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 exercise2c():
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
##### 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
#Frequency effect :
stim_frequency = np.array([0.05, 0.1, 0.5, 1, 5, 10, 50, 100, 500]) #in Hz
stim_amplitude = 1 # belongs to 0-1
phase = np.pi
frequency_pendelum = np.zeros(len(stim_frequency))
amplitude_pendelum = np.zeros(len(stim_frequency))
for j, frequency in enumerate(stim_frequency):
t_max = 5 / frequency # Maximum simulation time
time_step = 0.001 * (1 / frequency)
time = np.arange(0., t_max, time_step) # Time vector
act1 = np.zeros((len(time), 1))
act2 = np.zeros((len(time), 1))
act1[:,
0] = stim_amplitude * (1 +
np.sin(2 * np.pi * frequency * time)) / 2
act2[:, 0] = stim_amplitude * (
1 + np.sin(2 * np.pi * frequency * time + phase)) / 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 freuquency and amplitude
angular_position = res[:, 1]
#signal_stat = signal[index_start:len(signal)]
start_index = int(len(angular_position) / 2)
final_index = (len(angular_position))
index_zeros = np.where(
np.diff(np.sign(angular_position[start_index:final_index])))[
0] #np.where(signal_stat==0)[0]
deltas = np.diff(index_zeros)
delta = np.mean(deltas)
frequency_pendelum[j] = 1 / (2 * delta * time_step)
signal = angular_position[start_index:len(angular_position)]
amplitude = (np.max(signal) - np.min(signal)) / 2
amplitude_pendelum[j] = amplitude
plt.figure()
plt.subplot(121)
plt.loglog(stim_frequency, frequency_pendelum)
plt.grid()
plt.xlabel('Stimulation Frequency [Hz]')
plt.ylabel('Pendulum Oscillation Frequency [Hz]')
plt.subplot(122)
plt.loglog(stim_frequency, amplitude_pendelum)
plt.grid()
plt.xlabel('Stimulation Frequency [Hz]')
plt.ylabel('Pendulum Oscillation Amplitude [rad]')
plt.savefig('2c.png')
plt.show()
stim_frequency = 10 #in Hz
stim_amplitude = np.arange(0, 1.1, 0.1)
frequency_pendelum = np.zeros(len(stim_amplitude))
amplitude_pendelum = np.zeros(len(stim_amplitude))
for j, amplitude_ in enumerate(stim_amplitude):
t_max = 5 / stim_frequency # Maximum simulation time
time_step = 0.001 * (1 / stim_frequency)
time = np.arange(0., t_max, time_step) # Time vector
act1 = np.zeros((len(time), 1))
act2 = np.zeros((len(time), 1))
act1[:,
0] = amplitude_ * (1 +
np.sin(2 * np.pi * stim_frequency * time)) / 2
act2[:, 0] = amplitude_ * (
1 + np.sin(2 * np.pi * stim_frequency * time + phase)) / 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 freuquency and amplitude
angular_position = res[:, 1]
#signal_stat = signal[index_start:len(signal)]
start_index = int(len(angular_position) / 2)
final_index = (len(angular_position))
index_zeros = np.where(
np.diff(np.sign(angular_position[start_index:final_index])))[
0] #np.where(signal_stat==0)[0]
deltas = np.diff(index_zeros)
delta = np.mean(deltas)
frequency_pendelum[j] = 1 / (2 * delta * time_step)
signal = angular_position[start_index:len(angular_position)]
amplitude = (np.max(signal) - np.min(signal)) / 2
amplitude_pendelum[j] = amplitude
frequency_pendelum[0] = 0
plt.figure()
plt.subplot(121)
plt.plot(stim_amplitude, frequency_pendelum)
plt.grid()
plt.xlabel('Stimulation Amplitude [rad]')
plt.ylabel('Pendulum Oscillation Frequency [Hz]')
plt.subplot(122)
plt.plot(stim_amplitude, amplitude_pendelum)
plt.grid()
plt.xlabel('Stimulation Amplitude[rad]')
plt.ylabel('Pendulum Oscillation Amplitude [rad]')
plt.savefig('2c_amplitude.png')
plt.show()
def exercise2c():
""" Main function to run for Exercise 2c.
