End Point Force Control of a Flexible Timoshenko Arm

This paper discusses a force 
control problem for a flexible Timoshenko arm. The effect of shear deformation 
and the effect of rotary inertia are considered in Timoshenko beam theory. Most 
of the research about force control of the flexible arm is based on Euler 
Bernoulli beam theory. There are a few researches about force control of the 
flexible arm using Timoshenko beam theory. The aim of the force control is to 
control the contact force at the contact point. To solve this problem, we 
propose a simple controller using Timoshenko beam theory. Finally, we describe 
simulation results using a numerical inversion of Laplace transform carried out 
to investigate the validity of the proposed controller for the force control 
problem. The results of the time response show the transverse displacement, the 
angle of deflection, the slider position, the rotational angle and the contact 
force toward the desired their values.


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 bend 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. However, under author's knowledge, there has not yet been a study of force control of a flexible Timoshenko arm based on the infinite dimensional model. The effect of shear deformation and the effect of rotary inertia are considered in Timoshenko beam theory and thus the Timoshenko beam theory is modified for a non-slender beam and high-frequency response. This means that the Timoshenko beam theory has a wider application area than the Euler-Bernoulli beam theory. So we discuss the force control problem for the flexible Timoshenko arm. Figure 1 shows a constrained one-link flexible Timoshenko arm. One-end of the arm is clamped to control actuators consisting of the rotational motor and the translational slider, and the other end has a concentrated tip mass m . The tip mass makes contact with the surface of an object. The flexible arm translates and rotates in the horizontal plane (the XY plane in Figure 1) by control actuators; it is not affected by the acceleration of gravity. The flexible arm, with length l , mass per unit length  , mass moment of inertia I  , cross sectional area A , area moment of inertia I , Young's modulus E , shear modulus G , and shear coefficient  , satisfies the Timoshenko beam hypothesis.

Equations of Motion and Boundary Conditions
In Figure 1, XY is an absolute coordinate system and xy is a local coordinate system, whose origin is fixed at the rotor of the rotational motor. In addition, xy translates with the slider and rotates with the rotor of the motor.
and ( ) s t be the inertia moment of the motor, the torque generated by the motor at time t , the rotational angle of the motor, the mass of the slider, the force generated by the slider, and the translational position of the slider, respectively. Further, let ( , ) w x t and ( , ) x t  be the transverse displacement of the flexible arm at time t and spatial point x , and the rotation of the cross section due to bending deformation, respectively.
Since the tip mass makes contact with the surface of the object, we obtain the following geometric constraint: This constraint means that the Y-axis position of the tip mass is constrained on the surface of the object. The kinetic energy K E and the potential energy P E of the overall system are given by the following: , a dot denotes the time derivative, and a prime denotes the partial derivative with respect to x . Here the virtual work is given by δ Under the above preparation, we can obtain the following equations of motion by applying Hamilton's principle and Lagrange's multiplier, and using the procedure described in [13]: with the algebraic relation is Lagrange's multiplier and is equivalent to the contact force, i.e., the shear force at the tip of the flexible arm, which arises in the direction along the normal vector of the constraint surface.
The aim of this paper is to control the contact force at the tip of the flexible arm. In other words, the control objective is to construct a boundary controller satisfying ( ) , ( , ) 0, ( , ) 0, where d  is the constant desired contact force. At the desired equilibrium point  (1)-(6), we obtain: For this purpose, we propose the following controller: where feedback gain i k  , 1, ,8 i   , is a positive constant. In these controllers, (10) is the controller for the slider and (11) is the controller for the motor. In (10) Here, note that if we use the strain gauge, rotary encoder, and speed reference type servo amplifier of the motor and the slider, we can easily implement the controller.

Laplace Transform of the Equation of Motion
Taking the Laplace transform of (1)-(11), we can get where the constants      

Numerical 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 T. Hosono [12]. In the computer simulation study, we consider a typical arm whose parameters are given in the      It can be seen that the transverse displacement, the angle of deflection, the slider position, the rotational angle and the contact force toward the desired their values. With the adequate feedback gains there are no residual vibrations and no over shoot.

Conclusion
A contact-force control problem with regards to a constrained one-link flexible Timoshenko arm was described. The equations of motion and the boundary conditions of the overall system were derived. To solve the contact force control problem of such a system, we have proposed a simple controller, which is easy to implement. Several numerical simulations using a numerical inversion of Laplace transform were carried out. The simulation results showed the validity of the proposed controller for the contact-force control problem with no residual vibrations and no overshoot.