def generate_constraints_funcs(self):
"""
Generate callable functions which represent the algebraic constraints a(tt) and its first two derivatives
adot(tt, ttdot) and addot(tt, ttdot, ttddot).
:return: None
"""
if self.constraints_func is not None:
return
actual_symbs = self.constraints.atoms(sp.Symbol)
expected_symbs = set(self.mod.tt)
if not actual_symbs == expected_symbs:
msg = "Constraints can only converted to numerical func if all parameters are passed for substitution. " \
"Unexpected symbols: {}".format(actual_symbs.difference(expected_symbs))
raise ValueError(msg)
self.constraints_func = st.expr_to_func(self.mod.tt, self.constraints)
# now we need also the differentiated constraints (e.g. to calculate consistent velocities and accelerations)
self.constraints_d = st.time_deriv(self.constraints, self.mod.tt)
self.constraints_dd = st.time_deriv(self.constraints_d, self.mod.tt)
xx = st.row_stack(self.mod.tt, self.mod.ttd)
# this function depends on coordinates ttheta and velocities ttheta_dot
self.constraints_d_func = st.expr_to_func(xx, self.constraints_d)
zz = st.row_stack(self.mod.tt, self.mod.ttd, self.mod.ttdd)
# this function depends on coordinates ttheta and velocities ttheta_dot and accel
self.constraints_dd_func = st.expr_to_func(zz, self.constraints_dd)
def smooth_time_scaling(Tend, N, phase_fraction=.5):
"""
:param Tend:
:param N:
:param phase_fraction: fraction of Tend for smooth initial and end phase
"""
T0 = 0
T1 = Tend * phase_fraction
y0 = 0
y1 = 1
# for initial phase
poly1 = st.condition_poly(t, (T0, y0, 0, 0), (T1, y1, 0, 0))
# for end phase
poly2 = poly1.subs(t, Tend - t)
# there should be a phase in the middle with constant slope
deriv_transition = st.piece_wise((y0, t < T0), (poly1, t < T1), (y1, t < Tend - T1),
(poly2, t < Tend), (y0, True))
scaling = sp.integrate(deriv_transition, (t, T0, Tend))
time_transition = sp.integrate(deriv_transition * N / scaling, t)
# deriv_transition_func = st.expr_to_func(t, full_transition)
time_transition_func = st.expr_to_func(t, time_transition)
deriv_func = st.expr_to_func(t, deriv_transition * N / scaling)
deriv_func2 = st.expr_to_func(t, deriv_transition.diff(t) * N / scaling)
C = ipydex.Container(fetch_locals=True)
return C
def solve(problem_spec):
t = sp.Symbol('t')
planer_p2 = tp.Trajectory_Planning(problem_spec.YA_p2, problem_spec.YB_p2,
problem_spec.t0, problem_spec.tf,
problem_spec.tt)
mod = problem_spec.rhs(problem_spec.ttheta, problem_spec.tthetad,
problem_spec.u_F)
planer_p2.mod = mod
planer_p2.yy = problem_spec.output_func(problem_spec.ttheta,
problem_spec.u_F)
planer_p2.ff = mod.f # xd = f(x) + g(x)*u
planer_p2.gg = mod.g
yy = planer_p2.cal_li_derivative() # lie derivatives of the flat output
ploy_p2 = planer_p2.calc_trajectory() # planned trajectory of CuZn-ball
p2_func = st.expr_to_func(t, ploy_p2[0]) # trajectory to function
# find trajectory of Fe-ball
p1_p2 = planer_p2.ff[3].subs(problem_spec.ttheta[1], ploy_p2[0])
func_p1 = p1_p2 - ploy_p2[2]
ploy_p1 = sp.solve(func_p1, problem_spec.ttheta[0])
p1_func = st.expr_to_func(t, ploy_p1[0])
yy_4 = yy[4].subs([(problem_spec.ttheta[0], ploy_p1[0]),
(problem_spec.ttheta[1], ploy_p2[0])])
in_output_func = yy_4 - ploy_p2[4]
# input force trajectory
input_f_tra = sp.solve(in_output_func, problem_spec.u_F)
f_func = st.expr_to_func(t, input_f_tra)
# input current trajectory
f_c_func = problem_spec.force_current_function(problem_spec.ttheta,
problem_spec.u_i)
c_tra_func = (input_f_tra[0] - f_c_func).subs([(problem_spec.ttheta[0],
ploy_p1[0])])
input_c_tra = sp.solve(c_tra_func, problem_spec.u_i)
current_func = st.expr_to_func(t, input_c_tra[1])
# tracking controller
tracking_controller = tp.Tracking_controller(yy, mod.xx, problem_spec.u_F,
problem_spec.pol, ploy_p2)
control_law = tracking_controller.error_dynamics()[0] # control law
# simulate the system with control law
rhs = rhs_for_simulation(planer_p2.ff, planer_p2.gg, mod.xx, control_law)
# original initial values : [0.0008, 0.004, 0, 0]
res = odeint(rhs, problem_spec.xx0, problem_spec.tt2)
solution_data = SolutionData()
solution_data.res = res # output values of the system
solution_data.ploy_p1 = p1_func # desired full transition of p1
solution_data.ploy_p2 = p2_func # desired full transition of p2
solution_data.f_func = f_func # required magnet force input
solution_data.current_func = current_func # required current input
solution_data.coefficients = tracking_controller.coefficient # coefficients of error dynamics
solution_data.control_law = control_law # control law function
return solution_data
def solve(problem_spec):
s, t, T = sp.symbols("s, t, T")
transfer_func = problem_spec.transfer_func()
z_func, n_func = transfer_func.expand().as_numer_denom(
) # separate numerator and denominator
z_coeffs = [float(c)
for c in st.coeffs(z_func, s)] # coefficients of numerator
n_coeffs = [float(c)
for c in st.coeffs(n_func, s)] # coefficients of denominator
b_0 = z_func.coeff(s, 0)
# Boundary conditions for q and its derivative
q_a = [problem_spec.YA[0] / b_0, 0, 0, 0]
q_e = [problem_spec.YB[0] / b_0, 0, 0, 0]
# generate trajectory of q(t)
planer = tp.Trajectory_Planning(q_a, q_e, problem_spec.t0, problem_spec.tf,
problem_spec.tt)
planer.dem = n_func
planer.num = z_func
q_poly = planer.calc_trajectory()
# trajectory of input and output
u_poly, y_poly = planer.num_den_laplace(q_poly[0])
q_func = st.expr_to_func(t, q_poly[0])
u_func = st.expr_to_func(t, u_poly) # desired input trajectory function
y_func = st.expr_to_func(t, y_poly) # desired output trajectory function
# tracking controller
# numerator and denominator of controller
cd_res = cd.coprime_decomposition(z_func, n_func, problem_spec.pol)
u1, u2, fb = inputs('u1, u2, fb') # external force and feedback
SUM1 = Blockfnc(u1 - fb)
Controller = TFBlock(cd_res.f_func / cd_res.h_func, SUM1.Y)
SUM2 = Blockfnc(u2 + Controller.Y)
System = TFBlock(z_func / n_func, SUM2.Y)
loop(System.Y, fb)
t1, states = blocksimulation(6, {
u1: y_func,
u2: u_func
}) # simulate 10 seconds
t1 = t1.flatten()
bo = compute_block_ouptputs(states)
solution_data = SolutionData()
solution_data.u = u_func
solution_data.q = q_func
solution_data.yy = bo[System]
solution_data.y_func = y_func
solution_data.tt = t1
return solution_data
def test_conversion_all_funcs(self):
x1, x2, x3 = xx = st.symb_vector("x1:4")
u1, u2 = uu = st.symb_vector("u1:3")
