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Evaluation of Vehicle/Driver Performance Using Genetic Algorithms

James Bernard, James Gruening, and Kurt Hoffmeister

Iowa Center for Emerging Manufacturing Technology, Iowa State University

Copyright ? 1998 Society of Automotive Engineers, Inc.

ABSTRACT Simulation is often used to gain an understanding of vehicle directional response. Furthermore, it is widely agreed that, given an adequate set of parameters that model the vehicle and the surface it drives on, it is reasonable to simulate a particular vehicle with a view toward understanding and perhaps improving its performance. This is not the case with the vehicle/driver system. Rather, in terms of a particular vehicle and driver, simulations provide interesting but not particularly reliable results because of the routine variability of the human part of the system. Genetic algorithms and their derivatives are algorithms with their form drawn from the biological theory of evolution. This paper suggests that genetic algorithms may be useful to evaluate certain important facets of vehicle/driver performance. It supports this suggestion with an example that attempts to answer this question: What is the best a vehicle/driver system could do in the so-called Consumer Union short course? The example is challenging because the strategy the driver uses to drive through the course affects the result. The geneticalgorithm-based solution to this example problem provides evidence that the technique is promising. The paper concludes with speculation on the potential for applying genetic algorithms in a much less constrained set of circumstances, including determination of the possibility of untripped rollover on a smooth surface. INTRODUCTION Simulation is routinely used to gain an understanding of vehicle directional response. Furthermore, it is routinely agreed that, given an adequate set of parameters to model the vehicle and the surface it drives on, it is reasonable to simulate a particular vehicle with a view toward understanding and perhaps improving its performance. There are three ways that the simulation can be

configured. Figure 1 presents a simple schematic diagram of the most common form of simulation, traditional open loop simulation. The figure indicates that open loop simulation relies on a vehicle model and a roadway model. These models exhibit a wide range of complexity, depending on the needs and intentions of the analyst.

Figure 1. Traditional Open-Loop.

The input to open loop models includes initial conditions, which describe the vehicle’s velocity and orientation at the start of the simulation, and a time history of the steering and braking to be simulated. The equations of motion are then numerically integrated, providing output of the state of the vehicle model at discrete time increments during the course of the simulation. This output can be examined in numerical form, or it can be plotted or made into an animation for additional analysis. We refer to this process as vehicle simulation because the input from the driver, the steering and braking history, is chosen in advance.

Figure 2 presents a simple schematic diagram of an inverse model. In this case, the figure indicates that the path rather than the control history is the input to the simulation, and the simulation solves for the steering and throttle or braking controls to follow the path. Inverse modeling is directly related to traditional open loop modeling - if the steering and throttle or braking computed by an inverse model are used as input into an open loop model, the open loop model must produce as output the path that was input to the inverse model. As in the case of the open loop simulation, inverse simulation does not model the characteristics of the driver. Rather it computes the controls the driver would have to supply to exactly follow the input path. A shortcoming of the method is that the algorithm can call for steering beyond human ergonomic limits.

Figure 3. Driver Model. inadequacies inherent in the model of the driver. This paper uses the inverse procedure to study an important facet of driver/vehicle performance by addressing this question: What is the best (or worst) a vehicle/driver system can do in a particular set of circumstances. The technique we propose is based on genetic algorithms, so-called GAs, which are the subject of the next section of this paper. This is followed by an example in which GAs attempt to solve for the fastest speed a vehicle/driver system can traverse the so-called CU short course [1]. The last section of the paper comments on the utility of the use of GAs in this context and speculates on the potential of GAs to shed light on vehicle driver systems operating in less-tightlyconstrained scenarios. GENETIC ALGORITHMS Genetic Algorithms were introduced in the late 1970s [2], but have only recently begun to gain attention for optimization in engineering design applications [3]. An offshoot from the broad field of Artificial Life; a GA is an optimization technique that uses the process of biological evolution as a model and source of ideas [4-6]. The strength of GA’s lies in their ability to exploit nonobvious features in a system’s response to “evolve” toward an optimal solution. The basic steps in the use of a GA are [4]: 1. Encode design parameters into a “gene” string. 2. Initialize a “population” of candidate gene strings. 3. Loop for several “generations”: ? Evaluate “fitness” of candidate gene strings.

