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Fuzzy logic control of a novel robotic hanger for garment inspection: Modeling, simulation and experimental implementation
Expert Systems with Applications 38 (2011) 9924–9938
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Expert Systems with Applications journal homepage: www.elsevier.com/locate/eswa
Fuzzy logic control of a novel robotic hanger for garment inspection: Modeling, simulation and experimental implementation E.H.K. Fung a,⇑, Y.K. Wong b, X.Z. Zhang a, L. Cheng a, C.W.M. Yuen c, W.K. Wong c a
Department of Mechanical Engineering, The Hong Kong Polytechnic University, Hunghom, Kowloon, Hong Kong Department of Electrical Engineering, The Hong Kong Polytechnic University, Hunghom, Kowloon, Hong Kong c Institute of Textiles and Clothing, The Hong Kong Polytechnic University, Hunghom, Kowloon, Hong Kong b
a r t i c l e
i n f o
a b s t r a c t
Keywords: Fuzzy control Garment inspection Lagrangian method Robotic hanger
This paper presents a novel robotic hanger system for the garment inspection of knitting production. This 3-DOF hanger consists of three groups of linkages which can freely move in the 2D plane. In order to thoroughly understand the structure and motion of the robotic hanger so as to facilitate control simulations and experiments, the mathematical dynamic model of the hanger is analyzed and derived by using the Lagrangian energy approach. Since the dynamic model is complex, coupled and nonlinear, the fuzzy tuned PID (FT-PID) algorithm is employed to establish the controller. This FT-PID controller is found to be less susceptible to the sensor noise, the variation of garment material and size. Therefore, it possesses a certain degree of robustness and adaptability. The simulation and experimental results validate the dynamic model and the effectiveness of the FT-PID control method through comparisons with the conventional PID controller. Ó 2011 Elsevier Ltd. All rights reserved.
1. Introduction
the structure and motion of the robotic hanger and furthermore to provide the means for control simulation under the conditions that are hardly realized in practical experiments, the mathematical dynamic model of this robotic hanger is analyzed and derived using the energy approach. Lagrangian method (Craig, 1988) is applied to deduce a series of motion equations of linkages which are then used to establish the extension force exerted on the robotic hanger. To extend the garments perfectly, the force is an important controlled variable. It is computed from the derived complex, coupled and nonlinear motion equations, which makes it difficult to establish an accurate controller based on the model. Additionally, the serious noise in the environment creates extra difficulty in the design of controllers. In our previous simulation work, model reference adaptive control (MRAC) has been successfully applied to control the force of a 1-DOF mechanism (Fung, Yuen, Wong, Hau, & Chan, 2008). However, the MRAC system, when applied to the complex 3-DOF hanger, may easily lead to unstable results if inappropriate parameters are used. Coupling effects among the three links also make the parameter tuning process tedious and timeconsuming (Fung, Wong, Yuen, & Wong, 2009a). Practically, many control techniques for robot manipulators in industrial operations depend on proportional integral derivative (PID) control laws because of their simple structure, ease of design and low cost (Wai, Tu, & Hsieh, 2004). Since the model of the robotic hanger is complex, it is almost impossible to build an accurate PID controller based on the plant model. Fuzzy control utilizes the human control operator’s knowledge and experience
As an important stage of quality control in the knitting production, the garment inspection can be done either manually or automatically. This process alone is very time-consuming because of the variety of styles, sizes and fabrics used in the clothing. Another concern is the worker fatigue and quality standardization which requires the inspection to be repetitive to achieve a certain satisfaction. To tackle these problems, it is necessary to set up a feasible automatic intelligent hanger system (Patton, Swern, Tricamo, & Van der Veen, 1992; Yuen, Fung, Wong, Hau, & Chan, 2008) to simulate the manual operation to extend various knitted garments. The products currently in the market are found not suitable for these applications, as they usually have one or two degree of freedom (DOF) only. To extend the garment efficiently, a 6-axis robotic hanger with three DOF is now developed which can move in the two-dimensional (2D) plane. The proposed robotic hanger has three groups of linkages: body link, shoulder link and sleeve link which are designed to satisfy the conditions of inspection for various garments. The structural advantages of this robotic hanger are that the shoulder and sleeve link are lightweight; the body link always positions itself vertically during the movement and the gearing mechanism is simple. In order to comprehensively understand ⇑ Corresponding author. E-mail addresses: (E.H.K. Fung).
[email protected],
[email protected]
0957-4174/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.eswa.2011.02.037
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Nomenclature BMi BOi
dissipated energy coefficient for motor i proportional dissipated energy constant due to friction at output joint i c1i, c2i, c3i coefficients related to the garment materials in force equation for link i Fi normal force exerted on link i by the garment at the C.G. of link i IGi moment of inertia of gear i IMi moment of inertia of motor i iai, Lai, Rai current, armature winding inductance and resistance of motor i KLi kinetic energy of link i KGi kinetic energy of gear i KMi kinetic energy of motor i flexible coefficient of belt i ki l1 distance between the center of gear 1 and gear 3 l2, l3, lv length of link 2, 3 and vertical link mass of gear i mGi mLi mass of link i NGi number of teeth of gear i N1 reduction ratio between motor 1 and link 1
to intuitively construct controllers so that the designed controller can emulate the human control behaviors. A satisfactory nonlinear controller can be developed empirically in practice without complicated mathematics, which is established based on converting the human operation experiences to a series of control rules. The principle of the fuzzy controller makes it possess the properties of robustness and adaptability. Following these advantages, fuzzy tuned PID (FT-PID) controller has been proposed to regulate the controller parameters automatically and used successfully in many applications. Examples of applications are found in various consumer products and industrial systems such as hydraulic load simulator (Truong & Ahn, 2009), servo pneumatic position control (Situm, Pavkovic, & Novakovic, 2004), medical application (Ying, 2000), power system (Shayeghi, Shayanfar, & Jalili, 2007; Wang, Fu, & Zhang, 2008) and temperature control (Harinath & Mann, 2008; Soyguder, Karakose, & Alli, 2009). In this study, the FT-PID controller is also applied to implement force control and the controller performance is validated through simulations and experiments. The results obtained by FT-PID controller and conventional PID controller are compared. It is verified that FTPID can provide efficient performance even under the situation that the force sensor output is contaminated with noise; the garments are of different size and materials; and the controller possesses implementation errors. The rest of this paper is organized as follow: Section 2 discusses and analyzes the mathematical dynamic model of the robotic hanger in detail. Section 3 describes the design of the FT-PID control algorithm. In Section 4, simulation and experiment results are presented to validate the dynamic model and performance of the FT-PID control algorithm through comparison with the conventional PID controller. The paper is concluded in Section 5.
2. Dynamic model of the robotic hanger To fully extend the garment at a proper position, the forces exerted on the surface of the garments are the important variables which should be fed back to the controller. The relationship between the force exerted on the C.G. of the body, sleeve and shoulder links Fi and the corresponding garment extension xti can be expressed by
Pbi potential energy of belt i PGi potential energy of gear i potential energy of link i PLi r radius of the pulleys ~ r ci position vector of C.G. (center of gravity) of link i ~ r 0cv unextended position vector of C.G. of vertical link Vini, Kbi, Ksi input voltage, back emf constant and torque constant of motor i vGi velocity of the center of gear i xti extension of garment at the C.G. of link i xti maximum of xti for which the force equation of link i holds ai angular acceleration of link i ami angular acceleration of motor i h1i, h2i, h3i initial displacement of link 1, 2 and 3, respectively hmi angular displacement of motor i hi angular displacement of link i sMi torque of motor i xGi angular velocity of gear i xmi angular velocity of motor i xi angular velocity of link i
F i ¼ c1i x3ti þ c2i x2ti þ c3i xti
for 0 < xti 6 xti
ð1Þ
F i ¼ 0 for xti 6 0
It should be noted that, as shown in Fig. 1(a) and (b), only the forces acting on the right side of the hanger are described, the purpose of which is to make a simple explanation in the following derivation. In the actual robotic hanger system, the forces are found on both sides of the hanger. It can be seen from (1) that the force relates to the displacements of body, sleeve and shoulder links. According to Fig. 2, the displacement for each link is calculated as. Body link follows:
~ r cv ¼ l1 cos h1~i þ ðl1 sin h1 þ 0:5lv Þ~j ~ r 0 cv ¼ l1 cos hi1~i þ ðl1 sin h1i þ 0:5lv Þ~j
ð2Þ
)~ z1 ¼ ~ r cv ~ r 0 cv ¼ l1 ðcos h1 cos h1i Þ~i þ l1 ðsin h1 sin h1i Þ~j ) xt1 ¼ ~ z1 ~i ¼ l1 ðcos h1 cos h1i Þ Sleeve link:
xt2 ¼ 2
l2 Dh2 h2 h2i sin ¼ l2 sin 2 2 2
ð3Þ
for a small Dh2 ; xt2 12 l2 ðh2 h2i Þ Shoulder link:
xt3 ¼ 2
l3 Dh3 h3i h3 sin ¼ l3 sin 2 2 2
ð4Þ
for a small Dh3 ; xt3 12 l3 ðh3i h3 Þ From the derivation of xti, we can further find that hi correlates xti. To determine the governing equations of hi, the Lagrangian energy method is used.
d @L @L ¼ Q i; dt @ q_ i @qi
i ¼ 1; 2; . . . ; 5
where L ¼ K tot P tot ; Q i ¼ @W and q ¼ ½h1 @q i
ð5Þ h2
h3
hm2
hm3 .
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Fig. 1. The structure of robotic hanger. (a) Overview; (b) Schematic diagram (Motors, belts, pulleys, gears and force sensors are not shown).
Fig. 3. Kinetic energy computation of each link. (a) Body link; (b) Sleeve link; (c) Shoulder link.
Similar to the body link, the kinetic energies of sleeve and shoulder links are calculated as below. Kinetic energy of sleeve link (link 2) For an infinitesimal particle located by r on link 2,
position : x ¼ l1 cos h1 þ r sin h2 ;
y ¼ l1 sin h1 þ lv r cos h2
Fig. 2. Displacement for each link. (a) Body link; (b) Sleeve link; (c) Shoulder link.
velocity : x_ ¼ l1 sin h1 x1 þ r cos h2 x2 ; y_ ¼ l1 cos h1 x1 þ r sin h2 x2
The total kinetic energy Ktot is defined as
K tot ¼ ðK L1 þ K LV þ K L2 þ K L3 Þ þ ðK G1 þ K G2 þ K G3 Þ þ ðK M1 þ K M2 þ K M3 Þ
ð6Þ
v 2 ¼ x_ 2 þ y_ 2 ¼ l21 x21 þ r2 x22 2l1 rx1 x2 sinðh1 h2 Þ
It incorporates the energies caused by the motions of the links, gears and motors.
mass : dm ¼
Kinetic energy of body link (link 1 and link v) The body link consists of two links: link 1 and vertical link (link v). As shown in Fig. 3, for an infinitesimal particle located by r on link 1,
dK L1
1 1 mL1 dr ¼ v 2 dm ¼ ðrx1 Þ2 ) K L1 ¼ 2 2 l1 Z l1 1 mL1 2 2 1 ¼ x1 r dr ¼ mL1 l21 x21 2 6 l 1 0
In regard to link vv = x1l1. Thus
K Lv ¼
v,
Z
Thus,
K L2 ¼
dK L1 ¼
Z 0
l2
1 2 mL2 v dr 2 l2
1 2 2 mL ½3l x2 þ l2 x22 3l1 l2 x1 x2 sinðh1 h2 Þ 6 2 1 1
ð7Þ
since it does not rotate, the velocity is
1 1 2 mLv v 2v ¼ mLv l1 x21 2 2
mL2 dr l2
Kinetic energy of shoulder link (link 3) For an infinitesimal particle located by r on link 3,
ð8Þ
position : x ¼ l1 cos h1 r sin h3 ;
y ¼ l1 sin h1 þ lv þ r cos h3
ð9Þ
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velocity : x_ ¼ l1 sin h1 x1 r cos h3 x3 ; y_ ¼ l1 cos h1 x1 r sin h3 x3
Substitute (7)–(16) into (6),
v 2 ¼ x_ 2 þ y_ 2 ¼ l21 x21 þ r2 x23 þ 2l1 rx1 x3 sinðh1 h3 Þ mass : dm ¼
mL3 dr l3
Thus,
K L3 ¼
Z
l3
0
1 2 mL3 1 2 2 v dr ¼ mL3 ½3l1 x21 þ l3 x23 þ 3l1 l3 x1 x3 sinðh1 h3 Þ 2 6 l3 ð10Þ
The kinetic energies of gears KG are generated from the gear mounting which is installed on the link 1 (Fig. 4), where G1 is the pivot gear and cannot rotate. These energies are computed as kinetic energies of gears.
Gear 1 : xG1 ¼ v G1 ¼ 0 ) K G1 ¼
1 1 IG1 x2G1 þ mG1 v 2G1 ¼ 0 2 2
ð11Þ
sinðh1 h3 Þx1 x3 Therefore, it can be derived that the total kinetic energy
K tot ¼ A1 x21 þ A2 x22 þ A3 x23 þ A4 x2m2 þ A5 x2m3 þ ½A6 sinðh1 h2 Þx1 x2 þ ½A7 sinðh1 h3 Þx1 x3
NG1 1 Gear 2 : xG2 ¼ x1 1 þ ; v G2 ¼ x1 l1 2 NG2 1 1 ) K G2 ¼ IG2 x2G2 þ mG2 v 2G2 2 2 2 1 NG1 1 2 2 ¼ IG2 x1 1 þ þ mG2 x21 l1 2 8 NG2
ð17Þ
where
A1 ¼ ð12Þ
NG2 x xG2 ¼ 1 ðNG1 þ NG2 Þ; v G3 ¼ x1 l1 NG3 NG3 1 1 ) K G3 ¼ IG3 x2G3 þ mG3 v 2G3 2 2 1 ðNG1 þ NG2 Þ2 1 2 ¼ IG3 x21 þ mG3 x21 l1 2 2 N2G3
Gear 3 : xG3 ¼
1 1 1 2 2 mL2 l2 ; A3 ¼ mL2 l2 ; A4 ¼ IM2 ; 6 6 2 1 1 A6 ¼ mL2 l1 l2 ; A7 ¼ mL3 l1 l3 2 2
ð13Þ
1 ¼ IM1 x2m1 2
1 ¼ IM1 N21 x21 2 1 Motor 2 : K M2 ¼ IM2 x2m2 2 1 Motor 3 : K M3 ¼ IM3 x2m3 2
Fig. 4. Kinetic energy computation of each gear.
