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ocp.py
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import casadi as csd
import numpy as np
from quad_dynamics import PlanarDroneDynamics
class OptimalControlProblem:
def __init__(self, J, w, g, w0, lbw, ubw, lbg, ubg, p=None):
self.J = J
self.w = w
self.g = g
self.w0 = w0
self.lbw = lbw
self.ubw = ubw
self.lbg = lbg
self.ubg = ubg
self.p = p
class PlanarDroneFreeTimeOCP:
def __init__(self, x0 = [1, 1, 0, 0, 0, 0], xT = [9, 9, 0, 0, 0, 0], stage_cost_type = "quadratic",
obstacles = None):
self.N = 50
self.dt = 1/self.N
self.x0 = x0
self.xT = xT
self.stage_cost_type = stage_cost_type
self.obstacles = obstacles
self.Q = csd.diag([1, 1, 1, 1, 1, 1])
self.R = csd.diag([0.1, 0.1])
self.dynamics = PlanarDroneDynamics()
self.Fmin = 0
self.Fmax = 3*self.dynamics.m*self.dynamics.g/2
self.Tmax = 20.
self.define_integrator()
self.define_ocp()
self.define_solver()
def define_integrator(self):
""" An integrator for free time OCP"""
T = csd.MX.sym('T')
x_tilde = csd.vertcat(self.dynamics.x, T)
ode_tilde = csd.vertcat(T*self.dynamics.ode, 0)
if self.stage_cost_type == "quadratic":
self.L = (self.xT - self.dynamics.x).T @ self.Q @ (self.xT - self.dynamics.x) + \
self.dynamics.u.T @ self.R @ self.dynamics.u
elif self.stage_cost_type == "time":
self.L = T
dae = {'x': x_tilde, 'p': self.dynamics.u, 'ode': ode_tilde, 'quad': self.L}
opts = {'tf': self.dt, 'number_of_finite_elements': 4}
self.integrator = csd.integrator("integrator", "rk", dae, opts)
def formulate_initial_guess(self):
px_init = csd.linspace(self.x0[0], self.xT[0], self.N+1)
py_init = csd.linspace(self.x0[1], self.xT[1], self.N+1)
phi_init = csd.DM.zeros(self.N+1)
vx_init = csd.DM.zeros(self.N+1)
vy_init = csd.DM.zeros(self.N+1)
phidot_init = csd.DM.zeros(self.N+1)
x0 = csd.vertcat(px_init, py_init, phi_init, vx_init, vy_init, phidot_init)
x0 = csd.transpose(csd.reshape(x0, (self.N+1, 6)))
F1_init = csd.DM.ones(self.N) * self.dynamics.m*self.dynamics.g/2
F2_init = csd.DM.ones(self.N) * self.dynamics.m*self.dynamics.g/2
u0 = csd.vertcat(F1_init, F2_init)
u0 = csd.transpose(csd.reshape(u0, (self.N, 2)))
T0 = 5
return x0, u0, T0
def define_ocp(self):
# define lists for optimzation variables, constraints and bounds
w = []
w0 = []
lbw = []
ubw = []
J = 0
g = []
lbg = []
ubg = []
x0, u0, T0 = self.formulate_initial_guess()
# descion variables for initial state
xk = csd.MX.sym('xk_0', 7)
# Initial constraints
w += [xk]
w0 += [x0[:,0], T0]
lbw += [0, 0, -csd.inf, -csd.inf, -csd.inf, -csd.inf, 0.]
ubw += [10, 10, csd.inf, csd.inf, csd.inf, csd.inf, self.Tmax]
g += [self.x0 - xk[0:6]]
lbg += [0, 0, 0, 0, 0, 0]
ubg += [0, 0, 0, 0, 0, 0]
for k in range(self.N):
uk = csd.MX.sym('uk_'+str(k), 2)
w += [uk]
w0 += [u0[:,k]]
lbw += [self.Fmin, self.Fmin]
ubw += [self.Fmax, self.Fmax]
Fnext = self.integrator(x0=xk, p=uk)
xk = csd.MX.sym('xk_'+str(k+1), 7)
w += [xk]
w0 += [x0[:,k+1], 1]
lbw += [0, 0, -csd.inf, -csd.inf, -csd.inf, -csd.inf, 0.]