xxuusum = sum(xx) + sum(uu)
arg = sp.tanh(xxuusum) # limit the argument to (-1, 1)*0.99
# see mpc.CassadiPrinter.__init__ for exlanation
sp_func_names = mpc.CassadiPrinter().cs_func_keys.keys()
blacklist = ["atan2", ]
flist = [getattr(sp, name) for name in sp_func_names if name not in blacklist]
# create the test_matrix
expr_list = []
for func in flist:
if func is sp.acosh:
# only defined for values > 1
expr_list.append(func(1/arg))
else:
expr_list.append(func(arg))
expr_sp = sp.Matrix(expr_list + [arg, xxuusum])
func_cs = mpc.create_casadi_func(expr_sp, xx, uu)
xxuu = list(xx) + list(uu)
func_np = st.expr_to_func(xxuu, expr_sp)
argvals = np.random.rand(len(xxuu))
argvals_cs = (argvals[:len(xx)], argvals[len(xx):])
res_np = func_np(*argvals)
res_cs = func_cs(*argvals_cs).full().squeeze()
self.assertTrue(np.allclose(res_np, res_cs))
def solve(problem_spec):
s, t, T = sp.symbols("s, t, T")
planer = tp.Trajectory_Planning(problem_spec.YA, problem_spec.YB,
problem_spec.t0, problem_spec.tf,
problem_spec.tt)
mod = problem_spec.rhs(problem_spec.xx, problem_spec.uu)
planer.mod = mod
planer.yy = problem_spec.output_func(problem_spec.xx, problem_spec.uu)
planer.ff = mod.f # xd = f(x) + g(x)*u
planer.gg = mod.g
yy = planer.cal_li_derivative()
ploy_tem = planer.calc_trajectory()
tem_func = st.expr_to_func(t, ploy_tem[0])
# tracking controller
tracking_controller = tp.Tracking_controller(yy, mod.xx, problem_spec.uu,
problem_spec.pol, ploy_tem)
control_law = tracking_controller.error_dynamics()[0] # control law
rhs = rhs_for_simulation(planer.ff, planer.gg, mod.xx, control_law)
res = odeint(rhs, problem_spec.xx0, problem_spec.tt2)
solution_data = SolutionData()
solution_data.res = res
solution_data.p2_func = tem_func
return solution_data
def calc_eqlbr(self, parameter_values=None, ttheta_guess=None, etype='one_ep', display=False, **kwargs):
"""In the simplest case(RT1 and 2) only one of the equilibrium points of
a system is used for linearization and other researches. Such a equilibrium
point is calculated by minimizing the target function for a certain
input variable.
In other case(NT) all of the equilibrium points of a system are needed, which can be calculated
by using Slover in Sympy to solve the differential equations for certain input values.
:param: uu: certain input value defined by user
:param: parameter_values: unknown system parameters in list.(Default: all parameters are known)
:param: ttheta_guess: initial values for minimizing the target function.(Default: 0)
:param: etype: 'one_ep': one equilibrium point, 'all_ep': all of the equilibrium points.
:param: display: display all information of the result of the 'fmin' fucntion
:return: equilibrium point(s) in list
"""
if parameter_values is None:
parameter_values = []
if kwargs.get('uu') is None:
# if the system doesn't have input
assert self.tau is None
uu = []
all_vars = st.row_stack(self.tt)
uu_para = []
else:
uu = kwargs.get('uu')
all_vars = st.row_stack(self.tt, self.tau)
uu_para = list(zip(self.tau, uu))
class Type_equilibrium(Enum):
one_ep = auto()
all_ep = auto()
if etype == Type_equilibrium.one_ep.name:
# set all of the derivatives of the system states to zero
state_eqns = self.eqns.subz0(self.ttd, self.ttdd)
# target function for minimizing
state_eqns_func = st.expr_to_func(all_vars, state_eqns.subs(parameter_values))
if ttheta_guess is None:
ttheta_guess = st.to_np(self.tt * 0)
def target(ttheta):
"""target function for the certain global input values uu
"""
all_values = np.r_[ttheta, uu] # combine arrays
rhs = state_eqns_func(*all_values)
return np.sum(rhs ** 2)
self.eqlbr = fmin(target, ttheta_guess, disp=display)
elif etype == Type_equilibrium.all_ep.name:
state_eqns = self.eqns.subz0(self.ttd, self.ttdd)
all_vars_value = uu_para + parameter_values
self.eqlbr = sp.solve(state_eqns.subs(all_vars_value), self.tt)
def test_penalty_expression(self):
x1, = xx = st.symb_vector('x1:2')
pe = st.penalty_expression(x1, -2, 2)
pefnc = st.expr_to_func(x1, pe)
eps = 1e-2
self.assertTrue(pefnc(1) < eps)
self.assertTrue(pefnc(5) - 25 < eps)
self.assertTrue(pefnc(-3) - 9 < eps)
def test_conversion1(self):
x1, x2, x3 = xx = st.symb_vector("x1:4")
u1, u2 = uu = st.symb_vector("u1:3")
expr_sp = sp.Matrix([x1 + x2 + x3, sp.sin(x1)*x2**x3, 1.23, 0, u1*sp.exp(u2)])
func_cs = mpc.create_casadi_func(expr_sp, xx, uu)
xxuu = list(xx) + list(uu)
func_np = st.expr_to_func(xxuu, expr_sp)
argvals = np.random.rand(len(xxuu))
argvals_cs = (argvals[:len(xx)], argvals[len(xx):])
res_np = func_np(*argvals)
res_cs = func_cs(*argvals_cs).full().squeeze()
self.assertTrue(np.allclose(res_np, res_cs))
def calc_eqlbr_rt1(mod,
uu,
sys_paras,
ttheta_guess=None,
display=False,
debug=False):
"""
In the simplest case, only one of the equilibrium points of
a nonlinear system is used for linearization.Such a equilibrium
point is calculated by minimizing the target function for a certain
input variable.
:param mod: symbolic_model
:param uu: system inputs
:param sys_paras: system parameters
:param ttheta_guess: initial guess of the equilibrium points
:param display: Set to True to print convergence messages.
:param debug: output control for debugging in unittest(False:normal
output,True: output local variables and normal output)
:return: Parameter that minimizes function
"""
# set all of the derivatives of the system states to zero
stat_eqns = mod.eqns.subz0(mod.ttd, mod.ttdd)
all_vars = st.row_stack(mod.tt, mod.uu)
# target function for minimizing
mod.stat_eqns_func = st.expr_to_func(all_vars, stat_eqns.subs(sys_paras))
if ttheta_guess is None:
ttheta_guess = st.to_np(mod.tt * 0)
def target(ttheta):
"""target function for the certain global input values uu
"""
all_vars = np.r_[ttheta, uu] # combine arrays
rhs = mod.stat_eqns_func(*all_vars)
return np.sum(rhs**2)
res = fmin(target, ttheta_guess, disp=display)
if debug:
C = ipydex.Container(fetch_locals=True)
return C, res
return res
def add_element(self, points, init_fun, update_fun, **kwargs):
"""
Add a visualiser element
:param points: 2x? SymPy matrix or list of 2x1 SymPy vectors describing the defining points as symbolic
expressions w.r.t the visualisers free variables
:param init_fun: callable with args (matplotlib axes, 2x? numpy array of points, dict of kwargs) that returns a
list of matplotlib drawables. Will get called to create all drawables needed by this element.
