CC BY
2On the Problem of Optimal Control of the Motion
of Two-Link Planar Manipulator with Nonseparated Multipoint Intermediate Conditions
V.R. Barseghyan1,2*, S.V. Solodusha3, T.A. Simonyan2, Yu.A. Shaposhnikov4
1 Institute of Mechanics of National Academy of Sciences of the Republic of Armenia, Yerevan, Armenia
2 Yerevan State University, Yerevan, Armenia
3 Melentiev Energy Systems Institute of Siberian Branch of Russian Academy of Sciences, Irkutsk, Russia
4 Irkutsk State University, Irkutsk, Russia
Abstract — The study focuses on the problem of optimal control of the motion of a two-link planar manipulator on a fixed base with given initial and final conditions, nonseparated conditions for the values of the phase vector at intermediate times, and with a quality criterion given over the entire time interval. It is assumed that absolutely rigid links of the manipulator are interconnected by an ideal cylindrical hinge, and the similar hinge is used to attach the first link to the base. The optimal rules of changing the control moments are constructed, which allow the manipulator to move from a given initial state to a final one, satisfying nonseparated multipoint intermediate conditions. An application of the proposed approach is exemplified by constructed control functions and the corresponding motion with given nonseparated conditions for the values of the phase vector coordinates at some two intermediate times.
Index Terms: two-link manipulator, optimal control, nonseparated multipoint conditions, phase constraints.
I. Introduction Problems of control and optimal control of dynamical systems with given constraints on the values of the coordinates of the phase vector at intermediate times arise in a number of problems important for applications. Similar problems, in particular, are encountered in the case of control and optimal control of manipulation robots, aircraft, technological processes, energy-saving
* Corresponding author. E-mail: [email protected]
http://dx.doi.org/10.38028/esr.2022.04.0005
Received October 26, 2022. Revised November 11, 2022.
Accepted November 03, 2022. Available online January 30, 2023.
This is an open access article under a Creative Commons Attribution-NonCommercial 4.0 International License.
© 2022 ESI SB RAS and authors. All rights reserved.
control of thermal devices, and others [1-3]. Such a wide demand requires the development and design of modern (highly efficient) optimal control methods, which easily implement the control of the manipulator, leading to the desired movement. When studying the movements of manipulators and designing control systems, a mechanical model of a manipulator is usually used in the form of a system of absolutely rigid bodies (rods), which are connected with each other in series using ideal hinges [713]. Some important applied problems involve solving the problems of control and optimal control of the movement of manipulators as dynamic systems with nonseparated multipoint intermediate conditions. A characteristic feature of these problems is, along with the classical boundary (initial and final) conditions, the presence of nonseparated (nonlocal) conditions at several intermediate points of the considered interval. The study of these problems is of great importance for both theory and applications. Some issues of control and optimal control of linear dynamical systems with nonseparated multipoint intermediate conditions are examined, in particular, in [1-6].
This paper considers the problem of optimal control of the motion of a two-link planar manipulator on a fixed base with given initial and final conditions, nonseparated conditions for the values of the phase vector at intermediate times, and with a quality criterion given over the entire time interval. Based on the mathematical model of a two-link planar manipulator in the form of Lagrange equations of the second kind [14], in which the main moments are controls, we have constructed explicit forms of the optimal control action and the corresponding motion using the method of moment problems [15].
II. Mathematical Model of the Manipulator and Problem Statement
We consider a two-link manipulator (see Fig. 1) consisting of two absolutely rigid bodies (links) G1, G2 connected by hinge O2. Body Gj is connected with a fixed base by means of hinge O1. The hinges are ideal, cylindrical, and their axes are parallel to each other. The
Fig. 1. Two-link manipulator.
system moves in a horizontal plane perpendicular to the hinge axes O1, O2.
