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21
Нелинейная
динамика и нейронаука
Известия высших учебных заведений. Прикладная нелинейная динамика. 2021. Т. 29, № 3 Izvestiya Vysshikh Uchebnykh Zavedeniy. Applied Nonlinear Dynamics. 2021;29(3)
Article
DOI: 10.18500/0869-6632-2021-29-3-398-408
On trajectory tracking control of prismatic and revolute joined robotic manipulators
Abstract. The purpose of this paper is to construct a trajectory tracking feedback controller for prismatic and revolute joined multi-link robotic manipulators using a new form of sliding modes. Methods. In this paper, the Lyapunov functions method has been applied to establish the stability property of the closed-loop system. Results. Due to the presence of rotational joints, the motion equations of the manipulator are periodic in the angular coordinates of the corresponding links. A control law is constructed which is also periodic in the angular coordinates of the links. Thus, a closed-loop system has not one, but a whole set of equilibrium positions that differ from each other by a multiple of the system period. The dynamics mathematical model of a complex five-link manipulator with cylindrical and prismatic joints has been constructed on the basis of the Lagrange equations. Simulation results on a 5-degree-of-freedom robotic arm demonstrate the applicability of the proposed control scheme. Conclusion. We obtain a relay controller such that the set of all equilibrium positions of the closed-loop system is uniformly asymptotically stable. The novelty of the obtained control law is based on a new approach that takes into account the periodicity of the model in angular variables with the solution of the tracking problem in the cylindrical phase space. The simulation results for a 5-degree-of-freedom robotic manipulator clearly show the good performance of our controller. The applied significance of the results obtained in the paper is as follows. At present, in connection with the widespread introduction and mass production of manipulators, it seems important to develop the mathematical foundations for designing a control structure that has a universal character, namely, allowing to perform the required process without additional adjustment of control parameters with simple and convenient algorithms and programs of their implementation.
Keywords: trajectory tracking control, robot manipulator, revolute and prismatic joints, Lyapunov functions method, sliding mode, nonlinear system.
Acknowledgements. This work was supported by Russian Foundation for Basic Research, grants No. 18-41-730022, 19-01-00791.
For citation: Andreev AS, Peregudova OA, Petrovicheva YuV. On trajectory tracking control of prismatic and revolute joined robotic manipulators. Izvestiya VUZ. Applied Nonlinear Dynamics. 2021;29(3):398-408. DOI: 10.18500/0869-6632-2021-29-3-398-408
This is an open access article distributed under the terms of Creative Commons Attribution License (CC-BY 4.0).
A.S. Andreev, O.A. Peregudovaи, Yu. V. Petrovicheva
Ulyanovsk State University, Russia E-mail: asa5208@mail.ru, Elperegudovaoa@gmail.com, petrovichevayulia@yandex.ru Received 1.11.2020, accepted 22.03.2021, published 31.05.2021
Научная статья УДК 62-503.51
DOI: 10.18500/0869-6632-2021-29-3-398-408
Об управлении движением роботов-манипуляторов с призматическими и вращательными шарнирами
А. С. Андреев, О. А. Перегудоваи, Ю.В. Петровичева
Ульяновский государственный университет, Россия E-mail: asa5208@mail.ru, Elperegudovaoa@gmail.com, petrovichevayulia@yandex.ru Поступила в редакцию 1.11.2020, принята к публикации 22.03.2021, опубликована 31.05.2021
Аннотация. Цель настоящего исследования - с использованием новой формы скользящих режимов построить закон управления для отслеживания траектории многозвенных роботов-манипуляторов с призматическими и вращательными шарнирами. Методы. В данной работе для установления свойства устойчивости положений равновесия замкнутой системы применяется метод функций Ляпунова и его развитие для неавтономных систем. Результаты. Из-за наличия вращательных шарниров уравнения движения манипулятора являются периодическими по угловым координатам соответствующих звеньев. Построен закон управления, также являющийся периодическим по угловым координатам звеньев. Таким образом, замкнутая система имеет не одно, а целое множество положений равновесия, которые отличаются друг от друга на величину, кратную периоду системы. На основе уравнений Лагранжа построена математическая модель динамики сложного пятизвенного манипулятора с цилиндрическим и призматическим шарнирами. Результаты моделирования на примере роботизированной руки с 5 степенями свободы демонстрируют применимость предложенной схемы управления. Заключение. Для многозвенных роботов-манипуляторов с призматическими и вращательными шарнирами получен релейный закон управления, такой что множество всех положений равновесия замкнутой системы равномерно асимптотически устойчиво. Новизна полученного закона управления основана на новом подходе, учитывающем периодичность модели в угловых переменных с решением задачи слежения в цилиндрическом фазовом пространстве. Результаты моделирования робота-манипулятора с пятью степенями свободы ясно показывают хорошие характеристики нашего закона управления. Прикладное значение полученных в статье результатов состоит в следующем. В настоящее время в связи с повсеместным внедрением и массовым производством манипуляторов представляется актуальной разработка математических основ проектирования управляющей структуры, имеющей универсальный характер, а именно позволяющей без дополнительной настройки управляющих параметров выполнять требуемый процесс с помощью простых и удобных алгоритмов и программ их реализации.
