# Investigation on Sensorless Estimating Method and Characteristics of Friction for Ball Screw System
## 4 Friction Characteristics Analysis  
### Position-Dependent Friction Characteristics  
#### 4.1.1 Instantaneous Average Friction  
As mentioned above, the estimated friction oscillates around the IAFV, which also varies with the WP. Here, the IAFVs are assumed to be polynomial functions of the WP:  
\[F_{f}(z)=a_{m}z^{m}+a_{m-1}z^{m-1}+\ldots+a_{1}z+a_{0}. \tag{6}\]  
The coefficients of Equation (6) are determined by using the least squares polynomial fitting algorithm. Table 2 lists the average standard deviations for different orders of polynomials over the whole travel distance of the worktable (WTDOW). The quartic polynomial (\(m=4\)) exhibits the lowest average standard deviation for both the forward and backward feeds of all the polynomial orders and is therefore used to fit the IAFV. In Figure 6a,b, the estimated friction and its IAFV are compared by using dotted lines to show the polynomial fits over the WTDOW for the forward and backward feeds. The estimated friction exhibits different fluctuation patterns over the WTDOW; thus, a single quartic polynomial (as shown by the dotted lines) cannot fit the IAFVs well. To match the fluctuation patterns of the estimated friction, the WTDOW is divided into three intervals: Interval 1, which ranges from 0 to 85 mm; Interval 2, which ranges from 85 to 150 mm; and Interval 3, which ranges from 150 to 410 mm. The friction signals fluctuate smoothly over Intervals 1 and 3. Hence, the quartic polynomial (\(m=4\)) is used to fit the IAFVs for these intervals. The estimated friction exhibits anomalous fluctuations over Interval 2: Here, \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Order** & **1** & **2** & **3** & **4** & **5** & **6** \\ \hline Positive & 19.1 & 18.4 & 17.1 & 16.8 & 17.0 & 17.5 \\ Negative & 31.8 & 30.6 & 30.0 & 29.65 & 31.2 & 29. 6 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Values of standard deviation for different order polynomial fits. Figure 5: Variation in the estimated friction with WP for a feed velocity of 5 mm/min. a quintic polynomial (\(m=5\)) is used to describe the IAFVs. A global continuous curve fitting method in which the piecewise least squares method is combined with global continuity [17] is used for the continuous and smooth sections of the IAFV curve. The results are plotted as solid lines in Figure 6. The piecewise polynomials clearly describe the fluctuation characteristics of the IAFVs better than a single polynomial fit over the entire WTDOW. Figure 7 presents a comparison of the IAFV curves using linear, polynomial, and piecewise polynomial fits. For the linear fit, the absolute values of the IAFV increase as the worktable approaches the servomotor. The dashed-dotted lines and the solid lines fluctuate around the dashed lines. This phenomenon has been observed in studies on estimating friction [11] and on measuring the pretension in a double-nut [12]. The rigidity of the ball drive system affects the engagement of the balls and races and hence their contact forces, which causes the friction to vary. The torsional rigidity of the torque transmission segment of screw (TTSOS) decreases linearly as the distance between the worktable and the servomotor increases. The higher the torsional rigidity of the TTSOS is, the higher the contact force between the balls and the races becomes. As a result, the friction increases as the worktable approaches the servomotor. However, incorrect alignment of the linear guides and the ball screw also results in the unbalanced spread of the inner forces [12]. An experiment was performed where a check rod, 40 mm in diameter and 400 mm in length, was installed by using the top of the lathe tailstock and the chuck. The optoNCDT 2300 sensor, which is an optical system for performing measurements with micrometer accuracy, was installed on the turret to measure the distance between the surface of the check rod and the sensor during the motion of the worktable. The worktable was set to move from 25 mm to 370 mm at a velocity of 400 mm/min.