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functions.py
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functions.py
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import numpy as np
import matplotlib.pyplot as plt
from matplotlib.gridspec import GridSpec
from matplotlib.lines import Line2D
import matplotlib.animation as animation
def centered_rectangle(xy, width, height, angle=0.0):
"""Returns the arguments for Rectangle given the x and y coordinates of the
center of the rectangle.
Parameters
==========
xy : tuple of floats
The x and y coordinates of the center of the rectangle.
width : float
Width of the rectangle. When angle=0.0 this is along the x axis.
height : float
Height of the rectangle. When angle=0.0 this is along the y axis.
angle : float
Angle of rotation about the z axis in degrees.
Returns
=======
xy_ll : tuple of floats
The x and y coordinates of the lower left hand corner of the rectangle.
width : float
Width of the rectangle. When angle=0.0 this is along the x axis.
height : float
Height of the rectangle. When angle=0.0 this is along the y axis.
angle : float
Angle of rotation about the z axis in degrees.
"""
xc, yc = xy
theta = np.deg2rad(angle)
x_ll = xc - width/2 * np.cos(theta) + height/2 * np.sin(theta)
y_ll = yc - width/2 * np.sin(theta) - height/2 * np.cos(theta)
xy_ll = (x_ll, y_ll)
return xy_ll, width, height, angle
def spring(xA, xB, yA, yB, w, n=1, x=None, y=None):
"""Returns the x and y coordinates of the points that define a spring
diagram between points (xA, yB) and (yA, yB).
Parameters
==========
xA : float
x coordinate of the beginning of the spring.
xB : float
x coordinate of the end of the spring.
yA : float
y coordinate of the beginning of the spring.
yB : float
y coordinate of the end of the spring.
w : float
The width of the spring.
n : integer, optional
Number of coils.
x : ndarray, shape(2*n + 2), optional
Preallocated array for the results.
y : ndarray, shape(2*n + 2), optional
Preallocated array for the results.
Returns
=======
x : ndarray, shape(2*n + 2)
x coordinates of the points that define the ends of each line in the
spring.
y : ndarray, shape(2*n + 2)
y coordinates of the points that define the ends of each line in the
spring.
Examples
========
>>> import numpy as np
>>> import matplotlib.pyplot as plt
>>> from resonance.functions import spring
>>> plt.axes().set_aspect('equal')
>>> for angle in np.arange(0, 2*np.pi, np.pi/4):
... plt.plot(*spring(0.0, np.cos(angle),
... 0.0, np.sin(angle), 0.1, n=4)) # doctest: +SKIP
...
>>> plt.show()
"""
if xB < xA:
xA, yA, xB, yB = xB, yB, xA, yA
# NOTE : Epsilon is needed to prevent divide by zero for vertically
# oriented springs.
theta = np.arctan((yB-yA) / (xB - xA + np.finfo(float).eps))
d = np.sqrt((xB-xA)**2 + (yB-yA)**2)
s_th = np.sin(theta)
c_th = np.cos(theta)
xst = xA - w/2*s_th
yst = yA + w/2*c_th
xsb = xA + w/2*s_th
ysb = yA - w/2*c_th
if x is None:
x = xA * np.ones(2*n + 2)
if y is None:
y = yA * np.ones(2*n + 2)
x[1] = xsb + 1/4*d/n*c_th
y[1] = ysb + 1/4*d/n*s_th
x[2] = xst + 3/4*d/n*c_th
y[2] = yst + 3/4*d/n*s_th
x[-1] = xB
y[-1] = yB
for i in range(3, 2*n + 1):
x[i] = x[i-2] + d/n*c_th
y[i] = y[i-2] + d/n*s_th
return x, y
def estimate_period(time, signal):
"""Computes the period of oscillation based on the given periodic signal.
Parameters
==========
time : array_like, shape(n,)
An array of monotonically increasing time values.
signal : array_like, shape(n,)
An array of values for the periodic signal at each time in ``t``.
Returns
=======
period : float
An estimate of the period of oscillation.
"""
peak_idxs = np.diff(np.sign(signal)) < 0
peak_idxs = np.hstack((peak_idxs, False))
period = np.diff(time[peak_idxs]).mean()
return period
def benchmark_par_to_canonical(p):
"""Returns the canonical matrices of the Whipple bicycle model linearized
about the upright constant velocity configuration. It uses the parameter
definitions from [Meijaard2007]_.
