I ntroduction to Mobile Robotics Probabilistic Motion Models - - PowerPoint PPT Presentation

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Probabilistic Motion Models I ntroduction to Mobile Robotics

Wolfram Burgard

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Robot Motion

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

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Dynam ic Bayesian Netw ork for Controls, States, and Sensations

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Probabilistic Motion Models

• To implement the Bayes Filter, we need the

transition model .

• The term specifies a posterior

probability, that action ut carries the robot from xt-1 to xt.

• In this section we will discuss, how

can be modeled based on the motion equations and the uncertain

• utcome of the movements.

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Coordinate System s

• The configuration of a typical wheeled robot in 3D

can be described by six parameters.

• This are the three-dimensional Cartesian

coordinates plus the three Euler angles for roll, pitch, and yaw.

• For simplicity, throughout this section we consider

robots operating on a planar surface.

• The state space of such

systems is three- dimensional (x,y,θ).

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Typical Motion Models

• In practice, one often finds two types of

motion models:

• Odom etry-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.

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Exam ple W heel Encoders

These modules provide + 5V output when they "see" white, and a 0V

• utput 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/

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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.

• Historically used to log the position of ships.

[ Image source: Wikipedia, LoKiLeCh]

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Reasons for Motion Errors of W heeled Robots

bump ideal case different wheel diameters carpet and many more …

Odom etry Model

2 2

) ' ( ) ' ( y y x x

trans

− + − = δ θ δ − − − = ) ' , ' ( atan2

1

x x y y

rot 1 2

'

rot rot

δ θ θ δ − − =

• Robot moves from to .
• Odometry information .

θ , , y x ' , ' , ' θ y x

trans rot rot

u δ δ δ , ,

2 1

=

trans

δ

1 rot

δ

2 rot

δ

θ , , y x ' , ' , ' θ y x

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The atan2 Function

• Extends the inverse tangent and correctly

copes with the signs of x and y.

Noise Model for Odom etry

• The measured motion is given by the true

motion corrupted with noise.

| | | | 1 1

2 1 1

ˆ

trans rot

rot rot δ α δ α

ε δ δ

+

+ =

| | | | 2 2

2 2 1

ˆ

trans rot

rot rot δ α δ α

ε δ δ

+

+ =

|) | | | ( | |

2 1 4 3

ˆ

rot rot trans

trans trans δ δ α δ α

ε δ δ

+ +

+ =

Typical Distributions for Probabilistic Motion Models

2 2 2

2 1 2

2 1 ) (

σ σ

πσ ε

x

e x

−

=      − > =

2 2 2

6 | | 6 6 | x | if ) (

2

σ σ σ εσ x x

Normal distribution Triangular distribution

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Calculating the Probability Density ( zero-centered)

• For a normal distribution
• For a triangular distribution

1. Algorithm prob_ norm al_ distribution(a,b): 2. return 1. Algorithm prob_ triangular_ distribution(a,b): 2. return query point

• std. deviation

1. Algorithm m otion_ m odel_ odom etry( x, x’,u) 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. return p1 · p2 · p3

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Calculating the Posterior Given x, x’, and Odom etry

2 2

) ' ( ) ' ( y y x x

trans

− + − = δ θ δ − − − = ) ' , ' ( atan2

1

x x y y

rot 1 2

'

rot rot

δ θ θ δ − − =

2 2

) ' ( ) ' ( ˆ y y x x

trans

− + − = δ θ δ − − − = ) ' , ' ( atan2 ˆ

1

x x y y

rot 1 2

ˆ ' ˆ

rot rot

δ θ θ δ − − = ) | | , ˆ ( prob

trans 2 1 rot 1 1 rot 1 rot 1

δ α δ α δ δ + − = p |)) | | (| , ˆ ( prob

rot2 rot1 4 trans 3 trans trans 2

δ δ α δ α δ δ + + − = p ) | | , ˆ ( prob

trans 2 2 rot 1 2 rot 2 rot 3

δ α δ α δ δ + − = p

• dometry params (u)

values of interest (x,x’)

• dometry

hypotheses

Application

• Repeated application of the motion model

for short movements.