:param update_fun: callable with args (matplotlib axes, list of drawables, 2x? numpy array of points, dict of
kwargs) that returns a list of matplotlib drawables. Will get called every time the plot needs to be updated.
:param kwargs: arbitrary keyword arguments that get passed to init_fun and update_fun
"""
if not isinstance(points, sp.Matrix):
if isinstance(points, list):
points = st.col_stack(*points)
else:
raise Exception("'points' must be a SymPy matrix or a list of column vectors")
points_fun = st.expr_to_func(self.variables, points, keep_shape=True)
self.elements.append(VisualiserElement(points_fun, init_fun, update_fun, kwargs))
def test_conversion2(self):
x1, x2, x3 = xx = st.symb_vector("x1:4")
u1, u2 = uu = st.symb_vector("u1:3")
lmd1, lmd2 = llmd = st.symb_vector("lmd1:3")
xxuullmd = list(xx) + list(uu) + list(llmd)
expr_sp = sp.Matrix([x1 + x2 + x3, sp.sin(x1)*x2**x3, 1.23, 0, u1*sp.exp(u2), x1*lmd1 + lmd2**4])
func_cs = mpc.create_casadi_func(expr_sp, xxuullmd)
func_np = st.expr_to_func(xxuullmd, expr_sp)
argvals = np.random.rand(len(xxuullmd))
# unpack the array for lambdified function
res_np = func_np(*argvals)
# pass the whole array for casadi function
res_cs = func_cs(argvals).full().squeeze()
self.assertTrue(np.allclose(res_np, res_cs))
def test_num_trajectory_compatibility_test(self):
x1, x2, x3, x4 = xx = sp.Matrix(sp.symbols("x1, x2, x3, x4"))
u1, u2 = uu = sp.Matrix(sp.symbols("u1, u2")) # inputs
t = sp.Symbol('t')
# we want to create a random but stable matrix
np.random.seed(2805)
diag = np.diag(np.random.random(len(xx)) * -10)
T = sp.randMatrix(len(xx), len(xx), -10, 10, seed=704)
Tinv = T.inv()
A = Tinv * diag * T
B = B0 = sp.randMatrix(len(xx), len(uu), -10, 10, seed=705)
x0 = st.to_np(sp.randMatrix(len(xx), 1, -10, 10, seed=706)).squeeze()
tt = np.linspace(0, 5, 2000)
des_input = st.piece_wise((2 - t, t <= 1), (t, t < 2),
(2 * t - 2, t < 3), (4, True))
des_input_func_vec = st.expr_to_func(t,
sp.Matrix([des_input, des_input]))
mod2 = st.SimulationModel(A * xx, B, xx)
rhs3 = mod2.create_simfunction(input_function=des_input_func_vec)
XX = sc.integrate.odeint(rhs3, x0, tt)
UU = des_input_func_vec(tt)
res1 = mod2.num_trajectory_compatibility_test(tt, XX, UU)
self.assertTrue(res1)
# slightly different input signal -> other results
res2 = mod2.num_trajectory_compatibility_test(tt, XX, UU * 1.1)
self.assertFalse(res2)
def add_graph(self, content, subplot_pos=111, ax_kwargs=None, plot_kwargs=None, ax=None):
"""
Add a subplot containing an animated graph of some system variables.
:param content: Can be
- symbolic: SymPy expression, list of SymPy expressions or SymPy matrix, of some system variables or a
combination of them
- numeric: NumPy array of values to be plotted over time, rows are sample times, columns are separate plot
lines. The number of rows must match the length of the time vector.
:param subplot_pos: subplot position, this simply gets passed on to matplotlib's Figure.add_subplot() so every
valid type of specification should work.
:param ax_kwargs: keyword arguments to be passed on when creating Axes object
:param plot_kwargs: keyword arguments to be passed on when calling Axes.plot function
:param ax: optional, existing Axes object to be used for plotting, if omitted one will be created
:return: Axes object of the subplot
"""
# create Axes if necessary
if ax is None:
if ax_kwargs is None:
ax_kwargs = {}
ax = self.fig.add_subplot(subplot_pos, **ax_kwargs)
ax.grid()
assert isinstance(content, np.ndarray) or isinstance(content, sp.Expr) or isinstance(content, sp.Matrix)\
or isinstance(content, list)
if isinstance(content, np.ndarray):
assert content.ndim == 1 or content.ndim == 2, "Data must be one or two-dimensional"
assert content.shape[0] == self.n_sim_frames, "Data must have as many rows as there are entries in 't' vector"
# convert all types of symbolic content to a SymPy vector
if isinstance(content, sp.Expr):
content = sp.Matrix([content])
elif isinstance(content, list):
content = sp.Matrix(content)
# content is now a SymPy vector or an array of values
if isinstance(content, np.ndarray):
# We later expect all data to be two dimensional, so a vector must be converted to a ?x1 matrix
if content.ndim == 1:
data = np.reshape(content, (content.shape[0], 1))
else:
data = content
else:
# content is still symbolic, we need to generate the data vector ourselves
# instantiate a function that takes one row of x_sim and returns the values to plot at one time instance
expr_fun = st.expr_to_func(self.x_symb, content, keep_shape=True)
# allocate memory for the data to plot
data = np.zeros((self.n_sim_frames, len(content)))
# use the prepared function to fill the plotting data
for i in range(data.shape[0]):
data[i, :] = expr_fun(*self.x_sim[i, :]).flatten() # expr_fun returns a 2D column vector --> flatten
if plot_kwargs is None:
plot_kwargs = {}
self.axes.append((ax, data, plot_kwargs))
# adding a new subplot invalidates any cached animation we might have
self._cached_anim = None
return ax
fs = [140*mm*scale, 60*mm*scale]
fig1 = pl.figure(2, figsize=fs)
pl.plot(tt, w1fnc_v(tt), color=color_list[1], lw=1)
pl.plot(tt, bo[DI], color=color_list[0], lw=2)
pl.grid()
pl.xticks([0, 1, 2, 3])
pl.yticks([0, .5, 1])
pl.axis([0, 2, -.1, 1.3])
poly1 = st.trans_poly(t, 3, (T0, 0, 0, 0, 0), (T1, 1, 0, 0, 0))
pw_poly = st.create_piecewise(t, (T0, T1), (0, poly1, 1))
pw_poly_fnc = st.expr_to_func(t, pw_poly)
pl.plot(tt, pw_poly_fnc(tt), color=color_list[3], lw=3)
pl.savefig('pid_step2.pdf')
##
fs = [140*mm*scale, 60*mm*scale]
fig1 = pl.figure(3, figsize=fs)
pl.grid()
pl.xticks([0, 1, 2, 3])
def gen_leqs_for_acc_llmd(self, parameter_values=None):
"""
Create a callable function which returns A, bnum of the linear eqn-system
A*ww = bnum,
where ww := (ttheta_dd, llmnd).