Each link of the manipulator is an absolutely rigid homogeneous rod of length L. It is assumed that link G2 includes the executive body (grip), i.e., the mass of the gripper is neglected and the dynamic characteristics are not considered separately. Manipulator is controlled by two independent drives D1, D2. Drive D1 carries out the interaction of body G1 with the base, and drive D2 carries out the interaction between link G1 and link G2 of the manipulator. The main force vectors generated by drives D1, D2 are equal to zero, and the main moments relative to the hinge axes O1, O2 are equal to M1, M2, respectively. Values M1, M2 are taken as control functions in the considered model of the manipulator. It is also assumed that control functions belong to the class of piecewise continuous functions. We do not take into account the action of other forces.
Let us introduce a fixed Cartesian coordinate system O1XY with the origin on the hinge axis O1 in the considered plane. Let us denote by 91, 92 the angles between the horizontal axis and the first and second links, respectively; I1,12 are the moments of inertia of bodies G1, G2 relative to the corresponding axes; L1 = |O1O2| is the distance between hinge axes, L2 = |O2C2| is the distance from axis O2 to the center of gravity C2 for link G2. The kinetic energy of the two links is equal to
K =1 ( I
2V 1
-m.
A ) 91 +1 ( 12 + m2 L\ )
<P 2-
(I2 + m2L2)(p2 + m2L1L2 cos (( - p2 ) (pj -
(1.1)
first link, which corresponds to the static balance of the second link of the manipulator. In this case, assuming that |O2C2| = L2 = 0, equation (1.1) has the form
— , — UU, xx^ — X4 , xx 4 — u 2, (1.2)
where
X = (I + m2Ll) 9l, x2 = (I + m2L) <Pl ,
(I2 + m2LL2 ) 92 , x4 = (I2 + m2lL2 ) (p2.
Control functions u1 and u2 have the form u1 = M1 - M2, u2 = M2, where M1, M2 are the main moments relative to the hinge axes.
Let the initial and final states of system (1.2) be given
x(t0) = C^'oX •^('oX x3(toX X4(t0))T, x(T) = (xx(T), x2(T), x3(T), xA(T))T, (1.3)
and, at some fixed intermediate time instants
0 < to < '1 < '2 < '3 = T, nonseparated (nonlocal) multipoint intermediate conditions
Ë Fkx(tk ) = a
(1.4)
be given, where a is a two-dimensional column vector, Fk are (2 x 4)-dimensional matrices (k = 1, 2), whose elements are real numbers [4].
In general, for some cases, it can be assumed that at intermediate times tk (k = 1, 2) not all values of the coordinates of the phase vector x(tk) are present in (1.4), but only some of them. In such cases, we will assume the corresponding elements of the matrix Fk to equal zero.
System (1.2) with multipoint intermediate condition (1.4) on the time interval [t0, T] is completely controllable [2, 14].
The optimal control problem for system (1.2) with nonseparated multipoint intermediate conditions (1.4) can be formulated as follows.
Find the optimal control actions u10(t) and u°2(t), t e [t0, T], which transfer the solution to system (1.2) from the initial state x(t0) to the final state x(T), thereby ensuring satisfaction of the nonseparated multipoint intermediate condition (1.4) and having the smallest possible value of the quality criterion ^[u°]:
+m2LL2 cos (- 92) (pj(p2.
The equations of motion of the considered manipulator in the form of Lagrange differential equations of the second kind have the form:
(Ij + m2L) (f>j + m2LlL2 cos (( - p2) (p2 +
+m2LL2 sin ((j - p2) (p2 = Mj -M2,
œ[u ] =
1
j (uj2 + u?,)dt
1
ö 2
(1.5)
III. Solution to the Problem
-m2LLsin (< - <2) <2 = M2.