• Typical banana-shaped distributions
• btained for the 2d-projection of the 3d

posterior.

x u u x

Sam ple-Based Density Representation

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Sam ple-Based Density Representation

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How to Sam ple from a Norm al Distribution?

• Sampling from a normal distribution

1. Algorithm sam ple_ norm al_ distribution(b): 2. return

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Norm ally Distributed Sam ples

106 samples

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How to Sam ple from Norm al or Triangular Distributions?

• Sampling from a normal distribution
• Sampling from a triangular distribution

1. Algorithm sam ple_ norm al_ distribution(b): 2. return 1. Algorithm sam ple_ triangular_ distribution(b): 2. return

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For Triangular Distribution

103 samples 104 samples 106 samples 105 samples

How to Obtain Sam ples from Arbitrary Functions?

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Rejection Sam pling

• Sampling from arbitrary distributions
• Sample x from a uniform distribution from [-b,b]
• Sample c from [0, max f]
• if f(x) > c

keep the sample

• therwise

reject the sample

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c x f(x) c’ x’ f(x’)

OK

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Rejection Sam pling

• Sampling from arbitrary distributions

1. Algorithm sam ple_ distribution(f,b): 2. repeat 3. 4. 5. until ( ) 6. return

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Exam ple

• Sampling from

Sam ple Odom etry Motion Model

1. Algorithm sam ple_ m otion_ m odel(u, x): 1. 2. 3. 4. 5. 6. 7. Return

) | | sample( ˆ

2 1 1 1 1 trans rot rot rot

δ α δ α δ δ + + = |)) | | (| sample( ˆ

2 1 4 3 rot rot trans trans trans

δ δ α δ α δ δ + + + = ) | | sample( ˆ

2 2 1 2 2 trans rot rot rot

δ α δ α δ δ + + = ) ˆ cos( ˆ '

1 rot trans

x x δ θ δ + + = ) ˆ sin( ˆ '

1 rot trans

y y δ θ δ + + =

2 1

ˆ ˆ '

rot rot

δ δ θ θ + + = ' , ' , ' θ y x θ δ δ δ , , , , ,

2 1

y x x u

trans rot rot

= =

sam ple_ norm al_ distribution

Exam ples ( Odom etry-Based)

Sam pling from Our Motion Model

Start

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Velocity-Based Model

θ-90

Noise Model for the Velocity- Based Model

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• The measured motion is given by the true

motion corrupted with noise.

• Discussion: What is the disadvantage of this

noise model?

| | | |

2 1

ˆ

ω α α

ε

+

+ =

v

v v

| | | |

4 3

ˆ

ω α α

ε ω ω

+

+ =

v

Noise Model for the Velocity- Based Model

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• The -circle constrains the final
• rientation (2D manifold in a 3D space)
• Better approach:

| | | |

2 1

ˆ

ω α α

ε

+

+ =

v

v v

| | | |

6 5

ˆ

ω α α

ε γ

+

=

v | | | |

4 3

ˆ

ω α α

ε ω ω

+

+ =

v

Term to account for the final rotation

) ˆ , ˆ ( ω v

Motion I ncluding 3 rd Param eter

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Term to account for the final rotation

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Equation for the Velocity Model

Center of circle:

some constant (distance to ICC) (center of circle is orthogonal to the initial heading)

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Equation for the Velocity Model

Center of circle:

some constant some constant (the center of the circle lies

• n a ray half way between x and x’ and is
• rthogonal to the line between x and x’)

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Equation for the Velocity Model

Center of circle:

some constant

Allows us to solve the equations to:

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Equation for the Velocity Model

and

Equation for the Velocity Model

• The parameters of the circle:
• allow for computing the velocities as

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Posterior Probability for Velocity Model

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Sam pling from Velocity Model

Exam ples ( Velocity-Based)

Map-Consistent Motion Model

) , | ' ( x u x p

) , , | ' ( m x u x p ≠

) , | ' ( ) | ' ( ) , , | ' ( x u x p m x p m x u x p η =

Approximation:

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Sum m ary

• We discussed motion models for odometry-based

and velocity-based systems

• We discussed ways to calculate the posterior

probability p(x’| x, u).

• We also described how to sample from 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 how to improve this motion

model to take the map into account.