1 1 1 1 1 1 2 2 2 2 2 2 mL1 l1 þ mLv l1 þ mL2 l1 þ mL3 l1 þ mG2 l1 þ mG3 l1 6 2 2 2 8 2 2 1 NG1 1 ðNG1 þ NG2 Þ2 1 þ IG2 1 þ þ IG3 þ IM1 N21 2 2 2 NG2 N2G3
A2 ¼
There are three AC servo motors mounted on the base of the robotic hanger, and they drive three pairs of links to move in the 2D plane. Motor 1 is installed at the rear of the base and it provides the power for the motion of body links. Motor 2 & 3 are installed at the front of the base and drive the sleeve and shoulder links respectively. The motions of these motors also produce energies which are called kinetic energies of AC servo motors.
Motor 1 : hm1 ¼ N1 h1 ; xm1 ¼ N1 x1 ) K M1
K tot ¼ ðK L1 þ K LV þ K L2 þ K L3 Þ þ ðK G1 þ K G2 þ K G3 Þ 1 1 1 2 2 2 þ ðK M1 þ K M2 þ K M3 Þ ¼ mL1 l1 þ mLv l1 þ mL2 l1 6 2 2 2 1 1 1 1 NG1 2 2 2 þ mL3 l1 þ mG2 l1 þ mG3 l1 þ IG2 1 þ 2 8 2 2 NG2 # 2 1 ðNG1 þ NG2 Þ 1 1 2 2 2 2 I m þ IG3 þ N x þ l M1 1 L2 2 x2 1 2 2 6 N2G3 1 1 1 2 þ mL3 l3 x23 þ IM2 x2m2 þ IM3 x2m3 6 2 2 1 1 þ mL2 l1 l2 sinðh1 h2 Þx1 x2 þ mL3 l1 l3 2 2
ð16Þ
1 IM3 ; 2
The total potential energy Ptot is defined as
Ptot ¼ ðPL1 þ PLV þ PL2 þ PL3 Þ þ ðP G1 þ PG2 þ PG3 Þ þ ðPb2 þ Pb3 Þ
ð18Þ
Similarly, it includes three parts of energies which are generated from links, gears and belts. As shown in Fig. 5, these energies can be computed as below. Links:
PL1 ¼
1 mL1 gl1 sin h1 2
ð19Þ
P Lv ¼
1 mLv gðlv þ 2l1 sin h1 Þ 2
ð20Þ
ð14Þ ð15Þ
A5 ¼
1 PL2 ¼ mL2 gðl1 sin h1 þ lv l2 cos h2 Þ 2
Fig. 5. Potential energy computation. (a) Links and gears; (b) Belts.
ð21Þ
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1 PL3 ¼ mL3 gðl1 sin h1 þ lv þ l3 cos h3 Þ 2
ð22Þ
~ rc3 ¼ ðl1 cos h1 0:5l3 sin h3 Þ~i þ ðl1 sin h1 þ lv þ 0:5l3 cos h3 Þ~j
ð23Þ
d~ rc3 ¼ ðl1 sin h1 dh1 0:5l3 cos h3 dh3 Þ~i þ ðl1 cos h1 dh1 0:5l3
Gears:
PG1 ¼ 0 PG2
1 ¼ mG2 gl1 sin h1 2
PG3 ¼ mG3 gl1 sin h1
sin h3 dh3 Þ~j ð24Þ ð25Þ
Belts (assuming no slip):
Pb3
Therefore we have
dW ¼ Bo1 x1 dh1 Bo2 x2 dh2 Bo3 x3 dh3 þ ðsM1
1 k2 ðhm2 r h2 rÞ2 2
ð26Þ
1 ¼ k3 ðhm3 r h3 rÞ2 2
ð27Þ
Pb2 ¼
F 3 d~ rc3 ¼ F 3 l1 sinðh1 h3 Þdh1 þ 0:5F 3 l3 dh3 ) dW 3 ¼ ~
BM1 xM1 Þdhm1 þ ðsM2 BM2 xM2 Þdhm2 þ ðsM3 BM3 xM3 Þdhm3 þ F 1 l1 sin h1 dh1 þ F 2 l1 sinðh1 h2 Þdh1 0:5F 2 l2 dh2 þ F 3 l1 sinðh1 h3 Þdh1 þ 0:5F 3 l3 dh3
Substituting (17), (28) and (29) into (5) and using Lagrangian technique, we can derive the motion equations as follows: For i ¼ 1; q1 ¼ h1 ; q_ 1 ¼ x1
Substitute (19)–(27) into (18),
Ptot ¼ ðPL1 þ PLV þ PL2 þ PL3 Þ þ ðPG1 þ PG2 þ PG3 Þ þ ðPb2 þ Pb3 Þ 1 1 ¼ mL1 þ mLv þ mL2 þ mL3 þ mG2 þ mG3 gl1 sin h1 2 2 1 1 1 þ mL2 gl2 cos h2 þ mL3 gl3 cos h3 þ k2 r 2 ðhm2 2 2 2 1 1 mLv þ mL2 þ mL3 glv h2 Þ2 þ k3 r 2 ðhm3 h3 Þ2 þ 2 2
@L ¼ 2A1 x1 þ A6 sinðh1 h2 Þx2 þ A7 sinðh1 h3 Þx3 @ x1 d @L ¼ 2A1 a1 þ A6 sinðh1 h2 Þa2 þ A6 cosðh1 h2 Þðx1 dt @ x1 x2 Þx2 þ A7 sinðh1 h3 Þa3 þ A7 cosðh1 h3 Þðx1
Therefore, the total potential energy is
x3 Þx3
Ptot ¼ B1 sin h1 þ B2 cos h2 þ B3 cos h3 þ B4 ðhm2 h2 Þ2 þ B5 ðhm3 h3 Þ2 þ B6
ð29Þ
ð28Þ
@L ¼ A6 cosðh1 h2 Þx1 x2 þ A7 cosðh1 h3 Þx1 x3 B1 cos h1 @h1
where
1 1 mL1 þ mLv þ mL2 þ mL3 þ mG2 þ mG3 gl1 ; 2 2 1 1 B2 ¼ mL2 gl2 ; B3 ¼ mL3 gl3 ; 2 2 1 1 1 mLv þ mL2 þ mL3 glv B4 ¼ k2 r 2 ; B5 ¼ k3 r 2 ; B6 ¼ 2 2 2
B1 ¼
@W ¼ Bo1 x1 þ ðsM1 BM1 xm1 ÞN 1 þ F 1 l1 sin h1 þ F 2 l1 sinðh1 h2 Þ @h1 þ F 3 l1 sinðh1 h3 Þ Thus
2A1 a1 þ A6 sinðh1 h2 Þa2 þ A7 sinðh1 h3 Þa3 A6 cosðh1
There is another component dW in (5). It is the virtual work acting on the robotic hanger system defined by
h2 Þx22 A7 cosðh1 h3 Þx23 þ ðBo1 þ N21 BM1 Þx1 þ B1
dW ¼ BO1 x1 dh1 BO2 x2 dh2 BO3 x3 dh3 þ ðsM1 BM1 xm1 Þdhm1
¼ N1 sM1 þ F 1 l1 sin h1 þ F 2 l1 sinðh1 h2 Þ þ F 3 l1 sinðh1 h3 Þ
þ ðsM2 BM2 xm2 Þdhm2 þ ðsM3 BM3 xm3 Þdhm3 þ dW 1 þ dW 2 þ dW 3
where dWi is the virtual work done by the garment tension. They can be calculated as
dW 1 : ~ F 1 ¼ F 1~i;~ r cv ¼ l1 cos h1~i þ ðl1 sin h1 þ 0:5lv Þ~j; d~ r cv ¼ l1 sin h1 dh1~i þ l1 cos h1 dh1~j F 1 d~ r cv ¼ F 1 l1 sin h1 dh1 ) dW 1 ¼ ~ dW 2 : ~ F 2 ¼ F 2 cos h2~i F 2 sin h2~j
cos h1 ð30Þ
For i ¼ 2; q2 ¼ h2 ; q_ 2 ¼ x2
@L ¼ 2A2 x2 þ A6 sinðh1 h2 Þx1 @ x2 d @L ¼ 2A2 a2 þ A6 sinðh1 h2 Þa1 þ A6 cosðh1 h2 Þðx1 x2 Þx1 dt @ x2 @L ¼ A6 cosðh1 h2 Þx1 x2 þ B2 sin h2 þ 2B4 ðhm2 h2 Þ @h2
~ r c2 ¼ ðl1 cos h1 þ 0:5l2 sin h2 Þ~i þ ðl1 sin h1 þ lv 0:5l2 cos h2 Þ~j d~ r c2 ¼ ðl1 sin h1 dh1 þ 0:5l2 cos h2 dh2 Þ~i þ ðl1 cos h1 dh1 þ 0:5l2 sin h2 dh2 Þ~j F 2 d~ r c2 ¼ F 2 l1 sinðh1 h2 Þdh1 0:5F 2 l2 dh2 ) dW 2 ¼ ~ dW 3 : ~ F 3 ¼ F 3 cos h3~i F 3 sin h3~j
@W ¼ Bo2 x2 0:5F 2 l2 @h2 Thus
2A2 a2 þ A6 sinðh1 h2 Þa1 þ A6 cosðh1 h2 Þx21 þ Bo2 x2 B2 sin h2 þ 2B4 h2 2B4 hm2 ¼ 0:5F 2 l2
ð31Þ
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For i ¼ 3; q3 ¼ h3 ; q_ 3 ¼ x3
μ (e) /μ (ec)/μ (ΔK p)/μ (ΔK i )/μ (ΔK d)
@L ¼ 2A3 x3 þ A7 sinðh1 h3 Þx1 @ x3
NB
NM
NS
ZE
PS
PM
1/3
2/3
PB
1
d @L ¼ 2A3 a3 þ A7 sinðh1 h3 Þa1 þ A7 cosðh1 h3 Þðx1 x3 Þx1 dt @ x3 @L ¼ A7 cosðh1 h3 Þx1 x3 þ B3 sin h3 þ 2B5 ðhm3 h3 Þ @h3
-1
-2/3
-1/3
0
1 e/ec/ΔK p /ΔK i /ΔK d
Fig. 6. Membership functions for e, ec, DKp, DKi and DKd.
@W ¼ Bo3 x3 þ 0:5F 3 l3 @h3 Thus
Table 2 Fuzzy inference rules for DKp.
2A3 a3 þ A7 sinðh1 h3 Þa1 þ A7 cosðh1 h3 Þx21 þ Bo3 x3 B3 sin h3 þ 2B5 h3 2B5 hm3 ¼ 0:5F 3 l3 For i ¼ 4; q4 ¼ hm2 ; q_ 4 ¼ xm2
@L d @L ¼ 2A4 xm2 ; @ xm2 dt @ xm2 @L ¼ 2B4 ðhm2 h2 Þ; @hm2
ð32Þ
DKp
ec
¼ 2A4 am2 ;
e
NB NM NS ZE PS PM PB
NB
NM
NS
ZE
PS
PM
PB
PB PB PM PM PS PS ZE
PB PB PM PM PS ZE ZE
PM PM PM PS ZE NS NM
PM PS PS ZE NS NM NM
PS PS ZE NS NS NM NM
ZE ZE NS NM NM NM NB
ZE NS NS NM NM NB NB
@W ¼ sM2 BM2 xm2 @hm2 Thus
2A4 am2 þ BM2 xm2 þ 2B4 hm2 2B4 h2 ¼ sM2
ð33Þ
For i ¼ 5; q5 ¼ hm3 ; q_ 5 ¼ xm3
DKi
d @L ¼ 2A5 am3 ; dt @ xm3
@L
Table 3 Fuzzy inference rules for DKi.
¼ 2A5 xm3 ; @ xm3 @L ¼ 2B5 ðhm3 h3 Þ; @hm3
ec
@W ¼ sM3 BM3 xm3 @hm3
e
NB NM NS ZE PS PM PB
NB
NM
NS
ZE
PS
PM
PB
NB NB NB NM NM ZE ZE
NB NB NM NM NS ZE ZE
NM NM NS NS ZE PS PS
NM NS NS ZE PS PS PM
NS NS ZE PS PS PM PM
ZE ZE PS PM PM PB PB
ZE ZE PS PM PB PB PB
NB
NM
NS
ZE
PS
PM
PB
PS PS ZE ZE ZE PB PB
NS NS NS NS ZE NS PM
NB NB NM NS ZE PS PM
NB NM NM NS ZE PS PM
NB NM NS NS ZE PS PS
NM NS NS NS ZE PS PS
PS ZE ZE ZE ZE PB PB
Thus
2A5 am3 þ BM3 xm3 þ 2B5 hm3 2B5 h3 ¼ sM3
ð34Þ
It is obvious that Eqs. (30), (33) and (34) require the torque output from motors. Since in this robotic hanger system three AC servo motors are used to drive the linkages, it is inappropriate to use the DC motor model to compute the torque. Furthermore, the motor parameters listed in the manual are limited and we can hardly calculate the continuous torque output according to the AC servo motor model (Novotny, 1996). In this study, an alternative method is adopted to estimate the equivalent DC motor parameters corresponding to the AC servo motor. Variable voltage control under basic frequency is applied for the AC servo motor (Novotny, 1996) and the torque is
sM ¼ 3np
2
Us
x1
Table 4 Fuzzy inference rules for DKd.
