Passivity-Based Control of Rotational and Translational Timoshenko Arms

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.

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

Figure 1 shows a sketch of a translational Timoshenko arm. The equation of motions of the arm and mobile stage are 𝜕𝜌𝐴²𝑦𝜕𝑡²+𝜕²𝑊_𝑏𝜕𝑡²𝜕−𝜅𝐺𝐴²𝑦𝜕𝑥²−𝜕𝜙𝜕𝜕𝑥=0,𝜌𝐼²𝜙𝜕𝑡²𝜕−𝐸𝐼²𝜙𝜕𝑥²−𝜅𝐺𝐴𝜕𝑦𝑀𝜕𝑥−𝜙=0,_𝑝𝜕²𝑊_𝑏𝜕𝑡²()=−𝜅𝐺𝐴𝜕𝑦0,𝑡()𝜕𝑥−𝜙0,𝑡+𝑢. (1) with boundary conditions:𝑚𝜕²𝜕𝑡²𝑦(𝐿,𝑡)+𝑊_𝑏()+𝜅𝐺𝐴𝜕𝑦𝐿,𝑡()𝐼𝜕𝑥−𝜙𝐿,𝑡=0,_𝑐𝜕²𝜕𝑡²𝜙()()𝐿,𝑡+𝐸𝐼𝜕𝜙𝐿,𝑡𝑦()𝜙()𝜕𝑥=0,0,𝑡=0,0,𝑡=0. (2) Applying the Laplace transformation method with respect to 𝑡 to equations (1) and (2) gives𝜌𝐴𝑠²𝑌+𝑊_𝑏𝜕−𝜅𝐺𝐴²𝑌𝜕𝑥²−𝜕Θ𝜕𝑥=0,𝜌𝐼𝑠²𝜕Φ−𝐸𝐼²Φ𝜕𝑥²−𝜅𝐺𝐴𝜕𝑌𝑀𝜕𝑥−Φ=0,_𝑝𝑠²𝑊_𝑏()=−𝜅𝐺𝐴𝜕𝑌0,𝑡()𝜕𝑥−Φ0,𝑡+𝑈,𝑚𝑠²𝑌(𝑙,𝑠)+𝑊_𝑏()+𝜅𝐺𝐴𝜕𝑦𝐿,𝑠()𝐼𝜕𝑥−Φ𝐿,𝑠=0,_𝑐𝑠²Φ()()𝐿,𝑠+𝐸𝐼𝜕Φ𝐿,𝑠𝑌()Φ()𝜕𝑥=0,0,𝑠=0,0,𝑠=0. (3)(4)(5)(6)(7)(8)(9)

Figure 1 

Translational Timoshenko arm.

2.2. Rotational Arm

The rotational Timoshenko arm depicted in Figure 2 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 𝐼_𝑐 and on the other end a hub of inertia 𝐼_ℎ, which in turn is connected to an actuator that supplies a torque 𝑢. We obtain the following equations of motion with the boundary conditions.𝜕𝜌𝐴²𝑦𝜕𝑡²𝜕+𝑥²𝜗𝜕𝑡²𝜕−𝜅𝐺𝐴²𝑦𝜕𝑥²−𝜕𝜙𝜕𝜕𝑥=0,𝜌𝐼²𝜙𝜕𝑡²+𝜕²𝜗𝜕𝑡²𝜕−𝐸𝐼²𝜙𝜕𝑥²−𝜅𝐺𝐴𝜕𝑦𝐼𝜕𝑥−𝜙=0,_ℎ𝜕²𝜗𝜕𝑡²()=𝐸𝐼𝜕𝜙𝑡,0𝑚𝜕𝜕𝑥+𝑢,₂𝜕𝑡²𝑦()()𝐿,𝑡+𝐿𝜗=−𝜅𝐺𝐴𝜕𝑦𝐿,𝑡()𝐼𝜕𝑥−𝜙𝐿,𝑡,_𝑐𝜕²𝜕𝑡²𝜙()()𝐿,𝑡+𝜗=−𝐸𝐼𝜕𝜙𝐿,𝑡𝑦()𝜙()𝜕𝑥,0,𝑡=0,0,𝑡=0. (10)(11)(12)(13)(14)(15)(16)

Figure 2 

Rotational Timoshenko arm.

