Speed Profile Optimization for Enhanced Passenger Comfort: An Optimal Control Approach Yuyang Wang, Jean-Rémy Chardonnet, Frédéric Merienne To cite this version: Yuyang Wang, Jean-Rémy Chardonnet, Frédéric Merienne. Speed Profile Optimization for Enhanced Passenger Comfort: An Optimal Control Approach. 2018 21st International Conference on Intelligent Transportation Systems (ITSC), Nov 2018, Maui, Hawaii, United States. pp.723-728. hal-01963942 HAL Id: hal-01963942 https://hal.archives-ouvertes.fr/hal-01963942 Submitted on 21 Dec 2018 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Speed Profile Optimization for Enhanced Passenger Comfort: An Optimal Control Approach Yuyang Wangy, Jean-Remy´ Chardonnet and Fred´ eric´ Merienne Abstract— Autonomous vehicles are expected to start reach- We propose a novel approach in which we generate a glob- ing the market within the next years. However in practical ally lowest jerk trajectory with an optimized speed profile applications, navigation inside dynamic environments has to following also the same comfort constraints as defined in take many factors such as speed control, safety and comfort into consideration, which is more paramount for both passengers the ISO 2631-1 standard [5]. We achieve this by introducing and pedestrians. In this paper, a novel speed profile planner the optimal control theory into the problem, following a based on an optimal control approach considering passenger standard procedure for solving optimal problems, improving comfort is proposed. The approach is accomplished by mini- passenger comfort in a global manner. The proposed method mizing jerk under certain comfort constraints, which inherently complies with the speed, acceleration and jerk limitation gives a speed profile for the central nervous system to follow naturally. Imposed with the same conditions, the widely used and meanwhile provides higher order smoothness to the Jerk Limitation method is interpreted as an equivalent of the acceleration and speed. minimum time control method, the latter being used to verify that our method can ensure better continuity and smoothness II. PROBLEM DESCRIPTION of the speed profiles. A validation test was specifically designed Given a geometric trajectory generated from a path plan- and performed in order to show the feasibility of our method. ning algorithm for the vehicle to follow, the next problem I. INTRODUCTION is to design a smooth profile from the starting place to the Research on path planning for motion systems has been destination in the path. The speed planner should consider the primarily focusing on finding an optimal path through a cost speed, acceleration and jerk limits for passenger comfort, and function considering the physical road such as the length also be time efficient in order to keep real-time computing. and curvature of the road, or a combination of both [1], The theoretical passenger comfort is linked to the smoothness while the optimization of proper speed profiles with respect of the jerk profile which can be interpreted as a minimization to passenger comfort has received less attention [2]. problem [2]. Jerk is the temporal derivative of the acceleration during Gonzalez et al. propose a method to address such problem motion and thus is interpreted as a change of actuator forces. using a quintic Bezier´ curve to approximate a smooth change in the speed under given speed and acceleration limits, lead- Hogan [3] notes that the smoothness of speed profiles can be 2 formulated as a function of jerk. Using splines and clothoids, ing to a G continuous of the speed profile [2]. This method Labakhua et al. [4] propose smooth trajectories with low can efficiently compute an optimized speed profile, but the associated accelerations and jerk, trying to provide more physical meaning of its minimization process is unclear. comfort to the passenger: firstly, the global expressed discrete Also, since the jerk limit also holds the key to speed profile points are connected through high order polynomials, and generation [8], [9], the lack of such consideration about jerk in a second step, the speed must be computed according to may reduce comfort and safety [10]. Another method is the comfort constraints from the ISO 2631-1 standard [5], giving so-called Jerk Limitation method as implemented in [6], [11], rise to a smooth profile, although longitudinal and lateral in which the jerk can take three different values depending on accelerations may still be discontinuous at some points. the status of the vehicle: acceleration, cruise or deceleration. On the other hand, some authors separate the problem Once the vehicle decides to accelerate, it will choose the and consider motion variables such as the spatial