DKd
ec
e
NB NM NS ZE PS PM PB
sx1 R0r ðsRs þ R0r Þ2 þ s2 x21 ðLls þ Llr Þ2
Table 5 Parameters for conventional PID and FT-PID.
Table 1 Universe of discourse for input and output variables.
e ec DKp DKi DKd
⁄
Conventional PID
Body
Sleeve
Shoulder
[0.3, 0.3] [20, 20] [0.2, 0.2] [0.1, 0.1] [6e4, 6e4]
[0.2, 0.2] [2.5, 2.5] [0.05, 0.05] [0.01, 0.01] [6e4, 6e4]
[0.17, 0.17] [3.6, 3.6] [0.01, 0.01] [0.15, 0.15] [8e4, 8e4]
Kp (⁄initial value for FT-PID) Ki (⁄initial value for FT-PID) Kd (⁄initial value for FT-PID)
FT-PID
Body
Shoulder
Sleeve
Body
Shoulder
Sleeve
20
5
5.5
20
4
4
6
5
8
8
8.3
7.5
0.2
0.28
0.35
0.2
0.12
0.18
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where np = 1; the motor slip, s, is very small when the motor is running steadily. Therefore,
sM 3
Us
2
x1
sMet ¼
Us
x1
2
sx1 ¼ sM Ret
So
s x1 R0r
Ret ¼
Since the DC motor has only one ‘phase’, we assume the torque output of DC motor is equivalent to that of single phase AC motor and this torque has to maintain the same after equivalent transformation. That is
1 0 R; 3 r
and Let ¼
1 0 L 3 r
where L0r ¼ t e R0r ; t e is the electrical time constant of AC motor. The equivalent DC motor equation for the AC servo motor used in the dynamic model is
Force on the body link
a Force (N)
0.3
Set value PID 3% error band
0.2 0.1 0
0
0.1
0.2
0.3
0.4
0.5 0.6 Simulation time (s)
0.7
0.8
0.9
1
0.7
0.8
0.9
1
0.7
0.8
0.9
1
Force on the sleeve link
0.2 Force (N)
0.15 0.1 0.05 0
0
0.1
0.2
0.3
0.4
0.5 0.6 Simulation time (s)
Force on the shoulder link
Force (N)
0.2 0.15 0.1 0.05 0
0
0.1
0.2
0.3
b
0.4
0.5 0.6 Simulation time (s)
Force on the body link Set value FT-PID 3% error band
Force (N)
0.3 0.2 0.1 0
0
0.1
0.2
0.3
0.5 0.6 Simulation time (s)
0.7
0.8
0.9
1
0.7
0.8
0.9
1
0.7
0.8
0.9
1
Force on the sleeve link
0.2 Force (N)
0.4
0.15 0.1 0.05 0
0
0.1
0.2
0.3
0.4
0.5 0.6 Simulation time (s)
Force on the shoulder link
Force (N)
0.2 0.15 0.1 0.05 0
0
0.1
0.2
0.3
0.4
0.5 0.6 Simulation time (s)
Fig. 7. Results obtained with different controller (without noise signals). (a) PID; (b) FT-PID.
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In summary, the Eqs. (1) and (30)–(35) represent the dynamic model of the robotic hanger system. For these equations, the initial conditions when t = 0 are
Table 6 Indices of transient-response in force control (without noise). Conventional PID
Rise time tr (s) Settling time ts (s)
Let
FT-PID
Body
Shoulder
Sleeve
Body
Shoulder
Sleeve
0.023 0.040
0.11 0.19
0.14 0.20
0.024 0.041
0.058 0.10
0.11 0.20
diet þ Ret iet þ K b h_ m ¼ U s ; dt
sMet ¼ K s iet
ð35Þ
To maintain the consistency with the Eqs. (30), (33) and (34) we still call sMet as sM.
Force (N)
a
h2 ¼ h2i ) xt2 ¼ 0 ) F 2 ¼ 0;
h3 ¼ h3i ) xt3 ¼ 0 ) F 3 ¼ 0 x1 ¼ x2 ¼ x3 ¼ 0; a1 ¼ a2 ¼ a3 ¼ 0;
xm1 ¼ xm2 ¼ xm3 ¼ 0; am1 ¼ am2 ¼ am3 ¼ 0
sM1i ¼
B1 cos h1i ; N1
sM2i ¼ B2 sin h2i ; sM3i ¼ B3 sin h3i
Force on the body link Set value PID
0.3 0.15 0
0
0.1
0.2
0.3
0.4
0.5 0.6 Simulation time (s)
0.7
0.8
0.9
1
0.7
0.8
0.9
1
0.7
0.8
0.9
1
Force on the sleeve link
0.2 Force (N)
h1 ¼ h1i ) xt1 ¼ 0 ) F 1 ¼ 0;
0.15 0.1 0.05 0
0
0.1
0.2
0.3
0.4
0.5 0.6 Simulation time (s)
Force on the shoulder link Force (N)
0.2
0
0
0.1
0.2
0.3
0.4
0.5 0.6 Simulation time (s)
Force on the body link
b
Set value FT-PID 3% error band
Force (N)
0.3 0.2 0.1 0
0
0.1
0.2
0.3
0.5 0.6 Simulation time (s)
0.7
0.8
0.9
1
0.7
0.8
0.9
1
0.7
0.8
0.9
1
Force on the sleeve link
0.2 Force (N)
0.4
0.15 0.1 0.05 0
0
0.1
0.2
0.3
0.4
0.5 0.6 Simulation time (s)
Force on the shoulder link Force (N)
0.2 0.15 0.1 0.05 0
0
0.1
0.2
0.3
0.4
0.5 0.6 Simulation time (s)
Fig. 8. Results obtained with different controller (with white noise). (a) PID; (b) FT-PID.
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hm2i ¼
E.H.K. Fung et al. / Expert Systems with Applications 38 (2011) 9924–9938
B2 sin h2i þ 2B4 h2i ; 2B4
hm3i ¼
B3 sin h3i þ 2B5 h3i 2B5
of membership function for input and output variables. It is noted that Fig. 6 depicts the membership functions with the standard universe of discourse and for each input and output variable of different link the required universe of discourse can be obtained by simply multiplying it with a scale factor. The universe of discourse for e is determined from the desired force value. In practice, the desired force in this study never exceeds 1 N, therefore, e is also less than 1 N. As for the universe of discourse for ec, it is determined by simulation results. Based on the designed membership functions for input and output variables, it is possible to describe the fuzzy inference rules which are listed in Tables 2–4. The rules are deduced concurrently; therefore the number of rules is 49. In accordance with these rules, the fuzzy inference is implemented in a way as given by Negnevitsky (2005). As for the defuzzification process, the most commonly used formula, the center of gravity or centroid method, is employed. This is expressed as
The parameters required for simulations include A1 A7 ; B1 B6 ; BM1 BM3 ; Bo1 Bo3 ; c1i c3i ; l1 l3 ; R0r1 R0r3 ; te ; K b1 K b3 and Ks1 Ks3. Some of these parameters can be found in the manuals. Some parameters can be found by physical measurements while the remaining ones can be obtained by experiments. 3. Control strategy Instead of the conventional PID controller, the fuzzy tuned PID (FT-PID) controller is applied to implement the force control due to the difficulty to design an accurate PID controller based on the highly nonlinear and coupled dynamic model. Another reason for choosing the FT-PID is that the estimated parameters of the dynamic model obtained from experiments are not exact. The control algorithm of FT-PID can be expressed in the discrete incremental form:
R yf lðyf Þdyf yo ¼ R lðyf Þdyf
DuðkÞ ¼ K p ½eðkÞ eðk 1Þ þ K i eðkÞ þ K d ½eðkÞ 2eðk 1Þ þ eðk 2Þ ¼ K p ec ðkÞ þ K i eðkÞ þ K d ½ec ðkÞ ec ðk 1Þ ð36Þ
ð37Þ
where yf is the fuzzy value of the fuzzy output variable, l(yf) is the corresponding value of membership function, and yo is the defuzzified crisp output i.e. DKp, DKi or DKd. With these defuzzified outputs, the new PID parameters are Kp(k) = Kp(k 1) + DKp(k), Ki(k) = Ki(k 1) + DKi(k) and Kd(k) = Kd(k 1) + DKd(k). Furthermore the new Du(k) can be calculated according to Eq. (36).
where e is the force error and ec is the differential of force error. Kp, Ki and Kd are the PID parameters which are tuned by fuzzy logic with proper fuzzy rules. The input variables are the e and ec, and the universe of discourse for e and ec are enlisted in Table 1. Considering the complexity of the robotic hanger system, more fuzzy subsets (Carvajal, Chen, & Ogmen, 2000) need to be designed. The membership functions for e and ec in the fuzzification procedure have seven linguistic variables: NB (negative big), NM (negative medium), NS (negative small), ZE (zero), PS (positive small), PM (positive medium) and PB (positive big). The output variables are DKp, DKi and DKd. Universe of discourse for these variables are also listed in Table 1, and their membership functions have similar forms as the input variables. They also have seven linguistic variables: NB, NM, NS, ZE, PS, PM and PB. Fig. 6 illustrates the profiles
4. Case study In this section, we validate the dynamic model and FT-PID control strategy through simulations and experiments. To further investigate the performance of the FT-PID controller, the conventional PID controller is used for comparison where the parameters of PID controller are tuned by the Ziegler & Nichols method.
Force on the body link
0.6
PID FT-PID Set value
0.45 N
0.3 0.15 0
0
15
30
45
60
75
90
105
120
135
150
165
105
120
135
150
165
105
120
135
150
165
kTsf
Force on the shoulder link
0.6
N
0.4 0.2 0
0
15
30
45
60
75
90
kTsf
Force on the sleeve link
N
0.34
0.17
0
0
15
30
45
60
75
90
kTsf
Fig. 9. Force data on linkages in force control with different control algorithms.
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Before introducing the force control, we first discuss how to find the desired force (set value) for each link. The procedure of determining the set value for the body link is taken as an example as follow. After loading the garment on the robotic hanger, we manually controlled the body links to move to the position at which the garment was just tightly stretched on
a
the robotic hanger, and the force data was acquired for a defined period of time. Because there exists serious noise in the force data, the arithmetic average filter algorithm was applied to handle the force readings. The set value is the average of these measurements, so rbodyforce = 0.3 N, rshoulderforce = 0.2 N and rsleeveforce = 0.17 N.
Force on the body link
Force (N)
0.9
Set value PID 3% error band
0.6 0.3 0
0
0.1
0.2
0.3 0.4 0.5 Simulation time (s)
0.6
0.7
0.8
Force on the sleeve link
Force (N)
0.7
0.35
0
0
0.1
0.2
0.3 0.4 0.5 Simulation time (s)
0.6
0.7
0.6
0.7
0.8
Force on the shoulder link 1 Force (N)
0.8
0.4
0
0
0.1
0.2
b
0.3 0.4 0.5 Simulation time (s)
0.8
Force on the body link
Force (N)
0.9
Set value FT-PID 3% error band
0.6 0.3 0
0
0.1
0.2
0.3 0.4 0.5 Simulation time (s)
0.6
0.7
0.8
Force on the sleeve link
Force (N)
0.7
0.35
0
0
0.1
0.2
0.3 0.4 0.5 Simulation time (s)
0.6
0.7
0.8
0.6
0.7
0.8
Force on the shoulder link
Force (N)
0.8
0.4
0
0
0.1
0.2
0.3 0.4 0.5 Simulation time (s)
Fig. 10. Simulation results on garments of different sizes. (a) PID with small set point change; (b) FT-PID with small set point change; (c) PID with large set point change; (d) FT-PID with large set point change; (e) Force variation on body link at final state.
