Robot Calibration Without Scaling
M.S. Thesis — Mechanical Engineering — Texas A&M University — May 1995
Author: Thomas W. Ives
Chair of Advisory Committee: Dr. Louis J. Everett
Prior degree: B.S., The University of Texas at Austin
Abstract
Many engineering problems are studied through numerical schemes that use large matrices to mathematically describe the systems being analyzed. Some of the problems require iterative algorithms. In many cases the matrices used in these numerical schemes are found to be “ill-conditioned”, or close to singularity. The amount of ill-conditioning is measured with a condition number. Ill-conditioned matrices significantly slow, and usually prevent, convergence toward a solution when using iterative methods. Scaling is a common way of improving the condition number for a matrix. Researchers in other fields have developed specific methods of scaling matrices to improve the condition number. However, robotics researchers have not specifically addressed, in robotics literature, their approaches to improving the [conditioning of these matrices…]
Full abstract continues in the thesis PDF.
What this thesis is about
When robots are taught a task — picking parts from a fixture, welding a seam, applying paint — they're taught by being moved through the sequence and storing the positions in their program. That's how a robot's behavior gets calibrated to the physical workspace. But over time the calibration drifts: parts wear, fixtures move, the workspace shifts. Re-teaching every program by hand is expensive.
The mathematical machinery for re-calibrating a robot's model after these shifts depends on solving systems of equations — large matrices that describe the relationships between joint angles, link lengths, tool positions, and observed measurements. These matrices are often ill-conditioned (close to singularity), which makes iterative solvers slow or unable to converge.
The standard fix in other fields is scaling — pre-conditioning the matrices to spread their eigenvalues more usefully. Robotics researchers had borrowed scaling techniques without fully addressing whether they were the right approach for robot calibration specifically.
This thesis addresses that gap. It develops a robot calibration methodology that does not rely on matrix scaling — by structuring the calibration equations in a way that produces well-conditioned matrices from the start, the convergence problems that scaling tries to mask are sidestepped entirely. The work was validated with both simulated data and real-robot measurements using a sensor-based measurement system Thom co-developed with his advisor Dr. Louis Everett, leading to three peer-reviewed IEEE publications.
Peer-reviewed publications from this work
- L. J. Everett, T. W. Ives. “A sensor used for measurements in the calibration of production robots.” IEEE Transactions on Robotics and Automation, 12(1): 121–125 (1996).
- L. J. Everett, T. W. Ives. “Automatic maintenance of robot programs.” IEEE Transactions on Robotics and Automation, 11(4): 603–606 (1995).
- L. J. Everett, T. W. Ives. “A Sensor Used for Measurements in the Calibration of Production Robots.” IEEE International Conference on Robotics and Automation (ICRA), vol. 1: 174–179 (1993).
What this thesis demonstrates
- Numerical methods depth — ill-conditioning, condition numbers, iterative-solver behavior, scaling alternatives. The foundations that show up in every multi-physics simulation, every optimization problem, every modern ML pipeline involving matrix factorization
- Robotics and controls expertise — kinematic models, calibration mathematics, sensor integration, real-robot validation
- Methodology-over-recipe orientation — the thesis isn't “I scaled some matrices better”; it's “I re-structured the problem so scaling wasn't needed.” That kind of root-cause-first thinking carries through to every subsequent role
- NASA Space Shuttle Robot Arm relevance — the calibration work also fed into experiments serving NASA for Space Shuttle Robot Arm operations