Why and How to Avoid the Flipped Quaternion Multiplication

TL;DR

This paper clarifies the issues caused by two different quaternion multiplication conventions in robotics and aerospace, proposing a solution to unify practices and prevent errors in attitude representation.

Contribution

It explains the problem of flipped quaternion multiplication, offers an alternative compatible with Hamilton's multiplication, and provides methods for detecting and migrating between conventions.

Findings

Identifies the source of confusion in quaternion multiplication conventions.

Proposes an alternative quaternion multiplication method.

Provides practical guidelines for convention detection and migration.

Abstract

Over the last decades quaternions have become a crucial and very successful tool for attitude representation in robotics and aerospace. However, there is a major problem that is continuously causing trouble in practice when it comes to exchanging formulas or implementations: there are two quaternion multiplications in common use, Hamilton's original multiplication and its flipped version, which is often associated with NASA's Jet Propulsion Laboratory. We believe that this particular issue is completely avoidable and only exists today due to a lack of understanding. This paper explains the underlying problem for the popular passive world to body usage of rotation quaternions, and derives an alternative solution compatible with Hamilton's multiplication. Furthermore, it argues for entirely discontinuing the flipped multiplication. Additionally, it provides recipes for efficiently detecting relevant conventions and migrating formulas or algorithms between them.