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Advanced Mobile Robotics. Dr. J izhong Xiao Department of Electrical Engineering CUNY City College jxiao@ccny.cuny.edu. Probabilistic Robotics. Probabilistic Sensor Models Beam-based Model Likelihood Fields Model Feature-based Model. Bayes Filter Reminder. Prediction (Action)

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1. Advanced Mobile Robotics Dr. Jizhong Xiao Department of Electrical Engineering CUNY City College jxiao@ccny.cuny.edu

2. Probabilistic Robotics Probabilistic Sensor Models Beam-based Model Likelihood Fields Model Feature-based Model

3. Bayes Filter Reminder • Prediction (Action) • Correction (Measurement)

4. Sensors for Mobile Robots • Contact sensors:Bumpers • Internal sensors • Accelerometers (spring-mounted masses) • Gyroscopes (spinning mass, laser light) • Compasses, inclinometers (earth magnetic field, gravity) • 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

5. 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: Let’s try to explain a measurement.

6. Noise Issues Sensors are imperfect! Specular Reflection

7. Beam-based Sensor Model • Scan z consists of K measurements. • Individual measurements are independent given the robot position and the map of environment. A cyclic array of k ultrasound sensors

8. Beam-based Sensor Model

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

10. 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.

11. zexp zexp 0 zmax 0 zmax Beam-based Proximity Model Measurement noise Unexpected obstacles Likelihood of sensing unexpected objects decreased with range Gaussian Distribution Exponential Distribution

12. zexp zexp 0 zmax 0 zmax Beam-based Proximity Model Random measurement Max range/Failures If fail to detect obstacle, report maximum range Uniform distribution Point-mass distribution

13. Resulting Mixture Density Weighted average, and How can we determine the model parameters? System identification method: maximum likelihood estimator (ML estimator)

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

15. Approximation • The likelihood of the data Z is given by • Our goal is to identify intrinsic parameters that maximize this log likelihood • Mathematic derivation can be found pp163~167 • Psudo-code can be found in pp160 • An iterative procedure for estimating parameters • Deterministically compute the n-th parameter to satisfy normalization constraint.

16. Approximation Results Laser is more accurate than sonar, narrower Gaussians Laser The smaller range, the more accurate the measurement Sonar 400cm 300cm

17. Example z P(z|x,m) Laser scan, projected into a previously acquired map The likelihood, evaluated for all positions and projected into the map. The darker a position, the larger P(z|x,m)

18. Summary Beam-based Model • Assumes independence between beams. • Justification? • Overconfident!  Suboptimal result • Models physical causes for measurements. • Mixture of densities for these causes. • Assumes independence between causes. Problem? • Implementation • Learn parameters based on real data. • Different models should be learned for different angles at which the sensor beam hits the obstacle. • Determine expected distances by ray-tracing. • Expected distances can be pre-processed.

19. Likelihood Fields Model • Beam-based model is … • not smooth for small obstacles and at edges (small changes of a robot’s pose can have a tremendous impact on the correct range of sensor beam, heading direction is particularly affected). • not very efficient (computationally expensive in ray casting) • Idea: Instead of following along the beam, just check the end point.

20. Likelihood Fields Model • Project the end points of a sensor scan Zt into the global coordinate space of the map • 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.

21. Likelihood Fields Model Distance to the nearest obstacles. Max range reading ignored

22. Example Example environment Likelihood field The darker a location, the less likely it is to perceive an obstacle P(z|x,m) Sensor probability Oi : Nearest point to obstacles O1 O2 O3 Zmax

23. San Jose Tech Museum Occupancy grid map Likelihood field

24. Scan Matching • Extract likelihood field from scan and use it to match different scan. Sensor scan, 180 dots The darker a region, the smaller likelihood for sensing an object there

25. Properties of Likelihood Fields Model • Highly efficient, uses 2D tables only. • Smooth w.r.t. to small changes in robot position. • Allows gradient descent, scan matching. • Ignores physical properties of beams. • Will it work for ultrasound sensors?

26. Additional Models of Proximity Sensors • Map matching (sonar,laser): generate small, local maps from sensor data and match local maps against global model. (e.g., transform scans into occupancy maps) • Scan matching (laser): map is represented by scan endpoints, match scan into this map. (e.g. ICP algorithm to stitch scans together) • Features (sonar, laser, vision): Extract features such as doors, hallways from sensor data.

27. Features/Landmarks • Active beacons (e.g., radio, GPS) • Passive (e.g., visual, retro-reflective) • Standard approach is triangulation • Sensor provides • distance, or • bearing, or • distance and bearing.

28. Distance and Bearing

29. Feature-Based Measurement Model • Feature vector is abstracted from the measurement: • Sensors that measure range,bearing, & a signature (a numerical value, e.g., an average color) • Conditional independence between features • Feature-Based map: with i.e., a location coordinate in global coordinates & a signature • Robot pose: • Measurement model: Zero-mean Gaussian error variables with standard deviations

30. Sensor Model with Known Correspondence • A key problem for range/bearing sensors: data association, (arise when the landmarks cannot be uniquely identified) • Introduce a correspondence variable , between feature and landmark . If , then the i-th feature observed at time t corresponds to the j-th landmark in the map Algorithm for calculating the probability of a landmark measurement: inputs: feature , with know correspondence , the robot pose and the map, Its output is the numerical probability

31. Summary of Sensor Models • Explicitly modeling uncertainty in sensing is key to robustness. • In many cases, good models can be found by the following approach: • Determine parametric model of noise free measurement. • Analyze sources of noise. • Add adequate noise to parameters (eventually mix in densities for noise). • Learn (and verify) parameters by fitting model to data. • Likelihood of measurement is given by “probabilistically comparing” the actual with the expected measurement. • This holds for motion models as well. • It is extremely important to be aware of the underlying assumptions!

32. Thanks Homework 4, 5 posted

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