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WIP: plot based simulation

master
mcarfagno 2020-01-13 12:25:26 +00:00
parent 60f028e25b
commit 13935940b4
11 changed files with 115 additions and 404 deletions
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208
husky_mpc/CMakeLists.txt
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@ -1,208 +0,0 @@
cmake_minimum_required(VERSION 2.8.3)
project(husky_mpc)
## Compile as C++11, supported in ROS Kinetic and newer
# add_compile_options(-std=c++11)
## Find catkin macros and libraries
## if COMPONENTS list like find_package(catkin REQUIRED COMPONENTS xyz)
## is used, also find other catkin packages
find_package(catkin REQUIRED COMPONENTS
geometry_msgs
nav_msgs
roscpp
rospy
std_msgs
)
## System dependencies are found with CMake's conventions
# find_package(Boost REQUIRED COMPONENTS system)
## Uncomment this if the package has a setup.py. This macro ensures
## modules and global scripts declared therein get installed
## See http://ros.org/doc/api/catkin/html/user_guide/setup_dot_py.html
# catkin_python_setup()
################################################
## Declare ROS messages, services and actions ##
################################################
## To declare and build messages, services or actions from within this
## package, follow these steps:
## * Let MSG_DEP_SET be the set of packages whose message types you use in
## your messages/services/actions (e.g. std_msgs, actionlib_msgs, ...).
## * In the file package.xml:
## * add a build_depend tag for "message_generation"
## * add a build_depend and a exec_depend tag for each package in MSG_DEP_SET
## * If MSG_DEP_SET isn't empty the following dependency has been pulled in
## but can be declared for certainty nonetheless:
## * add a exec_depend tag for "message_runtime"
## * In this file (CMakeLists.txt):
## * add "message_generation" and every package in MSG_DEP_SET to
## find_package(catkin REQUIRED COMPONENTS ...)
## * add "message_runtime" and every package in MSG_DEP_SET to
## catkin_package(CATKIN_DEPENDS ...)
## * uncomment the add_*_files sections below as needed
## and list every .msg/.srv/.action file to be processed
## * uncomment the generate_messages entry below
## * add every package in MSG_DEP_SET to generate_messages(DEPENDENCIES ...)
## Generate messages in the 'msg' folder
# add_message_files(
# FILES
# Message1.msg
# Message2.msg
# )
## Generate services in the 'srv' folder
# add_service_files(
# FILES
# Service1.srv
# Service2.srv
# )
## Generate actions in the 'action' folder
# add_action_files(
# FILES
# Action1.action
# Action2.action
# )
## Generate added messages and services with any dependencies listed here
# generate_messages(
# DEPENDENCIES
# geometry_msgs# navigation_msgs# std_msgs
# )
################################################
## Declare ROS dynamic reconfigure parameters ##
################################################
## To declare and build dynamic reconfigure parameters within this
## package, follow these steps:
## * In the file package.xml:
## * add a build_depend and a exec_depend tag for "dynamic_reconfigure"
## * In this file (CMakeLists.txt):
## * add "dynamic_reconfigure" to
## find_package(catkin REQUIRED COMPONENTS ...)
## * uncomment the "generate_dynamic_reconfigure_options" section below
## and list every .cfg file to be processed
## Generate dynamic reconfigure parameters in the 'cfg' folder
# generate_dynamic_reconfigure_options(
# cfg/DynReconf1.cfg
# cfg/DynReconf2.cfg
# )
###################################
## catkin specific configuration ##
###################################
## The catkin_package macro generates cmake config files for your package
## Declare things to be passed to dependent projects
## INCLUDE_DIRS: uncomment this if your package contains header files
## LIBRARIES: libraries you create in this project that dependent projects also need
## CATKIN_DEPENDS: catkin_packages dependent projects also need
## DEPENDS: system dependencies of this project that dependent projects also need
catkin_package(
