Euler versus Runge-Kutta Comparison in Kinematic Simulation of a Guided Mobile Robot

Project info

• Field of studyRobotics and Mechatronics
• LevelMaster's thesis (S2)
• Method typeNumerical Simulation
• Estimated build time± 5 working days
• Toolsloops dan fungsi anonim; polyfit, loglog (MATLAB inti)
• PriceUS$25 - US$163

Research goal

Simulate the robot position using Euler and Runge-Kutta methods and examine how the time step size and the angular velocity influence its behaviour.

Variables

Example result

• 0.25Step size h
• 0.364Max error Euler
• 0Max error RK4
• 3089.505Error ratio
• 1.137Order Euler
• 4.164Order RK4

At step h=0.25 the maximum Euler error for Robot position is 0.3636 while RK4 reaches only 1.2e-04; the measured convergence orders of 1.14 (Euler) and 4.16 (RK4) match theory.

Computed from illustrative synthetic data, not from a real study.

About Euler versus Runge-Kutta Comparison

Comparing Euler and fourth-order Runge-Kutta shows how method order affects accuracy and the convergence rate of the error. It helps justify the choice of integration method and demonstrates numerical verification in a thesis.

Steps

1. 1Choose an equation with an analytic solution
2. 2Implement Euler and RK4
3. 3Run with several step sizes h
4. 4Compute the error against the exact solution
5. 5Estimate the order of convergence from the log-log slope
6. 6Plot error against h

Assumptions to check

• The exact solution is known as a reference
• Step sizes are small enough to be in the asymptotic regime
• The function is smooth so the theoretical order applies

Reading and reporting the results

Report a table of errors per h and the measured convergence order (about 1 for Euler and 4 for RK4). Explain the trade-off between cost per step and accuracy.

Example MATLAB code

A short starting script with this project's variable names. The delivered version is adapted to your data and fully commented.

f = @(t, y) -2*y + t;
exact = @(t) 1.25*exp(-2*t) + t/2 - 0.25;
T = 2;
time_step_size = [0.2 0.1 0.05 0.025 0.0125];             % step sizes
err = zeros(2, numel(time_step_size));

for k = 1:numel(time_step_size)
    h = time_step_size(k);
    n = round(T / h);
    ye = 1; yr = 1; t = 0;
    for i = 1:n
        ye = ye + h * f(t, ye);                    % Euler
        k1 = f(t, yr);
        k2 = f(t + h/2, yr + h/2*k1);
        k3 = f(t + h/2, yr + h/2*k2);
        k4 = f(t + h, yr + h*k3);
        yr = yr + h/6 * (k1 + 2*k2 + 2*k3 + k4);   % RK4
        t = t + h;
    end
    err(1, k) = abs(ye - exact(T));
    err(2, k) = abs(yr - exact(T));
end
robot_position = err;

pe = polyfit(log(time_step_size), log(err(1,:)), 1);
pr = polyfit(log(time_step_size), log(err(2,:)), 1);
fprintf('Euler order: %.2f, RK4 order: %.2f\n', pe(1), pr(1));
disp(array2table(err', 'VariableNames', {'Euler', 'RK4'}, 'RowNames', string(time_step_size)));

figure;
loglog(time_step_size, err(1,:), 'o-', time_step_size, err(2,:), 's-', 'LineWidth', 1.2);
legend('Euler', 'RK4'); xlabel('Step size h'); ylabel('Absolute error'); grid on;

Packages and prices

Choose how much help you need. Each higher package includes everything in the lower ones. Prices exclude 11% VAT.

• Payment: 50% down payment before work starts, balance after the deliverables are handed over
• Tax: Prices exclude 11% VAT
• Data and academic integrity: Results are computed from your own research data; you write and stand behind your manuscript, we help with the analysis and your understanding of it
• Revisions: Up to 2 revisions are included in the price (Package 3: 3 rounds)

Frequently asked questions