Open access peer-reviewed chapter

Trajectory Tracking Control of Parallel Manipulator with Integral Manifold and Observer

Written By

Zhengsheng Chen

Submitted: May 2nd, 2018 Reviewed: July 18th, 2018 Published: November 5th, 2018

DOI: 10.5772/intechopen.80375

Chapter metrics overview

902 Chapter Downloads

View Full Metrics

Abstract

In view of the problem that notable flexible displacement will occur for parallel manipulators when operating at high speed, the composite controller based on the integral manifold and high-gain observer is proposed for trajectory tracking and the 3RRR parallel manipulator is taken as the object. Based on the stiffness matrix, the small variable is introduced to decompose the rigid-flexible coupling dynamic model into slow and fast subsystem. For the slow subsystem, the backstepping control is applied for rigid motion tracking. In order to account for the links’ flexible displacement the corrective torque is deduced, and the compensation for the flexible displacement is realized. For the fast subsystem, the sliding mode control is utilized to suppress the vibration. The high-gain observer is designed to avoid the measurement of the curvature rate of flexible links. Also, the stability of the overall system is proven with the Lyapunov stability theorem and the upper bound of the small variable is obtained. At last, the proposed composite controller together with the singular perturbation control and the rigid body model-based backstepping control are simulated, and vibration suppression and tracking performances are compared to validate the proposed control scheme.

Keywords

  • parallel manipulator
  • integral manifold
  • high-gain observer
  • composite control
  • sliding mode control
  • backstepping control
  • vibration suppression

1. Introduction

Parallel manipulators (PMs) possess advantages of high precision, high stiffness, and large load-to-weight ratio; they have attracted wide attention and have been widely used in industries such as high-speed handling, motion simulation, and electronic manufacturing [1]. However, in order to increase efficiency, PMs are increasingly used in high-speed and heavy-duty operations. In order to reduce costs and energy consumption, the lightweight design of the mechanical body will be the inevitable choice. However, in the high-speed or heavy-duty application, the lightweight mechanical body will produce significant elastic deformation and vibration. Therefore, the end-effector’s movement consists of the rigid-body motion and the elastic displacement caused by elastic deformation and vibration. Using conventional control methods for rigid-body manipulators will not guarantee good tracking accuracy of flexible manipulator’s end-effector. Therefore, it is of great significance to improve the tracking accuracy of high-speed lightweight PMs by considering the flexibility of members to establish the dynamic model for rigid-flexible coupling and carrying out research on high-precision control algorithms.

Many scholars have conducted extensive and in-depth studies on modeling methods for manipulators with flexible links. Dwivedy et al. [2] reviewed the dynamic modeling of robots with flexible links. Due to the presence of link flexibility, the system will exhibit nonminimum phase characteristics when selecting the end-effector of the manipulator as the output. The literature [3, 4, 5] redefines the output of the manipulator’s end position by taking the link elasticity into account, and uses the control algorithm for the rigid-body manipulator to control the new output; however, this method can only realize the point-to-point position control and cannot guarantee tracking control of the end trajectory [6]. The singular perturbation method is another effective method to deal with the nonminimum phase characteristics of manipulators with elastic links. The small parameters are introduced to reduce the order of rigid-flexible coupling models, which are decomposed into two subsystems, the fast and the slow, and two subcontrollers are designed using compound control algorithm. The controller of the system realizes the control of the rigid body motion and the rapid suppression of the elastic vibration. However, as the deformation increases, the singularity perturbation algorithm shows a deficiency and the algorithm cannot compensate for the elastic displacement [7, 8, 9]. Khorasani [10] proposed an integral manifold method by high-order approximation of fast subsystem variables, which greatly improved the vibration suppression effect. By introducing the elastic displacement into the end of the manipulator and designing the corrective torque, Moallem et al. [11] realized the trajectory tracking precision control and vibration suppression of the two-degree-of-freedom serial robot. Based on the above method, Fotouhi et al. [12, 13, 14, 15, 16] studied the trajectory tracking control of the flexible joint robot, the flexible robot with the single link, the rigid-flexible hybrid robot, and the two-bar flexible robot by simplifying the selection of correction moments, and show good results.

