Luận văn design an adaptive controller and a state observer based on neural network for the 4dof parallel robot

IIANOI UNIVERSITY OF SCIENCE AND TECIINOLOGY MASTER THESIS Design an adaptive controller and a state observer based on neural network for the 4DOF parallel robot NGUYEN MANII CUONG Co

IIANOI UNIVERSITY OF SCIENCE AND TECIINOLOGY

MASTER THESIS

Design an adaptive controller and a state observer based on neural network for the

4DOF parallel robot

NGUYEN MANII CUONG Control Engineering and Automation

Supervisor: Assoc Prof Nguyen Tung Lam

School: School of Electrical and Electronic Engineering

IIA NOI, 2022

HANOI UNIVERSITY OF SCIENCE AND TECHNOLOGY

MASTER THESIS

Design an adaptive controller and a state

observer based on neural network for the

4DOF parallel robot

NGUYEN MANII CUONG Control Engineering and Automation

Supervisor: Assoc Prof Nguyen Tung Lam _

Supervisor's Signature

School: School of Electrical and Electronic Enginecring

HA NOT, 2022

CONG HOA XA HỘI CHỦ NGHĨA VIỆT NAM

Độc lập — Tự do— Hanh phic

BẢN XÁC NHẬN CHỈNH SỬA LUẬN VĂN THẠC SĨ

Ho va tén tac giả luận văn : Nguyễn Mạnh Cường,

Dé dải luận văn: Thiết kẻ bộ điêu khiển thích nghĩ và bộ quan sắt trạng thái dựa trên mạng nơ ron cho robot song song bên bậc tự do (Design an adaptive

controller and a stats observer based ơn neuai nolwork for thế 4DOE parallel robot)

Chuyên ngành: Kỳ thuật Điều khiển và Tự đông hóa

Mã số SV: 2020201 6M:

Tác giả, Người hưởng dẫn khoa học và Hội đông cham luận văn xác nhận

tác giả dã sửa chữa, bỏ sung luận văn theo biên bản họp Hội đồng ngày 04/05/2022 với các nội dụng sau:

TABLE OF CONTENT

CHAPTER 1 OVERVIEW

1.1 The four degrees of freedom parallel robot (ADOFPR) model

1.2 Trajectory trackuig controllers and state ObserVes

1.2.1 Trajectory tracking controllers

32 Controlter design for 4DOFPR - - 10

221 Backstepping aggregated with SMC (BASMC) - 10 2.22 RRFNN-based (RBFNNB) adaptive controller 13 2.2.3 High-gain observer for the adaptive controller - 7

23 Conelusion

CHAPTER 3 SIMULATION RESULT!

3.1 Results of the RBI'NN based adaptive controller (RIINNH)

3.2 Simulation results of the adaptive controller using the high-gain state

LIST OF TABLES

Table 3.1 Reference trajectory parameters

‘Table 3.2 Control parameters

‘Table 3.3 Trajectory reference parameters

„38

tờ

LIST OF FIGURES

Figure 1.1 Parallel robot applied in the car motion simtlator 2 Figure 1.2 Parallel robot applied in rehabilitation system [41 2 Figure 2.1 (a) Robot coordinate; (b) Vector diagram of ADOEP

Figure 2.2 Structure of BASMC controller

Figure 2.3 RBENN structure

Higure 2.4 Structure of the adaptive controller .essesessneeneenene -

Figure 3.1 Hxiemal fAT68 uc ceiiroreriiririrariirrrerroreuao T5 Tigure 3.2 Motion trajectory of p 34 Figure 3.3 racking erTor 0 g ào nen — - Figure 3.4 Approximated valies sccssssvestssessenesneenineeietonaeinte 26 Figure 3.5 Motion trajectory of 9 wasssssssuessssernenesneeneeietonaeinte 37 Figure 3.6 Tracking error of a7 Figure 3.7 Uncertain parts in the robot model 29 Figure 3.8 Observed values of q - 30 Tigure 3.9 Observed values of ¢ - - - - 30 Figure 3.10 Observational error of ý oiicenrororoee seo 3] Figure 3.11 Rstimated values from RBFNN 32 Figure 3.12 Robot’s trajectory: ssssssessenesersinessenssneeinete xaeseasao 9) Figure 3.13 Tracking exror „33 Figure 3.14 Observed position with diferent values øŸ sụ, saase34 Tigure 3.15 Observed velocity with diferent values of ø„ ¬—