:return: None, set self.leqs_acc_lmd_func
"""
if self.leqs_acc_lmd_func is not None and self.acc_of_lmd_func is not None:
return
if parameter_values is None:
parameter_values = []
ntt = self.ntt
nll = self.nll
self.generate_constraints_funcs()
# also respect those values, which have been passed to the constructor
parameter_values = list(self.parameter_values) + list(parameter_values)
# we use mod.eqns here because we do not want ydot-vars inside
eqns = st.concat_rows(self.mod.eqns.subs(parameter_values), self.constraints_dd)
ww = st.concat_rows(self.mod.ttdd, self.mod.llmd)
A = eqns.jacobian(ww)
b = -eqns.subz0(ww) # rhs of the leqs
Ab = st.concat_cols(A, b)
fvars = st.concat_rows(self.mod.tt, self.mod.ttd, self.mod.tau)
actual_symbs = Ab.atoms(sp.Symbol)
expected_symbs = set(fvars)
unexpected_symbs = actual_symbs.difference(expected_symbs)
if unexpected_symbs:
msg = "Equations can only converted to numerical func if all parameters are passed for substitution. " \
"Unexpected symbols: {}".format(unexpected_symbs)
raise ValueError(msg)
A_fnc = st.expr_to_func(fvars, A, keep_shape=True)
b_fnc = st.expr_to_func(fvars, b)
nargs = len(fvars)
# noinspection PyShadowingNames
def leqs_acc_lmd_func(*args):
"""
Calculate the matrices of the linear equation system for ttheta and llmd.
Assume args = (ttheta, theta_d, ttau)
:param args:
:return:
"""
assert len(args) == nargs
Anum = A_fnc(*args)
bnum = b_fnc(*args)
# theese arrays can now be passed to a linear equation solver
return Anum, bnum
self.leqs_acc_lmd_func = leqs_acc_lmd_func
def acc_of_lmd_func(*args):
"""
Calculate ttheta in dependency of args= (yy, ttau) = ((ttheta, ttheta_d, llmd), ttau)
:param args:
:return:
"""
ttheta = args[:ntt]
ttheta_d = args[ntt:2 * ntt]
llmd = args[2 * ntt:2 * ntt + nll]
ttau = args[2 * ntt + nll:]
args1 = np.concatenate((ttheta, ttheta_d, ttau))
Anum = A_fnc(*args1)
A1 = Anum[:ntt, :ntt]
A2 = Anum[:ntt, ntt:]
b1 = b_fnc(*args1)[:ntt]
ttheta_dd_res = np.linalg.solve(A1, b1 - np.dot(A2, llmd))
return ttheta_dd_res
self.acc_of_lmd_func = acc_of_lmd_func
def solve(problem_spec):
s, t, T = sp.symbols("s, t, T")
# transfer function of system
transfer_func = problem_spec.transfer_func()
z_func, n_func = transfer_func.expand().as_numer_denom() # separate numerator and denominator
z_coeffs = [float(c) for c in st.coeffs(z_func, s)] # coefficients of numerator
n_coeffs = [float(c) for c in st.coeffs(n_func, s)] # coefficients of denominator
b_0 = z_func.coeff(s, 0)
# Boundary conditions for q and its derivative
q_a = [problem_spec.YA[0] / b_0, 0]
q_e = [problem_spec.YB[0] / b_0, 0]
# generate trajectory of q(t)
planer = tp.Trajectory_Planning(q_a, q_e, problem_spec.t0, problem_spec.tf, problem_spec.tt)
planer.dem = n_func
planer.num = z_func
q_poly = planer.calc_trajectory()
# trajectory of input and output
u_poly, y_poly = planer.num_den_laplace(q_poly[0])
q_func = st.expr_to_func(t, q_poly[0])
u_func = st.expr_to_func(t, u_poly) # desired input trajectory function
y_func = st.expr_to_func(t, y_poly) # desired output trajectory function
# tracking controller
# numerator and denominator of controller
cd_res = cd.coprime_decomposition(z_func, n_func, problem_spec.poles)
# open_loop k(s) * P(s)
tf_k = (cd_res.f_func * z_func) / (cd_res.h_func * n_func)
z_o, n_o = sp.simplify(tf_k).expand().as_numer_denom()
# coefficients of controller
z_coeffs_c = [float(c) for c in st.coeffs(cd_res.f_func, s)] # coefficients of numerator
n_coeffs_c = [float(c) for c in st.coeffs(cd_res.h_func, s)] # coefficients of denominator
# coefficients of open loop
z_coeffs_o = [float(c) for c in st.coeffs(z_o, s)]
n_coeffs_o = [float(c) for c in st.coeffs(n_o, s)]
# In order to simulate the closed loop system with the controller,
# the system is divided into two subsystems. one of them with the y_ref as input
# and the other with u_ref
close_loop_1 = control.feedback(control.tf(z_coeffs_o, n_coeffs_o))
close_loop_2 = control.feedback(control.tf(z_coeffs, n_coeffs),
control.tf(z_coeffs_c, n_coeffs_c))
# subsystem 1 with y_ref
y_1 = control.forced_response(close_loop_1, problem_spec.tt2, y_func(problem_spec.tt2), problem_spec.x0_1)
# subsystem 2 with u_ref
y_2 = control.forced_response(close_loop_2, problem_spec.tt2, u_func(problem_spec.tt2), problem_spec.x0_2)
solution_data = SolutionData()
solution_data.u = u_func
solution_data.q = q_func
solution_data.y_1 = y_1[1]
solution_data.y_2 = y_2[1]
solution_data.y_func = y_func
return solution_data
if 0:
fs = [140 * mm * scale, 60 * mm * scale]
fig1 = pl.figure(2, figsize=fs)
pl.plot(tt, w1fnc_v(tt), color=color_list[1], lw=1)
pl.plot(tt, bo[DI], color=color_list[0], lw=2)
pl.grid()
pl.xticks([0, 1, 2, 3])
pl.yticks([0, .5, 1])
pl.axis([0, 2, -.1, 1.3])
poly1 = st.trans_poly(t, 3, (T0, 0, 0, 0, 0), (T1, 1, 0, 0, 0))
pw_poly = st.create_piecewise(t, (T0, T1), (0, poly1, 1))
pw_poly_fnc = st.expr_to_func(t, pw_poly)
pl.plot(tt, pw_poly_fnc(tt), color=color_list[3], lw=3)
pl.savefig('pid_step2.pdf')
##
fs = [140 * mm * scale, 60 * mm * scale]
fig1 = pl.figure(3, figsize=fs)
pl.grid()
pl.xticks([0, 1, 2, 3])
pl.yticks([0, .5, 1])
poly1 = st.trans_poly(t, 3, (T0, 0, 0, 0, 0), (T1, 1, 0, 0, 0))
def generate_eqns_funcs(self, parameter_values=None):
"""
Creates two callable functions.
The first of the form F_tilde(ww) which *internally* represents the lhs of F(t, yy, yyd) = 0
with ww = (yy, yydot, ttau). Note that the input tau will later be calculated by a controller ttau = k(t, yy).
The second: F itself with the signature as above.
