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139 lines
5.8 KiB
Markdown
139 lines
5.8 KiB
Markdown
# Aircraft Simulation
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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.
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### Simplifications
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The simulated aircraft moves on a map, but not in altitude.
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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.
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Control
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In this simulation, we can manually control the aircraft by specifying:
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1) a desired speed
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2) a desired heading.
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or automatically.
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## Building the Simulation
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In the ```SIM_aircrafte``` directory, type **```trick-CP```** to build the simulation executable. When it's complete, you should see:
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```
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=== Simulation make complete ===
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```
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Now **cd** into ```models/graphics/``` and type **make**. This builds the graphics client for the simulation.
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## Running the Simulation
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In the SIM_aircraft directory:
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```
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% S_main_*.exe RUN_test/input.py
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```
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The Sim Control Panel, and a graphics client called "Aircraft Display" should appear.
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Click Start on the Trick Sim Control Panel.
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q
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## Dynamics Model
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### General Strategy for Motion
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To determine the motion of an object we generally start with Newton’s Second Law: $F = ma$ That is, force equals mass times acceleration.
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This allows us to determine the rates that effect motion, that is: acceleration and velocity. Solving for acceleration, we get : $a = F/m$.
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This is the form of Newton’s Law that we generally use.
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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.
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### Total Force and Acceleration
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To calculate acceleration, we’ll use Newton’s Second Law as usual. The force is the total force acting on the aircraft.
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The total force on our aircraft is the sum of:
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* the thrust force produced by the aircraft’s engine and
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* the drag force produced by the aircraft pushing through the atmosphere.
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* the turning force, dictated by gravity and the desired turning rate.
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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.
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### Drag Force
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The magnitude of the drag force is proportional to the square of the aircraft’s speed. We’ll call our constant of proportionality $K_{drag}$ .
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By “speed”, we mean the magnitude of the velocity :
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### Thrust Force
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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.
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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.
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So, how much thrust do we need to reach and maintain a desired speed?
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* To increase speed, we must accelerate, in which case magnitude of the thrust must be greater than that of the drag.
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* To maintain the same speed acceleration must be zero, in which case the thrust and drag magnitudes must be equal.
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* To decrease speed, we must decelerate, in which case thrust magnitude must be less than that of drag.
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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.
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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.
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So, we must limit $thrust_{actual}$ to that which our engine can actually produce :
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So, the magnitude of the thrust force vector is $thrust_{actual}$ , and the direction is [1,0] in Body coordinates.
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### Aircraft Heading
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If the aircraft is moving directly north, $\psi = 0$ . Directly west is $\pi/2$ . Directly east is $-\pi/2$ .
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### Heading Control
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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.
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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?
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Do you ever have to turn more than 𝜋 radians (180 degrees) to face any direction? No, you don’t.
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Let’s now define $\pis_{error}$ as the actual direction of turning.
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This is how we will calculate our desired heading rate:
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That is, our desired heading rate is proportional to the amount and direction that we need to turn.
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$G_{\psi}$ is a proportionality constant.
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### Turning Force
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The turning force we needed to turn the desired rate is :
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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.
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But we need that total force in world coordinates so we can move our aircraft around in the world.
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So, if our heading of our aircraft is 𝜓, what is our direction vector expressed in world coordinates?
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