A Multivariate Analysis of Crash and Naturalistic Driving Data in Relation to Highway Factors (2013)

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63 A p p e n d i x d As shown in Chapter 3, the critical case occurs when the turn- ing radius R satisfies the equation φ =sin 2 (D.1) d R which is equivalent to the yaw rate condition = φ2 sin (D.2)r U d where U is vehicle speed. Thus Equation D.2 defines the maximum yaw rate of the vehicle to avoid conflict with right boundary point P. The idea is to use the vehicle itself, with known speed and yaw rate, to provide a reference for which to estimate the relative position of the lane boundary over time. From the variations in lateral lane position over time, the lane geometry is to be estimated, and variables such as f and d derived. We need to estimate the azimuth offset f0 for the direction of the velocity vector at the front wheel relative to the vehicle longitudinal axis (at low speed this is the steer angle, but in general it depends on the front axle cornering stiffness). In the case where steer angle and cornering stiffness is not avail- able, a simple estimate can be made on the basis of general vehicle dynamics properties φ = − tan (D.3)0 rL U rc where r is the yaw rate, L is the wheelbase, 2c is the front track, and U is the instantaneous vehicle speed—this equation is based on the assumption of near-zero slip angle at the rear axle but is expected to be reasonably accurate. Simple adjust- ments are to be made to this equation when considering left boundary points. Figure D.1 shows the modified geometry when boundary point B is offset from the vehicle path. For simplicity assume a fixed preview time T to the boundary point, and an approxi- mately constant curvature for the path of the reference point from Q to P. In the figure, f is the azimuth angle to the bound- ary point B, while q is now the critical azimuth angle corre- sponding to the motion from Q to P. (Again, both angles are defined relative to the velocity vector, not the vehicle longitudinal axis.) During the vehicle motion from Q to P, the heading angle and direction of velocity vector V change by 2 q, so numerically integrating the yaw rate over the time interval T we have ∑ ( )θ = δr t t (D.4)i i 1 2 The mean radius of turn, R, during the time interval can also be obtained from the yaw rate: θ = 2 (D.5) T U R where U __ is the mean vehicle speed during the interval, and both sides of this equation are estimates of the mean yaw rate during time interval T. Then, to determine f we consider triangle BPQ in Figure D.2. Angles at P and Q are known in terms of f and q and hence the angle at B is given by ( )β = − − θ + α = + θ − α = + θ − φ180 90 90 90 2 Then from the sine rule ( )β θ = φ − θsin 2 sin sin (D.6) R s which is a nonlinear implicit equation for f in terms of other known variables. For normal highway driving we expect Determination of Yaw Rate Error from Vehicle-Based Measurements

64 And hence ( )= − θ2 sin (D.7)d R s Equations D.4, D.5, D.6a, and D.7 then determine all the relevant terms in the critical yaw rate expression = φ2 sin (D.8)r U d c where U is the instantaneous vehicle speed at Q, and now rc denotes the critical yaw rate. Multiple calculations can be performed for point pairs (P, Q) for values of T in a range of say 0.5–2 seconds, and the results referenced on the initial point Q. We are then inter- ested in the minimum value of rc(Q) and its corresponding distance d from Q. The yaw rate error (YRE) is then ( ) ( ) ( )= − (D.9)YRE t r t r QQ Q c where r(tQ) is the vehicle yaw rate at time tQ, and r c  (Q) is the minimum critical yaw rate at Q. A second YRE for left boundary points also has to be found, making similar calculations with relevant shift of reference point (to the outside of the left front tire) together with rel- evant sign changes. The above equations are obtained for computing YRE but it is worth noting that, with a minor adjustment, they can be used to give a refined estimate of local road curvature from the onboard vehicle data (again assuming lane position, speed, and yaw rate are measured), thus removing the effects of vehicle lat- eral drift. The method is to estimate the critical yaw rate for a shifted point P that has the same lateral offset as current point Q; thus replace s = s(P) in the above, by s′ = s(P) - s(Q). The critical yaw rate r ′c is then the yaw rate that maintains equal lane deviation over time interval T, and hence provides the radius of curvature Re (referenced at the right lane boundary). We obtain = ′ − ′ (D.10)R U r se c a = f - q to be sufficiently small (less than around 5°) to allow the approximation sin a ≈ a, cos a ≈ 1. In this case, ( )( ) ( ) ( ) β = θ + − α = θ − α + θ − α = θ α + θ α = θ α + θ sin sin 90 sin cos 90 cos sin 90 sin sin cos cos sin cos Substituting this into Equation D.6 then gives θ α + θ θ = αsin cos 2 sinR s Hence ( )α − = θ1 12 12 tans R R giving the approximate expression for f (with all angles in radians) ( )φ = θ + − θ2 tan (D.6a) s R s Distance d = QB  is also found from the geometry of Figure D.2: ( )= α − − θ ≈ θ − θ cos cos 90 2 sin sin d QP BP R s 2R sin θ B P 90 − θ φ − θ = α β Figure D.2. Geometry to determine f. Figure D.1. Sketch of turning geometry for offset boundary point B.