APPENDIX
DYNAMICS IN MOVING
COORDINATE SYSTEMS
C
This appendix derives the equations of motion of a point in a coordinate system that translates and
rotates relative to a reference coordinate system. This set of equations is applied to the vehicular an-
gular rate sensor (ARS) presented in Chapter 5. We begin with a generic syst em and demonstrate the
origin in the motion of the so-called Coriolis acceleration. The general dynamic model then is applied
to the specific configuration of the ARS.
The coordinate systems and the dynamic motion of a point in the moving coordinate system are
depicted in Fig. C.1.
The reference or “fixed” coordinate system’s denoted X,Y,Z. The moving coordinate system is
denoted x,y,z, and the origin of this system is located at vector position
R in the X,Y,Z coordinates.
The moving coordinate system is translating at velocity
_
R and rotating about an axis at angular velocity
vector denoted
ω. The direction of ω is the axis about which the x,y,z coordinate system is rotating, and
the rate of rotation is given by
ω
kk
The point whose motion is being modeled, which is denoted P in Fig. C.1, is located at vector position
p
in the x,y,z coordinate system. This same point is located at vector position
r in the X,Y,Z coordinate
system. By vector addition,
r is given by
r ¼
R +
p
Fig. C.1 also depicted unit vectors
^
x,
^
y,^z in the moving coord inate system and unit vectors
^
X,
^
Y,
^
Z in the
fixed coordinate system such that
r is given by
r ¼ X
^
X + Y
^
Y + Z
^
Z + x
^
x + y
^
y + z^z
The calculation of the motion is done via time derivatives of the variables that are denoted by an over
dot. The vector velocity of the point P is the time derivative of
r, which is given by
_
r ¼
_
R +
_
x
^
x +
_
y
^
y + _z^z + x
_
^
x + y
_
^
y + z
_
^z
The time derivatives of the unit vectors are nonzero because the moving coordina te system has a ro-
tation expressed by
ω. These time derivatives are related to the angular velocity vector ω as given by
_
^
x ¼
ω
^
x
_
^
y ¼
ω
^
y
_
^z ¼
ω ^z
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