. 5 shows the longitudinal profiles taken from a terrain map of a dirt road.
It should be noted that this equation makes several assumptions about the experimental terrain profile and how it fits the fractal model. First, since it is based on the RMSE it is inherently assuming the roughness of the experimental profile is stationary. Also this equation was developed for the simulated terrain models which were generated, depending on the size and frequency distribution (c) of the profiles, different equations will be determined. 3. Terrain measurement In order to compare the generated fractal terrain to experimental terrains, a system must be developed to accurately map the terrains using sensors aboard the vehicle. A sick LMS-291 light detection and ranging (LiDAR) sensor is used to map the terrain. This sensor uses a scanning laser
Fig. 4. Experimental vehicle Prowler ATV.
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location of the CG is related to the track width given by the parameters tL and tR. The sprung mass is represented by m, and the un-sprung mass is represented by mi,j. The index i refers to the left or right side of the vehicle and the index j can be 1 or 2 referring to front or rear axle of the vehicle. The spring and damping coefficients are given by kij and cij respectively. The values for these parameters were determined from measurements and estimates of the experimental vehicle. The heave motion, roll motion and pitch motion are represented by z, /r and /p respectively. The motion of each un-sprung is represented by wij. For brevity the equations of motion have been omitted. 5. Discussion and results Fig. 5. Profiles extracted from terrain map for dirt road.
4. Vehicle model As a basis for comparison between the experimental and fractal terrains, a vehicle model can be simulated on each and the vehicle response can be compared. In this work, it is desired to obtain a high fidelity suspension model which models not only the vehicle heave, but also the roll and pitch motion. Thus, the seven degree of freedom (7-DOF) full suspension model is implemented to most accurately capture the motion. Shown in Fig. 6, the 7DOF model has a sprung mass connected to an un-sprung mass at each corner by a spring and damper. In this implementation the tires are represented by a linear spring. There are four inputs, one at each corner representing the terrain displacement on the tires. The origin of the model is located at the center of gravity (CG) relative to the wheelbase by parameters a and b. The
To test the effectiveness of the method, experimental tests were conducted on a dirt road and a gravel road. The two surfaces were mapped and terrain profiles for the wheel paths were extracted. The RMSE of the left and right wheel profiles were used to calculate two G values which were averaged together for the value used in the 3-D W–M function. The left and right wheel path profiles were then extracted from an artificial 3-D terrain surface and compared with the terrain profiles used to generate them. The comparison of the experimental terrain profiles and regenerated fractal terrain profiles for the dirt road and loose gravel terrains are shown in Figs. 7 and 8 respectively. It can be seen for both of these cases that the generated fractal terrain profiles are on a very similar scale as the original terrain profile. Following the top part of Fig. 7, there is a general elevation change in the right profile near the 30 m mark. The elevation change artificially inflates the RMSE causing fractal scaling parameter G to be inflated. When the inflated G is used to regenerate the surface, the increase is enough to cause the slightly higher overall
Fig. 6. 7 Degree of freedom full suspension model.
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Fig. 7. Dirt road longitudinal terrain profiles from terrain map (top) compared to longitudinal terrain profiles from a randomly generated 3-D surface using the Weierstrass–Mandelbrot function with parameters determined from the fractal parameter RMSE relationship (bottom).
Fig. 8. Loose gravel longitudinal terrain profiles from terrain map (top) compared to longitudinal terrain profiles from a randomly generated 3-D surface using the Weierstrass–Mandelbrot function with parameters determined from a fractal parameter RMSE relationship (bottom).
roughness of the regenerated profiles. The key distinction between the experimental terrain profiles and the fractal profiles is that the former tends to have the phases of the right and left profiles more aligned. Since the fractal surface is randomly generated there is no guarantee that this trait will be captured for a given profile. In the bottom of Fig. 8 it can be seen that for parts of the profile the left and right wheel paths are closer phase. To compare the frequency content of the experimentally measured and regenerated terrain profiles the power spectral densities for the profiles can be analyzed. Figs. 9 and 10 show the power spectral density (PSD) for the right and left profiles taken from the experimentally measured terrain and the fractal regenerated terrain. It can be seen
that the PSD of the experimental terrain profiles match the PSD of the fractal terrain profiles reasonably well for frequencies higher than 0.2 cycles/m. The magnitudes of the PSD for the fractal terrains are higher than their experimental counterparts for the frequency range of 0.02–0.1 cycles/m. This range is particularly important in determining the roughness of the terrain profile. It can also be seen that the deviation in this range is higher for the gravel surface than the dirt road. The oscillations of the PSD in the fractal surface (black markers) are caused by the discrete nature of the frequency scaling used in the fractal model. The locations of these spikes is related to the parameter c Recall, c determines the spacing of the frequency levels present in the fractal surface. Since the experimental surface has a continuous frequency distribution to the level which can be captured by the resolution, these spikes are not seen in its PSD. This implies there is no parameter c which directly accounts for the scaling of the frequency levels in the experimental surface, an observation also made by Wu [14]. To further compare the effectiveness of the terrain profile regeneration method, the 7-DOF model can be simulated on both the experimental and fractal terrains. The simulation was performed using MATLAB. Since the same model is used on both terrains the effects of the vehicle model error are reduced. This allows the response of the model to be analyzed to determine the similarity of the terrains with respect to the model dynamics. The heave, roll, and pitch from the simulated vehicle response on the experimental and regenerated loose gravel surfaces are shown in Figs. 11–13. It can be seen that the heave and pitch motions have oscillations of similar magnitudes. However, the vehicle simulated on the fractal surface experiences much more roll motion than for the experimental surface. This is caused by the left and right wheel paths being out of phase resulting in more vehicle roll. The experimental profiles tend to be more similar laterally, thus, the vehicle on these profiles does not exhibit the increased roll
Fig. 9. Power spectral densities for dirt road experimental surface and regenerated fractal surface.
