Model Predictive Control of Trajectory Tracking for a Differential-Drive Robot with Actuator Limits

Project info

• Field of studyRobotics and Mechatronics
• LevelDoctoral dissertation (S3)
• Method typeControl Systems
• Estimated build time± 7 working days
• Toolsquadprog, optimoptions (Optimization Toolbox); c2d (Control System Toolbox) opsional
• PriceUS$34 - US$209

Research goal

Design and simulate MPC so that the lateral tracking error tracks its target through the linear velocity, accounting for the angular velocity.

Variables

Example result

• 12Prediction horizon
• 0.211Tracking RMSE
• 1.099IAE
• 1.858Max |u|
• 1.429Saturated steps (%)
• 4.212Overshoot (%)

MPC with a 12-step horizon keeps Lateral tracking error tracking the reference at RMSE 0.211 and 4.2% overshoot while honouring the Linear velocity limits (1% of steps saturated); longer horizons reduce RMSE down to 0.210.

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

About Basic Model Predictive Control

Basic Model Predictive Control solves an optimisation problem at each step to find the best control sequence over a prediction horizon while respecting constraints. It suits systems with actuator or state limits that cannot be ignored.

Steps

1. 1Define the discrete state-space model
2. 2Choose the prediction horizon, weights Q and R, and control limits
3. 3Build the prediction matrices (Phi and Gamma)
4. 4Solve the quadratic program with quadprog at each step
5. 5Apply only the first control move (receding horizon)
6. 6Plot the states and control signal

Assumptions to check

• The prediction model is sufficiently accurate
• The optimisation problem is feasible at each step
• Computation time per step is shorter than the sampling period

Reading and reporting the results

Report the horizon, weights, limits, and state and control responses that satisfy the constraints. Compare with unconstrained LQR to show the added value of MPC.

Example MATLAB code

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

Ts = 0.1;
A = [1 Ts; 0 1];  B = [0.5*Ts^2; Ts];             % double integrator
nx = 2; Np = 15;
Q = diag([10 1]);  R = 0.1;  umax = 1;

Phi = zeros(nx*Np, nx);  Gam = zeros(nx*Np, Np);
for i = 1:Np
    Phi((i-1)*nx+(1:nx), :) = A^i;
    for j = 1:i
        Gam((i-1)*nx+(1:nx), j) = A^(i-j) * B;
    end
end
Qb = kron(eye(Np), Q);  Rb = R * eye(Np);
H = 2 * (Gam' * Qb * Gam + Rb);  H = (H + H') / 2;

Nsim = 80;
linear_velocity = [2; 0];                                  % initial state
X = zeros(nx, Nsim+1);  X(:,1) = linear_velocity;  U = zeros(1, Nsim);
opts = optimoptions('quadprog', 'Display', 'off');
for k = 1:Nsim
    f = 2 * Gam' * Qb * Phi * X(:,k);
    u = quadprog(H, f, [], [], [], [], -umax*ones(Np,1), umax*ones(Np,1), [], opts);
    U(k) = u(1);                                  % receding horizon
    X(:,k+1) = A * X(:,k) + B * U(k);
end
lateral_tracking_error = X(1,:);                                   % position trajectory
fprintf('Final position: %.4f, max |u|: %.3f\n', lateral_tracking_error(end), max(abs(U)));

t = (0:Nsim) * Ts;
figure;
subplot(2,1,1); plot(t, lateral_tracking_error); ylabel('Position'); grid on;
subplot(2,1,2); stairs(t(1:end-1), U); ylabel('Control u'); xlabel('Time (s)'); 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