def Zi_left_pinv_with_restrictions(P1i_rpinv, P1i_tilde_roc, Zi):
""" Given a matrix Zi, this function calculates a matrix such that:
Zi_left_pinv * Zi = I
Zi_left_pinv * P1i_tilde_roc = 0
Zi_left_pinv * P1i_rpinv = 0
"""
assert check_row_compatibility(P1i_rpinv, P1i_tilde_roc, Zi),\
"Matrices do not match in row dimension."
C = st.concat_cols(P1i_rpinv, P1i_tilde_roc, Zi)
assert is_regular_matrix(C), "C is not a regular matrix"
C_det = C.berkowitz_det()
C_inv = C.adjugate()/C_det
C_inv = custom_simplify(C_inv)
m, n = Zi.shape
Zi_left_pinv = sp.Matrix([])
for i in xrange(m-n,m):
Zi_left_pinv = st.concat_rows(Zi_left_pinv,C_inv.row(i))
o, p = Zi_left_pinv.shape
assert o==n and p==m, "There must have been a problem with the\
computation of Zi_left_pinv"
assert is_unit_matrix(Zi_left_pinv*Zi), "Zi_left_pinv is wrong"
assert is_zero_matrix(Zi_left_pinv*P1i_tilde_roc), "Zi_left_pinv is wrong"
assert is_zero_matrix(Zi_left_pinv*P1i_rpinv), "Zi_left_pinv is wrong"
return Zi_left_pinv
def right_pseudo_inverse(P):
""" Calculates a right pseudo inverse with as many zero entries as
possible. Given a [m x n] matrix P, the algorithm picks m
linearly independent column vectors of P to form a regular
block matrix A such that Q = P*R = (A B) with the permuation matrix
P. The right pseudo inverse of Q is
Q_rpinv = ( A^(-1) )
( 0 )
and the right pseudo inverse of P
P_rpinv = R*( A^(-1) )
( 0 ).
There is a degree of freedom in choosing the column vectors for
A. At the moment this can be done with respect to "count_ops"
(minimizes number of operations) or "free_symbols" (minimizes the
number of symbols).
"""
m0, n0 = P.shape
A, B, R = reshape_matrix_columns(P)
#apparently this is quicker than A_inv = A.inv() #[m x m]:
A_det = A.berkowitz_det()
A_inv = A.adjugate()/A_det
p = n0-m0
zero = sp.zeros(p,m0)
Q = custom_simplify(A_inv)
P_pinv = custom_simplify( R*st.concat_rows(Q,zero) )
assert is_unit_matrix(P*P_pinv), "Rightpseudoinverse is not correct."
return P_pinv
def right_pseudo_inverse(P):
""" Calculates a right pseudo inverse with as many zero entries as
possible. Given a [m x n] matrix P, the algorithm picks m
linearly independent column vectors of P to form a regular
block matrix A such that Q = P*R = (A B) with the permuation matrix
P. The right pseudo inverse of Q is
Q_rpinv = ( A^(-1) )
( 0 )
and the right pseudo inverse of P
P_rpinv = R*( A^(-1) )
( 0 ).
There is a degree of freedom in choosing the column vectors for
A. At the moment this can be done with respect to "count_ops"
(minimizes number of operations) or "free_symbols" (minimizes the
number of symbols).
"""
m0, n0 = P.shape
A, B, R = reshape_matrix_columns(P)
#apparently this is quicker than A_inv = A.inv() #[m x m]:
A_det = A.berkowitz_det()
A_inv = A.adjugate() / A_det
p = n0 - m0
zero = sp.zeros(p, m0)
Q = custom_simplify(A_inv)
P_pinv = custom_simplify(R * st.concat_rows(Q, zero))
assert is_unit_matrix(P * P_pinv), "Rightpseudoinverse is not correct."
return P_pinv
def calculate_H_matrix(self):
""" unimodular completion of P(d/dt) with H = P1i * ... * P11 * P10
"""
pc.print_line()
print("Exit condition satisfied\n")
H_relevant_matrices, H_tilde_relevant_matrices = self._myStack.get_H_relevant_matrices(
)
H1 = self.multiply_matrices_in_list(H_relevant_matrices)
if not len(H_tilde_relevant_matrices) == 0:
H2 = self.multiply_matrices_in_list(H_tilde_relevant_matrices)
else:
H2 = sp.Matrix([])
H = st.concat_rows(H1, H2)
m, n = H.shape
assert n == len(
self._myStack.vec_x), "Dimensions of H-Matrix do not fit."
print("H-matrix = ")
pc.print_nicely(H)
print("\n")
self.H = H
def test_ord(self):
x1, x2, x3 = xx = st.symb_vector("x1, x2, x3")
xdot1, xdot2, xdot3 = xxd = pc.st.time_deriv(xx, xx)
xxdd = pc.st.time_deriv(xx, xx, order=2)
XX = st.concat_rows(xx, xxd, xxdd)
XX, dXX = pc.setup_objects(XX)
dx1, dx2, dx3, dxdot1, dxdot2, dxdot3, dxddot1, dxddot2, dxddot3 = dXX
w0 = 0 * dx1
w1 = dx1 + dxdot3
w2 = 4 * x2 * dx1 - sp.sin(x3) * xdot1 * dx2
self.assertEqual(w0.ord, 0)
self.assertEqual(dx1.ord, 0)
self.assertEqual(dxdot1.ord, 1)
self.assertEqual(dxddot3.ord, 2)
self.assertEqual(w1.ord, 1)
self.assertEqual(w2.ord, 0)
self.assertEqual(w2.d.ord, 1)
w3 = w1 ^ w2
self.assertEqual(w3.ord, 1)
self.assertEqual(w3.dot().ord, 2)
def Zi_left_pinv_with_restrictions(P1i_rpinv, P1i_tilde_roc, Zi):
""" Given a matrix Zi, this function calculates a matrix such that:
Zi_left_pinv * Zi = I
Zi_left_pinv * P1i_tilde_roc = 0
Zi_left_pinv * P1i_rpinv = 0
"""
assert check_row_compatibility(P1i_rpinv, P1i_tilde_roc, Zi),\
"Matrices do not match in row dimension."
