# Passivity-Based Control of Rotational and Translational Timoshenko Arms

Authors: Minoru Sasaki; Takeshi Ueda; Yshihiro Inoue; Wayne J. Book
Journal: Advances in Acoustics and Vibration
 (2012)

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## Abstract

It is shown that the alternate passivity-based control schemes can be designed which explicitly exploit the passivity properties of the Timoshenko model. This approach has the advantage over the conventional methods in the respect that it allows one to deal directly with the system's partial differential equations without resorting to approximations. Numerical results for the tracking control of a translational and rotational flexible Timoshenko arm are presented and compared. They verify that the proposed control schemes are effective at controlling flexible dynamical systems.


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## Body

## 1. Introduction

In recent years, there has been a great deal of interest in the modeling and control of flexible arms [1–11]. This interest has been motivated by the prospect of fast, light, robot whose links, due to material characteristics, will flex under heavy loads. As a first step towards designing controllers for such robots, researchers have begun studying controllers for simple flexible links. These links, in most cases modeled as Euler-Bernoulli beams because of the small deflections involved, are often analyzed through an eigen-function series expansion of the solution to beam equation. The infinite term expansion is typically truncated after a finite number of terms and the controller designed using one of the various available linear control design techniques. Most practical controllers are designed based on models that are truncations of infinite-order transfer function models. There are several reasons for using the reduced-order models. First, these finite-order models agree very well with experimental results, assuming that an adequate number of modes have been retained. Second, there are no infinite-order transfer function models that coincide with experimental results past the first few modes. Finally, the use of a finite-order model greatly simplifies the controller design procedure. Most existing control design methods for flexible system, such as the Independent Modal Space Control, low-authority high-authority control design, frequency-weighted LQG, and positivity require modal truncation. One disadvantage of this kind of approach is that a controller, based on a truncated model, cannot guarantee stability of the closed loop system. This disadvantage arises because the model does not include the higher frequency terms. A second disadvantage is that the criteria for designing the controller for such a model are usually arbitrary and the design becomes more difficult as the truncated model order is increased.In this paper, it is shown that the alternate passivity-based control schemes can be designed, which explicitly exploit the passivity properties of the Timoshenko model [12, 13]. From standard nonlinear control theory, the passivity theorem tells us that any strictly passive controller with finite gain will stabilize the system in an L2 sense. This allows the design of very simple controllers for the systems. This approach has the advantage over the conventional methods in the respect that it allows one to deal directly with the system's partial differential equations without resorting to approximations. Moreover, it is potentially applicable to the control of nonlinear flexible systems. Numerical results for the tracking control of a translational and rotational flexible Timoshenko arm are presented and compared. They verify that the proposed control schemes are effective at controlling flexible dynamical systems.


## 2. Equation of Motions and Boundary Conditions

### 2.1. Translational Arm

Figure1 shows a sketch of a translational Timoshenko arm. The equation of motions of the arm and mobile stage are (1)ρA(∂2y∂t2+∂2Wb∂t2)-κGA(∂2y∂x2-∂ϕ∂x)=0,ρI∂2ϕ∂t2-EI∂2ϕ∂x2-κGA(∂y∂x-ϕ)=0,Mp∂2Wb∂t2=-κGA(∂y(0,t)∂x-ϕ(0,t))+u.
with boundary conditions:(2)m∂2∂t2(y(L,t)+Wb)+κGA(∂y(L,t)∂x-ϕ(L,t))=0,Ic∂2∂t2ϕ(L,t)+EI∂ϕ(L,t)∂x=0,y(0,t)=0,ϕ(0,t)=0.
Applying the Laplace transformation method with respect to t to equations (1) and (2) gives(3)ρAs2(Y+Wb)-κGA(∂2Y∂x2-∂Θ∂x)=0,(4)ρIs2Φ-EI∂2Φ∂x2-κGA(∂Y∂x-Φ)=0,(5)Mps2Wb=-κGA(∂Y(0,t)∂x-Φ(0,t))+U,(6)ms2(Y(l,s)+Wb)+κGA(∂y(L,s)∂x-Φ(L,s))=0,(7)Ics2Φ(L,s)+EI∂Φ(L,s)∂x=0,(8)Y(0,s)=0,(9)Φ(0,s)=0.Figure 1
Translational Timoshenko arm.


