Adaptive Model Learning of Neural Networks with UUB Stability for Robot Dynamic Estimation Pedram Agand

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Adaptive Model Learning of Neural Networks with

UUB Stability for Robot Dynamic Estimation

Pedram Agand

Advanced Robotics and Automation Systems (ARAS),

Department of Systems and Control,

K. N. Toosi University of Technology,

Tehran, Iran.

aagand@email.kntu.ac.ir

Mahdi Aliyari Shoorehdeli

Advanced Process Automation and Control (APAC),

Department of Mechatronics Engineering,

K. N. Toosi University of Technology,

Tehran, Iran.

aliyari@kntu.ac.ir

Abstract—Since batch algorithms suffer from lack of proﬁ-

ciency in confronting model mismatches and disturbances, this

contribution proposes an adaptive scheme based on continuous

Lyapunov function for online robot dynamic identiﬁcation. This

paper suggests stable updating rules to drive neural networks

inspiring from model reference adaptive paradigm. Network

structure consists of three parallel self-driving neural networks

which aim to estimate robot dynamic terms individually. Lya-

punov candidate is selected to construct energy surface for

a convex optimization framework. Learning rules are driven

directly from Lyapunov functions to make the derivative negative.

Finally, experimental results on 3-DOF Phantom Omni Haptic

device demonstrate efﬁciency of the proposed method.

Index Terms—Adaptive neural network, Lyapanov candidate,

robot dynamic, stable updating rule.

I. INTRODUCTION

Model-free identiﬁcation in robotic applications has been

extensively discussed in recent literature [1]. Since many

control architectures in robotics require meticulous model

of robot, exigency of efﬁcient identiﬁcation methodology is

inevitable [2]. Even by ignoring some high-frequency dynam-

ics and frictions, ﬁnding the dynamic of complex parallel

robots still somehow difﬁcult. Besides the incontinence along

with ﬁnding the physical equation, due to inaccuracy of the

model, robust approaches should be employed to guarantee

the stability of the overall system that is not satisfactory in

some delicate applications due to insufﬁcient performance [3].

To overcome this issue, data-driven identiﬁcation is suggested

for improvement of model quality.

It seems more likely that the breakthrough will come

through the use of other more ﬂexible and amenable nonlinear

system modeling tools such as the neural network in the

form of multilayer perceptron (MLP) and radial Basis function

(RBF), fuzzy, Local Linear Model (LLM), ARX, etc [4].

Among them, neural network proves to be a powerful yet

simple tool for the nonlinear identiﬁcation problems, that have

been used extensively in the areas of ﬁltering, prediction

(e.g. [5]), classiﬁcation and pattern recognition (e.g. [6]),

system modeling (e.g. [7]) and control (e.g. [8]). Worthiness of

nonlinear system identiﬁcation, especially in robotics systems

This work was published in 2019 International Joint Conference on Neural

Networks (IJCNN). More information contact: pagand@sfu.ca

is undeniable, since control systems encountered in practice

possess the property of linearity only over a certain range

of operation [9]. Identiﬁcation in robotics can be considered

as two different points of view. In the ﬁrst perspective,

identiﬁcation are accomplish for calibration of Kinematics

(e.g. [10]). In the other hand, identiﬁcation is utilized as a way

to render actual dynamic (e.g. [11]) and control applications

(e.g. [12], [13]). Based on this classiﬁcation, online and ofﬂine

identiﬁcation methodology has shade the world of science.

Stable learning rules were proposed for feedback lineariza-

tion network in a class of single-input-single-output systems

with continuous Lyapunov candidate. The idea of driving

adaptive rules directly from continuous time Lyapunov func-

tion was presented by [14], where there was no need for the

pre-assumption of network construction errors bounds, since

the rules are a smooth function of states. In [15], radial basis

function is used as an adaptive ﬁlter by constructing discrete

time Lyapunov functions. A robust modiﬁcation term is added

to the updating rules to reinforce identiﬁer against model

mismatches and runtime disturbances. New adaptive back-

propagation type algorithm is introduced by [16] to eliminate

disturbances. It is worth-mentioning to say that the utilized

Lyapunov candidate only includes networks error (V(k) =

Pβke2(k)). In [17], multilayer neural network is utilized

for classiﬁcation of multi-input-multi-output system. By using

Taylor expansion and discrete time Lyapunov candidate, the

stability of the learning rules were proven.

In this paper, an adaptive learning rule is adopted for the

network structure presented by [18] while preserving UUB

stability. By this end, nonlinear activation function in hidden

layer are linearized using Taylor expansion. Not only are

stability and convergence of the networks errors targeted in

this paper, but also the speed of error convergence is also

controlled by adjusting the deﬁned tuning parameters. The set

of three independent networks are guaranteed to converge by

deﬁning Weighted Augmentation Error (WAE). By proposing

a Lyapunov surface comprising set of augmentation errors

and parameter variation, a framework for converging global

minimum is established. Updating rules are driven directly

from Lyapunov function making its derivative negative. Since

neural network can never fully ﬁt the desired dynamic due to

arXiv:2210.15055v1 [cs.RO] 26 Oct 2022

network construction error, modeling mismatches and noises,

a robust modiﬁcation approach is utilized to avoid parameter

drifts.

The reminder of this paper is organized as follows. In

Sec. II, the problem is declared using mathematical relations.

