def doit():
# state vector is (x, y, theta)
x = cs.SX.sym('x', 3)
# control vector is (v, omega)
u = cs.SX.sym('u', 2)
# explicit variables for readability
v = u[0]
omega = u[1]
theta = x[2]
# unicycle kinematic model
xdot = cs.vertcat(v*cs.cos(theta),
v*cs.sin(theta),
omega)
# intermediate cost, final cost, final constraint
l = cs.sumsqr(u)
xf_des = np.array([0, 1, 0])
lf = 1000.0 * cs.sumsqr(x - xf_des)
hf = x - xf_des
# create solver
solver = ilqr.IterativeLQR(x=x,
u=u,
xdot=xdot,
dt=0.1,
N=100,
intermediate_cost=l,
final_cost=lf,
final_constraint=None)
# solve
solver.randomizeInitialGuess()
solver.solve(50)
plt.figure()
plt.title('Total cost')
plt.semilogy(solver._dcost, '--s')
plt.grid()
plt.figure()
plt.title('State trajectory')
plt.plot(solver._state_trj)
plt.legend(['x', 'y', 'theta'])
plt.grid()
plt.figure()
plt.title('Control trajectory')
plt.plot(solver._ctrl_trj)
plt.legend(['v', 'omega'])
plt.grid()
plt.show()
def BaseOptiDX():
pathpoints = 30
ref_path = {}
ref_path['x'] = 5 * np.sin(np.linspace(0, 2 * np.pi, pathpoints + 1))
ref_path['y'] = np.linspace(1, 2, pathpoints + 1)**2 * 10
wp = ca.horzcat(ref_path['x'], ref_path['y']).T
N = 5
N1 = N - 1
wpts = np.array(wp[:, 0:N])
x = ca.MX.sym('x', N)
dx = ca.MX.sym('dx', N - 1)
nlp = {\
'x': ca.vertcat(x, dx),
'f': ca.sumsqr(x - wpts[0, :]) + ca.sumsqr(dx),
'g': ca.vertcat(\
x[1:] - (x[:-1] + dx),
x[0] - wpts[0, 0],
)\
}
S = ca.nlpsol('S', 'ipopt', nlp)
# x0
x00 = wpts[0, :]
dx00 = (x00[1:] - x00[:-1])
x0 = ca.vertcat(x00, dx00)
lbx = [0] * N + [0] * N1
ubx = [ca.inf] * N + [10] * N1
lbg = [0] * N
ubg = [0] * N
# solve
r = S(x0=x0, lbx=lbx, ubx=ubx, lbg=lbg, ubg=ubg)
x_opt = r['x']
x = np.array(x_opt[:N])
xds = np.array(x_opt[N:N + N1])
print(f"xs: {x.T}")
print(f"dx's: {xds.T}")
plt.figure(1)
plt.plot(wpts[0, :], np.ones_like(wpts[0, :]), 'o', markersize=12)
plt.plot(x, np.ones_like(x), '+', markersize=20)
plt.show()
def smooth_track(track):
N = len(track)
xs = track[:, 0]
ys = track[:, 1]
th1_f = ca.MX.sym('y1_f', N-2)
th2_f = ca.MX.sym('y2_f', N-2)
x_f = ca.MX.sym('x_f', N)
y_f = ca.MX.sym('y_f', N)
real = ca.Function('real', [th1_f, th2_f], [ca.cos(th1_f)*ca.cos(th2_f) + ca.sin(th1_f)*ca.sin(th2_f)])
im = ca.Function('im', [th1_f, th2_f], [-ca.cos(th1_f)*ca.sin(th2_f) + ca.sin(th1_f)*ca.cos(th2_f)])
sub_cmplx = ca.Function('a_cpx', [th1_f, th2_f], [ca.atan(im(th1_f, th2_f)/real(th1_f, th2_f))])
d_th = ca.Function('d_th', [x_f, y_f], [ca.if_else(ca.fabs(x_f[1:] - x_f[:-1]) < 0.01 ,ca.atan((y_f[1:] - y_f[:-1])/(x_f[1:] - x_f[:-1])), 10000)])
x = ca.MX.sym('x', N)
y = ca.MX.sym('y', N)
th = ca.MX.sym('th', N-1)
B = 5
nlp = {\
'x': ca.vertcat(x, y, th),
'f': ca.sumsqr(sub_cmplx(th[1:], th[:-1])) + B* (ca.sumsqr(x-xs) + ca.sumsqr(y-ys)),
# 'f': B* (ca.sumsqr(x-xs) + ca.sumsqr(y-ys)),
'g': ca.vertcat(\
th - d_th(x, y),
x[0] - xs[0], y[0]- ys[0],
x[-1] - xs[-1], y[-1]- ys[-1],
)\
}
S = ca.nlpsol('S', 'ipopt', nlp)
# th0 = [lib.get_bearing(track[i, 0:2], track[i+1, 0:2]) for i in range(N-1)]
th0 = d_th(xs, ys)
x0 = ca.vertcat(xs, ys, th0)
lbx = [0] *2* N + [-np.pi]*(N -1)
ubx = [100] *2 * N + [np.pi]*(N-1)
r = S(x0=x0, lbg=0, ubg=0, lbx=lbx, ubx=ubx)
print(f"Solution found")
x_opt = r['x']
xs_new = np.array(x_opt[:N])
ys_new = np.array(x_opt[N:2*N])
track[:, 0] = xs_new[:, 0]
track[:, 1] = ys_new[:, 0]
return track
def init_ocp(self):
ocp = Ocp(T=FreeTime(20.0))
self.x = ocp.state()
self.y = ocp.state()
self.theta = ocp.state()
self.v = ocp.control()
self.theta_dot = ocp.control()
ocp.set_der(self.x, sin(self.theta) * self.v)
ocp.set_der(self.y, cos(self.theta) * self.v)
ocp.set_der(self.theta, self.theta_dot)
self.X_0 = ocp.parameter(self.nx)
X = vertcat(self.x, self.y, self.theta)
ocp.subject_to(ocp.at_t0(X) == self.X_0)
max_v = 5
ocp.subject_to(1 <= (self.v <= max_v))
th_dot_lim = 0.5
ocp.subject_to(-th_dot_lim <= (self.theta_dot <= th_dot_lim))
#ocp.subject_to( -0.3 <= (ocp.der(V) <= 0.3) )
# Minimal time
# ocp.add_objective(0.50*ocp.T)
self.waypoints = ocp.parameter(2, grid='control')
self.waypoint_last = ocp.parameter(2)
p = vertcat(self.x, self.y)
ocp.add_objective(ocp.sum(sumsqr(p - self.waypoints), grid='control'))
ocp.add_objective(sumsqr(ocp.at_tf(p) - self.waypoint_last))
options = {"ipopt": {"print_level": 5}}
options["expand"] = True
options["print_time"] = False
ocp.solver('ipopt', options)
ocp.method(
MultipleShooting(N=self.N,