Taking the Laplace transform of (10)–(16), we get𝜌𝐴𝑠²()𝜕𝑌+𝑥Θ−𝜅𝐺𝐴²𝑌𝜕𝑥²−𝜕Θ𝜕𝑥=0,𝜌𝐼𝑠²()𝜕Φ+Θ−𝐸𝐼²Φ𝜕𝑥²−𝜅𝐺𝐴𝜕𝑌𝐼𝜕𝑥−Φ=0,_ℎ𝑠²()Θ=𝐸𝐼𝜕Φ𝑠,0𝜕𝑥+𝑈,𝑚𝑠²(𝑌())()𝑙,𝑠+𝐿Θ=−𝜅𝐺𝐴𝜕𝑦𝐿,𝑠()𝐼𝜕𝑥−Φ𝐿,𝑠,_𝑐𝑠²(Φ())()𝐿,𝑠+Θ=−𝐸𝐼𝜕Φ𝐿,𝑠𝑌()Φ()𝜕𝑥,0,𝑠=0,0,𝑡=0. (17)(18)(19)(20)(21)(22)(23)

3. Stabilizing Feedback Controls

3.1. Translational Arm

The objective is to derive explicit expressions for feedback control 𝑢for controlling the arm. The approach taken here is to consider the total energy of the arm at any time 𝑡 given by1𝑉=2^𝐿₀𝜌𝐴𝜕𝑦+𝜕𝑡𝜕𝑤_𝑏𝜕𝑡²+𝜌𝐼𝜕𝜙𝜕𝑡²+1𝑑𝑥2𝑀_𝑝𝜕𝑤_𝑏𝜕𝑡²+12^𝐿₀𝐸𝐼𝜕𝜙𝜕𝑥²+𝜅𝐺𝐴𝜙−𝜕𝑦𝜕𝑥²𝑑𝑥, (24) that is positive definite with respect to 𝑦=0, 𝑑𝑦/𝑑𝑡=0.

The Time derivative of (24) is given by𝜕𝑉=𝜕𝑡^𝐿₀𝜌𝐴𝜕𝑦+𝜕𝑡𝜕𝑤_𝑏𝜕𝜕𝑡²𝑦𝜕𝑡²+𝜕²𝑤_𝑏𝜕𝑡²+𝜌𝐼𝜕𝜙𝜕𝜕𝑡²𝜙𝜕𝑡²𝑑𝑥+𝑀_𝑝𝜕𝑤_𝑏𝜕𝜕𝑡²𝑤_𝑏𝜕𝑡²+^𝐿₀𝐸𝐼𝜕𝜙𝜕𝜕𝑥²𝜙𝜕𝑥𝜕𝑡+𝜅𝐺𝐴𝜙−𝜕𝑦𝜕𝑥𝜕𝜙−𝜕𝜕𝑡²𝑦𝜕𝑥𝜕𝑡𝑑𝑥. (25)

By substituting (1) reduces to𝜕𝑉=𝜕𝑡^𝐿₀𝜕𝑦+𝜕𝑡𝜕𝑤_𝑏𝜕𝜕𝑡𝜅𝐺𝐴²𝑦𝜕𝑥²−𝜕𝜙+𝜕𝑥𝜕𝜙𝜕𝜕𝑡𝐸𝐼²𝜙𝜕𝑥²−𝜅𝐺𝐴𝜙−𝜕𝑦+𝜕𝑥𝑑𝑥𝜕𝑤_𝑏𝜙()−()𝜕𝑡𝜅𝐺𝐴0,𝑡𝜕𝑦0,𝑡+𝜕𝑥+𝑢_𝐿0𝐸𝐼𝜕𝜙𝜕𝜕𝑥²𝜙𝜙()−𝜕𝑥𝜕𝑡+𝜅𝐺𝐴𝑥,𝑡𝜕𝑦𝜕𝑥𝜕𝜙−𝜕𝜕𝑡²𝑦𝜕𝑥𝜕𝑡𝑑𝑥. (26)

We wish to derive stabilizing feedback controls 𝑢 such that the resulting feedback-controlled arm is dissipative and stable 𝜕𝑉=𝜕𝑥𝜕𝑤_𝑏𝜕𝑡𝑢. (27)

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:𝑢=−𝛾𝜕𝑤_𝑏𝜕𝑡𝛾>0, (28) which results in𝜕𝑉𝜕𝑡=−𝛾𝜕𝑤_𝑏𝜕𝑡²≤0. (29)

