Improve documentation for parameter identification

controls/parameterIdentification/MeasuringBatteryDischarge.md

+# Measuring Battery Discharge Rate[^1]
+
+## Average Battery Discharge Rate
+
+When flying the quadcopter, the voltage of the attached battery decreases throughout flight,
+which has nonnegligible effects on the motor output. Because of this, we want to be able
+to form a model for how the battery discharges. The actual battery discharge pattern is nonlinear,
+but a simple linearization puts our voltage approximation significantly closer than simply assuming
+a constant voltage, so that is the approach that is currently used.
+
+## Variables
+
+- $`V_0`$ - Modeled initial battery voltage
+- $`\delta_V`$ - Average battery discharge rate
+- $`t_i`$ - Time of $`i^{\text{th}}`$ voltage measurement
+- $`V_i`$ - Voltage value of $`i^{\text{th}}`$ measurement
+- $`{\mathbf t}`$ - Column vector of all $`t_i`$
+- $`{\mathbf V}`$ - Column vector of all $`V_i`$
+
+
+## Measurement Process
+
+1. Attach a multimeter to measure voltage in parallel with the quad's battery.
+1. Set the throttle to a value where the quad is nearly (or barely) hovering (because of the multimeter wires, this is as close as we can safely get to approximating flight).
+1. Regularly record time and voltage measurement pairs $`t_i`$ and $`V_i`$ (ideally this step is automated).
+1. Use MATLAB to solve the overdetermined system $`\begin{pmatrix}{\mathbf 1} & {\mathbf t}\end{pmatrix}\begin{pmatrix}V_0 \\ \delta_V\end{pmatrix} = {\mathbf V}`$ with the script given below.
+
+## MATLAB Script
+
+The below script assumes that you have your time (as seconds) and voltage (as volts) samples ($`t`$ and $`V`$) as column vectors (if you don't, transpose them before continuing).
+
+```matlab
+% A is matrix [1 t];
+A = [ones(size(t)), t];
+
+% Least squares to solve system
+x = A \ V;
+
+% Extract results for V_0 and delta_V
+V_0     = x(1)
+delta_V = x(2)
+
+% Plot the samples and linearization
+close all; figure; hold on;
+plot(t, V, 'bo');
+plot(t, V_0 + delta_V*V);
+xlabel('Time (s)'); ylabel('Batt. Voltage (V)');
+hold off;
+```
+
+This should output the proper values for $`V_0`$ and $`\delta_V`$, as well as plot the samples against
+the linearization so you can visualize the precision of the approximation.
+
+[^1]: Adapted from Section 5.4 of Matthew Rich's graduate  thesis "Model development, system identification, and control of a quadrotor helicopter"

controls/parameterIdentification/MeasuringEquivMoment.md

+# Measuring Equivalent Moment of Inertia[^1]
+
+## Variables
+
+- $`K_V`$ - Motor back-emf constant $`\left(\frac{\text{rad}}{Vs}\right)`$
+- $`R_m`$ - Motor resistance $`(\Omega)`$
+- $`K_Q`$ - Motor torque constant $`\left(\frac{Nm}{A}\right)`$
+- $`\tilde J_r`$ - Equivalent moment of inertia $`(kg\;m^2)`$
+- $`\tau`$ - Time constant of transient rotor response $`(s)`$
+
+## Equivalent Moment of Inertia
+
+In order to determine the response to a motor speed command, we need the combined moment of inertia for the rotor and the outer motor housing (the parts that spin).
+Note that this value is for a single motor its axis of rotation, not for all four combined.
+A larger moment of inertia corresponds to the rotor taking more time to reach its target angular velocity.
+The transient response of a rotor to a step change in input command should be approximately a negative exponential decay, specifically one where
+```math
+\tau = R_mK_VK_Q\tilde J_r
+```
+
+## Measurement Options
+
+The big-picture goal here is to measure the time constant of the rotor speed transition after a step change in input command
+(which when plotted with angular velocity as function of time should look like a negative exponential decay).
+The time constant is the time it takes the angular velocity to reach $`63.2\%`$ of its final value from its initial value.
