Using the matrix exponential solution to the direction cosine differential equation

Abstract

Most computation schemes for navigator updates assume constant rotation rates for each update interval. Given this, the matrix exponential approach is a possible solution. It becomes advantageous over the quaternion approach when one needs to calculate averaged direction cosine values, averaged linear with time values, etc., over an update interval. This allows easy computation of high order Coriolis body rotational corrections. One can furthermore compute angular derivatives of these values. These also supplement the error analysis of a navigation system already simplified by the exponential form itself. Additionally, these derivatives allow a mixed low-bandwidth direction cosine update to be performed, this case happening on a low-bandwidth wireline in a wellbore environment where the accelerometers, through gravity, furnish some of the orientation information.