def display_error(errors, method):
err = error(errors, n=1)
err_max = error(errors)
pylog.debug(
"Obtained an error of {} using {} method (max={}, len={})".format(
err, method, err_max, len(method)))
def exercise1():
if DEFAULT["1a"] is True:
exercise1a()
elif DEFAULT["1b"] is True:
exercise1b()
elif DEFAULT["1c"] is True:
exercise1c()
elif DEFAULT["1d"] is True:
exercise1d()
elif DEFAULT["1f"] is True:
exercise1f()
else:
exercise1a()
exercise1b()
exercise1c()
exercise1d()
exercise1f()
if DEFAULT["save_figures"] is False:
plt.show()
else:
figures = plt.get_figlabels()
print(figures)
pylog.debug("Saving figures:\n{}".format(figures))
for fig in figures:
plt.figure(fig)
save_figure(fig)
plt.close(fig)
def error(method_state, analytical_state, method):
""" Compute error of an ode method based on analytical solution """
err = np.sum(np.abs(analytical_state - method_state)) / len(method_state)
err_max = max(np.abs(analytical_state - method_state))
pylog.debug(
"Obtained an error of {} using {} method (max={}, len={})".format(
err, method, err_max, len(method_state)))
return err
def L(self, value):
""" Keyword Arguments:
value -- Set the value of pendulum's length [m] """
self.parameters['L'] = value
# ReCompute inertia
# Inertia = m*l**2
self.I = self.m * self.parameters['L']**2
pylog.debug('Changed pendulum length to {} [m]'.format(self.L))
def m(self, value):
"""
Set the mass of the pendulum.
Setting/Changing mass will automatically recompute the inertia.
"""
self.parameters['m'] = value
# ReCompute inertia
# Inertia = m*l**2
self.I = self.parameters['m'] * self.L**2
pylog.debug('Changed pendulum mass to {} [kg]'.format(self.m))
def exercise1(clargs):
""" Exercise 1 """
# Setup
pylog.info("Running exercise 1")
# Setup
time_max = 5 # Maximum simulation time
time_step = 0.2 # Time step for ODE integration in simulation
x0 = np.array([1.]) # Initial state
# Integration methods (Exercises 1.a - 1.d)
pylog.info("Running function integration using different methods")
# Example
pylog.debug("Running example plot for integration (remove)")
example = example_integrate(x0, time_max, time_step)
example.plot_state(figure="Example", label="Example", marker=".")
# Analytical (1.a)
time = np.arange(0, time_max, time_step) # Time vector
x_a = analytic_function(time)
analytical = Result(x_a, time) if x_a is not None else None
# Euler (1.b)
euler = euler_integrate(function, x0, time_max, time_step)
# ODE (1.c)
ode = ode_integrate(function, x0, time_max, time_step)
# ODE Runge-Kutta (1.c)
ode_rk = ode_integrate_rk(function_rk, x0, time_max, time_step)
# Euler with lower time step (1.d)
pylog.warning("Euler with smaller ts must be implemented")
euler_time_step = None
euler_ts_small = (euler_integrate(function, x0, time_max, euler_time_step)
if euler_time_step is not None else None)
# Plot integration results
plot_integration_methods(analytical=analytical,
euler=euler,
ode=ode,
ode_rk=ode_rk,
euler_ts_small=euler_ts_small,
euler_timestep=time_step,
euler_timestep_small=euler_time_step)
# Error analysis (Exercise 1.e)
pylog.warning("Error analysis must be implemented")
# Show plots of all results
if not clargs.save_figures:
plt.show()
return
def save_figure(figure, name=None, **kwargs):
""" Save figure """
for extension in kwargs.pop("extensions", ["png"]):
fig = figure.replace(" ", "_").replace(".", "dot")
if name is None:
name = "{}.{}".format(fig, extension)
else:
name = "{}.{}".format(name, extension)
name = save_folder + name
plt.figure(figure)
plt.savefig(name, bbox_inches='tight')
pylog.debug("Saving figure {}...".format(name))
def exercise3():
time_param = TimeParameters(time_start=0, time_stop=10., time_step=0.001)
exercise3a(time_param)
exercise3b(time_param)
if DEFAULT["save_figures"]:
figures = plt.get_figlabels()
pylog.debug("Saving figures:\n{}".format(figures))
for fig in figures:
plt.figure(fig)
save_figure(fig)
plt.show()
def save_figure(figure, name=None, **kwargs):
""" Save figure """
for extension in kwargs.pop("extensions", ["pdf"]):
fig = figure.replace(" ", "_").replace(".", "dot")
if name is None:
name = "{}.{}".format(fig, extension)
else:
name = "{}.{}".format(name, extension)
fig = plt.figure(figure)
size = plt.rcParams.get('figure.figsize')
fig.set_size_inches(0.7 * size[0], 0.7 * size[1], forward=True)
plt.savefig(name, bbox_inches='tight')
pylog.debug("Saving figure {}...".format(name))
fig.set_size_inches(size[0], size[1], forward=True)
def exercise1():
#exercise1a()
exercise1d()
if DEFAULT["save_figures"] is False:
plt.show()
else:
figures = plt.get_figlabels()
print(figures)
pylog.debug("Saving figures:\n{}".format(figures))
for fig in figures:
plt.figure(fig)
save_figure(fig)
plt.close(fig)
def exercise2():
""" Main function to run for Exercise 2.
"""
exercise2b()
if not DEFAULT["save_figures"]:
plt.show()
else:
figures = plt.get_figlabels()
pylog.debug("Saving figures:\n{}".format(figures))
for fig in figures:
plt.figure(fig)
save_figure(fig)
plt.close(fig)
def pendulum_set_position(x0, time=0.0, *args):
""" Function to analyse the pendulum spring damper system"""
pendulum = args[0]
pendulum.parameters.b1 = 1.
pendulum.parameters.b2 = 1.
pendulum.parameters.k1 = 50.0
pendulum.parameters.k2 = 50.0
pendulum.parameters.s_theta_ref1 = np.deg2rad(0.0)
pendulum.parameters.s_theta_ref2 = np.deg2rad(65.6)
pylog.info(
"1b. Running pendulum_system to set fixed position")
pylog.info(pendulum.parameters.showParameters())
title = "{} Pendulum Fixed Position (x0 = {})"
res = integrate(pendulum_integration, x0, time, args=args)
pylog.debug('Position : {}'.format(np.rad2deg(res.state[-1])))
res.plot_state(title.format("State", x0))
res.plot_phase(title.format("Phase", x0))
def save_figure(figure, name=None):
""" Save figure """
if os.path.isdir(DEFAULT["save_folder"]):
path = DEFAULT["save_folder"]
else:
pylog.warning(
"The DEFAULT save directory does not exist. Switching to current directory"
)
path = os.getcwd()
if DEFAULT["save_figures"]:
for extension in DEFAULT["save_extensions"]:
fig = figure.replace(" ", "_").replace(".", "dot")
if name is None:
name = "{}.{}".format(fig, extension)
else:
name = "{}.{}".format(name, extension)
plt.savefig(path + name, bbox_inches='tight')
pylog.debug("Saving figure {}...".format(name))
def exercise3(clargs):
""" Exercise 3 """
parameters = PendulumParameters() # Checkout pendulum.py for more info
pylog.info(parameters)
# Simulation parameters
time = np.arange(0, 30, 0.01) # Simulation time
x0 = [0.1, 0.0] # Initial state
# To use/modify pendulum parameters (See PendulumParameters documentation):
# parameters.g = 9.81 # Gravity constant
# parameters.m = 1. # Mass
# parameters.L = 1. # Length
# parameters.I = 1. # Inertia (Automatically computed!)
# parameters.d = 0.3 # damping
# parameters.sin = np.sin # Sine function
# parameters.dry = False # Use dry friction (True or False)
# Example of system integration (Similar to lab1)
# (NOTE: pendulum_equation must be imlpemented first)
pylog.debug("Running integration example")
res = integrate(pendulum_system, x0, time, args=(parameters,))
res.plot_state("State")
res.plot_phase("Phase")
# Evolutions
# Write code here (You can add functions for the different cases)
pylog.warning(
"Evolution of pendulum in normal conditions must be implemented"
)
pylog.warning(
"Evolution of pendulum without damping must be implemented"
)
pylog.warning(
"Evolution of pendulum with perturbations must be implemented"
)
pylog.warning(
"Evolution of pendulum with dry friction must be implemented"
)
# Show plots of all results
if not clargs.save_figures:
plt.show()
def exercise1():
plt.close("all")
pylog.info("Start exercise 1")
time_param = TimeParameters(time_start=0.0,
time_stop=0.2,
time_step=0.001,
time_stabilize=0.2)
exercise1a(time_param)
exercise1b(time_param)
exercise1c(time_param)
time_param.t_stop = 0.3 # change time parameters for the second part
exercise1d(time_param)
exercise1e(time_param)
if DEFAULT["save_figures"]:
figures = plt.get_figlabels()
pylog.debug("Saving figures:\n{}".format(figures))
for fig in figures:
plt.figure(fig)
save_figure(fig)
plt.show()
def two_li_ode(y, t, params):
""" Derivative function of a network of 2 leaky integrator neurons
y is the vector of membrane potentials (variable m in lecture equations)
yd the derivative of the vector of membrane potentials
"""
# Extract parameters
tau, D, b, w, exp = params.list()
# Update the firing rates:
y = np.array(y)
x = 1 / (1 + exp(-D * (y + b)))
# IMPLEMENT THE DIFFERENTIAL EQUATION FOR THE MEMBRANE POTENTIAL
# Compute the dentritic sums for both neurons
dend_sum = w[0] * x[0] + w[1] * x[1]
# Compute the membrane potential derivative:
yd = (-y + dend_sum) / tau
pylog.debug("x: {}\ndend_sum: {}\nyd: {}".format(x, dend_sum, yd))
return yd
def exercise2():
""" Main function to run for Exercise 2.
