# Investigation on Sensorless Estimating Method and Characteristics of Friction for Ball Screw System
## 4 Friction Characteristics Analysis
### Velocity-Dependent Friction Characteristics  
#### 4.2.1 Effect of the Feed Velocity on the IAFVs  
In the previous section, we discussed that for a constant feed velocity, the IAFV can be expressed in terms of a piecewise polynomial fitting function of the WP. To enhance the expression accuracy of the relationship between the IAFV and the WP, three idling tests over the WTDOW were conducted to estimate the friction at one feed velocity using three groups with forward feed friction and three groups with backward feed friction. Then, a piecewise polynomial fitting algorithm was applied to the estimated friction to determine three groups of coefficients for the forward and backward feeds. The mean parameter curve method [20] was applied to the three groups of coefficients for the piecewise polynomial fits to construct new piecewise polynomials of the coefficient means (PPOCM) to describe the relationship between the mean value of instantaneous average friction (MVOIAF) and the WP. Tests were conducted for 18 specific feed velocities, ranging from 5 to 560 mm/min, to determine the relationship of the friction to the feed velocity and the WP. In Figure 11, the IAFV curves are shown for the three tests and the PPOCM curves for a feed velocity of 5 mm/min. The IAFV curves for the three tests do not coincide, except for the case of the backward feed in the interval [25; 41]. In Figure 12, the variations in the MVOIAFs with the WP are shown for eight feed velocities. For the forward feed, the MVOIAF curve exhibits its maximum at a feed velocity of 5 mm/min and a secondary maximum at a feed velocity of 20 mm/min over the WTDOW. However, the MVOIAF curves exhibit minimum values for feed velocities of 560, 400, and 200 mm/min. For the backward feed, the highest MVOIAF curve is observed for a feed velocity of 5 mm/min, except over the interval [4.9, 9.5] where there is a higher MVOIAF for a feed velocity of 20 mm/min. The MVOIAF curves exhibit minima at the following locations: over the interval [1, 9.7], at a feed velocity of 200 mm/min, over the interval [9.7, 10.05], at a feed velocity of 400 mm/min, and over the interval [10.05, 41], at a feed velocity of 560 mm/min. Therefore, the effect of feed velocity on the estimated friction varies with the WP. In addition, the peaks of the IAFV curves, which result from the pitch errors, decreases as the feed velocity increases. To describe the effect of the feed velocity on the MVOIAFs, the average coefficients \(\overline{a}_{ijk}\) of the PPOCMs are assumed to be a quintic polynomial function of the feed velocity \(v\):  
\[\overline{a}_{ijk}=b_{ijk}v^{5}+b_{ijk}v^{4}+b_{ijk3}v^{3}+b_{ijk2}v^{2}+b_{ ijkl}v+b_{ijk0} \tag{8}\]  
Figure 10: Variations in the friction frequency spectrum with the WP at a backward feed of 5 mm/min. (**a**) Friction frequency spectrum of all the segments, (**b**) Friction frequency spectrum of segment No. 9; (**c**) Friction frequency spectrum of segment No. 1; (**d**) Amplitude of all segments for significant frequency components. where \(i\) denotes the feed direction, which is denoted by F or B for the forward or backward feeds, respectively; \(j\) denotes the interval \(j\), where \(j=1,2\), or \(3\);; and \(k\) denotes the \(k\)th coefficient of Equation (8), where \(k=0,1,2\), or \(4\). MATLAB software was used to apply the method of least squares to the coefficients of the MVOIAF curves for the abovementioned 18 experiments to determine the coefficients of Equation (8) for the three intervals. Figure 13 is a 3D graph that shows the variations in the MVOIAF with the WP and the feed velocity. Based on the variations in the MVOIAF with the WP in Figure 12, six specific positions, denoted by a-a to f-f, were chosen to study the variation in the friction with the feed velocity. These positions are listed in Table 3.  
Figure 11: IAFV curves and associated MVOIAF for three tests at a feed velocity of 5 mm/min. (**a**) Position direction; (**b**) Negative direction.