Due to the existence of the closed-chain structure, the dynamic model of PMs is complex when considering the flexibility of the links. Therefore, the research on the vibration suppression and trajectory tracking control is very limited. Zhang et al. [7] used assumption mode method and Lagrange equation to model 3PRR PMs with flexible passive links, and adopted singular perturbation compound control to suppress vibration. However, the influence of the elastic displacement of the links on the moving platform is not considered in the model, and the elastic displacement compensation and the rate of change of the elastic links are not processed when the algorithm is designed. Therefore, the trajectory tracking effect needs to be improved. Existing research has not yet been found for the above issues. In the research of trajectory tracking control based on integral manifold, no relevant research has been found for PMs. The control algorithms for the slow subsystem in the existing research are feedback linearization methods, and the fast subsystem is PD control or pole placement. In order to taking into accounts of the elastic deformation and vibration of high-speed PMs due to the flexibility of links and improve the tracking accuracy and dynamic performance, this chapter introduces the integral manifold based on the rigid-flexible coupling model of the 3RRR PM, the hypothesis of small deformation and the velocity mapping in the previous paper [17], and the high-order rigid-flexible coupling model is transformed into two subsystems, then a composite control algorithm based on sliding mode variable structure control and backstepping control is proposed. At the same time, a high-gain observer is introduced to the curvature rate caused by the flexibility. Finally, simulation studies are conducted to verify the feasibility of the algorithm.

Advertisement

2. The dynamic model of the 3RRR PM

The structure of the 3RRR parallel manipulator was shown in Figure 1, which consists of three branches, and each branch composed of one active link and passive link, the end of which is the moving platform. The coordinates and the parameters are given in Figure 2, OXY and GxGyG are the coordinate frames attached with the base and moving platform, with O and G as the origin, respectively. θi and βi are the angles of the active and passive links, i=1,2,3, the position and attitude of the moving platform are depicted as η=xyϕT in the base frame.

Figure 1.

The 3RRR PM.

Figure 2.

Coordinates of the 3RRR PM.

According to our previously published paper [17], the flexibility of passive links can be neglected, so only the deformation of active links is considered here, which can be expressed as δi=k=1nαikmik,i=1,2,3, where αik and mik are the shape function and the curvature of the kth point in the ith active link, respectively, where k=1. According to [17], after ignoring the deformation of the passive links and adding the parameters of the motors and reducers, the dynamic model of the PM can be expressed as:

M110+M111M12M12TM22η¨m¨+000Kηm+f10+Mf1m+Mf2ṁf20+Mf3ṁ=JTτ0E1

where Jm and Jg are the moment of inertia of the motor and the reducer, and K=diagksksks is the stiffness matrix, while ks and ig are the link’s stiffness and the reduction ratio, respectively, τ represents the driving torque, M110 and f10 are the mass matrix and quadratic terms in the dynamic equation derived from [17], while the item corresponding to m is neglected.

Advertisement

3. Integral manifold-based model reduction of the high-speed PM

From the dynamic model (1), the state variables are defined as below [15],

X1=η,X2=η̇z1=m/ε2,z2=ṁ/εE2

where z=z1z2T and X=X1X2T are state variables of the slow subsystem, εR is the small parameter larger than zero, which are used for subsequent model reduction and time scale transformation. From the state variables (2) and the system Eq. (1), the state equation of the perturbed form can be expressed as:

Ẋ1=X2,Ẋ2=J11JTτJ11f1J12f2J12k˜z1;E3
εż1=z2,εż2=J12TJTτJ12Tf1J22f2J22k˜z1.E4

where k˜=ksε2 is the stiffness coefficient, J=J11J12;J12TJ22 is the inverse matrix of the mass matrix M.

For Eq. (4), the integral manifold is defined as [15, 18],

ztε=hX1tεX2tετtεztε=haX1tεX2tετtεE5

Eq. (5) can be interpreted that if the fast subsystem variables arrive at the integral manifold trajectory at the moment t*, then for the moment t>t, the variable will always remain on the manifold trajectory. In order to ensure the above conditions valid, the additional control variables are added in the control system.

Due to the small variable ε close to 0, the integral manifold h and the moment τ are all functions of ε, Taylor expansion of the above variables is available as,

h1ah1=h10+εh11X1X2t++εph1pX1X2th2ah2=h20+εh21X1X2t++εph2pX1X2tττ0+ετ1X1X2t++εpτpX1X2t.E6

where h1 and h2 are the approximations of h1a and h2a, and hij=jhiaj!εjε=0 is the derivative of the integral manifold with respect to the small variable ε, while i=1,2j=0,1,2,p, and pN+ is the approximation order. Since the elastic displacement of the link is ε2 times of the state variable z of the fast subsystem, so p should be at least 2 when the elastic displacement can be accounted in the end trajectory, the p is selected 2 here.