It is assumed that the center of mass of the second link is located on the axis of hinge O2, connecting with the
To solve the problem, we write the solution to Eq. (1.2) following from the initial state x(t0), and by substituting the values x(tk) into (1.4) for the time instants t = tk (k = 1, 2), obtain the following relations:
£ FkX[tk, t0]x(t0) + £ j FkX[tk, t]Bu(T)dT = a. (2.1)
k=1 k=1 to
For a finite time t = T, we have
T
x(T) = X[T,t0]x(t0) + JX[T,T]Bu{T)dT, (2.2)
k=1
where X [t, t] denotes the normalized fundamental matrix of the solution to the homogeneous part of equation (1.2). The matrices B and X [t, t] have the following form:
x [t, t] =
œo 00
B = 1 0 0 0 ,0 10 ,
( t, t) X12 (t ,T) 0
0 X22 (t, T ) 0
0 0 X33 (t, T)
0 0 0
0 0
34 (t>T)
X44 (^ T)
X
where
xn(t, t) = x22(t, t) = x33(t, t) = x44(t, t) = 1;
x12(t, t) = x34(t, t) = t - t. (2.3)
Using the approaches given in [2, 4], from (2.1) and (2.2) we obtain the following integral relation
J H [t ] u(t )dt = n(t0,...., T)
to
where the following notation
H [t ] =
(2.4)
n(to,...., T ) =
F (t) B X [T, t]B a - Fx(t0 ) x(T ) - X [T, to]x(to)
F (t) = X Fk [t] =X FkX [tk, t],
k=1 k =1
F = £ F,X [t,, t0] = F (t0),
Fk [ t ] =
FkX[tk, t], for t0 £ t £ tk, 0 , for tk < t £ tm+x = T
k = 1,2, (2.5)
i.e., a = (ctj, a^7", F1 = F2 =
10 10 0 10 1
Substituting the expressions for matrices F1, F2 and the fundamental matrix of solution X [t, t] into formula (2.5), we have
F (t) = F X [tl,T} + F2 X [t2,t} =
f fil (t) fn (t) fn (t) fn (t) 1 , (2.7)
/21 (t) f22 (t) f23 (t) f24 (t)1
where
/H(T) = xn(t1,T) + xn(t2,T); /^(T) = x12(t1,T) + x12(t2,T);
/13(T) = x33(t1,T) + x33(t2,T); /14(T) = x34(t1,T) + x34(t2,T); /22(T = x22(t1 ,t) + x22(t2,T); /24(t) = x44(t1 ,t) + x44(t2,T);
/21(T) = /23(T) = 0. Therefore, matrix H[t] will be presented in the form: H [ t] =
(
(¿1, t) + ( ¿2 X22 (^ T) + X22 (¿2 X12 (T, T) X22 (T> T) 0 0
t) t)
X34 (¿1. T) + X34
(t7
(¿1 ,T) + X44 (¿2
.T) ö
,t)
is accepted. Here H[t] is a (6 x 2) block matrix, the known matrices F(t) and F have dimension (2 x 4), and n is a (6 x 1)-dimensional known column vector. For system (1.2) with nonseparated multipoint intermediate condition (1.4) to be completely controllable on the interval [t0, T], it is necessary and sufficient that the column vectors of the matrix H[t] be linearly independent on this interval. Let nonseparated intermediate values (1.4) have the form: xj(tj) + x3(tj) + xj(t2) + x3(t2) = ctj, x2(tj) + x4(tj) + x2(t2) + x4(t2) = a2, (2.6)
0 0
X34 (T, T) X44T (t2, T)
According to (2.4)-(2.6), we will have the following integral relations:
T
J[hjj (t)u1 + h12 (t)u2 Jdx = n 1,
»0
T
j[h2l (t) ul + h22 (t) u2 J dx = n2,
»0
T T
J hi (T) udx = n , j Ki(x)hi (x) dx = n 4,
10
T T
Jh52 (t)u2dx = n5, Jh62 (t)u2dx = n6, (2.8)
where the following notation
Au(t) = x^t^T) + xj2(t2,T);
h12(T) x34(t 1 ,t) + x34(t2,T);
^21 (T) =
- x22(t1 ,t) + x22(t2,T); h22(T) =
- x44(t1 ,t) + x44(t2,T);
h31(T) = X12(T,t); h41(T) = X22(T,t); h52(T) = X34(T,t); h62(T) = X44(T,t);
n1 = «1 - 2[x1(to) + x3(to)] -
- (t1 + t2 - 2to)[x2(to) + x4(to)]; (2.9)
n2 = 02 - 2[x2(to) + x4(to)];
n3 = x1(T) - x^) - (T - to) x2(to); n4 = x2(T) - x2(to);
n5 = x3(T) - x3(to) - (T - to) x4(to); n6 = x4(T) - x4(to)
is accepted.