Ключевые слова: отслеживание траектории, робот-манипулятор, вращательный и призматический шарниры, метод функций Ляпунова, скользящий режим, нелинейная система.
Благодарности. Работа выполнена при поддержке РФФИ, гранты №№ 18-41-730022, 19-01-00791.
Для цитирования : Андреев А. С., Перегудова О. А., Петровичева Ю. В. Об управлении движением роботов-манипуляторов с призматическими и вращательными шарнирами//Известия вузов. ПНД. 2021. T. 29, № 3. С. 398-408. DOI: 10.18500/0869-6632-2021-29-3-398-408
Статья опубликована на условиях лицензии Creative Commons Attribution License (CC-BY 4.0).
Introduction
The trajectory tracking control problem for robotic arms has been considered in great detail in the literature (see the survey [1] and the monograph [2]). Many control schemes have been proposed by the researches to solve the control problem for serial robot manipulators. A feedback linearization method has been used in [1,3], a sliding mode control methodology has been considered in [4-8], a passivity-based technique has been applied in [9], and a robust adaptive control design has been implemented in [10]. In [11], a robust control scheme has been proposed based on the combination of both proportional derivative (PD) and sliding mode terms for the global trajectory tracking of robotic manipulators with uncertain dynamics. In [12], a PD with a feedforward scheme has been proposed to provide the global tracking of robot-manipulators under the bounded constantly acting disturbances.
The main contribution of this paper consists in the following. We propose a solution to the trajectory tracking control problem for prismatic and revolute joined robotic manipulators using a discontinuous bounded controller without a feedback linearization.
We denote by || ■ ||, both the Euclidean vector norm and the operator matrix norm. We use the symbol lgn|| ■ || as a logarithmic matrix norm which corresponds to the Euclidean vector norm and can be calculated as lgn||A|| = A,max(AT + A)/2 y A e Rraxra, where Xmax(-) is the maximum eigenvalue. The symbol E e Rraxra denotes the identity matrix.
1. Robot Model and Preliminaries
The dynamics of an n-degree-of-freedom (DOF) serial rigid robotic manipulator with revolute and prismatic joints is defined by the following equations
A(q)q + C(q, q)q + g(q) + d(q, q) = u, (1)
where the vector of joint positions is represented by q e Rra, the inertia matrix A(q) e Rraxra is positive definite, the Coriolis and centrifugal forces are expressed by the term C(q, q)q with C(q, q) e Rraxra, the gravitational forces are expressed by the vector g(q) e Rra, the vector d(q, q) e Rraxra is the viscous damping forces of the manipulator links, u e Rra is the vector of control forces.
Proposition 1. Let Jr and Jp be two subsets of the set ,Jn = {1,2,..., n} such that Jr U Jp = ,Jn and Jr n Jp = 0 . Let the generalized coordinates qi Vi e Jr be the angular displacements of the revolute joints, and let the others qi Vi e Jp be the linear displacements of the prismatic joints.
Remark 1. The matrices A(q) and C (q, q) have some properties such as follows [2].
1. The following inequalities hold
X?|M|2 < qTA(q)q < ^|M|2 Vq e Rra, (2)
where Xi and \2 are some positive constants.
