Control of mechatronic drives considering the type and number of the nonlinearities of a mechanical converter

. In this article the particularities of the dynamics study of mechatronic drives that operate at low velocities are discussed; the origins of frictional self-exited oscillations to develop an algorithm for nonlinearity correction are analyzed; a drive mathematical model and the qualitative picture of the self-exited oscillations regimes were presented for the three cases: a "system with clearance and friction-free"; a system without clearance with "dry friction" and a system with clearance and "dry friction". Therefore, an algorithm to compensate the frictional self-excited oscillations based on the feedback introduction for the sliding velocity of the linear motion output link was proposed.


Introduction
The concept of designing an electromechanical drive as a single mechatronic device is widely known and has been realized during the creation of a mechatronic single-block drive, that is a whole structure based on a contactless torque motor to the hollow rotor built with a planetary screw or roller screw mechanism which provides either a linear or rotational output motion Fig. 1 [1][2][3][4]. The software implementation and hardware execution of the motion controller by the drives largely determine the quality of the output motion [5][6][7][8].
The single-block design which distinguishes by its compactness, increased stiffness, reliability, energyefficiency and a lower vibroactivity [3] is engineeringly manufacturable and can be unified with other elements. The transition to the single-block principle of construction provides an improvement of the drive operation and control, an increased reliability, even during failure of individual elements, a reduction in time and cost to get the object being controlled ready for operation, and a mass large saving, which is reduced not only by a reduction in the mass of the drives themselves, but due to the reduction of auxiliary fixing units and mechanically connected blocks. The use of self-locking actuators ensures a reliable neutralization of the failure effects of locking devices [9][10].
The design of mechatronic actuators which operate at low velocities is not possible without knowing the dynamic, computable and other functioning regularities of its actuating elements. In this sense, the study of the mechatronic modules dynamics seems to be an inherent stage in the design of modern systems for the implementation of a time-optimal motion control algorithm [5,6]. One of the most difficult obstacles to eliminate for ensuring the specified quality is the manufacturing and wear errors, and, therefore, the clearance in the transmission and its kinematic and static mechanical properties instability. The instability of the specified output displacement principle can also be caused by factors such as self-locking in the transmission, control torque pulsations and load oscillations which are caused by nonlinear frictional properties and a high drive vibroactivity.

Materials and methods
The enhancement of the functionality and the improvement of the dynamic mechanical properties of electromechanical drives is possible only on the basis of a new approach to their design and development. In this case, in addition to the kinematic requirements, the dynamic ones are brought to the forefront. It is necessary to consider the dynamic interaction of the actuating mechanism and the electric motor.
To make the electromechanical drive the source of a smooth and uniform motion, it is necessary to minimize the causes of irregular running in the mechanical transmission and in the engine. No linear motion drive can provide an "ideal" smooth and uniform motion. All the time there are some external perturbing influences that generate an irregular stroke running of the actuating body. This is like the disturbances caused by the electric a) b) Fig. 1. a) External view of the mechatronic electromechanical module with an incorporated roller screw mechanism. b) Roller screw mechanism motor, and the disturbances caused by the actuating mechanism and the working process executed in the machine.
It is assumed that the mechanical drive system has an infinitely high stiffness, and the entire inertia of the electromechanical system is concentrated on the motor shaft as the given moment of inertia. Increasing it, we reduce the motion deviation magnitude of the uniform motion. In the practice, this is usually achieved by installing on the output engine shaft an additional mass, i.e. a flywheel which has a large given torque.
The rotation unevenness of the output link is dangerous, because it causes loads in the transmission mechanism, in addition, it leads to the appearance of a variable component of the driving torque, which can adversely affect the engine operation and cause a reduction in its efficiency.
By installing the flywheel on the output engine shaft, we reduce the torque portion that is associated with the disturbance from the engine. This increases the load caused by the disturbance from the actuator. This can be avoided by installing the flywheel on the output shaft of the transmission device. However, this way is generally unacceptable, since the output shaft is running at low velocity and the proper moment of inertia of the flywheel mounted on it must be і2 times greater than the moment of inertia, given to the motor shaft. The installation of such a large flywheel mass is most often unprofitable for design reasons. The influence of the flywheel on the dynamic loads of the drive, on the one hand, is positive, but on the other the installation of additional mass leads to the increment of the rotation unevenness.
The problem to chose the location of the flywheel mass for a drive traditionally actuating (the engine is the actuator) with high requirements for the smoothness of the output displacement is very delicate. In order to achieve a smooth displacement of the actuating mechanism by its additional weighting, we are obliged to operate the engine at a larger value of the rating torque due to the permissible overload. It is possible to eliminate this negative phenomenon only by using special methods (forcing the engine by increasing the voltage, torque and similar techniques; covering the engine with a tachometer feedback; using engines with increased sliding motion or load regulators). In the end, the aim to eliminate the noted limitations of the flywheel electromechanical drive will lead to a significant increment of its dimensions.
So, the physically existing inertia can be eliminated only by removing an element from the drive that has this inertia, which is not always feasible. But it is possible to compensate the drive inertia influence, using its special arrangement, without involving additional masses.
The DC motors of the DBM series are well known, whose design allows to place the actuator inside the hollow rotor. Thus, the electromechanical drive takes the monoblock form: The motion source (engine) is as close as possible to the actuating link. As a result, the quality of the output displacement will be mainly determined by the mechanical part of the drive; and the rotor rotation unevenness will be partially "smoothed out" (or "damped") by its weighting with the mechanical transmission. This design peculiarity should reduce the load and torque variables in the actuating mechanism and the engine; and stabilize the drive's output motion. In addition, the transmission mechanism is simultaneously an actuator: there is a significant reduction in the kinematic chain

