#!/usr/bin/env python
"""
==========================================================
2 - Multiple Sensor Management
==========================================================
"""
# %%
#
# This tutorial follows on from the Single Sensor Management tutorial and further explores how existing
# Stone Soup features can be used to build simple sensor management algorithms for tracking and
# state estimation. This second tutorial demonstrates the limitations of the brute force optimisation
# method introduced in the previous tutorial by increasing the number of sensors used in the scenario.
#
# Introducing multiple sensors
# ----------------------------
# The example in this tutorial considers the same sensor management methods as in Tutorial 1 and applies them to the
# same set of ground truths in order to observe the difference in tracks. The scenario simulates 3
# targets moving on nearly constant velocity trajectories and in this case an adjustable number of sensors.
# The sensors are
# simple radar with a defined field of view which can be pointed in a particular direction in order
# to make an observation.
#
# The first method, using the class :class:`~.RandomSensorManager` chooses a target for each sensor to
# observe randomly with equal probability.
#
# The second method, uses the class :class:`~.BruteForceSensorManager` and aims to reduce the total
# uncertainty of the track estimates at each
# time step. To achieve this the sensor manager considers all possible configurations of directions for the sensors
# to point in. The sensor manager chooses the configuration for which the sum of estimated
# uncertainties (as represented by the Frobenius norm of the covariance matrix) can be reduced the most by observing
# the targets within the chosen sensing configuration.
#
# The introduction of multiple sensors means an increase in the possible combinations of action choices
# that the brute force sensor manager must consider. This brute force optimisation method of looking at every possible
# combination of actions becomes very slow as more sensors are introduced, demonstrating the
# limitations of using this method in more complex scenarios.
#
# As in the first tutorial the OSPA [#]_, SIAP [#]_ and uncertainty metrics are used to assess the performance of the
# sensor managers.
# %%
# Sensor Management example
# -------------------------
#
# Setup
# ^^^^^
#
# First a simulation must be set up using components from Stone Soup. For this the following imports are required.
import numpy as np
import random
from ordered_set import OrderedSet
from datetime import datetime, timedelta
start_time = datetime.now()
from stonesoup.models.transition.linear import CombinedLinearGaussianTransitionModel, ConstantVelocity
from stonesoup.types.groundtruth import GroundTruthPath, GroundTruthState
# %%
# Generate ground truths
# ^^^^^^^^^^^^^^^^^^^^^^
#
# Generate transition model and ground truths as in Tutorial 1.
#
# The number of targets in this simulation is defined by `ntruths` - here there are 3 targets travelling
# in different directions. The time the
# simulation is observed for is defined by `time_max`.
#
# We can fix our random number generator in order to probe a particular example repeatedly. This can be undone by
# commenting out the first two lines in the next cell.
np.random.seed(1990)
random.seed(1990)
# Generate transition model
transition_model = CombinedLinearGaussianTransitionModel([ConstantVelocity(0.005),
ConstantVelocity(0.005)])
yps = range(0, 100, 10) # y value for prior state
truths = OrderedSet()
ntruths = 3 # number of ground truths in simulation
time_max = 50 # timestamps the simulation is observed over
xdirection = 1
ydirection = 1
# Generate ground truths
for j in range(0, ntruths):
truth = GroundTruthPath([GroundTruthState([0, xdirection, yps[j], ydirection], timestamp=start_time)],
id=f"id{j}")
for k in range(1, time_max):
truth.append(
GroundTruthState(transition_model.function(truth[k - 1], noise=True, time_interval=timedelta(seconds=1)),
timestamp=start_time + timedelta(seconds=k)))
truths.add(truth)
xdirection *= -1
if j % 2 == 0:
ydirection *= -1
# %%
# Plot the ground truths. This is done using the :class:`~.Plotterly` class from Stone Soup.
from stonesoup.plotter import Plotterly
# Stonesoup plotter requires sets not lists
plotter = Plotterly()
plotter.plot_ground_truths(truths, [0, 2])