M=1,
intg='rk',
grid=FreeGrid(min=0.05, max=2)))
ocp.set_initial(self.x, 0)
ocp.set_initial(self.y, 0)
ocp.set_initial(self.theta, 0)
ocp.set_initial(self.theta_dot, 0)
ocp.set_initial(self.v, 2)
self.ocp = ocp
def generate_brent(n=2, a=10., b=10., func_opts={}, data_type=cs.SX):
if n != 2:
raise ValueError("brent is only defined for n=2")
x = data_type.sym("x", n)
f = (x[0] + a)**2 + (x[1] + b)**2 + cs.exp(-cs.sumsqr(x))
func = cs.Function("brent", [x], [f], ["x"], ["f"], func_opts)
return func, [[-20., 0.]] * n, [[-10.] * n]
def robust_control_stages(ocp, delta, t0):
"""
Returns
-------
stage: :obj: `~rockit.stage.Stage
x : :obj: `~casadi.MX
"""
stage = ocp.stage(t0=t0, T=1)
x = stage.state(2)
u = stage.state()
der_state = vertcat(x[1],
-0.1 * (1 - x[0]**2 + delta) * x[1] - x[0] + u + delta)
stage.set_der(x, der_state)
stage.set_der(u, 0)
stage.subject_to(u <= 40)
stage.subject_to(u >= -40)
stage.subject_to(x[0] >= -0.25)
L = stage.variable()
stage.add_objective(L)
stage.subject_to(L >= sumsqr(x[0] - 3))
bound = lambda t: 2 + 0.1 * cos(10 * t)
stage.subject_to(x[0] <= bound(stage.t))
stage.method(MultipleShooting(N=20, M=1, intg='rk'))
return stage, x, stage.at_t0(u)
def generate_egg_crate(n=2, a=25., func_opts={}, data_type=cs.SX):
if n != 2:
raise ValueError("egg_crate is only defined for n=2")
x = data_type.sym("x", n)
f = cs.sumsqr(x) + a * (cs.sin(x[0])**2 + cs.sin(x[1])**2)
func = cs.Function("egg_crate", [x], [f], ["x"], ["f"], func_opts)
return func, [[-5., 5.]] * n, [[0., 0.]]
def generate_griewank(n=2, a=1., b=4000., func_opts={}, data_type=cs.SX):
x = data_type.sym("x", n)
product_part = 1.
for i in range(n):
product_part = product_part * cs.cos(x[i] / cs.sqrt(i + 1))
f = a + (1. / b) * cs.sumsqr(x) - product_part
func = cs.Function("griewank", [x], [f], ["x"], ["f"], func_opts)
return func, [[-600., 600.]] * n, [[0.] * n]
def generate_deckkers_aarts(n=2, a=1e5, b=1e-5, func_opts={}, data_type=cs.SX):
if n != 2:
raise ValueError("deckkers_aarts is only defined for n=2")
x = data_type.sym("x", n)
sqr = cs.sumsqr(x)
f = a * x[0]**2 + x[1]**2 - sqr**2 + b * sqr**4
func = cs.Function("deckkers_aarts", [x], [f], ["x"], ["f"], func_opts)
minima = [[0., 15.], [0., -15.]]
return func, [[-20., 20.], [-20., 20.]], minima
def generate_bartelsconn(n=2, func_opts={}, data_type=cs.SX):
if n != 2:
raise ValueError("bartelsconn is only defined for n=2")
x = data_type.sym("x", n)
f = cs.norm_1(cs.sumsqr(x) + x[0] * x[1])
f += cs.norm_1(cs.sin(x[0]))
f += cs.norm_1(cs.cos(x[1]))
func = cs.Function("bartelsconn", [x], [f], ["x"], ["f"], func_opts)
return func, [[-500., 500.]] * n, [[0.] * n]
def create_objective(self, model):
self.obj_1 = sum(
w / len(ov) * ca.sumsqr(self.densities * (ov - cm @ ca.interp1d(
model.observation_times,
om(model.observation_times, model.ps, *
(model.xs[j] for j in oj)), self.observation_times)))
for om, oj, ov, w, cm in zip(
self.observation_model, self.observation_vector,
self.observations, self.weightings, self.collocation_matrices))
self.obj_2 = sum(
ca.sumsqr((model.xdash[:, i] - model.model(
model.observation_times, *model.cs, *model.ps)[:, i]))
for i in range(model.s)) / model.n
self.regularisation = ca.sumsqr(
(ca.vcat(model.ps) - ca.vcat(self.regularisation_vector)) /
(1 + ca.vcat(self.regularisation_vector)))
self.objective = self.obj_1 + self.rho * self.obj_2 + self.alpha * self.regularisation
def init_mpc(self):
# Specify ODE
self.ocp.set_der(self.x, self.V * cos(self.theta))
self.ocp.set_der(self.y, self.V * sin(self.theta))
self.ocp.set_der(self.theta, self.V / L * tan(self.delta))
# Initial constraints
self.ocp.subject_to(self.ocp.at_t0(self.X) == self.X_0)
# Initial guess
self.ocp.set_initial(self.x, 0)
self.ocp.set_initial(self.y, 0)
self.ocp.set_initial(self.theta, 0)
self.ocp.set_initial(self.V, 0.5)
# Path constraints
max_v = 5
self.ocp.subject_to(0 <= (self.V <= max_v))
#ocp.subject_to( -0.3 <= (ocp.der(V) <= 0.3) )
self.ocp.subject_to(-pi / 6 <= (self.delta <= pi / 6))
# Minimal time
self.ocp.add_objective(0.50 * self.ocp.T)
self.ocp.add_objective(
self.ocp.sum(sumsqr(self.p - self.waypoints), grid='control'))
self.ocp.add_objective(
sumsqr(self.ocp.at_tf(self.p) - self.waypoint_last))
# Pick a solution method
options = {"ipopt": {"print_level": 0}}
options["expand"] = True
options["print_time"] = False
self.ocp.solver('ipopt', options)
