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Modeling hand glider trajectory

Description
Maximize the total horizontal distance a hang glider flies subject to different configurable wind conditions.

Further explanation of this example:


Source Files

Data Files





glidert.mos

(!*********************************************************************
   Mosel NL examples
   =================
   file glidert.mos
   ````````````````

   Maximize the total horizontal distance a hang glider flies.
 
   Configure the wind conditions by setting the parameter WIND:
   WIND = 0 : no wind
   WIND = 1 : thermal uplift 250m ahead (positive in a neighborhood
          of 250m but drops to zero exponentially away from this point)
   WIND = 2 : wind shear (a rapidly diminishing head wind that may cause
          the glider to stall)

   Trapezoidal discretization.

   Based on AMPL models hang_midpt.mod, shear_midpt.mod, stableair_midpt.mod
   Source: http://www.orfe.princeton.edu/~rvdb/ampl/nlmodels/hang/ 
   Reference: 
   R. Bulirsch, E. Nerz, H.J. Pesch, and O. von Stryk, "Combining Direct and 
   Indirect Methods in Optimal Control: Range Maximization of a Hang Glider",
   in "Optimal Control: Calculus of Variations, Optimal Control Theory and 
   Numerical Methods", ed. by R. Bulirsch, A. Miele, J. Stoer, and K.H. Well 
   (1993) Birkhauser
   
   (c) 2013 Fair Issac Corporation
       author: S. Heipcke, Mar. 2013
*********************************************************************!)

    
model "glider"
  uses "mmxnlp"

  parameters
    WIND = 1              ! Wind conditions; possible values: 0, 1, 2
    N = 150               ! Number of discretization points
  end-parameters
    
  declarations
    POS_NODES = 0..N      ! Positions
    VEL_NODES = 1..N      ! Velocity measure points

    X_0,Y_0 : real        ! Initial position (x,y) of glider
    VX_0,VY_0: real       ! Initial velocity (x',y') of glider
    CL_0: real            ! Initial lift coefficient
    X_N,Y_N : real        ! n_th step position (x,y) of glider
    VX_N,VY_N: real       ! n_th step velocity (x',y') of glider
    CL_N: real            ! n_th step Lift coefficient

    CL_MIN,CL_MAX: real   ! Minimum/maximum value of lift coefficient
    UM  : real            ! Upward wind-velocity coefficient (m/sec)
    R   : real            ! Upward wind-velocity ratio (m)
    C0,K  : real          ! Drag coefficient constants
    MASS: real            ! Mass of glider & pilot
    SPAN: real            ! Wing area
    RHO : real            ! Air density
    GRAV: real            ! Acceleration due to gravity
    WEIGHT: real          ! Weight of glider & pilot (= m*g)

    dur: mpvar                          ! Total time for the flight

  ! State variables:
    x,y: array(POS_NODES) of mpvar      ! Position at time t
    vx,vy: array(POS_NODES) of mpvar    ! Velocity at time t
    vx_dot,vy_dot: array(POS_NODES) of mpvar   ! Derivatives of vx,vy

    cLift: array(POS_NODES) of mpvar    ! Lift coefficient (control variable)

  ! Auxiliary expressions
    CDrag: array(POS_NODES) of nlctr    ! Drag coefficient
    X: array(POS_NODES) of nlctr
    UpAcc: array(POS_NODES) of nlctr    ! Upward wind velocity
    HorAcc: array(POS_NODES) of nlctr   ! Horizontal air motion
    Vx,Vy: array(POS_NODES) of nlctr
    Vr: array(POS_NODES) of nlctr       ! Velocity relative to the air
    Drag: array(POS_NODES) of nlctr     ! Drag force (opposite to Vr)
    Lift: array(POS_NODES) of nlctr     ! Lift force (perpendicular to Vr)
