Operations of
controlled displacement of objects of large mass, large dimensions, complex
geometry, or other characteristics that are inconvenient for practice are often
encountered in technological processes.
In
some cases, such operations have to be performed by two or more robotic devices
interacting with each other. The factor of interaction of several drive and
control units significantly complicates the process of synthesis of such
systems, and, moreover, the optimization of their structure and parameters,
taking into account the requirements, criteria and constraints imposed on the
system. Visualization of the expected solutions, especially at the initial
stages of design, can be of great help here. Talking about visualization, we mean a
graphical representation of a complex of results of computer simulation of the
system's dynamics in the dialogue mode. A visual analysis of the information
provided makes it possible to quickly obtain visual estimates of the options
and use them as source material in the subsequent stages of finding the optimal
solution. As an example, in this paper we consider the purposeful process of
visualizing the dynamics of a manipulator designed to lift heavy loads. The
lifting force is created by two parallel, synchronously working hydraulic
actuators with a common control system. The visualization procedure was used
for a preliminary assessment of the general properties of such a system and its
ultimate capabilities. At the same time, the ability of the system to provide
synchronous motion of the drives with the inequality of their loads,
sensitivity to short-term load jumps and the impact on the process of basic
parameters were studied. The problem of forming a block of criteria and
constraints, including those that are not amenable to formalization was solved.
The procedure is based on computer modeling of the system dynamics, represented
by a dimensionless model, which contributes to the generalization of the data
obtained.
As practice shows us, in such systems, actuators with different types of
engines, but mainly with hydraulic and electric motors, are used.
When moving especially heavy and bulky objects, preference is given to
hydraulic motors, which are distinguished by high power density and do not
require the inclusion of additional reinforcement mechanisms in the design. A
number of examples of systems with parallel operating drives are given in
[1-4]. However, there is little information about the methods of synthesis and
optimization of the structure and parameters of such systems. To solve such
problems, it is proposed to use the approaches developed by the authors based
on an interactive procedure for the synthesis of dynamic systems,
rationalization of their mathematical models, methods of matching the processes
of dynamics and control of individual blocks, as well as the active use of
similarity theory and, in particular, the analogy theory, in order to summarize
the results [5-8]. In present work, this problem is solved by the example
of controlling the movement of a heavy object moved by two synchronously
working piston hydraulic engines. Mass load is assumed to be distributed
between the engines unequally. We consider the initial stage of the synthesis
of such a system, which is based on the analysis of visual images of computer
modeling of its dynamics.
The basic mathematical model is obtained under the
following conditions. The drives are hydraulic cylinders 1 and 2 of the same
parameters and powered from a common source (Fig. 1). Synchronously operating
drives move an object of mass
m = m
1
+ m
2
,
where
m
1,2
are the mass loads related to the drives. The law of motion is realized by
controlled pressure change in the lower cavities, which are connected to a
power source (with pressure
p
M
)
through control valve 3, intermediate cavity 4 (with volume
V
) and
control valve 5.
Fig.
1.
The equations of motion of the drives are:
|
,
|
(1)
|
here,
x
is piston displacement,
p
is pressure in the lower cavity,
F
is effective piston area,
m
1,2
g
,
P
L
1,2
are weight and force loads on the rod,
k
1,2
are coefficients of fluid friction in the drive.
The changes of the pressure
p
1,2
in the lower cavities of the drives and the pressure
p
in the intermediate cavity are related by dependencies:
|
,
|
(2)
|
here,
;
;
;
,
;
β
,
β
1
and
β
2
are opening degrees of the channels
f
,
f
1
and
f
2
,
; E is bulk
modulus of working fluid;
x
ν
is the length of the
intermediate cavity;
ρ
is working fluid density.
