FURTHER DEVELOPMENTS IN THE SIMULATION OF AUTOMOBILE HANDLING by D. L. Wilson R. A. Scott Technical Report, Department of Mechanical Engineering and Applied Mechanics UM-MEAM- 8 2- 2

TABLE OF CONTENTS Page Chapter 1. Introduction........................... 1 References for Chapter 1....................... 6 Chapter 2. Three-Degree-of-Freedom Vehicle Model..... 7 Section 2.1. Equations of Motion for the Model..... 7 Section 2.2. Solution Procedure For the Equations for the Three-Degree-of-Freedom Model. 27 Section 2.3. Fortran Computer Program for the Three-Degree-of-Freedom-Model....... 32 Section 2.4. Program Variable Definitions for Three-Degree-of-Freedom Model......... 51 Section 2.5. Three-Degree-of-Freedom Vehicle Data for Base Configuration............... 62 Figures for Chapter 2........................... 66 References for Chapter 2.......................... 73 List of Symbols for Chapter 2............... 74 Chapter 3. Modification Required in IDSFC To Simulate Asymmetric Vehicles.............. 81 Section 3.1. Changes in the Equations of Motion.... 81 Section 3.2. Running the IDSFCAS Simulation....... 90 References for Chapter 3..................... 97 Chapter 4. Modifications in Driver Module............ 98 Section 4.1. Technical Changes Required to Implement Extended Cross-over Model.... 98 Section 4.2. Programming Chages Required to Implement Extended CROSS-OVER Model...100 Section 4.3. Programming Changes Required to Interface Driver Module With the Three-Degree-of-Freedom Model........106 List of Symbols for Chapter 4.......................112 References for Chapter 4............................ 114

1 Chapter 1. Introduction Recently, under contract No. DOT-HS7-01715, the authors were involved in the creation of some all-digital simulations for both open and closed-loop automobile maneuvers. Specific achievements of that project were the development of: (1) A vehicle module IDSFC (Improved Digital Simulation Fully Comprehensive). (2) A driver module DRIVER, which involved several mathematical models of the human driver. Such a module is required for closed loop maneuvers Overall features of these modules are as follows. Full details can be found in Refs. [1.1] to [1.9]. (i) IDSFC. The vehicle model IDSFC involves the following degrees of freedom: Sprung Mass. Specification of the sprung mass requires 3 translational and 3 rotational degrees of freedom. Front Unsprung Masses. The degrees of freedom allowed are 2 wheel hops, 2 wheel spins, 2 wheel rotations about the kingpins and 1 steering connecting rod displacement. To reduce costs, the steering system is handled statically. Rear Unsprung Masses. A. Solid Rear Axle. The degrees of freedom allowed are 1 rear suspension deflection, 1 rear axle roll and 2 wheel spins. B. Independent Rear Suspensions. The degrees of

2 freedom allowed are 2 rear suspension deflections and 2 wheel spins. The mathematical representation of the vehicle model involves 30 first order nonlinear differential equations and approximately 250 algebraic equations. The digital program contains 30 subroutines and both single precision and double precision versions are available. The vehicle simulation capabilities are basically as follows: (1) Straight line braking/acceleration, cornering without braking/acceleration and cornering with braking/ acceleration are allowed. (2) Maneuvers up to and including the limit range can be studied in that (i) Nonlinear terms in the kinematics are retained. (ii) These terms are activated by model level switches and can be deleted for less severe maneuvers, thereby decreasing running costs. These switches can also be employed if the user wishes to do studies on the effects of various nonlinearities. The tire and suspension forces and moments are modeled into the nonlinear range. (3) For system and user flexibility, two methods are provided for computing tire forces and moments, namely (i) The APL-CALSPAN model which is based on curves fitted to the measured data. (ii) A Partial Data Deck model which directly uses the measured data. (4) An antilock capacity, which can be activated by a model level switch, is available.

3 (5) Both solid rear axle and independent rear suspensions are allowed. (6) Front wheel drive, rear wheel drive and four wheel drive are available. (7) Separate braking at each wheel is permissible. (8) An interactive capability is provided, which is activated by a model level switch. (ii) A driver module (DRIVER), involving several mathematical models of human driving behavior, was also developed. The main features of DRIVER are as follows: (1) DRIVER controls steering, braking and drive torque inputs to the vehicle model. (2) There are 5 pre-programmed open-loop maneuvers available, namely: (a) Sinusoidal steer with trapezoidal braking. (b) Trapezoidal steer with trapezoidal braking. (c) Double trapezoidal steer with trapezoidal braking. (d) Trapezoidal steering with a sinusoidal perturbation with trapezoidal braking. (e) Sinusoidal steering sweep with no braking. In addition the driver module will accept: (i) Any open-loop maneuver supplied by the user in tabular form. (ii) Any open-loop maneuver specified by a user supplied subroutine. (3) The driver module can operate in a closed-loop

4 mode following a desired path. Four control strategies are available, namely: (a) A "cross-over" model for a straight line path. (b) A "cross-over" model for an arbitrary path. (c) A preview-predictor model which uses a geometric predictor. (d) A preview-predictor model which uses a threedegree-of-freedom vehicle model as a predictor. (4) The driver module permits a mixed-mode operation which allows combined open and closed loop control. (5) An obstacle avoidance strategy using the previewpredictor models is available. Since the completion of that contract, additional work has been done in the areas of: (i) developing a simpler vehicle model, (ii) vehicle asymmetry, and (iii) driver modeling. More specifically: (a) A three-degree-of-freedom vehicle model incorporating certain asymmetries has been developed and a digital program has been written for it. (b) IDSFC has been modified to take into account certain vehicle asymmetries. (c) The module DRIVER has been modified to allow it to interface with the three-degree-of-freedom vehicle model. Also, an improved cross-over driver model has been implemented in it. The purpose of this report is to provide documentation to enable users of IDSFC and DRIVER to incorporate the above additions/changes. In Chapter 2 a description is

5 given of the mechanical modeling involved in the three-degree-of-freedom vehicle model. Then a discussion of the numerical strategy used is given, as well as a Fortran program listing of the computer code. In Chapter 3, the alterations in the seventeen-degree-of-freedom vehicle model IDSFC that are required to allow for certain asymmetries are presented. Both equation and program changes are given. Chapter 4 is concerned with the module DRIVER. The program alterations required to interface it with the three-degree-of-freedom vehicle model are documented. Also, an extended cross-over model of human driving is described and the program changes required to implement it are detailed.

6 REFERENCES FOR CHAPTER 1 1.1 "Improvement of Mathematical Models for Simulation of Vehicle Handling, Vol. 4. User's Guide for the General Simulation", W. R. Garrott, D. L Wilson A. M. White and R. A. Scott. Final Report DOT-HS7-01715, July 1979. 1.2 "Improvement of Mathematical Models for Simulation of Vehicle Handling, Vol. 5, Programmer's Guide for the Vehicle Model, Part 1. Documentation Through Subroutine IOUT. Part 2. Documentation for Subroutines PRTCID through KNMTC. Part 3. Documentation for Subroutines SLPCLC Through OUTPUT and a Double Precision Listing", W. R. Garrott, A. M. White and R. A. Scott. Final Report DOT-HS-7-01715, December, 1979. 1.3 "Improvement of Mathematical Models for Simulation of Vehicle Handling, Vol. 6. Programmer's Guide for the Driver Module", W. R. Garrott, D. Wilson, A. M. White and R. A. Scott. Final Report DOT-HS-7-01715, December 1979. 1.4 "Improvement of Mathematical Models for Simulation of Vehicle Handling, Vol. 7. Technical Manual for the General Simulation", W. R. Garrott and R. A. Scott. Final Report DOT-HS-7-01715, March 1980. 1.5 "Improvement of Mathematical Models for Simulation of Vehicle Handling, Vol. 8. Summary Report", W. R. Garrott and R. A. Scott. Final Report DOT-HS-7 -01715, April 1980. 1.6 "A Note on the Calculation of Tire Side Forces", W. R. Garrott and R. A. Scott. Journal of Sound and Vibration, Vol. 78, 1981, pp. 306-309. 1.7 "An All-Digital Simulation for Open and Closed-Loop Automobile Maneuvers", W. R. Garrott, D. L. Wilson and R. A. Scott. Simulation, Sept. 1981, pp. 83-91. 1.8 "Closed-Loop Automobile Maneuvers Using Describing Function Methods", W. R. Garrott, D. L Wilson and R. A. Scott. SAE Paper, No. 820305, Feb. 1982. 1.9 "Closed-Loop Automobile Maneuvers Using PreviewPredictor Models", W. R. Garrott D. L. Wilson and R. A. Scott. SAE Paper, No. 820306, Feb. 1982.

7 Chapter 2. Three-Degree-of-Freedom Vehicle Model 2.1 Equations of Motion for the Model A relatively simple mathematical model of a four-wheel vehicle has been developed as an inexpensive and easily manipulated simulation tool. It can be applied to both symmretric, and certain types of unsymmetric, vehicles. The model possesses three degrees of freedom; namely, translation in a plane (X-Y plane) and rotation about an axis (the Z axis) perpendicular to that plane. The vehicle is considered to be acted upon by gravity, by air resistance, and by contact forces and moments at the four tire-road contact points. The following asymmetries can be treated: (1) Addition of a "payload" at any arbitrary location in the vehicle. (2) Independent tire properties at each wheel. (3) Torque transfer from the chassis to the driven axle(s). (4) Asymmetrical brake-torque and drive-torque distribution. (5) Independent steering system compliance at each front wheel. In addition, the model includes the following features which are particularly important to the asymmetry mechanisms. (i) Non-linear tire models. (ii) Variable lateral weight transfer at front and rear (corresponding to front-rear roll

8 stiffness ratio). Payloads are incorporated by considering a system consisting of two rigidly connected arbitrary rigid bodies. Body 1 will ultimately be interpreted as the nominal (unloaded) vehicle, and Body 2 will be interpreted as a payload. Let Body 1 be a rigid body (mass m1) and let xyz be a set of axes fixed in Body 1 with the origin 0 at the center of mass.* Let [I1] denote the inertia tensor of Body 1 with respect to these axes. Let Body 2 be another rigid body of mass mn2 which is rigidly fastened to Body 1 such that the center of mass of Body 2 lies at point P which has coordinates (xp,ypzp). The position of P with respect to 0 is given by Op x ( pi+y +zpk) (2.1) Define a set of axes x2y2z2 with origin at P, which will be parallel to xyz as shown in Fig. 2.1. Let [I2] denote the inertia tensor of Body 2 with respect to axes X2Y2Z2 ~ Assume that the only forces and moments acting on Body -. 2 are gravity and the forces and moments F21 and M21 exerted by the connection to Body 2 from Body 1. Then, Newton's laws give *A List of Symbols is given at the end of the chapter.

9 m2~ 2 m2g + 9F21 (2.2) ~-> H2 M21 (2.3) The forces and moments acting on Body 1 are gravity, the forces and moments F12 and M12 exerted by the connection from Body 2, and F0 and M0 which are the resultants at 0 of all other forces and moments acting on Body 1 as shown in Fig. 2.2. Then 4.- 4 -. + 4 -m1a1 m1g + F12 + F0 (2.4) H. 12 + M + ppXF12 (2.5) Noting that F12 x -F21 m2a2 + M - M12 + 21 (2. (2.4) and (2.5) become, using (2.2), mlal = (ml+m - ma2 + (2.6) p+ P X(-m2a2+m2g) + M0 (2.7) The equation of translational motion will be developed

10 first, from (2.6). The equation of rotational motion will then be developed from (2.7). Let t be the angular velocity of Body 1. Then + -+ ++ + -* a2 = a1 + wxp( + wx(WxP ) (2.8) Substitution of (2.8) into (2.6) gives the equation of translational motion (m1+m2a1+ m2 xp + m2Wx(W xp) (m+m2)a.1 + m2pxp p m2oXOXp) Z (m +m2)g + F0. (2.9) Now specialization is made to the three-degree-offreedom model, for which O z rk (2.10) g = gk (2.11) Recalling that (t) = (trel + wx (2.12) where "rel" stands for relative to the moving frame, it follows that

11 ~1 (Vl)rel + W x 1 (2.13) where vl is the velocity of the point 0, namely, v1 = ul + vJ (2.14) Using (2.10), (2.11), (2.13) and (2.14), (2.9) gives, in component form, u(mt+m2) - rypm2 (m1+m2)vr +m2r2x + F 2- p x (2.15) v(m1+m2) + rx pm2 X -(m1+m2)ur 2 + m2ryp + F 0 (m+m2)g + F 0 z (m1+M2)g + Fz (2.16) (2.17) For the model at hand, angular momenta are given by Hy z IYzw z Izr, y1,2,nz1,2,3 (2.18) Noting this and the relation (212), the coponent version Noting this and the relation (2.12), the component version of (2.7) is, using (2.10) and (2.11)

12 Is Is r2 z 2( (Zp+iXpZp+urz xzr - yz p -r2ypZp+ypg) + M IS xr2 + ISy rz 2(ypp-uz +vrz xz yz +r2xpzp-xpg) + My Izzr M2(-vx p-rx 2+p y 2 zz (-VX p +uY -rY +urxP -vry ) + Mz where s I + 2 mn mn +mn (2.19) (2.20) (2.21) (2.22) The forces and moments acting on the vehicle are tire forces and moments, aerodynamic effects, and gravity (which is already included in the formulation). Then 4 Z F F z ixl xi + F x xaero 4 F iXZ i FF F ~ iml yi + F y yaero (2.23) (2.24) 4 Z F F - i z1 zi + Fz z zaero 4 MxL i1 (tiFZi-Z Fyi+xi) + M ixl(YtiztFyi +Mxi xaero (2.25) (2.26)

13 M E (z tiF -x F z+Myi) + My (2.27) y i1 i xiti ziyi yaero M C(x F i-yti F i+Mi) + M o (2.28) z Zi1 ti yi i xi zi zaero where Fi, Fyi F, Mx, Myi Mzi are the tire forces at, and moments about, the tire contact points; Faero x aero' Fyaero Fzaero Mxaero Myaero Mzaero are the forces at, and moments about, the vehicle c.m. due to aerodynamic effects, and xti, Yti' zti are the coordinates of the tire contact points. The tire is assumed to operate with zero camber angle in the three-degree-of-freedom model, so Fxi, Fyi M i, Myi Mzi are functions only of Fzi, tire drive torque, and slip angle. Fxaero, Mz are functions of vehicle xaerol zaero velocity. Equations (2.15),(2.16),(2.17),(2.19),(2.20) and (2.21) thus provide six equations in the seven variables (Ui,v,rFz1,Fz2,Fz3,Fz4). A seventh equation is obtained by the following argument. Consider a system consisting of a rigid body which pivots on two solid axles, with torsional springs at the pivots, as in Fig. 2.3. Assume that the axles are parallel, and that there is some angular deflection (roll) of the body with respect to the axles. In addition to the moments caused by the springs, let there be torques T1 and T2 transmitted between the body and axles which are independent of 8.

