Estimator OMNE is a Maximum Likelihood

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OMNE is a Maximum Likelihood Estimator

With application to robotics mobile mapping

J. Nicola and L. Jaulin, SWIM 2016 {jeremy.nicola,luc.jaulin}@ensta-

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• Context

• Problem

• Solution intuition

• Classical method

• Proposed method

• Application

• Conclusion

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• Context

• Problem

• Solution intuition

• Classical method

• Proposed method

• Application

• Conclusion

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Obstacle

Beacon 1

Beacon 2

Beacon 5 Beacon 3

Beacon 4

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Obstacle

Beacon 1

Beacon 2

Beacon 5 Beacon 3

Beacon 4

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Obstacle

Beacon 1

Beacon 2

Beacon 5 Beacon 3

Beacon 4

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Obstacle

Beacon 1

Beacon 2

Beacon 5 Beacon 3

Beacon 4

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Obstacle

Beacon 1

Beacon 2

Beacon 5 Beacon 3

Beacon 4

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Obstacle

Beacon 1

Beacon 2

Beacon 5 Beacon 3

Beacon 4

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Obstacle

Beacon 1

Beacon 2

Beacon 5 Beacon 3

Beacon 4

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Obstacle

Beacon 1

Beacon 2

Beacon 5 Beacon 3

Beacon 4

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Obstacle

Beacon 1

Beacon 2

Beacon 5 Beacon 3

Beacon 4

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Obstacle

Beacon 1

Beacon 2

Beacon 5 Beacon 3

Beacon 4

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Obstacle

Beacon 1

Beacon 2

Beacon 5 Beacon 3

Beacon 4

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Obstacle

Beacon 1

Beacon 2

Beacon 5 Beacon 3

Beacon 4

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• Simultaneous Localization and Mapping

(SLAM) is a typical state estimation problem

Evolution equation

Observation equation

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• Evolution equation

– Models the system dynamics

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• Evolution equation

– Models the system dynamics

Example: the motion of a robot

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• Evolution equation

– Models the system dynamics

Example: the motion of a robot

J. Nicola and L. Jaulin, SWIM 2016

{jeremy.nicola,luc.jaulin}@ensta- 19

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• Evolution equation

– Models the system dynamics

Example: the motion of a robot

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• Evolution equation

– Models the system dynamics

Example: the motion of a robot

Kalman Contractor

J. Nicola, L. Jaulin

SWIM 2014

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• Observation equation

– Models the measurements made on or by the system

Example: measuring distances to a beacon

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• Observation equation

– Models the measurements made on or by the system

Example: measuring distances to a beacon

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• Observation equation

– Models the measurements made on or by the system

Example: measuring distances to a beacon

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• Observation equation

– Models the measurements made on or by the system

Example: measuring distances to a beacon

Nonlinear Gaussian set-inversion

J. Nicola, L. Jaulin

SWIM 2015

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• Context

• Problem

• Solution intuition

• Classical method

• Proposed method

• Application

• Conclusion

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• Practicality of a set-estimation

– Design a pipe joining beacon 1 & beacon 2

B1

B2

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• Practicality of a set-estimation

– Design a pipe joining beacon 1 & beacon 2

B1

B2

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• Practicality of a set-estimation

– Design a pipe joining beacon 1 & beacon 2

B1

B2

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• Practicality of a set-estimation

– Design a pipe joining beacon 1 & beacon 2

B1

B2

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• Practicality of a set-estimation

– Design a pipe joining beacon 1 & beacon 2

B1

B2

Center of the box-hull

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• Practicality of a set-estimation

– Design a pipe joining beacon 1 & beacon 2

B1

B2

Chebychev center

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• Practicality of a set-estimation

– Design a pipe joining beacon 1 & beacon 2

B1

B2

Center of the minimum trace/volume enclosing ellipsoid

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• Practicality of a set-estimation

– Design a pipe joining beacon 1 & beacon 2

B1

B2

Center of mass

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• Context

• Problem

• Solution intuition

• Classical method

• Proposed method

• Application

• Conclusion

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• Practicality of a set-estimation

