Ballistic Flight Trajectory Optimization

Problem Description

Considering a two-dimensional longitudinal profile missile trajectory optimization problem, the ballistic dynamic equation can be simplified as follows:

$\begin{align} m \dot{V}&= -F_l + F_y \sin \alpha - m g \sin \gamma \label{1.1} \tag{1.1} \\ m V \dot{\gamma}&= -F_y \cos \alpha - m g \cos \gamma \label{1.2} \tag{1.2} \\ \dot{l}&= V \cos \gamma \label{1.3} \tag{1.3} \\ \dot{y}&= V \sin \gamma \label{1.4} \tag{1.4} \end{align}$

Among them $F_l = Q S C_{x_0}, F_y = - Q S C_{N\alpha} \alpha$ are lift and drag, and coefficients $C_{x_0}, C_{N\alpha}$ related to aerodynamic environment.

气动参数

参数	$m$	$L$	$D$	$S$	$C_{x_0}$	$C_{N\alpha}$
值	497 (kg)	7.6 (m)	0.3(m)	0.0707( $m^2$ )	0.53	11.78

Trajectory Optimization

The optimization objective comprehensively considers overload (least squares) integration and maximizing end velocity, and the OCP model is defined as:

$\begin{array}{cll} \underset{V(\cdot), \gamma(\cdot), x(\cdot), z(\cdot)}{\operatorname{minimize}} & -V(t_f) + \int_{0}^{T} \ell(a(t)) \mathrm{d} t \\ \text { subject to } & \text{Eq.} (1). & \\ & V(0)=\bar{V}_{0}, \gamma(0)=\bar{\gamma}_{0}, x(0)=\bar{x}_{0}, z(0)=\bar{z}_{0} \\ & \gamma(t_f)=\bar{\gamma}_{t_f}, z(t_f)=\bar{z}_{t_f} \end{array}$

Modeling

Python

NN = 100
prob = multi_stage_problem('ballistic_flight', NN + 1)

""" Provide Auxiliary Data for Problem """
D = 0.3
S = 0.25 * pi * D ** 2 # Vehicle Reference Area (mˆ2)
rho0 = 1.225570827014494 # Sea Level Atmospheric Density (kg/mˆ3)
beta = 1.208e-4
mass0 = 812.0
m_pulse = 333.0
mass = mass0 - m_pulse # Vehicle Mass (kg)
Cx0_s = 0.53
CN_alpha_s = 11.78

""" Boundary Conditions """
speed0, fpa0, range0, alt0 = prob.parameters(['speed0', 'fpa0', 'range0', 'alt0'],  stage_dependent=False)
fpaf, rangef, altf = prob.parameters(['fpaf', 'rangef', 'altf'],  stage_dependent=False)
tfWeight, vfWeight, uWeight = prob.parameters(['tfWeight', 'vfWeight', 'uWeight'],  stage_dependent=False)

""" Limits on Variables """
tfMin, tfMax = 40.0, 880.0
rangeMin, rangeMax = 0, 200000
altMin, altMax = -40000.0, 0.0
speedMin, speedMax = 250.0, 2500.0
anMin, anMax = -20.0, 20.0
fpaMin, fpaMax = -90.0 * pi / 180.0, 90.0 * pi / 180
aoaMin, aoaMax = -10.0 * pi / 180.0, 10.0 * pi / 180.0


V = prob.variable('V', hard_lowerbound=speedMin, hard_upperbound=speedMax)
gamma = prob.variable('gamma', hard_lowerbound=fpaMin, hard_upperbound=fpaMax)
l = prob.variable('y', hard_lowerbound=rangeMin, hard_upperbound=rangeMax)
y = prob.variable('l', hard_lowerbound=altMin, hard_upperbound=altMax)
tf = prob.variable('tf', hard_lowerbound=tfMin, hard_upperbound=tfMax)
aoa = prob.variable('aoa', hard_lowerbound=aoaMin, hard_upperbound=aoaMax) # aoa

""" Compute the Aerodynamic Quantities """
CD = 0.5 * rho0 * S * Cx0_s
CN = -0.5 * rho0 * S * CN_alpha_s # lift coefficient
Fl = CD * exp(-beta * y) * V * V
Fy = CN * exp(-beta * y) * V * V * aoa
gravity = 9.81

""" Evaluate Right-Hand Side of the Dynamics """
vdot = (-Fl + Fy * aoa) / mass - gravity * sin(gamma)  
gammadot = (-Fy - mass * gravity * cos(gamma)) / (mass * V)
ldot = V * cos(gamma)
ydot = V * sin(gamma)
dynamics = [vdot, gammadot, ldot, ydot, 0]

ts = tf / NN
ode = differential_equation(
    state=[V, gamma, l, y, tf],
    state_dot=dynamics, 
    stepsize=ts, 
    discretization_method='forward_euler')
prob.equality(ode)

obj_u = general_objective(uWeight * aoa ** 2)
prob.objective(obj_u)
obj = general_objective(-vfWeight * V + tfWeight * tf)
# maximize range
prob.end_objective(obj)

seq = general_equality([V - speed0, gamma - fpa0, l - range0, y - alt0])
prob.start_equality(seq)

eeq = general_equality([gamma - fpaf, l - rangef, y - altf])
prob.end_equality(eeq)

# amin <= a <= a_max
an = V * gammadot / gravity
ineq = general_inequality(
    expr  =  [an, an],
    sign  =  ['>=',  '<='],
    bound =  [anMin,  anMax])
prob.inequality(ineq)

option = codegen_option()
codegen = code_generator()
codegen.codegen(prob, option)

Solution

[1]. 聂万胜, 冯必鸣, 李柯. 高速远程精确打击飞行器方案设计方法与应用. 国防工业出版社, 2014.