# INCLUDE_DIRS include
# LIBRARIES husky_mpc
# CATKIN_DEPENDS geometry_msgs navigation_msgs roscpp rospy std_msgs
# DEPENDS system_lib
)
###########
## Build ##
###########
## Specify additional locations of header files
## Your package locations should be listed before other locations
include_directories(
# include
${catkin_INCLUDE_DIRS}
)
## Declare a C++ library
# add_library(${PROJECT_NAME}
# src/${PROJECT_NAME}/husky_mpc.cpp
# )
## Add cmake target dependencies of the library
## as an example, code may need to be generated before libraries
## either from message generation or dynamic reconfigure
# add_dependencies(${PROJECT_NAME} ${${PROJECT_NAME}_EXPORTED_TARGETS} ${catkin_EXPORTED_TARGETS})
## Declare a C++ executable
## With catkin_make all packages are built within a single CMake context
## The recommended prefix ensures that target names across packages don't collide
# add_executable(${PROJECT_NAME}_node src/husky_mpc_node.cpp)
## Rename C++ executable without prefix
## The above recommended prefix causes long target names, the following renames the
## target back to the shorter version for ease of user use
## e.g. "rosrun someones_pkg node" instead of "rosrun someones_pkg someones_pkg_node"
# set_target_properties(${PROJECT_NAME}_node PROPERTIES OUTPUT_NAME node PREFIX "")
## Add cmake target dependencies of the executable
## same as for the library above
# add_dependencies(${PROJECT_NAME}_node ${${PROJECT_NAME}_EXPORTED_TARGETS} ${catkin_EXPORTED_TARGETS})
## Specify libraries to link a library or executable target against
# target_link_libraries(${PROJECT_NAME}_node
# ${catkin_LIBRARIES}
# )
#############
## Install ##
#############
# all install targets should use catkin DESTINATION variables
# See http://ros.org/doc/api/catkin/html/adv_user_guide/variables.html
## Mark executable scripts (Python etc.) for installation
## in contrast to setup.py, you can choose the destination
# install(PROGRAMS
# scripts/my_python_script
# DESTINATION ${CATKIN_PACKAGE_BIN_DESTINATION}
# )
## Mark executables for installation
## See http://docs.ros.org/melodic/api/catkin/html/howto/format1/building_executables.html
# install(TARGETS ${PROJECT_NAME}_node
# RUNTIME DESTINATION ${CATKIN_PACKAGE_BIN_DESTINATION}
# )
## Mark libraries for installation
## See http://docs.ros.org/melodic/api/catkin/html/howto/format1/building_libraries.html
# install(TARGETS ${PROJECT_NAME}
# ARCHIVE DESTINATION ${CATKIN_PACKAGE_LIB_DESTINATION}
# LIBRARY DESTINATION ${CATKIN_PACKAGE_LIB_DESTINATION}
# RUNTIME DESTINATION ${CATKIN_GLOBAL_BIN_DESTINATION}
# )
## Mark cpp header files for installation
# install(DIRECTORY include/${PROJECT_NAME}/
# DESTINATION ${CATKIN_PACKAGE_INCLUDE_DESTINATION}
# FILES_MATCHING PATTERN "*.h"
# PATTERN ".svn" EXCLUDE
# )
## Mark other files for installation (e.g. launch and bag files, etc.)
# install(FILES
# # myfile1
# # myfile2
# DESTINATION ${CATKIN_PACKAGE_SHARE_DESTINATION}
# )
#############
## Testing ##
#############
## Add gtest based cpp test target and link libraries
# catkin_add_gtest(${PROJECT_NAME}-test test/test_husky_mpc.cpp)
# if(TARGET ${PROJECT_NAME}-test)
# target_link_libraries(${PROJECT_NAME}-test ${PROJECT_NAME})
# endif()
## Add folders to be run by python nosetests
# catkin_add_nosetests(test)

74
husky_mpc/package.xml
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@ -1,74 +0,0 @@
<?xml version="1.0"?>
<package format="2">
<name>husky_mpc</name>
<version>0.0.0</version>
<description>The husky_mpc package</description>
<!-- One maintainer tag required, multiple allowed, one person per tag -->
<!-- Example: -->
<!-- <maintainer email="jane.doe@example.com">Jane Doe</maintainer> -->
<maintainer email="marcello@todo.todo">marcello</maintainer>
