Appendix.

Nomenclature and Definition

Symbol	Definition
{W}	World frame (or fixed frame)
{B}	Body frame (or moving frame)
θ	Orientation of the stage
$\stackrel{.}{\theta }$	Angular velocity of the stage
Sθ	sinθ
Cθ	cosθ
Tθ	tanθ
${\mathbf{R}}_B^W$	Rotation transformation matrix from {B} to {W}
${\stackrel{.}{\mathbf{R}}}_B^W$	Time derivative of ${\mathbf{R}}_B^W$
O	Origin of {B}	Ω
P	Idealized point of contact of PZT2 with the stage
Q	Idealized point of contact of PZT1 with the stage
M	Midpoint of $\overline{PQ}$
A	CoR of the stage
C	CoM of the stage
ΩW	Position vector of Ω in {W} $\left({[x_{\Omega }^W,y_{\Omega }^W,z_{\Omega }^W]}^T\right)$
${\stackrel{.}{\Omega }}^W$	Velocity vector of Ω in {W} $\left({[{\stackrel{.}{x}}_{\Omega }^W,{\stackrel{.}{y}}_{\Omega }^W,{\stackrel{.}{z}}_{\Omega }^W]}^T\right)$
ΩB	Position vector of Ω in {B} $\left({[x_{\Omega }^B,y_{\Omega }^B,z_{\Omega }^B]}^T\right)$
${\stackrel{.}{\Omega }}^B$	Velocity vector of Ω in {B} $\left({[{\stackrel{.}{x}}_{\Omega }^B,{\stackrel{.}{y}}_{\Omega }^B,{\stackrel{.}{z}}_{\Omega }^B]}^T\right)$
α	Angle of ZQPO
u1	Linear input velocity by PZT1 along the xW-axis
u2	Linear input velocity by PZT2 along the yW-axis
${\mathbf{u}}_1^W$	Velocity input vector by PZT1 in {W}
${\mathbf{u}}_2^W$	Velocity input vector by PZT2 in {W}
Dx	x-coordinate of the geometric center of the stage in {W}
Dy	y-coordinate of the geometric center of the stage in {W}
IC	Moment of inertia of the stage at the CoM
m	Mass of the stage
dw	Moment arm vector in {W}
${\mathbf{v}}_C^W$	Velocity vector of the CoM in {W}
L	Angular momentum vector of the stage in {W}
Cx	x-coordinate of the CoM of the stage in {B}
Cy	y-coordinate of the CoM of the stage in {B}
Jc	Ratio of ${\stackrel{.}{y}}_O^W$ to ${\stackrel{.}{x}}_O^W$ under the constraint of $\stackrel{.}{\theta }=0$
Jd	Value of Jc at the desired position and orientation
xd	Desired x-coordinate position of the stage in {W}
yd	Desired y-coordinate position of the stage in {W}
θd	Desired orientation of the stage
X	State vector of the stage
Ẋ	Time derivative of X
ex	Position tracking error along the xW axis
ey	Position tracking error along the yW axis
eθ	Orientation tracking error