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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, …

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**Probabilistic Robotics**

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Robotics Today

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**Overcoming the uncanny valley**

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

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Humanoid Robots [Courtesy by Sven Behnke]

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DARPA Grand Challenge [Courtesy by Sebastian Thrun]

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DCG 2007

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**Robot Projects: Interactive Tour-guides**

Rhino: Albert: Minerva:

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Minerva

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**Robot Projects: Acting in the Three-dimensional World**

Herbert: Zora: Groundhog:

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Nature of Data Range Data Odometry Data

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**Probabilistic Techniques in Robotics**

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

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**Axioms of Probability Theory**

Pr(A) denotes probability that proposition A is true.

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A Closer Look at Axiom 3 B

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Using the Axioms

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**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. .

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**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

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**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)

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**Law of Total Probability, Marginals**

Discrete case Continuous case

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Bayes Formula

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Normalization Algorithm:

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Conditioning Law of total probability:

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**Bayes Rule with Background Knowledge**

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Conditioning Total probability:

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**Conditional Independence**

Equivalent to and

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**Simple Example of State Estimation**

Suppose a robot obtains measurement z What is P(open|z)?

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**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!

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**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.

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**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 )?

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**Recursive Bayesian Updating**

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

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**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.

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**A Typical Pitfall Two possible locations x1 and x2 P(x1)=0.99**

P(z|x2)=0.09 P(z|x1)=0.07

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**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?

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**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.

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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.

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**Example: Closing the door**

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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.

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**Integrating the Outcome of Actions**

Continuous case: Discrete case:

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**Example: The Resulting Belief**

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**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:

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**Markov Assumption Underlying Assumptions Static world**

Independent noise Perfect model, no approximation errors

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**Bayes Filters z = observation u = action x = state Bayes Markov**

Total prob. Markov Markov

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**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)

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**Bayes Filters are Familiar!**

Kalman filters Discrete filters Particle filters Hidden Markov models Dynamic Bayesian networks Partially Observable Markov Decision Processes (POMDPs)

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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.

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**Dimensions of Mobile Robot Navigation**

SLAM localization mapping integrated approaches active localization exploration motion control

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Outline Localization Mapping Exploration

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**Probabilistic Localization**

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**Localization with Bayes Filters**

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

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**Localization with Sonars in an Occupancy Grid Map**

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Resulting Beliefs

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**What is the Right Representation?**

Kalman filters Multi-hypothesis tracking Grid-based representations Topological approaches Particle filters

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**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

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**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]

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**Monte Carlo Localization (MCL) Represent Density Through Samples**

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Importance Sampling Weight samples:

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**Mobile Robot Localization with Particle Filters**

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MCL: Sensor Update

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PF: Robot Motion

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**Particle Filter Algorithm**

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

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**Beam-based Sensor Model**

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**Typical Measurement Errors of an Range Measurements**

Beams reflected by obstacles Beams reflected by persons / caused by crosstalk Random measurements Maximum range measurements

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**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.

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**Beam-based Proximity Model**

Measurement noise Unexpected obstacles zexp zmax zexp zmax

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**Beam-based Proximity Model**

Random measurement Max range zexp zmax zexp zmax

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**Resulting Mixture Density**

How can we determine the model parameters?

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**Measured distances for expected distance of 300 cm.**

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

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**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.

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**Approximation Results**

Laser Sonar 400cm 300cm

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Example z P(z|x,m)

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Odometry Model Robot moves from to Odometry information

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**Sampling from a Motion Model**

Start

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**MCL: Global Localization (Sonar)**

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**Using Ceiling Maps for Localization**

[Dellaert et al. 99]

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**Vision-based Localization**

h(x) s P(s|x)

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Under a Light Measurement: Resulting density:

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Next to a Light Measurement: Resulting density:

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Elsewhere Measurement: Resulting density:

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**MCL: Global Localization Using Vision**

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**Vision-based Localization**

Odometry only: Vision: Laser:

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**Vision-based Localization**