The inverse matrix of the mass matrix, the Coriolis force and the centrifugal force terms are functions of the small variable ε, the Taylor expansion of the inverse matrix about ε can be expressed as,

J11=J110,J12=J120J22=J220+J222ε2/2E7

The centrifugal and inertial force after the expansion of Eq. (1) can be expressed as,

f1=f10+f120h10+f121ḣ10ε2/2,f2=f20+ε2f221ḣ10/2.E8

Substituting Eqs. (6) through (8) into Eq. (4), we can obtain,

{ h10=(J22k˜)01(J12TJpθTτ0J12T(f1)0(J22)0(f2)0),h11=(J22k˜)01(J12TJpθTτ1h˙20),h12=(J22k˜)01(J12TJpθTτ2h˙21J12T((f1)20h10+(f1)21h˙10)/2)(J22)2 ((f2)0+h10)/2(J22)0(f2)21h˙10/2 ),h20=0,h21=h˙10,h22=h˙11. E9

When the flexibility of the links is ignored, the small variable ε=0 is valid. Substituting h1 into Eq. (3), the differential equation of the slow subsystem can be obtained as,

X¯̇1=X¯2X¯̇2=M1101JTτ0M1101f10E10

where X¯1 and X¯2 represent variables of the slow subsystem, for the convenience of description, X¯1 and X¯2 are replaced by X1 and X2 in the following expressions,

According to the integral manifold, the deviation of the fast subsystem variable can be expressed as,

Xf1=z1h10εh11ε2h12Xf2=z2h20εh21ε2h22E11

Multiply the Eq. (11) with ε, derive and substitute it into Eq. (6). According to Eq. (9), the fast subsystem equation can be obtained by substituting hij,

εẊf1=Xf2,εẊf2=J12TJTτfJ220k˜Xf1ε2J222+J12Tf120Xf1/2εJ12Tf121+J220f221Xf2/2.E12

For the slow and fast subsystems represented by Eqs. (10) and (12), the composite control algorithm is designed as shown in Figure 3. For the slow subsystem, the backstepping control is used to achieve the tracking control of the rigid body motion. At the same time, according to the velocity mapping relationship, the mapping relationship between the elastic deformation of the links and the elastic displacement of the moving platform is established. The motion of the moving platform is obtained according to the rigid-body motion and the elastic displacement, and the elastic torque compensation is realized by designing the correction torque τ1 and τ2. For fast subsystems, the sliding mode control is used to ensure the manifold valid. Considering the difficulty of measuring the rate of curvature change of the links, a high-gain observer will be designed to estimate the rate of curvature change based on the curvature value. The algorithm design will be based on the control structure shown in Figure 3.

Figure 3.

Scheme of the controller.

Advertisement

4. The backstepping algorithm-based slow subsystem control

The backstepping control is a recursive control algorithm for complex nonlinear systems. The original system is decomposed into subsystems that do not exceed the system order. The control design is realized by establishing Lyapunov functions step by step for each subsystem, and the stability of the system is ensured [19]. First, define the position error as,

E13

where Xd is the command signal, define the amount of virtual control as,

E14

where c1 is a constant greater than zero and the velocity error e2 can be defined as,

E15

Based on the position error, define the Lyapunov function as,

E16

Deriving Eq. (16) can be obtained,

E17

According to the velocity error (15) in conjunction with Eq. (17), the Lyapunov function is defined as,

E18

Deriving the above formula and substituting the relevant parameters, the derivative of the Lyapunov function can be expressed as,

V̇2=c1e1Te1+e1Te2+e2Tė2=e2TM1101JTτ0M1101f10+c1ė1X¨dc1e1Te1+e1Te2E19

According to Eq. (19), the control torque of the slow subsystem is

τ0=JT1f10+M110c1ė1+X¨dc2e2e1=JT1f10+M110X¨dc1+c2ė1c1c2+1e1E20

where c2 is a positive real number, and substitute Eq. (20) into Eq. (19), the derivative of the Lyapunov function of the slow subsystem can be expressed as:

V̇=c1e1Te1+c2e2Te20E21

Therefore, according to the Lyapunov stability principle, the slow subsystem is stable with the torque τ0. Due to the existence of the elasticity, the end position of the PM can be expressed as:

r=X1+f3ηh10h11h12ε.E22

where f3 is the elastic displacement of the center G of the moving platform induced by the elastic deformation and vibration of the links, which is the elastic displacement of the end-effector of the moving platform.