14 A free-body-diagram of the front axle is shown in Fig. 2.4. If the axle is taken to be massless, the equation of rotational motion about the x-axis becomes Fzlytl + F 2Yt2 - FlhF - F2hF + x2 - T 1= -k1 (2.29) Note that y < 0. The analogous equation for the rear axle is Fz3Yt3 + Fz4Yt4 - F3hR - F4hR + 3 + M4- = -k2 (2.30) Define K z k1 + k2 (2.31) Then XFS k1/K k 1 z RSK k2 ( 1- XRs)K (2.32) (2.33) (2.34) Substitution of (2.33) into (2.30) and (2.34) into (2.31) yields (FzlYtl+F z2Yt2-F ylhFhF+Mx+M 2-T1) /Rs = -KO ylF Y2 2 1 R (2.35)

15 (F3Yt3+Fz4Yt4-Fy3hR-F4hR+Mx3+M -x4T)/ (- -K (2.36) Equating (2.35) and (2.36) yields (1-R) (F Ytl+F Y ) - (F Y +F) (1 RS) 1(zitR zl z 2 x RS z3Yt3+Fz4Yt4) = (- RS) (T +FlhF+Fy hF -M 1 yF y2F xl xz RS (T2+Fy3hR+Fy3hR+Fy4hR-Mx3-M4) (2.37) Equation (2.37) holds for all values of K. In the limit as K becomes infinitely large, the two-axle/body system will behave as a rigid body in plane motion (i.e. no rolling). Eq. (2.37) can thus be used to remove the indeterminacy in Fz for the three-degree-of-freedom model. The steering system will now be addressed. The front tire forces depend in part on the steer angles of the wheels. The steering system is shown schematically in Fig. 2.5. It is assumed to be massless, so that the deflection of the compliant members can be computed statically. The input to the system is the steering-wheel displacement. The linkage geometry can be specified to yield either parallel steer angles or Ackerman steer angles, as shown in Fig. 2.6. In either case, the simplifying assumption is made that the torque acting at the pitman arm is the sum of the kingpin torques. The steering system is described by the following equations, where 6S is the steering-wheel displacement, 6p is the pitman arm displacement, 61 and 62 are the front

16 wheel steer angles, TQST! and TQST2 are the steering torques about the kingpins, NG is the steering ratio, CST, CS2, and CST3 are the steering-system compliances as shown in Fig 2., and 6TOE AND 6TOE2 are toe-in angles. P 6Sw+CST3 (TQsT +TQT2 )/NG /NG (2.38) For parallel steer linkage geometry, 61 Z 6p TQ + 6TTOE1 (2.39) 62 S6p + CST2TQST2 + 6TOE2 (2.40) For Ackerman steering linkage geometry (assuming a perfectly stiff linkage) the following relations are obtained from geometry, where the wheelbase L and rearaxle center turning radius Rr are defined as in Fig. 2.6. 6p x tan1(L/R ) (2.41) r 61 ) tan (L/(Rr'Yti)) (2.42) 1 r-Ytl24 62 x tan1 (L/(R-Y t2)) (2.43) (note that yt2 < 0) Substitution of (2.41) into (2.42) and (2.43) and addition of initial toe-in angle and steering compliance

17 effects yields 61 tan1 [Ltan6p/(L-ytltan6p)] + CST1TQST1+6TOE1 (2.44) 62 x tan1 [Ltan/(-yttan6p) + CT2TQT2+6TOE2 4) 2 K p t.p + ST2 ST2 TOE2 where 6p is given by (2.38). Tire force and moment equations will now be considered. It is assumed that the tires are capable of generating side forces, circumferential forces (along the intersection of the ground plane with the wheel plane), and aligning torques (moments about the z-axis) in addition to the normal forces (perpendicular to the ground plane). The tire forces are calculated using a simplified CALSPAN tire model in which the tire forces and moments depend only on the normal force, slip angle, and drive/brake torque [2.1]. The slip angle at each tire is computed by the following equations. oti Bi - 6i ix1,2 (2.46) (Oi I ai iz3,14 (2.47) where i x tan-1 (v+xtir)/(u-ytir)] (2.48)

18 The tire circumferential and side forces and aligning torques are resolved into the vehicle axis system by the following equations, where FCi, FSi, and TQALi refer to circumferential force, side force, and aligning torque, F xi yi z -FSisin6i + FCicos6i x FSicos6i + FCisin6i i1,2 (2.49) i1,2 (2.50) x FCi iz3,4 (2.51) yi x FSi iz3,4 (2.52) zi TQALi i1,2,3,4 (2.53) The steering following equation, in Fig. 2.7. torques TQSTi are computed where Cx is the caster offset, by the as shown TQT TQ Li -CFSi STi ALl X 1 i1,2 (2.54) Tire modeling will be addressed now. The force and. moment generating capabilities of the tires in the vehicle model in this report are represented by a tire model developed by investigators at the CALSPAN Corporation. The model has evolved over the last 15 years; in its current form it supplies a fairly complete representat:n of the

19 non-linear steady-state t automobile tire [2.3, 2.4]. to be necessary even with behavior of the pneumatic Such a representation is felt simple vehicle models, if realistic results are to be obtained. It includes the following features: (1) non-linear relation (2) non-linear (3) non-linear including (4) non-linear relation (5) side force slip cornering stiffness/normal force camber stiffness/normal force relation side force/slip angle relation side force saturation circumferential force/normal force roll-off as a function of longitudinal (6) aligning moments and overturning moments as nonlinear functions of normal load, side force, and camber angle. The CALSPAN tire model is based on a representation of experimentally measured data curves with polynomial expressions. A total of 29 descriptive parameters plus a side force roll-off versus longitudinal slip table are required. The independent variables for the model are slip angle, camber angle, radial deflection, and longitudinal slip. The dependent variables are radial force, side force, circumferential force, aligning moment, and overturning moment A simplified version of the CALSPAN tire model for use

20 with the 3-DOF vehicle model will now be described. The simplifications arise since no tire deflection is allowed and wheel-spin dynamics are not included. As a result, tire deflection and longitudinal slip cannot be independent variables. Instead, the normal force and drive/brake torque are taken as input variables. Also, the camber angle is always zero, so the camber angle dependence is removed. In addition, the overturning moment was felt to be unimportant for this vehicle model, so it was taken to be zero. In surrmmary, for this simplified model the independent variables are slip angle, normal force, and drive/brake torque; the dependent variables are side force, circumferential force, and aligning moment. Longitudinal slip is calculated as an intermediate variable so that the standard CALSPAN side force roll-off calculation, which depends on longitudinal slip, can be used. The simplified CALSPAN tire model "can be broken down into the following steps. Details of the calculations are given later, (1) The circumferential force necessary to give the existing drive/brake torque is calculated. (2) The corresponding longitudinal slip is computed, which requires the calculation of several frictional properties of the tire and road surface. (3) The circumferential force computed in (1) is compared with the maximum tire capabilities and

21 reduced if necessary. (4) Rolling resistance effects are added. (5) The side force is calculated for the existing slip angle and normal force as if the tire were freerolling (no longitudinal slip). (6) The side force computed in (5) is modified if the longitudinal slip computed in (2) is non-zero. (7) The aligning moment is computed based on the normal force and side force. These steps are carried out by the following sequence of calculations. The circumferential force for the ith tire is first assumed to have the following form FCi TQti/RTi (2.55) The coefficient of sliding friction in braking (at longitudinal slip 1.0) as a function of normal load is represented by JSi i SO + SiFNi + S2FN (2.56) The peak coefficient of friction for zero slip angle as a function of normal load is represented by Pi P + PiFN + P2iFNi (2.57) Pi Oi ii i 2i~~~~

22 The longitudinal slip at which VPi occurs is given by SIi -ROi - RiFNi (2.58) Note that in the CALSPAN data ROi AND Rli have negative values so that (2.58) gives positive value for SI. An effective coefficient of sliding friction (longitudinal slip z 1) for the existing slip angle and road surface skid number is given by 11i x WSicoS(i)SNi (2.59) An effective peak coefficient of friction for the existing slip angle and road surface skid number is given by Mi z Pi(1-57.3BciCi)SSNi (2.60a) provided this expression is greater than p1i. Otherwise, uMi x Wi (2.60b) The longitudinal slip corresponding to this level of FNi and FCOi is given by Si z 1.0, for FCOi < -PMiFNi (2.61a)

23 O i EOi i i MN f M i Si -(FCi/FNi)(SIi/VMi), for FCOi<MiFNi ( Si -1.0, for FCoi >pMiFNi (2.61c) Note that if | FCoi i MiFNi, a value of slip Si will be obtained such that -SI < Si < SIi. The circumferential force generated by the tire is then given by FCAi FCi, for FCoi <MiFNi (2.62a) iFNi' for FCi >MiFNi (2.62b) Finally, an additional circumferential force, proportional to the normal load, is added opposing the direction of motion to simulate the effects of rolling resistance [2.5]. Thus, FC FC - KRiFNi (2.63) where K-yi is the rolling resistance proportionality factor. Some remarks should be made concerning the possibility of tire spin due to drive torque (S.i-1), even though maneuvers involving tire spin were not included in this investigation. The limit on circumferential force in this tire model leads to a corresponding limit on the drive

24 torque which can be utilized, given by TLi - FCAiRTi (2.64) This utilized drive torque would be less than TQti in the case of Si z -1. For a two-wheel-drive vehicle with a standard differential in the driven axle the drive torque applied to the wheels on that axle is the same and is limited by the torque which can be utilized by either tire. To accurately simulate post-tire-spin behavior of such a vehicle, TLi should be taken as the input drive torque to the non-spinning driven wheel, rather than TQti. This provision is included in the 3-DOF model. The side force generatd by the tire is computed by first calculating the force which would be generated by a free-rolling tire operating at the same slip angle, then modidying this to account for the longitudinal slip. The side force for a free-rolling tire is calculated by FSOi -G 'ai yiFNi (2.65) where yi is the peak lateral friction coefficient for the existing normal load, given by 'yi Z (BiFNi+B3i+B4iFN2SNi (2.66) G i is a side force shaping function, given by

25 Gai x 1.0 for ci > 3.0 (2.67) - l/3di i+ + 1/27)3 for a < 3.0(2.68) -1.0, for ai < -3.0 (2.69) where ai is a.an-dimensional 'slip angle defined by i. z -iC a/( yiFNi) (2.70) 1 i 1 i and C i is the low-slip angle cornering stiffness, represented by Ci A AOi + AiFNi - (Ai/A2i)FN2 for FNi ( A2i (2.71) AOi, for FN > A2i (2.72) It may be noted that at low slip angles this freerolling tire model behaves like a linear tire, reducing to FSOi C iai; at extremely high slip angles it saturates and behaves like a sliding tire, reducing to FSOi pyi FNi. The effects of longitudinal slip are accounted for by assuming that the side force of a side-slipping longitudinally-slipping tire can be broken down into two components: a "rolling" side force and a "sliding" side

26 force. A side force roll-off factor fi is defined where fi z 0 corresponds to a free-rolling tire (Si x 0), and fi 1 corresponds to a sliding tire (Si z 1.0). fi is given by linear interpolation on Si in a lookup table. The final value of the side force is then given by FSi FS i(l fi) + FNi Si sinai fisgn(FS i)SNi (2.73) where the first term is the "rolling" component and the second term is the "sliding" component. The aligning torque is assumed to be a function of both normal load and side force, and is given by TQALi (K iFNi+K 2i FS )FSi (2.74) Aerodynamic effects are treated in the standard fashion. They are represented by the following equations, where all forces are taken to act at the center of mass of the vehicle. The longitudinal drag, which affects the drive thrust requirement, is given by (2.75) where CD is the drag coefficient, ApF is the projected frontal area, pA is the density of air, and u is the forward velocity. This assumes the vehicle is moving through still air with constant velocity. Fxaero x CDApFPAU Ju./2 (2.75)

27 F = F =M =M = M =0(2.76) yaero zaero xaero yaero zaero The terms which are taken to be zero in (2.76) will be retained in the model for completeness. The drive-brake torque at tire i is given by TQ i = TQD RA Qi - PFL BRKi (2.77) where TQD is drive-line torque, RA is the drive axle ratio, XTQi are torque distribution parameters, PFL is brake-line pressure, and BRKi are brake torque coefficients. Chassis-drive axle torque transfer is given by Ti = QD XDTi i = 1,2 (2.78) Differential equations for inertial coordinates X, Y, and heading angle i are given by X = u cosi - v sin l (2.79) Y = u sin + v cost (2.80) ) = r (2.81) 2.2 Solution Procedure for the Equations for the ThreeDegree-of-Freedom Model.

28 Equations (2.15), (2.16), (2.17), (2.19), (2.20), (2.21) and (2.37) describe a set of coupled non-linear first order differential equations in the variables u, v, and r. To integrate these equations, it is convenient to put them into the form v = v(u,v,r) r = r(u,v,r) This cannot be done immediately due to the implicit nature of the tire force relations, the presence of compliance in the steering system, and the fact that the normal forceside force relation is not one-to-one. The desired form is obtained by the following method. Equations (2.23) through (2.28) are substituted into (2.15), (2.16), (2.17), (2.19) and (2.21). In matrix form, they, plus (2.37), (2.44), and (2.45) can then be written [C] la} = IbI (2.82) where a = u

29 a2 = a3 = a4 = FZ1 a5 = Fz2 a6 = F3 Fz3 a7 = F 7 =F z4 a8 = 6 p (2.83) and C.. = 0, except: 1] C11 = 22 = m + m 1 2 C13 = C61 = -m2 C23 = C62 = m2p C34 = C35 = C36 = C37 = -1 C42 = -m2p

30 C43 = I 43 =Ixz C44 =-Ytl ' C45 -Yt2' C46 -Yt3 ' C7 = t4 C51 = m2z C53 =S C53 =I yz -m2y Zp 54 Xtl C55 t2, C56 xt3 57 x C63 = Iz + 2(x +y2) C74 = Ytl (1-RS) C75 = Yt2 (1- RS) C76 = -Yt3RS C77 = -Yt4 RS C88 1 4 b1 = (ml+m2)vr+m2r 2x + a + b 1 2 2sp i=lfxi+Fxaero 4 = (m +m )r+m2 i Fyi+Fyaero (2.84)

31 b3 = (ml+m2)g+Fzaero 4 S 2 2 p b4 = Iyr +m2(urzp-r ypZp+ypg)-i=lztiFyi+Mxaero S 2 2 4 b -Izr +m2(vrz +r x -x)+ i z F +M +XZrp P P p i=l ti xi yaero 4 b6 = m2(urxp-vryp)+il[(xtiFyi-t iFxi )+Mzi+Mzaer b7 = (1-XRS)Tl RST2 8 = Sw + CST3 (TQsTl + TQT2)/NG /NG (2.85) {b} is a function of u, v, r, Fi Fy MZ and TQSTi (Faero, Maero, are functions of u, v, r). Fi, Fyi Mzi and TQTi depend on Fzi, which are elements of [a). It is thus STi. ZiJ necessary to solve (2.82) simultaneously with the following non-linear equation {bi = {b}(u,v,r,{a}) (2.86) The solution to (2.82) and (2.86) is obtained iteratively. At a given time, u, v, and r are known. An estimate of FZ is made from a previous time step. (The first estimate is made using the static weight distribution.) An initial {b0} can be computed, and the following algorithm applied.

32 {aj = [C]- bj (2.87) [bj+1) = {b}(u,v,r,{aj}) (2.88) This process is continued until ai, j-ai, 11 <(. i=l,...8 (2.89) where s. are assigned convergence parameters whose magnitudes are related to the expected magnitudes of a.. 2.3 Fortran Computer Program for the Three-Degree-ofFreedom Model The differential equations presented in 2.1 were integrated by a fourth-order predictor-corrector method. A pre-programmed code, the HPCG subroutine in the IBM SSP package [2.6] was used to implement the scheme. The source code for HPCG is not given in the following program listing since it is widely available. The 3-DOF model, as programmed, requires a set of "driver" subroutines, named DRINPT, DRINIT, DRIOUT, and DRIVER. This was done to interface this model with the driver module described in Reference [2.7] and Chapter 4. The call statements for these subroutines may be removed if alternate provisions are made for supplying values of

33 DELSW, TQD, and PFL in subroutine FCT. A Fortran listing of the program is given on the following pages, followed by a list of program variables and a typical data set.

34 0001 0002 0003 0004 0005 0007 0008 Lono 0010 0011 0912 0013 0)!3 0D14 ) 15 0016 0017 0319 00211 0022 0023 0024 0027 OO2 7 0028 002 0 0030 0031 0932 0033 0034 0035 0036 003 7 0030 )3 43: MAIN SUBROUTINE TRANS35 C MAIN PROGRAM F'R THF 3DOF MODEL. MAIN READS VEHICLE AND TIRE C DATA AND CONTROLS.THE ITERATIVE SOLUTION O= TIE STEADY-STATE EOUATI3NS C THIS VERSION OF MAIN REQUIRES SUBRDUTIES F35ITIrE3, 4INV, HPCG, FCT, OUTn. (OUT2 INCLUDED FOR EXTENDED OUTPUTI C DEVELOPED BY DOUSLAS L. WILSON, 8/30/81 LOGICAL*1 VEHCDN(6),TIRC3 (6),ICSET(6) REAL KIK2, KD REAL*8 TDTTTDTPRJT DIMENSION C (77) PRMT(51,Y(6),DEPY(6),AUX(16, 6),JUNK(7)t,JUNK2(7) EXTERNAL FCT, OUTI COMMON /T3DATA/ FRD(4,10,2), AO(4), A14), A2t4), 81(4), B3( 4) 1 B4(4), RT(41, P0(4), P1(4), P2(41, SO(4), S1(4), S2(4), 2 P0(4). R1(4), K1(41) K2(4), BC(4), SN(4), FRR(4) COMMON V3D/ CI(7,7),ALAMT(4) EC( 5iXT(41,YT(4,DTDE( 2),TAXL( 2, 1 AXLR,VC,G,ALAMRS,VMVIZZ,VIYZVIYVZXXPL,YPLZPLPLMPLIZZPLIfZ 2 PLIZX,CST1,CST2, CST3, SR, XC(21,BRK(4) HF HR,:),PA,RHOA,I ACKER COMMON /FOUT! FN(4),ALPHAT(4 ),TOT(4 ),FS(41,FC( 4,S(4,DOELT(2), 1 TOST(2) COMMON /OUTPT/ DSWOUT,TODOUT,PFLnUT COMMON /PRNT/ 3TPRNT,Tl COMMON /VPR/ DSWMAX, TODMAX, PIFLUAXKD,DSWOTO),PFLO COMMON /FINFO! IFIR; r C C REA) VEHTCLF AND TIRE DATA READ (5,105) VEHCON RFAD (5,101) (XT( I), YT I), 1=1,4) READ (5,101) DTOE(l) DT3E(2) READ (5,101) ALAMT(l), ALAMT(2) READ (5,101 )ALAMT(3), ALAMT(4) READ (5,1011 TAXL(1), TAXL(2) READ (5,100) VC. VII, VI YZ, VIZX, VM, AXLR, ALAMRS, G RFAD (5,100) XPL, YPL, ZPL, PLM, PLIZZ, PLIYZ, P.IZX RFAD (5,100) CST1, CST2, CST, SR, HF, HR READ (5,101 )XC(1), XC(2) RFAD (5,101) BRK(1), BQK(2) RPEA) (5,101) BRK(3), BRK(4) READ (5,100) CD, PFA, RHOA PFAD (5,99) IACKEF PFAD (5,100) DSWMAX,TQDMAX,PFLMAXKD READ RFAn RFAD READ READ READ R EAD PFAD READ READ READ R FD (5,105) (5,13)2 (5,102) ( 5,102) (5,10 2 (5,102) (5, 10 2) (~,102 (5,102) (5,10 2) (,102) (5,102) (5,102) TIRCON (PT(I), (A (I), A1( I), (A2 (I ) (R1( ), (B3( 11 (R4(I ) (PO 1) (P1 I), (P?(I), (SO( T), (Sl( 1, 1=1,4) I=1,4) =1, 4) I=1,4) I=1,4) 1=1,4).1=1,4 1=1,4) I=1,4) 1=1,4) 1=1,4) I=1,4 )