In addition, the structure of neural network to solve it is

presented. Sec. III is devoted to learning algorithm and weights

adjustment. Some discussion about convergence and stability

are outlined in Sec. IV. Experimental results on a 3-DOF serial

manipulator, Phantom Omni Haptic device is done in Section

V. Finally, the paper is concluded in Sec. VI.

II. PROBLEM STATEMENT

By a class of Euler-Lagrange equations for robot dynamic

we have

M(X)¨

X+C(X, ˙

X)˙

X+G(X) = τ=JTF(1)

where, M, C, G are the Inertia, the Coriolis and centrifugal

and the gravity. τ, Fdenotes task-space and workspace forces,

respectively. Jacobian of the robot is denoted by J. The ﬁnal

aim is to obtain dynamics terms (namely M, C, G) individually

by a set of excitatory inputs including X, ˙

X, ¨

X(motion

variables, namely position, velocity, and acceleration) using

a gray-box identiﬁcation framework. Three parallel MLP-

network is considered for each term as following:

ˆyn(Wh, W o) =

j=1 Wo

nj FPn

i=1

jixi+Wh

j0+Wo

n0(2)

where Wh, W oare the hidden and output layer weights

respectively, n={1,2,· · · , N},Nis number of robot degrees

of freedom, Pnis equal to number of hidden layers in each

network. Inputs for each network (xi) is illustrated in Fig. 1.

The term ˆyncan be presented in the form of ˆ

Mn,ˆ

Cn,ˆ

Gn. Each

of these targets construct their own local error as follows:

n−ˆyj

n=ej

n∀j∈ {M, C, G}, n ∈ {1,2, ..., N}.(3)

This error can not be calculated in each step, since no target

output exist for each subnetwork. Therefore, the error is

conveyed to secondary layer as follows:

τn−ˆ

Mn¨

X−ˆ

Cn˙

X−ˆ

Gn=e1n∀n∈ {1,2, ..., N}.(4)

The relation presented in Eq. (4) is not sufﬁcient to optimize

whole networks. Moreover, admission of other criteria are

taken into account to conﬁne feasible set of parameters. By

other words, representation of the error is conveyed to the

secondary layer which resolve inherent interconnectivity of

the structure. Transparency is an intrinsic speciﬁc of this

structure, since dynamics terms are included explicitly by

separate parallel networks. Relations presented in (5) and (6)

are driven from the skew-symmetric property of (˙

M−2C).

Mnn −2ˆ

Cnn =e2n∀n∈ {1,2, ..., N}.(5)

Mni +˙

Min−2( ˆ

Cni +ˆ

Cin) = e3m

∀i6=n, m ∈ {1,2, ..., (N2−N)/2}.(6)

Furthermore, according to mass property that states inertia

matrix is not rank deﬁcient and should satisﬁes λIn×n≤

M(X)≤¯

λIn×nfor 0≤λ≤¯

λ, another error will be deﬁned

as follows:

|ˆ

M−λ In×n|=e4n, λ =eig(ˆ

M)eλ0/t ∀n∈ {1,2, ..., N}

(7)

where |.|denotes matrix determinate. To drive the updating

rules, a multi-objective optimization problem with prescribed

equality conditions is considered as

Min{J}=eT

1e1, s.t e2=e3=e4= 0 (8)

where,

j(k)=[et

j1(k), et

j2(k), ..., et

jn(k)]T,∀j∈ {1,2,3,4}(9)

in which the superscript (t)shows the order of data and

(k)depicts epoch’s number. The cost function in (8) can be

optimized by adding Lagrangian weights as

H=eT

1e1+λ1eT

2e2+λ2eT

3e3+λ3eT

4e4.(10)

Therefore encapsulated error in a form of WAE is constructed

as follows representing a target solution:

εt(k) = [eT

1(k), λ1eT

2(k), λ2eT

3(k), λ1eT

4(k)]T, λi>1.(11)

III. LEARNING ALGORITHM

In this section, an adaptive learning rule is driven with

Lyapunov stability rules. The trajectory of a dynamic system

with an equilibrium point at origin is said to be uniformly

ultimated bounded (UUB stable) with respect to Sas any

Lyapunov level surface of V(system Lyapunov candidate), if

the derivative of Vis strictly negative outside of S. Hence, all

trajectory outside Smust be converged toward it. However, the

asymptotic convergence of the trajectory can not be inferred.

Theorem 1. If the updating rules of the prescribed network

in (2) are considered as

i=−γi1εξi;˙

i=−γi2εζi∀i∈ {M, C, G}(12)

where εis obtained from encapsulated error in (11) and,

ζi=∂ˆτ

∂W o

;ξi=Wo

∂ζi

∂W h

;i∈ {M, C, G}(13)

then the system has UUB stability in conﬁned subset

S:{x|kx−ˆxk ≤ γν0}(14)

where γ > 1is a tuning parameter which controls the stability

margin of the system and ν0is the minimum inevitable error

in a network that can be conﬁned by

ν0= sup

x∈B

|f0−ν|<∞(15)

where f0is Taylor expansion error and νdenotes the network

reconstruction error.

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AdaptiveModelLearningofNeuralNetworkswithUUBStabilityforRobotDynamicEstimationPedramAgandAdvancedRoboticsandAutomationSystems(ARAS),DepartmentofSystemsandControl,K.N.ToosiUniversityofTechnology,Tehran,Iran.aagand@email.kntu.ac.irMahdiAliyariShoorehdeliAdvancedProcessAutomationandControl(APAC),Depart...

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Adaptive Model Learning of Neural Networks with UUB Stability for Robot Dynamic Estimation Pedram Agand

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