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comparison_of_braking_performance.m
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comparison_of_braking_performance.m
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%% Comparison of braking performance
% Simulation and animation of two vehicles 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]
air_Density = 1; % Air density [kg/m3]
% Vehicle 1
vehicle_1_Length = 4.65; % Length of the vehicle [m]
vehicle_1_Width = 1.78; % Width of the vehicle [m]
vehicle_1_Init_Speed = 72/3.6; % Initial speed of the vehicle [m/s]
vehicle_1_Mass = 1145; % Mass of the vehicle [kg]
vehicle_1_Area = 2.5; % Frontal area of the vehicle [m2]
vehicle_1_Cd = 0.35; % Drag coefficient [-]
vehicle_1_Lat_Pos = 2.5; % Lateral position [m]
% Lumped air drag coefficient [N(s/m)2]
vehicle_1_C = 0.5 * vehicle_1_Area * vehicle_1_Cd * air_Density;
% Vehicle 1 struct
vehicle_1.C = vehicle_1_C;
vehicle_1.M = vehicle_1_Mass;
% Vehicle 2
vehicle_2_Length = 4.65; % Length of the vehicle [m]
vehicle_2_Width = 1.78; % Width of the vehicle [m]
vehicle_2_Init_Speed = 72/3.6; % Initial speed of the vehicle [m/s]
vehicle_2_Mass = 1145; % Mass of the vehicle [kg]
vehicle_2_Area = 2.5; % Frontal area of the vehicle [m2]
vehicle_2_Cd = 0.35; % Drag coefficient [-]
vehicle_2_Lat_Pos = -2.5; % Lateral position [m]
% Lumped air drag coefficient [N(s/m)2]
vehicle_2_C = 0.5 * vehicle_2_Area * vehicle_2_Cd * air_Density;
% Vehicle 2 struct
vehicle_2.C = vehicle_2_C;
vehicle_2.M = vehicle_2_Mass;
% Parameters
playback_speed = 1; % Speed of playback
tf = 100; % Final time [s]
% OBS: tf must be larger than the stopping time.
fR = 30/playback_speed; % Frame rate [fps]
dt = 1/fR; % Time resolution [s]
TSPAN = linspace(0,tf,tf*fR); % Time [s]
%% Braking force
% Vehicle 1
% Brake
vehicle_1_brake_Time_Constant = 1; % Brake time constant [s]
vehicle_1_brake_Force_Max = 3000; % Brake max. force [N]
% Braking force during time span [N]
vehicle_1_FbBrakingForce = vehicle_1_brake_Force_Max*(1-exp(-vehicle_1_brake_Time_Constant*TSPAN));
% Brake struct
vehicle_1_brake.time = TSPAN;
vehicle_1_brake.force = vehicle_1_FbBrakingForce;
% Vehicle 2
% Brake
vehicle_2_brake_Time_Constant = 1; % Brake time constant [s]
vehicle_2_brake_Force_Max = 2000; % Brake max. force [N]
% Braking force during time span [N]
vehicle_2_FbBrakingForce = vehicle_2_brake_Force_Max*(1-exp(-vehicle_2_brake_Time_Constant*TSPAN));
% Brake struct
vehicle_2_brake.time = TSPAN;
vehicle_2_brake.force = vehicle_2_FbBrakingForce;
%% Simulation
% Initial conditions [position speed]
vehicle_1_Z0 = [0 vehicle_1_Init_Speed];
vehicle_2_Z0 = [0 vehicle_2_Init_Speed];
% Options:
% Simulation ends when v=0 (See auxiliary function)
options = odeset('events',@vehicleAtRest);
% Integration vehicle 1
[TOUT_1,ZOUT_1] = ode45(@(t,z) vehicle_braking_dynamics(t,z,vehicle_1,vehicle_1_brake),TSPAN,vehicle_1_Z0,options);
% Integration vehicle 2
[TOUT_2,ZOUT_2] = ode45(@(t,z) vehicle_braking_dynamics(t,z,vehicle_2,vehicle_2_brake),TSPAN,vehicle_2_Z0,options);
% Vehicle 1 States
vehicle_1_position = ZOUT_1(:,1);
vehicle_1_speed = ZOUT_1(:,2);
% Vehicle 1 Acceleration
% Vehicle 1 Preallocating
vehicle_1_accel = zeros(length(TOUT_1),1);
for i=1:length(TOUT_1)
dz = vehicle_braking_dynamics(TOUT_1(i),ZOUT_1(i,:),vehicle_1,vehicle_1_brake);
vehicle_1_accel(i) = dz(2);
end
% Vehicle 2 States
vehicle_2_position = ZOUT_2(:,1);
vehicle_2_speed = ZOUT_2(:,2);
% Vehicle 2 Acceleration
% Vehicle 2 Preallocating
vehicle_2_accel = zeros(length(TOUT_2),1);
for i=1:length(TOUT_2)
dz = vehicle_braking_dynamics(TOUT_2(i),ZOUT_2(i,:),vehicle_2,vehicle_2_brake);
vehicle_2_accel(i) = dz(2);
end
% Typically, length(TOUT_1) different than length(TOUT_2). Adjusting:
if TOUT_1(end) > TOUT_2(end)
vehicle_2_position = [vehicle_2_position ; vehicle_2_position(end)*ones(length(TOUT_1)-length(TOUT_2),1)];
vehicle_2_speed = [vehicle_2_speed ; vehicle_2_speed(end)*ones(length(TOUT_1)-length(TOUT_2),1)];
