def expansion_ratio_from_pressure(chamber_pressure, exit_pressure, gamma,
oxamide_fraction):
"""Find the nozzle expansion ratio from the chamber and exit pressures.
See :ref:`expansion-ratio-tutorial-label` for a physical description of the
expansion ratio.
Reference: Rocket Propulsion Elements, 8th Edition, Equation 3-25
Arguments:
chamber_pressure (scalar): Nozzle stagnation chamber pressure [units: pascal].
exit_pressure (scalar): Nozzle exit static pressure [units: pascal].
gamma (scalar): Exhaust gas ratio of specific heats [units: dimensionless].
Returns:
scalar: Expansion ratio :math:`\\epsilon = A_e / A_t` [units: dimensionless]
"""
chamber_pressure = cas.fmax(
chamber_pressure, dubious_min_combustion_pressure(oxamide_fraction))
chamber_pressure = cas.fmax(chamber_pressure, exit_pressure * 1.5)
AtAe = ((gamma + 1) / 2) ** (1 / (gamma - 1)) \
* (exit_pressure / chamber_pressure) ** (1 / gamma) \
* cas.sqrt((gamma + 1) / (gamma - 1) * (1 - (exit_pressure / chamber_pressure) ** ((gamma - 1) / gamma)))
er = 1 / AtAe
return er
def Cd_wave_rae2822(Cl, mach, sweep=0.):
"""
A curve fit I did to RAE2822 airfoil data.
Within -0.4 < CL < 0.75 and 0 < mach < ~0.9, has R^2 = 0.9982.
See: C:\Projects\GitHub\firefly_aerodynamics\MSES Interface\analysis\rae2822
:param Cl: Lift coefficient
:param mach: Mach number
:param sweep: Sweep angle, in deg
:return: Wave drag coefficient.
"""
mach = cas.fmax(mach, 0)
sweep_rad = np.pi / 180 * sweep
mach_perpendicular = mach * cas.cos(
sweep_rad) # Relation from FVA Eq. 8.176
Cl_perpendicular = Cl / cas.cos(
sweep_rad)**2 # Relation from FVA Eq. 8.177
# Coeffs
c2 = 4.5776476424519119e+00
mc0 = 9.5623337929607111e-01
mc1 = 2.0552787101770234e-01
mc2 = 1.1259268018737063e+00
mc3 = 1.9538856688443659e-01
m = mach_perpendicular
l = Cl_perpendicular
Cd_wave = cas.fmax(m - (mc0 - mc1 * cas.sqrt(mc2 +
(l - mc3)**2)), 0)**2 * c2
return Cd_wave
def Cd_wave_e216(Cl, mach, sweep=0.):
"""
A curve fit I did to Eppler 216 airfoil data.
Within -0.4 < CL < 0.75 and 0 < mach < ~0.9, has R^2 = 0.9982.
See: C:\Projects\GitHub\firefly_aerodynamics\MSES Interface\analysis\e216
:param Cl: Lift coefficient
:param mach: Mach number
:param sweep: Sweep angle, in deg
:return: Wave drag coefficient.
"""
mach = cas.fmax(mach, 0)
sweep_rad = np.pi / 180 * sweep
mach_perpendicular = mach * cas.cos(
sweep_rad) # Relation from FVA Eq. 8.176
Cl_perpendicular = Cl / cas.cos(
sweep_rad)**2 # Relation from FVA Eq. 8.177
# Coeffs
c0 = 7.2685945744797997e-01
c1 = -1.5483144040727698e-01
c3 = 2.1305118052118968e-01
c4 = 7.8812272501525316e-01
c5 = 3.3888938102072169e-03
l0 = 1.5298928303149546e+00
l1 = 5.2389999717540392e-01
m = mach_perpendicular
l = Cl_perpendicular
Cd_wave = (cas.fmax(m - (c0 + c1 * cas.sqrt(c3 +
(l - c4)**2) + c5 * l), 0) *
(l0 + l1 * l))**2
return Cd_wave
def apply_finish_line_constraints(
self, opti, endx, endy, endtheta, finish_line, direction, backwards=False
):
# Transform robot geometry to pose
# Only need to consider front or back edge, depending on direction
p1, p2 = self.GEOMETRY[1] if backwards else self.GEOMETRY[0]
x1, y1 = rotate_around_origin(p1, endtheta)
x2, y2 = rotate_around_origin(p2, endtheta)
x1 += endx
y1 += endy
x2 += endx
y2 += endy
# Enforce that at least one corner crosses the line
(x1_fin, y1_fin), (x2_fin, y2_fin) = finish_line
if direction in ["right", "left"]:
assert x1_fin == x2_fin
if direction == "right":
opti.subject_to(ca.fmax(x1, x2) > x1_fin)
else:
opti.subject_to(ca.fmin(x1, x2) < x1_fin)
elif direction in ["up", "down"]:
assert y1_fin == y2_fin
if direction == "up":
opti.subject_to(ca.fmax(y1, y2) > y1_fin)
else:
opti.subject_to(ca.fmin(y1, y2) < y1_fin)
else:
raise Exception(f"Unknown direction '{direction}")
def Cl_rae2822(alpha, Re_c):
# A curve fit I did to a RAE2822 airfoil, 2D XFoil data. Incompressible flow.
# Within -2 < alpha < 12 and 10^4 < Re_c < 10^6, has R^2 = 0.9857
# Likely valid from -6 < alpha < 12 and 10^4 < Re_c < 10^6.
# See: C:\Projects\GitHub\firefly_aerodynamics\Gists and Ideas\XFoil Drag Fitting\rae2822
Re_c = cas.fmax(Re_c, 1)
log10_Re = cas.log10(Re_c)
# Coeffs
a1l = 5.5686866813855172e-02
a1t = 9.7472055628494134e-02
a4l = -7.2145733312046152e-09
a4t = -3.6886704372829236e-06
atr = 8.3723547264375520e-01
atr2 = -8.3128119739031697e-02
c0l = -4.9103908291438701e-02
c0t = 2.3903424824298553e-01
ctr = 1.3082854754897108e+01
rtr = 2.6963082864300731e+00
a = alpha
r = log10_Re
Cl = (c0t + a1t * a + a4t * a**4) * 1 / (
1 + cas.exp(ctr - rtr * r - atr * a - atr2 * a**2)) + (
c0l + a1l * a + a4l * a**4) * (
1 - 1 / (1 + cas.exp(ctr - rtr * r - atr * a - atr2 * a**2)))
return Cl
def Cd_profile_rae2822(alpha, Re_c):
# A curve fit I did to a RAE2822 airfoil, 2D XFoil data. Incompressible flow.
# Within -2 < alpha < 12 and 10^4 < Re_c < 10^6, has R^2 = 0.9995
# Likely valid from -6 < alpha < 12 and 10^4 < Re_c < Inf.
# see: C:\Projects\GitHub\firefly_aerodynamics\Gists and Ideas\XFoil Drag Fitting\e216
Re_c = cas.fmax(Re_c, 1)
log10_Re = cas.log10(Re_c)
# Coeffs
at = 8.1034027621509015e+00
c0l = -8.4296746456429639e-01
c0t = -1.3700609138855402e+00
kart = -4.1609994062600880e-01
kat = 5.9510959342452441e-01
krt = -7.1938030052506197e-01
r1l = 1.1548628822014631e-01
r1t = -4.9133662875044504e-01
rt = 5.0070459892411696e+00
a = alpha
r = log10_Re
log10_Cd = (c0t + r1t * (r - 4)) * (
1 / (1 + cas.exp(kat * (a - at) + krt * (r - rt) + kart * (a - at) * (r - rt)))) + (
c0l + r1l * (r - 4)) * (
1 - 1 / (1 + cas.exp(kat * (a - at) + krt * (r - rt) + kart * (a - at) * (r - rt))))
Cd = 10 ** log10_Cd
return Cd
def Cd_profile_e216(alpha, Re_c):
# A curve fit I did to a Eppler 216 (e216) airfoil, 2D XFoil data. Incompressible flow.
# Within -2 < alpha < 12 and 10^4 < Re_c < 10^6, has R^2 = 0.9995
# Likely valid from -6 < alpha < 12 and 10^4 < Re_c < 10^6.
# see: C:\Projects\GitHub\firefly_aerodynamics\Gists and Ideas\XFoil Drag Fitting\e216
Re_c = cas.fmax(Re_c, 1)
log10_Re = cas.log10(Re_c)
# Coeffs
a1l = 4.7167470806940448e-02
a1t = 7.5663005080888857e-02
a2l = 8.7552076545610764e-04
a4t = 1.1220763679805319e-05
atr = 4.2456038382581129e-01
c0l = -1.4099657419753771e+00
c0t = -2.3855286371940609e+00
ctr = 9.1474872611212135e+01
rtr = 3.0218483612170434e+01
rtr2 = -2.4515094313899279e+00
a = alpha
r = log10_Re
log10_Cd = (c0t + a1t * a + a4t * a**4) * 1 / (
1 + cas.exp(ctr - rtr * r - atr * a - rtr2 * r**2)) + (
c0l + a1l * a + a2l * a**2) * (
1 - 1 / (1 + cas.exp(ctr - rtr * r - atr * a - rtr2 * r**2)))
Cd = 10**log10_Cd
return Cd
def Cd_profile_2412(alpha, Re_c):
# A curve fit I did to a NACA 2412 airfoil in incompressible flow.
# Within -2 < alpha < 12 and 10^5 < Re_c < 10^7, has R^2 = 0.9713
print(
"Warning: Cd_profile_e216() recommended over Cd_profile_2412(); those are MUCH more accurate fits."