In previous papers by the authors [5–8], it was
shown that in complex, multi-parameter mechanical systems, it is recommended to
use dimensionless models in which the actual parameters and other
characteristics of the system are represented as dimensionless complexes. This
reduces the number of basic parameters and allows us to obtain generalized
characteristics, which facilitates the analysis and synthesis of the drive
(mechatronic) complex. The transition to dimensionless parameters is based on
the method of the analogy theory [9], equations (1, 2) are transformed into a
dimensionless form by replacing variables with their dimensionless analogs
λ
,
τ
,
σ
according to the relations
,
,
. As a result of
this replacement, in the first approximation, as well as
,
, and simple transformations, we obtain the
transformed system (3):
Here, the
following designations are introduced:
a) dimensionless
variables:
– movements of drives;
–
cavity pressures;
– differential pressures between the cavities;
b)
dimensionless parameters: distribution
c
1,2
of mass load between drives; total mass loading of the manipulator
; additional resistance forces
; stiffness of
the intermediate cavity
;
reduced initial volumes of actuating drives cavities
;
liquid friction forces
,
where
; the ratio
between the dimensions of the flow areas of the channels
of the valves 3
and 5.
The initial conditions are the actual time
τ
S
of the mass
m
movement and the magnitude
x
S
of its stroke. Since the scale
q
1
of measurement of dimensionless displacement is the working stroke of the
manipulator, it is accepted that
.
The choice of manipulator parameters is based on the results of computer
modeling of the transformed systems (3) and valve control systems.
The parameters of the manipulator are selected
according to the following data: the working stroke of the object; set or
minimum travel time; the mass of the object being moved and the distribution of
weight loads between the drives. The objects of choice are the sizes of
hydraulic cylinders, the operating pressure in the hydraulic system, the flow
areas of the valves and the parameters of their flow characteristics, as well
as the parameters of the valve control systems.
The transformed
computer model (
3) is used in this case to
investigate a relatively little studied dynamic system with two hydraulic
execution units that together solve the problem of linear movement of an object
with large dimensions and mass. With the continuous interaction of the drives
and the external object, quite complex mechanical and hydraulic cross-links
arise, affecting the dynamics of the system. Accounting for these links is also
hampered by the fact that in the course of the movement, unpredictable
additional disturbances in the form of random forces of resistance, instability
of parameters and other factors may arise. The nodes of the system involved in
the movement are connected to each other through hydraulic and mechanical
circuits, which leave some mutual freedom in the movements, which also affects
the quality of the operation.
There are
several methods for finding and choosing the optimal solution in the synthesis
of a complex system, including methods of multicriteria optimization with the
introduction of weight criteria [5, 7] or taking into account preferences. This
work is devoted to the stage of data preparation for the use of these methods,
for which the results of computer simulation, presented in a visual for
viewing, are interactively compared [10, 11].
To present the
received information in a visual form, a special program was developed using Delphi, which facilitates a visual comparison of the obtained solutions quality. The
program interface is shown in Figures 2-10. In the left part of the program window are fields for entering 20 main parameters of the model. The parameters used
and their designations in the program are presented in table 1. To the right is
a panel for entering the parameters of the numerical Runge-Kutta method for
solving the system of equations (3). The user of the program can choose the
accuracy order of the method: 2nd or 4th, and also set the grid step
h
on the scale
x
of the piston displacement. Here, the user can specify
the control value of the parameter
x
, for which in the text box in the
program window the intermediate calculation results will be displayed in
numerical form. The meaning of the graphical representation of the results
located on the right side of the program window will be explained later in the
analysis of specific examples.
Table 1. System model parameters
|
Designations
|
Designations in the program
|
Ranges of change
|
Comments
|
|
c
1
|
c1
|
0,4 – 0,6
|
Resistance
to weight imbalance drives
|
|
|
χ
L
|
|
Del
|
0,4 – 2
|
The
relative total workload on the drives, which also serves as a measure of
their dimensions
|
|
λ
V
|
|
0,2 – 1,0
|
The
measure of the volume of the intermediate chamber
|
|
β
0
|
be0
|
0,3 – 0,7
|
The
share of the opening of the common channel in the line leading to the drives
that relates to the first drive.