LIST OF ABBREVIATIONS

Abbreviation Definition

4DOFPR Four Degrees of Freedom Parallel Robot DOF Degrees of Freedom

SMC Sliding Mode Control

DSC Dynamic Surface Control

RBENN Radius Basis !’unction Neural Network

BASMC Backstepping aggregated with Sliding Mode

PTER t OVERVIEW

1.1 The four degrees of freedom parallel robot (4D0FPR) model

Nowatlays, robotic systems are being increasingly rapidly developed and

applied in several economic and social life fields because they are designed for particularly complex and dangerous tasks or repetitive jobs and require high accuracy Morcover, apart from being almost precise and consistoril, with their

flexible operating ability, robots are capable of working in hazardous

environments In addition, the robot can perform tasks with heavy loads and toxic substances and can adapt to particular environmental conditions Thus, these

advantages have signifieanily contributed to produclivity and qualily

improvement, preventing accidents and saving labor costs

In state-of-the-art technology, parallel robots are increasingly prevalent in the industry, military, medical, and entertainment Various numbers of parallel structures in |1 |, [2], |3], [4], and [5] have been taken into account, including the six degrees of freedoms (DOF) robot in [1 J, which is capable of applied in medical surgery, as well as rehabilitation in [1], and some other structures applied into flight and automobile simulation Most of these models have been implemented based on the advantages of parallel structure, namely low inertia moment, high

load, and stnooth iansmission vapacity [6] From reality-based car models, lo

assist trainees and drivers have an alternative approach to getting familiar with the automobile’s movements, it is necessary to construct a driving simulation model based on a class of parallel architeclures and motion plai forms developed recently

[3] Moreover, car driving simulation models are also constructed with the purpose

of mitigating unexpected forces impacting drivers in practical and virtual reality cuviomments wi relalion to health care and rehabililation 14], L5]

In order to describe the movement of the robot system, the demand for robot

modeling is imperative Several studies [6], [7] showed the geometrical analysis

of a six DOF constrained parallel robot Regarding the construction of the mathematical model, a forward and inverse kinematics medel of Quanser’s Hexapod robot has been illustrated in [8] In addition, the six DOF parallel robots have a positive advantage of high accuracy movements However, the complexity

of six actuators’ interaction and coordination gives the rising complexity in

designing trajectory tracking controllers of parallel robots, especially in the presence of massive uncortaintics Therefore, the configuration with fewer joints and DOF is able to mitigate the inevitable hysteresis and redundancy of actuators shown in [9], [10], and [11], thereby, it would be more convenient in particular practical applications and controller design considered uncertain elements In addition, in the attempt to reduce computation complexity and redundant constraints, the group of authors has constructed the four DOF platform, comprising the movements of rotating and translating along the vertical axis OZ,

rotating about the OX and OY axis.

Figure 1.1 Parallel robot applied in the car motion simulator

Figure 1.2 Parallel robot applied in rehabilitation system [4]

From the reference and analysis of the above scientific works, moreover,

intending to reduce the computational complexity and redundant constraints while still ensuring the necessary motion, the thesis puts focus on the four degrees of freedom parallel robot platform with the movements of rotational and translational

movements along the OZ axis, rotation in the OX axis and the OY axis

1.2 Trajectory tracking controllers and state observers

1.2.1 Trajectory tracking controllers

In robot control, especially in orbital tracking control problems, modern

methods specially put focus on designing control algorithms capable of handling

problems related to uncertainties, perturbations, and unknown structural

components in the system model while still ensuring stability and tracking quality The 4DOFPR parallel robot model is considered to be a model being commonly

affected by nonlinear uncertain elements in practical applications, especially

external forces acting in different directions on the system

The parallel structures are considered a nonlinear model in the control design field, therefore, a control issue has attracted significant attention in the scientific

community One of these designed methodologies for nonlinear control systems

-thai have been inlorested in is the Backstepping technique as in [12], [13], [14],