# Make it concrete for this ocp
self.ocp.method(
MultipleShooting(N=N,
M=1,
intg='rk',
grid=FreeGrid(min=0.05, max=2)))
def generate_happy_cat(n=2,
a=0.125,
b=0.5,
c=0.5,
func_opts={},
data_type=cs.SX):
x = data_type.sym("x", n)
nx2 = cs.dot(x, x)
f = ((nx2 - n)**2)**a + (1. / n) * (b * nx2 + cs.sumsqr(x)) + c
func = cs.Function("happy_cat", [x], [f], ["x"], ["f"], func_opts)
return func, [[-2., 2.]] * n, [[-1] * n]
def generate_salomon(n=2,
a=1.,
b=2 * cs.np.pi,
c=0.1,
func_opts={},
data_type=cs.SX):
x = data_type.sym("x", n)
sumsq = cs.sumsqr(x)
f = a - cs.cos(b * cs.sqrt(sumsq)) + c * cs.sqrt(sumsq)
func = cs.Function("salomon", [x], [f], ["x"], ["f"], func_opts)
return func, [[-10., 10.]] * n, [[0.] * n]
def optCtrl(optQueue,opt2Queue,P,MU,S,H,tgt,optdt):
sold=None
opt=qnTransient()
opt.buildOpt(MU, P, S,optdt,tgt,H)
while(True):
q=optQueue.get()
opt.model.set_value(opt.initX,q)
obj=0
for h in range(H):
obj+=(opt.stateVar[1,h]-tgt/MU[1,0]*opt.tVar[1,h])**2
#obj+=opt.E_abs[0,h]
if(sold is not None):
opt.model.minimize(obj+0.0*casadi.sumsqr(opt.sVar-sold)+0.000*casadi.sumsqr(opt.sVar))
else:
opt.model.minimize(obj)
sol=opt.model.solve()
opt2Queue.put(sol.value(opt.sVar))
def add_target_orbit_constraint(mocp, p, phase_insertion):
start = lambda a: mocp.start(a)
end = lambda a: mocp.end(a)
# Target orbit - soft constraints
# 3 parameter orbit: SMA,ECC,INC
#
# h_target_magnitude - h_final_magnitude == 0
# h_target_z - h_final_z == 0
# c_target - c_final == 0
r_final_ECEF = casadi.vertcat(end(phase_insertion.rx),
end(phase_insertion.ry),
end(phase_insertion.rz))
v_final_ECEF = casadi.vertcat(end(phase_insertion.vx),
end(phase_insertion.vy),
end(phase_insertion.vz))
# Convert to ECI (z rotation is omitted, because LAN is free)
r_f = r_final_ECEF
v_f = (
v_final_ECEF +
casadi.cross(casadi.SX([0, 0, p.body_rotation_speed]), r_final_ECEF))
h_final = casadi.cross(r_f, v_f)
h_final_mag = casadi.norm_2(h_final)
h_final_z = h_final[2]
c_final = casadi.sumsqr(v_f) / 2 - 1.0 / casadi.norm_2(r_f)
defect_h_magnitude = h_final_mag - p.target_orbit_h_magnitude
defect_h_z = h_final_z - p.target_orbit_h_z
defect_c = c_final - p.target_orbit_c
slack_h_magnitude = mocp.add_variable('slack_h_magnitude', init=3.0)
slack_h_z = mocp.add_variable('slack_h_z', init=3.0)
slack_c = mocp.add_variable('slack_c', init=3.0)
mocp.add_constraint(slack_h_magnitude > 0)
mocp.add_constraint(slack_h_z > 0)
mocp.add_constraint(slack_c > 0)
mocp.add_constraint(defect_h_magnitude < slack_h_magnitude)
mocp.add_constraint(defect_h_magnitude > -slack_h_magnitude)
mocp.add_constraint(defect_h_z < slack_h_z)
mocp.add_constraint(defect_h_z > -slack_h_z)
mocp.add_constraint(defect_c < slack_c)
mocp.add_constraint(defect_c > -slack_c)
mocp.add_objective(100.0 * slack_h_magnitude)
mocp.add_objective(100.0 * slack_h_z)
mocp.add_objective(100.0 * slack_c)
def solve(self, x0):
xs = ca.vcat([state for state in self.states])
us = ca.vcat([control for control in self.controls])
dyns = ca.vcat([
var[1:] - (var[:-1] + self.state_der[var] * self.dt)
for var in self.states
])
cons = ca.vcat(
[state[0] - x0[i] for i, state in enumerate(self.states)])
obs = ca.vcat([(o - self.objectives[o]) * self.o_scales[o]
for o in self.objectives.keys()])
nlp = {\
'x': ca.vertcat(xs, us, self.dt),
'f': ca.sumsqr(obs),
'g': ca.vertcat(dyns, cons)
}
n_g = nlp['g'].shape[0]
self.lbg = [0] * n_g
self.ubg = [0] * n_g
x00 = ca.vcat([self.initial[state] for state in self.states])
u00 = ca.vcat([self.initial[control] for control in self.controls])
x0 = ca.vertcat(x00, u00, self.initial[self.dt])
# S = ca.nlpsol('S', 'ipopt', nlp, {'ipopt':{'print_level':0}})
S = ca.nlpsol('S', 'ipopt', nlp, {'ipopt': {'print_level': 5}})
r = S(x0=x0, lbx=self.lbx, ubx=self.ubx, lbg=self.lbg, ubg=self.ubg)
x_opt = r['x']
n_state_vars = len(self.states) * self.n
n_control_vars = len(self.controls) * self.n
states = np.array(x_opt[:n_state_vars])
controls = np.array(x_opt[n_state_vars:n_state_vars + n_control_vars])
times = np.array(x_opt[-self.n + 1:])
for i, state in enumerate(self.states):
self.set_inital(state, states[i * self.n:self.n * (i + 1)])
for i, control in enumerate(self.controls):
self.set_inital(control,
controls[(self.n - 1) * i:(i + 1) * (self.n - 1)])
self.set_inital(self.dt, times)
return states, controls, times
def unit_quat(q):
"""
Normalizes a quaternion to be unit modulus.