Parameters
----------
None
Returns
-------
None
"""
# Define and Setup your pendulum model here
# Check PendulumSystem.py for more details on Pendulum class
pendulum_params = PendulumParameters() # Instantiate pendulum parameters
pendulum_params.L = 0.5 # To change the default length of the pendulum
pendulum_params.m = 1. # To change the default mass of the pendulum
pendulum = PendulumSystem(pendulum_params) # Instantiate Pendulum object
#### CHECK OUT PendulumSystem.py to ADD PERTURBATIONS TO THE MODEL #####
pylog.info('Pendulum model initialized \n {}'.format(
pendulum.parameters.showParameters()))
# Define and Setup your pendulum model here
# Check MuscleSytem.py for more details on MuscleSytem class
M1_param = MuscleParameters() # Instantiate Muscle 1 parameters
M1_param.f_max = 1500 # To change Muscle 1 max force
M2_param = MuscleParameters() # Instantiate Muscle 2 parameters
M2_param.f_max = 1500 # To change Muscle 2 max force
M1 = Muscle(M1_param) # Instantiate Muscle 1 object
M2 = Muscle(M2_param) # Instantiate Muscle 2 object
# Use the MuscleSystem Class to define your muscles in the system
muscles = MuscleSytem(M1, M2) # Instantiate Muscle System with two muscles
pylog.info('Muscle system initialized \n {} \n {}'.format(
M1.parameters.showParameters(), M2.parameters.showParameters()))
# Define Muscle Attachment points
m1_origin = np.array([-0.17, 0.0]) # Origin of Muscle 1
m1_insertion = np.array([0.0, -0.17]) # Insertion of Muscle 1
m2_origin = np.array([0.17, 0.0]) # Origin of Muscle 2
m2_insertion = np.array([0.0, -0.17]) # Insertion of Muscle 2
# Attach the muscles
muscles.attach(np.array([m1_origin, m1_insertion]),
np.array([m2_origin, m2_insertion]))
# Create a system with Pendulum and Muscles using the System Class
# Check System.py for more details on System class
sys = System() # Instantiate a new system
sys.add_pendulum_system(pendulum) # Add the pendulum model to the system
sys.add_muscle_system(muscles) # Add the muscle model to the system
##### Time #####
t_max = 3 # Maximum simulation time
time = np.arange(0., t_max, 0.004) # Time vector
##### Model Initial Conditions #####
x0_P = np.array([0., 0.]) # Pendulum initial condition
# Muscle Model initial condition
x0_M = np.array([0., M1.L_OPT, 0., M2.L_OPT])
x0 = np.concatenate((x0_P, x0_M)) # System initial conditions
##### System Simulation #####
# For more details on System Simulation check SystemSimulation.py
# SystemSimulation is used to initialize the system and integrate
# over time
sim = SystemSimulation(sys) # Instantiate Simulation object
# Add muscle activations to the simulation
# Here you can define your muscle activation vectors
# that are time dependent
'''
#act1 = np.ones((len(time), 1)) * 1.
#act2 = np.ones((len(time), 1)) * 0.05
act1 = (np.sin((time/t_max)*10*np.pi)+1)/2
act2 = (np.sin((time/t_max)*10*np.pi + np.pi)+1)/2
act1 = np.reshape(act1, (len(time),1))
act2 = np.reshape(act2, (len(time),1))
activations = np.hstack((act1, act2))
# Plotting the results
plt.figure('Activations')
plt.title('Muscle activations')
plt.plot(time, act1, label = 'Activation muscle 1')
plt.plot(time, act2, label = 'Activation muscle 2')
plt.xlabel('Time [s]')
plt.ylabel('Activation')
plt.legend()
plt.grid()
# Method to add the muscle activations to the simulation
sim.add_muscle_activations(activations)
'''
max_amplitude = np.zeros([10, 10])
i = 0
j = 0
# Simulate the system for given time
for activation_max in np.arange(0, 1, 0.9):
i = 0
for frequency in np.arange(1, 10, 4):
act1 = ((np.sin(
(time / t_max) * frequency * np.pi) + 1) / 2) * activation_max
act2 = ((np.sin((time / t_max) * frequency * np.pi + 1) + 1) /
2) * activation_max
act1 = np.reshape(act1, (len(time), 1))
act2 = np.reshape(act2, (len(time), 1))
activations = np.hstack((act1, act2))
sim.add_muscle_activations(activations)
sim.initalize_system(x0, time) # Initialize the system state
#: If you would like to perturb the pedulum model then you could do
# so by
sim.sys.pendulum_sys.parameters.PERTURBATION = False
# The above line sets the state of the pendulum model to zeros between
# time interval 1.2 < t < 1.25. You can change this and the type of
# perturbation in
# pendulum_system.py::pendulum_system function
# Integrate the system for the above initialized state and time
sim.simulate()
# Obtain the states of the system after integration
# res is np.array [time, states]
# states vector is in the same order as x0
res = sim.results()
# In order to obtain internal states of the muscle
# you can access the results attribute in the muscle class
muscle1_results = sim.sys.muscle_sys.Muscle1.results
muscle2_results = sim.sys.muscle_sys.Muscle2.results
max_amplitude[i, j] = np.max(np.abs(res[:, 1]))
i += 1
# Plotting the results
plt.figure('Pendulum')
plt.title('Pendulum Phase')
plt.plot(res[:, 1],
res[:, 2],
label='activation %.2f - frequency %f' %
(activation_max, frequency))
plt.xlabel('Position [rad]')
plt.ylabel('Velocity [rad.s]')
plt.grid()
j += 1
plt.figure('Amplitude')
fig, ax1 = plt.subplots(1, 1)
ax1.set_xticklabels(np.array([0, 0, 0.2, 0.4, 0.8, 1]))
ax1.set_yticklabels(np.array([0, 1, 3, 5, 7, 9]))
plt.title('Ampliudes')
plt.imshow(max_amplitude, aspect='equal', origin='lower')
plt.xlabel('Activation')
plt.ylabel('Frequncy')
# To animate the model, use the SystemAnimation class
# Pass the res(states) and systems you wish to animate
simulation = SystemAnimation(res, pendulum, muscles)
# To start the animation
if DEFAULT["save_figures"] is False:
simulation.animate()
if not DEFAULT["save_figures"]:
plt.show()
else:
figures = plt.get_figlabels()
pylog.debug("Saving figures:\n{}".format(figures))
for fig in figures:
plt.figure(fig)
save_figure(fig)
plt.close(fig)
def exercise2():
""" Main function to run for Exercise 2.
"""
# Define and Setup your pendulum model here
# Check PendulumSystem.py for more details on Pendulum class
pendulum_params = PendulumParameters() # Instantiate pendulum parameters
pendulum_params.L = 0.5 # To change the default length of the pendulum
pendulum_params.m = 1. # To change the default mass of the pendulum
pendulum = PendulumSystem(pendulum_params) # Instantiate Pendulum object
#### CHECK OUT PendulumSystem.py to ADD PERTURBATIONS TO THE MODEL #####
pylog.info('Pendulum model initialized \n {}'.format(
pendulum.parameters.showParameters()))
# Define and Setup your pendulum model here
# Check MuscleSytem.py for more details on MuscleSytem class
M1_param = MuscleParameters() # Instantiate Muscle 1 parameters
M1_param.f_max = 1500 # To change Muscle 1 max force
M2_param = MuscleParameters() # Instantiate Muscle 2 parameters
M2_param.f_max = 1500 # To change Muscle 2 max force
M1 = Muscle(M1_param) # Instantiate Muscle 1 object
M2 = Muscle(M2_param) # Instantiate Muscle 2 object
# Use the MuscleSystem Class to define your muscles in the system
muscles = MuscleSytem(M1, M2) # Instantiate Muscle System with two muscles
pylog.info('Muscle system initialized \n {} \n {}'.format(
M1.parameters.showParameters(),
M2.parameters.showParameters()))
# 2a : set of muscle 1 attachment points
m1_origin = np.array([[-0.17, 0.0]]) # Origin of Muscle 1
m1_insertion = np.array([[0.0, -0.17], [0.0, -0.3], [0.0, -0.4], [0.0, -0.5]]) # Insertion of Muscle 1
theta = np.linspace(-np.pi/2,np.pi/2)
m_lengths = np.zeros((len(m1_insertion),len(theta)))
m_moment_arms = np.zeros((len(m1_insertion),len(theta)))
leg=[]
for i in range(0,len(m1_insertion)):
m_lengths[i,:]=np.sqrt(m1_origin[0,0]**2 + m1_insertion[i,1]**2 +
2 * np.abs(m1_origin[0,0]) * np.abs(m1_insertion[i,1]) * np.sin(theta))
m_moment_arms[i,:]= m1_origin[0,0] * m1_insertion[i,1] * np.cos(theta) / m_lengths[i,:]
leg.append('Origin: {}m, Insertion: {}m'.format(m1_origin[0,0],m1_insertion[i,1]))
# Plotting
plt.figure('2a length')
plt.title('Length of M1 with respect to the position of the limb')
for i in range(0,len(m_lengths)):
plt.plot(theta*180/np.pi, m_lengths[i,:])
plt.plot((theta[0]*180/np.pi,theta[len(theta)-1]*180/np.pi),(0.11,0.11), ls='dashed')
leg.append('l_opt')
plt.plot((theta[0]*180/np.pi,theta[len(theta)-1]*180/np.pi),(0.13,0.13), ls='dashed')
leg.append('l_slack')
plt.xlabel('Position [deg]')
plt.ylabel('Muscle length [m]')
plt.legend(leg)
plt.grid()
plt.savefig('2_a_length.png')
plt.figure('2a moment')
plt.title('Moment arm over M1 with respect to the position of the limb')
for i in range(0,len(m_moment_arms)):
plt.plot(theta*180/np.pi, m_moment_arms[i,:])
plt.xlabel('Position [deg]')
plt.ylabel('Moment arm [m]')
plt.legend(leg)
plt.grid()
plt.savefig('2_a_moment.png')
# 2b : simple activation wave forms
# Muscle 2 attachement point
m2_origin = np.array([0.17, 0.0]) # Origin of Muscle 2
m2_insertion = np.array([0.0, -0.17]) # Insertion of Muscle 2
# Attach the muscles
muscles.attach(np.array([m1_origin[0,:], m1_insertion[0,:]]),
np.array([m2_origin, m2_insertion]))
# Create a system with Pendulum and Muscles using the System Class
# Check System.py for more details on System class
sys = System() # Instantiate a new system
sys.add_pendulum_system(pendulum) # Add the pendulum model to the system
sys.add_muscle_system(muscles) # Add the muscle model to the system
##### Time #####
t_max = 2.5 # Maximum simulation time
time = np.arange(0., t_max, 0.001) # Time vector
##### Model Initial Conditions #####
x0_P = np.array([np.pi/4, 0.]) # Pendulum initial condition
# Muscle Model initial condition
x0_M = np.array([0., M1.L_OPT, 0., M2.L_OPT])
x0 = np.concatenate((x0_P, x0_M)) # System initial conditions