35 0041 On4? 0343 0044 0045 0046 0047 0048 0049 0050 001 o 0052 035 0054 00 54 READ (5,102 REFD ( 5,1021 READ (5,102) READ (5,102) READ (5,1021 READ (5,1021 READ (5,1021 READ (5,102) READ (5,104) 1S2()1, 1I1,4) (ROll), 1 I,4 (Ri I), I=1,4) (K1(l),I-1,4) ((2( ), I 1,41 (BC(l), 1=1,41 (SN(I), 1=1,4) (FR({ I)1, =,4) ( FRD( 1J,1),FRO(I,J,2), I=1,4),J1,'13) 0055 0056 0057 00o; 0959 0060 0061 0062 on62 Cn64 0065 0066 '0n67 0068 006c 0070 OD71 0072 0Q7? 0074 0075 0076 0077 0078 007 3ORO 00nop I 0082 0083 OOP4 c READ (4,1051 ICSET READ (4,100) XO,YO,PSIOUOVOPSIDO,SWO D),PFL 3,DT, DTPRNT READ (4,100) TMAX READ (4,103) (EC(I), 1=1,5) C W ITE f 1,1~c) VE4CON, TIRCON, ICSET 199 FORMAT ('IVFHICLF CONFIGURATION:'.6AU1/' TIRE tnNFIGURATIDN: &,Al/ *' INITIAL CCNDITIONS SET:,6A1/ *' Ts,llX, DELSWt'7X.'X'tllX ',X 11X,'PST',9X,'V',lX, *'p.llX,' AY' ) oq FORMAT (16) 133 FORMAT (F12.5) 101 FORMAT (2F12.5) 102 FORMAT (4E16.6) 133 FOR MAT (5F12.5) 104 FORMAT (RF12, 51 105 FORMAT (6A1 ) C C SET UP THE COEFFICIENT MATRIX 00 10 1=1,7 DO 10 J=1,7 10 r.(I,J) = 0.0 C C C c C(1,1) C( 1 3) C(2,? C 2,r3) = VM + PLM = -YPL*PLM = C(lf,1 = XPL*PLM C(t,4) = 1.0 Cfc3,5) = 1.0 Ci3,6) = 1.0 C(3,7) = 1.0 C(4,?) C( 4,3) C(4,4) C(4,95) C(4,6) C(4,7 ) C(5,1 ) C(15,4) C(5,5 ) C(5,61 C(5,7) = -ZFL*Pt = -PLM*XPL*ZPL + VTZX + = -YT( 1) = -YT!2) = -YT(3) = -YT(4) = ZPL*PLM = VIYZ + PLIf Z - fPL*ZPL*PLM = XTfl) = XT 2) = XT(3) = XT (4)

36 C 0085 0086 OOR 7 0088 OR 9 0090 ro091 C(6,l) = C(t,3) C(6,2) C(2,3) C(5,3) = (PLM*(XPL**2 + YPL**2) + PLIZZ + VIZZ) C C(7,4) C(7,5) C(7,6) C(7,7) = YT(1)*10.O- AL&MRS) = YT(2)*(l.f ) - ALAMRS) = -YT(3)*ALAMRS = -YT(4)*ALAM S C c OOP 2 0002 0004 0005 Ong9 Or)Q7 OnQ 8 OO9 0130 0101 0102 0103 0105 0106 0107 0108 0139 0110 0111 0112 11 I 0114 0115 0116 3117 011 8 0119 0120 0121 0122 0123 C c c c r, r CALCULATE THE INVERSE Dn 20 I=1,7 00 20 J=l,7 20 CI(I,J) = C(T,J) CALL MINV(CI,7,D,JJ NK1,JJNK2) IF (ABS(D).LT, 0.001) G3 TO 70 Tl = -DTPRNT IFT RST = 1 PPMT(I) 1 0.0 Or4T(2) = rMAX RMT( 3) = OT PRMT(4) = 0.1 YV 1) = XO Y(2) = Y9 Y(3) = PSIO Y(4) = UO Yt(5) = VO Y(6) = PSIDO OERY(l) = 0.16666 nERY(2 = 0.16565 r)EYt3) = 0.16667 OERY(4) = 0.16667 OERY(S) = 0.16667 )OF.Y(6) = 0.16667 CALL ORINPT CALL DRINIT CALL DRIOUT(It CALL HPCG (PRMT, YV DERY, 6, IHLF, FCT, OUT1, A'JXI IF (IHLF.LT. Ill GO TO 30 ERR3) RETURN FROM -PC3 WRITE (6,1511 IHLF 151 FORMAT ('OERROR RETURN FROM HPCG; IHLF -=' t3) STOP 30 IF (PRMT(51.NE. 0.0) GO TO 40 0124 0125 Ml?6 TM4AX HAS BEEN FXCEEDED WPITE ( 6, 1521 152 FRMAT ('OTMAX HAS BEEN EXCEEDED') ST OP Cr C VEHICLE HAS STOPPED

37 0127 40 WQITE (6,1513 0128 153 FORMAT('OVEHICLE HAS STOPPED' ) 012? ST3P C C COEFFICIENT MATRIX SINGULAR 0130 70 WITE (6,210) 3 0131 210 FORMAT ('OCCEFFICIENT MATRIX SINGULAR; D =',;20.1D 0132 STOP 0133 END *OPTIONS IN EFFECT* lDEBCOICSOJRCENOLISTNODECKLOADNOMAP *OPTIO4S 14 EFFECT* NAME = MAIN * LINECNT a 57 *STATISTICS* SOURCE STATEMENTS = 133,PR3GRAM SIZE - 5078 *STATISTI:S* NO DIAGNOSTICS GENERATED

38 C C SUBROUTINE OUTI CHECKS THE VEHICLE STOPPING CRITERI3N C AND WRITFS DUT INTERME3IATE VALUES. C 0001 SURPRUTINE OUT1 (TY,DY,IHLF,NFO,PaMT) 0002 REAL*8 DTPRNTT1 0003 DIMFNSICN PP4T 5), Y(6), DY(61 0004 COMMDN /OUTPT/ OSWOUT TOOUT PFLIUT 0005 COMMON /PRNT/ OTPRYTT1 C 0006 IF (((Y(41**2* Y(5)**2).GT. 0.11.OR. (TOD.NE. 3.0)) GO O'10 0007 PPMT(5) = 1.0 0008 RFTURN C C PPINT OUT INTERMEDIATE VALUES AT INTERVALS DTPRNT 0009 10 IT (T-Tl+O. COO1.LT. OTPRNTI GO TO 20 0010 T1 = T 0011 VEL = SORT(Y(3 )**2 Y(41**2) 0012 AY = DY(5) + Y(4t *Y(6 C WRITF(1,201) TDSf1JUTY(I),Y(2),Y(3),VEL,TODDUTYI4),Y(5),tY6), C 1 AY, PFLOUT DY(4),ODY 5),DY (6 0013 201 FORMAT( 'OT =',F12.4, 5X,' ELSW=',F12.4,5X,'X =',F12.,5X, ** Y =',F12.4 5X,' PSI =',F12.4,5X, 'VEL = ', F12.4/ *21X, 'TOD =',F12.4,5X,'U =',F12.4,5X,'V =',F12.4,5X, *'p =',F2.4,5X,' AY =, F12.44/ *21X,'PFL =',F12.4,5X,'UOOT =',F12.4,5X,'VDOr =',F12.45X, *,OO OT = ',F2.4) 0014 WR TF (2,?37? T,DS3Jr,f ( 1) Y( 2 ), Y(3), YI5), Y( 6), 4Y,Y (4) 0015 202 F3JRMAT( F12.5) 0016 20 RETURN 0017 FND *OPTI34S 14 FFFECT* ID,EBCOtC,SOJRCENOLIST,NODECK,LOAD,NOMAP *OPTIONS IN EFFECT* NAME = WT1, L INFCNT = 57 *STATISTI:* SOURCF STATFMENTS = 17,PRPOGRAM S IE 936 *STATTSTICS* NO DIAGNOSTICS GENERATED

39 C C SUBROUTINE OUT2 PRINTS OUT TIRE FORCES AND SLIPS C OUT2 IS NOT CALLED IN THIS VERSION OF MAIN PROGRAM' C 0001 SU3ROUTINE OUT2 0002 COMMON /FOUT/ FN(4),ALPHAT(4),TOT(4),FS(41,F: (4),S( 4) DELT(2), I TO ST(2) C 20 WRITE (1,2051 C WRITE (1,206) (,1FN( I,IFS(I),IFC( I ), 1,4 C WRITE (1,207) C WRITE (1,208) ( I,S(I) IALPHAT(I,I=,4) C WRITE (1.209) DELT(1,TOST(l),OELT(2),TOST(2) 0003 205 FORMAT ('ONORMAL FORCES AT TIRES',BX,'SIDE FORCES AT TIRES',13X, *'CIRCUMFERENTIAL FORCES AT TIRES') 0004 206 FORMAT I' FN(',I,'l) =',F12.5,11X,'FS ( ',i,' =e',F12.5,11X, *'FC(', Il,' ) t',F12.5) O95 207 FORMAT ('OSLIPS AT TIRES,16X,'SLIP ANGLES AT TIRES') 0006 208 FORMAT (' S(',I1,') =' 12.5,12X,'ALPHAT(,I1,' ) ='.F12.5) 0007 209 FORMAT( OFPCNT TIRE STFFR ANGLES: DELT(1) =',F12.5,8X, * 'KINGPIN STFERING TOROUES: TOST(l) =',FI2.5/27X, * ')ELT(2) =',F12. 5, t5X,' TOST(2) =',F12.5) 03D0 RET'JRN C009 ENI *OPTTONS IN EFFFCT* ID, EBCDIC,SOURCE. NOLIST,NODECKLOAD,NOMAP *OPTITNS IN EFFECT* NAME = OUT2, LINECNT = 57 *STATISTrIS* SOURCE STATE:NTS = 9,PROGRAM SIZE = 573 *STATISTICS* NO DIAGNnSTICS GENERATED

40 c c C SU'ROUTINE FCT SERVES AS AN INTERFACE BETWEEN HPCG A4D F35 C 0001 SUBROUTINE FCT(TYDYI 0002 DIMENSION Y161) DYt61 0003 CODMON /FINFO/ Ic!RST 0004 COMMON /FOUT/ FNI4, ALPHAT41,QTOT(4,FS( 41,F:(4),S(4),DELT 2), 1 TQST(21 0005 COMMON /OUTPT/ DSWDUT,TDOUTPFLOUT C 0006 nELl = DELTfII 0307 DEL2 a OELT(2) 0008 CALL DRIVER(DELSW,PFLtTOODD3,D49DYVTY,DEL1,DEL2) 000o CALL F35(DELSWtTOODPFLtY(4,tY(s5)Y(6) DY(4)!,Y(5)D Y(6),IFIRSTtT) 0310 DY(1) = Y(41*COS(Y(3)) - Y(5)*SIN(Y(3)) 0011 DY( 2) = Y(4)*SIN(Y(3)) + Y(51*COSY(3 )) 0012 DY(31 = Y(61 0013 DSWOUT = OELSW 0014 TODOJT = TOO 0015 PFLOUT = PFL 0016 RETURN 0017 END *OPTIONS I EFFFCT* I0, EBCDIC, tJ RCE, NOLIST r NODEC(, 3 AN 4JA) *OPTIONS T4 EFFECT* MAVE = FCT, LINECNT = 57 *STATISTICS* SOURCE STATEMENTS = 17,PROGRAM SIZE = 766 *STATISTI:S* NO DIAGNOSTICS GENERATED

41 C SURROUTINE F35 COMPUTES THE TIME DERIVATIVES FOR A SIMPLE 3-DOF C VEHICLE MODEL. THE ONLY FORCES ACTING ON THF VEHICLE ARE TIRE C FORCES, GRAVITY, AND AIR RESISTANCE C F35 REOUIRES SUBRnUTINF TIRE3 C DEVELOPED BY DOUGLAS L. WILSON 8/30/81 0001 SUBROUTINE F35(DELSWTOOrPFL UtV RUDOT VDOT RDOTI FIRSTT ) 0002 DIMENSICN B(I), 80(7), FX(4), FY(41, *FZ(4), THETAT(4), *TOAL(4 ), UT(4), VT(4),TL(4! 0003 COMMON /V30D CI(7,7),ALAMT(4),C(5),XT(4),YT(4),DTOE (2)TAXLI2), 1 AXLRVC,G, LAM SVM, VIZZVIYZ,V ZX,XPLYPLZPL,PLM PLIZZPLIYZ, 2 PLIZXCSTl,CST2,CST3,SRtXC(2),BRK(4,HFHRCD,PFA,RHOA,IACKER 0004 COMMnN /FOUT/ FN(41,ALPHAT(4),TQT(4),FS(4),FC(4),S(4),DELT(21, 1 TQST(21 C C INITIALIZE FZ AND TOST ON FIRST CALL 3F F35 0005 IF (IFIRST.NE. 1) GO TO 10 0006 FZ(1) = (VM*XT(3) + PLM*(XT(3)-XPL )/(XT(1)-XT(3))*G/2.0 0007 FZ(2) = FZ(ll 0008 FZ( 3) -(VM*XT(1) + PLM ( XT(1i-XPL) /(XT( )-XT(3 )*G/2.) 0039 FZ(4) = FZ (3) 0010 TOST( ) = 0.0 0011 TOST(2) O0.0 0012 IFIRST = 0 C C COMPUTE THE PART OF B WHICH IS INDEPENDENT OF TIRE FORCES 0013 10 80BO( = (VM+PLM)*V*R + PLM*XPL*R**2 - CD*PFA*P.H0OA*05*U*ABS(U) 0014 0( 2) = -(VM + PLM)*U*R + PLM*YPL*R**2 0315 B0(3) = -(VP + PLM)*G 0016 0( 4) = (VIYZ + PLIYZ)*R**2 + PLM*(U*R*ZPL - YPL*ZPL*R**2 1 + PLQ*YPL*@ 0017 B)(5) = -(VIZX + PLIZKI*R**2 + PLM*(V*R*ZPt + XP. *ZL*R**2) 1 -PL MXPL*G 0018 B0(6) = PLM*R*(XPL*U - YPL*V) 0019 80(7) = (1.0 - ALAMRS)*TAXL(l*TOD - ALAMRS*TAXL(2)*TOD C C CMPJTE T IRE PATCH VELOCITIES 0020 PI 02 = 1. 57079633 0021 00 20 I=1,4 0322 UT(I) = U - YT(I)*R 0023 VT(I) = V + XT(I)*R 0024 IF (ABS (JT(I.LT. 0.001) G3 T3 15 0025 THFTAT(I = ATAN(VT(I)/UT(I)) 0026 GO TO 20 0027 15 THETAT(I) = P102 0028 IF (VT(I.LT. 0.0) THETAT(It = -PI02 0029 IC (ABS(VT(I)).LT. 3.001) THETAT(I) = 0.0 00'30 O CONTINUE C COMPUT_ DRIVE TORQJE DISTRIBUTION 0031 00 30 1=1,4 0032 TOT (I) = TOD*ALAMT(I)*AXLR 003 3) TOT(I) = TOT(I) - BRK(I)*PFL C ITERATE TO FIND SOLUTION TO AX=B C