)
Re_c = cas.fmax(Re_c, 1)
log_Re = cas.log(Re_c)
CD0 = -5.249
Re0 = 15.61
Re1 = 15.31
alpha0 = 1.049
alpha1 = -4.715
cx = 0.009528
cxy = -0.00588
cy = 0.04838
log_CD = CD0 + cx * (alpha - alpha0)**2 + cy * (log_Re - Re0)**2 + cxy * (
alpha - alpha1) * (log_Re - Re1
) # basically, a rotated paraboloid in logspace
CD = cas.exp(log_CD)
return CD
def Cl_e216(alpha, Re_c):
# A curve fit I did to a Eppler 216 (e216) airfoil, 2D XFoil data. Incompressible flow.
# Within -2 < alpha < 12 and 10^4 < Re_c < 10^6, has R^2 = 0.9994
# Likely valid from -6 < alpha < 12 and 10^4 < Re_c < Inf.
# See: C:\Projects\GitHub\firefly_aerodynamics\Gists and Ideas\XFoil Drag Fitting\e216
Re_c = cas.fmax(Re_c, 1)
log10_Re = cas.log10(Re_c)
# Coeffs
a1l = 3.0904412662858878e-02
a1t = 9.6452654383488254e-02
a4t = -2.5633334023068302e-05
asl = 6.4175433185427011e-01
atr = 3.6775107602844948e-01
c0l = -2.5909363461176749e-01
c0t = 8.3824440586718862e-01
ctr = 1.1431810545735890e+02
ksl = 5.3416670116733611e-01
rtr = 3.9713338634462829e+01
rtr2 = -3.3634858542657771e+00
xsl = -1.2220899840236835e-01
a = alpha
r = log10_Re
Cl = (c0t + a1t * a + a4t * a**4) * 1 / (
1 + cas.exp(ctr - rtr * r - atr * a - rtr2 * r**2)) + (
c0l + a1l * a + asl / (1 + cas.exp(-ksl * (a - xsl)))) * (
1 - 1 / (1 + cas.exp(ctr - rtr * r - atr * a - rtr2 * r**2)))
return Cl
def _include_statistics_eqs_of_stochastic_variables(
self, exp_phi, ls_factor, model, problem):
self.stochastic_variables = vertcat(*self.stochastic_variables)
for i in range(self.stochastic_variables.numel()):
var = self.stochastic_variables[i]
if var.is_symbolic():
name = var.name()
else:
name = 'stoch_var_' + str(i)
_, _, _ = self._include_statistics_of_expression(
var, name, exp_phi, ls_factor, model, problem)
for i in range(self.socp.n_g_stochastic):
var = self.socp.g_stochastic_ineq[i]
rhs = self.socp.g_stochastic_rhs[i]
name = 'stoch_constr_' + str(i)
[stoch_ineq_mean, stoch_ineq_var,
_] = self._include_statistics_of_expression(
var, name, exp_phi, ls_factor, model, problem)
stoch_cosntr_viol_prob = problem.create_optimization_theta(
name + '_viol_prob', new_theta_opt_max=0.0)
k_viol = sqrt(self.socp.g_stochastic_prob[i] /
(1 - self.socp.g_stochastic_prob[i]))
problem.include_time_equality(
stoch_cosntr_viol_prob -
(k_viol * sqrt(fmax(1e-6, stoch_ineq_var)) + stoch_ineq_mean -
rhs),
when='end')
def vec_max(x):
""" Compute (symbolically) the maximum element in a vector
Parameters
----------
x : nx1 or array
The symbolic input array
"""
n, _ = x.shape
if n == 1:
return x[0]
c = fmax(x[0], x[1])
if n > 2:
for i in range(1, n - 1):
c = fmax(c, x[i + 1])
return c
def atmos(h):
"""Compute press, density and temperature at a given altitude.
Args:
h (float or ndarray): Altitude (in meters).
Returns:
(float, float, float) or (ndarray, ndarray, ndarray):
Air pressure (Pa), density (kg/m3), and temperature (K).
"""
T = ca.fmax(288.15 - 0.0065 * h, 216.65)
rhotrop = 1.225 * (T / 288.15)**4.256848030018761
dhstrat = ca.fmax(0.0, h - 11000.0)
rho = rhotrop * np.exp(-dhstrat / 6341.552161)
p = rho * R * T
return p, rho, T
def burn_rate_coefficient(oxamide_fraction):
"""Burn rate vs oxamide content model.
Valid from 0% to 15% oxamide. # TODO IMPLEMENT THIS
Returns:
a: propellant burn rate coefficient
[units: pascal**(-n) meter second**-1].
"""
oxamide_fraction = cas.fmax(oxamide_fraction, 0)
return a_0 * (1 - oxamide_fraction) / (1 + lamb * oxamide_fraction)
def Cf_flat_plate(Re_L):
"""
Returns the mean skin friction coefficient over a flat plate. Don't forget to double it (two sides) if you want a drag coefficient.
:param Re_L: Reynolds number, normalized to the length of the flat plate.
:return: Mean skin friction coefficient over a flat plate.
"""
Re_L = cas.fabs(Re_L)
# return 0.074 / Re_L ** 0.2 # Turbulent flat plate
# return 0.02666 * Re_L ** -0.139 # Schlichting's model, roughly accounts for laminar part ("Boundary Layer Theory" 7th Ed., pg. 644)
# return smoothmax(0.074 / Re_L ** 0.2 - 1742 / Re_L, 1.33 / Re_L ** 0.5, 1000)
return cas.fmax(0.074 / Re_L**0.2 - 1742 / Re_L, 1.33 / Re_L**0.5)
def predict():
Xc = cs.SX.sym('control', self.n_control, 1)
Xd = cs.SX.sym('disturbance', self.n_disturbance, 1)
X1 = cs.vertcat(Xd, Xc).T
X2 = cs.DM(self.model.X_train_)
RBF_length_scale = self.model.kernel_.get_params(
)['k1__k1__k2__length_scale'].reshape(1, -1)
RBF_constant = self.model.kernel_.get_params(
)['k1__k1__k1__constant_value']
Constant_constant = self.model.kernel_.get_params(
)['k1__k2__constant_value']
RQ_constant = self.model.kernel_.get_params(
)['k2__k1__constant_value']
RQ_alpha = self.model.kernel_.get_params()['k2__k2__alpha']
RQ_length_scale = self.model.kernel_.get_params(
)['k2__k2__length_scale']
# mean
K1 = RBF_constant * self.RBFn(X1, RBF_length_scale)
K2 = Constant_constant * Constant(self.n_samples)
K3 = RQ_constant * self.RationalQuadraticn(X1, RQ_alpha,
RQ_length_scale)
K = (K1 + K2) * K3
y_mean = cs.mtimes(K, self.model.alpha_) + self.model.y_train_mean
y_mean = 0.5 * (y_mean + 1) * (
self.normalizer_y.data_max_ -
self.normalizer_y.data_min_) + self.normalizer_y.data_min_
# variance: solve L L^T x = K^T
# v = linsolve_lower(self.model.L_, K.T)
# x = linsolve_uppper(self.model.L_.T, v)
x = cs.mtimes(self.LLinv, K.T)
K1 = RBF_constant * self.RBF1(X1, X1, RBF_length_scale)
K2 = Constant_constant * Constant(1)
K3 = RQ_constant * self.RationalQuadratic1(X1, X1, RQ_alpha,
RQ_length_scale)
K_ = (K1 + K2) * K3
y_var = K_ - cs.mtimes(K, x)
y_var = cs.fmax(0, y_var)
y_std = y_var**0.5
y_std = y_std * (self.normalizer_y.data_max_ -
self.normalizer_y.data_min_) / 2
predict = cs.Function('predict', [Xc, Xd], [y_mean, y_std])
self.predict_ = predict
def scattering_factor(elevation_angle):
"""
Calculates a scattering factor (a factor that gives losses due to atmospheric scattering at low elevation angles).
Source: AeroSandbox/studies/SolarPanelScattering
:param elevation_angle: Angle between the horizon and the sun [degrees]
:return: Fraction of the light that is not lost to scattering.
"""
elevation_angle = cas.fmin(cas.fmax(elevation_angle, 0), 90)
theta = 90 - elevation_angle # Angle between panel normal and the sun, in degrees
theta_rad = theta * cas.pi / 180
# # Model 1
# c = (
# 0.27891510500505767300438719757949,
# -0.015994330894744987481281839336589,
# -19.707332432605799255043166340329,
# -0.66260979582573353852126274432521
# )
# scattering_factor = c[0] + c[3] * theta_rad + cas.exp(
# c[1] * (
# cas.tan(theta_rad) + c[2] * theta_rad
# )
# )
# Model 2
c = (
-0.04636,
-0.3171
)
scattering_factor = cas.exp(
c[0] * (
cas.tan(theta_rad * 0.999) + c[1] * theta_rad
)
)
# # Model 3
# p1 = -21.74
# p2 = 282.6
# p3 = -1538
# p4 = 1786
# q1 = -923.2
# q2 = 1456
# x = theta_rad
# scattering_factor = ((p1*x**3 + p2*x**2 + p3*x + p4) /
# (x**2 + q1*x + q2))
# Keep this:
# scattering_factor = cas.fmin(cas.fmax(scattering_factor, 0), 1)
return scattering_factor
def find_closest_point_on_line(posex, posey, startx, starty, endx, endy):
#A to Position = position - line_start
a2px = posex - startx
a2py = posey - starty
#A to B = line_end - line_start
a2bx = endx - startx
a2by = endy - starty
sq_a2b = ca.fmax(0.001, a2bx**2 +
a2by**2) #max, to make sure that we are not deviding by 0
a2p_dot_a2b = a2px * a2bx + a2py * a2by
diff = a2p_dot_a2b / sq_a2b
diff = ca.if_else(diff < 0, 0, diff)
diff = ca.if_else(diff > 1, 1, diff)
#diff = ca.fmax(0, ca.fmin(1, a2p_dot_a2b/sq_a2b))
closest_point = ca.vertcat(startx + a2bx * diff, starty + a2by * diff)
return closest_point
def Cd_wave_Korn(Cl, t_over_c, mach, sweep=0, kappa_A=0.95):
"""
Wave drag_force coefficient prediction using the (very) low-fidelity Korn Equation method; derived in "Configuration Aerodynamics" by W.H. Mason, Sect. 7.5.2, pg. 7-18
:param Cl: (2D) lift_force coefficient
:param t_over_c: thickness-to-chord ratio
:param sweep: sweep angle, in degrees
:param kappa_A: Airfoil technology factor (0.95 for supercritical section, 0.87 for NACA 6-series)
:return: Wave drag coefficient
"""
mach = cas.fmax(mach, 0)
sweep_rad = np.pi / 180 * sweep
Mdd = kappa_A / cas.cos(sweep_rad) - t_over_c / cas.cos(sweep_rad) ** 2 - Cl / (10 * cas.cos(sweep_rad) ** 3)
Mcrit = Mdd - (0.1 / 80) ** (1 / 3)
# Cd_wave = 20 * cas.ramp(mach - Mcrit) ** 4
Cd_wave = 20 * cas.if_else(mach > Mcrit, mach - Mcrit, 0) ** 4
return Cd_wave
def smoothmax(value1, value2, hardness):
"""
A smooth maximum between two functions.
Useful because it's differentiable and convex!
Great writeup by John D Cook here:
https://www.johndcook.com/soft_maximum.pdf
:param value1: Value of function 1.
:param value2: Value of function 2.
:param hardness: Hardness parameter. Higher values make this closer to max(x1, x2).
:return: Soft maximum of the two supplied values.
"""
value1 = value1 * hardness
value2 = value2 * hardness
max = cas.fmax(value1, value2)
min = cas.fmin(value1, value2)
out = max + cas.log(1 + cas.exp(min - max))
out /= hardness
return out
def solar_elevation_angle(latitude, day_of_year, time):
"""
Elevation angle of the sun [degrees] for a local observer.