plotter.fig
# %%
# Create sensors
# ^^^^^^^^^^^^^^
# Create a set of sensors for each sensor management algorithm. As in Tutorial 1 this tutorial uses the
# :class:`~.RadarRotatingBearingRange` sensor with the
# number of sensors initially set as 2. Each sensor is positioned along the line :math:`x=10`, at distance
# intervals of 50.
#
# Increasing the number of sensors above 2 significantly increases the run time of the sensor manager due to the
# increase in combinations to be considered by the :class:`~.BruteForceSensorManager`.
# Note that in Tutorial 1 we did not set the resolution for the dwell centre whereas here we are setting it to 30
# degrees. This is because for the brute force algorithm with multiple sensors, using the default resolution of 1
# degree is not practical. These limitations due to combinatorics are discussed further later.
total_no_sensors = 2
from stonesoup.types.state import StateVector
from stonesoup.sensor.radar.radar import RadarRotatingBearingRange
from stonesoup.types.angle import Angle
sensor_setA = set()
for n in range(0, total_no_sensors):
sensor = RadarRotatingBearingRange(
position_mapping=(0, 2),
noise_covar=np.array([[np.radians(0.5) ** 2, 0],
[0, 1 ** 2]]),
ndim_state=4,
position=np.array([[10], [n * 50]]),
rpm=60,
fov_angle=np.radians(30),
dwell_centre=StateVector([0.0]),
max_range=np.inf,
resolutions={'dwell_centre': Angle(np.radians(30))}
)
sensor_setA.add(sensor)
for sensor in sensor_setA:
sensor.timestamp = start_time
sensor_setB = set()
for n in range(0, total_no_sensors):
sensor = RadarRotatingBearingRange(
position_mapping=(0, 2),
noise_covar=np.array([[np.radians(0.5) ** 2, 0],
[0, 1 ** 2]]),
ndim_state=4,
position=np.array([[10], [n * 50]]),
rpm=60,
fov_angle=np.radians(30),
dwell_centre=StateVector([0.0]),
max_range=np.inf,
resolutions={'dwell_centre': Angle(np.radians(30))}
)
sensor_setB.add(sensor)
for sensor in sensor_setB:
sensor.timestamp = start_time
# %%
# Create the Kalman predictor and updater
# ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
#
# Construct a predictor and updater using the :class:`~.KalmanPredictor` and :class:`~.ExtendedKalmanUpdater`
# components from Stone Soup. The measurement model for the updater is `None` as it is an attribute of the sensor.
from stonesoup.predictor.kalman import KalmanPredictor
predictor = KalmanPredictor(transition_model)
from stonesoup.updater.kalman import ExtendedKalmanUpdater
updater = ExtendedKalmanUpdater(measurement_model=None)
# measurement model is added to detections by the sensor
# %%
# Run the Kalman filters
# ^^^^^^^^^^^^^^^^^^^^^^
#
# Create priors which estimate the targets' initial states - these are the same as in the first
# sensor management tutorial.
from stonesoup.types.state import GaussianState
priors = []
xdirection = 1.2
ydirection = 1.2
for j in range(0, ntruths):
priors.append(GaussianState([[0], [xdirection], [yps[j]+0.1], [ydirection]],
np.diag([0.5, 0.5, 0.5, 0.5]+np.random.normal(0,5e-4,4)),
timestamp=start_time))
xdirection *= -1
if j % 2 == 0:
ydirection *= -1
# %%
# Initialise the tracks by creating an empty list and appending the priors generated. This needs to be done
# separately for both sensor manager methods as they will generate different sets of tracks.
from stonesoup.types.track import Track
# Initialise tracks from the RandomSensorManager
tracksA = {Track([prior]) for prior in priors}
# Initialise tracks from the BruteForceSensorManager
tracksB = {Track([prior]) for prior in priors}
# %%
# Create sensor managers
# ^^^^^^^^^^^^^^^^^^^^^^
#
# Next we create our sensor management classes. As in Tutorial 1, two sensor manager classes are used -
# :class:`~.RandomSensorManager` and :class:`~.BruteForceSensorManager`.
#
# Random sensor manager
# """""""""""""""""""""
#
# The first method :class:`~.RandomSensorManager`, randomly chooses the action(s) for the sensors to
# take to make an observation. To do this the
# :meth:`choose_actions` function uses :meth:`random.choice()` to choose a direction for each
# sensor to observe from the possible actions it can take. It returns the chosen configuration of sensors and
# actions to be taken as a mapping.
from stonesoup.sensormanager import RandomSensorManager
# %%
# Brute force sensor manager
# """"""""""""""""""""""""""
#
# The second method :class:`~.BruteForceSensorManager` chooses the configuration of sensors and actions which results
# in the greatest reward as calculated by the reward function.
#
# In this example this is the largest difference between the uncertainty covariances of the target
# predictions and posteriors
# assuming a predicted measurement corresponding to that prediction. This means the sensor manager chooses
# to point the sensors in directions such that the uncertainty will be reduced the most by
# making observations in those directions.
from stonesoup.sensormanager import BruteForceSensorManager
# %%
# Reward function
# """""""""""""""
# The :class:`UncertaintyRewardFunction` calculates the uncertainty reduction by computing the difference between the
# covariance matrix norms of the
# prediction and the posterior assuming a predicted measurement corresponding to that prediction. The sum
# of these differences is returned as a metric for that configuration.
from stonesoup.sensormanager.reward import UncertaintyRewardFunction
# %%
# Initiate sensor managers
# ^^^^^^^^^^^^^^^^^^^^^^^^
#
# Create an instance of each sensor manager class. Both sensor managers take in a `sensor_set`.
# The :class:`~.BruteForceSensorManager` also requires a callable reward function which is initiated here
# from the :class:`UncertaintyRewardFunction`.
randomsensormanager = RandomSensorManager(sensor_setA)
# initiate reward function
reward_function = UncertaintyRewardFunction(predictor, updater)
bruteforcesensormanager = BruteForceSensorManager(sensor_setB,
reward_function=reward_function)
# %%
# Run the sensor managers
# ^^^^^^^^^^^^^^^^^^^^^^^
#
# Both sensor management methods require a timestamp and a list of tracks at each time step when calling
# the function :meth:`choose_actions`. This returns a mapping of sensors and actions to be taken by each
# sensor, decided by the sensor managers.
#
# For both sensor management methods, at each time step the chosen action is given to the sensors and then
# measurements taken. The tracks are updated based on these measurements with predictions made for tracks
# which have not been observed.
#
# First a hypothesiser and data associator are required for use in both trackers.
from stonesoup.hypothesiser.distance import DistanceHypothesiser
from stonesoup.measures import Mahalanobis
hypothesiser = DistanceHypothesiser(predictor, updater, measure=Mahalanobis(), missed_distance=5)
from stonesoup.dataassociator.neighbour import GNNWith2DAssignment
data_associator = GNNWith2DAssignment(hypothesiser)
# %%
# Run random sensor manager
# """""""""""""""""""""""""
#
# Here the chosen target for observation is selected randomly using the method :meth:`choose_actions()` from the class
# :class:`~.RandomSensorManager`. This returns a mapping of sensors to actions where actions are a
# :class:`~.ChangeDwellAction`, selected at random.
from ordered_set import OrderedSet
# Generate list of timesteps from ground truth timestamps
timesteps = []
for state in truths[0]:
timesteps.append(state.timestamp)
for timestep in timesteps[1:]:
# Generate chosen configuration
chosen_actions = randomsensormanager.choose_actions(tracksA, timestep)
# Create empty dictionary for measurements
measurementsA = set()
for chosen_action in chosen_actions:
for sensor, actions in chosen_action.items():
sensor.add_actions(actions)
for sensor in sensor_setA:
sensor.act(timestep)
# Observe this ground truth
measurementsA |= sensor.measure(OrderedSet(truth[timestep] for truth in truths), noise=True)
hypotheses = data_associator.associate(tracksA,
measurementsA,
timestep)
for track in tracksA:
hypothesis = hypotheses[track]
if hypothesis.measurement:
post = updater.update(hypothesis)
track.append(post)
else: # When data associator says no detections are good enough, we'll keep the prediction
track.append(hypothesis.prediction)
# %%
# Plot ground truths, tracks and uncertainty ellipses for each target. The positions of the sensors are indicated
# by black x markers.
plotterA = Plotterly()
plotterA.plot_sensors(sensor_setA)
plotterA.plot_ground_truths(truths, [0, 2])
plotterA.plot_tracks(tracksA, [0, 2], uncertainty=True)