Figure 2. Inverse Model. Figure 3 presents a simple schematic diagram of the simulation of a driver/vehicle system. The figure indicates that there exists a path that the driver would like to follow, a model for the driver, a model for the vehicle, and a model for the roadway. Typically the driver model, by comparing measures of the present state of the vehicle and the desired path, “decides” on the appropriate steering and perhaps throttle or braking. The model does not follow the path exactly. Rather it attempts to compute how well the driver can do in trying to follow the path. The results of this kind of simulation are usually viewed as speculative because the details needed to adequately model the driver are not very well known. None of the three simulation techniques have been very useful for assessing the performance of a particular driver/vehicle combination. The first two do not attempt to model the driver, and the third suffers because of

Select “parent” solutions for “breeding” new candidate gene strings. ? Create “offspring” from parents’ genes with “crossover” and “mutation” operators. ? Replace old gene strings that are the least fit with the new offspring. 4. Repeat until gene strings are fully “evolved” (i.e., converged) or for a specified amount of analysis time. The solution is then the final generation of gene strings. Since GAs are a fairly young technique, there is a good deal of ongoing research regarding each of the above concepts [5]. In practice, it is often useful to try several of these variations and select a set that works well for a given problem. EXAMPLES This section presents two examples, each in the context of the CU short course. The CU Short Course is a double lane change maneuver, as shown in Figure 4. The first lane change takes place in the longitudinal distance of 50 feet and the return lane change takes place in the subsequent 60 feet. The vehicle coasts through the course, which is driven with steering input only. The performance

?

between the cones as indicated by Figure 4. The numerical constraints on the parameters of the gene strings are: -35.0 -3.0 110.0 -3.0 < < < < Xs Ys Xe Ye < 0.0 < 3.0 < 165.0 < 3.0

where the subscript s indicates at the start, and the subscript e indicates at the end. Figure 5 shows how the algorithm works. The vehicle model begins the simulated short course at a given speed, U. The model uses inverse techniques to follow the path, as indicated by Figure 2. The inverse procedure calls for lateral acceleration as input, and computes the steer necessary to follow the path as part of the output. The lateral acceleration used as input to the inverse model is a function only of the current speed along the path and the characteristics of the path. (See Appendix B.)

Figure 4. Consumer Union Short Course. measure is the speed at which the vehicle enters the course, and the goal is to negotiate the course with the highest possible initial speed. This speed is measured 35 feet before the end of the first set of lane cones. EXAMPLE 1, MAXIMUM SPEED PATH In this example, the GA evolves a population of paths through the simulated CU short course. The paths, which are specified as B-splines, evolve in a quasibiological fashion. The characteristics of the paths through the course make up the gene string. The gene string includes the starting and ending points of the path, and a measure of the position and first and second derivatives of the path at a point near the middle set of cones. Appendix A presents the details of the representation of the paths. For our purposes here it is important to know that the vehicle model is constrained to start and finish with its velocity vector in the longitudinal direction and without any lateral acceleration. The global X and Y coordinates of the start and finish are constrained to be Figure 5. Flow Chart for this Implementation of the Genetic Algorithm.

The path derived from a gene string is preprocessed to determine if a vehicle model could negotiate the path through the short course. If the path itself causes the vehicle to hit any of the cones and thereby fails the course, its fitness is set to a low value so that the gene string does not survive to the next generation. Appendix C presents details of the Genetic Algorithm used here. Admissible paths are supplied to the inverse vehicle dynamics model. If the vehicle model successfully negotiates the path at an initial speed of 30 ft/s, the initial speed is incremented by 0.1 ft/s, and the course is re-

