nonlinear model predictive controller for a franka emika panda robot in matlab

Nonlinear model predictive controllers are widely used in robotics to control the movement of robots such as the Franka Emika Panda robot. Here's an overview of how to implement an NMPC for controlling a Franka Emika Panda robot in MATLAB.

1. First, define the dynamics of the robot using the dynamic equations of the Panda robot and write it in state-space form using the state equation:

main.m
xdot = f(x,u);
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and the output equation:

main.m
y = g(x,u);
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2. Next, linearize the state and output equations around the current operating point of the system. This will allow you to approximate the nonlinear system with a linear system which makes it easier to design a controller. You can use the linearize function in MATLAB to perform the linearization.

3. Once you have obtained the linearized state and output equations, you can design a linear controller using techniques such as linear quadratic regulator (LQR) or pole placement. You can use MATLAB's Control System Toolbox to design the controller.

4. After designing the controller, implement the NMPC algorithm. The basic idea behind the NMPC is to solve an optimization problem at every time step to find the optimal control input based on the current state of the system.

5. The NMPC algorithm involves formulating an objective function that includes a cost function and constraints. The cost function is a measure of how well the system is tracking the desired trajectory, while the constraints are limits on the state and control inputs of the system.

6. Finally, use an optimization solver such as the nonlinear programming solver 'fmincon' to solve the optimization problem and find the optimal control inputs for the next time step.

Here's an example code that shows how to implement an NMPC for a Franka Emika Panda robot in MATLAB:

main.m
% Define the state-space dynamics of the robot
A = [0 1 0 0; 0 0 -K2/m2 0; 0 0 0 1; 0 0 -K1*K2/(m1*m2) -D/(m1*m2)];
B = [0; 0; 0; K1/(m1*m2)];
C = [1 0 0 0; 0 0 1 0];

% Linearize the system
x0 = [0; 0; pi/4; 0];
u0 = 0;
[Alin, Blin, Clin, Dlin] = linearize(@pendulum_cart_dynamics, x0, u0);

% Design the LQR controller
Q = diag([10 1 10 1]);
R = 0.1;
[K, S, E] = lqr(Alin, Blin, Q, R);

% Implement the NMPC algorithm
N = 30; % prediction horizon
T = 0.01; % sampling time
x = x0; % initial state
for i = 1:500
    % Define the optimization problem
    cost = @(u) nmpc_cost_function(u, x, K, T, N);
    nonlcon = @(u) nmpc_constraint_function(u, x, T, N);
    lb = -10;
    ub = 10;
    u0 = zeros(N,1); % initial guess for optimal control input
    options = optimoptions('fmincon','OptimalityTolerance',1e-5);

    % Solve the optimization problem
    [u, fval] = fmincon(cost, u0, [], [], [], [], lb, ub, nonlcon, options);
    fprintf('Iteration %d: fval = %f\n', i, fval)

    % Apply the optimal control input to the system
    x = pendulum_cart_dynamics(x, K(1,:), u(1), T);
end
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