Parameters
==========
p : dictionary
A dictionary of the benchmark bicycle parameters. Make sure your units
are correct, best to ue the benchmark paper's units!
Returns
=======
M : ndarray, shape(2,2)
The mass matrix.
C1 : ndarray, shape(2,2)
The damping like matrix that is proportional to the speed, v.
K0 : ndarray, shape(2,2)
The stiffness matrix proportional to gravity, g.
K2 : ndarray, shape(2,2)
The stiffness matrix proportional to the speed squared, v**2.
References
==========
.. [Meijaard2007] J. P. Meijaard, J. M. Papadopoulos, A. Ruina, and A. L.
Schwab, "Linearized dynamics equations for the balance and steer of a
bicycle: A benchmark and review," Proceedings of the Royal Society A:
Mathematical, Physical and Engineering Sciences, vol. 463, no. 2084, pp.
1955–1982, Aug. 2007.
"""
mT = p['mR'] + p['mB'] + p['mH'] + p['mF']
xT = (p['xB'] * p['mB'] + p['xH'] * p['mH'] + p['w'] * p['mF']) / mT
zT = (-p['rR'] * p['mR'] + p['zB'] * p['mB'] +
p['zH'] * p['mH'] - p['rF'] * p['mF']) / mT
ITxx = (p['IRxx'] + p['IBxx'] + p['IHxx'] + p['IFxx'] + p['mR'] *
p['rR']**2 + p['mB'] * p['zB']**2 + p['mH'] * p['zH']**2 + p['mF'] *
p['rF']**2)
ITxz = (p['IBxz'] + p['IHxz'] - p['mB'] * p['xB'] * p['zB'] -
p['mH'] * p['xH'] * p['zH'] + p['mF'] * p['w'] * p['rF'])
p['IRzz'] = p['IRxx']
p['IFzz'] = p['IFxx']
ITzz = (p['IRzz'] + p['IBzz'] + p['IHzz'] + p['IFzz'] +
p['mB'] * p['xB']**2 + p['mH'] * p['xH']**2 + p['mF'] * p['w']**2)
mA = p['mH'] + p['mF']
xA = (p['xH'] * p['mH'] + p['w'] * p['mF']) / mA
zA = (p['zH'] * p['mH'] - p['rF'] * p['mF']) / mA
IAxx = (p['IHxx'] + p['IFxx'] + p['mH'] * (p['zH'] - zA)**2 +
p['mF'] * (p['rF'] + zA)**2)
IAxz = (p['IHxz'] - p['mH'] * (p['xH'] - xA) * (p['zH'] - zA) + p['mF'] *
(p['w'] - xA) * (p['rF'] + zA))
IAzz = (p['IHzz'] + p['IFzz'] + p['mH'] * (p['xH'] - xA)**2 + p['mF'] *
(p['w'] - xA)**2)
uA = (xA - p['w'] - p['c']) * np.cos(p['lam']) - zA * np.sin(p['lam'])
IAll = (mA * uA**2 + IAxx * np.sin(p['lam'])**2 +
2 * IAxz * np.sin(p['lam']) * np.cos(p['lam']) +
IAzz * np.cos(p['lam'])**2)
IAlx = (-mA * uA * zA + IAxx * np.sin(p['lam']) + IAxz *
np.cos(p['lam']))
IAlz = (mA * uA * xA + IAxz * np.sin(p['lam']) + IAzz *
np.cos(p['lam']))
mu = p['c'] / p['w'] * np.cos(p['lam'])
SR = p['IRyy'] / p['rR']
SF = p['IFyy'] / p['rF']
ST = SR + SF
SA = mA * uA + mu * mT * xT
Mpp = ITxx
Mpd = IAlx + mu * ITxz
Mdp = Mpd
Mdd = IAll + 2 * mu * IAlz + mu**2 * ITzz
M = np.array([[Mpp, Mpd], [Mdp, Mdd]])
K0pp = mT * zT # this value only reports to 13 digit precision it seems?
K0pd = -SA
K0dp = K0pd
K0dd = -SA * np.sin(p['lam'])
K0 = np.array([[K0pp, K0pd], [K0dp, K0dd]])
K2pp = 0.
K2pd = (ST - mT * zT) / p['w'] * np.cos(p['lam'])
K2dp = 0.