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Case 1: Simulation and experiment on force control with and without sensor noise The initial parameters of FT-PID controller and conventional PID parameters for simulation are listed in Table 5. Fig. 7 shows the control results without noise signals.
c
Based on the results of PID and FT-PID control, it can be seen that the response speed in FT-PID control is in general faster than that in PID control. For the sleeve link, the FT-PID control yields a smoother response when compared to PID control. Some indices on response time are listed in Table 6 where the rise time tr is defined as the time required for the step response to rise from 10% to
Force on the body link Set value PID
Force (N)
3 2 1 0
0
0.1
0.2
0.3 0.4 0.5 Simulation time (s)
0.6
0.7
0.8
0.6
0.7
0.8
0.6
0.7
0.8
Force on the sleeve link
Force (N)
1.7 1.275 0.85 0.425 0
0
0.1
0.2
0.3 0.4 0.5 Simulation time (s) Force on the shoulder link
2.5
Force (N)
2 1.5 1 0.5 0
0
0.1
0.2
0.3 0.4 0.5 Simulation time (s)
Force on the body link
d
3.5
Set value FT-PID
Force (N)
3 2.5 2 1.5 1 0.5 0 0
0.1
0.2
0.3
0.4 0.5 Simulation time (s)
0.6
0.7
0.8
0.6
0.7
0.8
0.6
0.7
0.8
Force on the sleeve link
Force (N)
1.7 1.275 0.85 0.425 0 0
0.1
0.2
0.3
0.4 0.5 Simulation time (s)
Force on the shoulder link 2.5
Force (N)
2 1.5 1 0.5 0 0
0.1
0.2
0.3
0.4 0.5 Simulation time (s)
Fig. 10 (continued)
E.H.K. Fung et al. / Expert Systems with Applications 38 (2011) 9924–9938
90% of the change; ts is defined as the time required for the response curve to reach and stay within ±3% of the set value change. It is obvious that tr of shoulder link under FT-PID control is much smaller than that of the conventional PID control which states that the FT-PID control algorithm provides faster response; ts in the FTPID control is also less than that of the conventional PID controller which indicates that the FT-PID algorithm makes the system reach the steady state at a quicker pace. In practice, the sensor always has noise signals and additionally there exists distinct environment noise. All of them have an adverse effect on the control performance. In the simulation study, it is impossible to retrieve the real noise signals. Alternatively, three white noise signals are incorporated into the feedback force data. These noises have zero mean with variance 0.3, 0.2 and 0.2 for body, sleeve and shoulder forces respectively. The results are shown in Fig. 8. It is obvious that the PID controller cannot provide a good performance especially for body link, but the FT-PID controller still yields a smooth response in the presence of noise. In addition, FT-PID controller can also yields a fast response. The indices on response time are trbody = 0.045 s, trsleeve = 0.15 s, trshoulder = 0.096 s and tsbody = 0.13 s, tssleeve = 0.27 s, tsshoulder = 0.20 s. To verify the FT-PID controller can give a better control performance, the FTPID controller is also tested in the practical experiments. In practical control system, the controller is a PC based controller with a D/A converter I/O card. The actuators are the AC servo motors accompanied with AC servo amplifiers. The forces are measured by force sensors and sent back to the controller via the amplifier, the RC filter circuit and an A/D card. The control algorithms are implemented by C++ and API functions of the I/O acquisition cards. The sampling time Tsf for force control is 12 ms. The garment is a piece of gray cotton large size T-shirt. During the experiments, the force sensor information is severely contaminated by the unknown noise, i.e. the signal-to-noise ratio (SNR) is very small. Although the digital filter algorithm has been applied, noises are still present in the force data. Usually these noises affect adversely the conventional PID controller which has no robust and adaptive properties. Since the fuzzy control technique possesses some adaptive capabilities, the FT-PID controller can still perform well in the presence of noise. Fig. 9 shows the force data on each pair of links after implementing force control. It is clear that the force measurements are contaminated with noise irrespective of the type of the controller employed. The force curves obtained under PID control have heavier oscillation than
e
Force on the body link
Force (N)
3.1 3 2.9
Set value PID
2.8
0.6
0.62
0.64
0.66
0.68 0.7 0.72 Simulation time (s)
0.74
0.76
0.78
0.8
Force on the body link Force (N)
3.2 3 Set value FT-PID
2.8
0.6
0.62
0.64
0.66
0.68 0.7 0.72 Simulation time (s)
Fig. 10 (continued)
0.74
0.76
0.78
0.8
9935
those under FT-PID control, especially for the sleeve and shoulder links. The oscillation induced by the noise made the conventional PID controller fail to compute the correct control output as seen by the force curve of sleeve link around 120Tsf in Fig. 9. It also reveals that the controllers themselves cannot filter out the noise if no extra filter, either hardware or software, is adopted. Although the corresponding indices such as tr and ts cannot easily be determined, the trend of the force curve can be tracked. It can be seen from Fig. 9 that the FT-PID controller provides a faster response than conventional PID, and at the same time it is less susceptible to noise. Especially for the sleeve link, the force with the conventional PID is much larger than the set value due to the noise, however the FT-PID still maintains the force stable around the set value. In addition, for the force control of sleeve link the noise makes the conventional PID failed around 120Tsf and the garment cannot be fully extended, but the FT-PID controller performs well. The effectiveness of the FT-PID controller is manifested in the experimental results confirming the validity of the simulation results. Case 2: Simulation on garments of different size (different desired force value) In Case 1, we tested the FT-PID controller via simulation and experiment on a cotton garment of size L. Force of different magnitude is required to extend various size garments. In order to interpret the process of extending the garments with diverse sizes and assess the adaptability of the FT-PID controller, we examine the same material but different size garments by changing the desired force value. Hence the desired forces on the body, sleeve and shoulder links are increased from 0.3 to 0.9, 0.17 to 0.7, and 0.2 to 0.8 respectively, and the same FT-PID and conventional PID controller are used to implement the force control. Fig. 10 illustrates the results with different controller. It can been seen from Fig. 10 that by using conventional PID and FT-PID controller the force curves on the links show very little difference except the sleeve. With PID controller, there exist more oscillations in the transient response (Fig. 10(a)) while with FT-PID the curve has less oscillations yielding a smoother response (Fig. 10(b)), and concurrently the settling time is more or less the same (trPID = 0.218 s, trFTPID = 0.223 s). As stated in Case 1, the desired force for garment of different size is determined from a special category of cotton T-shirt. Actually, the force set value may exceed 1 N for different sizes of other types of fabrics. Therefore, we further increased the force values on the body, sleeve and shoulder links from 0.3 to 3, 0.17 to 1.7 and 0.2 to 2 respectively, and again tested the FT-PID controller for the new desired force. The simulation results are illustrated in Fig. 10(c) and (d) for application to other fabrics. The FT-PID yields better performance than the conventional PID. The force oscillations on sleeve link are smaller which is similar to the results in Fig. 10(a) and (b), and the force curve on the shoulder link still converges under the FT-PID control but with conventional PID control it diverges. As for the force curves of the body link, both controllers give similar results. However, after the curves are enlarged (Fig. 10(e)) there exist several ripples at the final state by using conventional PID controller; the FT-PID controller can eliminate these ripples. Hence, the simulation results demonstrate that the FT-PID control algorithm has a higher adaptability than conventional PID and can handle different size clothes. Case 3: Simulation on garments of different materials (different c1i, c2i and c3i) One of the purposes of designing this robotic hanger is to extend different material garments. In the dynamic model, three
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E.H.K. Fung et al. / Expert Systems with Applications 38 (2011) 9924–9938
a
Force on the body link Set value PID
Force (N)
0.3
0.15
0
0
0.1
0.2
0.3 0.4 0.5 Simulation time (s)
0.6
0.7
0.8
Force on the sleeve link 0.2
Force (N)
0.15 0.1 0.05 0
0
0.1
0.2
0.3 0.4 0.5 Simulation time (s)
0.6
0.7
0.8
Force on the shoulder link 0.4
Force (N)
0.3 0.2 0.1 0
0
0.1
0.2
b
0.3 0.4 0.5 Simulation time (s)
0.6
0.7
0.8
Force on the body link Set value FT-PID
Force (N)
0.3
0.15
0
0
0.1
0.2
0.3 0.4 0.5 Simulation time (s)
0.6
0.7
0.8
Force on the sleeve link
Force (N)
0.2 0.15 0.1 0.05 0
0
0.1
0.2
0.3 0.4 0.5 Simulation time (s)
0.6
0.7
0.6
0.7
0.8
Force on the shoulder link
Force (N)
0.2 0.15 0.1 0.05 0
0
0.1
0.2
0.3 0.4 0.5 Simulation time (s)
0.8
Fig. 11. Simulation results on garments of different materials. (a) PID; (b) FT-PID.
parameters c1i, c2i and c3i are used to model the stiffness of various materials used for garments. In the current study, we obtained c1i, c2i and c3i through the repetitive experiments on the cotton garments (Fung et al., 2009). However, it is impossible and impracticable to test the properties of all materials by tedious
experiments. Fortunately, with our derived model, we can easily and conveniently test the implementation of the force control on the garments of various materials. On the basis of the cotton parameters, in simulation we multiplied c1i, c2i and c3i by 10, 8 and 5 times respectively i.e. c1inew = 10c1i, c2inew = 8c2i and c3inew =
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E.H.K. Fung et al. / Expert Systems with Applications 38 (2011) 9924–9938
Force on the shoulder link
Force on the shoulder link
b
0.2
0.2
0.18
0.18
0.16
0.16
0.14
0.14 Force (N)
Force (N)
a
0.12 0.1
0.12 0.1
0.08
0.08
0.06
0.06
0.04 0.02 0
0.04
Set value PID
0
0.1
0.2
0.3
0.4 0.5 0.6 Simulation time (s)
0.7
0.8
0.9
Set value FT-PID
0.02
1
0
0
0.1
0.2
0.3
0.4 0.5 0.6 Simulation time (s)
0.7
0.8
0.9
1
Fig. 12. Simulation results on implementation error. (a) PID; (b) FT-PID.
5c3i, which indicates that the garment to be extended is made by a different material other not cotton. We further tested the adaptability of FT-PID controller through comparison with conventional PID controller. Fig. 11 displays the simulation results. It is obvious that after changing the material coefficients the conventional PID controller cannot yield a stable response where the force curves of the body and shoulder links diverge (Fig. 11(a)). In contrast, FT-PID controller still retains excellent performance characteristics, and the force curve on each link is as smooth as before and the response speed is also fast. This simulation manifests that the FT-PID controller has adaptive property for various material garments.
the noise in the sensor output. The above factors lead to the difficulty of designing the controller using the model-based method. To tackle these problems, a fuzzy tuned PID control algorithm has been employed to control the robotic hanger. This controller adaptively adjusts the Kp, Ki and Kd values according to the force error and error differential. The simulations and experiments have validated the dynamic model and the effectiveness of the FT-PID controller, and demonstrated that the FT-PID controller gives no overshoot, shorter settling time and rise time in the force response. Besides, it is also found to possess acceptable degree of adaptability and robustness when compared to the conventional PID controller. The future work will be focused on the testing of the hanger for stretching garments of different size and materials.
Case 4: Simulation on implementation error in controller Acknowledgement Another case which possibly happens in practical control system is so-called implementation error. That is the situation that the controller has a fault that makes the controller parameters change. Similar to Case 3, for this case we also could not do the real experiments on the robotic hanger, which could cause the unpredictable damage for the hanger. Therefore, we still perform the simulation study using the derived dynamic model. To analyze the robustness of the conventional PID and FT-PID controller, as an example we reduced the value of the integral gain for shoulder link by 80% which is larger than the case presented by Carvajal et al. (2000), and the control trajectories for the shoulder link are shown in Fig. 12. When the implementation error occurs, the PID controller causes steady-state error for the response curve; however the FT-PID controller automatically adjusts the parameters and makes the force reach the desired point at around 0.7 s. This implies that FT-PID controller is more tolerant to the implementation error occurred in controller. 5. Conclusion In this paper, a novel and flexible 6-axis robotic hanger system has been proposed for the inspection of knitted garments with various styles, sizes and cloth fabrics. It has three groups of linkages including body, shoulder and sleeve links and can move in the 2D vertical plane. To comprehensively explain the structure and motion of the robotic hanger and furthermore to provide the means for control simulation under the conditions that are hardly realized in practical experiments, the dynamic model of the robotic hanger is derived using the Lagrangian energy approach. This complex, nonlinear and coupled mathematical model also considers
The work described in this paper was supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. PolyU5170/06E). References Carvajal, J., Chen, G., & Ogmen, H. (2000). Fuzzy PID controller: Design, performance evaluation, and stability analysis. Information Sciences, 123, 249–270. Craig, J. J. (1988). Adaptive control of mechanical manipulators/ John J. Craig. Reading, Mass.: Addison-Wesley. Fung, E. H. K., Yuen, C. W. M., Wong, W. K., Hau, L. C. & Chan, L. K. (2008). A robot system for the control of fabric tension for inspection. In Paper presented at the ASME international mechanical engineering congress and exposition, Seattle, WA, USA. Fung, E. H. K., Cheng, L., Wong, Y. K., Zhang, X. Z., Yuen, C. W. M. & Wong, W. K. (2009). Modeling, design and analysis of a robot system for garment inspection. In Paper presented at the 2009 ASME international mechanical engineering congress and exposition, Orlando, FL, USA. Fung, E. H. K., Wong, Y. K., Yuen, C. W. M. & Wong, W. K. (2009a). Model reference adaptive control of a three DOF garment hanger. In Paper presented at the 22nd canadian congress of applied mechanics, Halifax, NS, Canada. Harinath, E., & Mann, G. K. I. (2008). Design and tuning of standard additive model based fuzzy PID controllers for multivariable process systems. IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics, 38(3), 667–674. Negnevitsky, M. (2005). Artificial intelligence: A guide to intelligent systems/ Michael Negnevitsky (2nd ed.). Harlow, England: Addison-Wesley. Novotny, D. W. (1996). Vector control and dynamics of AC drives. Oxford: New York: Clarendon Press; Oxford University Press. Patton, R., Swern, F., Tricamo, S., & Van der Veen, A. (1992). Automated cloth handling using adaptive force feedback. Journal of Dynamic Systems, Measurement, and Control, 114(4), 731–733. Shayeghi, H., Shayanfar, H. A., & Jalili, A. (2007). Multi stage fuzzy PID load frequency controller in a restructured power system. Journal of Electrical Engineering-Elektrotechnicky Casopis, 58(2), 61–70. Situm, Z., Pavkovic, D., & Novakovic, B. (2004). Servo pneumatic position control using fuzzy PID gain scheduling. Journal of Dynamic Systems, Measurement, and Control, 126(2), 376–387.
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Soyguder, S., Karakose, M., & Alli, H. (2009). Design and simulation of self-tuning PID-type fuzzy adaptive control for an expert HVAC system. Expert Systems with Applications, 36(3, Part 1), 4566–4573. Truong, D. Q., & Ahn, K. K. (2009). Force control for hydraulic load simulator using self-tuning grey predictor – fuzzy PID. Mechatronics, 19(2), 233–246. Wai, R. J., Tu, C. Y., & Hsieh, K. Y. (2004). Adaptive tracking control for robot manipulator. International Journal of Systems Science, 35(11), 615–627.