+This can then be plugged into the above equation and rearranged such that $`\tilde J_r = \frac{\tau}{R_mK_VK_Q}`$.  
+
+There are two[^1][^2] approaches that have been used in the lab in the past. The first is to record the sound of rotors spinning,
+view the spectrogram of the sound with MATLAB (the sound should transition to a higher pitch at higher rotor speeds), and measure the time constant.
+This is done by the approximation that the time that the rotor takes to reach $`95\%`$ of its final velocity (pitch) is equal to $`3\tau`$.  
+
+The other option is to continuously measure angular velocity with a photointerrupter or optical tachometer, from which data can be used to determine the
+actual time when the velocity crosses the $`63.2\%`$ threshold. Note that past time constants have been on the order of $`10^{-2}`$s, so if this approach is used,
+the setup has to be able to take sufficiently fast samples.
+
+## Procedure
+
+1. Disable gyroscope feedback and secure or arrange the quadcopter so that it will not move when thrust is applied
+1. Set up the chosen measurement device to be recording data
+1. Apply an initial (non-zero) throttle command to the quadcopter
+1. After a few seconds, step the throttle input to a larger value (there should be a fairly significant difference in the two command values)
+1. After a few more seconds, return the throttle to 0 and stop recording data
+1. Using the *Measurement Options* Section above, compute the time constant and moment of inertia
+
+[^1]: Adapted from subsection 5.5.5 of Matthew Rich's graduate  thesis "Model development, system identification, and control of a quadrotor helicopter"
+[^2]: The photointerrupter approach can be found in subsection 6.4.4 of Ian McInerney's graduate thesis "Development of a multi-agent quadrotor research platform with distributed computational capabilities"
\ No newline at end of file

controls/parameterIdentification/MeasuringMomentOfInertia.md

+# Measuring Moment of Inertia[^1]
+
+## Bifilar Pendulum
+
+A bifilar pendulum consists of a mass affixed to a pair of vertical wires.
+An angular displacement in the plane perpendicular to the wires creates an oscillation that is characterized by the moment of inertia of the mass.
+We can measure this oscillation to determine the moments of intertia of the quadcoopter
+about all three axes.
+
+## Variables
+
+- $`\theta`$ - Angular position relative to equilibrium
+- $`J`$ - Moment of inertia
+- $`D`$ - Distance between the wires
+- $`m`$ - Mass of the quadcopter
+- $`h`$ - Height of wire attachment from quadcopter center of mass
+- $`\omega_n`$ - Frequency of oscillations
+
+## Computing the Moment of Inertia
+
+1. Suspend the quadcopter from a pair of wires to form a bifilar pendulum such that the $`z`$-axis of the quad is vertical
+1. Introduce and angular displacement (around the vertical axis) to the quadcopter
+1. Repeatedly measure the angular position with the VRPN camera system over a number of oscillations
+1. Use MATLAB to compute the average frequency of the oscillations, $`\omega_n`$, and use it to compute the moment of inertia from the equation[^2] $`J_{zz} = \frac{mgD^2}{4h\omega_n^2}`$
+1. Repeat for the $`x`$- and $`y`$-axes to compute $`J_{xx}`$ and $`J_{yy}`$
+
+## Notes on Setup
+
+- The wires should set up to be parallel (i.e. vertical, so the spacing between them on the quad matches the space where they are affixed above)
+- The center of mass should be positioned halfway in between the two wires
+- The value of $`D`$ can affect the precision and stability of the measurement, so try spacing the wires differently if the measurements seem inaccurate
+- The value of $`h`$ should be maximized for best results, so it is recommend to affix the top of the wires as close to the ceiling as possible
+
+[^1]: Adapted from Section 6.3 of Ian McInerney's graduate thesis "Development of a multi-agent quadrotor research platform with distributed computational capabilities"
+
+[^2]: The derivation of this is shown in part in [this][matlab] article about computing moment of inertia measurements with MATLAB, which also contains some details regarding implementation
+
+[matlab]: https://www.mathworks.com/company/newsletters/articles/improving-mass-moment-of-inertia-measurements.html
+

controls/parameterIdentification/MeasuringThrustAndDragConstants.md

+# Measuring Thrust and Drag Constants[^1]
+
+## Introduction
+
+For proper modeling of the quadrotor we need to go through the identification process.
