As widely analyzed, the approximations made in the simple inverse kinematics solution are not accurate enough to meet the requirements for the SCU (Superconducting Undulator).

To enhance the overall accuracy of the solution, two approaches are followed:

Error function

This approach aims to improve the accuracy of the inverse kinematics (IK) by identifying an error function that models the IK errors as a function of the quadrupole orientation and position. By determining this ideal function for all degrees of freedom, we can apply corrections to the IK solution, thereby increasing its accuracy.

Concept

1. Error Mapping:

   • Map the joint space at specific positions.

   • Compute the error between the actual quadrupole position and the desired position.

2. Multidimensional Matrix:

   • For each degree of freedom (DOF), create a multidimensional matrix of size m×n×u×v×w where:

     • m is the number of sampled positions on the x-axis.

     • n is the number of sampled positions on the y-axis.

     • u, v, and ware the number of sampled positions for pitch, roll, and yaw, respectively.

3. Optimization:

   • Execute an optimization algorithm to find the coefficients of an arbitrarily defined error function that minimizes the error function's error.

Implementation Steps

1. Initialization:

   • Define simulation parameters and geometric properties.
   • Set test positions for all DOFs.

2. Simulation Loop:

   • Iterate over all combinations of test positions.
   • For each combination, solve the inverse kinematics.
   • Run the simulation model and capture the output.
   • Calculate the average positions and orientations.
   • Store the results for analysis.

3. Optimization:

   • Use the collected data to optimize the error function.
   • Apply the optimized error function to correct the IK solution.

This algorithm is repeated two times: one to find an error function that corrects for X and Y and then a final one that computes the fine error for pitch roll and yaw.