stands for normal distribution and iid for independent identically    dis-tributed. The signal x[t] corresponds to the velocity Vd of the robot arm and the y[t] to the induced force F at the robot’s end-effector (Fig.   3),both collected during the desired PtP motion, which in this study is the fast motion from point A to B (see Fig.   1).

]𝑇 is a vector including the controller gains and ̂𝐤

The identification of an ARX model involves parameter estima- tion and model order (structure) determination. The parameter  vector1 𝜽 = [𝛼1 𝛼2 … 𝛼na | b0 b1 … bnb]T is estimated using the acquired data via typical  Least  Squares  (LS)  [20,  pp.  203–204],while  model  structure  se- lection referring to the determination of the AR and X orders  is  achieved  by  fitting  increasingly  higher  order  models  to  the data   until   no   further  improvement  is  observed.  Model  order  selection is based on typical cri- teria,  such  as  the  Bayesian  Information Criterion  (BIC)  that  penalizes  model  over-parametrization  and  the  RSS/SSS  (Residual  Sum  of Squares / Signal Sum of Squares) [20, pp. 503–505]. Model validation is then achieved based on formal verification of the residual sequence (white- ness) hypothesis criterion [20, pp. 512–513].

Using the backshift operator 𝔅 (𝔅i u[t] = u[t − i]) the ARX model of Eq. (1) may be written in a transfer function form:

the corresponding estimate that minimizes the mean square error.  This

optimization procedure is achieved in Matlab via the fminsearch.m func- tion that performs the Nelder–Mead simplex algorithm [25] allowing for various termination criteria such as iterations number, tolerance of the estimated parameters (TolX) that minimize the objective function as well as the objective function evaluations number and tolerance (Tol- Fun).

It is worth stressing that in principle the above control system design via either of the two approaches is accomplished once for each different beam and PtP motion based exclusively on the ARX model that repre- sents the dynamics of the investigated system. Then, the outer control system may be directly implemented in the industrial robot as described in Section 4.4 and thus there is no need for interruptions of the produc- tion line or for costly long-term occupation of the robot.  Additionally, it is noted that the stability of the controller is inherently tested during the design via the above approaches as the controller that leads to theminimum, finite, ef  is selected.