The article develops an engineering method for tuning (or reconfiguring during operation) the regulators of electric drives of manipulator mobility units, which takes into account the presence of significant - l system of electric drives of the manipulator's mobility units, which stimulate the emergence of resonant elastic vibrations and self-oscillations (the effect of autoelasticity). The proposed method allows not only to eliminate the cause of the autoelasticity effect, but also to do so at the engineering level of mastery of the mathematical apparatus, computer mathematics systems, and programming skills. The manifestation of the self-elasticity effect is associated with the presence of such factors as: dynamic properties of the drive of mobility units; elastic flexibility of manipulators; significant nonlinearities of a structural and technological nature or those that arise during operation in mechanical and electrical devices. The engineering simplicity and convenience of the method is expressed in the fact that the adjustment of the regulators of electric drives of the mobility units during the manufacture of the manipulator or their reconfiguration during operation does not require specialized scientific research, but can be performed by a specialist with an engineering level of mathematical training in an interactive mode in a short time.
In the aerospace industry, multi-link manipulators are used to ensure smooth and insensitive
manipulators must usually be lightweight, have at least three degrees of mobility (i.e., mobility
nodes), and cover a large working area [
take into account the effect of significant nonlinearities when setting up the controllers of electric
Once again, we emphasize that, in terms of physical content, a significant nonlinearity is either
inherent in the design and technological execution of the manipulator, or arises during normal
operation as a result of wear and runout in the bearings of the electric drive, or is caused by external
mechanical (physical) shocks, or is specifically set by the control algorithm of the manipulator's
mobility unit [
At present, there are well-developed methods for analyzing and synthesizing the operating modes
of nonlinear automatic control systems (ACS) in general and, in particular, significantly nonlinear
ACS, which are aimed at: first, finding out the conditions for the occurrence of self-oscillations or
instability; second, finding ways to stabilize and (or) eliminate self-oscillations: frequency methods
(Popov frequency method of stability analysis; Goldfarb-Popov harmonic linearization method);
phase plane method (phase trajectories); fitting method (Andronov point transformation method);
statistical linearization method; methods of catastrophe theory; method of Lyapunov functions
[47]. Existing methods for the analysis and synthesis of nonlinear ACS are based on the fundamental
theoretical mathematical works of A.M. Lyapunov, which outlines the necessary and sufficient
conditions for the stability of nonlinear systems. These methods are aimed at mathematicians and
scientists, not at operational engineers. In aerospace engineering, manipulators are manufactured
and used en masse. The problem of AEE is of a massive nature. Controllers need to be adjusted and
reconfigured in the course of operation on a massive scale. This means that a method is needed that
is accessible at the theoretical level to operating engineers and is focused on the use of modern and
advanced computer mathematics systems with advanced specialized software aimed at solving static
and dynamic optimization problems. In other words, we need an engineering method (EM) for
engineering practice. In the MATLAB computer mathematics system, there is a so-called Nonlinear
Control Design (NCD Blockset) package that implements a method of dynamic optimization with
user-defined time constraints [
The engineering method allows: algorithmically calculating the quantitative values of the criterion for the given values of the controller parameter vector in order to obtain a quantitative integrated assessment of the efficiency of the ACS functioning; visually observing the change in time of the variables characterizing the quality of the ACS functioning and interactively analyzing and synthesizing the controller algorithm. When using the engineering method, a nonlinear element (NE), which is a nonlinear link of directed action, is modeled as a static link with a static characteristic that corresponds to the physical content of the action of the nonlinear element in the ACS. The engineering method can be used for parametric synthesis of the controller (PSC) and for controller tuning (CT) during operation. The initial data for the use of engineering method PSC is a mathematical model of a prototype ACS, i.e., a structural diagram of a prototype ACS with known mathematical models of all links of directed action, as well as circuits that include nonlinear elements. The initial data for the application of the engineering method CT is a real ACS.