|
|
α
1
|
a1
|
0,25 – 1
|
The
ratio between the flow sections of the common supply channel and the channel
leading to the first drive
|
|
α
2
|
a2
|
0,25 – 1
|
The
same for the channel leading to the second drive
|
|
κ
1
|
nu1
|
0,05 – 0,1
|
Coefficient
of friction of the first drive
|
|
κ
2
|
nu2
|
0,05 – 0,1
|
Coefficient
of friction of the second drive
|
|
ϑ
1
|
kd1
|
25 – 50
|
Position
feedback ratio
|
|
ϑ
2
|
kv1
|
0 – 50
|
Speed
feedback ratio
|
|
ϑ
D
|
Kd
|
25 – 100
|
Position
feedback ratio
|
|
ϑ
V
|
Kv
|
0 – 5
|
Speed
feedback ratio
|
|
|
χ
L
1
|
|
dell1
|
0 – 0,1
|
Additional
short-term intermittent drag force acting on the first drive
|
|
|
χ
L
2
|
|
dell2
|
0 – 0,1
|
Additional
short-term intermittent drag force acting on the second drive
|
|
Ε
|
Nux
|
500
|
The
dimensionless analogue of the elastic modulus of a liquid (does not vary)
|
|
λ
01
|
L1
|
0,05
– 1
|
The
measure of the initial (harmful) volume of the first drive
|
|
λ
02
|
L2
|
0,05
– 1
|
The
measure of the initial (harmful) volume of the second drive
|
Dynamics of the
four system options is analyzed: 1) with the same load on the drives; 2) the
first case plus additional short-term loads; 3) with unequal loads on the
drives; 4) with different relative loads of the drives, which is equivalent to
their different sizes
.
Option
1. Fig. 2 shows the motion characteristics of the system with the same mass
loads on the drives
and
in the absence of other power loads. In the lower part of the column a) the
displacement curves and speeds of the first drive are presented, in the upper
part of this column the pressures in its working cavity; in column b) there are
similar indicators of the second drive. Column c) shows the channel opening
curves, changes in the mismatch criterion
in the motion
of the actuators and the pressure in the intermediate cavity above. The
designations of these quantities were explained in the previous section. The
curves in the subsequent figures are arranged in the same order. In addition,
to the left of the graphs on each figure there is a table of the values of the
parameters selected for this version of the model. The scale of the displacement
λ
is doubled relative to the pressure
σ
.
The scale in speed
is
ten times relative to pressure. The scale of
β
(the size of the flow area) is increased five times relative to the pressure.
From
the graphs it follows that in the conditions of option 1, despite the high load
level of the drives (|
χ
L
|
= 1,6), the given laws of motion of the drives are implemented with good
accuracy, and the pressures after a short initial disturbance
stabilize quickly in all cavities. The functioning of the system under the conditions of option
1 is distinguished by a very low sensitivity to parameter variations within the
entire selected range.
Fig.
2.
Option 2. In this option, the drives are under the additional influence of random short-term resistance
forces (|
χ
L
1,2
|
= 0,1). The periods of their action are marked with the labels below in the
graphs presented in Fig. 3. Although discontinuous perturbations influence the
process, it can be seen that the process is restored after the termination of
the action of the perturbations. It should be borne in mind that the results,
presented in Fig. 2 and Fig. 3, are obtained under the condition that there is
no imbalance of mass loads on the drives. This explains the possibility of
choosing such a high value of |
χ
L
|
= 1.6, close to the upper limit (the minimum size of the drives).
Fig.
3.
Option
3. Imbalance of mass loads of drives (
) has a
significant impact on the process. The study showed that with the values of the
parameters indicated in Fig. 6, rather narrow limits of variation
are
allowed; moreover, the process is also disturbed here (see Fig. 6). Outside
these limits, the system is practically inoperable, as evidenced by the graphs
presented in Fig. 4 and Fig. 5: in the first case
(Fig.
4), i.e. the imbalance is
, in the second
case (Fig. 5), the imbalance is the same but with opposite sign.
Fig
.