[15], and [16] in order to ensure the quality of trajectory tracking control, Ilowever, when uncertainties or unmodeled components exist in the system model, the

“oxplosion of lms” phenomena adversely allects the control quality Another

prominent control methad is sliding made control (SMC) which las been widely

used because of its robust characteristic as in [17], [1B], and [19] when considering the existence of unknown elements However, the chattering phenomenon wgonorated by the SMC controller is ikely Lo demolish the aystom [20], as well ax

the computational burden with the high order systems Combining the two

aforementioned controllers is an approach to improving control performance because it takes advantage of them Then, the robustness characteristic is enhanced, and the computational cost is reduced as in [20], [21], [22], and [23] Nevertheless, the combined controller cannot cope with the chattering and

“explosion of terms” phenomena

On the other hand, by taking advantage of the multiple sliding surface controller and Backstepping technique, dynamic surface control (DSC} has been proposed to address the problem “explosion of terms” in [24] and [25] by using a low-pass filter for each computation step 1Iowever, the errors of the low-pass filter

in the DSC controller are a dilemma, majorly depending ou a filter time constant and being proven by complex malhemalical conditions in (24), which may correlate with the frequency of experimental devices Alternatively, a more efficient method in this paper handling mathematics difficulty is utilizing a neural network to approximate virtual signals and alleviate the chattering phenomena

Tu control Iheory, noise components are commonly considered to be an

tnevilable part of the whole system, and analyzing noise is the key Lo finding a way that assists the (DOFPR system to be more stable and accurate To be more specific, stochastic disturbances are problematic, impacting the 4DOI'PR system

Tn teras of non-Gaussian noises, the modified extended Masrclicy—Martin filter

constructed in [26] is an efficient approach to handle nonlinear systems when environmental disturbances influence the whole system Besides, stochastic parameters have been taken into consideration in [27] by estimating stochastic

nonlinear systems By laking into cautious consideration published in [28] and

[29], it is assumed that some stochastic disturbances as to an unknown varying force from the input system act on actuators of the 4DOFPR system along the vertical direction because of body weight arc moment disturbance as well as unknown parts However, there have been several kinds of noises in external and intemal stochastic disturbances because of all range elements [30], from frictions, vibrations, and changes of sudden forces to the shuft in environmental conditions,

which are considered uncertamies Tn this thesis, we assume Lhal the 4Q0FPR is

the model prone ta the impact of stochastic uncertainty elements

As mentioned above, for many conventional nonlinear controllers such as SMC or Backstepping, there have been drawbacks in improving control performances whon it is challenging to identify the accurate model because of the

existeuee of the two kinds of unknown elements, meluding matched and unmatched uncertainties In addition, in order to achieve high control quality, almost conventional methodologies in control designs require the identification of imaihemalical models thal have uncerlainty elements in advance Thus, adaplive comrollers, in which the updated law is designed based on mathematical characteristics to adapt unknown components, were considered as in [31], [32], and |33[ However, the convergence of these controllers majorly depends on the system model ckuragloristics and control parameters, and thus the wide range of