:return: None, set self.eq_func
"""
if self.eq_func is not None and self.model_func is not None:
return
if parameter_values is None:
parameter_values = []
# also respect those values, which have been passed to the constructor
parameter_values = list(self.parameter_values) + list(parameter_values)
# create needed helper function:
self.gen_leqs_for_acc_llmd(parameter_values=parameter_values)
# avoid dot access in the internal function below
ntt, nll, acc_of_lmd_func = self.ntt, self.nll, self.acc_of_lmd_func
fvars = st.concat_rows(self.yy, self.yyd, self.mod.tau)
# keep self.eqns unchanged (maybe not necessary)
eqns = self.eqns.subs(parameter_values)
actual_symbs = eqns.atoms(sp.Symbol)
expected_symbs = set(fvars)
unexpected_symbs = actual_symbs.difference(expected_symbs)
if unexpected_symbs:
msg = "Equations can only be converted to numerical func if all parameters are passed for substitution. " \
"Unexpected symbols: {}".format(unexpected_symbs)
raise ValueError(msg)
# full equations in classical formulation (currently not needed internally)
self.eq_func = st.expr_to_func(fvars, eqns)
# only the ode part
self.deq_func = st.expr_to_func(fvars, eqns[:2 * self.ntt, :])
def model_func(t, yy, yyd):
"""
This function is intended to be passed to a DAE solver like IDA.
The model consists of two coupled parts: ODE part F_ode(yy, yydot)=0 and algebraic C(yy)=0 part.
Problem is, that for mechanical systems with constraints we have differential index=3,
i.e. C and C_dot not depend on llmd. C_ddot can be formulated to depend on llmd (if F_ode is plugged in).
Idea: instead fo returning just C we return C**2 + C_dot**2 + C_ddot**2
:param t:
:param yy:
:param yyd:
:return: F(t, yy, yyd) (should be 0 with shape (2*ntt + nll,))
"""
# to use a controller, this needs to be more sophisticated
external_forces = self.input_func(t)
args = np.concatenate((yy, yyd, external_forces))
ttheta = yy[:ntt]
ttheta_d = yy[ntt:2 * ntt]
# not needed, just for comprehension
# llmd = yy[2*ntt:2*ntt + nll]
ode_part = self.deq_func(*args)
# now calculate the accelerations in depencency of yy (and thus in dependency of llmd)
# Note: signature: acc_of_lmd_func(yy, ttau)
ttheta_dd = acc_of_lmd_func(*np.concatenate((yy, external_forces)))
c2 = self.constraints_dd_func(*np.concatenate((ttheta, ttheta_d, ttheta_dd)))
c2 = np.atleast_1d(c2)
res = np.concatenate((ode_part, c2))
return res
self.model_func = model_func
def solve(problem_spec):
# problem_spec is dummy
t = sp.Symbol('t') # time variable
np = 2
nq = 2
ns = 2
n = np + nq + ns
p1, p2 = pp = st.symb_vector("p1:{0}".format(np + 1))
q1, q2 = qq = st.symb_vector("q1:{0}".format(nq + 1))
s1, s2 = ss = st.symb_vector("s1:{0}".format(ns + 1))
ttheta = st.row_stack(qq[0], pp[0], ss[0], qq[1], pp[1], ss[1])
qdot1, pdot1, sdot1, qdot2, pdot2, sdot2 = tthetad = st.time_deriv(ttheta, ttheta)
tthetadd = st.time_deriv(ttheta, ttheta, order=2)
ttheta_all = st.concat_rows(ttheta, tthetad, tthetadd)
c1, c2, c3, c4, c5, c6, m1, m2, m3, m4, m5, m6, J1, J2, J3, J4, J5, J6, l1, l2, l3, l4, l5, l6, d, g = params = sp.symbols(
'c1, c2, c3, c4, c5, c6, m1, m2, m3, m4, m5, m6, J1, J2, J3, J4, J5, J6, l1, l2, l3, l4, l5, l6, d, g')
tau1, tau2, tau3, tau4, tau5, tau6 = ttau = st.symb_vector("tau1, tau2, tau3, tau4, tau5, tau6 ")
# unit vectors
ex = sp.Matrix([1, 0])
ey = sp.Matrix([0, 1])
# coordinates of centers of mass and joints
# left
G0 = 0 * ex ##:
C1 = G0 + Rz(q1) * ex * c1 ##:
G1 = G0 + Rz(q1) * ex * l1 ##:
C2 = G1 + Rz(q1 + p1) * ex * c2 ##:
G2 = G1 + Rz(q1 + p1) * ex * l2 ##:
C3 = G2 + Rz(q1 + p1 + s1) * ex * c3 ##:
G3 = G2 + Rz(q1 + p1 + s1) * ex * l3 ##:
# right
G6 = d * ex ##:
C6 = G6 + Rz(q2) * ex * c6 ##:
G5 = G6 + Rz(q2) * ex * l6 ##:
C5 = G5 + Rz(q2 + p2) * ex * c5 ##:
G4 = G5 + Rz(q2 + p2) * ex * l5 ##:
C4 = G4 + Rz(q2 + p2 + s2) * ex * c4 ##:
G3b = G4 + Rz(q2 + p2 + s2) * ex * l4 ##:
# time derivatives of centers of mass
Sd1, Sd2, Sd3, Sd4, Sd5, Sd6 = st.col_split(st.time_deriv(st.col_stack(C1, C2, C3, C4, C5, C6), ttheta))
# Kinetic Energy (note that angles are relative)
T_rot = (J1 * qdot1 ** 2) / 2 + (J2 * (qdot1 + pdot1) ** 2) / 2 + (J3 * (qdot1 + pdot1 + sdot1) ** 2) / 2 + \
(J4 * (qdot2 + pdot2 + sdot2) ** 2) / 2 + (J5 * (qdot2 + pdot2) ** 2) / 2 + (J6 * qdot2 ** 2) / 2
T_trans = (
m1 * Sd1.T * Sd1 + m2 * Sd2.T * Sd2 + m3 * Sd3.T * Sd3 + m4 * Sd4.T * Sd4 + m5 * Sd5.T * Sd5 + m6 * Sd6.T * Sd6) / 2
T = T_rot + T_trans[0]
# Potential Energy
V = m1 * g * C1[1] + m2 * g * C2[1] + m3 * g * C3[1] + m4 * g * C4[1] + m5 * g * C5[1] + m6 * g * C6[1]
parameter_values = list(dict(c1=0.4 / 2, c2=0.42 / 2, c3=0.55 / 2, c4=0.55 / 2, c5=0.42 / 2, c6=0.4 / 2,
m1=6.0, m2=12.0, m3=39.6, m4=39.6, m5=12.0, m6=6.0,
J1=(6 * 0.4 ** 2) / 12, J2=(12 * 0.42 ** 2) / 12, J3=(39.6 * 0.55 ** 2) / 12,
J4=(39.6 * 0.55 ** 2) / 12, J5=(12 * 0.42 ** 2) / 12, J6=(6 * 0.4 ** 2) / 12,
l1=0.4, l2=0.42, l3=0.55, l4=0.55, l5=0.42, l6=0.4, d=0.26, g=9.81).items())
external_forces = [tau1, tau2, tau3, tau4, tau5, tau6]
dir_of_this_file = os.path.dirname(os.path.abspath(sys.modules.get(__name__).__file__))
fpath = os.path.join(dir_of_this_file, "7L-dae-2020-07-15.pcl")
if not os.path.isfile(fpath):
# if model is not present it could be regenerated
# however this might take long (approx. 20min)
mod = mt.generate_symbolic_model(T, V, ttheta, external_forces, constraints=[G3 - G3b], simplify=False)
mod.calc_state_eq(simplify=False)
mod.f_sympy = mod.f.subs(parameter_values)
