Применение машинного обучения для навигации и управления роботами

Machine Learning Applications forRobot Navigation and Control

E. Burnaev

Skoltech

E. Burnaev (Skoltech) ML&Robotics 1 / 33

Outline

1 Predictive Modelling in Industrial EngineeringPredictive ModellingCustomer ExpectationsSurrogate ModelsDimension Reduction

2 Robot navigation

3 ML problem statement

4 Robot control

5 Conclusions


Predictive Modelling

Traditional approach based on the first principles

A first principlesphysics modelX Y

Example: Aircraft Aerodynamics Prediction model

NumericalPDE solver

(Euler, Navier-Stokes)

Mach numbers,Reynolds numbers,angles of attack,contact angles, ...3D surface modelof an aircraft

Lift,tension,

drag, et c.


PM: Customer Expectations

Airbus: Software for computation of Reserve Factors of aircraftstructural elements (stringers) for given geometry, material parameters,loads and applied forces (∼ 150 parameters)

Expectations (Airbus): A ∼ 100-fold drop in the running time ofthis software shortens the full cycle of structure optimization fromseveral days to several hours


SM in Engineering

PM is used in:“What-If” and Sensitivity Analysis, Design Space Exploration;Design Optimization with respect to specified efficiency criteria

Prohibitive volumes and/or run-time costs:from thousands to millions of computational experiments;running time of an experiment ranges form seconds to days;

Surrogate models: fast approximations, which substitute the originalmodels without a significant loss in accuracy


Constructing Surrogate Models

The original model f0 : X 7→ Y is based on the“first principles”

Main stages of SM construction:

1 Initialization: carry out experiments to get the initial sampleS0 =

(xi, f0(xi)

)mi=1

;2 Construction: learn a “fast” approximation f ≈ f0 over some

domain U ⊂ X ;3 Assessment: measure the accuracy of f ;4 Exploration: pick an x ∈ U at which to evaluate f0, and updateS0 with (x, f0(x));

5 Repeat steps 2-4 until satisfactory accuracy is achieved, or thecomputational budget is exhausted


Quick Aerodynamic Design of Passenger Aircraft Layout

Objective: Quickly perform trade-off studiesof passenger aircraft layouts at early designstageChallenges: CFD calculation is slow toquickly analyze thousands of aircraft layoutsin various flight regimesData:— Inputs: Geometry description, flight

conditions— Outputs: Aerodynamic characteristics

Solution: Surrogate models for global (CL,CD, . . .) and local (Spanwise lift distribution,. . .) aerodynamic characteristicsResult: Surrogate models have averagerelative error ∼ 1% and are 360 000 timesfaster compared to CFD solver


Geometry Description

A wing section is described by ∼ 60 coordinates

Number of parameters— ∼ 7 parameters to define a wing planform— airfoil in each of a cross-section:∼ 60× 7 = 420 parameters

In total we have 420 parameters


Dimension Reduction

Efficiency of surrogate models is largely due to efficient dimensionreduction

Object O → description X(O) ∈ Rp

Original airfoil descriptionX = (x1, x2, . . . , xp) ∈ Rp, p� 1

We need a reduced description Xred ∈ Rq,q � p

Using a sample of objects descriptions X1, . . . , Xn we construct a DRprocedure Π : X → Xred such that for any new X = X(O)

X(Xred(O)) ≈ X(O)

It turned out that we can reduce the dimension from p ∼ 60 to q ∼ 6thereby reducing the total number of parameters to ∼ 50


DR and SM: Challenges

Dimension Reduction procedure Π, such that

X ≈ gΠ(Xred) for all X ∈ X ,

where gΠ(Xred) is a reconstruction transformation, should satisfy therequirements:

functional proximity F (X) ≈ F (gΠ(Xred)) for all X ∈ Xtangential proximity T (X) ≈ TΠ(gΠ(Xred)) for all X ∈ X

Low-dimensional sub-manifold, defined bya “physical” model, should beincorporated inside Π

Surrogate Modeling procedures should be able to process VariableFidelity Data, Specific Data Structures, etc.


DR and SM: New Developments

Manifold Learning based on Grassmann & Stiefel EigenmapsSurrogate Modeling on manifoldsDeveloped methods allows to provide both functional andtangential proximity, as well as to incorporate submanifolds,defined by “physical” modelsIn case DR is realized by a Deep Neural Network, a physical modelcan be easily incorporated inside the corresponding computationalgraph



2 Robot navigation


4 Robot control

5 Conclusions


Robot localization

Fundamentalto practical mobile robotics, requiresa reliable model of the environment

Appearance-based localizationusing visual information:

360 degreespanorama from the mountedomnidirectional imaging system,narrow field of view images formthe mounted reorientable camera

Robot position and relative camera orientation determine the capturedimages


Regression on images

Let Θ – the localization (orientation) parameter space

At θ the imaging system captures a p-pixel imageX = φ(θ) given by an unknown map φ : Θ 7→ Rp

The Appearance space of all possible images is

M = {φ(θ) : θ ∈ Θ}

Goal: Given the training set Sm = (θi, Xi)ni=1 estimate:

an unknown Localization mapping F :M 7→ Θ;an unknown Image modeling mapping φ : Θ 7→M

in the current (fixed) environment

The estimate φ can be used to detect changes in the environment


Regression on images

The Localization mapping F , and the Image modeling mapping φ sufferfrom the “curse of dimensionality”:

instability due to collinearity or “near-collinearity” of p-dimensionalinputs;regression error can not tend to zero faster than O(n−

s2s+p ) when

an unknown function is at least s times differentiable


Regression on Image Manifold

The Appearance space

M = {φ(θ) : θ ∈ Θ ⊂ Rq} ⊂ Rp ,

is a low-dimensional manifold (Appearance manifold) with smallintrinsic dimension q embedded in p-dimensional Euclidean space andcovered by a single chart φ