run. This exercise is repeated until an initial velocity is reached that leads to failure to complete the course. Failure occurs in one of two ways: 1) The front wheels of the vehicle model cannot steer enough to provide the lateral acceleration needed to continue to follow the path at the current speed. 2) A wheel lifts. (We are using a fairly simple vehicle model that assumes the sum of the normal loads on the tires adds up to the weight of the vehicle. The model loses validity at or near the wheel lift condition, so the runs are stopped when wheel lift occurs.) For this example, the highest initial velocity with which the vehicle model can negotiate the course is the fitness of the gene string for the given path. (The vehicle model slows down during the course of the run due to lateral forces that have a component along its path. Rolling resistance, engine braking, and air resistance are neglected.) As indicated in Table 1, two vehicle models were used in the simulations. One of the models is a generic passenger car, the other is a generic sport utility vehicle (SUV). The table indicates that important differences between the models are in the wheelbase, the moments of inertia, and the cg height of the sprung mass. The vehicle model has nine degrees of freedom, five for the sprung mass, which has no vertical degree of freedom, and wheel spin for each tire. See [7,8] for more details. Both of the vehicle models use the same tire model, a semi-empirical model given by [9]. Appendix D presents the input data to the tire model and a carpet plot. Table 1. Key Vehicle Parameters. PARAMETERS CG to Front Wheel (ft) CG to Rear Wheel (ft) Torsional Roll Stiff. (ft·lb/rad) Torsional Roll Damping (ft·lb·sec/rad) 2 Roll Moment of Inertia (slug·ft ) Roll Axis Hgt, Frt. & Rear (ft) Sprung Mass Height (ft) Vehicle Weight (lb) Front Track (ft) Rear Track (ft) 2 Yaw Moment of Inertia (slug·ft ) Unsprung Yaw Moment of 2 Inertia (slug·ft ) CG Height (ft) Pitch Stiffness (ft·lb/rad) Pitch Damping (ft·lb·sec/rad) 2 Pitch Inertia (slug·ft ) PASS. CAR 3.01 5.82 58652.0 1867.0 371.42 1.0 1.94 3405.0 5.13 5.03 2064.1 82.56 1.83 163361.0 8933.0 1839.11 SUV 3.8 4.2 58652.0 1867.0 371.42 1.0 2.3 3405.0 4.42 4.42 1630.8 82.56 2.10 163361.0 8933.0 1839.11 This fitness function is ‘penalized’ at high speeds, thus we expect to evolve genes which lead to high load transfer at low speeds. Again the vehicles start out on a given path at a low velocity (30.0 ft/s) and continue to increment the initial velocity by 0.1 ft/s until failure occurs. Figure 14 presents the maximum load transfer paths for the two vehicles. Again the passenger car model is Figure 6. Maximum Speed Paths. maximum speed paths for the two vehicle models. The limiting factor for the passenger car is that, for higher speeds, the model cannot deliver enough lateral acceleration to stay on the given path. The limiting factor for the SUV is wheel lift. Figures 7-13 present time histories for the highest speed runs. The figures indicate that the passenger car’s best initial velocity was 53.0 ft/s with a peak lateral acceleration of about 0.80 g. The SUV’s best initial speed was 48.1 ft/s with a peak lateral acceleration of about 0.76 g. EXAMPLE 2, MAXIMUM LOAD TRANSFER PATH In this case, we wanted to generate paths that generated high load transfer at low speed. The fitness of the pathgenerating gene string is: F = ? – 5(u–30.0) where: F - fitness ? - absolute value of the peak total load transfer u – beginning vehicle velocity (1)

Evolution led to a population whose best paths have similar characteristics, clearly attempting to minimize curvature so as to allow the highest possible ratio of speed to lateral acceleration. Figure 6 presents the

Figure 7. Example 1, Speed.

Figure 8. Example 1, Lateral Acceleration.

Figure 11. Example 1, Yaw Rate.

Figure 9. Example 1, Total Load Transfer.

Figure 12. Example 1, Road Wheel Steer Angle.

Figure 10. Example 1, Roll Angle. limited by its potential to supply higher lateral acceleration, and the SUV model is limited by wheel lift. Figures 15-21 present related time histories. Note that the initial speed of both models on the limit path is much lower than in the previous example (31.1 ft/s vs. 53.0 ft/s for the passenger car and 30.9 ft/s vs. 48.1 ft/s for the SUV). This SUV result is remarkable because it indicates that wheel lift occurs on one path at a much lower speed than on another path. The clear implication is that driver strategy has a first order effect on whether a run at a given initial speed leads to a wheel

Figure 13. Example 1, Sideslip Angle.

