Fig. 1.  The investigated industrial  problem.

Fig. 2.  Schematic diagram of the ARX-based vibration control method for flexible  beams.

transfer of a flexible metallic beam from point A to point B and its in- sertion with safety and precision into a slot (see Fig. 1). Yet, the very fast PtP motion leads to a vibration with significant amplitude at the free-end of the beam which does not allow the immediate insertion of the beam into the slot. Thus, the design of a control system that quickly attenuates the amplitude of this vibration, once the robot completes its trajectory and stops in front of the slot (Point B), is necessary.

The general concept of the vibration control method that is devel- oped in this study is schematically presented in Fig. 2. More specifically, the left part of this figure corresponds to the actions of the robot internal controller, while the right part to the design of an outer control system that “supplies” the robot controller with the appropriate corrective ac- tions so as to minimize the residual vibration at the beam’s free-end. The outer control system is designed for each different type of beam based on a single pair of signals that consists of the end-effector velocity and the force obtained by the robot’s embedded sensor on its wrist. These are used for the identification of an ARX model that represents the  dy- namics of the robot-beam system, which is used for the determination of a PID-type controller. The optional use of a synthetic (simulation) environment may be adopted for testing and fine-tuning the designed control system before its final implementation in the industrial robot. All details for each phase of the methodology are described in the next section.

3. The  ARX-based  vibration  control method

The proposed method consists of: (i) The data-based ARX modeling for the robot-beam system dynamics representation, (ii) the outer control system design and (iii) the synthetic (simulation) environment for further (optional) assessment of the developed control system through simula- tions. Once the design of the control system has been completed the controller parameters may be directly used in the actual robot-beam system (see Fig. 2) without further testing or fine tuning.

Fig. 3.  Schematic representation of the ARX modeling.

force at the wrist of the robot leading thus to the suppression of the vibration at the beam’s free-end. Once the robot has completed the de- sired PtP motion and it is in front of the slot, where the beam should be inserted, the outer control loop is activated. As it is shown in Fig. 4 the controller receives as input the error ef  between the force reference

∼

F (set to zero2) and the measured force Fc  as obtained from the ARX

model. By using this closed-loop, the gains of the controller that mini- mize the ef are obtained based on either of the two methods which are subsequently described.

Among various types of controllers, a typical discrete PID-type is presently selected as it achieves adequately fast minimization of the force Fc  (see Section 4.1) providing also simple and fast   implementa-

tion in industrial robot-programming environments. The controller is of

the well-known Backward-Euler form [21]:

3.1. ARX-based modeling of the robot-beam  system

The stochastic modeling of the robot-beam system dynamics is achieved by using a single-input single-output AutoRegressive with eX- ogenous (ARX) data-based model. An ARX(na,nb) model representation has the following form [20, pp.    81–82]:

where KP, KI and KD are the corresponding gains of the Proportional, Integral and Derivative parts to be determined, z designates z-transform and Ts  is the sampling period.

The controller gains in the closed loop of Fig. 4 may be easily deter-

mined based on the PID auto-tuning block in Matlab/Simulink environ- ment [22,23]. By setting the sampling period equal to the one used  for the collection of the force measurements via the robot’s embedded sen-sor, a PID tuning algorithm for a bounded input bounded output closed loop is activated [24] aiming at a good balance between    performance with t = 1, …,N designating the normalized discrete time, x[t], y[t]   the input-output signals, respectively, na, nb the corresponding AR and X model orders, 𝛼i, bi the ith AR and X parameters, w[t] the model residual  (one-step-ahead  prediction  error)  that  is  a  white (uncorrelated)  Gaus- and robustness allowing also the fine-tuning of the controller with re- spect to its response time.Alternatively, the use of a nonlinear optimization algorithm for  the determination  of  the  controller  gains  that  minimize  the  force mean  sian   zero-mean   with   variance 𝜎2 sequence  uncorrelated  with  x[t], N square error ef  (see Fig. 4) may be used: