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Recognition of Electrical Control System of Flexible Manipulator Based on Transfer Function Estimation Method

Data publikacji: 15 Jul 2022
Tom & Zeszyt: AHEAD OF PRINT
Zakres stron: -
Otrzymano: 12 Jan 2022
Przyjęty: 29 Mar 2022
Informacje o czasopiśmie
License
Format
Czasopismo
eISSN
2444-8656
Pierwsze wydanie
01 Jan 2016
Częstotliwość wydawania
2 razy w roku
Języki
Angielski
Abstract

This article proposes a resonance suppression method for a flexible load servo drive system based on a flexible manipulator's pose transformation. We establish a flexible load servo drive system for the robotic arm based on the continuum vibration theory and the transfer function estimation method. The controller consists of two parts: compensation control and dynamic feedback. The transfer function of the active feedback part is strictly positive and real. At the end of the thesis, the asymptotic stability of the closed-loop system in the neighborhood of the desired position is proved through the linear operator semigroup theory and the LaSalle invariant set principle.

Keywords

MSC 2010

Introduction

Modern electromechanical systems are constantly developing in the direction of high speed and lightweight. Its requirements for system accuracy and stability are getting higher and higher. The assembly robot experiences frequent high-acceleration starts and stops during the system's operation, which can easily cause the broadband resonance of the system. In this way, it is easy to reduce the system's motion accuracy and limit the system's working speed [1]. There is a mutual coupling between mechanical structure parameters and control parameters. We should model the electromechanical system uniformly and optimize the electromechanical design parameters.

Some scholars proposed a design method of electromechanical fusion based on PD control and single-arm and double-arm lever mechanism. It establishes the electromechanical coupling model of the flexible rod mechanism and the transfer function of the system. They used the maximum value of the real part of the dominant pole as the objective function to optimize the design of parameters such as the cross-sectional size of the mechanism, the acting position of the driving force, and the control gain [2]. In this way, the vibration stabilization adjustment time of the entire system is the shortest. Some scholars have proposed a set of electromechanical system design methods based on electromechanical coupling. Some scholars have studied the matching relationship between the electromechanical parameters of the system by taking the transmission mechanism composed of the motor-screw-moving table as an example. At the same time, some criteria for component selection are put forward. Some scholars have proposed the use of the regression experiment optimization method for electromechanical fusion design. The optimization design of the electromechanical parameters of the system is realized through the iterative process of test-analysis-improvement of the physical model. Some scholars have proposed an electromechanical coupling method for analyzing fixed structure and control. A solution to the problem of electromechanical coupling is introduced. The main idea is to develop a communication interface between the structural analysis software and the control system analysis and design software. In this way, data transmission between analysis programs and system coupling analysis is realized. At the same time, it can also realize the calculation and response simulation of some simple control system closed-loop eigenvalues.

The above kinds of literature are generally single-objective models in system modeling or use structure-control separate modeling, structure-control loop iterative optimization, and other methods. These are difficult to reflect the parallel requirements for multiple targets of the system in practical applications. Multi-objective models and their parallel optimization methods are particularly important in mechanical/control fusion design. This paper proposes an optimization model that includes structural parameters and control parameter performance configuration based on a single-arm manipulator with point-to-point motion [3]. Configure the feasible region of the closed-loop control transfer function pole according to the requirements of the control system overshoot σp and adjustment time ts. At the same time, we studied the multi-objective electromechanical fusion modeling and parallel optimization design of the mechanical/control system with the system's moment of inertia and control energy index as the objective function.

Establishment of the dynamic model of flexible manipulator

We simplified the manipulator into a flexible beam, as shown in Figure 1. Assume that its cross-sectional shape is rectangular. In the figure, the beam is taken as a rod whose mass is evenly distributed along the length [4]. We assume that the length of the beam is L. The bending stiffness of the section to the Y axis is EIy. The cross-sectional area of ρ is S = HB. C is the distance from the point of application of torque T to the center of rotation O.

