RI 16-735, Howie Choset, with slides from George Kantor, G.D. Hager, and D. Fox

The Problem •

What is the world around me (mapping) – sense from various positions – integrate measurements to produce map – assumes perfect knowledge of position

•

Where am I in the world (localization) – – – –

•

sense relate sensor readings to a world model compute location relative to model assumes a perfect world model

Together, these are SLAM (Simultaneous Localization and Mapping)

RI 16-735, Howie Choset, with slides from George Kantor, G.D. Hager, and D. Fox

Localization

Tracking: Known initial position Global Localization: Unknown initial position Re-Localization: Incorrect known position (kidnapped robot problem)

SLAM

Challenges – – – – – – – –

Sensor processing Position estimation Control Scheme Exploration Scheme Cycle Closure Autonomy Tractability Scalability

Mapping while tracking locally and globally

RI 16-735, Howie Choset, with slides from George Kantor, G.D. Hager, and D. Fox

Representations for Robot Localization Kalman filters (late-80s?) Discrete approaches (’95) • Topological representation (’95) • uncertainty handling (POMDPs) • occas. global localization, recovery • Grid-based, metric representation (’96) • global localization, recovery

Particle filters (’99) • sample-based representation • global localization, recovery

AI

• Gaussians • approximately linear models • position tracking

Robotics Multi-hypothesis (’00) • multiple Kalman filters • global localization, recovery

RI 16-735, Howie Choset, with slides from George Kantor, G.D. Hager, and D. Fox

Three Major Map Models

Grid-Based:

Feature-Based:

Topological:

Collection of discretized obstacle/free-space pixels

Collection of landmark locations and correlated uncertainty

Collection of nodes and their interconnections

Elfes, Moravec, Thrun, Burgard, Fox, Simmons, Koenig, Konolige, etc.

Smith/Self/Cheeseman, Durrant–Whyte, Leonard, Nebot, Christensen, etc.

Kuipers/Byun, Chong/Kleeman, Dudek, Choset, Howard, Mataric, etc.

RI 16-735, Howie Choset, with slides from George Kantor, G.D. Hager, and D. Fox

Three Major Map Models

Grid-Based Resolution vs. Scale

Computational Complexity Exploration Strategies

Feature-Based

Topological

Discrete localization

Arbitrary localization

Localize to nodes

Grid size and resolution

Landmark covariance (N2)

Minimal complexity

Frontier-based exploration

No inherent exploration

Graph exploration

RI 16-735, Howie Choset, with slides from George Kantor, G.D. Hager, and D. Fox

Atlas Framework

•

Hybrid Solution: – Local features extracted from local grid map. – Local map frames created at complexity limit. – Topology consists of connected local map frames.

Authors: Chong, Kleeman; Bosse, Newman, Leonard, Soika, Feiten, Teller

RI 16-735, Howie Choset, with slides from George Kantor, G.D. Hager, and D. Fox

H-SLAM

RI 16-735, Howie Choset, with slides from George Kantor, G.D. Hager, and D. Fox

What does a Kalman Filter do, anyway?

Given the linear dynamical system:

x ( k + 1) = F ( k ) x ( k ) + G ( k )u( k ) + v ( k ) y ( k ) = H ( k ) x ( k ) + w( k ) x ( k ) is the n - dimensional state vector (unknown) u( k ) is the m - dimensional input vector (known) y ( k ) is the p - dimensional output vector (known, measured) F ( k ), G ( k ), H ( k ) are appropriately dimensioned system matrices (known) v ( k ), w( k ) are zero - mean, white Gaussian noise with (known) covariance matrices Q( k ), R ( k )

the Kalman Filter is a recursion that provides the “best” estimate of the state vector x. RI 16-735, Howie Choset, with slides from George Kantor, G.D. Hager, and D. Fox

What’s so great about that? x ( k + 1) = F ( k ) x ( k ) + G ( k )u( k ) + v ( k ) y ( k ) = H ( k ) x ( k ) + w( k )

