Probabilistic Robotics: Motion Model/EKF Localization

transcript

City College of New York

Dr. Jizhong Xiao

Department of Electrical Engineering

CUNY City College

jxiao@ccny.cuny.edu

Probabilistic Robotics: Motion Model/EKF Localization

Advanced Mobile Robotics

Robot Motion• Robot motion is inherently uncertain.• How can we model this uncertainty?

• Prediction (Action)

• Correction (Measurement)

Bayes Filter Revisit

111 )(),|()( tttttt dxxbelxuxpxbel

)()|()( tttt xbelxzpxbel

111 )(),|()|()( tttttttt dxxBelxuxPxzPxBel

Probabilistic Motion Models

• To implement the Bayes Filter, we need the transition model p(xt | xt-1, u).

• The term p(xt | xt-1, u) specifies a posterior probability, that action u carries the robot from xt-1 to xt.

• In this section we will specify, how p(xt | xt-1, u) can be modeled based on the motion equations.

Coordinate Systems• In general the configuration of a robot can be described

by six parameters.

• Three-dimensional cartesian coordinates plus three Euler angles pitch, roll, and tilt.

• Throughout this section, we consider robots operating on a planar surface.

• The state space of such systems is three-dimensional (x, y, ).

Typical Motion Models• In practice, one often finds two types of motion

models:

– Odometry-based

– Velocity-based (dead reckoning)

• Odometry-based models are used when systems are equipped with wheel encoders.

• Velocity-based models have to be applied when no wheel encoders are given.

• They calculate the new pose based on the velocities and the time elapsed.

Example Wheel EncodersThese modules require +5V and GND to power them, and provide a 0 to 5V output. They provide +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 very crisp black to white transition. This enables a wheel encoder sensor to easily see the transitions.

Source: http://www.active-robots.com/

Dead Reckoning• Derived from “deduced reckoning.”• Mathematical procedure for determining the present

location of a vehicle.• Achieved by calculating the current pose of the

vehicle based on its velocities and the time elapsed.

• Odometry tends to be more accurate than velocity model,

• But, Odometry is only available after executing a motion command, cannot be used for motion planning

Reasons for Motion Errors

ideal casedifferent wheeldiameters

carpetand many more …

Odometry Model

22 )'()'( yyxxtrans

)','(atan21 xxyyrot

12 ' rotrot

• Robot moves from to . • Odometry information .

,, yx ',',' yx

transrotrotu ,, 21

trans1rot

',',' yx

Relative motion information, “rotation” “translation” “rotation”

The atan2 Function• Extends the inverse tangent and correctly copes with the signs of x and y.

Noise Model for Odometry

• The measured motion is given by the true motion corrupted with independent noise.

||||11 211

ˆtransrotrotrot

||||22 221

ˆtransrotrotrot

|||| 2143

ˆrotrottranstranstrans

Typical Distributions for Probabilistic Motion Models

6|x|if0)(2

Normal distribution Triangular distribution

Calculating the Probability (zero-centered)

• For a normal distribution

• For a triangular distribution

1. Algorithm prob_normal_distribution(a, b):

2. return

1. Algorithm prob_triangular_distribution(a,b):

2. return

Calculating the Posterior Given xt, xt-1, and u

22 )'()'( yyxxtrans )','(atan21 xxyyrot

12 ' rotrot 22 )'()'(ˆ yyxxtrans )','(atan2ˆ

1 xxyyrot

12ˆ'ˆrotrot

)ˆ|ˆ|,ˆ(prob trans21rot11rot1rot1 p|))ˆ||ˆ(|ˆ,ˆ(prob rot2rot14trans3transtrans2 p

)ˆ|ˆ|,ˆ(prob trans22rot12rot2rot3 p

1. Algorithm motion_model_odometry (xt, xt-1, u)

11. return p1 · p2 · p3

odometry values (u)

values of interest (xt-1, xt)

Ttt xxu 1

Tt yxx )(1

Tt yxx )(

An initial pose Xt-1

A hypothesized final pose Xt

A pair of poses u obtained from odometry

),( baprobImplements an error distribution over a with zero mean and standard deviation b),( 1ttt xuxp

Application• Repeated application of the sensor model for

short movements.• Typical banana-shaped distributions obtained for

2d-projection of 3d posterior.

x’ u

p(xt| u, xt-1)

Posterior distributions of the robot’s pose upon executing the motion command illustrated by the solid line. The darker a location, the more likely it is.