Wang, J., Fu, C., & Zhang, Y. (2008). SVC control system based on instantaneous reactive power theory and fuzzy PID. IEEE Transactions on Industrial Electronics, 55(4), 1658–1665. Ying, H. (2000). Theory and application of a novel fuzzy PID controller using a simplified Takagi–Sugeno rule scheme. Information Sciences, 123, 281–293. Yuen, C. W. M., Fung, E. H. K., Wong, W. K., Hau, L. C., & Chan, L. K. (2008). Application of smart system to textile industry: Preliminary design of a smart hanger for garment inspection. Journal of the Textile Institute, 99, 569–580.
Contents lists available at ScienceDirect
Expert Systems with Applications journal homepage: www.elsevier.com/locate/eswa
Fuzzy logic control of a novel robotic hanger for garment inspection: Modeling, simulation and experimental implementation E.H.K. Fung a,⇑, Y.K. Wong b, X.Z. Zhang a, L. Cheng a, C.W.M. Yuen c, W.K. Wong c a
Department of Mechanical Engineering, The Hong Kong Polytechnic University, Hunghom, Kowloon, Hong Kong Department of Electrical Engineering, The Hong Kong Polytechnic University, Hunghom, Kowloon, Hong Kong c Institute of Textiles and Clothing, The Hong Kong Polytechnic University, Hunghom, Kowloon, Hong Kong b
a r t i c l e
i n f o
a b s t r a c t
Keywords: Fuzzy control Garment inspection Lagrangian method Robotic hanger
This paper presents a novel robotic hanger system for the garment inspection of knitting production. This 3-DOF hanger consists of three groups of linkages which can freely move in the 2D plane. In order to thoroughly understand the structure and motion of the robotic hanger so as to facilitate control simulations and experiments, the mathematical dynamic model of the hanger is analyzed and derived by using the Lagrangian energy approach. Since the dynamic model is complex, coupled and nonlinear, the fuzzy tuned PID (FT-PID) algorithm is employed to establish the controller. This FT-PID controller is found to be less susceptible to the sensor noise, the variation of garment material and size. Therefore, it possesses a certain degree of robustness and adaptability. The simulation and experimental results validate the dynamic model and the effectiveness of the FT-PID control method through comparisons with the conventional PID controller. Ó 2011 Elsevier Ltd. All rights reserved.
1. Introduction
the structure and motion of the robotic hanger and furthermore to provide the means for control simulation under the conditions that are hardly realized in practical experiments, the mathematical dynamic model of this robotic hanger is analyzed and derived using the energy approach. Lagrangian method (Craig, 1988) is applied to deduce a series of motion equations of linkages which are then used to establish the extension force exerted on the robotic hanger. To extend the garments perfectly, the force is an important controlled variable. It is computed from the derived complex, coupled and nonlinear motion equations, which makes it difficult to establish an accurate controller based on the model. Additionally, the serious noise in the environment creates extra difficulty in the design of controllers. In our previous simulation work, model reference adaptive control (MRAC) has been successfully applied to control the force of a 1-DOF mechanism (Fung, Yuen, Wong, Hau, & Chan, 2008). However, the MRAC system, when applied to the complex 3-DOF hanger, may easily lead to unstable results if inappropriate parameters are used. Coupling effects among the three links also make the parameter tuning process tedious and timeconsuming (Fung, Wong, Yuen, & Wong, 2009a). Practically, many control techniques for robot manipulators in industrial operations depend on proportional integral derivative (PID) control laws because of their simple structure, ease of design and low cost (Wai, Tu, & Hsieh, 2004). Since the model of the robotic hanger is complex, it is almost impossible to build an accurate PID controller based on the plant model. Fuzzy control utilizes the human control operator’s knowledge and experience
As an important stage of quality control in the knitting production, the garment inspection can be done either manually or automatically. This process alone is very time-consuming because of the variety of styles, sizes and fabrics used in the clothing. Another concern is the worker fatigue and quality standardization which requires the inspection to be repetitive to achieve a certain satisfaction. To tackle these problems, it is necessary to set up a feasible automatic intelligent hanger system (Patton, Swern, Tricamo, & Van der Veen, 1992; Yuen, Fung, Wong, Hau, & Chan, 2008) to simulate the manual operation to extend various knitted garments. The products currently in the market are found not suitable for these applications, as they usually have one or two degree of freedom (DOF) only. To extend the garment efficiently, a 6-axis robotic hanger with three DOF is now developed which can move in the two-dimensional (2D) plane. The proposed robotic hanger has three groups of linkages: body link, shoulder link and sleeve link which are designed to satisfy the conditions of inspection for various garments. The structural advantages of this robotic hanger are that the shoulder and sleeve link are lightweight; the body link always positions itself vertically during the movement and the gearing mechanism is simple. In order to comprehensively understand ⇑ Corresponding author. E-mail addresses: (E.H.K. Fung).
[email protected],
[email protected]
0957-4174/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.eswa.2011.02.037
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Nomenclature BMi BOi
dissipated energy coefficient for motor i proportional dissipated energy constant due to friction at output joint i c1i, c2i, c3i coefficients related to the garment materials in force equation for link i Fi normal force exerted on link i by the garment at the C.G. of link i IGi moment of inertia of gear i IMi moment of inertia of motor i iai, Lai, Rai current, armature winding inductance and resistance of motor i KLi kinetic energy of link i KGi kinetic energy of gear i KMi kinetic energy of motor i flexible coefficient of belt i ki l1 distance between the center of gear 1 and gear 3 l2, l3, lv length of link 2, 3 and vertical link mass of gear i mGi mLi mass of link i NGi number of teeth of gear i N1 reduction ratio between motor 1 and link 1
to intuitively construct controllers so that the designed controller can emulate the human control behaviors. A satisfactory nonlinear controller can be developed empirically in practice without complicated mathematics, which is established based on converting the human operation experiences to a series of control rules. The principle of the fuzzy controller makes it possess the properties of robustness and adaptability. Following these advantages, fuzzy tuned PID (FT-PID) controller has been proposed to regulate the controller parameters automatically and used successfully in many applications. Examples of applications are found in various consumer products and industrial systems such as hydraulic load simulator (Truong & Ahn, 2009), servo pneumatic position control (Situm, Pavkovic, & Novakovic, 2004), medical application (Ying, 2000), power system (Shayeghi, Shayanfar, & Jalili, 2007; Wang, Fu, & Zhang, 2008) and temperature control (Harinath & Mann, 2008; Soyguder, Karakose, & Alli, 2009). In this study, the FT-PID controller is also applied to implement force control and the controller performance is validated through simulations and experiments. The results obtained by FT-PID controller and conventional PID controller are compared. It is verified that FTPID can provide efficient performance even under the situation that the force sensor output is contaminated with noise; the garments are of different size and materials; and the controller possesses implementation errors. The rest of this paper is organized as follow: Section 2 discusses and analyzes the mathematical dynamic model of the robotic hanger in detail. Section 3 describes the design of the FT-PID control algorithm. In Section 4, simulation and experiment results are presented to validate the dynamic model and performance of the FT-PID control algorithm through comparison with the conventional PID controller. The paper is concluded in Section 5.
2. Dynamic model of the robotic hanger To fully extend the garment at a proper position, the forces exerted on the surface of the garments are the important variables which should be fed back to the controller. The relationship between the force exerted on the C.G. of the body, sleeve and shoulder links Fi and the corresponding garment extension xti can be expressed by
Pbi potential energy of belt i PGi potential energy of gear i potential energy of link i PLi r radius of the pulleys ~ r ci position vector of C.G. (center of gravity) of link i ~ r 0cv unextended position vector of C.G. of vertical link Vini, Kbi, Ksi input voltage, back emf constant and torque constant of motor i vGi velocity of the center of gear i xti extension of garment at the C.G. of link i xti maximum of xti for which the force equation of link i holds ai angular acceleration of link i ami angular acceleration of motor i h1i, h2i, h3i initial displacement of link 1, 2 and 3, respectively hmi angular displacement of motor i hi angular displacement of link i sMi torque of motor i xGi angular velocity of gear i xmi angular velocity of motor i xi angular velocity of link i
F i ¼ c1i x3ti þ c2i x2ti þ c3i xti
for 0 < xti 6 xti
ð1Þ
F i ¼ 0 for xti 6 0
It should be noted that, as shown in Fig. 1(a) and (b), only the forces acting on the right side of the hanger are described, the purpose of which is to make a simple explanation in the following derivation. In the actual robotic hanger system, the forces are found on both sides of the hanger. It can be seen from (1) that the force relates to the displacements of body, sleeve and shoulder links. According to Fig. 2, the displacement for each link is calculated as. Body link follows:
~ r cv ¼ l1 cos h1~i þ ðl1 sin h1 þ 0:5lv Þ~j ~ r 0 cv ¼ l1 cos hi1~i þ ðl1 sin h1i þ 0:5lv Þ~j
ð2Þ
)~ z1 ¼ ~ r cv ~ r 0 cv ¼ l1 ðcos h1 cos h1i Þ~i þ l1 ðsin h1 sin h1i Þ~j ) xt1 ¼ ~ z1 ~i ¼ l1 ðcos h1 cos h1i Þ Sleeve link:
xt2 ¼ 2
l2 Dh2 h2 h2i sin ¼ l2 sin 2 2 2
ð3Þ
for a small Dh2 ; xt2 12 l2 ðh2 h2i Þ Shoulder link:
xt3 ¼ 2
l3 Dh3 h3i h3 sin ¼ l3 sin 2 2 2
ð4Þ
for a small Dh3 ; xt3 12 l3 ðh3i h3 Þ From the derivation of xti, we can further find that hi correlates xti. To determine the governing equations of hi, the Lagrangian energy method is used.
d @L @L ¼ Q i; dt @ q_ i @qi
i ¼ 1; 2; . . . ; 5
where L ¼ K tot P tot ; Q i ¼ @W and q ¼ ½h1 @q i
ð5Þ h2
h3
hm2
hm3 .
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Fig. 1. The structure of robotic hanger. (a) Overview; (b) Schematic diagram (Motors, belts, pulleys, gears and force sensors are not shown).
Fig. 3. Kinetic energy computation of each link. (a) Body link; (b) Sleeve link; (c) Shoulder link.
Similar to the body link, the kinetic energies of sleeve and shoulder links are calculated as below. Kinetic energy of sleeve link (link 2) For an infinitesimal particle located by r on link 2,
position : x ¼ l1 cos h1 þ r sin h2 ;
y ¼ l1 sin h1 þ lv r cos h2
Fig. 2. Displacement for each link. (a) Body link; (b) Sleeve link; (c) Shoulder link.
velocity : x_ ¼ l1 sin h1 x1 þ r cos h2 x2 ; y_ ¼ l1 cos h1 x1 þ r sin h2 x2
The total kinetic energy Ktot is defined as
K tot ¼ ðK L1 þ K LV þ K L2 þ K L3 Þ þ ðK G1 þ K G2 þ K G3 Þ þ ðK M1 þ K M2 þ K M3 Þ
ð6Þ
v 2 ¼ x_ 2 þ y_ 2 ¼ l21 x21 þ r2 x22 2l1 rx1 x2 sinðh1 h2 Þ
It incorporates the energies caused by the motions of the links, gears and motors.
mass : dm ¼
Kinetic energy of body link (link 1 and link v) The body link consists of two links: link 1 and vertical link (link v). As shown in Fig. 3, for an infinitesimal particle located by r on link 1,
dK L1
1 1 mL1 dr ¼ v 2 dm ¼ ðrx1 Þ2 ) K L1 ¼ 2 2 l1 Z l1 1 mL1 2 2 1 ¼ x1 r dr ¼ mL1 l21 x21 2 6 l 1 0
In regard to link vv = x1l1. Thus
K Lv ¼
v,
Z
Thus,
K L2 ¼
dK L1 ¼
Z 0
l2
1 2 mL2 v dr 2 l2
1 2 2 mL ½3l x2 þ l2 x22 3l1 l2 x1 x2 sinðh1 h2 Þ 6 2 1 1
ð7Þ
since it does not rotate, the velocity is
1 1 2 mLv v 2v ¼ mLv l1 x21 2 2
mL2 dr l2
Kinetic energy of shoulder link (link 3) For an infinitesimal particle located by r on link 3,
ð8Þ
position : x ¼ l1 cos h1 r sin h3 ;
y ¼ l1 sin h1 þ lv þ r cos h3
ð9Þ
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velocity : x_ ¼ l1 sin h1 x1 r cos h3 x3 ; y_ ¼ l1 cos h1 x1 r sin h3 x3
Substitute (7)–(16) into (6),
v 2 ¼ x_ 2 þ y_ 2 ¼ l21 x21 þ r2 x23 þ 2l1 rx1 x3 sinðh1 h3 Þ mass : dm ¼
mL3 dr l3
Thus,
K L3 ¼
Z
l3
0
1 2 mL3 1 2 2 v dr ¼ mL3 ½3l1 x21 þ l3 x23 þ 3l1 l3 x1 x3 sinðh1 h3 Þ 2 6 l3 ð10Þ
The kinetic energies of gears KG are generated from the gear mounting which is installed on the link 1 (Fig. 4), where G1 is the pivot gear and cannot rotate. These energies are computed as kinetic energies of gears.
