Probabilistic Fundamentals in Robotics. DAUIN Politecnico di Torino July 2010

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Gianmarco Corradini
4 anni fa
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1 Probabilistic Fundamentals in Robotics Probabilistic Models of Mobile Robots Robot Motion Basilio Bona DAUIN Politecnico di Torino July 2010

2 Course Outline Basic mathematical framework Probabilistic models of mobile robots Mobile robot localization problem Robotic mapping Probabilistic planning and control Reference textbook Thrun, Burgard, Fox, Probabilistic Robotics, MIT Press, robotics.org/ Basilio Bona July

3 Probabilistic models of mobile robots Robot motion Kinematics Velocity motion model Odometry motion model Robot perception Maps Beam model of laser range finders Correlation based measurement models Feature based measurement models Basilio Bona July

4 Introduction Basilio Bona July

5 Kinematic states yt () orientation θ () t xt () Basilio Bona July

6 Probabilistic kinematics State (pose or location) control From Wikipedia: In applications, controls are sometimes provided by rover odometry Odometry is the use of data from the movement of actuators to estimate change in position over time. Odometry is used by some robots to estimate their position relative to a starting location. The method is sensitive to errors due to the integration of velocity measurements over time to give position estimates. Rapid and accurate data collection, equipment calibration, i and processing are required din most cases for odometry to be used effectively. Basilio Bona July

7 Example y y x x darker points show higher probabilities of being there the orientation is not shown, but contributes to the uncertainty of the final location Basilio Bona July

Motion models Velocity model: the simplest one, assumes that the control is given as a velocity command to the motors; velocity remain constant in the sampling interval [t 1, t) Acceleration model:

8 Motion models Velocity model: the simplest one, assumes that the control is given as a velocity command to the motors; velocity remain constant in the sampling interval [t 1, t) Acceleration model: is slightly more complicated, assuming a constant acceleration motion, i.e., a linearly increasing velocity Odometric model: assumes the accessibility to odometric information, usually provided by wheel sensors, but often also by other means (i.e., visual odometry) Basilio Bona July

9 Motion models Odometric models are usually more accurate than velocity models, but odometry is available only after the motion command has been executed, while velocity commands are available before performing the actual motion Odometric models are good for estimation, while velocity models dlare btt better suited for path planning Basilio Bona July

10 Velocity motion model Basilio Bona July

11 Velocity motion model: noise free x u = ( x y θ ) t t t t = ( v ω ) t t t T T y t r t θ t y c θ t 90 x c x t Basilio Bona July

12 Velocity motion model: noise free x t x t t 11 Δθ Δθ r t = v t ω c is negative t Basilio Bona July

13 Exact velocity model Basilio Bona July

14 Velocity models Exact Euler Runge Kutta Basilio Bona July

15 Velocity models Exact Euler Runge Kutta Basilio Bona July

16 Odometry errors Basilio Bona July

17 Error noise Basilio Bona July

18 Velocity model with error noise Basilio Bona July

19 Velocity motion model algorithm Basilio Bona July

20 Example Basilio Bona July

21 Odometry motion model Odometry is obtained integrating sensor reading from wheel encoders,or or from other sources (e.g., visual odometry) Odometry provides the information of the relative motion of the robot. Odometry is more accurate than velocity Odometry measurements are available only AFTER a control has been supplied to the robot, then they should be better considered as measurements, but usually the are included as control signals u t For this reason odometry cannot be used for planning and control Basilio Bona July

22 Odometry motion model Odometry model considers the motion in the time interval 1. A first rotation 2. A translation 3. A second rotation Each of them is noisy Basilio Bona July

23 Odometry model Basilio Bona July

24 Example Repeated application of the sensor model for short movements Typical banana shaped distributions obtained for 2D projection of 3D posterior Basilio Bona July

25 Example Basilio Bona July

26 Sampling One can use normal (Gaussian) distributions or triangular distributions for describing uncertainty and for sampling Normal distribution Triangular distribution Basilio Bona July

27 How to Sample from Normal or Triangular Distributions? Sampling from a normal distribution Sampling from a triangular distribution Basilio Bona July

28 Normally distributed samples Basilio Bona July

29 Triangular distributed samples 10 3 samples 10 4 samples 10 5 samples 10 6 samples Basilio Bona July

30 Sample odometry motion model Sample normaldistribution Basilio Bona July

31 Example Start Basilio Bona July

32 Motion and maps In many cases we have a map m that represents the environment wherethe robot moves Occupacy maps distinguish free (traversable) from occupied space: robot pose shall be always in freespace A motion model that takes into consideration a map computes Map based motion model If the map m carries information relevant to pose estimation Basilio Bona July

33 Approximation Map free estimate Consistency on the pose with the map This is the result of checking model consistency at the final pose, instead of verifying it on the entire path to the goal Basilio Bona July

34 Thank you. Any question? Basilio Bona July

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