<!-- One license tag required, multiple allowed, one license per tag -->
<!-- Commonly used license strings: -->
<!-- BSD, MIT, Boost Software License, GPLv2, GPLv3, LGPLv2.1, LGPLv3 -->
<license>TODO</license>
<!-- Url tags are optional, but multiple are allowed, one per tag -->
<!-- Optional attribute type can be: website, bugtracker, or repository -->
<!-- Example: -->
<!-- <url type="website">http://wiki.ros.org/husky_mpc</url> -->
<!-- Author tags are optional, multiple are allowed, one per tag -->
<!-- Authors do not have to be maintainers, but could be -->
<!-- Example: -->
<!-- <author email="jane.doe@example.com">Jane Doe</author> -->
<!-- The *depend tags are used to specify dependencies -->
<!-- Dependencies can be catkin packages or system dependencies -->
<!-- Examples: -->
<!-- Use depend as a shortcut for packages that are both build and exec dependencies -->
<!-- <depend>roscpp</depend> -->
<!-- Note that this is equivalent to the following: -->
<!-- <build_depend>roscpp</build_depend> -->
<!-- <exec_depend>roscpp</exec_depend> -->
<!-- Use build_depend for packages you need at compile time: -->
<!-- <build_depend>message_generation</build_depend> -->
<!-- Use build_export_depend for packages you need in order to build against this package: -->
<!-- <build_export_depend>message_generation</build_export_depend> -->
<!-- Use buildtool_depend for build tool packages: -->
<!-- <buildtool_depend>catkin</buildtool_depend> -->
<!-- Use exec_depend for packages you need at runtime: -->
<!-- <exec_depend>message_runtime</exec_depend> -->
<!-- Use test_depend for packages you need only for testing: -->
<!-- <test_depend>gtest</test_depend> -->
<!-- Use doc_depend for packages you need only for building documentation: -->
<!-- <doc_depend>doxygen</doc_depend> -->
<buildtool_depend>catkin</buildtool_depend>
<build_depend>geometry_msgs</build_depend>
<build_depend>nav_msgs</build_depend>
<build_depend>roscpp</build_depend>
<build_depend>rospy</build_depend>
<build_depend>std_msgs</build_depend>
<build_export_depend>geometry_msgs</build_export_depend>
<build_export_depend>nav_msgs</build_export_depend>
<build_export_depend>roscpp</build_export_depend>
<build_export_depend>rospy</build_export_depend>
<build_export_depend>std_msgs</build_export_depend>
<exec_depend>geometry_msgs</exec_depend>
<exec_depend>nav_msgs</exec_depend>
<exec_depend>roscpp</exec_depend>
<exec_depend>rospy</exec_depend>
<exec_depend>std_msgs</exec_depend>
<!-- The export tag contains other, unspecified, tags -->
<export>
<!-- Other tools can request additional information be placed here -->
</export>
</package>

84
husky_mpc/scripts/mpc_node.py
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@ -1,84 +0,0 @@
#! /usr/bin/env python
import rospy
import numpy as np
from nav_msgs.msg import Odometry
from geometry_msgs.msg import Twist
from utils import compute_path_from_wp
from cvxpy_mpc import optimize
# classes
class Node():
def __init__(self):
rospy.init_node('mpc_node')
N = 5 #number of state variables
M = 2 #number of control variables
T = 20 #Prediction Horizon
dt = 0.25 #discretization step
# State for the robot mathematical model
self.state = None
# starting guess output
self.opt_u = np.zeros((M,T))
self.opt_u[0,:] = 1 #m/s
self.opt_u[1,:] = np.radians(0) #rad/s
# Interpolated Path to follow given waypoints
self.path = compute_path_from_wp([0,20,30,30],[0,0,10,20])
self._cmd_pub = rospy.Publisher(rospy.get_namespace() + 'husky_velocity_controller/cmd_vel', Twist, queue_size=10)
self._odom_sub = rospy.Subscriber(rospy.get_namespace() +'husky_velocity_controller/odom', Odometry, self._odom_cb, queue_size=1)
def run(self):
while 1:
if self.state is not None:
#optimization loop
self.opt_u = optimize(self.state,
self.opt_u,
self.path)
msg = Twist()
msg.linear.x=self.opt_u[0,1]
msg.angular.z=self.opt_u[0,1]
self._cmd_pub(msg)
def _odom_cb(self,odom):
'''
Updates state with latest odometry.