Building the structure of the modernized ACS. The type of controller with a known structure of control laws is selected and the controller is connected to the structural diagram of the computer mathematical model of the ACS prototype. The parameters of the controller are to be determined by numerical optimization of an algorithmically specified criterion.
Search for the first approximation to the optimal values of the controller parameters, which will
be further calculated by numerical optimization of an algorithmically specified criterion.
Selection of a reference mathematical model for modeling the desired change in time of those
variables that are selected for tuning in the mathematical model of the ACS. Selection of the criterion
for evaluating the difference (tuning criterion) in time between the corresponding output variables
(coordinates) of the reference mathematical model (desired change in time) and the mathematical
model of the ACS (change in time of variables at the given values of the controller parameters).
Selection of the method of numerical optimization of the controller parameters. Connecting the
reference model and the algorithm for calculating the quantitative value of the criterion for assessing
the discrepancy (tuning criterion) to the structural scheme of the modernized ACS. Application of
the numerical method for parametric synthesis (tuning), i.e., search for optimal values of the
controller parameters. The stages of EM CT consist of 3 - 8 stages of EM PCS, where mathematical
models of the ACS elements - the prototype and the modernized ACS - are replaced by full-scale
models (i.e., real equipment and devices). Let us consider the use of EM PCS to adjust the control of
the ACS of the electric drive of the link mobility unit, on which the actuating element with a fixed
directional antenna is located. This antenna is designed to transmit sensing signals in the mode of
-sized objects (i.e., scanning a selected area when searching for actively
maneuvering group objects and their group tracking (capturing the movement of a group object)).
The input signal u(t) sets the angular velocity of the actuating element of the manipulator x(t), which
is the output signal [
The mathematical model of the ACS prototype consists of mathematical models: 1 - angular velocity sensor of the actuating element; 2, 3, Relay, 4 - electronic converter (EC) and electrical part of the electric drive; 5 - mechanical part of the electric drive with the actuating element; 6 - force sensor.
The block diagram of the computer mathematical model of the prototype L ACS allows modeling the prototype ACS with a linear mathematical model of the ES (switch P in the lower position), and the prototype R ACS models the ES with a power amplifier with a relay characteristic ([-10;10], switch P in the upper position). It should be emphasized that in both computer mathematical models of the prototype L and R, a significant nonlinearity of the mechanical part of the electric drive with the link on which the actuating element is located is modeled in the form of a zone of insensitivity [-1;1] and saturation [-5;5]. Let's proceed to the implementation of the stages of the EM PID.
Stage 1. As controllers for the internal and external circuits, we choose proportional, integral,
and differentiating controllers (PID controllers) (Fig. 2). The mathematical models of the regulators
take into account the physical realization of these regulators (a real differentiating link and the
limitation of the type of saturation zone ([-10;10]) for the output signal of PID regulators) [
Stage 3. For tuning, we select the output coordinate of the modernized ACS L and ACS R, i.e.,
the angular velocity of the manipulator's actuating element x(t). We assume that the duration of the
transient process should be reduced to 2 s, and the overshoot should be approximately halved. Taking
into account the form of the transient function (Fig. 3) and analyzing the physical content of the
processes occurring in the modernized ACS L and ACS R, we choose the standard 5th-order
Butterworth form as a reference model for changing the output coordinate of the modernized ACS
L and ACS R over time [
is the duration of the normalized reference transient. ( = 7) reference models:
, desired reference models, respectively
- the time of the transient process duration in the normalized and
Stage 4. As a criterion for assessing the time difference between the corresponding output variables (coordinates) of the reference mathematical model (desired change in time) and the mathematical model of the ACS (change in time of variables at the given values of the parameters of the PID-1 and PID-2 controllers), that is, the tuning criterion, we choose the weighted square error integral:
= ∫ 0 ( ) ∙ ( ( ) − ( ))2 , (4) where we assume that the weighting factor ( ) = 1, 0 = 0, = 10 .