### 2.2. Rotational Arm

The rotational Timoshenko arm depicted in Figure2 of total length L, area moment of inertia I, cross-sectional area A, density ρ, Young’s modulus E, shear modulus G, and shear coefficient κ, is attached at one end to a payload of mass m and inertia Ic and on the other end a hub of inertia Ih, which in turn is connected to an actuator that supplies a torque u. We obtain the following equations of motion with the boundary conditions.(10)ρA(∂2y∂t2+x∂2ϑ∂t2)-κGA(∂2y∂x2-∂ϕ∂x)=0,(11)ρI(∂2ϕ∂t2+∂2ϑ∂t2)-EI∂2ϕ∂x2-κGA(∂y∂x-ϕ)=0,(12)Ih∂2ϑ∂t2=EI∂ϕ(t,0)∂x+u,(13)m∂2∂t2(y(L,t)+Lϑ)=-κGA(∂y(L,t)∂x-ϕ(L,t)),(14)Ic∂2∂t2(ϕ(L,t)+ϑ)=-EI∂ϕ(L,t)∂x,(15)y(0,t)=0,(16)ϕ(0,t)=0.Figure 2
Rotational Timoshenko arm.Taking  the Laplace transform of (10)–(16), we get(17)ρAs2(Y+xΘ)-κGA(∂2Y∂x2-∂Θ∂x)=0,(18)ρIs2(Φ+Θ)-EI∂2Φ∂x2-κGA(∂Y∂x-Φ)=0,(19)Ihs2Θ=EI∂Φ(s,0)∂x+U,(20)ms2(Y(l,s)+LΘ)=-κGA(∂y(L,s)∂x-Φ(L,s)),(21)Ics2(Φ(L,s)+Θ)=-EI∂Φ(L,s)∂x,(22)Y(0,s)=0,(23)Φ(0,t)=0.


## 2.1. Translational Arm

Figure1 shows a sketch of a translational Timoshenko arm. The equation of motions of the arm and mobile stage are (1)ρA(∂2y∂t2+∂2Wb∂t2)-κGA(∂2y∂x2-∂ϕ∂x)=0,ρI∂2ϕ∂t2-EI∂2ϕ∂x2-κGA(∂y∂x-ϕ)=0,Mp∂2Wb∂t2=-κGA(∂y(0,t)∂x-ϕ(0,t))+u.
with boundary conditions:(2)m∂2∂t2(y(L,t)+Wb)+κGA(∂y(L,t)∂x-ϕ(L,t))=0,Ic∂2∂t2ϕ(L,t)+EI∂ϕ(L,t)∂x=0,y(0,t)=0,ϕ(0,t)=0.
Applying the Laplace transformation method with respect to t to equations (1) and (2) gives(3)ρAs2(Y+Wb)-κGA(∂2Y∂x2-∂Θ∂x)=0,(4)ρIs2Φ-EI∂2Φ∂x2-κGA(∂Y∂x-Φ)=0,(5)Mps2Wb=-κGA(∂Y(0,t)∂x-Φ(0,t))+U,(6)ms2(Y(l,s)+Wb)+κGA(∂y(L,s)∂x-Φ(L,s))=0,(7)Ics2Φ(L,s)+EI∂Φ(L,s)∂x=0,(8)Y(0,s)=0,(9)Φ(0,s)=0.Figure 1
Translational Timoshenko arm.


