fuzzy adaptive control for wireless optimal charging

FUZZY ADAPTIVE CONTROL FOR WIRELESS OPTIMAL CHARGINGGANTRY ROBOT SYSTEM

WEN-SHYONG YU, YU-FENG LIN

Department of Electrical Engineering, Tatung University, 40 Chung-Shan North Rd. 3rd. Sec., Taipei, 10451 TaiwanE-MAIL: [email protected]

Abstract:

This paper mainly studies the realization of the wireless opti-mal charging gantry robot system using type-2 fuzzy adaptive con-trol for mobile rechargeable devices. The wireless charging sys-tem is based on the energy management systems using the adap-tive control algorithm to achieve the maximum charging powercontrol. The type-2 fuzzy dynamic model is used to approximatethe charging system in accordance with current standards with-out constructing sector dead-zone inverse, where the parametersof the fuzzy model are obtained both from the fuzzy inference andonline update laws. The tracking trajectory tore chargeable de-vices including forward/inverse kinematics written by C# in Vi-sual Studio is used for obtaining the joint angles of the xyz tablecorresponding to the desired trajectory. By feedback the charg-ing current from the coil to detect position of the mobile devices,the optimal charging device tracking algorithm is given for ob-taining the shortest distance and maximum power transmissionbetween the induction coil and the rechargable device. Based onthe Lyapunov criterion and Riccati-inequality, the control schemeis derived to stabilize the closed-loop system such that all statesof the system are guaranteed to be bounded due to uncertainties,dead-zone nonlinearities, and external disturbances. The advan-tage of the proposed control scheme is that it can better handle thevagueness or uncertainties inherent in linguistic words using fuzzyset membership functions with adaptation capability by linear an-alytical results instead of estimating non-linear system functionsas the system parameters are unknown. Finally, both simulationand experimental results are provided to verify the validity of thewireless optimal charging system.

Keywords:

Type-2 fuzzy adaptive control algorithm forward kinematics;Inverse kinematics; Path planning; Maximum power transmis-sion; Uncertainties

1. Introduction

In order to alleviate the global world warming problem, thedevelopment of electric-powered vehicles, motorcycles, and bi-cycles with smart portable, networked, modular rechargeablebatteries should be urgent and can immediately tackle the prob-lem effectively. As a promising and convenient energy trans-mission technology, wireless energy transmission systems haveshown application prospects in many aspects. It can now beachieved through three mechanisms, namely electromagneticinduction, magnetic resonance, and microwave. Wherein mag-netic resonance is particularly sensitive to the relative positionof the transmitting coil and the inductive coil, since a slight po-sitional shift will result in a large difference in transmission effi-ciency. Charging performance, compatibility between differentcharging technologies, and reducction of power consumptionwill be an important consideration for various manufacturersand even academic circles to develop related charging technol-ogy. The problems of rechargeable battery life, charging time,and even battery size and power consumption should be consid-ered for currently mobile vehicles such as electric vehicles, etc.,wearable vehicles such as smart sports watches and bracelets,and handheld devices such as mobile phones, digital devices,and cameras, etc.. Compared with the wired charging systems,the charging speed of wireless charging technology is reallyslow at this stage, and it is frequently used in low-power elec-tronic devices with the distance between the charging deviceand the power transmitting device not to exceed several me-ters. As for the trend of smart charging development, wirelesscharging is connected to the terminal equipment that needs tobe charged without using a conventional charging power line,and the electric energy can be transmitted by using the mag-netic field generated between the coils, e.g., implantable medi-cal devices, portable electronic devices, solar cells for charging[1], and wireless charging for electric vehicles, and system ef-

410

Proceedings of the 2019 International Conference on Machine Learning and Cybernetics, Kobe, Japan, 7-10 July, 2019