According to the velocity mapping relationship, the acceleration of the moving platform generated by the elastic motion can be expressed as,

f¨3=ε2J1ϕlh10+εh11+ε2h12/l1+ε2J1¯·ϕlḣ10+εḣ11+ε2ḣ12/l1.E23

where J1¯· is the time derivative of J1.

The flexibility examined in this chapter is within a small deformation range, and the elastic displacement f3 of the end-effector of the moving platform due to the elastic displacement of the rod can be simplified as,

E24

Make the second derivative of Eq. (22), when considering the rigid-flexible coupling motion, the acceleration of the end-effector of the moving platform can be expressed as,

r¨=X¨d+c1+c2Ẋdη̇+c1c2+1Xdη+M111JTετ1+ε2τ2+ε2J1ϕlh¨10+J1¯·ϕlḣ10/l1M111f120h10+f121ḣ10ε2/2+J12J2201J222f20+h10ε2/2+J12J2201h¨10ε2+J12f221ḣ10ε2/2.E25

Defining the position error e3=Xdr and velocity error e4=ė3 of the end-effector of the moving platform, Eq. (25) can be transformed as,

{ e˙3=e4e˙4=(c1+c2)e4(c1c2+1)e3M111JpθT(ετ1+ε2τ2)ε2J12(J22)01h¨10ε2(Jpθ1ϕlh¨10+Jpθ1¯·ϕlh˙10)/l1+ε2M111((f1)20h10+(f1)21h˙10)/2ε2(c1c2+1)Jpθ1ϕlh10/l1ε2J12(J22)01(J22)2((f2)0+h10)/2ε2(c1+c2)Jpθ1ϕlh˙10/l1ε2J12(f2)21h˙10/2 E26

According to Eq. (26), define the Lyapunov function,

E27

Derivative of Eq. (27) with respect to time can be obtained as,

E28

Let the coefficient of ε and ε2 be zero, and the corrective torque is,

τ1=0,τ2=(JpθT)1M11((c1+c2)(Jpθ1ϕlh˙10)/l1+(c1c2+1)Jpθ1ϕlh10/l1M111((f1)20h10+(f1)21h˙10)/2+J12(J22)01(J22)2(f2+h10)/2+J12(J22)01h¨10+(Jpθ1ϕlh¨10+Jpθ1¯·ϕlh˙10)/l1+J12(f2)21h˙10/2)E29

At this time, V̇=e4Tc1+c2e40 is valid, and the system is stable, which means the elastic displacement compensation for the end-effector’s pose is realized by designing the corrective torque.

Advertisement

5. Sliding mode variable structure-based fast subsystem control

Define a new time scale tf=t/ε, and the fast subsystem differential Eq. (12) can be expressed as,

dXf1dtf=Xf2,dXf2dtf=J12TJTτfJ220Xf1ε2J222+J12Tf120Xf1/2εJ12Tf121Xf2/2E30

The latter two terms of the second equation contain small parameter ε, and the control amount is small compared to other terms, which can be regarded as the disturbance, so the disturbance term can be expressed as,

Δ1=ε2J222+J12Tf120Xf1/2εJ12Tf121+J220f221Xf2/2.E31

Due to the existence of the disturbance term, the fast subsystem adopts sliding mode variable structure control, and the sliding mode surface is selected as,

St=KfXf1+Xf2.E32

where K1 is a positive number, the derivation of the upper sliding surface can be obtained,

Ṡt=KfX+J12TJTτfJ220Xf1Δ1.E33

According to the sliding surface, the Lyapunov function is defined as,

V4=1/2STS.E34

Derivative of the above equation with respect to time can be obtained as,

V̇4=STṠ=STKfXf2+J12TJTτfJ220Xf1Δ1E35

According to Eq. (35), the fast subsystem control law designed as,

τf=J12TJT1KfXf2+J220Xf1KfS+Δ1sgnS.E36

where sgn· is the sign function, substituting Eq. (36) into (35) can be obtained,

V̇4=STKfX+J12TJTτfJ220Xf1Δ1=Δ1SΔ1SSTKfSSTKfS0E37

Therefore, according to the Lyapunov stability principle, the fast subsystem is convergent with torque of (36). The symbolic function will cause jitter to the system. To reduce the generation of jitter, the saturation function sat· is substituted for the symbol function. The saturation function can be defined as [20],

sats1=1,s1>Δ2;s1/Δ2,s1Δ2;1,s1<Δ2.E38

where Δ2 is the buffer layer.

