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polynomials.py
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polynomials.py
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'''
This script contains classes and functions to support the approximation of
muscle-tendon lengths, velocities, and moment arms using polynomial
functions of joint positions and velocities.
'''
# %% Import packages.
import numpy as np
import matplotlib.pyplot as plt
# %% This class evaluates polynomial approximations given the coefficients,
# the dimension, and the order.
class polynomials:
def __init__(self, coefficients, dimension, order):
self.coefficients = coefficients
self.dimension = dimension
self.order = order
nq = [0, 0, 0, 0]
NCoeff = 0
for nq[0] in range(order + 1):
if (dimension < 2):
nq2_s = 0
else:
nq2_s = order - nq[0]
for nq[1] in range(nq2_s + 1):
if (dimension < 3):
nq3_s = 0
else:
nq3_s = order - nq[0] - nq[1]
for nq[2] in range(nq3_s + 1):
if (dimension < 4):
nq4_s = 0
else:
nq4_s = order - nq[0] - nq[1] - nq[2]
for nq[3] in range(nq4_s + 1):
NCoeff += 1
if len(coefficients) != NCoeff:
raise Exception('Expected: {}'.format(NCoeff), 'coefficients',
'but got: {}'.format(len(coefficients)))
def calcValue(self, x):
nq = [0, 0, 0, 0]
coeff_nr = 0
value = 0
for nq[0] in range(self.order + 1):
if (self.dimension < 2):
nq2_s = 0
else:
nq2_s = self.order - nq[0]
for nq[1] in range(nq2_s + 1):
if (self.dimension < 3):
nq3_s = 0
else:
nq3_s = self.order - nq[0] - nq[1]
for nq[2] in range(nq3_s + 1):
if (self.dimension < 4):
nq4_s = 0
else:
nq4_s = self.order - nq[0] - nq[1] - nq[2]
for nq[3] in range(nq4_s + 1):
valueP = 1
for d in range(self.dimension):
valueP *= pow(x[d], nq[d])
value += valueP * self.coefficients[coeff_nr]
coeff_nr += 1
return value
def calcDerivative(self, x, derivComponent):
nq = [0, 0, 0, 0]
coeff_nr = 0
value = 0
for nq[0] in range(self.order + 1):
if (self.dimension < 2):
nq2_s = 0
else:
nq2_s = self.order - nq[0]
for nq[1] in range(nq2_s + 1):
if (self.dimension < 3):
nq3_s = 0
else:
nq3_s = self.order - nq[0] - nq[1]
for nq[2] in range(nq3_s + 1):
if (self.dimension < 4):
nq4_s = 0
else:
nq4_s = self.order - nq[0] - nq[1] - nq[2]
for nq[3] in range(nq4_s + 1):
if (derivComponent == 0):
nqNonNegative = nq[0] - 1
if (nqNonNegative < 0):
nqNonNegative = 0
valueP = nq[0] * pow(x[0], nqNonNegative);
for d in range(self.dimension):
if (d == derivComponent):
continue
valueP *= pow(x[d], nq[d])
value += valueP * self.coefficients[coeff_nr]
elif (derivComponent == 1):
nqNonNegative = nq[1] - 1
if (nqNonNegative < 0):
nqNonNegative = 0
valueP = nq[1] * pow(x[1], nqNonNegative);
for d in range(self.dimension):
if (d == derivComponent):
continue
valueP *= pow(x[d], nq[d])
value += valueP * self.coefficients[coeff_nr]
elif (derivComponent == 2):
nqNonNegative = nq[2] - 1
if (nqNonNegative < 0):
nqNonNegative = 0
valueP = nq[2] * pow(x[2], nqNonNegative);
for d in range(self.dimension):
if (d == derivComponent):
continue
valueP *= pow(x[d], nq[d])
value += valueP * self.coefficients[coeff_nr]
elif (derivComponent == 3):
nqNonNegative = nq[3] - 1
if (nqNonNegative < 0):
nqNonNegative = 0
valueP = nq[3] * pow(x[3], nqNonNegative);
for d in range(self.dimension):
if (d == derivComponent):
continue
valueP *= pow(x[d], nq[d])
value += valueP * self.coefficients[coeff_nr]
