A Lane Keeping System with a Weighted Preview Measurement Kıvanc ¸ Sarac ¸o˘ glu Electronic and Communication Engineering C ¸ ankaya ¨ Universitesi, Ankara Email: kivancsaracoglu@gmail.com Bu˘ gra ¨ Ules ¸ Electronic Research Deptartment Bozankaya A. S ¸. Email: bugraules@bozankaya.com Klaus Werner Schmidt Electrical and Electronics Engineering Middle East Technical University, Ankara Email: schmidt@metu.edu.tr Abstract—This paper proposes a new lane keeping system (LKS) based on a weighted preview measurement. The paper first identifies shortcomings of existing methods that use the sole measurement of lateral displacement error at a preview distance or at the center of gravity. Then, the novel idea of computing a weighted average of both measurements is proposed. The stability of the resulting LKS is analyzed and the improved performance of the resulting LKS is supported by simulation experiments. Index Terms—Intelligent transportation systems, lane keeping system, preview, robustness. I. I NTRODUCTION Lane keeping systems (LKS) constitute an important appli- cation for autonomous or semi-autonomous vehicles with the aim of improving traffic comfort and safety [1], [2]. LKS must ensure that vehicles stay close to the lane centerline, while avoiding undesired effects such as oscillations [3]. Moreover, LKS need to be applicable at different velocities [4], [5]. LKS in the recent literature employ different control met- hods and signal measurements [6]. [7], [8] apply linear model predictive control (MPC), whereby [7] uses a time-varying prediction model depending on the velocity and [8] extends the objective function by a smoothness term. Several approaches are based on the measurement of the lateral displacement error at a preview distance (LDEPD) and the yaw rate. [9] proposes to track a reference yaw rate using sliding mode control and [4] designs two nested PID controllers to control the LDEPD and a desired yaw rate. Differently, [3], [5] suggest to employ measurements of the lateral displacement error at the center of gravity (LDECOG) and the yaw rate. [3] empirically determines a velocity-dependent controller based on driving experiments and [5] uses the concept of immersion and invariance to design a PID controller with additional feedback of yaw rate, side slip angle and road curvature. This paper focuses on lane keeping systems that do not require optimization. In our study, two shortcomings of exis- ting methods are determined. On the one hand, approaches using the LDEPD measurement (at a preview distance) do not necessarily achieve small values of the actual LDECOG (at preview distance zero). On the other hand, methods that di- rectly measure the LDECOG have a deteriorated lane keeping performance especially at higher velocities [10]. Accordingly, the paper proposes the usage of a combined measurement of the LDEPD and LDECOG and performs a stability analysis based on an extended linearized vehicle model. The improved performance of the proposed method is supported by simula- tion experiments with a nonlinear vehicle model. The remainder of the paper is organized as follows. Section II gives background information on modeling and lane keeping control. The proposed modified LKS is described in Section III and evaluated in Section IV. Section V gives conclusions. II. BACKGROUND A. Nonlinear Vehicle Model We use the dynamic bicycle model [11] in Fig. 1. G \ \ E U O U ) OU ) FU ) OI ) FI Y O I S ¸ekil 1: Dynamic bicycle model. x, y, ψ are the local vehicle coordinates, v is the vehicle velocity, β denotes the side-slip angle, δ is the steering angle and r = ˙ ψ represents the yaw rate. The tire distances from the center of gravity (COG) are given by l f and l r . The forces in the tire directions are F lf , F lr and the cornering forces are F cf , F cr at the front and rear tire, respectively. Assuming a rear- wheel drive vehicle such as a bus, F rf is the engine traction force and F lf =0. Using the vehicle mass m and the moment of inertia J , the mathematical model of the vehicle is ˙ β = − ˙ ψ + F cr cos(β)+ F cf cos(β − δ) − F lr sin(β) mv (1) ¨ ψ = (F lf sin(δ)+ F cf cos(δ)) a − F cr b I (2) ˙ v = F cr sin(β)+ F lr cos(β)+ F cf sin(β − δ) m . (3) Hereby, the forces F cf and F cr depend on the lateral tire slip angles α cf (front) and α cr (rear) of the respective tire: α cf = tan −1 ( ˙ y + a ˙ ψ ˙ x ) − δ and α cr = tan −1 ( ˙ y − b ˙ ψ ˙ x ) (4) 978-1-5090-1679-2/16/$31.00 2016 IEEE