##### System Simulation #####
# For more details on System Simulation check SystemSimulation.py
# SystemSimulation is used to initialize the system and integrate
# over time
sim = SystemSimulation(sys) # Instantiate Simulation object
# Add muscle activations to the simulation
# Here you can define your muscle activation vectors
# that are time dependent
sin_frequency = 2 #Hz
amp_stim = 1
phase_shift = np.pi
act1 = np.zeros((len(time),1))
act2 = np.zeros((len(time),1))
for i in range(0,len(time)):
act1[i,0] = amp_stim*(1+np.sin(2*np.pi*sin_frequency*time[i]))/2
act2[i,0] = amp_stim*(1+ np.sin(2*np.pi*sin_frequency*time[i] + phase_shift))/2
plt.figure('2b activation')
plt.plot(time,act1)
plt.plot(time,act2)
plt.legend(["Activation for muscle 1", "Activation for muscle 2"])
plt.title('Activation for muscle 1 and 2 with simple activation wave forms')
plt.xlabel("Time [s]")
plt.ylabel("Activation")
plt.savefig('2_b_activation.png')
plt.show()
activations = np.hstack((act1, act2))
# Method to add the muscle activations to the simulation
sim.add_muscle_activations(activations)
# Simulate the system for given time
sim.initalize_system(x0, time) # Initialize the system state
#: If you would like to perturb the pedulum model then you could do
# so by
sim.sys.pendulum_sys.parameters.PERTURBATION = True
# The above line sets the state of the pendulum model to zeros between
# time interval 1.2 < t < 1.25. You can change this and the type of
# perturbation in
# pendulum_system.py::pendulum_system function
# Integrate the system for the above initialized state and time
sim.simulate()
# Obtain the states of the system after integration
# res is np.array [time, states]
# states vector is in the same order as x0
res = sim.results()
# Plotting the results
plt.figure('2b phase')
plt.title('Pendulum Phase')
plt.plot(res[:, 1], res[:, 2])
plt.xlabel('Position [rad]')
plt.ylabel('Velocity [rad/s]')
plt.grid()
plt.savefig('2_b_phase.png')
plt.show()
plt.figure('2b oscillations')
plt.title('Pendulum Oscillations')
plt.plot(time,res[:, 1])
plt.xlabel('Time [s]')
plt.ylabel('Position [rad]')
plt.grid()
plt.savefig('2_b_oscillations.png')
plt.show()
# To animate the model, use the SystemAnimation class
# Pass the res(states) and systems you wish to animate
simulation = SystemAnimation(res, pendulum, muscles)
# To start the animation
if DEFAULT["save_figures"] is False:
simulation.animate()
if not DEFAULT["save_figures"]:
plt.show()
else:
figures = plt.get_figlabels()
pylog.debug("Saving figures:\n{}".format(figures))
for fig in figures:
plt.figure(fig)
save_figure(fig)
plt.close(fig)
# 2c : relationship between stimulation frequency and amplitude
# Effect of frequency
stim_frequency_range = np.array([0.05,0.1,0.5,1,5,10,50,100,500]) #Hz
stim_amp = 1
phase_shift = np.pi
frequency_pend=np.zeros(len(stim_frequency_range))
amplitude_pend=np.zeros(len(stim_frequency_range))
for j,stim_frequency in enumerate(stim_frequency_range):
period = 1/stim_frequency
t_max = 10*period # Maximum simulation time
time = np.arange(0., t_max, 0.001*period) # Time vector
act1 = np.zeros((len(time),1))
act2 = np.zeros((len(time),1))
act1[:,0] = stim_amp*(1 + np.sin(2*np.pi*stim_frequency*time))/2
act2[:,0] = stim_amp*(1+ np.sin(2*np.pi*stim_frequency*time + phase_shift))/2
activations = np.hstack((act1, act2))
sim.add_muscle_activations(activations)
sim.initalize_system(x0, time) # Initialize the system state
sim.simulate()
res = sim.results()
# computing the frequency and amplitude
angular_position = res[:,1]
signal_stat = angular_position[int(len(angular_position)/2):len(angular_position)]
index_zeros = np.where(np.diff(np.sign(signal_stat)))[0]
deltas = np.diff(index_zeros)
delta = np.mean(deltas)
period = 2*delta*0.001*period
frequency_pend[j] = 1/period
amplitude_pend[j] = (np.max(signal_stat)-np.min(signal_stat))/2
# Plotting
plt.figure('2c : effect of frequency')
plt.subplot(121)
plt.loglog(stim_frequency_range,frequency_pend)
plt.grid()
plt.xlabel('Stimulation Frequency in Hz')
plt.ylabel('Pendulum Oscillation Frequency [Hz]')
plt.subplot(122)
plt.loglog(stim_frequency_range,amplitude_pend)
plt.grid()
plt.xlabel('Stimulation Frequency in Hz')
plt.ylabel('Pendulum Oscillation Amplitude [rad]')
plt.savefig('2c_frequency.png')
plt.show()
# Effect of amplitude
stim_frequency = 10 #Hz
stim_amp_range = np.arange(0,1.1,0.1)
phase_shift = np.pi
frequency_pend=np.zeros(len(stim_amp_range))
amplitude_pend=np.zeros(len(stim_amp_range))
for j,stim_amp in enumerate(stim_amp_range):
period = 1/stim_frequency
t_max = 5*period # Maximum simulation time
time = np.arange(0., t_max, 0.001*period) # Time vector
act1 = np.zeros((len(time),1))
act2 = np.zeros((len(time),1))
act1[:,0] = stim_amp*(1 + np.sin(2*np.pi*stim_frequency*time))/2
act2[:,0] = stim_amp*(1+ np.sin(2*np.pi*stim_frequency*time + phase_shift))/2
activations = np.hstack((act1, act2))
sim.add_muscle_activations(activations)
sim.initalize_system(x0, time) # Initialize the system state
sim.simulate()
res = sim.results()
# computing the frequency and amplitude
angular_position = res[:,1]
signal_stat = angular_position[int(len(angular_position)/2):len(angular_position)]
index_zeros = np.where(np.diff(np.sign(signal_stat)))[0]
deltas = np.diff(index_zeros)
delta = np.mean(deltas)
period = 2*delta*0.001*period
frequency_pend[j] = 1/period
amplitude_pend[j] = (np.max(signal_stat)-np.min(signal_stat))/2
frequency_pend[0] = 0.0;
# Plotting
plt.figure('2c : effect of amplitude')
plt.subplot(121)
plt.plot(stim_amp_range,frequency_pend)
plt.grid()
plt.xlabel('Stimulation Amplitude')
plt.ylabel('Pendulum Oscillation Frequency [Hz]')
plt.subplot(122)
plt.plot(stim_amp_range,amplitude_pend)
plt.grid()
plt.xlabel('Stimulation Amplitude')
plt.ylabel('Pendulum Oscillation Amplitude [rad]')
plt.savefig('2c_amplitude.png')
plt.show()
def exercise3a():
""" Main function to run for Exercise 3.
Parameters
----------
None
Returns
-------
None
"""
# Define and Setup your pendulum model here
# Check Pendulum.py for more details on Pendulum class
P_params = PendulumParameters() # Instantiate pendulum parameters
P_params.L = 0.5 # To change the default length of the pendulum
P_params.m = 1. # To change the default mass of the pendulum
P_params.PERTURBATION = True
pendulum = PendulumSystem(P_params) # Instantiate Pendulum object
#### CHECK OUT Pendulum.py to ADD PERTURBATIONS TO THE MODEL #####
pylog.info('Pendulum model initialized \n {}'.format(
pendulum.parameters.showParameters()))
# Define and Setup your pendulum model here
# Check MuscleSytem.py for more details on MuscleSytem class
M1_param = MuscleParameters() # Instantiate Muscle 1 parameters
M1_param.f_max = 1500 # To change Muscle 1 max force
M2_param = MuscleParameters() # Instantiate Muscle 2 parameters
M2_param.f_max = 1500 # To change Muscle 2 max force
M1 = Muscle(M1_param) # Instantiate Muscle 1 object
M2 = Muscle(M2_param) # Instantiate Muscle 2 object
# Use the MuscleSystem Class to define your muscles in the system
muscles = MuscleSytem(M1, M2) # Instantiate Muscle System with two muscles
pylog.info('Muscle system initialized \n {} \n {}'.format(
M1.parameters.showParameters(), M2.parameters.showParameters()))
# Define Muscle Attachment points
m1_origin = np.array([-0.17, 0.0]) # Origin of Muscle 1
m1_insertion = np.array([0.0, -0.17]) # Insertion of Muscle 1
m2_origin = np.array([0.17, 0.0]) # Origin of Muscle 2
m2_insertion = np.array([0.0, -0.17]) # Insertion of Muscle 2
# Attach the muscles
muscles.attach(np.array([m1_origin, m1_insertion]),
np.array([m2_origin, m2_insertion]))
##### Neural Network #####
# The network consists of four neurons
N_params = NetworkParameters() # Instantiate default network parameters
N_params.tau = [0.02, 0.02, 0.1, 0.1]
N_params.b = [3.0, 3.0, -3.0, -3.0]
N_params.D = 1.0 # To change a network parameter
N_params.w = np.asarray([[0.0, -5.0, -5.0, 0.0], [-5.0, 0.0, 0.0, -5.0],
[5.0, -5.0, 0.0, 0.0], [-5.0, 5.0, 0.0, 0.0]])
# Similarly to change w -> N_params.w = (4x4) array
print(N_params.w)
############################# Exercise 3A ######################
N_params.w = np.transpose(
np.asarray([[0, -1, 1, -1], [-1, 0, -1, 1], [-1, 0, 0, 0],
[0, -1, 0, 0]])) * 5
print(N_params.w, N_params.D, N_params.tau, N_params.b, N_params.exp)
# Create a new neural network with above parameters
neural_network = NeuralSystem(N_params)
pylog.info('Neural system initialized \n {}'.format(
N_params.showParameters()))
# Create system of Pendulum, Muscles and neural network using SystemClass
# Check System.py for more details on System class
sys = System() # Instantiate a new system
sys.add_pendulum_system(pendulum) # Add the pendulum model to the system
sys.add_muscle_system(muscles) # Add the muscle model to the system
sys.add_neural_system(
neural_network) # Add the neural network to the system
##### Time #####
t_max = 2. # Maximum simulation time
time = np.arange(0., t_max, 0.001) # Time vector
##### Model Initial Conditions #####
x0_P = np.array([[-0.5, 0], [-0.25, -0.25], [0., 0.],
[0.5, 0]]) # Pendulum initial condition
for i in x0_P:
# Muscle Model initial condition
x0_M = np.array([0., M1.L_OPT, 0., M2.L_OPT])
x0_N = np.array([-1.5, 1, 2.5, 1]) # Neural Network Initial Conditions
x0 = np.concatenate((i, x0_M, x0_N)) # System initial conditions
##### System Simulation #####
# For more details on System Simulation check SystemSimulation.py
# SystemSimulation is used to initialize the system and integrate
# over time
sim = SystemSimulation(sys) # Instantiate Simulation object