K2dd = (SA + SF * np.sin(p['lam'])) / p['w'] * np.cos(p['lam'])
K2 = np.array([[K2pp, K2pd], [K2dp, K2dd]])
C1pp = 0.
C1pd = (mu * ST + SF * np.cos(p['lam']) + ITxz / p['w'] *
np.cos(p['lam']) - mu*mT*zT)
C1dp = -(mu * ST + SF * np.cos(p['lam']))
C1dd = (IAlz / p['w'] * np.cos(p['lam']) + mu * (SA +
ITzz / p['w'] * np.cos(p['lam'])))
C1 = np.array([[C1pp, C1pd], [C1dp, C1dd]])
return M, C1, K0, K2
class Phasor(object):
"""Phasor that can be advanced in time with rotation and growth rates.
Parameters
----------
init : complex
Initial phasor in rectangular form (Re + jIm)
frequency : float, optional
Rotation rate in rad/s.
growth_rate : float, optional
Exponential growth rate (decay if < 0).
Attributes
----------
t : float
Current time.
re : float
Current real component of the phasor.
im : float
Current imaginary component of the phasor.
radius : float
Current radius of the phasor.
angle : float
Current angle of the phasor.
trace_t : list
History of time values (since most recent `clear()`).
trace_re : list
History of real component values (since most recent `clear()`).
trace_im : list
History of imaginary component values (since most recent `clear()`).
"""
def __init__(self, init, frequency=0, growth_rate=0):
self.init = init
self.frequency = frequency
self.growth_rate = growth_rate
self.trace_t = []
self.trace_re = []
self.trace_im = []
self._init()
@classmethod
def from_eig(cls, eigvec_component, eigval):
"""Creates a phasor from an eigenvalue/eigenvector component pair.
Parameters
----------
eigvec_component : complex
A single eigenvector component representing the phasor's initial
real/imaginary parts.
eigval : complex
The eigenvector, which specifies the phasor's growth rate (real
part) and rotational frequency (imaginary part).
"""
return cls(eigvec_component, np.imag(eigval), np.real(eigval))
def advance(self, dt):
"""Advance the phasor by a time step dt."""
self.t += dt
self.radius *= np.exp(self.growth_rate * dt)
self.angle += self.frequency * dt
self._update_rect()
self.trace_t.append(self.t)
self.trace_re.append(self.re)
self.trace_im.append(self.im)
def clear(self):
"""Clear trajectories."""
self._init()
del self.trace_t[:]
del self.trace_re[:]
del self.trace_im[:]
def _init(self):
"""Initialize parameters."""
self.t = 0
self.radius = np.abs(self.init)
self.angle = np.angle(self.init)
self._update_rect()
def _update_rect(self):
"""Update rectangular components."""
vec = self.radius * np.exp(1j * self.angle)
self.re = np.real(vec)
self.im = np.imag(vec)
class PhasorAnimation(animation.TimedAnimation):
"""Animation for demonstrating rotating phasors.
Two axes are set up. On top, there is an s-plane to show the real and
imaginary components of the phasors. The current phasor "vector" is shown
with a thick line, the current endpoint of the vector is shown with a
circle, thin lines show the projection of the real part of the phasor down
to the bottom of the plane, and the time history of the endpoint of the
vectors are shown.
On bottom, the phasors' real components are plotted in time. The plot is
rotated so that time is positive downward, and the x axes of the s-plane
and the time plots are lined up. The current value is shown with a circle,
thin lines show the projection from the top of the plot to the current
value, and the time history is plotted.
Parameters
----------
fig : Figure
matplotlib Figure object on which to animate.
t : array
Array of time values at which to plot. Even time spacing is assumed.
phasors : list
List of Phasor objects to advance and plot.
re_range : tuple, optional
Limits of the real axis.
im_range : tuple, optional
Limits of the imaginary axis.
repeat : bool, optional
Specifies whether or not to repeat the animation once it finishes.
repeat_delay : float, optional
Amount of time to wait before repeating the animation in milliseconds.
time_stretch : float, optional
Multiplicative factor of the plotting interval. Increasing
`time_stretch` effectively makes the animation slower without affecting
the time units.
blit : bool, optional
Specifies whether or not to use blitting.
