Mobile Robotics Probabilistic Sensor Models PDF

Introduction to Mobile Robotics Probabilistic Sensor Models 8 Bayes Filters are Familiar! Bel ( x t )  P ( z t | x t )  P ( x t | u t , x t 1 )Bel ( x t 1 )dx t 1  Kalman filters  Particle filters  Hidden Markov models  Dynamic Bayesian net...

Introduction to Mobile Robotics Probabilistic Sensor Models 8 Bayes Filters are Familiar! Bel ( x t )  P ( z t | x t )  P ( x t | u t , x t 1 )Bel ( x t 1 )dx t 1  Kalman filters  Particle filters  Hidden Markov models  Dynamic Bayesian networks  Partially Observable Markov Decision Processes (POMDPs) 9 Sensors for Mobile Robots  Contact sensors: Bumpers Proprioceptive sensors measure values internal to the system  Proprioceptive sensors  Accelerometers  Gyroscopes , laser light)  Proximity sensors  Sonar (time of flight)  Radar (phase and frequency)  Laser range-finders (triangulation, tof, phase)  Infrared (intensity)  Visual sensors: Cameras  Satellite-based sensors: GPS 10 Proximity Sensors  The central task is to determine P(z|x), i.e., the probability of a measurement z given that the robot is at position x.  Question: Where do the probabilities come from?  Approach: Letstry to explain a measurement. 11 Beam-based Sensor Model  Scan z consists of K measurements. z  { z1 , z 2 ,..., z K }  Individual measurements are independent given the robot position. K P ( z | x , m )   P ( z k | x, m k 1 ) 12 Beam-based Sensor Model K P ( z | x , m )   P (z k | x , m k 1 ) m is a map of the environment , which provides us with the list of objects in the environment and their locations. 13  Typical Measurement Errors of an Range Measurements 1. Beams reflected by obstacles 2. Beams reflected by persons / caused by crosstalk 3. Random measurements 4. Maximum range measurements Robot is traveling in a corridor 14 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. 15 1. Beam-based Proximity Model Our model incorporates four types of measurement errors, all of which are essential to making this model work: small measurement noise, errors due to unexpected objects, errors due to failures to detect objects, and random unexplained noise. Beam-based Proximity Model Measurement noise Unexpected obstacles 0 zexp zmax 0 zexp zmax 1 ( z  z exp ) 2  e  z z  exp  1  Phit ( z | x , m )  e 2 b Punexp ( z | x , m )  z  2b   0 otherwise  19 Beam-based Proximity Model Random measurement Max range 0 zexp zmax 0 zexp zmax Prand ( z | x , m )   1 𝑃max 𝑧𝑥, = 1 𝑧=𝑧max z max 0 otherwise 20 Resulting Mixture Density T  hit   Phit ( z | x , m )         Punexp ( z | x , m )   P ( z | x , m )   unexp   P ( z | x , m )   max   max     P   rand   rand ( z | x , m )  How can we determine the model parameters? 21 return p Raw Sensor Data Measured distances for expected distance of 300 cm. Sonar Laser 23 Approximation  Maximize log likelihood of the data P ( z | z exp )  Search space of n-1 parameters.  Hill climbing  Gradient descent  Genetic algorithms  …  Deterministically compute the n-th parameter to satisfy normalization constraint. 24 Approximation Results Laser Sonar 300cm 400cm 2 Example 6 z P(z|x,m) 1 9 Influence of Angle to Obstacle "sonar-0" 0.25 0.2 0.15 0.1 0.05 0 70 60 40 50 0 10 20 30 20 30 40 50 10 60 70 0 2 0 Influence of Angle to Obstacle "sonar-1" 0.3 0.25 0.2 0.15 0.1 0.05 0 70 60 40 50 0 10 20 30 20 30 40 50 10 60 70 0 2 1 Influence of Angle to Obstacle "sonar-2" 0.3 0.25 0.2 0.15 0.1 0.05 0 70 60 40 50 0 10 20 30 20 30 40 50 10 60 70 0 2 2 Influence of Angle to Obstacle "sonar-3" 0.25 0.2 0.15 0.1 0.05 0 70 60 40 50 0 10 20 30 20 30 40 50 10 60 70 0 2 4 2. Scan-based Model  Beam-based model is …  not smooth for small obstacles and at edges.  not very efficient since evaluating p(zt | xt, m) for each single sensor measurement zt is computationally expensive.  Idea: Instead of following along the beam, just check the end point. 2 5 Scan-based Model  Probability is a mixture of …  a Gaussian distribution with mean at distance to closest obstacle,  a uniform distribution for random measurements, and  a small uniform distribution for max range measurements.  Again, independence between different components is assumed. 2 6 Example Likelihood field Map m P(z|x,m) 2 7 San Jose Tech Museum Occupancy grid map Likelihood field 2 8 Scan Matching  Extract likelihood field from scan and use it to match different scan. 3 0 Properties of Scan-based Model  Highly efficient, uses 2D tables only.  Distance grid is smooth w.r.t. to small changes in robot position.  Allows gradient descent, scan matching.  Ignores physical properties of beams. End