def homog_2d(xi, theta):
"""
Computes a 3x3 homogeneous transformation matrix given a 2D twist and a
joint displacement
Args:
xi - (3,) ndarray: the 2D twist
(2,1,3)
theta: the joint displacement
0.658
ret_desired = np.array([[-0.3924, -0.9198, 0.1491],
[ 0.9198, -0.3924, 1.2348],
[ 0. , 0. , 1. ]])
Returns:
g - (3,3) ndarray: the resulting homogeneous transformation matrix
"""
if not xi.shape == (3,):
raise TypeError('xi must be a 3-vector')
#YOUR CODE HERE
g = np.eye(3)
r = rotation_2d(xi[2]*theta)
w = xi[2]
p1 = np.array([[1 - cos(w*theta), sin(w*theta)], [-sin(w*theta), 1 - cos(w*theta)]])
p2 = np.array([[0, -1], [1, 0]])
p3 = np.array([[xi[0]/w], [xi[1]/w]])
p = linalg.dot(linalg.dot(p1,p2),p3)
g[0:2,0:2] = r
g[0:2,2] = np.transpose(p)
#print g
#print omega
#g = expm(omega*theta)
return g
def fit_allgasdata( E, conservative=True ) : A = [] # design matrix Y = [] # target values for the fit for e in E : for d in e['data'] : A.append( [ e['D47eq']/d['sD47'], d['d47']/d['sD47'], 1./d['sD47'] ] ) Y.append( d['D47'] / d['sD47'] ) A,Y = numpy.array(A), numpy.array(Y) f = linalg.lstsq(A,Y.T)[0] # best-fit parameters CM = linalg.inv(linalg.dot(A.T,A)) # covariance matrix of fit parameters if conservative : # Scale up uncertainties in the fit parameters if the goodnes-of-fit is worse than average. # To some extent, this helps account for external errors in the gas line data. chi2 = sum( ( Y - linalg.dot( A, f ) )**2 ) nf = sum([len(e['data']) for e in E]) -3 if chi2 > nf : CM = CM * chi2 / nf return f,CM
def fit_gaslines( E, conservative=True ) : A = [] # design matrix Y = [] # target values for the fit nE = len(E) N = sum([len(e['data']) for e in E]) r = 0 for e in E : v = [0.]*nE v[r] = 1. r = r + 1 for d in e['data'] : A.append( scipy.array( [ d['d47'] ] + v ) / d['sD47'] ) Y.append( d['D47'] / d['sD47'] ) A,Y = scipy.array(A), scipy.array(Y) f = linalg.lstsq(A,Y.T)[0] # best-fit parameters C = linalg.inv(linalg.dot(A.T,A)) # covariance matrix of fit parameters if conservative : # Scale up uncertainties in the fit parameters if the goodnes-of-fit is worse than average. # To some extent, this helps account for external errors in the gas line data. chi2 = sum( ( Y - linalg.dot( A, f ) )**2 ) nf = N-3 if chi2 > nf : C = C * chi2 / nf return f,C
Dc = E[1]['D47eq'] a,b,c =f u = (y - a*x - b) / (c-b) dudx = -a/(c-b) dudy = 1/(c-b) duda = -x/(c-b) dudb = (u-1)/(c-b) dudc = -u/(c-b) g = scipy.array([ dudx, dudy, duda, dudb, dudc ]).T Cov = scipy.zeros((5,5)) Cov[1,1] = sy**2 Cov[2:,2:] = C su = (linalg.dot( g.T, linalg.dot( Cov, g ) ))**.5 s['D47corrected'] = E[0]['D47eq'] + (E[1]['D47eq']-E[0]['D47eq']) * u s['sD47corrected'] = (E[1]['D47eq']-E[0]['D47eq']) * su # Although the above formula holds true for linear combinations of normal variables # (Tellinghuisen, 2001) [http://dx.doi.org/10.1021/jp003484u], # Our Monte-Carlo simulations showed that distributions of D47 values # estimated using this code are undistinguishable from normal distributions, # so that 95% confidence limits correspond to +/- 2*sD47corrected. # These limits account for internal and external uncertainties on gas line data, # and for internal errors in sample measurements, but not for external uncertainties in the latter. # It is thus always recommended to check external reproducibility for # duplicate measurements of homogeneous samples. print '%s %.4f %.4f %s' %( s['id'], s['D47corrected'], s['sD47corrected'], s['label'] )
sy = 0 z = x / a - b/a * y - c/a s['trueD47'] = z dzdx = a ** -1 dzdy = -b / a dzda = -(x-b*y-c) * a ** -2 dzdb = -y / a dzdc = -a ** -1 v = numpy.array([ dzdx, dzdy, dzda, dzdb, dzdc ]) C = numpy.zeros((5,5)) C[0,0] = sx ** 2 C[1,1] = sy ** 2 C[2:,2:] = CM sz = (linalg.dot( v, linalg.dot( C, v.T ) ))**.5 s['strueD47'] = sz # Although the above formula holds true for linear combinations of normal variables # (Tellinghuisen, 2001) [http://dx.doi.org/10.1021/jp003484u], # Our Monte-Carlo simulations showed that distributions of D47 values # estimated using this code are undistinguishable from normal distributions, # so that 95% confidence limits correspond to +/- 2*sD47corrected. # These limits account for internal and external uncertainties on gas line data, # and for internal errors in sample measurements, but not for external uncertainties in the latter. # It is thus always recommended to check external reproducibility for # duplicate measurements of homogeneous samples. print '%s %.4f %.4f %s' %( s['id'], s['trueD47'], s['strueD47'], s['label'] ) # xmin = numpy.floor(