Figure 1

Schematic diagram of flexible beam deformation

According to the Bernoulli-Euler equation of beam lateral vibration, the differential equation of motion of its lateral free vibration is: EIy4zx4+ρS2zt2=0 E{I_y}{{{\partial ^4}z} \over {\partial {x^4}}} + \rho S{{{\partial ^2}z} \over {\partial {t^2}}} = 0

We use the variable separation method to obtain the first three modes of the flexible arm as:

φ1 (x) = − sin(3.9266x / L) + 0.027875 sinh(3.9266x / L)

ϕ2(x)=sin7.0686xL+1.2041×103sinh7.0686xL {\phi _2}(x) = \sin {{7.0686x} \over L} + 1.2041 \times {10^{- 3}}\sinh {{7.0686x} \over L}

ϕ3(x)=sin10.2102xL+5.2032×105sinh10.2102xL {\phi _3}(x) = - \sin {{10.2102x} \over L} + 5.2032 \times {10^{- 5}}\sinh {{10.2102x} \over L}

We set the mode shape function of the rigid displacement as φ0 (x) = x. At this time ω0 =0. According to the Lagrangian method, the dynamic equation of the flexible arm in the movement process is as follows: Mq+Kq=DT Mq + Kq = DT

Among them: M=(m0000mnn) {\rm{M}} = \left({\matrix{{m00} & \cdots & 0 \cr \vdots & \ddots & \vdots \cr 0 & \cdots & {mnn} \cr}} \right) , K=(k0000knn) {\rm{K}} = \left({\matrix{{k00} & \cdots & 0 \cr \vdots & \ddots & \vdots \cr 0 & \cdots & {knn} \cr}} \right) , T = (T0T1 L Tn)T, q = (q0q1qn)T, D=(ϕ0(C)ϕ1(C)ϕn(C))T D = {(\phi _0^{'}(C)\phi _1^{'}(C) \cdots \phi _n^{'}(C))^T} .

Among them: mii=01ρSϕi2dx, kii=01EI(ϕi)2dx mii = \int_0^1 {\rho S\phi _i^2dx},\,\,kii = \int_0^1 {EI{{(\phi _i^{''})}^2}dx} .

T represents torque. C represents the distance from the point of application of the driving torque to the center of arm rotation. The Laplace transform of the total displacement at the end of the flexible arm is ZL(s)=i=0nϕi(L)qi(s) {Z_L}(s) = \sum\limits_{i = 0}^n {{\phi _i}(L){q_i}(s)} . The transfer function of the input torque T and the displacement of the moving end is: G0(s)=ZL(s)T(s)=i=0nϕi(L)ϕi(C)mii(s2+ωi2)=Dns2n+Dn1s2n2+L+D1s2+D0(s2+ωi2)(s2+ω12)L(s2+ωn2) {G_0}(s) = {{{Z_L}(s)} \over {T(s)}} = \sum\limits_{i = 0}^n {{{{\phi _i}(L)\phi _i^{'}(C)} \over {mii({s^2} + \omega _i^2)}} = {{{D_n}{s^{2n}} + {D_{n - 1}}{s^{2n - 2}} + L + {D_1}{s^2} + {D_0}} \over {({s^2} + \omega _i^2)({s^2} + \omega _1^2)L({s^2} + \omega _n^2)}}}

In: Dn=i=0nϕi(L)ϕi(C)mii,Dn1=i=0nϕi(L)ϕi(C)miijiωi2, {D_n} = \sum\limits_{i = 0}^n {{{{\phi _i}(L)\phi _i^{'}(C)} \over {mii}}},{D_{n - 1}} = \sum\limits_{i = 0}^n {{{{\phi _i}(L)\phi _i^{'}(C)} \over {mii}}\sum\limits_{j \ne i} {\omega _i^2}}, Dn2=i=0nϕi(L)ϕi(C)miij,ki,jiωj2ωk2,,D=i=0nϕi(L)ϕi(C)miijinωj2=ϕ0(L)ϕ0(C)m00j0nωj2 {D_{n - 2}} = \sum\limits_{i = 0}^n {{{{\phi _i}(L)\phi _i^{'}(C)} \over {mii}}} \sum\limits_{j,k \ne i,j \ne i} {\omega _j^2\omega _k^2, \ldots,} D = \sum\limits_{i = 0}^n {{{{\phi _i}(L)\phi _i^{'}(C)} \over {mii}}} \prod\limits_{j \ne i}^n {\omega _j^2 = {{{\phi _0}(L)\phi _0^{'}(C)} \over {m\,00}}} \prod\limits_{j \ne 0}^n {\omega _j^2}

