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\begin{document}
\title{MIT-4031 Lab Exercise 1 pre-report:\\Inertial sensors}

\author{Group 1: \\
    Petteri Aimonen, petteri.aimonen@tut.fi \\
    Michael Braun, Michael.Braun86@gmx.de \\
    Timo Pöhö, timo.poho@tut.fi
    }

\maketitle

\begin{abstract}
\end{abstract}

\section{Introduction}
In this laboratory exercise, we study the principles and functionality of inertial sensors, namely, accelerometers and gyroscopes.
For the laboratory measurements we use \textbf{LIS3L06AL} 3-axis accelerometer\cite{lis3l06al} and \textbf{IDG300} 2-axis gyroscope\cite{idg300}.

Both sensors give an analog output, which is sampled using a 16 bit AD-converter.
The supply voltage of the sensors and the ADC reference voltage is 3~V.
The accelerometer is used on 6g mode, in which it has a sensitivity of $0.2\unit{\frac{V}{g}} \pm 10\%$.
The gyroscope has a sensitivity of $0.004\unit{\frac{V}{\degree/s}} \pm 7\%$.

\section{Theoretical background}

\subsection{Pendulums}

The laboratory coordinates are 61.4508N, 23.8579E, and elevation 150 meters. The exact gravitational acceleration at the laboratory can be calculated using Helmert's equation \cite{helmert}:
\begin{align*}
g = [& 9.80616 - 0.025928 \cos 2 \phi + (6.9 \cdot 10^{-5}) \cos^2 2 \phi \\
& - (3.086 \cdot 10^{-6})H ] \unit{m/s^2}
\end{align*}
where $\phi$ is the latitude (61.4508°) and H is the height above sea level.
In this case, the acceleration is $9.8198\unit{m/s^2}$.

In this experiment, the sensors are attached to the end of a 2 meter long pendulum.
We are primarily interested in the acceleration in the radial direction, i.e. centripetal acceleration, and the rotational speed on the oscillation plane.
The maximum occur at the middle point of the swing, where the speed is the largest.

If the pendulum is released at an angle $\theta$, the initial potential energy is
\begin{equation*}
E_p = m g (1 - \cos \theta) r
\end{equation*}
where $r$ is the length of the pendulum, 2 meters in this case.
At the middle point, all of this potential energy has become kinetic energy and the velocity is
\begin{equation*}
v = \sqrt{\frac{2E}{m}} = \sqrt{2 g (1 - \cos \theta) r}
\end{equation*}
From the velocity, we can get the centripetal acceleration $a_r$ and angular rate of rotation $\omega$:
\begin{equation*}
a_r = \frac{v^2}{r} , \quad
\omega = 360\degree \cdot \frac{v}{2 \pi r}
\end{equation*}

For $\theta = 90\degree$, we get
$v = 6.2673 \unit{m/s}$,
$a_r = 19.640 \unit{m/s^2}$ and
$\omega = 179.54 \unit{\degree/s}$.
The actual maximum reading of the accelerometer will be $a_r + g = 29.459 \unit{m/s^2}$.

Each step of the 16 bit ADC equals $3 \unit{V} / 2^{16}$.
Using the nominal sensitivities given earlier, the maximum readings are
\begin{align*}
\mathrm{accelerometer: }&\; \frac{(a_r + g)/g \cdot 0.2\unit{\frac{V}{g}}}{3 \unit{V} / 2^{16}} = 13~120 \\
\mathrm{gyroscope: }&\; \frac{\omega \cdot 0.004\unit{\frac{V}{\degree/s}}}{3 \unit{V} / 2^{16}} = 15~688
\end{align*}

