given the forward kinematics fk and the given desired end-effector location in matlab

To compute the forward kinematics (FK) for a given robot, we need to determine the position and orientation of the end-effector in the workspace, given the joint angles. This can be done using the FK transformation matrix (T_fk) which relates the end-effector frame to the base frame.

We can use the following steps to compute the FK for a given robot:

1. Define the robot model using the Robotics System Toolbox in Matlab. This includes specifying the number of joints, joint types, and joint limits.

2. Initialize the joint angles to a desired configuration.

3. Use the forwardKinematics() function to compute the FK. This function takes the robot model and the joint angles as inputs, and outputs the end-effector pose (position and orientation) in a 4x4 homogeneous transformation matrix.

4. Extract the position and orientation of the end-effector from the transformation matrix.

Here's an example code snippet that shows how to perform these steps:

main.m
% Define robot model using Robotics System Toolbox
robot = robotics.RigidBodyTree();

% Define joint parameters for the robot
joint1 = robotics.Joint('joint1', 'revolute');
joint1.setLimits(deg2rad(-180), deg2rad(180));
joint2 = robotics.Joint('joint2', 'revolute');
joint2.setLimits(deg2rad(-180), deg2rad(180));
joint3 = robotics.Joint('joint3', 'revolute');
joint3.setLimits(deg2rad(-180), deg2rad(180));

% Add joint to robot model
robot.addBody(robot.BaseName, 'joint1');
robot.addBody('joint1', 'joint2');
robot.addBody('joint2', 'joint3');

% Set the initial joint angles
q0 = [0, 0, 0];

% Compute the forward kinematics
t_fk = forwardKinematics(robot, q0);

% Extract the position and orientation of the end-effector
pos = t_fk(1:3, 4);
rot = t_fk(1:3, 1:3);
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In this example, we define a robot with three revolute joints, set the initial joint angles to zero, and compute the forward kinematics using the forwardKinematics() function. We then extract the position and orientation of the end-effector from the resulting transformation matrix.

Note that the forwardKinematics() function returns the pose of the end-effector in the robot's frame of reference. To get the pose of the end-effector in the world frame, we need to apply a transformation that takes into account the position and orientation of the robot's base.