mod.G_sympy = mod.g.subs(parameter_values)
with open(fpath, "wb") as pfile:
pickle.dump(mod, pfile)
else:
with open(fpath, "rb") as pfile:
mod = pickle.load(pfile)
# calculate DAE equations from symbolic model
dae = mod.calc_dae_eq(parameter_values)
dae.generate_eqns_funcs()
torso1_unit = Rz(q1 + p1 + s1) * ex
torso2_unit = Rz(q2 + p2 + s2) * ex
neck_length = 0.12
head_radius = 0.1
body_width = 15
neck_width = 15
H1 = G3 + neck_length * torso1_unit
H1r = G3 + (neck_length - head_radius) * torso1_unit
H2 = G3b + neck_length * torso2_unit
H2r = G3b + (neck_length - head_radius) * torso2_unit
vis = vt.Visualiser(ttheta, xlim=(-1.5, 1.5), ylim=(-.2, 2))
# get default colors and set them explicitly
# this prevents color changes in onion skin plot
default_colors = plt.get_cmap("tab10")
guy1_color = default_colors(0)
guy1_joint_color = "darkblue"
guy2_color = default_colors(1)
guy2_joint_color = "red"
guy1_head_fc = guy1_color # facecolor
guy1_head_ec = guy1_head_fc # edgecolor
guy2_head_fc = guy2_color # facecolor
guy2_head_ec = guy2_head_fc # edgecolor
# guy 1 body
vis.add_linkage(st.col_stack(G0, G1, G2, G3).subs(parameter_values),
color=guy1_color,
solid_capstyle='round',
lw=body_width,
ms=body_width,
mfc=guy1_joint_color)
# guy 1 neck
# vis.add_linkage(st.col_stack(G3, H1r).subs(parameter_values), color=head_color, solid_capstyle='round', lw=neck_width)
# guy 1 head
vis.add_disk(st.col_stack(H1, H1r).subs(parameter_values), fc=guy1_head_fc, ec=guy1_head_ec, plot_radius=False,
fill=True)
# guy 2 body
vis.add_linkage(st.col_stack(G6, G5, G4, G3b).subs(parameter_values),
color=guy2_color,
solid_capstyle='round',
lw=body_width,
ms=body_width,
mfc=guy2_joint_color)
# guy 2 neck
# vis.add_linkage(st.col_stack(G3b, H2r).subs(parameter_values), color=head_color, solid_capstyle='round', lw=neck_width)
# guy 2 head
vis.add_disk(st.col_stack(H2, H2r).subs(parameter_values), fc=guy2_head_fc, ec=guy2_head_ec, plot_radius=False,
fill=True)
eq_stat = mod.eqns.subz0(tthetadd).subz0(tthetad).subs(parameter_values)
# vector for tau and lambda together
ttau_symbols = sp.Matrix(mod.uu) ##:T
mmu = st.row_stack(ttau_symbols, mod.llmd) ##:T
# linear system of equations (and convert to function w.r.t. ttheta)
K0_expr = eq_stat.subz0(mmu) ##:i
K1_expr = eq_stat.jacobian(mmu) ##:i
K0_func = st.expr_to_func(ttheta, K0_expr)
K1_func = st.expr_to_func(ttheta, K1_expr, keep_shape=True)
def get_mu_stat_for_theta(ttheta_arg, rho=10):
# weighting matrix for mu
K0 = K0_func(*ttheta_arg)
K1 = K1_func(*ttheta_arg)
return solve_qlp(K0, K1, rho)
def solve_qlp(K0, K1, rho):
R_mu = npy.diag([1, 1, 1, rho, rho, rho, .1, .1])
n1, n2 = K1.shape
# construct the equation system for least squares with linear constraints
M1 = npy.column_stack((R_mu, K1.T))
M2 = npy.column_stack((K1, npy.zeros((n1, n1))))
M_coeff = npy.row_stack((M1, M2))
M_rhs = npy.concatenate((npy.zeros(n2), -K0))
mmu_stat = npy.linalg.solve(M_coeff, M_rhs)[:n2]
return mmu_stat
ttheta_start = npy.r_[0.9, 1.5, -1.9, 2.1, -2.175799453493845, 1.7471971159642905]
mmu_start = get_mu_stat_for_theta(ttheta_start)
connection_point_func = st.expr_to_func(ttheta, G3.subs(parameter_values))
cs_ttau = mpc.casidify(mod.uu, mod.uu)[0]
cs_llmd = mpc.casidify(mod.llmd, mod.llmd)[0]
controls_sp = mmu
controls_cs = cs.vertcat(cs_ttau, cs_llmd)
coords_cs, _ = mpc.casidify(ttheta, ttheta)
# parameters: 0: value of y_connection, 1: x_connection_last,
# 2: y_connection_last, 3: delta_r_max, 4: rho (penalty factor for 2nd persons torques),
# 5:11: ttheta_old[6], 11:17: ttheta:old2
#
P = SX.sym('P', 5 + 12)
rho = P[10]
# weightning of inputs
R = mpc.SX_diag_matrix((1, 1, 1, rho, rho, rho, 0.1, 0.1))
# Construction of Constraints
g1 = [] # constraints vector (system dynamics)
g2 = [] # inequality-constraints
closed_chain_constraint, _ = mpc.casidify(mod.dae.constraints, ttheta, cs_vars=coords_cs)
connection_position, _ = mpc.casidify(list(G3.subs(parameter_values)), ttheta, cs_vars=coords_cs) ##:i
connection_y_value, _ = mpc.casidify([G3[1].subs(parameter_values)], ttheta, cs_vars=coords_cs) ##:i
stationary_eqns, _, _ = mpc.casidify(eq_stat, ttheta, controls_sp, cs_vars=(coords_cs, controls_cs)) ##:i
g1.extend(mpc.unpack(stationary_eqns))
g1.extend(mpc.unpack(closed_chain_constraint))
# force the connecting joint to a given hight (which will be provided later)
g1.append(connection_y_value - P[0])
ng1 = len(g1)
# squared distance from the last reference should be smaller than P[3] (delta_r_max):
# this will be a restriction between -inf and 0
r = connection_position - P[1:3]
g2.append(r.T @ r - P[3])
# change of angles should be smaller than a given bound (P[5:11] are the old coords)
coords_old = P[5:11]
coords_old2 = P[11:17]
pseudo_vel = (coords_cs - coords_old) / 1
pseudo_acc = (coords_cs - 2 * coords_old + coords_old2) / 1
g2.extend(mpc.unpack(pseudo_vel))
g2.extend(mpc.unpack(pseudo_acc))
g_all = mpc.seq_to_SX_matrix(g1 + g2)
### Construction of objective Function
obj = controls_cs.T @ R @ controls_cs + 1e5 * pseudo_acc.T @ pseudo_acc + 0.3e6 * pseudo_vel.T @ pseudo_vel
OPT_variables = cs.vertcat(coords_cs, controls_cs)
# for debugging
g_all_cs_func = cs.Function("g_all_cs_func", (OPT_variables, P), (g_all,))
nlp_prob = dict(f=obj, x=OPT_variables, g=g_all, p=P)
ipopt_settings = dict(max_iter=5000, print_level=0,
acceptable_tol=1e-8, acceptable_obj_change_tol=1e-6)
opts = dict(print_time=False, ipopt=ipopt_settings)