‘uncertain elements can not be taken into account Therefore, these controllers are

not adaptable to the change of unknown uncertainties and the diversity of the model specifications in the 4DOFPR model Hence, another approach using

approximate (he uncerlainties by using neural networks is imvestigaled in this

in [38 [has solved the problem of the dual-arm rebot containing the noises

Moreover, ihe idea to use the nolable technique in approximating the noulinear function of the Radial Basis Funclion neural network was couducled in [39] and [40] By online tuning network parameters, the updated law of the RBENN shown in [40], [41], and [42] was derived based on the Lyapunov theory, guarantecing the minimization of the stochastic disuurbance in terms of the overall

system Besides using RBFNN, the fuzzy logic in which the weight value is based

on the adaptive law has been designed [43] ta effectively deal with the uncertain dynamics problem by estimating the appropriate values for the controller

By analyzing the above studies, solutions, and approaches, the thesis proposes an adaplive controller based on Backstopping-SMC using RBFNN 1o

guarantee the tracking performance of the four degrees of freedoms parallel robot,

which contributes:

- The method focuses on the problem of controller design for the uncertain model of the ADOFPR Dissimilar Lo the paper [31 |, [32], and [33], the approach

based ou RBFNN does iol majorly depend on the mathematical characteris

parallel robots Thus, the developed controller can cope with the larger range of uncertam clements and bounded disturbances because of its outstanding characteristic in approximating nonlingar paris and compensating for the external

noises

sof

- The proposed adaplive controller is designed based on the Backslepping technique aggregated with SMC Therefore, it takes advantage of these controllers

in eliminating nonlinear parts, as well as enhancing the robustness of the system Moreover, the adverse impacts of the wo controllers, including “explosion of

terms” and challering, are remarkably alleviated because the unknown elements,

which are the main factors causing the phenomena, are approximated and compensated by the neural network Thus, the developed controller not only overcomes Lhe disadvantage of the above-mentioned techniques bul also increases

the robustness and adaptive behavior of the overall system

- ‘The RBENN is designed to estimate values for all unknown components of the dynamic system and compensate for the disturbances of the input torque and

extemal forces Thus the controller can guarantee the control quality ever when

only the inaccurate information about the system is known Different from [37],

1381, [39], [40], and [43], neural network outputs are shown in comparison to nominal uncertain values in order lo verify Ue approximaling performane

as the influence of the neural network on the system performance as well

The purpose of the controller design is to construct an adaptive controller based on RBFNN for the ADOFPR to approximate uncertain components and compensate for the disturbances in the model when these factors exist in the robot model Tn this study, the approach focuses on constructing an adaptive control algorithm for the ADOFPR, which is affected by unknown extemal forces and the uncertain components in the mathematical model ‘I'he 3ASMC supported by the RBFNN demonstrates well-being tracking performance and closed-loop stability Given the sell-leaming capacity of the neural network, such considerable information associated with complicated driving car modeling is not required for neural controllers ‘hus, newal-besed controllers are able to cope with a more extensive seope of une

In robotic systems, system state estimation plays a vital role in controller

calculation or simply collecting information for visualization and monitoring, Tn

addition, system signals, including position and velocity, apparently need to be accurately measured in the feedback control system architecture to ensure control quality In fact, improving the accuracy of the state measurement and information about the system is continuously focused on research A conventional way to solve this problem is to use measurement sensors to obtain the necessary information, including both direct and indirect data about the required quantities Then, those values are used to design observation algorithms to process this information to generate reliable information about the entire system states Ilowever, these algorithms can only exist if the sensors can obtain enough information to determine the systom’s stato In practice, the number and quality of sensors used are often

limited by cost and physical constraints, so cbscrvers play an essential role in many applications

Commonly, linear observers and sliding mode observers are constructed for linear systems Llowever, the fact that the system contains nonlinear components, even uncerlain components, il is challenging for these observers to guarantee the quality of observation and control of the system Therefore, nonlinear observers have been researched and designed with specific structures and advantages [44], [46], [47], 1481, [49], and [50] ‘the most prominent aud prevalent observers for the class of nonlinear systems are the high-gain observers [44], [52], the sliding mode observers [46], [17], [18], and extended observers The four-degree-of- freedom parallel robot model is an object that contains highly nonlinear components and is also affected by noise and unknown components Besides, although it is robust to noise, the sliding observer has the well-known chattering phenomenon Therefore, extended or adapted observers or techniques were studied

in [46], [47], [48], and [49] to improve the quality and handle this phenomenon However, this also increases the complexity of the computation and implementation of these observables Therefore, due to its ability to minimize the impact of model variation and uncertainty components along with a simple design structure and ease of implementation in practice, the high-gain observer is chosen 1o design (or the robot object in the thesis

