ISSN 1052-6188, Journal of Machinery Manufacture and Reliability, 2019, Vol. 48, No. 5, pp. 408–415. © Allerton Press, Inc., 2019.
Russian Text © The Author(s), 2019, published in Problemy Mashinostroeniya i Nadezhnosti Mashin, 2019, No. 5, pp. 34–42.
MECHANICS OF MACHINES
Mathematical Modeling of Copying Control by a Robotic
Unit with a Linear Electric Drive with an Elastic Link
A.V. Mal’chikova, S. F. Yatsuna,*, and A. S. Yatsuna
aSouthwest State University, Kursk, 305040 Russia
*e-mail: teormeh@inbox.ru
Received September 21, 2018; revised June 25, 2019; accepted June 25, 2019
Abstract—This article is about developing the mathematical model and studying the copying control
system of a robotic unit with a linear electric drive with an elastic link. Particular emphasis is placed
on modeling the nonlinear electric drive and measurement system devices. A comprehensive criterion
is elaborated for evaluating the quality indicators of the control system. Opportunities for minimizing
the influence of nonlinearities on the stability of the copying control system by optimization of parameters are studied.
Keywords: mathematical model, copying control system, linear electric drive, elastic link, nonlinear
effects.
DOI: 10.3103/S1052618819050054
INTRODUCTION: ANALYZING THE CURRENT STATE OF THE PROBLEM
Bioengineering control systems in which a robot arm copies the movements of an operator’s hand or
leg have gained widespread use in Russia and abroad since the 1960s. This control principle, referred to as
copying, significantly expands the operator’s functionalities because the human operator can make movements he is unable to make without a robotic unit. In addition, control can be exercised by the remote
supervision of a robot at a fairly large distance from the workplace. It is also important that tasks can be
fulfilled at scale, e.g., the movement of an operator’s hand by a centimeter is equal to the shift of the robot
arm by a millimeter, etc. [1].
We should separately highlight the application of copying control systems in so-called portable robotics
tasks, where a robot arm moves together with human extremities by intensifying or accelerating the muscular work.
However, despite several publications [2–7] and implemented systems with a copying control system,
the techniques of synthesizing parameters and analyzing the behavior of CCSs with an elastic member in
the drive still remain insufficiently elaborated.
TOPICALITY OF THE PROBLEM
One of the ways to improve the performance and increase the robustness of copying control systems
(CCSs) is to introduce elastic members in the electric drive: these robotic systems have been referred to as
elastic or soft robots [8–13]. First of all, this engineering solution allows making more efficient and
expanding the application range of portable robotic units called power armor, which can be put on several
body parts or on a particular extremity [14, 15]. However, the further advancement of these devices is
impeded by the fact that the copying control applied in them often fails to ensure the necessary performance quality. This has to do with the fact that the introduction of elastic members in CCSs with nonlinear elements typical of electric drives and measurement systems in certain modes changes the behavior of
the automated control system, compromises accuracy, and extends transitional processes, i.e., downgrades the performance quality and stability of the control system [16]. To identify these modes, it is necessary to develop a mathematical model of the copying control system and the stability investigation procedure.
It is convenient to test an automated control system with several nonlinear elements for stability by
probing the region of variable parameters and defining the transitional process duration by numerically
solving nonlinear differential equations at each point of the hyperspace. The multidimensional space is
408
MATHEMATICAL MODELING OF COPYING CONTROL BY A ROBOTIC UNIT
1
2
MM
409
MG1 3
5
6
7
MG2
c32
Fact
4
Fig. 1. CCS linear electric drive circuit.
y
b3
b2
x3
Fact
P23 c
Ffr
m3
23
b1
x2
b23
P32 P12
P2
m2
b12
x1
P21
FM
c12
m1
x30
x
Fig. 2. Design model of the device with a CCS.
probed using irregular meshes, the projections of which to any facet of the hypercube of smaller dimensionality do not lead to a major reduction in the number of points. Sequences of random points are used
as a sequence of points. A random search procedure is presented in several works [17, 18]. The variable
space dimensionality is defined by variable parameter vector a ∈ R N , where N is the number of parameters defining the properties of the CCS.
Essentials of the Problem Solution and Design Model
One of the drives broadly used in robotics is linear electric drives based on a lead screw (LSCR), a
reducer, and an electric motor. Depending on the parameters of this transmission and, first of all, the
LSCR step, it is possible to ensure a running mode implementing the self-braking effect as well as dry friction. In addition, the electric motor itself has nonlinear properties like a dead zone and saturation. The
copying control system must have a human–robot communication system so as to ensure almost synchronous movements of the operator and the machine. For this purpose, the CCS has displacement or force
sensors and, sometimes, muscular activity probes that transmit information to an onboard computer that
forms control commands supplied to the drives. The CCS measurement system with these sensors also has
nonlinear properties like an air gap, allowance, and saturation. For the layout of the electric drive with an
elastic member putting the robot link in motion, see Fig. 1, where 1 is the electric drive producing rotation
torque M M ; 2 is the reducer with gear ratio kG1 ; 3 is the cylindrical reducer with gear ratio kG 2 ; 4 is the
LSCR bolt; 5 is the LSCR nut; 6 is the drive actuating link, and 7 is the additional elastic member.
Mathematical Models of Control System Parts
Let us consider the CCS design model (Fig. 2) of three solids interconnected by viscoelastic members.
The agreed set of indices used in Fig. 2 is as follows: mass m1 models the robot’s biological link, m2 is
the robot’s actuating link, m3 is the drive’s actuating link, c12, b12 are the stiffness and viscosity coefficients
of the measurement system; c23 , b23 are the stiffness and viscosity coefficients of the elastic member that
connects the electric drive with the robot’s actuating link; x1, x2 , x3 are the absolute mass coordinates; b1 ,
b2 , b3 are the absolute viscous resistance coordinates; Fact is the force actuated by the electric drive; Ffr is
the friction in the mechanical gear of the electric drive; FM is the driving force that affects m1 on the part
of the operator; P12 = P21 is the force measured by the measurement system sensors; P23 = P32 is the force
of the elastic member embedded between the electric drive’s actuating link and the robot’s link; P2 is the
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force to be overcome in motion. Link m1 is affected by FM , P12 . Link m2 is affected by P12 , P32 , P2 . Link m3
is affected by Fact , P23 , Ffr . In addition, all the bodies are affected by viscous resistance.
The mathematical model of the force produced by the electric drive is found as
0, at I < I 0,

