7c05355c-4f18-4383-91e1-b43dc622ec51 12 Emilio Rosales 4 5 false false false 0 3 decimal

&[DATE] &[TIME] - &[FILENAME]

Problem Description

This project studies the longitudinal dynamics of a platoon of autonomous vehicles traveling along a single lane in an urban or congested traffic environment. Lane changes are not considered; thus, each vehicle interacts only with the vehicle directly ahead. The central objective is to model how follower vehicles adapt their motion based on the behavior of the leading vehicle, ensuring safe spacing and smooth traffic flow.

Each vehicle is assumed to be equipped with sensors that allow it to measure its own velocity as well as the velocity of the preceding vehicle. The control strategy adopted is based on relative velocity feedback, whereby a vehicle accelerates or decelerates proportionally to the difference between its velocity and that of the vehicle in front.

The leading vehicle follows a prescribed velocity profile over time, which acts as an external input to the system. This velocity perturbation propagates through the platoon via the interaction law governing the follower vehicles.

Mathematical Model

Notation and Variables

Consider a platoon consisting of (N+1) vehicles indexed by

n = 1,2,3,…,N,

where vehicle (n=1) is the leader and vehicles n≥2 are followers.

The following variables and parameters are defined:

Xn (t) : longitudinal position of vehicle n
Vn (t) : velocity of vehicle n
an(t)=Vn (t): acceleration of vehicle n
k>0: proportional control constant that determines how strongly each follower reacts to relative velocity differences

Governing Equations

Follower vehicle dynamics
For each follower vehicle n=1,…,N, the acceleration is assumed to be proportional to the relative velocity with respect to the vehicle immediately ahead:
Velocity with respect to the vehicle immediately ahead:
vn (t)=a_n (t)=k(v(n-1) (t)-v_n (t)).

The kinematic relationship between position and velocity is given by:
xn (t)=v_n (t).

Thus, the dynamics of each follower vehicle are described by the system:

\[REGION[f1f40b7b2189417fb7e50e957e976500]]\

Leader Vehicle Input
The leader vehicle does not follow the interaction law. Instead, its velocity is prescribed as a piecewise function of time:
\[REGION[c5df4a014fd54513bd7755f9a74fdd74]]\

This function represents different driving regimes of the leader (acceleration, constant speed, and deceleration) and serves as the external excitation of the platoon.
Initial Conditions

To fully define the problem, initial conditions must be specified for all vehicles:

xn(0)=xn,0,
vn(0)=vn,0,
n=1,2,…,N.

Here, Xn,0 represents the initial longitudinal positions, and Vn,0 represents the initial velocities.

PARAMETERS

N 4 :

n.0 1 :

N = NUMBER OF VEHICHLES

Δ.t 1 s * :

TIME STEP

t.0 0 s * :

INITIAL TIME

t.f 500 s * :

FINAL TIME

k 0.8 1 s / * :

FOLLOWER RESPONSIVENESS (GAIN)

d.0 10 m * :

INITIAL SPACING

d.s 6 m * :

SAFTEY MINIMUM GAP

TIME VECTOR

K t.f t.0 - (Δ.t / 1 round :

NUMBER TIME STEPS

n 1 K range :

INDEX VECTOR

t t.0 n 1 - (Δ.t * (+ :

TIME VECTOR (S)

Leader Velocity v0

VECTOR

t V.0 t 16 s * ≤ t (1 s * / sqrt 1 m s / * (* t 30 s * ≤ 4 m s / * t 50 s * ≤ 0.2 m s 2^/ (* t * - 10 m s / * + (0 m s / * if :

V.0 t V.0 vectorize :

INITIAL CONDITIONS

n.1 1 :

V.1 t V.0 vectorize eval :

X.2 10 m * :

X.3 20 m * :

x.4 30 m * :

X.1 0 m * :

Velocities

V.1 01 m s / * 1.4142 m s / * 31 mat

FOLLOWER 1 STARTS AT LEADER SPEED

ODE Integration Euler

RULE: RHS USES (J), LHS ASSIGNS (W) NEXT. NO REDEFINITIONS

j 1 K (range :

CURRENT INDEX RANGE

w j 1 + :

NEXT INDEX VECTOR

EULER VELOCITIES

V.1 01 m s / * 1.4142 m s / * 31 mat

X.1 Δ.t V.1 j el * :

V.2 w el V.2 j el Δ.t + (k * V.1 j el V.2 j el - (* :

V.3 w el V.3 j el Δ.t + (k * V.2 j el V.3 j el - (* :

V.4 w el V.4 j el Δ.t + (k * V.3 j el V.4 j el - (* :

EULER POSITIONS

x.2 w el Δt V2 j el * + #

x.3 w el Δt V3 j el * + #

x.4 w el Δt V4 j el * + #

Distances

GAPS + SAFETY CHECK

D2 x2 x1 - :

GAP BETWEEN 2 AND 1

D3 x3 x2 - :

GAP BETWEEN 3 AND 2

D4 x4 x3 - :

GAP BETWEEN 4 AND 3

df 0 t * ds + :

CONSTANT VECTOR FOR PLOTTING

Plots

Plots should look like this, (made in excel)