Reinforcement Learning Sensor Manager

This example looks at how to interface a reinforcement learning framework with a Stone Soup sensor manager.

Making a Reinforcement Learning Sensor Manager

This example introduces using a Deep Q Network (DQN) reinforcement learning (RL) sensor management algorithm in Stone Soup. This is compared to the performance of a brute force algorithm using the same metrics shown in the sensor management tutorials. This example is similar to the sensor management tutorials, simulating 3 targets and a RadarRotatingBearingRange sensor which can be actioned to point in different directions.

Tensorflow-agents is used as the reinforcement learning framework. This is a separate python package that can be found at


This currently only works on Linux based OSes, or via Windows Subsystem for Linux (WSL). MacOS users may be able to make use of a Linux VM to run this example. See Tensorflow instructions for creating Python virtual environments (with GPU support if applicable) [1].

To run this example, in a clean environment, do pip install stonesoup, followed by pip install tf-agents[reverb].

# Some general imports and set up
import numpy as np
import random
from datetime import datetime, timedelta

    import reverb
except ImportError:
    raise ImportError('To run this example, reverb must be installed. Please read the warning'
                      'and instructions at the top of this notebook.')

start_time =

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 sensor management tutorials, 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. To produce random examples, comment out the next two lines.


# Generate transition model
# i.e. fk(xk|xk-1)
transition_model = CombinedLinearGaussianTransitionModel([ConstantVelocity(0.005),

yps = range(0, 100, 10)  # y value for prior state
truths = []
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],

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

    # 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])

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 take at a given time step and can also be given an action to take before measuring. 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]]),
    position=np.array([[10], [0]]),
sensorA.timestamp = start_time

sensorB = RadarRotatingBearingRange(
    position_mapping=(0, 2),
    noise_covar=np.array([[np.radians(0.5) ** 2, 0],
                          [0, 1 ** 2]]),
    position=np.array([[10], [0]]),
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. A hypothesiser and data associator are required for use in both trackers.

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

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)

Generate Priors

First create ntruths priors which estimate the targets’ initial states, one for each target. In this example each prior is offset by 0.1 in the y direction meaning the position of the track is initially not very accurate. The velocity is also systematically offset by +0.2 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)),
    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 = []
for j, prior in enumerate(priors):

tracksB = []
for j, prior in enumerate(priors):

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

reward_function = UncertaintyRewardFunction(predictor=predictor, updater=updater)

Reinforcement Learning

Reinforcement learning involves intelligent agents making decisions to maximise a cumulative reward. The agent must train in an environment in order to create a policy, which later determines the actions it will take. During training, the agent makes decisions and receives rewards, which it uses to optimise the policy.

Illustration of sequential actions and measurements

Illustration of an RL algorithm taking actions during training. The state and reward it receives are used to determine the best actions.

Once training has completed, the policy can be exploited to gain rewards.

Design Environment

An environment is needed for the RL agent to learn in. There are resources online for how to design these [2].

In this example, the action space is equal to the number of targets in the simulation, so at each time step, the sensor can select one target to look at. For the environment, we make a copy of the sensor that we will pass to the sensor manager later on. This is so the agent can train in the environment without altering the sensor itself. The UncertaintyRewardFunction is used to calculate the reward obtained for each step in the environment. The trace of the covariances for each object is used as the observation for the agent to learn from - it should learn to select targets with a larger covariance (higher uncertainty).

from abc import ABC
import numpy as np
import copy
from ordered_set import OrderedSet

from stonesoup.sensor.action.dwell_action import DwellActionsGenerator
from stonesoup.functions import mod_bearing

from tf_agents.environments import py_environment
from tf_agents.specs import array_spec
from tf_agents.trajectories import time_step as ts
from tf_agents.environments import utils

class StoneSoupEnv(py_environment.PyEnvironment, ABC):
    """Example reinforcement learning environment. Environments must contain __init__, _reset,
    _step, and generate_action methods

