## Monte Carlo Localization Algorithm

### Overview

The Monte Carlo Localization (MCL) algorithm is used to estimate the position and orientation of a robot. The algorithm uses a known map of the environment, range sensor data, and odometry sensor data. To see how to construct an object and use this algorithm, see `monteCarloLocalization`.

To localize the robot, the MCL algorithm uses a particle filter to estimate its position. The particles represent the distribution of the likely states for the robot. Each particle represents a possible robot state. The particles converge around a single location as the robot moves in the environment and senses different parts of the environment using a range sensor. The robot motion is sensed using an odometry sensor.

The particles are updated in this process:

1. Particles are propagated based on the change in the pose and the specified motion model, `MotionModel`.

2. The particles are assigned weights based on the likelihood of receiving the range sensor reading for each particle. This reading is based on the sensor model you specify in `SensorModel`.

3. Based on these weights, a robot state estimate is extracted based on the particle weights. The group of particles with the highest weight is used to estimate the position of the robot.

4. Finally, the particles are resampled based on the specified `ResamplingInterval`. Resampling adjusts particle positions and improves performance by adjusting the number of particles used. It is a key feature for adjusting to changes and keeping particles relevant for estimating the robot state.

The algorithm outputs the estimated pose and covariance. These estimates are the mean and covariance of the highest weighted cluster of particles. For continuous tracking, repeat these steps in a loop to propagate particles, evaluate their likelihood, and get the best state estimate.

For more information on particle filters as a general application, see Particle Filter Workflow.

### State Representation

When working with a localization algorithm, the goal is to estimate the state of your system. For robotics applications, this estimated state is usually a robot pose. For the `monteCarloLocalization` object, you specify this pose as a three-element vector. The pose corresponds to an x-y position, `[x y]`, and an angular orientation, `theta`.

The MCL algorithm estimates these three values based on sensor inputs of the environment and a given motion model of your system. The output from using the `monteCarloLocalization` object includes the `pose`, which is the best estimated state of the ```[x y theta]``` values. Particles are distributed around an initial pose, `InitialPose`, or sampled uniformly using global localization. The pose is computed as the mean of the highest weighted cluster of particles once these particles have been corrected based on measurements.

This plot shows the highest weighted cluster and the final robot pose displayed over the samples particles in green. With more iterations of the MCL algorithm and measurement corrections, the particles converge to the true location of the robot. However, it is possible that particle clusters can have high weights for false estimates and converge on the wrong location. If the wrong convergence occurs, resample the particles by resetting the MCL algorithm with an updated `InitialPose`.

### Initialization of Particles

When you first create the `monteCarloLocalization` algorithm, specify the minimum and maximum particle limits by using the `ParticleLimits` property. A higher number of particles increases the likelihood that the particles converge on the actual location. However, a lower particle number is faster. The number of particles adjusts dynamically within the limits based on the weights of particle clusters. This adjustment helps to reduce the number of particles over time so localization can run more efficiently.

