Linear Quaternion Differential Equations: Basic Theory and Fundamental Results

Abstract

Quaternion‐valued differential equations (QDEs) are a new kind of differential equations which have many applications in physics and life sciences. The largest difference between QDEs and ordinary differential equations (ODEs) is the algebraic structure. Due to the noncommutativity of the quaternion algebra, the set of all the solutions to the linear homogenous QDEs is completely different from ODEs. It is actually a right‐free module, not a linear vector space. This paper establishes a systematic frame work for the theory of linear QDEs, which can be applied to quantum mechanics, fluid mechanics, Frenet frame in differential geometry, kinematic modeling, attitude dynamics, Kalman filter design, spatial rigid body dynamics, etc. We prove that the set of all the solutions to the linear homogenous QDEs is actually a right‐free module, not a linear vector space. On the noncommutativity of the quaternion algebra, many concepts and properties for the ODEs cannot be used. They should be redefined accordingly. A definition of Wronskian is introduced under the framework of quaternions which is different from standard one in the ODEs. Liouville formula for QDEs is given. Also, it is necessary to treat the eigenvalue problems with left and right sides, accordingly. Upon these, we studied the solutions to the linear QDEs. Furthermore, we present two algorithms to evaluate the fundamental matrix. Some concrete examples are given to show the feasibility of the obtained algorithms. Finally, a conclusion and discussion ends the paper.

Laplace transform: a new approach in solving linear quaternion differential equations

“…which can be written in terms of the piecewise functions as follows: Because of the non-commutative property of quaternion, there is another kind of QDEs that has two-sided coefficients. For the QDEs with two-sided coefficients, the methods in literature [18][19][20] become invalid, while LTM shows its powerful ability to solve the problem. Using the approach given by Remark 4.1, we can get…”

Floquet Theory for Quaternion-Valued Differential Equations

“…Proof. Let M (t) be a fundamental matrix of (4.1), by Liouville's formula of QDEs (see [14]), we have…”

“…theorem of algebra, Vieta's formulas of quaternions, it is difficult to solve QDEs. In [13,14,15,16], the authors proposed several new methods to construct the fundamental matrices of linear QDEs.As a generalization, QDEs have many properties similar to ODEs. At the same time, for the relatively complicated algebraic structure of quaternion, one may encounter various new difficulties when studying QDEs.1.…”

“…Recently, the Floquet theory has been extensively explored for dynamic systems on time scales (see e. g. [31,32,33,34]). As a continuation of [13,14,15], we generalize the Floquet theory to QDEs in this paper. Specifically, the contributions of this paper are summarized as follows.1.…”

A Modified Cartesian Space DMPs Model for Robot Motion Generation

“…In fact, the Eq. 22 is a quaternions differential equation (QDE), the solution space is mathematically right free module, and the joint space differential equation of the solution space is actually linear vector space [14]. The algebraic structure of the solutions to the QDEs is completely different from ODEs, since the non-commutativity of the quaternion algebra.…”