Advertisement

6. The high-gain observer for the curvature change rate

The curvature can be obtained by strain gage measurement of the stress of the links, and the change rate of curvature is directly related to the rate of change of stress, and generally cannot be directly measured. In order to avoid direct measurement of the change rate of curvature, this chapter will design a high-gain observer to observe the curvature change rate by measuring the curvature. It can be known from Eq. (11) that the fast subsystem variable Xf1 corresponds to the curvature, which can be directly converted by measuring the stress. Xf2 corresponds to observed curvature change rate. According to the literature [21, 22] and the formula (4), the observer can be expressed as,

εX̂̇f1=X̂f2+1ε1HpXf1X̂f1,εX̂̇f2=1ε12HvXf1X̂f1.E39

where X̂f1 and X̂f2 represent the estimated values of Xf1 and Xf2, respectively, ε1 is the minimum positive number, Hp and Hv are the constant matrix, the observer tracking error is defined as,

X˜f1=X̂f1Xf1,X˜f2=X̂f2Xf2.E40

To prove the stability of the system, new variables of error are defined as,

Z˜f1=X˜f1,Z˜f2=ε1X˜f2.E41

Substitute the above equation into (39), the state observer can be expressed as,

εε1Z˜̇f1=Z˜f2HpZ˜f1,εε1Z˜̇f2=HvZ˜f1+εε12J12TJTτfJ220Xf1Δ1E42

The Eq. (42) can be rewritten as,

εε1Z˜̇f=A0Z˜f+εε12B0J12TJTτfJ220Xf1Δ1.E43

where A0=HpI3×3Hv03×3 and B0=03×3I3×3. All eigenvalues of A0 can be guaranteed negative by selecting Hp and Hv, which means that A0 is the Hurwitz matrix. Define a new Lyapunov function as,

V6=Z˜fTP1Z˜f.E44

where P1 is the positive definite symmetry matrix, the derivation is expressed as,

V̇6=1εε1Z˜fTA0TP1+P1A0Z˜f+2εε12·J12TJTτfJ220Xf1Δ1TB0TP1Z˜f.E45

Since A0 is a Hurwitz matrix, there is a positive definite matrix P1, which makes,

A0TP1+P1A0=I3×3.E46

V̇6 can be rewritten as,

V̇61εε1Z˜f2+2ε1J12TJTτfJ220Xf1Δ1TB0TP1Z˜f.E47

According to Eq. (47), when ε12 satisfied the following relationship, V̇60 is established, which means the high-gain observer gradually converges,

ε122J12TJTτfJ220Xf1Δ1TB0TP1εZ˜f.E48

Therefore, according to Eq. (48), the upper bound of the small parameter can be obtained, and the fast subsystem torque can be expressed as,

τf=J12TJT1KfX̂f2+J220X̂f1KfŜ+Δ1satŜ.E49

where Ŝ=KfX̂f1+X̂f2. According to Eq. (12) and (42), the error equation of the fast subsystem can be expressed as,

εξ̇=Aξξ+hξE50

where

ξ=XfZ˜fT, Xf=Xf1Xf2T, Aξ=Aξ11Aξ120A0/ε1,Aξ11=03×3I3×3Kf22Kf,

Aξ12=03×303×3J220Kf22Kf, hξ=Δ1satŜΔ1εε1B0J12TJTτfJ220Xf1Δ1.

According to Eq. (50), the Lyapunov function can be defined as:

V5=εξTPξξE51

where Pξ is the symmetric positive definite matrix, Eq. (51) is derived as,

V5=εξTAξTPξ+PξTAξξ+2hξTPξξ+εξTṖξξ.E52

Since Aξ11 and A0 are Hurwitz matrix, for a given symmetric positive definite matrix Sξ, there is a symmetric positive definite matrix Pξ that satisfies the following conditions,

AξTPξ+PξTAξ=Sξ.E53

According to the Rayleigh-Ritz inequality,

ξTSξξλminSξξ2,E54
hξTPξξχ0+χ1ε1ξ, Ṗξχ2.E55

where λmin· represents the minimum eigenvalues of the corresponding matrix. χ0, χ1, and χ2 are positive real numbers. According to Eqs. (53) through (55), Eq. (52) can be expressed as,

V̇5λminSξξ2+εχ2ξ2+2χ0+χ1ε1ξ.E56

According to Eq. (56), when V̇50, the small parameters in the high-gain observer satisfied 0ε1ε1max, the fast subsystem based on the high-gain observer is stable, and the upper bound of the small parameter satisfies the following requirements,