xx_guess = npy.r_[ttheta_start, mmu_start]
# note: g1 contains the equality constraints (system dynamics) (lower bound = upper bound)
delta_phi = .05
d_delta_phi = .02
eps = 1e-9
lbg = npy.r_[[-eps] * ng1 + [-inf] + [-delta_phi] * n, [-d_delta_phi] * n]
ubg = npy.r_[[eps] * ng1 + [0] + [delta_phi] * n, [d_delta_phi] * n]
# ubx = [inf]*OPT_variables.shape[0]##:
# lower and upper bounds for decision variables:
# lbx = [-inf, -inf, -inf, -inf, -inf, -inf, -inf, -inf]*N1 + [tau1_min, tau4_min, -inf, -inf]*N
# ubx = [inf, inf, inf, inf, inf, inf, inf, inf]*N1 + [tau1_max, tau4_max, inf, inf]*N
rho = 3
P_init = npy.r_[connection_point_func(*ttheta_start)[1],
connection_point_func(*ttheta_start), 0.01, rho, ttheta_start, ttheta_start]
args = dict(lbx=-inf, ubx=inf, lbg=lbg, ubg=ubg, # unconstrained optimization
p=P_init, # initial and final state
x0=xx_guess # initial guess
)
solver = cs.nlpsol("solver", "ipopt", nlp_prob, opts)
sol = solver(**args)
global_vars = ipydex.Container(old_sol=xx_guess, old_sol2=xx_guess)
def get_optimal_equilibrium(y_value, rho=3):
ttheta_old = global_vars.old_sol[:n]
ttheta_old2 = global_vars.old_sol2[:n]
opt_prob_params = npy.r_[y_value, connection_point_func(*ttheta_old),
0.01, rho, ttheta_old, ttheta_old2]
args.update(dict(p=opt_prob_params, x0=global_vars.old_sol))
sol = solver(**args)
stats = solver.stats()
if not stats['success']:
raise ValueError(stats["return_status"])
XX = sol["x"].full().squeeze()
# save the last two results
global_vars.old_sol2 = global_vars.old_sol
global_vars.old_sol = XX
return XX
y_start = connection_point_func(*ttheta_start)[1]
N = 100
y_end = 1.36
y_func = st.expr_to_func(t, st.condition_poly(t, (0, y_start, 0, 0), (1, y_end, 0, 0)))
def get_qs_trajectory(rho):
pseudo_time = npy.linspace(0, 1, N)
yy_connection = y_func(pseudo_time)
# reset the initial guess
global_vars.old_sol = xx_guess
global_vars.old_sol2 = xx_guess
XX_list = []
for i, y_value in enumerate(yy_connection):
# print(i, y_value)
XX_list.append(get_optimal_equilibrium(y_value, rho=rho))
XX = npy.array(XX_list)
return XX
rho = 30
XX = get_qs_trajectory(rho=rho)
def smooth_time_scaling(Tend, N, phase_fraction=.5):
"""
:param Tend:
:param N:
:param phase_fraction: fraction of Tend for smooth initial and end phase
"""
T0 = 0
T1 = Tend * phase_fraction
y0 = 0
y1 = 1
# for initial phase
poly1 = st.condition_poly(t, (T0, y0, 0, 0), (T1, y1, 0, 0))
# for end phase
poly2 = poly1.subs(t, Tend - t)
# there should be a phase in the middle with constant slope
deriv_transition = st.piece_wise((y0, t < T0), (poly1, t < T1), (y1, t < Tend - T1),
(poly2, t < Tend), (y0, True))
scaling = sp.integrate(deriv_transition, (t, T0, Tend))
time_transition = sp.integrate(deriv_transition * N / scaling, t)
# deriv_transition_func = st.expr_to_func(t, full_transition)
time_transition_func = st.expr_to_func(t, time_transition)
deriv_func = st.expr_to_func(t, deriv_transition * N / scaling)
deriv_func2 = st.expr_to_func(t, deriv_transition.diff(t) * N / scaling)
C = ipydex.Container(fetch_locals=True)
return C
N = XX.shape[0]
Tend = 4
res = smooth_time_scaling(Tend, N)
def get_derivatives(XX, time_scaling, res=100):
"""
:param XX: Nxm array
:param time_scaling: container for time scaling
:param res: time resolution of the returned arrays
"""
N = XX.shape[0]
Tend = time_scaling.Tend
assert npy.isclose(time_scaling.time_transition_func([0, Tend])[-1], N)
tt = npy.linspace(time_scaling.T0, time_scaling.Tend, res)
NN = npy.arange(N)
# high_resolution version of index arry
NN2 = npy.linspace(0, N, res, endpoint=False)
# time-scaled verion of index-array
NN3 = time_scaling.time_transition_func(tt)
NN3d = time_scaling.deriv_func(tt)
NN3dd = time_scaling.deriv_func2(tt)
XX_num, XXd_num, XXdd_num = [], [], []
# iterate over every column
for col in XX.T:
spl = splrep(NN, col)
# function value and derivatives
XX_num.append(splev(NN3, spl))
XXd_num.append(splev(NN3, spl, der=1))
XXdd_num.append(splev(NN3, spl, der=2))
XX_num = npy.array(XX_num).T
XXd_num = npy.array(XXd_num).T
XXdd_num = npy.array(XXdd_num).T
NN3d_bc = npy.broadcast_to(NN3d, XX_num.T.shape).T
NN3dd_bc = npy.broadcast_to(NN3dd, XX_num.T.shape).T
XXd_n = XXd_num * NN3d_bc
# apply chain rule
XXdd_n = XXdd_num * NN3d_bc ** 2 + XXd_num * NN3dd_bc
C = ipydex.Container(fetch_locals=True)
return C
C = XX_derivs = get_derivatives(XX[:, :], time_scaling=res)
expr = mod.eqns.subz0(mod.uu, mod.llmd).subs(parameter_values)
dynterm_func = st.expr_to_func(ttheta_all, expr)
def get_torques(dyn_term_func, XX_derivs, static1=False, static2=False):
ttheta_num_all = npy.c_[XX_derivs.XX_num[:, :n], XX_derivs.XXd_n[:, :n], XX_derivs.XXdd_n[:, :n]] ##:S
if static1:
# set velocities to 0
ttheta_num_all[:, n:2 * n] = 0
if static2:
# set accelerations to 0
ttheta_num_all[:, 2 * n:] = 0
res = dynterm_func(*ttheta_num_all.T)
return res
lhs_static = get_torques(dynterm_func, XX_derivs, static1=True, static2=True) ##:i
lhs_dynamic = get_torques(dynterm_func, XX_derivs, static2=False) ##:i
mmu_stat_list = []
for L_k_stat, L_k_dyn, ttheta_k in zip(lhs_static, lhs_dynamic, XX_derivs.XX_num[:, :n]):
K1_k = K1_func(*ttheta_k)
mmu_stat_k = solve_qlp(L_k_stat, K1_k, rho)
mmu_stat_list.append(mmu_stat_k)
mmu_stat_all = npy.array(mmu_stat_list)
solution_data = SolutionData()
solution_data.tt = XX_derivs.tt
solution_data.xx = XX_derivs.XX_num
solution_data.mmu = mmu_stat_all
solution_data.vis = vis
