# Investigation on Sensorless Estimating Method and Characteristics of Friction for Ball Screw System
## 3 Sensorless Friction Estimation
### Algorithm of Sensorless Friction Estimation  
In the operation of a feed drive servomechanism, the friction and the screw pretension vary instantaneously [10; 11; 12]. As a result, the velocity of the worktable also varies instantaneously [15; 16]. If the velocity and acceleration of the worktable are experimentally measured, the friction \(F_{f}\) can be determined by solving Equation (5). However, only data for the WP versus time are obtained in Section 2. The experimental data are sampled as given below.  
The data knots for the WP are  
\[(0,z_{0}),(t_{z1},z_{1}),\ \ldots\,,(t_{zi},z_{i}),\ \ldots\,,(t_{zn},z_{n}).\]  
The data knots for the SMTC are  
\[(0,i_{0}),(t_{11},i_{1}),\ \ldots\,,(t_{li},i_{i}),\ \ldots\,,(t_{ln},i_{n}).\]  
In the experimental data, \(t_{zi}\
eq t_{li}\), \(t_{zi}-t_{z(i-1)}\), and \(t_{li}-t_{l(i-1)}\) range from 70 to 75 ms. The experimental data are interpolated as time-continuous functions of the WP and the SMTC using a cubic spline algorithm. The velocity and acceleration of the worktable are then determined by finding the first and second derivatives of the time-continuous function of the WP, respectively. For a feed servomechanism, the friction force and the screw pretension vary periodically with the WP, and their respective spectra include multiple frequency components of the positioning frequency [10; 11]. To overcome the effect of the worktable velocity on the estimation of the friction force, the velocity, and acceleration for the worktable, the SMTC are resampled with one mth of the screw lead as the sampling period. The experimental data are processed using MATLAB, as described below.  
1. Cubic spline functions are used to interpolate between all of pairs of knots, \((t_{z(i-1)},z_{i-1})\) and \((t_{zi},z_{i})\), and between \((t_{l(i-1)},t_{l(i-1)})\) and \((t_{li},t_{li})\), to yield time-continuous functions for the WP and the SMTC, \(z(t)\) and \(i(t)\), respectively.
2. The first and second derivatives of the time spline function of the WP are calculated to yield the velocity and acceleration functions of the worktable, \(z^{\prime}\,_{t}(t)\) and \(z^{\prime\prime}\,_{t}(t)\), respectively.
3. All of the knots, \((z_{i-1},t_{z(i-1)})\) and \((z_{i},t_{zi})\), are interpolated to yield the position-dependent run time of the worktable, \(t(z)\).
4. The WP is sampled with one mth of the screw lead as the sampling period to yield two arrays of position sampling points, \(Z=[0,p/m,2p/m,\ \ldots\,,z_{n}]\) for the forward experiment and \(Z=[z_{n},z_{n}-p/m,\ \ldots\,,p/m,0]\) for the backward experiment.
5. The resampling time points, \(T_{\rm Re}\), are determined by using the function \(t(z)\) and the position sampling point, \(Z\).
6. The velocity and acceleration for the worktable and the SMTC that correspond to the resampling time points \(T_{\rm Re}\) are determined by using the functions \(\tilde{z}(t)\), \(\tilde{z}(t)\) and \(i(t)\).
7. The friction force is calculated from Equation (5).  
The starting position of the worktable is manually adjusted to the absolute zero of the lathe. The end point is fixed at 410 mm of the absolute position of the worktable, and the feed speed is 5 mm/min. To estimate the friction of the servomechanism, a NC program automatically controls the worktable to move from the start point to the end point with a residence time of one second, after which the worktable returns to the start point. The sampling cycle of the screw length is set at p/1024 to resample the velocity and acceleration of the worktable and the SMTC. In Figure 5, the variation in the estimated friction with the WP is shown for both the forward and backward motion of the worktable. The estimated friction fluctuates significantly around the IAFVs with the WP, and the IAFVs also vary with the WP.