Abstract: This paper derives various well-known strapdown inertial system attitude algorithms using a geometric viewpoint based on the Goodman-Robinson theorem. This theorem describes three-dimensional rotation kinematics of a rigid body. The attitude algorithms derived are the third-order quaternion and direction cosine matrix, the third-order quaternion modified to compute the cross-product term at twice the update rate and the computationally partitioned algorithm using the rotation vector and an associated quaternion. The distinguishing features of these algorithms are readily apparent from the unified derivations. In particular, the assumptions made in the computation of areas on a unit sphere using the Goodman-Robinson theorem illuminates the performance limitations of these algorithms and should be of use in deriving more efficient algorithms.

Abstract: Four algorithms for computing the attitude matrix in a strapdown navigation system are described. The algorithms studied are Bortz 9-direction cosine, Bortz-quaternion, Lie algebra and Cayley transform. Each algorithm has a fast cycle and a slow cycle. In the fast cycle short term attitude information is computed in a three parameter array. This information is then used to compute the attitude matrix in the slow cycle and the 3 parameter array is reinitialized at the start of the slow cycle. In the Bortz-quaternion and Lie algebra quaternion algorithms, the attitude information is resident in the quaternion and the attitude matrix then computed from the quaternion, whereas in the Bortz 9-direction and Cayley transform algorithms, the attitude matrix is directly updated. The attitude matrix computed from the Bortz 9-direction cosine algorithm is orthonormalized in a separate routine, whereas the Bortz quaternion and Lie algebra quaternion algorithms provide an orthogonal matrix which is normalized by dividing by the length of the quaternion. The attitude matrix computed from the Cayley transform algorithm is inherently orthonormal. The overall computer efficiency is considered.

Abstract: Design and performance evaluations of an onboard inertial attitude determination system for a dual-spin planetary spacecraft are presented. A quaternion integration algorithm and a rate estimator sequentially determine the scan platform's inertial attitude and rate from the drift and misalignment compensated gyro outputs. A least-squares estimator algorithm processes the star transit data from a rotor-mounted star scanner, and provides periodic updates of the scan platform's attitude as a means of correcting the drift of the quaternion integration algorithm. Computer simulated algorithm performance in the presence of nutation and sensor noise are presented.

Abstract: (1981). Equivalence of an Identity in Vector Analysis to Quaternion Associativity, and Ramifications. The American Mathematical Monthly: Vol. 88, No. 6, pp. 441-443.

Abstract: : This report explores the algebraic and measure-theoretic properties of the Euler parameter (quaternion) representation of the rotation group A family of probability densities on this group is examined, and schemes for numerical integration and estimation of satellite attitude are presented with numerical examples An effort has been made wherever possible to deal with nonlinear problems directly rather than linearizing them Several questions for further research are raised regarding convolutions of densities on the group, Brownian motion, and the statistics of spin and torque vectors considered as stochastic variables

Abstract: A quaternion manifold (or quaternion Kaehlerian manifold [10]) is defined as a Riemannian manifold whose holonomy group is a subgroup of Sp(l). Sp(m)=Sp(l)xSp(m)/{±1}. The quaternion projective space QP, its noncompact dual and the quaternion number space Q are three important examples of quaternion manifolds. It is well-known that on a quaternion manifold M, there exists a 3-dimensional vector bundle E of tensors of type (1, 1) with local cross-section of almost Hermitian structures satisfying certain conditions (see § 2 for details). A submanifold N in a quaternion manifold M is called a quaternion (respectively, totally real) submanifold if each tangent space of N is carried into itself (respectively, the normal space) by each section in E. It is known that every quaternion submanifold in any quaternion manifold is always totally geodesic. So it is more interesting to study a more general class of submanifolds than quaternion submanifolds. The main purpose of this paper is to establish the general theory of quaternion Ci?-submanifolds in a quaternion manifold which generalizes the theory of quaternion submanifolds and the theory of totally real submanifolds. It is proved in section 3 that such submanifolds are characterized by a simple equation in terms of the curvature tensor of a quaternion-space-form. In section 4 we shall study the integrability of the two natural distributions on a quaternion Ci?-submanifold. In section 5 we obtain some basic lemmas for quaternion Ci?-submamfolds. In particular, we shall obtain two fundamental lemmas which play important role in this theory. Several applications of the fundamental lemmas are given in section 6. In section 7 we study quaternion C7?-submanifolds which are foliated by totally geodesic, totally real submanifolds. In the last section we give an example of a quaternion Ci?-submanifold of an almost quaternion metric manifold on which the totally real distribution is not integrable.