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 = 15 # Maximum simulation time
time = np.arange(0., t_max, 0.005) # 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
sim = SystemSimulation(sys) # Instantiate Simulation object
plt.figure('Pendulum with different stimulation frequencies')
frequencies = [0.25, 0.5, 0.75, 1.0, 2.0, 3.0, 4.0, 5.0]
for freq in frequencies:
act1 = np.array([np.sin(freq * time)]).T
act2 = np.array([-np.sin(freq * time)]).T
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
# 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.plot(res[:, 1], res[:, 2])
plt.title('Pendulum Phase')
plt.xlabel('Position [rad]')
plt.ylabel('Velocity [rad.s]')
plt.legend((
'0.25',
'0.5',
'0.75',
'1.0',
'2.0',
'3.0',
'4.0',
'5.0',
))
plt.grid()
plt.figure('Pendulum with different stimulation amplitudes')
amplitudes = [0.25, 0.5, 0.75, 1.0, 2.0, 3.0, 4.0, 5.0]
for amp in amplitudes:
act1 = np.array([amp * np.sin(time)]).T
act2 = np.array([amp * (-np.sin(time))]).T
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
# 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.plot(res[:, 1], res[:, 2])
plt.title('Pendulum Phase')
plt.xlabel('Position [rad]')
plt.ylabel('Velocity [rad.s]')
plt.legend((
'0.25',
'0.5',
'0.75',
'1.0',
'2.0',
'3.0',
'4.0',
'5.0',
))
plt.grid()
def exercise2b():
""" Main function to run for Exercise 2b.
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 = 20 # Maximum simulation time
time = np.arange(0., t_max, 0.005) # 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
sim1 = 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.array([np.sin(time)]).T
act2 = np.array([-np.sin(time)]).T
activations = np.hstack((act1, act2))
# Method to add the muscle activations to the simulation
sim1.add_muscle_activations(activations)
# Simulate the system for given time
sim1.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
sim1.simulate()
# Obtain the states of the system after integration
# res is np.array [time, states]
# states vector is in the same order as x0
res1 = sim1.results()
sim2 = SystemSimulation(sys) # Instantiate Simulation object
sim2.add_muscle_activations(activations)
# Simulate the system for given time
sim2.initalize_system(x0, time) # Initialize the system state
#add perturbation
sim2.sys.pendulum_sys.parameters.PERTURBATION = True
# Integrate the system for the above initialized state and time
sim2.simulate()
# Obtain the states of the system after integration
# res is np.array [time, states]
# states vector is in the same order as x0
res2 = sim2.results()
# In order to obtain internal states of the muscle
# you can access the results attribute in the muscle class
muscle1_results = sim1.sys.muscle_sys.Muscle1.results
muscle2_results = sim1.sys.muscle_sys.Muscle2.results
# Plotting the results
plt.figure('Pendulum')
plt.title('Pendulum Phase')
plt.plot(res1[:, 1], res1[:, 2])
plt.xlabel('Position [rad]')
plt.ylabel('Velocity [rad.s]')
plt.grid()
plt.figure('Pendulum with perturbation')
plt.title('Pendulum Phase')
plt.plot(res2[:, 1], res2[:, 2])
plt.xlabel('Position [rad]')
plt.ylabel('Velocity [rad.s]')
plt.grid()
plt.figure('Activation Wave Forms')
plt.title('Activation Wave Forms')
plt.plot(time, act1)
plt.plot(time, act2)
plt.xlabel('Time [s]')
plt.ylabel('Activation')
plt.legend(('Actication muscle 1', 'Activation muscle 2'))
plt.grid
poincare_crossings(res1, 0.5, 1, "poincare_cross")
# To animate the model, use the SystemAnimation class
# Pass the res(states) and systems you wish to animate
simulation1 = SystemAnimation(res1, pendulum, muscles)
simulation2 = SystemAnimation(res2, pendulum, muscles)
# To start the animation
if DEFAULT["save_figures"] is False:
simulation1.animate()
simulation2.animate()
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 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
"""
# 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.
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