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Fig. 10. Power spectral densities for loose gravel experimental surface and regenerated fractal surface.
Fig. 11. Heave motion comparison of 7-DOF model vehicle response on longitudinal profiles taken from experimental loose gravel profile and longitudinal profiles taken from regenerated fractal surface.
Fig. 12. Roll motion comparison of 7-DOF model vehicle response on longitudinal profiles taken from experimental loose gravel profile and longitudinal profiles taken from regenerated fractal surface.
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Fig. 13. Pitch motion comparison of 7-DOF model vehicle response on longitudinal profiles taken from experimental loose gravel profile and longitudinal profiles taken from regenerated fractal surface.
motion. Another observation is that the fractal surface results in a homogenous vehicle response as compared to the real surface. Due to the non-stationary nature of the experimentally measured terrain, there are bumps and discontinuities which cause dynamics to change over the length of the run. There is a peak in each of the signals at near the 32 s mark for the experimental terrain which was caused by a bump. The vehicle simulations were conducted on each of the terrains at four speeds between 5 and 20 mph. As a measure of how similar the vehicle responses were, the RMS of the time response of the various vehicle states was calculated for each of the simulations. The results for the simulations on the experimental and fractal terrains at various speeds for the dirt road and loose gravel are summarized in Tables 1 and 2 respectively. For the loose gravel terrain the fractal generation provides good agreement with the experimental terrain except for in the roll motion for the reasons explained previously. The fractal generated surface for the dirt road terrain performs well, but the values are slightly elevated relative to their experimental counter parts. Again this is caused by the abrupt increase in the elevation of the experimental profile resulting in an increased value of RMSE. Although this method provides a framework for generating fractal terrains which resemble experimental terrains, there are several areas which must be addressed to increase the realism of the terrains. One observation is that the fractal model is not effective in capturing large scale undulations while preserving the detail of the smaller scales. To more accurately represent real terrain profiles, this method needs to be combined with a methodology for generating large scale elevation changes. Additionally, the relationship used in the terrain generation methodology relies on the RMSE which assumes stationarity of the terrain and can be inflated by the large scale changes in elevation. This method could be improved by using a similar methodology
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Table 1 Summary of RMS values for 7-DOF model simulated on longitudinal terrain profiles from dirt off-road. 5 mph
RMS RMS RMS RMS RMS RMS
heave (m) heave rate (m/s) roll (deg) roll rate (deg/s) pitch (deg) pitch rate (deg/s)
10 mph
15 mph
20 mph
Exper.
Fractal
Exper.
Fractal
Exper.
Fractal
Exper.
Fractal
0.06 0.36 5.33 32.08 0.52 4.10
0.07 0.50 5.08 32.11 0.71 6.18
0.08 0.43 4.88 32.46 0.38 4.48
0.07 0.51 5.23 30.18 0.88 8.83
0.06 0.47 3.74 30.38 0.29 4.29
0.07 0.50 4.55 30.24 0.78 9.51
0.07 0.43 3.32 27.00 0.21 3.29
0.07 0.54 4.81 33.39 0.61 8.46
Table 2 Summary of RMS values for 7-DOF model simulated on longitudinal terrain profiles from loose gravel. 5 mph
RMS RMS RMS RMS RMS RMS
heave (m) heave rate (m/s) roll (deg) roll rate (deg/s) pitch (deg) pitch rate (deg/s)
10 mph
15 mph
20 mph
Exper.
Fractal
Exper.
Fractal
Exper.
Fractal
Exper.
