# Modified Smith Input-Shaper Crane-Controller for Position Control and Sway Reduction
## 4 Experimental and Simulation Results
### Estimation of Parameters of the Laboratory Crane System  
A priori knowledge gained from measurements and the technical data provided by the manufacturer is detailed in Table 1. These measurements were essential for capturing and further evaluating the identified dynamics of the cart's position and the cargo sway. The LS (Last Square) offline method was employed to identify the system dynamics. The parametric model used for this purpose was the ARX (AutoRegressive with eXogenous inputs) model, from which the continuous model parameters were obtained (Equations (9), (12), and (13)). All modes of the crane's dynamics were excited using a PRBS (Pseudo-Random Binary Sequence) signal, shown in Figure 8.  
By use of this signal, the PWM cycle of the crane motor was controlled. Identification of the crane parameters was made partially by model fitting and partially by measurement. Parameters which were obtained by measurement are given in Table 1. The response of the position and the angle of the cargo, together with estimated responses, which were calculated with estimated parameters, are given in Figures 9-11. Because the dynamics of bridge position has astatism of the first order, the minor error in the estimate of the mechanic time constant of the cart will cause deviation in the estimated cart position from the current cart position over time, which can be seen in Figure 9. Thanks to the error controller of the proposed algorithm, this is not a problem because this error controller will inject the addition signal, bringing the cargo to the goal location within the given position tolerance. A good fit of the cargo sway and estimated cargo sway obtained by Equations (9) and (13) is shown in Figure 10, while the speed of the cart and estimated speed of the cart obtained by Equation (12) are shown in Figure 11. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline \(m_{r}\) & 19 kg \\ \(m_{p}\) & 3.8 kg \\ \(l\) & 1.2 m \\ \(R_{a,em}\) & 0.6-1.3 \(\Omega\) \\ \(L_{a,em}\) & 151 mH \\ \(K_{v,em}\) & 0.006 Vs \\ \(D_{cw}/2\) & 0.015 m \\ \(\omega_{1}/\omega_{2}\) & 100 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of the laboratory bridge crane. Figure 8: PRBS control signal of the duty cycle of the DC motor. A deviation occurs in the angle and speed responses from the estimated ones because of the difference between the estimated and actual parameters. The DC motor's armature resistance varies from 0.6 \(\Omega\) to 1.3 \(\Omega\) according to temperature changes while the motor is working. Another reason for the response deviation is the dynamics, which are not incorporated into the mathematical model. Considering this fact, it is concluded that the  
Figure 11: The cart speed response \(v\) and estimated cart speed response \(\overline{v}\) for PRBS signal. Figure 10: The sway angle response \(\Phi\) and estimated sway angle response \(\overline{\Phi}\) for PRBS signal. Figure 9: Position cart response \(x\) and estimated cart response \(\overline{x}\) for PRBS signal. estimated parameters describe the laboratory crane well and were used in the proposed control structure. Additionally, the difference between the dynamics described by the model which was used for the proposed control structure and the identified dynamics of the laboratory crane can be seen in the Bode amplitude and phase diagram of the sway dynamics, which takes into account the cart, the DC motor, and the sway dynamics, shown in Figures 12 and 13, respectively. The diagrams illustrate the differences between the model descriptions and real-time crane dynamics at low frequencies (below 1.5 rad/s) and higher frequencies (above 7 rad/s). The discrepancies observed at lower frequencies are attributed to friction and the cables that transfer energy for cargo transportation, which the cart's motor must pull. At frequencies above 7 rad/s, the influence of sensors on cargo sway measurements becomes evident. Sway was measured using two different methods.