– Design a pipe joining beacon 1 & beacon 2

𝛾 = 0,99

B1

B2

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• Practicality of a set-estimation

– Design a pipe joining beacon 1 & beacon 2

𝛾 = 0,95

B1

B2

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• Practicality of a set-estimation

– Design a pipe joining beacon 1 & beacon 2

𝛾 = 0,90

B1

B2

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• Practicality of a set-estimation

– Design a pipe joining beacon 1 & beacon 2

𝛾 = 0,85

B1

B2

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B1

B2

• Practicality of a set-estimation

– Design a pipe joining beacon 1 & beacon 2

𝛾 = 0,85

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• Context

• Problem

• Solution intuition

• Classical method

• Proposed method

• Application

• Conclusion

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• Find the parameter that maximizes the Likelihood function

– 𝐿 𝒚 𝒑 = 𝜋(𝑦 _𝑖 |𝒑)

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• Find the parameter that maximizes the Likelihood function

– 𝐿 𝒚 𝒑 = 𝜋(𝑦 _𝑖 |𝒑) ∝ 𝑒 ^−(𝜓 ^𝑖 ^{𝒑 −𝑦} ^𝑖 ⁾ ²

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• Find the parameter that maximizes the Likelihood function

– 𝐿 𝒚 𝒑 = 𝜋(𝑦 _𝑖 |𝒑) ∝ 𝑒 ^−(𝜓 ^𝑖 ^{𝒑 −𝑦} ^𝑖 ⁾ ²

– arg 𝑚𝑖𝑛 ( 𝜓 _𝑖 𝒑 − 𝑦 _𝑖 ² )

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• Find the parameter that maximizes the Likelihood function

– 𝐿 𝒚 𝒑 = 𝜋(𝑦 _𝑖 |𝒑) ∝ 𝑒 ^−(𝜓 ^𝑖 ^{𝒑 −𝑦} ^𝑖 ⁾ ² – arg 𝑚𝑖𝑛 ( 𝜓 _𝑖 𝒑 − 𝑦 _𝑖 ² )

• This is a classical nonlinear least squares problem

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• Why the Gaussian assumption?

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• Why the Gaussian assumption?

– The distribution is easy to parametrize

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• Why the Gaussian assumption?

– The distribution is easy to parametrize

– It conveniently reduces the Maximum Likelihood

Estimation problem to a (nonlinear) least-squares

estimation

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• Is the noise really Gaussian?

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80% of errors

in the « main » mode of the quasi-uniform

distribution

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20% outside

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• Context

• Problem

• Solution intuition

• Classical method

• Proposed method

• Application

• Conclusion

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• Modeling the problem

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• Modeling the problem

• Consider the following static parameter estimation model

– 𝒚 = 𝝍 𝒑 + 𝒆

• 𝒚 is the measurement vector

• 𝝍 is the model

• 𝒑 is the parameter vector to be estimated

• 𝒆 is the noise

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• Modeling the problem

• Consider the following static parameter estimation model

– 𝒚 = 𝝍 𝒑 + 𝒆

• 𝒚 is the measurement vector

• 𝝍 is the model

• 𝒑 is the parameter vector to be estimated

• 𝒆 is the noise

– Define the function

• 𝒇 𝒑 = 𝒆 = 𝒚 − 𝝍(𝒑)

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• Modeling the problem

• Given an error interval [𝑒] that is supposed to contain the error 𝑒 _𝑖 if the corresponding data 𝑦 _𝑖 is an inlier

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• Modeling the problem

• Given an error interval [𝑒] that is supposed to contain the error 𝑒 _𝑖 if the corresponding data 𝑦 _𝑖 is an inlier

• The OMNE estimator returns the set of all 𝒑

such as the property 𝑓 _𝑖 𝒑 ∈ [𝑒] is satisfied for

a maximal number of data

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• Modeling the noise by quasi-uniform distributions

– Assume that the error vector e is white (the 𝑒 _𝑖 ’s are independant), the 𝑒 _𝑖 ’s are identically

distributed with the probability density function 𝜋 _𝑒 which is quasi uniform

• 𝜋 _𝑒 𝑒 _𝑖 = 𝑎 if 𝑒 _𝑖 ∈ [𝑒], 𝜋 _𝑒 𝑒 _𝑖 = 𝑏 < 𝑎 otherwise

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• Modeling the noise by quasi-uniform distributions