ubw += [10, 10, csd.inf, csd.inf, csd.inf, csd.inf, self.Tmax]
J += Fnext['qf']
g += [Fnext['xf'] - xk]
lbg += [0, 0, 0, 0, 0, 0, 0]
ubg += [0, 0, 0, 0, 0, 0, 0]
if self.obstacles != None:
# Collision avoidance constraint
pos = Fnext['xf'][0:2]
for obst in self.obstacles:
p_obst = obst[0]
r_obst = obst[1]
dpos = pos - p_obst
g += [dpos.T @ dpos - (self.dynamics.L + r_obst)**2]
lbg += [0]
ubg += [csd.inf]
# terminal state constraint
if self.stage_cost_type == "time":
g += [self.xT - xk[0:6]]
lbg += [0, 0, 0, 0, 0, 0]
ubg += [0, 0, 0, 0, 0, 0]
self.ocp = OptimalControlProblem(J, csd.vertcat(*w), csd.vertcat(*g), csd.vertcat(*w0), lbw, ubw, lbg, ubg)
def define_solver(self):
prob = {'f': self.ocp.J, 'x': self.ocp.w, 'g': self.ocp.g}
opts = {"expand": False,
"verbose": False,
"print_time": True,
"error_on_fail": True,
"ipopt": {"linear_solver": "mumps",
"max_iter": 1000,
'print_level': 5,
'sb': 'yes', # Suppress IPOPT banner
'tol': 1e-9,
# 'warm_start_init_point': 'yes',
# 'warm_start_bound_push': 1e-8,
# 'warm_start_mult_bound_push': 1e-8,
# 'mu_init': 1e-5,
# 'hessian_approximation': 'limited-memory'
}}
# opts = {}
self.solver = csd.nlpsol('solver', 'ipopt', prob, opts)
def solve_ocp(self):
sol = self.solver(x0=self.ocp.w0, lbx=self.ocp.lbw, ubx=self.ocp.ubw, lbg=self.ocp.lbg, ubg=self.ocp.ubg)
w_opt = sol['x'].full().flatten()
px_opt = w_opt[0::9]
py_opt = w_opt[1::9]
phi_opt = w_opt[2::9]
vx_opt = w_opt[3::9]
vy_opt = w_opt[4::9]
phidot_opt = w_opt[5::9]
T_opt = w_opt[6]
F1_opt = w_opt[7::9]
F2_opt = w_opt[8::9]
x_opt = np.array([px_opt, py_opt, phi_opt, vx_opt, vy_opt, phidot_opt])
u_opt = np.array([F1_opt, F2_opt])
return x_opt, u_opt, T_opt
class PlanarDroneFixedTimeOCP:
def __init__(self, x0 = [1, 1, 0, 0, 0, 0], xT = [9, 9, 0, 0, 0, 0], dt = 0.01, N = 50,
stage_cost_type = "quadratic", obstacles = None):
self.N = N
self.dt = dt
self.x0 = x0
self.xT = xT
self.stage_cost_type = stage_cost_type
self.obstacles = obstacles
self.Q = csd.diag([1e+4, 1e+4, 1e-2, 1e-3, 1e-3, 1e-3])
self.R = csd.diag([1, 1])
self.dynamics = PlanarDroneDynamics()
self.L = (self.xT - self.dynamics.x).T @ self.Q @ (self.xT - self.dynamics.x) + \
self.dynamics.u.T @ self.R @ self.dynamics.u
self.Fmin = 0
self.Fmax = 3*self.dynamics.m*self.dynamics.g/2
self.define_integrator()
self.define_ocp()
self.define_solver()
def define_integrator(self):
""" An integrator for fixed time OCP"""
dae = {'x': self.dynamics.x, 'p': self.dynamics.u, 'ode': self.dynamics.ode, 'quad': self.L}
opts = {'tf': self.dt, 'number_of_finite_elements': 5}
self.integrator = csd.integrator("integrator", "rk", dae, opts)
def formulate_initial_guess(self):
px_init = csd.linspace(self.x0[0], self.xT[0], self.N+1)
py_init = csd.linspace(self.x0[1], self.xT[1], self.N+1)