• • •

noise smoothing (improve noisy measurements) state estimation (for state feedback) recursive (computes next estimate using only most recent measurement)

RI 16-735, Howie Choset, with slides from George Kantor, G.D. Hager, and D. Fox

How does it work? x ( k + 1) = F ( k ) x ( k ) + G ( k )u( k ) + v ( k ) y ( k ) = H ( k ) x ( k ) + w( k ) 1. prediction based on last estimate:

xˆ ( k + 1 | k ) = F (k ) xˆ (k | k ) + G (k )u (k ) yˆ (k ) = H (k ) xˆ (k + 1 | k ) 2. calculate correction based on prediction and current measurement:

Δx = f ( y (k + 1), xˆ (k + 1 | k ) ) 3. update prediction:

xˆ ( k + 1 | k + 1) = xˆ (k + 1 | k ) + Δx

RI 16-735, Howie Choset, with slides from George Kantor, G.D. Hager, and D. Fox

Finding the correction (no noise!)

y = Hx Given prediction xˆ (k + 1 | k ) and output y, find Δx so that xˆ = xˆ (k + 1 | k ) + Δx is the " best" estimate of x. Want the best estimate to be consistent with sensor readings

xˆ (k+1|k) •

Ω = {x | Hx = y}

“best” estimate comes from shortest Δx

RI 16-735, Howie Choset, with slides from George Kantor, G.D. Hager, and D. Fox

Finding the correction (no noise!)

y = Hx Given prediction xˆ (k + 1 | 1) and output y, find Δx so that xˆ = xˆ (k + 1 | 1) + Δx is the " best" estimate of x. “best” estimate comes from shortest Δx shortest Δx is perpendicular to Ω

xˆ (k + 1 | k ) •

Δx

Ω = {x | Hx = y}

RI 16-735, Howie Choset, with slides from George Kantor, G.D. Hager, and D. Fox

Some linear algebra a is parallel to Ω if Ha = 0

Null ( H ) = {a ≠ 0 | Ha = 0}

Ω = {x | Hx = y}

a is parallel to Ω if it lies in the null space of H

for all v ∈ Null ( H ), v ⊥ b if b ∈ column( H T ) Weighted sum of columns means

b = Hγ , the weighted sum of columns

RI 16-735, Howie Choset, with slides from George Kantor, G.D. Hager, and D. Fox

Finding the correction (no noise!)

y = Hx Given prediction xˆ (k + 1 | k ) and output y, find Δx so that xˆ = xˆ (k + 1 | k ) + Δx is the " best" estimate of x. “best” estimate comes from shortest Δx shortest Δx is perpendicular to Ω

xˆ (k + 1 | k )

T ⊥ ⇒ Δx ∈ null(H ) ⇒ Δx ∈ column(H )

•

Δx

Ω = {x | Hx = y}

⇒ Δx = H T γ

RI 16-735, Howie Choset, with slides from George Kantor, G.D. Hager, and D. Fox

Finding the correction (no noise!)

y = Hx Given prediction xˆ (k + 1 | k ) and output y, find Δx so that xˆ = xˆ (k + 1 | k ) + Δx is the " best" estimate of x.

“best” estimate comes from shortest Δx shortest Δx is perpendicular to Ω

xˆ (k + 1 | k )

T ⊥ ⇒ Δx ∈ null(H ) ⇒ Δx ∈ column(H )

•

Δx

Ω = {x | Hx = y}

⇒ Δx = H T γ

Real output – estimated output

ν = y − H ( xˆ (k + 1 | k )) = H ( x − xˆ (k + 1 | k )) innovation RI 16-735, Howie Choset, with slides from George Kantor, G.D. Hager, and D. Fox

Finding the correction (no noise!)

y = Hx Given prediction xˆ (k + 1 | k ) and output y, find Δx so that xˆ = xˆ (k + 1 | k ) + Δx is the " best" estimate of x.

