Add README to SIM_aircraft.

trick_sims/SIM_aircraft/README.md

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+# Aircraft Simulation
+
+SIM_aircraft is a 2-dimensional aircraft simulation. It uses the forces of engine thrust, and aerodynamic drag, and a turning force, as described in the following pages to compute the position and velocity of an aircraft.
+
+### Simplifications
+
+The simulated aircraft moves on a map, but not in altitude. 
+
+In this simulation, we’ll assume that the aircraft always points in the direction of flight. In the aircraft body coordinates, thrust is directed in the +x direction and drag in the –x direction.
+
+Control
+In this simulation, we can manually control the aircraft by specifying:
+1) a desired speed
+2) a desired heading.
+
+or automatically.
+
+## Building the Simulation
+In the ```SIM_aircrafte``` directory, type **```trick-CP```** to build the simulation executable. When it's complete, you should see:
+
+```
+=== Simulation make complete ===
+```
+Now **cd** into ```models/graphics/``` and type **make**. This builds the graphics client for the simulation.
+
+## Running the Simulation
+In the SIM_aircraft directory:
+
+```
+% S_main_*.exe RUN_test/input.py
+```
+The Sim Control Panel, and a graphics client called "Aircraft Display" should appear.
+
+Click Start on the Trick Sim Control Panel.
+q
+![](images/GraphicsClient.png)
+
+## Dynamics Model
+
+### General Strategy for Motion
+To determine the motion of an object we generally start with Newton’s Second Law: $F = ma$ That is, force equals mass times acceleration.
+This allows us to determine the rates that effect motion, that is: acceleration and velocity. Solving for acceleration, we get : $a = F/m$. 
+
+This is the form of Newton’s Law that we generally use.
+So, if we have a force acting on a mass, we can determine its acceleration. Then, we can then numerically integrate that acceleration to get velocity and then integrate the velocity to get position.
+
+### Total Force and Acceleration
+
+To calculate acceleration, we’ll use Newton’s Second Law as usual. The force is the total force acting on the aircraft.
+
+![](images/Equation_1.png)
+
+The total force on our aircraft is the sum of:
+
+* the thrust force produced by the aircraft’s engine and
+* the drag force produced by the aircraft pushing through the atmosphere.
+* the turning force, dictated by gravity and the desired turning rate.
+
+![](images/Equation_2.png)
+
+If we affix a “Body” coordinate system to the aircraft as shown, then the thrust force will act in the +x direction, and the drag force in the –x direction. That is, thrust and drag that point in opposite directions.
+
+![](images/BodyCoordinates.png)
+
+### Drag Force
+
+The magnitude of the drag force is proportional to the square of the aircraft’s speed. We’ll call our constant of proportionality $K_{drag}$ .
+
+![](images/Equation_3.png)
+
+By “speed”, we mean the magnitude of the velocity :
+
+![](images/Equation_4.png)
+
+### Thrust Force 
+
+The magnitude of the thrust force produced by our engine can be anywhere between 0 and some value that we specify as the maximum ( $thrust_{MAX}$ ). For our aircraft. 
+
+We said in the beginning that we want to control our aircraft by setting a “desired speed”. The amount of thrust produced obviously determines the speed.
+So, how much thrust do we need to reach and maintain a desired speed?
+
+* To increase speed, we must accelerate, in which case magnitude of the thrust must be greater than that of the drag.
+* To maintain the same speed acceleration must be zero, in which case the thrust and drag magnitudes must be equal.
+* To decrease speed, we must decelerate, in which case thrust magnitude must be less than that of drag.
+
+Therefore, the thrust we require to reach and maintain a desired speed must be the same magnitude as the drag force at that desired speed.
+
+![](images/Equation_5.png)
+
+If our engine can produce the desired thrust, then we can set the throttle to produce that thrust ( $thrust_{actual}$ ) and we’ll speed up to that speed. But what if we can’t? The most thrust we can produce is $thrust_{MAX}$ , when we’re at full throttle.
+So, we must limit $thrust_{actual}$ to that which our engine can actually produce :
+
+![](images/Equation_6.png)
+![](images/Equation_7.png)
+
+So, the magnitude of the thrust force vector is $thrust_{actual}$ , and the direction is [1,0] in Body coordinates. 
+
+### Aircraft Heading
+
+If the aircraft is moving directly north, $\psi = 0$ . Directly west is $\pi/2$ . Directly east is $-\pi/2$ .
+
+![](images/AircraftHeading.png)
+
+### Heading Control
+
+To control our heading, we’d like to calculate a heading rate that is proportional to the difference between our current heading and our desired heading. 
+
+![](images/HeadingControl.png)
+
+![](images/Equation_8.png)
+
+Notice (in the figure) that $\psi_{diff}$ is in the direction that we want to turn. But, what if ( $\left| \psi_{diff} \right| > \pi$ )? That is, it’s greater than 180 degrees?
+
+Do you ever have to turn more than 𝜋 radians (180 degrees) to face any direction? No, you don’t.
+
+Let’s now define $\pis_{error}$ as the actual direction of turning.
+
+![](images/Equation_9.png)
+
+This is how we will calculate our desired heading rate:
+
+![](images/Equation_10.png)
+
+That is, our desired heading rate is proportional to the amount and direction that we need to turn.
+
+$G_{\psi}$ is a proportionality constant.
+
+### Turning Force
+
+The turning force we needed to turn the desired rate is :
+
+![](images/Equation_11.png)
+
+Like our desired speed, just we want something doesn’t mean that the aircraft is capable of it. In our case we need to limit the magnitude of our force to 1 G ( force of gravity), that is 9.8 m/s2 * m. So if our mass is 5000 kg, we need to limit our turning force to between -49000 .. 49000 Newtons.
+
+But we need that total force in world coordinates so we can move our aircraft around in the world.
+So, if our heading of our aircraft is 𝜓, what is our direction vector expressed in world coordinates? 
+
+![](images/Equation_12.png)