Tn [44], [45], and [51], an averview of high-gain observers was introduced for nonlinear systems and feedback control systems In addition, the observer design structure for the multi-input, multi-output nonlinear systems was also constructed in [51] in two approaches , including a general method (or the observed

uoulinear system and the olher which structure is decomposable into linear and

nonlinear parts It can be seen that the high-gain observer has the advantage of observing systems with high-order nonlinear components as well as with

distisbances [52], [53], and [54] Typically with the Quadrolor drone [$2], the

high-gain nonlinear observer was used ta monitor, analyze, and compensate for

the actuatar’s deviation and the measurement errors In addition, the controller designed m [52] reduced the use of sensors and actuators when using an observer

When using a high-gain observer with feedback controllers for a nonlinear system,

especially a sliding mode controller, the system’s overall stability was ensured even if the system contains uncertain components [48] Besides, the study [54] proposed a structure of the high-gain observer combined with control law [or arr

electro-hydraulic servo steering system because of its complex structure and

variable load The automatic steering system plays a decisive role in the stability

of self-driving car systerns ‘The flexible combination of the observer and the controller not only reduces the complexity of using the sensors bul also ensures

the stability of the system In addition, an adaptive controller using a high-gain observer was also designed for the mobile robot to solve the problem of orbital

tracking [$5]

From the reference, atalysis, and evaluation, the high-gain observer will be

constructed for the neural network-based adaptive controller for the uncertain amodel of 4DO¥P affected by exogenous noise ‘thanks to the ability to quickly converge as well as handle matheratical model errors, the high-gain obsorver can

ensure observation quality to guaranlee the quality of the trajectory tracking

contro! problem ‘he observer design and its use for the adaptive controller wilt

be presented m Chapter 2, along with the results accompanied by the analysis and evaluation of [he whole system quality in Chapter 3

13 Conclusion

In Chapter 1, an overview of the four-degree-of-freedom parallel robot was presented along with various applications in several fields From the study and

analysis of revent studies on nonlinear controllers in solving the trajectory tracking

problem, an adaptive controller based on a radial neural network is proposed to solve the problem of robot model uncertainty as well as the disadvantages of model-based controllers Promiment representatives of slaic-observer seis were

algo reviewed From the tracking control problem for a parallel robot model with

four degrees of freedorn combined with the analysis of related scientific publications, the scope and research objectives of the thesis have been proposed

and presented.

CHAPTER 2 DESIGN AN ADAPTIVE CONTROLLER AND A STATE

OBSERVER FOR THE FOUR DEGREES OF FREEDOM

PARALLEL ROBOT This chapter presents the design of an adaptive controller anda state observer

for a four-degree-of-freedom parallel robot First, the kinematics and dynamics

equations are described to serve as a premise for controller and observer design

Then, an adaptive controller will be designed using the combination of the Backstepping controller and the sliding mode control technique to take advantage

of the two controllers for nonlinear systems Next, the radial radius neural network

and the update rule designed based on the Lyapunov stability criterion are

constructed to approximate the uncertain components to overcome the

disadvantages of the above controllers and enhance the control quality for the robot system Finally, a high-gain state observer is used to reduce the complexity of

using sensors to measure the motion velocity quantities of the robot

2.1 Mathematical model of the 4DOFPR

Figure 2.1 (a) Robot coordinate; (b) Vector diagram of 4DOFPR

This section describes the robot’s kinematic and dynamic models followed by

a global problem in the control design procedure Figure 2.1 (a) depicts the overall