:param q: 4-dimensional numpy array or CasADi object
:return: the unit quaternion in the same data format as the original one
"""
if isinstance(q, np.ndarray):
# if (q == np.zeros(4)).all():
# q = np.array([1, 0, 0, 0])
q_norm = np.sqrt(np.sum(q**2))
else:
q_norm = cs.sqrt(cs.sumsqr(q))
return 1 / q_norm * q
def generate_drop_wave(n=2,
a=1.,
b=12.,
c=0.5,
d=2.,
func_opts={},
data_type=cs.SX):
if n != 2:
raise ValueError("drop_wave is only defined for n=2")
x = data_type.sym("x", n)
sqr = cs.sumsqr(x)
f = -(a + cs.cos(b * cs.sqrt(sqr))) / (c * sqr + d)
func = cs.Function("drop_wave", [x], [f], ["x"], ["f"], func_opts)
return func, [[-5.2, 5.2]] * n, [[0., 0.]]
def ModelTraining(model,
data,
initializations=10,
BFR=False,
p_opts=None,
s_opts=None):
results = []
for i in range(0, initializations):
# in first run use initial model parameters (useful for online
# training when only time for one initialization)
if i > 0:
model.Initialize()
# Estimate Parameters on training data
new_params = ModelParameterEstimation(model, data, p_opts, s_opts)
# Assign new parameters to model
model.Parameters = new_params
# Evaluate on Validation data
u_val = data['u_val']
y_ref_val = data['y_val']
init_state_val = data['init_state_val']
# Loop over all experiments
e = 0
for j in range(0, u_val.shape[0]):
# Simulate Model
y = model.Simulation(init_state_val[j], u_val[j])
y = np.array(y)
e = e + cs.sumsqr(y_ref_val[j] - y)
# Calculate mean error over all validation batches
e = e / u_val.shape[0]
e = np.array(e).reshape((1, ))
# save parameters and performance in list
results.append([e, new_params])
results = pd.DataFrame(data=results, columns=['loss', 'params'])
return results
def gen(ocpVars):
constrList = []
M = ocpVars['u'].shape[1]
for i in range(M):
quat = ocpVars['u'][1:5, i]
# norm has to be one
g = ca.sumsqr(quat) - 1
lbg = np.zeros(g.shape)
ubg = np.zeros(g.shape)
# create constraint tuple
constrList += [(g, lbg, ubg)]
return constrList
def solve(self):
xs = ca.vcat([state for state in self.states])
us = ca.vcat([control for control in self.controls])
dyns = ca.vcat([var[1:] - (var[:-1] + self.state_der[var] * self.dt) for var in self.states])
cons = ca.vcat([cons[0] - self.constraints[cons] for cons in self.constraints.keys()])
obs = ca.vcat([(o - self.objectives[o]) * self.o_scales[o] for o in self.objectives.keys()])
nlp = {\
'x': ca.vertcat(xs, us, self.dt),
'f': ca.sumsqr(obs),
'g': ca.vertcat(dyns, cons)
}
n_g = nlp['g'].shape[0]
lbg = [0] * n_g
ubg = [0] * n_g
x00 = ca.vcat([self.initial[state] for state in self.states])
u00 = ca.vcat([self.initial[control] for control in self.controls])
x0 = ca.vertcat(x00, u00, self.initial[self.dt])
x_mins = [self.min_lims[state] for state in self.states] * self.n
x_maxs = [self.max_lims[state] for state in self.states] * self.n
u_mins = [self.min_lims[control] for control in self.controls] * (self.n - 1)
u_maxs = [self.max_lims[control] for control in self.controls] * (self.n - 1)
lbx = x_mins + u_mins + list(np.zeros(self.n-1))
ubx = x_maxs + u_maxs + list(np.ones(self.n-1) * self.max_t)
S = ca.nlpsol('S', 'ipopt', nlp)
r = S(x0=x0, lbx=lbx, ubx=ubx, lbg=lbg, ubg=ubg)
x_opt = r['x']
n_state_vars = len(self.states) * self.n
n_control_vars = len(self.controls) * self.n
states = np.array(x_opt[:n_state_vars])
controls = np.array(x_opt[n_state_vars:n_state_vars + n_control_vars])
times = np.array(x_opt[-self.n+1:])
return states, controls, times
def solve_launch_direction_problem(p):
# Find the approximate launch direction vector.
# The launch direction vector is horizontal (orthogonal to the launch site position vector).
# The launch direction vector points (approximately) in the direction of the orbit insertion velocity.
mocp = MOCP()
rx = mocp.add_variable('rx', init=p.launch_site_x)
ry = mocp.add_variable('ry', init=p.launch_site_y)
rz = mocp.add_variable('rz', init=p.launch_site_z)
vx = mocp.add_variable('vx', init=0.01)
vy = mocp.add_variable('vy', init=0.01)
vz = mocp.add_variable('vz', init=0.01)
mocp.add_objective((rx - p.launch_site_x)**2)
mocp.add_objective((ry - p.launch_site_y)**2)
mocp.add_objective((rz - p.launch_site_z)**2)
r = casadi.vertcat(rx, ry, rz)
v = casadi.vertcat(vx, vy, vz)
h = casadi.cross(r, v)
h_mag = casadi.norm_2(h)
h_z = h[2]
c = casadi.sumsqr(v) / 2 - 1.0 / casadi.norm_2(r)
mocp.add_objective((h_mag - p.target_orbit_h_magnitude)**2)
mocp.add_objective((h_z - p.target_orbit_h_z)**2)
mocp.add_objective((c - p.target_orbit_c)**2)
mocp.solve()
v_soln = DM([mocp.get_value(vx), mocp.get_value(vy), mocp.get_value(vz)])
launch_direction = normalize(v_soln)
up_direction = normalize(
DM([p.launch_site_x, p.launch_site_y, p.launch_site_z]))
launch_direction = launch_direction - (
launch_direction.T @ up_direction) * up_direction
return launch_direction
def eval(self, x, w):
'''
evaluate the cost function, with casadi operation
input:
x: primal decision var
w: RV, randomness, in ml: it's the data
'''
if len(x.shape)==1:
x=x.reshape(-1,1)
elif len(x.shape)==2:
pass
else:
raise NotImplementedError
if self.method =='boyd':
'''
Boyd & Vandenberghe book. Figure 6.15.
min_x || A(w) - b0 ||^2 where
A(w) := A0 + w * B0 and w is a scalar.