ε1maxλminSξξεχ2ξ2χ0/χ1.E57
Advertisement

7. Stability proof of the system

The abovementioned integral manifold is used to reduce the rigid-flexible coupling system of high-speed PM, and the complex high-order system is decomposed into a slow subsystem describing the rigid body motion and a fast subsystem of elastic deformation, and the backstepping control and sliding mode variable structure control are adopted for two subsystems, respectively, and designed a high-gain observer to solve the problem that the elastic displacement change rate is difficult to measure, and proved the stability of each subsystem. However, the stability of each subsystem does not guarantee the stability of the overall system. Therefore, it is necessary to synthesize the subsystems to prove the stability of the overall system. Substituting Eqs. (9), (20), and (29) into kinetic Eq. (3), the systematic error equation can obtained,

ės=Ases+hs,εξ̇=Aξξ+hξ.E58

where

es=X1XdẊ1ẊdT, hs=0hs1, As=03×3I3×3c1c2+1I3×3c1+c2I3×3,

hs1=J11JTτfJ12Xf1ε2J11M11c1+c2J1ϕlḣ10/l1+c1c2+1J1ϕlh10/l1+J1ϕlh¨10+J1¯·ϕlḣ10/l1.

According to the error equation, define the Lyapunov function that contains the overall system as,

V6=esTPses+εξTPξξ.E59

where Ps and Pξ are the symmetric positive definite matrix, the derivative of Eq. (59) can be obtained,

V̇6=esTAsTPs+PsTAses+ξTAξTPξ+PξTAξξ+2hsTPses+2hξTPξξ+εξTṖξξ.E60

Since As is a Hurwitz matrix, for a given symmetric positive definite matrix Ss, there is a symmetric positive definite matrix Ps that satisfies the following conditions,

AsTPs+PsTAs=Ss.E61

According to Eqs. (53) and (61), V̇6 can be rewritten as,

V̇6=esTSsesξTSξξ+2hsTPses+2hξTPξξ+εξTṖξξE62

According to the Rayleigh-Ritz inequality, we can obtain,

esTSsesλminSses2,E63
ξTSξξλminSξξ2,E64
hsTPsesχ3+χ4ε+χ5ε2esξ,E65
hξTPξξχ6+χ7ε+χ8ε2ξ2.E66

where χii=016 is positive. According to the inequality relationship shown by Eqs. (63) to (66), V̇6 satisfied the following relationship,

V̇6esξ·λminSsχ3+χ4ε+χ5ε2χ3+χ4ε+χ5ε2λminSξ2χ6+χ7ε+χ8ε2χ2εesξ.E67

The condition that the closed-loop system is asymptotically stable is V̇60, from the above equation, the condition of V̇60 is that the coefficient matrix is positive, that is,

λminSsλminSξ2χ6+χ7ε+χ8ε2χ2εχ3+χ4ε+χ5ε220.E68

Ignoring the influence of high-order terms of Oε2, when the maximum value of the small parameter ε satisfied,

εmax=λb+λb2+4λaλc2λa.E69

V̇60 is valid, where

λa=λminSsχ8+χ42+2χ3χ5,λb=2λminSsχ7λminSsχ22χ3χ4,λc=λminSsλminSξ2λminSsχ6χ32.E70

According to Eq. (67), when the value of ε satisfied 0<εεmax, the overall system is stable.

Advertisement

8. Algorithm simulations

When the Taylor expanding order p=0 is valid, the integral manifold (IM) is equivalent to the singular perturbation (SP). In order to verify the composite control proposed in this chapter, this section compares it with the singular perturbation control and the backstepping (BS) control considering only the rigid-body dynamic model. The above algorithm simulation will be carried out under the SIMULINK module of the MATLAB software, and the ode15s integral will be selected. According to formula (29), in the composite control algorithm based on the integral manifold and observer, the desired trajectory of the end-effector of the moving platform needs to satisfy the fourth derivative continuous, and at the same time to reduce the impact to the system at the beginning and end point of the desired trajectory. The nine-order polynomial shown in Eq. (71) is used to ensure that the velocity, acceleration, and the third and fourth derivatives at the start and end points are zero.

{ px=A0( 125t5/td5420t6/td6+540t7/td7 315t8/td8+70t9/td9 )+px0,py=py0,ϕ=0. E71

where the running time td is 0.06 s, the starting position px0=187.5, py0=187.5/3, and the amplitude A0=30 of the desired trajectory. Take ε2=1/ks, Δ1=1×103, c1=c2=50, Δ2=0.05, Hp=diag404040, Hv=diag400400400, Kf=diag606060. According to Eq. (57), take ε1=0.001. The parameters added and modified in [17] are as follows: the height and thickness of the links are 30 and 5 mm, respectively, the reduction ratio is 20, and the moment of inertia between the motor and the reducer is 284.1kg·mm2.