2. The matrix A(q) — 2C(q, q) is skew symmetric.
Define the set of the reference trajectories for the robot-manipulator (1) as follows
X = {q(0)(t):[0, +^ Rra : ||^°)(i)|| < 9l, ||$(°>(i)|| < 92}, (3)
where the time functions (t) are twice continuously differentiable, g1 and g2 are some constants.
The problem of trajectory tracking control consists in constructing a feedback controller u = u(t,q,q) such that the reference trajectory q(°"l(t) e X of the manipulator ( ) is uniformly asymptotically stable.
2. Problem Solution
For some reference trajectory q(°\t) e X, define the feedforward term of the controller
u = u(t, q, q) as follows
u(0l(t) = A(q(°l(t))q(°l(t)+ C(q(°l(t),q(°l(t))q(°l(t) + g(q(°)(t))+ d(q(0)(t), q(0)(t)). (4)
The tracking errors are given by x = q — q(°\t) and x = q — q(°\t). The error dynamics equations can be expressed as
A(1)(t, x)x + C(1\t, x, 2q(°l(t) + x)x + Ri(t, x) + R2(t, x, x) = u(1), (5)
where A(1)(t,x) = A(q(0)(t) + x), C(1)(t,x,x) = C(q(0)(t) + x,x), Rl(t,x) = (A(q(0)(t) + x) -—A(q(0\t)))q(0\t) + (C(1)(t,x,q(0)(t)) - C(1)(t, 0, q(0) (t)))q(0) (t) + g(q(0)(t) + x) - g(q(0)(t)) + +d(q(0)(t)+x,q(0)(t))-d(q(0)(t),q(0)(t)), R2(t,x,x)=d(q(0)(t)+x,q(0)(t)+x)-d(q(0)(t)+x,q(0)(t)), u(1) = u - u(0)(t).
Proposition 2. Assume that there exist positive reals kA, kc, kg, and kd such that V(t, x) G R+ x Rra the following conditions hold
||A(g(0)(i) + x) - A(g(0)(i))|| < kAllp(x)ll, \\C(1)(t, x, q(0)(t)) - C(1)(t, 0, q(0)(t))|| < kcUp(x)||,
Mé0)(t) + x) - g(é0)(tm < k9Mx^, ||d(q(0)(t)+ x,q(0)(t)) - d(q(0)(t),q(0)(tm < Mp^H,
(6)
where the function p : Rra ^ Rra is such that p(0) = 0 and p = (pl,p2,... ,Pn)T, Pi = Pi(xi) (i = 1,2,... ,n) are continuously differentiable functions, and the following holds
Hp(x)H = Hp(x + 2nZ)||VZ e Zn : k = 0 Vi e JP, Vx e Rra, Hp(x)H ^ Pmax = constant > 0.
Also assume that the functions Rl(t, x) and R2(t, x, y) satisfy the equalities
Ri(t,x) = F(t,x)p(x) V(t,x) e R+ x Rra, R2(t, x, y) = B(t, x, y)y V(t, x, y) e R+ x Rra x Rra,
where F : R+ x Rra ^ Rraxra B : R+ x Rra x Rra ^ Rraxra are some continuous matrix functions.
(7)
(8)
Remark 2. Note that the inequalities in ( ) with p(x) = x represent the well-known important properties of the robot-manipulators [2]. We generalize these inequalities by introducing a periodic function p(x) satisfying the conditions (7). Thus, we took into account the following property of the robotic manipulators with revolute joints: they are described by the equations with periodic functions of x with period 2n. For example, the motion equation of the pendulum is given by ^ + bdp + a sin ^ = cU, where ^ is the angle between the pendulum bar and the vertical axis, a = g/L, b = k/m, c = 1/(mL2), L is the pendulum length, m is the pendulum mass, k is the coefficient of the damping torque, U is the torque applied to the pendulum. The function considered as a controller is given by u = cU. The aim of the controller is to stabilize the desired motion (t) of the pendulum. The equality that the feedforward controller satisfies has the form u(0\t) = q>(0\t) + bq>(0\t) + a sin q>(0\t). The error dynamics equation is given by x + bx + a sin(^ + ^^(t)) — a sin(^(0)(i)) = u(l\ where x = ^ — <f(°Xt). This equation can be represented as follows x + bx + 2a cos(x/2 + q>(0\t)) sin(x/2) = u(l\ If one chooses p(x) = sin(x/2), then the constant k0 of (6) is such as k0 = 2a. Note that the function p(x) has a period of 4n, and the function lp(x)l has a period of 2n.
