Gravity Turn Maneuver (Casadi)
This page contains a discretized version of the OCP Gravity Turn Maneuver in CasADi 2.4.2 format. The continuous model was discretized using a direct multiple shooting approach with 300 shooting nodes distributed equidistantly over a variable time horizon. The control grid is equal to the DMS grid. An initial solution is chosen by linearly interpolating between initial and terminal state.
CasADi
# ----------------------------------------------------------------
# Gravity Turn Maneuver with direct multiple shooting using CVodes
# (c) Mirko Hahn
# ----------------------------------------------------------------
from casadi import *
# Artificial model parameters
N = 300 # Number of shooting intervals
vel_eps = 1e-6 # Initial velocity (km/s)
# Vehicle parameters
m0 = 11.3 # Launch mass (t)
m1 = 1.3 # Dry mass (t)
g0 = 9.81e-3 # Gravitational acceleration at altitude zero (km/s^2)
r0 = 6.0e2 # Radius at altitude zero (km)
Isp = 300.0 # Specific impulse (s)
Fmax = 600.0e-3 # Maximum thrust (MN)
# Atmospheric parameters
cd = 0.021 # Drag coefficients
A = 1.0 # Reference area (m^2)
H = 5.6 # Scale height (km)
rho = (1.0 * 1.2230948554874) # Density at altitude zero
# Target orbit parameters
h_obj = 75 # Target altitude (km)
v_obj = 2.287 # Target velocity (km/s)
q_obj = 0.5 * pi # Target angle to vertical (rad)
# Create symbolic variables
x = SX.sym('[m, v, q, h, d]') # Vehicle state
u = SX.sym('u') # Vehicle controls
T = SX.sym('T') # Time horizon (s)
# Introduce symbolic expressions for important composite terms
Fthrust = Fmax * u
Fdrag = 0.5e3 * A * cd * rho * exp(-x[3] / H) * x[1]**2
r = x[3] + r0
g = g0 * (r0 / r)**2
vhor = x[1] * sin(x[2])
vver = x[1] * cos(x[2])
# Build symbolic expressions for ODE right hand side
mdot = -(Fmax / (Isp * g0)) * u
vdot = (Fthrust - Fdrag) / x[0] - g * cos(x[2])
hdot = vver
ddot = vhor / r
qdot = g * sin(x[2]) / x[1] - ddot
# Build the DAE function
ode = [
mdot,
vdot,
qdot,
hdot,
ddot
]
quad = u
dae = SXFunction("dae", daeIn(x=x, p=vertcat([u, T])), daeOut(ode=T*vertcat(ode), quad=T*quad))
I = Integrator("I", "cvodes", dae, {'t0': 0.0, 'tf': 1.0 / N})
# Specify upper and lower bounds as well as initial values for DAE parameters,
# states and controls
p_min = [120.0]
p_max = [600.0]
p_init = [120.0]
u_min = [0.0]
u_max = [1.0]
u_init = [0.5]
x0_min = [m0, vel_eps, 0.0, 0.0, 0.0]
x0_max = [m0, vel_eps, 0.5 * pi, 0.0, 0.0]
x0_init = [m0, vel_eps, 0.05 * pi, 0.0, 0.0]
xf_min = [m1, v_obj, q_obj, h_obj, 0.0]
xf_max = [m0, v_obj, q_obj, h_obj, inf]
xf_init = [m1, v_obj, q_obj, h_obj, 0.0]
x_min = [m1, vel_eps, 0.0, 0.0, 0.0]
x_max = [m0, inf, pi, inf, inf]
x_init = [0.5 * (m0 + m1), 0.5 * v_obj, 0.5 * q_obj, 0.5 * h_obj, 0.0]
# Useful variable block sizes
np = 1 # Number of parameters
nx = x.size1() # Number of states
nu = u.size1() # Number of controls
ns = nx + nu # Number of variables per shooting interval
# Introduce symbolic variables and disassemble them into blocks
V = MX.sym('X', N * ns + nx + np)
P = V[0]
X = [V[np+i*ns:np+i*ns+nx] for i in range(0, N+1)]
U = [V[np+i*ns+nx:np+(i+1)*ns] for i in range(0, N)]
# Nonlinear constraints and Lagrange objective
G = []
F = 0.0
# Build DMS structure
x0 = p_init + x0_init
for i in range(0, N):
Y = I({'x0': X[i], 'p': vertcat([U[i], P])})
G = G + [Y['xf'] - X[i+1]]
F = F + Y['qf']
frac = float(i+1) / N
x0 = x0 + u_init + [x0_init[i] + frac * (xf_init[i] - x0_init[i]) for i in range(0, nx)]
# Lower and upper bounds for solver
lbg = 0.0
ubg = 0.0
lbx = p_min + x0_min + u_min + (N-1) * (x_min + u_min) + xf_min
ubx = p_max + x0_max + u_max + (N-1) * (x_max + u_max) + xf_max
# Solve the problem using IPOPT
nlp = MXFunction("nlp", nlpIn(x=V), nlpOut(f=m0 - X[-1][0], g=vertcat(G)))
S = NlpSolver("S", "ipopt", nlp, {'tol': 1e-5})
r = S({
'x0' : x0,
'lbx': lbx,
'ubx': ubx,
'lbg': lbg,
'ubg': ubg
})
# Extract state sequences and parameters from result
x = r['x']
f = r['f']
T = x[0]
t = [i * (T / N) for i in range(0, N+1)]
m = x[np ::ns].get()
v = x[np+1::ns].get()
q = x[np+2::ns].get()
h = x[np+3::ns].get()
d = x[np+4::ns].get()
u = x[np+nx::ns].get() + [0.0]
Results
The solution is calculated using IPOPT (3.11.9, default settings with tolerance 1e-5, using linear solver MUMPS in sequential mode on Ubuntu 15.10 for x86-64 with Linux 4.2.0-19-generic, Intel(R) Core(TM) i7-4790K CPU @ 4.00GHz, 16 GB RAM). The objective function value is . IPOPT finds the solution in 84 iterations (77.466 seconds proc time).