    def __init__(self):
        # Action size is number of targets
        self._action_spec = array_spec.BoundedArraySpec(
            shape=(), dtype=np.int32, minimum=0, maximum=ntruths - 1, name='action')
        # Observation size is also number of targets
        self.obs_size = ntruths
        self._observation_spec = array_spec.BoundedArraySpec(
            shape=(self.obs_size,), dtype=np.float32, name='observation')
        self._episode_ended = False
        self.max_episode_length = time_max
        self.current_step = 0
        self.start_time = start_time
        # Use deepcopy to prevent the original sensor/tracks being changed each time an episode is run
        self.sensor = copy.deepcopy(sensorA)
        self.sensor.timestamp = start_time
        self.tracks = copy.deepcopy(tracksA)

    def action_spec(self):
        """Return action_spec."""
        return self._action_spec

    def observation_spec(self):
        """Return observation_spec."""
        return self._observation_spec

    def _reset(self):
        """Restarts the environment from the first step, resets the initial state
        and observation values, and returns an initial observation
        self._episode_ended = False
        self.current_step = 0
        self.sensor = copy.deepcopy(sensorA)
        self.sensor.timestamp = start_time
        self.tracks = copy.deepcopy(tracksA)
        return ts.restart(np.zeros((self.obs_size,), dtype=np.float32))

    def _step(self, action):
        """Apply action and take one step through environment, and return new time_step.

        reward = 0
        if self._episode_ended:
            # The last action ended the episode. Ignore the current action and start
            # a new episode.
            return self.reset()

        uncertainty = []
        for i, target in enumerate(self.tracks):
            # Calculate the bearing of the chosen target from the sensor
            if i == action:
                x_target = target.state.state_vector[0] - self.sensor.position[0]
                y_target = target.state.state_vector[2] - self.sensor.position[1]
                bearing_target = mod_bearing(np.arctan2(y_target, x_target))


        current_timestep = self.start_time + timedelta(seconds=self.current_step)
        next_timestep = self.start_time + timedelta(seconds=self.current_step + 1)

        # Create action generator which contains possible actions
        action_generator = DwellActionsGenerator(self.sensor,

        # Action the environment's sensor to point towards the chosen target
        current_action = [action_generator.action_from_value(bearing_target)]
        config = ({self.sensor: current_action})
        reward += reward_function(config, self.tracks, next_timestep)


        # Calculate a measurement from the sensor
        measurement = set()
        measurement |= self.sensor.measure(OrderedSet(truth[current_timestep] for truth in truths), noise=True)
        hypotheses = data_associator.associate(self.tracks,

        for track in self.tracks:
            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

        # Set the observation as the prior uncertainty of each target
        observation = np.array(uncertainty, dtype=np.float32)

        self.current_step += 1

        if self.current_step >= self.max_episode_length - 1:
            self._episode_ended = True
            return ts.termination(observation, reward)
            return ts.transition(observation, reward=reward, discount=1.0)

    def generate_action(action, tracks, sensor):
        """This method is used to convert a tf-agents action into a Stone Soup action"""
        for i, target in enumerate(tracks):
            if i == action:
                x_target = target.state.state_vector[0] - sensor.position[0]
                y_target = target.state.state_vector[2] - sensor.position[1]
                action_bearing = mod_bearing(np.arctan2(y_target, x_target))

        action_generators = DwellActionsGenerator(sensor,
                                                  end_time=sensor.timestamp + timedelta(seconds=1))

        current_action = [action_generators.action_from_value(action_bearing)]
        return current_action

# Validate the environment to ensure that the environment returns the expected specs
train_env = StoneSoupEnv()
utils.validate_py_environment(train_env, episodes=5)

RL Sensor Manager

To be able to use the RL environment we have designed, we need to make a ReinforcementSensorManager class, which inherits from SensorManager.

We introduce some additional methods that are used by tensorflow-agents: compute_avg_return(), dense_layer(), and train(). compute_avg_return() is used to find the average reward by using a given policy. This is used to evaluate the training. dense_layer() is used when generating the Q-Network, a neural network model that learns to predict Q-Values. train() is used to generate the policy by running a large number of episodes through the Q-Network to work out which actions are best. An episode in RL refers to a single run or instance of the learning process, where the agent interacts with the environment.

