  Tolerance estimation problems are prevailing in engineering applications. For
example, in modern robotics, it remains challenging to efficiently estimate
joint tolerance, \ie the maximal allowable deviation from a reference robot
state such that safety constraints are still satisfied. This paper presented an
efficient algorithm to estimate the joint tolerance using sum-of-squares
programming. It is theoretically proved that the algorithm provides a tight
lower bound of the joint tolerance. Extensive numerical studies demonstrate
that the proposed method is computationally efficient and near optimal. The
algorithm is implemented in the JTE toolbox and is available at
\url{https://github.com/intelligent-control-lab/Sum-of-Square-Safety-Optimization}.