The closed-loop transfer function of PD control is: G(s)=(kp+kυs)i=0nDis2ii=0n(s2+ωj2)+(kp+kυs)i=0nDis2i G(s) = {{({k_p} + {k_\upsilon}s)\sum\limits_{i = 0}^n {{D_i}{s^{2i}}}} \over {\prod\limits_{i = 0}^n {({s^2} + \omega _j^2) + ({k_p} + {k_\upsilon}s)} \sum\limits_{i = 0}^n {{D_i}{s^{2i}}}}}

To facilitate the discussion, we take the first-order model of the system to study, then: G(s)=(kp+kυs)(D1s2+D0)s2(s2+ω12)+(kp+kυs)(D1s2+D0) G(s) = {{({k_p} + {k_\upsilon}s)({D_1}{s^2} + {D_0})} \over {{s^2}({s^2} + \omega _1^2) + ({k_p} + {k_\upsilon}s)({D_1}{s^2} + {D_0})}}

The corresponding characteristic equation is: s2(s2+ω12)+(kp+kυs)(D1s2+D0)=0 {s^2}({s^2} + \omega _1^2) + ({k_p} + {k_\upsilon}s)({D_1}{s^2} + {D_0}) = 0

The dominant pole is the farthest from the imaginary axis when the system has two pairs of very complex roots. The system also has the best stability. To make the characteristic equation (16) have two repeated roots, we set: kρ=D0D12 k\rho = {{{D_0}} \over {D_1^2}} kυ=2D1ω12D0D1 k\upsilon = {2 \over {{D_1}}}\sqrt {\omega _1^2 - {{{D_0}} \over {{D_1}}}}

At this time, the root of the characteristic equation is s12=kυD14±kυ2D1216kρD14 s12 = - {{{k_\upsilon}{D_1}} \over 4} \pm \sqrt {{{k_\upsilon ^2D_1^2 - 16k\rho {D_1}} \over 4}} . The real and imaginary parts are: Real(λeig)=kυD14=12ω12D0D1 {\mathop{\rm Re}\nolimits} al({\lambda _{eig}}) = {{{k_\upsilon}{D_1}} \over 4} = - {1 \over 2}\sqrt {\omega _1^2 - {{{D_0}} \over {{D_1}}}} Image(λeig)±16kpD1kυ2D14=±125×D0D1ω12 {\mathop{\rm Im}\nolimits} age({\lambda _{eig}}) \pm \sqrt {{{16{k_p}{D_1} - k_\upsilon ^2{D_1}} \over 4}} = \pm {1 \over 2}\sqrt {5 \times {{{D_0}} \over {{D_1}}} - \omega _1^2}

System performance requirements and pole configuration

The system's poles largely determine the various characteristics and quality indicators of electromechanical system dynamics. Therefore, the pole position can be selected in advance according to the performance requirements of the system. Then design other parameters of the system by optimizing the system [5]. Suppose the comprehensive indicators of the system are as follows: σp represents the output overshoot. ts represents the adjustment time. According to the overshoot σp and the adjustment time ts, the damping coefficient ξ and the circular frequency ω of the system can be approximately calculated: ξ=|lnσp|(lnσp)2+π2ωT=3tsξ \matrix{{\xi = {{\left| {\ln {\sigma _p}} \right|} \over {\sqrt {{{(\ln {\sigma _p})}^2} + {\pi ^2}}}}} \hfill \cr {{\omega _T} = {3 \over {{t_s}\xi}}} \hfill \cr} . Then its dominant pole is defined as follows: s12T=ξωT±jωT1ξ2 s_{12}^T = - \xi {\omega ^T} \pm j{\omega ^T}\sqrt {1 - {\xi ^2}} .

Usually, we can draw the area where the dominant pole of the closed-loop system that meets this requirement is located on the complex plane according to the requirements of the performance index. It is shown in the shaded area in Figure 2. In the figure σ0 = −ξωT is the real part of S12, the damping angle β = arccos ξ. We must make the dominant pole of the control system located in the shaded area in Figure 2. To make the dominant pole S12 located in the shaded area in Figure 2, the following equations can be obtained from equations (6) and (7): 12ω1D0D1ξωT - {1 \over 2}\sqrt {{\omega _1} - {{{D_0}} \over {{D_1}}} \le - \xi {\omega ^T}} ω125D0D1<0 \omega _1^2 - 5{{{D_0}} \over {{D_1}}} < 0 arctg|Image(λeig)Real(λeig)|β arctg\left| {{{{\mathop{\rm Im}\nolimits} age({\lambda _{eig}})} \over {{\mathop{\rm Re}\nolimits} al({\lambda _{eig}})}}} \right| \le \beta

