The study on the control of a flexible arm manipulator started as a part of the space robots' research. A space manipulator should be as light as possible to reduce its launching costs. However, the light weight of the manipulator causes a low stiffness. But as the manipulator is needed to handle the objects heavier than itself even, hence the low stiffness creates different problems. For these reasons, we need to consider the flexibility of manipulator's arms and joints while controlling it. Not only the flexible manipulators but also the industrial manipulators face a problem of arm vibrations during high speed motions. Because of these reasons, the research on flexible manipulators is getting more and more important.
To achieve own goals, we developed a flexible manipulator with two flexible links and three joints, named as FLEBOT II. The overview of FLEBOT II is shown in Fig. 1.
Fig. 1 Overview of FLEBOT II.
It is very important to consider what type of models should be constructed for the dynamic analysis and control of a flexible manipulator. For our manipulator, we have used the lumped mass and spring model used in the Holzer's method for the analysis of twisting vibrations of a rotor, and extended to the beam deflection in the Myklestad's method [1]. Fig. 2 shows the concept of lumped mass and spring model, while Fig. 3 shows the result of application of the dynamic modeling we proposed for FLEBOT II.
The modeling method we proposed is realistic and easier than the distributed parameter modeling been used until now. Moreover, this method has clear relations with the rigid manipulators and preserves their modeling too.
Fig. 2 The lumped mass and spring model.
Fig. 3 Dynamic model of FLEBOT II.
In the flexible manipulator's control, the most difficult task is to control the manipulator's motions and the vibrations of its elastic links consecutively. Moreover, it has also been verified that, for a 3D flexible manipulator if the elastic vibrations are feedback with a fixed vibration's control gain the manipulator's control system becomes unstable depending on its configuration.
Keeping this point in mind, we developed the dynamic equations of the flexible manipulator based on the above-mentioned modeling method. Moreover, we developed a vibration suppression control scheme which calculates the configuration dependent variable gains in real time using the inertial matrix, and a control system using an optimal regulator for the calculation of vibration control gains, and applied them to FLEBOT II [2]-[7]. The vibration control gains calculated with the optimal regulator have some discontinuities depending on the manipulator configuration. These configurations correspond to the uncontrollable configurations for elastic vibrations [8]，[9].
The horizontal vibration mode shapes in the arm extended configuration are represented in a simple diagram shown in Fig. 4. Fig. 5 shows a rough sketch of the vibration mode based on Fig. 4. Modal accessibility indices of the first and second vibration modes of the manipulator at various configurations are plotted in Figs. 6 and 7. The configurations in which the value of the index is 0 are modal inaccessible configurations.
 |
 |
| Fig. 4 Horizontal vibration mode shapes. | Fig. 5 Rough sketch of vibration mode. |
|---|---|
 |
 |
| Fig. 6 Inaccessible configurations in first mode ( part of a1 = 0 ). |
Fig. 7 Inaccessible configuration in second mode (part of a2 = 0 ). |
To make a flexible manipulator execute various tasks efficiently, it is necessary to control its end-effector's trajectory. However, due to elastic deflections of the links, the inverse kinematics of the flexible manipulator is different from the usual rigid manipulators, and thus it is very difficult to solve it analytically. Therefore, we have proposed a method for solving the inverse kinematics using the idea of iterative calculation method in learning controls [10], [11]. The experimental results of this method are shown in Fig. 8. The main features of the inverse kinematics solution achieved are the commencement of the joints's motions even earlier than the end-effector's one, and the angular velocity of the joints and the maximum velocity even lesser than those for a rigid manipulator etc.
a) Joint acceleration
b) Position of end-effector
c) Error of end-effector position
d) Deflection of link
Fig. 8 Inverse kinematics solution of flexible manipulator.
(A solid line is in case of inverse kinematics solution we proposed, a dashed line is in case of the rigid body)