:param odom: nav_msgs.msg.Odometry
'''
state = np.zeros(3)
# Update current position
state[0] = odom.pose.pose.position.x
state[1] = odom.pose.pose.position.y
# Update current orientation
_, _, state[2] = euler_from_quaternion(
[odom.pose.pose.orientation.x,
odom.pose.pose.orientation.y,
odom.pose.pose.orientation.z,
odom.pose.pose.orientation.w])
self.state = state
def main():
ros_node=Node()
try:
ros_node.run()
except rospy.exceptions.ROSException as e:
sys.exit(e)
if __name__ == '__main__':
main()

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36
husky_mpc/scripts/cvxpy_mpc.py → mpc_demo/cvxpy_mpc.py
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@ -19,16 +19,16 @@ def get_linear_model(x_bar,u_bar):
theta = x_bar[2] theta = x_bar[2]
psi = x_bar[3] psi = x_bar[3]
cte = x_bar[4] cte = x_bar[4]
v = u_bar[0] v = u_bar[0]
w = u_bar[1] w = u_bar[1]
A = np.zeros((N,N)) A = np.zeros((N,N))
A[0,2]=-v*np.sin(theta) A[0,2]=-v*np.sin(theta)
A[1,2]=v*np.cos(theta) A[1,2]=v*np.cos(theta)
A[4,3]=v*np.cos(-psi) A[4,3]=v*np.cos(-psi)
A_lin=np.eye(N)+dt*A A_lin=np.eye(N)+dt*A
B = np.zeros((N,M)) B = np.zeros((N,M))
B[0,0]=np.cos(theta) B[0,0]=np.cos(theta)
B[1,0]=np.sin(theta) B[1,0]=np.sin(theta)
@ -36,10 +36,10 @@ def get_linear_model(x_bar,u_bar):
B[3,1]=-1 B[3,1]=-1
B[4,0]=np.sin(-psi) B[4,0]=np.sin(-psi)
B_lin=dt*B B_lin=dt*B
f_xu=np.array([v*np.cos(theta),v*np.sin(theta),w,-w,v*np.sin(-psi)]).reshape(N,1) f_xu=np.array([v*np.cos(theta),v*np.sin(theta),w,-w,v*np.sin(-psi)]).reshape(N,1)
C_lin = dt*(f_xu - np.dot(A,x_bar.reshape(N,1)) - np.dot(B,u_bar.reshape(M,1))) C_lin = dt*(f_xu - np.dot(A,x_bar.reshape(N,1)) - np.dot(B,u_bar.reshape(M,1)))
return A_lin,B_lin,C_lin return A_lin,B_lin,C_lin
def calc_err(state,path): def calc_err(state,path):
@ -58,7 +58,7 @@ def calc_err(state,path):
try: try:
v = [path[0,nn_idx+1] - path[0,nn_idx], v = [path[0,nn_idx+1] - path[0,nn_idx],
path[1,nn_idx+1] - path[1,nn_idx]] path[1,nn_idx+1] - path[1,nn_idx]]
v /= np.linalg.norm(v) v /= np.linalg.norm(v)
d = [path[0,nn_idx] - state[0], d = [path[0,nn_idx] - state[0],
@ -74,17 +74,17 @@ def calc_err(state,path):
path_ref_vect = [np.cos(path[2,target_idx] + np.pi / 2), path_ref_vect = [np.cos(path[2,target_idx] + np.pi / 2),
np.sin(path[2,target_idx] + np.pi / 2)] np.sin(path[2,target_idx] + np.pi / 2)]
#heading error w.r.t path frame #heading error w.r.t path frame
psi = path[2,target_idx] - state[2] psi = path[2,target_idx] - state[2]
# the cross-track error is given by the scalar projection of the car->wp vector onto the faxle versor # the cross-track error is given by the scalar projection of the car->wp vector onto the faxle versor
#cte = np.dot([dx[target_idx], dy[target_idx]],front_axle_vect) #cte = np.dot([dx[target_idx], dy[target_idx]],front_axle_vect)
cte = np.dot([dx[target_idx], dy[target_idx]],path_ref_vect) cte = np.dot([dx[target_idx], dy[target_idx]],path_ref_vect)
return target_idx,psi,cte return target_idx,psi,cte
def optimize(starting_state,u_bar,track); def optimize(starting_state,u_bar,track):
''' '''
:param starting_state: :param starting_state:
:param u_bar: :param u_bar:
@ -100,7 +100,7 @@ def optimize(starting_state,u_bar,track);
M = 2 #number of control variables M = 2 #number of control variables
T = 20 #Prediction Horizon T = 20 #Prediction Horizon
dt = 0.25 #discretization step dt = 0.25 #discretization step
#Starting Condition #Starting Condition