coeff_nr += 1
return value
# %% This class evaluates the terms of the polynomial approximations, given
# the dimension and the order. It is used when fitting the coefficients.
class polynomial_estimation:
def __init__(self, dimension, order):
self.dimension = dimension
self.order = order
nq = [0, 0, 0, 0]
self.NCoeff = 0
for nq[0] in range(order + 1):
if (dimension < 2):
nq2_s = 0
else:
nq2_s = order - nq[0]
for nq[1] in range(nq2_s + 1):
if (dimension < 3):
nq3_s = 0
else:
nq3_s = order - nq[0] - nq[1]
for nq[2] in range(nq3_s + 1):
if (dimension < 4):
nq4_s = 0
else:
nq4_s = order - nq[0] - nq[1] - nq[2]
for nq[3] in range(nq4_s + 1):
self.NCoeff += 1
def calcValue(self, x):
nq = [0, 0, 0, 0]
coeff_nr = 0
value = np.zeros((x.shape[0], self.NCoeff))
for nq[0] in range(self.order + 1):
if (self.dimension < 2):
nq2_s = 0
else:
nq2_s = self.order - nq[0]
for nq[1] in range(nq2_s + 1):
if (self.dimension < 3):
nq3_s = 0
else:
nq3_s = self.order - nq[0] - nq[1]
for nq[2] in range(nq3_s + 1):
if (self.dimension < 4):
nq4_s = 0
else:
nq4_s = self.order - nq[0] - nq[1] - nq[2]
for nq[3] in range(nq4_s + 1):
valueP = 1
for d in range(self.dimension):
valueP *= pow(x[:,d], nq[d])
value[:,coeff_nr ] = valueP
coeff_nr += 1
return value
def calcDerivative(self, x, derivComponent):
nq = [0, 0, 0, 0]
coeff_nr = 0
value = np.zeros((x.shape[0], self.NCoeff))
for nq[0] in range(self.order + 1):
if (self.dimension < 2):
nq2_s = 0
else:
nq2_s = self.order - nq[0]
for nq[1] in range(nq2_s + 1):
if (self.dimension < 3):
nq3_s = 0
else:
nq3_s = self.order - nq[0] - nq[1]
for nq[2] in range(nq3_s + 1):
if (self.dimension < 4):
nq4_s = 0
else:
nq4_s = self.order - nq[0] - nq[1] - nq[2]
for nq[3] in range(nq4_s + 1):
if (derivComponent == 0):
nqNonNegative = nq[0] - 1
if (nqNonNegative < 0):
nqNonNegative = 0
valueP = nq[0] * pow(x[:,0], nqNonNegative);
for d in range(self.dimension):
if (d == derivComponent):
continue
valueP *= pow(x[:,d], nq[d])
value[:,coeff_nr ] = valueP
elif (derivComponent == 1):
nqNonNegative = nq[1] - 1
if (nqNonNegative < 0):
nqNonNegative = 0
valueP = nq[1] * pow(x[:,1], nqNonNegative);
for d in range(self.dimension):
if (d == derivComponent):
continue
valueP *= pow(x[:,d], nq[d])
value[:,coeff_nr ] = valueP
elif (derivComponent == 2):
nqNonNegative = nq[2] - 1
if (nqNonNegative < 0):
nqNonNegative = 0
valueP = nq[2] * pow(x[:,2], nqNonNegative);
for d in range(self.dimension):
if (d == derivComponent):
continue
valueP *= pow(x[:,d], nq[d])
value[:,coeff_nr ] = valueP
elif (derivComponent == 3):
nqNonNegative = nq[3] - 1
if (nqNonNegative < 0):
nqNonNegative = 0
valueP = nq[3] * pow(x[:,3], nqNonNegative);
for d in range(self.dimension):
if (d == derivComponent):
continue
valueP *= pow(x[:,d], nq[d])
value[:,coeff_nr ] = valueP
coeff_nr += 1
return value
# %% This functions fits the polynomial coefficients.
def getPolynomialCoefficients(pathCoordinates, pathMuscleAnalysis, joints,
muscles, order_min=3, order_max=9,
threshold=0.003):
# Get joint coordinates.
from utilities import getIK
jointCoordinates = (getIK(pathCoordinates, joints)[0]).to_numpy()[:,1::]
# Get muscle-tendon lengths.
from utilities import getFromStorage
pathMuscleTendonLengths = pathMuscleAnalysis + 'Length.sto'
muscleTendonLengths = getFromStorage(
pathMuscleTendonLengths, muscles).to_numpy()[:,1::]
# Get moment arms.
momentArms = np.zeros((jointCoordinates.shape[0], len(muscles),
len(joints)))
for i, joint in enumerate(joints):
pathMomentArm = pathMuscleAnalysis + 'MomentArm_' + joint + '.sto'
# getFromStorage outputs time vector as well, so [:,1::].
momentArms[:, :, i] = getFromStorage(
pathMomentArm, muscles).to_numpy()[:,1::]