phi_init = csd.DM.zeros(self.N+1)
vx_init = csd.DM.zeros(self.N+1)
vy_init = csd.DM.zeros(self.N+1)
phidot_init = csd.DM.zeros(self.N+1)
x0 = csd.vertcat(px_init, py_init, phi_init, vx_init, vy_init, phidot_init)
x0 = csd.transpose(csd.reshape(x0, (self.N+1, 6)))
F1_init = csd.DM.ones(self.N) * self.dynamics.m*self.dynamics.g/2
F2_init = csd.DM.ones(self.N) * self.dynamics.m*self.dynamics.g/2
u0 = csd.vertcat(F1_init, F2_init)
u0 = csd.transpose(csd.reshape(u0, (self.N, 2)))
return x0, u0
def define_ocp(self):
# define lists for optimzation variables, constraints and bounds
w = []
w0 = []
lbw = []
ubw = []
J = 0
g = []
lbg = []
ubg = []
x0, u0 = self.formulate_initial_guess()
# descion variables for initial state
xk = csd.MX.sym('xk_0', 6)
# initial state (will be a parameter of the optimization problem)
x0_bar = csd.MX.sym('x0_bar', 6)
# Initial constraints
w += [xk]
w0 += [x0[:,0]]
lbw += [0, 0, -csd.inf, -csd.inf, -csd.inf, -csd.inf]
ubw += [10, 10, csd.inf, csd.inf, csd.inf, csd.inf]
g += [x0_bar - xk]
lbg += [0, 0, 0, 0, 0, 0]
ubg += [0, 0, 0, 0, 0, 0]
for k in range(self.N):
uk = csd.MX.sym('uk_'+str(k), 2)
w += [uk]
w0 += [u0[:,k]]
lbw += [self.Fmin, self.Fmin]
ubw += [self.Fmax, self.Fmax]
Fnext = self.integrator(x0=xk, p=uk)
xk = csd.MX.sym('xk_'+str(k+1), 6)
w += [xk]
w0 += [x0[:,k+1]]
lbw += [0, 0, -csd.inf, -csd.inf, -csd.inf, -csd.inf]
ubw += [10, 10, csd.inf, csd.inf, csd.inf, csd.inf]
J += Fnext['qf']
g += [Fnext['xf'] - xk]
lbg += [0, 0, 0, 0, 0, 0]
ubg += [0, 0, 0, 0, 0, 0]
if self.obstacles is not None:
# Collision avoidance constraint
pos = Fnext['xf'][0:2]
for obst in self.obstacles:
p_obst = obst[0]
r_obst = obst[1]
dpos = pos - p_obst
g += [dpos.T @ dpos - (self.dynamics.L + r_obst)**2]
lbg += [0]
ubg += [csd.inf]
self.ocp = OptimalControlProblem(J, csd.vertcat(*w), csd.vertcat(*g), csd.vertcat(*w0), lbw, ubw, lbg, ubg, x0_bar)
def define_solver(self):
prob = {'f': self.ocp.J, 'x': self.ocp.w, 'g': self.ocp.g, 'p':self.ocp.p}
opts = {"expand": False,
"verbose": False,
"print_time": True,
"error_on_fail": True,
"ipopt": {"linear_solver": "mumps",
"max_iter": 1000,
'print_level': 5,
'sb': 'yes', # Suppress IPOPT banner
'tol': 1e-9,
# 'warm_start_init_point': 'yes',
# 'warm_start_bound_push': 1e-8,
# 'warm_start_mult_bound_push': 1e-8,
# 'mu_init': 1e-5,
# 'hessian_approximation': 'limited-memory'
}}
# opts = {}
self.solver = csd.nlpsol('solver', 'ipopt', prob, opts)
def solve_ocp(self, x0_bar):
# self.ocp.w0[0:6] = x0_bar
sol = self.solver(x0=self.ocp.w0, lbx=self.ocp.lbw, ubx=self.ocp.ubw, lbg=self.ocp.lbg, ubg=self.ocp.ubg, p=x0_bar)
w_opt = sol['x'].full().flatten()
# set current solution of the current ocp as initial guess of the next OCP
self.ocp.w0 = w_opt
return w_opt[6:8]