transform inverse kinematics to a root-finding problem, newton method in matlab

To transform inverse kinematics to a root-finding problem, we need to define a function that computes the difference between the desired end-effector pose and the actual end-effector pose. In other words, we want to find the root of a function that takes the joint angles as input and outputs the error in end-effector position and orientation.

We can use the Newton-Raphson method to find the root of this function, which involves computing the Jacobian matrix of the function and iteratively improving our estimate of the joint angles until we converge to a solution.

Here's an example implementation of this approach in MATLAB:

main.m
function [q,error] = inverseKinematicsNewton(robot,pose,q0,maxIter,tol)

% Define the function that computes the error in end-effector pose
f = @(q) pose - robot.fk(q);

% Define the Jacobian function
J = @(q) robot.jacobian(q);

% Initialize the joint angles
q = q0;

% Iterate until convergence
for i = 1:maxIter
    % Compute the error in end-effector pose and its derivative (Jacobian)
    error = f(q);
    Jq = J(q);
    
    % Check for convergence
    if norm(error) < tol
        break
    end
    
    % Update the joint angles using the Newton-Raphson method
    dq = Jq \ error;
    q = q + dq;
end

% Check for convergence and return the joint angles and error
if norm(error) >= tol
    warning('Newton-Raphson did not converge')
end
end
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In this code, robot is a MATLAB class that defines the forward kinematics and Jacobian of the robot, pose is the desired end-effector pose, q0 is an initial guess for the joint angles, maxIter is the maximum number of iterations, and tol is the tolerance for convergence. The function returns the joint angles q that correspond to the desired end-effector pose, as well as the final error in end-effector position and orientation.