1. Advanced Mobile Robotics
Dr. Jizhong Xiao
Department of Electrical Engineering
CUNY City College

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

12. Beam-based Proximity Model
Random measurement
Max range/Failures
zexp
zmax
zexp
zmax
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. Measured distances for expected distance of 300 cm.
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