How can I plot a point which is product of two functions in time domain with changing parameter?

Hello, I'm making one assignment to my class and i hit the wall and can't proceed.

We're supposed to program movement of a car. ( look at the attached picture )

and the equations are: x' = cos( θ) *cos( ϕ)*u1 | y' = cos (θ) sin( ϕ)*u1 | ϕ' = (sin( θ)/L)*u1 | θ' = u2

where x,y are the position of the rear axle of the car and u1 is longitudinal speed and u2 is wheels turning speed, θ is the wheel steering angle, ϕ is the orientation of the car ( look at the attached picture ).

Written in Matlab function as :

function carDerivatives = car2(t, car, l, u1, u2)
    % car is in form:
    %[x, y, phi, theta]
    % carDerivatives = zeros(1, 4);
    dxdt = cos(car(4)) * cos(car(3)) * u1;
    dydt = cos(car(4)) * sin(car(3)) * u1;
    dphidt = sin(car(4)) * u1 / l;
    dthetadt = u2;
    carDerivatives = [dxdt; dydt; dphidt; dthetadt];
end

with defined variables and initial conditions and the ode45:

l = 1;  %length
u1 = 1; %speed
u2 = 0; %turning speed(movement=[u1;u2]=[1;0]
%inits :
x0 = 0;     %rear axle x position
y0 = 0;     %rear axle y position
phi0 = 0;   %car orientation angle
theta0 = 0; %steering wheel angle
car0 = [x0; y0; phi0; theta0];

tspan = [0 40];

[t, car] = ode45(@(t, car)(car2(t, car, l, u1, u2)), tspan, car0);

so as an output we have x and y values which are car(:,1) and car(:,2), when plotted they correctly represent the rear axle movement, with relation to the distance not time.

i want to plot the position of the car but in time domain, so i made

rear=[car(:,1), car(:,2)];
figure(3)
plot(t,rear)

and the output is 2 graph in 1 figure, what can i do to express in the time domain?

And the second question is:

i would like to program the cars movement with changing the values of u1 and u2, using a piecewise function, for example:

for t<1 u1=1, u2=0;

for 1<t<2 u1=0, u2=1;

for 2<t<3 u1=1, u2=0;

for 3<t<4 u1=0,u2=-1;

how can i express this in the time domain ? How can i write piecewise function for these controls ?

Thank You for your time!

Answer 1

[Star Strider]

Try this:

l = 1;  %length
u1 = 1; %speed
u2 = 0; %turning speed(movement=[u1;u2]=[1;0]
%inits :
x0 = 0;     %rear axle x position
y0 = 0;     %rear axle y position
phi0 = 0;   %car orientation angle
theta0 = 0; %steering wheel angle
car0 = [x0; y0; phi0; theta0];
tspan = [0 40];

u1v = [1; 0; 1; 0];
u2v = [0; 1; 0; 1];
tm = [0 1; 1 2; 2 3; 3 4];
for k1 = 1:size(tm,1)
    k1
    tspan = tm(k1,:);
    u1 = u1v(k1);
    u2 = u2v(k1);
    [t, car] = ode45(@(t, car)(car2(t, car, l, u1, u2)), tspan, car0);
    car0 = car(end,:);
    tv{k1,:} = t;
    carc{k1,:} = car;
end

t_all = cell2mat(tv);
car_all = cell2mat(carc);

figure
plot(t_all, car_all)
grid

figure
plot3(car_all(:,1), car_all(:,2), t_all)
grid on
xlabel('car(:,1)')
ylabel('car(:,2)')
zlabel('t')

This code illustrates the general idea.

Experiment to get the result you want.

Your ‘car2’ function remains unchanged, so there is no reason to post it again.

[Mikolaj Krakowski]

Thank You for the answer.

Sorry, i didn't make it precise. as an output i try to get 2D graph which reflects t(y)=y(x). Sorry for the confusion. Would you know how can i get there ?

[Star Strider]

Probably the easiest way is to use the view function on the second plot:

figure
plot3(car_all(:,1), car_all(:,2), t_all)
grid on
xlabel('car(:,1)')
ylabel('car(:,2)')
zlabel('t')
view(0,90)

Another option is to color-code the segments, if you want to distinguish them. Otherwise, use the same colors for all of them. A third plot is then:

c = 'brgm';
figure
hold all
for k1 = 1:size(tm,1)
    plot(carc{k1}(:,1), carc{k1}(:,2), 'Color',c(k1), 'LineWidth',2)
end
hold off
grid

Experinent to get the result you want.

[Mikolaj Krakowski]

Okay, everything clear, thank you so much for helping my stupid a*s go through this, THANK YOU again!

[Star Strider]

As always, my pleasure!