4.
Fig
.
5.
Fig.
6.
However,
as mentioned above, this conclusion was obtained with the drive sizes too low.
The study showed that the allowable imbalance may exceed
if
= 1, which
corresponds to a nearly twofold increase in the size of the drives. If
necessary, the allowable load imbalance can be increased even more at the
expense of drives. This problem should be solved in accordance with the
requirements for the designed system.
Option
4. The latter conclusion is also confirmed by the simulation results presented
in Fig. 8, which are obtained with a value of
= 0.2. A
significant decrease in pressure levels in all cavities indicates redundancy of
the driving forces of the drives, which easily cope with the task. And from the
comparison of the curves shown in Fig. 6 (
=
1.6) and in Fig. 7 (
=
1.8), we see a process of gradual destruction of the driving mode as the
driving force of the drives decreases.
Fig
.
7.
In conclusion,
let us additionally consider the effect of the time factor on the process
quality, which in our model is set by the parameter
τ
S
,
chosen in the interval 15-40. The speed of the drive directly depends on the
value of this parameter. The minimum value of
τ
S
,
i.e. drive speed limit is constrained. The constraint condition can be
expressed by the inequality
that
must be satisfied when
and
. From the
comparison of the simulation results presented in Fig. 9 and Fig. 10, it
follows that an increase in
τ
S
,
(in this case from 17 to 40) stabilizes the process. At the same time, a
further increase in
τ
S
,
may cause instability at the end of the turn due to a significant decrease in
the speed of movement.
Fig
.
8.
Fig
.
9.
Fig.
10.
The paper
presents an analytical interactive procedure for the formation and processing
of information about the dynamics of the designed object, which can further
serve as a basis for its multi-parameter, multi-criteria optimization by
various methods. Information is obtained by computer simulation of the system
behavior. A characteristic feature of the procedure is the visual form of
presenting the results obtained in an interactive mode with a consistent and
focused choice of the values of model parameters when moving from
one option to another. To achieve greater generality of conclusions, the model
of the system with all its characteristic indicators (criteria, constraints) is
reduced to a dimensionless form.
The following
problems that can be solved using the proposed procedure should be noted. First
of all, this is the re-formation (narrowing) of the search space for a good
solution due to the exclusion of clearly unpromising areas from it, and an
assessment of the feasibility of the system with each set of parameters. In the
example with the manipulator it was found, in particular, that the synchronous
movement of the drives can in principle be implemented in a fairly wide range
of parameters, with deviations of no more than 5% of the full stroke.
It is relatively
simple to distinguish the model parameters, which have little effect on the
operation of the system and, therefore, we can choose fixed values for them.
Such values in the model in question include, for example, the parameter
characterizing the volume of the intermediate cavity.
Visualizing
simulation results is useful for formalizing complex relationships between
several parameters. As an example, it is possible to point out the relationship
between the unbalance of loads
Δ
c,
parameters
characterizing
the sizes of flow sections of the hydraulic system channels, and the size
of the drive.
By adapting these values, it is possible to significantly increase the
allowable imbalance. A certain problem arose when formalizing the estimate of
the sensitivity of the manipulator to sharp, short-term relatively small power
disturbances. As a result, the choice fell on the use of visual information
only.
A relatively
simple condition was obtained which explicitly gives an estimate of the
feasibility of the required speed of drives operating in the mode of a given
law of motion.
The proposed
scheme for coordinated control of two parallel operating hydraulic actuators
proved to be very effective and can be recommended for use in industry. The
system provides a sufficiently high synchronism in the movement of the drives
when the imbalance of mass loads
ñ
1
(
ñ
2
)
is in the range of 0.4-0.6 and higher under the conditions of short-term power
disturbances. The procedure for visualizing the results of the analysis and
synthesis of the system simplifies preparation for finding the optimal solution
and, therefore, can be recommended for use in the educational process of
technical universities when studying the operation of drive systems and
mechatronic complexes.
The study was
financially
supported by the RFBR project No. 18-29-10072.
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