called the fundamental equation is satisfied. Then an invariant R-module is defined for any diagram of a (virtual) knot or link. Solutions in the classic quaternion case have been found by Bartholomew, Budden and Fenn. Solutions in the generalised quaternion case have been found by Fenn in an earlier paper. These latter solutions are only partial in the case of 2×2 matrices and the aim of this ...

A new spacecraft attitude estimation approach based on the Unscented Filter is derived. For nonlinear systems the Unscented Filter uses a carefully selected set of sample points to more accurately map the probability distribution than the linearization of the standard Extended Kalman Filter, leading to faster convergence from inaccurate initial conditions in attitude estimation problems. The fi...

We establish necessary and sufficient conditions for the existence of and the expressions for the general real and complex Hermitian solutions to the classical system of quaternion matrix equationsA1X C1, XB1 C2, and A3XA3 C3. Moreover, formulas of the maximal andminimal ranks of four real matrices X1, X2, X3, and X4 in solution X X1 X2i X3j X4k to the system mentioned above are derived. As app...

This paper presents the overall mathematical model and results from pseudo linear recursive estimators of attitude and rate for a spinning spacecraft. The measurements considered are vector measurements obtained by sun-sensors, fixed head star trackers, horizon sensors, and three axis magnetometers. Two filters are proposed for estimating the attitude as well as the angular rate vector. One fil...

The general linear quaternion function of degree one is a sum of terms with quaternion coefficients on the left and right. The paper considers the canonic form of such a function, and builds on the recent work of Todd Ell, who has shown that any such function may be represented using at most four quaternion coefficients. In this paper, a new and simple method is presented for obtaining these co...

To solve the problem of base frame calibration for coordinated multi-robot system, a new method is proposed in this paper. It is carried out through a series of ‘‘handclasp’’ manipulations between two coordinated robots, then a preliminary result can be reached by the calibrating equation. After that, in order tomake sure that the calibrated rotationmatrix is orthonormal, an optimal estimation ...