+This involves measuring parameters of the specific quadrotor in use, including both the thrust and drag constants.
+This document describes the setup and procedure for measuring the thrust and drag constants of a quadrotor.
+
+## Setup 
+
+The materials required are the following:
+* Function generator
+* Power supply capable of outputting ~30A
+* Digital scale
+* Optical tachometer
+* Quadrotor
+
+The first thing needed is to set the quadrotor onto the scale facing upside down.
+This directs the force generated by the quad directly onto the scale
+(note that you may need to place an object in between the quad and the scale to eliminate the possibility of a rotor hitting the scale).
+After this is done the scale needs to be zeroed out so that the only force being measured is produced by the quads thrust.
+
+Next we need to power the quad with the power supply.
+This will replace the LiPO battery so that we can accurately measure the voltage being supplied at all times.
+The voltage from the power supply should be ~12V.
+
+The next step is to connect the ESC[^2] signal lines to the function generator to produce a known PWM output.
+This PWM output has the following characteristics:
+* $`f`$ = 400 Hz Square Wave
+* $`V_{pp}`$ = 3.2 V
+* $`V_{\text{offset}}`$ = 1.6 V
+* Needs a duty cycle of 40% (1 millisecond pulse width) to initialize the ESCs
+
+In the case that the onboard ESCs have changed,
+the frequency and duty cycle required for initialization may change.
+Simply reference the ESC documentation for the compatible signal frequencies
+and choose a frequency within that range.
+From there the calculation of the duty cycle for initialization,
+denoted by $`P_{\text{min}}`$,  is as follows:
+
+```math
+P_{\text{min}} = 0.001f
+```
+
+To calculate the maximum duty cycle that the ESC can handle, denoted by $`P_{\text{max}}`$, we use the following equation:
+
+```math
+P_{\text{max}} = 0.002f
+```
+
+These ESC signal lines are connected to a ribbon cable,
+which then connects to the zybo board
+through an insulation-displacement connector (IDC).
+The function generator inputs go directly to the ESC
+through this IDC connector and ribbon cable.
+The pinout of the IDC connector is as follows:
+
+| Pin 1[^3] | Pin 3 | Pin 5 | Pin 7 | Pin  9 | Pin 11 |
+|:---------:|:-----:|:-----:|:-----:|:------:|:------:|
+| Pin 2     | Pin 4 | Pin 6 | Pin 8 | Pin 10 | Pin 12 |
+
+| Pin Number | Purpose                |
+|:----------:|:-----------------------|
+| Pin 1      | Throttle               |
+| Pin 2      | ESC 0 signal line      |
+| Pin 3      | Aileron                |
+| Pin 4      | ESC 1 signal line      |
+| Pin 5      | Elevator               |
+| Pin 6      | ESC 2 signal line      |
+| Pin 7      | Rudder                 |
+| Pin 8      | ESC 3 signal line      |
+| Pin 9      | Ground RC Receiver     |
+| Pin 10     | Ground                 |
+| Pin 11     | Not used               |
+| Pin 12     | $`V_{cc}`$ RC Receiver |
+
+After the ESC signal lines are properly connected
+and the ESCs are being properly powered by the power supply,
+we can begin collecting data.
+
+## Data Collection
+
+The first thing we need to do is increase the duty cycle of the function generator
+from the calculated initialization value until the motors begin to turn.
+We will denote this duty cycle value as $`P_i`$ for reference.
+If the rotors do not begin to turn,
+verify that the wave being sent from the function generator is correct
+and that the ESCs are being powered properly from the power supply.
+
+The data collection process is as follows:
+
+1. Increase the input duty cycle by 1% (starting at $`P_i`$) and record its current value.
+1. Using the optical tachometer, record the approximate steady state speed reached by each motor/rotor in RPM.
+1. Record the corresponding input voltage from the power supply.
+1. Record the corresponding digital scale reading in grams.
+1. Record the current draw from the power supply.[^4]
+1. Repeat 1-5 until reaching $`P_{\text{max}}`$.