# sim.add_external_inputs_to_network(np.ones((len(time), 4)))
# wave_h1 = np.sin(time*3)*2 #makes a sinusoidal wave from 'time'
# wave_h2 = np.sin(time*3 + np.pi)*1 #makes a sinusoidal wave from 'time'
#
# wave_h1[wave_h1<0] = 0 #formality of passing negative values to zero
# wave_h2[wave_h2<0] = 0 #formality of passing negative values to zero
#
# act1 = wave_h1.reshape(len(time), 1) #makes a vertical array like act1
# act2 = wave_h2.reshape(len(time), 1) #makes a vertical array like act1
# column = np.ones((len(time), 1))
# ext_in = np.hstack((act1, column, act2, column))
# sim.add_external_inputs_to_network(ext_in)
sim.initalize_system(x0, time) # Initialize the system state
sim.sys.pendulum_sys.parameters.PERTURBATION = False
# Integrate the system for the above initialized state and time
sim.simulate()
# Obtain the states of the system after integration
# res is np.array [time, states]
# states vector is in the same order as x0
res = sim.results()
# In order to obtain internal states of the muscle
# you can access the results attribute in the muscle class
muscle1_results = sim.sys.muscle_sys.Muscle1.results
muscle2_results = sim.sys.muscle_sys.Muscle2.results
# Plotting the results: Position(phase) vs time
plt.figure('Pendulum Phase')
plt.title('Pendulum Phase')
plt.plot(res[:, 0], res[:, 1]) #to plot pendulum Position (phase)
# plt.plot(res[:, 0], time) #to plot position
# plt.plot(res[:, 0], res[:, -5:-1]) # to Plot neurons' states
plt.xlabel('time [s]')
plt.ylabel('Position [rad]')
plt.grid()
# Plotting the results: Velocity vs Position (phase)
plt.figure('Pendulum Vel v.s. Phase')
plt.title('Pendulum Vel v.s. Phase')
plt.plot(res[:, 1], res[:, 2]) #to plot Velocity vs Position (phase)
plt.xlabel('Position [rad]')
plt.ylabel('Velocity [rad.s]')
plt.grid()
# Plotting the results: Velocity vs time
plt.figure('Pendulum Velocity')
plt.title('Pendulum Velocity')
plt.plot(res[:, 0], res[:, 2]) #to plot Velocity vs Position
plt.xlabel('time [s]')
plt.ylabel('Velocity [rad.s]')
plt.grid()
# Plotting the results: Output of the network
plt.figure('Network output')
plt.title('Network output')
plt.plot(res[:, 0], res[:, -1],
label='neuron1') #to plot Velocity vs Position
plt.plot(res[:, 0], res[:, -2], label='neuron2')
plt.plot(res[:, 0], res[:, -3], label='neuron3')
plt.plot(res[:, 0], res[:, -4], label='neuron4')
plt.xlabel('time [s]')
plt.ylabel('Stimulation ')
plt.legend(loc='upper right')
plt.grid()
if DEFAULT["save_figures"] is False:
plt.show()
else:
figures = plt.get_figlabels()
pylog.debug("Saving figures:\n{}".format(figures))
for fig in figures:
plt.figure(fig)
save_figure(fig)
plt.close(fig)
# To animate the model, use the SystemAnimation class
# Pass the res(states) and systems you wish to animate
simulation = SystemAnimation(res, sim.sys.pendulum_sys, sim.sys.muscle_sys,
sim.sys.neural_sys)
# To start the animation
simulation.animate()
@property
def mass(self):
"""Get the value of mass in the mass system """
return self.parameters["mass"]
@mass.setter
def mass(self, value):
""" Keyword Arguments:
value -- Set the value of mass"""
if value <= 0.00001:
pylog.error(
"Mass you are trying to set is too low!. Setting to 1.")
value = 1.0
self.parameters["mass"] = value
def showParameters(self):
return self.msg(self.parameters, self.units)
if __name__ == '__main__':
P = PendulumParameters(g=9.81, L=1.)
pylog.debug(P.showParameters())
M = MuscleParameters()
pylog.debug(M.showParameters())
Mass = MassParameters()
pylog.debug(Mass.showParameters())
def exercise3():
""" Main function to run for Exercise 3.
Parameters
----------
None
Returns
-------
None
"""
# Define and Setup your pendulum model here
# Check Pendulum.py for more details on Pendulum class
P_params = PendulumParameters() # Instantiate pendulum parameters
P_params.L = 0.5 # To change the default length of the pendulum
P_params.m = 1. # To change the default mass of the pendulum
pendulum = PendulumSystem(P_params) # Instantiate Pendulum object
#### CHECK OUT Pendulum.py to ADD PERTURBATIONS TO THE MODEL #####
pylog.info('Pendulum model initialized \n {}'.format(
pendulum.parameters.showParameters()))
# Define and Setup your pendulum model here
# Check MuscleSytem.py for more details on MuscleSytem class
M1_param = MuscleParameters() # Instantiate Muscle 1 parameters
M1_param.f_max = 1500 # To change Muscle 1 max force
M2_param = MuscleParameters() # Instantiate Muscle 2 parameters
M2_param.f_max = 1500 # To change Muscle 2 max force
M1 = Muscle(M1_param) # Instantiate Muscle 1 object
M2 = Muscle(M2_param) # Instantiate Muscle 2 object
# Use the MuscleSystem Class to define your muscles in the system
muscles = MuscleSytem(M1, M2) # Instantiate Muscle System with two muscles
pylog.info('Muscle system initialized \n {} \n {}'.format(
M1.parameters.showParameters(),
M2.parameters.showParameters()))
# Define Muscle Attachment points
m1_origin = np.array([-0.17, 0.0]) # Origin of Muscle 1
m1_insertion = np.array([0.0, -0.17]) # Insertion of Muscle 1
m2_origin = np.array([0.17, 0.0]) # Origin of Muscle 2
m2_insertion = np.array([0.0, -0.17]) # Insertion of Muscle 2
# Attach the muscles
muscles.attach(np.array([m1_origin, m1_insertion]),
np.array([m2_origin, m2_insertion]))
##### Neural Network #####
# The network consists of four neurons
N_params = NetworkParameters() # Instantiate default network parameters
# Similarly to change w -> N_params.w = (4x4) array
# From lecture 4, slide 85 -> Generate oscillations !!
N_params.D = 2.
N_params.tau = [0.02,0.02,0.1,0.1]
N_params.b = [3.0,3.0,-3.0,-3.0]
N_params.w = [[0,-5,-5,0], # 1 <- 2
[-5,0,0,-5],
[5,-5,0,-5],
[-5,5,0,0]]
# Create a new neural network with above parameters
neural_network = NeuralSystem(N_params)
pylog.info('Neural system initialized \n {}'.format(
N_params.showParameters()))
# Create system of Pendulum, Muscles and neural network using SystemClass
# Check System.py for more details on System class
sys = System() # Instantiate a new system
sys.add_pendulum_system(pendulum) # Add the pendulum model to the system
sys.add_muscle_system(muscles) # Add the muscle model to the system
# Add the neural network to the system
sys.add_neural_system(neural_network)
##### Time #####
t_max = 2.5 # Maximum simulation time
time = np.arange(0., t_max, 0.001) # Time vector
##### Model Initial Conditions #####
x0_P = np.array([0., 0.]) # Pendulum initial condition
# Muscle Model initial condition
x0_M = np.array([0., M1.L_OPT, 0., M2.L_OPT])
x0_N = np.array([-0.5, 1, 0.5, 1]) # Neural Network Initial Conditions
x0 = np.concatenate((x0_P, x0_M, x0_N)) # System initial conditions
##### System Simulation #####
# For more details on System Simulation check SystemSimulation.py
# SystemSimulation is used to initialize the system and integrate
# over time
sim = SystemSimulation(sys) # Instantiate Simulation object
# Add external inputs to neural network
# sim.add_external_inputs_to_network(np.ones((len(time), 4)))
#ext_in = np.ones((len(time), 4))
#ext_in[:,2] = 0.2
#ext_in[:,3] = 0.2
#sim.add_external_inputs_to_network(ext_in)
sim.initalize_system(x0, time) # Initialize the system state
# Integrate the system for the above initialized state and time
sim.simulate()
# Obtain the states of the system after integration
# res is np.array [time, states]
# states vector is in the same order as x0
res = sim.results()
# Obtain the states of the system after integration
# res is np.array [time, states]
# states vector is in the same order as x0
#res = sim.results()
# In order to obtain internal states of the muscle
# you can access the results attribute in the muscle class
muscle1_results = sim.sys.muscle_sys.Muscle1.results
muscle2_results = sim.sys.muscle_sys.Muscle2.results
# Plotting the phase
fig = plt.figure('Pendulum')
plt.title('Pendulum Phase')
plt.plot(res[:, 1], res[:, 2])
plt.xlabel('Position [rad]')
plt.ylabel('Velocity [rad/s]')
plt.grid()
fig.tight_layout()
fig.savefig('graphs/PendulumPhase.png')
# Plotting the neuronal activation
# Access the neurons outputs:
# [t] theta theta. A1 lCE1 A2 lCE2 m1 m2 m3 m4
fig = plt.figure('Neuron output')
plt.title('Membrane potentials')
plt.plot(res[:, 0], res[:, 7],label='m1')
plt.plot(res[:, 0], res[:, 8],label='m2')
plt.plot(res[:, 0], res[:, 9],label='m3')
plt.plot(res[:, 0], res[:, 10],label='m4')
plt.xlabel('Time [s]')
plt.ylabel('Potential')
plt.legend()
plt.grid()
fig.tight_layout()
fig.savefig('graphs/MembranePotentials.png')
if DEFAULT["save_figures"] is False:
plt.show()
else:
figures = plt.get_figlabels()
pylog.debug("Saving figures:\n{}".format(figures))
for fig in figures:
plt.figure(fig)
save_figure(fig)
plt.close(fig)
# To animate the model, use the SystemAnimation class
# Pass the res(states) and systems you wish to animate
simulation = SystemAnimation(
res,
sim.sys.pendulum_sys,
sim.sys.muscle_sys,
sim.sys.neural_sys)
# To start the animation
simulation.animate()
# 3.b
ext_in = np.ones((len(time), 4))*0.0
plotExternalDrive(sys,x0,ext_in,typ='low')
ext_in = np.ones((len(time), 4))
plotExternalDrive(sys,x0,ext_in,typ='high')
ext_in = np.ones((len(time), 4))
ext_in[:,0] *= 0.1
ext_in[:,1] *= 0.1
plotExternalDrive(sys,x0,ext_in,typ='asymmetric')
def exercise1(clargs):
""" Exercise 1 """
# Setup
pylog.info("Running exercise 1")
# Setup
time_max = 5 # Maximum simulation time
time_step = 0.2 # Time step for ODE integration in simulation
x0 = np.array([1.]) # Initial state
# Integration methods (Exercises 1.a - 1.d)
pylog.info("Running function integration using different methods")
# Example
pylog.debug("Running example plot for integration (remove)")
example = example_integrate(x0, time_max, time_step)
example.plot_state(figure="Example", label="Example", marker=".")