Figure 14. Maximum Load Transfer Paths.

Figure 15. Example 2, Speed.

Figure 18. Example 2, Roll Angle.

Figure 16. Example 2, Lateral Acceleration.

Figure 19. Example 2, Yaw Rate. Clock time for all the runs for one vehicle in one example is about 2 hours. Clock times vary depending on other demands on the multi-processor computer. CONCLUSIONS In the context of a model of the CU short course, the paper has illustrated the power of genetic algorithms to model driver/vehicle behavior. In particular, the GA provides an answer to the question: how fast could a given vehicle model be driven through the short course without failure by hitting a cone or lifting wheels?

Figure 17. Example 2, Total Load Transfer. lift. Figures 22 and 23 superimpose the path and lateral acceleration generated from the SUV maximum speed run of Figure 7 and the SUV load-transfer-generated run of Figure 14. Recall that, in either case, starting at a slightly higher initial speed on the same path will yield wheel lift. NUMERICAL DETAILS These results were computed using four R10k processors on SGI Rack ONYX computer. On this machine, the vehicle model runs at about five times faster than real time. Approximately 100,000 separate runs were simulated for each vehicle in each example.

Consider first the high-speed runs. Although the theory of genetic algorithms does not guarantee that the results are the maximum speeds of the simulated vehicles through the short course, our experience leads us to expect that it is very near the maximum for this model within the limitations imposed by our representation of the path. Since the model of the path is quite robust (see Appendix A), we expect that the highest computed velocity, as indicated by the fitness function associated with the highest speed path, is likely to be very near the maximum for the vehicle models in the short course. The load-transfer-based results presented here indicate that the SUV model has a low speed path through the

course which leads to wheel lift, contrasting the

previously computed higher speed path through the course which does not lead to wheel lift. Thus the calculations verify that the strategy the driver uses to go through the course has a first order influence on load transfer. Finally, it is important to reiterate that the simulation which produced this data is an inverse model, not a driver model. Thus, it is clearly possible for the algorithm to call for steering at frequencies outside the range of typical driver performance. In the next section, we will suggest ways to meet this challenge. FUTURE WORK Figure 20. Example 2, Road Wheel Steer Angle. We believe the work presented here indicates that Genetic Algorithms have remarkable potential for providing insight into the potential behaviors of the driver/vehicle system. Although the vehicle models used here have sophisticated tire models, the remainder of the vehicle model is quite rudimentary. We believe it will be worthwhile to continue our work with better vehicle models. Particularly important are additional degrees of freedom for sprung mass bounce and for wheel hop, and the addition of rolling resistance and engine braking. In the context of the CU simulation, we expect the addition of engine braking and rolling resistance to increase the initial speeds with which simulated vehicles can negotiate a given path. The Conclusions section of this paper pointed out that the steer needed to traverse the paths called for by the GA can be outside ergonomic limits. We are looking at two ways to meet this challenge 1) low-pass filtering of the steer called for to follow the path, and 2) using a measure of the frequency content of the steering as part of the fitness function. Filtering will allow the vehicle model to deviate from the path called for by the gene string, yielding ergonomically reasonable steering. Using the frequency content of the steering as part of the fitness function will continue to cause the vehicle to follow the path called for by the gene string, but will favor ergonomically reasonable paths. We believe GA-based analysis can also address more ambitious challenges. For example, we are planning work on a GA-based simulation attempting to answer this question: Given a model of a vehicle and a surface plus an initial speed of the vehicle, is it possible to find a combination of steer and braking that will lead to rollover? We will attempt to answer this question using traditional open loop simulations rather than inverse calculations, and we expect the gene string to include measures of the steering and braking commands. We expect this challenge to call for a much longer gene string than sufficed for our work on the CU short course. Since the length of the gene string determines a population size (hence the number of runs needed for evolution) [5], we expect this problem to be taxing on

Figure 21. Example 2, Sideslip Angle.