:param latitude: Latitude [degrees]
:param day_of_year: Julian day (1 == Jan. 1, 365 == Dec. 31)
:param time: Time after local solar noon [seconds]
:return: Solar elevation angle [degrees] (angle between horizon and sun). Returns 0 if the sun is below the horizon.
"""
# Solar elevation angle (including seasonality, latitude, and time of day)
# Source: https://www.pveducation.org/pvcdrom/properties-of-sunlight/elevation-angle
declination = declination_angle(day_of_year)
solar_elevation_angle = cas.asin(
sind(declination) * sind(latitude) + cosd(declination) *
cosd(latitude) * cosd(time / 86400 * 360)) * 180 / cas.pi # in degrees
solar_elevation_angle = cas.fmax(solar_elevation_angle, 0)
return solar_elevation_angle
def calculate_optimal_control(self):
dd_h_dudu, d_h_du = hessian(self.problem.H, self.model.u)
if is_equal(dd_h_dudu, DM.zeros(self.model.n_u, self.model.n_u)):
# TODO: Implement the case where the controls are linear on the Hamiltonian ("Bang-Bang" control)
raise Exception(
'The Hamiltonian "H" is not strictly convex with respect to the control "u". '
+ 'The obtained hessian d^2 H/du^2 = 0')
# if not ddH_dudu.is_constant():
# raise NotImplementedError('The Hessian of the Hamiltonian with respect to "u" is not constant,
# this case has not been implemented')
u_opt = -mtimes(inv(dd_h_dudu), substitute(d_h_du, self.model.u, 0))
for i in range(self.model.n_u):
if not self.problem.u_min[i] == -inf:
u_opt[i] = fmax(u_opt[i], self.problem.u_min[i])
if not self.problem.u_max[i] == inf:
u_opt[i] = fmin(u_opt[i], self.problem.u_max[i])
return u_opt
def loadGPModel(name, model, xscaler, yscaler, kernel='RBF'):
""" GP mean and variance as casadi.SX variable
"""
X = model.X_train_
x = cs.SX.sym('x', 1, X.shape[1])
# mean
if kernel == 'RBF':
K1 = CasadiRBF(x, X, model)
K2 = CasadiConstant(x, X, model)
K = K1 + K2
elif kernel == 'Matern':
K = CasadiMatern(x, X, model)
else:
raise NotImplementedError
y_mu = cs.mtimes(K, model.alpha_) + model._y_train_mean
y_mu = y_mu * yscaler.scale_ + yscaler.mean_
# variance
L_inv = solve_triangular(model.L_.T,np.eye(model.L_.shape[0]))
K_inv = L_inv.dot(L_inv.T)
if kernel == 'RBF':
K1_ = CasadiRBF(x, x, model)
K2_ = CasadiConstant(x, x, model)
K_ = K1_ + K2_
elif kernel == 'Matern':
K_ = CasadiMatern(x, x, model)
y_var = cs.diag(K_) - cs.sum2(cs.mtimes(K, K_inv)*K)
y_var = cs.fmax(y_var, 0)
y_std = cs.sqrt(y_var)
y_std *= yscaler.scale_
gpmodel = cs.Function(name, [x], [y_mu, y_std])
return gpmodel
def apply_obstacle_constraints(self, opti, xpos, ypos, theta, obstacles):
for obx, oby, obr in obstacles:
for p1, p2 in self.GEOMETRY:
# Transform robot geometry to pose
x1, y1 = rotate_around_origin(p1, theta)
x2, y2 = rotate_around_origin(p2, theta)
x1 += xpos
y1 += ypos
x2 += xpos
y2 += ypos
# Compute the closest distance between a point and a line segment
px = x2 - x1
py = y2 - y1
norm = px * px + py * py
u = ((obx - x1) * px + (oby - y1) * py) / norm
u = ca.fmax(ca.fmin(u, 1), 0)
x = x1 + u * px
y = y1 + u * py
dx = x - obx
dy = y - oby
dist = np.sqrt((dx * dx + dy * dy))
opti.subject_to(dist > obr + self.OBSTACLE_BUFFER)
def opt_mintime(reftrack: np.ndarray,
coeffs_x: np.ndarray,
coeffs_y: np.ndarray,
normvectors: np.ndarray,
pars: dict,
tpamap_path: str,
tpadata_path: str,
export_path: str,
print_debug: bool = False,
plot_debug: bool = False) -> tuple:
"""
Created by:
Fabian Christ
Documentation:
The minimum lap time problem is described as an optimal control problem, converted to a nonlinear program using
direct orthogonal Gauss-Legendre collocation and then solved by the interior-point method IPOPT. Reduced computing
times are achieved using a curvilinear abscissa approach for track description, algorithmic differentiation using
the software framework CasADi, and a smoothing of the track input data by approximate spline regression. The
vehicles behavior is approximated as a double track model with quasi-steady state tire load simplification and
nonlinear tire model.
Please refer to our paper for further information:
Christ, Wischnewski, Heilmeier, Lohmann
Time-Optimal Trajectory Planning for a Race Car Considering Variable Tire-Road Friction Coefficients
Inputs:
reftrack: track [x_m, y_m, w_tr_right_m, w_tr_left_m]
coeffs_x: coefficient matrix of the x splines with size (no_splines x 4)
coeffs_y: coefficient matrix of the y splines with size (no_splines x 4)
normvectors: array containing normalized normal vectors for every traj. point [x_component, y_component]
pars: parameters dictionary
tpamap_path: file path to tpa map (required for friction map loading)
tpadata_path: file path to tpa data (required for friction map loading)
export_path: path to output folder for warm start files and solution files
print_debug: determines if debug messages are printed
plot_debug: determines if debug plots are shown
Outputs:
alpha_opt: solution vector of the optimization problem containing the lateral shift in m for every point
v_opt: velocity profile for the raceline
"""
# ------------------------------------------------------------------------------------------------------------------
# PREPARE TRACK INFORMATION ----------------------------------------------------------------------------------------
# ------------------------------------------------------------------------------------------------------------------
# spline lengths
spline_lengths_refline = tph.calc_spline_lengths.calc_spline_lengths(
coeffs_x=coeffs_x, coeffs_y=coeffs_y)
# calculate heading and curvature (numerically)
kappa_refline = tph.calc_head_curv_num. \
calc_head_curv_num(path=reftrack[:, :2],
el_lengths=spline_lengths_refline,
is_closed=True,
stepsize_curv_preview=pars["curv_calc_opts"]["d_preview_curv"],
stepsize_curv_review=pars["curv_calc_opts"]["d_review_curv"],
stepsize_psi_preview=pars["curv_calc_opts"]["d_preview_head"],
stepsize_psi_review=pars["curv_calc_opts"]["d_review_head"])[1]
# close track
kappa_refline_cl = np.append(kappa_refline, kappa_refline[0])
w_tr_left_cl = np.append(reftrack[:, 3], reftrack[0, 3])
w_tr_right_cl = np.append(reftrack[:, 2], reftrack[0, 2])
# step size along the reference line
h = pars["stepsize_opts"]["stepsize_reg"]
# optimization steps (0, 1, 2 ... end point/start point)
steps = [i for i in range(kappa_refline_cl.size)]
# number of control intervals
N = steps[-1]
# station along the reference line
s_opt = np.linspace(0.0, N * h, N + 1)
# interpolate curvature of reference line in terms of steps
kappa_interp = ca.interpolant('kappa_interp', 'linear', [steps],
kappa_refline_cl)
# interpolate track width (left and right to reference line) in terms of steps
w_tr_left_interp = ca.interpolant('w_tr_left_interp', 'linear', [steps],
w_tr_left_cl)
w_tr_right_interp = ca.interpolant('w_tr_right_interp', 'linear', [steps],
w_tr_right_cl)
# describe friction coefficients from friction map with linear equations or gaussian basis functions
if pars["optim_opts"]["var_friction"] is not None:
w_mue_fl, w_mue_fr, w_mue_rl, w_mue_rr, center_dist = opt_mintime_traj.src. \
approx_friction_map.approx_friction_map(reftrack=reftrack,
normvectors=normvectors,
tpamap_path=tpamap_path,
tpadata_path=tpadata_path,
pars=pars,
dn=pars["optim_opts"]["dn"],
n_gauss=pars["optim_opts"]["n_gauss"],
print_debug=print_debug,
plot_debug=plot_debug)
# ------------------------------------------------------------------------------------------------------------------
# DIRECT GAUSS-LEGENDRE COLLOCATION --------------------------------------------------------------------------------
# ------------------------------------------------------------------------------------------------------------------
# degree of interpolating polynomial
d = 3
# legendre collocation points
tau = np.append(0, ca.collocation_points(d, 'legendre'))
# coefficient matrix for formulating the collocation equation
C = np.zeros((d + 1, d + 1))
# coefficient matrix for formulating the collocation equation
D = np.zeros(d + 1)
# coefficient matrix for formulating the collocation equation
B = np.zeros(d + 1)
# construct polynomial basis
for j in range(d + 1):
# construct Lagrange polynomials to get the polynomial basis at the collocation point
p = np.poly1d([1])
for r in range(d + 1):
if r != j:
p *= np.poly1d([1, -tau[r]]) / (tau[j] - tau[r])
# evaluate polynomial at the final time to get the coefficients of the continuity equation
D[j] = p(1.0)
# evaluate time derivative of polynomial at collocation points to get the coefficients of continuity equation
p_der = np.polyder(p)
for r in range(d + 1):
C[j, r] = p_der(tau[r])
# evaluate integral of the polynomial to get the coefficients of the quadrature function
pint = np.polyint(p)
B[j] = pint(1.0)
# ------------------------------------------------------------------------------------------------------------------
# STATE VARIABLES --------------------------------------------------------------------------------------------------
# ------------------------------------------------------------------------------------------------------------------
# number of state variables
nx = 5
# velocity [m/s]
v_n = ca.SX.sym('v_n')
v_s = 50
v = v_s * v_n
# side slip angle [rad]
beta_n = ca.SX.sym('beta_n')
beta_s = 0.5
beta = beta_s * beta_n
# yaw rate [rad/s]
omega_z_n = ca.SX.sym('omega_z_n')
omega_z_s = 1
omega_z = omega_z_s * omega_z_n
# lateral distance to reference line (positive = left) [m]
n_n = ca.SX.sym('n_n')
n_s = 5.0
n = n_s * n_n
# relative angle to tangent on reference line [rad]
xi_n = ca.SX.sym('xi_n')
xi_s = 1.0
xi = xi_s * xi_n
# scaling factors for state variables
x_s = np.array([v_s, beta_s, omega_z_s, n_s, xi_s])
# put all states together
x = ca.vertcat(v_n, beta_n, omega_z_n, n_n, xi_n)
# ------------------------------------------------------------------------------------------------------------------
# CONTROL VARIABLES ------------------------------------------------------------------------------------------------
# ------------------------------------------------------------------------------------------------------------------
# number of control variables
nu = 4
# steer angle [rad]
delta_n = ca.SX.sym('delta_n')
delta_s = 0.5
delta = delta_s * delta_n
# positive longitudinal force (drive) [N]
f_drive_n = ca.SX.sym('f_drive_n')
f_drive_s = 7500.0
f_drive = f_drive_s * f_drive_n
# negative longitudinal force (brake) [N]
f_brake_n = ca.SX.sym('f_brake_n')
f_brake_s = 20000.0
f_brake = f_brake_s * f_brake_n
# lateral wheel load transfer [N]
gamma_y_n = ca.SX.sym('gamma_y_n')
gamma_y_s = 5000.0
gamma_y = gamma_y_s * gamma_y_n
# scaling factors for control variables
u_s = np.array([delta_s, f_drive_s, f_brake_s, gamma_y_s])
# put all controls together
u = ca.vertcat(delta_n, f_drive_n, f_brake_n, gamma_y_n)
# ------------------------------------------------------------------------------------------------------------------
# MODEL EQUATIONS --------------------------------------------------------------------------------------------------
# ------------------------------------------------------------------------------------------------------------------
# extract vehicle and tire parameters
veh = pars["vehicle_params_mintime"]
tire = pars["tire_params_mintime"]
# general constants
g = pars["veh_params"]["g"]
rho = pars["veh_params"]["rho_air"]
mass = pars["veh_params"]["mass"]
# curvature of reference line [rad/m]
kappa = ca.SX.sym('kappa')
# drag force [N]
f_xdrag = pars["veh_params"]["dragcoeff"] * v**2
# rolling resistance forces [N]
f_xroll_fl = 0.5 * tire["c_roll"] * mass * g * veh["wheelbase_rear"] / veh[
"wheelbase"]
f_xroll_fr = 0.5 * tire["c_roll"] * mass * g * veh["wheelbase_rear"] / veh[
"wheelbase"]
f_xroll_rl = 0.5 * tire["c_roll"] * mass * g * veh[
"wheelbase_front"] / veh["wheelbase"]
f_xroll_rr = 0.5 * tire["c_roll"] * mass * g * veh[
"wheelbase_front"] / veh["wheelbase"]
f_xroll = tire["c_roll"] * mass * g
# static normal tire forces [N]
f_zstat_fl = 0.5 * mass * g * veh["wheelbase_rear"] / veh["wheelbase"]
f_zstat_fr = 0.5 * mass * g * veh["wheelbase_rear"] / veh["wheelbase"]
f_zstat_rl = 0.5 * mass * g * veh["wheelbase_front"] / veh["wheelbase"]
f_zstat_rr = 0.5 * mass * g * veh["wheelbase_front"] / veh["wheelbase"]
# dynamic normal tire forces (aerodynamic downforces) [N]
f_zlift_fl = 0.5 * 0.5 * veh["clA_front"] * rho * v**2
f_zlift_fr = 0.5 * 0.5 * veh["clA_front"] * rho * v**2
f_zlift_rl = 0.5 * 0.5 * veh["clA_rear"] * rho * v**2
f_zlift_rr = 0.5 * 0.5 * veh["clA_rear"] * rho * v**2
# dynamic normal tire forces (load transfers) [N]
f_zdyn_fl = (-0.5 * veh["cog_z"] / veh["wheelbase"] *
(f_drive + f_brake - f_xdrag - f_xroll) -
veh["k_roll"] * gamma_y)
f_zdyn_fr = (-0.5 * veh["cog_z"] / veh["wheelbase"] *
(f_drive + f_brake - f_xdrag - f_xroll) +
veh["k_roll"] * gamma_y)
f_zdyn_rl = (0.5 * veh["cog_z"] / veh["wheelbase"] *
(f_drive + f_brake - f_xdrag - f_xroll) -
(1.0 - veh["k_roll"]) * gamma_y)
f_zdyn_rr = (0.5 * veh["cog_z"] / veh["wheelbase"] *
(f_drive + f_brake - f_xdrag - f_xroll) +
(1.0 - veh["k_roll"]) * gamma_y)
# sum of all normal tire forces [N]
f_z_fl = f_zstat_fl + f_zlift_fl + f_zdyn_fl
f_z_fr = f_zstat_fr + f_zlift_fr + f_zdyn_fr
f_z_rl = f_zstat_rl + f_zlift_rl + f_zdyn_rl
f_z_rr = f_zstat_rr + f_zlift_rr + f_zdyn_rr
# slip angles [rad]
alpha_fl = delta - ca.atan(
(v * ca.sin(beta) + veh["wheelbase_front"] * omega_z) /
(v * ca.cos(beta) - 0.5 * veh["track_width_front"] * omega_z))
alpha_fr = delta - ca.atan(
(v * ca.sin(beta) + veh["wheelbase_front"] * omega_z) /
(v * ca.cos(beta) + 0.5 * veh["track_width_front"] * omega_z))
alpha_rl = ca.atan(
(-v * ca.sin(beta) + veh["wheelbase_rear"] * omega_z) /
(v * ca.cos(beta) - 0.5 * veh["track_width_rear"] * omega_z))
alpha_rr = ca.atan(
(-v * ca.sin(beta) + veh["wheelbase_rear"] * omega_z) /
(v * ca.cos(beta) + 0.5 * veh["track_width_rear"] * omega_z))
# lateral tire forces [N]
f_y_fl = (pars["optim_opts"]["mue"] * f_z_fl *
(1 + tire["eps_front"] * f_z_fl / tire["f_z0"]) *
ca.sin(tire["C_front"] *
ca.atan(tire["B_front"] * alpha_fl - tire["E_front"] *
(tire["B_front"] * alpha_fl -
ca.atan(tire["B_front"] * alpha_fl)))))
f_y_fr = (pars["optim_opts"]["mue"] * f_z_fr *
(1 + tire["eps_front"] * f_z_fr / tire["f_z0"]) *
ca.sin(tire["C_front"] *
ca.atan(tire["B_front"] * alpha_fr - tire["E_front"] *
(tire["B_front"] * alpha_fr -
ca.atan(tire["B_front"] * alpha_fr)))))
f_y_rl = (pars["optim_opts"]["mue"] * f_z_rl *
(1 + tire["eps_rear"] * f_z_rl / tire["f_z0"]) *
ca.sin(tire["C_rear"] *
ca.atan(tire["B_rear"] * alpha_rl - tire["E_rear"] *
(tire["B_rear"] * alpha_rl -
ca.atan(tire["B_rear"] * alpha_rl)))))
f_y_rr = (pars["optim_opts"]["mue"] * f_z_rr *
(1 + tire["eps_rear"] * f_z_rr / tire["f_z0"]) *
ca.sin(tire["C_rear"] *
ca.atan(tire["B_rear"] * alpha_rr - tire["E_rear"] *
(tire["B_rear"] * alpha_rr -
ca.atan(tire["B_rear"] * alpha_rr)))))
# longitudinal tire forces [N]
f_x_fl = 0.5 * f_drive * veh["k_drive_front"] + 0.5 * f_brake * veh[
"k_brake_front"] - f_xroll_fl
f_x_fr = 0.5 * f_drive * veh["k_drive_front"] + 0.5 * f_brake * veh[
"k_brake_front"] - f_xroll_fr
f_x_rl = 0.5 * f_drive * (1 - veh["k_drive_front"]) + 0.5 * f_brake * (
1 - veh["k_brake_front"]) - f_xroll_rl
f_x_rr = 0.5 * f_drive * (1 - veh["k_drive_front"]) + 0.5 * f_brake * (
1 - veh["k_brake_front"]) - f_xroll_rr
# longitudinal acceleration [m/s²]
ax = (f_x_rl + f_x_rr + (f_x_fl + f_x_fr) * ca.cos(delta) -
(f_y_fl + f_y_fr) * ca.sin(delta) -
pars["veh_params"]["dragcoeff"] * v**2) / mass
# lateral acceleration [m/s²]
ay = ((f_x_fl + f_x_fr) * ca.sin(delta) + f_y_rl + f_y_rr +
(f_y_fl + f_y_fr) * ca.cos(delta)) / mass
# time-distance scaling factor (dt/ds)
sf = (1.0 - n * kappa) / (v * (ca.cos(xi + beta)))
# model equations for two track model (ordinary differential equations)
dv = (sf / mass) * (
(f_x_rl + f_x_rr) * ca.cos(beta) +
(f_x_fl + f_x_fr) * ca.cos(delta - beta) +
(f_y_rl + f_y_rr) * ca.sin(beta) -
(f_y_fl + f_y_fr) * ca.sin(delta - beta) - f_xdrag * ca.cos(beta))
dbeta = sf * (
-omega_z +
(-(f_x_rl + f_x_rr) * ca.sin(beta) +
(f_x_fl + f_x_fr) * ca.sin(delta - beta) +
(f_y_rl + f_y_rr) * ca.cos(beta) +
(f_y_fl + f_y_fr) * ca.cos(delta - beta) + f_xdrag * ca.sin(beta)) /
(mass * v))
domega_z = (sf / veh["I_z"]) * (
(f_x_rr - f_x_rl) * veh["track_width_rear"] / 2 -
(f_y_rl + f_y_rr) * veh["wheelbase_rear"] +
((f_x_fr - f_x_fl) * ca.cos(delta) +
(f_y_fl - f_y_fr) * ca.sin(delta)) * veh["track_width_front"] / 2 +
((f_y_fl + f_y_fr) * ca.cos(delta) +