42 0034 K = 0 0035 40 K=<+1 C C SAVE THE OLD VALUES OF UDOT, VOCT, RDOT, FZ( fI DELT(I) C 3036 UOOTO = UOCT 0037 VnrTO = VDOT Or)3 8 ROOTO = ROOT 0q39 FZ1O = FZ(1) 0040 FZ20 = FZ(2) 00! FZ30 = FZ(3) 0042 FZ40 = FZ(4) 0043 OELTIO = DELT(1) 30044 DELT20 = DELT(2) C C COMPLOTE TIRE FORCES 3145 r00 50 1=1,4 0046 50 F4( I) = -FZ(I C C CHECK FOR TIRE LIFTOFF 0047 On 60 I=1,4 0014R IF (FN(I).GF. 0.001 G3 T3 60 0r09 WRITE (6,202) ( J,FNt(J)J=1,4) 0050 STOP 0051 63 CfNTI NUE C COMPUTE STEER ING ANGLES 9352 OmLTO = DEL SW/SR + CST3*( TQST(1) + T:QST(21 /SR* 2 C CHECK FOR STEERING TYPE 0053 IF (IACKER.EQ. t) GO TO 70 COMPUTE WAGON-TYPE STEER ING ANGLES 0054 nELT(l) = DELTP + CST *TQST( ) 0055 DELT(2) = DELTP + CST2*T-OST(2) 0056 "O TO 75 C C COMPUTE ACKERMAN STEER ANGLES 0057 70 XL = XT(11 - XT(3) o058 TO - TAN(DELTP) 0059 DELT(1 ) = DTOE(1) +ATAN(XL*TP/ (XL-TP4YT(11 ) +CSTl*rtST( 1) nO50 r ELT(2) = DTnF (?)+ATAV(XL*TP/(XL-TP*VT( 2)))4+:ST2#TQST(2I C C COMPUTE TIRE SLIP ANGLES 0061 75 00 80 =1, 2 0052 ALPHAT(I) = THETAT( I) - OELT( I) 0063 80 ALPHAT(I +2) = THETAT(1+21 C C COMPUTE TIRE FORCES 0064 00 85 1 = 4 0065 CALL T RE3(FNALPHATTQTFS,FCTQALStTLt C CHECK FOR TIRE SpI4 0066 IF (SMI) *GT. -0.999) GO TO 85 0067 IF (I.EO. 1) OT4ER ~ 2 0068 IF (I.EO.? IOT HER = 1 0069 IF (I.EO. 3) IOTHER = 4

43 0070 IF (I. EQ 41 IOTHER - 3 0071 TOT(IOTHER) = TLfI) 0372 IF (IOTHER.EQ. I.IQ. IOTHER.EQ. 3) * CALL TIRE3(FN,ALPHAT,TAT,FS,FC,TQALSTLIOTHER) 0073 85 CONTINUE C C RESnLVE TIRE FOR.ES ALONG VEHICLE AXES 0074 FX(1) = -FS(l)*SIN(DELT(l ) + FC (1 *COS(OELT 1 ) 0075 FYf ) = FSf*1 C3S(DELT(I) + FC( l)*SIN()ELT( 1) 0076 FX(21 = -FS(2)*SIN(ELT(2) ) + FC(2)*COS(OELT( 2 ) 0077 FV( 2) = FS(7)*COS(DELT(2) + FC(2)*SlN(rELTt( ) 0078 FX(3) = FC(3) 0079 FY( 31 = FS() 0080 FX(4) = FC(4) 0081 FY(4) = FS(4) 0082 TQST(l) = TOAL(l) - XC(11*FS(l) 0083 TOST(2) = TQALt() - XC(2)*FS(2) C C C COMPUTE THE TIRE-)IRCE DEPENDENT PART 3F 8 C 0084 8f 1 = 80(1) + FX(l) + FX(2) + FX(3) + FX(4) 0085 B(21 = 83(2) + FY(1) F FY(2) + FY( 3) FY( 4) 0086 8(61 = 80(6) - FX(1)*YTtfl-FX(2)*YT(2)-FX(3}*fT(3)-FX(4)*YT(4) 1 + FY(I)*XT(t)+FY(2)*XT( 2)+FY(3) XT(3)+FY(4 *XT(4) 2 + TOAL I1)+TQAL( 2+TOAL(3) +TOAL(4) t007 8(3) = 80(3) 0088 3(4) = 8O(4 - VC*(FY(1)+FY(2)+FY(3)+FY(41) 0089 8(5) = 80(5) + VC*(FX(1 +FX2 )+FX(3)+FX(4)) OOQO (7t1 = BO(7)+(1.0-ALAMRS)*HF*(CY( 1+FY(2))-ALAMRSiIR* (FY(3)+FY(4 ) C C rOMPUTE NEW VALUES FOR UOnT,Vn)OT RDOT, FZ(I) C 0091 UOOT = ).O 0092 V03T = 0.0 0093 ROOT = 0.0 0094 FZ(1) = 0.0 0095 97(21 = 0.0 0096 FZ(31 = 0.0 0097 FZ(4) = 0.0 009R 00 90 J = 1,7 0099 JDOT = UOOT + CT(1,J)*8(J! 0100 V)OT = VDOT + CI(2,J)*tB(J) 0101 ROOT = ROOT + CI(3, J*89(J 0102 FZi 1) FZ(l) + CI(4,J)*l8(J) 0103 FZ(2) FZ(2) + CI(5,J)B*(J) 0104 FZ(3) = FZ(3) + CI(6,J)*B(J) 0105 90 FZt 4) - FZ(4) + CI(T,J)*B(fJ C CHECK FOR CONVERGENCE C 0106 IF((UOOT-UOOTO)**2.LE. EC(l).AND. (VDOT-V03TO0**2.LE. EC(2) 1.AND. (RDOT-ROOTO1**2.LE. EC(3).AND. (FZ(1-FZ10)**2.LE. EC(41 2.ANO. (FZ(2)-FZ20)**2.LE. EC(4).AND. (FZ(3)-FZ30)**2.LE. EC(4)

44 3.AND. (FZ(4)-FZ4r.I**2.IF. EC( 4.4AND ( 3ELTf 1-3EL TlOl)2.LTT 4 fCC(5).AND. (OELT( 2-DELT20)**2.LT. EC(5 1 RETUR4 C C CHECK WHETHER ITERATION LIMIT HAS BEEN EXCFEOE) 0107 IF (K.LT. 20) GO TO 40 0108 WRfITE (6,201) T 3109 201 FORMAT ('ONO CONVERGENCE AFTER 20 ITERATIfrNS IN SUBROUTINE F32't * * AT T=',Ft2.5) 01.10 202 FORMAT (ONEGATIVE NORMAL TIRE FORCE —TIRE LIFTOFF'/ * f * FN,, t[1~ ' =',F12.5)) C 0111 RqETURN 0112 END *OPTIONS IN EFFECT* TD EBCDICSOURCENOLIST,NCDECK, LOADO 34A> *OPTIONS 14 cFFECT* 4AME = F35, LINECNT = 57 *STATTSTTCS* SOURCE STATEMENTS = 1l2,PROGRAS SIZE = 4088 *STATISTI:S* NO OIAGNOSTICS GENERATED

45 C SUBPOUTINE TIRE3 COMPUTES TIRE SIDE FORCE USIN3 THE CALSPAN MODEL. C CIRCUMFERENTIAL FORCES ARE COMPUTED BY FC=TQT/RT. C THIS M'DEL INCLU3ES SIDE-FORCE FRICTION ROLL-OCF AS A FUNCTION C OF SLIP, WHICH IS COMPUTED FROM THE CIRCUMFEqRETIAL FORCE. DEVELOPED BY DOUGLAS L. WILSON, 8/30/91 0001 SJRROUTINE TIRE3( FY ALPHAT TOT,FS,F, TOALtS,TL, tI) 00_2 REAL K1,K2 0003 r)IENSION FN( 4),ALP-iAT(4,TOT (4) tFS(4 ),FC (4 ) TQAL (4 I S (4) TLt(4 0004 COMMON IT3DAT A FRD (4Itl2), AO(4) Al(4)t, 2(41t B1 ( 4), B3(4) 1 B4(4), RT(4) P0(4), Pl(4 ) P2 (4 S0(4 ) 5S(4,, S 2(4), 2 R0(4), Ri(4i, Kl(4), K2(4), BC(4)~ SN(4), FRR(4) C 0005 FMAX = (Sl(II*FN(I) * 83(11 + B4(I)IFN( 1**2)*FN(II*SN(I) 3006 CALPHA = AOII) + A I )*N( I - A A (I ) FN( I) *2/ 2( I) 0007 IF (FN(I).GT. A?(I)) CALPHA = AO(I) 0008 ALFBAR = -CALPHA * ALPHAT(I) / FMAX 0009 DALF = ABS(ALFBAR) 0010 G = 1.0 0011 IF (ALFBAR.LT. 0.01 G = -G 3312 IF (DALE.LT. 3.3) G = ALFRAR - ALFBAR * DALF / 3.0 1+ ALFRAP**3! 27.0 0013 FS( II = G * FMAX 0014 FC(I) = TOT(I~/PT(I) 0015 UP = P0(I) + PlfI)*FN(I) 4 P2(It*FNI1)**2 0016 US = SO (I) + St ( )*FN(I) + S2( I )*FN( I )**2 0017 SI = -RO(I) - Rl(I)*FN(I) 001 Ul1 = US*ABS COS(ALPHAT(If ) /SN(I) C XM1 IS THE SLOPE AT LOW SLIP NUMBERS 0019 XM1 = UP*( 1.0-57. 3*C(I)IABS(ALPHAT(II))*SN(I)/SI 0020 IF ( XM.GE. Jl /SI G3 T3 20 0021 XM1 = Ul/SI C S IS THE SLIP NUMiER 0022?0 S([) = -(FC(It/FN(II /XM1 C023 IF (S(I).GT. SlI Sl(1 = 10. 0024 IF (S (.LT. -SI) S I = -1.0 0025 IF (ABS(S(I) I.LE. 3.999) GO TO 30 0026 FC(I) = U1*FN(I) 0027 IF (S(I) *GT. 0.3) FC(I) = -FC( I 0028 IF (S(IT.LE. -0.999) TLII) = FC(II)RT(I) ADD ROLLING RESISTANC. FORCES 33)29 0 FC I) = FC(I) - FP II)*FN( I) C CCMPUTE SIDE-FDRCE FRICTION ROLL-OFF000 D00 40 Jl~,9 0031 IF (ABS(S(II).LE. FRO(IJ+l,1 ) GO TO 50 0032 40 CONTINUE 0033 F = 1.3 0034 53 T3 60 0035 50 F = FRO(IJ,2 + (ABSISII) - FROII,Jl)) 1 *(FRO(IJ+1,2) - FRO(I,J,2 /(FrRO(IJ+,19)-FRO(ItJt,1 0036 60 L = l 0037 IF (FS( I.LT. 0.01 L t -1 0038 70 FS(I = FS( ) *(I*.O-F) + FN( I)*US*ARS (SIN(ALPHAT (I)) 1 *F*LtSN(I C nO039 80 TOL( II) = (K(1 ( FN(Il), K2 (I ) ABS FS( Il )FS (I

46 0040 R ETUN 0041 END *nPTIONS IN =FFECT* ID,EBCDIC,SOURCE,NOL ST,NCDECK,LOAD,NOMAP *OPTI3NS T^! EFFECT* NAME = TTRE, LIECNT = 57 *STATISrICS* S JRCE STATEMENTS - 41-ROGRAM SIZE = 2056 *STATISTICS* MD nOlG'OSTICS GENERATE3 NO STATEMENrTS FLAGGED IN THE ABOVE COMPILATIONS.

47 C NAASA 2.1.020 MINV FTN 06-24-75 THE UNIV OF MICH COMP CTR C C C SUBROUTINE MINV C C PJRPCSF INVERT A MATRIX C C USAGE C CALL MINV(A,ND,L,M) C C DESCRIPTION OF PARAMETERS C A - INJPUT MATRIX, DESTROYED IN COMPUTATION AND REPLACED BY C FRFSULTANT INVERSE. N - ORDER OF MATRIX A C D - RESULTANT DETERMINANT C L - WORK VECT3R OF LENGTH N M - WORK VECTOR OF LENGTH N C C REMARKS C MATRIX A MUST BE A GENERAL MATRIX C SUBROUTINES AND FJNCTICN SUBPROGRAMS RE')JIED C C.ONE C C METHOD C THE STANDARD GAUISS-JODAN METHOD IS USED. THE DETERMINANT C IS ALSO CALCULATED, A DETERMINANT OF ZERO INr)ICATES THAT C THE MATRIX IS SINGULAR. C C 0301 SUBRUT I E I ( A, N, D.L, M ) 0002 DTMFNSICN A(l),L(1, M( 1) C C IF A DOJBLE PPECISION VFRSI3N 3F THIS ROUTINE IS DESIRED, THE C IN COLUMN I SHOULD BE RFMOVED FROM THE D3JBLE )RECISION C STATEMENT WHI-H F3LL3WS. C DOUBLE PRECISION AD,BIGAHOLD C C THE C MUST ALSO BE REMOVED FROM DOUBLE PRECISI3N STATEMENTS C 4ADEARING IN OTHER ROUTINES USED IN CONJUNCTION WITH THIS C ROUTINE. C C THE DOUBLE PRECISION VERSInN OF THIS SUBP3UTINE MUST ALSO C CONTAIN DOUBLE PRECISIlnO FORTRAN FUNCTIONS. ABS IN STATSMENT C 10 MUST LE CHNGfOn-TO DABS. C C C SEARCH FCR LARGEST ELEMENT

48 c 0003 0004 O005 0006 0007 0008 000Q 0010 0011 0012 0013 0014 001 5 3316 0017 0018 0019 D=l.0 NK —N DO 83 K1, N NK=NK+ N L(K)=K M(K)=K KK= NK K BIGA= A(KK) DO 20 J=K,N IZ=N*( J-1) DO 20 I =KN IJ=IZ+I 10 IF( AS (BIGA)15 RIGA=A(IJ) LIK)=I M(K =J 20 CONTINUE ABS(A(I J I )) 15,20,20 0020 3021 0022 0023 0024 0025 0026 002 8 C C 0030 0031 0032 0032 0033 0034 0035 0036 0337 INTERCHANGE R3)S J=L(K I IF(J-K) 35,35,25 25 KI=K-N DO 30 I=1,N KT=KI+N OL D=-A (KI ) JI=KI -K+J AfKI =A(JI) 3D AfJII =HOLD INTERCHANGE COLUMNS 35 I=M(K) IF(I-K) 45,45,3 3R JP=N (1-1) DO 40 J=1,N JK =NK+J JI=JP+J HOLD=-A(JKI A(JK)=A(J I) 40 A(JIl =HOLD DIVIDE COLUMN BY MINJS PIVOT (VALUE OF PIVO CONTAINED IN RIGA) 45 IF(BIGA) 48,46,48 45 0=0.0 Q FTURN 48 Dn 55 I=1,N IF( I-KE 50,55,'0 50 IK=NK +I A( TK)=A(IK)/(-BIGA) 55 C TTINUE C C C C 0038 009 0040 0'41 0042 0043 0344 0045 C

49 C REDUCE MATRIX C 0046 00 65 l=lN 0047 IK NK+I 0048 HOLD=A(IK) 0049 IJ= I-N 0050 DO 65 J=1N 0051 IJ=IJ+N 0052 IF(I-K) 60,65,60 0053 60 IF(J-K) 62,65,62 0054 62 KJ=IJ-I+K 0055 A( IJ)=HCLD+A(KJ )+A(IJ) 0056 65 CONTINUE C C DIVIDE ROW BY PIVOT C 0057 KJ=K-N 0058 00 75 J=1,N 005 KJ=KJ+N 0060 IF(J-K) 70,75,70 0061 70 A(KJ) =A (KJ) /IGA 0062 75 CDOTINUE C PRODLCT OF PIVOTS C 0063 D=D*BIGA C C RFPLACE PIVOT BY RFCIPROCAL C 00 64 A(KK)=1.0/B IGA 0065 8' CONTINUE C C FINAL ROW AND COLUMN INTERCHANGE C 0066 K=N 0067 100 K=(K-1) 0068 IF(K) 150,150,105 006q 105 I=L(K) 0070 IF(I-K) 2120129108 0071 108 JO=N*(K-1) 0072 JR=N*( -1) 0073 DO 110 J=1,N 0074 JK=JO+J 00 7 5 HOL D= A (JK) 0076 JI=JR+J 0077 A(JK)=-A(J I 0078 110 A(J!) =HOLD 0)79 120 J=M(K) 0080 IF(J-K) 100,100o125 0301 125 KI=K- N 0092 DO 130 1=1,N 00Pt KI=KT+N O 8 4 HOLD=A(KI 0095 JI=KI-K+J 0096 aK I) =-A(J I

50 0087 130 A(JI) =HOLD 00 8 GO TO 100 0089 150 RETURN 0090 END *OPTIONS IN EFFECT* ID,EBCDIC,SOURCE,NOLIST,NODECK,LOAD,NOMAP *OPTIONS IN EFFECT* NAMF = MINV, LINECNT = 57 *STATISTICS* SOURCE STATEMENTS ~ 90PROGRAM SIZE E 2084 *STATISTICS* NO DIAGNOSTICS GENERATED NO STATEMENTS FLAGGED IN THE ABOVE COMPILATIONS.