“best” estimate comes from shortest Δx shortest Δx is perpendicular to Ω

xˆ (k + 1 | k )

T ⊥ ⇒ Δx ∈ null(H ) ⇒ Δx ∈ column(H )

•

Δx

Ω = {x | Hx = y}

⇒ Δx = H T γ assume γ is a linear function of ν

⇒ Δx = H T K ν Guess, hope, lets face it, it has to be some function of the innovation

for some m × m matrix K

RI 16-735, Howie Choset, with slides from George Kantor, G.D. Hager, and D. Fox

Finding the correction (no noise!)

y = Hx Given prediction xˆ− and output y, find Δx so that xˆ = xˆ (k + 1 | k ) + Δx

is the " best" estimate of x. we require

H ( xˆ (k + 1 | k ) + Δx) = y

xˆ (k + 1 | k ) •

Δx

Ω = {x | Hx = y}

RI 16-735, Howie Choset, with slides from George Kantor, G.D. Hager, and D. Fox

Finding the correction (no noise!)

y = Hx Given prediction xˆ− and output y, find Δx so that xˆ = xˆ (k + 1 | k ) + Δx

is the " best" estimate of x. we require

H ( xˆ (k + 1 | k ) + Δx) = y

xˆ (k + 1 | k )

⇒ HΔx = y − Hxˆ (k + 1 | k ) = H ( x − xˆ (k + 1 | k )) = ν

•

Δx

Ω = {x | Hx = y}

RI 16-735, Howie Choset, with slides from George Kantor, G.D. Hager, and D. Fox

Finding the correction (no noise!)

y = Hx Given prediction xˆ− and output y, find Δx so that xˆ = xˆ (k + 1 | k ) + Δx

is the " best" estimate of x. we require

H ( xˆ (k + 1 | k ) + Δx) = y

xˆ (k + 1 | k )

⇒ HΔx = y − Hxˆ (k + 1 | k ) = H ( x − xˆ (k + 1 | k )) = ν

•

Δx

Ω = {x | Hx = y}

substituting Δx = H T Kν yields

HH T Kν = ν

Δx must be perpendicular to Ω b/c anything in the range space of HT is perp to Ω

RI 16-735, Howie Choset, with slides from George Kantor, G.D. Hager, and D. Fox

Finding the correction (no noise!)

y = Hx Given prediction xˆ− and output y, find Δx so that xˆ = xˆ (k + 1 | k ) + Δx

is the " best" estimate of x. we require

H ( xˆ (k + 1 | k ) + Δx) = y

xˆ (k + 1 | k )

⇒ HΔx = y − Hxˆ (k + 1 | k ) = H ( x − xˆ (k + 1 | k )) = ν

•

Δx

Ω = {x | Hx = y}

substituting Δx = H T Kν yields

HH T Kν = ν

(

⇒ K = HH T

)

−1

The fact that the linear solution solves the equation makes assuming K is linear a kosher guess RI 16-735, Howie Choset, with slides from George Kantor, G.D. Hager, and D. Fox

Finding the correction (no noise!)

y = Hx Given prediction xˆ− and output y, find Δx so that xˆ = xˆ (k + 1 | k ) + Δx

is the " best" estimate of x. we require

H ( xˆ (k + 1 | k ) + Δx) = y

xˆ (k + 1 | k )

⇒ HΔx = y − Hxˆ (k + 1 | k ) = H ( x − xˆ (k + 1 | k )) = ν

•

Δx

Ω = {x | Hx = y}

substituting Δx = H T Kν yields

HH T Kν = ν

(

⇒ K = HH T

Δx = H

T

)

−1

(HH )

T −1

ν

RI 16-735, Howie Choset, with slides from George Kantor, G.D. Hager, and D. Fox

A Geometric Interpretation Ω = {x | Hx = y}

xˆ (k + 1 | k ) ••

Δx

RI 16-735, Howie Choset, with slides from George Kantor, G.D. Hager, and D. Fox

A Simple State Observer

System:

x(k + 1) = Fx(k ) + Gu (k ) y (k ) = Hx(k )