system and system coordinates The global coordinate is chosen as in Figure 2.1

(b) System states are rotational angular about Ox, Oy, Oz axes, and translational

position along the vertical axis of the upper platform’s panel Robot movements

are performed by simultaneous vertical motions of three pistons combined with

the rotation of a rotating shaft attached below the base platform The kinematic

model demonstrates a transformation between the piston’s movement and P’s

position, shown in the following subsection,

Model-based control of robots requires a computationally efficient

formulation of motion equations achieved by expressing the dynamic model in

terms of a set of independent joint coordinates, referred to as minimal coordinates Minimal coordinates formulations have been widely used for nonlinear motion control of robot manipulators To this end, the joint coordinates of a robot

8

imanipulator thal, is subjected to kinematic loop constraints are spl into a scl of

dependent and independent coordinates

2.1.1 Kinematic modet

The system state vector is denoted as p—[p., œ, f, 7], where p is

vertical position and a, Z, and 7 are rotational angular about OX, OY, and OZ

axes, respectively Besides, vector q=|, / 4 yf describes the piston’s

lengths and QZ rotational angular with J is the piston length ¿ (—1,2,3) ¢ is the distance between origin O and base platform’s O,, while a and 6 are the radius

of upper and base platforms, respectively According to the vector diagram, a vector cquation of 4DOFPR is computed as:

where /=1,2,3 The coordinates of 1:, A2, and A; in the local coordinate PY,

Ao =[ @ 0 Of, and Aan =| ia) aeo( ) 0]

Assuming that a—4, ihe coordinates of 2: Bz, and Br in the local coordinate

;

Buyer =| asin(=) -a cost) 0| Besides, the center points of the base and upper

panel are O,-|0 0 cf and y,-|0 0 p[ in the global axis, respectively Then, we obtain fixed coordinates of 4 and 8 are 4 —1).4,.,+0,, and

foos #cosy cosysinasinf—cosasiny sinesiny+cosacos;sin J

T,-| cosPsiny cosxcosy tsinasinfsiny cosasinfsiny coszsine |

Finally, the length of the robot's legs is computed as

where A.B, —8—4 From (2), q can be computed from p

2.1.2 Dynamic model

Commonly, to achieve more precise movement, Ihe controller lakes the

dynamic model, which additionally considers the kinetics energy of the system, into account ‘therefore, the elements containing forces, acceleration, mass, inertia, and other system parameters are pul effort (o accuralely identified TLis evident thal with the controller designed based on a dynamic model the more reliable the system is, the better control quality can obtain Nevertheless, it is challenging to identify these parameters exactly, and thus, there has been errors between the

actual and mathematical madel Therefore, asstaning thal the dynamic robo maded

is a nominal model with bounded uncertain elements and noises

According to [25], the dynamic model of the robot based on the Ruler-Lagrange

tà (M),m, (Âg), mạ (Âu ),rn, (Âg 3,12, (Man), 1ạ„ (pm), and 1, (gam) are the mass

of the covering of pistons, the mass of pistons, the mass of motors on the cylinders, the mass of the mobile panels, and the inertia of the mobile panels about Ox,Op,

g298(n/5") is the gravity neccleration F=[25 6, 4 +]

control signal vector

2.2 Controller design for 4DOFPR

2.2.1 Backstepping aggregated with SMC (BASMC)

‘The controller designed by combining the two controllers can ensure that the robot tends to track the desired trajectory with the identified mathematical model

By combining the Backsleppiug tachnique and SMC method, the controller lakes

10

advantage of the {wo control algorithms, which means il can eliminate the systetn”s nonlinear elements and increase robust behavior to external noises