For a given performance criterion s[m], the optimal control problem with integral condition (2.4) is a conditional extremum problem, where the minimum of the functional s[m] must be determined under conditions (2.4).
The left-hand side of condition (2.4) is a linear operation generated by function u(t) on the time interval [to, T], and the functional is the norm of a normed linear space. Then
k =1
the optimal control action u0(t), [t0, T], minimizing the functional œ[u] and satisfying condition (2.4) must be constructed according to the algorithm for solving optimal control problems using the moment problem method [15]. To solve the problem of moments (1.5) and ((2.8)), following [15], we need to find the quantities l,, i = 1,...,6, related by condition
6
^ l n = 1, (2.10)
for which
where
L
(p0 )2 = min j [h (t) + h22 (t)] dx, (2.11)
¿1(1) = /1^11(1) + /2^21(1) + /3^31(1) + /4^41(1), h2(x) = /1^12(1) + /2^22(1) + /5^52(1) + /6h62(x), (2.12)
To determine the quantities /°, i = 1,...,6, minimizing (2.11), we apply the method of indefinite Lagrange multipliers. Let us introduce the function
T
f = J[ ( hi(T))2 +( h2 (T))2 yz +1
E l. n -1
where 1 is the indefinite Lagrange multiplier. Based on this method, calculating the derivatives of function f with respect to /i, i = 1,.. .,6 and equating them to zero, we obtain the following system of integral relations:
T ^
j [hii (x)hi (t) + hi2 (x)h- (t)]A = - -n,
À
j [ h2i(t)hi (t) + h22 (T)h2 (t^t = - - n2
J h3i(x)hi (x)dx = - - n, J h4i(x)hi (x)dx = - - n, (2.13)
J h52(x)h2 (x)dx = - - n5, J h62(x)h2 (x)dx = - - n.
t 2 t 2
'0 '0
Given the notation (2.12), equations (2.13) can be written in the form of the following algebraic equations:
a11/1 + a12/2 + a13/3 + a14/4 + a15/5 + a16/6 = - (1/2)nl,
a21/1 + a22/2 + a23/3 + a24/4 + a25/5 + a26/6 = - (1/2)n2, a31/1 + a32/2 + a33/3 + a34/4 = - (1/2)n3, a41/1 + a42/2 + a43/3 + a44/4 = - (1/2)n4, (2.14)
a51/1 + a52/2 + a55/5 + a56/6 = - (1/2)n5, a61/1 + a62/2 + a65/5 + a66/6 = - (1/2)%
The following notation is used here:
T
au = J[(hu(x))2 +(hi2 (x))2 ~\dx =
te
= J[(^"12 (t, t))2 + 2Xi2 (te, t) x2 (t2, t) +
'0
+ (x34 (tt, t))2 + 2X34 (t ,t) X34 (t 2 , t) Ot +
12 r -,
+J |_(X12 (t2 > T))2 + (X34 (t2 , T))2 J 0T
t'
T
au = a21 = J [âu(x)â21(x) + h12(x)h22(x)]J t =
<0
<1
= J [ X12 (<1 >t ) X22 (<1 >t ) + X12 (<1 >t ) X22 (<2 >t ) +
<0
+X12 (<2>t ) 22 (<1, t) + x34
(1, t ) X44 (<1, t ) + + X34 (<1>t ) X44 (<2> t) + X34 (<2> t) X44
(<1, t)] dt +
<2
+J [X12 (<2 .t) X22 (<2 .t) + X34 (<2 .t) X44 (<2 't)] dt t0
T
a13 = a31 = J h11(x)h31(x)d t =
to