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 control 𝑢 such that the arm’s total energy given by1𝑉=2^𝐿₀𝜌𝐴𝜕𝑦𝜕𝑡+𝑥𝜕𝜃𝜕𝑡²+𝜌𝐼𝜕𝜙+𝜕𝑡𝜕𝜃𝜕𝑡²+1𝑑𝑥2𝐼_ℎ𝜕𝜃𝜕𝑡²+12^𝐿₀𝐸𝐼𝜕𝜙𝜕𝑥²+𝜅𝐺𝐴𝜙−𝜕𝑦𝜕𝑥²𝑑𝑥. (30)

The time rate-of-change of𝑉is given by𝜕𝑉=𝜕𝑡_𝐿0𝜌𝐴𝜕𝑦𝜕𝑡+𝑥𝜕𝜃𝜕𝜕𝑡²𝑦𝜕𝑡²𝜕+𝑥²𝜃𝜕𝑡²+𝜌𝐼𝜕𝜙+𝜕𝑡𝜕𝜃𝜕𝜕𝑡²𝜙𝜕𝑥²+𝜕²𝜃𝜕𝑡²𝑑𝑥+𝐼_ℎ𝜕𝜃𝜕𝜕𝑡²𝜃𝜕𝑡²+^𝐿₀𝐸𝐼𝜕𝜙𝜕𝜕𝑥²𝜙𝜕𝑥𝜕𝑡+𝜅𝐺𝐴𝜙−𝜕𝑦𝜕𝑥𝜕𝜙−𝜕𝜕𝑡²𝑦𝜕𝑥𝜕𝑡𝑑𝑥. (31)

By substituting (10)-(11) reduces to𝜕𝑉=𝜕𝑡^𝐿₀𝜕𝑦𝜕𝑡+𝑥𝜕𝜃𝜕𝜕𝑡𝜅𝐺𝐴²𝑦𝜕𝑥²−𝜕𝜙+𝜕𝑥𝜕𝜙+𝜕𝑡𝜕𝜃𝜕𝜕𝑥𝐸𝐼²𝜙𝜕𝑥²−𝜅𝐺𝐴𝜙−𝜕𝑦+𝜕𝑥𝑑𝑥𝜕𝜃()𝜕𝑡𝐸𝐼𝜕𝜙0,𝑡+𝜕𝑥+𝑢_𝐿0𝐸𝐼𝜕𝜙𝜕𝜕𝑥²𝜙𝜙()−𝜕𝑥𝜕𝑡+𝜅𝐺𝐴𝑥,𝑡𝜕𝑦𝜕𝑥𝜕𝜙−𝜕𝜕𝑡²𝑦𝜕𝑥𝜕𝑡𝑑𝑥. (32)

The equation (32) is then integrated by parts and boundary conditions, (12)–(16) are substituted into the resulting equation. The result is 𝜕𝑉=𝜕𝑡𝜕𝜃𝜕𝑡𝑢. (33)

Taking into consideration of the fact that there is a feedback control law which gives nonpositive 𝜕𝑉/𝜕𝑡 is𝑢=−𝛾𝜕𝜃𝜕𝑡𝛾>0. (34)

Substituting (34) into (33) leads to𝜕𝑉𝜕𝑡=−𝛾𝜕𝜃𝜕𝑡²≤0, (35) 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 from 0to 𝑡, we get the following restatement of energy balance principle:(1)translational arm ^𝑡₀𝜕𝑤_𝑏𝜕𝜏𝑢𝑑𝜏=^𝑡₀𝜕𝑉(𝑡)(0)(0)𝜕𝜏𝑑𝜏=𝑉−𝑉≥−𝑉=−𝛾₀², (36)(2)rotational arm ^𝑡₀𝜕𝜃𝜕𝜏𝑢𝑑𝜏=^𝑡₀𝜕𝑉(𝑡)(0)(0)𝜕𝜏𝑑𝜏=𝑉−𝑉≥−𝑉=−𝛾₀^′2, (37)

where the left-hand side is the supplied energy and right-hand side is the energy at time 𝑡 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.

Figure 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.