# Analytical (1.a)
time = np.arange(0, time_max, time_step) # Time vector
x_a = analytic_function(time)
analytical = Result(x_a, time) if x_a is not None else None
# Euler (1.b)
euler = euler_integrate(function, x0, time_max, time_step)
display_error(euler.state - analytical.state, "Euler")
# ODE (1.c)
ode = ode_integrate(function, x0, time_max, time_step)
display_error(ode.state - analytical.state, "LSODA")
# ODE Runge-Kutta (1.c)
ode_rk = ode_integrate_rk(function_rk, x0, time_max, time_step)
display_error(ode_rk.state - analytical.state, "Runge-Kutta")
# Euler with lower time step (1.d)
euler_time_step = 0.05
euler_ts_small = (euler_integrate(function, x0, time_max, euler_time_step)
if euler_time_step is not None else None)
# New analytical time step
time_step_small = 0.05
time_small = np.arange(0, time_max, time_step_small) # Time vector
x_a_small = analytic_function(time_small)
analytical_small = Result(x_a_small,
time_small) if x_a_small is not None else None
display_error(euler_ts_small.state - analytical_small.state, "Euler small")
# Plot integration results
plot_integration_methods(analytical=analytical_small,
euler=euler,
ode=ode,
ode_rk=ode_rk,
euler_ts_small=euler_ts_small,
euler_timestep=time_step_small,
euler_timestep_small=euler_time_step)
# Error analysis (Exercise 1.e)
pylog.warning("Error analysis must be implemented")
dt_list = np.logspace(-3, 0, 20) # List of timesteps (powers of 10)
integration_errors = [["L1", 1], ["L2", 2], ["Linf", 0]]
methods = [["Euler", euler_integrate, function],
["Lsoda", ode_integrate, function],
["RK", ode_integrate_rk, function_rk]]
for error_name, error_index in integration_errors:
for name, integration_function, f in methods:
compute_error(f,
analytic_function,
integration_function,
x0,
dt_list,
time_max=time_max,
figure=error_name,
label=name,
n=error_index)
# Show plots of all results
if not clargs.save_figures:
plt.show()
return
def exercise2():
""" Main function to run for Exercise 2.
Parameters
----------
None
Returns
-------
None
"""
# Define and Setup your pendulum model here
# Check PendulumSystem.py for more details on Pendulum class
pendulum_params = PendulumParameters() # Instantiate pendulum parameters
pendulum_params.L = 0.5 # To change the default length of the pendulum
pendulum_params.m = 1. # To change the default mass of the pendulum
pendulum = PendulumSystem(pendulum_params) # Instantiate Pendulum object
#### CHECK OUT PendulumSystem.py to ADD PERTURBATIONS TO THE MODEL #####
pylog.info('Pendulum model initialized \n {}'.format(
pendulum.parameters.showParameters()))
# Define and Setup your pendulum model here
# Check MuscleSytem.py for more details on MuscleSytem class
M1_param = MuscleParameters() # Instantiate Muscle 1 parameters
M1_param.f_max = 1500 # To change Muscle 1 max force
M2_param = MuscleParameters() # Instantiate Muscle 2 parameters
M2_param.f_max = 1500 # To change Muscle 2 max force
M1 = Muscle(M1_param) # Instantiate Muscle 1 object
M2 = Muscle(M2_param) # Instantiate Muscle 2 object
# Use the MuscleSystem Class to define your muscles in the system
muscles = MuscleSytem(M1, M2) # Instantiate Muscle System with two muscles
pylog.info('Muscle system initialized \n {} \n {}'.format(
M1.parameters.showParameters(), M2.parameters.showParameters()))
# Define Muscle Attachment points
m1_origin = np.array([-0.17, 0.0]) # Origin of Muscle 1
m1_insertion = np.array([0.0, -0.17]) # Insertion of Muscle 1
m2_origin = np.array([0.17, 0.0]) # Origin of Muscle 2
m2_insertion = np.array([0.0, -0.17]) # Insertion of Muscle 2
# Attach the muscles
muscles.attach(np.array([m1_origin, m1_insertion]),
np.array([m2_origin, m2_insertion]))
############Exercise 2A ###############################################
# rigth after creating and attaching both muscles:
print(m1_origin, m2_origin)
m1a1 = abs(abs(m1_origin[0]) - abs(m1_origin[1]))
m1a2 = abs(abs(m1_insertion[0]) - abs(m1_insertion[1]))
m1a1 = m1_origin[0] - m1_origin[1]
m1a2 = m1_insertion[0] - m1_insertion[1]
m2a1 = m2_origin[0] - m2_origin[1]
m2a2 = m2_insertion[0] - m2_insertion[1]
print(m1a1, m1a2)
fromtheta(M1, m1a1, m1a2, 1)
fromtheta(M2, m2a1, m2a2, 2)
# Create a system with Pendulum and Muscles using the System Class
# Check System.py for more details on System class
sys = System() # Instantiate a new system
sys.add_pendulum_system(pendulum) # Add the pendulum model to the system
sys.add_muscle_system(muscles) # Add the muscle model to the system
##### Time #####
t_max = 5 # Maximum simulation time
time = np.arange(0., t_max, 0.002) # Time vector
##### Model Initial Conditions #####
x0_P = np.array([np.pi / 4, 0.]) # Pendulum initial condition
x0_P = np.array([0., 0.]) # Pendulum initial condition
# Muscle Model initial condition
x0_M = np.array([0., M1.L_OPT, 0., M2.L_OPT])
x0 = np.concatenate((x0_P, x0_M)) # System initial conditions
##### System Simulation #####
# For more details on System Simulation check SystemSimulation.py
# SystemSimulation is used to initialize the system and integrate
# over time
sim = SystemSimulation(sys) # Instantiate Simulation object
# Add muscle activations to the simulation
# Here you can define your muscle activation vectors
# that are time dependent
wave_h1 = np.sin(time * 3) * 1 #makes a sinusoidal wave from 'time'
wave_h2 = np.sin(time * 3 +
np.pi) * 1 #makes a sinusoidal wave from 'time'
wave_h1[wave_h1 < 0] = 0 #formality of passing negative values to zero
wave_h2[wave_h2 < 0] = 0 #formality of passing negative values to zero
act1 = wave_h1.reshape(len(time), 1) #makes a vertical array like act1
act2 = wave_h2.reshape(len(time), 1) #makes a vertical array like act1
# Plotting the waveforms
plt.figure('Muscle Activations')
plt.title('Muscle Activation Functions')
plt.plot(time, wave_h1, label='Muscle 1')
plt.plot(time, wave_h2, label='Muscle 2')
plt.xlabel('Time [s]')
plt.ylabel('Muscle Excitation')
plt.legend(loc='upper right')
plt.grid()
activations = np.hstack((act1, act2))
# Method to add the muscle activations to the simulation
sim.add_muscle_activations(activations)
# Simulate the system for given time
sim.initalize_system(x0, time) # Initialize the system state
#: If you would like to perturb the pedulum model then you could do
# so by
sim.sys.pendulum_sys.parameters.PERTURBATION = False
# The above line sets the state of the pendulum model to zeros between
# time interval 1.2 < t < 1.25. You can change this and the type of
# perturbation in
# pendulum_system.py::pendulum_system function
# Integrate the system for the above initialized state and time
sim.simulate()
# Obtain the states of the system after integration
# res is np.array [time, states]
# states vector is in the same order as x0
res = sim.results()
# In order to obtain internal states of the muscle
# you can access the results attribute in the muscle class
muscle1_results = sim.sys.muscle_sys.Muscle1.results
muscle2_results = sim.sys.muscle_sys.Muscle2.results
# Plotting the results
plt.figure('Pendulum_phase')
plt.title('Pendulum Phase')
plt.plot(res[:, 1], res[:, 2])
plt.xlabel('Position [rad]')
plt.ylabel('Velocity [rad.s]')
plt.grid()
# Plotting the results: Amplidute stimulation
plt.figure('Amplidute stimulation')
plt.title('Amplidute stimulation')
plt.plot(time, res[:, 1], label='Stimul. 0.2')
plt.xlabel('time [s]')
plt.ylabel('Position [rad]')
plt.legend(loc='upper left')
plt.grid()
# Plotting the results: frequency stimulation
plt.figure('Frequency stimulation')
plt.title('Frequency stimulation')
plt.plot(time, res[:, 1], label='w: 3 rad/s')
plt.xlabel('time [s]')
plt.ylabel('Position [rad]')
plt.legend(loc='upper left')
plt.grid()
poincare_crossings(res, -2, 1, "Pendulum")
# To animate the model, use the SystemAnimation class
# Pass the res(states) and systems you wish to animate
simulation = SystemAnimation(res, pendulum, muscles)
# To start the animation
if DEFAULT["save_figures"] is False:
simulation.animate()
if not DEFAULT["save_figures"]:
plt.show()
else:
figures = plt.get_figlabels()
pylog.debug("Saving figures:\n{}".format(figures))
for fig in figures:
plt.figure(fig)
save_figure(fig)
plt.close(fig)
def exercise2():
""" Main function to run for Exercise 2.