51 2.4 Program Variable Model Definitions for Three-Degree-of-Freedom Program Variable Analytic Symbol Definition ALAMRS ALAMT(I) ALFBAR ALPHAT (I) 1=1,4 XRS xTQi 'ai i Fraction of total roll stiffness at front axle Fraction of total drive torque applied at wheel I. Note that ZX = 1 TQi Normalized slip angle Slip angle at tire i AUX AXLR AY RA Auxiliary variable required by subroutine HPCG Axle drive ratio, same at front and rear if both driven Lateral acceleration AO(I ),A1 (I), A2(I) I=1,4 Coefficients in expression for low-slip cornering stiffness B(I) I=1, 7 BC(I) 1=1,4 {b} BCi Vector of force-type quantities in iterative solution to {a} Tire parameters which give the influence of slip angles on circumferential force *Proaram variables are not listed for Subroutines MINV or HPCG

52 Program Variable Definitions for Three-Degree-of-Freedom Model Program Variable Analytic ymbo Definition BRK(I) I=1,4 BRKi Brake torque coefficient for wheel i Part of {b} which does not depend on tire forces BO(I) I=1,7 B1(I),B3(I) B4(I),I=1,4 C(I,J) I=1,7,J=1,7 CALPHA CD Coefficients in peak lateral friction coefficient expression cij Cci CD Matrix of inertia-type quantities in iterative solution to {a} Low-slip angle tire cornering stiffness Aerodynamic drag coefficient Inverse of C.. 13 CI(I,J) I=l, 7,J=1, 7 CST1, CST2 CST3 ST1 ' CST2 Steering system compliances (right, left, steering column) CST3 D DALF Determinant of C.. 13 Absolute value of a DELSW 6Sw Steering wheel angle

53 ProgramVariable Definitions for Three-Degree-of-Freedom Model Program Variable Analytic Symbol Definition DELT(I) 6 6 1' 2 Front wheel steering angles (right, left) Pitman arm angle DELTP p DELT10,DELT20 Values of DELT(I) from previous iteration DELl, DEL2 Arguments for DRIVER; correspond to DELT(I) DERY(I) 1=1,6 Input variable for HPCG DSWMAX DSWOUT Maximum allowable value for DELSW; required by DRIVER Output variable; corresoonds to DELSW DSWO DT Initial value of DELSW; reauired by DRIVER Integration time step DTOE(I) I=1,2 STOE1 STOE2 Toe-in angles of right and left front wheels, positive for positive rotation about z - axis DTPRNT Time increment at which output is printed Dummy arquments; required by DRIVER D3,D4

54 Program Variable Model Definitions for Three-Deqree-of-Freedom Program Variable Analytic Symbol Definition EC(I) 1=1,5 ei 1 F fi 1 FC(I) I=1,4 FC. 1 Convergence criteria for the iterative solution to ~a1[ Side force roll-off factor Circumferential force at tire i Intermediate variable in tire side force calculation Normal force at tire i FIA X FN(I) I=1, 4 FRO(I J,K) 1=1,4 J=l,10 K=l,2 FRR(I) I=1,4 FS(I) I=1,4 FN. 1 Lookup table for computation of f. K=l gives S, *=2 gives fi. I=tire number, J gives tabular values KRRi FSi 1 Rolling resistance proportionality factors Side force at tire I FX(I),FY(I), FZ(I) I=1, 4 Fxi Fyi F zi 1 Components of tire forces at tire contact points FZ10,FZ20 FZ30,FZ40 Values of FZ(I) from previous iteration

55 Program Variable Definitions for Three-Degree-of-Freedom Model Program Variable Analytic Symbol Def inition G g Gravity G (subroutine TIRE3) Ga. 1 Tire side force saturation function Height of roll center above ground plane, front and rear axles HF, HR IACKER IC SET IFIRST hF'hR Steering type indicator; IACKER = 0: Parallel IACKER = 1: Ackerman Label for initial conditions data set (6 characters) Indicator for first integration time step IHLF Error indicator for HPCG IOTHER Index identifying laterally opposite tire JUK1(I),JUNK2(I) 1=1,7 Dummy vectors recuired by 1,MI: NV KD Understeer factor; reauired by DRIVER K1(I),K2(I) 1=1,4 Ki,K2i Coefficients in tire aligning torque calculations

56 Program Variable Model Definitions for Three-Degree-of-Freedom Program Variable Analytic Symbol Definition NEQ PFA APF PFL FL PFLMAX Number of equations integrated by HPCG Projected frontal area of vehicle Brake-line pressure Maximum allowable value of PFL; required by DRIVER Output variable; corresponds to PFL Initial value of PFL; required by DRIVER 7T/ 2 PFLOUT PFLU PI02 PLIZZPLIYZ, PLIZX I2 2 I zzI yz yz 2 zx Movements of inertia of payload w.r.t. x2y2z2 axes PLM m2 PRMT(I) I=1,4 Mass of payload Control parameters reauired by HPCG Initial value of PSIO P0(I),P(I) P2(I) 1=1,4 Po i P1 i P2i Coefficients in the peak coefficient of friction versus normal force expression

57 Program Variable Model Definitions for Three-Degree-of-Freedom Program Variable Analytic Symbol Definition R r RDOT r RHOA RT(I) I=1,4 0RO(I),P1(I) 1=1,4 S(I) I=1,4 SI RT. OiRli 1 Si SI. 1 SN. 1 Angular velocity of vehicle about z-axis d(r)/dt Density of air Rolling radius of tire i Coefficients in longitudinal slip versus normal load expression Longitudinal slip at tire I Longitudinal slip at which peak coefficient of friction occurs Skid number ratio: present surface/ measurement surface Steering ratio SN(I) 1=1,4 SR NG S(I),S1(I) S2(I) I=1,4 T 0i sli S2i t Coefficients in expression for the sliding friction coefficient Time TAXL (I) DT1' DT2 Drive toroue transfer

58 Program Variable oI c del Definitions for Three-Decree-of-Preedom Program Variable Analytic Symbol Definition I=1,2 parameter: fraction of drive torcue acting at axle I (front,rear) which would cause axle roll relative to chassis. TAXLtO corresponds to negative axle roll about x-axis THETAT(I) I=1,4 i Angle between x-axis and tire contact point velocity vector at tire i TIRCON Label for tire data set (6 characters) TL(I) 1=1,4 TLi 1 Drive torque limit. When driven wheel i.spins, only generating TL., then TL. is drive to cue input to wheel IOTHER TMAX Simulation stopping time TP tan (6 p) A l g i g toPe a TOAL(I) 1=1,4 TALi TQD TQDi-AX Aligning toroue at tire i Drive torcue Maximum allowable value of TOD; recuired by DRIVER TQDOUT Output variable; cor

59 Program Variable Model Definitions for Three-Degree-of-Freedom Program Variable Analytic Symbol Definition TQDO TQST(I) I=1,2 TQST1, TQST2 T ST2 responds to TQD Initial value of TQD; reouired by DRIVER Steering toraue about the kingpin due to tire forces and moments (right,left) Drive/brake toroue at tire i Time at which output was most recently printed Components of vehicle velocity along x and y axes TQT(I) I=1,4 TQti Tl U,V u,v UDOTVDOT UP US u,v uPi Si du/dt, dv/odt Peak coefficient of friction Coefficient of sliding friction Components of tire contact point velocity along x and y axes UT(I),VT(I) I=1,4 uo,vO Initial values of u, v Effective coefficient of sliding friction Ul li

60 Program Variable Definitions for Three-Degree-of-Freedom Model Program Variable Analytic Symbol Definition VC ti Height of vehicle center of mass above ground plane Label for vehicle data set (6 characters) Speed of vehicle VEHCON VEL VIZZ,VIYZ VIZX 1 1 I,I zz yz zx Moments of inertia of vehicle w.r.t. xyz axes VM XC(I) I=1,2 C x Vehicle mass Caster offset Wheelbase Effective maximum coefficient of friction XL L XM1 Mi XPL,YPL,ZPL Xpr yp' Coordinates of center of mass of payload w.r.t. xyz axes zP XT(I),YT(I) 1=1,4 Xti ryti Coordinates of tire contact points w.r.t. xyz axes XO,YO Initial val'ues of inertial position X and Y

61 Program Variable Definitions for Three-Degree-of-Freedom Model Program Variable Analytic Symbol Definition Y(I) State vector: 1=1,6 Y(I) = X Y(2) = Y Y(3) = Y(4) = u Y(5) = v Y(6) = r

62 2.5. Three-Degree-of-Freedom Vehicle Data For Base Configuration Xti'Yti'zti 48.0 30.75 21.74 (coordinates of tire 48.0 -30.75 21.74 contact points), in -61.0 30.50 21.74 -61.0 -30.50 21.74 6 0.0 0.0 TOE1 ' TOE2 (toe angles), rad drive torque distri- 0.0 0.0 bution parameters 0.5 0.5 (fraction of TQD at each wheel) ATT (drive torque 0.0 0.0 distribution parameter: front, rear) A (roll-stiffness 0.66 RS ratio, front/total)

63 rear axle ratio g, in/sec2 2 m" Ib-sec -in 2.79 386.4 9.403 I1 Izz I1 yz' I1 zxI ZX~* lb-sec -in lb 2 Ib-sec -in 22500..0 0.0 -230.0 Xp,ypZp (payload coordinates), in 2 m2, lb-sec /in 2 2 I2 lb-sec -in ZZI -6.0 0.0 -6.0 0.83 0.0 0.0 2yz lb-se2in lb-sec -in lb-sec Ib-sec -in 0.0

64 CST1 (right steering linkage compliance), rad/(in-lb) CST2 (left steering linkage compliance), rad/(in-lb) CST3 (steering column compliance), rad/ ( in-lb) SR (overall steering ratio) 0.000005404 0.000005404 0.001389 17.5 0.0 hF (height of roll center), hR (height of roll center), front in rear in 0.0 Cxi (caster offset), in BK. (brake torque coefficient), in-lb/(lb/in2) = in3 0.0 0.0 30.0 20.0 30.0 20.0 CD (drag coefficient) 0.45

65 Ap (projected frontal 3100.0 area), in2 PA (air density), 0.000000115 2 4 lb-sec /in Convergence criteria used in solution: C 0.05 convg 5 2 c1 = 0.001 in/sec c2 = ~0.001 in/sec2 2 ~3 0.0001 rad/sec2 ~4 = 5 = 6 - ~7 = 0.1 lb 68 = 0.001 rad

66 BODY 1 BODY 2 X2 x z2 z FIGURE 2.1 BODY-FIXED AXIS SYSTEMS

67 MM 21 BODY 2 F12' Tmlg BY 1 BODY 1 FIGURE 2.2 FREE-BODY DIAGRAMS OF BODY 1 AND BODY 2

68 N N Torsional Spring, 01 Stiffness k2 - Torsional Spring, Stiffness k FIGURE 2.3 CONCEPTUAL MODEL FOR NORMAL FORCE DISTRIBUTION

Y 69 z x 1 xl zl 1 Z1 FIGURE 2.4 FREE-BODY DIAGRAM OF FRONT AXLE OF CONCEPTUAL MODEL

70 FSw FRONT CST2 CST1 LEFT RIGHT FIGURE 2.5 SCHEMATIC DIAGRAM OF STEERING SYSTEM, 3-DOF MODEL

71 PARALLEL STEER I- ~ 11 6P 2= 6p ACKERMAN STEER I I /// 6.2// II.// 621 61 R Ir 1 ~~ /// I 1 / _ ' 4 J -. + FIGURE 2.6 STEERING LINKAGE GEOMETRIES: PARALLEL AND ACKERMAN

72 FRONT Looll A = Tire contact point B = Intersection of kingpin axis with ground plane C = Caster Offset X FIGURE 2.7 SCHEMATIC DIAGRAM OF CASTER OFFSET

73 REFERENCES FOR CHAPTER 2 2.1 "Reserch on the Influence of Tire Properties on Vehicle Handling", Calspan Corporation, Rept. No. DOT-HS-053-3-727, 1976. 2.2 "Tire Parameter Determination", D. J. Schuring, DOT-HS-4-00923, 1975. 2.3 "The Dynamics of Pneumatic Tires", Motor Vehicle Performance - Measurement and Prediction", Highway Safety Research Institute, Univ. of Michigan, 1974. 2.4 "Analysis of Tire Properties", H. Pacejka, Chapter 9, Mechanics of Pneumatic Tires (S. K. Clark, Ed.) NHTSA, 1982. 2.5 "A Handbook for the Rolling Resistance of Pneumatic Tires", S. K. Clark and R. N. Dodge, DOT-TSC-1031, 1978. 2.6 System/360 Scientific Subroutine Package, Version III, Programmer's Manual, IBM Corporation, 1970. 2.7 "Improvement of Mathematical Models for Simulation of Vehicle Handling", Vol. 6. Programmer's Guide for the DRIVER MODULE, W. R. Garrott, D. L. Wilson, A. M. White and R. A. Scott. Final Report DOT-HS7-01715, December 1979.

74 List of Symbols The nomenclature used in Chapter 2 is presented below. Symbol Definition,1 a2 Acceleration of body 1 and body 2, respectively. Body 1 is the vehicle. Body 2 is the payload. {a),a. Vector of kinematic derivatives, normal tire forces, and steering angle. APF Projected frontal area of vehicle. Oi' li'A2i Coefficients in expression for low-slip-angle cornering stiffness. B lil B3i Bqi Coefficients of curves fitted to the peak lateral friction coefficients. BCi Tire parameters which give the influence of slip angles on circumferential tire forces. BRKi [b} bi Brake torque coefficient for wheel i. Vector of force type terms. Aerodynamic drag coefficient. CD C.i 1) Matrix of inertia-type quantities in iterative solution to {a}. ST1 ' CST2' CST3 Steering system compliances. CX Caster offset.

75 Coi Low-slip-angle tire cornering stiffness. Side force roll-off factor for tire i. Net force acting on body 1, excluding gravity and F12 F Fx Fy Fz Components of F O xaero' yaero' zaero Components of the aerodynamic force that acts at the vehicle center of mass. These are designated as F, ZF ys F in Ref [1.1]. F12 Force on body 1 exerted by the connection from body 2. F21 Force on body 2 exerted by the connection from body 1. Fxi yi zi Components of the tire forces at the tire contact points. FC. 1 Circumferential force at tire i. FCAi FCoi Intermediate variables in tire circumferential force calcultions. FN. 1 Normal force at tire i. FS 1 g Side force at tire i. The acceleration due to gravity. Function that arises in tire side force saturation.

76 hFhR Height of roll center above ground plane, front and rear axles, respectively. +1 +2 H,H Angular momentum of body 1 and body 2, respectively. HY Angular momentum components. n I 1,[I2] Inertial tensor of body 1 and body 2, respectively. i1 2 I 1,2I Moments of inertia of bodies 1 and 2 ~nz' nz ww.r.t. the z-axis 2 2 2 Note: IxIyz Ixz are defined such that the inertial tensor [I ] is given by 2 2 2 2 [I2] = Ixx -Ixy -I 2 2 2 I2y Iyy Iyz I -I I -xz yz zz S ~I mn~ ~Sum of corresponding moments of inertia of bodies 1 and 2. K Sum of torsional spring constants kl and k2. kl,k2 Roll stiffness used in determining the normal tire force distribution. KliK2i Coefficients in tire aligning torque calculation. KRRi Rolling resistance proportionality factors. L Wheel base. +^ Mo Resultant external moment acting

77 Mx My M z Mxl1 Mx2' x3' Mx4 M12 M21 ml, m2 xaero' yaero' zaero NG Poi'Pli'P2i FL r RA R r R0i, Rli on body 1, excluding m12 Components of Mo0 X components of moments generated by the tires. Moment on body 1 exerted by the connection from body 2. Moment on body 2 exerted by the connection from body 1. Mass of body 1 and body 2, respectively. Components of the aerodynamic moment at the vehicle center of mass. Steering ratio. Coefficients in the peak coefficient of friction versus normal force relations. Brake line pressure. Angular velocity of body 1 about the z-axis. Axle drive ratio. Rear axle center turning radius. Coefficients in longitudinal slip versus normal load relations. Rolling radius of tire i. RT. 1

78 Si Si. 1 SOi'li'S2i SI. SN. 1 t T1T2 TLi 1 Longitudinal slip of wheel i. Coefficients in the sliding friction coefficient versus normal load relations. Longitudinal slips for which the peak coefficients of friction occur. Skid number. ratio: present surface/measurement surface. Time. Torque about the x-axis transmitted from the vehicle body to the solid front and rear axles, respectively, as a result of drive torque. Drive torque which can be utilized by tire i. Tire aligning moments. Drive line torque. Steering torque about the kingpin due to tire forces and moments (right, left). Drive/brake torque at tire i. Component of v along the x-axis. Component of v1 along the y-axis. Inertial coordinate system. Z is directed downward TALi QD TQST1, TOST2 TQti u v X,Y,Z

79 xy, z Xpyp zp Body-fixed coordinate system. x and z are directed towards the front and bottom of the vehicle, respectively. Coordinates of P (center of the payload) with respect to the xyz axes. xti',Yti Coordinates of tire contact points with respect to the xyz axes. x2'Y2,Z2 Body-fixed coordinate system with origin at P (center of mass of payload), parallel to the xyz axes. aC. 1i a. 1 Slip angle at tire i. Normalized slip angle at tire i. Angle between x-axis and tire contact point velocity vector at tire i. ri 6P Pitman arm angle. 6 Sw Steering wheel angle. 6TOE1'6TOE2 Toe-in angles of right and left front wheels, positive for positive (i.e., right-hand) rotation about the Z-axis. 61'62 Front wheel steering angles (right, left). Y(=1,2) An index designating body 1 or body 2.

