Improving the 1D search algorithm in IK-Geo to achieve high precision and success rate. Testing is performed on the Motoman SIA50D 7-DOF robot arm parameterized by conventional SEW angle with shoulder placed on joint 2.

The goal was to achieve 100% empirical success rate over 10,000 random test cases (random joint angles) while minimizing execution time based on the following milestones:

• 20 Hz, 50 ms – GUI display
• 100 Hz, 10 ms – Mid-range control loop
• 500 Hz, 2 ms – Real-time control loop (stretch goal)

Testing was performed using MATLAB MEX on a computer running Windows 10 on an Intel Core i7-3770K CPU at 3.50 GHz and 16 GB memory.

With just a few improvements, 1D search performance was significantly improved. Success rate went from 89% to 100%, and computation time was better than the fastest milestone.

The following improvements were made:

• Search over only 2 of 4 error function branches. Due to robot symmetry, the four error function branches come in two identical pairs.
• Use least-squares solutions. This guarantees the error function is always defined, meaning the bracketing method does not fail near endpoints.
• Search not just for zeros but also for minima / maxima. This improves performance near robot singularities where the error function touches but only barely passes zero.

There are still a number of strategies that may improve 1D search performance even further:

• Use multiple unique error functions
• Hard-code kinematic parameters into code
  • Or hard code error function with tan⁡(𝜃/2)
• Use intelligent global optimization algorithms (not uniform sampling)
• Search for endpoints
  • Or find endpoints analytically