## 2.2. Rotational Arm

The rotational Timoshenko arm depicted in Figure2 of total length L, area moment of inertia I, cross-sectional area A, density ρ, Young’s modulus E, shear modulus G, and shear coefficient κ, is attached at one end to a payload of mass m and inertia Ic and on the other end a hub of inertia Ih, which in turn is connected to an actuator that supplies a torque u. We obtain the following equations of motion with the boundary conditions.(10)ρA(∂2y∂t2+x∂2ϑ∂t2)-κGA(∂2y∂x2-∂ϕ∂x)=0,(11)ρI(∂2ϕ∂t2+∂2ϑ∂t2)-EI∂2ϕ∂x2-κGA(∂y∂x-ϕ)=0,(12)Ih∂2ϑ∂t2=EI∂ϕ(t,0)∂x+u,(13)m∂2∂t2(y(L,t)+Lϑ)=-κGA(∂y(L,t)∂x-ϕ(L,t)),(14)Ic∂2∂t2(ϕ(L,t)+ϑ)=-EI∂ϕ(L,t)∂x,(15)y(0,t)=0,(16)ϕ(0,t)=0.Figure 2
Rotational Timoshenko arm.Taking  the Laplace transform of (10)–(16), we get(17)ρAs2(Y+xΘ)-κGA(∂2Y∂x2-∂Θ∂x)=0,(18)ρIs2(Φ+Θ)-EI∂2Φ∂x2-κGA(∂Y∂x-Φ)=0,(19)Ihs2Θ=EI∂Φ(s,0)∂x+U,(20)ms2(Y(l,s)+LΘ)=-κGA(∂y(L,s)∂x-Φ(L,s)),(21)Ics2(Φ(L,s)+Θ)=-EI∂Φ(L,s)∂x,(22)Y(0,s)=0,(23)Φ(0,t)=0.


## 3. Stabilizing Feedback Controls

### 3.1. Translational Arm

The objective is to derive explicit expressions for feedback controlufor controlling the arm. The approach taken here is to consider the total energy of the arm at any time t given by(24)V=12∫0L[ρA{∂y∂t+∂wb∂t}2+ρI{∂ϕ∂t}2]dx+12Mp{∂wb∂t}2+12∫0L[EI{∂ϕ∂x}2+κGA{ϕ-∂y∂x}2]dx,
that is positive definite with respect to y=0, dy/dt=0.The Time derivative of (24) is given by(25)∂V∂t=∫0L[ρA{∂y∂t+∂wb∂t}{∂2y∂t2+∂2wb∂t2}+ρI{∂ϕ∂t}{∂2ϕ∂t2}]dx+Mp{∂wb∂t}{∂2wb∂t2}+∫0L[EI{∂ϕ∂x}{∂2ϕ∂x∂t}+κGA{ϕ-∂y∂x}{∂ϕ∂t-∂2y∂x∂t}]dx.By substituting (1) reduces to(26)∂V∂t=∫0L[{∂y∂t+∂wb∂t}{κGA(∂2y∂x2-∂ϕ∂x)}+{∂ϕ∂t}{EI∂2ϕ∂x2-κGA(ϕ-∂y∂x)}]dx+∂wb∂t{κGA(ϕ(0,t)-∂y(0,t)∂x)+u}+∫0L[EI{∂ϕ∂x}{∂2ϕ∂x∂t}+κGA{ϕ(x,t)-∂y∂x}{∂ϕ∂t-∂2y∂x∂t}]dx.We wish to derive stabilizing feedback controlsu such that the resulting feedback-controlled arm is dissipative and stable (27)∂V∂x=∂wb∂tu.Since our goal is to design a stabilizing control input, the most natural choice is to make the time derivative of the Lyapunov function negative in such a way. The following stabilizing feedback control is chosen:(28)u=-γ∂wb∂t(γ>0),
which results in(29)∂V∂t=-γ(∂wb∂t)2≤0.This means that a control input should be chosen with as large as possible and should generate a base velocity that ensures asymptotic stability of the tip bending motion of the arm.