Figure 22. Comparison of SUV Model Paths.

Figure 23. Comparison of SUV Model Accelerations.

numerical resources. Nevertheless we remain optimistic about prospects for taking on at least a large subset of this challenge in the near future. Our optimism derives from the continuing rapid increase in computing resources per unit cost, and from the fact that GA-based work is inherently parallelizable. ACKNOWLEDGEMENT Thanks to Professor Dan Ashlock for his help with genetic algorithms. REFERENCES [1] “Not Acceptable: Isuzu Trooper/Acura SLX,” Consumer Reports, October, 1996. [2] Holland, J. 1975, Adaptation in Natural and Artificial Systems, MIT Press, Cambridge, Mass. [3] Goldberg, D.E., 1989. Genetic Algorithms in Search, Optimization, and Machine Learning, Addison-Wesley, Reading, Mass. [4] Walker, John F., “Evolution of Simple Virtual Robots Using Genetic Algorithms.” Master’s Thesis, Iowa State

University, Ames, Iowa, 1995. [5] Ashlock, Daniel A., “Optimization and Modeling with Artificial Life.” Manuscript, Iowa State University, Department of Mathematics, 1997. [6] Hoffmeister, Kurt M. and Bernard, James E., “Tread Pitch Arrangement Optimization Through the Use of a Genetic Algorithm,” Presented for publication to Tire Science and Technology, March, 1997. [7] Bernard, J.E., Bhatnager, A., Clover, C.L., “Evaluation of Select Vehicle Dynamics Models – Phase II Final Report,” Motor Vehicle Manufactures Association, Contract 9114-C11302, 1992. [8] Bernard, J.E., Clover, C.L., “Tire Modeling for LowSpeed and High-Speed Calculations,” SAE Paper 950311, 1995. [9] Allen, R.W, Szostak, H.T., Rosenthal, T.J., et al., “Vehicle Dynamic Stability and Rollover,” Systems Technology, Inc., TR-1268-1, NHTSA Contract DTNH2288-C-07384. [10] Shames, I. H., “Engineering Mechanics”, PrenticeHall, Inc., 1960.

APPENDIX APPENDIX A: THE PATH THROUGH THE CU SHORT COURSE The path through the short course includes two segments. On each segment, X and Y are functions of the dimensionless parameter s, 0≤s≤1 X(s) is straightforward. For the first segment of the path X(s) = L1s1 for 0≤s1≤1 And for the second X = L1 + L2s2 for 0≤s2≤1 This indicates that the horizontal dimension of the first segment of the path will be L1 and the horizontal dimension of the second segment of the path will be L2. Y is given by the sum of six fifth order polynomials in s such that: Yj(s) = Sum yjifi(sj) Where j indicates the segment and the yji are constants. The properties of the fij are: fj1(0) = 1; fj1’(0) = fj1”(0) = fj1(1) = fj1’(1) = fj1”(1) = 0 fj2’(0) = 1; fj2(0) = fj2”(0) = fj2(1) = fj2’(1) = fj2”(1) = 0 fj3”(0) = 1; fj3(0) = fj3’(0) = fj3(1) = fj3’(1) = fj3”(1) = 0 fj4(1) = 1; fj4(0) = fj4’(0) = fj4”(0) = fj4’(1) = fj4”(1) = 0 In detail, the fij are: F1 F2 F3 F4 F5 F6 1 0 0 0 0 0 0 0 -10.0 15.0 -6 1 0 -6.0 8.0 -3 0 0.5 -1.5 1.5 -0.5 0 0 10.0 -15.0 6 0 0 -4.0 7.0 -3 0 0 0.5 -1.0 0.5 1 s 2 s 3 s 4 s 5 s

