Real-Time Calibration of Gyro-Magnetometer Misalignment

Abstract

This letter considers a common calibration problem in robotics: determining the orientation of a three-axis gyro relative to a three-axis magnetometer. This letter first forms the calibration problem as <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$R_i X m_i = R_j X m_j$</tex-math></inline-formula> and then presents two methods to solve the equation on <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$SO(3)$</tex-math></inline-formula> . The first method is to solve for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$X$</tex-math></inline-formula> directly by regarding the equation as a homogeneous linear equation with respect to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$X$</tex-math> </inline-formula> and subsequently enforce the solution onto <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$SO(3)$ </tex-math></inline-formula> . The second approach, utilizing geometrical properties of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$SO(3)$</tex-math></inline-formula> , transforms the equation into an equivalent linear equation in the form of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$Ax = b$</tex-math></inline-formula> and invokes a recursive least-squares algorithm to solve. The proposed calibration algorithms succeed in their computational and memory efficiency and suitability for the real-time implementation on embedded systems. Both simulation and experiments on large-scale bulky unmanned aerial vehicles (UAVs) are provided to show that the proposed formulation along with the two solvers can successfully calibrate large misalignments under small and casual movements of the UAV.