978-1-7281-2816-0/19/$31.00 ©2019 IEEE

ficiency for electric vehicles wireless charging system [2]. In[3], the authors introduce a conceptual approach to the possi-bility that electric vehicles can be charged. There are two typesof wireless charging: near field transmission and far field trans-mission. As a far-field transmission, radio waves are a rela-tively mature technology. Although its transmission distance islong, it is quite difficult to transmit high energy for charging,and may even generate harmful electromagnetic waves. In con-trast, near-field energy transmission utilizes electromagnetic in-duction, although relatively close, but also transmits higher en-ergy relative to radio waves. In [4], the authors propose powertransfer efficiency of resonant inductive coupling based wire-less EV charging system by analyzing the effect of design pa-rameters. In [5], the authors propose a reconfigurable multi-ratio charge pump with wide input/output voltage range forwireless energy harvesting system. In [6], the authors proposean automatic current control by self-inductance variation fordynamic wireless EV charging. Among them, resonant induc-tive coupling or magnetic resonance coupling [7] is the focusof research in the field of wireless charging. In [8], the authorspropose that the resonance characteristic can be used for en-cryption during wireless charging, and even multi-point charg-ing can be realized. As mentioned earlier, many researchershave a lot of fruitful results in the field of wireless chargingsystems, circuit topology and transmission efficiency. How-ever, there is little research related to the actively tracking therechargeable devices to be charged. Therefore, this paper pro-poses the realization of the wireless optimal charging gantryrobot system using type-2 fuzzy adaptive control to optimizethe wireless charging system for mobile rechargeable devices.

2. System analysis

The gantry robot illustrated in Fig. 1 for wireless powertransmission system has affixed base (Module y) and 4 links(Module x, Module z, Arm, and End-effector) with 3DOF intranslation movements, connected as an open kinematics chainin the joints space coordinates. From Fig. 2, the design param-eters based on gantry robot model are shown in Table 1. Byusing D-H method from Table 1, we can found the homoge-neous transfer matrix.

TABLE 1. D-H parameters.

Link (i) ai (cm) di (cm) αi θi1 a1 = 5 d1 (variable) α1 = π/2 θ1 = 0

2 a2 = 5 d2 (variable) α2 = π/2 θ2 = π/2

3 a3 = 0 d3 (variable) α3 = 0 θ3 = 0

In order to obtain the forward kinematics of three degrees

y

x

z

FIGURE 1. Gantry robot powertransmission system.

x

x0

y0

d1

Link 0

Link 1

Link 2

Link 3

d2

d3

x1

y1

x2

y2

joint 1

joint 2

joint 3

y3

x3

a1

a2

y

base

reference

1 0q

FIGURE 2. DH parameters forsimplified gantry robot modelfor reference frame.

of freedom cartesian robot we need to draw a system diagramfrom Fig. 2, where q = [d1 d2 d3]> are join displacement vec-tor. As it is observed, translation is the unique movement thatrealizes this kind of robots, then the position of end-effector(3q = [qu qv qw]>) with respect to the base coordinate sys-tem (0q = [qx qy qz]

>) can be found and the final equationswhich describe the kinematics of the gantry robot are shown asfollows:

0q =

1 0 0 −a2 + d1

0 1 0 d2

0 0 1 −a1 + d3

0 0 0 1

= 0A33q (1)

The Jacobian of the gantry robot is given by c = J(q)q, wherec = [x y z]> and J(q) = I3×3. Since det(J(q)) = 1,the determinant in the Jacobian matrix is not undefined in anypoint which indicates the workspace for the cartesian robot iscomplete. Thus, the dynamic model of the cartesian robot La-grange’s equation for a conservative system under friction in-fluence is given by

M(q)q + G(q)g + d = u (2)

where u = [u1 u2 u3]> is the vector of applied torque, m1,m2, m3 are joint mass of each joint, g is the vector of gravitytorque, d is the friction influence, and

M=

m1 0 00 m1 +m2 00 0 m1 +m2 +m3

, G(q)=

00

m1 +m2 +m3

(3)

Then, we haveq = F∆ + Gu (4)

411


where F∆ = −M(q)−1G(q)g − M(q)−1d and G =M(q)−1. Considering the induction current from both coilsaccording to the Biansha law and assuming that the tiny lineelement d`′ with source position r′ has current I , d`′ acts onthe field position r, we have

dBc =µ0I

4πd`′ × r − r′

|r − r′|3(5)

where dBc is the tiny magnetic field and µ0 is the vacuum per-meability. According to (5), the magnetic field generated bythe symmetrical current can be used to calculate the distanceof the magnetic field in z-direction from the central axis of thecircular current-carrying coil as follows:

Bc(z) =µ0iR

2

2(R2 + z2)3/2=µ0i

2R· 1

[1 + (z/R)2]3/2(6)

Setting the circular coil in radius R and current i, we can cal-culate the magnetic field of any point P (x, y, z) near the coil inthe end-effector. By placing the coil on the xy plane, the centerof the coil can be set as the origin of the coordinates. By takinga small length of d`(Rcosθ,Rsinθ, 0) for current on the coil,the magnetic field generated by P is given by

d ~Bc =µ0

4π

id~× rr3

(7)

where r = (x − Rcosθ, y − Rsinθ, z), d¯ =(−Rsinθdθ,Rcosθdθ, 0). Hence, we have

dBc =µ0iR

4π· [zcosθ, zsinθ,R− (xcosθ + ysinθ)]

