# Documentation

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## Model Motion of Circling Airplane

Start with an airplane moving at 150 kmh in a circle of radius 10 km and descending at the same time at a rate of 20 m/sec. Compute the motion of the airplane from its instantaneous acceleration as an argument to the `step` method. Set the initial orientation of the platform to the identity, coinciding with the global coordinate system.

### Set up the scenario

Specify the initial position and velocity of the airplane. The airplane has a ground range of 10 km and an altitude of 20 km.

```range = 10000; alt = 20000; initPos = [cosd(60)*range;sind(60)*range;alt]; originPos = [1000,1000,0]'; originVel = [0,0,0]'; vs = 150.0; phi = atan2d(initPos(2)-originPos(2),initPos(1)-originPos(1)); phi1 = phi + 90; vx = vs*cosd(phi1); vy = vs*sind(phi1); initVel = [vx,vy,-20]'; platform = phased.Platform('MotionModel','Acceleration',... 'AccelerationSource','Input port','InitialPosition',initPos,... 'InitialVelocity',initVel,'OrientationAxesOutputPort',true,... 'InitialOrientationAxes',eye(3)); relPos = initPos - originPos; relVel = initVel - originVel; rel2Pos = [relPos(1),relPos(2),0]'; rel2Vel = [relVel(1),relVel(2),0]'; r = sqrt(rel2Pos'*rel2Pos); accelmag = vs^2/r; unitvec = rel2Pos/r; accel = -accelmag*unitvec; T = 0.5; N = 1000;```

### Compute the trajectory

Specify the acceleration of an object moving in a circle in the x-y plane. The acceleration is v^2/r towards the origin.

```posmat = zeros(3,N); r1 = zeros(N); v = zeros(N); for n = 1:N [pos,vel,oax] = platform(T,accel); posmat(:,n) = pos; vel2 = vel(1)^2 + vel(2)^2; v(n) = sqrt(vel2); relPos = pos - originPos; rel2Pos = [relPos(1),relPos(2),0]'; r = sqrt(rel2Pos'*rel2Pos); r1(n) = r; accelmag = vel2/r; accelmag = vs^2/r; unitvec = rel2Pos/r; accel = -accelmag*unitvec; end```

Display the final orientation of the local coordinate system.

`disp(oax)`
``` -0.3658 -0.9307 -0.0001 0.9307 -0.3658 -0.0010 0.0009 -0.0005 1.0000 ```

### Plot the trajectory and the origin position

```posmat = posmat/1000; figure(1) plot3(posmat(1,:),posmat(2,:),posmat(3,:),'b.') hold on plot3(originPos(1)/1000,originPos(2)/1000,originPos(3)/1000,'ro') xlabel('X (km)') ylabel('Y (km)') zlabel('Z (km)') grid hold off```