  This paper details an accessible geometric derivation of the forward and
inverse kinematics of a parallel robotic linkage known as the Canfield joint,
which can be used for pointing applications. The original purpose of the
Canfield joint was to serve as a human wrist replacement, and it can be
utilized for other purposes such as the precision pointing and tracking of
antennas, telescopes, and thrusters. We build upon previous analyses, and
generalize them to include the situation where one of the three legs freezes;
the kinematics are also substantially generalized beyond failure modes,
detailed within. The core of this work states and clarifies the assumptions
necessary to analyze this type of parallel robotic linkage. Specific guidance
is included for engineering use cases.