Fractal
0.07 0.31 2.77 18.00 0.75 4.64
0.06 0.46 5.45 31.82 0.31 4.29
0.07 0.32 3.32 19.78 0.58 5.55
0.06 0.46 5.27 30.90 0.33 4.51
0.08 0.34 2.96 20.23 0.39 4.78
0.05 0.46 5.34 31.35 0.34 4.88
0.07 0.24 2.50 18.09 0.23 2.94
0.05 0.44 5.52 31.91 0.30 4.53
to relate the fractal parameters to other metrics which do not have these limitations. It was also observed in this analysis that the experimental terrain profiles tend to have similar phases for much of the data collected. To improve the terrain generation methodology it would be helpful to include more coupling between the lower frequency content of the W–M function. This could potentially be implemented by defining the phase shifts of the surface for lower frequencies while randomizing them for higher frequencies. Lastly, the experimental terrains have transients and discontinuities which cannot easily be modeled using the fractal model. Methods need to be developed which will better capture these events.
paths of experimentally measured terrains tend to be more in phase than their fractal counterparts. There are also other non-periodic events present in actual terrains such as bumps, which the W–M function fails to capture. These discontinuities could be implemented by maintaining a database of the geometry of common terrain features such as rocks and sharp undulations. These features could then be randomly distributed on the fractal surface based on the density of the features found in the experimental data. Considering the phases of the lower frequency content and adding discontinuities could improve the fidelity of the simulated terrain with respect to the experimental terrain as determined by the simulated vehicle response.
6. Conclusions
References
The 3-D Weierstrass–Mandelbrot fractal function can be used to generate terrain surfaces for the simulation of vehicle motion. An analysis of the behavior of this function for different parameters was conducted. Longitudinal profiles were extracted from the surface and analyzed. By generating surfaces using the fractal function the relationship between the fractal parameters and the root mean square elevation (RMSE) can be determined. The RMSE of the modeled terrain profiles exhibits an exponential relationship with fractal scaling parameter (G) for a given fractal dimension (D) Additionally, the exponential relationship between RMSE and G can be used to relate experimentally measured terrain profiles to the W–M function to generate terrains similar to experimentally measured terrains, but the equations are limited to surfaces of similar length and resolution. This method can produce terrains which provide comparable results to the experimental terrain based on a vehicle simulation. However, the left and right wheel
[1] Li L, Sandu C. Modeling of 1-D and 2-D terrain profiles using a polynomial chaos approach. In: Proceedings of the 16th international conference of international society for terrain-vehicle systems (ISTVS), Turin, Italy; 2008. [2] Li L, Sandu C. Modeling and simulation of 2D ARMA terrain models for vehicle dynamics applications. J Comm Veh 2007;116(7). [3] Lee R, Sandu C. Terrain profile modeling using stochastic partial differential equations. Int J Veh Syst 2009;4(4). [4] Kern JV, Ferris JB. Preliminary results for model identification in characterizing 2-D topographic road profiles. In: Proceedings of SPIEthe International Society for Optical Engineering, Orlando, FL; 2006. [5] Mandelbrot BB. The fractal geometry of nature. W.H. Freeman and Company; 1982. [6] Howe JG, Lee D-C, Chrstos JP, Myers TT, Allen RW, Gorsich DJ, et al. Analysis of potential road/terrain characterization rating metrics. In: SAE commercial vehicle engineering congress and exibition, Rosemount, IL; 2004. [7] Dawkins JJ, Bevly DM, Jackson RL. Multiscale terrain characterization using fourier and wavelet transforms for unmanned ground vehicles. In: ASME dynamic systems and controls conference, Hollywood, CA; 2009. [8] Wang Q, Guo J, Chen Y. Fractal modeling of off-road terrain oriented to vehicle virtual test. J Zhejiang Univ Sci A 2006;7:287–92.
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[12] Kogut L, Jackson RL. A comparison of contact modeling utilizing statistical and fractal approaches. ASME J Tribol January 2006;128:213–7. [13] Majumdar A, Tien CL. Fractal characterization and simulation of rough surfaces. Wear 1990;136:313–27. [14] Wu J. Characterization of fractal surfaces. Wear 2000;239:36–47.
