Error Analysis of Euler Angle Transformations

Abstract

Error analyses of Euler angle transformations arise in the design of precision pointing systems, guidance systems and other systems containing gimbals. The generalized problem, including nonorthogonality of nominally orthogonal coordinate axes as well as errors in the Euler angles, is treated. The usual approach by tedious matrix techniques is simplified to yield a vector solution by application of a similarity transformation to the skew- symmetric error matrices. A vector solution is obtained directly by invoking the vector property of infinitesimal rotations. Piograms, symbolic representations of Euler angle transformations, are used to formulate the problem and to develop the vector solution. EQUENCES of angular rotations are universally used in the analysis of rigid-body dynamics. Examples can be widely found in analysis of precision pointing systems and aircraft, missiles, and space vehicles. As shown by Euler, a minimum of three rotations are required to specify the relative orientation of two orthogonal coordinate systems with arbitrary attitudes. Any physical realization of coordinate transformations, as with gimbals, for example, will introduce angular errors so that each coordinate axis is perturbed from its idealized position. The most general error analysis of an Euler angle sequence must include the effects of nonorthogonal ity of the nominally orthogonal coordinate axes as well as errors in the angles of rotation. An example problem is presented to illustrate the various approaches available for error analysis of an Euler angle sequence. The usual but tedious solution by matrix techniques is given, A vector solution is obtained by applying a similarity transformation to the matrix solution. In a second approach a vector solution to the problem is obtained directly by invoking the vector property of infinitesimal rotations. Piograms, symbolic representations of rotational coordinate transformations,1'2 are used to define the problem and are shown to give the vector solution in a particularly compact way. Three different Euler angle sequences are commonly used in the literature. One of these three is used as the example problem in this paper. The method of solution presented in the paper is applicable to any of the twelve possible three-angle Euler sequences. The method is applicable to rotational sequences of any length by an obvious extension of the technique.