#### Particle Distribution

Particles must be sampled across a specified distribution. To initialize particles in the state space, you can use either an initial pose or global localization. With global localization, you can uniformly distribute particles across your expected state space (pulled from the `Map` property of your`SensorModel` object). In the default MCL object, set the `GlobalLocalization` property to `true`.

```mcl = monteCarloLocalization; mcl.GlobalLocalization = true;```

Global localization requires a larger number of particles to effectively sample particles across the state space. More particles increase the likelihood of successful convergence on the actual state. This large distribution greatly reduces initial performance until particles begin to converge and particle number can be reduced.

By default, global localization is set to `false`. Without global localization, you must specify the `InitialPose` and `InitialCovariance` properties, which helps to localize the particles. Using this initial pose, particles are more closely grouped around an estimated state. A close grouping of particles enables you to use fewer of them, and increases the speed and accuracy of tracking during the first iterations.

These images were taken from the Localize TurtleBot Using Monte Carlo Localization example, which shows how to use the MCL algorithm with the TurtleBot® in a known environment.

### Resampling Particles and Updating Pose

To localize your robot continuously, you must resample the particles and update the algorithm. Use the `UpdateThreshold` and `ResamplingInterval` properties to control when resampling and updates to the estimated state occur.

The `UpdateThreshold` is a three-element vector that defines the minimum change in the robot pose, `[x y theta]`, to trigger an update. Changing a variable by more than this minimum triggers an update, causing the object to return a new state estimate. This change in robot pose is based on the odometry, which is specified in the functional form of the object. Tune these thresholds based on your sensor properties and the motion of your robot. Random noise or minor variations greater than your threshold can trigger an unnecessary update and affect your performance. The `ResamplingInterval` property defines the number of updates to trigger particle resampling. For example, a resampling interval of `2` resamples at every other update.

The benefit of resampling particles is that you update the possible locations that contribute to the final estimate. Resampling redistributes the particles based on their weights and evolves particles based on the Motion Model. In this process, the particles with lower weight are eliminated, helping the particles converge to the true state of the robot. The number of particles dynamically changes to improve speed or tracking.

The performance of the algorithm depends on proper resampling. If particles are widely dispersed and the initial pose of the robot is not known, the algorithm maintains a high particle count. As the algorithm converges on the true location, it reduces the number of particles and increases the speed of performance. You can tune your `ParticleLimits` property to limit the minimum and maximum particles used to help with the performance.

### Motion and Sensor Model

The motion and sensor models for the MCL algorithm are similar to the `StateTransitionFcn` and `MeasurementLikelihoodFcn` functions for the `stateEstimatorPF` object, which are described in Particle Filter Parameters. For the MCL algorithm, these models are more specific to robot localization. After calling the object, to change the `MotionModel` or `SensorModel` properties, you must first call `release` on your object.

#### Sensor Model

By default, the `monteCarloLocalization` uses a `likelihoodFieldSensorModel` object as the sensor model. This sensor model contains parameters specific to the range sensor used, 2-D map information for the robot environment, and measurement noise characteristics. The sensor model uses the parameters with range measurements to compute the likelihood of the measurements given the current position of the robot. Without factoring in these parameters, some measurement errors can skew the state estimate or increase weight on irrelevant particles.

The range sensor properties are:

• `SensorPose` – The pose of the range sensor relative to the robot location. This pose is used to transform the range readings into the robot coordinate frame.

• `SensorLimits` – The minimum and maximum range limits. Measurement outside of these ranges are not factored into the likelihood calculation.

• `NumBeams` – Number of beams used to calculate likelihood. You can improve performance speed by reducing the number of beams used.

Range measurements are also known to give false readings due to system noise or other environmental interference. To account for the sensor error, specify these parameters:

• `MeasurementNoise` – Standard deviation for measurement noise. This deviation applies to the range reading and accounts for any interference with the sensor. Set this value based on information from your range sensor.

• `RandomMeasurementWeight` — Weight for probability of random measurement. Set a low probability for random measurements. The default is 0.05.

• `ExpectedMeasurementWeight` — Weight for probability of expected measurement. Set a high probability for expected measurements. The default is 0.95.

The sensor model also stores a map of the robot environment as an occupancy grid. Use `binaryOccupancyMap` to specify your map with occupied and free spaces. Set any unknown spaces in the map as free locations. Setting them to free locations prevents the algorithm from matching detected objects to these areas of the map.

Also, you can specify `MaximumLikelihoodDistance`, which limits the area for searching for obstacles. The value of `MaximumLikelihoodDistance` is the maximum distance to the nearest obstacle that is used for likelihood computation.

#### Motion Model

The motion model for robot localization helps to predict how particles evolve throughout time when resampling. It is a representation of robot kinematics. The motion model included by default with the MCL algorithm is an odometry-based differential drive motion model (`odometryMotionModel`). Without a motion model, predicting the next step is more difficult. It is important to know the capabilities of your system so that the localization algorithm can plan particle distributions to get better state estimates. Be sure to consider errors from the wheel encoders or other sensors used to measure the odometry. The errors in the system define the spread of the particle distribution.

You can specify the error expected based on the motion of your robot as a four-element vector, `Noise`. These four elements are specified as weights on the standard deviations for [1]:

• Rotational error due to rotational motion

• Rotational error due to translational motion

• Translational error due to translational motion

• Translational error due to rotational motion

For differential drive robots, when a robot moves from a starting pose to a final pose, the change in pose can be treated as:

1. Rotation to the final position

2. Translation in a direct line to the final position

3. Rotation to the goal orientation

Assuming these steps, you can visualize the effect of errors in rotation and translation. Errors in the initial rotation result in your possible positions being spread out in a C-shape around the final position.

Large translational errors result in your possible positions being spread out around the direct line to the final position.

Large errors in both translation and rotation can result in wider-spread positions.

Also, rotational errors affect the orientation of the final pose. Understanding these effects helps you to define the Gaussian noise in the `Noise` property of the `MotionModel` object for your specific application. As the images show, each parameter does not directly control the dispersion and can vary with your robot configuration and geometry. Also, multiple pose changes as the robot navigates through your environment can increase the effects of these errors over many different steps. By accurately defining these parameters, particles are distributed appropriately to give the MCL algorithm enough hypotheses to find the best estimate for the robot location.

## References

[1] Thrun, Sebastian, and Dieter Fox. Probabilistic Robotics. 3rd ed. Cambridge, Mass: MIT Press, 2006. p.136.