Velocity-Based Model

ucontrol v

r Rotation radius

Equation for the Velocity ModelInstantaneous center of curvature (ICC) at (xc , yc)

sinrxx c cosryy c

Initial pose Tt yxx 1

Keeping constant speed, after ∆t time interval, ideal robot will be at T

)cos(cos

)sin(sin Corrected, -90

Velocity-based Motion Model

With and are the state vectors at time t-1 and t respectively

)ˆcos(ˆ

)ˆsin(ˆ

Tt yxx 1 Tt yxx '''

The true motion is described by a translation velocity and a rotational velocity

tv tMotion Control with additive Gaussian noise

221 )(

t Mvvv

Tttt vu )(

Circular motion assumption leads to degeneracy ,2 noise variables v and w 3D poseAssume robot rotates when arrives at its final pose

Velocity-based Motion ModelMotion Model:

)ˆcos(ˆ

)ˆsin(ˆ

221 )(

1 to 4 are robot-specific error parameters determining the velocity control noise

5 and 6 are robot-specific error parameters determining the standard deviation of the additional rotational noise

Probabilistic Motion Model

Center of circle:

How to compute ?),( 1ttt xuxp Move with a fixed velocity during ∆t resulting in a circular trajectory from to

Tt yxx 1 T

Radius of the circle:

2*2*2*2** )()()()( yyxxyyxxr

Change of heading direction: ),(2tan),(2tan **** xxyyaxxyya

(angle of the final rotation)

Posterior Probability for Velocity Model

Motion error: verr ,werr and

Center of circle

Radius of the circle

Change of heading direction

Examples (velocity based)

Map-Consistent Motion Model

)',|( xuxp

)',|()|(),',|( xuxpmxpmxuxp Approximation:

),',|( mxuxp)',|( xuxp

Map free estimate of motion model

)|( mxp“consistency” of pose in the map

“=0” when placed in an occupied cell

Obstacle grown by robot radius

Summary

• We discussed motion models for odometry-based and velocity-based systems

• We discussed ways to calculate the posterior probability p(x| x’, u).

• Typically the calculations are done in fixed time intervals t.

• In practice, the parameters of the models have to be learned.

• We also discussed an extended motion model that takes the map into account.

Localization, Where am I??

• Given – Map of the environment.– Sequence of measurements/motions.

• Wanted– Estimate of the robot’s position.

• Problem classes– Position tracking (initial robot pose is known)– Global localization (initial robot pose is unknown)– Kidnapped robot problem (recovery)

Markov Localization

Markov Localization: The straightforward application of Bayes filters to the localization problem

• Prediction (Action)

• Correction (Measurement)

Bayes Filter Revisit

111 )(),|()( tttttt dxxbelxuxpxbel

)()|()( tttt xbelxzpxbel

111 )(),|()|()( tttttttt dxxBelxuxPxzPxBel

• Prediction:

• Correction:

EKF Linearization

)(),(),(

),(),(

ttttttt

tttttt

xGugxug

ugugxug

)()()(

First Order Taylor Expansion

EKF Algorithm 1. Extended_Kalman_filter( t-1, t-1, ut, zt):

2. Prediction:3. 4.

5. Correction:6. 7. 8.

9. Return t, t

),( 1 ttt ug

tTtttt RGG 1

1)( tTttt

Tttt QHHHK

))(( ttttt hzK

tttt HKI )(

ttttt uBA 1

tTtttt RAA 1

1)( tTttt

Tttt QCCCK

)( tttttt CzK

tttt CKI )(

1. EKF_localization ( t-1, t-1, ut, zt, m):

Prediction:

),( 1 ttt ug T

tttTtttt VMVGG 1

,1,1,1

Motion noise covariance

Matrix from the control

Jacobian of g w.r.t location

Predicted mean

Predicted covariance

Jacobian of g w.r.t control

With and are the state vectors at time t-1 and t respectively

)ˆcos(ˆ

)ˆsin(ˆ

Tt yxx 1 Tt yxx '''