1. prediction:

xˆ (k + 1 | k ) = Fxˆ (k | k ) + Gu (k )

Observer:

2. compute correction:

Δx = H

T

(HH )

T −1

( y (k + 1) − Hxˆ (k + 1 | k ))

3. update:

xˆ ( k + 1 | k + 1) = xˆ (k + 1 | k ) + Δx

RI 16-735, Howie Choset, with slides from George Kantor, G.D. Hager, and D. Fox

Caveat #1

Note: The observer presented here is not a very good observer. Specifically, it is not guaranteed to converge for all systems. Still the intuition behind this observer is the same as the intuition behind the Kalman filter, and the problems will be fixed in the following slides. It really corrects only to the current sensor information, so if you are on the hyperplane but not at right place, you have no correction…. I am waiving my hands here, look in book

RI 16-735, Howie Choset, with slides from George Kantor, G.D. Hager, and D. Fox

Estimating a distribution for x

Our estimate of x is not exact! We can do better by estimating a joint Gaussian distribution p(x).

p( x) =

(

(

1 (2π ) n / 2 P

where P = E ( x − xˆ )( x − xˆ ) T

)

1/ 2

−1 ( x − xˆ )T P −1 ( x − xˆ ) e2

)

is the covariance matrix

RI 16-735, Howie Choset, with slides from George Kantor, G.D. Hager, and D. Fox

Finding the correction (geometric intuition) Given prediction xˆ (k + 1 | k ), covariance P, and output y , find Δx so that xˆ = xˆ (k + 1 | k ) + Δx is the " best" (i.e. most probable) estimate of x.

Ω = {x | Hx = y}

xˆ•(k + 1 | k )

Δx

p( x) =

(

1 (2π ) n / 2 P

1/ 2

−1 ( x − xˆ )T P −1 ( x − xˆ ) e2

)

The most probable Δx is the one that : 1. satisfies xˆ ( k + 1 | k + 1) = xˆ (k + 1 | k ) + Δx 2. minimizes ΔxT P −1Δx

RI 16-735, Howie Choset, with slides from George Kantor, G.D. Hager, and D. Fox

A new kind of distance

Suppose we define a new inner product on R n to be :

x1 , x2 = x1T P −1 x 2

(this replaces the old inner product x1T x2 )

Then we can define a new norm x

2

= x , x = x T P −1 x

The xˆ in Ω that minimizes Δx is the orthogonal projection of xˆ (k + 1 | k ) onto Ω, so Δx is orthogonal to Ω.

⇒ ω , Δx = 0 for ω in TΩ = null ( H )

ω , Δx = ω T P −1Δx = 0 iff Δx ∈ column ( PH T )

RI 16-735, Howie Choset, with slides from George Kantor, G.D. Hager, and D. Fox

Finding the correction (for real this time!)

Assuming that Δx is linear in ν = y − Hxˆ ( k + 1 | k ) Δx = PH T Kν

The condition y = H ( xˆ (k + 1 | k ) + Δx ) ⇒ HΔx = y − Hxˆ (k + 1 | k ) = ν Substituti on yields : HΔx = ν = HPH T Kν

(

⇒ K = HPH ∴

Δx = PH

T

T

)

−1

(HPH ) T

−1

ν

RI 16-735, Howie Choset, with slides from George Kantor, G.D. Hager, and D. Fox

A Better State Observer

x(k + 1) = Fx(k ) + Gu (k ) + v(k ) y (k ) = Hx(k )

Sample of Guassian Dist. w/ COV Q

We can create a better state observer following the same 3. steps, but now we must also estimate the covariance matrix P. We start with x(k|k) and P(k|k) Where did noise go? Expected value…

Step 1: Prediction xˆ (k + 1 | k ) = Fxˆ (k | k ) + Gu (k ) What about P? From the definition:

(

P ( k | k ) = E ( x (k ) − xˆ (k | k ))( x(k ) − xˆ (k | k ))T and

)

