Probabilistic Techniques for Mobile Robot Navigation

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Probabilistic Techniques for Mobile Robot Navigation
Wolfram Burgard University of Freiburg Department of Computer Science Autonomous Intelligent Systems Special Thanks to Frank Dellaert, Dieter Fox, Giorgio Grisetti, Dirk Haehnel, Cyrill Stachniss, Sebastian Thrun, …

Probabilistic Robotics

Robotics Today

Overcoming the uncanny valley
“Humanoids” Overcoming the uncanny valley [Courtesy by Hiroshi Ishiguro]

Humanoid Robots [Courtesy by Sven Behnke]

DARPA Grand Challenge [Courtesy by Sebastian Thrun]

DCG 2007

Robot Projects: Interactive Tour-guides
Rhino: Albert: Minerva:

Minerva

Robot Projects: Acting in the Three-dimensional World
Herbert: Zora: Groundhog:

Nature of Data Range Data Odometry Data

Probabilistic Techniques in Robotics
Perception = state estimation Action = utility maximization Key Question How to scale to higher-dimensional spaces

Axioms of Probability Theory
Pr(A) denotes probability that proposition A is true.

A Closer Look at Axiom 3 B

Using the Axioms

Discrete Random Variables
X denotes a random variable. X can take on a countable number of values in {x1, x2, …, xn}. P(X=xi), or P(xi), is the probability that the random variable X takes on value xi. P( ) is called probability mass function. E.g. .

Continuous Random Variables
X takes on values in the continuum. p(X=x), or p(x), is a probability density function. E.g. p(x) x

Joint and Conditional Probability
P(X=x and Y=y) = P(x,y) If X and Y are independent then P(x,y) = P(x) P(y) P(x | y) is the probability of x given y P(x | y) = P(x,y) / P(y) P(x,y) = P(x | y) P(y) If X and Y are independent then P(x | y) = P(x)

Law of Total Probability, Marginals
Discrete case Continuous case

Bayes Formula

Normalization Algorithm:

Conditioning Law of total probability:

Bayes Rule with Background Knowledge

Conditioning Total probability:

Conditional Independence
Equivalent to and

Simple Example of State Estimation
Suppose a robot obtains measurement z What is P(open|z)?

Causal vs. Diagnostic Reasoning
P(open|z) is diagnostic. P(z|open) is causal. Often causal knowledge is easier to obtain. Bayes rule allows us to use causal knowledge: count frequencies!

Example P(z|open) = 0.6 P(z|open) = 0.3 P(open) = P(open) = 0.5
z raises the probability that the door is open.

Combining Evidence Suppose our robot obtains another observation z2.
How can we integrate this new information? More generally, how can we estimate P(x| z1...zn )?

Recursive Bayesian Updating
Markov assumption: zn is independent of z1,...,zn-1 if we know x.

Example: Second Measurement
P(z2|open) = 0.5 P(z2|open) = 0.6 P(open|z1)=2/3 z2 lowers the probability that the door is open.

A Typical Pitfall Two possible locations x1 and x2 P(x1)=0.99
P(z|x2)=0.09 P(z|x1)=0.07

Often the world is dynamic since
Actions Often the world is dynamic since actions carried out by the robot, actions carried out by other agents, or just the time passing by change the world. How can we incorporate such actions?

Typical Actions The robot turns its wheels to move
The robot uses its manipulator to grasp an object Plants grow over time… Actions are never carried out with absolute certainty. In contrast to measurements, actions generally increase the uncertainty.

Modeling Actions To incorporate the outcome of an action u into the current “belief”, we use the conditional pdf P(x|u,x’) This term specifies the pdf that executing u changes the state from x’ to x.

Example: Closing the door

State Transitions P(x|u,x’) for u = “close door”: If the door is open, the action “close door” succeeds in 90% of all cases.