11 t2 = J x12 (t1, t) x12 (T, t)dt + J x12 (t2, t) x12 (T, t)dt,
to t0
T
au = a 4i = j hu(x)h4i(x)d t =
'0
= j x12 ('1,T) x22 (T, t) dt + j x12 (t2, t) x22 (T, t) dt,
'0 'o
T
ai5 = a5i = j h12 (t)h52 (t)d t =
to
ti <2 = j X34 (<1, t) X34 (T, t)dt + j X34 (<2, t) X34 (T, t)dt,
<0
T
ai6 = a6l = j h12 (t)h62 (t)d^ =
to
ti t2 = j x34 (t1, t) x44 (T, t)dt + j x34 (t2, t) x44 (T, t)dt,
T
a22 = J[( h2i (t) )2 +( h22 (t) )2 t = <0
t
= J [(*22 (<1 >t))2 + 2^22 (<1 >t) ^22 (<2 >t) +
<0
+ (x44 (<1, t))2 + 2x44 (<1, t) x44 (<2, t) dt
+Jl( X22 (<2 't))2 +(
X44
(<2, t ))2
<0
T
a23 = a32 = j h21(x)h31(x)d t =
d t,
*1 2 = | x22 (t1, t) x12 (T, t)dt +1 x22 (t2, t) x12 (T, t)dt,
t0 t0 T
«24 = «42 = j h21(t)h41(x)dt =
to
k h = j x22 (t1, t) x22 (T, t)dt + j x22 (t2, t) x22 (T, t)dt,
to to
T
a25 = a52 = J h22 (t)h52 (t)d t =
to
k t2 = J x44 (t1, t) X34 (T, t)dt + J x44 (t2, t) X34 (T, t)dt,
to to
T
a26 = a62 = j h22(t)h62(t)dT =
to
k h = j x44 (^, t) x44 (T, t)dt + j x44 (t2, t) x44 (T, t)dt,
io t0
7 T 2
a33 = | (A31(t))2dt = | (x12 (T, t)) dt,
to to
T T
= a43 = J h31(x)h41(x)dt = J x12 (T, t) x22 (T, t)dt,
to t0
T T
a44 = j (MT))2dT = j (X22 (T> T)) dt
«55 = j (MT))2dT = j (X34 (T, T)) dt,
to t0
T T
a56 = a65 = Jh52(x)h62(x)dt = JX34 (T,t)x44 (T,t)dt,
'o 'o
T T
a66 = j ( MT) )2 d T = j ( X44 (T > T ^
'o 'o
Adding condition (2.1o) to the obtained equations (2.14), we obtain a closed system of seven algebraic equations for the same number of unknown quantities l,, i = 1,.. .,6, and X. Let the quantities l,, i = 1,.. .,6, and Xo be the solution to this closed system of algebraic equations. Then, according to (2.11), (2.12), we have
hi (t) = 10hn(t) +10h2l(t) +10h31 (t) +10hAl(t), h? (t) = 10h^ (t) + /20A22 (t) +150h52 (t) +160h62 (t), (2.15)
< (t ) =
0 (t ) =
[li° ((^ x) + (t2, x)) + l2° (X22 (ti, x) + x22 (i2, x)) +
Po
+l°Xi2 (T, t) + l>22 (T, t )], t Î [t°, ti ),
P2 [li°Xi2 ( t2 , t ) + l°°X22 (t2, t ) + Xi2 (T, t ) + lO*22 (T, t )] , P0
t Î [t1, t2 ),
P2 [lO Xi2 (T, t) + lO X22 (T, t )] , t Î [t2,T ],
P2
-12[/10 (^34 ('l> + ^34 (Î2, x)) + (^44 (il, + *44 ^)) +
Po
+/>34 (T, t) + 10X44 (T, t ), t Î [to, ti ),
P2 [I" X34 (t2, t) + /2 X44 (t2, t) + l50 X34 (T, t) + l60 X44 (T, t)], po
t Î [t1, t2 ),
-2[/0X34 (T, t)+/0x44 (T, t)], t g [t2,T],
Po
or, given (2.3), they can have the form:
-I[I,0(/, + i2 -2t) + 2/0 +10(T-t) + /0
t G [to,)
«0 (t ) =
^ [/0 (t2-t)+ /0 + /0 (T-t)+ /0 ] , t e [to, t2
-Or [/30 (T-t)+ /0 ],
t e [t2,T],
0 (' ) =
[/0(i, +12 -2t) + 2/0 +10(7-t) +10
Po
i G [io, t, )
P_ [/0 (t2 -t) + /0 + /0 (T -t) + /0 ], t G [t,, t2 P0
-2 [/0 (T-t) + /0 ], P0
t G [t2,T].
