slide1 l.
Download
Skip this Video
Download Presentation
Simultaneous Localization and Mapping

Loading in 2 Seconds...

play fullscreen
1 / 27

Simultaneous Localization and Mapping - PowerPoint PPT Presentation


  • 247 Views
  • Uploaded on

Simultaneous Localization and Mapping. Presented by Lihan He Apr. 21, 2006. Outline. Introduction SLAM using Kalman filter SLAM using particle filter Particle filter SLAM by particle filter My work : searching problem. Introduction: SLAM. SLAM: Simultaneous Localization and Mapping.

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about 'Simultaneous Localization and Mapping' - cedric


Download Now An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
slide1

Simultaneous Localization and Mapping

Presented by Lihan He

Apr. 21, 2006

slide2

Outline

  • Introduction
  • SLAM using Kalman filter
  • SLAM using particle filter
    • Particle filter
    • SLAM by particle filter
  • My work : searching problem
slide3

Introduction: SLAM

SLAM: Simultaneous Localization and Mapping

A robot is exploring an unknown, static environment.

  • Given:
  • The robot’s controls
  • Observations of nearby features

The controls and observations are both noisy.

  • Estimate:
  • Location of the robot -- localization

where I am ?

  • Detail map of the environment – mapping

What does the world look like?

slide4

a2

at

a1

at-1

x1

x2

xt

xt-1

x0

o2

o1

ot

ot-1

m

Introduction: SLAM

Markov assumption

State transition:

Observation function:

slide5

p(xt)

p(mt) or mt

p(x1)

p(m1) or m1

p(xt-1)

p(mt-1) or mt-1

Prior distribution on xt after taking action at

Introduction: SLAM

Method: Sequentially estimate the probability distribution p(xt) and update the map.

Prior: p(x0)

slide6

Introduction: SLAM

Map representations

1. Landmark-based map representation

Track the positions of a fixed number of predetermined sparse landmarks.

Observation: estimated distance from each landmark.

2. Grid-based map representation

Map is represented by a fine spatial grid, each grid square is either occupied or empty.

Observation: estimated distance from an obstacle using a laser range finder.

slide7

Represent the distribution of robot location xt(and map mt) by a Normal distribution

Introduction: SLAM

Methods:

The robot’s trajectory estimateis a tracking problem

1. Parametric method – Kalman filter

Sequentially update μt and Σt

2. Sample-based method – particle filter

Represent the distribution of robot location xt(and map mt)by a large amount of simulated samples.

Resample xt (and mt) at each time step

slide8

Location error

Map error

Introduction: SLAM

Why is SLAM a hard problem?

Robot location and map are both unknown.

  • The small error will quickly accumulated over time steps.
  • The errors come from inaccurate measurement of actual robot motion (noisy action) and the distance from obstacle/landmark (noisy observation).

When the robot closes a physical loop in the environment, serious misalignment errors could happen.

slide9

SLAM: Kalman filter

Update equation:

Assume:

Prior p(x0) is a normal distribution

Observation function p(o|x) is a normal distribution

Then:

Posterior p(x1), …, p(xt) are all normally distributed.

Mean μtand covariance matrix Σt can be derived analytically.

Sequentially update μt and Σt for each time step t

slide10

Assume:

State transition

Observation function

Kalman filter:

Propagation (motion model):

Update (sensor model):

SLAM: Kalman filter

slide11

localization

mapping

SLAM: Kalman filter

The hidden state for landmark-based SLAM:

Map with N landmarks: (3+2N)-dimentional Gaussian

State vector xt can be grown as new landmarks are discovered.

slide12

Idea:

  • Normal distribution assumption in Kalman filter is not necessary
  • A set of samples approximates the posterior distribution and will be used at next iteration.
  • Each sample maintains its own map; or all samples maintain a single map.
  • The map(s) is updated upon observation, assuming that the robot location is given correctly.