plotterA.fig
# %%
# In comparison to Tutorial 1 the performance of the :class:`~.RandomSensorManager` has improved. This is
# because a greater number of sensors means each target is more likely to be observed. This means the uncertainty
# of the track does not increase as much because the targets are observed more often.
# %%
# Run brute force sensor manager
# """"""""""""""""""""""""""""""
#
# Here the direction for observation is selected based on the difference between the covariance matrices of the
# prediction and posterior, for targets which could be observed by the sensor
# pointing in the given direction.
#
# Within the sensor manager a dictionary is created of sensors and all the possible actions they can take.
# When the :meth:`choose_actions` function is called (at each time step), for each track in the tracks list:
#
# * A prediction is made for each track and the covariance matrix norms stored
# * For each possible action a sensor could take, a synthetic detection is made using this sensor configuration
# * A hypothesis is generated based on the stored prediction and synthetic detection
# * This hypothesis is used to do an update and the covariance matrix norms of the update are stored
# * The difference between the covariance matrix norms of the update and the prediction is calculated
#
# The overall reward is calculated as the sum of the differences between these covariance matrix norms
# for the tracks observed by the possible action configuration. The sensor manager identifies the
# configuration which results in the largest value of this reward and therefore
# largest reduction in uncertainty. It returns the optimum sensors/actions configuration as a dictionary.
#
# The actions are given to the sensors, measurements made and
# the tracks updated based on these measurements. Predictions are made for tracks
# which have not been observed by the sensors.
for timestep in timesteps[1:]:
# Generate chosen configuration
chosen_actions = bruteforcesensormanager.choose_actions(tracksB, timestep)
# Create empty dictionary for measurements
measurementsB = set()
for chosen_action in chosen_actions:
for sensor, actions in chosen_action.items():
sensor.add_actions(actions)
for sensor in sensor_setB:
sensor.act(timestep)
# Observe this ground truth
measurementsB |= sensor.measure(OrderedSet(truth[timestep] for truth in truths), noise=True)
hypotheses = data_associator.associate(tracksB,
measurementsB,
timestep)
for track in tracksB:
hypothesis = hypotheses[track]
if hypothesis.measurement:
post = updater.update(hypothesis)
track.append(post)
else: # When data associator says no detections are good enough, we'll keep the prediction
track.append(hypothesis.prediction)
# %%
# Plot ground truths, tracks and uncertainty ellipses for each target.
plotterB = Plotterly()
plotterB.plot_sensors(sensor_setB)
plotterB.plot_ground_truths(truths, [0, 2])
plotterB.plot_tracks(tracksB, [0, 2], uncertainty=True)
plotterB.fig
# %%
# The smaller uncertainty ellipses in this plot suggest that the :class:`~.BruteForceSensorManager` provides a much
# better track than the :class:`~.RandomSensorManager`.
# The tracking is also improved from Tutorial 1 due to the increased number of sensors.
# %%
# Combinatorics
# -------------
#
# The following graph demonstrates how the number of possible sensor-action configurations increases with the number
# of sensors and number of actions. The number of configurations which are considered by the sensor manager
# for :math:`M` actions and :math:`N` sensors is :math:`M^N`.
#
# With a resolution of 1 degree there are 360 possible dwell centres for a sensor so the number of
# possible configurations should be :math:`360^N`
# where :math:`N` is the number of sensors. This exponential increase means that as the number of
# sensors increase, the run time of the sensor manager slows down significantly because there are so many more
# iterations to consider.
#
# In this scenario :math:`N=2` so with a resolution of 1 degree that would 129,600
# actions to consider.
# Even with a resolution of 30 degrees, changing the number of sensors to :math:`N\geq 2` leads to a
# much longer run time.
# This highlights a practical limitation of using this brute force optimisation method for multiple
# sensors. In this example we have dealt with this by introducing a run time limit to the brute force sensor manager.
import matplotlib.pyplot as plt
nsensors = np.arange(1, 10)
nactions = np.arange(1, 360.0)
nsensors, nactions = np.meshgrid(nsensors, nactions)
ncombinations = nactions**nsensors
fig = plt.figure()
ax = fig.add_subplot(projection="3d")
ax.plot_surface(nsensors, nactions, np.log10(ncombinations), cmap='coolwarm')
ax.set_xlabel("No. sensors")
ax.set_ylabel("No. actions")
ax.set_zlabel("log of no. combinations")
# %%
# Metrics
# -------
#
# Metrics can be used to compare how well different sensor management techniques are working.
# As in Tutorial 1 the metrics used here are the OSPA, SIAP and uncertainty metrics.
from stonesoup.metricgenerator.ospametric import OSPAMetric
ospa_generator = OSPAMetric(c=40, p=1)
from stonesoup.metricgenerator.tracktotruthmetrics import SIAPMetrics
from stonesoup.measures import Euclidean
siap_generator = SIAPMetrics(position_measure=Euclidean((0, 2)),
velocity_measure=Euclidean((1, 3)))
from stonesoup.dataassociator.tracktotrack import TrackToTruth
associator = TrackToTruth(association_threshold=30)
from stonesoup.metricgenerator.uncertaintymetric import SumofCovarianceNormsMetric
uncertainty_generator = SumofCovarianceNormsMetric()
# %%
# Generate a metrics manager for each sensor management method.
from stonesoup.metricgenerator.manager import SimpleManager
metric_managerA = SimpleManager([ospa_generator, siap_generator, uncertainty_generator],
associator=associator)
metric_managerB = SimpleManager([ospa_generator, siap_generator, uncertainty_generator],
associator=associator)
