Data processing: measuring – calibrating – or testing – Measurement system – Orientation or position
Reexamination Certificate
2000-02-03
2002-04-23
Hoff, Marc S. (Department: 2857)
Data processing: measuring, calibrating, or testing
Measurement system
Orientation or position
C702S150000, C702S152000, C702S153000, C033S321000, C701S004000, C701S124000, C244S164000, C244S171000
Reexamination Certificate
active
06377906
ABSTRACT:
This invention relates to tracking and control of tiltable bodies.
It is known to use quaternions to represent the orientation in space of an object. Quaternion notation is, in general, more computationally efficient to use than the more widely employed Euler data representation. Furthermore, quaternion notation is not subject to singularities which can occur when using Euler notation. The following U.S. patents disclose using quaternions to control, determine and/or display the orientation of an object in space: Nos. 5,875,993; 5,212,480; 4,797,836; 4,742,356; and 4,737,794.
GENERAL DISCUSSION OF QUATERNIONS
A quaternion is a four-element, hypercomplex number first conceived by Sir William Rowan Hamilton in 1843. A quaternion consists of a scalar part and a complex vector part. The vector part consists of an ordered triple (vector) of three real components which are assigned direction by three orthogonal complex unit vectors: i, j, k. An example of a general quaternion Q is shown below:
Q=q
0
+iq
1
+jq
2
+kq
3
(Eq. 1)
Addition of quaternions is performed by adding components in like directions. Multiplication is performed by noting the following products of the unit basis vectors:
i
2
=j
2
=k
2
=ijk=−1 (Eq. 2)
ij=−ji=k (Eq. 3)
jk=−kj=i (Eq. 4)
ki=−ik=j (Eq. 5)
Since the quaternion is hypercomplex, it also has a complex conjugate in which the direction of the vector part is reversed. An example is shown below:
Q*=q
0
−iq
1
−jq
2
−kq
3
(Eq. 6)
The square magnitude of the quaternion can be computed by forming the product of the quaternion with its complex conjugate as shown:
Q
2
=QQ*=q
0
2
+q
1
2
+q
2
2
+q
3
2
(Eq. 7)
A quaternion with unit magnitude (Q
2
=1) has special significance. Specifically, it acts as a two-sided rotation operator. It is worthy of note that Hamilton discovered quaternions through his efforts to develop a three dimensional extension of the rotational effect produced in the complex plane when a complex number is multiplied by a unit complex number of the form exp (i&thgr;). The rotational effect of exp (i&thgr;) results because multiplication of complex numbers requires the multiplication of their respective magnitudes and the addition of their respective phases. Since exp (i&thgr;) has unit magnitude, it can only affect the phase of the product. In the complex plane, this manifests itself as a rotation about the origin by an angle &thgr;. In trying to generalize this effect to vector rotations, Hamilton originally tried three-element hypercomplex numbers. It was not until he realized that four elements are required to account for “phase” changes in three-dimensional space that he successfully produced the desired result.
Typically, a vector rotation is accomplished using a one-sided rotation operator, R, which in three-dimensional space can be represented as a real, 3×3 orthogonal matrix. This transformation matrix rotates a vector, x, into a vector, x′, by multiplication on the left as shown:
x′=Rx where x &egr; R
3×1
and R &egr; R
3×3
(Eq. 8)
A two-sided operator must be applied using both left and right multiplication. In the case of the quaternion operator, the rotation is accomplished when a given quaternion (X) with a null scalar part (i.e. a vector) is pre- and post-multiplied by the unit quaternion and its conjugate as shown:
X′=QXQ* (Eq. 9)
The resulting vector, X′, is rotated about a general axis by a specific angle, both of which are determined by the unit quaternion, Q. If the axis of rotation is denoted by a unit vector, n, and the angle of rotation is denoted by an angle, &thgr;, then the unit quaternion components can be written as:
q
0
=cos(&thgr;/2) and q=nsin(&thgr;/2) where q=(q
1
, q
2
,q
3
) (Eq. 10)
These components satisfy the normalization condition:
1=q
0
2
+q
1
2
+q
2
2
+q
3
2
(Eq. 11)
The quaternion components thus defined are also called the Euler Parameters. These parameters contain all of the information necessary to derive the axis and angle of rotation. The rotation axis defined by unit vector, n, is also called the eigenaxis since it is the eigenvector of the one-sided rotation matrix, R, corresponding to the eigenvalue, &lgr;=+1. This occurs because the axis of rotation must be common to both the original and rotated frames and therefore must be unchanged by the rotation operator. Note that the so-called eigenaxis rotation is a single rotation about a general axis as compared to the Euler angle rotation which accomplishes the same transformation by performing three separate rotations: yaw, pitch, and roll, about the z, y, and x axes, respectively.