(f_x_fl + f_x_fr) * ca.sin(delta)) * veh["track_width_front"])
dn = sf * v * ca.sin(xi + beta)
dxi = sf * omega_z - kappa
# put all ODEs together
dx = ca.vertcat(dv, dbeta, domega_z, dn, dxi) / x_s
# ------------------------------------------------------------------------------------------------------------------
# CONTROL BOUNDARIES -----------------------------------------------------------------------------------------------
# ------------------------------------------------------------------------------------------------------------------
delta_min = -veh["delta_max"] / delta_s # min. steer angle [rad]
delta_max = veh["delta_max"] / delta_s # max. steer angle [rad]
f_drive_min = 0.0 # min. longitudinal drive force [N]
f_drive_max = veh[
"f_drive_max"] / f_drive_s # max. longitudinal drive force [N]
f_brake_min = -veh[
"f_brake_max"] / f_brake_s # min. longitudinal brake force [N]
f_brake_max = 0.0 # max. longitudinal brake force [N]
gamma_y_min = -np.inf # min. lateral wheel load transfer [N]
gamma_y_max = np.inf # max. lateral wheel load transfer [N]
# ------------------------------------------------------------------------------------------------------------------
# STATE BOUNDARIES -------------------------------------------------------------------------------------------------
# ------------------------------------------------------------------------------------------------------------------
v_min = 1.0 / v_s # min. velocity [m/s]
v_max = pars["optim_opts"]["v_max"] / v_s # max. velocity [m/s]
beta_min = -0.5 * np.pi / beta_s # min. side slip angle [rad]
beta_max = 0.5 * np.pi / beta_s # max. side slip angle [rad]
omega_z_min = -0.5 * np.pi / omega_z_s # min. yaw rate [rad/s]
omega_z_max = 0.5 * np.pi / omega_z_s # max. yaw rate [rad/s]
xi_min = -0.5 * np.pi / xi_s # min. relative angle to tangent on reference line [rad]
xi_max = 0.5 * np.pi / xi_s # max. relative angle to tangent on reference line [rad]
# ------------------------------------------------------------------------------------------------------------------
# INITIAL GUESS FOR DECISION VARIABLES -----------------------------------------------------------------------------
# ------------------------------------------------------------------------------------------------------------------
v_guess = 20.0 / v_s
# ------------------------------------------------------------------------------------------------------------------
# HELPER FUNCTIONS -------------------------------------------------------------------------------------------------
# ------------------------------------------------------------------------------------------------------------------
# continuous time dynamics
f_dyn = ca.Function('f_dyn', [x, u, kappa], [dx, sf], ['x', 'u', 'kappa'],
['dx', 'sf'])
# longitudinal tire forces [N]
f_fx = ca.Function('f_fx', [x, u], [f_x_fl, f_x_fr, f_x_rl, f_x_rr],
['x', 'u'], ['f_x_fl', 'f_x_fr', 'f_x_rl', 'f_x_rr'])
# lateral tire forces [N]
f_fy = ca.Function('f_fy', [x, u], [f_y_fl, f_y_fr, f_y_rl, f_y_rr],
['x', 'u'], ['f_y_fl', 'f_y_fr', 'f_y_rl', 'f_y_rr'])
# vertical tire forces [N]
f_fz = ca.Function('f_fz', [x, u], [f_z_fl, f_z_fr, f_z_rl, f_z_rr],
['x', 'u'], ['f_z_fl', 'f_z_fr', 'f_z_rl', 'f_z_rr'])
# longitudinal and lateral acceleration [m/s²]
f_a = ca.Function('f_a', [x, u], [ax, ay], ['x', 'u'], ['ax', 'ay'])
# ------------------------------------------------------------------------------------------------------------------
# FORMULATE NONLINEAR PROGRAM --------------------------------------------------------------------------------------
# ------------------------------------------------------------------------------------------------------------------
# initialize NLP vectors
w = []
w0 = []
lbw = []
ubw = []
J = 0
g = []
lbg = []
ubg = []
# initialize ouput vectors
x_opt = []
u_opt = []
dt_opt = []
tf_opt = []
ax_opt = []
ay_opt = []
ec_opt = []
# initialize control vectors (for regularization)
delta_p = []
F_p = []
# boundary constraint: lift initial conditions
Xk = ca.MX.sym('X0', nx)
w.append(Xk)
n_min = (-w_tr_right_interp(0) + pars["optim_opts"]["width_opt"] / 2) / n_s
n_max = (w_tr_left_interp(0) - pars["optim_opts"]["width_opt"] / 2) / n_s
lbw.append([v_min, beta_min, omega_z_min, n_min, xi_min])
ubw.append([v_max, beta_max, omega_z_max, n_max, xi_max])
w0.append([v_guess, 0.0, 0.0, 0.0, 0.0])
x_opt.append(Xk * x_s)
# loop along the racetrack and formulate path constraints & system dynamic
for k in range(N):
# add decision variables for the control
Uk = ca.MX.sym('U_' + str(k), nu)
w.append(Uk)
lbw.append([delta_min, f_drive_min, f_brake_min, gamma_y_min])
ubw.append([delta_max, f_drive_max, f_brake_max, gamma_y_max])
w0.append([0.0] * nu)
# add decision variables for the state at collocation points
Xc = []
for j in range(d):
Xkj = ca.MX.sym('X_' + str(k) + '_' + str(j), nx)
Xc.append(Xkj)
w.append(Xkj)
lbw.append([-np.inf] * nx)
ubw.append([np.inf] * nx)
w0.append([v_guess, 0.0, 0.0, 0.0, 0.0])
# loop over all collocation points
Xk_end = D[0] * Xk
sf_opt = []
for j in range(1, d + 1):
# calculate the state derivative at the collocation point
xp = C[0, j] * Xk
for r in range(d):
xp = xp + C[r + 1, j] * Xc[r]
# interpolate kappa at the collocation point
kappa_col = kappa_interp(k + tau[j])
# append collocation equations (system dynamic)
fj, qj = f_dyn(Xc[j - 1], Uk, kappa_col)
g.append(h * fj - xp)
lbg.append([0.0] * nx)
ubg.append([0.0] * nx)
# add contribution to the end state
Xk_end = Xk_end + D[j] * Xc[j - 1]
# add contribution to quadrature function
J = J + B[j] * qj * h
# add contribution to scaling factor (for calculating lap time)
sf_opt.append(B[j] * qj * h)
# calculate used energy
dt_opt.append(sf_opt[0] + sf_opt[1] + sf_opt[2])
ec_opt.append(Xk[0] * v_s * Uk[1] * f_drive_s * dt_opt[-1])
# add new decision variables for state at end of the collocation interval
Xk = ca.MX.sym('X_' + str(k + 1), nx)
w.append(Xk)
n_min = (-w_tr_right_interp(k + 1) +
pars["optim_opts"]["width_opt"] / 2.0) / n_s
n_max = (w_tr_left_interp(k + 1) -
pars["optim_opts"]["width_opt"] / 2.0) / n_s
lbw.append([v_min, beta_min, omega_z_min, n_min, xi_min])
ubw.append([v_max, beta_max, omega_z_max, n_max, xi_max])
w0.append([v_guess, 0.0, 0.0, 0.0, 0.0])
# add equality constraint
g.append(Xk_end - Xk)
lbg.append([0.0] * nx)
ubg.append([0.0] * nx)
# get tire forces
f_x_flk, f_x_frk, f_x_rlk, f_x_rrk = f_fx(Xk, Uk)
f_y_flk, f_y_frk, f_y_rlk, f_y_rrk = f_fy(Xk, Uk)
f_z_flk, f_z_frk, f_z_rlk, f_z_rrk = f_fz(Xk, Uk)
# get accelerations (longitudinal + lateral)
axk, ayk = f_a(Xk, Uk)
# path constraint: limitied engine power
g.append(Xk[0] * Uk[1])
lbg.append([-np.inf])
ubg.append([veh["power_max"] / (f_drive_s * v_s)])
# get constant friction coefficient
if pars["optim_opts"]["var_friction"] is None:
mue_fl = pars["optim_opts"]["mue"]
mue_fr = pars["optim_opts"]["mue"]
mue_rl = pars["optim_opts"]["mue"]
mue_rr = pars["optim_opts"]["mue"]
# calculate variable friction coefficients along the reference line (regression with linear equations)
elif pars["optim_opts"]["var_friction"] == "linear":
# friction coefficient for each tire
mue_fl = w_mue_fl[k + 1, 0] * Xk[3] * n_s + w_mue_fl[k + 1, 1]
mue_fr = w_mue_fr[k + 1, 0] * Xk[3] * n_s + w_mue_fr[k + 1, 1]
mue_rl = w_mue_rl[k + 1, 0] * Xk[3] * n_s + w_mue_rl[k + 1, 1]
mue_rr = w_mue_rr[k + 1, 0] * Xk[3] * n_s + w_mue_rr[k + 1, 1]
# calculate variable friction coefficients along the reference line (regression with gaussian basis functions)
elif pars["optim_opts"]["var_friction"] == "gauss":
# gaussian basis functions
sigma = 2.0 * center_dist[k + 1, 0]
n_gauss = pars["optim_opts"]["n_gauss"]
n_q = np.linspace(-n_gauss, n_gauss,
2 * n_gauss + 1) * center_dist[k + 1, 0]
gauss_basis = []
for i in range(2 * n_gauss + 1):
gauss_basis.append(
ca.exp(-(Xk[3] * n_s - n_q[i])**2 / (2 * (sigma**2))))
gauss_basis = ca.vertcat(*gauss_basis)
mue_fl = ca.dot(w_mue_fl[k + 1, :-1],
gauss_basis) + w_mue_fl[k + 1, -1]
mue_fr = ca.dot(w_mue_fr[k + 1, :-1],
gauss_basis) + w_mue_fr[k + 1, -1]
mue_rl = ca.dot(w_mue_rl[k + 1, :-1],
gauss_basis) + w_mue_rl[k + 1, -1]
mue_rr = ca.dot(w_mue_rr[k + 1, :-1],
gauss_basis) + w_mue_rr[k + 1, -1]
else:
raise ValueError("No friction coefficients are available!")