vehicle_2_accel = [vehicle_2_accel ; zeros(length(TOUT_1)-length(TOUT_2),1)];
TOUT_2 = TOUT_1;
elseif TOUT_1(end) < TOUT_2(end)
vehicle_1_position = [vehicle_1_position ; vehicle_1_position(end)*ones(length(TOUT_2)-length(TOUT_1),1)];
vehicle_1_speed = [vehicle_1_speed ; vehicle_1_speed(end)*ones(length(TOUT_2)-length(TOUT_1),1)];
vehicle_1_accel = [vehicle_1_accel ; zeros(length(TOUT_2)-length(TOUT_1),1)];
TOUT_1 = TOUT_2;
end
%% Animation
figure
set(gcf,'Position',[50 50 1280 720]) % 720p
% set(gcf,'Position',[50 50 854 480]) % 480p
% Create and open video writer object
v = VideoWriter('Comparison_of_braking_performance.avi');
v.Quality = 100;
% v.FrameRate = fR;
open(v);
for i=1:length(TOUT_1)
subplot(3,2,1)
hold on ; grid on
set(gca,'xlim',[0 max([TOUT_1(end) TOUT_2(end)])],'ylim',[0 1.2*max([vehicle_1_position(end) vehicle_2_position(end)])])
cla
% Vehicle 1
plot(TOUT_1,vehicle_1_position,'m')
% Vehicle 2
plot(TOUT_2,vehicle_2_position,'c')
plot([TOUT_1(i) TOUT_1(i)],[0 1.2*max([vehicle_1_position(end) vehicle_2_position(end)])],'k--')
xlabel('Time [s]')
ylabel('Position [m]')
title('Position')
subplot(3,2,2)
hold on ; grid on
set(gca,'xlim',[0 max([TOUT_1(end) TOUT_2(end)])],'ylim',[0 1.2*max([vehicle_1_speed(1) vehicle_2_speed(1)])])
cla
% Vehicle 1
plot(TOUT_1,vehicle_1_speed,'m')
% Vehicle 2
plot(TOUT_2,vehicle_2_speed,'c')
plot([TOUT_1(i) TOUT_1(i)],[0 1.2*max([vehicle_1_speed(1) vehicle_2_speed(1)])],'k--')
xlabel('Time [s]')
ylabel('Speed [m/s]')
title('Speed')
subplot(3,2,3)
hold on ; grid on
set(gca,'xlim',[0 max([TOUT_1(end) TOUT_2(end)])],'ylim',[1.2*min([min(vehicle_1_accel) min(vehicle_2_accel)]) 1.2*max([max(vehicle_1_accel) max(vehicle_2_accel)])])
cla
% Vehicle 1
plot(TOUT_1,vehicle_1_accel,'m')
% Vehicle 2
plot(TOUT_2,vehicle_2_accel,'c')
plot([TOUT_1(i) TOUT_1(i)],[1.2*min([min(vehicle_1_accel) min(vehicle_2_accel)]) 1.2*max([max(vehicle_1_accel) max(vehicle_2_accel)])],'k--')
xlabel('Time [s]')
ylabel('Acceleration [m/s2]')
title('Acceleration')
subplot(3,2,4)
hold on ; grid on
set(gca,'xlim',[0 max([TOUT_1(end) TOUT_2(end)])],'ylim',[0 1.2*max([max(vehicle_1_FbBrakingForce) max(vehicle_2_FbBrakingForce)])])
cla
% Vehicle 1
plot(TSPAN,vehicle_1_FbBrakingForce,'m')
% Vehicle 2
plot(TSPAN,vehicle_2_FbBrakingForce,'c')
plot([TOUT_1(i) TOUT_1(i)],[0 1.2*max([max(vehicle_1_FbBrakingForce) max(vehicle_2_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 1 instant [m]
vehicle_1_position_inst = vehicle_1_position(i);
% Position of the vehicle 1 instant [m]
vehicle_2_position_inst = vehicle_2_position(i);
road_Length = 1.2*max([max(vehicle_1_position) max(vehicle_2_position)]); % Road length [m]
sideMarkingsX = [-1.5*vehicle_1_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,[0 0],'k--') % Center marking
plot(sideMarkingsX,[-road_Width/2 -road_Width/2],'k--') % Right marking
% Vehicle 1 Dimensions
vehicle_1_dimension_X = [vehicle_1_position_inst vehicle_1_position_inst vehicle_1_position_inst-vehicle_1_Length vehicle_1_position_inst-vehicle_1_Length];
vehicle_1_dimension_Y = [+vehicle_1_Width/2+vehicle_1_Lat_Pos -vehicle_1_Width/2+vehicle_1_Lat_Pos -vehicle_1_Width/2+vehicle_1_Lat_Pos +vehicle_1_Width/2+vehicle_1_Lat_Pos];
% Vehicle 1 Plotting
fill(vehicle_1_dimension_X,vehicle_1_dimension_Y,'m')
% Vehicle 2 Dimensions
vehicle_2_dimension_X = [vehicle_2_position_inst vehicle_2_position_inst vehicle_2_position_inst-vehicle_2_Length vehicle_2_position_inst-vehicle_2_Length];
vehicle_2_dimension_Y = [+vehicle_2_Width/2+vehicle_2_Lat_Pos -vehicle_2_Width/2+vehicle_2_Lat_Pos -vehicle_2_Width/2+vehicle_2_Lat_Pos +vehicle_2_Width/2+vehicle_2_Lat_Pos];
% Vehicle 2 Plotting
fill(vehicle_2_dimension_X,vehicle_2_dimension_Y,'c')
xlabel('Lon. distance [m]')
ylabel('Lat. distance [m]')
title(strcat('Time=',num2str(TOUT_1(i),'%.3f'),' s'))
% title(strcat('Time=',num2str(TOUT_1(i),'%.3f'),' s (Playback speed=',num2str(playback_speed),')'))
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