The quaternion complex unit vectors (i, j, k) are related to the Pauli spin matrices as shown:
i=−i&sgr;
1
(Eq. 12)
j=−i&sgr;
2
(Eq. 13)
k=−i&sgr;
3
(Eq. 13A)
where
ⅈ
=
-
1
(
Eq
.
14
)
σ
0
=
[
1
0
0
1
]
(
Eq
.
15
)
σ
1
=
[
0
1
1
0
]
(
Eq
.
16
)
σ
2
=
[
0
-
ⅈ
ⅈ
0
]
(
Eq
.
17
)
σ
3
=
[
1
0
0
-
1
]
(
Eq
.
18
)
(
See The Theory of Spinors
by E. Cartan.)
Using the definitions of the Pauli spin matrices and the unit quaternion coefficients, the unit quaternion can be written as:
Q=&sgr;
0
cos(&thgr;/2)−i(n·&sgr;)sin(&thgr;/2) (Eq. 19)
where
n·&sgr;=n
1
&sgr;
1
+n
2
&sgr;
2
+n
3
&sgr;
3
(Eq. 20)
This can also be shown to be equivalent to the following matrix exponential:
Q=e
−i&sgr;·&thgr;/2
where &thgr;=n&thgr; (Eq. 21)
Note the similarity between this form of the quaternion and the exponential form of a unit complex number discussed previously. This form suggests the three-dimensional change in “phase” that Hamilton was originally seeking. The appearance of the half angle is accounted for by the fact that Q is a two-sided transformation. Thus the left and right factors, Q and Q*, each contribute half of the desired spatial phase shift.
It is often more convenient to use the Pauli spin matrix form rather than the traditional Hamiltonian form of the quaternion. For example, a vector can be represented as a matrix by forming the inner product as shown:
X=x·&sgr;=x
1
&sgr;
1
+x
2
&sgr;
2
+x
3
&sgr;
3
(Eq. 22)
which yields:
X
=
[
x
3
x
1
-
ⅈ
x
2
x
1
+
ⅈ
x
2
-
x
3
]
(
Eq
.
23
)
This matrix form has many useful properties. For example, it can be shown that the reflection of the vector, x, through a plane defined by the unit normal, a, is easily produced using the matrix form of the vectors as shown:
X′=−AXA where A=a·&sgr; (Eq. 24)
It can also be shown that any rotation can be produced by two reflections. If the planes of reflection intersect at an angle of &thgr;/2 and the line of intersection is defined by the unit vector, n, then the resulting transformation will rotate any vector, x, about the eigenaxis, n, by an angle, &thgr;. This is illustrated below where the unit normals to the planes are vectors a and b.
X′=BAXAB (Eq. 25)
This two-sided operator, which performs a rotation, closely resembles the quaternion rotation described previously. In fact, it can be shown that Q=BA. The following multiplicative identity results from the properties of the matrix form of a vector:
AB=&sgr;
0
a·b+i(a×b)·&sgr; (Eq. 26)
In the case of a rotation by angle &thgr;,the unit normal vectors, a and b, must intersect at angle &thgr;/2. Therefore, their dot and cross products produce a·b=cos(&thgr;/2) and a×b=nsin &thgr;/2) where n is paral
Hoff Marc S.
Independence Technology, L.L.C.
Vo Hien
LandOfFree
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