[x2 + y2 + z2 +R2 − 2R(xcosθ + ysinθ)]3/2dθ

(8)Because the circular coil is symmetric about the z axis, one cancalculate the magnetic field change in the xy diameter planeon the xy plane. Here we take the x axis and let y = 0. It isassumed that the component of the magnetic field on the threeaxes is

Bcx =µ0iR

4π·∫ 2π

0

zcosθ

[x2 + z2 +R2 − 2Rxcosθ]3/2dθ (9)

Bcy =µ0iR

4π·∫ 2π

0

zsinθ

[x2 + z2 +R2 − 2Rxcosθ]3/2dθ (10)

Bcz =µ0iR

4π·∫ 2π

0

(R− xcosθ)[x2 + z2 +R2 − 2Rxcosθ]3/2

dθ (11)

As for the distance between both induction current coils, themethod of least squares interpolation shown in Fig. 7 is usedhere to approximate the distance between the charging coil and

Φi(u

i)

ui

α

α

ǫ−i

ǫ+i

κi

κi

FIGURE 3. The sector dead-zone with uncertainty.

the load coil in the case of noise in both the x, y, and z axessince it is biased even for the case of straight line fitting asfollows:

2× 10−5x2 − 0.0029x+ 0.4394 (12)

where the least squares error is 0.739506654.Actuator nonlinearities, e.g. dead-zone, saturation, and hys-

teresis characteristics in magnetic coils, always have break-points so that they are non-differentiable but can be parameter-ized [9, 10]. The nonlinear continuous functions Φ

i(u

i) with

inputs ui , i = 1, 2, 3, can be intervalized as follows:

Φi(ui)=

Φi(ui)=

α(u

i− ε+

i) + κ

i, u

i≥ ε+

iand Φ

i(ui) ≥ κ

i

κi, ε−

i<u

i< ε+

iand 0< Φ

i(ui)< κ

i

α(ui− ε−

i) + κ

i, u

i≤ ε−

iand Φ

i(ui) ≤ κ

i

Φ′i(ui)=

α′(u

i− ε+

i), u

i≥ ε+

i

0, ε−i< u

i< ε+

i

α′(ui− ε−

i), u

i≤ ε−

i

Φi(ui)=

α(u

i− ε+

i) + κ

i, u

i≥ ε+

iand Φ

i(ui) ≥ κ

i

κi, ε−

i<u

i< ε+

iand κ

i<Φ

i(ui) < 0

α(ui− ε−

i) + κ

i, u

i≤ ε−

iand Φ

i(ui) ≤ κ

i(13)

where Φi(ui) = [Φi(ui) Φi(ui)], for which Φ′

i(ui), Φ

i(ui),

and Φi(u

i) are the median, lower, and upper bounds, respec-

tively, ε−i

, ε+i

, κi, and κ

iare sector dead-zone envelopes and

are constants, and α′, α, and α stand for the median, lower,and upper slopes of the sector dead-zone characteristic, respec-tively.

In order to obtain the key features of actuator nonlinearitiesin the control problems, we have the following assumptions:

Assumption 1: The sector dead-zone outputs Φi(u

i), i =

1, 2, 3, are not available for measurement.Assumption 2: The actuator nonlinearities slopes in positiveand negative regions are the same, i.e., α = α′ = α = α.Assumption 3: There exist known constants ε−

imax, ε−

imin,

ε+imax

, ε+imin

, κimax

, κimin

, κimax

, κimin

, α′max

, α′min

, αmax

,αmin

, αmax

, and αmin

such that the unknown sector dead-zone parameters ε−

i, ε+

i, κ

i, κi , α

′, α, and α satisfy ε−i∈

ε−imin

, ε−imax, ε+

i∈ ε+

imin, ε+imax, κ

i∈ κ

imin, κ

imax,

κi ∈ κimin , κimax , α′ ∈ α′

min, α′

max, α ∈ α

min, α

max,

and α ∈ αmin

, αmax, for which ε+

imax, ε−

imin, κ

imax, and

412


κimin

, i = 1, 2, are negative values and α′min

, αmin

, and αmin

are nonzero.

Assumption 4: The outputs of the actuator nonlinearities,Φi(u

i), are not accessible for measurements, but the state vec-

tor of the charging system is supposed to be completely mea-surable.