# Detect which muscles actuate which joints.
spanningInfo = np.sum(momentArms, axis=0)
spanningInfo = np.where(np.logical_and(
spanningInfo<=0.0001, spanningInfo>=-0.0001), 0, 1)
polynomialData = {}
for i, muscle in enumerate(muscles):
muscle_momentArms = momentArms[:, i, spanningInfo[i, :]==1]
muscle_dimension = muscle_momentArms.shape[1]
muscle_muscleTendonLengths = muscleTendonLengths[:, i]
is_fullfilled = False
order = order_min
while not is_fullfilled:
polynomial = polynomial_estimation(muscle_dimension, order)
mat = polynomial.calcValue(
jointCoordinates[:, spanningInfo[i, :]==1])
diff_mat = np.zeros(
(jointCoordinates.shape[0], mat.shape[1], muscle_dimension))
diff_mat_sq = np.zeros(
(jointCoordinates.shape[0]*(muscle_dimension), mat.shape[1]))
for j in range(muscle_dimension):
diff_mat[:,:,j] = polynomial.calcDerivative(
jointCoordinates[:, spanningInfo[i, :]==1], j)
diff_mat_sq[
jointCoordinates.shape[0]*j:
jointCoordinates.shape[0]*(j+1),:] = -(
diff_mat[:,:,j]).reshape(-1, diff_mat.shape[1])
A = np.concatenate((mat,diff_mat_sq),axis=0)
B = np.concatenate((muscle_muscleTendonLengths,
(muscle_momentArms.T).flatten()))
# Solve least-square problem .
coefficients = np.linalg.lstsq(A,B,rcond=None)[0]
# Compute difference with model data.
# Muscle-tendon lengths.
muscle_muscleTendonLengths_poly = np.matmul(mat,coefficients)
muscleTendonLengths_diff_rms = np.sqrt(np.mean(
muscle_muscleTendonLengths -
muscle_muscleTendonLengths_poly)**2)
# Moment-arms.
muscle_momentArms_poly = np.zeros((jointCoordinates.shape[0],
muscle_dimension))
for j in range(muscle_dimension):
muscle_momentArms_poly[:,j] = np.matmul(
-(diff_mat[:,:,j]).reshape(-1, diff_mat.shape[1]),
coefficients)
momentArms_diff_rms = np.sqrt(np.mean((
muscle_momentArms - muscle_momentArms_poly)**2, axis=0))
# Check if criterion is satisfied.
if (muscleTendonLengths_diff_rms <= threshold and
np.max(momentArms_diff_rms) <= threshold):
is_fullfilled = True
elif order == order_max:
is_fullfilled = True
print("Max order (" + str(order_max) + ") for " + muscle)
else:
order += 1
polynomialData[muscle] = {
'dimension': muscle_dimension, 'order': order,
'coefficients': coefficients, 'spanning': spanningInfo[i, :]}
return polynomialData
# %% This function plots muscle-tendon lengths and moment arms. Note that this
# is limited to 3D, so muscles actuating more than 2 DOFs are not displayed.
def testPolynomials(pathCoordinates, pathMuscleAnalysis, joints, muscles,
f_polynomial, polynomialData, momentArmIndices,
trunkMomentArmPolynomialIndices=[]):
# Get joint coordinates.
from utilities import getIK
jointCoordinates = (getIK(pathCoordinates, joints)[0]).to_numpy()[:,1::]
# Get muscle-tendon lengths.
from utilities import getFromStorage
pathMuscleTendonLengths = pathMuscleAnalysis + 'Length.sto'
muscleTendonLengths = getFromStorage(
pathMuscleTendonLengths, muscles).to_numpy()[:,1::]
# Get moment arms.
momentArms = np.zeros((jointCoordinates.shape[0], len(muscles),
len(joints)))
for i, joint in enumerate(joints):
pathMomentArm = pathMuscleAnalysis + 'MomentArm_' + joint + '.sto'
# getFromStorage outputs time vector as well, so [:,1::].
momentArms[:, :, i] = getFromStorage(
pathMomentArm, muscles).to_numpy()[:,1::]