We also need to re-define the choose_actions() method from SensorManager to be able to interface Stone Soup actions with tensorflow-agent actions.

from stonesoup.sensormanager.base import SensorManager
from stonesoup.base import Property
from tf_agents.environments import tf_py_environment

class ReinforcementSensorManager(SensorManager):
    """A sensor manager that employs reinforcement learning algorithms from tensorflow-agents.
    The sensor manager trains on an environment to find an optimal policy, which is then exploited
    to choose actions.
    env: py_environment.PyEnvironment = Property(doc="The environment which the agent learns the policy with.")

    def __init__(self, *args, **kwargs):
        super().__init__(*args, **kwargs)
        self.tf_env = tf_py_environment.TFPyEnvironment(self.env)
        self.test_env = tf_py_environment.TFPyEnvironment(self.env)
        self.agent = None

    def compute_avg_return(environment, policy, num_episodes=10):
        """Used to calculate the average reward over a set of episodes.

            tf-agents environment for evaluating policy on

            tf-agents policy for choosing actions in environment

        num_episodes: int
            Number of episodes to sample over

        : int
            average reward calculated over num_episodes

        time_step = None
        episode_return = None
        total_return = 0.0
        for _ in range(num_episodes):
            time_step = environment.reset()
            episode_return = 0.0

            while not time_step.is_last():
                action_step = policy.action(time_step)
                time_step = environment.step(action_step.action)
                episode_return += time_step.reward
            total_return += episode_return

        avg_return = total_return / num_episodes
        return avg_return.numpy()[0]

    def dense_layer(num_units):
        """Method for generating fully connected layers for use in the neural network.

        num_units: int
            Number of nodes in dense layer

        : tensorflow dense layer

        # Define a helper function to create Dense layers configured with the right
        # activation and kernel initializer.
        return tf.keras.layers.Dense(
                scale=2.0, mode='fan_in', distribution='truncated_normal'))

    def train(self, hyper_parameters):
        """Trains a DQN agent on the specified environment to learn a policy that is later
        used to select actions.

        hyper_parameters: dict
            Dictionary containing hyperparameters used in training. See tutorial for
            necessary hyperparameters.

        if self.env is not None:

            train_py_env = self.env
            eval_py_env = self.env
            self.train_env = tf_py_environment.TFPyEnvironment(train_py_env)
            self.eval_env = tf_py_environment.TFPyEnvironment(eval_py_env)

            fc_layer_params = hyper_parameters['fc_layer_params']
            action_tensor_spec = tensor_spec.from_spec(self.env.action_spec())
            num_actions = action_tensor_spec.maximum - action_tensor_spec.minimum + 1

            # QNetwork consists of a sequence of Dense layers followed by a dense layer
            # with `num_actions` units to generate one q_value per available action as
            # its output.

            dense_layers = [self.dense_layer(num_units) for num_units in fc_layer_params]
            q_values_layer = tf.keras.layers.Dense(
                    minval=-0.03, maxval=0.03),
            q_net = sequential.Sequential(dense_layers + [q_values_layer])

            optimizer = tf.keras.optimizers.Adam(hyper_parameters['learning_rate'])

            train_step_counter = tf.Variable(0)

            self.agent = dqn_agent.DdqnAgent(


            random_policy = random_tf_policy.RandomTFPolicy(self.train_env.time_step_spec(),

            # See also the metrics module for standard implementations of different metrics.

            self.compute_avg_return(self.eval_env, random_policy,

            table_name = 'uniform_table'
            replay_buffer_signature = tensor_spec.from_spec(
            replay_buffer_signature = tensor_spec.add_outer_dim(

            table = reverb.Table(

            reverb_server = reverb.Server([table])

            replay_buffer = reverb_replay_buffer.ReverbReplayBuffer(

            rb_observer = reverb_utils.ReverbAddTrajectoryObserver(

                    random_policy, use_tf_function=True),

            # Dataset generates trajectories with shape [Bx2x...]
            dataset = replay_buffer.as_dataset(

            iterator = iter(dataset)