The quantitative value of the criterion is shown in the display D (see Fig. 5).
Stage 5. As a method of numerical optimization of the parameters of the PID-1 and PID-2
controllers, it is proposed to choose a numerical method among the methods described in [
As an example, in step 7 we will consider the use of the Gauss-Seidel numerical method.
Stage 6. Fig. 5 shows the connection of the reference model 7 and the algorithm for calculating the quantitative value of criterion I (integral of the weighted square of the error, the value of which is displayed on the display D) to the computer mathematical model of the modernized ACS. connected reference model 7 and the algorithm for calculating the quantitative value of the tuning criterion displayed on the display D: G - vibration (oscillation) generator for vibration linearization
The block diagram shown in Fig. 5 allows modeling the types of modernized ACS depending on the position of the switches. If all switches are in the position indicated in the diagram (Fig. 5), except for switches P and PG, which are moved to the lower position, then the modernized ACS L is modeled. If all switches are in the position indicated in the diagram, except for switch PG, which is moved to the lower position, then the modernized ACS R1 is modeled. If all the switches are in the up position (as shown in the diagram in Fig. 5), then the modernized ACS R2 is modeled.
The peculiarity of the modernized ACS R1 and R2 compared to the modernized ACS L is the presence of a significantly nonlinear relay-type element in their composition. Significant nonlinearity of the relay type is inherent in some types of electronic converters used in electric drives of manipulator mobility units.
The feasibility of using such converters is determined by the relevant design and technological features of the electric drive and manipulator. The peculiarity of the modernized ACS R2 compared to R1 is the use of vibration linearization to reduce the influence of a nonlinear relay-type element on the effective functioning of the ACS.
The physical meaning of the effect of vibration linearization on improving the static characteristics of an electronic converter is as follows. In the presence of a smooth useful signal at the input of a link with a significant nonlinearity of the relay type (the useful signal and the signal from the vibration generator G are added), a smooth dependence of the average value of the output signal of this link during the period of operation of the vibration generator G on the average value of the input signal will be ensured.
Stage 7. Let us apply the Gauss-Seidel numerical method to optimize (tune) the parameters of the PID-1 and PID-2 controllers. We will use a computer mathematical model (Fig. 5), taking into account a different set of significant nonlinearities for each option of the ACS modernization. The option of modernization of the ACS is denoted by the letters L or R1 or R2.
The position of the switches (Fig. 5) for the variant of the modernized ACS: variant L corresponds to the upper position of the switches P0, P1, P2 and the lower position of the switches P and PG; variant R1 corresponds to the upper position of the switches P, P0, P1, P2 and the lower position of the switch PG (i.e., vibration linearization is not applied); variant R2 corresponds to the upper position of all switches, i.e., vibration linearization is applied).
The tuning (optimization) criterion depends on the six parameters of the PID-1 and PID-2 controllers: ( ) → min ,
∈ where = [ 1, 2, 3, 4, 5, 6] is a vector composed of the parameters of the PID-1 and PID-2 controllers;
1 = 1, 2 = 1, 3 = 1, 4 = 2, 5 = 2, 6 = 1
The DFD is the domain of admissible solutions for finding the optimal values of the parameters of the PID-1 and PID-2 controllers, which is set by the physical content of the problem (in particular, the features of the implementation of the PID-1 and PID-2 controllers in a particular problem). We will denote the tuning criterion for the corresponding variant of the ACS modernization by the letters L or R1, or R2, i.e.: ( ), 1( ), 2( ).
We consider the parameters of the controllers taud1 and taud2 to be constant parameters. These parameters are equal to the values that were obtained in step 2 of the method, i.e., during the initial tuning of the PID-1 and PID-2 controllers: taud1 = 0.0150 and taud2 = 0.0225.
We emphasize once again that to begin applying the numerical Gauss-Seidel method, the first approximation to the optimal values of the parameters of the PID-1 and PID-2 controllers, obtained at stage 2 by the Ziegler-Nichols oscillation method for the SISO LTI mathematical model of the modernized ACS L and R1, R2 (all switches in Fig. 2 are in the lower position), is used.