To facilitate the control design procedure, initially, assuming, that M, C, and

D are accurately known, which means all the information conccming the robot system is clearly identified Therefore, the controller is designed under ideal conditions with all information available Subsequently, a reference trajectory for

the dynamic model qO=[4,0 bj fe) 7,0] is compuled from the

vobot’s reference trajectory and is deGned as pyO=[p,@ «(A 2,0 voy

using the forward kinematic equations ‘To clearly describe the tracking process, in the begining, the differcuce between q and q, in cach control period is derioted

by

With the tacking ovor vector &, the control objective is now equivalent to Loree

&, tend to the vieinity of vero A virtual control which 1s chosen as the derivative

of q, which satisfics the following Lyapunov candidate function

& is also considered as a sliding manifold to compute the system control input

1

Sliding mode contra

Figure 2.2 Structure of BASMC controtler

Figure 2.3 illustrates the structure diagram of the BASCM controller The

tracking error is considered as an input for the Backstepping controller, and the virtual control signal is calculated In fact, the physical value of this signal is the ideal velocity of the robot’s lengths and the rotational about Oz axis After the desired control signal is derived, the virtual signal error is fed into the sliding mode control to calculate the control signal for the system

Taking derivative of &, results in

‘The control signal F comprises two parts: an equivalent control signal and

a switching control signal having the form of the SMC controller The equivalent control KE is chosen as follows

which holds the system state on the defined sliding manifold Besides, the switch control K,, drives the ayslem state to the stiding surface

where Ki, = dicag ly, Rayong hog) and Ky = diag kyp oq hy) are diagonal positive

definite matrices One of the main factors causing the chattering phenomena is the

presence of the signum function Commonly, to decrease the impact of this issuc, the satlins function is selected, and the switching control signal (2.14) is transformed into

Eventually, the control signal is obtained by

P-R+R, (2.16)

Llowever, there will be a trade-off between the robustness of the system and the smoothness of the control signal In addition, this phenomenon can be climinated only in a certain range using this method Theorelically, the chattering phenomenon is majorly caused by the existence of unmodeled dynamic elements

in the systems ‘he RBKNN can estimate appropriate values for unmodeled clements, thus reducing the impacts of the chatfcring phonomenon,

Substituting (2.16) into (2.12), ?, tums into

Remark 1 BASMC controller is constructed based on the Backstepping

technique; therefore, the phenomenon called “explosion of terms” is caused by iteratively and incrementally taking derivative of the virtual control signal (2.7)

Especially when the syslem contains umcerlain elements such as discontinuous

functions, this caloulation can have an adverse effect on system performance Additionally, even though the satlins function in (2.15) has the ability to reduce the chattering’s impact, there will be a trade-off in the system’s robust

characteristic since this model-based controller cannot capture all the dynamics in the ADOFPR model, which is often subjected to uncertainties, noises, and external forces,

2.2.2 RBFNN-based (RBFNNB) adaptive controller

In practice, it is difficult to precisely identify M,C, D, and external

disturbances t, Therefore, with a model-based controller such as BASMC, the

control Jaw only uses nominal values of Lhese matrices Hence, BASMC may nol

meet the tracking quality because of the “explosion of terms” phenomena

accompanied by the chattering problem This section proposes a controller which can solve the problems stemming from uncertain elements and extemal

disturbances

M,C, and D are considered as the actual values of system parameter

AD are denoted as the differences between ideal and nominal values of these cuncertainties Without loss of generality, it can be shown that

M_M+AM,C_@+AC, and D-D+AD, the robot system (2.3) is transformed into

To facilitate the control design steps, (2.18) is rewritten as

—M'F—Ñ'(Cq+ D)+M{x¡—(AMR + ACủ + AD)) (2.19)

13

Finally, system dynamig equalions lum into

The iterative calculation of @ causes “explosion of terms”, therefore, to

overcame this problem, proposing an RBFNW to estimate the uncertainties and

provide the appropriate value for the derivalive of @ Moreover, estimated values

generated by RBENN for the uncertain dynamical parts can significantly reduce chattering impacts The RBFNN has the ability of online learning to approximate

highly-nonlincar functions, and thus, the controller dees nol require full or even

any prior accurate knowledge about the system Assuming that @ is unknown and

respoolively In (2.23), @CR™ is an out veclor of the neural network Tn addition,

6 is denoted as the idea value for 6, and thus, RRFNN is designed ta guarantee

6 approaches « with an arbitrarily small error or estimate an appropriate value for o to ensure the system’s stahilily

where is approximation ciror and |t| %$ with ¿ is an arbitrarily small positive constant W and W are ideal weight and updated weight matrices, respectively