Gear 1 : xG1 ¼ v G1 ¼ 0 ) K G1 ¼
1 1 IG1 x2G1 þ mG1 v 2G1 ¼ 0 2 2
ð11Þ
sinðh1 h3 Þx1 x3 Therefore, it can be derived that the total kinetic energy
K tot ¼ A1 x21 þ A2 x22 þ A3 x23 þ A4 x2m2 þ A5 x2m3 þ ½A6 sinðh1 h2 Þx1 x2 þ ½A7 sinðh1 h3 Þx1 x3
NG1 1 Gear 2 : xG2 ¼ x1 1 þ ; v G2 ¼ x1 l1 2 NG2 1 1 ) K G2 ¼ IG2 x2G2 þ mG2 v 2G2 2 2 2 1 NG1 1 2 2 ¼ IG2 x1 1 þ þ mG2 x21 l1 2 8 NG2
ð17Þ
where
A1 ¼ ð12Þ
NG2 x xG2 ¼ 1 ðNG1 þ NG2 Þ; v G3 ¼ x1 l1 NG3 NG3 1 1 ) K G3 ¼ IG3 x2G3 þ mG3 v 2G3 2 2 1 ðNG1 þ NG2 Þ2 1 2 ¼ IG3 x21 þ mG3 x21 l1 2 2 N2G3
Gear 3 : xG3 ¼
1 1 1 2 2 mL2 l2 ; A3 ¼ mL2 l2 ; A4 ¼ IM2 ; 6 6 2 1 1 A6 ¼ mL2 l1 l2 ; A7 ¼ mL3 l1 l3 2 2
ð13Þ
1 ¼ IM1 x2m1 2
1 ¼ IM1 N21 x21 2 1 Motor 2 : K M2 ¼ IM2 x2m2 2 1 Motor 3 : K M3 ¼ IM3 x2m3 2
Fig. 4. Kinetic energy computation of each gear.
1 1 1 1 1 1 2 2 2 2 2 2 mL1 l1 þ mLv l1 þ mL2 l1 þ mL3 l1 þ mG2 l1 þ mG3 l1 6 2 2 2 8 2 2 1 NG1 1 ðNG1 þ NG2 Þ2 1 þ IG2 1 þ þ IG3 þ IM1 N21 2 2 2 NG2 N2G3
A2 ¼
There are three AC servo motors mounted on the base of the robotic hanger, and they drive three pairs of links to move in the 2D plane. Motor 1 is installed at the rear of the base and it provides the power for the motion of body links. Motor 2 & 3 are installed at the front of the base and drive the sleeve and shoulder links respectively. The motions of these motors also produce energies which are called kinetic energies of AC servo motors.
Motor 1 : hm1 ¼ N1 h1 ; xm1 ¼ N1 x1 ) K M1
K tot ¼ ðK L1 þ K LV þ K L2 þ K L3 Þ þ ðK G1 þ K G2 þ K G3 Þ 1 1 1 2 2 2 þ ðK M1 þ K M2 þ K M3 Þ ¼ mL1 l1 þ mLv l1 þ mL2 l1 6 2 2 2 1 1 1 1 NG1 2 2 2 þ mL3 l1 þ mG2 l1 þ mG3 l1 þ IG2 1 þ 2 8 2 2 NG2 # 2 1 ðNG1 þ NG2 Þ 1 1 2 2 2 2 I m þ IG3 þ N x þ l M1 1 L2 2 x2 1 2 2 6 N2G3 1 1 1 2 þ mL3 l3 x23 þ IM2 x2m2 þ IM3 x2m3 6 2 2 1 1 þ mL2 l1 l2 sinðh1 h2 Þx1 x2 þ mL3 l1 l3 2 2
ð16Þ
1 IM3 ; 2
The total potential energy Ptot is defined as
Ptot ¼ ðPL1 þ PLV þ PL2 þ PL3 Þ þ ðP G1 þ PG2 þ PG3 Þ þ ðPb2 þ Pb3 Þ
ð18Þ
Similarly, it includes three parts of energies which are generated from links, gears and belts. As shown in Fig. 5, these energies can be computed as below. Links:
PL1 ¼
1 mL1 gl1 sin h1 2
ð19Þ
P Lv ¼
1 mLv gðlv þ 2l1 sin h1 Þ 2
ð20Þ
ð14Þ ð15Þ
A5 ¼
1 PL2 ¼ mL2 gðl1 sin h1 þ lv l2 cos h2 Þ 2
Fig. 5. Potential energy computation. (a) Links and gears; (b) Belts.
ð21Þ
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1 PL3 ¼ mL3 gðl1 sin h1 þ lv þ l3 cos h3 Þ 2
ð22Þ
~ rc3 ¼ ðl1 cos h1 0:5l3 sin h3 Þ~i þ ðl1 sin h1 þ lv þ 0:5l3 cos h3 Þ~j
ð23Þ
d~ rc3 ¼ ðl1 sin h1 dh1 0:5l3 cos h3 dh3 Þ~i þ ðl1 cos h1 dh1 0:5l3
Gears:
PG1 ¼ 0 PG2
1 ¼ mG2 gl1 sin h1 2
PG3 ¼ mG3 gl1 sin h1
sin h3 dh3 Þ~j ð24Þ ð25Þ
Belts (assuming no slip):
Pb3
Therefore we have
dW ¼ Bo1 x1 dh1 Bo2 x2 dh2 Bo3 x3 dh3 þ ðsM1
1 k2 ðhm2 r h2 rÞ2 2
ð26Þ
1 ¼ k3 ðhm3 r h3 rÞ2 2
ð27Þ
Pb2 ¼
F 3 d~ rc3 ¼ F 3 l1 sinðh1 h3 Þdh1 þ 0:5F 3 l3 dh3 ) dW 3 ¼ ~
BM1 xM1 Þdhm1 þ ðsM2 BM2 xM2 Þdhm2 þ ðsM3 BM3 xM3 Þdhm3 þ F 1 l1 sin h1 dh1 þ F 2 l1 sinðh1 h2 Þdh1 0:5F 2 l2 dh2 þ F 3 l1 sinðh1 h3 Þdh1 þ 0:5F 3 l3 dh3
Substituting (17), (28) and (29) into (5) and using Lagrangian technique, we can derive the motion equations as follows: For i ¼ 1; q1 ¼ h1 ; q_ 1 ¼ x1
Substitute (19)–(27) into (18),
Ptot ¼ ðPL1 þ PLV þ PL2 þ PL3 Þ þ ðPG1 þ PG2 þ PG3 Þ þ ðPb2 þ Pb3 Þ 1 1 ¼ mL1 þ mLv þ mL2 þ mL3 þ mG2 þ mG3 gl1 sin h1 2 2 1 1 1 þ mL2 gl2 cos h2 þ mL3 gl3 cos h3 þ k2 r 2 ðhm2 2 2 2 1 1 mLv þ mL2 þ mL3 glv h2 Þ2 þ k3 r 2 ðhm3 h3 Þ2 þ 2 2
@L ¼ 2A1 x1 þ A6 sinðh1 h2 Þx2 þ A7 sinðh1 h3 Þx3 @ x1 d @L ¼ 2A1 a1 þ A6 sinðh1 h2 Þa2 þ A6 cosðh1 h2 Þðx1 dt @ x1 x2 Þx2 þ A7 sinðh1 h3 Þa3 þ A7 cosðh1 h3 Þðx1
Therefore, the total potential energy is
x3 Þx3
Ptot ¼ B1 sin h1 þ B2 cos h2 þ B3 cos h3 þ B4 ðhm2 h2 Þ2 þ B5 ðhm3 h3 Þ2 þ B6
ð29Þ
ð28Þ
@L ¼ A6 cosðh1 h2 Þx1 x2 þ A7 cosðh1 h3 Þx1 x3 B1 cos h1 @h1
where
1 1 mL1 þ mLv þ mL2 þ mL3 þ mG2 þ mG3 gl1 ; 2 2 1 1 B2 ¼ mL2 gl2 ; B3 ¼ mL3 gl3 ; 2 2 1 1 1 mLv þ mL2 þ mL3 glv B4 ¼ k2 r 2 ; B5 ¼ k3 r 2 ; B6 ¼ 2 2 2
B1 ¼
@W ¼ Bo1 x1 þ ðsM1 BM1 xm1 ÞN 1 þ F 1 l1 sin h1 þ F 2 l1 sinðh1 h2 Þ @h1 þ F 3 l1 sinðh1 h3 Þ Thus
2A1 a1 þ A6 sinðh1 h2 Þa2 þ A7 sinðh1 h3 Þa3 A6 cosðh1
There is another component dW in (5). It is the virtual work acting on the robotic hanger system defined by
h2 Þx22 A7 cosðh1 h3 Þx23 þ ðBo1 þ N21 BM1 Þx1 þ B1
dW ¼ BO1 x1 dh1 BO2 x2 dh2 BO3 x3 dh3 þ ðsM1 BM1 xm1 Þdhm1
¼ N1 sM1 þ F 1 l1 sin h1 þ F 2 l1 sinðh1 h2 Þ þ F 3 l1 sinðh1 h3 Þ
þ ðsM2 BM2 xm2 Þdhm2 þ ðsM3 BM3 xm3 Þdhm3 þ dW 1 þ dW 2 þ dW 3
where dWi is the virtual work done by the garment tension. They can be calculated as
dW 1 : ~ F 1 ¼ F 1~i;~ r cv ¼ l1 cos h1~i þ ðl1 sin h1 þ 0:5lv Þ~j; d~ r cv ¼ l1 sin h1 dh1~i þ l1 cos h1 dh1~j F 1 d~ r cv ¼ F 1 l1 sin h1 dh1 ) dW 1 ¼ ~ dW 2 : ~ F 2 ¼ F 2 cos h2~i F 2 sin h2~j
cos h1 ð30Þ
For i ¼ 2; q2 ¼ h2 ; q_ 2 ¼ x2
@L ¼ 2A2 x2 þ A6 sinðh1 h2 Þx1 @ x2 d @L ¼ 2A2 a2 þ A6 sinðh1 h2 Þa1 þ A6 cosðh1 h2 Þðx1 x2 Þx1 dt @ x2 @L ¼ A6 cosðh1 h2 Þx1 x2 þ B2 sin h2 þ 2B4 ðhm2 h2 Þ @h2
~ r c2 ¼ ðl1 cos h1 þ 0:5l2 sin h2 Þ~i þ ðl1 sin h1 þ lv 0:5l2 cos h2 Þ~j d~ r c2 ¼ ðl1 sin h1 dh1 þ 0:5l2 cos h2 dh2 Þ~i þ ðl1 cos h1 dh1 þ 0:5l2 sin h2 dh2 Þ~j F 2 d~ r c2 ¼ F 2 l1 sinðh1 h2 Þdh1 0:5F 2 l2 dh2 ) dW 2 ¼ ~ dW 3 : ~ F 3 ¼ F 3 cos h3~i F 3 sin h3~j
@W ¼ Bo2 x2 0:5F 2 l2 @h2 Thus
2A2 a2 þ A6 sinðh1 h2 Þa1 þ A6 cosðh1 h2 Þx21 þ Bo2 x2 B2 sin h2 þ 2B4 h2 2B4 hm2 ¼ 0:5F 2 l2
ð31Þ
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For i ¼ 3; q3 ¼ h3 ; q_ 3 ¼ x3
μ (e) /μ (ec)/μ (ΔK p)/μ (ΔK i )/μ (ΔK d)
@L ¼ 2A3 x3 þ A7 sinðh1 h3 Þx1 @ x3
NB
NM
NS
ZE
PS
PM
1/3
2/3
PB
1
d @L ¼ 2A3 a3 þ A7 sinðh1 h3 Þa1 þ A7 cosðh1 h3 Þðx1 x3 Þx1 dt @ x3 @L ¼ A7 cosðh1 h3 Þx1 x3 þ B3 sin h3 þ 2B5 ðhm3 h3 Þ @h3
-1
-2/3
-1/3
0
1 e/ec/ΔK p /ΔK i /ΔK d
Fig. 6. Membership functions for e, ec, DKp, DKi and DKd.
@W ¼ Bo3 x3 þ 0:5F 3 l3 @h3 Thus
Table 2 Fuzzy inference rules for DKp.
2A3 a3 þ A7 sinðh1 h3 Þa1 þ A7 cosðh1 h3 Þx21 þ Bo3 x3 B3 sin h3 þ 2B5 h3 2B5 hm3 ¼ 0:5F 3 l3 For i ¼ 4; q4 ¼ hm2 ; q_ 4 ¼ xm2
@L d @L ¼ 2A4 xm2 ; @ xm2 dt @ xm2 @L ¼ 2B4 ðhm2 h2 Þ; @hm2
ð32Þ
DKp
ec
¼ 2A4 am2 ;
e
NB NM NS ZE PS PM PB
NB
NM
NS
ZE
PS
PM
PB
PB PB PM PM PS PS ZE
PB PB PM PM PS ZE ZE
PM PM PM PS ZE NS NM
PM PS PS ZE NS NM NM
PS PS ZE NS NS NM NM
ZE ZE NS NM NM NM NB
ZE NS NS NM NM NB NB
@W ¼ sM2 BM2 xm2 @hm2 Thus
2A4 am2 þ BM2 xm2 þ 2B4 hm2 2B4 h2 ¼ sM2
ð33Þ
For i ¼ 5; q5 ¼ hm3 ; q_ 5 ¼ xm3
DKi
d @L ¼ 2A5 am3 ; dt @ xm3
@L
Table 3 Fuzzy inference rules for DKi.
¼ 2A5 xm3 ; @ xm3 @L ¼ 2B5 ðhm3 h3 Þ; @hm3
ec
@W ¼ sM3 BM3 xm3 @hm3
e
NB NM NS ZE PS PM PB
NB
NM
NS
ZE
PS
PM
PB
NB NB NB NM NM ZE ZE
NB NB NM NM NS ZE ZE
NM NM NS NS ZE PS PS
NM NS NS ZE PS PS PM
NS NS ZE PS PS PM PM
ZE ZE PS PM PM PB PB
ZE ZE PS PM PB PB PB
NB
NM
NS
ZE
PS
PM
PB
PS PS ZE ZE ZE PB PB
NS NS NS NS ZE NS PM
NB NB NM NS ZE PS PM
NB NM NM NS ZE PS PM
NB NM NS NS ZE PS PS
NM NS NS NS ZE PS PS
PS ZE ZE ZE ZE PB PB
Thus
2A5 am3 þ BM3 xm3 þ 2B5 hm3 2B5 h3 ¼ sM3
ð34Þ
It is obvious that Eqs. (30), (33) and (34) require the torque output from motors. Since in this robotic hanger system three AC servo motors are used to drive the linkages, it is inappropriate to use the DC motor model to compute the torque. Furthermore, the motor parameters listed in the manual are limited and we can hardly calculate the continuous torque output according to the AC servo motor model (Novotny, 1996). In this study, an alternative method is adopted to estimate the equivalent DC motor parameters corresponding to the AC servo motor. Variable voltage control under basic frequency is applied for the AC servo motor (Novotny, 1996) and the torque is
sM ¼ 3np
2
Us
x1
Table 4 Fuzzy inference rules for DKd.
