Motion Models (cont) 1 2/17/2017

Transcription

1 Motion Models (cont) 1 /17/017

2 Sampling from the Velocity Motion Model suppose that a robot has a map of its enironment and it needs to find its pose in the enironment this is the robot localization problem seeral ariants of the problem the robot knows where it is initially the robot does not know where it is initially kidnapped robot: at any time, the robot can be teleported to another location in the enironment a popular solution to the localization problem is the particle filter uses simulation to sample the state density p x u, x ) ( t t t1 /17/017

3 Sampling from the Velocity Motion Model sampling the conditional density is easier than computing the density because we only require the forward kinematics model gien the control u t and the preious pose x t-1 find the new pose x t 3 /17/017

4 Sampling from the Velocity Motion Model /17/017 4 c c y x y x x t 1? y x x t t r cos sin y y x x c c Eqs 5.7, 5.8

5 Sampling from the Velocity Motion Model /17/017 5 t t t y x t t y t x y x c c ) cos( cos ) sin( sin ) cos( ) sin( Eqs 5.9 *we already deried this for the differential drie!

6 Sampling from the Velocity Motion Model /17/017 6 as with the original motion model, we will assume that gien noisy elocities the robot can also make a small rotation in place to determine the final orientation of the robot t t t t y x y x ) cos( cos ) sin( sin

7 Sampling from the Velocity Motion Model 7 /17/017

8 Sampling from the Velocity Motion Model the function sample(b ) generates a random sample from a zero-mean distribution with ariance b Matlab is able to generate random numbers from many different distributions help randn help stats 8 /17/017

9 How to Sample from Normal or Triangular Distributions? Sampling from a normal distribution 1. Algorithm sample_normal_distribution(b):. return Sampling from a triangular distribution 1. Algorithm sample_triangular_distribution(b):. return 9

10 Normally Distributed Samples samples

11 For Triangular Distribution 10 3 samples 10 4 samples samples 10 6 samples

12 Rejection Sampling Sampling from arbitrary distributions 1. Algorithm sample_distribution(f,b):. repeat until ( ) 6. return 1

13 Examples 13 /17/017

14 Odometry Motion Model many robots make use of odometry rather than elocity odometry uses a sensor or sensors to measure motion to estimate changes in position oer time typically more accurate than elocity motion model, but measurements are aailable only after the motion has been completed technically a measurement rather than a control but usually treated as control to simplify the modeling odometry allows a robot to estimate its pose but no fixed mapping from odometer coordinates and world coordinates in wheeled robots the sensor is often a rotary encoder 14 /17/017

15 Example Wheel Encoders These modules require +5V and GND to power them, and proide a 0 to 5V output. They proide +5V output when they "see" white, and a 0V output when they "see" black. These disks are manufactured out of high quality laminated color plastic to offer a ery crisp black to white transition. This enables a wheel encoder sensor to easily see the transitions. 15 Source:

16 Odometry Model bar indicates odometer coordinates Robot moes from x, y, to x', y', '. x', y', ' x, y,

17 Odometry Model bar indicates odometer coordinates Robot moes from x, y, to x', y', '. Step 1: rotate in place by δ rot1 x', y', ' x, y, rot1

18 Odometry Model bar indicates odometer coordinates Robot moes from x, y, to x', y', '. Step 1: rotate in place by δ rot1 Step : moe to x, y x', y', ' x, y, rot1 trans

19 Odometry Model bar indicates odometer coordinates Robot moes from x, y, to x', y', '. Step 1: rotate in place by δ rot1 Step : moe to x, y Step 3: rotate in place δ rot rot x', y', ' x, y, rot1 trans

20 Odometry Model bar indicates odometer coordinates Robot moes from x, y, to x', y', '. Odometry information. u,, rot1 rot trans trans ( x' x) ( y' y atan( y' y, x' ) rot1 x rot ' rot1 ) rot x', y', ' x, y, rot1 trans

21 Noise Model for Odometry The measured motion is gien by the true motion corrupted with noise. rot1 trans rot rot1 trans rot rot 1 rot trans 4 trans ( rot 1 trans rot )

22 Sample Odometry Motion Model 1. Algorithm sample_motion_model(u, x): u,, rot1 rot trans, x x, y, rot1 rot 1 sample( 1 rot 1 trans) sample( ( trans trans 3 trans 4 rot 1 trans rot rot sample( 1 rot trans) )) x x cos( trans y y sin( ' rot1 ' trans rot1 ' rot 1 rot ) ) sample_normal_distribution 7. Return x', y', '

23 Sampling from Our Motion Model Start