Construct the feedback term of the controller u as follows
u(l) = K Sign (x + r(x)), (9)
where K e Rraxra is the constant gain matrix, lgn||^^|| ^ —k0 = constant < 0, the vector function
r : Rra ^ Rra is continuous, r(0) = 0, and r(x) = (rl(xl),r2(x2),... ,rn(xn))T Vx e Rra, Ti(y) = r0d(p2(y))/dy Vy e R, Vi e {1,2,..., n}, r0 = const > 0, the vector function Sign (x) is given by Sign (x) = (sign (x\), sign (x2),..., sign (xn))^ Vx e Rra.
Remark 3. One can easily see that n(y) = ri(y + 2nl) Vi e Jr, Vy e R, VI e Z. Proposition 3. There does not exist a vector x e Rra such that R1(t, x) = 0 Vt e R+ and p(x) = 0. By substituting (9) into (6) and using (8), we get the closed-loop system such as
A(1) (t, x)x + C(1) (t, x, 2q(0) (t) + x)x + F(t, x)p(x) + B(t, x, x)x - K Sign(x + r(x)) =0. (10)
Note that under Assumption 3 the closed-loop system (10) has the set of all equilibrium positions such as
H = {(y, y) e Rra x Rra : p(y) = 0, y = 0}. (11)
One can easily see that the set H can be written as
H = {(y, y) e Rra x Rra : yi = 2nk, Vj = 0, y = 0, h e Z, i e Jr, j e JP}. (12)
Theorem 1. Consider the closed-loop system (10) under Propositions 1, 2 and 3. Let the positive constants m1, m2, and ô0 exist such that the following holds
к a + kc + kg + kd + ro
(CW(t,x,r(x) - 2q(°\t)) - AM(t,x)^
С(1)(t, x, q(0\t) - r(x)) + A(1\t, x)
^ m2,
(13)
lgn
mPmax + m2&°/h < k°.
Then, the set (11) of (10) is uniformly asymptotically stable. Proof 1. Consider the Lyapunov function candidate V = V (t, x, x) as follows
V (t,x,x) = y/(x + r(x))TAW (t, x)(x + r(x)). (14)
Note that the function V(t, x, x) is periodic on Xi, i e Jr with period 2n. Note also that the Lyapunov vector function candidate V is continuous but not continuously differentiate. We will use the right hand time derivative of V along the trajectories of the closed-loop system (10). The time derivative of the function V 2(t,x,x)/2 is given by
VV = (x + r(x))TA(1l(t,x)(x + r(x)) + (x + r(x))TA(1l(t,x)(x + r(x)) = = (C (1)(t,x,x + 2q(°l(t)) + B(t,x,x)x + F (t,x)p(x) + +K Sign (x + r(x)))T(A(1)(t, x))-1A(1)(t, x)(x + r(x)) +
+ (^~ ^ A(1)(t,x)(x + r(x)) + (x + r(x))TA(1l(t,x)(x + r(x)) =
= (x + r(x))T ^С(1)(t,x,x + q(0)(t)) + 1 À(1)(t,x)^ (x + r(x)) + +(x + r(x))T (f(t, x) - ro (B(t, x, x) + С(1) (t, x, 2q(0) (t) - r(x))+
^ **>+
+(x + r(x)) T (B(t,x,x)+ С(1)(t, x, q(0\t) - r(x)) +
+A(1) (t, x)-^pj (% + r(x)) + (% + r(x))TK Sign (x + r(x)).