To describe the control performance of the end-effector, an average error is introduced, and is defined as,

tM=1td0tdCR12+CR22dtrM=1td0tdCR32dtE72

where CR represents the performance index of the three directions of the moving platform, trM and rM are the average error of the translation direction and the rotation direction.

According to Eq. (24), the elastic displacement f3 of the moving platform can be calculated. vi and vm represent the maximum elastic displacement and the average elastic displacement in all directions of the moving platform during operation, vend indicates the elastic displacement at the end point (residual vibration). For the same expected input, the magnitude of the elastic displacement of the moving platform can reflect the vibration suppression effect of the three control algorithms. The elastic displacements in all directions are shown in Figures 4 and 5, which shows that the maximum elastic displacement amplitude in all directions is reduced by more than 28% compared with the backstepping control, and the composite control is reduced by 4.75, 33.42, and 33.52% compared with the singular perturbation. The average elastic displacement for the translational direction decreases from 1.579 and 1.112 mm for backstepping control and singular perturbation to 0.970 mm for composite control. For the rotational direction, 0.0014 and 9.863 × 10−4 rad from backstepping control and singular perturbation drops to 6.872 × 10−4 rad of the composite control. Compared with the above algorithm, the elastic displacement of the composite control decreases by more than 14% in both directions. Compared with the backstepping control, when the composite control and the singular perturbation algorithm are used, the residual vibration is greatly reduced, and both algorithms are close to zero.

Figure 4.

Flexible displacement of moving platform. (a) Displacement of X direction. (b) Displacement of Y direction. (c) Displacement of rotational direction.

Figure 5.

Vibration of the moving platform. (a) Flexible displacement of all directions. (b) Residue vibration of all directions.

The tracking error is the difference between the actual output and the desired output of the end of the moving platform. tr indicates the maximum tracking error in all directions of the moving platform during the whole running process, trm indicates the average tracking error of the translational and rotational directions, tend is the tracking error at the end point. As shown in Figures 6 and 7, compared with the singular perturbation and backstepping control, the composite control based on the integral manifold and the observer has obvious advantages in trajectory tracking. For the maximum tracking error, the X direction decreased by 85.56 and 91.41%, and the Y direction decreased by 57.55 and 90.57%, while the rotation direction decreased by 53.34 and 61.5%, respectively. For the average tracking error, the translation direction decreased by 88.2 and 92.62%, the rotational direction decreased by 37.26 and 49.57%, respectively; in the tracking error of the end point, the X direction decreased by 92.8 and 72.34%, and the Y direction decreased by 89.73 and 83.62%, respectively, while the rotational direction decreased by 85.96 and 70.85%, respectively. For the tracking error at the end point, the singular perturbation method is significantly worse than the backstepping controller in all directions. This is mainly because the singular perturbation algorithm only considers the vibration suppression, and the cost of the vibration suppression is at the cost of sacrificing the trajectory tracking due to the delay of the adjustment. It can be seen from the above analysis that in the aspect of trajectory tracking accuracy, the composite control based on integral manifold and observer has significant advantages.

Figure 6.

Trajectory error of directions. (a) Trajectory error of X direction. (b) Trajectory error of Y direction. (c) Trajectory error of rotational direction.

Figure 7.

Tracking error of the moving platform. (a) Trajectory error of directions. (b) Residue error of directions.

Advertisement

9. Conclusions

  1. Decompose the rigid-flexible coupling dynamic model into fast and slow subsystems based on the integral manifold, and employ the sliding mode control and backstepping control to design the fast and slow subsystem controllers, respectively, and compensate the elastic displacement at the end of the manipulator. A high-gain observer estimates the rate of change of curvature, which in turn enables trajectory tracking control of high-speed PM.

  2. The Lyapunov function is selected to prove the asymptotic stability of the slow subsystem, fast subsystem, high-gain observer, and the overall system. The conditions for selecting the integral manifold and the small parameters of the observer are given.

  3. Apply MATLAB-SIMULINK to establish a comparison simulation to verify the performance of the proposed compound control algorithm. The simulation results show that the composite control algorithm has obvious advantages in vibration suppression and trajectory tracking.

Advertisement

Acknowledgments

This research was supported in part by the Natural Science Foundation of Zhejiang under Grant No. LY18E050019 and the Excellent Talent Cultivation Foundation under Grant No. ZSTUME02B09.