DKd
ec
e
NB NM NS ZE PS PM PB
sx1 R0r ðsRs þ R0r Þ2 þ s2 x21 ðLls þ Llr Þ2
Table 5 Parameters for conventional PID and FT-PID.
Table 1 Universe of discourse for input and output variables.
e ec DKp DKi DKd
⁄
Conventional PID
Body
Sleeve
Shoulder
[0.3, 0.3] [20, 20] [0.2, 0.2] [0.1, 0.1] [6e4, 6e4]
[0.2, 0.2] [2.5, 2.5] [0.05, 0.05] [0.01, 0.01] [6e4, 6e4]
[0.17, 0.17] [3.6, 3.6] [0.01, 0.01] [0.15, 0.15] [8e4, 8e4]
Kp (⁄initial value for FT-PID) Ki (⁄initial value for FT-PID) Kd (⁄initial value for FT-PID)
FT-PID
Body
Shoulder
Sleeve
Body
Shoulder
Sleeve
20
5
5.5
20
4
4
6
5
8
8
8.3
7.5
0.2
0.28
0.35
0.2
0.12
0.18
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where np = 1; the motor slip, s, is very small when the motor is running steadily. Therefore,
sM 3
Us
2
x1
sMet ¼
Us
x1
2
sx1 ¼ sM Ret
So
s x1 R0r
Ret ¼
Since the DC motor has only one ‘phase’, we assume the torque output of DC motor is equivalent to that of single phase AC motor and this torque has to maintain the same after equivalent transformation. That is
1 0 R; 3 r
and Let ¼
1 0 L 3 r
where L0r ¼ t e R0r ; t e is the electrical time constant of AC motor. The equivalent DC motor equation for the AC servo motor used in the dynamic model is
Force on the body link
a Force (N)
0.3
Set value PID 3% error band
0.2 0.1 0
0
0.1
0.2
0.3
0.4
0.5 0.6 Simulation time (s)
0.7
0.8
0.9
1
0.7
0.8
0.9
1
0.7
0.8
0.9
1
Force on the sleeve link
0.2 Force (N)
0.15 0.1 0.05 0
0
0.1
0.2
0.3
0.4
0.5 0.6 Simulation time (s)
Force on the shoulder link
Force (N)
0.2 0.15 0.1 0.05 0
0
0.1
0.2
0.3
b
0.4
0.5 0.6 Simulation time (s)
Force on the body link Set value FT-PID 3% error band
Force (N)
0.3 0.2 0.1 0
0
0.1
0.2
0.3
0.5 0.6 Simulation time (s)
0.7
0.8
0.9
1
0.7
0.8
0.9
1
0.7
0.8
0.9
1
Force on the sleeve link
0.2 Force (N)
0.4
0.15 0.1 0.05 0
0
0.1
0.2
0.3
0.4
0.5 0.6 Simulation time (s)
Force on the shoulder link
Force (N)
0.2 0.15 0.1 0.05 0
0
0.1
0.2
0.3
0.4
0.5 0.6 Simulation time (s)
Fig. 7. Results obtained with different controller (without noise signals). (a) PID; (b) FT-PID.
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In summary, the Eqs. (1) and (30)–(35) represent the dynamic model of the robotic hanger system. For these equations, the initial conditions when t = 0 are
Table 6 Indices of transient-response in force control (without noise). Conventional PID
Rise time tr (s) Settling time ts (s)
Let
FT-PID
Body
Shoulder
Sleeve
Body
Shoulder
Sleeve
0.023 0.040
0.11 0.19
0.14 0.20
0.024 0.041
0.058 0.10
0.11 0.20
diet þ Ret iet þ K b h_ m ¼ U s ; dt
sMet ¼ K s iet
ð35Þ
To maintain the consistency with the Eqs. (30), (33) and (34) we still call sMet as sM.
Force (N)
a
h2 ¼ h2i ) xt2 ¼ 0 ) F 2 ¼ 0;
h3 ¼ h3i ) xt3 ¼ 0 ) F 3 ¼ 0 x1 ¼ x2 ¼ x3 ¼ 0; a1 ¼ a2 ¼ a3 ¼ 0;
xm1 ¼ xm2 ¼ xm3 ¼ 0; am1 ¼ am2 ¼ am3 ¼ 0
sM1i ¼
B1 cos h1i ; N1
sM2i ¼ B2 sin h2i ; sM3i ¼ B3 sin h3i
Force on the body link Set value PID
0.3 0.15 0
0
0.1
0.2
0.3
0.4
0.5 0.6 Simulation time (s)
0.7
0.8
0.9
1
0.7
0.8
0.9
1
0.7
0.8
0.9
1
Force on the sleeve link
0.2 Force (N)
h1 ¼ h1i ) xt1 ¼ 0 ) F 1 ¼ 0;
0.15 0.1 0.05 0
0
0.1
0.2
0.3
0.4
0.5 0.6 Simulation time (s)
Force on the shoulder link Force (N)
0.2
0
0
0.1
0.2
0.3
0.4
0.5 0.6 Simulation time (s)
Force on the body link
b
Set value FT-PID 3% error band
Force (N)
0.3 0.2 0.1 0
0
0.1
0.2
0.3
0.5 0.6 Simulation time (s)
0.7
0.8
0.9
1
0.7
0.8
0.9
1
0.7
0.8
0.9
1
Force on the sleeve link
0.2 Force (N)
0.4
0.15 0.1 0.05 0
0
0.1
0.2
0.3
0.4
0.5 0.6 Simulation time (s)
Force on the shoulder link Force (N)
0.2 0.15 0.1 0.05 0
0
0.1
0.2
0.3
0.4
0.5 0.6 Simulation time (s)
Fig. 8. Results obtained with different controller (with white noise). (a) PID; (b) FT-PID.
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hm2i ¼
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B2 sin h2i þ 2B4 h2i ; 2B4
hm3i ¼
B3 sin h3i þ 2B5 h3i 2B5
of membership function for input and output variables. It is noted that Fig. 6 depicts the membership functions with the standard universe of discourse and for each input and output variable of different link the required universe of discourse can be obtained by simply multiplying it with a scale factor. The universe of discourse for e is determined from the desired force value. In practice, the desired force in this study never exceeds 1 N, therefore, e is also less than 1 N. As for the universe of discourse for ec, it is determined by simulation results. Based on the designed membership functions for input and output variables, it is possible to describe the fuzzy inference rules which are listed in Tables 2–4. The rules are deduced concurrently; therefore the number of rules is 49. In accordance with these rules, the fuzzy inference is implemented in a way as given by Negnevitsky (2005). As for the defuzzification process, the most commonly used formula, the center of gravity or centroid method, is employed. This is expressed as
The parameters required for simulations include A1 A7 ; B1 B6 ; BM1 BM3 ; Bo1 Bo3 ; c1i c3i ; l1 l3 ; R0r1 R0r3 ; te ; K b1 K b3 and Ks1 Ks3. Some of these parameters can be found in the manuals. Some parameters can be found by physical measurements while the remaining ones can be obtained by experiments. 3. Control strategy Instead of the conventional PID controller, the fuzzy tuned PID (FT-PID) controller is applied to implement the force control due to the difficulty to design an accurate PID controller based on the highly nonlinear and coupled dynamic model. Another reason for choosing the FT-PID is that the estimated parameters of the dynamic model obtained from experiments are not exact. The control algorithm of FT-PID can be expressed in the discrete incremental form:
R yf lðyf Þdyf yo ¼ R lðyf Þdyf
DuðkÞ ¼ K p ½eðkÞ eðk 1Þ þ K i eðkÞ þ K d ½eðkÞ 2eðk 1Þ þ eðk 2Þ ¼ K p ec ðkÞ þ K i eðkÞ þ K d ½ec ðkÞ ec ðk 1Þ ð36Þ
ð37Þ
where yf is the fuzzy value of the fuzzy output variable, l(yf) is the corresponding value of membership function, and yo is the defuzzified crisp output i.e. DKp, DKi or DKd. With these defuzzified outputs, the new PID parameters are Kp(k) = Kp(k 1) + DKp(k), Ki(k) = Ki(k 1) + DKi(k) and Kd(k) = Kd(k 1) + DKd(k). Furthermore the new Du(k) can be calculated according to Eq. (36).
where e is the force error and ec is the differential of force error. Kp, Ki and Kd are the PID parameters which are tuned by fuzzy logic with proper fuzzy rules. The input variables are the e and ec, and the universe of discourse for e and ec are enlisted in Table 1. Considering the complexity of the robotic hanger system, more fuzzy subsets (Carvajal, Chen, & Ogmen, 2000) need to be designed. The membership functions for e and ec in the fuzzification procedure have seven linguistic variables: NB (negative big), NM (negative medium), NS (negative small), ZE (zero), PS (positive small), PM (positive medium) and PB (positive big). The output variables are DKp, DKi and DKd. Universe of discourse for these variables are also listed in Table 1, and their membership functions have similar forms as the input variables. They also have seven linguistic variables: NB, NM, NS, ZE, PS, PM and PB. Fig. 6 illustrates the profiles
4. Case study In this section, we validate the dynamic model and FT-PID control strategy through simulations and experiments. To further investigate the performance of the FT-PID controller, the conventional PID controller is used for comparison where the parameters of PID controller are tuned by the Ziegler & Nichols method.
Force on the body link
0.6
PID FT-PID Set value
0.45 N
0.3 0.15 0
0
15
30
45
60
75
90
105
120
135
150
165
105
120
135
150
165
105
120
135
150
165
kTsf
Force on the shoulder link
0.6
N
0.4 0.2 0
0
15
30
45
60
75
90
kTsf
Force on the sleeve link
N
0.34
0.17
0
0
15
30
45
60
75
90
kTsf
Fig. 9. Force data on linkages in force control with different control algorithms.
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Before introducing the force control, we first discuss how to find the desired force (set value) for each link. The procedure of determining the set value for the body link is taken as an example as follow. After loading the garment on the robotic hanger, we manually controlled the body links to move to the position at which the garment was just tightly stretched on
a
the robotic hanger, and the force data was acquired for a defined period of time. Because there exists serious noise in the force data, the arithmetic average filter algorithm was applied to handle the force readings. The set value is the average of these measurements, so rbodyforce = 0.3 N, rshoulderforce = 0.2 N and rsleeveforce = 0.17 N.
Force on the body link
Force (N)
0.9
Set value PID 3% error band
0.6 0.3 0
0
0.1
0.2
0.3 0.4 0.5 Simulation time (s)
0.6
0.7
0.8
Force on the sleeve link
Force (N)
0.7
0.35
0
0
0.1
0.2
0.3 0.4 0.5 Simulation time (s)
0.6
0.7
0.6
0.7
0.8
Force on the shoulder link 1 Force (N)
0.8
0.4
0
0
0.1
0.2
b
0.3 0.4 0.5 Simulation time (s)
0.8
Force on the body link
Force (N)
0.9
Set value FT-PID 3% error band
0.6 0.3 0
0
0.1
0.2
0.3 0.4 0.5 Simulation time (s)
0.6
0.7
0.8
Force on the sleeve link
Force (N)
0.7
0.35
0
0
0.1
0.2
0.3 0.4 0.5 Simulation time (s)
0.6
0.7
0.8
0.6
0.7
0.8
Force on the shoulder link
Force (N)
0.8
0.4
0
0
0.1
0.2
0.3 0.4 0.5 Simulation time (s)
Fig. 10. Simulation results on garments of different sizes. (a) PID with small set point change; (b) FT-PID with small set point change; (c) PID with large set point change; (d) FT-PID with large set point change; (e) Force variation on body link at final state.