80 Ei eR xRS Convergence criteria for the iterative solution to {a}. Roll of vehicle body about an axis parallel to the x-axis. Fraction of total roll stiffness at front axle in conceptual model for normal tire force distribution. Fraction of drive torque applied at wheel i. XTQi XDT1 XDT2 Drive torque transfer parameter (front, rear). UMi UPi Effective maximum coefficient of friction at tire i. Peak coefficient of friction for tire i. Coefficient of sliding friction at tire i. Peak lateral friction coefficents for tire i. y-yi Pli Effective coefficient of sliding friction at tire i. Density of air. pp Position vector from vehicle center of mass to payload center of mass. 14 Heading angle; angle between inertial X axis and vehicle x axis.

81 + Angular velocity of body 1.

82 Chapter 3. Modifications Required in IDSFC to Simulate Asymmetric Vehicles 3.1 Changes in the Equations of Motion The equations of motion of the vehicle in IDSFC are written [M]{x} = {F} (3.1) where {x} is the state vector consisting of the ten degrees of freedom described by differential equations (the remaining seven "degrees of freedom" are handled using algebraic approximations), [M] is a matrix of inertial-like quantities, which depends on {x} through suspension deflections, and {F} is a vector of forcing functions which also depends on {x}, and is found through a numerical iterative procedure. IDSFC was modified to include the opti ns of adding a rigidly attached payload of arbitrary inertial properties at an arbitrary point in the vehicle sprung mass, and of transmitting an arbitrary portion of the drive torque from the chassis to the rear axle in the case of a driven solid rear axle. The equations below detail the changes necessary to implement the options above; the new terms are underlined. The "Technical Manual" referred to below is 2 X Ref. [3.1]. Variables m2, I in' XP, y, ZP, and XTT are 29 mn' Xpi yp, pI a nd defined in section 2.1.

83 Changes Necessary in {F} Equations (2.2S) and (2.21) in Technical Manual: F(1) z (vr-wq-gsinO) (M+m2) + EFs. + EFX + m2(q2x +r2x -qpy-rpzp) 2 P p-.p + terms involving assumption switches 0ijkl (3.2) Equations (2.3S) and (2.3I) in Technical Manual: F(2) 3 (wp-ur+gcosesin )) (EM+m2) + 2( r2yp+p2yp-rqzp-pqx ) - 2 P P P P + ZFys + EFyu + terms involving 0ijkl Equations (2.4S) and (2.4I) in Technical Manual: F () (uq- p+gcosOcos ) (MS+m2) + mr2(p2zp+q2zp-prxp-qryp) 4 + ZF - ZSi zs i= 1 (3.3) (3.4) Equation (2.5S) in Technical Manual:

84 F(4) z y2(ur-wp-gcosesinc) + ENgs + ENu + N p2 + T2 + terms involving 0 ijkl Equation (2.51) in Technical Manual: (3.5) F(4) Z y2(ur-wp-gcosesinc) + EN s + EN u + N,2 + terms involving 0ijkl (3.6) wh e r e N ~ ~2 2 2 2 2 I2pq + q - I rq- I pr + I qr - I r X2 xz L yz zz xy YY yz + m2(uqy-vpyp-prxyp+q z y -qry 2 + urz -wpz +rqz 2-r2ypz +pqxpx +ypgcosecosc-z gcosesinc) (3.7) and T2 x TQDXTT Equations (2.6S) and (2.6I) in Technical Manual: F(5) z y2(vr-wq-gsine) + EN s + ENe + (3.8)

85 + terms involving 0ijkl where _ 2 2 2 2 2 2 Ne2 s -IxxPr qr x + I r - p -I yqp xyxz x z yz 2zz + vrz qzp qpypz r2 +vpX -uqXp+prxp2-p Zp xpqrypx P P P P p 9r px (3.9) -z gsinE-x gcosCcos) p p (3.10) Equations (2.7S) and (2.7I) in Technical Manual: F(6) x Y'1 (wp-ur+gcosesinc) + ENgs N+ ZN.+ N 2 + terns involving 0ijkl ijkl (3.11) wh e r e 2 2 + wqy2(wpxp-rqZpXp+p2ypXp-pqXp 2 +wqyp-vryp+qpyp-qxpyp+rpzpyp

86 +xpgcose sinp+ypgsine) Equation (2.11S) in Technical Manual: F(10) = ZN R - T2 + terms involving 0ijkl Changes Necessary in [M] Equation (2.16) in Technical Manual: (3.12) (3.13) M(1,1) = EM + m2 (3.14) M(2,2) = ZM + m2 (3.15) Equation (2.22) in Technical Manual: M(3,3) = M + m2 (3.16) Equation (2.15) in Technical Manual: M(3,4) = m2p M(3,5) = m2xp M(4,3) m2y M(4,5) = I2 xy (3.17) (3.18) (3.19) (3.20)

87 M(5,3) = -m2xp (3.21) M(5,4) = -I2 xy - m2 yp 2p p (3.22) Equations (2.17S) and (2.171) in Technical Manual: M(5,1) Y2 + m2zp + terms involving Oijkl (3.23) Equations (2.18S) and (2.181) in Technical Manual: M(6,1) = -m2 + terms involving ijkl (3 Equations (2.19) in Technical Manual: M(4,2) = - Y m2Zp + terms involving Oijk (3 Equation (2.20) in Technical Manual:.24).25) M(2,6) = Y1 + m2xp (3.26) M(6,2) = Y + m2xp (3 Equations (2.26S) and (2.26I) in Technical Manual: M(2,4) = - 2 - m2Zp + terms involving 0ijkl (3.27).28)

88 M(1,5) = Y2 + m2Zp + terms involving 0ijkl 2.2....k (3.29) Equations (2.27S) and (2.27I) in Technical Manual: M(44) 2 2 2 M(4,4) = Ix+, + Ixx+ m2(Yp + xx 2 p p + terms involving 0ijkl ijkl (3.30) Equations (2.28S) and (2.281) in Technical Manual: M(6, 4) = -I ' ~~ XZ - Ixz - I2z - m2x z P P + terms involving 0ijkl ijkl (3.31) Equations (2.31S) and (2.311) in Technical Manual: ' 2 M(5,5) = I + I + I2 Y. yy + m2(p +Zp2) +zp + terms involving 0ijkl ijkl (3.32) Equations (2.32S) and (2.321) in Technical Manual: M(6,5) = -I yz - m2Yp z + terms involving 0ijkl 2 PY p ijkl (3.33) Equations (2.36S) and (2.361) in Technical Manual: M(1,6) = -m2p + terms involving 0ijkl (3.34)

89 Equations (2.37S) and (2.371) in Technical Manual: M(4,6) = - Ix - Ixz m2xpzp + terms involving 0ijkl ijkl (3.35) Equations (2.38S) and (2.381) in Technical Manual: M(5,6) 2- = -Iyz YZ - m2ypzp + terms involving Oijkl p (3.36) Equation (2.39S) in Technical Manual: M(6,6) = I + I + MF (a2+T2/4 ) + M b2 2 I+ m (x 2+yp2) zz 2 p p + terms involving 0. j Eauation (2.39) in Technical Manual: Eauation (2.391) in Technical Manual: (3.37) M(6,6) = I + M (a2+I2/4) + M (b2+I2/4) z. uF +F uR +R + I2Z + m2 (xD2 + y 2) zz 2 o o (3.38)

90 3.2 Running the IDSFCAS Simulation The modified version of IDSFC described by the equations above is referred to as IDSFCAS (IDSFC-Asymmetric). The following changes in simulation operating instructions are necessary as a result of the modifications: 1) Payload data must be read in on I/O unit 9 in the format and order described below. Format for payload data deck: (card number, data) in format (I4,1X,F20.10) Order of payload data deck: Card Number Variable to be Read 1 m2(lb-sec2/in) 2 xp(in) 3 yp(in) ~~~~4 zp(in) 2 2 5 I (lb-sec2 in) 6 I2 (lb-sec2-in) 7 I2z(lb-sec2-in) 8 T2ylb-sec2-in) 9 2z(lb-sec2-in) 10 I2 (lb-sec2-in) 10 Ixz 11 (TT(dimensionless) 2) Initial suspension deflections (which are required by the simulation) corresponding to the static equilibrium position of the vehicle plus payload must be determined by

91 one of two procedures: a calculation based on the given force/suspension-deflection data, or a brief simulation run. Suspension forces in IDSFCAS are given by interpolation from tabular data as a function of suspension deflection. For payloads which are sufficiently small that the suspension deflections remain within the initial linear segment of the force/deflection table the equations below can be used to compute the initial deflections. After the deflections have been computed they should be checked against the force/deflection table to insure that the deflections remained within the initial segment. Equations for the loaded equilibrium position were derived by minimizing a potential energy expression in terms of suspension and tire deflections. Tire compliances are in general large enough that they must be included in the formulation. The potential energy of the loaded vehicle with respect to the unloaded equilibrium position is given by equation (3.39), where 6si is the suspension deflection for wheel i, measured at the wheel for an independent suspension and at the spring mount for a solid axle, 6ti is the deflection of tire i, ki is the stiffness of the suspension for wheel i, effective at the wheel for independent suspension and at the spring mount for solid axle suspension, kti is the tire stiffness, kF and kR are auxiliary front and rear roll stiffness, TF and TR are front and rear track widths, and TsF and TSR are the front

92 and rear spring mount spacings (which are equal to TF and TR, respectively, for independent suspensions). Deflections are taken to be positive in elongation. PE = (k sii +ktii )/2 + kF(6sl 2 /TsF i=l ' si s tii /2 F 2 2 + kR(6s36s4 /TsR + m2gAhp (3.39) where Ahp ( s1+6s2+6t+6t2)(b+x )/[2(a+b)] + (6s3+6s4+ 6t3+6t4) (a-p )/[2(a+b)] + [( sl-6s2)/TsF+( 6tl6t2)/TF]Yp (3.40) The frame and body are assumed to be rigid, so that the body roll is the same at the front and rear. This constraint is expressed by equation (3.41). (6sl-6s2)/TsF + (6t1-6t2)/TF (6 - 64/T R + (6t36t3)/TR 0 (3.41) A constrained potential energy expression PE is formed by multiplying the left-hand side of (3.41) by a Lagrange multiplier X and adding it to (3.39). The

93 equilibrium equations are then given by PE / /a = 0 i = 1,9 (3.42) where Fi = (6 6 6 6 6 6 6 6 X) 1i = (6 sl ' s2' s3' s49 tl' t2' t3' t4' After some algebra the following expressions obtained for the suspension deflections 6 i. 2i are 6s [K] 6s3 = -m2g 2 A+CF/TsF+2AkF/ (ks2Ts ) B+F/TsR+2BkR/(ks4TsR 2 ) (3.43) 6s2 s4 wher =(-2m2gA-k 1 6s 1)/ks2 (-2m2gB-ks36s )/K3 4 e ks + (kF+E)(l+ksl/k2)/TsF2 -E(1+ks3/ks4)/(TsFTsR) (3.45) (3.44) K1 1 K12 K21 = -E(l+kl /k2 )/(T sFT - ~~~s2 sF sR)

94 K22 = 3 + (kR+E)(+k3/ks4)/TsR A = (b+x )/(2a+2b) p B = (a-x )/(2a+2b) P C -Yp/TF p sF D = -Yp/TF E 1/[/(ktlTF )+1/(kt2T ) 2 2 +1 /(kt3TR )+l/(kt4TR ) F = E ( 1/kt 1-l/kt2)A/TF( 1/k3-l/kt4)B/TR +(1/kt +1/kt2)D/TF-2A/(ks2TsF)+2B/(ks4TsR)] For solid rear axle suspensions the initial conditons which must be specified are 6R and CR' These are computed by the following equations. 6R = (6 +6s4)/2 R- s3 s (3. 46) cR - ( s3 6 )/Ts R s3- s4 sR (3.47)

95 Initial conditions can also be determined by performing a brief simulation run with either zero steering angle input or closed-loop driver control. The zero steering angle procedure is appropriate for symmetric vehicle configurations with payloads which are sufficiently large that the suspension operates outside of the initial linear range (or simply as an alternative to Equations (3.43)-(3.47)). A three-to-five second run at moderate speed (e.g., 400 in/sec) should be sufficient to determine the loaded equilibrium position. Extremely low speeds (below 100 in/sec) are not recommended because the wheel spin equations are very sensitive at such speeds. Some asymmetric configurations, such as those involving geometrically asymmetric center of gravity location in a vehicle with significant roll steer, will not travel in a straight line for a zero steer angle. A closed-loop driver model, such as the "Straight Line Crossover Model" in the IDSFC Driver Module, may be used to control the steering angle to obtain straight line motion. (Driver time delay may be set to zero to improve convergence to a straight path.) This method will provide the steering wheel path angle necessary for straight line motion and the resultant vehicle side-slip angle as well as suspension deflections. It should be noted that initial conditions computed from equations (3.43) - (3.47) will in general not agree

96 exactly with those obtained from a simulation run. This discrepancy is largely due to coulomb friction in the suspension (which is not included in the static analysis), with small contributions coming from approximations to the trigonometric functions and other minor sources.

97 REFERENCE FOR CHAPTER 3 3.1 "Improvement of Mathematical Models for Simulation of Vehicle Handling, Vol. 7: Technical Manual", W. R. Garrott and R. A. Scott. Final Report DOT-HS-7 -01715. March 1980

98 Chapter 4. Modifications in Driver Module 4.1. Technical Changes Required to Implement Extended Cross-Over Model Based on a cross-over model of human driving, Garrott et al [4.1], implemented, in the Driver Module, the following control law for steering regulation while maintaining a constant forward speed. 6Sw(t) = KpIerr + K Teq'kerr + KKyYerr + K KTeqYerr However, this control law is inadequate for curved paths. If the vehicle is perfectly positioned, i.e.pe = *rr err Yerr Yerr ~, then = 0, which is not correct. err err err Sw To remedy this, following Garrott et al [4.2], it is assumed that the driver is able to perceive path curvature and vehicle speed u and is able to determine and implement the steering angle necessary to cause a steady turn of that curvature. This is achieved by the addition of the term, (NG) L1+(KD)u2] L K(t-T) A further extension has also been performed, following a proposal of Allen [4.3]. This extension involves the *A list of symbols is given at the end of the chapter.

99 addition of an "integral trim" term. Thus, the latest version of the control law in the Driver Module, is 6Sw(t) = K 1e (t-T) + KT eq err(t T) + K KyYerr(t-T) + K Ky Teq Yerr (t-) t-T( + K / K err() + KTeq 'er.r() + KWKyYerr( ) + K1KyTeq Yerr () + (GR) [1 + (KD)u2] L K(t-T) (4.2) The following change was made in the desired heading Yle computation in the Driver module. des = linear interpolation between ~i and.j, where.i and j are the average path angle at the two end points of the closest road segment (j = i+ 1), defined by 'i^ = (' + )/2 (4.3). = ( B + C)/2 (4.4)

100 A = tan YRi -YRPi-1)/(XRPi -XRPi1) (4.5) B = tan-1 (Y - pY )/(X -X (4.6) B RPj RPi RPj -RPi = tan-1 (4.7) = tan (YRPj+1 YRPj )/RPi+1 -YR If i = 1A (4.8) f j = RDPT C - B (4.9) 4.2 Prc gramming Changes Required to Implement Extended Cross-over Model The extended cross-over model and improved path curvature calculation described in section 4.1 were programmed into the Driver Module. Alterations were made in subroutines DRINPT, DRIOUT, DRINIT, DCROV, and DCRERR. Listings of the modified Fortran code for these subroutines are given on the followina pages. For subroutines DRINPT, DRIOUT, and DRI!HIT only the portions c fo the ccde affected by the chlanges are listed. 'et lines arc "enc ed hvy " > ".. For subr-outines DRCROV and DCRERR the entire subroutine is listed.