C = st.concat_cols(P1i_rpinv, P1i_tilde_roc, Zi)
assert is_regular_matrix(C), "C is not a regular matrix"
C_det = C.berkowitz_det()
C_inv = C.adjugate() / C_det
C_inv = custom_simplify(C_inv)
m, n = Zi.shape
Zi_left_pinv = sp.Matrix([])
for i in range(m - n, m):
Zi_left_pinv = st.concat_rows(Zi_left_pinv, C_inv.row(i))
o, p = Zi_left_pinv.shape
assert o == n and p == m, "There must have been a problem with the\
computation of Zi_left_pinv"
assert is_unit_matrix(Zi_left_pinv * Zi), "Zi_left_pinv is wrong"
assert is_zero_matrix(Zi_left_pinv *
P1i_tilde_roc), "Zi_left_pinv is wrong"
assert is_zero_matrix(Zi_left_pinv * P1i_rpinv), "Zi_left_pinv is wrong"
return Zi_left_pinv
def calculate_H_matrix(self):
""" unimodular completion of P(d/dt) with H = P1i * ... * P11 * P10
"""
pc.print_line()
print("Exit condition satisfied\n")
H_relevant_matrices, H_tilde_relevant_matrices = self._myStack.get_H_relevant_matrices()
H1 = self.multiply_matrices_in_list( H_relevant_matrices )
if not len(H_tilde_relevant_matrices)==0:
H2 = self.multiply_matrices_in_list( H_tilde_relevant_matrices )
else:
H2 = sp.Matrix([])
H = st.concat_rows(H1, H2)
m, n = H.shape
assert n==len(self._myStack.vec_x), "Dimensions of H-Matrix do not fit."
print "H-matrix = "
pc.print_nicely(H)
print "\n"
self.H = H
def vec_x(self, value):
self._vec_x = value
# calculate vec_xdot
self.vec_xdot = st.time_deriv(self.vec_x, self.vec_x)
# vector for differentiation
self.diffvec_x = st.concat_rows(self.vec_x, self.vec_xdot, self.diff_symbols)
def vec_x(self, value):
self._vec_x = value
# calculate vec_xdot
self.vec_xdot = st.time_deriv(self.vec_x, self.vec_x)
# vector for differentiation
self.diffvec_x = st.concat_rows(self.vec_x, self.vec_xdot, self.diff_symbols)
def alternative_right_ortho_complement(P):
m0, n0 = P.shape
A, B, R = reshape_matrix_columns(P)
#apparently this is quicker than A_inv = A.inv() #[m x m]:
A_det = A.berkowitz_det()
A_inv = A.adjugate()/A_det
minusAinvB = custom_simplify(-A_inv*B)
p = n0-m0
unit_matrix = sp.eye(p)
Q = custom_simplify(A_inv)
P_roc = custom_simplify( R*st.concat_rows(minusAinvB, unit_matrix) )
assert is_zero_matrix(P*P_roc), "Right orthocomplement is not correct."
return P_roc
def alternative_right_ortho_complement(P):
m0, n0 = P.shape
A, B, R = reshape_matrix_columns(P)
#apparently this is quicker than A_inv = A.inv() #[m x m]:
A_det = A.berkowitz_det()
A_inv = A.adjugate() / A_det
minusAinvB = custom_simplify(-A_inv * B)
p = n0 - m0
unit_matrix = sp.eye(p)
Q = custom_simplify(A_inv)
P_roc = custom_simplify(R * st.concat_rows(minusAinvB, unit_matrix))
assert is_zero_matrix(P * P_roc), "Right orthocomplement is not correct."
return P_roc
def generate_basis(self):
# TODO: not the most elegant way?
# check highest order of derivatives
highest_order = 0
for n in self._myStack.transformation.H.atoms():
if hasattr(n, "difforder"):
if n.difforder>highest_order:
highest_order = n.difforder
# generate vector with vec_x and its derivatives up to highest_order
new_vector = self._myStack.vec_x
for index in xrange(1, highest_order+1):
vec_x_ndot = st.time_deriv(self._myStack.vec_x, self._myStack.vec_x, order=index)
new_vector = st.concat_rows( new_vector, vec_x_ndot )
# generate basis_1form up to this order
basis, basis_1form = ct.diffgeo_setup(new_vector)
# store
self._myStack.basis = basis
self._myStack.basis_1form = basis_1form
def generate_basis(self):
# TODO: not the most elegant way?
# check highest order of derivatives
highest_order = 0
for n in self._myStack.transformation.H.atoms():
if hasattr(n, "difforder"):
if n.difforder > highest_order:
highest_order = n.difforder
# generate vector with vec_x and its derivatives up to highest_order
new_vector = self._myStack.vec_x
for index in range(1, highest_order + 1):
vec_x_ndot = st.time_deriv(self._myStack.vec_x,
self._myStack.vec_x,
order=index)
new_vector = st.concat_rows(new_vector, vec_x_ndot)
# generate basis_1form up to this order
basis, basis_1form = ct.diffgeo_setup(new_vector)
# store
self._myStack.basis = basis
self._myStack.basis_1form = basis_1form
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 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