(

P ( k + 1 | k ) = E ( x(k + 1) − xˆ (k + 1 | k ))( x(k + 1) − xˆ (k + 1 | k ))T

RI 16-735, Howie Choset, with slides from George Kantor, G.D. Hager, and D. Fox

)

Continuing Step 1

To make life a little easier, lets shift notation slightly:

(

Pk−+1 = E ( xk +1 − xˆ k−+1 )( xk +1 − xˆ k−+1 )T

)

( = E ((F (x − xˆ ) + v )(F (x − xˆ ) + v ) ) = E (F (x − xˆ )(x − xˆ ) F + 2 F (x − xˆ )v = FE ((x − xˆ )(x − xˆ ) )F + E (v v )

= E (Fx k + Gu k + vk − ( Fxˆ k + Gu k ) )(Fx k + Gu k + vk − ( Fxˆ k + Gu k ) )T T

k

k

k

k

T

k

k

k

k

k

k

T

k

k

T

k

k

k

T

k

T k

+ v k v Tk

)

T

k

k

= FPk F T + Q

P (k + 1 | k ) = FP (k | k ) F T + Q

RI 16-735, Howie Choset, with slides from George Kantor, G.D. Hager, and D. Fox

)

Step 2: Computing the correction From step 1 we get xˆ ( k + 1 | k ) and P (k + 1 | k ). Now we use these to compute Δx :

(

Δx = P ( k + 1 | k ) H HP ( k + 1 | k ) H

T

)

−1

( y (k + 1) − Hxˆ (k + 1 | k ) )

For ease of notation, define W so that

Δx = Wν

RI 16-735, Howie Choset, with slides from George Kantor, G.D. Hager, and D. Fox

Step 3: Update

xˆ ( k + 1 | k + 1) = xˆ ( k + 1 | k ) + Wν

( = E (( x

Pk +1 = E ( x k +1 − xˆ k +1 )( x k +1 − xˆ k +1 )T

)

− − T ˆ ˆ − x − W ν )( x − x − W ν ) k +1 k +1 k +1 k +1

)

(just take my word for it…)

P (k + 1 | k + 1) = P (k + 1 | k ) − WHP (k + 1 | k ) H T W T

RI 16-735, Howie Choset, with slides from George Kantor, G.D. Hager, and D. Fox

Just take my word for it… ( = E (( x = E ⎛⎜ (( x ⎝

Pk +1 = E ( x k +1 − xˆ k +1 )( x k +1 − xˆ k +1 )T k +1

) ) ) − Wν ) ⎞⎟ ⎠

− xˆ k−+1 − Wν )( xk +1 − xˆ k−+1 − Wν )T

)(

ˆ k−+1 ) − Wν ( xk +1 − xˆ k−+1 k +1 − x

T

(

= E ( xk +1 − xˆ k−+1 )( xk +1 − xˆ k−+1 )T − 2Wν ( xk +1 − xˆ k−+1 )T + Wν (Wν )T

)

(

= Pk−+1 + E − 2WH ( xk +1 − xˆ k−+1 )( xk +1 − xˆ k−+1 )T + WH ( xk +1 − xˆ k−+1 )( xk +1 − xˆ k−+1 )T H T W T = Pk−+1 − 2WHPk−+1 + WHPk−+1 H T W T

( (HP

= Pk−+1 − 2 Pk−+1 H T HPk−+1 H T = Pk−+1 − 2 Pk−+1 H T

) HP ) (HP −1

− T −1 H k +1

− k +1

+ WHPk−+1 H T W T

− T k +1 H

)(HP

)

− T −1 H HPk−+1 k +1

+ WHPk−+1 H T W T

= Pk−+1 − 2WHPk−+1 H T W T + WHPk−+1 H T W T

RI 16-735, Howie Choset, with slides from George Kantor, G.D. Hager, and D. Fox

)

Better State Observer Summary

System: 1. Predict

x(k + 1) = Fx(k ) + Gu (k ) + v(k ) y (k ) = Hx(k )

xˆ (k + 1 | k ) = Fxˆ (k | k ) + Gu (k )