The true motion is described by a translation velocity and a rotational velocity

Motion Control with additive Gaussian noise

221 )(

t Mvvv

Tttt vu )(

),0()cos(cos

)sin(sin

),0(),( 1 tttt RNxugx

)(),(),(

),(),(

ttttttt

tttttt

xGugxug

ugugxug

Motion Model:

),0()cos(cos

)sin(sin

,1,1,1

),(),(

Derivative of g along x’ dimension, w.r.t. x at

Jacobian of g w.r.t location

),0()cos(cos

)sin(sin

Derivative of g w.r.t. the motion parameters, evaluated at and

ttvtvt

)sin())cos((cos)cos(cos

)cos())sin((sin)sin(sin

Ttttt VMVGG 1

Mapping between the motion noise in control space to the motion noise in state space

Jacobian of g w.r.t control

1. EKF_localization ( t-1, t-1, ut, zt, m):

Correction:

)ˆ( ttttt zzK

tttt HKI

,2atanˆ

txtxyty

ytyxtxt

tTtttt QHHS

1 tTttt SHK

Predicted measurement mean

Pred. measurement covariance

Kalman gain

Updated mean

Updated covariance

Jacobian of h w.r.t location

Feature-Based Measurement Model

)),(2tan

)()( ttt

),0(),,( ttit QNmjxhz

• Jacobian of h w.r.t location

Is the landmark that corresponds to the measurement of

EKF Localizationwith known

correspondences

EKF Localizationwith unknown

correspondences

Maximum likelihood estimator

EKF Prediction Step

EKF Observation Prediction Step

EKF Correction Step

Estimation Sequence (1)

Estimation Sequence (2)

Comparison to Ground Truth

UKF Localization?

• Given – Map of the environment.– Sequence of measurements/motions.

• Wanted– Estimate of the robot’s position.

• UKF localization

Unscented Transform

2,...,1for )(2

)1( 2000

Sigma points Weights

)( ii g

Pass sigma points through nonlinear function

Recover mean and covarianceFor n-dimensional Gaussianλ is scaling parameter that determine how far the sigma points are spread from the meanIf the distribution is an exact Gaussian, β=2 is the optimal choice.

UKF_localization ( t-1, t-1, ut, zt, m):

Prediction:

at 000011

at 111111

xt ug 1,

Motion noise

Measurement noise

Augmented state mean

Augmented covariance

Sigma points

Prediction of sigma points

Predicted mean

Predicted covariance

UKF_localization ( t-1, t-1, ut, zt, m):

Correction:

imt wz

Measurement sigma points

Predicted measurement mean

Pred. measurement covariance

Cross-covariance

Kalman gain

Updated mean

Updated covariance

ict zzwS ˆˆ ,

zxt zw ˆ,

1, tzx

)ˆ( ttttt zzK

Tttttt KSK

UKF Prediction Step

UKF Observation Prediction Step

UKF Correction Step

EKF Correction Step

Estimation Sequence

EKF PF UKF

Estimation Sequence

EKF UKF

Prediction Quality

EKF UKF

• [Arras et al. 98]:

• Laser range-finder and vision

• High precision (<1cm accuracy)

Kalman Filter-based System

[Courtesy of Kai Arras]

Multi-hypothesisTracking

• Belief is represented by multiple hypotheses

• Each hypothesis is tracked by a Kalman filter

• Additional problems:

• Data association: Which observation

corresponds to which hypothesis?

• Hypothesis management: When to add / delete

hypotheses?

• Huge body of literature on target tracking, motion

correspondence etc.

Localization With MHT

• Hypotheses are extracted from Laser Range Finder

(LRF) scans• Each hypothesis has probability of being the correct

• Hypothesis probability is computed using Bayes’ rule

• Hypotheses with low probability are deleted.

• New candidates are extracted from LRF scans.

MHT: Implemented System (1)

)}(,,ˆ{ iiii HPxH

},{ jjj RzC

)()|()|(

HPHsPsHP ii

[Jensfelt et al. ’00]

MHT: Implemented System (2)

Courtesy of P. Jensfelt and S. Kristensen

MHT: Implemented System (3)Example run

Map and trajectory

# hypotheses

#hypotheses vs. time

P(Hbest)

Courtesy of P. Jensfelt and S. Kristensen

Thank You

Probabilistic Robotics: Motion Model/EKF Localization

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