Based on the above assumptions, (13) can be shown in Fig.3 and represented as

Φi(ui) = αui + ψi(ui), i = 1, 2, 3 (14)

where

ψi(ui)=

−αε+

i+κi , ui≥ε+i , and Φi(ui)≥ κi

−αui+κi , ε−i <ui < ε+i, and κ

i<Φi(ui)<κi

−αε−i

+κi, ui ≤ ε−i , and Φi(ui)≤κi

.

Then, (14) can be rewritten as

Φ(u) = αu +ψ(u) (15)

where ψ(u)=[ψ1(u

1) ψ

2(u

2)]>.

Assumption 5: From the actuator nonlinearities propertiesmentioned above, ψ

i(u

i) in (14) is bounded, i.e., there exists

a positive constant ρi

chosen as

ρi = max|αmaxε+imax

+ κimax |, |αminε−imin

+ κimin|

such that |ψi(u

i)| ≤ ρ

i, i = 1, 2, 3.

Let qm

= [qm1qm

2qm

3]> be the given tracking reference

signal satisfying qm

(t) ∈ Ur for all t ≥ 0 and for some desiredcompact set Ur. Define the tracking error as

e=q − qm

=[q1 − qm1q2 − qm2

q3 − qm3]>=[e1 e2 e3 ]> (16)

Then from (4) and (15), the error dynamic equation can be ob-tained as

e = Ae+ B(F∆(x) + αGu + Gψ(u)

)(17)

where xd

=[xm2

xm3

]> and

A=

[0 10 0

], B=

[01

](18)

FIGURE 4. Structure of the type-2 FLS.

3. Interval type-2 fuzzy system

In this section, type-2 fuzzy logic system (T2FLS) shownin Fig. 4 is used to approximate the unknown functions F

∆

of the controlled system (4), which is very similar to type-1fuzzy logic system (T1FLS) but with the defuzzifier replacedby the output processing block consisting of the type reductionfollowed by defuzzification. Consider the T2FLS with inputsxp

and output yf∆i

for p = 1, 2, 3 and i = 1, 2, 3, where theIF-THEN rules in the rule base are like that in the T1FLS case.Let the `th rule be given by:

R`f∆i

: IF x1 is F `f∆1

AND x2 is F `f∆2

AND x3 is F `f∆3

THEN yf∆i

is C`f∆i

for ` = 1, . . . , n, where F `f∆p

and C`f∆i

, i = 1, 2, p = 1, 2, 3,are inputs and outputs of the T2 fuzzy sets, respectively, andtheir corresponding membership functions are µF `

f∆p

(xp) and

µC`f∆i

(yf∆i

), respectively, for which n is the total number of

the T2 fuzzy rules.The bounds for footprint of uncertainty (FOU) can be

divided into upper and lower T1 membership functions,µC`

f∆i

(yf∆i

) and µC`f∆i

(yf∆i

), respectively. The fuzzy infer-

ence engine is the decision making logic which uses the fuzzyrules to determine the mapping from input to output of the T2fuzzy sets. The firing interval of the `th rule for f `

f∆i∈ C`

f∆i

in (??) are interval T2 fuzzy sets and not crisp values, andf `f∆i

= [f `f∆i

f `f∆i

] are intervalized by the left-most f `f∆i

and

right-most f `f∆i

points, respectively. By using the singletonfuzzifier and product inference, we obtain

f `f∆i

= µC`f∆i

(yf∆i

)×Π3p=1µF `

f∆p

(xp), (19)

f `

f∆i

= µC`f∆i

(yf∆i

)×Π3p=1µF `

f∆p

(xp) (20)

413


−6 −4 −2 0 2 4 60

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Me

mb

ersh

ip fu

nctio

n

−µ

Fl

fp

−µ

Fl

fp

0

2

4

6

0

0.5

10

0.2

0.4

0.6

0.8

1

Membership function

FIGURE 5. Membership function and fuzzy sets of the IT2FAC for xp ,p = 1, 2, 3.

Then, we adopt the Gaussian membership functions for xp

ofthe IT2FAC, p = 1, 2, 3, shown in Fig. 5, and they are given by

µF `f∆p

(xp)=e−(

xp−mf∆p

σf∆p

)2

, µF `f∆p

(xp)=af∆pe−(

xp−mf∆p

σf∆p

)2

(21)

where σf∆p

and σf∆p

are fixed standard deviations of the lowerand upper membership functions which characterize the shapeof µF `

f∆p

(xp), respectively, a

f∆p

are the FOU width coeffi-

cients between the upper and lower membership function, re-spectively, and m

f∆p

are the means of the membership func-tions of the T2FS. These parameters are subject to 0<a

f∆p<1

and σf∆p

< σf∆p

, p = 1, 2, 3. For the consequent fuzzy

set membership functions µC`f∆i

(yf∆i

), because y`f∆i

are the

points of the T2 fuzzy system output variables yf∆i

at whichµC`

f∆i

(yf∆i

), i=1, 2, 3, achieves its maximum values, respec-

tively.After defuzzifying the type-reduced sets, F

∆(x) can be ap-

proximated by the crisp outputs from the T2FLS and expressedas:

F∆(x, Θf∆

) = [f∆1(x, Θf ) f∆2(x, Θf )]> = Ξ>f∆

(x)Θf∆

(22)

respectively, where Ξf∆

(x) =diagξf∆1

(x) ξf∆2

(x)>, Θf∆

=

[θf∆1

θf∆2

]>, for which θf∆i

are adjustable parameter vectorsand ξ>

f∆i

(x) are vectors of the fuzzy basis functions.Furthermore, according to the universal approximation the-

orem, there exist optimal approximation parameters Θ∗f∆

suchthat

Θ∗f∆=argmin ˆΘ

f∆∈Ω

f∆

supx∈R3‖F∆(x, Θf∆

)−F∆(x)‖ (23)

which lead to minimum approximation errors for F∆(x) asfollows:

F∆ = F∆(x)− F∆(x,Θ∗f∆

) (24)

where F∆

= [f∆1f

∆2]> are the minimum approximation errors

assumed to be bounded, and the estimated parameter errors aregiven by

Θf∆

= Θ∗f∆− Θf

∆(25)

Hence, the T2FLS can be principally used to approximate theunknown functions F

∆of the nonlinear controlled system in

(4).

Assumption 6: There exists a positive constant Me > 0 suchthat |(F∆)i | ≤Me , where |(·)i | is the absolute value of the ithelement of F

∆, i = 1, 2.

Let the controller be given by

u = ~Φ>Θσ (26)

where u = [uf1uf2uf3

]>, ~ is scalar, σ = [χ> q>m]>1×6, andχ is the normalized firing strengths.

4. Experimental results

The the wireless optimal charging gantry robot system isshown in Fig. 6 with power supply voltage 29V, where thetransmissin coil and feed back coil are shown in Figs. 8 and 9,respectively. Because the charging induction coil will generatea cut-off area, in order to enable the Arduino in the charging de-vice to smoothly control the induced voltage and current, andthen control the power and the self-control platform to com-plete the charging device quickly. This cut-off area needs tobe included in the dynamic analysis and the tracking responsesare shown in Fig. 7 In the experiments, the parameters of thesector dead-zone nonlinearities for the system are ε+

1= 0.9,

ε−1

= −0.9, ε+2

= 0.5, ε−2

= −0.5, κ1 = 0.9, κ1

= −0.9,κ2 = 0.5, κ

2= −0.5, and α′ = α = α = α = 1. Then,

the bounds of them are chosen as αmax = 1.25, αmin = 0.85,ε+

1imax=ε+

2imax=1, ε−

1imin=ε−

2imin=−1, κ

1imax= κ

2imax=1,

κ1imin

=κ2imin

=−1, and

ρ1

= max|αmax

ε+1max

+κ1max|, |α

minε−

1min+κ

1min| = 2.25

ρ2

= max|αmax

ε+2max

+κ2max|, |α

minε−

2min+κ

2min| = 2.25

The tracking error responses are shown in Figs. 10. It is seenthat the tracking errors for the charging system is satisfied.

5. Conclusion

A complete analytical solution to the forward kinematics ofgantry Robot is derived in this Paper and the problems wasmentioned above was fixed. The forward kinematics analysis

414


FIGURE 6. Charging test sys-tem.

0 100 200 300 400 500 600 700

Area

0

1

2

3

4

5

6

7

8

9

Ro

ot

me

an

sq

ua

re

Regression Analysis of Coil-1

R2 = 0.9325

First dataRegression Analysis

FIGURE 7. Regression analysisfor induced current of coil.

FIGURE 8. Transmissin Coil.

FIGURE 9. Feed Back Coil.

of gantry robot is investigated. The mathematical model is pre-pared and solved for positioning of the end-effector by prepar-ing a program in MATLAB. Furthermore GUI is done using vi-sual basic to simplify gantry robot kinematics. Analysis Basedon D.H Method and now it is become easy for researchers andstudents to study the kinematics of gantry robot and changingthe parameters to see the behavior of the system for getting bestdesign and configuration of robot.

AcknowledgementsFinancial supports of this research by Ministry of Science

and Technology, Taiwan, under the grant MOST 107-2221-E-036-019- is gratefully acknowledged.

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