### 3.2. Rotational Arm

The problem is to find feedback controlu such that the arm’s total energy given by(30)V=12∫0L[ρA{∂y∂t+x∂θ∂t}2+ρI{∂ϕ∂t+∂θ∂t}2]dx+12Ih{∂θ∂t}2+12∫0L[EI{∂ϕ∂x}2+κGA{ϕ-∂y∂x}2]dx.The time rate-of-change ofVis given by(31)∂V∂t=∫0L[ρA{∂y∂t+x∂θ∂t}{∂2y∂t2+x∂2θ∂t2}+ρI{∂ϕ∂t+∂θ∂t}{∂2ϕ∂x2+∂2θ∂t2}]dx+Ih{∂θ∂t}{∂2θ∂t2}+∫0L[EI{∂ϕ∂x}{∂2ϕ∂x∂t}+κGA{ϕ-∂y∂x}{∂ϕ∂t-∂2y∂x∂t}]dx.By substituting (10)-(11) reduces to(32)∂V∂t=∫0L[{∂y∂t+x∂θ∂t}{κGA(∂2y∂x2-∂ϕ∂x)}+{∂ϕ∂t+∂θ∂x}{EI∂2ϕ∂x2-κGA(ϕ-∂y∂x)}]dx+∂θ∂t{EI∂ϕ(0,t)∂x+u}+∫0L[EI{∂ϕ∂x}{∂2ϕ∂x∂t}+κGA{ϕ(x,t)-∂y∂x}{∂ϕ∂t-∂2y∂x∂t}]dx.The equation (32) is then integrated by parts and boundary conditions, (12)–(16) are substituted into the resulting equation. The result is (33)∂V∂t=∂θ∂tu.Taking into consideration of the fact that there is a  feedback control law  which gives nonpositive∂V/∂t is(34)u=-γ∂θ∂t(γ>0).Substituting (34) into (33) leads to(35)∂V∂t=-γ(∂θ∂t)2≤0,
which is negative semidefinite. This implies that the closed-loop system is energy dissipative and, hence, stable.


## 3.1. Translational Arm

The objective is to derive explicit expressions for feedback controlufor controlling the arm. The approach taken here is to consider the total energy of the arm at any time t given by(24)V=12∫0L[ρA{∂y∂t+∂wb∂t}2+ρI{∂ϕ∂t}2]dx+12Mp{∂wb∂t}2+12∫0L[EI{∂ϕ∂x}2+κGA{ϕ-∂y∂x}2]dx,
that is positive definite with respect to y=0, dy/dt=0.The Time derivative of (24) is given by(25)∂V∂t=∫0L[ρA{∂y∂t+∂wb∂t}{∂2y∂t2+∂2wb∂t2}+ρI{∂ϕ∂t}{∂2ϕ∂t2}]dx+Mp{∂wb∂t}{∂2wb∂t2}+∫0L[EI{∂ϕ∂x}{∂2ϕ∂x∂t}+κGA{ϕ-∂y∂x}{∂ϕ∂t-∂2y∂x∂t}]dx.By substituting (1) reduces to(26)∂V∂t=∫0L[{∂y∂t+∂wb∂t}{κGA(∂2y∂x2-∂ϕ∂x)}+{∂ϕ∂t}{EI∂2ϕ∂x2-κGA(ϕ-∂y∂x)}]dx+∂wb∂t{κGA(ϕ(0,t)-∂y(0,t)∂x)+u}+∫0L[EI{∂ϕ∂x}{∂2ϕ∂x∂t}+κGA{ϕ(x,t)-∂y∂x}{∂ϕ∂t-∂2y∂x∂t}]dx.We wish to derive stabilizing feedback controlsu such that the resulting feedback-controlled arm is dissipative and stable (27)∂V∂x=∂wb∂tu.Since our goal is to design a stabilizing control input, the most natural choice is to make the time derivative of the Lyapunov function negative in such a way. The following stabilizing feedback control is chosen:(28)u=-γ∂wb∂t(γ>0),
which results in(29)∂V∂t=-γ(∂wb∂t)2≤0.This means that a control input should be chosen with as large as possible and should generate a base velocity that ensures asymptotic stability of the tip bending motion of the arm.