Integrating the Outcome of Actions
Continuous case: Discrete case:

Example: The Resulting Belief

Bayes Filters: Framework
Given: Stream of observations z and action data u: Sensor model P(z|x). Action model P(x|u,x’). Prior probability of the system state P(x). Wanted: Estimate of the state X of a dynamical system. The posterior of the state is also called Belief:

Markov Assumption Underlying Assumptions Static world
Independent noise Perfect model, no approximation errors

Bayes Filters z = observation u = action x = state Bayes Markov
Total prob. Markov Markov

Bayes Filter Algorithm
Algorithm Bayes_filter( Bel(x),d ): h=0 If d is a perceptual data item z then For all x do Else if d is an action data item u then Return Bel’(x)

Bayes Filters are Familiar!
Kalman filters Discrete filters Particle filters Hidden Markov models Dynamic Bayesian networks Partially Observable Markov Decision Processes (POMDPs)

Summary Bayes rule allows us to compute probabilities that are hard to assess otherwise. Under the Markov assumption, recursive Bayesian updating can be used to efficiently combine evidence. Bayes filters are a probabilistic tool for estimating the state of dynamic systems.

Dimensions of Mobile Robot Navigation
SLAM localization mapping integrated approaches active localization exploration motion control

Outline Localization Mapping Exploration

Probabilistic Localization

Localization with Bayes Filters
p(z|x) observation x laser data p(o|s,m) s’ a p(x|u,x’)

Localization with Sonars in an Occupancy Grid Map

Resulting Beliefs

What is the Right Representation?
Kalman filters Multi-hypothesis tracking Grid-based representations Topological approaches Particle filters

Monte-Carlo Localization
Set of N samples {<x1,w1>, … <xN,wN>} containing a state x and an importance weight w Initialize sample set according to prior knowledge For each motion u do: Sampling: Generate from each sample a new pose according to the motion model For each observation z do: Importance sampling: weigh each sample with the likelihood Re-sampling: Draw N new samples from the sample set according to the importance weights

Particle Filters Represent density by random samples
Estimation of non-Gaussian, nonlinear processes Monte Carlo filter, Survival of the fittest, Condensation, Bootstrap filter, Particle filter Filtering: [Rubin, 88], [Gordon et al., 93], [Kitagawa 96] Computer vision: [Isard and Blake 96, 98] Dynamic Bayesian Networks: [Kanazawa et al., 95]

Monte Carlo Localization (MCL) Represent Density Through Samples

Importance Sampling Weight samples:

Mobile Robot Localization with Particle Filters

MCL: Sensor Update

PF: Robot Motion

Particle Filter Algorithm
1. Draw from 3. Importance factor for 2. Draw from 4. Re-sample

Beam-based Sensor Model

Typical Measurement Errors of an Range Measurements
Beams reflected by obstacles Beams reflected by persons / caused by crosstalk Random measurements Maximum range measurements

Proximity Measurement
Measurement can be caused by … a known obstacle. cross-talk. an unexpected obstacle (people, furniture, …). missing all obstacles (total reflection, glass, …). Noise is due to uncertainty … in measuring distance to known obstacle. in position of known obstacles. in position of additional obstacles. whether obstacle is missed.

Beam-based Proximity Model
Measurement noise Unexpected obstacles zexp zmax zexp zmax

Beam-based Proximity Model
Random measurement Max range zexp zmax zexp zmax

Resulting Mixture Density
How can we determine the model parameters?

Measured distances for expected distance of 300 cm.
Raw Sensor Data Measured distances for expected distance of 300 cm. Sonar Laser

Approximation Maximize log likelihood of the data
Search space of n-1 parameters. Hill climbing Gradient descent Genetic algorithms Deterministically compute the n-th parameter to satisfy normalization constraint.