Parameters
----------
None
Returns
-------
None
"""
# Define and Setup your pendulum model here
# Check PendulumSystem.py for more details on Pendulum class
pendulum_params = PendulumParameters() # Instantiate pendulum parameters
pendulum_params.L = 0.5 # To change the default length of the pendulum
pendulum_params.m = 1. # To change the default mass of the pendulum
pendulum = PendulumSystem(pendulum_params) # Instantiate Pendulum object
#### CHECK OUT PendulumSystem.py to ADD PERTURBATIONS TO THE MODEL #####
pylog.info('Pendulum model initialized \n {}'.format(
pendulum.parameters.showParameters()))
# Define and Setup your pendulum model here
# Check MuscleSytem.py for more details on MuscleSytem class
M1_param = MuscleParameters() # Instantiate Muscle 1 parameters
M1_param.f_max = 1500 # To change Muscle 1 max force
M2_param = MuscleParameters() # Instantiate Muscle 2 parameters
M2_param.f_max = 1500 # To change Muscle 2 max force
M1 = Muscle(M1_param) # Instantiate Muscle 1 object
M2 = Muscle(M2_param) # Instantiate Muscle 2 object
# Use the MuscleSystem Class to define your muscles in the system
muscles = MuscleSytem(M1, M2) # Instantiate Muscle System with two muscles
pylog.info('Muscle system initialized \n {} \n {}'.format(
M1.parameters.showParameters(), M2.parameters.showParameters()))
# Define Muscle Attachment points
m1_origin = np.array([-0.17, 0.0]) # Origin of Muscle 1
m1_insertion = np.array([0.0, -0.17]) # Insertion of Muscle 1
m2_origin = np.array([0.17, 0.0]) # Origin of Muscle 2
m2_insertion = np.array([0.0, -0.17]) # Insertion of Muscle 2
# Attach the muscles
muscles.attach(np.array([m1_origin, m1_insertion]),
np.array([m2_origin, m2_insertion]))
# Create a system with Pendulum and Muscles using the System Class
# Check System.py for more details on System class
sys = System() # Instantiate a new system
sys.add_pendulum_system(pendulum) # Add the pendulum model to the system
sys.add_muscle_system(muscles) # Add the muscle model to the system
##### Time #####
t_max = 3 # Maximum simulation time
time = np.arange(0., t_max, 0.001) # Time vector
##### Model Initial Conditions #####
x0_P = np.array([0., 0.]) # Pendulum initial condition
# Muscle Model initial condition
x0_M = np.array([0., M1.L_OPT, 0., M2.L_OPT])
x0 = np.concatenate((x0_P, x0_M)) # System initial conditions
##### System Simulation #####
# For more details on System Simulation check SystemSimulation.py
# SystemSimulation is used to initialize the system and integrate
# over time
simsin = SystemSimulation(sys) # Instantiate Simulation object
#simsquare = SystemSimulation(sys)
# Add muscle activations to the simulation
# Here you can define your muscle activation vectors
# that are time dependent
label_test = []
"""" definition of different kinds of activation for each muscle.
Amplitude1 and amplitude2 allows to play with the amplitude of activation on each muscle (RMS value for the sinus activation)
act1 and act2 activates the muscle all the time.
actsin activates with sin(wi) if sin(wi)>0 (no negative activation). The 2 muscles are in opposition of phase.
actsquare does the same with a square signal.
"""
amplitude1 = 1.
amplitude2 = 1.
#declaration of the activations
act1 = np.ones((len(time), 1)) * amplitude1
act2 = np.ones((len(time), 1)) * amplitude2
actsin = np.ones((len(time), 1))
actsin2 = np.ones((len(time), 1))
actsquare = np.ones((len(time), 1))
actsquare2 = np.ones((len(time), 1))
wlist = [0.1, 0.05, 0.01, 0.005]
k = 0
for w in wlist:
#generation of the signals at pulsation w
for i in range(len(actsin)):
if math.sin(w * i) <= 0:
actsin[i] = 0
actsin2[i] = abs(amplitude2 * math.sqrt(2) * math.sin(w * i))
else:
actsin[i] = abs(amplitude1 * math.sqrt(2) * math.sin(w * i))
actsin2[i] = 0
for i in range(len(actsquare)):
if i % (2 * math.pi / w) <= math.pi / w:
actsquare[i] = amplitude1
actsquare2[i] = 0
else:
actsquare[i] = 0
actsquare2[i] = amplitude2
""" uncomment this to plot the activation signals"""
# #Plot of the activation through time
# plt.figure
# plt.plot(actsquare)
# plt.plot(actsin)
# plt.title("Activations wave forms used")
# plt.xlabel("Time (s)")
# plt.ylabel("Activation amplitude (.)")
""" put as parameters the activation you want (act1/2, actsin1/2 or actsquare1/2)"""
activationssin = np.hstack((actsquare, actsquare2))
#activationssquare = np.hstack((actsquare, actsquare2))
# Method to add the muscle activations to the simulation
simsin.add_muscle_activations(activationssin)
#simsquare.add_muscle_activations(activationssquare)
# Simulate the system for given time
simsin.initalize_system(x0, time) # Initialize the system state
#simsquare.initalize_system(x0, time)
#: If you would like to perturb the pedulum model then you could do
# so by
"""perturbation of the signal"""
simsin.sys.pendulum_sys.parameters.PERTURBATION = False
#simsquare.sys.pendulum_sys.parameters.PERTURBATION = True
# The above line sets the state of the pendulum model to zeros between
# time interval 1.2 < t < 1.25. You can change this and the type of
# perturbation in
# pendulum_system.py::pendulum_system function
# Integrate the system for the above initialized state and time
simsin.simulate()
#simsquare.simulate()
# Obtain the states of the system after integration
# res is np.array [time, states]
# states vector is in the same order as x0
ressin = simsin.results()
#ressquare = simsquare.results()
# In order to obtain internal states of the muscle
# you can access the results attribute in the muscle class
muscle1_results = simsin.sys.muscle_sys.Muscle1.results
muscle2_results = simsin.sys.muscle_sys.Muscle2.results
# Plotting the results
plt.figure('Pendulum')
plt.title('Pendulum Phase')
plt.plot(ressin[:, 1], ressin[:, 2])
label_test.append('w=' + str(wlist[k]))
k = k + 1
#plt.plot(ressquare[:, 1], ressquare[:, 2])
plt.xlabel('Position [rad]')
plt.ylabel('Velocity [rad.s]')
plt.legend(label_test)
plt.grid()
# To animate the model, use the SystemAnimation class
# Pass the res(states) and systems you wish to animate
simulationsin = SystemAnimation(ressin, pendulum, muscles)
#simulationsquare = SystemAnimation(ressquare, pendulum, muscles)
# To start the animation
if DEFAULT["save_figures"] is False:
simulationsin.animate()
#simulationsquare.animate()
if not DEFAULT["save_figures"]:
plt.show()
else:
figures = plt.get_figlabels()
pylog.debug("Saving figures:\n{}".format(figures))
for fig in figures:
plt.figure(fig)
save_figure(fig)
plt.close(fig)
def exercise2():
""" Main function to run for Exercise 2.
Parameters
----------
None
Returns
-------
None
"""
#----------------# Exercise 2a #----------------#
theta = np.linspace(-np.pi/4, np.pi/4,num=50)
h1=[]
a1= 1
a2a1=np.linspace(0.5,2,num=4)
plt.figure('2a_Muscle_Length_vs_Theta')
plt.title('Muscle Length vs Theta')
plt.xlabel('Position [rad]')
plt.ylabel('Muscle length [m]')
plt.grid()
plt.figure('2a_Moment_arm_vs_Theta')
plt.title('Moment arm vs Theta')
plt.xlabel('Position [rad]')
plt.ylabel('Moment arm [m]')
plt.grid()
for i in range(0,len(a2a1)):
a2=a2a1[i]*a1
L1=(np.sqrt(a1**2+a2**2+2*a1*a2*np.sin(theta)))
h1=((a1*a2*np.cos(theta))/L1)
plt.figure('2a_Muscle_Length_vs_Theta')
plt.plot(theta,L1,label=('a2/a1 = %.1f' %(a2a1[i])))
plt.figure('2a_Moment_arm_vs_Theta')
plt.plot(theta,h1,label=('a2/a1= %.1f' %(a2a1[i])))
plt.figure('2a_Muscle_Length_vs_Theta')
plt.legend()
plt.figure('2a_Moment_arm_vs_Theta')
plt.legend()
#----------------# Exercise 2a finished #----------------#
# Define and Setup your pendulum model here
# Check PendulumSystem.py for more details on Pendulum class
pendulum_params = PendulumParameters() # Instantiate pendulum parameters
pendulum_params.L = 0.5 # To change the default length of the pendulum
pendulum_params.m = 1. # To change the default mass of the pendulum
pendulum = PendulumSystem(pendulum_params) # Instantiate Pendulum object
#### CHECK OUT PendulumSystem.py to ADD PERTURBATIONS TO THE MODEL #####
pylog.info('Pendulum model initialized \n {}'.format(
pendulum.parameters.showParameters()))
# Define and Setup your pendulum model here
# Check MuscleSytem.py for more details on MuscleSytem class
M1_param = MuscleParameters() # Instantiate Muscle 1 parameters
M1_param.f_max = 1500 # To change Muscle 1 max force
M2_param = MuscleParameters() # Instantiate Muscle 2 parameters
M2_param.f_max = 1500 # To change Muscle 2 max force
M1 = Muscle(M1_param) # Instantiate Muscle 1 object
M2 = Muscle(M2_param) # Instantiate Muscle 2 object
# Use the MuscleSystem Class to define your muscles in the system
muscles = MuscleSytem(M1, M2) # Instantiate Muscle System with two muscles
pylog.info('Muscle system initialized \n {} \n {}'.format(
M1.parameters.showParameters(),
M2.parameters.showParameters()))
# Define Muscle Attachment points
m1_origin = np.array([-0.17, 0.0]) # Origin of Muscle 1
m1_insertion = np.array([0.0, -0.17]) # Insertion of Muscle 1
m2_origin = np.array([0.17, 0.0]) # Origin of Muscle 2
m2_insertion = np.array([0.0, -0.17]) # Insertion of Muscle 2
# Attach the muscles
muscles.attach(np.array([m1_origin, m1_insertion]),
np.array([m2_origin, m2_insertion]))
# Create a system with Pendulum and Muscles using the System Class
# Check System.py for more details on System class
sys = System() # Instantiate a new system
sys.add_pendulum_system(pendulum) # Add the pendulum model to the system
sys.add_muscle_system(muscles) # Add the muscle model to the system
##### Time #####
t_max = 5 # Maximum simulation time
time = np.arange(0., t_max, 0.001) # Time vector
##### Model Initial Conditions #####
x0_P = np.array([np.pi/4, 0.]) # Pendulum initial condition