Journal of Terramechanics Journal of Terramechanics 49 (2012) 299–307 www.elsevier.com/locate/jterra
Evaluation of fractal terrain model for vehicle dynamic simulations Jeremy J. Dawkins ⇑, David M. Bevly 1, Robert L. Jackson 2 1418 Wiggins Hall, Department of Mechanical Engineering, Auburn University, AL 36849, USA Received 2 March 2012; received in revised form 22 August 2012; accepted 5 October 2012
Abstract Fractals are a popular method for modeling terrains that include various scales. This paper investigates the effectiveness of using fractals for generating artificial terrains which can be used for vehicle simulations. The 3-D Weierstrass–Mandelbrot function was used to generate surfaces based on experimentally measured terrains. There is an exponential relationship between the root means squared elevation of the surfaces and the fractal scaling parameter. This relationship was used to determine the required fractal parameters to generate a surface with a desired roughness. A light detection and ranging (LiDAR) sensor coupled with a global positioning system (GPS) and inertial navigation system (INS) was used to measure two off road surfaces. The experimental terrain was then compared to the simulated terrain. Based on the comparison, the fractal model can capture the general roughness of the experimentally measured terrains as determined by the dynamic response of a suspension model. However, the fractal model fails to capture some of the nuances and nonperiodic events observed in experimental terrains. Ó 2012 ISTVS. Published by Elsevier Ltd. All rights reserved. Keywords: Fractal terrain models; Terrain vehicle interaction; Root mean squared elevation; Off-road vehicle dynamics
1. Introduction The roughness of the terrain will have a significant impact on the dynamic behavior of a vehicle driving off-road. It is desirable to conduct experiments on off-road terrains to better understand the vehicle dynamics under varying conditions. Unfortunately, experimental testing of off road vehicle dynamic maneuvers can prove to be problematic for various reasons. It is difficult to find testing locations which accurately represent the terrain on which the vehicle will be required to operate. Additionally, off-road vehicle dynamic experiments can be more dangerous than their on-road counterparts. Developing these methodologies in simulation allows one to cover a wider range of scenarios while avoiding the danger and expense of running numerous
⇑ Corresponding author. Tel.: +1 334 844 3267.
E-mail addresses: [email protected] (J.J. Dawkins), bevlydm@au burn.edu (D.M. Bevly), [email protected] (R.L. Jackson). 1 Tel.: +1 334 844 3446. 2 Tel.: +1 334 844 3340.
vehicle tests. Thus, it is desirable to develop methods for generating artificial terrains for the purpose of vehicle dynamic simulation. There have been several works which have generated terrain profiles for the purpose of vehicle modeling. Li and Sandu have developed methods of modeling profiles using polynomial chaos methods [1] and using ARMA models [2], both of which are effective in generating profiles which can be used for simulation. A work by Lee and Sandu used stochastic partial differential equations to model profiles [3]. The ARIMA models used by Kern and Ferris can also be used to generate rough terrain profiles accounting for the non-stationary nature of the terrain profiles [4]. Another common method for modeling terrain roughness is by using fractals. A fractal profile is one which has the property of self-similarity, which is to say that smaller scales are a reduced size copy of the whole [5]. The most popular uses for fractal terrain models have been for large scale elevation data. However, fractals have also been used for modeling smaller scales of terrain roughness relevant to the vehicle dynamic response. Howe et al. have
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generated 2-D profiles using fractal Brownian motion, implemented with the midpoint displacement method [6]. A previous work has demonstrated how fractal terrain profiles of varying roughness can be generated with the 2-D Weierstrass–Mandelbrot function [7]. There are several methods for using 2-D terrain profiles for vehicle simulations. Depending on the desired fidelity of the model, a quarter car model, a planar heave and pitch model (half car), or a full suspension model assuming the left and right wheel paths have the same terrain input. It is also possible to generate two separate profiles for the left and right wheel paths of the vehicle for the full car simulation. However, for higher fidelity vehicle models it is beneficial for the vehicle simulation to use fully 3-D surface models. Although it can be assumed that there is an effective 2-D profile for any trajectory the vehicle may travel, this effective 2-D profile may be difficult to determine. The 3-D surface also allows the vehicle response to be more accurately determined with regards to lateral dynamics. Several of the 2-D methods can be extended quite nicely to 3-D. Li and Sandu extended the ARMA model to create 3-D terrain surfaces [2]. They simulated quarter car suspension models on the terrains for validation with the international roughness index. Additionally Wang et al. developed methods of generating 3-D fractal terrains using Brownian motion for vehicle testing in simulation, but the implications on the vehicle simulation aspect was not thoroughly discussed [8]. It has also been shown that the Weierstrass–Mandelbrot function can be extended to 3D which can be used for vehicle simulations [9]. Durst et al. [10] introduced a related work which developed a method to predict the small scale RMS roughness of a 2-D terrain profile based on a lower resolution scan by using the fractal dimension. Using a linear regression between the fractal parameters, the RMS roughness, and the PSD fit parameters. The predicted RMS roughness could then be related to maximum vehicle speeds based on empirical data. While fractals have been used to model the roughness of a terrain in various works, few efforts have discussed the fidelity of the fractal model for vehicle dynamic simulations, especially with respect to experimental terrain data. When predicting or generating the roughness of a terrain surface using a fractal model, one inherently assumes the surface is self-similar. While this may be valid for certain scales of a terrain surface it may break down for others. The primary purpose of this paper is to determine the effectiveness of using the fractal model for capturing the vehicle dynamic behavior of an experimental vehicle driving on a rough terrain. The approach here differs from that presented by Durst et al. since here the roughness is compared by generating surfaces for comparison. Additionally, here the similarity of the specific vehicle dynamic response model is desired. 2. Methodology One implementation of creating a fractal surface is to use the 3-D Weierstrass–Mandelbrot (W–M) function as
