The Quaternion Solution CORRECTION METHOD

2.4 The Quaternion Solution

The quaternion is the useful element in the four-dimensional space, which defines a non-commutative multiplication, i.e., a quaternion q can be considered as a four-dimensional vector, also can be considered as ,   ,  is a real number equals to  , similar to the real part of plural,  is a vector of 1 2 3 , , T    , similar to the imaginary part of plural. The multiplication of two quaternions is defined as: , q q               4 The relationship between the bias matrix R and the unit quaternion 1 2 3 , , , T q      is: 2 2 2 2 1 2 3 1 2 3 1 3 2 2 2 2 2 1 2 3 1 2 3 2 3 0 1 2 2 2 2 1 3 2 2 3 0 1 1 2 3 2 2 2 2 2 2 R                                                               5 Product of the matrix R and the vector v is Rv q v q    6 Then we have 2 2 1 2 2 3 2 2 1 2 3 4 1 1 n n i i i i i i i i n i i i i n n n T T i i i i i i i i i i i n T i i i i T T i f R w RU U w q U q U v w q U U q V v q v f R w q U U q w Aq w q A Aq A w A A q Aq q q q                                                            7 If there is more than one control point solvingequation , you need to introduce the method of least squares, take the appropriate R with minimum constraints: 2 2 2 n n i i i i i i i i n i i i i f R w RU U w q U q U w q U U q                 8 By definition: i i i q U U q Aq     9 i A is a 4 4 antisymmetric matrix: T i i i i i i i U U A U U U U             10 Then, the smallest solution of f R becomes constrained minimization with 1  q : 2 2 n n i i i i i i i n T T T i i i i f R w q U U q w Aq w q A Aq q Aq              11 Where A n T i i i i A w A A    is the symmetric matrix . Using the method of Lagrange factor,the minimization problems solutions can be written as: 1 T T q Aq q q q       12 Acording to the eigenvectors and eigenvalues of the matrix formula as: Aq q   ,then  is an eigenvalue of A , and q is the corresponding eigenvector. But A is a four-order square matrix, so there are four possible solutions. Assuming four eigenvalues of the matrix as 1  , 2  , 3  , 4  with the following sorting : 1 2 3 4        13 if i    ˄ i=1,2,3,4 ˅ ,then the solution of following formula is the A ’s smallest unit eigenvector of the corresponding eigenvalue. 1 T T T i i i i q Aq q q q Aq q           14 Then we can construct the quaternion rotation matrix with the eigenvector,which is the final bias matrix.

3. EXPERIMENTS