# %%
# For each time step, data is added to the metric manager on truths and tracks.
# The metrics themselves can then be generated from the metric manager.
metric_managerA.add_data(truths, tracksA)
metric_managerB.add_data(truths, tracksB)
metricsA = metric_managerA.generate_metrics()
metricsB = metric_managerB.generate_metrics()
# %%
# OSPA metric
# ^^^^^^^^^^^
#
# First we look at the OSPA metric. This is plotted over time for each sensor manager method.
ospa_metricA = metricsA['OSPA distances']
ospa_metricB = metricsB['OSPA distances']
fig = plt.figure()
ax = fig.add_subplot(1, 1, 1)
ax.plot([i.timestamp for i in ospa_metricA.value],
[i.value for i in ospa_metricA.value],
label='RandomSensorManager')
ax.plot([i.timestamp for i in ospa_metricB.value],
[i.value for i in ospa_metricB.value],
label='BruteForceSensorManager')
ax.set_ylabel("OSPA distance")
ax.set_xlabel("Time")
ax.legend()
# %%
# The OSPA distance for the :class:`~.BruteForceSensorManager` is generally smaller than for the random
# observations of the :class:`~.RandomSensorManager`.
#
# SIAP metrics
# ^^^^^^^^^^^^
#
# Next we look at SIAP metrics. We are only interested in the positional accuracy (PA) and
# velocity accuracy (VA). These metrics can be plotted to show how they change over time.
fig, axes = plt.subplots(2)
times = metric_managerA.list_timestamps()
pa_metricA = metricsA['SIAP Position Accuracy at times']
va_metricA = metricsA['SIAP Velocity Accuracy at times']
pa_metricB = metricsB['SIAP Position Accuracy at times']
va_metricB = metricsB['SIAP Velocity Accuracy at times']
axes[0].set(title='Positional Accuracy', xlabel='Time', ylabel='PA')
axes[0].plot(times, [metric.value for metric in pa_metricA.value],
label='RandomSensorManager')
axes[0].plot(times, [metric.value for metric in pa_metricB.value],
label='BruteForceSensorManager')
axes[0].legend()
axes[1].set(title='Velocity Accuracy', xlabel='Time', ylabel='VA')
axes[1].plot(times, [metric.value for metric in va_metricA.value],
label='RandomSensorManager')
axes[1].plot(times, [metric.value for metric in va_metricB.value],
label='BruteForceSensorManager')
axes[1].legend()
# %%
# Similar to the OSPA distance the :class:`~.BruteForceSensorManager` generally results in both a better
# positional accuracy and velocity accuracy than the random observations of the :class:`~.RandomSensorManager`.
#
# Uncertainty metric
# ^^^^^^^^^^^^^^^^^^
#
# Finally we look at the uncertainty metric which computes the sum of covariance matrix norms of each state at each
# time step. This is plotted over time for each sensor manager method.
uncertainty_metricA = metricsA['Sum of Covariance Norms Metric']
uncertainty_metricB = metricsB['Sum of Covariance Norms Metric']
fig = plt.figure()
ax = fig.add_subplot(1, 1, 1)
ax.plot([i.timestamp for i in uncertainty_metricA.value],
[i.value for i in uncertainty_metricA.value],
label='RandomSensorManager')
ax.plot([i.timestamp for i in uncertainty_metricB.value],
[i.value for i in uncertainty_metricB.value],
label='BruteForceSensorManager')
ax.set_ylabel("Sum of covariance matrix norms")
ax.set_xlabel("Time")
ax.legend()
# sphinx_gallery_thumbnail_number = 7
# %%
# This metric shows that the uncertainty in the tracks generated by the :class:`~.RandomSensorManager` is
# generally greater than for those generated by the :class:`~.BruteForceSensorManager`.
# This is also reflected by the uncertainty ellipses
# in the initial plots of tracks and truths.
# %%
# References
# ----------
#
# .. [#] *D. Schuhmacher, B. Vo and B. Vo*, **A Consistent Metric for Performance Evaluation of
# Multi-Object Filters**, IEEE Trans. Signal Processing 2008
# .. [#] *Votruba, Paul & Nisley, Rich & Rothrock, Ron and Zombro, Brett.*, **Single Integrated Air
# Picture (SIAP) Metrics Implementation**, 2001