'''
# boy data set
A0, B0, b0 = self.model # model of the optimization problem
if self.mode== 'casadi':
from casadi import sumsqr
cost_val = sumsqr((A0 + w * B0) @ x - b0)
elif self.mode== 'cvxpy':
import cvxpy as cp
cost_val = cp.sum_squares((A0 + w * B0) @ x - b0)
elif self.mode=='numpy':
cost_val = np.sum(((A0 + w * B0) @ x - b0)**2)
else:
raise NotImplementedError
else:
raise NotImplementedError
return cost_val
q_torso_dot_sol = var_dot_sol[16]
print("Solver time is = " + str(hlp.time_taken))
comp_time.append(hlp.time_taken)
con_viol1 = hlp.optimal_slacks[1]
con_viol2 = hlp.optimal_slacks[2]
con_viol3 = hlp.optimal_slacks[3]
con_viol4 = hlp.optimal_slacks[4]
constraint_violations = cs.horzcat(constraint_violations, cs.vertcat(con_viol1, con_viol2, con_viol3, con_viol4))
# print(con_viols)
# print(q_opt)
# print(s2_opt)
number_non_zero = sum(cs.fabs(q_dot1_sol) >= 1e-3) + sum(cs.fabs(q_dot2_sol) >= 1e-3)
q_zero_integral += number_non_zero*ts
q_2_integral += (cs.sumsqr(q_dot1_sol) + cs.sumsqr(q_dot2_sol))*ts
q_1_integral += sum(cs.fabs(q_dot1_sol).full() + cs.fabs(q_dot2_sol).full())*ts
#compute q_1_integral
# print(number_non_zero)
#Update all the variables
q_dot_opt = cs.vertcat(q_dot1_sol, q_dot2_sol, s_dot1_sol, s_dot2_sol, q_torso_dot_sol)
q_opt[0:17] += ts*q_dot_opt
s1_opt = q_opt[14]
if s1_opt >=1:
print("Robot1 reached it's goal. Terminating")
cool_off_counter += 1
# break
# if cool_off_counter >= 100: #ts*100 cooloff period for the robot to exactly reach it's goal
# break
def compute_mprim(state_i, lattice, L, v, u_max, print_level):
N = 75
nx = 3
Nc = 3
mprim = []
for state_f in lattice:
x = ca.MX.sym('x', nx)
u = ca.MX.sym('u')
F = ca.Function('f', [x, u],
[v*ca.cos(x[2]), v*ca.sin(x[2]), v*ca.tan(u)/L])
opti = ca.Opti()
X = opti.variable(nx, N+1)
pos_x = X[0, :]
pos_y = X[1, :]
ang_th = X[2, :]
U = opti.variable(N, 1)
T = opti.variable(1)
dt = T/N
opti.set_initial(T, 0.1)
opti.set_initial(U, 0.0*np.ones(N))
opti.set_initial(pos_x, np.linspace(state_i[0], state_f[0], N+1))
opti.set_initial(pos_y, np.linspace(state_i[1], state_f[1], N+1))
tau = ca.collocation_points(Nc, 'radau')
C, _ = ca.collocation_interpolators(tau)
Xc_vec = []
for k in range(N):
Xc = opti.variable(nx, Nc)
Xc_vec.append(Xc)
X_kc = ca.horzcat(X[:, k], Xc)
for j in range(Nc):
fo = F(Xc[:, j], U[k])
opti.subject_to(X_kc@C[j+1] == dt*ca.vertcat(fo[0], fo[1], fo[2]))
opti.subject_to(X_kc[:, Nc] == X[:, k+1])
for k in range(N):
opti.subject_to(U[k] <= u_max)
opti.subject_to(-u_max <= U[k])
opti.subject_to(T >= 0.001)
opti.subject_to(X[:, 0] == state_i)
opti.subject_to(X[:, -1] == state_f)
alpha = 1e-2
opti.minimize(T + alpha*ca.sumsqr(U))
opti.solver('ipopt', {'expand': True},
{'tol': 10**-8, 'print_level': print_level})
sol = opti.solve()
pos_x_opt = sol.value(pos_x)
pos_y_opt = sol.value(pos_y)
ang_th_opt = sol.value(ang_th)
u_opt = sol.value(U)
T_opt = sol.value(T)
mprim.append({'x': pos_x_opt, 'y': pos_y_opt, 'th': ang_th_opt,
'u': u_opt, 'T': T_opt, 'ds': T_opt*v})
return np.array(mprim)
def construct_problem(self):
"""Formulate optimal control problem"""
dt = self.sample_rate
# Create an casadi.Opti instance.
opti = casadi.Opti('conic')
d0 = opti.parameter()
th0 = opti.parameter()
v = opti.parameter()
curvature = opti.parameter(self.N)
X = opti.variable(2, self.N + 1)
proj_error = X[0, :]
head_error = X[1, :]
# Control variable (steering angle)
Delta = opti.variable(self.N)
# Goal function we wish to minimize
### YOUR CODE HERE ###
# Use the casadi.sumsqr function to compute sum-of-squares
d_sum = casadi.sumsqr(proj_error)
th_sum = casadi.sumsqr(head_error)
u_sum = casadi.sumsqr(Delta)
J = self.gamma_d * d_sum + self.gamma_theta * th_sum + self.gamma_u * u_sum
opti.minimize(J)
# Simulate the system forwards using RK4 and the implemented error model.
for k in range(self.N):
k1 = self.error_model(X[:, k], v, Delta[k], curvature[k])
k2 = self.error_model(X[:, k] + dt / 2 * k1, v, Delta[k],
curvature[k])
k3 = self.error_model(X[:, k] + dt / 2 * k2, v, Delta[k],
curvature[k])
k4 = self.error_model(X[:, k] + dt * k3, v, Delta[k], curvature[k])
x_next = X[:, k] + dt / 6 * (k1 + 2 * k2 + 2 * k3 + k4)
opti.subject_to(X[:, k + 1] == x_next)
# Problem constraints.
opti.subject_to(proj_error[0] == d0)
opti.subject_to(head_error[0] == th0)
opti.subject_to(
opti.bounded(-self.steer_limit, Delta, self.steer_limit))
