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vehicle_braking.m
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vehicle_braking.m
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%% Vehicle braking to a stop
% Simulation and animation of a road vehicle braking to a stop with
% position, speed, acceleration and braking force plots.
%
%%
clear ; close all ; clc
%% Scenario
% Road
road_Width = 10; % Road width [m]
road_Margin = 2; % Road margin [m]
% Vehicle
vehicle_Length = 4.65; % Length of the vehicle [m]
vehicle_Width = 1.78; % Width of the vehicle [m]
vehicle_Initial_Speed = 72/3.6; % Initial speed of the vehicle [m/s]
vehicle_Mass = 1145; % Mass of the vehicle [kg]
vehicle_Area = 2.5; % Frontal area of the vehicle [m2]
vehicle_Cd = 0.35; % Drag coefficient [-]
air_Density = 1; % Air density [kg/m3]
% Lumped air drag coefficient [N(s/m)2]
C = 0.5 * vehicle_Area * vehicle_Cd * air_Density;
% Vehicle struct
vehicle.C = C;
vehicle.M = vehicle_Mass;
% Parameters
tf = 100; % Final time [s]
% OBS: tf must be larger than the stopping time.
fR = 30; % Frame rate [fps]
dt = 1/fR; % Time resolution [s]
TSPAN = linspace(0,tf,tf*fR); % Time [s]
%% Braking force
% Brake
brake_Time_Constant = 1; % Brake time constant [s]
brake_Force_Max = 5000; % Brake max. force [N]
% Braking force during time span [N]
FbBrakingForce = brake_Force_Max*(1-exp(-brake_Time_Constant*TSPAN));
% Brake struct
brake.time = TSPAN;
brake.force = FbBrakingForce;
%% Simulation
% Initial conditions [position speed]
Z0 = [0 vehicle_Initial_Speed];
% Options:
% Simulation ends when v=0 (See auxiliary function)
options = odeset('events',@vehicleAtRest);
% Integration
[TOUT,ZOUT] = ode45(@(t,z) vehicle_braking_dynamics(t,z,vehicle,brake),TSPAN,Z0,options);
% States
vehicle_position = ZOUT(:,1);
vehicle_speed = ZOUT(:,2);
% Acceleration
% Preallocating
vehicle_accel = zeros(1,length(TOUT));
for i=1:length(TOUT)
dz = vehicle_braking_dynamics(TOUT(i),ZOUT(i,:),vehicle,brake);
vehicle_accel(i) = dz(2);
end
%% Animation
figure
set(gcf,'Position',[50 50 1280 720]) % YouTube: 720p
% set(gcf,'Position',[50 50 854 480]) % YouTube: 480p
% set(gcf,'Position',[50 50 640 640]) % Social
% Create and open video writer object
v = VideoWriter('braking_dynamics.mp4','MPEG-4');
v.Quality = 100;
v.FrameRate = fR;
open(v);
for i=1:length(TOUT)
subplot(3,2,1)
hold on ; grid on
set(gca,'xlim',[0 TOUT(end)],'ylim',[0 1.2*max(vehicle_position)])
cla
plot(TOUT,vehicle_position)
plot([TOUT(i) TOUT(i)],[0 1.2*max(vehicle_position)],'k--')
xlabel('Time [s]')
ylabel('Position [m]')
title('Position')
subplot(3,2,2)
hold on ; grid on
set(gca,'xlim',[0 TOUT(end)],'ylim',[0 1.2*max(vehicle_speed)])
cla
plot(TOUT,vehicle_speed)
plot([TOUT(i) TOUT(i)],[0 1.2*max(vehicle_speed)],'k--')
xlabel('Time [s]')
ylabel('Speed [m/s]')
title('Speed')
subplot(3,2,3)
hold on ; grid on
set(gca,'xlim',[0 TOUT(end)],'ylim',[1.2*min(vehicle_accel) 1.2*max(vehicle_accel)])
cla
plot(TOUT,vehicle_accel)
plot([TOUT(i) TOUT(i)],[1.2*min(vehicle_accel) 1.2*max(vehicle_accel)],'k--')
xlabel('Time [s]')
ylabel('Acceleration [m/s2]')
title('Acceleration')
subplot(3,2,4)
hold on ; grid on
set(gca,'xlim',[0 TOUT(end)],'ylim',[0 1.2*max(FbBrakingForce)])
cla
plot(TSPAN,FbBrakingForce)
plot([TOUT(i) TOUT(i)],[0 1.2*max(FbBrakingForce)],'k--')
xlabel('Time [s]')
ylabel('Braking force [N]')
title('Braking force')
subplot(3,2,5:6)
hold on ; axis equal
cla
% Position of the vehicle instant [m]
vehicle_position_inst = vehicle_position(i);
road_Length = 1.2*max(vehicle_position); % Road length [m]
sideMarkingsX = [-1.5*vehicle_Length road_Length];
set(gca,'xlim',sideMarkingsX,'ylim',[-road_Width/2-road_Margin +road_Width/2+road_Margin])
plot(sideMarkingsX,[+road_Width/2 +road_Width/2],'k--') % Left marking
plot(sideMarkingsX,[-road_Width/2 -road_Width/2],'k--') % Right marking
% Dimensions
vehicle_dimension_X = [vehicle_position_inst vehicle_position_inst vehicle_position_inst-vehicle_Length vehicle_position_inst-vehicle_Length];
vehicle_dimension_Y = [+vehicle_Width/2 -vehicle_Width/2 -vehicle_Width/2 +vehicle_Width/2];
% Plotting
fill(vehicle_dimension_X,vehicle_dimension_Y,'r')
xlabel('Lon. distance [m]')
ylabel('Lat. distance [m]')
frame = getframe(gcf);
writeVideo(v,frame);
end
% Additional frames repeating the last frame
% Parameters
tf_add = 5; % Final time [s]
fR_add = 30; % Frame rate [fps]
dt_add = 1/fR; % Time resolution [s]
time_add = linspace(0,tf_add,tf_add*fR_add); % Time [s]
for i=1:length(time_add)
frame = getframe(gcf);
writeVideo(v,frame);
end
close(v);
%% Auxiliary functions
function dz = vehicle_braking_dynamics(t,z,vehicle,brake)
% States
% z1 = z(1);
z2 = z(2);
% Parameters
C = vehicle.C;
M = vehicle.M;
% Brake force
timeBraking = brake.time;
forceBraking = brake.force;
Fb = interp1(timeBraking,forceBraking,t);
% State Equations
dz(1,1) = z2;
dz(2,1) = -(Fb + C*z2^2)/M;
end
function [speed,isterminal,direction] = vehicleAtRest(~,z)
speed = z(2);
isterminal = 1;
direction = 0;
end