            # (Optional) Optimize by wrapping some code in a graph using TF function.
            self.agent.train = common.function(self.agent.train)

            # Reset the train step.

            # Evaluate the agent's policy once before training.
            avg_return = self.compute_avg_return(self.eval_env, self.agent.policy,
            returns = [avg_return]

            # Reset the environment.
            time_step = train_py_env.reset()

            # Create a driver to collect experience.
            collect_driver = py_driver.PyDriver(
                    self.agent.collect_policy, use_tf_function=True),

            for _ in range(hyper_parameters['num_iterations']):
                # Collect a few steps and save to the replay buffer.
                time_step, _ =

                # Sample a batch of data from the buffer and update the agent's network.
                experience, unused_info = next(iterator)
                train_loss = self.agent.train(experience).loss

                step = self.agent.train_step_counter.numpy()

                if step % hyper_parameters['log_interval'] == 0:
                    print('step = {0}: loss = {1}'.format(step, train_loss))

                if step % hyper_parameters['eval_interval'] == 0:
                    # Agent Policy Output
                    avg_return = self.compute_avg_return(self.eval_env, self.agent.policy,
                    print('step = {0}: Average Return = {1}'.format(step, avg_return))
                    if ('max_train_reward' in hyper_parameters) and \
                            (avg_return > hyper_parameters['max_train_reward']):

            print('\n-----\nTraining complete\n-----')

    def choose_actions(self, tracks, sensors, timestamp, nchoose=1, **kwargs):
        """Returns a chosen [list of] action(s) from the action set for each sensor.
        Chosen action(s) is selected by exploiting the reinforcement learning agent's
        policy that was found during training.

        tracks: set of :class:`~Track`
            Set of tracks at given time. Used in reward function.
        sensors: :class:`~Sensor`
            Sensor(s) used for observation
        timestamp: :class:`tf_agents.trajectories.TimeSpec`
            Timestep of environment at current time
        nchoose : int
            Number of actions from the set to choose (default is 1)

        : dict
            The pairs of :class:`~.Sensor`: [:class:`~.Action`] selected

        configs = [dict() for _ in range(nchoose)]
        for sensor_action_assignment in configs:
            for sensor in sensors:
                chosen_actions = []
                action_step = self.agent.policy.action(timestamp)
                action = action_step.action
                stonesoup_action = self.env.generate_action(action, tracks, sensor)
                sensor_action_assignment[sensor] = chosen_actions

            return configs

Create Sensor Managers

We initiate our reinforcement learning sensor manager with the environment we have designed

from stonesoup.sensormanager import BruteForceSensorManager

reinforcementsensormanager = ReinforcementSensorManager({sensorA}, env=StoneSoupEnv())
bruteforcesensormanager = BruteForceSensorManager({sensorB}, reward_function=reward_function)

Train RL agent

To generate a policy, we need to train the reinforcement learning agent using the environment we created above. Some hyperparameters are created that the agent uses to train with.

To train the agent, the hyperparameters are passed to the train method in the ReinforcementSensorManager.

import tensorflow as tf
import reverb
from tf_agents.agents.dqn import dqn_agent
from tf_agents.drivers import py_driver
from tf_agents.networks import sequential
from tf_agents.policies import py_tf_eager_policy, random_tf_policy
from tf_agents.replay_buffers import reverb_replay_buffer, reverb_utils
from tf_agents.specs import tensor_spec
from tf_agents.utils import common

num_iterations = 10000

initial_collect_steps = 100
collect_steps_per_iteration = 1
replay_buffer_max_length = 100000

batch_size = 64
learning_rate = 1e-4
log_interval = 500

num_eval_episodes = 10
eval_interval = 1000
fc_layer_params = (100, 50)

# ---- Optional ----
max_train_reward = 250

hyper_parameters = {'num_iterations': num_iterations,
                    'initial_collect_steps': initial_collect_steps,
                    'collect_steps_per_iteration': collect_steps_per_iteration,
                    'replay_buffer_max_length': replay_buffer_max_length,
                    'batch_size': batch_size,
                    'learning_rate': learning_rate,
                    'log_interval': log_interval,
                    'num_eval_episodes': num_eval_episodes,
                    'eval_interval': eval_interval,
                    'fc_layer_params': fc_layer_params,
                    'max_train_reward': max_train_reward}