# Approximate muscle-tendon lengths and moment-arms.
lMT = np.zeros((len(muscles),muscleTendonLengths.shape[0]))
dM = np.zeros((len(muscles),len(joints),muscleTendonLengths.shape[0]))
dM_all = {}
for k in range(muscleTendonLengths.shape[0]):
Qsin = jointCoordinates[k, :].T
Qdotsin = np.zeros((1,Qsin.shape[0]))
lMT[:,k] = f_polynomial(Qsin, Qdotsin)[0].full().flatten()
dM[:,:,k] = f_polynomial(Qsin, Qdotsin)[2].full()
for j, joint in enumerate(joints):
if joint[-1] == 'r' or joint[-1] == 'l':
dM_all[joint] = dM[momentArmIndices[joint[:-1] + 'l'], j, :]
else:
dM_all[joint] = dM[trunkMomentArmPolynomialIndices, j, :]
ny_0 = (np.sqrt(len(muscles)))
ny = np.floor(np.sqrt(len(muscles)))
ny_a = int(ny)
ny_b = int(ny)
if not ny == ny_0:
ny_b = int(ny+1)
fig = plt.figure()
fig.suptitle('Muscle-tendon lengths')
for i in range(len(muscles)):
muscle_obj = muscles[i][:-1] + 'r'
if polynomialData[muscle_obj]['dimension'] == 1:
temp = polynomialData[muscle_obj]['spanning']==1
y = (i for i,v in enumerate(temp) if v == True)
x1 = next(y)
ax = fig.add_subplot(ny_a, ny_b, i+1)
ax.scatter(jointCoordinates[:,x1],lMT[i,:])
ax.scatter(jointCoordinates[:,x1],muscleTendonLengths[:,i],c='r')
ax.set_title(muscles[i])
ax.set_xlabel(joints[x1])
elif polynomialData[muscle_obj]['dimension'] == 2:
ax = fig.add_subplot(ny_a, ny_b, i+1, projection='3d')
temp = polynomialData[muscle_obj]['spanning']==1
y = (i for i,v in enumerate(temp) if v == True)
x1 = next(y)
x2 = next(y)
ax.scatter(jointCoordinates[:,x1],jointCoordinates[:,x2],lMT[i,:])
ax.scatter(jointCoordinates[:,x1],jointCoordinates[:,x2],
muscleTendonLengths[:,i],c='r')
ax.set_title(muscles[i])
ax.set_xlabel(joints[x1])
ax.set_ylabel(joints[x2])
for i, joint in enumerate(joints):
fig = plt.figure()
fig.suptitle('Moment arms: ' + joint)
NMomentarms = len(momentArmIndices[joint])
ny_0 = (np.sqrt(NMomentarms))
ny = np.round(ny_0)
ny_a = int(ny)
ny_b = int(ny)
if (ny == ny_0) == False:
if ny_a == 1:
ny_b = NMomentarms
if ny < ny_0:
ny_b = int(ny+1)
for j in range(NMomentarms):
if joint[-1] == 'r' or joint[-1] == 'l':
muscle_obj_r = (
muscles[momentArmIndices[joint[:-1] + 'l'][j]][:-1] + 'r')
muscle_obj = muscles[momentArmIndices[joint[:-1] + 'l'][j]]
else:
muscle_obj_r = muscles[trunkMomentArmPolynomialIndices[j]]
muscle_obj = muscles[trunkMomentArmPolynomialIndices[j]]
if polynomialData[muscle_obj_r]['dimension'] == 1:
temp = polynomialData[muscle_obj_r]['spanning']==1
y = (i for i,v in enumerate(temp) if v == True)
x1 = next(y)
ax = fig.add_subplot(ny_a, ny_b, j+1)
ax.scatter(jointCoordinates[:,x1],dM_all[joint][j,:])
if joint[-1] == 'r' or joint[-1] == 'l':
ax.scatter(
jointCoordinates[:,x1],
momentArms[:,momentArmIndices[joint[:-1] + 'l'][j],i],
c='r')
else:
ax.scatter(
jointCoordinates[:,x1],
momentArms[:,trunkMomentArmPolynomialIndices[j],i],
c='r')
ax.set_title(muscle_obj)
ax.set_xlabel(joints[x1])
if polynomialData[muscle_obj_r]['dimension'] == 2:
temp = polynomialData[muscle_obj_r]['spanning']==1
y = (i for i,v in enumerate(temp) if v == True)
x1 = next(y)
x2 = next(y)
ax = fig.add_subplot(ny_a, ny_b, j+1, projection='3d')
ax.scatter(jointCoordinates[:,x1],jointCoordinates[:,x2],
dM_all[joint][j,:])
if joint[-1] == 'r' or joint[-1] == 'l':
ax.scatter(
jointCoordinates[:,x1],jointCoordinates[:,x2],
momentArms[:,momentArmIndices[joint[:-1] + 'l'][j],i],
c='r')
else:
ax.scatter(
jointCoordinates[:,x1],jointCoordinates[:,x2],
momentArms[:,trunkMomentArmPolynomialIndices[j],i],
c='r')
ax.set_title(muscle_obj)
ax.set_xlabel(joints[x1])
ax.set_ylabel(joints[x2])