# The cost function is quadratic and the problem is linear by design,
# this is utilized when choosing solver.
# Other possible solvers provided by CasADi include: 'qpoases' and 'ipopt'...
opts_dict = {
"print_iter": False,
"print_time": 0,
#"max_iter": 100
}
opti.solver('qrqp', opts_dict)
return opti.to_function('f', [d0, th0, v, curvature], [Delta])
def solve_opti(self, abs_t, x_0, x_ss, N, last_u, x_guess=None, u_guess=None, expl_constraints=None, verbose=False):
opti = ca.Opti()
x = opti.variable(self.n_x, N+1)
u = opti.variable(self.n_u, N)
slack = opti.variable(self.n_x)
if x_guess is not None:
opti.set_initial(x, x_guess)
if u_guess is not None:
opti.set_initial(u, u_guess)
opti.set_initial(slack, np.zeros(self.n_x))
da_lim = self.agent.da_lim
ddf_lim = self.agent.ddf_lim
opti.subject_to(x[:,0] == np.squeeze(x_0))
opti.subject_to(opti.bounded(ddf_lim[0]*self.dt, u[0,0]-last_u[0], ddf_lim[1]*self.dt))
opti.subject_to(opti.bounded(da_lim[0]*self.dt, u[1,0]-last_u[1], da_lim[1]*self.dt))
stage_cost = 0
for i in range(N):
stage_cost = stage_cost + 1
beta = ca.atan2(self.l_r*ca.tan(u[0,i]), self.l_f+self.l_r)
opti.subject_to(x[0,i+1] == x[0,i] + self.dt*x[3,i]*ca.cos(x[2,i] + beta))
opti.subject_to(x[1,i+1] == x[1,i] + self.dt*x[3,i]*ca.sin(x[2,i] + beta))
opti.subject_to(x[2,i+1] == x[2,i] + self.dt*x[3,i]*ca.sin(beta))
opti.subject_to(x[3,i+1] == x[3,i] + self.dt*u[1,i])
if self.F is not None:
opti.subject_to(ca.mtimes(self.F, x[:,i]) <= self.b)
if self.H is not None:
opti.subject_to(ca.mtimes(self.H, u[:,i]) <= self.g)
if expl_constraints is not None:
V = expl_constraints[i][0]
w = expl_constraints[i][1]
opti.subject_to(ca.mtimes(V, x[:2,i]) + w <= np.zeros(len(w)))
if i < N-1:
opti.subject_to(opti.bounded(ddf_lim[0]*self.dt, u[0,i+1]-u[0,i], ddf_lim[1]*self.dt))
opti.subject_to(opti.bounded(da_lim[0]*self.dt, u[1,i+1]-u[1,i], da_lim[1]*self.dt))
opti.subject_to(x[:,N] - x_ss == slack)
slack_cost = 1e6*ca.sumsqr(slack)
total_cost = stage_cost + slack_cost
opti.minimize(total_cost)
solver_opts = {
"mu_strategy" : "adaptive",
"mu_init" : 1e-5,
"mu_min" : 1e-15,
"barrier_tol_factor" : 1,
"print_level" : 0,
"linear_solver" : "ma27"
}
# solver_opts = {}
plugin_opts = {"verbose" : False, "print_time" : False, "print_out" : False}
opti.solver('ipopt', plugin_opts, solver_opts)
sol = opti.solve()
slack_val = sol.value(slack)
if la.norm(slack_val) > 1e-6:
print('Warning! Solved, but with slack norm of %g is greater than 1e-8!' % la.norm(slack_val))
if sol.stats()['success'] and la.norm(slack_val) <= 1e-6:
print('Solve success, with slack norm of %g!' % la.norm(slack_val))
feasible = True
x_pred = sol.value(x)
u_pred = sol.value(u)
sol_cost = sol.value(stage_cost)
else:
# print(sol.stats()['return_status'])
# print(opti.debug.show_infeasibilities())
feasible = False
x_pred = None
u_pred = None
sol_cost = None
# pdb.set_trace()
return x_pred, u_pred, sol_cost
def solve(self, abs_t, x_0, x_ss, N, last_u, x_guess=None, u_guess=None, expl_constraints=None, verbose=False):
x = ca.SX.sym('x', self.n_x*(N+1))
u = ca.SX.sym('y', self.n_u*N)
slack = ca.SX.sym('slack', self.n_x)
z = ca.vertcat(x, u, slack)
# Flatten candidate solutions into 1-D array (- x_1 -, - x_2 -, ..., - x_N+1 -)
if x_guess is not None:
x_guess_flat = x_guess.flatten(order='F')
else:
x_guess_flat = np.zeros(self.n_x*(N+1))
if u_guess is not None:
u_guess_flat = u_guess.flatten(order='F')
else:
u_guess_flat = np.zeros(self.n_u*N)
slack_guess = np.zeros(self.n_x)
z_guess = np.concatenate((x_guess_flat, u_guess_flat, slack_guess))
lb_slack = [-1000.0]*self.n_x
ub_slack = [1000.0]*self.n_x
# Box constraints on decision variables
lb_x = x_0.tolist() + self.state_lb*(N) + self.input_lb*(N) + lb_slack
ub_x = x_0.tolist() + self.state_ub*(N) + self.input_ub*(N) + ub_slack
# Constraints on functions of decision variables
lb_g = []
ub_g = []
stage_cost = 0
constraint = []
for i in range(N):
stage_cost += 1
# Formulate dynamics equality constraints as inequalities
beta = ca.atan2(self.l_r*ca.tan(u[self.n_u*i+0]), self.l_f+self.l_r)
constraint = ca.vertcat(constraint, x[self.n_x*(i+1)+0] - (x[self.n_x*i+0] + self.dt*x[self.n_x*i+3]*ca.cos(x[self.n_x*i+2] + beta)))
constraint = ca.vertcat(constraint, x[self.n_x*(i+1)+1] - (x[self.n_x*i+1] + self.dt*x[self.n_x*i+3]*ca.sin(x[self.n_x*i+2] + beta)))
