Optimal Lap Time for a Race Car: A Planar Multibody Dynamics Approach

Abstract

The optimal lap time for a race vehicle in a race track represents the minimum possible time for a vehicle to negotiate a complete round about the racetrack. In this work, a 2D multibody dynamic analysis program is developed to allow modelling and simulating the vehicle and racetrack scenario by implementing all the necessary kinematic constraints, which includes a steering constraint for a 4-wheel vehicle with a front steering axle and the necessary force elements including traction and braking and tyre-road contact. A trajectory optimization, on a given track with a prescribed geometry, which is obtained by a mix of the shortest and the least curvature paths with a speed profile optimized within limits for the longitudinal and lateral vehicle accelerations. A controller is developed to enforce that the vehicle follows the optimal path and the speed profile. This controller uses a preview distance, which allows for the vehicle to find its way even when it starts or goes off-track. The controller and the dynamic analysis program are demonstrated in a scenario in which the behavior of a race car in a realistic racetrack is analyzed.

Appendices

Appendix 1.1: Pacejka Magic Formula Parameters

The Pacekja Magic Formula coefficients appearing in Eqs. (1.5) through (1.10) need to be evaluated based on the tyre specific data. The normalized vertical load increment, df_z, calculated as

$$ df_{z} = \left( {f_{z} - f_{z0}^{{\prime }} } \right)/f_{z0}^{{\prime }} $$

(1.20)

For the longitudinal force, in pure slip conditions, evaluate the parameters:

$$ \begin{aligned} \mu_{x} & = \left( {p_{Dx1} + p_{Dx2} \,df_{z} } \right)\uplambda_{\mu x}^{*} \\ K_{x} & = f_{z} \,\left( {p_{Kx1} + p_{Kx2} \,df_{z} } \right)\exp \left( {p_{Kx3} \,df_{z} } \right)\,\uplambda_{Kx} \\ C_{x} & = p_{Cx1} \,\uplambda_{Cx} \\ D_{x} & = \mu_{x} \,f_{z} \,\zeta_{1} \\ B_{x} & = {{K_{x} } \mathord{\left/ {\vphantom {{K_{x} } {\left( {C_{x} D_{x} } \right)}}} \right. \kern-0pt} {\left( {C_{x} D_{x} } \right)}} \\ E_{x} & = \left( {p_{Ex1} + p_{Ex2} \,df_{z} + p_{Ex3} \,df_{z}^{2} } \right)\left( {1 - p_{Ex4} \,\text{sgn} s_{x} } \right)\uplambda_{Ex} \\ S_{Hx} & = \left( {p_{Hx1} + p_{Hx2} \,df_{z} } \right)\uplambda_{Hx}^{{}} \\ S_{Vx} & = f_{z} \left( {p_{Vx1} + p_{Vx2} \,df_{z} } \right)\uplambda_{Vx}^{{}} \,\uplambda_{\mu x}^{{\prime }} \,\zeta_{1} \\ \end{aligned} $$

(1.21)

For the lateral force, in pure slip conditions, evaluate the parameters:

$$ \begin{aligned} \mu_{y} & = \left( {p_{Dy1} + p_{Dy2} \,df_{z} } \right)\uplambda_{\mu y}^{*} \\ K_{y} & = p_{Ky1} \,f^{\prime}_{z0} \,\,\sin \left[ {p_{Ky4} \arctan \left( {{{f_{z} } \mathord{\left/ {\vphantom {{f_{z} } {\left( {p_{Ky2} \,f^{\prime}_{z0} } \right)}}} \right. \kern-0pt} {\left( {p_{Ky2} \,f^{\prime}_{z0} } \right)}}} \right)} \right]\uplambda_{Ky\alpha } \,\zeta_{3} \\ C_{y} & = p_{Cy1} \,\uplambda_{Cy} \\ D_{y} & = \mu_{y} \,f_{z} \,\zeta_{2} \\ B_{y} & = {{K_{y} } \mathord{\left/ {\vphantom {{K_{y} } {\left( {C_{y} D_{y} } \right)}}} \right. \kern-0pt} {\left( {C_{y} D_{y} } \right)}}\, \\ E_{y} & = \left( {p_{Ey1} + p_{Ey2} \,df_{z} } \right)\left( {1 - p_{Ey3} \,\text{sgn} \alpha_{y} } \right)\uplambda_{Ey} \\ S_{Hy} & = \left( {p_{Hy1} + p_{Hy2} \,df_{z} } \right)\uplambda_{Hy}^{{}} + \zeta_{4} - 1 \\ S_{Vy} & = f_{z} \left( {p_{Vy1} + p_{Vy2} \,df_{z} } \right)\uplambda_{Vy}^{{}} \,\uplambda_{\mu y}^{{\prime }} \,\zeta_{4} \\ \end{aligned} $$

(1.22)