Appendix

In the case of the translational Timoshenko arm, the resulting solution to (3), (4), and (5) is given by𝑌(𝑥,𝑠)=Δ𝑘₁Δ𝑒^𝜆1𝑥+Δ𝑘₂Δ𝑒^𝜆2𝑥−Δ𝑘₃Δ𝑒^−𝜆1𝑥−Δ𝑘₄Δ𝑒^−𝜆2𝑥−𝑊_𝑏, (A.1)

where 𝑎₁₁=𝑚𝑠²+𝜌𝐴𝑠²𝜆₁𝑒^𝜆1𝐿,𝑎₁₂=𝑚𝑠²+𝜌𝐴𝑠²𝜆₂𝑒^𝜆2𝐿,𝑎₁₃=𝑚𝑠²−𝜌𝐴𝑠²𝜆₁𝑒^−𝜆1𝐿,𝑎₁₄=𝑚𝑠²−𝜌𝐴𝑠²𝜆₂𝑒^−𝜆2𝐿,𝑎₂₁=𝜆₁−𝜌𝑠²𝜅𝐺𝜆₁𝐼_𝑐𝑠²+𝐸𝐼(1+𝐶𝑠)𝜆₁𝑒^𝜆1𝐿,𝑎₂₂=𝜆₂−𝜌𝑠²𝜅𝐺𝜆₂𝐼_𝑐𝑠²+𝐸𝐼(1+𝐶𝑠)𝜆₂𝑒^𝜆2𝐿,𝑎₂₃=−𝜆₁−𝜌𝑠²𝜅𝐺𝜆₁𝐼_𝑐𝑠²−𝐸𝐼(1+𝐶𝑠)𝜆₁𝑒^−𝜆1𝐿,𝑎₂₄=−𝜆₂−𝜌𝑠²𝜅𝐺𝜆₂𝐼_𝑐𝑠²−𝐸𝐼(1+𝐶𝑠)𝜆₂𝑒^−𝜆2𝐿,𝑎₃₁=𝜆₁−𝜌𝑠²𝜅𝐺𝜆₁,𝑎₃₂=𝜆₂−𝜌𝑠²𝜅𝐺𝜆₂,𝑎₃₃=−𝜆₁−𝜌𝑠²𝜅𝐺𝜆₁,𝑎₃₄=−𝜆₂−𝜌𝑠²𝜅𝐺𝜆₂,𝑎₄₁=𝑀_𝑝𝑠²−𝜌𝐴𝑠²𝜆₁,𝑎₄₂=𝑀_𝑝𝑠²−𝜌𝐴𝑠²𝜆₂,𝑎₄₃=𝑀_𝑝𝑠²+𝜌𝐴𝑠²𝜆₁,𝑎₄₄=𝑀_𝑝𝑠²+𝜌𝐴𝑠²𝜆₂,𝑎Δ=¹¹𝑎₁₂𝑎₁₃𝑎₁₄𝑎₂₁𝑎₂₂𝑎₂₃𝑎₂₄𝑎₃₁𝑎₃₂𝑎₃₃𝑎₃₄𝑎₄₁𝑎₄₂𝑎₄₃𝑎₄₄,Δ𝑘₁=0𝑎₁₂𝑎₁₃𝑎₁₄0𝑎₂₂𝑎₂₃𝑎₂₄0𝑎₃₂𝑎₃₃𝑎₃₄𝑈𝑎₄₂𝑎₄₃𝑎₄₄,Δ𝑘₂=𝑎₁₁0𝑎₁₃𝑎₁₄𝑎₂₁0𝑎₂₃𝑎₂₄𝑎₃₁0𝑎₃₃𝑎₃₄𝑎₄₁𝑈𝑎₄₃𝑎₄₄,Δ𝑘₃=𝑎₁₁𝑎₁₂0𝑎₁₄𝑎₂₁𝑎₂₂0𝑎₂₄𝑎₃₁𝑎₃₂0𝑎₃₄𝑎₄₁𝑎₄₂𝑈𝑎₄₄,Δ𝑘₄=𝑎₁₁𝑎₁₂𝑎₁₃0𝑎₂₁𝑎₂₂𝑎₂₃0𝑎₃₁𝑎₃₂𝑎₃₃0𝑎₄₁𝑎₄₂𝑎₄₃𝑈,1𝜆=±21𝐸()+11+𝐶𝑠𝜅𝐺𝜌𝑠²±𝜌²𝑠⁴𝐸²(1+𝐶𝑠)²−2𝜌²𝑠⁴𝐸()+𝜌1+𝐶𝑠𝜅𝐺₂𝑠⁴𝜅²𝐺²−4𝜌𝐴𝑠²𝐸𝐼(1+𝐶𝑠)^1/21/2,𝜆=±𝜆₁,±𝜆₂. (A.2)(A.3)(A.4) In the case of the rotational Timoshenko arm, the resulting solution to (17), (18), and (19) can be obtained (omitted details for want of space).