# path constraint: Kamm's Circle for each wheel
g.append(((f_x_flk / (mue_fl * f_z_flk))**2 + (f_y_flk /
(mue_fl * f_z_flk))**2))
g.append(((f_x_frk / (mue_fr * f_z_frk))**2 + (f_y_frk /
(mue_fr * f_z_frk))**2))
g.append(((f_x_rlk / (mue_rl * f_z_rlk))**2 + (f_y_rlk /
(mue_rl * f_z_rlk))**2))
g.append(((f_x_rrk / (mue_rr * f_z_rrk))**2 + (f_y_rrk /
(mue_rr * f_z_rrk))**2))
lbg.append([0.0] * 4)
ubg.append([1.0] * 4)
# path constraint: lateral wheel load transfer
g.append((
(f_y_flk + f_y_frk) * ca.cos(Uk[0] * delta_s) + f_y_rlk + f_y_rrk +
(f_x_flk + f_x_frk) * ca.sin(Uk[0] * delta_s)) * veh["cog_z"] /
((veh["track_width_front"] + veh["track_width_rear"]) / 2) -
Uk[3] * gamma_y_s)
lbg.append([0.0])
ubg.append([0.0])
# path constraint: f_drive * f_brake == 0 (no simultaneous operation of brake and accelerator pedal)
g.append(Uk[1] * Uk[2])
lbg.append([-20000.0 / (f_drive_s * f_brake_s)])
ubg.append([0.0])
# path constraint: actor dynamic
if k > 0:
sigma = (1 - kappa_interp(k) * Xk[3] * n_s) / (Xk[0] * v_s)
g.append((Uk - w[1 + (k - 1) * nx]) /
(pars["stepsize_opts"]["stepsize_reg"] * sigma))
lbg.append([
delta_min / (veh["t_delta"]), -np.inf,
f_brake_min / (veh["t_brake"]), -np.inf
])
ubg.append([
delta_max / (veh["t_delta"]), f_drive_max / (veh["t_drive"]),
np.inf, np.inf
])
# path constraint: safe trajectories with acceleration ellipse
if pars["optim_opts"]["safe_traj"]:
g.append((ca.fmax(axk, 0) / pars["optim_opts"]["ax_pos_safe"])**2 +
(ayk / pars["optim_opts"]["ay_safe"])**2)
g.append((ca.fmin(axk, 0) / pars["optim_opts"]["ax_neg_safe"])**2 +
(ayk / pars["optim_opts"]["ay_safe"])**2)
lbg.append([0.0] * 2)
ubg.append([1.0] * 2)
# append controls (for regularization)
delta_p.append(Uk[0] * delta_s)
F_p.append(Uk[1] * f_drive_s / 10000.0 + Uk[2] * f_brake_s / 10000.0)
# append outputs
x_opt.append(Xk * x_s)
u_opt.append(Uk * u_s)
tf_opt.extend([f_x_flk, f_y_flk, f_z_flk, f_x_frk, f_y_frk, f_z_frk])
tf_opt.extend([f_x_rlk, f_y_rlk, f_z_rlk, f_x_rrk, f_y_rrk, f_z_rrk])
ax_opt.append(axk)
ay_opt.append(ayk)
# boundary constraint: start states = final states
g.append(w[0] - Xk)
lbg.append([0.0] * nx)
ubg.append([0.0] * nx)
# path constraint: limited energy consumption
if pars["optim_opts"]["limit_energy"]:
g.append(ca.sum1(ca.vertcat(*ec_opt)) / 3600000.0)
lbg.append([0])
ubg.append([pars["optim_opts"]["energy_limit"]])
# formulate differentiation matrix (for regularization)
diff_matrix = np.eye(N)
for i in range(N - 1):
diff_matrix[i, i + 1] = -1.0
diff_matrix[N - 1, 0] = -1.0
# regularization (delta)
delta_p = ca.vertcat(*delta_p)
Jp_delta = ca.mtimes(ca.MX(diff_matrix), delta_p)
Jp_delta = ca.dot(Jp_delta, Jp_delta)
# regularization (f_drive + f_brake)
F_p = ca.vertcat(*F_p)
Jp_f = ca.mtimes(ca.MX(diff_matrix), F_p)
Jp_f = ca.dot(Jp_f, Jp_f)
# formulate objective
J = J + pars["optim_opts"]["penalty_F"] * Jp_f + pars["optim_opts"][
"penalty_delta"] * Jp_delta
# concatenate NLP vectors
w = ca.vertcat(*w)
g = ca.vertcat(*g)
w0 = np.concatenate(w0)
lbw = np.concatenate(lbw)
ubw = np.concatenate(ubw)
lbg = np.concatenate(lbg)
ubg = np.concatenate(ubg)
# concatenate output vectors
x_opt = ca.vertcat(*x_opt)
u_opt = ca.vertcat(*u_opt)
tf_opt = ca.vertcat(*tf_opt)
dt_opt = ca.vertcat(*dt_opt)
ax_opt = ca.vertcat(*ax_opt)
ay_opt = ca.vertcat(*ay_opt)
ec_opt = ca.vertcat(*ec_opt)
# ------------------------------------------------------------------------------------------------------------------
# CREATE NLP SOLVER ------------------------------------------------------------------------------------------------
# ------------------------------------------------------------------------------------------------------------------
# fill nlp structure
nlp = {'f': J, 'x': w, 'g': g}
# solver options
opts = {
"expand": True,
"verbose": print_debug,
"ipopt.max_iter": 2000,
"ipopt.tol": 1e-7
}
# solver options for warm start
if pars["optim_opts"]["warm_start"]:
opts_warm_start = {
"ipopt.warm_start_init_point": "yes",
"ipopt.warm_start_bound_push": 1e-9,
"ipopt.warm_start_mult_bound_push": 1e-9,
"ipopt.warm_start_slack_bound_push": 1e-9,
"ipopt.mu_init": 1e-9
}
opts.update(opts_warm_start)
# load warm start files
if pars["optim_opts"]["warm_start"]:
try:
w0 = np.loadtxt(os.path.join(export_path, 'w0.csv'))
lam_x0 = np.loadtxt(os.path.join(export_path, 'lam_x0.csv'))
lam_g0 = np.loadtxt(os.path.join(export_path, 'lam_g0.csv'))
except IOError:
print('\033[91m' + 'WARNING: Failed to load warm start files!' +
'\033[0m')
sys.exit(1)
# check warm start files
if pars["optim_opts"]["warm_start"] and not len(w0) == len(lbw):
print(
'\033[91m' +
'WARNING: Warm start files do not fit to the dimension of the NLP!'
+ '\033[0m')
sys.exit(1)
# create solver instance
solver = ca.nlpsol("solver", "ipopt", nlp, opts)
# ------------------------------------------------------------------------------------------------------------------
# SOLVE NLP --------------------------------------------------------------------------------------------------------
# ------------------------------------------------------------------------------------------------------------------
# start time measure
t0 = time.perf_counter()
# solve NLP
if pars["optim_opts"]["warm_start"]:
sol = solver(x0=w0,
lbx=lbw,
ubx=ubw,
lbg=lbg,
ubg=ubg,
lam_x0=lam_x0,
lam_g0=lam_g0)
else:
sol = solver(x0=w0, lbx=lbw, ubx=ubw, lbg=lbg, ubg=ubg)
# end time measure
tend = time.perf_counter()
# ------------------------------------------------------------------------------------------------------------------
# EXTRACT SOLUTION -------------------------------------------------------------------------------------------------
# ------------------------------------------------------------------------------------------------------------------
# helper function to extract solution for state variables, control variables, tire forces, time
f_sol = ca.Function(
'f_sol', [w], [x_opt, u_opt, tf_opt, dt_opt, ax_opt, ay_opt, ec_opt],
['w'],
['x_opt', 'u_opt', 'tf_opt', 'dt_opt', 'ax_opt', 'ay_opt', 'ec_opt'])
# extract solution
x_opt, u_opt, tf_opt, dt_opt, ax_opt, ay_opt, ec_opt = f_sol(sol['x'])
# solution for state variables
x_opt = np.reshape(x_opt, (N + 1, nx))
# solution for control variables
u_opt = np.reshape(u_opt, (N, nu))
# solution for tire forces
tf_opt = np.append(tf_opt[-12:], tf_opt[:])
tf_opt = np.reshape(tf_opt, (N + 1, 12))
# solution for time
t_opt = np.hstack((0.0, np.cumsum(dt_opt)))
# solution for acceleration
ax_opt = np.append(ax_opt[-1], ax_opt)
ay_opt = np.append(ay_opt[-1], ay_opt)
atot_opt = np.sqrt(np.power(ax_opt, 2) + np.power(ay_opt, 2))
# solution for energy consumption
ec_opt_cum = np.hstack((0.0, np.cumsum(ec_opt))) / 3600.0
# ------------------------------------------------------------------------------------------------------------------
# EXPORT SOLUTION --------------------------------------------------------------------------------------------------
# ------------------------------------------------------------------------------------------------------------------
# export data to CSVs
opt_mintime_traj.src.export_mintime_solution.export_mintime_solution(
file_path=export_path,
s=s_opt,
t=t_opt,
x=x_opt,
u=u_opt,
tf=tf_opt,
ax=ax_opt,
ay=ay_opt,
atot=atot_opt,
w0=sol["x"],
lam_x0=sol["lam_x"],
lam_g0=sol["lam_g"])
# ------------------------------------------------------------------------------------------------------------------
# PLOT & PRINT RESULTS ---------------------------------------------------------------------------------------------
# ------------------------------------------------------------------------------------------------------------------
if plot_debug:
opt_mintime_traj.src.result_plots_mintime.result_plots_mintime(
pars=pars,
reftrack=reftrack,
s=s_opt,
t=t_opt,
x=x_opt,
u=u_opt,
ax=ax_opt,
ay=ay_opt,
atot=atot_opt,
tf=tf_opt,
ec=ec_opt_cum)
if print_debug:
print("INFO: Laptime: %.3fs" % t_opt[-1])
print("INFO: NLP solving time: %.3fs" % (tend - t0))
print("INFO: Maximum abs(ay): %.2fm/s2" % np.amax(ay_opt))
print("INFO: Maximum ax: %.2fm/s2" % np.amax(ax_opt))
print("INFO: Minimum ax: %.2fm/s2" % np.amin(ax_opt))
print("INFO: Maximum total acc: %.2fm/s2" % np.amax(atot_opt))
print('INFO: Energy consumption: %.3fWh' % ec_opt_cum[-1])
return -x_opt[:-1, 3], x_opt[:-1, 0]
def _convert(self, symbol, t, y, y_dot, inputs):
""" See :meth:`CasadiConverter.convert()`. """
if isinstance(
symbol,
(
pybamm.Scalar,
pybamm.Array,
pybamm.Time,
pybamm.InputParameter,
pybamm.ExternalVariable,
),
):
return casadi.MX(symbol.evaluate(t, y, y_dot, inputs))
elif isinstance(symbol, pybamm.StateVector):
if y is None:
raise ValueError("Must provide a 'y' for converting state vectors")
return casadi.vertcat(*[y[y_slice] for y_slice in symbol.y_slices])
elif isinstance(symbol, pybamm.StateVectorDot):
if y_dot is None:
raise ValueError("Must provide a 'y_dot' for converting state vectors")
return casadi.vertcat(*[y_dot[y_slice] for y_slice in symbol.y_slices])
elif isinstance(symbol, pybamm.BinaryOperator):
left, right = symbol.children
# process children
converted_left = self.convert(left, t, y, y_dot, inputs)
converted_right = self.convert(right, t, y, y_dot, inputs)
if isinstance(symbol, pybamm.Minimum):
return casadi.fmin(converted_left, converted_right)
if isinstance(symbol, pybamm.Maximum):
return casadi.fmax(converted_left, converted_right)