– Assume that the error vector e is white (the 𝑒 _𝑖 ’s are independant), the 𝑒 _𝑖 ’s are identically

distributed with the probability density function 𝜋 _𝑒 which is quasi uniform

• 𝜋 _𝑒 𝑒 _𝑖 = 𝑎 if 𝑒 _𝑖 ∈ [𝑒], 𝜋 _𝑒 𝑒 _𝑖 = 𝑏 < 𝑎 otherwise

– Then the Maximum Likelihood Estimator is OMNE

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• Modeling the noise by quasi-uniform distributions

– Compatible with an infinity of p.d.f

• Including heavy-tailed distributions

– Well suited for interval methods

• Using the q-relaxed intersection we are able to

efficiently represent a confidence region of the quasi- uniformly distributed error

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– The probability to have more than q measurements outside of the « main mode » of the quasi-uniform distribution is

• 𝛾 𝑞, 𝑁, 𝑣 = ¹

2 (1 + erf ( ^{𝑁 1−𝑣 −𝑞−1}

2𝑁𝑣(1−𝑣) ))

– N: number of measurements (N>>1)

– v: probability of having a measurement in the main mode

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– The probability to have more than q measurements outside of the « main mode » of the quasi-uniform distribution is

• 𝛾 𝑞, 𝑁, 𝑣 = ¹

2 (1 + erf ( ^{𝑁 1−𝑣 −𝑞−1}

2𝑁𝑣(1−𝑣) ))

– N: number of measurements (N>>1)

– v: probability of having a measurement in the main mode

– An expression for q is:

𝑞 𝑁, 𝑣, 𝛾 = 𝑁 1 − 𝑣 − 1 − 2𝑁𝑣 1 − 𝑣 erf(2𝛾 − 1) ⁻¹

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0-relaxed box

[𝑥]

^{0}

= 𝑥

₁

× 𝑥

₂

× 𝑥

₃

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1-relaxed box

[𝑥]

^{1}

= −∞, ∞ × 𝑥

₂

× 𝑥

₃

∪ 𝑥

₁

× −∞, ∞ × 𝑥

₃

∪ 𝑥

₁

× 𝑥

₂

× −∞, ∞

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2-relaxed box (truncated)

[𝑥]

^{2}

= −∞, ∞

^×2

× 𝑥

₃

∪ 𝑥

₁

× −∞, ∞

^×2

∪ −∞, ∞ × 𝑥

₂

× −∞, ∞

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• Context

• Problem

• Solution intuition

• Classical method

• Proposed method

• Application

• Conclusion

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Trajectory known with a centimetric precision

(High quality INS+GPS RTK)

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Trajectory known with a centimetric precision (High quality INS+GPS RTK)

4 beacons we want to localize

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Range signals received for each beacon

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We’ll be looking

for this beacon

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– For easier illustration, we assume that the z- position of the beacon is exactly known

• In practice, the z-component is easily found in an underwater context

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Real beacon

position

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Real beacon position

50%

confidence

region

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25%

confidence

region

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12.5%

confidence

region

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6.25%

confidence

region

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3.125%

confidence

region

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1.5625%

confidence

region

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~0.8%

confidence

region

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Less than 10 cm error

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• Context

• Problem

• Solution intuition

• Classical method

• Proposed method

• Application

• Conclusion

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• If the measurement error is assumed to be distributed quasi-uniformly, OMNE / GOMNE is a Maximum Likelihood Estimator

• Maximum Likelihood Estimation in an interval framework enables us to compute estimates precise enough for industrial applications

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• Probabilistic and set-membership methods don’t have to be considered as different, incompatible representations of uncertainties

• Probabilistic uncertainties can be cast as geometrical constraints, and we can take advantage of them in a set-membership framework

– Without approximations (reliability) – With fault tolerant tools (robustness)

– With a precision suitable for industrial applications

(precision)

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