Observer

P (k + 1 | k ) = FP (k | k ) F T + Q 2. Correction

3. Update

(

W = P (k + 1 | k ) H HP (k + 1 | k ) H T Δx = W ( y ( k + 1) − Hxˆ (k + 1 | k ) )

)

−1

xˆ ( k + 1 | k + 1) = xˆ ( k + 1 | k ) + Wν

P (k + 1 | k + 1) = P (k + 1 | k ) − WHP (k + 1 | k ) H T W T

RI 16-735, Howie Choset, with slides from George Kantor, G.D. Hager, and D. Fox

•Note: there is a problem with the previous slide, namely the covariance matrix of the estimate P will be singular. This makes sense because with perfect sensor measurements the uncertainty in some directions will be zero. There is no uncertainty in the directions perpendicular to Ω P lives in the state space and directions associated with sensor noise are zero. In the step when you do the update, if you have a zero noise measurement, you end up squeezing P down. In most cases, when you do the next prediction step, the process covariance matrix Q gets added to the P(k|k), the result will be nonsingular, and everything is ok again. There’s actually not anything wrong with this, except that you can’t really call the result a “covariance” matrix because “sometimes” it is not a covariance matrix Plus, lets not be ridiculous, all sensors have noise.

RI 16-735, Howie Choset, with slides from George Kantor, G.D. Hager, and D. Fox

Finding the correction (with output noise)

y = Hx + w

Ω = {x | Hx = y}

The previous results require that you know which hyperplane to aim for. Because there is now sensor noise, we don’t know where to aim, so we can’t directly use our method. If. we can determine which hyperplane aim for, then the previous result would apply.

xˆ (k +• 1 | k )

We find the hyperplane in question as follows: 1. 2. 3.

project estimate into output space find most likely point in output space based on measurement and projected prediction the desired hyperplane is the preimage of this point

RI 16-735, Howie Choset, with slides from George Kantor, G.D. Hager, and D. Fox

Projecting the prediction (putting current state estimates into sensor space)

xˆ (k + 1 | k ) → yˆ = Hxˆ (k + 1 | k ) P (k + 1 | k ) → Rˆ = HP(k + 1 | k ) H T P(k + 1 | k )

xˆ (k + 1 | k )

•

state space (n-dimensional)

Rˆ

yˆ

•

output space (p-dimensional)

RI 16-735, Howie Choset, with slides from George Kantor, G.D. Hager, and D. Fox

Finding most likely output ence independent, so multiply them because we want them both to be true at the same time

The objective is to find the most likely output that results from the independent Gaussian distributions ( y, R) measurement and associate covariance ( yˆ , Rˆ ) projected prediction (what you think your measurements should be and how confident you are)

Fact (Kalman Gains): The product of two Gaussian distributions given by mean/covariance pairs (x1,C1) and (x2,C2) is proportional to a third Gaussian with mean

x3 = x1 + K ( x2 − x1 )

and covariance where

C3 = C1 − KC1

K = C1 (C1 + C2 )

−1

Strange, but true, this is symmetric

RI 16-735, Howie Choset, with slides from George Kantor, G.D. Hager, and D. Fox

Most likely output (cont.)

Using the Kalman gains, the most likely output is

(

y = yˆ + ⎛⎜ Rˆ Rˆ + R ⎝ *

(

(

)

−1

⎞⎟( yˆ − y ) ⎠

= Hxˆ (k + 1 | k ) + HPH HPH + R T

T

)

−1

)(Hxˆ(k + 1 | k ) − y)

RI 16-735, Howie Choset, with slides from George Kantor, G.D. Hager, and D. Fox

Finding the Correction

Now we can compute the correction as we did in the noiseless case, this time using y* instead of y. In other words, y* tells us which hyperplane to aim for.