(P0)2
-( Ä20(t))2
^.
Following [15], the optimal control actions can be represented as:
U (t) = -1 h? (t) ; «0 (t) = -1 h (t).
Po Po
Taking into account the notation (2.15), the optimal control actions are represented as follows:
Substituting the expression for the optimal control action into (1.2) and integrating these equations, we obtain the optimal motion on each time interval.
IV. Example
Let some fixed intermediate times o < to < tj < t2 = T, and to = o; t1 = 2; t2 = 3; T = 4 be given. The initial and final states for phase vector x = (x1, x2, x3, x4)T will be x(o) = (o, o, o, o)T, x(4) = (5, o, 4, 1)T.
According to formula (2.9), assuming that a1 = 3, a2 = 2, we obtain the following value for the constant vector n:
n = (n1, n2, n3, n4, n5, n6)T = (3, 2, 5, o, 4, 1)T.
Further, carrying out the corresponding calculations of the integrals for the coefficients of the system of equations (2.14), we obtain
an = 42, a12 = a21 = 25, a13 = a31 = a15 = a51 = 121/6,
^14 = a41 = a16 = a61 = 13/2, a22 = 18, a 23 a32 a 25 a 52 27/2, a24 a42 a26 a62 5,
0
0
Fig. 2. Graphs of the vectorfunction of th e optimalmovement x0(t) at t e [0,4] by coordinates: a) x1 (t) ;b) x2(t) ; c) x3 ( t); d) x°(t).
a33 = 64/3, a44 = 4, a34 = a43 = a56 = a65 = 8, a55 = 644/3, a66 = 4. Solving the system of algebraic equations (2.14) with the obtained numerical values of the coefficients for l,0, i = 1,.,6, and X0, we obtain the following values:
w =-
18528
243299
l4 = -
L = -
24702 243299 22015
243499 '
21616 423229
9 15
48965 2—3299
I5 =
1 = -
39867 243299 : 3924
243295'
Based onthe notation (2.5), the optimal functions /^"(x), hi (x), o e[t0 ,T( are represented as follows:
h0(T) =
— 2 +t2— 2x) 1- 2l\ 2—152-%
h0(t) =
/0 (t2-t) + /0 +130 (r-t) + /0, /3° (r-t)+/;,
/" (t, +12 - 2t) + 2/20 + /50 (T -t
/0 (t2-T)+/0 + /5° (T-t)+/; /5° (t-t)+/;0,
+1 °
^ t1 ) , = K t2 ) , [t2 ,
+ / 0
;
t2 ) , ; [t2, T ].
(Po)2 =
1952 54832^^ '
Further, for the components of the vector of optimal control action u0(t), t e [t0,T], we will haw explicit expressions in the following form:
'1.6014-2.0026/, ie[0,2], (0 = | 31.6998-11.49442, t e(2, 3],
71.2899 - 20.9862/, / e (3,4],
0.7264-L940K, h e[0,2], a, (0 = [.^0.822t! -10.96 [9t,/ e (2, 3.