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Adaptive Sampling

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**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:

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**MCL: Adaptive Sampling (Sonar)**

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**MCL: Adaptive Sampling (Laser)**

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**Performance Comparison**

Grid-based localization Monte Carlo localization

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**Dimensions of Mobile Robot Navigation**

SLAM localization mapping integrated approaches active localization exploration motion control

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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

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**Updating Occupancy Grid Maps**

Update the map cells using the inverse sensor model Or use the log-odds representation

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**Typical Sensor Model for Occupancy Grid Maps**

Combination of a linear function and a Gaussian:

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**Incremental Updating of Occupancy Grids (Example)**

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**Resulting Map Obtained with Ultrasound Sensors**

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**Resulting Occupancy and Maximum Likelihood Map**

The maximum likelihood map is obtained by clipping the occupancy grid map at a threshold of 0.5

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**Occupancy Grids: From scans to maps**

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Tech Museum, San Jose CAD map occupancy grid map

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**Dimensions of Mobile Robot Navigation**

SLAM localization mapping integrated approaches active localization exploration motion control

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**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”

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Outline Localization Mapping Exploration

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**Why SLAM is Hard: Raw Odometry**

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**Why SLAM is Hard: Ambiguity**

End Same position Start [Courtesy of Eliazar & Parr]

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**Probabilistic Formulation of SLAM**

three dimensions n=a£b dimensions map m

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

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**A Graphical Model for SLAM**

x z u 2 ... t 1 t-1

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**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)

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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

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Scan Matching Example

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**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]

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**Factorization Underlying Rao-Blackwellization**

Mapping with known poses Particle filter representing trajectory hypotheses

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**Example 3 particles map of particle 3 map of particle 1**

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**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.

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**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)

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**Motion Model for Scan Matching**

Raw Odometry Scan Matching

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**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

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Application Example

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**The Optimal Proposal Distribution**

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

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**Resulting Proposal Distribution**

Gaussian approximation:

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**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

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**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

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**Computing the Importance Weight**

Sampled points around the maximum of the observation likelihood

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Improved Proposal The proposal adapts to the structure of the environment

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**Incorporating the Measurements**

End of a corridor: Free space: Corridor:

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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?

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**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

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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]

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**Typical Evolution of Neff**

visiting new areas closing the first loop visiting known areas second loop closure

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**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

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**Outdoor Campus Map 30 particles 250x250m2 1.75 km (odometry)**

20cm resolution during scan matching 30cm resolution in final map

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MIT Kilian Court

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MIT Kilian Court

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**Dimensions of Mobile Robot Navigation**

SLAM localization mapping integrated approaches active localization exploration motion control

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Outline Localization Mapping Exploration

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**Exploration The approaches seen so far are purely passive.**

By reasoning about control, the mapping process can be made much more effective.

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**Decision-Theoretic Formulation of Exploration**

reward (expected information gain) cost (path length)

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**Exploration with Known Poses**

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

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**Exploration With Known Poses**

Real robot, ultrasounds only:

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**Dimensions of Mobile Robot Navigation**

SLAM localization mapping integrated approaches active localization exploration motion control

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Where to Move Next?

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**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.

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**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.

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**Application Example True map and trajectory**

Path estimated by the particle filter

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**Map and Pose Uncertainty**

map uncertainty

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**Goal Integrated approach that considers to control the robot.**

exploratory actions, place revisiting actions, and loop closing actions to control the robot.

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**Dual Representation for Loop Detection**

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Application Example

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**Real Exploration Example**

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Comparison Map uncertainty only: Map and pose uncertainty:

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**Example: Entropy Evolution**

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**map and pose uncertainty**

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

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Corridor Exploration

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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.

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**Questions? No Arrows Left … localization mapping motion control SLAM**

active localization exploration integrated approaches

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3D Mapping

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3D Map Example

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**The Autonomous Blimp Project**

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**Navigation in Environments with Deformable Objects**

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**Navigation in Environments with Deformable Objects**

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Autonomous Cars

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