# _binary_evaluate defined in derived classes for specific rules
return symbol._binary_evaluate(converted_left, converted_right)
elif isinstance(symbol, pybamm.UnaryOperator):
converted_child = self.convert(symbol.child, t, y, y_dot, inputs)
if isinstance(symbol, pybamm.AbsoluteValue):
return casadi.fabs(converted_child)
return symbol._unary_evaluate(converted_child)
elif isinstance(symbol, pybamm.Function):
converted_children = [
self.convert(child, t, y, y_dot, inputs) for child in symbol.children
]
# Special functions
if symbol.function == np.min:
return casadi.mmin(*converted_children)
elif symbol.function == np.max:
return casadi.mmax(*converted_children)
elif symbol.function == np.abs:
return casadi.fabs(*converted_children)
elif symbol.function == np.sqrt:
return casadi.sqrt(*converted_children)
elif symbol.function == np.sin:
return casadi.sin(*converted_children)
elif symbol.function == np.arcsinh:
return casadi.arcsinh(*converted_children)
elif symbol.function == np.arccosh:
return casadi.arccosh(*converted_children)
elif symbol.function == np.tanh:
return casadi.tanh(*converted_children)
elif symbol.function == np.cosh:
return casadi.cosh(*converted_children)
elif symbol.function == np.sinh:
return casadi.sinh(*converted_children)
elif symbol.function == np.cos:
return casadi.cos(*converted_children)
elif symbol.function == np.exp:
return casadi.exp(*converted_children)
elif symbol.function == np.log:
return casadi.log(*converted_children)
elif symbol.function == np.sign:
return casadi.sign(*converted_children)
elif isinstance(symbol.function, (PchipInterpolator, CubicSpline)):
return casadi.interpolant("LUT", "bspline", [symbol.x], symbol.y)(
*converted_children
)
elif symbol.function.__name__.startswith("elementwise_grad_of_"):
differentiating_child_idx = int(symbol.function.__name__[-1])
# Create dummy symbolic variables in order to differentiate using CasADi
dummy_vars = [
casadi.MX.sym("y_" + str(i)) for i in range(len(converted_children))
]
func_diff = casadi.gradient(
symbol.differentiated_function(*dummy_vars),
dummy_vars[differentiating_child_idx],
)
# Create function and evaluate it using the children
casadi_func_diff = casadi.Function("func_diff", dummy_vars, [func_diff])
return casadi_func_diff(*converted_children)
# Other functions
else:
return symbol._function_evaluate(converted_children)
elif isinstance(symbol, pybamm.Concatenation):
converted_children = [
self.convert(child, t, y, y_dot, inputs) for child in symbol.children
]
if isinstance(symbol, (pybamm.NumpyConcatenation, pybamm.SparseStack)):
return casadi.vertcat(*converted_children)
# DomainConcatenation specifies a particular ordering for the concatenation,
# which we must follow
elif isinstance(symbol, pybamm.DomainConcatenation):
slice_starts = []
all_child_vectors = []
for i in range(symbol.secondary_dimensions_npts):
child_vectors = []
for child_var, slices in zip(
converted_children, symbol._children_slices
):
for child_dom, child_slice in slices.items():
slice_starts.append(symbol._slices[child_dom][i].start)
child_vectors.append(
child_var[child_slice[i].start : child_slice[i].stop]
)
all_child_vectors.extend(
[v for _, v in sorted(zip(slice_starts, child_vectors))]
)
return casadi.vertcat(*all_child_vectors)
else:
raise TypeError(
"""
Cannot convert symbol of type '{}' to CasADi. Symbols must all be
'linear algebra' at this stage.
""".format(
type(symbol)
)
)
def f_inv_neg_relu(x):
return csd.fmax(0, -x)
def f_inv_relu(x):
return csd.fmax(0, x)
if load_dual:
with open(dual_location, 'r') as infile:
dual_vals = json.load(infile)
if len(dual_vals) != opti.ng:
raise Exception(
"Couldn't load the dual, since your problem has %i cons and the cached problem has %i cons."
% (opti.ng, len(dual_vals)))
for i in tqdm(range(opti.ng), desc="Loading dual variables:"):
opti.set_initial(opti.lam_g[i], dual_vals[i])
sind = lambda theta: cas.sin(theta * cas.pi / 180)
cosd = lambda theta: cas.cos(theta * cas.pi / 180)
tand = lambda theta: cas.tan(theta * cas.pi / 180)
atan2d = lambda y_val, x_val: cas.atan2(y_val, x_val) * 180 / np.pi
clip = lambda x, min, max: cas.fmin(cas.fmax(min, x), max)
def smoothmax(value1, value2, hardness):
"""
A smooth maximum between two functions.
Useful because it's differentiable and convex!
Great writeup by John D Cook here:
https://www.johndcook.com/soft_maximum.pdf
:param value1: Value of function 1.
:param value2: Value of function 2.
:param hardness: Hardness parameter. Higher values make this closer to max(x1, x2).
:return: Soft maximum of the two supplied values.
"""
value1 = value1 * hardness
value2 = value2 * hardness
def simplify(self, options):
if options.get('replace_parameter_expressions', False):
logger.info("Replacing parameter expressions")
simple_parameters, symbols, values = [], [], []
for p in self.parameters:
value = ca.MX(p.value)
if value.is_constant():
simple_parameters.append(p)
else:
symbols.append(p.symbol)
values.append(value)
self.parameters = simple_parameters
if len(values) > 0:
# Resolve expressions that include other, non-simple parameter
# expressions.
converged = False
while not converged:
new_values = ca.substitute(values, symbols, values)
converged = ca.is_equal(ca.veccat(*values), ca.veccat(*new_values), CASADI_COMPARISON_DEPTH)
values = new_values
if len(self.equations) > 0:
self.equations = ca.substitute(self.equations, symbols, values)
if len(self.initial_equations) > 0:
self.initial_equations = ca.substitute(self.initial_equations, symbols, values)
# Replace parameter expressions in metadata
for variable in itertools.chain(self.states, self.alg_states, self.inputs, self.parameters, self.constants):
for attribute in ast.Symbol.ATTRIBUTES:
value = getattr(variable, attribute)
if isinstance(value, ca.MX) and not value.is_constant():
[value] = ca.substitute([value], symbols, values)
setattr(variable, attribute, value)
if options.get('replace_constant_expressions', False):
logger.info("Replacing constant expressions")
simple_constants, symbols, values = [], [], []
for c in self.constants:
value = ca.MX(c.value)
if value.is_constant():
simple_constants.append(c)
else:
symbols.append(c.symbol)
values.append(value)
self.constants = simple_constants
if len(values) > 0:
# Resolve expressions that include other, non-simple parameter
# expressions.
converged = False
while not converged:
new_values = ca.substitute(values, symbols, values)
converged = ca.is_equal(ca.veccat(*values), ca.veccat(*new_values), CASADI_COMPARISON_DEPTH)
values = new_values
if len(self.equations) > 0:
self.equations = ca.substitute(self.equations, symbols, values)
if len(self.initial_equations) > 0:
self.initial_equations = ca.substitute(self.initial_equations, symbols, values)
# Replace constant expressions in metadata
for variable in itertools.chain(self.states, self.alg_states, self.inputs, self.parameters, self.constants):
for attribute in ast.Symbol.ATTRIBUTES:
value = getattr(variable, attribute)
if isinstance(value, ca.MX) and not value.is_constant():
[value] = ca.substitute([value], symbols, values)
setattr(variable, attribute, value)
if options.get('eliminate_constant_assignments', False):
logger.info("Elimating constant variable assignments")
alg_states = OrderedDict([(s.symbol.name(), s) for s in self.alg_states])
reduced_equations = []
for eq in self.equations:
if eq.is_symbolic() and eq.name() in alg_states:
constant = alg_states.pop(eq.name())
constant.value = 0.0
self.constants.append(constant)
# Skip this equation
continue
if eq.n_dep() == 2 and (eq.is_op(ca.OP_SUB) or eq.is_op(ca.OP_ADD)):
if eq.dep(0).is_symbolic() and eq.dep(0).name() in alg_states and eq.dep(1).is_constant():
variable = eq.dep(0)
value = eq.dep(1)
elif eq.dep(1).is_symbolic() and eq.dep(1).name() in alg_states and eq.dep(0).is_constant():
variable = eq.dep(1)
value = eq.dep(0)
else:
variable = None
value = None
if variable is not None:
constant = alg_states.pop(variable.name())
if eq.is_op(ca.OP_SUB):
constant.value = value
else:
constant.value = -value
self.constants.append(constant)
# Skip this equation
continue
# Keep this equation
reduced_equations.append(eq)
# Eliminate alias variables
self.alg_states = list(alg_states.values())
self.equations = reduced_equations
if options.get('replace_parameter_values', False):
logger.info("Replacing parameter values")
# N.B. Any parameter expression elimination must be done first.
unspecified_parameters, symbols, values = [], [], []
for p in self.parameters:
if ca.MX(p.value).is_constant() and ca.MX(p.value).is_regular():
symbols.append(p.symbol)
values.append(p.value)
else:
unspecified_parameters.append(p)
if len(self.equations) > 0:
self.equations = ca.substitute(self.equations, symbols, values)
if len(self.initial_equations) > 0:
self.initial_equations = ca.substitute(self.initial_equations, symbols, values)
self.parameters = unspecified_parameters
# Replace parameter values in metadata
for variable in itertools.chain(self.states, self.alg_states, self.inputs, self.parameters, self.constants):
for attribute in ast.Symbol.ATTRIBUTES:
value = getattr(variable, attribute)
if isinstance(value, ca.MX) and not value.is_constant():
[value] = ca.substitute([value], symbols, values)
setattr(variable, attribute, value)
if options.get('replace_constant_values', False):
logger.info("Replacing constant values")