For the aligning torque, in pure slip conditions, evaluate the parameters:

$$ \begin{aligned} C_{r} & = \zeta_{7} \\ D_{r} & = f_{z} \,\left( {q_{Dz6} + q_{Dz7} \,df_{z} } \right)R_{0} \,\uplambda_{r} \, + \zeta_{8} \, - 1 \\ B_{r} & = \left( {{{q_{Bz9} \,\uplambda_{Ky} } \mathord{\left/ {\vphantom {{q_{Bz9} \,\uplambda_{Ky} } {\uplambda_{\mu y} }}} \right. \kern-0pt} {\uplambda_{\mu y} }} + q_{Bz10} \,B_{y} C_{y} } \right)\zeta_{6} \\ C_{t} & = q_{Cz1} \\ D_{t} & = f_{z} \left( {q_{Dz1} + q_{Dz2} \,df_{z} } \right){{R_{0} \,\uplambda_{t} \,\zeta_{5} } \mathord{\left/ {\vphantom {{R_{0} \,\uplambda_{t} \,\zeta_{5} } {f^{\prime}_{z0} }}} \right. \kern-0pt} {f^{\prime}_{z0} }} \\ B_{t} & = \left( {q_{Bz1} + q_{Bz2} \,df_{z} + q_{Bz3} \,df_{z}^{2} } \right){{\uplambda_{Ky} } \mathord{\left/ {\vphantom {{\uplambda_{Ky} } {\uplambda_{\mu y} }}} \right. \kern-0pt} {\uplambda_{\mu y} }} \\ E_{t} & = \left( {q_{Ez1} + q_{Ez2} \,df_{z} + q_{Ez3} \,df_{z}^{2} } \right)\left[ {1 + \tfrac{2}{\pi }\,q_{Ez4} \,\arctan \left( {B_{t} C_{t} \,\alpha_{t} } \right)} \right] \\ S_{Ht} & = q_{Hz1} + q_{Hz2} \,df_{z} \\ S_{Hf} & = S_{Hy} + {{S_{Vy} } \mathord{\left/ {\vphantom {{S_{Vy} } {K_{y} }}} \right. \kern-0pt} {K_{y} }} \\ \end{aligned} $$

(1.23)

For the longitudinal force, in combined slip conditions, evaluate the parameters:

$$ \begin{aligned} B_{x\alpha } & = r_{Bx1} \,\cos \left[ {\arctan \left( {r_{Bx2} \,s} \right)} \right]\uplambda_{x\alpha } \\ C_{x\alpha } & = r_{Cx1} \\ E_{x\alpha } & = r_{Ex1} + r_{Ex2} \,df_{z} \\ S_{Hx\alpha } & = r_{Hx1} \\ G_{x\alpha 0} & = \cos \left\{ {C_{x\alpha } \arctan \left[ {B_{x\alpha } S_{Hx\alpha } - E_{x\alpha } \left( {B_{x\alpha } S_{Hx\alpha } - \arctan (B_{x\alpha } S_{Hx\alpha } )} \right)} \right]} \right\} \\ G_{x\alpha } & = {{\cos \left\{ {C_{x\alpha } \arctan \left[ {B_{x\alpha } \alpha_{s} - E_{x\alpha } \left( {B_{x\alpha } \alpha_{s} - \arctan (B_{x\alpha } \alpha_{s} )} \right)} \right]} \right\}} \mathord{\left/ {\vphantom {{\cos \left\{ {C_{x\alpha } \arctan \left[ {B_{x\alpha } \alpha_{s} - E_{x\alpha } \left( {B_{x\alpha } \alpha_{s} - \arctan (B_{x\alpha } \alpha_{s} )} \right)} \right]} \right\}} {G_{x\alpha 0} }}} \right. \kern-0pt} {G_{x\alpha 0} }} \\ \end{aligned} $$

(1.24)

For the lateral force, in combined slip conditions, evaluate the parameters:

$$ \begin{aligned} B_{ys} & = r_{By1} \,\cos \left[ {\arctan \left( {r_{By2} \left( {\alpha - r_{By3} } \right)} \right)} \right]\uplambda_{ys} \\ C_{ys} & = r_{Cy1} \\ E_{ys} & = r_{Ey1} + r_{Ey2} \,df_{z} \\ D_{Vys} & = \mu_{y} \,f_{z\,} \,\left( {r_{Vy1} + r_{Vy2} \,df_{z} } \right)\cos \left[ {\arctan \left( {r_{Vy4} \,\alpha } \right)} \right]\zeta_{2} \\ S_{Hys} & = r_{Hy1} + r_{Hy2} \,df_{z} \, \\ S_{Vys} & = D_{Vys} \,\sin \left[ {r_{Vy5} \arctan \left( {r_{Vy6} \,s} \right)} \right]\uplambda_{Vys} \\ s_{s} & = s + S_{Hys} \\ G_{ys0} & = \cos \left\{ {C_{ys} \arctan \left[ {B_{ys} S_{Hys} - E_{ys} \left( {B_{ys} S_{Hys} - \arctan (B_{ys} S_{Hys} )} \right)} \right]} \right\} \\ G_{ys} & = {{\cos \left\{ {C_{ys} \arctan \left[ {B_{ys} s_{s} - E_{ys} \left( {B_{ys} s_{s} - \arctan (B_{ys} s_{s} )} \right)} \right]} \right\}} \mathord{\left/ {\vphantom {{\cos \left\{ {C_{ys} \arctan \left[ {B_{ys} s_{s} - E_{ys} \left( {B_{ys} s_{s} - \arctan (B_{ys} s_{s} )} \right)} \right]} \right\}} {G_{ys0} }}} \right. \kern-0pt} {G_{ys0} }} \\ \end{aligned} $$

(1.25)

In the current use of the Pacejka Magic Formula it is assumed that the camber angle of the tyre is null and that the spin has small values. Consequently the turn-slip and parking parameters are ζ_i = 1, i = 1,…,8. It is also assumed that no modification of the tires properties are used and, consequently, the scaling factors for pure slip are all unitary, i.e., all λ = 1.

Appendix 1.2: Hoosier Slick Racing Tyre Coefficients

See Tables 1.4, 1.5.

Table 1.4 Basic Hoosier slick tire parameters

Table 1.5 Tyre force coefficients for Hoosier slick racing tyre