+
+To utilize the later provided MATLAB scripts to analyze the collected data and calculate the thrust and drag constants, the data needs to be stored in a specific way. Firstly, Excel should be used for storing the data. The required header layout of this Excel file should be as follows:
+
+
+|Duty Cycle|Rotor 0 RPM|Rotor 1 RPM|Rotor 2 RPM|Rotor 3 RPM|Voltage|Scale Reading|Current|
+|:--------:|:---------:|:---------:|:---------:|:---------:|:-----:|:-----------:|:-----:|
+
+This is due to the fact that the functions created for calculating
+the thrust and drag constants take in data as a table data type in MATLAB.
+How these functions access the data in this table
+is through specific indexing shown below
+
+|Index 1|Index 2|Index 3|Index 4|Index 5|Index 6|Index 7|Index 8|
+|:-----:|:-----:|:-----:|:-----:|:-----:|:-----:|:-----:|:-----:|
+
+where each index value corresponds to its respective header.
+Therefore, the name of the header doesn’t matter
+as long as its associated data is in the same column as its referenced index.
+
+## MATLAB Functions and Script
+
+### Calculate Thrust Constant Function
+
+```matlab
+function [ Kt ] = calc_thrust_constant( data )
+%CALC_THRUST_CONSTANT
+%   Calculates the thrust constant (Kt) given experimental data. The thrust
+%   constant is described in detail in sections 4.2.1.1 and 5.5.1 of 
+%   "Model development, system identification, and control of a quadrotor
+%   helicopter" by Matt Rich.
+
+%   Input Arguments:
+%   data: experimental data
+
+% Convert RPM to angular speed of each rotor.
+rotor_speed_0 = data.(2) * (pi/30);
+rotor_speed_1 = data.(3) * (pi/30);
+rotor_speed_2 = data.(4) * (pi/30);
+rotor_speed_3 = data.(5) * (pi/30);
+
+% Define the A matrix as the sum of each rotors speed squared.
+A = (rotor_speed_0.^2 + rotor_speed_1.^2 + rotor_speed_2.^2 + rotor_speed_3.^2);
+
+% Convert weight (g) to thrust force (N) by converting grams to kilograms and
+% multiplying by the acceleration of gravity.
+T = (data.(7)/1000)*9.8;
+
+% Calculate the thrust constant (Kt) through the following least squares
+% approximation: Kt = (A'A)^-1.*A'.*T as defined on page 65 of "Model
+% development, system identification, and control of a quadrotor
+% helicopter" (Matt Rich's Thesis).
+Kt = ((A'*A)^(-1))*(A'*T) 
+
+end
+```
+
+### Calculate Drag Constant Function 
+
+```matlab
+function [ Kd ] = calc_drag_constant( data, Pmin, Pmax, Rm, Kv, Kq, If )
+%CALC_DRAG_CONSTANT 
+%   Calculates the drag constant (Kd) given experimental data. The drag
+%   constant is described in detail in sections 4.2.5, 4.2.6, and 5.5.4.1 
+%   of "Model development, system identification, and control of a 
+%   quadrotor helicopter" by Matt Rich.
+
+%   Input Arguments:
+%   data: experimental data
+%   Pmax: Calculated maximum duty cycle of the function generators PWM wave that the ESC can  %   handle.
+%   Pmin: Calculated minimum duty cycle of the function generators PWM wave for initialization %   of the ESC.
+%   Rm: Motor resistance
+%   Kv: Back-EMF constant of the motor
+%   Kq: Torque constant of the motor
+%   If: No-Load (friction) current of the motor
+
+
+% Convert RPM to angular speed of each rotor.
+rotor_speed_0 = data.(2) * (pi/30);
+rotor_speed_1 = data.(3) * (pi/30);
+rotor_speed_2 = data.(4) * (pi/30);
+rotor_speed_3 = data.(5) * (pi/30);
+
+% Refer to the sections described in the header of this function
+% for a better understanding of what this loop is doing.