# Muscle Model initial condition
x0_M = np.array([0., M1.L_OPT, 0., M2.L_OPT])
x0 = np.concatenate((x0_P, x0_M)) # System initial conditions
##### System Simulation #####
# For more details on System Simulation check SystemSimulation.py
# SystemSimulation is used to initialize the system and integrate
# over time
sim = SystemSimulation(sys) # Instantiate Simulation object
# Add muscle activations to the simulation
# Here you can define your muscle activation vectors
# that are time dependent
activationFunction = ['sin','square']
for idx, act in enumerate(activationFunction):
#----------------# Exercise 2c #----------------#
w = np.linspace(0.2,4,4)
# w = 0.5
# a = np.linspace(0.1,1,4)
plt.figure('2c_LimitCycle_'+str(act))
# plt.figure('2c_LimitCycle_Amplitude_'+str(act))
plt.title('Pendulum Phase')
plt.figure('2c_Amplitude_'+str(act))
# plt.figure('2c_Amplitude_Amplitude_'+str(act))
plt.title('Amplitude vs. Frequency')
# plt.title('Amplitude vs. Stimulation Amplitude')
for i in range(0,len(w)):
# for i in range(0,len(a)):
# plt.figure('2c_LimitCycle_Amplitude_'+str(act))
plt.figure('2c_LimitCycle_'+str(act))
print('Running simulation %d out of %d'%(i+1,len(w)))
# print('Running simulation %d out of %d'%(i+1,len(a)))
if act == 'sin':
sinAct = np.sin(2*np.pi*w[i]*time).reshape(len(time),1)
# sinAct = a[i]*np.sin(2*np.pi*w*time).reshape(len(time),1)
else:
sinAct = signal.square(2*np.pi*w[i]*time).reshape(len(time),1)
# sinAct = a[i]*signal.square(2*np.pi*w*time).reshape(len(time),1)
sinFlex = sinAct.copy()
sinFlex[sinAct<0] = 0
sinExt = sinAct.copy()
sinExt[sinAct>0] = 0
sinExt = abs(sinExt)
sinAct1 = np.ones((len(time),1))
sinAct2 = np.ones((len(time),1))
sinAct1 = sinFlex
sinAct2 = sinExt
sinActivations = np.hstack((sinAct1,sinAct2))
# Method to add the muscle activations to the simulation
sim.add_muscle_activations(sinActivations)
# Simulate the system for given time
sim.initalize_system(x0, time) # Initialize the system state
#: If you would like to perturb the pedulum model then you could do
# so by
sim.sys.pendulum_sys.parameters.PERTURBATION = False
# The above line sets the state of the pendulum model to zeros between
# time interval 1.2 < t < 1.25. You can change this and the type of
# perturbation in
# pendulum_system.py::pendulum_system function
# Integrate the system for the above initialized state and time
sim.simulate()
# Obtain the states of the system after integration
# res is np.array [time, states]
# states vector is in the same order as x0
res = sim.results()
# In order to obtain internal states of the muscle
# you can access the results attribute in the muscle class
muscle1_results = sim.sys.muscle_sys.Muscle1.results
muscle2_results = sim.sys.muscle_sys.Muscle2.results
# Plotting the results
plt.plot(res[:, 1], res[:, 2], label='Act. $%s(2\cdot{}\\pi\cdot{}%.1f\cdot{}t)$'%(act,w[i]))
# plt.plot(res[:, 1], res[:, 2], label='Act. $%.1f\cdot{}%s(2\cdot{}\\pi\cdot{}0.5\cdot{}t)$'%(a[i],act))
plt.figure('2c_Amplitude_'+str(act))
plt.plot(time,res[:, 1], label='Frequency = %.1f'%(w[i]))
# plt.figure('2c_Amplitude_Amplitude_'+str(act))
# plt.plot(time,res[:, 1], label='Amplitude = %.1f'%(a[i]))
plt.figure('2c_LimitCycle_'+str(act))
# plt.figure('2c_LimitCycle_Amplitude_'+str(act))
plt.xlabel('Position [rad]')
plt.ylabel('Velocity [rad/s]')
plt.grid()
plt.legend()
plt.figure('2c_Amplitude_'+str(act))
# plt.figure('2c_Amplitude_Amplitude_'+str(act))
plt.xlabel('Time [s]')
plt.ylabel('Amplitude [rad]')
plt.grid()
plt.legend()
#----------------# Exercise 2c finished #----------------#
#----------------# Exercise 2b #----------------#
w = 0.5
if act == 'sin':
sinAct = np.sin(2*np.pi*w*time).reshape(len(time),1)
else:
sinAct = signal.square(2*np.pi*w*time).reshape(len(time),1)
sinFlex = sinAct.copy()
sinFlex[sinAct<0] = 0
sinExt = sinAct.copy()
sinExt[sinAct>0] = 0
sinExt = abs(sinExt)
sinAct1 = np.ones((len(time),1))
sinAct2 = np.ones((len(time),1))
sinAct1 = sinFlex
sinAct2 = sinExt
activations = np.hstack((sinAct1,sinAct2))
# Method to add the muscle activations to the simulation
sim.add_muscle_activations(activations)
# Simulate the system for given time
sim.initalize_system(x0, time) # Initialize the system state
#: If you would like to perturb the pedulum model then you could do
# so by
sim.sys.pendulum_sys.parameters.PERTURBATION = True
# The above line sets the state of the pendulum model to zeros between
# time interval 1.2 < t < 1.25. You can change this and the type of
# perturbation in
# pendulum_system.py::pendulum_system function
# Integrate the system for the above initialized state and time
sim.simulate()
# Obtain the states of the system after integration
# res is np.array [time, states]
# states vector is in the same order as x0
res = sim.results()
# In order to obtain internal states of the muscle
# you can access the results attribute in the muscle class
muscle1_results = sim.sys.muscle_sys.Muscle1.results
muscle2_results = sim.sys.muscle_sys.Muscle2.results
# Plotting the results
plt.figure('2b_LimitCycle_'+str(act))
plt.title('Pendulum Phase')
plt.plot(res[:, 1], res[:, 2], label='Act. $%s(2\cdot{}\\pi\cdot{}%.1f\cdot{}t)$, Pert. ($t=3.2,\\theta = 1, \dot{\\theta} = -0.5$)' %(act,w))
plt.xlabel('Position [rad]')
plt.ylabel('Velocity [rad/s]')
plt.grid()
plt.legend()
plt.figure('2b_ActivationFunction_'+str(act))
plt.title('Activation Function')
plt.plot(time, sinAct1, label='Flexor')
plt.plot(time, sinAct2, label='Extensor')
plt.xlabel('Time [s]')
plt.ylabel('Activation')
plt.grid()
plt.legend()
#----------------# Exercise 2b finished #----------------#
# To animate the model, use the SystemAnimation class
# Pass the res(states) and systems you wish to animate
simulation = SystemAnimation(res, pendulum, muscles)
# To start the animation
if DEFAULT["save_figures"] is False:
simulation.animate()
if not DEFAULT["save_figures"]:
plt.show()
else:
figures = plt.get_figlabels()
pylog.debug("Saving figures:\n{}".format(figures))
for fig in figures:
plt.figure(fig)
save_figure(fig)
#plt.close(fig)
plt.show
def save_figures(**kwargs):
"""Save_figures"""
figures = [str(figure) for figure in plt.get_figlabels()]
pylog.debug("Other files:\n - " + "\n - ".join(figures))
for name in figures:
save_figure(name, extensions=kwargs.pop("extensions", ["pdf"]))
def exercise2():
""" Main function to run for Exercise 2.
Parameters
----------
None
Returns
-------
None
"""
# Define and Setup your pendulum model here
# Check PendulumSystem.py for more details on Pendulum class
pendulum_params = PendulumParameters() # Instantiate pendulum parameters
pendulum_params.L = 0.5 # To change the default length of the pendulum
pendulum_params.m = 1. # To change the default mass of the pendulum
pendulum = PendulumSystem(pendulum_params) # Instantiate Pendulum object
#### CHECK OUT PendulumSystem.py to ADD PERTURBATIONS TO THE MODEL #####
pylog.info('Pendulum model initialized \n {}'.format(
pendulum.parameters.showParameters()))
# Define and Setup your pendulum model here
# Check MuscleSytem.py for more details on MuscleSytem class
M1_param = MuscleParameters() # Instantiate Muscle 1 parameters
M1_param.f_max = 1500 # To change Muscle 1 max force
M2_param = MuscleParameters() # Instantiate Muscle 2 parameters
M2_param.f_max = 1500 # To change Muscle 2 max force
M1 = Muscle(M1_param) # Instantiate Muscle 1 object
M2 = Muscle(M2_param) # Instantiate Muscle 2 object
# Use the MuscleSystem Class to define your muscles in the system
muscles = MuscleSytem(M1, M2) # Instantiate Muscle System with two muscles
pylog.info('Muscle system initialized \n {} \n {}'.format(
M1.parameters.showParameters(), M2.parameters.showParameters()))
# Define Muscle Attachment points
m1_origin = np.array([-0.17, 0.0]) # Origin of Muscle 1
m1_insertion = np.array([0.0, -0.17]) # Insertion of Muscle 1
m2_origin = np.array([0.17, 0.0]) # Origin of Muscle 2
m2_insertion = np.array([0.0, -0.17]) # Insertion of Muscle 2
# Attach the muscles
muscles.attach(np.array([m1_origin, m1_insertion]),
np.array([m2_origin, m2_insertion]))
# Create a system with Pendulum and Muscles using the System Class
# Check System.py for more details on System class
sys = System() # Instantiate a new system
sys.add_pendulum_system(pendulum) # Add the pendulum model to the system
sys.add_muscle_system(muscles) # Add the muscle model to the system
##### Time #####
t_max = 2.5 # Maximum simulation time
time = np.arange(0., t_max, 0.001) # Time vector
##### Model Initial Conditions #####
x0_P = np.array([np.pi / 6, 0.]) # Pendulum initial condition
# Muscle Model initial condition
x0_M = np.array([0., M1.L_OPT, 0., M2.L_OPT])
x0 = np.concatenate((x0_P, x0_M)) # System initial conditions
##### System Simulation #####
# For more details on System Simulation check SystemSimulation.py
# SystemSimulation is used to initialize the system and integrate
# over time
sim = SystemSimulation(sys) # Instantiate Simulation object
# Add muscle activations to the simulation
# Here you can define your muscle activation vectors
# that are time dependent
sin_freq = 1 #hz
ampl_sin = 1
phase_difference_1_2 = np.pi
act1 = np.ones((len(time), 1))
act2 = np.ones((len(time), 1))
for i in range(len(time)):
act1[i, 0] = ampl_sin * (1 + np.sin(2 * np.pi * sin_freq * time[i]))
act2[i, 0] = ampl_sin * (
1 + np.sin(2 * np.pi * sin_freq * time[i] + phase_difference_1_2))
activations = np.hstack((act1, act2))
# Method to add the muscle activations to the simulation
sim.add_muscle_activations(activations)
# Simulate the system for given time
sim.initalize_system(x0, time) # Initialize the system state
#: If you would like to perturb the pedulum model then you could do