Problem formulation
In an electromechanical drive which operates at low velocities the frictional self-excited oscillations (SEO) arise. The presence of clearance in the transmission also leads to self-excited oscillations. The analysis of the selfexcited oscillating regimes is a necessary stage for the development of a microprocessor algorithm to correct the nonlinearity.
The compensation of the clearance negative influence can be carried out by introducing a correcting link or using a microprocessor control (a software). The use of a microprocessor control provides increased opportunities to compensate the clearance influence on the drive quality indices. A significant advantage of this compensation method is the relative simplicity of the software implementation.
The task of the control in the system with clearance is to get fastest backlash elimination in such a way that at the moment of the connection the links do not impact each other as a result of this elimination. That is more strictly formulated as follows. Let us assume that at the disconnection time, the 1st link was in the  

Mathematical model
The dynamics of the servo drive which consists of a brushless DC electric motor and a roller screw actuating mechanism (RSM) is described by a two-mass model. The first mass is rigidly connected to the motor shaft and is described by the phase variables 1 x and 1 1 x v   ; the state of the second mass (load) is determined by the variables 2 x and The qualitative picture of the self-excited oscillating regimes in system (1) depends on the type and number of nonlinearities. Therefore, the analysis was carried out for three cases: system with a clearance of 5 , 0   mm and friction-free (system I); system without clearance with "dry friction" (2) (system II); system with a clearance and "dry friction" (system III). System I operates in stabilization mode 0 2 x x  (= 10 mm in Fig. 2 a); systems II and III in stabilization mode of the sliding velocity 0 2 v v  ( = 1...6 mm/min in Fig. 3 and 4).
The analysis of the self-excited oscillations occurring in system I showed that sustained relaxational (sawtoothed) oscillations are established when there is not damping (solid curve in Fig. 2a). When the masses are disconnected, their motion continues independently one of the other and the load is displaced at the constant velocity that they had at the disconnection time.
During the clearance elimination, the masses collide and they are again disconnected. In a system with damping, the masses can move together after the impact or lose some of their kinetic energy: we can see the sinusoidal zones which correspond to the combined motion of the masses in Fig. 2b. Under these conditions, the backlash elimination time is increased due to the lower motion velocity (horizontal sections in Fig. 2b). The amplitude of the steady-state self-excited oscillations is primarily determined by the clearance size and the damping coefficient. The frequency of the SEO is directly proportional to the stiffness coefficient and inversely proportional to the links mass ( 1 J and 2 m ) and the damping coefficient.
The task of the control in system I is the fastest backlash elimination in such a way that at the end of the elimination there won't be impact at the joining links time. The algorithm works as follows. Let us suppose that at the moment of the disconnection, the first link was in   The elimination time  is implicitly found from the intersection condition of the In the numerical study of system II, the following different regimes were qualitatively identified (Fig. 3). At The frequency of the frictional SEO primarily depends on the load mass and the RSM stiffness. In a mode with a significant stopping rate, a frequency reduction proportional to the increment in the frictional force jump or a sliding velocity reduction is observed.
The dependence on the frictional SEO amplitude is determined by the self-excited oscillating regime: the amplitude is the maximum one in the transitional mode when . When the self-excited oscillations change as in the harmonic mode as in the relaxational oscillations tightening, the amplitude decreases uniformly to zero. In this sense, we should choose a change in the parameters, which not only reduces the frictional SEO amplitude, but leads the system out of the region of relaxational oscillations. This occurs with a reduction in the frictional force jump and an increment in the sliding velocity, the load mass and the damping.
The algorithm (Fig. 5) to compensate the frictional self-excited oscillations is based on the feedback introduction for the sliding velocity of the output link 2 v . Unlike the linear feedback, when the control voltage is proportional to the magnitude of the disparity, the algorithm works according to the nonlinear law. If the algorithm past along the path c, then, at the backlash elimination time  , the links have the same speed and no shock occurs. During the execution of the path a or b, the drive input link may not be able to reach the required speed 0 2 v at the time  . Then, during the links engagement, a shock occurs and a new links disconnection is possible. In this case, the program go back to step 1 and determines to chose whether the clearance occurs or not. If necessary, the above steps sequence is repeated.
During the research of a possibility to compensate the clearance influence on the drive output displacement by using digital control, the proposed model did not take into account the delays and errors related to the operation of the microprocessor and as well as the ones introduced by the measuring sensors.
The exact dependencies  