    sin_eta: array(POS_NODES) of nlctr
    cos_eta: array(POS_NODES) of nlctr
  end-declarations

  initializations from "glider.dat"
    X_0 Y_0 VX_0 VY_0 Y_N VX_N VY_N
    UM R C0 K MASS SPAN RHO GRAV CL_MIN CL_MAX
  end-initializations
  if WIND=2 then
    Y_0 := 100; Y_N := 0; VY_N := 0;
  end-if
  
  WEIGHT := MASS*GRAV

! Start values
  forall(j in POS_NODES) do
    setinitval(vx(j), VX_0)
    setinitval(vy(j), VY_0)
  end-do

  forall(j in POS_NODES) do
    setinitval(x(j), 5000*j/N)   ! 1000*(j-1)/N)
    setinitval(y(j), -100*j/N+1000)
  end-do

  forall(j in POS_NODES)  setinitval(cLift(j), (CL_MAX+CL_MIN)/2) 

  ! Trapezoidal discretization
  forall(j in VEL_NODES)  do
    x(j) = x(j-1) + 0.5*(dur/N)*(vx(j) + vx(j-1))
    y(j) = y(j-1) + 0.5*(dur/N)*(vy(j) + vy(j-1))
    vx(j) = vx(j-1) + 0.5*(dur/N)*(vx_dot(j) + vx_dot(j-1))
    vy(j) = vy(j-1) + 0.5*(dur/N)*(vy_dot(j) + vy_dot(j-1))
    -100 <= vx(j); vx(j) <= 100
    -100 <= vy(j); vy(j) <= 100
  end-do

  ! Wind formulae
  forall(j in POS_NODES)
    if WIND=1 then              ! Upward wind velocity formula
      X(j) := (x(j)/R - 2.5)^2  
      UpAcc(j) := UM*(1-X(j))*exp(-X(j)) 
    elif WIND = 2 then          ! Horizontal wind formula
      HorAcc(j) := -5*(1+arctan((y(j)-30)/10))
    end-if

! Definition of bounds and auxiliary expressions
  forall(j in POS_NODES) do
    cLift(j) >= CL_MIN
    cLift(j) <= CL_MAX
    CDrag(j) := C0 + K*cLift(j)^2    ! The drag coeff. depends on the lift coeff.

    if WIND=1 then
      Vy(j) := vy(j) - UpAcc(j)
    else
      Vy(j) := vy(j) 
    end-if  
    if WIND=2 then
      Vx(j) := vx(j) - HorAcc(j)
    else
      Vx(j) := vx(j) 
    end-if

    Vr(j) := sqrt(Vx(j)^2 + Vy(j)^2)
    Drag(j) := 0.5*CDrag(j)*RHO*SPAN*Vr(j)^2
    Lift(j) := 0.5*cLift(j)*RHO*SPAN*Vr(j)^2
    sin_eta(j) := Vy(j)/Vr(j)
    cos_eta(j) := Vx(j)/Vr(j)

    vx_dot(j) is_free
    vy_dot(j) is_free
  end-do

  dur >= 10; dur <= 200

! Initial and final states
  x(0) = X_0
  y(0) = Y_0
  vx(0) = VX_0 - if(WIND=2, 2.5*(1+arctan((Y_0-30)/10)), 0)
  vy(0) = VY_0
  y(N) = Y_N   !or: >=
  vx(N) = VX_N   !or: >=
  vy(N) = VY_N   !or: >=

! Equations of motion: F = m*a
! (eta denotes the angle between Vr and the horizontal plane)
  forall(j in VEL_NODES) do
    newt_x(j) := vx_dot(j) = (-Lift(j)*sin_eta(j) - Drag(j)*cos_eta(j))/MASS
    newt_y(j) := vy_dot(j) = (Lift(j)*cos_eta(j) - Drag(j)*sin_eta(j) - WEIGHT)/MASS
  end-do

! Monotonicity condition
  forall(j in VEL_NODES)  x(j) >= x(j-1)

! Limit velocity to levels that are supportable for the pilot
  forall(j in POS_NODES) do
    -3 <= vx_dot(j); vx_dot(j) <= 3     
    -3 <= vy_dot(j); vy_dot(j) <= 3
  end-do 
    
! Setting up and solving the problem
  setparam("xnlp_verbose", true)
 
 ! Check the presence of Knitro; this example needs the interior point solver
  if hasfeature("XPKNTLib") then
    setparam("xnlp_solver", 1)    ! Preselect Knitro as solver
  else
    writeln("This example requires Knitro")
    exit(1)    	
  end-if
 
  maximize(x(N))

! Test whether a solution is available
  nlstat:=getparam("XNLP_STATUS")
  if nlstat<>XNLP_STATUS_LOCALLY_OPTIMAL and nlstat<>XNLP_STATUS_OPTIMAL then
    writeln("No solution found.")
    exit(0)
  end-if   

! Solution display
  forall(i in POS_NODES) 
    writeln("xy(",i,") = (", getsol(x(i)), ",", getsol(y(i)), ")")
  writeln("Max distance flown: ", getsol(x(N)))
  writeln("Total flight time : ", getsol(dur))
  

end-model

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