# A Human-Inspired Control Strategy for Improving Seamless Robot-To-Human Handovers
## 3 Results and Discussion
### Optimization of the Robotic PID Position Control  
To successfully adopt an appropriate PID controller, the control gains (\(k_{p}\), \(k_{i}\) and \(k_{d}\)) can be estimated using the Ziegler-Nichols tuning method incorporated with the robot transfer function of the proposed dynamic model in Section 2.7. However, it is complicated to estimate each model parameter in the dynamic equations and then the MATLAB system identification was consequently applied. One of the effective parametric models identifying system transfer functions which are recommended by [PERSON] et al. [52] is the Autoregressive Moving Average eXogenous (ARMAX). The ARMAX is one of the most robust techniques and is normally used to study complex dynamical systems in time-series analysis. The Z-transform is reasonable for analysis of practical model identification, which presents the relationship between the eigenvalues and the poles of the ARMAX-based estimated model. The model [53] can be formulated as follows:  
\[ \begin{array}{l}y(t)+a_{1}y(t-1)+\dots+a_{n_{a}}y(t-n_{a})=b_{1}u(t-1)+\dots +b_{n_{b}}u(t-n_{b})+e(t)\\ \hskip 14.226378 pt+c_{1}e(t-1)+\dots+c_{n_{c}}e(t-n_{c})\end{array}  \tag{25}\]  
where the observed input and output are represented as \(u(k)\) and \(y(k)\). The noise signal moderated into the system is \(e(k)\). The Matlab Identification Toolbox was implemented to appropriately estimate the model unknown ARMAX parameters by selecting a set of a polynomial of \(n_{a}\), \(n_{b}\) and \(n_{c}\) to minimize the forecasting errors and to obtain an effective model. A set of preliminary experiments involving a robot position control system has been conducted. The time-series information required in the system identification is made up of (1) robot position target, namely the ARMAX input and (2) the measured robot position, namely the ARMAX output. Presumably, the range of the robot forward translation movement is approximately 1 meter. Additionally, to present the system-parameter estimation technique under consideration adequately, the model validation finally verifies a proposed identified model. The discrete-time ARMAX model was identified and its results are given in Equation (26). The set of minimized polynomial functions is made up of \(n_{a}\) : 2, \(n_{b}\) : 2 and \(n_{c}\) : 2, respectively. \[ \begin{array}{l}A(z)=1+a_{1}z^{-1}+\dots+a_{n_{a}}z^{-n_{a}}=1-1.993\ z^{-1}-0.99 34z^{-2}\\ B(z)=b_{0}+b_{1}z^{-1}+\dots+b_{n_{b}}z^{-n_{b}}=0.0001068z^{-1}-9.555\text{e}- 05z^{-2}\\ C(z)=1+c_{1}z^{-1}+\dots+a_{n_{a}}z^{-n_{c}}=1-0.9241z^{-1}-0.02802z^{-2}\end{array}  \tag{26}\]  
It is to be noted that the MATLAB discrete-time to continuous-time model conversion was then applied to determine the robotic transfer function into a continuous-time transfer function (\(HSR_{TF}\)) as formulated in Equation (27). \[HSR_{TF}=\frac{1.036\text{e}-05z^{2}+0.002733s+0.03113}{s^{2}+1.666s+0.7003} \tag{27}\]  
Model validation was carried out in the final stage of the model identification process to verify the simulated model. Several model validation techniques were then employed and the results express as follows: (1) percentage of best fit (\(R^{2}\)) was approximately 100%, (2) mean squared error (MSE) was 1.069 \(\times\) 10\({}^{-10}\) and finally, (3) final prediction error (FPE) was 1.077 \(\times\) 10\({}^{-10}\), respectively. Using the predicted model indicates that the actual and estimated data have high similarity in terms of the system responses. Then, it is considered acceptable for the HSR representation model. After executing [PERSON] PID tuning in Matlab by providing the numerator and denominator coefficients of the HSR mathematical model in the Laplace domain, it gives gain parameters the critical gain \(k_{c}\) and ultimate period \(\tau_{c}\). Consequently, it can compute the gain values of \(k_{p}\), \(k_{i}\) and \(k_{d}\). Nevertheless, the tuning technique still needs an operator's experience for fine-tuning until the system specification is achieved.