developed by Ausloos and Berman [11]. In this method, the surface is built from stacking sinusoids of increasing frequencies up to a maximum frequency level. Fig. 1 shows a simplified graphical representation of four iterations of this process for a 2-D profile. Note this specific example is not intended to be representative any particular terrain profile but simply to demonstrate the concept of the W–M function. This surface elevation is a function of the position in two perpendicular directions. The 3-D W–M function uses superimposed ridges to build the surface. The surface height (z(x, y)) can be described by the following, nmax M X X zðx;yÞ ¼ A cðD3Þn cosð/m;n Þ m¼1 n¼0
( )# 1=2 2pcn ðx2 þ y 2 Þ Y pm þ /m;n cos cos atan X M L
where D2 1=2 G lnc A¼L L M
ð1Þ
ð2Þ
The roughness of the surface is governed by the parameters D and G, the fractal dimension and scaling coefficient respectively. G is a measure of the amplitude roughness which effectively scales the surface, and L is the transverse width of the profile. Since the vehicle will typically travel along a trajectory, the roughness of the terrain to the sides of the vehicle will not generally affect the dynamics. Thus, it is assumed the path is much longer than it is wide. In order to not improperly scale the surface, the transverse width of the profile is used for the parameter L rather than the longitudinal length. The parameter c controls the distribution of the frequency levels in the surface. Depending on the desired surface roughness, c can range between 1.2 and 1.7. M is the number of superimposed ridges used to construct the surface profile and the parameter n is a frequency index with nmax being some upper cutoff frequency. /m,n are random phases among the sinusoids which create the surface. For a 3-D surface 2 < D < 3, and in general the surface will appear more flat when D is close to 2 and become jagged as D approaches 3. Typically the values of these parameters will be in the middle of the range, not near the extremes. The overall roughness of the surface is defined by the combination of G and D. Additionally by randomly selecting the phase shift, unique surfaces can be created using the same fractal parameters. This can be beneficial for simulating a vehicle on multiple scenarios with the same statistical roughness. Note as the frequencies in the surface increase, the amplitude decreases. Thus, at a certain frequency level the changes in the surface will no longer affect the dynamics of the vehicle. Fig. 2 shows four surfaces which were generated using this method, two surfaces with D = 2.25 and two with D = 2.5, each surface has a different G value. It can be seen that surfaces with a large range of roughness can be gener-
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fractal dimension D can be determined from the power spectral density of an experimental surface. However, the accuracy of D determined from this method will be dependent on the appropriateness of the fractal assumption for the surface. The root mean squared elevation (RMSE) or RMS roughness is a common method of describing the roughness of any type of surface, thus it is a popular way to characterize terrain profiles as well. The RMSE can be calculated with the following expression, rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 XN 2 RMSE ¼ Z ð3Þ i¼0 i N Fig. 1. Simplified 2-D graphical representation of generating a fractal rough surface profile using Weierstrass–Mandelbrot function.
ated using this method. In general as G is increased the surface will become rougher for a given D. However, the particular magnitude of G required to yield a desired roughness can change significantly based on the current D. This relationship can be seen by examining Eq. (2) which describes the overall scaling of the surface. When D is closer to 2 the exponent is closer to zero making the quantity less sensitive to variations in G. As D approaches 3 the scaling factor becomes more dependent on G. The effect of this on generating a desired terrain surface is that it requires the correct combination of D and G to be selected in order to yield the desired roughness. It should be noted that the roughness of the surfaces created using the W–M function are not strictly isotropic, although any variations in roughness along different directions are negligible from a vehicle dynamics standpoint. In general the
where Z is the profile height in meters and N is the number of samples. One of the drawbacks to this method is that in order for it to accurately represent the profile’s roughness the profile must be stationary. That is to say, the statistics do not change over the length of the profile. If one considers a profile for which the first half is smooth and the last half is rough, the RMSE for the entire profile will be an average of the two, which does not accurately represent the true profile. Since the profiles being tested here are non-stationary the RMSE can vary for subsections of the profiles. It has also been shown that the measured RMSE is dependent on the length of the profile [12,13]. If the roughness (RMSE) of the generated terrains can be related to the fractal parameters based on the W–M function, the appropriate fractal parameters can be determined to yield the desired surface roughness. To determine the relationship between the RMSE and the fractal parameters several surfaces were generated with various fractal parameters. Fig. 3 shows the RMSE as a function of scaling coefficient G for each of the terrain surfaces generated. The data points are the calculated RMSE values for each
Fig. 2. Examples of surfaces generated using Weierstrass–Mandelbrot function with various values of D and G.
