Influence of the gap and the friction on trajectory reproduction accuracy in a multiaxis machine with CNC

3.1. Backlash in the mechanical system not covered by position feedback loop

A “free travel” (backlash) is understood in mechanics (Fig. 4(a)) as a gap, i.e., the minimal magnitude of the travel of the specifying element of the mechanical system, mass М1, at which no motion of the output element of the system mass М2 is observed.

Fig. 4Backlash in the mechanical system not covered by position feedback loop: a) the interaction model of two bodies М1 and M2, b) reproduction deviations of the reference circle in the polar system of coordinates, and c) travel of body M2 in the quasi-static mode

a)

b)

c)

Fig. 4(c) shows how travel X $=sin\left(\omega t\right)$ (input signal $X_1$, displacement of mass M1 on the X axis during movement over the reference circle), is transformed into travel M2 (output signal $X_2$) in the backlash presence. The backlash magnitude equals the BE segment, $\omega =$ 2π/10 rad/s.

The shape of segment AB of the output signal coincides with the input signal and is shifted downwards by backlash magnitude. Segment BC corresponds to the motion of mass М1 in the backlash. After the backlash is taken up, the shape of segment CD of the output signal coincides by the input one and is shifted upwards by the backlash magnitude. Segment DE corresponds to the travel of mass М1 in backlash. After backlash is taken up, the shape of segment EA’B’ of the output signal coincides with the input signal and is shifted downwards by the backlash magnitude, etc. If we consider the execution of two sinusoidal signals X1 $=sin\left(\omega t\right)$ without backlash and Y1 $=sin(\omega t+\pi /2)$ with backlash according to the diagram of Figs. 4(a) and 4(b), we will acquire the pattern presented in Fig. 4(c) (found using modeling). Here, input signals X1 and Y1 are depicted by a dotted line (reference circle), while output signals X2 and Y2 are depicted by the solid line (actual trajectory). The considered case corresponds to the execution of the “circle” trajectory by the drive over X without backlash and drive over Y with backlash not covered by the position feedback.

3.2. Backlash in the mechanical system covered by position feedback loop

The pattern of the influence of backlash on the execution of the input signal in the mechanical system covered by the position feedback loop changes in principle.

The deviation of the actual position of the actuator from the specified one as applied to modeling the carrying system of the drive over the linear coordinate (Fig. 3) is shown in Fig. 5. The following curves are displayed in the oscillograph (“Scope backlash” unit, Fig. 3): (1) setpoint for the rotary angle (2) readings of the rotary angle sensor for the travel of the backlash system, and (3) the mismatch signal (the difference between the specified and actual rotary angle of the output shaft-actuator, entering the controller input. It is seen that the variation in the direction of the drive travel is accompanied by a characteristic “projection” (curve 2) when the signal from the rotary angle sensor shows a constant value for a certain time. If we consider actuator motion over the reference circle implemented by two linear drives arranged orthogonally relative to one another, then the deviations of the actual trajectory from the specified one take the form shown in Fig. 5(b), where the radius of the reference circle is 10 mm, the gap is 0.4 mm, the traversal is counterclockwise, 1 is the reference circle, and 2 is the actual trajectory. To construct these deviations in the polar system of coordinates, Simulink blocks U_zad and U_delta (Fig. 3) are used.

Fig. 5Reproduction accuracy of the reference travel: a) deviations when reproducing the reference sinusoidal travel in the presence of the FB-enveloped backlash, where (1) is the input signal (specified value), (2) is the output signal (actual value), and (3) is the error signal (the difference between the actual and specified values); b) reproduction deviations of the reference circle in the polar coordinate system

a)

b)

The backlash in the mechanical part of the drive in principle differently affects the reversal accuracy of the “circle” trajectory depending on whether the mechanical part of the drive is covered by the position feedback loop.

Backlash magnitude in the mechanical part covered by position feedback loop directly determines the “burst” size during the reversal of the reference circle with the help of two orthogonal linear drives under the assumption of the absence of other nonlinearities as “friction”, “elasticity hysteresis”, etc.

4.1. Model of the friction

Dry friction in the model implemented by “Dry friction” unit, generating disturbance torque $M_{fr}$ depending by the actual speed value ${\omega }_{act}$ (Fig. 6(a)). Fig. 6(b) shows an implementation of the “dry friction” unit in the Simulink environment.

Fig. 7 shows a simulation model of the electrical drive with friction in mechanical subsystem, covered by position feedback loop. In order to improve the moving accuracy, feedforward control was implemented.

Total compensation signal consists of the output signal of FFC unit and output signal of dry friction compensation unit.

Dry friction compensation unit performs correction signal when the drive changes its moving direction. Shape of the correction signal determined by signal amplitude value $\Delta n$ and by time constant $T_{comp}$, which determines the rate of decay of the output signal [3].

Fig. 6Implementation of dry friction in the model: a) Mω curve; b) unit “Dry friction” implemented in Simulink environment

a)

b)

Fig. 7Simulation model of the electrical drive with friction in mechanical subsystem covered by position feedback loop

4.2. Modeling system with friction

Fig. 8Simulation results of modeling system with friction: a) reproducing the reference sinusoidal signal by one drive; b) reproducing the reference circle in the polar coordinate system

a)

b)

Fig. 8(a) shows results of modeling of reproducing the reference sinusoidal signal by one servo drive with friction. The following curves are depicted on the chart: (1) setpoint for the rotary angle, (2) readings of the rotary angle sensor for the travel of the system with dry friction, (3) disturbance torque, (4) drive torque on the motor shaft, (5) speed of the motor shaft, (6) error signal.

Fig. 8(b) shows results of modeling actuator motion over the reference circle implemented by two linear drives arranged orthogonally relative to one another. It is seen, that the form of the burst is close to that produced by system with backlash covered by position feedback loop.

It is seen, how friction changes motion: when drive changes its moving direction, we can see characteristic “stagnation”, i.e. time period when the motor shaft is not moving. This effect is due to a disturbance torque jump when drive speed sign changes. The system will remain fixed until the torque on the motor shaft exceeds the value of friction torque.

If friction compensation unit is “on”, speed setpoint signal is summed with additional correction signal when drive changes its moving direction. In this case difference between reference and actual speed values is increasing, as a result current controller reference signal is also increasing, therefore drive produces required torque much quicker.

The values of $\Delta n$ and $T_{comp}$ corresponding to the minimum of the burst value for a real machine tool can be obtained by measuring burst values during experiments with ballbar equipment by trial-and-error method. Burst can be compensated by about 80 % with optimally adjusted values [5]. It should be noted that these values are optimal only for feed speed, at which burst measuring and values selection was made.