Fact = kact I , at I 0 < I < I max ,
kact I max , at I > I max .

(1)
kact is the power coefficient of the electric drive defined as
kG × 2πCM
(2)
η,
n
where kG is the aggregate gear ratio of the planetary and parallel-shaft reducer, CM is the torque constant
of the electric motor, n is the lead screw step, and η is the reduced aggregate efficiency of the unit.
In formula (1) I 0 is the minimal current in the electric drive and I max is the maximal starting current of
the electric drive.
The dry friction force is found as
kact =

 max
−Ffr sign( x3 ), at x3 ≠ 0;
n
 n m3
max
m3
Ffr = − F j , at x3 = 0 and
F j ≤ Ffr ;
j =0
 j =0
n
n



−Ffrmax sign  F jm3  , at x3 = 0
F jm3 > Ffrmax .

j =0
 j =0





(3)

n
F jm3 is the sum of the external forces affecting m3 but for Ffr ; Ffrmax is the limit friction force found by
the experiment. The modeled self-braking effect works as follows: the actuating link with mass m3 can
move only when affected by the force produced by electric drive Fact on the condition it exceeds the aggregate forces that affect m3.
Let us record the differential equation of motion for m3 as
j =0
n
mx3 + bx3 = Р32 − Ffr −
n
x3 = 0,
where

n
j =0
at

j =0
G mj 3
> Fact ,
G ,
j =0
n
m3
j
(4)


sign  G mj 3  ≠ sign(Fact ),
 j =0


G mj3 are the aggregate external forces that affect m3 but for the force of electric drive Fact . In
addition, there are restrictions for operational stroke x3 conditioned by the electric drive design. Let us
express these restrictions as 0 ≤ x3 ≤ x30 .
Now we shall consider the layout of the CCS measurement system (Fig. 3).
The measurement system consists of two piezoresistive sensitive elements each of which is mounted on
I
II
an elastic base with stiffnesses c12
и c12
. The circuit for clamping the sensitive elements allows adjusting gap
Δ12 between them and mass m1 (Fig. 3).
The strain of each resistive strain gage is evaluated using the half-bridge circuit from Fig. 3, where R0
is the measuring resistive strain gage; R1 , R2 , R3 are the compensating resistors. Specialized microcircuit
chip HX711 is used as an analog-to-digital converter. Then the digital signals from both ADCs are supplied
to the microcontroller that forms the input command for the electric drive control system in which case
the output quantity used in ACS P12 is defined as the difference in the readings from both ADC sensors (5):
I
II
(5)
P12 = P21I − P12II = c12
( x1 − x2 ) − c12
( x2 − x1).
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x
m2
x2
Measuring
path
Measuring path
R2
R1
x1
R0
I
P12
I
c12
II
c12
A+
A
E+
E
R3
FM
HX711
*
P21
P12
*
P12
II
P12
2'12
m1
x
Fig. 3. CCS measurement system circuit.
If the measurement system sensors are mounted with a gap, then, taking into account the stiffness of
II
both sensitive elements found as c I12 = c12
= c12 , we have
0, at x1 − x2 ⇐ Δ12
P12 = 
c12(( x1 − x2 ) − Δ12sign( x1 − x2 )),
at
(6)
x1 − x2 > Δ12,
where 2Δ12 is the gap size.
At Δ12 = 0 P12 = c12( x1 − x2 ); therefore, any movement of the operator produces measurable quantity
P12 found as
P12 = c12( x1 − x2 ).
(7)
To probe the space of parameters of vector a ∈ R N , we record the differential equations describing the
motion of the mechanical system considered as
MX + Φ( X , X ) = Q,
(8)
where
 m1 0 0 
M =  0 m2 0  ;