WARNING:tensorflow:From /home/sgboakes/venvs/stonesoup-rl/lib/python3.10/site-packages/tensorflow/python/util/ calling foldr_v2 (from tensorflow.python.ops.functional_ops) with back_prop=False is deprecated and will be removed in a future version.
Instructions for updating:
back_prop=False is deprecated. Consider using tf.stop_gradient instead.
Instead of:
results = tf.foldr(fn, elems, back_prop=False)
results = tf.nest.map_structure(tf.stop_gradient, tf.foldr(fn, elems))
step = 500: loss = 1354.437255859375
step = 1000: loss = 19099.845703125
step = 1000: Average Return = 65.86003875732422
step = 1500: loss = 7924.47900390625
step = 2000: loss = 44789.375
step = 2000: Average Return = 117.9991226196289
step = 2500: loss = 94972.765625
step = 3000: loss = 14028.505859375
step = 3000: Average Return = 65.43219757080078
step = 3500: loss = 30156.640625
step = 4000: loss = 4416.0810546875
step = 4000: Average Return = 106.593994140625
step = 4500: loss = 30297.248046875
step = 5000: loss = 6728.6787109375
step = 5000: Average Return = 67.36097717285156
step = 5500: loss = 2970.46826171875
step = 6000: loss = 4445.5810546875
step = 6000: Average Return = 99.21797180175781
step = 6500: loss = 2844.02392578125
step = 7000: loss = 20083.6875
step = 7000: Average Return = 574.1416625976562

Training complete

Run the sensor managers

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.

Run reinforcement learning sensor manager

To be able to exploit the policy generated by the reinforcement sensor manager, it must be passed appropriate ‘timesteps’. These are distinct from the timesteps in Stone Soup, and is of the form time_step_spec from tf-agents.

from itertools import chain

sensor_history_A = dict()

timesteps = []
for state in truths[0]:

tf_timestep = reinforcementsensormanager.test_env.reset()
for timestep in timesteps[1:]:

    # Generate chosen configuration
    # i.e. {a}k
    # Need to make our own "timestamp" that matches tensorflow time_step_spec
    observation = []
    uncertainty = []
    for target in tracksA:
        x_target = target.state.state_vector[0] - sensorA.position[0]
        y_target = target.state.state_vector[2] - sensorA.position[1]
        bearing_target = mod_bearing(np.arctan2(y_target, x_target))

        # observation.append(np.degrees(bearing_target))

    observation = np.array(uncertainty, dtype=np.float32)
    # observation = np.array(observation, dtype=np.float32)

    chosen_actions = reinforcementsensormanager.choose_actions(tracksA, [sensorA], tf_timestep)

    # Create empty dictionary for measurements
    measurementsA = []

    for chosen_action in chosen_actions:
        # chosen_action is a pair of {sensor, action}
        for sensor, actions in chosen_action.items():


    # Store sensor history for plotting
    sensor_history_A[timestep] = copy.copy(sensorA)

    # Observe this ground truth
    # i.e. {z}k
    measurements = sensorA.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

    # Propagate environment
    action_step = reinforcementsensormanager.agent.policy.action(tf_timestep)
    tf_timestep = reinforcementsensormanager.test_env.step(action_step.action)

Plot ground truths, tracks and uncertainty ellipses for each target.

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_history):
    # Plot sensor field of view
    trace_base = len(

    for frame in fig.frames:
        traces_ = list(frame.traces)
        data_ = list(
        x = [0, 0]
        y = [0, 0]
        timestring =
        timestamp = datetime.strptime(timestring, "%Y-%m-%d %H:%M:%S")

        if timestamp in sensor_history:
            sensor = sensor_history[timestamp]
            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]],
        frame.traces = traces_ = data_

plot_sensor_fov(plotterA.fig, sensor_history_A)