constraint = ca.vertcat(constraint, x[self.n_x*(i+1)+2] - (x[self.n_x*i+2] + self.dt*x[self.n_x*i+3]*ca.sin(beta)))
constraint = ca.vertcat(constraint, x[self.n_x*(i+1)+3] - (x[self.n_x*i+3] + self.dt*u[self.n_u*i+1]))
lb_g += [0.0]*self.n_x
ub_g += [0.0]*self.n_x
# Steering rate constraints
if self.ddf_lim is not None:
if i == 0:
# Constrain steering rate with respect to previously applied input
constraint = ca.vertcat(constraint, u[self.n_u*(i)+0]-last_u[0])
if i < N-1:
# Constrain steering rate along horizon
constraint = ca.vertcat(constraint, u[self.n_u*(i+1)+0]-u[self.n_u*(i)+0])
lb_g += [self.ddf_lim[0]*self.dt]
ub_g += [self.ddf_lim[1]*self.dt]
# Throttle rate constraints
if self.da_lim is not None:
if i == 0:
# Constrain throttle rate
constraint = ca.vertcat(constraint, u[self.n_u*i+1]-last_u[1])
if i < N-1:
constraint = ca.vertcat(constraint, u[self.n_u*(i+1)+1]-u[self.n_u*(i)+1])
lb_g += [self.da_lim[0]*self.dt]
ub_g += [self.da_lim[1]*self.dt]
# Exploration constraints on predicted positions of agent
if expl_constraints is not None:
V = expl_constraints[i][0]
w = expl_constraints[i][1]
for j in range(V.shape[0]):
constraint = ca.vertcat(constraint, V[j,0]*x[self.n_x*i+0] + V[j,1]*x[self.n_x*i+1] + w[j])
lb_g += [-1e9]
ub_g += [0]
# Formulate terminal soft equality constraint as inequalities
constraint = ca.vertcat(constraint, x[self.n_x*N:] - x_ss + slack)
lb_g += [0.0]*self.n_x
ub_g += [0.0]*self.n_x
slack_cost = 1e6*ca.sumsqr(slack)
total_cost = stage_cost + slack_cost
opts = {'verbose' : False, 'print_time' : 0,
'ipopt.print_level' : 5,
'ipopt.mu_strategy' : 'adaptive',
'ipopt.mu_init' : 1e-5,
'ipopt.mu_min' : 1e-15,
'ipopt.barrier_tol_factor' : 1,
'ipopt.linear_solver': 'ma27'}
nlp = {'x' : z, 'f' : total_cost, 'g' : constraint}
solver = ca.nlpsol('solver', 'ipopt', nlp, opts)
sol = solver(lbx=lb_x, ubx=ub_x, lbg=lb_g, ubg=ub_g, x0=z_guess)
pdb.set_trace()
if solver.stats()['success']:
if la.norm(slack_val) <= 1e-8:
print('Solve success for safe set point', x_ss)
feasible = True
z_sol = np.array(sol['x'])
x_pred = z_sol[:self.n_x*(N+1)].reshape((N+1,self.n_x)).T
u_pred = z_sol[self.n_x*(N+1):self.n_x*(N+1)+self.n_u*N].reshape((N,self.n_u)).T
slack_val = z_sol[self.n_x*(N+1)+self.n_u*N:]
cost_val = sol['f']
else:
print('Warning! Solved, but with slack norm of %g is greater than 1e-8!' % la.norm(slack_val))
feasible = False
x_pred = None
u_pred = None
cost_val = None
else:
print(solver.stats()['return_status'])
feasible = False
x_pred = None
u_pred = None
cost_val = None
# pdb.set_trace()
# pdb.set_trace()
return x_pred, u_pred, cost_val
def create_rocket_stage_phase(mocp, phase_name, p, **kwargs):
is_powered = kwargs['is_powered']
has_heating = kwargs['has_heating']
drag_area = kwargs['drag_area']
maxQ = kwargs['maxQ']
init_mass = kwargs['init_mass']
init_position = kwargs['init_position']
init_velocity = kwargs['init_velocity']
if is_powered:
init_steering = kwargs['init_steering']
engine_type = kwargs['engine_type']
engine_count = kwargs['engine_count']
mdot_max = kwargs['mdot_max']
mdot_min = kwargs['mdot_min']
if engine_type == 'vacuum':
effective_exhaust_velocity = kwargs['effective_exhaust_velocity']
else:
exhaust_velocity = kwargs['exhaust_velocity']
exit_area = kwargs['exit_area']
exit_pressure = kwargs['exit_pressure']
duration = mocp.create_phase(phase_name, init=0.001, n_intervals=2)
# Total vessel mass
mass = mocp.add_trajectory(phase_name, 'mass', init=init_mass)
# ECEF position vector
rx = mocp.add_trajectory(phase_name, 'rx', init=init_position[0])
ry = mocp.add_trajectory(phase_name, 'ry', init=init_position[1])
rz = mocp.add_trajectory(phase_name, 'rz', init=init_position[2])
r_vec = casadi.vertcat(rx, ry, rz)
# ECEF velocity vector
vx = mocp.add_trajectory(phase_name, 'vx', init=init_velocity[0])
vy = mocp.add_trajectory(phase_name, 'vy', init=init_velocity[1])
vz = mocp.add_trajectory(phase_name, 'vz', init=init_velocity[2])
v_vec = casadi.vertcat(vx, vy, vz)
if is_powered:
# ECEF steering unit vector
ux = mocp.add_trajectory(phase_name, 'ux', init=init_steering[0])
uy = mocp.add_trajectory(phase_name, 'uy', init=init_steering[1])
uz = mocp.add_trajectory(phase_name, 'uz', init=init_steering[2])
u_vec = casadi.vertcat(ux, uy, uz)
# Engine mass flow for a SINGLE engine
mdot = mocp.add_trajectory(phase_name, 'mdot', init=mdot_max)
mdot_rate = mocp.add_trajectory(phase_name, 'mdot_rate', init=0.0)