Assuming that the whole setup is frictionless and the complete mass of the pendulum is concentrated in one point, i.e. at the end of the thread at a distance $r$ from the pivot point of the pendulum, the time for one double stroke can be determined in the following way:

\begin{equation*}
I \theta'' = -r F_t = -r m g \sin \theta
\end{equation*}

This is the basic Newtonian equation of motion for the pendulum, where $I$ is the moment of inertia and $F_t$ describes the accelerating tangential force as a function of the angle $\theta$. The moment of inertia of a point mass at a distance r from the axis of rotation is given as:

\begin{equation*}
I = m r^2
\end{equation*}

If we consider only small swinging angles, we can use the following simplification:

\begin{equation*}
\sin \theta \approx \theta
\end{equation*}

Thus the initial equation can be reduced to:
\begin{equation*}
\theta'' = -\frac{g}{r} \theta
\end{equation*}
Or:
\begin{equation*}
0 = \theta'' + \frac{g}{r} \theta
\end{equation*}
The equation shows the form of the standard differential equation for a free undamped oscillation. In this case the cycle duration - according to \cite{hering} - is given as:
\begin{equation*}
T = 2 \pi \sqrt{\frac{l}{g}} = 2.8356\unit{s}
\end{equation*}

\subsection{Gyroscopes}
A gyroscope is a device to measure and verify the rotation of a body due to external forces and thus makes it possible to stabilize respectively maintain the orientation of that body in space. Traditional gyroscopes take advantage of the fact that the angular momentum of a closed system is constant. A fast spinning wheel within the closed system will tend to remain in its current position. If the orientation in space of the closed system is changed due to the influence of external forces a momentum is created which will try to keep the spinning wheel in its current orientation in space by tilting the axle of the wheel perpendicular to the applied external force. Constructive  arrangements (e.g. cardanic suspension) must be taken to guarantee the necessary degrees of freedom for the gyroscope. Assuming that the rotation axle of the spinning wheel is constant, suitable measurement devices can be used to determine the motion of the closed system.

Gyroscopes are often used in navigation systems and position control for aerospace applications.

\subsubsection{Mechanical gyroscopes}
Mechanical gyroscopes nowadays are still working on the same principle as described previously. There are various types of setups and measurement systems available.  

\subsubsection{Optical gyroscopes}
Optical gyroscopes are based on a different physical effect and function principle. Two beams of light (usually emitted by a laser due to the constant wave length) are sent on paths of equal length around the same axle but in different directions. If the gyroscope is spinning around this axle, the beam which is travelling against the rotation experiences a shorter path delay than the other beam. Therefore the phasing of the two beams is different at the end of the paths and can be measured utilizing interferometry. The unequal phase shift of the beams is used to determine the rotation speed around that axle of the gyroscope. Different setups for example optical fibers or mirror systems can be used to guide the light on its ways around the axle.

\subsubsection{Vibrating structure gyroscopes}
A vibrating gyroscope consists of a vibrating element or resonator and an elastic suspension. The vibrating element is driven to oscillate. When the gyroscope is rotated around a certain axle, the Coriolis effect will cause a change in the oscillation which can be measured and thus the rate of turn can be determined. Since the piezo-electric effect is often used for the vibration excitation as well as for the detection of the changes in the vibration patterns this gyroscopes are often called \emph{piezoelectric gyroscopes}. Other implementations of vibrating gyroscopes are the \emph{Wine Glass Resonator} gyroscope, the \emph{Tuning Fork} gyroscope and the \emph{Vibrating Wheel} gyroscope which use similar function principles.   

% \section{Research methods}
% \section{Results}
% \section{Conclusions}

\vspace{2cm}

\begin{thebibliography}{9}
% \bibitem{duda}
%   Duda, Hart, Stork,
%   \emph{Pattern Classification},
%   2nd ed, 2001.
% 

\bibitem{lis3l06al}
    STMicroelectronics, \emph{LIS3L06AL datasheet}, 2006, 
    \url{http://www.st.com/internet/com/TECHNICAL_RESOURCES/TECHNICAL_LITERATURE/DATASHEET/CD00068498.pdf}

\bibitem{idg300}
    InvenSense, \emph{IDG300 datasheet}, 2008,
    \url{http://invensense.com/mems/gyro/documents/DS_IDG1004.pdf}

\bibitem{helmert}
    Simpson, D.G., \emph{Physics Recreations: Helmert's Equation}, 2009,
    \url{http://www.pgccphy.net/rec/rec002-helmert.pdf}

\bibitem{hering}
    Hering E., Martin R., Stohrer M., \emph{Physik für Ingenieure}, 2007,
    Springer Verlag Berlin Heidelberg, p. 449.

\end{thebibliography}

\end{document}


