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 RandomSensorManager chooses a target for each sensor to observe randomly with equal probability.

The second method, uses the 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 [1], SIAP [2] and uncertainty metrics are used to assess the performance of the sensor managers.

Sensor Management example


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().replace(microsecond=0)

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.


# Generate transition model
transition_model = CombinedLinearGaussianTransitionModel([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
timesteps = [start_time + timedelta(seconds=k) for k in range(time_max)]

xdirection = 1
ydirection = 1

# Generate ground truths
for j in range(0, ntruths):
    truth = GroundTruthPath([GroundTruthState([0, xdirection, yps[j], ydirection],
                                              timestamp=timesteps[0])], id=f"id{j}")

    for k in range(1, time_max):
            GroundTruthState(transition_model.function(truth[k - 1], noise=True, time_interval=timedelta(seconds=1)),

    xdirection *= -1
    if j % 2 == 0:
        ydirection *= -1

Plot the ground truths. This is done using the AnimatedPlotterly class from Stone Soup.

from stonesoup.plotter import AnimatedPlotterly

# Stonesoup plotter requires sets not lists

plotter = AnimatedPlotterly(timesteps, tail_length=1)
plotter.plot_ground_truths(truths, [0, 2])

Create sensors

Create a set of sensors for each sensor management algorithm. As in Tutorial 1 this tutorial uses the RadarRotatingBearingRange sensor with the number of sensors initially set as 2. Each sensor is positioned along the line \(x=10\), 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 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]]),
        position=np.array([[10], [n * 50]]),
        resolutions={'dwell_centre': Angle(np.radians(30))}
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]]),
        position=np.array([[10], [n * 50]]),
        resolutions={'dwell_centre': Angle(np.radians(30))}

for sensor in sensor_setB:
    sensor.timestamp = start_time

Create the Kalman predictor and updater

Construct a predictor and updater using the KalmanPredictor and 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)),
    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 - RandomSensorManager and BruteForceSensorManager.

Random sensor manager

The first method RandomSensorManager, randomly chooses the action(s) for the sensors to take to make an observation. To do this the choose_actions() function uses 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 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 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 BruteForceSensorManager also requires a callable reward function which is initiated here from the UncertaintyRewardFunction.

Run the sensor managers

Both sensor management methods require a timestamp and a list of tracks at each time step when calling the function 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 choose_actions() from the class RandomSensorManager. This returns a mapping of sensors to actions where actions are a ChangeDwellAction, selected at random.

from ordered_set import OrderedSet
from collections import defaultdict
import copy

sensor_history_A = defaultdict(dict)
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():

    for sensor in sensor_setA:
        sensor_history_A[timestep][sensor] = copy.copy(sensor)

        # Observe this ground truth
        measurementsA |= sensor.measure(OrderedSet(truth[timestep] for truth in truths), noise=True)

    hypotheses = data_associator.associate(tracksA,
    for track in tracksA:
        hypothesis = hypotheses[track]
        if hypothesis.measurement:
            post = updater.update(hypothesis)
        else:  # When data associator says no detections are good enough, we'll keep the prediction

Plot ground truths, tracks and uncertainty ellipses for each target. The positions of the sensors are indicated by black x markers. This uses the Stone Soup AnimatedPlotterly, with added code to plot the field of view of the sensor.

import plotly.graph_objects as go
from stonesoup.functions import pol2cart

plotterA = AnimatedPlotterly(timesteps, tail_length=1, sim_duration=10)
plotterA.plot_ground_truths(truths, [0, 2])
plotterA.plot_tracks(tracksA, [0, 2], uncertainty=True, plot_history=False)

def plot_sensor_fov(fig, sensor_set, sensor_history):
    # Plot sensor field of view
    trace_base = len(fig.data)
    for _ in sensor_set:

    for frame in fig.frames:
        traces_ = list(frame.traces)
        data_ = list(frame.data)

        timestring = frame.name
        timestamp = datetime.strptime(timestring, "%Y-%m-%d %H:%M:%S")

        for n, sensor_ in enumerate(sensor_set):
            x = [0, 0]
            y = [0, 0]

            if timestamp in sensor_history:
                sensor = sensor_history[timestamp][sensor_]
                for i, fov_side in enumerate((-1, 1)):
                    range_ = min(getattr(sensor, 'max_range', np.inf), 100)
                    x[i], y[i] = pol2cart(range_,
                                          sensor.dwell_centre[0, 0]
                                          + sensor.fov_angle / 2 * fov_side) \
                                 + sensor.position[[0, 1], 0]

            data_.append(go.Scatter(x=[x[0], sensor.position[0], x[1]],
                                    y=[y[0], sensor.position[1], y[1]],
            traces_.append(trace_base + n)

        frame.traces = traces_
        frame.data = data_

plot_sensor_fov(plotterA.fig, sensor_setA, sensor_history_A)