=

It follows that: y11 is the starting Y coordinate y12 = dY1/ds1 at the start. Note dY/dX at the start is 0, therefore y12 = 0. 2 2 y13 = d Y1/ds1 at the start. Since we call for no lateral acceleration at the start, y13 = 0 y14 is the Y coordinate at the end of the first segment. y15 is the derivative with respect to s1 at the end of the first segment. Note dY/dX at this point is (dY/ds)/dX/ds = y15/L1 2 2 2 2 2 y16 is the second derivative with respect to s1 at the end of the first segment. Note that d Y/dX at this point is (d Y/ds )/L1 The y2j can be found in a similar way. Continuous lateral acceleration at the junction of the two splines demands second order continuity at s1 = 1 and s2 = 0. This calls for: y14 = y21 L2y15 = L1y22 L2 y16 = L1 y23

2 2

APPENDIX B: VEHICLE PATH DYNAMICS

Y, j

eN

eT

X, i Figure B1, Example Path with Tangent and Normal Vectors Assume arc length is given by q and that x and y are functions of s. The unit tangent is given by: eT = (dx/dq)i + (dy/dq)j (B-1) And we have: dx/dq = (dx/ds)(ds/dq) dy/dq = (dy/ds)(ds/dq) The unit tangent is now: eT = ds/dq {x′i + y′j} Where the prime indicates differentiation with respect to s. Noting that: 2 2 ds/dq = 1 / √[(x′) + (y′) ] yields: eT = (x′i + y′j) / L Where: L = √[(x′) + (y′) ]

2 2

(B-2) (B-3)

(B-4)

(B-5)

(B-6)

(B-7)

The acceleration along the path can be shown to be [10]: 2 2 2 A = d q/dt eT + (dq/dt) /ReN Where R is the local turn radius. Furthermore the unit normal en is given by: eN/R = deT/dq Noting that: deT/dq = 1/LdeT/ds and carrying out the differentiation yields: 2 2 2 eN/R = ( x” – [(x’) x” – x’y’y”]/L ) / L i + 2 2 2 ( y” – [x’y’x” – (y’) y”]/L ) /L j Now consider equation B-8. Summing forces normal to the path yields: 2 (ΣF ? eN) = M(dq/dt) /R

(B-8)

(B-9)

(B-10)

(B-11)

Where ΣF is the vector sum of all the forces on all the tires. Equation B-11 is the basis for the algorithm that finds the steer at each point on the path.

APPENDIX C: GENETIC ALGORITHM DETAILS The points below describe the Genetic Algorithm (GA) used in this paper. 1. The gene string, {Xs Ys A4 A5 A6 L1 Xe Ye}, is a list of parameters describing a vehicle path as discussed in this paper. The path accomplishes a double lane change maneuver within bounds defined by the Consumer Union’s Short Course. The GA used here operates on a string of eight double precision real numbers in the order given above. Where: Xs = beginning X position of the path Ys = beginning Y position of the path A4 A5 A6 = weights for the path segment blending functions L1 = switching point from segment one to segment two of the mathematical formulation for the path Xe = ending X position of the path Ye = ending Y position of the path

2. A population of candidate gene strings is created by randomly generating a value for each loci of the gene strings. In order to insure reasonable gene strings, bounding limits are set for each parameter. This insures the candidate paths defined by the gene string begin and end on the course and lead to a reasonable path. A population of 160 gene strings is used. 3. The following steps represent one generation of the GA: ? Fitness Evaluation: All gene strings are assigned a fitness metric. For example the one in the text the fitness metric is the maximum speed the simulated vehicle can negotiate the path defined by the gene string. Before starting the vehicle dynamics simulation, the GA first checks the path against placement of traffic cones bounding the course. If a gene string defines a path that violates the constraints of the course, it is randomly assigned a fitness (speed) less than a preset minimum speed. Otherwise the vehicle dynamics simulation is used to “run” the vehicle along the path at successively higher speeds until the maximum possible speed for each gene string (i.e. path) is reached. If the path is valid, but the vehicle cannot stay on the path at the minimum speed; then the gene string is also assigned a random fitness less than the minimum speed. ? Tournament Selection: Groups of four gene strings are randomly drawn from the population until the entire population has been assigned a group. For each group of four a “tournament” consists of determining the two genes with the highest fitness (ties are decided by random draw). These two gene strings are deemed “parents” and survive to the next generation. The parent gene strings create two “offspring” gene strings with a crossover operation (described below). The offspring are mutated and replace the two gene strings from the original group of four who “lost” their tournament. In this fashion one half of each generation (after the first generation) consists of new members. ? Crossover Operator: This operation is repeated to create two offspring from each set of parents from the above tournaments. For each set of parents one location in the gene string is randomly selected for use with both parent gene strings. Each offspring gene string matches a different parent gene string from its beginning up to the crossover location and then matches the other parent gene string from the crossover location to the end of the gene string. The crossover operator is illustrated in Figure C-1. This operation, formally known as Single Point Crossover, generally insures that the new gene strings are created different from each other and still are cross combinations of both parent gene strings. Parents: Xs_1 Ys_1 A4_1 A5_1 A6_1 Xs_2 Ys_2 A4_2 A5_2 A6_2 Crossover point Children: Xs_1 Ys_1 A4_1 A5_1 A6_1 Xs_2 Ys_2 A4_2 A5_2 A6_2 L1_2 L1_1 Xe_2 Xe_1 Ye_2 Ye_1 L1_1 L1_2 Xe_1 Xe_2 Ye_1 Ye_2

