Subsections

• Formulation
• Performance

11 Hang Glider

Maximize the final horizontal position of a hang glider while in the presence of a thermal updraft.

Formulation

The formulation of this problem follows [9]. The equations of motion for the hang glider are

$$x'' = \frac{1}{m}(-L\sin (\eta )-D\cos (\eta )) , \qquad y'' = \frac{1}{m}(L\cos (\eta )-D\sin (\eta )) - g ,$$ (8)

where $( x, y )$ is the position of the glider, $m$ is the mass of the glider, $g$ is the gravitational constant, and the function $\eta$ is defined by

$$\sin (\eta ) = \frac{w( x,y')}{v(x,x',y')} , \qquad \cos (\eta ) = \frac{x'}{v(x,x',y')} ,$$

where

$$v(x,x',y') = \sqrt{x'^{2}+w(x,y')^{2}}, \qquad w(x,y') = y'- u (x) ,$$

$$u (x) = u_{c}(1-r(x)) \exp ( -r(x) ), \qquad r(x) = \left ( \frac{x}{r_c}-2.5 \right ) ^{2} ,$$

and constants $u_{c} = 2.5$ and $r_c = 100$. The updraft function $u$ is positive in a neighborhood of $x = 2.5 \, r_c$ but drops to zero exponentially away from $x = 2.5 \, r_c$. The functions $D$ and $L$ are defined by

$$D(x,x',y',c_{L}) = \frac{1}{2} \left ( c_{0} + c_1 c_{L}^{2}... ... L(x,x',y',c_{L}) = \frac{1}{2}c_{L} \rho S v (x,x',y') ^{2} ,$$

where $S$ is the wing area, $\rho$ is the air density, $c_{L}$ is the aerodynamic lift coefficient, and $c_{0} + c_1 c_{L}^{2}$ is the drag coefficient. For this glider

$$c_{0} = 0.034 , \quad c_1 = 0.069662 , \qquad S = 14 , \quad \rho = 1.13 .$$

The aerodynamic lift coefficient $c_L$ must satisfy the bounds

$$0 \leq c_{L} (t) \leq c_{\max} ,$$

and we also impose the natural bounds $x \ge 0$ and $x' \ge 0$. In this problem $c_{\max} = 1.4$, $m = 100$, $g = 9.81$, and the boundary conditions are

$$x(0) = 0 , \qquad y(0) = 1000 , \qquad y(t_{f}) = 900,$$

$$x'(0) = x'(t_{f}) = 13.23 , \qquad y'(0) = y'(t_{f}) = -1.288 .$$

Discretization is done with a uniform time step and the trapezoidal rule over intervals. Data for this problem is shown in Table 11.1.

Table 11.1: Hang glider problem data

Variables
Constraints
Bounds
Linear equality constraints	0
Linear inequality constraints	0
Nonlinear equality constraints
Nonlinear inequality constraints	0
Nonzeros in	0
Nonzeros in

Performance

Results for the AMPL implementation are shown in Table 11.2. For starting points we use $t_f = 1$ and the functions $x' = x'(0)$, $y' = y'(0)$, and

$$x(t) = x(0) + x'(0) \left ( \frac{t}{t_f} \right ), \qquad y... ...left ( y(t_f) - y(0) \right ) \left ( \frac{t}{t_f} \right ) ,$$

evaluated at the grid points. The initial value for the control is $c_L(t) = c_{\max}$.

MINOS fails to produce a solution for any of the problem versions we present it, declaring each an infeasible problem (or bad starting guess).

Table 11.2: Performance on hang glider problem

Solver	$n_h=50$	$n_h=100$	$n_h=200$	$n_h=400$
LANCELOT	$\ddagger$	211.76 s	693.74 s	$\ddagger$
$f$	$\ddagger$	1.25461e+03	1.24889e+03	$\ddagger$
$c$ violation	$\ddagger$	9.32090e-08	2.86060e-07	$\ddagger$
iterations	$\ddagger$	383	539	$\ddagger$
LOQO	$\ddagger$	2174.8 s	2601.83 s	$\ddagger$
$f$	$\ddagger$	1.25461e+03	1.24880e+03	$\ddagger$
$c$ violation	$\ddagger$	2.1e-11	1.6e-12	$\ddagger$
iterations	$\ddagger$	14257	7206	$\ddagger$
MINOS	28.06 s	95.8 s	206.62 s	732.06 s
$f$	2.12853e+04$\dagger$	4.55001e+05$\dagger$	7.47275e+03$\dagger$	5.93037e+03$\dagger$
$c$ violation	3.2e+03$\dagger$	6.0e+06$\dagger$	6.4e+02$\dagger$	6.1e+02$\dagger$
iterations	83	124	118	161
SNOPT	11.14 s	44.04 s	240.36 s	1268.67 s
$f$	1.28239e+03	1.25461e+03	1.24889e+03	1.24797e+03
$c$ violation	1.7e-10	1.8e-10	1.2e-11	5.1e-11
iterations	73	94	125	166
$\dagger$ Errors or warnings. $\ddagger$ Timed out.

Figure 11.1 shows the altitude and control function $c_L$ as a function of time. The glider starts at an altitude of $y(0) = 1000$ and descends until the glider meets the updraft centered at $x = 250$. As a result the glider climbs and then descends to the desired final altitude of $y(t_f) = 900$ at time $t_f = 105$.

Figure 11.1: Altitude and control $c_L$ for the hang glider problem

Figure 11.2 shows velocities $x'$ and $y'$ as a function of time. Note, in particular, the erratic behavior of the velocities while the control is in the bang-region where $c_L(t) = c_{\max}$.

Figure 11.2: Velocities $x'$ and $y'$ for the hang glider problem