Approximation Results
Laser Sonar 400cm 300cm

Example z P(z|x,m)

Odometry Model Robot moves from to Odometry information

Sampling from a Motion Model
Start

MCL: Global Localization (Sonar)

Using Ceiling Maps for Localization
[Dellaert et al. 99]

Vision-based Localization
h(x) s P(s|x)

Under a Light Measurement: Resulting density:

Next to a Light Measurement: Resulting density:

Elsewhere Measurement: Resulting density:

MCL: Global Localization Using Vision

Vision-based Localization
Odometry only: Vision: Laser:

Vision-based Localization

KLD-sampling Idea: Observation: Assume we know the true belief.
Represent this belief as a multinomial distribution. Determine number of samples such that we can guarantee that, with probability (1- d), the KL-distance between the true posterior and the sample-based approximation is less than e. Observation: For fixed d and e, number of samples only depends on number k of bins with support:

MCL: Adaptive Sampling (Sonar)

MCL: Adaptive Sampling (Laser)

Performance Comparison
Grid-based localization Monte Carlo localization

Dimensions of Mobile Robot Navigation
SLAM localization mapping integrated approaches active localization exploration motion control

Occupancy Grid Maps Store in each cell of a discrete grid the probability that the corresponding area is occupied. Approximative Do not require features All cells are considered independent Their probabilities are updated using the binary Bayes filter

Updating Occupancy Grid Maps
Update the map cells using the inverse sensor model Or use the log-odds representation

Typical Sensor Model for Occupancy Grid Maps
Combination of a linear function and a Gaussian:

Incremental Updating of Occupancy Grids (Example)

Resulting Map Obtained with Ultrasound Sensors

Resulting Occupancy and Maximum Likelihood Map
The maximum likelihood map is obtained by clipping the occupancy grid map at a threshold of 0.5

Occupancy Grids: From scans to maps

Tech Museum, San Jose CAD map occupancy grid map

Dimensions of Mobile Robot Navigation
SLAM localization mapping integrated approaches active localization exploration motion control

Simultaneous Localization and Mapping (SLAM)
To determine its position, the robot needs a map. During mapping, the robot needs to know its position to learn a consistent model Simultaneous localization and mapping (SLAM) is a “chicken and egg problem”

Outline Localization Mapping Exploration

Why SLAM is Hard: Raw Odometry

Why SLAM is Hard: Ambiguity
End Same position Start [Courtesy of Eliazar & Parr]

Probabilistic Formulation of SLAM
three dimensions n=a£b dimensions map m

Key Question/Problem How to maintain multiple map and pose hypotheses during mapping? Ambiguity caused by the data association problem.

A Graphical Model for SLAM
x z u 2 ... t 1 t-1

Techniques for Generating Consistent Maps
Scan matching (online) Probabilistic mapping with a single map and a posterior about poses Mapping + Localization (online) EKF SLAM (online, mostly landmarks or features only) EM techniques (offline) Lu and Milios (offline)

Scan Matching Maximize the likelihood of the i-th pose and map relative to the (i-1)-th pose and map. To compute consistent maps, we apply a recursive scheme. At each point in time we compute the most likely position of the robot, given the map constructed so far. Based on the position hat l-t, we then extend the map an incorporate the scan obtained at time t. current measurement map constructed so far robot motion

Scan Matching Example

Rao-Blackwellized Particle Filters for SLAM
Observation: Given the true trajectory of the robot, we can efficiently compute the map (mapping with known poses). Idea: Use a particle filter to represent potential trajectories of the robot. Each particle carries its own map. Each particle survives with a probability that is proportional to the likelihood of the observation given that particle and its map. [Murphy et al., 99]

Factorization Underlying Rao-Blackwellization
Mapping with known poses Particle filter representing trajectory hypotheses

Example 3 particles map of particle 3 map of particle 1

Limitations A huge number of particles is required.
This introduces enormous memory and computational requirements. It prevents the application of the approach in realistic scenarios.