{

Ω = x | Hx = y *

} The result is:

xˆ − •

Δx

(

Δx = PH HPH T

T

) (y −1

*

− Hxˆ ( k + 1 | k )

ot going all the way to y, but splitting the difference between how confident you are with your Sensor and process noise RI 16-735, Howie Choset, with slides from George Kantor, G.D. Hager, and D. Fox

)

Finding the Correction (cont.)

(

) (y − Hxˆ (k + 1 | k )) ) (Hxˆ + HPH (HPH + R ) ( y − Hxˆ (k + 1 | k )) − Hxˆ (k + 1 | k ))

Δx = PH HPH T

(

= PH T HPH T

(

−1

T

*

−1

= PH HPH + R T

T

T

T

−1

) ( y − Hxˆ (k + 1 | k )) −1

For convenience, we define

(

W = PH HPH + R T

T

)

−1

So that

Δx = W ( y − Hxˆ ( k + 1 | k ) ) RI 16-735, Howie Choset, with slides from George Kantor, G.D. Hager, and D. Fox

Correcting the Covariance Estimate

The covariance error estimate correction is computed from the definition of the covariance matrix, in much the same way that we computed the correction for the “better observer”. The answer turns out to be:

(

)

P (k + 1 | k + 1) = P (k + 1 | k ) − W HP (k + 1 | k ) H W

RI 16-735, Howie Choset, with slides from George Kantor, G.D. Hager, and D. Fox

T

T

LTI Kalman Filter Summary

System:

Kalman Filter

1. Predict

x(k + 1) = Fx(k ) + Gu (k ) + v(k ) y (k ) = Hx(k ) + w(k )

xˆ (k + 1 | k ) = Fxˆ (k | k ) + Gu (k ) P (k + 1 | k ) = FP (k | k ) F T + Q S = HP ( k + 1 | k ) H T + R

2. Correction

3. Update

W = P ( k + 1 | k ) H T S −1

Δx = W ( y ( k + 1) − Hxˆ (k + 1 | k ) ) xˆ ( k + 1 | k + 1) = xˆ ( k + 1 | k ) + Wν

P (k + 1 | k + 1) = P (k + 1 | k ) − WSW T

RI 16-735, Howie Choset, with slides from George Kantor, G.D. Hager, and D. Fox

Kalman Filters

RI 16-735, Howie Choset, with slides from George Kantor, G.D. Hager, and D. Fox

Kalman Filter for Dead Reckoning •

Robot moves along a straight line with state x = [xr, vr]T

•

u is the force applied to the robot

•

Newton tells us

or

Process noise from a zero mean Gaussian V

•

Robot has velocity sensor Sensor noise from a zero mean Gaussian W

RI 16-735, Howie Choset, with slides from George Kantor, G.D. Hager, and D. Fox

Set up

Assume

At some time k

PUT ELLIPSE FIGURE HERE

RI 16-735, Howie Choset, with slides from George Kantor, G.D. Hager, and D. Fox

Observability Recall from last time

Actually, previous example is not observable but still nice to use Kalman filter

RI 16-735, Howie Choset, with slides from George Kantor, G.D. Hager, and D. Fox

Extended Kalman Filter •

Life is not linear

•

Predict

RI 16-735, Howie Choset, with slides from George Kantor, G.D. Hager, and D. Fox

Extended Kalman Filter •

Update

RI 16-735, Howie Choset, with slides from George Kantor, G.D. Hager, and D. Fox

EKF for Range-Bearing Localization •

State

position and orientation

•

Input

forward and rotational velocity

•

Process Model

• •

nl landmarks can only see p(k) of them at k

Association map

RI 16-735, Howie Choset, with slides from George Kantor, G.D. Hager, and D. Fox

Be wise, and linearize..