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Case 1: Simulation and experiment on force control with and without sensor noise The initial parameters of FT-PID controller and conventional PID parameters for simulation are listed in Table 5. Fig. 7 shows the control results without noise signals.
c
Based on the results of PID and FT-PID control, it can be seen that the response speed in FT-PID control is in general faster than that in PID control. For the sleeve link, the FT-PID control yields a smoother response when compared to PID control. Some indices on response time are listed in Table 6 where the rise time tr is defined as the time required for the step response to rise from 10% to
Force on the body link Set value PID
Force (N)
3 2 1 0
0
0.1
0.2
0.3 0.4 0.5 Simulation time (s)
0.6
0.7
0.8
0.6
0.7
0.8
0.6
0.7
0.8
Force on the sleeve link
Force (N)
1.7 1.275 0.85 0.425 0
0
0.1
0.2
0.3 0.4 0.5 Simulation time (s) Force on the shoulder link
2.5
Force (N)
2 1.5 1 0.5 0
0
0.1
0.2
0.3 0.4 0.5 Simulation time (s)
Force on the body link
d
3.5
Set value FT-PID
Force (N)
3 2.5 2 1.5 1 0.5 0 0
0.1
0.2
0.3
0.4 0.5 Simulation time (s)
0.6
0.7
0.8
0.6
0.7
0.8
0.6
0.7
0.8
Force on the sleeve link
Force (N)
1.7 1.275 0.85 0.425 0 0
0.1
0.2
0.3
0.4 0.5 Simulation time (s)
Force on the shoulder link 2.5
Force (N)
2 1.5 1 0.5 0 0
0.1
0.2
0.3
0.4 0.5 Simulation time (s)
Fig. 10 (continued)
E.H.K. Fung et al. / Expert Systems with Applications 38 (2011) 9924–9938
90% of the change; ts is defined as the time required for the response curve to reach and stay within ±3% of the set value change. It is obvious that tr of shoulder link under FT-PID control is much smaller than that of the conventional PID control which states that the FT-PID control algorithm provides faster response; ts in the FTPID control is also less than that of the conventional PID controller which indicates that the FT-PID algorithm makes the system reach the steady state at a quicker pace. In practice, the sensor always has noise signals and additionally there exists distinct environment noise. All of them have an adverse effect on the control performance. In the simulation study, it is impossible to retrieve the real noise signals. Alternatively, three white noise signals are incorporated into the feedback force data. These noises have zero mean with variance 0.3, 0.2 and 0.2 for body, sleeve and shoulder forces respectively. The results are shown in Fig. 8. It is obvious that the PID controller cannot provide a good performance especially for body link, but the FT-PID controller still yields a smooth response in the presence of noise. In addition, FT-PID controller can also yields a fast response. The indices on response time are trbody = 0.045 s, trsleeve = 0.15 s, trshoulder = 0.096 s and tsbody = 0.13 s, tssleeve = 0.27 s, tsshoulder = 0.20 s. To verify the FT-PID controller can give a better control performance, the FTPID controller is also tested in the practical experiments. In practical control system, the controller is a PC based controller with a D/A converter I/O card. The actuators are the AC servo motors accompanied with AC servo amplifiers. The forces are measured by force sensors and sent back to the controller via the amplifier, the RC filter circuit and an A/D card. The control algorithms are implemented by C++ and API functions of the I/O acquisition cards. The sampling time Tsf for force control is 12 ms. The garment is a piece of gray cotton large size T-shirt. During the experiments, the force sensor information is severely contaminated by the unknown noise, i.e. the signal-to-noise ratio (SNR) is very small. Although the digital filter algorithm has been applied, noises are still present in the force data. Usually these noises affect adversely the conventional PID controller which has no robust and adaptive properties. Since the fuzzy control technique possesses some adaptive capabilities, the FT-PID controller can still perform well in the presence of noise. Fig. 9 shows the force data on each pair of links after implementing force control. It is clear that the force measurements are contaminated with noise irrespective of the type of the controller employed. The force curves obtained under PID control have heavier oscillation than
e
Force on the body link
Force (N)
3.1 3 2.9
Set value PID
2.8
0.6
0.62
0.64
0.66
0.68 0.7 0.72 Simulation time (s)
0.74
0.76
0.78
0.8
Force on the body link Force (N)
3.2 3 Set value FT-PID
2.8
0.6
0.62
0.64
0.66
0.68 0.7 0.72 Simulation time (s)
Fig. 10 (continued)
0.74
0.76
0.78
0.8
9935
those under FT-PID control, especially for the sleeve and shoulder links. The oscillation induced by the noise made the conventional PID controller fail to compute the correct control output as seen by the force curve of sleeve link around 120Tsf in Fig. 9. It also reveals that the controllers themselves cannot filter out the noise if no extra filter, either hardware or software, is adopted. Although the corresponding indices such as tr and ts cannot easily be determined, the trend of the force curve can be tracked. It can be seen from Fig. 9 that the FT-PID controller provides a faster response than conventional PID, and at the same time it is less susceptible to noise. Especially for the sleeve link, the force with the conventional PID is much larger than the set value due to the noise, however the FT-PID still maintains the force stable around the set value. In addition, for the force control of sleeve link the noise makes the conventional PID failed around 120Tsf and the garment cannot be fully extended, but the FT-PID controller performs well. The effectiveness of the FT-PID controller is manifested in the experimental results confirming the validity of the simulation results. Case 2: Simulation on garments of different size (different desired force value) In Case 1, we tested the FT-PID controller via simulation and experiment on a cotton garment of size L. Force of different magnitude is required to extend various size garments. In order to interpret the process of extending the garments with diverse sizes and assess the adaptability of the FT-PID controller, we examine the same material but different size garments by changing the desired force value. Hence the desired forces on the body, sleeve and shoulder links are increased from 0.3 to 0.9, 0.17 to 0.7, and 0.2 to 0.8 respectively, and the same FT-PID and conventional PID controller are used to implement the force control. Fig. 10 illustrates the results with different controller. It can been seen from Fig. 10 that by using conventional PID and FT-PID controller the force curves on the links show very little difference except the sleeve. With PID controller, there exist more oscillations in the transient response (Fig. 10(a)) while with FT-PID the curve has less oscillations yielding a smoother response (Fig. 10(b)), and concurrently the settling time is more or less the same (trPID = 0.218 s, trFTPID = 0.223 s). As stated in Case 1, the desired force for garment of different size is determined from a special category of cotton T-shirt. Actually, the force set value may exceed 1 N for different sizes of other types of fabrics. Therefore, we further increased the force values on the body, sleeve and shoulder links from 0.3 to 3, 0.17 to 1.7 and 0.2 to 2 respectively, and again tested the FT-PID controller for the new desired force. The simulation results are illustrated in Fig. 10(c) and (d) for application to other fabrics. The FT-PID yields better performance than the conventional PID. The force oscillations on sleeve link are smaller which is similar to the results in Fig. 10(a) and (b), and the force curve on the shoulder link still converges under the FT-PID control but with conventional PID control it diverges. As for the force curves of the body link, both controllers give similar results. However, after the curves are enlarged (Fig. 10(e)) there exist several ripples at the final state by using conventional PID controller; the FT-PID controller can eliminate these ripples. Hence, the simulation results demonstrate that the FT-PID control algorithm has a higher adaptability than conventional PID and can handle different size clothes. Case 3: Simulation on garments of different materials (different c1i, c2i and c3i) One of the purposes of designing this robotic hanger is to extend different material garments. In the dynamic model, three
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E.H.K. Fung et al. / Expert Systems with Applications 38 (2011) 9924–9938
a
Force on the body link Set value PID
Force (N)
0.3
0.15
0
0
0.1
0.2
0.3 0.4 0.5 Simulation time (s)
0.6
0.7
0.8
Force on the sleeve link 0.2
Force (N)
0.15 0.1 0.05 0
0
0.1
0.2
0.3 0.4 0.5 Simulation time (s)
0.6
0.7
0.8
Force on the shoulder link 0.4
Force (N)
0.3 0.2 0.1 0
0
0.1
0.2
b
0.3 0.4 0.5 Simulation time (s)
0.6
0.7
0.8
Force on the body link Set value FT-PID
Force (N)
0.3
0.15
0
0
0.1
0.2
0.3 0.4 0.5 Simulation time (s)
0.6
0.7
0.8
Force on the sleeve link
Force (N)
0.2 0.15 0.1 0.05 0
0
0.1
0.2
0.3 0.4 0.5 Simulation time (s)
0.6
0.7
0.6
0.7
0.8
Force on the shoulder link
Force (N)
0.2 0.15 0.1 0.05 0
0
0.1
0.2
0.3 0.4 0.5 Simulation time (s)
0.8
Fig. 11. Simulation results on garments of different materials. (a) PID; (b) FT-PID.
parameters c1i, c2i and c3i are used to model the stiffness of various materials used for garments. In the current study, we obtained c1i, c2i and c3i through the repetitive experiments on the cotton garments (Fung et al., 2009). However, it is impossible and impracticable to test the properties of all materials by tedious
experiments. Fortunately, with our derived model, we can easily and conveniently test the implementation of the force control on the garments of various materials. On the basis of the cotton parameters, in simulation we multiplied c1i, c2i and c3i by 10, 8 and 5 times respectively i.e. c1inew = 10c1i, c2inew = 8c2i and c3inew =
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E.H.K. Fung et al. / Expert Systems with Applications 38 (2011) 9924–9938
Force on the shoulder link
Force on the shoulder link
b
0.2
0.2
0.18
0.18
0.16
0.16
0.14
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Fig. 12. Simulation results on implementation error. (a) PID; (b) FT-PID.
5c3i, which indicates that the garment to be extended is made by a different material other not cotton. We further tested the adaptability of FT-PID controller through comparison with conventional PID controller. Fig. 11 displays the simulation results. It is obvious that after changing the material coefficients the conventional PID controller cannot yield a stable response where the force curves of the body and shoulder links diverge (Fig. 11(a)). In contrast, FT-PID controller still retains excellent performance characteristics, and the force curve on each link is as smooth as before and the response speed is also fast. This simulation manifests that the FT-PID controller has adaptive property for various material garments.
the noise in the sensor output. The above factors lead to the difficulty of designing the controller using the model-based method. To tackle these problems, a fuzzy tuned PID control algorithm has been employed to control the robotic hanger. This controller adaptively adjusts the Kp, Ki and Kd values according to the force error and error differential. The simulations and experiments have validated the dynamic model and the effectiveness of the FT-PID controller, and demonstrated that the FT-PID controller gives no overshoot, shorter settling time and rise time in the force response. Besides, it is also found to possess acceptable degree of adaptability and robustness when compared to the conventional PID controller. The future work will be focused on the testing of the hanger for stretching garments of different size and materials.
Case 4: Simulation on implementation error in controller Acknowledgement Another case which possibly happens in practical control system is so-called implementation error. That is the situation that the controller has a fault that makes the controller parameters change. Similar to Case 3, for this case we also could not do the real experiments on the robotic hanger, which could cause the unpredictable damage for the hanger. Therefore, we still perform the simulation study using the derived dynamic model. To analyze the robustness of the conventional PID and FT-PID controller, as an example we reduced the value of the integral gain for shoulder link by 80% which is larger than the case presented by Carvajal et al. (2000), and the control trajectories for the shoulder link are shown in Fig. 12. When the implementation error occurs, the PID controller causes steady-state error for the response curve; however the FT-PID controller automatically adjusts the parameters and makes the force reach the desired point at around 0.7 s. This implies that FT-PID controller is more tolerant to the implementation error occurred in controller. 5. Conclusion In this paper, a novel and flexible 6-axis robotic hanger system has been proposed for the inspection of knitted garments with various styles, sizes and cloth fabrics. It has three groups of linkages including body, shoulder and sleeve links and can move in the 2D vertical plane. To comprehensively explain the structure and motion of the robotic hanger and furthermore to provide the means for control simulation under the conditions that are hardly realized in practical experiments, the dynamic model of the robotic hanger is derived using the Lagrangian energy approach. This complex, nonlinear and coupled mathematical model also considers
The work described in this paper was supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. PolyU5170/06E). References Carvajal, J., Chen, G., & Ogmen, H. (2000). Fuzzy PID controller: Design, performance evaluation, and stability analysis. Information Sciences, 123, 249–270. Craig, J. J. (1988). Adaptive control of mechanical manipulators/ John J. Craig. Reading, Mass.: Addison-Wesley. Fung, E. H. K., Yuen, C. W. M., Wong, W. K., Hau, L. C. & Chan, L. K. (2008). A robot system for the control of fabric tension for inspection. In Paper presented at the ASME international mechanical engineering congress and exposition, Seattle, WA, USA. Fung, E. H. K., Cheng, L., Wong, Y. K., Zhang, X. Z., Yuen, C. W. M. & Wong, W. K. (2009). Modeling, design and analysis of a robot system for garment inspection. In Paper presented at the 2009 ASME international mechanical engineering congress and exposition, Orlando, FL, USA. Fung, E. H. K., Wong, Y. K., Yuen, C. W. M. & Wong, W. K. (2009a). Model reference adaptive control of a three DOF garment hanger. In Paper presented at the 22nd canadian congress of applied mechanics, Halifax, NS, Canada. Harinath, E., & Mann, G. K. I. (2008). Design and tuning of standard additive model based fuzzy PID controllers for multivariable process systems. IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics, 38(3), 667–674. Negnevitsky, M. (2005). Artificial intelligence: A guide to intelligent systems/ Michael Negnevitsky (2nd ed.). Harlow, England: Addison-Wesley. Novotny, D. W. (1996). Vector control and dynamics of AC drives. Oxford: New York: Clarendon Press; Oxford University Press. Patton, R., Swern, F., Tricamo, S., & Van der Veen, A. (1992). Automated cloth handling using adaptive force feedback. Journal of Dynamic Systems, Measurement, and Control, 114(4), 731–733. Shayeghi, H., Shayanfar, H. A., & Jalili, A. (2007). Multi stage fuzzy PID load frequency controller in a restructured power system. Journal of Electrical Engineering-Elektrotechnicky Casopis, 58(2), 61–70. Situm, Z., Pavkovic, D., & Novakovic, B. (2004). Servo pneumatic position control using fuzzy PID gain scheduling. Journal of Dynamic Systems, Measurement, and Control, 126(2), 376–387.
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Soyguder, S., Karakose, M., & Alli, H. (2009). Design and simulation of self-tuning PID-type fuzzy adaptive control for an expert HVAC system. Expert Systems with Applications, 36(3, Part 1), 4566–4573. Truong, D. Q., & Ahn, K. K. (2009). Force control for hydraulic load simulator using self-tuning grey predictor – fuzzy PID. Mechatronics, 19(2), 233–246. Wai, R. J., Tu, C. Y., & Hsieh, K. Y. (2004). Adaptive tracking control for robot manipulator. International Journal of Systems Science, 35(11), 615–627.
Wang, J., Fu, C., & Zhang, Y. (2008). SVC control system based on instantaneous reactive power theory and fuzzy PID. IEEE Transactions on Industrial Electronics, 55(4), 1658–1665. Ying, H. (2000). Theory and application of a novel fuzzy PID controller using a simplified Takagi–Sugeno rule scheme. Information Sciences, 123, 281–293. Yuen, C. W. M., Fung, E. H. K., Wong, W. K., Hau, L. C., & Chan, L. K. (2008). Application of smart system to textile industry: Preliminary design of a smart hanger for garment inspection. Journal of the Textile Institute, 99, 569–580.