Note

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# 1 - Single Sensor Management

This tutorial introduces the sensor manager classes in Stone Soup which can be used to build simple sensor management algorithms for tracking and state estimation. The intention is to further build on these base classes to develop more complex sensor management algorithms.

## Background

Sensor management is the process of deciding and executing the actions that a sensor, or group of sensors will take in a specific scenario and with a particular objective, or objectives in mind. The process involves using information about the scenario to determine an appropriate action for the sensing system to take. An observation of the state of the system is then made using the sensing configuration decided by the sensor manager. The observations are used to update the estimate of the collective states and this update is used (if necessary) to determine the next action for the sensing system to take.

A simple example can be imagined using a sensor with a limited field of view which must decide which direction it should point in at each time step. Alternatively, we might construct an objective based example by imagining that the desired target is fast moving and the sensor can only observe one target at a time. If there are multiple targets which could be observed the sensor manager could choose to observe the target that had the greatest estimated velocity at the current time.

The example in this notebook considers two simple sensor management methods and applies them to the same ground truths in order to quantify the difference in behaviour. The scenario simulates 3 targets moving on nearly constant velocity trajectories and a radar with a specified field of view, which can be pointed in a particular direction.

The first method, using the class `RandomSensorManager`

chooses a direction to point in randomly
with equal probability. The
second method, using the class `BruteForceSensorManager`

considers every possible direction the
sensor could point in and uses a
reward function to determine the best choice of action.
In this example the reward function aims to reduce the total uncertainty of the track estimates at each time step.
To achieve this the sensor manager chooses to look in the direction which results in the greatest reduction in
uncertainty - as represented by
the Frobenius norm of the covariance matrix.

### Sensor management as a POMDP

Sensor management problems can be considered as Partially Observable Markov Decision Processes (POMDPs) where observations provide information about the current state of the system but there is uncertainty in the estimate of the underlying state due to noisy sensors and imprecise models of target evaluation.

- POMDPs consist of:
\(X_k\), the finite set of possible states for each stage index \(k\).

\(A_k\), the finite set of possible actions for each stage index \(k\).

\(R_k(x, a)\), the reward function.

\(Z_k\), the finite set of possible observations for each stage index \(k\).

\(f_k(x_{k}|x_{k-1})\), a (set of) state transition function(s). (Note that actions are excluded from the function at the moment. It may be necessary to include them if prior sensor actions cause the targets to modify their behaviour.)

\(h_k(z_k | x_k, a_k)\), a (set of) observation function(s).

\(\{x\}_k\), the set of states at \(k\) to be estimated.

\(\{a\}_k\), a set of actions at \(k\) to be chosen.

\(\{z\}_k\), the observations at \(k\) returned by the sensor.

\(\Psi_{k-1}\), denotes the complete set of ‘intelligence’ available to the sensor manager before deciding on an action at \(k\). This includes the prior set of state estimates \(\{x\}_{k-1}\), but may also encompass contextual information, sensor constraints or mission parameters.

Figure 1: Illustration of sequential actions and measurements. [1]

\(\Psi_k\) is the intelligence available to the sensor manager at stage index \(k\), to help select the action \(a_k\) for the system to take. An observation \(z_k\) is made by the sensing system, giving information on the state \(x_k\). The action \(a_k\) and observation \(z_k\) are added to the intelligence set to generate \(\Psi_{k+1}\), the intelligence available at stage index \(k+1\).

### Comparing sensor management methods using metrics

The performance of the two sensor management methods explored in this tutorial can be assessed using metrics available from the Stone Soup framework. The metrics used to assess the performance of the different methods are the OPSA metric [2], SIAP metrics [3] and an uncertainty metric. Demonstration of the OSPA and SIAP metrics can be found in the Metrics Example.

The uncertainty metric computes the covariance matrices of all target states at each time step and calculates the sum of their norms. This gives a measure of the total uncertainty across all tracks at each time step.