RI 16-735, Howie Choset, with slides from George Kantor, G.D. Hager, and D. Fox

Data Association

•BIG PROBLEM Ith measurement corresponds to the jth landmark innovation

where

Pick the smallest RI 16-735, Howie Choset, with slides from George Kantor, G.D. Hager, and D. Fox

Kalman Filter for SLAM (simple) state

Inputs are commands to x and y velocities, a bit naive Process model

RI 16-735, Howie Choset, with slides from George Kantor, G.D. Hager, and D. Fox

Kalman Filter for SLAM

RI 16-735, Howie Choset, with slides from George Kantor, G.D. Hager, and D. Fox

Range Bearing

Inputs are forward and rotational velocities

RI 16-735, Howie Choset, with slides from George Kantor, G.D. Hager, and D. Fox

Greg’s Notes: Some Examples •

Point moving on the line according to f = m a – state is position and velocity – input is force – sensing should be position

•

Point in the plane under Newtonian laws

•

Nonholonomic kinematic system (no dynamics) – state is workspace configuration – input is velocity command – sensing could be direction and/or distance to beacons

• •

Note that all of dynamical systems are “open-loop” integration Role of sensing is to “close the loop” and pin down state

RI 16-735, Howie Choset, with slides from George Kantor, G.D. Hager, and D. Fox

Kalman Filters Bel ( xt ) = N ( μ t , σ t2 ) − ⎧ μ t +1 = μt + But − Bel(xt +1 ) = ⎨ − 2 2 2 2 σ = A σ + σ t act ⎩ t +1

⎧ μt +1 = μt−+1 + K t +1 ( μ zt +1 − μt−+1 ) Bel(xt +1 ) = ⎨ σ t2+1 = (1 − K t +1 )σ − t2+1 ⎩ − ⎧ μ t + 2 = μt +1 + But +1 − Bel(xt + 2 ) = ⎨ − 2 2 2 2 σ = A σ + σ t +1 act ⎩ t +2

RI 16-735, Howie Choset, with slides from George Kantor, G.D. Hager, and D. Fox

Kalman Filter Algorithm 1.

Algorithm Kalman_filter( , d ):

2. 3. 4. 5.

If d is a perceptual data item y then

6. 7. 8.

Else if d is an action data item u then μ = Aμ + Bu

9.

Return

(

K = ΣC CΣC + Σ obs μ = μ + K ( z − Cμ ) T

T

)

−1

Σ = ( I − KC )Σ

Σ = AΣAT + Σ act

RI 16-735, Howie Choset, with slides from George Kantor, G.D. Hager, and D. Fox

Limitations •

Very strong assumptions: – Linear state dynamics – Observations linear in state

•

What can we do if system is not linear? – Non-linear state dynamics – Non-linear observations

X t−+1 = AX t + But + Σ act

Z t = CX t + Σ obs X t−+1 = f ( X t , ut , Σ act )

Z t = c( X t , Σ obs )

• Linearize it! • Determine Jacobians of dynamics f and observation function c w.r.t the current state x and the noise.

Aij =

∂c ∂f i ∂ci ∂f i ( xt , ut ,0) Cij = i ( xt ,0) Wij = ( xt , ut ,0) Vij = ( xt ,0) ∂x j ∂x j ∂Σ act j ∂Σ obs j RI 16-735, Howie Choset, with slides from George Kantor, G.D. Hager, and D. Fox

Extended Kalman Filter Algorithm 1.

Algorithm Extended_Kalman_filter(, d):

2. 3. 4. 5.

If d is a perceptual data item z then

6. 7. 8.

Else if d is an action data item u then μ = f ( μ , u , 0) μ = Aμ + Bu

9.

Return

(

K = ΣC CΣC + VΣ obsV μ = μ + K (z − c( μ ,0) ) Σ = ( I − KC )Σ T

T

Σ = AΣAT + WΣ actW T

)

T −1

(

K = ΣC CΣC + Σ obs μ = μ + K ( z − Cμ ) T

T

Σ = ( I − KC )Σ

Σ = AΣAT + Σ act

RI 16-735, Howie Choset, with slides from George Kantor, G.D. Hager, and D. Fox

)

−1

Kalman Filter-based Systems (2) • [Arras et al. 98]: • Laser range-finder and vision • High precision (