SLAM: particle filter

Update equation:

slide13

Particle filter:

Assume it is difficult to sample directly from

But get samples from another distribution is easy.

We sample from , with normalized weight for each xit as

The set of (particles) is an approximation of

Resamplextfrom ,with replacement, to get a sample set with uniform weights

SLAM: particle filter

slide14

Particle filter (cont’d):

0.4

0.3

0.2

0.1

SLAM: particle filter

slide15

Choose appropriate

Transition probability

Choose

Then

Observation function

SLAM: particle filter

slide16

Algorithm:

Let state xt represent the robot’s location,

1. Propagate each state through the state transition probability

. This samples a new state given the previous state.

2. Weight each new state according to the observation function

3. Normalize the weights, get .

4. Resampling: sample Ns new states from

are the updated robot location from

SLAM: particle filter

slide17

are the expected robot moving distance (angle) by taking action at.

Measured distance (observation) for sensor k

Map distance from location xtto the obstacle

SLAM: particle filter

State transition probability:

Observation probability:

slide18

SLAM: particle filter

  • Lots of work on SLAM using particle filter are focused on:
  • Reducing the cumulative error
  • Fast SLAM (online)
  • Way to organize the data structure (saving robot path and map).

Maintain single map: cumulative error

Multiple maps: memory and computation time

  • In Parr’s paper:
  • Use ancestry tree to record particle history
  • Each particle has its own map (multiple maps)
  • Use observation tree for each grid square (cell) to record the map corresponding to each particle.
  • Update ancestry tree and observation tree at each iteration.
  • Cell occupancy is represented by a probabilistic approach.
slide20

Searching problem

Assumption:

  • The agent doesn’t have map, doesn’t know the underlying model, doesn’t know where the target is.
  • Agent has 2 sensors:
    • Camera: tell agent “occupied” or “empty” cells in 4 orientations, noisy sensor.
    • Acoustic sensor: find the orientation of the target, effective only within certain distance.
  • Noisy observation, noisy action.
slide21

Searching problem

  • Similar to SLAM
  • To find the target, agent need build map and estimate its location.
  • Differences from SLAM
  • Rough map is enough; an accurate map is not necessary.
  • Objective is to find the target. Robot need to actively select actions to find the target as soon as possible.
slide22

Searching problem

  • Model:
  • Environment is represented by a rough grid;
  • Each grid square (state) is either occupied or empty.
  • The agent moves between the empty grid squares.
  • Actions: walk to any one of the 4 directions, or “stay”. Could fail in walking with certain probability.
  • Observations: observe 4 orientations of its neighbor grid squares: “occupied” or “empty”. Could make a wrong observation with certain probability.
  • State, action and observation are all discrete.
slide23

Searching problem

In each step, the agent updates its location and map:

  • Belief state: the agent believes which state it is currently in. It is a distribution over all the states in the current map.
  • The map: The agent thinks what the environment is .
    • For each state (grid square), a 2-dimentional Dirichlet distribution is used to represent the probability of “empty” and “occupied”.
    • The hyperparameters of Dirichlet distribution are updated based on current observation and belief state.
slide24

Belief state update:

the set of neighbor states of s

where

Probability of successful moving from sjto s when taking action a

From map representation

and

with

Neighbor of s in orientation j

Searching problem

slide25

Assumeat step t-1, the hyperparameter for state S is

At step t, the hyperparameter for state S is updated as

and are the posterior after observing o given that the agent is in the neighbor of state s. If the probability of wrong observation for any orientation is p0,then

p0 if o is “occupied”

1-p0 if o is “empty”

prior

can be computed similarly.

Searching problem

Map update (Dirichlet distribution):

slide26

Searching problem

Belief state update:

a=“up”

a=“right”

Map representation update:

a=“right”

a=“up”

slide27

Searching problem

Choose actions:

Each state is assigned a reward R(s) according to following rules:

Less explored grids have higher reward.

Try to walk to the “empty” grid square.

Consider neighbor of s with priority.

x

x