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Fig. 3. Exponential relationship between RMSE and Scaling parameter, for various fractal dimensions D.
of the surfaces. Since there is an exponentially increasing trend in RMSE as G is increased, an exponential equation was fit to each set of data. The regression carried out here is similar to that presented by Durst et al. [10] except for it is applied to simulated fractal surfaces rather than experimental surfaces. It is interesting to observe that there are several combinations of G and D that will yield a surface with the same RMSE. This figure also shows how the range of G values can change based on D. It should be noted that the RMSE will vary based on the length of the profile and the parameter c. The equations shown here were determined assuming the length of the profile is 100 m and c = 1.2. These equations can be used to determine the appropriate G value to generate a fractal terrain surface with the measured surface roughness. The equation determined for the relationship when D = 2.25 can be solved to yield an expression for G as a function of the profile RMSE as seen in 1 Z rms 0:244 G¼ ð4Þ 2:62
beam to measure ranges to objects at which the sensor is pointed. The LiDAR was mounted to the front of the experimental vehicle, shown in Fig. 4, scanning the ground in front of the vehicle. Since the LiDAR is mounted to the vehicle, the motion of the vehicle will affect the measurements taken by the sensor. One of the key challenges in accurately mapping the terrain is to estimate the states of the vehicle in order to account for the change in the LiDAR position and orientation. The experimental vehicle was equipped with a Novatel Propak v3 and a Crossbow 440 six degree of freedom inertial measurement unit for the global positioning system (GPS) and inertial navigation system (INS) respectively. The measurements from these sensors were used in a loosely coupled GPS/INS Extended Kalman filter to estimate vehicle position, velocity and attitudes. The estimated current states from the GPS/INS filter are used to correct the position and orientation of the LiDAR. The result of the blended solution of the LiDAR and GPS/INS system is a 3-D point cloud of the terrain. However, in order to compare the experimental terrain with the simulated terrain it is beneficial to do so using 2-D profiles extracted from the terrain. To determine the longitudinal profile from the terrain map the values which are within a narrow range ±0.05 m of a desired lateral position are determined. If the value for this range is too large, the profiles will have increased noise from the laterally neighboring points. If the value is too small the profile will have gaps where no points exist within the tolerance. Three profiles were extracted from each of terrain maps. Two of the profiles are from the left and right wheel paths and one is from center of the vehicle. It should be noted that the further away from the center of the vehicle the profiles are taken the more error which will be present from lack of resolution and the increased effects of roll dynamics. Longitudinally the profiles have a resolution of approximately 5 cm although this will change based on vehicle velocity. Fig. 5 shows the longitudinal profiles taken from a terrain map of a dirt road.
It should be noted that this equation makes several assumptions about the experimental terrain profile and how it fits the fractal model. First, since it is based on the RMSE it is inherently assuming the roughness of the experimental profile is stationary. Also this equation was developed for the simulated terrain models which were generated, depending on the size and frequency distribution (c) of the profiles, different equations will be determined. 3. Terrain measurement In order to compare the generated fractal terrain to experimental terrains, a system must be developed to accurately map the terrains using sensors aboard the vehicle. A sick LMS-291 light detection and ranging (LiDAR) sensor is used to map the terrain. This sensor uses a scanning laser
Fig. 4. Experimental vehicle Prowler ATV.
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location of the CG is related to the track width given by the parameters tL and tR. The sprung mass is represented by m, and the un-sprung mass is represented by mi,j. The index i refers to the left or right side of the vehicle and the index j can be 1 or 2 referring to front or rear axle of the vehicle. The spring and damping coefficients are given by kij and cij respectively. The values for these parameters were determined from measurements and estimates of the experimental vehicle. The heave motion, roll motion and pitch motion are represented by z, /r and /p respectively. The motion of each un-sprung is represented by wij. For brevity the equations of motion have been omitted. 5. Discussion and results Fig. 5. Profiles extracted from terrain map for dirt road.