Using Remark 1, one can easily obtain the following estimate
V <
F(t, x) — ro (B(t, x, x) + С(1)(t, x, 2q(0\t) — r(x)) +
+ A(i\t,x)^) Щ
OX J ox
+ lgn
\\p(x)\\/l(t,x,x) + B(t,x,x) + C (l)(t,x,q(0)(t) — r(x)) +
V/(l(t, x, x)2) + lgn \\K\\ /l(t, x, x),
where the function \(t, x, x) satisfies the following relationship
\(t,x,x)\\x + r(x)\\ = "kiV, Xl ^ \(t,x,x) ^ \2. From (15) using (13), we get the following inequality
V(t,x,x) ^ \b(^)\\ + w,m2 .,2V(t,x,x) — ^0
X(t, x, x)
X(t, x, x)2
X(t, x, x)
From (17) using (13), we get the following inequality
V(t, x,x) < — e < 0 V(t, x, x) e R^0 x Rra x Rra,
where £ > 0 is some constant.
Using (18), one can obtain
V(t,x(t),x(t)) < V(t0,x(t0),x(t0)) — £t Vt ^ t0.
From (19) one can easily see that there exists the time moment tl > t0 such that
V(t,x,x) = 0 Vt ^ tl.
Consider now the Lyapunov function candidate W = W (x) such as
W = \\p(x)\\.
The time derivative of the function W2/2 is given by
dn(x) dv(x)
WW = p(x) ——— x = p(x) ———(x + r(x) — r(x)) = ox ox
(dv(x)\2 dv(x)
dx j p(x) + p(x) ^ (x + r(x)).
From (22) one can easily obtain the following estimate
W(t, x, x) ^ lgn
-4
W (x) +
1
X(t, x, x)
dp(x)
dx
V(t, x, x).
(16)
(17)
(18)
(19)
(20) (21)
(22)
(23)
Using (20) and (23), one can obtain that the estimation of the right hand time derivative of the function W (x) is given by
W(t, x, x) ^ lgn
—-"( 9jt)
W(x) < 0 Ш ^ ti.
(24)
Using Theorem from [13], we get that the positive limit set Q+(t°, y°) of the solution (x(t, t° ,x°, x°), x(t, t°,x°, ¿°))T of (10) is such that the following holds
Q+(t°,y°) C{^*(t,x,x)=0} = A, (25)
where y° = (x°, x°)T, W *(t, x, x) is a limiting function to W (t, x, x).
From (25) one can conclude that the set of all equilibrium points (11) of the closed-loop system (10) is uniformly attractive.
Using Assumptions 1 and 2 note now that the function V (t,x,x) satisfies the inequality
m1(||(r(a:),a:)T||) < V(t,x,x) < MHM^), ¿)T||), (26)
where and rn2 are Hahn functions.
Then, using (18) and (26) we obtain that the set (11) is uniformly stable.
Thus, the set of all equilibrium points (11) of the closed-loop system (10) is uniformly asymptotically
stable.
3. Solution to the trajectory tracking for a 5-DOF robotic manipulator
In this section, the simulation test for a 5-DOF serial robotic manipulator (see Fig. 1) is illustrated. The robot has one prismatic joint and four revolute ones.
Each link of the manipulator is represented as a solid body. The kinematic pairs of the manipulator are assumed to be single-link, their geometric centers are denoted by Ok (k = 1,2,..., 5). The centers of mass Ck (k = 1, 2,..., 5) of the links lie on the axes OkOk+1, the axes OkOk+1 (k = 1, 2,..., 5) are their axes of symmetry. The first base link is vertical, it rotates around 0102, its rotation angle is 61. The second kinematic pair allows rotation of the second link around the horizontal axis passing through 02. The third kinematic pair allows rectilinear movement of the third link along 0204 (03 e 0104), let us introduce the notation for the displacement of the third link x = d3 = 0104. The fourth link can rotate around a horizontal axis passing through 04 with the angle of rotation 64. The fifth link simulating
the gripper allows the rotation around 0405, its rotation angle is denoted by 65. We will assume that the centers of mass Ck of the links lie on the axes OkOk+1 and these axes are the axes of symmetry of the corresponding links (k = = 1,2,..., 5). Let's introduce the main central axes CkXk,Ckyk,CkZk of the links. We will assume that for links 1 and 5 the axes C1z1 and C5z5 are the axes of symmetry. For links 2, 3 and 4 such axes are C2x2, C3x3 and C4x4 respectively. Let us assume that the axes C2z2, C3z3 and C4z4 are horizontal. The masses of the links are denoted by m,k (k = 1,2,..., 5), and their main central moments of inertia are
denoted by
I(k) and 1)^'. Accordingly, we
Л®
have Ii1
(1'
T (5'
1x
r(5' r(2'
j-v 1 J-V
=
IV
(3'_T(3'
T(3' r
1z
(4'
V
№. Let 's introduce the lengths
O2C2 = I2, C3O4 = Is, O4O5 = 2O4C4 = 2I4, and 05C5 = l5.