Manifold nature of the input space avoids the curse of dimensionality


Regression on Image Manifold

A typical “feature-based approach”:the Principal component analysis (PCA) to find a q-dimensionallinear manifold L with q � p, and satisfying projection proximityproperty X ≈ πLX;the features of an image X are the coefficients of πLX;localization θ = F (X) is defined as

θopt = argminθ‖πL(X)− πLφ(θ)‖2 ;

Advanced method use Kernel PCA features or other techniques basedon Kernel Density Estimation, Ridge regression, Locally LinearProjection, Bayesian filtering, etc.

The Appearance manifold is curved, making estimated dimension q isusually much larger than the “true” intrinsic dimension of M (equal to∼ 2− 4)



2 Robot navigation


4 Robot control

5 Conclusions


Regression on Appearance manifold

q-dimensional Extended Appearance manifold (EAM) in Rp+q

EM ={Z(X) =

(X

F (X)

): X ∈M

}={Z(X) =

(φ(θ)θ

): θ ∈ Θ ⊂ Rq

},

consists of ‘inputs-outputs’ of the unknown mappings F and φ

Training set Sn = (Xi, θi)ni=1 determines a sample

Zn ={Zi =

(Xi

θi

)i = 1, . . . ,m

},

from the manifold

Goal: Estimate the unknown EAM from the given training dataset Zn


Manifold learning problem

Grassmann & Stiefel Eigenmaps with Regression on manifold methods

produce:

an embedding mapping h :M 7→ Rq that parameterizes the M anddetermines its “minimal-dimensional” feature space Y = h(M);

a mapping g : y ∈ Y 7→ (gX(y), gθ(y))′ ∈ Rp+q with gX(y) and

gθ(y) satisfying the proximity conditions

gX(h(X)) ≈ X and gθ(h(X)) ≈ F (X)

respectively


Parameterizations of Mthe “natural” θ = F (X), andthe recovered y = h(X)

are linked by an unknown one-to-one reparameterization: θ = u(y),and y = v(θ) for v = u−1.

Estimates u and v of the mappings are based on the dataset

Su,v = (θi = u(yi), yi = h(Xi) = v(θi))ni=1 .

Therefore parameterizations are:θ = F (X) = u(h(X));X = φ(θ) = gX(v(θ)).



2 Robot navigation


4 Robot control

5 Conclusions


Human-Robot control

Overwhelmingnumber of variables makesanthropomorphic manipulatiorswith high DoF difficult to control

Solution:infer a lower-dimensional spacehosting valid hand poses basedon captured images of validreaching and grasping motionsin controlled environmentconstructa control space and lawfrom this low-dimensional space



Manifold learning approach

The Image space of hand configurations M consists of all possibleimages with valid poses during reaching and grasping motions

Robotics and neuroscience research show that the intrinsic dimension ofthe Image space does not exceed DoF of human hand (≈ 20)


Low-dimensional image features in Human hand poses

Training set (Visual database) Mn of capturedimages (Xi)

ni=1 of human hand motions in controlled

environment;DR techniques infer a low-dimensional structure ofM , and

the intrinsic dimension q of M ;the q-dimensional image features which describep-dimensional images;

The space of q-dimensional image features can betaken as prototype of the control space


Low-dimensional image features in Human hand poses

Physically simulated robot hand attempts to achievethe captured human hand configurationThe set of all Actuator-state vectors correspondingto valid human hand poses has smaller dimensionthan DoF (∼ q)

The image features can be associated withmanipulator’s actuators states, to learn thestate-gesture map more directly

Captured Image → Marker-vectors →→ Features (DR) Regression−→ Actuator-state vector

A kinematic model can be used to provide areference “trend” to robustify a control law


Eigengrasps

PCA determines EigengraspsChanging a value of one parameter from minimal to maximalvalues under fixed value of another parameter:


Nonlinear Grasping Manifold

Nonlinear 2D subspace allows describing the grasping hand motion andis sufficient for controlling high DOF robotic systems

2D nonlinear features are used as control variables to form ControlspaceThe control function f : y → X maps 2D Control space to theDOF-dimensional hand poses



2 Robot navigation


4 Robot control

5 Conclusions


Conclusions

Efficient approaches to Predictive Modeling are developedThese approaches allows to take into account specific requirementsof Robot navigation and control problemsIn particular, we can efficiently incorporate a sub-manifold, definedby a physical model, into a predictive model, used for robot controlFurther applications motivate new specific Machine vision taskssuch as Regression on Images, Nonlinear dimension reduction onImage manifolds, etc.



PM: Potential

∂v

∂t+ (v · ∇)v = −1

ρ∇p+ ν∇2v + f ,

∇ · v = 0 .

Predictive modelling in engineering:1990-s Typical volume of experiments (around 10-100) is enough

to compare solutions, but not enough to carry out fullyfledged optimization;

2000-s Advances in High Performance Computing makeengineering optimization economically feasible

Fact: The demands of the industry grow much faster than thecomputational capacity


Technology

Применение машинного обучения для навигации и управления роботами