henry carrillo josé a. castellanos ian reid (university of oxford) on the comparison of uncertainty...

Henry CarrilloJosé A. Castellanos

Ian Reid (University of Oxford)

On the Comparison of Uncertainty Criteria for Active SLAM

On the Comparison of Uncertainty Criteria for Active SLAM

Contents

Active SLAM Uncertainty criteria Computation of uncertainty criteria Experiments Conclusions

1

Preliminaries – Active SLAM (I) Active SLAM => To integrate path planning into

a SLAM process. To explorer more area. Navigate safely. Reduce uncertainty.

Algorithms 1º Alg. [Feder, Leonard](99)

Active perception [Bajacksy](86) Control theroy and MPC [Leung, Dissanayake](06)

2

Preliminaries – Active SLAM (II)

Pseudo-code:

Set of trajectories Assign a score to each trajectory

Uncertainty of map+robot Trajectory constraints

Execute the trajectory with the optimum .

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Require: A priori partially known map.

Preliminaries – Active SLAM (II)

Pseudo-code:

Set of trajectories Assign a score to each trajectory

Uncertainty of map+robot Trajectory constraints

Execute the trajectory with the optimum .

2

Preliminaries – Active SLAM (II)

Pseudo-code:

Set of trajectories Assign a score to each trajectory

Uncertainty of map+robot Trajectory constraints

Execute the trajectory with the optimum .

2

J1 J2 J3 J4 J5

1 1,5 1,9 0,8 3

Preliminaries – Active SLAM (II)

Pseudo-code:

Set of trajectories Assign a score to each trajectory

Uncertainty of map+robot Trajectory constraints

Execute the trajectory with the optimum .

2

J1 J2 J3 J4 J5

1 1,5 1,9 0,8 3

Uncertainty Criteria for Active SLAM (I) Uncertainty/Inform. Criteria =>

In the TOED, a design (i.e. ), is better than a design, if:

The above does not allow to quantify the improvement, therefore is desirable to:

It permits to quantify the uncertainty in .

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• Theory of Optimal Experiment Design (A-opt, D-opt, E-opt…).

• Information Theory ( Entropy, MI…).

Uncertainty Criteria for Active SLAM (II) Some possible uncertainty criteria for active SLAM

are:

Previous works ([Sim and Roy, 2005], [Mihaylova and De Schutter, 2003]) report A-opt as the best criterion and that D-opt gives null values. A-opt, widely used: [Kollar2008] [Martinez-

Cantin2008] [Meger2008] [Dissanayake2006]. Although D-opt is commonly used in the TOED

because it is optimal.4

Determinant (D-opt)

Trace (A-opt)

max (𝜆1 ,…,𝜆𝑘)

Max (E-opt)

trace (Σ )= ∑𝑘=1 ,… , 𝑙

𝜆𝑘det (Σ )= ∏𝑘=1 ,…, 𝑙

𝜆𝑘

Uncertainty Criteria for Active SLAM (III) It is indeed possible to use D-opt in the

Active SLAM context: The structure of the problem needs to be taken into

account (i.e. The covariance matrix varies with time). It is not informative to compare the determinant of a matrix

l x l with a m x m. det(l x l) is homogeneous of grade l.

The computation of the determinant of a highly correlated matrix (e.g. SLAM) is prone to round-off errors. Processing in the logarithm space

D-opt for a l x l covariance matrix:

Stem from [Kiefer, 1974] :5

First experiment First experiment: on the computation

Is it possible to compute D-opt from a robot doing SLAM?

Execute a SLAM algorithm (e.g. EKF-SLAM, iSAM). Compute in each step: A-opt, E-opt , D-opt,

Determinant, entropy and mutual Information.

• Simulated Robot indoor environment : MRPT/C++

• Real Robot indoor environment : Pioneer 3 DX - Ad-hoc

• Real Robot indoor environment : DLR dataset• Real Robot outdoor environment : Victoria

Park dataset6

1E - Simulated Robot indoor environment (I)

Scenario: Area of 25x25 m 2D EKF-SLAM Sensor: Odometry +

Camera (360º - 3m range)

180 landmarks - DA Known.

Gaussian errors: Odometry + Sensors

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1E-Simulated Robot indoor environment (II) Qualitative results

(a)-(f) A-opt, E-opt, D-opt, determinant, entropy and MI.8

1E-Real Robot indoor environment @ DLR

Scenario: Area 60x40 m Sensor: Odometry + Camera

2D EKF-SLAM 576 landmarks – DA known.

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1E-Real Robot indoor environment @ DLR Qualitative results

(a)-(f) A-opt, E-opt, D-opt, determinant, entropy and MI.10

First experiment – Quantitative analysis Average correlation between the uncertainty

criteria:

Variance: A-E (0,0002) / A-D (0,0540) / D-E (0,0481).

A-opt y E-opt => High correlation. E-opt is guided by a single eigenvalue.

A-opt y D-opt => Medium correlation. H0: D-opt take into account more components than A-opt.

A-opt E-opt D-opt

A-opt 1 0,9872 0,6003

E-opt 0,9872 1 0,5903

D-opt 0,6003 0,5903 1

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Second Experiment Second experiment: Active SLAM

What is the effect of the uncertainty criteria in active SLAM?

