Solving the Euler Angle Singularity Problem Using Vector or Quaternion Algebra, without Algorithm Switching

Author(s):

Mark Bundy

Abstract:

This report reviews the mathematical origins of, and gives context to, what is traditionally called the “Euler angle singularity problem,” which arises when seeking to track or simulate the motion of a rotating body relative to an earth-fixed frame of reference, based on input from body-fixed angular rate sensors. The singularity problem centers around division by zero in the solution for the earth-referenced angular rotation rates of the body when one of the Euler tracking angles reaches ±𝜋𝜋/2. The novelty of this report is the demonstration of a methodology for solving the singularity problem for any possible Euler angle 1) without incurring the infinity-causing, division-by-zero condition, and 2) without using quaternion algebra or changing the angle-axes sequencing (a.k.a. algorithm switching). The solution centers around making the iteration time step for solving the nonlinear kinematic equations of motion proportional to, and thus canceling out, the factor that would otherwise produce division by zero in the angular rate equations. Numerical simulations are used to validate this Euler-angle, vector algebra-based solution, versus solutions formulated using 1) quaternion algebra, 2) the vector angular velocity cross product equation (introduced in physics texts), and 3) algorithm switching. Moreover, the report shows how the various formulations, vector, quaternion, and algorithm switching are derived and relatable to each other, explaining what often appears to be plus or minus sign differences in referenced equations, which can be disconcerting to “students” of the subject. A discussion of computational efficiency is also included, with numerical simulations showing that it may be possible, depending on how trigonometric functions and power laws are evaluated (e.g., using memory-based look-up tables), for vector algebra solutions to be faster than quaternion algebra solutions. An overview of the highlights and findings is provided in the summary and conclusions.