Thesis - Leigh Moody.pdf - Bad Request - Cranfield University

Glossary
_ _
0.14.5 General State Vectors
The term vector is used in the context of state observers and controllers to
describe an ordered-array of (n,1) elements. The following notation is
reserved for state vectors and state observers. For a Position-Velocity-
Acceleration (PVA) tracking filter, with the obvious reduction for PV
filters,
X
: =
xxxvi
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎝
X
X
P
L
XV
L
0.15 Three Dimensional Cartesian Vector Notation
0.15.1 Euclidean Space
Euclidean space (E 3 ) is essentially a set of points each described by an
ordered triplet. Cartesian vectors in Euclidean space have all the properties
of the general n-dimensional vectors described previously, however, they
have additional structure and may represent geometrical objects. Position
triplets conform to the axioms of a vector space can be treated as vectors
and no distinction is required between the two.
0.15.2 Vector Components
All 3D Cartesian vectors have components in some Frame of Reference, i.e.
they are in a “Cartesian Frame”. The Cartesian axes can be numbered
(1,2,3) or labelled (X,Y,Z). Hence the components of a general 3D
Cartesian vector (V) are written,
V
≡
A
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎠
X Y Z
( V , V , V )
T
≡
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎝
V
V
V
( 1)
( 2 )
⎞
( ) ⎟⎟⎟⎟⎟⎟
3
As for a general n-vector, a superscript “T” is used to indicate vector
transposition thereby maintaining a clear distinction between row and
column vectors.
0.15.3 Points and Position Vectors
The position vector of a point (b) relative to point (a) is represented
geometrically by a directed line segment (arrow) with its head at (b) and its
tail at (a). The notation for this vector, in the “Cartesian Frame” (C) is
⎠