Figure 2

Schematic diagram of the range of the poles of the closed-loop transfer function

To make the system a minimum phase system, we must make Di≥ 0. which is: D0=ϕ0(L)ϕ0(C)m00ω120 {D_0} = {{{\phi _0}(L)\phi _0^{'}(C)} \over {m00}}\omega _1^2 \ge 0 D1=ϕ0(L)ϕ0(C)m00+ϕ1(L)ϕ1(C)m110 {D_1} = {{{\phi _0}(L)\phi _0^{'}(C)} \over {m00}} + {{{\phi _1}(L)\phi _1^{'}(C)} \over {m11}} \ge 0

From equations (4)(12), it can be seen that the real and imaginary parts of the poles of ω1, D0, D1 and the closed-loop control transfer function are related to the parameter and L, H, B, C, Kp, Kυ. And there is a strong mutual coupling between these several parameters [6]. Therefore, to make the system have the best all-around performance, the above parameters must be optimized simultaneously.

Design optimization calculation

To achieve the high-speed and high-precision requirements of the system for the point-to-point reciprocating motion arm, the moment of inertia of the system needs to be small [7]. At the same time, we also need to ensure that the control energy of the control system is small and the moment of inertia of the flexible beam is: W=ρHBL33 W = {{\rho HB{L^3}} \over 3}

The state equation of the PD feedback control system that only considers the first-order mode shape of the beam is: X=A0X+BuY=CX \matrix{{X = {A_0}X + {B_u}} \hfill \cr {Y = CX} \hfill \cr}

Among them: X = [θ θ γ γ]T, A0=(01000000000100k11m110)B=(0ϕ0(C)m000ϕ1(l)φ1(C)m11) {A_0} = \left({\matrix{0 & 1 & 0 & 0 \cr 0 & 0 & 0 & 0 \cr 0 & 0 & 0 & 1 \cr 0 & 0 & {- {{{k_{11}}} \over {{m_{11}}}}} & 0 \cr}} \right)B = \left({\matrix{0 \cr {{{\phi _0^{'}(C)} \over {m00}}} \cr 0 \cr {{{{\phi _1}(l)\varphi _1^{'}(C)} \over {m11}}} \cr}} \right) , C={ϕ0(l)0100ϕ0(l)01} C = \left\{{\matrix{{{\phi _0}(l)} \hfill & 0 \hfill & 1 \hfill & 0 \hfill \cr 0 \hfill & {{\phi _0}(l)} \hfill & 0 \hfill & 1 \hfill \cr}} \right\} .

θ, θ is the angular displacement and angular velocity of the beam, respectively. γ, γ is the elastic displacement and velocity of the beam endpoint, respectively. Y is the total displacement and velocity of the beam endpoint. According to the principle of PD feedback control, the control law is u = − KY = − KCX. Where K = (kpkυ). kp, kυ is calculated by formulas (4) and (5). We substitute the above formula into (14) to get: X=(A0BKC)XA=A0BKC \matrix{{X = ({A_0} - BKC)X} \hfill \cr {A = {A_0} - BKC} \hfill \cr}

X=eAtX0u=KCX=KCAtX0 \matrix{{X = {e^{At}}{X_0}} \hfill \cr {u = KCX = K{C^{At}}{X_0}} \hfill \cr} can be solved by formula (15). Then the energy performance index of the vibration consumption of the control system is: J=120uTudt=X0T(120eATtCTKTKCeAtdt)X0 J = {1 \over 2}\int_0^\infty {{u^T}udt = X_0^T\left({{1 \over 2}\int_0^\infty {{e^{{A^T}t}}{C^T}{K^T}KC{e^{At}}dt}} \right){X_0}} .