 0 0 m3 
 x1 
X =  x2  ;
 
 x3 
(9)
P21 + b1 x1



Φ( X , X ) =
−P12 + P32 + b2 x2  ;


 −Fact − P23 + Ffr + b3 x3 
(10)
 FM 
(11)
Q =  P2  .
 
 0 
To generalize the research and optimization results, we reduce system of equations (8) to a dimensionless form for which purpose we shall introduce such quantities as τ = t defined as the dimensionless time
T
xi
(T is the time scale), and xi = defined as the dimensionless movement ( L is the movement scale).
L
Then the expressions true for P12 and P23 are
P12 = c12( x1 − x2 ) = c12L( x1 − x2 );
(12)
P23 = c23( x2 − x3 ) = c23L( x2 − x3 ).
(13)
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Cont
AMP
*
P12
U
Dr
ADC
Fact
m3
C
P23
m2
P12
М
FM
x1
m1
P12
Fig. 4. Function chart of the electric drive control system.
The result of transforming system (8) is
2
 x1 = fM − ξ12 ( x1 − x2 ) − ζ1 x1,

2
2
 x2 = fP − m32 ξ32 ( x2 − x3 ) + m12 ξ12 ( x1 − x2 ) − ζ2 x2,
2
  x3 = fact − ffr + ξ32 ( x2 − x3 ) − ζ3 x3,
(14)
F T2
c
FMT 2
PT 2
F T2 2
c
2
2
2 2
, fact = act , fP = 2 , fP = fr , ξ12
= k122 T 2 = 12 T 2 , ξ32
= k32
T = 32 T , ζi =
m1L
m3L
m2 L
m2L
m1
m3
bi
m3
m1
= T ; m12 = , m32 = .
2nT
i
mi
m2
m2
where fM =
Function Chart of the Copying Control System
Let us consider the function chart of the electric drive copying control system presented in Fig. 4. This
system consists of a controller, an amplifier, a drive, an actuating drive link, an actuating robot link, a biological link, and a converter.
Force FM generated by the operator’s muscles leads to the movement of biological link m1 . The controlling voltage formed by the power proportional plus derivative (PPD) controller is
u
k +1
 P k − P12k −1 
k
= k p (P12 ) + kd  12
,
Δt


(15)
where P12k is the effort of the sensor-measured sleeve gasket at step k of the sensor’s scanning, P12k −1 are the
sensor’s readings at the previous step (sensor’s scanning), Δt is the time section between neighboring measurements, and k p and kd are the controller’s coefficients. The amplifier transforms the controller’s voltage U = kampu that is supplied to the electric drive forming force Fact , which results in the movement of
masses m3 and m2 .
MATHEMATICAL MODELING: DISCUSSION OF RESULTS
IN THE SCIENTIFIC AND APPLIED SENSE
The resulting mathematical model of the controlled electromechanical system allows conducting the
optimal synthesis of the parameters of the controlled system as exposed below. Having found x1 and x2 ,
we find the position error as ε = x1 − x2 and set the electric drive’s and the controller’s parameters that
allow minimizing ε. To evaluate the system’s performance for efficiency, we shall introduce criteria such
T
as K I = 1
x1 − x2 dt defined as the integral criterion of error; K A = max( x1 − x2 ), t ∈ [0,T ] defined as
T 0
the amplitude criterion of error; KT = ts defined as the duration of the transitional process defined by the
pertinence of an error to a five percent range of deviation from the steady-state value. We shall take as the
target function the criterion that takes into account the integral and the amplitude error, as well as the
duration of the transitional process, and record the function as