Comparative modeling of the transient functions of the reference model 7 and the mathematical model of the modernized ACS L, R1, and R2 (Fig. 5) for the first approximation (5) (6) 1 = [ to the optimal values of the parameters of the PID-1 and PID-2 controllers obtained at stage 2 is shown in Fig. 6 (a, b, c, d). a) (Fig. 5) are in the lower position (the influence of significant nonlinearities is not taken into account); b) - switches (Fig. 5) P0, P1, P2 - in the upper position, and switches P and PG - in the lower position; c) - switches (Fig. 5) P, P0, P1, P2 - in the upper position, and switch PG - in the lower position; d) all switches (Fig. 5) in the upper position
When using the first approximation r1 to the optimal values of the parameters of the PID-1 and PID-2 controllers, which were obtained in step 2, the following values of the tuning criterion are displayed on the display D: ( 1) = 0.5871, 1( 1) = 61.21, 2( 1) = 60.51 (8) in accordance with the computer mathematical models of the modernized ACS L or R1 or R2 (Fig. 5).
Using the Gauss-Seidel method, six parameters of the PID-1 and PID-2 controllers were tuned for
The Gauss- optimization for each coordinate (parameter of the PID-1 and PID-2 controllers), a direct interactive method of cycle consists of passing through six coordinates, i.e., from 1 = 1 to 6 = 2 inclusive (see Step 7). After optimization for a particular coordinate, the minimum point at that coordinate was memorized. The optimization process did not aim to achieve a global minimum.
Since we are talking about an engineering tuning method, the goal was to show that even without criterion), and even in manual mode, it is possible to achieve the desired result of tuning a significantly nonlinear ACS. It should be emphasized that during the optimization process, a visual comparison of the quality of the transient of the modernized ACS with the quality of the transient of the reference model was performed using a Scope 2 oscilloscope (Fig. 5).
As a result of performing one complete cycle in the interactive mode of coordinate descent (the time of one complete cycle was less than 30 minutes), the following quantitative values of the parameters of the PID-1, PID-2 controllers and the tuning criterion for the modernized ACS were obtained: : С = [Kp1 = 2.5; Ki1 = 3.2; Kd1 = 0.14; Kp2 = 2.1; Ki2 = 2; Kd2 = 0], ( С ) = 0.06487; 1( С 1) = 0.2387; 1: С 1 = [ 1 = 10; 1 = 4; 1 = 0.4; 2 = 2.1; 2 = 1.5; 2 = 0], 2: С 2 = [ 1 = 5, 1 = 4.5, 1 = 0, 2 = 2.1, 2 = 1,
2 = 0], 2( С 2) = 0.2895.
Transition functions for the computer mathematical models of modernized ACS L, R1, and R2 for the vector of parameters of PID-1 and PID-2 5. Conclusions
application of nonlinear controllers (nonlinear corrective devices) of electric drives of manipulator mobility units with significant nonlinearities; expanding the vector of adjustable parameters from six ( = [ 1, 2, 3, 4, 5, 6], where 1 = 1, 2 = 1, 3 = 1, 4 = 2, 5 = 2, 6 = 1 (see step 7)) to ten components due to the inclusion of the parameters of the mathematical model of the real differentiating link taud1, taud2 and the parameters of the vibration generator (Fig. 5), which are the amplitude and angular frequency of vibrations.
The engineering method proposed in this paper for tuning the electric drive regulators of manipulator mobility units with significant nonlinearities allows us to: to obtain quantitative values of the controller parameters for a ACS with a given structure and a known type of significant nonlinearities based on the use of known, practically tested, and available software tools of modern computer mathematics systems; to be fully programmed using any numerical optimization method, which will allow full automation of parametric optimization of the ACS with significant nonlinearities.
The author(s) have not employed any Generative AI tools.