+% u: Defined in section 4.2.5
+% omega: vector of each rotors speed in rad/s
+% A: column vector of each rotors speed squared
+% b: Defined in section 5.5.4.1
+% Kd_vector: Vector containing all experimental Kd values
+for i = 1:length(rotor_speed_1)
+    u = ((data.(1)(i)/100) - Pmin)/(Pmax - Pmin);
+    omega = [rotor_speed_0(i), rotor_speed_1(i), rotor_speed_2(i), rotor_speed_3(i)];
+    A = omega'.^2;
+    b = ((u*data.(6)(i))/(Rm*Kq))-omega'/(Rm*Kq*Kv)-If/Kq;
+    Kd_vector(i) = ((A'*A)^(-1))*(A'*b);
+end
+
+% Calculate the average of the Kd_vector to obtain the drag constant.
+Kd = sum(Kd_vector)/length(Kd_vector)
+
+end
+```
+
+Note that the function for calculating the drag constant
+is dependant upon the following motor parameters:
+
+| Parameter | Units                | Description                |
+|:---------:|:--------------------:|:---------------------------|
+| $`R_m`$   | $`\Omega`$           | Motor Resistance           |
+| $`K_V`$   | $`\frac{rad}{V\,s}`$ | Back-emf constant          |
+| $`K_Q`$   | $`\frac{A}{N\,m}`$   | Torque constant            |
+| $`i_f`$   | $`A`$                | No-Load (friction) current |
+
+### Script for Calculating both Thrust and Drag Constants
+
+```matlab
+% Import data as a table.
+data = readtable(‘file location’);
+
+% Define Pmin and Pmax, as well as the motor parameters Rm, Kv, Kq, and If
+Pmin = ‘insert calculated Pmin value here’;
+Pmax = ‘insert calculated Pmax value here’;
+Rm = ‘insert Rm value here’;
+Kv = ‘insert Kv value here’;
+Kq = ‘insert Kq value here’;
+If = ‘insert If value here’;
+
+% Call the calc_thrust_constant() function.
+Kt = calc_thrust_constant(data);
+
+% Call the calc_drag_constant() function.
+Kd = calc_drag_constant(data, Pmin, Pmax, Rm, Kv, Kq, If);
+```
+
+[^1]: This document was written in October 2016 by Andy Snawerdt, Tara Mina, and Brendan Bartels.
+
+[^2]: This document's specifications references the current ESCs onboard, which are the [DJI 30A opto](https://www.dji.com/en/index.php/ESC).
+
+[^3]: Pin 1 is specified by a trinagle on the body of the IDC connector.
+
+[^4]: This isn’t actually necessary for the thrust or drag constant calculations, however this information may be useful later on for power requirements of the rotors.

controls/parameterIdentification/ModelingParameters.md

+# Modeling Parameters
+
+This table was moved from a Word document that was last pushed to Git on Feb 5, 2017.
+If the date listed for "Last Updated" is this, then the actual update date may be before that.
+
+If the same procedure or document is used for multiple consecutive parameters,
+it is given for the first and listed as "same" for subsequent rows. 