# so by
sim.sys.pendulum_sys.parameters.PERTURBATION = True
# The above line sets the state of the pendulum model to zeros between
# time interval 1.2 < t < 1.25. You can change this and the type of
# perturbation in
# pendulum_system.py::pendulum_system function
# Integrate the system for the above initialized state and time
sim.simulate()
# Obtain the states of the system after integration
# res is np.array [time, states]
# states vector is in the same order as x0
res = sim.results()
# In order to obtain internal states of the muscle
# you can access the results attribute in the muscle class
muscle1_results = sim.sys.muscle_sys.Muscle1.results
muscle2_results = sim.sys.muscle_sys.Muscle2.results
# Plotting the results
plt.figure('Pendulum')
plt.title('Pendulum Phase')
plt.plot(res[:, 1], res[:, 2])
plt.xlabel('Position [rad]')
plt.ylabel('Velocity [rad.s]')
plt.grid()
plt.figure('Activations')
plt.title('Sine wave activations for both muscles')
plt.plot(time, act1)
plt.plot(time, act2)
plt.legend(("activation muscle1", "activation muscle2"))
# To animate the model, use the SystemAnimation class
# Pass the res(states) and systems you wish to animate
simulation = SystemAnimation(res, pendulum, muscles)
# To start the animation
if DEFAULT["save_figures"] is False:
simulation.animate()
if not DEFAULT["save_figures"]:
plt.show()
else:
figures = plt.get_figlabels()
pylog.debug("Saving figures:\n{}".format(figures))
for fig in figures:
plt.figure(fig)
save_figure(fig)
plt.close(fig)
def plotExternalDrive(sys,x0,ext_in,typ='low'):
##### Time #####
t_max = 2.5 # Maximum simulation time
time = np.arange(0., t_max, 0.001) # Time vector
sim = SystemSimulation(sys) # Instantiate Simulation object
# Add external inputs to neural network
# sim.add_external_inputs_to_network(np.ones((len(time), 4)))
#ext_in = np.ones((len(time), 4))
#ext_in[:,2] = 0.2
#ext_in[:,3] = 0.2
sim.add_external_inputs_to_network(ext_in)
sim.initalize_system(x0, time) # Initialize the system state
# Integrate the system for the above initialized state and time
sim.simulate()
# Obtain the states of the system after integration
# res is np.array [time, states]
# states vector is in the same order as x0
res = sim.results()
# Obtain the states of the system after integration
# res is np.array [time, states]
# states vector is in the same order as x0
#res = sim.results()
# In order to obtain internal states of the muscle
# you can access the results attribute in the muscle class
muscle1_results = sim.sys.muscle_sys.Muscle1.results
muscle2_results = sim.sys.muscle_sys.Muscle2.results
# Plotting the phase
fig = plt.figure('Pendulum Phase, {} drive'.format(typ))
plt.title('Pendulum Phase, {} drive'.format(typ))
plt.plot(res[:, 1], res[:, 2])
plt.xlabel('Position [rad]')
plt.ylabel('Velocity [rad/s]')
plt.grid()
fig.tight_layout()
fig.savefig('graphs/PendulumPhase{}Drive.png'.format(typ))
# Plotting the state evolution
fig = plt.figure('Pendulum State, {} drive'.format(typ))
plt.title('Pendulum State, {} drive'.format(typ))
plt.plot(res[:, 0], res[:, 1], label='position [rad]')
plt.plot(res[:, 0], res[:, 2], label='speed [rad/s]')
plt.xlabel('Time [s]')
plt.ylabel('')
plt.legend()
plt.grid()
fig.tight_layout()
fig.savefig('graphs/PendulumState{}drive.png'.format(typ))
# Plotting the neuronal activation
# Access the neurons outputs:
# [t] theta theta. A1 lCE1 A2 lCE2 m1 m2 m3 m4
fig = plt.figure('Neuron output, {} drive'.format(typ))
plt.title('Membrane potentials')
plt.plot(res[:, 0], res[:, 7],label='m1')
plt.plot(res[:, 0], res[:, 8],label='m2')
plt.plot(res[:, 0], res[:, 9],label='m3')
plt.plot(res[:, 0], res[:, 10],label='m4')
plt.xlabel('Time [s]')
plt.ylabel('Potential')
plt.legend()
plt.grid()
fig.tight_layout()
fig.savefig('graphs/MembranePotentials{}drive.png'.format(typ))
if DEFAULT["save_figures"] is False:
plt.show()
else:
figures = plt.get_figlabels()
pylog.debug("Saving figures:\n{}".format(figures))
for fig in figures:
plt.figure(fig)
save_figure(fig)
plt.close(fig)
# To animate the model, use the SystemAnimation class
# Pass the res(states) and systems you wish to animate
simulation = SystemAnimation(
res,
sim.sys.pendulum_sys,
sim.sys.muscle_sys,
sim.sys.neural_sys)
# To start the animation
simulation.animate()
def exercise3():
""" Main function to run for Exercise 3.
Parameters
----------
None
Returns
-------
None
"""
# Define and Setup your pendulum model here
# Check Pendulum.py for more details on Pendulum class
P_params = PendulumParameters() # Instantiate pendulum parameters
P_params.L = 0.5 # To change the default length of the pendulum
P_params.m = 1. # To change the default mass of the pendulum
pendulum = PendulumSystem(P_params) # Instantiate Pendulum object
#### CHECK OUT Pendulum.py to ADD PERTURBATIONS TO THE MODEL #####
pylog.info('Pendulum model initialized \n {}'.format(
pendulum.parameters.showParameters()))
# Define and Setup your pendulum model here
# Check MuscleSytem.py for more details on MuscleSytem class
M1_param = MuscleParameters() # Instantiate Muscle 1 parameters
M1_param.f_max = 1500 # To change Muscle 1 max force
M2_param = MuscleParameters() # Instantiate Muscle 2 parameters
M2_param.f_max = 1500 # To change Muscle 2 max force
M1 = Muscle(M1_param) # Instantiate Muscle 1 object
M2 = Muscle(M2_param) # Instantiate Muscle 2 object
# Use the MuscleSystem Class to define your muscles in the system
muscles = MuscleSytem(M1, M2) # Instantiate Muscle System with two muscles
pylog.info('Muscle system initialized \n {} \n {}'.format(
M1.parameters.showParameters(), M2.parameters.showParameters()))
# Define Muscle Attachment points
m1_origin = np.array([-0.17, 0.0]) # Origin of Muscle 1
m1_insertion = np.array([0.0, -0.17]) # Insertion of Muscle 1
m2_origin = np.array([0.17, 0.0]) # Origin of Muscle 2
m2_insertion = np.array([0.0, -0.17]) # Insertion of Muscle 2
# Attach the muscles
muscles.attach(np.array([m1_origin, m1_insertion]),
np.array([m2_origin, m2_insertion]))
##### Neural Network #####
# The network consists of four neurons
N_params = NetworkParameters() # Instantiate default network parameters
N_params.D = 2. # To change a network parameter
# Similarly to change w -> N_params.w = (4x4) array
# Create a new neural network with above parameters
neural_network = NeuralSystem(N_params)
pylog.info('Neural system initialized \n {}'.format(
N_params.showParameters()))
# Create system of Pendulum, Muscles and neural network using SystemClass
# Check System.py for more details on System class
sys = System() # Instantiate a new system
sys.add_pendulum_system(pendulum) # Add the pendulum model to the system
sys.add_muscle_system(muscles) # Add the muscle model to the system
# Add the neural network to the system
sys.add_neural_system(neural_network)
##### Time #####
t_max = 2.5 # Maximum simulation time
time = np.arange(0., t_max, 0.001) # Time vector
##### Model Initial Conditions #####
x0_P = np.array([0., 0.]) # Pendulum initial condition
# Muscle Model initial condition
x0_M = np.array([0., M1.L_OPT, 0., M2.L_OPT])
x0_N = np.array([-0.5, 1, 0.5, 1]) # Neural Network Initial Conditions
x0 = np.concatenate((x0_P, x0_M, x0_N)) # System initial conditions
##### System Simulation #####
# For more details on System Simulation check SystemSimulation.py
# SystemSimulation is used to initialize the system and integrate
# over time
sim = SystemSimulation(sys) # Instantiate Simulation object
# Add external inputs to neural network
# sim.add_external_inputs_to_network(np.ones((len(time), 4)))
# sim.add_external_inputs_to_network(ext_in)
sim.initalize_system(x0, time) # Initialize the system state
# Integrate the system for the above initialized state and time
sim.simulate()
# Obtain the states of the system after integration
# res is np.array [time, states]
# states vector is in the same order as x0
res = sim.results()
# Obtain the states of the system after integration
# res is np.array [time, states]
# states vector is in the same order as x0
res = sim.results()
# In order to obtain internal states of the muscle
# you can access the results attribute in the muscle class
muscle1_results = sim.sys.muscle_sys.Muscle1.results
muscle2_results = sim.sys.muscle_sys.Muscle2.results
# Plotting the results
plt.figure('Pendulum')
plt.title('Pendulum Phase')
plt.plot(res[:, 0], res[:, :2])
plt.xlabel('Position [rad]')
plt.ylabel('Velocity [rad.s]')
plt.grid()
if DEFAULT["save_figures"] is False:
plt.show()
else:
figures = plt.get_figlabels()
pylog.debug("Saving figures:\n{}".format(figures))
for fig in figures:
plt.figure(fig)
save_figure(fig)
plt.close(fig)
# To animate the model, use the SystemAnimation class
# Pass the res(states) and systems you wish to animate
simulation = SystemAnimation(res, sim.sys.pendulum_sys, sim.sys.muscle_sys,
sim.sys.neural_sys)
# To start the animation
simulation.animate()
self.parameters['b1'] = value
pylog.info(
'Changed damping constant for damper 1 to {} [N-s/rad]'.format(
self.parameters['b1']))
@property
def b2(self):
""" Get the value of damping constant for damper 2. [N-s/rad]
Default is 0.5"""
return self.parameters['b2']
@b2.setter
def b2(self, value):
""" Keyword Arguments:
value -- Set the value of damping constant for damper 2. [N-s/rad] """
if (value < 0.0):
pylog.warning('Setting bad damping values. Should be positive!')
else:
self.parameters['b2'] = value
pylog.info(
'Changed damping constant for damper 2 to {} [N-s/rad]'.format(
self.parameters['b2']))
def showParameters(self):
return self.msg(self.parameters, self.units)
if __name__ == '__main__':
P = PendulumParameters(g=9.81, L=1.)
pylog.debug(P.showParameters())