## 3.2. Rotational Arm

The problem is to find feedback controlu such that the arm’s total energy given by(30)V=12∫0L[ρA{∂y∂t+x∂θ∂t}2+ρI{∂ϕ∂t+∂θ∂t}2]dx+12Ih{∂θ∂t}2+12∫0L[EI{∂ϕ∂x}2+κGA{ϕ-∂y∂x}2]dx.The time rate-of-change ofVis given by(31)∂V∂t=∫0L[ρA{∂y∂t+x∂θ∂t}{∂2y∂t2+x∂2θ∂t2}+ρI{∂ϕ∂t+∂θ∂t}{∂2ϕ∂x2+∂2θ∂t2}]dx+Ih{∂θ∂t}{∂2θ∂t2}+∫0L[EI{∂ϕ∂x}{∂2ϕ∂x∂t}+κGA{ϕ-∂y∂x}{∂ϕ∂t-∂2y∂x∂t}]dx.By substituting (10)-(11) reduces to(32)∂V∂t=∫0L[{∂y∂t+x∂θ∂t}{κGA(∂2y∂x2-∂ϕ∂x)}+{∂ϕ∂t+∂θ∂x}{EI∂2ϕ∂x2-κGA(ϕ-∂y∂x)}]dx+∂θ∂t{EI∂ϕ(0,t)∂x+u}+∫0L[EI{∂ϕ∂x}{∂2ϕ∂x∂t}+κGA{ϕ(x,t)-∂y∂x}{∂ϕ∂t-∂2y∂x∂t}]dx.The equation (32) is then integrated by parts and boundary conditions, (12)–(16) are substituted into the resulting equation. The result is (33)∂V∂t=∂θ∂tu.Taking into consideration of the fact that there is a  feedback control law  which gives nonpositive∂V/∂t is(34)u=-γ∂θ∂t(γ>0).Substituting (34) into (33) leads to(35)∂V∂t=-γ(∂θ∂t)2≤0,
which is negative semidefinite. This implies that the closed-loop system is energy dissipative and, hence, stable.


## 4. Passivity of Timoshenko Arm

Taking the time derivative of the total energy, and then integrating from0to t, we get the following restatement of energy balance principle:(1)
translational arm(36)∫0t∂wb∂τudτ=∫0t∂V∂τdτ=V(t)-V(0)≥-V(0)=-γ02,(2)
rotational arm(37)∫0t∂θ∂τudτ=∫0t∂V∂τdτ=V(t)-V(0)≥-V(0)=-γ0′2,where the left-hand side is the supplied energy and right-hand side is the energy at timet minus the initial energy. Passivity of the operator u relative to the function V follows from the equation above and positivity of the total energy.


## 5. Simulation Results

Numerical inversion of Laplace transform is used to obtain the results in the time domain. The computation of the inverse Laplace transform is based on the paper of Hosono [14]. Numerical solutions are obtained by the Bromwich integrating the system corresponding to (A.1) and so forth. In the computer simulation study, we consider a typical arm whose parameters are given in the Table 1. Figures 3–6 show the step response of the arm-tip displacement with feedback gains variations. Figures 3 and 5 show the results of the Euler arm. Figures 4 and 6 show the results of the Timoshenko arm. There are few differences with each other in the case of this system parameters. It can be seen that the arm-tip displacement toward the desired position. With increasing the feedback gains, the settling times are longer and the residual vibrations are more slightly damped. In this case, shear deformations are small and negligible, so that the shear angle is almost zero and the rotary inertia is small and negligible, the so-called Euler-Bernoulli beam equations result. If the beam is long and skinny (aspect ratio, say greater than 10), the Euler-Bernoulli assumptions are appropriate.Table 1
System parameters.
ParameterNomenclatureValuesUnitLength of the armL1.2mArm cross-sectional areaA2.5×10-4m2Volume mass densityρ2.7667Kg/m3Young’s modulusE6.8944×1010PaInternal structural dampingC0Kg/secPayload moment of inertiaIc5×10-4Kg·m2Payload massm0.2KgShear correction factorκ5/6–Shear modulusG2.6517×1010PaSlider massMp0.5KgHub moment of inertiaIh0.5Kg·m2Figure 3
Step response of the translational Euler arm.Figure 4
Step response of the translational Timoshenko arm.Figure 5
Step response of the rotational Euler arm.Figure 6
Step response of the rotational Timoshenko arm.


## 6. Conclusion

In this paper, the passivity-based control of rotational and translational Timoshenko arms is studied. This approach has the advantage over the conventional methods in the respect that it allows one to deal directly with the system's partial differential equations without resorting to approximations. Numerical results for the control of a translational and rotational flexible Timoshenko arm are presented and compared in graphical plots. They verify that the proposed control schemes are effective at controlling flexible dynamical systems and vibration control of mechanical systems.



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