To calculate J, you must first specify the initial value X0. For the convenience of calculation, we assume that X0 is randomly distributed on the 2n dimensional unit sphere. And use the average performance function Jμ of Levine and Athans to express J: J=12tr[0eATtCTKTKCeAtdt]=12tr[P] J = {1 \over 2}tr\left[{\int_0^\infty {{e^{{A^T}t}}{C^T}{K^T}KC{e^{At}}dt}} \right] = {1 \over 2}tr[P] . Where [P]=0eATtCTKTKCeAtdt [P] = \int_0^\infty {{e^{{A^T}t}}{C^T}{K^T}KC{e^{At}}dt} . We can obtain PA + ATP + CT KT KC = 0 by solving the following Lyapunov equation. For this reason, this article takes the first-order natural frequency and energy performance index as the objective function [8]. The optimized expression is as follows: FindX=[HBC]TXRn \matrix{{FindX = {{[HBC]}^T}} \hfill & {X \in {R^n}} \hfill \cr} MinF=λ1W/W0+λ2J/J0 MinF = {\lambda _1}W/{W_0} + {\lambda _2}J/{J_0}

From equations (8)(12), the constraints are: g1=2ξωTω12D0D10 {g_1} = 2\xi {\omega ^T} - \sqrt {\omega _1^2 - {{{D_0}} \over {{D_1}}}} \le 0 g2=ω125D0D10 {g_2} = \omega _1^2 - 5{{{D_0}} \over {{D_1}}} \le 0 g3=arctg5×D0ω12D1ω12D1D0β {g_3} = arctg\sqrt {{{5 \times {D_0} - \omega _1^2{D_1}} \over {\omega _1^2{D_1} - {D_0}}}} \le \beta g4=cos(3.9266×CL)>0 {g_4} = - \cos (3.9266 \times {C \over L}) > 0 g4=δυ=3ρgL42EH2δυmax {g_4} = {\delta _\upsilon} = {{3\rho g{L^4}} \over {2E{H^2}}} \le {\delta _{\upsilon \max}}

W0, J0 is the initial value of the system moment of inertia and the initial value of the energy index, respectively. δυ is the deformation of the flexible arm in the vertical direction. δυmax is the maximum allowable deformation of the flexible arm in the vertical direction.

Optimization results and analysis

The electromechanical coupling design of a single flexible arm is transformed into an optimization problem of the objective function equation (17) with constraints (18)~(22). We use a genetic algorithm to optimize this optimization model [9]. The initial value is L = 0.73m, H = 0.019m, B = 0.0032m, C = 0m, δυmax = 5 mm, σp = 15%, ts = 0.15s. The calculation results are shown in Table 1.

Design results before and after optimization

parameter Initial value Final optimization value
Design Parameters L(m) 0.75 0.75
H(m) 0.019 0.0057
B(m) 0.0032 0.0067
C(m) 0 0.3302
Kp 30.8363 250.1
Kv 0.824 3.6327
Performance parameter Inertia (kgm) 0.0206 0.0134
Mass (kg) 0.1231 0.0773
ω(rad/s) 138.36 285.88
F(1/s) 22.02 45.5
σp 15% 12%
ts(s) 0.15 0.09
δv(mm) 9.72 4.9
δvmax(mm) 5 5

It can be seen from Table 1 that the entire system is the minimum phase-stable system after the optimized design. The moment of inertia is significantly reduced while meeting the requirements of the stability performance indicators σp and ts. This is much lower than the initial value, and the first-order natural frequency increases [10]. As the moment of inertia is reduced, the acceleration performance of the system has been greatly improved. Due to the increase of the first-order natural frequency, the system's operating speed and start-stop frequency can be higher.

Conclusion

This paper takes a single-arm manipulator as an example and proposes a mechanical/control multi-object fusion modeling theory based on system performance configuration. The research results show that the system's dynamic characteristics can be improved while meeting the system performance indicators. This can make the control energy smaller and get better overall performance. This is very necessary for the design of high-performance electromechanical systems. This article provides a feasible solution to meet the requirements of high-tech equipment for lightweight, high-speed, and high-precision.

Figure 1

Schematic diagram of flexible beam deformation
Schematic diagram of flexible beam deformation

Figure 2

Schematic diagram of the range of the poles of the closed-loop transfer function
Schematic diagram of the range of the poles of the closed-loop transfer function

Design results before and after optimization

parameter Initial value Final optimization value
Design Parameters L(m) 0.75 0.75
H(m) 0.019 0.0057
B(m) 0.0032 0.0067
C(m) 0 0.3302
Kp 30.8363 250.1
Kv 0.824 3.6327
Performance parameter Inertia (kgm) 0.0206 0.0134
Mass (kg) 0.1231 0.0773
ω(rad/s) 138.36 285.88
F(1/s) 22.02 45.5
σp 15% 12%
ts(s) 0.15 0.09
δv(mm) 9.72 4.9
δvmax(mm) 5 5

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