position, maximum jerk in order to reach the acceleration and speed speed and acceleration independently. For example, Liu [6] limits in a minimum time, and vice versa, therefore leaving and Piazzi and Visioli [7] solve the problem by setting a discontinuous jerk profiles. Based on this idea, we interpret constant jerk in different time phases and then integrate this method as a minimum-time problem using the optimal for the acceleration and speed. Therefore the corresponding control theory as a baseline to verify our new proposition. smoothness of the acceleration and speed are C0 and C1 In order to address these issues, a new speed planner based continuous respectively. on the optimal control theory is proposed and validated. The resulting profiles are continuous and provide globally *This work was supported by the China Scholarship Coun- minimum jerk. Our contribution are described as follows: cil(No.201708390014) Yuyang Wang, Graduate Student Member IEEE, Jean-Remy´ • Propose a general speed planning method based on Chardonnet, Member IEEE, and Fred´ eric´ Merienne are with the universality and generality of the optimal control LISPEN EA7515, Arts et Metiers,´ HESAM, UBFC, Institut Image, theory. Depending on different metrics to be optimized, 71100 Chalon-sur-Saone,ˆ France [email protected], [email protected], categorize the speed optimization method with three [email protected] different metrics: minimum time, minimum norm of jerk, minimum square of jerk. This will be presented dimensions. Three major components of optimal control exist in Sec. IV. to solve optimal control problems [13]: differential equations • Compare these different speed planning methods. In and integration of equations, systems of nonlinear algebraic either case, the algorithm is able to compute a desired equations and nonlinear optimization problems. speed profile, considering passenger comfort, but only Algorithm 1 shows how the direct method is used to solve the last metric gives the speed profile with higher optimal control problems. First, the solver makes an initial continuity, as we will show in Sec. V. guess and solves the performance index and the dynamic • Finally, apply the minimum square of jerk to solve a constraints; then the solver converts the problem into a real case considering constraints like acceleration limits nonlinear optimization problem to be solved with gradient- according to the ISO 2631-1 standard. Besides, we based methods; the procedure ends once the desired accuracy impose other constraints to the control method in order is achieved. to enhance passenger comfort. This will be addressed in Sec. VI. Algorithm 1: Solving optimal control problems with To better show the effectiveness of our approach, the same direct methods conditions are imposed to all of these methods. Input: Initial guess with zero for all unknown nodes Output: State and control profiles w.r.t time: x, u III. OPTIMAL CONTROL BASICS 1 Initialize the solver; A. General Formulation of an Optimal Control Problem 2 while Error is larger than the accepted tolerance do The optimal control problem can be posed in the following 3 Read the current solution of x and u; n 4 Convert the problem to a constrained nonlinear Bolza form [12]: Find the motion states x 2 R , the vector q m optimization problem; of static parameters p 2 R , the control u 2 R , the initial 5 Calcuate the numerical solution in a differential time t0 2 R, and the final time tf 2 R, that optimize form (2) and in an integral form (1); 6 Optimize and update the solution with the Z tf gradient-based method; J = Φ (x; t0; x; tf ; p) + L ([x; u; t; p]) dt (1) t0 7 Compute the error and update the solution; subject to the dynamic constraints (representing the system 8 end dynamics) _x = f[x; u; t; p] (2) 1) Numerical Solution of Differential Equations: Several the state constraints methods can be used to solve differential equations, among which are time-marching methods. Time-marching methods Cmin ≤ C[x; u; t; p] ≤ Cmax (3) are designed to solve differential equations as (2) at each time and the boundary conditions step tk based on the current and/or previous condition for the solution [14]. The most common methods are usually θ- φmin ≤ φ [x; t0; x; tf ; p] ≤ φmax (4) family methods (the forward/backward Euler method and the Crank-Nicolson method) and Runge-Kutta methods. Recall J represents a performance index that measures the quality that the objective is to numerically solve an optimal control of the path. The optimal control problem can be decomposed problem, more specifically the one in (1). Integration can be in phases, in this case the performance index J is the sum addressed by many numerical schemes [14]. of each performance index over the number of phases p.