Figure C1, Illustration of a Single Point Crossover Operator. ? Mutation Operator: The mutation operator is used to increase diversity in the values available for each gene string loci. (Without mutation the only values available for each loci would be those of the original population and there would be no ability to explore small numerical changes from the original values.) The operation used here randomly selects four loci from each of the newly created offspring gene strings and modifies their values by a randomly selected percent (±15%).

4. After several generations we expect to see improvement in the best fitness and the average fitness of the upper half

of the population. Note that the tournament selection method insures at least the two best gene strings always survive to the next generation. If there isn’t any limit on the analysis time available the GA would continue indefinitely. Therefore a limit is set on the number of generations that pass without an improvement in the best fitness value. At this time the population is considered fully evolved (or at least the evolution has stalled within a limited number of generations) and the algorithm is terminated. 5. The population can be ranked by gene string fitness values and the several best gene strings should be considered evolved solutions. In example 1, these gene strings will represent possible high speed paths through the course and may all be slightly different solutions.

APPENDIX D: PARAMETERS FOR THE TIRE MODEL Reference [8] presents the details of the tire model. The input parameters used for the simulation runs of figures 7 to 21 are shown below: TWIDTH = 5.8 A0 = 6375.57 A1 = 8.5360 A2 = 44592.58 A3 = -0.08038 A4 = 322.8 KA = 0.0 KMU = .234 TPRESS = 35.0 B1 = -0.81615e-04 B3 = 1.132 B4 = -0.7229e-07 KGAMMA = .9 CSFZ = 16.246 MUNOM = .95 FZTRL = 1400.0 KK1 = -0.18629e-03 C1 = 1.0 C2 = .34 C3 = .57 C4 = .32 G1 = 1.0 G2 = 1.0 SNT = 0.85 KLT = 3.8462E-05 Where: TWIDTH A0..A2 A3..A4 KA KMU TPRESS B1,B3,B4 KGAMMA CSFZ MUNOM FZTRL KK1 C1..C4 G1, G2 SNT KLT - Tire tread width - Calspan tire test coefficients for small slip angle side forces - Calspan coefficients for camber thrust - Proportion of tread length that changes with lateral load - Proportion which the friction coefficient decays as sideslip and circumferential slip increase - Tire pressure - Calspan coefficients that account for effective friction coefficient changes with normal loading variations due to load transfer - Defines the radial tire effect - Calspan parameter for unit longitudinal stiffness of the tire tread - Pavement skid number - Tire design load at operating pressure - Aligning torque coefficient from Calspan data - Coefficients to calculate the force saturation function - Shaping coefficients for aligning torque - Skid number from tire test machine - Lateral tire spring rate

Figure D1 presents a carpet plot using the above parameters.

Carpet Plot

1600 1400 1200 -Fy (lb) 1000 800 600 400 200 0 Slip Angle (deg)

4 2 6 8 10 2000 1500 1250 1000

750

500

250 12

Figure D1. Tire Carpet Plot.

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