Additionally, W = W —W is considered as the error weight matrices RII’ function

is employed Lo calculate the outpul of the hidden layer as follows

where the system state q is chosen as an input vector, @ is a center point, and 7

presents the width value of the RBF function h Substitute the control signal in

(2.23) and (2.24) into (2.22) we obtain

Sliding mode

control

Figure 2 4 Structure of the adaptive contraller

The structure of the proposed controller based on RBFXN is shown in

Figure 2.4 In comparison to BASMC’s structure, the RBFNN with the input of

robot states is constituted to estimate the appropriate values for ở with online

weigh! updated law in the adaptive controller,

The second Lyapunov candidate fimetion is rescleeted as

iy 51 (A + With 8k, sign(S,)— 5, )+ OWI WY)

= ks, GhuøG,) STs, es Whe WW) With the updated law (2.29), Y, is rendered as

Applying the Cauchy-Schwarz inequality results in

Tt can be shown that

Pee Eh, Al l6! ls.!z l9,IWI, JWMẸ)

(2.35) where k,,,, and A;,,, are the minimum values of k, and ky If the bounded condition in the following inequality is satisfied eas,

Remark 2 With the uncertain model of the system, the RBFNN plays a

key role in estimating the appropriale values for the uncerlainties to ensure control

oe <-gle§ ~alk|G IWI,~|WI

quality W(0) is an initial value of a weight matrix W , and it is chosen based on

the scope of the input data, especially the system's states Moreover, the updated

law is to calculate an appropriate value for W., which is proven based on the Lyapunov standard to ensure convergence In the radial basis function activation

Bh oof the hidden layer: @ is the cemer poinl, and y represents the width valuc, and

they should be moderately chosen based on knowledge concerning the system and input and output data The Center point @ has an impact on the sensitivity of the RBF function to the input vector’s scope, and it opts in the range of the system state vector q Besides, 9 expresses the covering scope of the network input If

these parameters are chosen inappropriately, the RBF will not be effectively

mapped, and RBFNN will not take effect

16

2.2.3 High-gain observer for the adaptive contraller

Typically, a combination of measurement and data processing from sensors

is synthesized to obtain robot states’ position and velocity values However, in many circumstances, the construction of the sensor system and the calculation of the necessary values are significanlly complicaled and costly to satisfy the

acceptable accuracy due to the influence of enviranmental conditions, noises, and

sensor quality Therefore, the observer is constructed to estimate the system states,

namely the velocity values of the parallel robot system On the other hand, a four-

degree-of-freedom parallel robot system is fast responsive, hence, the observed

velocity signal requires a fast convergence rate with the actual velocity value Thus, the high-gain observer is selected due to its advantages of convergence speed, as well as resistance to model errors [44], [51], and [55]

‘To facilitate the observer design process, the auxiliary variables are defined as follows,

|x 'ía aie -ciaya-v)] [ican]

with assuming that the control signal # is able to cnsure the system's stability From that, the ideal values for the observers are generated by

8-40.48), #1, em)

with ca, 1,2,3,4) are the ideal values of 2, (¢=1,2,3,4), respectively,

Next, defining 4,4 @=12,3,4) as the high-gain observer outputs The equations

of the high-gain observer are described by

with g—@—-&, ky,k,>00<2, 1) 1 The parameters „6c, are chosen to satisly

that s* 14.5 1 4,, is the Hurwilz polynomial

In (2.41), the robot’s ideal dynamic model is taken into account to update the

observer's oulpuls However, uncerlam components exist evermore in the system

model A considerable advantage of a high-gain observer 1s that it can minimize

17