4. Vehicle model As a basis for comparison between the experimental and fractal terrains, a vehicle model can be simulated on each and the vehicle response can be compared. In this work, it is desired to obtain a high fidelity suspension model which models not only the vehicle heave, but also the roll and pitch motion. Thus, the seven degree of freedom (7-DOF) full suspension model is implemented to most accurately capture the motion. Shown in Fig. 6, the 7DOF model has a sprung mass connected to an un-sprung mass at each corner by a spring and damper. In this implementation the tires are represented by a linear spring. There are four inputs, one at each corner representing the terrain displacement on the tires. The origin of the model is located at the center of gravity (CG) relative to the wheelbase by parameters a and b. The
To test the effectiveness of the method, experimental tests were conducted on a dirt road and a gravel road. The two surfaces were mapped and terrain profiles for the wheel paths were extracted. The RMSE of the left and right wheel profiles were used to calculate two G values which were averaged together for the value used in the 3-D W–M function. The left and right wheel path profiles were then extracted from an artificial 3-D terrain surface and compared with the terrain profiles used to generate them. The comparison of the experimental terrain profiles and regenerated fractal terrain profiles for the dirt road and loose gravel terrains are shown in Figs. 7 and 8 respectively. It can be seen for both of these cases that the generated fractal terrain profiles are on a very similar scale as the original terrain profile. Following the top part of Fig. 7, there is a general elevation change in the right profile near the 30 m mark. The elevation change artificially inflates the RMSE causing fractal scaling parameter G to be inflated. When the inflated G is used to regenerate the surface, the increase is enough to cause the slightly higher overall
Fig. 6. 7 Degree of freedom full suspension model.
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Fig. 7. Dirt road longitudinal terrain profiles from terrain map (top) compared to longitudinal terrain profiles from a randomly generated 3-D surface using the Weierstrass–Mandelbrot function with parameters determined from the fractal parameter RMSE relationship (bottom).
Fig. 8. Loose gravel longitudinal terrain profiles from terrain map (top) compared to longitudinal terrain profiles from a randomly generated 3-D surface using the Weierstrass–Mandelbrot function with parameters determined from a fractal parameter RMSE relationship (bottom).
roughness of the regenerated profiles. The key distinction between the experimental terrain profiles and the fractal profiles is that the former tends to have the phases of the right and left profiles more aligned. Since the fractal surface is randomly generated there is no guarantee that this trait will be captured for a given profile. In the bottom of Fig. 8 it can be seen that for parts of the profile the left and right wheel paths are closer phase. To compare the frequency content of the experimentally measured and regenerated terrain profiles the power spectral densities for the profiles can be analyzed. Figs. 9 and 10 show the power spectral density (PSD) for the right and left profiles taken from the experimentally measured terrain and the fractal regenerated terrain. It can be seen
that the PSD of the experimental terrain profiles match the PSD of the fractal terrain profiles reasonably well for frequencies higher than 0.2 cycles/m. The magnitudes of the PSD for the fractal terrains are higher than their experimental counterparts for the frequency range of 0.02–0.1 cycles/m. This range is particularly important in determining the roughness of the terrain profile. It can also be seen that the deviation in this range is higher for the gravel surface than the dirt road. The oscillations of the PSD in the fractal surface (black markers) are caused by the discrete nature of the frequency scaling used in the fractal model. The locations of these spikes is related to the parameter c Recall, c determines the spacing of the frequency levels present in the fractal surface. Since the experimental surface has a continuous frequency distribution to the level which can be captured by the resolution, these spikes are not seen in its PSD. This implies there is no parameter c which directly accounts for the scaling of the frequency levels in the experimental surface, an observation also made by Wu [14]. To further compare the effectiveness of the terrain profile regeneration method, the 7-DOF model can be simulated on both the experimental and fractal terrains. The simulation was performed using MATLAB. Since the same model is used on both terrains the effects of the vehicle model error are reduced. This allows the response of the model to be analyzed to determine the similarity of the terrains with respect to the model dynamics. The heave, roll, and pitch from the simulated vehicle response on the experimental and regenerated loose gravel surfaces are shown in Figs. 11–13. It can be seen that the heave and pitch motions have oscillations of similar magnitudes. However, the vehicle simulated on the fractal surface experiences much more roll motion than for the experimental surface. This is caused by the left and right wheel paths being out of phase resulting in more vehicle roll. The experimental profiles tend to be more similar laterally, thus, the vehicle on these profiles does not exhibit the increased roll
Fig. 9. Power spectral densities for dirt road experimental surface and regenerated fractal surface.
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Fig. 10. Power spectral densities for loose gravel experimental surface and regenerated fractal surface.
Fig. 11. Heave motion comparison of 7-DOF model vehicle response on longitudinal profiles taken from experimental loose gravel profile and longitudinal profiles taken from regenerated fractal surface.
Fig. 12. Roll motion comparison of 7-DOF model vehicle response on longitudinal profiles taken from experimental loose gravel profile and longitudinal profiles taken from regenerated fractal surface.