x0 = np.zeros(N) x0 = np.zeros(N)
x0[0] = starting_state[0] x0[0] = starting_state[0]
@ -141,7 +141,7 @@ def optimize(starting_state,u_bar,track);
idx,_,_ = calc_err(x_bar[:,t],track) idx,_,_ = calc_err(x_bar[:,t],track)
delta_x = track[:,idx]-x[0:3,t] delta_x = track[:,idx]-x[0:3,t]
cost+= cp.quad_form(delta_x,10*np.eye(3)) cost+= cp.quad_form(delta_x,10*np.eye(3))
# Tracking last time step # Tracking last time step
if t == T: if t == T:
idx,_,_ = calc_err(x_bar[:,t],track) idx,_,_ = calc_err(x_bar[:,t],track)
@ -151,26 +151,26 @@ def optimize(starting_state,u_bar,track);
# Actuation rate of change # Actuation rate of change
if t < (T - 1): if t < (T - 1):
cost += cp.quad_form(u[:, t + 1] - u[:, t], 25*np.eye(M)) cost += cp.quad_form(u[:, t + 1] - u[:, t], 25*np.eye(M))
# Actuation effort # Actuation effort
cost += cp.quad_form( u[:, t],1*np.eye(M)) cost += cp.quad_form( u[:, t],1*np.eye(M))
# Constrains # Constrains
A,B,C=get_linear_model(x_bar[:,t],u_bar[:,t]) A,B,C=get_linear_model(x_bar[:,t],u_bar[:,t])
constr += [x[:,t+1] == A*x[:,t] + B*u[:,t] + C.flatten()] constr += [x[:,t+1] == A*x[:,t] + B*u[:,t] + C.flatten()]
# sums problem objectives and concatenates constraints. # sums problem objectives and concatenates constraints.
constr += [x[:,0] == x_sim[:,sim_time]] # starting condition constr += [x[:,0] == x0] # starting condition
constr += [u[0, :] <= MAX_SPEED] constr += [u[0, :] <= MAX_SPEED]
constr += [u[0, :] >= MIN_SPEED] constr += [u[0, :] >= MIN_SPEED]
constr += [cp.abs(u[1, :]) <= MAX_STEER_SPEED] constr += [cp.abs(u[1, :]) <= MAX_STEER_SPEED]
# Solve # Solve
prob = cp.Problem(cp.Minimize(cost), constr) prob = cp.Problem(cp.Minimize(cost), constr)
solution = prob.solve(solver=cp.ECOS, verbose=False) solution = prob.solve(solver=cp.ECOS, verbose=False)
#retrieved optimized U and assign to u_bar to linearize in next step #retrieved optimized U and assign to u_bar to linearize in next step
u_bar=np.vstack((np.array(u.value[0, :]).flatten(), u_bar=np.vstack((np.array(u.value[0, :]).flatten(),
(np.array(u.value[1, :]).flatten()))) (np.array(u.value[1, :]).flatten())))
return u_bar return u_bar

91
mpc_demo/main.py Executable file
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@ -0,0 +1,91 @@
#! /usr/bin/env python
import numpy as np
import matplotlib.pyplot as plt
from matplotlib import animation
from utils import compute_path_from_wp
from cvxpy_mpc import optimize
import sys
import time
# classes
class MPC():
def __init__(self):
# State for the robot mathematical model [x,y,heading]
self.state = np.zeros(3)
# Sim step
self.dt = 0.25
# starting guess output
N = 5 #number of state variables
M = 2 #number of control variables
T = 20 #Prediction Horizon
self.opt_u = np.zeros((M,T))
self.opt_u[0,:] = 1 #m/s
self.opt_u[1,:] = np.radians(0) #rad/s
# Interpolated Path to follow given waypoints
self.path = compute_path_from_wp([0,20,30,30],[0,0,10,20])
#Initialise plot
# First set up the figure, the axis, and the plot element we want to animate
plt.style.use("ggplot")
self.fig = plt.figure()
plt.ion()
plt.show()
def run(self):
'''
'''
while 1:
if self.state is not None:
#optimization loop
start=time.time()
self.opt_u = optimize(self.state,
self.opt_u,
self.path)
print("CVXPY Optimization Time: {:.4f}s".format(time.time()-start))
self.update_sim(self.opt_u[0,1],self.opt_u[1,1])
self.plot_sim()
def update_sim(self,lin_v,ang_v):
'''
Updates state.
