﻿ 东南大学学报(自然科学版)

(东南大学机械工程学院,南京 210096)

Cooperative merging control of multiple connected and automated vehicles on freeway ramp
Liu Chang,Zhuang Weichao,Yin Guodong,Huang Zehao,Liu Haoji
(School of Mechnical Engineering, Southeast University, Nanjing 210096, China)

To improve the safety and the efficiency of vehicle merge at freeway ramp and reduce the fuel consumption, an optimal longitudinal trajectory planning method for multiple connected and automated vehicles facing freeway ramp was proposed to realize the vehicle cooperative merge. First, the vehicle longitudinal dynamics model was established, and the cost function of energy efficiency and ride comfort was considered to construct the optimal vehicle speed control problem on the on-ramp. Based on the first-in, first-out(FIFO)merging sequence, the time and the time interval of each adjacent vehicle arriving at merging point were designed to realize safe and efficient cooperative merge. The optimal vehicle speed control problem was solved by using Pontryagin's minimum principle, and the optimal analytical solution of each vehicle longitudinal speed was derived. The simulation results show that compared with uncontrolled natural merge, the traffic time and the fuel consumption of the proposed control method are reduced by 41.64% and 12.25%, respectively. Compared with the existing control method based on virtual queue, the traffic efficiency difference is 1.67% and the fuel consumption is reduced by 4.52%.

1 入口匝道协同合流场景与控制问题构建1.1 入口匝道合流

1.2 协同合流控制问题建模

(·overx)i=f[xi(t),ui(t),t]=Axi(t)+Bui(t)(1)

0 0 1

0 0 0]; B=[0

0

1]; 状态变量pi(t)、vi(t)、ai(t)及输入变量ui(t)分别表示车辆i在t时刻的位置、速度、加速度及加速度变化率(控制输入).xi(t)具有初始状态xi(tei)={0, vi(tei), ai(tei)}T,其中tei表示车辆i首次进入协同区域的时刻.

Ji=1/2∫tm</sub>itei1a2i(t)+ω2u2i(t))dt(2)

si(t)=pj(t)-pi(t)≥s(t)t∈[tei, tmi](3)

s(t)=γ0+ζvi(t)t∈[tei, tmi](4)

vmin≤vi(t)≤vmax, amin≤ai(t)≤amax,

umin≤ui(t)≤umax t∈[tei, tmi](5)

vi(tmi)=vmer(6)

1)情况1.如图3(a)所示,当车辆i+1首次进入协同区域时,i+1∈Si(t),即车辆i+1和其前一个进入协同区域的车辆i在相同车道上.此时规定车辆i在达到合流点后将以合流速度vmer匀速行驶长度为s1的一段距离以保证行驶安全,防止两车发生追尾.为了最小化车辆间距以缩短通行时长,车辆i+1到达合流点的时刻将遵循

tmi+1=tmi+hi+1(7)

2)情况2.如图3(b)所示,当车辆i+1首次进入协同区域时,i+1∈Di(t),即车辆i+1和其前一个进入协同区域的车辆i在不同车道上.规定车辆i在达到合流点后将以合流速度vmer匀速行驶长度为s2的距离.为防止两车发生横向碰撞,车辆i到达合流点后匀速行驶的距离s2应不小于s1,车辆i+1到达合流点的时刻将遵循

tmi+1=tmi+h'i+1(8)

3)情况3.定义预期行驶时间,即

ttrai+1=tmi+1-tei+1(9)

tmi+1=tei+1+(2L)/(vmer+vi+1(tei+1))(10)

2 车辆最优轨迹规划求解

Hi[xi(t),ui(t),t]=L[xi(t),ui(t),t]+

λTif[xi(t),ui(t),t](11)

Hi1ivi2iai3iui+1/2ω1a2i+1/2ω2u2i(12)

u*(t)和x*(t)分别为最优控制输入和最优轨迹,其中x*(t)={p*(t),v*(t),a*(t)}T,p*(t)、v*(t)、a*(t)分别为最优位置、最优速度及最优加速度轨迹.根据庞特里亚金极小值原理[16],在适当选取的拉格朗日乘子下,如下方程和等式成立.

(∂Hi)/(∂ui)=λ3i2u*i=0(13)

(·overλ)1i=-(∂Hi)/(∂pi)=0(14)

(·overλ)2i=-(∂Hi)/(∂vi)=-λ1i(15)

(·overλ)3i=-(∂Hi)/(∂ai)=-ω1a*i2i(16)

ω2(·overu)*i1a*i+k1it-k2i=0(17)

u*i(t)=k3iAeAt-k4ie-At+(k1i)/(ω1)(18)

p*i(t)=(k3i)/(A2)eAt+(k4i)/(A2)e-At+(k1i)/(6ω1)t3-(k2i)/(2ω1)t2+k5it+k6i(19)

v*i(t)=(k3i)/AeAt+(k4i)/Ae-At+(k1i)/(2ω1)t2-(k2i)/(ω1)t+k5i(20)

a*i(t)=k3ieAt+k4ie-At+(k1i)/(ω1)t-(k2i)/(ω1)(21)

Ki(pi(t),vi(t),ai(t),t)=M-1i×Ni(t,pi(t),vi(t))(22)

Ni={pi(t), pi(tmi), vi(t), vi(tmi), ai(t), ai(tmi)}T(23)

Mi=[(eAt)/(A2)(e-At)/(A2)(t3)/(6ω1)-(t2)/(2ω1)t 1

(eAtmi)/(A2)(e-Atmi)/(A2)((tmi)3)/(6ω1)-((tmi)2)/(2ω1)tmi 1

(eAt)/A -(e-At)/A(t2)/(2ω1)t/(ω1)1 0

(eAtmi)/A -(e-Atmi)/A((tmi)2)/(2ω1)(tmi)/(ω1)1 0

eAt e-At t/(ω1)-1/(ω1)0 0

eAtmi eAtmi(tmi)/(ω1)-1/(ω1)0 0](24)

3 仿真验证3.1 代价函数权重因子选取

3.2 协同合流车辆轨迹仿真

4 结论

1)本文针对高速匝道路口通行效率低下、燃油经济性差以及存在安全隐患等问题,在V2X背景下,通过中央控制器协调受控车辆的合流时刻并规划各个车辆的纵向运动,提出了一种基于最优控制的车辆合流轨迹规划方法,并应用庞特里亚金原理给出该控制问题解析解的计算方法.

2)代价函数中权重因子ω2在0.01～100之间变化时,最优速度轨迹受到的影响较小,而加速度和控制输入所受影响显著,当ω2取值过小时,车辆在刚进入协同区域及到达合流点前控制输入将突增,推荐ω2取1～10.

3)该方法与无控制自然合流相比入口匝道通行时长缩短41.64%,燃油消耗降低12.25%,同时车辆的加速度变化平滑,提升了乘坐舒适性.与现有基于虚拟队列的控制方法相比,在通行时长上仅相差1.67%,然而燃油消耗降低4.52%.