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Fig. 13. Pitch motion comparison of 7-DOF model vehicle response on longitudinal profiles taken from experimental loose gravel profile and longitudinal profiles taken from regenerated fractal surface.
motion. Another observation is that the fractal surface results in a homogenous vehicle response as compared to the real surface. Due to the non-stationary nature of the experimentally measured terrain, there are bumps and discontinuities which cause dynamics to change over the length of the run. There is a peak in each of the signals at near the 32 s mark for the experimental terrain which was caused by a bump. The vehicle simulations were conducted on each of the terrains at four speeds between 5 and 20 mph. As a measure of how similar the vehicle responses were, the RMS of the time response of the various vehicle states was calculated for each of the simulations. The results for the simulations on the experimental and fractal terrains at various speeds for the dirt road and loose gravel are summarized in Tables 1 and 2 respectively. For the loose gravel terrain the fractal generation provides good agreement with the experimental terrain except for in the roll motion for the reasons explained previously. The fractal generated surface for the dirt road terrain performs well, but the values are slightly elevated relative to their experimental counter parts. Again this is caused by the abrupt increase in the elevation of the experimental profile resulting in an increased value of RMSE. Although this method provides a framework for generating fractal terrains which resemble experimental terrains, there are several areas which must be addressed to increase the realism of the terrains. One observation is that the fractal model is not effective in capturing large scale undulations while preserving the detail of the smaller scales. To more accurately represent real terrain profiles, this method needs to be combined with a methodology for generating large scale elevation changes. Additionally, the relationship used in the terrain generation methodology relies on the RMSE which assumes stationarity of the terrain and can be inflated by the large scale changes in elevation. This method could be improved by using a similar methodology
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Table 1 Summary of RMS values for 7-DOF model simulated on longitudinal terrain profiles from dirt off-road. 5 mph
RMS RMS RMS RMS RMS RMS
heave (m) heave rate (m/s) roll (deg) roll rate (deg/s) pitch (deg) pitch rate (deg/s)
10 mph
15 mph
20 mph
Exper.
Fractal
Exper.
Fractal
Exper.
Fractal
Exper.
Fractal
0.06 0.36 5.33 32.08 0.52 4.10
0.07 0.50 5.08 32.11 0.71 6.18
0.08 0.43 4.88 32.46 0.38 4.48
0.07 0.51 5.23 30.18 0.88 8.83
0.06 0.47 3.74 30.38 0.29 4.29
0.07 0.50 4.55 30.24 0.78 9.51
0.07 0.43 3.32 27.00 0.21 3.29
0.07 0.54 4.81 33.39 0.61 8.46
Table 2 Summary of RMS values for 7-DOF model simulated on longitudinal terrain profiles from loose gravel. 5 mph
RMS RMS RMS RMS RMS RMS
heave (m) heave rate (m/s) roll (deg) roll rate (deg/s) pitch (deg) pitch rate (deg/s)
10 mph
15 mph
20 mph
Exper.
Fractal
Exper.
Fractal
Exper.
Fractal
Exper.
Fractal
0.07 0.31 2.77 18.00 0.75 4.64
0.06 0.46 5.45 31.82 0.31 4.29
0.07 0.32 3.32 19.78 0.58 5.55
0.06 0.46 5.27 30.90 0.33 4.51
0.08 0.34 2.96 20.23 0.39 4.78
0.05 0.46 5.34 31.35 0.34 4.88
0.07 0.24 2.50 18.09 0.23 2.94
0.05 0.44 5.52 31.91 0.30 4.53
to relate the fractal parameters to other metrics which do not have these limitations. It was also observed in this analysis that the experimental terrain profiles tend to have similar phases for much of the data collected. To improve the terrain generation methodology it would be helpful to include more coupling between the lower frequency content of the W–M function. This could potentially be implemented by defining the phase shifts of the surface for lower frequencies while randomizing them for higher frequencies. Lastly, the experimental terrains have transients and discontinuities which cannot easily be modeled using the fractal model. Methods need to be developed which will better capture these events.
paths of experimentally measured terrains tend to be more in phase than their fractal counterparts. There are also other non-periodic events present in actual terrains such as bumps, which the W–M function fails to capture. These discontinuities could be implemented by maintaining a database of the geometry of common terrain features such as rocks and sharp undulations. These features could then be randomly distributed on the fractal surface based on the density of the features found in the experimental data. Considering the phases of the lower frequency content and adding discontinuities could improve the fidelity of the simulated terrain with respect to the experimental terrain as determined by the simulated vehicle response.
6. Conclusions
References
The 3-D Weierstrass–Mandelbrot fractal function can be used to generate terrain surfaces for the simulation of vehicle motion. An analysis of the behavior of this function for different parameters was conducted. Longitudinal profiles were extracted from the surface and analyzed. By generating surfaces using the fractal function the relationship between the fractal parameters and the root mean square elevation (RMSE) can be determined. The RMSE of the modeled terrain profiles exhibits an exponential relationship with fractal scaling parameter (G) for a given fractal dimension (D) Additionally, the exponential relationship between RMSE and G can be used to relate experimentally measured terrain profiles to the W–M function to generate terrains similar to experimentally measured terrains, but the equations are limited to surfaces of similar length and resolution. This method can produce terrains which provide comparable results to the experimental terrain based on a vehicle simulation. However, the left and right wheel
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