Challenge Reduction of the number of particles. Approaches:
Focused proposal distributions (keep the samples in the right place) Adaptive re-sampling (avoid depletion of relevant particles)

Motion Model for Scan Matching
Raw Odometry Scan Matching

Graphical Model for Mapping with Improved Odometry
z k x 1 u' u k-1 ... k+1 2k-1 2k 2 n n·k (n+1)·k-1 n·k+1

Application Example

The Optimal Proposal Distribution
[Doucet, 98] For lasers is extremely peaked and dominates the product. We can safely approximate by a constant:

Resulting Proposal Distribution
Gaussian approximation:

Resulting Proposal Distribution
Approximate this equation by a Gaussian: maximum reported by a scan matcher Gaussian approximation Draw next generation of samples Sampled points around the maximum

Estimating the Parameters of the Gaussian for each Particle
xj are a set of sample points around the point x* the scan matching has converged to.  is a normalizing constant

Computing the Importance Weight
Sampled points around the maximum of the observation likelihood

Improved Proposal The proposal adapts to the structure of the environment

Incorporating the Measurements
End of a corridor: Free space: Corridor:

Selective Resampling Resampling is dangerous, since important samples might get lost (particle depletion problem) In case of suboptimal proposal distributions resampling is necessary to achieve convergence. Key question: When should we resample?

Effective Number of Particles
Empirical measure of how well the goal distribution is approximated by samples drawn from the proposal Neff describes “the variance of the particle weights” Neff is maximal for equal weights. In this case, the distribution is close to the proposal

Resampling with Neff If our approximation is close to the proposal, no resampling is needed We only resample when Neff drops below a given threshold (N/2) See [Doucet, ’98; Arulampalam, ’01]

Typical Evolution of Neff
visiting new areas closing the first loop visiting known areas second loop closure

Map of the Intel Lab 15 particles
four times faster than real-time (P4, 2.8GHz) 5cm resolution during scan matching 1cm resolution in final map

Outdoor Campus Map 30 particles 250x250m2 1.75 km (odometry)
20cm resolution during scan matching 30cm resolution in final map

MIT Kilian Court

MIT Kilian Court

Dimensions of Mobile Robot Navigation
SLAM localization mapping integrated approaches active localization exploration motion control

Outline Localization Mapping Exploration

Exploration The approaches seen so far are purely passive.
By reasoning about control, the mapping process can be made much more effective.

Decision-Theoretic Formulation of Exploration
reward (expected information gain) cost (path length)

Exploration with Known Poses
Move to the place that provides you with the best tradeoff between information gathered and cost of reaching that point.

Exploration With Known Poses
Real robot, ultrasounds only:

Dimensions of Mobile Robot Navigation
SLAM localization mapping integrated approaches active localization exploration motion control

Where to Move Next?

Naïve Approach to Combine Exploration and Mapping
Learn the map using a Rao-Blackwellized particle filter. Apply an exploration approach that minimizes the map uncertainty.

Disadvantage of the Naïve Approach
Exploration techniques only consider the map uncertainty for generating controls. They avoid re-visiting known areas. Data association becomes harder. More particles are needed to learn a correct map.

Application Example True map and trajectory
Path estimated by the particle filter

Map and Pose Uncertainty
map uncertainty

Goal Integrated approach that considers to control the robot.
exploratory actions, place revisiting actions, and loop closing actions to control the robot.

Dual Representation for Loop Detection

Application Example

Real Exploration Example

Comparison Map uncertainty only: Map and pose uncertainty:

Example: Entropy Evolution

map and pose uncertainty
Quantitative Results Localization error: avg. localization error [m] map and pose uncertainty map uncertainty

Corridor Exploration

Summary Probabilistic techniques are a powerful tool to deal with a variety of problems in mobile robot navigation. By actively controlling mobile robots one can more effectively solve high-dimensional state estimation problems.

Questions? No Arrows Left … localization mapping motion control SLAM
active localization exploration integrated approaches

3D Mapping

3D Map Example

The Autonomous Blimp Project

Navigation in Environments with Deformable Objects

Navigation in Environments with Deformable Objects

Autonomous Cars