References

  1. 1. Pietsch I, Krefft M, Becker O, et al. How to reach the dynamic limits of parallel robots? An autonomous control approach. IEEE Transactions on Automation Science and Engineering. 2005;2(4):369-380
  2. 2. Dwivedy S, Eberhard P. Dynamic analysis of flexible manipulators, a literature review. Mechanism and Machine Theory. 2006;41(7):749-777
  3. 3. Yongming LI, Tong S, Tieshan LI. Adaptive fuzzy output feedback control for a single-link flexible robot manipulator driven DC motor via backstepping. Nonlinear Analysis: Real World Applications. 2013;14(1):483-494
  4. 4. Vakil M, Fotouhi R, Nikiforuk P. Causal end-effector inversion of a flexible link manipulator. Mechatronics. 2009;19(7):1197-1210
  5. 5. Moallem M, Patel R, Khorasani K. Nonlinear tip-position tracking control of a flexible-link manipulator, theory and experiments. Automatica. 2001;37(11):1825-1834
  6. 6. Lizarraga I, Etxebarria V. Combined PD-H approach to control of flexible link manipulators using only directly measurable variables. Cybernetics and Systems. 2003;34(1):19-31
  7. 7. Zhang Q, Mills J, Cleghorn W, et al. Trajectory tracking and vibration suppression of a 3-PRR parallel manipulator with flexible links. Multibody System Dynamics. 2015;33(1):27-60
  8. 8. El-Badawy A, Mehrez MW, Ali AR. Nonlinear modeling and control of flexible-link manipulators subjected to parametric excitation. Nonlinear Dynamics. 2010;62(4):769-779
  9. 9. Subudhi B, Morris A. Soft computing methods applied to the control of a flexible robot manipulator. Applied Soft Computing. 2009;9(1):149-158
  10. 10. Khorasani K. Adaptive control of flexible-joint robots. IEEE Transactions on Robotics and Automation. 1992;8(2):250-267
  11. 11. Moallem M, Khorasani K, Patel R. An integral manifold approach for tip-position tracking of flexible multi-link manipulators. IEEE Transactions on Robotics and Automation. 1997;13(6):823-837
  12. 12. Salmasi H, Fotouhi R, Nikiforuk P. A manoeuvre control strategy for flexible-joint manipulators with joint dry friction. Robotica. 2010;28(04):621-635
  13. 13. Vakil M, Fotouhi R, Nikiforuk P. Application of the integral manifold concept for the end-effector trajectory tracking of a flexible link manipulator. In: Proceedings of the 26th American Control Conference. New York: IEEE; 2007. pp. 741-747
  14. 14. Vakil M, Fotouhi R, Nikiforuk P. End-effector trajectory tracking of a class of flexible link manipulators. In: Proceedings of 32nd Annual Mechanisms and Robotics Conference. New York: ASME; 2008. pp. 1085-1094
  15. 15. Vakil M, Fotouhi R, Nikiforuk P. End-effector trajectory tracking of a flexible link manipulator using integral manifold concept. International Journal of Systems Science. 2011;42(12):2057-2069
  16. 16. Vakil M, Fotouhi R, Nikiforuk P. Maneuver control of the multilink flexible manipulators. International Journal of Non-Linear Mechanics. 2009;44(8):831-844
  17. 17. Chen Z, Kong M, Ji C, et al. An efficient dynamic modelling approach for high-speed planar parallel manipulator with flexible links. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science. 2015;229(4):663-678
  18. 18. Gorius T, Seifried R, Eberhard P. Approximate end-effector tracking control of flexible multibody systems using singular perturbations. Journal of Computational and Nonlinear Dynamics. 2013;9(1):011017
  19. 19. Ge Z, Lee T, Ge S. Tip tracking control of a single-link flexible robot, a backstepping approach. Dynamics and Control. 1997;7(4):341-360
  20. 20. Lee S, Lee C. Hybrid control scheme for robust tracking of two-link flexible manipulator. Journal of Intelligent and Robotic Systems. 2001;32(4):389-410
  21. 21. Heredia J, Wen Y. A high-gain observer-based PD control for robot manipulator. In: Proceedings of the 2000 American Control Conference. Chicago: IEEE; 2000. pp. 2518-2522
  22. 22. Mosayebi M, Ghayour M, Sadigh M. A nonlinear high gain observer based input-output control of flexible link manipulator. Mechanics Research Communications. 2012;45:34-41

Written By

Zhengsheng Chen

Submitted: May 2nd, 2018 Reviewed: July 18th, 2018 Published: November 5th, 2018