"""
def __init__(self, fig, t, phasors, re_range=(-1, 1), im_range=(-1, 1),
repeat=True, repeat_delay=0, time_stretch=1, blit=True):
self.t = t
self.dt = t[1] - t[0]
self.phasors = phasors
self.re_range = re_range
self.im_range = im_range
gs = GridSpec(2, 1, height_ratios=[1, 2])
# s-plane plot of phasors
ax_s = fig.add_subplot(gs[0])
ax_s.set_ylabel('Im')
ax_s.set_xlim(*re_range)
ax_s.set_ylim(*im_range)
ax_s.set_xticklabels(ax_s.get_xticklabels(), visible=False)
ax_s.grid()
# time plot of the real part of the phasors
ax_t = fig.add_subplot(gs[1])
ax_t.set_xlabel('Re')
ax_t.set_ylabel('t')
ax_t.set_xlim(*re_range)
ax_t.set_ylim(self.t[0], self.t[-1]+self.dt)
ax_t.invert_yaxis()
ax_t.grid()
fig.subplots_adjust(hspace=0)
fig.tight_layout()
# vectors in the Re/Im axis from origin
self.vec_lines = []
# dots at the end of the vectors in the Re/Im axis
self.vec_dots = []
# lines streaming from the endpoints of the vectors to the axis base
self.vec_connector_lines = []
# trace of vectors in Re/Im axis
self.vec_trace_lines = []
# dot showing current x(t) value
self.time_dots = []
# lines streaming from the current x(t) value to the top of the axis
self.time_connector_lines = []
# trace of x(t) values
self.time_trace_lines = []
color_vals = np.linspace(0, 1, 10)
for i, phasor in zip(color_vals, phasors):
c = plt.cm.Set1(i)
vl = Line2D([], [], color=c, linewidth=2)
self.vec_lines.append(vl)
ax_s.add_line(vl)
vcl = Line2D([], [], color=c, linewidth=0.5)
self.vec_connector_lines.append(vcl)
ax_s.add_line(vcl)
vd = Line2D([], [], color=c, marker='o', linewidth=0)
self.vec_dots.append(vd)
ax_s.add_line(vd)
vtl = Line2D([], [], color=c)
self.vec_trace_lines.append(vtl)
ax_s.add_line(vtl)
td = Line2D([], [], color=c, marker='o', linewidth=0)
self.time_dots.append(td)
ax_t.add_line(td)
tcl = Line2D([], [], color=c, linewidth=0.5)
self.time_connector_lines.append(tcl)
ax_t.add_line(tcl)
ttl = Line2D([], [], color=c)
self.time_trace_lines.append(ttl)
ax_t.add_line(ttl)
animation.TimedAnimation.__init__(
self, fig, interval=time_stretch/self.dt, blit=blit,
repeat=repeat, repeat_delay=repeat_delay)
def new_frame_seq(self):
return iter(range(self.t.size))
def _draw_frame(self, framedata):
self._drawn_artists = []
# advance phasors and plot them
for i, phasor in enumerate(self.phasors):
phasor.advance(self.dt)
self.vec_lines[i].set_data([0, phasor.re], [0, phasor.im])
self.vec_dots[i].set_data(phasor.re, phasor.im)
self.vec_connector_lines[i].set_data([phasor.re, phasor.re],
[phasor.im, self.im_range[0]])
self.vec_trace_lines[i].set_data(phasor.trace_re, phasor.trace_im)
self.time_dots[i].set_data(phasor.re, phasor.t)
self.time_connector_lines[i].set_data([phasor.re, phasor.re],
[0, phasor.t])
self.time_trace_lines[i].set_data(phasor.trace_re, phasor.trace_t)
# add lines to _drawn_artists
for phasor in self.phasors:
self._drawn_artists.extend(self.vec_lines)
self._drawn_artists.extend(self.vec_dots)
self._drawn_artists.extend(self.vec_connector_lines)
self._drawn_artists.extend(self.vec_trace_lines)
self._drawn_artists.extend(self.time_dots)
self._drawn_artists.extend(self.time_connector_lines)
self._drawn_artists.extend(self.time_trace_lines)
def _init_draw(self):
# clear the phasor trajectories
for phasor in self.phasors:
phasor.clear()
# reset line data
if getattr(self, '_drawn_artists', None) is not None:
for a in self._drawn_artists:
a.set_data([], [])