## 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().replace(microsecond=0)
from stonesoup.models.transition.linear import CombinedLinearGaussianTransitionModel, ConstantVelocity
from stonesoup.types.groundtruth import GroundTruthPath, GroundTruthState
```

### Generate ground truths

Following the methods from previous Stone Soup tutorials we generate a series of combined linear Gaussian transition models and generate ground truths. Each ground truth is offset in the y-direction by 10.

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
# i.e. fk(xk|xk-1)
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
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):
truth.append(
GroundTruthState(transition_model.function(truth[k - 1], noise=True, time_interval=timedelta(seconds=1)),
timestamp=timesteps[k]))
truths.add(truth)
# alternate directions when initiating tracks
xdirection *= -1
if j % 2 == 0:
ydirection *= -1
```

Plot the ground truths. This is done using the `Plotterly`

class from Stone Soup.

```
from stonesoup.plotter import AnimatedPlotterly
plotter = AnimatedPlotterly(timesteps, tail_length=1)
plotter.plot_ground_truths(truths, [0, 2])
plotter.fig
```

### Create sensors

Create a sensor for each sensor management algorithm. This tutorial uses the
`RadarRotatingBearingRange`

sensor. This sensor is an `Actionable`

so
is capable of returning the actions it can possibly
take at a given time step and can also be given an action to take before taking
measurements.
See the Creating an Actionable Sensor Example for a more detailed explanation of actionable sensors.

The `RadarRotatingBearingRange`

has a dwell centre which is an `ActionableProperty`

so in this case the action is changing the dwell centre to point in a specific direction.

```
from stonesoup.types.state import StateVector
from stonesoup.sensor.radar.radar import RadarRotatingBearingRange
sensorA = 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], [0]]),
rpm=60,
fov_angle=np.radians(30),
dwell_centre=StateVector([0.0]),
max_range=np.inf
)
sensorA.timestamp = start_time
sensorB = 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], [0]]),
rpm=60,
fov_angle=np.radians(30),
dwell_centre=StateVector([0.0]),
max_range=np.inf
)
sensorB.timestamp = start_time
```

### Create the Kalman predictor and updater

Construct a predictor and updater using the `KalmanPredictor`

and `ExtendedKalmanUpdater`

components from Stone Soup. The `ExtendedKalmanUpdater`

is used because it can be used for both linear
and nonlinear measurement models.

```
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

First create ntruths priors which estimate the targets’ initial states, one for each target. In this example each prior is offset by 0.5 in the y direction meaning the position of the track is initially not very accurate. The velocity is also systematically offset by +0.5 in both the x and y directions.

```
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.

### Create sensor managers

Next we create our sensor manager classes. Two sensor manager classes are used in this tutorial
- `RandomSensorManager`

and `BruteForceSensorManager`

.

#### Random sensor manager

The first method `RandomSensorManager`

, randomly chooses the action(s) for the sensor to take
to make an observation. To do this the `choose_actions()`

function uses `random.sample()`

to draw a random sample from all possible directions the sensor could point in
at each time step.

```
from stonesoup.sensormanager import RandomSensorManager
```

#### Brute force sensor manager

The second method `BruteForceSensorManager`

iterates through every possible action a sensor can take at a
given time step and selects the action(s) which give the maximum reward as calculated by the reward function.
In this example the reward function is used to select a direction for the sensor to point in
such that the total uncertainty of the tracks will be
reduced the most by making an observation in that direction.

```
from stonesoup.sensormanager import BruteForceSensorManager
```

#### Reward function

A reward function is used to quantify the benefit of sensors taking a particular action or set of actions. This can be crafted specifically for an example in order to achieve a particular objective. The function used in this example is quite generic but could be substituted for any callable function which returns a numeric value that the sensor manager can maximise.

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.

```
from stonesoup.sensormanager.reward import UncertaintyRewardFunction
```

### Initiate the sensor managers

Create an instance of each sensor manager class. Each class takes in a sensor_set, for this example
it is a set of one sensor.
The `BruteForceSensorManager`

also requires a callable reward function which here is the
`UncertaintyRewardFunction`

.

```
randomsensormanager = RandomSensorManager({sensorA})
# initiate reward function
reward_function = UncertaintyRewardFunction(predictor, updater)
bruteforcesensormanager = BruteForceSensorManager({sensorB},
reward_function=reward_function)
```

### Run the sensor managers

For both methods the `choose_actions()`

function requires a time step and a tracks list as inputs.

For both sensor management methods, the chosen actions are added to the sensor and measurements made. Tracks which have been observed by the sensor are updated and those that haven’t are predicted forward. These states are appended to the tracks list.

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`

.

```
for timestep in timesteps[1:]:
# Generate chosen configuration
# i.e. {a}k
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)
sensorA.act(timestep)
# Observe this ground truth
# i.e. {z}k
measurementsA |= sensorA.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.

```
plotterA = AnimatedPlotterly(timesteps, tail_length=1)
plotterA.plot_sensors(sensorA)
plotterA.plot_ground_truths(truths, [0, 2])
plotterA.plot_tracks(tracksA, [0, 2], uncertainty=True, plot_history=False)
plotterA.fig
```