60.4249-00.42470, 6e(4,4], I2 w,e substitute the obtained expiesiions for the optimal coilrol inti (4.22) and integrate these equations, then we obtain the optimal motion on each time interval in the following form:
x0(t ) =
0.8007t2 -0.3338t3,
t e[0,2],
15.8499t2 - 1.9157t3 + 34.8852 - 41.2131t,
ie(2,3],
35.2429222349774 3-127.6147 - 117.2705t,
Now, calculating the value of (p0)2 according to formula (2.15), we obtain
x0(t ) =
1.60141 -1.001312
t e (3,4], t e [0,2],
31.69981-5.747212 -41.2131, t e (2,3], 71.28991 -10.493012 -117.2705, t e (3,4],
x0(t ) =
0.3632t2 - 0.2400t3,
t î [0,2],
15.4124t2 - 1.8219t3 + 34.8852 - 41.2131t,
t î (2,3],
35.2075t2 -3.4039t3 +127.6147 -
- 117.2705t,
t î (3,4],
Kit ) =
0.72641 - 0.720011
t e [0,2],
30.82481-5.465912 -41.2131, t e (2,3], 70.41491 -10.211812 -117.2705, t e (3,4].
Figure 2 shows a graphical view of the vector-function of the optimal movement x0(t) at t e [0,4] by coordinates
X°(0 , x02(t), x0(t), x°4(t).
Note, by direct substitution, we can verify that the obtained optimal motions satisfy condition (1.4), i.e.,
FlX° h )+F2 x (t2 )=œ3
Thus, we have obtained explicit expressions for the optimal control and the corresponding optimal motion for system (1.2) with given initial and final values of the phase vector and nonseparated intermediate conditions (1.4).
V. Conclusion
The problem of optimal control of the motion of a two-link planar manipulator on a fixed base with given initial and final conditions and nonseparated conditions for the values of the phase vector at intermediate times is solved. The application of the proposed approach is exemplified by the construction of the functions of optimal control and the corresponding optimal motion with given nonseparated conditions for the values of the phase vector coordinates at some two intermediate times. The constructed corresponding graphs for the coordinates of the phase vector of the manipulator confirm the results obtained.
Acknowledgment
The research of S. Solodusha was carried out within the state assignment of Ministry of Science and Higher Education of the Russian Federation (Project FWEU-2021-0006, theme no. AAAA-A21-121012090034-3).
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as
Vanya Rafaelovich Barseghyan received the PhD. degree from Leningrad State University in 1988. In 2000 he re-ceived the Dr. Eng. degrees in mathematical cybernetics and mathematical logic and in theoretical mechanics from the Institute of Informatics and Automation Problems of the National Academy of Sciences of the Republic of Armenia in the specialties. In 2002, he was awarded the academic title of professor. Currently, he is a leading researcher at the Insti-tute of Mechanics of the National Academy of Sciences of the Republic of Armenia and a professor at Yerevan State University. He is a laureate of the 2012 Presidential Prize in the field of development of natural sciences. He is an au-thor of more than 170 publications, including the monograph "Control of Composite Dynamical Systems and Systems with Multipoint Intermediate Conditions" (Nauka Publ., Moscow, 2016), and chapters and sections in five collective monographs.
Yuri Andreevich Shaposhnikov
entered Irkutsk State University in 2019. Currently he is a 4th year student majoring in Applied Mathematics and Computer Science. His research interests include the study of computer and mathematical modeling in engineering.
Svetlana Vitalievna Solodusha received the Ph.D. degree from Irkutsk State University in 1996. In 2019, she received the Dr. Sc. degree in mathematical modeling, numerical methods and software packages (engineering) from the Melentiev Energy Systems Institute SB RAS. Currently, she is the head of the Laboratory of Unstable Problems of Computational Mathematics. Her research interests include mathematical methods for identifying dynamics based on integral models. She is an author and co-author of more than 100 scientific papers.
Tamara Alexanovna Simonyan
received the Ph.D. degree in Physics and Mathematics from Yerevan State University in 1999. She is an Associate Professor of Department of Mechanics, Faculty of Mathematics and Mechanics, at Yerevan State University. Her scientific interests include optimal control and stabilization of systems, differential games, and their applications. She is an author and coauthor of 32 scientific papers.