# N.B. Any parameter expression elimination must be done first.
symbols = self._symbols(self.constants)
values = [v.value for v in self.constants]
if len(self.equations) > 0:
self.equations = ca.substitute(self.equations, symbols, values)
if len(self.initial_equations) > 0:
self.initial_equations = ca.substitute(self.initial_equations, symbols, values)
self.constants = []
# Replace constant values in metadata
for variable in itertools.chain(self.states, self.alg_states, self.inputs, self.parameters, self.constants):
for attribute in ast.Symbol.ATTRIBUTES:
value = getattr(variable, attribute)
if isinstance(value, ca.MX) and not value.is_constant():
[value] = ca.substitute([value], symbols, values)
setattr(variable, attribute, value)
if options.get('eliminable_variable_expression', None) is not None:
logger.info("Elimating variables that match the regular expression {}".format(options['eliminable_variable_expression']))
p = re.compile(options['eliminable_variable_expression'])
alg_states = OrderedDict([(s.symbol.name(), s) for s in self.alg_states])
variables = []
values = []
reduced_equations = []
for eq in self.equations:
if eq.is_symbolic() and eq.name() in alg_states and p.match(eq.name()):
variables.append(eq)
values.append(0.0)
del alg_states[eq.name()]
# Skip this equation
continue
if eq.n_dep() == 2 and (eq.is_op(ca.OP_SUB) or eq.is_op(ca.OP_ADD)):
if eq.dep(0).is_symbolic() and eq.dep(0).name() in alg_states and p.match(eq.dep(0).name()):
variable = eq.dep(0)
value = eq.dep(1)
elif eq.dep(1).is_symbolic() and eq.dep(1).name() in alg_states and p.match(eq.dep(1).name()):
variable = eq.dep(1)
value = eq.dep(0)
else:
variable = None
value = None
if variable is not None:
del alg_states[variable.name()]
variables.append(variable)
if eq.is_op(ca.OP_SUB):
values.append(value)
else:
values.append(-value)
# Skip this equation
continue
# Keep this equation
reduced_equations.append(eq)
# Eliminate alias variables
self.alg_states = list(alg_states.values())
self.equations = reduced_equations
if len(self.equations) > 0:
self.equations = ca.substitute(self.equations, variables, values)
if len(self.initial_equations) > 0:
self.initial_equations = ca.substitute(self.initial_equations, variables, values)
if options.get('expand_vectors', False):
logger.info("Expanding vectors")
symbols = []
values = []
for l in ['states', 'der_states', 'alg_states', 'inputs', 'parameters', 'constants']:
old_vars = getattr(self, l)
new_vars = []
for old_var in old_vars:
if old_var.symbol.numel() > 1:
rows = []
for i in range(old_var.symbol.size1()):
cols = []
for j in range(old_var.symbol.size2()):
if old_var.symbol.size1() > 1 and old_var.symbol.size2() > 1:
component_symbol = ca.MX.sym('{}[{},{}]'.format(old_var.symbol.name(), i, j))
elif old_var.symbol.size1() > 1:
component_symbol = ca.MX.sym('{}[{}]'.format(old_var.symbol.name(), i))
elif old_var.symbol.size2() > 1:
component_symbol = ca.MX.sym('{}[{}]'.format(old_var.symbol.name(), j))
else:
raise AssertionError
component_var = Variable(component_symbol, old_var.python_type)
for attribute in ast.Symbol.ATTRIBUTES:
value = ca.MX(getattr(old_var, attribute))
if value.size1() == old_var.symbol.size1() and value.size2() == old_var.symbol.size2():
setattr(component_var, attribute, value[i, j])
elif value.size1() == old_var.symbol.size1():
setattr(component_var, attribute, value[i])
elif value.size2() == old_var.symbol.size2():
setattr(component_var, attribute, value[j])
else:
assert value.size1() == 1
assert value.size2() == 1
setattr(component_var, attribute, value)
cols.append(component_var)
rows.append(cols)
symbols.append(old_var.symbol)
values.append(ca.vertcat(*[ca.horzcat(*[v.symbol for v in row]) for row in rows]))
new_vars.extend(itertools.chain.from_iterable(rows))
else:
new_vars.append(old_var)
setattr(self, l, new_vars)
if len(self.equations) > 0:
self.equations = ca.substitute(self.equations, symbols, values)
self.equations = list(itertools.chain.from_iterable(ca.vertsplit(ca.vec(eq)) for eq in self.equations))
if len(self.initial_equations) > 0:
self.initial_equations = ca.substitute(self.initial_equations, symbols, values)
self.initial_equations = list(itertools.chain.from_iterable(ca.vertsplit(ca.vec(eq)) for eq in self.initial_equations))
# Replace values in metadata
for variable in itertools.chain(self.states, self.alg_states, self.inputs, self.parameters, self.constants):
for attribute in ast.Symbol.ATTRIBUTES:
value = getattr(variable, attribute)
if isinstance(value, ca.MX) and not value.is_constant():
[value] = ca.substitute([value], symbols, values)
setattr(variable, attribute, value)
if options.get('detect_aliases', False):
logger.info("Detecting aliases")
states = OrderedDict([(s.symbol.name(), s) for s in self.states])
der_states = OrderedDict([(s.symbol.name(), s) for s in self.der_states])
alg_states = OrderedDict([(s.symbol.name(), s) for s in self.alg_states])
inputs = OrderedDict([(s.symbol.name(), s) for s in self.inputs])
all_states = OrderedDict()
all_states.update(states)
all_states.update(der_states)
all_states.update(alg_states)
all_states.update(inputs)
alias_rel = AliasRelation()
# For now, we only eliminate algebraic states.
do_not_eliminate = set(list(der_states) + list(states) + list(inputs))
reduced_equations = []
for eq in self.equations:
if eq.n_dep() == 2 and (eq.is_op(ca.OP_SUB) or eq.is_op(ca.OP_ADD)):
if eq.dep(0).is_symbolic() and eq.dep(1).is_symbolic():
if eq.dep(0).name() in alg_states:
alg_state = eq.dep(0)
other_state = eq.dep(1)
elif eq.dep(1).name() in alg_states:
alg_state = eq.dep(1)
other_state = eq.dep(0)
else:
alg_state = None
other_state = None
# If both states are algebraic, we need to decide which to eliminate
if eq.dep(0).name() in alg_states and eq.dep(1).name() in alg_states:
# Most of the time it does not matter which one we eliminate.
# The exception is if alg_state has already been aliased to a
# variable in do_not_eliminate. If this is the case, setting the
# states in the default order will cause the new canonical variable
# to be other_state, unseating (and eliminating) the current
# canonical variable (which is in do_not_eliminate).
if alias_rel.canonical_signed(alg_state.name())[0] in do_not_eliminate:
# swap the states
other_state, alg_state = alg_state, other_state
if alg_state is not None:
# Check to see if we are linking two entries in do_not_eliminate
if alias_rel.canonical_signed(alg_state.name())[0] in do_not_eliminate and \
alias_rel.canonical_signed(other_state.name())[0] in do_not_eliminate:
# Don't do anything for now, we only eliminate alg_states
pass
else:
# Eliminate alg_state by aliasing it to other_state
if eq.is_op(ca.OP_SUB):
alias_rel.add(other_state.name(), alg_state.name())
else:
alias_rel.add(other_state.name(), '-' + alg_state.name())
# To keep equations balanced, drop this equation
continue
# Keep this equation
reduced_equations.append(eq)
# Eliminate alias variables
variables, values = [], []
for canonical, aliases in alias_rel:
canonical_state = all_states[canonical]
python_type = canonical_state.python_type
start = canonical_state.start
m, M = canonical_state.min, canonical_state.max
nominal = canonical_state.nominal
fixed = canonical_state.fixed
for alias in aliases:
if alias[0] == '-':
sign = -1
alias = alias[1:]
else:
sign = 1
alias_state = all_states[alias]
variables.append(alias_state.symbol)
values.append(sign * canonical_state.symbol)
# If any of the aliases has a nonstandard type, apply it to
# the canonical state as well
if alias_state.python_type != float:
python_type = alias_state.python_type
# If any of the aliases has a nondefault start value, apply it to
# the canonical state as well
alias_state_start = ca.MX(alias_state.start)
if alias_state_start.is_regular() and not alias_state_start.is_zero():
start = sign * alias_state.start
# The intersection of all bound ranges applies
m = ca.fmax(m, alias_state.min if sign == 1 else -alias_state.max)
M = ca.fmin(M, alias_state.max if sign == 1 else -alias_state.min)
# Take the largest nominal of all aliases
nominal = ca.fmax(nominal, alias_state.nominal)
# If any of the aliases is fixed, the canonical state is as well
fixed = ca.fmax(fixed, alias_state.fixed)
del all_states[alias]
canonical_state.aliases = aliases
canonical_state.python_type = python_type
canonical_state.start = start
canonical_state.min = m
canonical_state.max = M
canonical_state.nominal = nominal
canonical_state.fixed = fixed
self.states = [v for k, v in all_states.items() if k in states]
self.der_states = [v for k, v in all_states.items() if k in der_states]
self.alg_states = [v for k, v in all_states.items() if k in alg_states]
self.inputs = [v for k, v in all_states.items() if k in inputs]
self.equations = reduced_equations
if len(self.equations) > 0:
self.equations = ca.substitute(self.equations, variables, values)
if len(self.initial_equations) > 0:
self.initial_equations = ca.substitute(self.initial_equations, variables, values)
if options.get('reduce_affine_expression', False):
logger.info("Collapsing model into an affine expression")
for equation_list in ['equations', 'initial_equations']:
equations = getattr(self, equation_list)
if len(equations) > 0:
states = ca.veccat(*self._symbols(itertools.chain(self.states, self.der_states, self.alg_states, self.inputs)))
constants = ca.veccat(*self._symbols(self.constants))
parameters = ca.veccat(*self._symbols(self.parameters))
equations = ca.veccat(*equations)
Af = ca.Function('Af', [states, constants, parameters], [ca.jacobian(equations, states)])
bf = ca.Function('bf', [states, constants, parameters], [equations])
# Work around CasADi issue #172
if len(self.constants) == 0 or not ca.depends_on(equations, constants):
constants = 0
else:
logger.warning('Not all constants have been eliminated. As a result, the affine DAE expression will use a symbolic matrix, as opposed to a numerical sparse matrix.')
if len(self.parameters) == 0 or not ca.depends_on(equations, parameters):
parameters = 0
else:
logger.warning('Not all parameters have been eliminated. As a result, the affine DAE expression will use a symbolic matrix, as opposed to a numerical sparse matrix.')
A = Af(0, constants, parameters)
b = bf(0, constants, parameters)
# Replace veccat'ed states with brand new state vectors so as to avoid the value copy operations induced by veccat.
self._states_vector = ca.MX.sym('states_vector', sum([s.numel() for s in self._symbols(self.states)]))
self._der_states_vector = ca.MX.sym('der_states_vector', sum([s.numel() for s in self._symbols(self.der_states)]))
self._alg_states_vector = ca.MX.sym('alg_states_vector', sum([s.numel() for s in self._symbols(self.alg_states)]))
self._inputs_vector = ca.MX.sym('inputs_vector', sum([s.numel() for s in self._symbols(self.inputs)]))
states_vector = ca.vertcat(self._states_vector, self._der_states_vector, self._alg_states_vector, self._inputs_vector)
equations = [ca.reshape(ca.mtimes(A, states_vector), equations.shape) + b]
setattr(self, equation_list, equations)
if options.get('expand_mx', False):
logger.info("Expanding MX graph")
if len(self.equations) > 0:
self.equations = ca.matrix_expand(self.equations)
if len(self.initial_equations) > 0:
self.initial_equations = ca.matrix_expand(self.initial_equations)
logger.info("Finished model simplification")
def trim_and_square(x):
return cd.fmax(x, 0)**2