save_plot(problem_spec, solution_data)
return solution_data
def test_create_simfunction(self):
x1, x2, x3, x4 = xx = sp.Matrix(sp.symbols("x1, x2, x3, x4"))
u1, u2 = uu = sp.Matrix(sp.symbols("u1, u2")) # inputs
p1, p2, p3, p4 = pp = sp.Matrix(
sp.symbols("p1, p2, p3, p4")) # parameter
t = sp.Symbol('t')
A = A0 = sp.randMatrix(len(xx), len(xx), -10, 10, seed=704)
B = B0 = sp.randMatrix(len(xx), len(uu), -10, 10, seed=705)
v1 = A[0, 0]
A[0, 0] = p1
v2 = A[2, -1]
A[2, -1] = p2
v3 = B[3, 0]
B[3, 0] = p3
v4 = B[2, 1]
B[2, 1] = p4
par_vals = lzip(pp, [v1, v2, v3, v4])
f = A * xx
G = B
fxu = (f + G * uu).subs(par_vals)
# some random initial values
x0 = st.to_np(sp.randMatrix(len(xx), 1, -10, 10, seed=706)).squeeze()
# Test handling of unsubstituted parameters
mod = st.SimulationModel(f, G, xx, model_parameters=par_vals[1:])
with self.assertRaises(ValueError) as cm:
rhs0 = mod.create_simfunction()
self.assertTrue("unexpected symbols" in cm.exception.args[0])
# create the model and the rhs-function
mod = st.SimulationModel(f, G, xx, par_vals)
rhs0 = mod.create_simfunction()
self.assertFalse(mod.compiler_called)
self.assertFalse(mod.use_sp2c)
res0_1 = rhs0(x0, 0)
dres0_1 = st.to_np(fxu.subs(lzip(xx, x0) + st.zip0(uu))).squeeze()
bin_res01 = np.isclose(res0_1, dres0_1) # binary array
self.assertTrue(np.all(bin_res01))
# difference should be [0, 0, ..., 0]
self.assertFalse(np.any(rhs0(x0, 0) - rhs0(x0, 3.7)))
# simulate
tt = np.linspace(0, 0.5,
100) # simulation should be short due to instability
res1 = sc.integrate.odeint(rhs0, x0, tt)
# create and try sympy_to_c bridge (currently only works on linux
# and if sympy_to_c is installed (e.g. with `pip install sympy_to_c`))
# until it is not available for windows we do not want it as a requirement
# see also https://stackoverflow.com/a/10572833/333403
try:
import sympy_to_c
except ImportError:
# noinspection PyUnusedLocal
sympy_to_c = None
sp2c_available = False
else:
sp2c_available = True
if sp2c_available:
rhs0_c = mod.create_simfunction(use_sp2c=True)
self.assertTrue(mod.compiler_called)
res1_c = sc.integrate.odeint(rhs0_c, x0, tt)
self.assertTrue(np.all(np.isclose(res1_c, res1)))
mod.compiler_called = None
rhs0_c = mod.create_simfunction(use_sp2c=True)
self.assertTrue(mod.compiler_called is None)
# proof calculation
# x(t) = x0*exp(A*t)
Anum = st.to_np(A.subs(par_vals))
Bnum = st.to_np(G.subs(par_vals))
# noinspection PyUnresolvedReferences
xt = [np.dot(sc.linalg.expm(Anum * T), x0) for T in tt]
xt = np.array(xt)
# test whether numeric results are close within given tolerance
bin_res1 = np.isclose(res1, xt, rtol=2e-5) # binary array
self.assertTrue(np.all(bin_res1))
# test handling of parameter free models:
mod2 = st.SimulationModel(Anum * xx, Bnum, xx)
rhs2 = mod2.create_simfunction()
res2 = sc.integrate.odeint(rhs2, x0, tt)
self.assertTrue(np.allclose(res1, res2))
# test input functions
des_input = st.piece_wise((0, t <= 1), (t, t < 2), (0.5, t < 3),
(1, True))
des_input_func_scalar = st.expr_to_func(t, des_input)
des_input_func_vec = st.expr_to_func(t,
sp.Matrix([des_input, des_input]))
# noinspection PyUnusedLocal
with self.assertRaises(TypeError) as cm:
mod2.create_simfunction(input_function=des_input_func_scalar)
rhs3 = mod2.create_simfunction(input_function=des_input_func_vec)
# noinspection PyUnusedLocal
res3_0 = rhs3(x0, 0)
rhs4 = mod2.create_simfunction(input_function=des_input_func_vec,
time_direction=-1)
res4_0 = rhs4(x0, 0)
self.assertTrue(np.allclose(res3_0, np.array([119., -18., -36.,
-51.])))
self.assertTrue(np.allclose(res4_0, -res3_0))
def test_expr_to_func(self):
x1, x2 = xx = sp.Matrix(sp.symbols("x1, x2"))
t, = sp.symbols("t,")
r_ = np.r_
f1 = st.expr_to_func(x1, 2 * x1)
self.assertEqual(f1(5.1), 10.2)
XX1 = np.r_[1, 2, 3.7]
res1 = f1(XX1) == 2 * XX1
self.assertTrue(res1.all)
f2 = st.expr_to_func(x1, sp.Matrix([x1 * 2, x1 + 5, 4]))
res2 = f2(3) == r_[6, 8, 4]
self.assertTrue(res2.all())
res2b = f2(r_[3, 10, 0]) == np.array([[6, 8, 4], [20, 15, 4],
[0, 5, 4]])
self.assertTrue(res2b.all())
f3 = st.expr_to_func(xx, sp.Matrix([x1 * 2, x2 + 5, 4]))
res3 = np.allclose(f3(-3.1, 4), r_[-6.2, 9, 4])
self.assertTrue(res3)
# test compatibility with Piecewise Expressions
des_input = st.piece_wise((0, t <= 1), (t, t < 2), (0.5, t < 3),
(1, True))
f4s = st.expr_to_func(t, des_input)
f4v = st.expr_to_func(t, sp.Matrix([des_input, des_input]))
self.assertEqual(f4s(2.7), 0.5)
sol = r_[0, 1.6, 0.5, 1, 1]
res4a = f4s(r_[0.3, 1.6, 2.2, 3.1, 500]) == sol
self.assertTrue(res4a.all())
res4b = f4v(r_[0.3, 1.6, 2.2, 3.1, 500])
col1, col2 = res4b.T
self.assertTrue(np.array_equal(col1, sol))
self.assertTrue(np.array_equal(col2, sol))
spmatrix = sp.Matrix([[x1, x1 * x2], [0, x2**2]])
fnc1 = st.expr_to_func(xx, spmatrix, keep_shape=False)
fnc2 = st.expr_to_func(xx, spmatrix, keep_shape=True)
res1 = fnc1(1.0, 2.0)
res2 = fnc2(1.0, 2.0)
self.assertEqual(res1.shape, (4, ))
self.assertEqual(res2.shape, (2, 2))
# noinspection PyTypeChecker
self.assertTrue(np.all(res1 == [1, 2, 0, 4]))
# noinspection PyTypeChecker
self.assertTrue(np.all(res1 == res2.flatten()))
fnc = st.expr_to_func(xx, x1 + x2)
self.assertEqual(fnc(1, 3), 4)
xx_res = np.array([1, 3, 1.1, 3, 1.2, 3.0]).reshape(3, -1)
self.assertTrue(np.allclose(fnc(*xx_res.T), np.array([4, 4.1, 4.2])))
fnc1 = st.expr_to_func(xx, 3 * xx)
fnc2 = st.expr_to_func(xx, 3 * xx, allow_kwargs=True)
self.assertTrue(np.allclose(fnc1(10, 100), fnc2(x2=100, x1=10)))