101 Changes in subroutine DRINPT 4 1 '42 50 9 5C) 53.'1 S'l > 17 '9 14 172 171 174 17q 167 1 71 175 1 71 i, ************************* **********$** *C C SUB3OUTINE DPINPT THIS SnU3PUTINE READS THE DRIVER DATADECK INTO THE SIMULATION. S U30RUTINE DRINPT IMPLICIT RrAT.(^ - H,0 - Z) ZTiEGEr DR9RODE LOGICAL AT~Nl, PASTOB, VIEWOB CO~.V0ON /DPDX.\P/ RDPT(3,300), IPT1, NRDPT CO aION /DROLCrw/ LCOM(6,300), OLTIM(2,10), PPPARM(14), I OLCO- (300), NINT, N'AX, NOLP CO 8ON /DROPTmD/ DEMODE COM.ON /DRPAv1/ DB?R1(9), PE(12), WTAC(1O) WTST(10), 'PpR E D, NOPERR COM^"ON /DRPA72/ DGAIN(7), TAU, VDES, DELINT, ACCINT, TLCR Cn..fON /'DnE?.R3/ ACCSW, ACPRAC(2) rn"lCCN /DOBST/ OPCLP, OBBEK, TPPTO, IOD.MOn, IOP, IVP, 1 'ASTOB, VIEWO3 I.rTNSTION LIT1 (), LIT2(4), LIT3 (8), NAME(4) nAT A I3LNK /' nOQC' / CRn CnlFo 0ui)ci n9RIVFo DAPAM;TEF SFCfTION 7'? LTT? (4 = IRLNK nD 210 K = l, 5 o AO fr)3 4?.1, OGA4IM (K ~ 7'1.tLLt r)QOQOD(I J, I, NAMF) oFAO (,420 ) J, DGAII ( 6) CAr L flnnor)( I,1,1,NAMl4 ) QFAr) f3,420) J, ' ^1 7 C (LL DOnP0( I,J, I, NAF! OQO f(3,4?20 J, TAtl CAl.L DRORr)(, J, 1, NAMF) PFr) (3, 4?7O J, VOES C LtL ORQ rOt I, J, 1, I IAM } rC1 I O FNDS ( I NA'4 F r,r TO 60 Po1DVT:W-DREnTCT0C MOgr)L DrPIVgF PARAMCTTCP SFrTIrj r

384 385 386 387 388 389 390 391 3q2 393 394 3qq 395 396 3Q7 399 399 4 00 '01 402 40a 4()C 4 OC, 406 ( 7 U06 409 010 411 401 4A Q 46? 40 401 40c 407 600 102 Changes in subroutine DRIOUT C SUBROUTINE DPIOtUJT c?PRPOSE TO PRINT OUT DRIVER MODULE PARAMETERS SUBROUTINE DPIOUrT (PRNTCN) I'PLICIT REAL(A - H,O - Z) T TEGER DR.OOFE, PRNTCN LOGICAL PAlTOB, VIEWOB CO.MON /DOBST/ OPCLR, OBBPK, TPPTO, I09MOD, IOP, IVP, 1 PASTOS, VIEWOq CO3MMON /DROMAP/ RDPT(3,300), IPT1, NRDPT COMMON /DROLCM/ OLCO (6,300), OLTIM (2, 10), PPPARM(14), 1 IOLCCM (300), NINT, NMAX, NOLP COM'.ON /DROPMD/ DR.MODE COMMON /DRPAPl/ DRPB1(9), PE(12), WTAC(10), WTST(10), 1 NPPED, NOPERR COMMON /DRPAP2/ DGAIN(7), TAIl, VDES, DELINT, ACCINT, TLCR CO.MON /DRnAF3/ ACCSW, ACFRAC(2) IF (PFNTCN.LT. 1) RETURN TF (DR.MODE.GT. 0) GO TO 40?PfINT OUT OPEN-LOOP COMMAND PARAMETERS WRITE (1,270) WRITE (1,2901 L = 0 10 I = 1 J = qr)'AnnE - 10 * I IF (J.t=.?) GO T3 213 PRI T gUT DESCPr I INT-=UNT TTOI( CQOSScVECZ UDOEL o4RAMErE S IF ( J.FO. '3) WRITC (i,c;651) IC (J. F1. 4) WOTTcI (lS71) Wm TTF (1,901l r r) rIG (, a^&T (2} WPITC (l,5)90 DAT'IJ ), Or'A)IAT 4 W IT T: f Il 6,t ) DGAIf^l ), TtAI WoT9 ( 1, o05 ) D)GA\I (^A) WRTTC (,60 61 nOGAINf) TI r J,.O. 3) rn TO?23 WP IT ( 1,6 ) V f) Q_ TIJPN P QTMT 'JT OQPFVIFW PRQ^3I'CTR M1O7) PA:AMc TpS 1 IF (j F:r). ) W IT- (1F, l.? ) fT (J,F.?l) VWIT=. (1,630 W IT: ( 4 ) ( nI, O P ( ( K <),K= l, 5 WOTT= (!, trS ) ( n PO Il (K),K=,0 ) }, hI ORQf! WqITC ( 1,660 ACF( 6f P\() 1, Arpa&r(2, ArrSW 597 0o 507 504 507 FCQR OQ8 ^,nn; 7 1 0'0, t) _l ) 1^ n FOO AT ('0 ', 'DRT V E D AQ AFT rc 5 FIn THF STRAI -IT LI N: 1,'CROSSOVF 'An')cL' ) FO:V AT ('f ' ' DOTVFP GATN nN Y =, G12. 5,! ', DPTVFP ' 'GAIN ON V =', G12.5)! ^lTA (9 o, QTV'DIVE ATN nv PST =, G12.5, I ', DPIV F ', 'GAI'G rN PSIOOT ='. G12.5) cOnPMAT (9, DRIVPo GAIN ON I) =e, r?12.5 1 '. DPIVCQ ', TIM"C LAG TA'I -=', 012.5~; FnfMAT ( ')RIVFP GAIN rMN CTFPCDIN, FPno INtFN^PL t- =',r,1?. 5 Fi 1.- 4AT 1( O)RIVCP ^r8t N ^~ V=l. rrTTY FTPv E INTFPR L =' I,.l I Ft ouAr (I, ')OFS IQ O FLDIT =*,.l?. 5) ) FORMAT I ('0, 9DQTV PAOAMFTFOS FCO THF OPEVIEW-OQFD,, tI TCTOR DOIVC= Mr)OFL IJSINr, TC= ^,=OuFTCIt POC), I TCTCPI

103 Changes in subroutine DRINIT 661- CP'u M! /Or INTl/T TLAST 6-^7 COM^-iP^N /OrINT7/ NDQINT ^6 A3 rA'Af.unN /OrPINT?/ "CC5), n5LSWO, Gr,' TSTAOT 664 COU"nN /nQOLCu/ nLCnlM (6,0 n), 1-TT%(.1 ),1, PPPA~oA(14), 665 1 InLCOM( 300), NINT, Nm4X, NOLO 666 Cn,,.#nN tnonPmr)/ O)QuRn)D 667 CnmrNuu /9qulX/ TI, 1,tLt rO,66.C C3'J'4n /)oo L/ )RPR1l fr 0, PE ( 12), yTar (, w T FTITi. ), 6ACN tI NPRFPC, NO0ro: 670 COMMOM /r)DARP / r,4TN1 f7),TAU,VOFS.rL ITNT, ACCT!NT, r CR A71 CrfM^rn /fnP4AR3/ ^trSW, ACFZAC( 2) 677 CnrM"SCN /DPST/ CrCLQ, nRqPK, TPPTOI, iTMOr, T, T, IP, 17~? 1 PASTOB, V!FW(R' 674 NTNT = 1 57'5 6'6 r lHCCK FQD oDIRcI r O)D-_nLo CONTRcOL 677 678 Ic ( DPnr)E. LEo 10) Q'TIJRN A70!T' = 1 659 NPQ T N = 6R1 TF ( 'o 0On.EO. 14 rC TO?7 5";R TF (DP'4OnE.EO. 24) G1 T'?n 69q' 64 CHE'K -0Ra ERRORS TN ROA'-P-'ITNT nA&T 6Q(z 6 6 Tf (~NPTPT. LT. 2) GO TO 93 4A7 r 6PR rO 1) I = 2, NROPT 68P. TI STS' = (RDPT(1,T - - OPT(,T)) ** 2 + (P OPT f2, 6o0 1 - 1 ) - )PT(2, T ) -- 7 61:l FTP (0IST2.LT. 1.E-O19 GO TO 90 607 10 r3 'TI UE 603 r 604 O CONTTNUJ 605 r CHcK FOR NN —ZqERO DoQrcCTIn) clRpP 60A. NOD:9RP = 1 607 ^9o n 30n T = I, 12 600 TF (PF(T).GF. l.E-4, Gn rn 40 700 (' CnNTTNUF. 701 7?9 N7nPFRP = f 703 Gr TrO 50 7n14 I NIT IALIZE PEPCOT ION-EqRR R)r 4Nnr)mTfZATI^N SCHEUE 70s; 0 CALL r)RANr)l 7n A nRT AIN Ncr'SSAPY VTrELC oQ OAM1ETRPS 7^7 7) CAIL DRV=H 70n8 TC:T OMT NE CLOSFO-LOOP OrIVFP mOnFL 700 T = DORnnE / 10 710 J = OP'400E - 10 * T "11 IF ( J.L5. 2) GO TO A0 71?2 NfTTIALIt7F OFSSCTIRIING-PJNCTION MnOFL P.ESmONSF UTl I X 713 NUfS< = n 714 nOLITNT =.0 7! 5 arC.CNT = n.0 71t TI rt = TSTAPT 717 RF ( l,1 ) = TST ART 71f QFS(,[ = O WELSW) 710R F.I(3',I = A.CCO 770 (P; (l.,2) = TfTAPT + TAJ - 1.F-04

104 17TP 1 2^n 1261 12F,' 12A4 1 2f4!2A5 A. _67 1 27n 1 271 1271 1 271 _.2" 1 776 1277 1 27 l37q 127Q I2Q1 1?q3 120 4 1297 12R7 1 2R9 12o0 1291 12q? 1 203 1295! 205!20o 1 3n?! 3q3 r C. SrRo JUTTNF ORCRnV - nCQln IS THE CONTROL SUBPOITINE FOR THE OFSCRIBING FJNCTION 00LE.L r icjROQntjT o E CRC.RCV&CC v(c,.t SW, T, YV TLot IrlT OEALtA - 4,0 - 7) I T; FRQ: nPMoFf QC: L V. rn'v,. nm1 /'3noMO / q'5mqEn ren rMN / n opR/ RO.fA T! (7, T Al), Vr)FS,nCL I r, AC,r T NV, TL P u"-Irr, N /")CQV=l-/ a, '_,,, K rY F^.'S SION: P (5 ), Y(6 r)NFtLW = o.o T1r (nQfnO.E.EO. 13) Gn Tn 10 IF C(OQprE.FO.??) G3 'r i ' TQ: IG- T LTN': OEECR I INcG FPJrTTrC cQRnR C rMDJT T T IYC E c (1) = -V (2) tQQ( ) = -Y(4) * SIhN(Y(1) ) - Y() * Cns( (y ) ) cqO( 'l = -Y(3) RP (4) = -Y (6) cF sq ) = V)ES - Y ('+ G' Tn?0 GCNtRAL PATH DESCRIBING FUNCTIrN Foono CnuOlJTAT'I.N 1' CALL DCR1 cPQEPR, T, Y, RDCRV) Cu"P!ITE= ST;ADY-TI)J0 ING STcFP ANGLE PO9S = Y(6) + FR (4) VEL2 = Yf 4**? + Y(V 6**7 FELNEW = (I.D+KlO*VLt) )*(+ +);ronS/SORT (VFtL?)/G: AOr) IN FRCRR COQ=?CCTTONC l nOELC Q = 0.0 nn ) I = I, 4 'O3 I'L'CO = r),AIN(T ) cQfr T) + OFLCnu DELNtW = nFLNFW + rOFL Cr + OFL TNT'OG, I1( Ah C CFMUTC STECPTIN FRRnQ INTEGR4L )ELIN2T = nr LINT + 3)LCCR* (T-TLC ) CO'"JrT E ACC FL EP Ar IO C3 1 A3J) A') EIS = O ATlN(5)*Co0 (51 4Arr= j = ArCnrES + )AT '( 7)1 *&CC INT CnOnJTIT ACCELER ATION INlT FGRAL ACCTNT = ACCINT + ACCnE:S* T-TLCP!JPn)AT TLCQ TLCQ = T CALL nCRP4F(A CC, AC'N=W, 3ELN= W, OELSW, r, TA'J) pF 'tJ N FN?)

105 1r304 r* *I*** ****************************************************4c * *- * 1305 1306 SUJRQUJT TN CCPFRP 1H7 nCrp R!P CALCULATES THC FEcFOBACK cRQ.RS FOR THF PESCP IR1I4G-FUNCTION 1308 MC ntL 139g 110 SUJRQnOUTINE 'lCRE PREPQ, T, Y, RDCQV) 1311 IMPLICIT REAL(A -,O - 7) 1312 rnMmfN /DRPTFR/ UTNT, XTNT, Y INT 17, TPT 1313 COMMCN /ORDMAP/ ROPT(3,100), IPT1, NRDPT 1314 niMCNSION RQQ(5), Y( 6), YP(6, 1) 1316 )On 10 1 = 1, 6 1" VP(T,t) = VYT)!318 ' nFTP-RTN: POSITION A^n VELOCITY EPRCPS FnR C!.JRPET P3S IT ION 13 q CALL nOTERR(DTST, 'J2EQQ, YP, 1, T) 13"9) r FR(l) IS THE LATERAL P3SITI:N EQP3R 1371 EQ(1) = -OIST 13? C IPT AND T12 APF TH'. INDICES OF THC ROAD POTI!TS l,?3 _ CN P!TT-CQ PEN O0 THE NE&OCST RCAD SrG4'4^JIT 1374 TF (IPT.LT. 12 1 I = IPT 13?71 = ( IPT.GT. I? ) I = T? 13" J = I + 1 13?7 OD!ST = SORT((XI^T-.rp)OT(1,l))':? + (YIJT-PnOT(?, I)**-) 12')q nIT STCT = SORT ( Pr)pT (,J) - OnPT(,I ) I**7 + (P or(2, J 1 2c! - On oTf 2,I ))**2) 133n rC CrnO'AJTE Dn=c IR ED oATH AN,.c oY L IN FA IJrcRPOLATTI3N 1331 DSI8 = ATAN_?( (R)PTf,J-OPD'D( _,[I ), (RnT(fI,J)-Or)P T( 1,T)I.3??IF (I.NF. 1) GO TO 7n 13' uPSIA = PSI8 13.4 r.n TO ~20 1335 ~7 K = t - I.336 PITA = A;AN2f(POPT(2,I)- QOT(? f,),f(PT (1f,I)- Prtl,< ) ) 1 337 3) IF I(J.N, NPnPT) GO T' 40 *2 2 SoI'r = PKT8 1 C,1 TO 9C) 1*340n K = J + 1 1241 PSTC = ATANP?(( PnPT(7,K)-PflOT (?,J ), RrofT (1,K l-)orDT (.J ) I 1 3'? q) DS DI = (~SIA + PSOTB)*O.. 1343, O I? = (PSD B + PSTC*'^). 1344 PSIlnc = SI 1 + rnIST1*(PSI? - PSIll/DISTOT 1345 - E0(?) Ic THE LATr4AL VELOCITY copnp t _13 I?) = -Y(4z*STN(Y('A)-oDS )I=" ) - V(fS *COS(Y(3- S I'DOES 1l-7 I = '.1415q265359 134P An IF (Y(3).LF. PI) GO TO 70 1349 Y( ) = Y(3) - 2.O*PI 1'2O5,3 T n o0 1351 7? IF fY C).GT. (-Pit ) GO TnO 1.52 Y(3 = Yt3) + 2.0*PI 1353 GO T 70 134 r cQP(3) IS THE HEADING ANrLP cqpnr 1355 'RPt 3) = PSIOES - f 3) 13q6 IF (crPQ(3.GT. PI) POR () = EP~(3) -?.0*0I 1357 Ir ({ER(3).LT. (-Pi) crP{(3 I = =DP(ll + 2.3*) 13p r COMP:ITE ROMA) CUPVAT1RE ' AT POINT OF PEQPEN TIC.ILAP INTERSECTION 1 c '. RY I NTCRPOLATI NG BETWF-.N CIJPVATIIRE AT SEGM'ENT C.)POrINrS 13A. IF f (T.NF. 1) GOr TO n 11361 PrOrPVl =.q.0.i?.: r, TO 3') 136' I): K = I - 1 1364 CALL nQCfJQV (RDPT(l,K), PDPT(?, K, POPT(1,I), 136. RI rPTf?,I), RO P T 1UP((,JJ, R OnT(,DCPV1, FLG)!.366 IF (PSi nos.LT. PSP 1 ) I P CRV1 = -o- cPV 1367 1n 00 IP f J N. NO)PT) Gn TO 11) 136q ROrRv? = 0.0 136 r.0 TO 12') 1370 110 K = J J + 1 1371 CALL nr)r11Rv (RDPT PT( f,I), "O u )PT lJ), 1377 t 9r)PT (2,J, QnPT (.K,Pr)P 2IK, R DCV2, IFL.')I. 1372 TP (PSI 2.LT. PSTIr)ES IPCPV' = -r' CrPV 1374 12 ) PrCvRV R- CRV * DISTI * (Rr)CPv? - QnOCQv l / O)TSTOT t37c VF~. = S;QPTCY{(4,*2 + Yf **l 1376 r E PR f4l T THE YAW-RATE FRPno 1377 FRP(4) - VEL * QrCRV - Y(6) 1378 ERR( )' IS THF FCPWAP) f LOCIT Y eRROR 1379 FP(5I = JITNT - VEL 13 8 RQ F TURN 13A1 ENf) l3

106 4.3 Programming Changes Required to Interface Driver Module with The Three-Degree-of-Freedom Model Four subroutines in the Driver Module are specific for the vehicle model being used, namely, DRFAC 1, DRFAC 2, (which includes an entry for DRFAC 3), DRFAC 4, and DRVEH. Versions of these subroutines have been written to interface the Driver Module with the three-degree-cf-freedom vehicle model describe in Chapter 2. The Fortran code for these subroutines is listed on the following pages.

