11/12/2007 11/12/2007 CSE-571 Probabilistic Robotics 2 Outputs - - PowerPoint PPT Presentation

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11/12/2007 11/12/2007 CSE-571 Probabilistic Robotics 2 Outputs - - PowerPoint PPT Presentation

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11/12/2007 11/12/2007 CSE-571 Probabilistic Robotics 2 Outputs are noisy function of inputs: Non-parametric regression model Distribution over functions y f ( x ) i i Fully specified by training data and

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 Non-parametric regression model  Distribution over functions  Fully specified by training data and kernel

function

 Output variables are jointly Gaussian  Covariance given by distance of inputs in

kernel space

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 Outputs are noisy function of inputs:  Function values are jointly Gaussian:  Considering noise:

( )

i i

y f    x

 

2 2 2

1 co v ( , ( ( , ) e ) ) x p | | 2

p q p q f p q

f f k l            x x x x x x

 

2 2

c o v , ( , ) ( | ) ( , ( , ) )

n p q p q p q n

y y k p K         x x Y X X X I

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 Training data:  Prediction given training samples:

     

* * 2 1 * 1 2 * * 2 * 2 *

( | , , ) , ( , ) ( , ) ( , ) ( , ) ( , ) ( , )

n n

p y N K K K K K K      

 

      x y X X X X X y x x x X X X X x I I

11/12/2007 5 CSE-571 Probabilistic Robotics 1 1 2 2

{( , ), ( , ), , ( , )} ( , )

n n

D y y y    x x x X y

 Maximize data log likelihood:  Compute derivatives wrt. params  Optimize using conjugate gradient  

1 2 2

log ( | ) 1 1 ( , ) log ( , ) log 2 2 2 2

T n n

p n K K   

      y I I y X X X y X X

2 2

, ,

n f

l      

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Mean Variance

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[Ferris-Haehnel-Fox: RSS-06]

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 GP-LVM: GP with latent / unobserved

variables (locations)

 Can incorporate motion constraints

[Ferris-Fox-Lawrence: IJCAI-07]

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 Controller development benefits from

accurate model

 Two approaches to system modeling

  • Parametric / physics-based models
  • Non-parametric / data-driven models

 Combining these two approaches yields

superior model

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 12-D state=[pos,rot,transvel,rotvel]  Describes evolution of state as ODE  Forces / torques considered: buoyancy, gravity, drag, thrust  16 parameters are learned by optimization on ground truth

motion capture data

                             

 

 

) * ( ) * ( ) (

1 1

      J Torques J Mv Forces M H v R v p dt d s

e b

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 ODE can be used to generate a step ahead

prediction function )) ( ), ( ( ) ( ) 1 ( k u k s f k s k s    Problems

 Limited accuracy  Noise not explicit in model

f

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) ( )], ( ), ( [ k T k u k s D

gt

GP 

) ( ) 1 ( ) ( k s k s k T

gt gt

  

State transition learned directly

)]) ( ), ( ([ ) ( ) 1 ( k u k s g k s k s   

GP prediction Training data for GP: Problems:

 Generalizes poorly  Full coverage of state space difficult

Grimes et al 2006 Kocijan et al. 2005

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f )]) ( ), ( ([ )) ( ), ( ( ) ( ) 1 ( k u k s g k u k s f k s k s

EGP

   

Enhanced-GP model equation

)) ( ), ( ( ) ( ) 1 ( ) ( k u k s f k s k s k T

gt gt gt

   

Target output takes parametric model into account Better accuracy Less training data necessary Noise incorporated into system

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cameras

Mocap system Mocap system

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 State: xyz pitch yaw  Observation: xc yc width height theta  Models: Parametric using computer graphics/vision, GP, EGP

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Propagation method pos(mm) rot(deg) vel(mm/s) rotvel(deg/s) Param 3.3 0.5 14.6 1.5 GPonly 1.8 0.2 9.8 1.1 EGP 1.6 0.2 9.6 1.3

  • Single step prediction error
  • ¼ sec timesteps
  • Avg over ~1000 test points

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Modeling method pos(pix) Major axis(pix) Minor axis(pix) Theta(deg) Param 7.1 2.9 5.7 9.2 Gponly 4.7 3.2 1.9 9.1 EGP 3.9 2.4 1.9 9.4

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 Sequential state estimation  Prediction step:  Correction step:

u(k-1) s(k-1) z(k-1) u(k) s(k) z(k) u(k+1) s(k+1) z(k+1)

) , | (

1 t t t

u s s p

) | (

t t

s z p

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Process model Observation model

 Linear dynamical system  Extended-KF / Unscented-KF: Locally linearized

state propagation and observation models

u(k-1) s(k-1) s(k) z(k) z(k-1) u(k) u(k+1) s(k+1) z(k+1) R(k) Q(k)  

) 1 ( ) ( ), (   t s t u t s g

 

) ( ) ( t z t s h 

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 

 

 

           

n i T i i i c i n i i m i i

u g n

2 ' ' ' 2 '

) )( ( ) , ( : 2 ... i for                

 Determine sigma points based on covariance  Propagate using nonlinear function g  Use propagated sigma points to recreate mean and

covariance

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Q(k) R(k) GP R(k) GP Q(k) Process model Observation model GP Process model GP Observation model

 Use GP process and observation models  Replace static noise parameters with

uncertainty from GP

u(k-1) s(k-1) s(k) z(k)

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, , ,   u z

 GP-UKF( ):

  • Determine sigma points
  • Recover new mean and sigma
  • Determine sigma points
  • Recover new mean and sigma
  • Perform correction

 

1 1,  

k i k g i k

X GP X u

 

1 1,   

k k g k

GP R u 

 

i k h i k

X GP Z ˆ ˆ

 

k h k

GP Q  ˆ

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Propagation method pos(mm) rot(deg) vel(mm/s) rotvel(deg/s) Param 3.3 0.5 14.6 1.5 GPonly 1.8 0.2 9.8 1.1 EGP 1.6 0.2 9.6 1.3 Modeling method pos(pix) Major axis(pix) Minor axis(pix) Theta(deg) Param 7.1 2.9 5.7 9.2 GPonly 4.7 3.2 1.9 9.1 EGP 3.9 2.4 1.9 9.4

Prediction error Observation error

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Tracking algorithm pos(mm) rot(deg) vel(mm/s) rotvel(deg/s) MLL UKF 141 9.6 141.5 8.1 2.1 GP-UKF (GPonly) 107.9 10.2 71.7 5.9 5.1 GP-UKF (EGP) 86 6.1 57.1 5.7 12.9

  • Average tracking error
  • Trajectory ~12 min long
  • 0.5 sec timesteps

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Full process model tracking No right turn process model tracking

  • Training data for right turns removed
  • Uncertainty increases appropriately

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 GPs provide flexible modeling framework  Take data noise and uncertainty due to data

sparsity into account

 Combination with parametric models

increases accuracy and reduces need for training data

 Seamless integration into Bayes filters  Compexity is problem:

  • Training: Prediction

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) (

3

n O

2

( ) O n

 Complexity can be reduced by removing / ignoring data

points (sparse GP)

 Input dependent signal noise (heteroscedastic GP)  Input dependent kernel parameters  Can be used for dimensionality reduction (e.g. GP-LVM)  Uncertainty provides means for active exploration and

  • ptimal sensor placement

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