How to correctly use ode45 function in MATLAB for differential drive robot?

Question

I am trying to determine the pose (x,y,theta) of a differential drive robot using ode45. The code I have below solves for the x position, but I am having an issue with the initial condition. I set it to 0 since at time 0 the robot is assumed to be at the origin, but I get an error. How do I set the initial condition for ode45 such that I get the expected output?

I tried to make the initial conditions vector of the same length as dxdt by setting initial conditions as a 41x1 zeros matrix, but I didn't understand the output and I don't believe I did it correctly.

% Given conditions for a differential drive robot:
B = 20; % (cm) distance along axle between centers of two wheels
r = 10; % (cm) diameter of both wheels
w_l = 5*sin(3*t); % (rad/s) angular rate of left wheel
w_r = 5*sin(3*t); % (rad/s) angular rate of right wheel
v_l = r*w_l; % (cm/s) velocity of left wheel
v_r = r*w_r; % (cm/s) velocity of right wheel
v = (v_r+v_l)/B; % (cm/s) velocity of robot
theta = 90; % constant orientation of robot since trajectory is straight

% Solve differential equation for x:
dxdt = v*cos(theta); % diff equaition for x
tspan = [0 20]; % time period to integrate over
x0 = 0; % initial condition since robot begins at origin
[t,x] = ode45(@(t,y) dxdt, tspan, x0);

I want to solve the differential equation dxdt for 0 to 20 seconds with an initial condition of 0. I expect the output to give me a vector of time from 0 to 20 and an array of for x. The problem I believe lies with the initial condition. MATLAB gives me an error in the live editor telling me, " @(t,y)dxdt returns a vector of length 69, but the length of initial conditions vector is 1. The vector returned by @(t,y)dxdt and the initial conditions vector must have the same number of elements."

Jamie Parkinson Jamie Parkinson · Accepted Answer · 2019-02-11T10:18:26

The issue is not the initial condition. You need to define dxdt as a function i.e.

% Define differential equation
function derivative = dxdt(t,y)

% Given conditions for a differential drive robot:
B = 20; % (cm) distance along axle between centers of two wheels
r = 10; % (cm) diameter of both wheels
w_l = 5*sin(3*t); % (rad/s) angular rate of left wheel
w_r = 5*sin(3*t); % (rad/s) angular rate of right wheel
v_l = r*w_l; % (cm/s) velocity of left wheel
v_r = r*w_r; % (cm/s) velocity of right wheel
v = (v_r+v_l)/B; % (cm/s) velocity of robot
theta = 90; % constant orientation of robot since trajectory is straight

derivative = v*cos(theta); % diff equation for x

end

Then when you use ode45 you should tell it to pass the t and y variables as arguments to dxdt like

[t,x] = ode45(@(t,y) dxdt(t,y), tspan, x0);

This should then work. In this case, as dxdt only takes the default arguments, you could also write

[t,x] = ode45(@dxdt, tspan, x0);

The error you got indicates that at some point you made dxdt into a vector of length 69, whilst MATLAB was expecting to get back 1 value for dxdt when it passed one t and one y to your dxdt 'function'. Whenever you get errors like this, I'd recommend putting

~~clear all~~

`clearvars` % better than clear all - see am304's comment below

at the top of your script to avoid polluting your workspace with previously defined variables.

How to correctly use ode45 function in MATLAB for differential drive robot?

1 Answers