107 107nr 1 071 r 1 97? r StPQ OpJ INE ORF AC4 in7? > T-4IS SURR3UTINE COMPU!TES )ELTA &ND afC n 74. FopO VFHIC.LE CONTRnL VARIABLFS 107T C7tJRO n l[IJTI E DRFAC 4(f., ELT T, nFt SW, rnUTr,.0'IT7, 1'-77 t r)CUT3, DrjT4, InLCO M) 107R8!T PoCT PALtA - H,O - Z) 1 9 r3Mr1n 4 /nRV H4/ R < KIN, nRV IN, R, R QKTA8(?,1 ) lf).r " CONVVRT STFERING WHccL ANGLr TO FRONT WHFEL ANGLE 1Q1 onFLTT = *ELtW * sR 1R0, TF (JOLCOM.EO. 1 ) PPT JRN )08',ONVERT DRI IE TORQU ANWD RPRAK' LINF PRcSSUoE r) ACCE-F4rI3AT 1~)Q4 3 10 I =?, 10?^s Tic fnrOIT.LF. RPT AR(, T) O) G Tr?r) 10IRD13 I T IN11 lnlc )I oAf: = no < T9q(f 1, ) - ')mnT1) / fQRKTPfl,,T) - naKTTAR 1I~O i111! - i}) 1 ) 1 T QO = 9 KTA ( 7, T) - FRA4 * ( RKT 8(7,I) - B KTAR 2, - [ ~'. ' t i *q & 1Afr = TOq / RoKCON + )n.j T? / oVrrVtn!no; MNn! 06 t Q6.1 o 7 f dr ** S ***^ t *c *csae ^< cbB cijc^4c *c * tlc 40c * <tA t - c c *it iic*** t ** ******e * ^*^ **^*** 1 09' SVUROUTINC DIRFACI 1on 0 r nRFAr.l CONVCRTS VEHICLE <IMcuATIC VA TIARLES TO )RIVFQ FORMAT 11 -2 cl)R'?011TINE DPFACl(oY, nYVFH, v, VVFw, r)tLl, rEL?, JOo, 112 ' J OL CO ) 1. t? TMl TCIT RCAL(A - H, l - 7) 11OE nrm lNc ITn\ nV( 1 ) YV-( ), Yf 1), YVF-(1) 1106 Y( ) = YVF () 1107 Y(7) = YVEH(2) 1lOR Y( ) = YVVE(f3) 113Q Y(&) = YVFH(4).111 Y(' YI = YVFH ( 5) 1111 Y(6) = YVFH(^) 1112 DY(1) = DYVEH( ) 1113 Of 7?) = OYVFH(2) tt1114 (. ) = nY/FH( 3) It, 1.5 r) Y() = DYV-Hf4) 11 6 Liy5 = nYVEH( 5 ) 1117 nY~ A = nYVcH(6 ) ll QFTIJPN 1 9 E ND 117t Ll"****^^^^^**^*^**~*f*^;<r<34A**<Ac~*4c( ^irr^^^ c

108 256' f $******************************<*************************t~kA tt 2564 C 2,565 ' SUlBRQ UTINE ORFACt '5f r nOFAC7 CONVEPTS THE r)ROVFR 'UTPUT TO VEHICLE F=RMAT 7567 25_6R SUJRQOJTITNE ORFAC7tACC, DELTA, OELSW, DOUTI, r)OUT2. 756Q 1 DOUT3, OOUT4, JOLCOMI 2570 IMPLICIT RFAL(A - HO - Z) 7571 Cn'MO'J /OR)VEH4/ RtR<CO4, DRVC'N, GR, RRKTAB(2,Il) 2572 COMMON /FOUT/ FN(4),ALPHAT(4),TOT (4), FS( 4 ),FC(41,S (4,3EEtT(21. 7573 I TQST(2)?574 C.nMMO /V 30/ CI(7,7),aLAMT(4),EC(5),XT(4) tYT(4),DTOEF(?,TAXL(2), 757 1 AXt.,VrG.rALAMRS,VM,VT7Z,VTY7,VIZXXPLYPL, ZPL,PL'4,PLIZ 7,PLITY7,?576 7 PL T7X, ST1,CST7r.ST3,SR, XC f 7) RRK (4 )CD,PFAtRHOA, IA ER ^;77 r.IrrrM.M EO. R tcANS PqRnTCTOR nPFPFATING I r7 ' WHILE STTLL TN TOTAL OPEN-LOOP CCNTROL?570 TF (JnLCOME.EO. fR QETtIRN 25P) IF (JOLCM.EO. 1 ) CO T3 10 25R81 CfnMP _ 4A fc FOR FLXTi L TL NT F aSTEE I'N', SYST= 25? Al = TO<T (1 )*CSTl 25RI A? T = TOST(?)*CST7 q4 TEUP MPJ D 1 *( A4+A?)/2.0 58'5 TCMU? = (TOST(lt + TOST(?I)*rST$/SP 7?56 ncL W = nrLTA*R - T MIP - T TMP2 7?57 13 CCT INLIF ~?;n.' RO4AK=/r)qTVc TOROIJE COUTP'JT CCU%&ANn CrCMUTATIMN?500a cMTVY )RFAC3fACC,)nIJT 1,'QIJT?,')Of T T, rCUT4, JOLCrtn?5Q'P IF f JntCnL.NF. 5) O',JT = ).) _;Q1 Tc (JLC.n N.NF. 61 ')OIJT4 = 1.,9 25q Ic ( JOLCOM.FQO.?) QRTJPR?:oTq!: (JOtnTM.cO. 3 ) GO TO 59 7sc4 TF fJ nLrn'.EO. 4) rn Tn 8o?505 IF (Arc.GT. 9.0 SO Tn 4,. 2506f < S;CTF Y nOUTI (eRAKFI TN.= PQrccUoF) PY I TE9PnlLIrTIO^ 2507 FRQnM BqAKF TABLE?508 TOP = RPKCON * ACC 2500?^AO n 'r 7 I =, 10. 760! T f TQR, LE. RP KTABf?,T) ) GO O0 30 260' 2) r3 n TN JF 26"3 r?6n 7604 1 = 11 260n B0 FRAC = (BRKTAB( 2, [) - T9 T / (R9 KTA (?.I) - RPKTAB(2, 76f65 IT - i ) >607 OnUTl' = qF3KTAR(,I) - FRPC * (BQKTArP(l,I - RkKTAR(1 27hA I! - 1)) ^6S09) OOUT2 =. 0 2 ^ 0 RE TIJP U 2611 CruMrIE nr'JT2 FOR CLOSFrf-L).' 1WnFS 2612 &" 3O 'JTl = 1. 0 2^13 nOUT? = OVCnN -* ACC 761 4 Q PFTIlRN? A^1 r C'1u 'JT r OUT Wr4 ) OnUT! I S OPEJN-LOOP rCMTROLLEO 26!6A ) 00n r T = 2, 13 21t7 Tc (OjUTI.LE. BRKTAB(lIT I r,C TO 70?61t I'? r T TNJC 26?1 7' FPAC = (RPKTAB( 1 I) - rnltTt) / (RRKTAR( 1, I - 9RKTAB( 2627 i? I. -! 1) 26? TOR = RQKTB( 7,I) - FRA" * ( P.KTARf 2,I) - PP KTABr, - 2674 t I)) 7,7 5 ArCCO = TOR / RKCrON 2626 ArCn0) = ArC- Arrpo 7677 rnUr 2 = r)PVCON * ACCvOr) 7A78 tF (fOUT?. LT. 0. 0) OOUT7 = O.n 76?70 R CT; QN 2 % ~ (r)COMPIJTF nOUTe WHtN nnUT2 IS CPEFN-LCCP CONTRnLLED _1] 0 A rrCCTR = o01IT2 / ORVCON 763 7 ACCuOO = ACC - ACCPQ 2?6 ITF ( AC MOD.LT. 0.01 rn TO O 2634 ml'r i r.O 762^5 PTTIRPN 2?616 on TOR = RtQC.rN * ACCMOn

109 26-7 V?,q nn 100 t = 2, 10 26,Q T{:: fTOt.LF. m~KTA',12, i(1,n TO 110 ^40 t 0C) r. nOIN UE?641 247_2! = 11?64 1! r FQAr = (RqKTAR(?,T) - ORn) (8KTR(B2. T - B<TBt' 2, 7644 1 - il)?645 rnJT = rRRKTAB'(1 I - FPAC * (!<TB 1B(1, I - BRVTTB( 1, 746 1 T - 1)) 7647 R =TItRN? 64P FNn

110 7^0 L, **** **** ***** ** ^^ <c * ***** * *^********* *******$*** **** *******t 761 C SURQ OUT IE r RVFH 76F? r rnV: 3FRT&INS NcrFSS.AR VEHiCLE P&RAMFTCRS FOR THE DRIVER wnUJtLE?65' DRVFH IS SPECIFIC FnR THF TA NSS?' 1-DOF VEHICLE U9)EL 7654 3_R 'SJPOOJTrJ iqF nRVEH?65. IMPLICIT RFAL(A-H,n-Z) 7?67 TNTC.FGR P'M0E, TMqO)?ASR OF ALt KO 26 59 ItrGTCAL FIPST 2660 C3M4n l4l /r.RVEH/ CRA, CRr, "^QR, CRKn s $1. CrfMMCN /IF)INT3/ ACC3O OELS'7?, GPC, TSTART 26t2 COMMOnN /OROPMnr/ DRMODE 26A3 CnaMMMN /rORPAR3/ ACCSW, ACFRA4(?7 26564 COMc'CN /DRTrlAT/ nRTR(?4) 26A5 COMMON /O)RVFEHl/ tRV (3) _266 COM01ON /DRPVEH2/ ACCUAX, ACCU YI, OEL4AX?6f7 COMmCN /DRVEH3/ SIDACC 766R CIMMAqN1 /'RVEH-4/ RRKCN, ODRVCON, rGP, fRR KTfr(2,11 76f9 CO'Mn^J1 /DRVEH5/ A9M, Al, A4, C, DPC,, Tn=( 2), XA(4,?570 XW (4) 76? 71 nM4^oN /VPP / nSWMAXTOQDMAX,PFLMAX,Ko),DSWo,TO)n,Pot1).?677 COUmNN /Vr0/ CI (7,7),ALAMT(4),EC(5),XTf4 iYT(4), DT)F(?,,TAXL ( ), 767 1I AXLR,VC,G,ALAMRS,VM.VI77,VTYZV tZ V XPL,YPL,7PL,PL PtL? 7, PLIYZ, 7A74? Pt L IX,CST TCST2,CST3,SQ,XC(?),RRK(4),C,PFA,RHOA,IACKF R?675 COMMON /T3nATA/ FR0(4,1),2), AO(4), A(4, A2 4), 2 1(41 4, B3(41, 2?'T6 1 P4( 4 tT( 4, 0TI) o)1, P(41, P2(4), S )(4), S1 I), S? (I,?.h7 P 0(4), Pl(41, Kl4l, '(4?, Rr(4), S N(4), FRR(4) 7678 DIMCNSION FQ(4),FCMN( 4), F SMt (4)?7AAo XT 96q.) a - T( )1 'f'1q 9 = -YT(3) tACq AI = V1/77 + tLI27 + PLM*t(XPt*,2 + VPL*'i)?6R3 AM = VM+PtLM?6Q4 ARM = AM/IA +R ) '?6qc; C = (VCr*, + PL*(/C - ZPL 1/AM 7q6 ")=-tmAX = nSWM'AX/SR 26a7 ODLSWO = OSWO f6qR nRlC = G,6qqq &oo0 - ASSIGN TIRE PROPERTIES Ff=R 3DOF MODFL PQEDICTOP,6r!L r)O I=t 1,4?6a7 rRTR (I) = A0( I) 7 93 )R TR ( +4) = A ( 1) 7604 r)PTQ (TJ.R) = A?2! I '0<C O DO PTr(T+t12) Rl (I)?76a~ nrio l -PTO(.1 3 R3( ) 7507 QI )QTO(T4 20) = R4(I) P 'QR r oMDjTF= MAX tIJmI CIRCU4MFFR NTIA AND) SI;TF FnRCFS 76Qq IFCMNT = 0,0 7704 FSMXT = 0.0?7nr FO (It) = A*ARIM 27"? FQ(7) = B*.B4 77nf O0n 70! = t, 4 74 J = { ( + 1L /??7n =F'"MN( I) = -_ (.D) * ( (! + FRfJ)*( PI ( I + F FPj)* 7-70 t1 D (ifl) I SN( ) 7n17 F MNT = FCMNT 4 FC,4N(T ) 7 7q FPS Xf II = FR f J * fR f T I 4 F fj l*( tl( ) I I + (J) * 4( _7^c T 1' * FN(I)?71 n 73 F!AXTr = FS'4XT + FSMYXfT?711

11 71T rC CMPUTC MAXIMUM ALLOWABLF RAKTING &CC ER4ATOID A D._ATERAL 4::=Lt TIrN 7lt 3. ACCO 1I = 4CFQAC(7t * FT..NT / AM 2714 STOACC = FSMXT / AM 2715 ^OMPT'JTC '^AXTMUM ALLOWARLE ACCELERATION DUE TO DPIVE TOR'3JF 2716 DRVrC O = AM/AXLR/(ALAMT( l)/T(]) + ALA4T(2)/QT(?) 7? 7 1 + ALA.MT 3)/PT ( I + AL A4T( 4)/ T( 4 )1 2719 qa"cAX = ACFRACf 1)TOMr)AX*)RVC0' 271. DORVI(1) = A 277n nR VL (?) = = 7721 rRV1}3) = KD 7?723 I TSTPArRT = 3.0 277? -R = 1.O/SR 7724 r? = GR 27?77D 10 n 1=1,4 2776 XA(I) = XTf 1) 277.7 XW )( = YT ) '7')R?77 7 C%4cD)(TrJ NTR TES FOR )3IVEQ-,MO)EL R4AKE TABLE 273n B!PKCrrN = AM*(RT(l) + RTf2)+RTf(31+RT(4))*,O25 7731 RQKOTA( 1,1) = 0.) 773 RRKTA3( 2,.) = 0.0 2733 BRKTAR(J,2) = PFL'4AX + 1.3 2774 SRKTABf 2, 2 = RRKTaRF( 1,?2 *(9RK(1 )+BPK (7)-RKf3)+RRKY(4+ *0.25 27156 ACrr. = TOOO/DRVCON - PFLO/(PFLMAX+1.0 )*BPKTAR(2?)/BRKC3N 7717 TO(f 1) = r)TOf 1) 27.1 TnF(: ) = rDTOE(2) 277o0 -^ = A 2 74^ r, F = B 7 741 (PRGo = GR?74? CPKR = Kr?74 PET F!TJQ '744 FND

112 List Of Symbols For Chapter 4 Symbol GR K' K. KD L NRDPT T eq Definition Steering gear box ratio Driver trim integrator gain Factor converting position error to heading angle error Driver gain on heading angle error Understeer factor for steady turning Vehicle wheelbase Number of road points Equivalent yaw rate time constant Velocity component along the x-axis Distance from vehicle position to desired path Coordinates of the road point i Steering wheel angle Intermediate variables defined by eqs. (4.5), (4.6), (4.7) u Y err XRPi YRPi 6Sw.A#' B' C

113 Symbol Definition ides Tangent angle to the desired path at the closest point on the road path 4%rr yDifference between current and desired heading angle P~i, -4j ~Average path angles K Curvature of the desired path T Driver time delay

114 REFERENCES FOR CHAPTER 4 4.1 "Improvement of Mathematical Models for Simulation of Vehicle Handling, Vol. 7. Technical Manual for the General Simulation", W. R. Garrott and R. A. Scott. Final Report DOT-HS-7-01715, March 1980. 4.2 "Closed-Loop Automobile Maneuvers Using Describing Function Methods", W. R. Garrott, D. L. Wilson and R. A. Scott. SAE Paper, No. 820305, Feb. 1982. 4.3 "Stability and Performance Analysis of Automobile Driver Steering Control," R. W. Allen, SAE Paper No. 820303, Feb. 1982.