+
+| Symbol                   | Nominal Value           | Units                     | Brief Description                                             | Last Updated | Measurement Procedure |
+|:------------------------:|:-----------------------:|:-------------------------:|:--------------------------------------------------------------|:-------------|:----------------------|
+| $`m_q`$                  | $`0.986`$               | $`kg`$                    | Quadcopter mass                                               | Feb 5, 2017  | Measured with a scale |
+| $`m_b`$                  | $`0.204`$               | $`kg`$                    | Battery mass                                                  | Feb 5, 2017  | same                  |
+| $`m`$                    | $`1.19`$                | $`kg`$                    | Quadcopter + battery mass                                     | Feb 5, 2017  | same                  |
+| $`g`$                    | $`9.81`$                | $`\frac{m}{s^2}`$         | Acceleration of gravity                                       | [constant]   | [constant]            |
+| $`J_{xx}`$               | $`0.0218`$              | $`kg\,m^2`$               | Quadrotor + battery moment of inertia around $`b_x`$          | Feb 5, 2017  | [Measuring Moment of Inertia][3] |
+| $`J_{yy}`$               | $`0.0277`$              | $`kg\,m^2`$               | Quadrotor + battery moment of inertia around $`b_y`$          | Feb 5, 2017  | same                  |
+| $`J_{zz}`$               | $`0.0332`$              | $`kg\,m^2`$               | Quadrotor + battery moment of inertia around $`b_z`$          | Feb 5, 2017  | same                  |
+| $`J_{\text{req}}`$       | $`4.2012\times10^{-5}`$ | $`kg\,m^2`$               | Rotor + motor moment of inertia around motor axis of rotation | Feb 5, 2017  | [Measuring Equivalent Moment of Inertia][5] |
+| $`K_T`$                  | $`8.6519\times10^{-6}`$ | $`\frac{kg\,m}{rad^2}`$   | Rotor thrust constant                                         | Feb 5, 2017  | [Measuring Thrust and Drag Constants][1] |
+| $`K_d`$                  | $`1.0317\times10^{-7}`$ | $`\frac{kg\,m^2}{rad^2}`$ | Rotor drag constant                                           | Feb 5, 2017  | same                  |
+| $`K_H`$                  |                         | $`\frac{kg}{rad}`$        | Rotor in-plane drag constant                                  | Feb 5, 2017  | _Matt's thesis 5.5.3, needs doc_ |
+| $`\delta_{T_z}`$         |                         | $`\frac{kg}{rad}`$        | Rotor velocity thrust adjustment factor                       | Feb 5, 2017  | _Matt's thesis 5.5.2, needs doc_ |
+| $`\lvert r_{h_x}\rvert`$ | $`0.16`$                | $`m`$                     | $`x`$-axis distance from center of mass to a rotor hub        | Feb 5, 2017  | Measure by hand       |
+| $`\lvert r_{h_y}\rvert`$ | $`0.16`$                | $`m`$                     | $`y`$-axis distance from center of mass to a rotor hub        | Feb 5, 2017  | same                  |
+| $`\lvert r_{h_z}\rvert`$ | $`0.03`$                | $`m`$                     | $`z`$-axis distance from center of mass to a rotor hub        | Feb 5, 2017  | same                  |
+| $`R_m`$                  | $`0.2308`$              | $`\Omega`$                | Motor resistance                                              | Feb 5, 2017  | [Measuring Motor Resistance][2] |
+| $`K_Q`$                  | $`96.3422`$             | $`\frac{A}{N\,m}`$        | Motor torque constant                                         | Feb 5, 2017  | Specified by motor    |
+| $`K_V`$                  | $`96.3422`$             | $`\frac{rad}{V\,s}`$      | Motor back-emf constant                                       | Feb 5, 2017  | Specified by motor    |
+| $`i_f`$                  | $`0.511`$               | $`A`$                     | Motor internal friction (no-load) current                     | Feb 5, 2017  | Remove rotor and measure with power supply and ammeter |
+| $`P_\bot`$               | $`117000`$              | [none]                    | ESC turn-on duty cycle command                                | Feb 5, 2017  | [Measuring Thrust and Drag Constants][1] |
+| $`P_\bot`$               | $`100000`$              | [none]                    | Minimum Zybo output duty cycle command                        | Feb 5, 2017  |                       |
+| $`P_\top`$               | $`100000`$              | [none]                    | Maximum Zybo output duty cycle command                        | Feb 5, 2017  |                       |
+| $`\delta_V`$             |                         | $`\frac{V}{s}`$           | Approximate constant battery discharge rate                   | Feb 5, 2017  | [Measuring Battery Discharge Rate][4]  |
+| $`T_C`$                  | $`0.01`$                | $`s`$                     | Camera system sampling period                                 | Feb 5, 2017  |                       |
+| $`\tau_C`$               |                         | $`s`$                     | Camera system total latency                                   | Feb 5, 2017  |                       |
+
+* 0.175 ms single trip latency between camera system and ground station (ping)
+
+[1]: MeasuringThrustAndDragConstants.md
+[2]: ../controls/documentation/MeasuringMotorResistance.pdf
+[3]: MeasuringMomentOfInertia.md
+[4]: MeasuringBatteryDischarge.md
+[5]: MeasuringEquivMoment.md