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Marginal Particle and Multirobot Slam: SLAM=‘SIMULTANEOUS LOCALIZATION AND MAPPING’

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### Marginal Particle and Multirobot Slam: SLAM=‘SIMULTANEOUS LOCALIZATION AND MAPPING’

By Marc Sobel

(Includes references to Brian Clipp

Comp 790-072 Robotics)

The SLAM Problem

- Given
- Robot controls
- Nearby measurements
- Estimate
- Robot state (position, orientation)
- Map of world features

Outline

- Sensors
- SLAM
- Full vs. Online SLAM
- Marginal Slam
- Multirobot marginal slam
- Example Algorithms
- Extended Kalman Filter (EKF) SLAM
- FastSLAM (particle filter)

Types of Sensors

- Odometry
- Laser Ranging and Detection (LIDAR)
- Acoustic (sonar, ultrasonic)
- Radar
- Vision (monocular, stereo etc.)
- GPS
- Gyroscopes, Accelerometers (Inertial Navigation)
- Etc.

Sensor Characteristics

- Noise
- Dimensionality of Output
- LIDAR- 3D point
- Vision- Bearing only (2D ray in space)
- Range
- Frame of Reference
- Most in robot frame (Vision, LIDAR, etc.)
- GPS earth centered coordinate frame
- Accelerometers/Gyros in inertial coordinate frame

A Probabilistic Approach

- Notation:

Full vs. Online classical SLAM

- Full SLAM calculates the robot pose over all time up to time t given the signal and odometry:

- Online SLAM calculates the robot pose for the current time t

Classical Fast and EKF Slam

- Robot Environment:
- (1) N distances: mt ={d(xt,L1),….,d(xt,LN)};

m measures distances from landmarks at

time t.

- (2) Robot pose at time t: xt.
- (3) (Scan) Measurements at time t: zt
- Goal: Determine the poses x1:T

given scans z1:t,odometry u1:T, and map

measurements m.

EKF SLAM (Extended Kalman Filter)

- As the state vector moves, the robot pose moves according to the motion function, g(ut,xt). This can be linearized into a Kalman Filter via:
- The Jacobian J depends on translational and rotational velocity. This allows us to assume that the motion and hence distances are Gaussian. We can calculate the mean μ and covariance matrix Σ for the particle xt at time t.

Outline of EKF SLAM

- By what preceded we assume that the map vectors m (measuring distance from landmarks) are independent multivariate normal: Hence we now have:

Conditional Independence

- For constructing the weights associated with the classical fast slam algorithm, under moderate assumptions, we get:
- We use the notation,
- And calculate that:

Problems With EKF SLAM

- Uses uni-modal Gaussians to model non-Gaussian probability density function

Particle Filters (without EKF)

- The use of EKF depends on the odometry (u’s) and motion model (g’s) assumptions in a very nonrobust way and fails to allow for multimodality in the motion model. In place of this, we can use particle filters without assuming a motion model by modeling the particles without reference to the parameters.

Particle Filters (an alternative)

- Represent probability distribution as a set of discrete particles which occupy the state space

Particle Filters

- For constructing the weights associated with the classical fast slam algorithm, under moderate assumptions, we get: (for x’s simulated by q)

Resampling

- Assign each particle a weight depending on how well its estimate of the state agrees with the measurements
- Randomly draw particles from previous distribution based on weights creating a new distribution

Particle Filter Update Cycle

- Generate new particle distribution
- For each particle
- Compare particle’s prediction of measurements with actual measurements
- Particles whose predictions match the measurements are given a high weight
- Resample particles based on weight

Particle Filter Advantages

- Can represent multi-modal distributions

Problems with Particle Filters

- Degeneracy: As time evolves particles increase in dimensionality. Since there is error at each time point, this evolution typically leads to vanishingly small interparticle (relative to intraparticle) variation.
- We frequently require estimates of the ‘marginal’ rather than ‘conditional’ particle distribution.
- Particle Filters do not provide good methods for estimating particle features.

Marginal versus nonmarginal Particle Filters

- Marginal particle filters attempt to update the X’s using their marginal (posterior) distribution rather than their conditional (posterior) distribution. The update weights take the general form,

Marginal Particle update

- We want to update by using the old weights rather than conditioning on the old particles.

Marginal Particle Filters

- We specify the proposal distribution ‘q’ via:

Marginal Particle Algorithm

- (1)
- (2) Calculate the importance weights:

Updating Map features in the marginal model

- Up to now, we haven’t assumed any map features. Let θ={θt} denote the e.g., distances of the robot at time t from the given landmarks. We then write
- for the probability associated with scan Zt given the position x1:t.
- We’d like to update θ. This should be based, not on the gradient , but rather on the gradient, .

Taking the ‘right’ derivative

- The gradient, is highly non-robust; we are essentially taking derivatives of noise.
- By contrast, the gradient, is robust and represents the ‘right’ derivative.

Estimating the Gradient of a Map

- We have that,

Simplification

- We can then show for the first term that:

Simplification II

- For the second term, we convert into a discrete sum by defining ‘derivative weights’
- And combining them with the standard weights.

Estimating the Gradient

- We can further write that:

Gradient (continued)

- We can therefore update the gradient weights via:

Parameter Updates

- We update θ by:

Normalization

- The β’s are normalized differently from the w’s. In effect we put:
- And then compute that:

The Bayesian Viewpoint

- Retain a posterior sample of θ at time t-1.
- Call this (i=1,…,I)
- At time t, update this sample:

Multi Robot Models

- Write for the poses and scan statistics for the r robots.
- At each time point the needed weights have r indices:
- We also need to update the derivative weights – the derivative is now a matrix derivative.

Multi-Robot SLAM

- The parameter is now a matrix (with time being the row values and robot index being the column. Updates depend on derivatives with respect to each timepoint and with respect to each robot.

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