:param lin_v: float
:param ang_v: float
'''
self.state[0] = self.state[0] +lin_v*np.cos(self.state[2])*self.dt
self.state[1] = self.state[1] +lin_v*np.sin(self.state[2])*self.dt
self.state[2] = self.state[2] +ang_v*self.dt
def plot_sim(self):
plt.clf()
self.ax = plt.axes(xlim=(np.min(self.path[0,:])-1, np.max(self.path[0,:])+1),
ylim=(np.min(self.path[1,:])-1, np.max(self.path[1,:])+1))
self.track, = self.ax.plot(self.path[0,:],self.path[1,:], "g-", label="reference track")
self.vehicle, = self.ax.plot([self.state[0]], [self.state[1]], "r*", label="vehicle path")
plt.legend()
plt.draw()
plt.pause(0.1)
def do_sim():
sim=MPC()
try:
sim.run()
except Exception as e:
sys.exit(e)
if __name__ == '__main__':
do_sim()

12
husky_mpc/scripts/utils.py → mpc_demo/utils.py
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@ -1,12 +1,10 @@
import numpy as np import numpy as np
from scipy.integrate import odeint
from scipy.interpolate import interp1d from scipy.interpolate import interp1d
import cvxpy as cp
def compute_path_from_wp(start_xp, start_yp, step = 0.1): def compute_path_from_wp(start_xp, start_yp, step = 0.1):
""" """
Interpolation range is computed to assure one point every fixed distance step [m]. Interpolation range is computed to assure one point every fixed distance step [m].
:param start_xp: array_like, list of starting x coordinates :param start_xp: array_like, list of starting x coordinates
:param start_yp: array_like, list of starting y coordinates :param start_yp: array_like, list of starting y coordinates
:param step: float, interpolation distance [m] between consecutive waypoints :param step: float, interpolation distance [m] between consecutive waypoints
@ -21,13 +19,15 @@ def compute_path_from_wp(start_xp, start_yp, step = 0.1):
section_len = np.sum(np.sqrt(np.power(np.diff(start_xp[idx:idx+2]),2)+np.power(np.diff(start_yp[idx:idx+2]),2))) section_len = np.sum(np.sqrt(np.power(np.diff(start_xp[idx:idx+2]),2)+np.power(np.diff(start_yp[idx:idx+2]),2)))
interp_range = np.linspace(0,1,section_len/delta) interp_range = np.linspace(0,1,section_len/delta)
fx=interp1d(np.linspace(0,1,2),start_xp[idx:idx+2],kind=1) fx=interp1d(np.linspace(0,1,2),start_xp[idx:idx+2],kind=1)
fy=interp1d(np.linspace(0,1,2),start_yp[idx:idx+2],kind=1) fy=interp1d(np.linspace(0,1,2),start_yp[idx:idx+2],kind=1)
final_xp=np.append(final_xp,fx(interp_range)) final_xp=np.append(final_xp,fx(interp_range))
final_yp=np.append(final_yp,fy(interp_range)) final_yp=np.append(final_yp,fy(interp_range))
dx = np.append(0, np.diff(final_xp)) dx = np.append(0, np.diff(final_xp))
dy = np.append(0, np.diff(final_yp)) dy = np.append(0, np.diff(final_yp))
theta = np.arctan2(dy, dx) theta = np.arctan2(dy, dx)
return np.vstack((final_xp,final_yp,theta))

7
notebooks/.ipynb_checkpoints/MPC_cvxpy-checkpoint.ipynb
View File

@ -1158,13 +1158,6 @@
"plt.tight_layout()\n", "plt.tight_layout()\n",
"plt.show()" "plt.show()"
] ]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": []
} }
], ],
"metadata": { "metadata": {

7
notebooks/MPC_cvxpy.ipynb
View File

@ -1158,13 +1158,6 @@
"plt.tight_layout()\n", "plt.tight_layout()\n",
"plt.show()" "plt.show()"
] ]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": []
} }
], ],
"metadata": { "metadata": {