if has_heating:
heat = mocp.add_trajectory(phase_name, 'heat', init=0.0)
## Dynamics
r_squared = rx**2 + ry**2 + rz**2 + 1e-8
v_squared = vx**2 + vy**2 + vz**2 + 1e-8
airspeed = v_squared**0.5
r = r_squared**0.5
altitude = r - 1
r3_inv = r_squared**(-1.5)
mocp.add_path_output(
'downrange_distance',
casadi.asin(
casadi.norm_2(
casadi.cross(
r_vec / r,
casadi.SX(
[p.launch_site_x, p.launch_site_y,
p.launch_site_z])))))
mocp.add_path_output('altitude', altitude)
mocp.add_path_output('airspeed', airspeed)
mocp.add_path_output('vertical_speed', (r_vec.T / r) @ v_vec)
mocp.add_path_output(
'horizontal_speed_ECEF',
casadi.norm_2(v_vec - ((r_vec.T / r) @ v_vec) * (r_vec / r)))
# Gravitational acceleration (mu=1 is omitted)
gx = -r3_inv * rx
gy = -r3_inv * ry
gz = -r3_inv * rz
# Atmosphere
atmosphere_fraction = casadi.exp(-altitude / p.scale_height)
air_pressure = p.air_pressure_MSL * atmosphere_fraction
air_density = p.air_density_MSL * atmosphere_fraction
dynamic_pressure = 0.5 * air_density * v_squared
mocp.add_path_output('dynamic_pressure', dynamic_pressure)
if is_powered:
lateral_airspeed = casadi.sqrt(
casadi.sumsqr(v_vec - ((v_vec.T @ u_vec) * u_vec)) + 1e-8)
lateral_dynamic_pressure = 0.5 * air_density * lateral_airspeed * airspeed # Same as Q * sin(alpha), but without trigonometry
mocp.add_path_output('lateral_dynamic_pressure',
lateral_dynamic_pressure)
mocp.add_path_output(
'alpha',
casadi.acos((v_vec.T @ u_vec) / airspeed) * 180.0 / pi)
mocp.add_path_output(
'pitch', 90.0 - casadi.acos((r_vec.T / r) @ u_vec) * 180.0 / pi)
# Thrust force
if is_powered:
if engine_type == 'vacuum':
engine_thrust = effective_exhaust_velocity * mdot
else:
engine_thrust = exhaust_velocity * mdot + exit_area * (
exit_pressure - air_pressure)
total_thrust = engine_thrust * engine_count
mocp.add_path_output('total_thrust', total_thrust)
Tx = total_thrust * ux
Ty = total_thrust * uy
Tz = total_thrust * uz
else:
Tx = 0.0
Ty = 0.0
Tz = 0.0
# Drag force
drag_factor = (-0.5 * drag_area) * air_density * airspeed
Dx = drag_factor * vx
Dy = drag_factor * vy
Dz = drag_factor * vz
# Coriolis Acceleration
ax_Coriolis = 2 * vy * p.body_rotation_speed
ay_Coriolis = -2 * vx * p.body_rotation_speed
az_Coriolis = 0.0
# Centrifugal Acceleration
ax_Centrifugal = rx * p.body_rotation_speed**2
ay_Centrifugal = ry * p.body_rotation_speed**2
az_Centrifugal = 0.0
# Acceleration
ax = gx + ax_Coriolis + ax_Centrifugal + (Dx + Tx) / mass
ay = gy + ay_Coriolis + ay_Centrifugal + (Dy + Ty) / mass
az = gz + az_Coriolis + az_Centrifugal + (Dz + Tz) / mass
if is_powered:
mocp.set_derivative(mass, -engine_count * mdot)
mocp.set_derivative(mdot, mdot_rate)
else:
mocp.set_derivative(mass, 0.0)
mocp.set_derivative(rx, vx)
mocp.set_derivative(ry, vy)
mocp.set_derivative(rz, vz)
mocp.set_derivative(vx, ax)
mocp.set_derivative(vy, ay)
mocp.set_derivative(vz, az)
# Heat flow
heat_flow = 1e-4 * (air_density**0.5) * (airspeed**3.15)
mocp.add_path_output('heat_flow', heat_flow)
if has_heating:
mocp.set_derivative(heat, heat_flow)
## Constraints
# Duration is positive and bounded
mocp.add_constraint(duration > 0.006)
mocp.add_constraint(duration < 10.0)
# Mass is positive
mocp.add_path_constraint(mass > 1e-6)
# maxQ soft constraint
weight_dynamic_pressure_penalty = mocp.get_parameter(
'weight_dynamic_pressure_penalty')
slack_maxQ = mocp.add_trajectory(phase_name, 'slack_maxQ', init=10.0)
mocp.add_path_constraint(slack_maxQ > 0.0)
mocp.add_mean_objective(weight_dynamic_pressure_penalty * slack_maxQ)
mocp.add_path_constraint(dynamic_pressure / maxQ < 1.0 + slack_maxQ)
if is_powered:
# u is a unit vector
mocp.add_path_constraint(ux**2 + uy**2 + uz**2 == 1.0)
# Do not point down
mocp.add_path_constraint((r_vec / r).T @ u_vec > -0.2)
# Engine flow limits
mocp.add_path_constraint(mdot_min < mdot)
mocp.add_path_constraint(mdot < mdot_max)
# Throttle rate reduction
mocp.add_mean_objective(1e-6 * mdot_rate**2)
# Lateral dynamic pressure penalty
weight_lateral_dynamic_pressure_penalty = mocp.get_parameter(
'weight_lateral_dynamic_pressure_penalty')
mocp.add_mean_objective(
weight_lateral_dynamic_pressure_penalty *
(lateral_dynamic_pressure / p.maxQ_sin_alpha)**8)
variables = locals().copy()
RocketStagePhase = collections.namedtuple('RocketStagePhase',
sorted(list(variables.keys())))
return RocketStagePhase(**variables)