In = In(a ) = α1K I + α2K A + α3KТ ,
(16)
where α1, α2 , and α3 are the weight coefficients.
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k~d
0.4
~
xi
0.04
~
kp = 4
~
kd = 0.37
0.02
0.3
0
0.02
0.2
0.1
0
0.4
20
40
0.06
0.04
[21 = 400,
[32 = 200
0.02
0
60
80
~
kp = 44
~
kd = 0.17
0.04
~
kp = 4
~
kd = 0.37
0.3
0.02
0
0.2
0.1
0
413
[21 = 400,
[32 = 800
20
40
60
80
k~p
~
kp = 75
~
kd = 0.21
0.04
0.02
0
~
kp = 75
~
kd = 0.15
0.02
0
0
0.4
0.8
1.2
1.6
W
Fig. 5. Comprehensive parameter of CCS efficiency in the parameter region of the PPD controller and the set of transitional characteristics at various parameters.
The parameters that define the nonlinear properties of the CCS can be divided in unchangeable, e.g.,
masses, friction coefficient, etc., and variable components that form vector a = (kact k p kd c12c32 ) . We conducted several computational experiments in the space of these parameters to find response surface In .
Let us assume that at In < In0 the system meets the requirements on stability and strength. The respective
values of the motor’s and the outfit’s unchangeable parameters used in the modeling for a 60 W rotational
DC motor are kact = 6.8 ; R = 1.2; L = 0.008 and m1 = 10; m2 = 25; m3 = 2; b1 = b2 = b3 = 100, Δ12 = 0.002.
We used as the input effort the rectangular function described as
0, t ∈ [0, 0.1),
100, t ∈ [0.1, 0.3),
fM = 
−100, t ∈ [0.3, 0.5),
0, t ∈ [0.5, + ∞).
(17)
For the curves of the relation of comprehensive criterion In(a ) to the values of kp, kd, see Fig. 5, where
the regions of modes satisfactory and unsatisfactory for In are outlined and examples of transitional processes for steady and unsteady modes are shown.
To evaluate the size of the system’s satisfiability (stability) region, we used the control criterion
λ = S1 /S0 defined by the relation of the number of experiments with the steady-state mode to the overall
number of computational experiments (Fig. 6). That said, we treat as satisfactory the mode in which tranJOURNAL OF MACHINERY MANUFACTURE AND RELIABILITY
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S1/S0
1.0
(a)
0.8
4
0.6
0.4
0.2
0
(b)
min(S1/S0)
0.50
0.45
0.40
0.35
0.30
0.25
0.20
0.15
0.10
0.05
1 2
3
100 200 300 400 500 600 700 800
[32
0
200
400
600
800
1000
[11
Fig. 6. Evaluation of the influence of the stiffness of the drive’s elastic link and the measurement system on the system
stability: 1 is the results for ξ12 = 150; 2 is the results for ξ12 = 200 ; 3 is the results for ξ12 = 400; 4 is the results for
ξ12 = 800 .
sitional process duration ts is shorter than the target. We also introduce parameter λ m = min(S1 /S0 ) that
corresponds to the minimum of the relation ( S1 /S0 ) in the range of values of dimensionless stiffness ξ32 .
The behavior of this quantity as a function of dimensionless stiffness ξ21 is shown in Fig. 6b.
As shown by analyzing the diagrams presented in Fig. 6a, the size of the steady-state modes region relative to the total number λ = S1 /S0 depends significantly on the values of ξ32 and ξ21 . Monotonic changes
in criteria λ and λ m are observed. The region of parameters with the most unsatisfactory modes is defined
by ξ32 = 70 −350 and ξ21 = 350 at Δ12 = 0.002.
CONCLUSIONS
We have developed here a mathematical model of the nonlinear electric drive of the copying control
system, taking into account nonlinear effects with unchangeable and variable parameters, and suggested
the comprehensive control quality criterion, taking into account the integral and the maximal error as well
as the transitional process duration. Particular attention is paid to studying the influence on the quality
performance of the control system, the stiffness of the additional elastic member in the electric drive, and
the stiffness of the measurement system. It is shown that control quality criterion λ = S1 /S0 depends on
the stiffness of the elastic member. The diagram for evaluating the influence of the stiffness of the drive’s
elastic link and the measurement system on the system’s stability has three distinct regions: the first one
corresponds to high level λ = 1 at low values of ξ32 and expands with an increase in ξ21 ; the second one
corresponds to low level λ = 0.3−0.4 at average values of ξ32 and shifts to the right with an increase in ξ21 ;
the third one corresponds to high level λ = 1 at high values of ξ32 . The minimal control quality criterion
value λ m nonmonotonically depends on ξ21 and reaches the minimum at ξ21 = 350 . In practical terms we
have identified the region of parameters of the elastic suspension that allows ensuring a satisfactory quality
of the designed device.
FUNDING
This article was supported by the Russian Foundation for Basic Research, project no. 18-08-00773А “Studying
Regularities in the Interaction of Power Armor Feet with a Rough Bearing Area.”
CONFLICT OF INTERESTS
The authors declare that they have no conflict of interest.
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Translated by S. Kuznetsov
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