Active SLAM => Unitary horizon (greedy). Uncertainty criteria => A-opt, D-opt and

Entropy. Effect => MSE y .• Simulated Robot with unitary horizon: MRPT /

C++

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2E-Simulated Robot indoor environment (I)

Scenario: Area of 20x20m and

30x30m 2D EKF-SLAM Sensor: Odometry +

Camera (360º - 3m range)

Gaussian errors: Odometry + sensors.

Path planner: Discrete (A*) and continuous (Attract-Repel).1

3

2E-Simulated Robot indoor environment (II)

Resulting paths for each uncertainty criterion: (a) D-opt, (b) A-opt y (c) Entropy. Each colour represents an executed path. 20 x 20 m map.

• Qualitative analysis

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2E-Simulated Robot indoor environment (III)

Resulting trajectories for 10000 steps active SLAM simulation. (a). Initial trajectory. (b) A-opt. (c). D-opt.

• Qualitative analysis.

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2E – Quantitative Analysis 30x30 m

Evolution of MSE ((a)-(c)) y chi2 ((d)-(f)) ratio. Average of 10 MC simulations.

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Take home message D-opt is the optimum criterion to measure

uncertainty according to the TOED (i.e. better than A-opt (Trace)).

It is possible to obtain useful information regarding the uncertainty of a SLAM process with D-opt.

D-opt shows better performance than A-opt in our simulated experiments of active SLAM.

To compute D-opt in the context of a SLAM process => use the formulation presented here.

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On the Comparison of UncertaintyCriteria for Active SLAM

Thanks!!!hcarri@unizar.es

http://webdiis.unizar.es/~hcarri

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Preliminaries – SLAM H0: A model of the operative environment is an

essential requirement for an autonomous mobile robot.

Three basic tasks: Where am I? What does the world look like? Where do I go?

SLAM => Joint of two tasks. SLAM => Does not define

the path-trajectory of the robot. Integrated approach => On the way to autonomy.4 Exploration and Mapping with Mobile Robots. Cyrill Stachniss. 2006.

Experimentos Primer experimento : acerca del cálculo

Segundo experimento : SLAM activo

• Robot simulado ambiente interior : MRPT / C++

• Robot real ambiente interior : Pioneer 3 DX - Ad-hoc

• Robot real ambiente interior : DLR dataset• Robot real ambiente exterior : Victoria Park

dataset

• Robot simulado con horizonte unitario : MRPT / C++

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1E-Robot en ambiente exterior @ VP (I)

Escenario: Área de 350 x 350 m iSAM Sensor: Odometría +

Laser 150 landmarks – DA

conocida.

13

1E-Robot en ambiente exterior @ VP (II) – Resultados cualitativos

(a)-(f) A-opt, E-opt, D-opt, determinante, entropía y MI.14

1E-Robot en ambiente interior ad-hoc (I)

Escenario:

Área 6x4 m 2D EKF-SLAM Sensor: Odometría +

Kinect 5 landmarks – DA

conocida

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1E-Robot en ambiente interior ad-hoc (II) – Resultados cualitativos

(a)-(f) A-opt, E-opt, D-opt, determinante, entropía y MI.16

2E - Análisis cuantitativo 20x20 m

Evolución del MSE ((a)-(c)) y chi2 ((d)-(f)). Promedio de 10 MC.

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DeterminanteOperación algebraica que transforma una matriz en un escalar. Propiedades (matriz n x n)

Geométrica: Volumen del paralelepípedo definido en el espacio n-dimensional.

Homogéneo de grado n. Si,

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What have I done?

Metrical: an example using D-

opt17

FaMUS: Fast Minimum Uncertainty Search

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• Minimum uncertainty path between A to B in a graph.

• Exhaustive search.

FaMUS: Fast Minimum Uncertainty Search

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• Minimum uncertainty path between A to B in a graph.

• Exhaustive search.

FaMUS: Fast Minimum Uncertainty Search

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Experiment: Are the minimum uncertainty path and the shortest path necessarily equal? Select two points A and B, and compare the final

uncertainty. 1000 times x 4 datasets. (Biccoca, Intel , New

colleges and Manhattan).

FaMUS: Fast Minimum Uncertainty Search

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Examples of paths.

FaMUS: Fast Minimum Uncertainty Search

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Summary of results

• Improvement of a least 50% in timing respect to the state of the art. [Valencia2011]

What have I done?

Topological

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Topological Guiding question:

Where should I go in order to improve my topological map?

Challenges: well-posed and egocentric images. Execute a SLAM algorithm (e.g. EKF-SLAM, iSAM). Compute in each step: A-opt, E-opt , D-opt,

Determinant, entropy and mutual Information.

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Topological One solution:

Textons (a.k.a gist)- Undelaying Structure- Probabilistic decision

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What have I done?

TBD

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TBD Which are the confidence intervals in the active

predictions? When do I stop the active behaviour?

Find a relationship between uncertainty and metrical error.

Use other constraints other than uncertainty. Speed up the decision process.

Real experiments.

31

Active SLAM : a FrameworkMy, on-going, PhD Research

Thanks!!!hcarri@unizar.es

http://webdiis.unizar.es/~hcarri

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Artículos “Experimental Comparison of Optimum

Criteria for Active SLAM”. Oral presentation in the “III Workshop de Robótica: Robótica Experimental (ROBOT’11)”.

“On the Comparison of Uncertainty Criteria for Active SLAM”. Submitted to ICRA’12.

“Planning Minimum Uncertainty Paths Over Pose/Feature Graphs Constructed Via SLAM” . Submitted to ICRA’12.

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FaMUS: Fast Minimum Uncertainty Search

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