Fig. 1. Model of a 5-DOF robotic manipulator
By using Koenig theorem we can find the kinetic energy T of each link i = 1,2,..., 5 as the kinetic energy of an absolutely rigid body.
ti=2 № l,
T2 = \m2iKsin2 6261 + 6 2) + 2( №6 2 cos2 62 + ii2)(e 2 sin2 62 + 6 2)),
T3 = 1m3(x2 + (x — ls)262 + (x — Is) sin2 62 • 62) + 2(i3)61 cos2 62 + if (62 sin2 62 + 62)),
T4 = 1m4((x + l464 sin 64)2 + (x62 — l464 cos 64)2 + (x cos 64 — l4 sin(62 + 64))262+
+ 2( li4)61 cos2 (62 + 64) + I(4)(61 sin2(62 + 64) + (6 2 + 6 4)2)),
T5 = 2m5((x + (214 + l5)64 sin 64)2 + (x62 — (214 + k)64 cos 64^ + + (x cos 6 — (214 + I5) sin(62 + 64))2(62 +
+ 2li5)(61 cos(62 + 64) + 6^)2 + 11(5) (61 cos2(62 + 64) + (62 + 64)2). The kinetic energy of the manipulator is given by
T = Tl +T2 +Ts + T4 + T5. The potential energy of the manipulator has the form n = —m2gl2 cos 62 — m3g(x — l3) cos 62 — m4g(x cos 62 + l4 cos(62 + 64)) —
— m,5g(x cos 62 + (214 + 15) cos(62 + 64)). The motion equations of the manipulator are as follows
d (дТ\ дТ _ Ш dt \дq J dq dq '
where q = (6l, 62, x, 64,65)T.
The dynamics for the manipulator can be expressed by (1). The vector functions p = p(x) and r = r(x) are chosen such as
/ . x1 . x2 . x4 . x5 у p = sin —, sin — ,x3, sin —, sin —
1 \ 2' 2 ' 2 ' 2 J
T
2 2 '— 2
r = (sin«l, sinx2,x3, sin^4, sin^5)T. Choose the reference trajectory of the manipulator as
ql^ (t) = ^ rad, q^ (t) = 2 cos ^ rad,
(27)
q3\t) = 0.5+ 0.1 sin ^ m, (28)
qf° (t) = 4 cos ^ rad, q5° (t) = 3 sin ^ rad.
We fixed the control gains to K = —k0E, k0 = 600. Let the initial position of the manipulator be such as
ql(0) = —3.1 rad, q2(0) = —2.8 rad, q3(0) = 0.6 m
(29)
q4(0) = —3.1 rad, q5(0) = 2.9 rad
In Figures 2-6, one can see the time evolution of the link positions as well as the references. From the simulation tests, it can be seen that the controller
u = u(0)(t) + K sign (q - q(0)(t) + r(q - q(0)(t))) (30)
provides the asymptotic stability property of the reference trajectory.
--;
Л-
— Ф) "-Л)
----
— <й№ "-Л) ■ ■ ■ ■ 1 ■ ■ ■ ■ 1 ■ ■ ■ ■ 1 ■ ■ ■ ■
■ ■ ■ ■ i ■ ■ ■ ■ ■ ■ ■ ■ 1
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
Fig. 2. Time evolution of the first link angular position and reference
Fig. 3. Time evolution of the second link angular position and reference
-0.5--
-1--
-Я
— 54 (t)
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
Fig. 4. Time evolution of the third link linear position and reference
Fig. 5. Time evolution of the forth link angular position and reference
4 3.5 3
— <ф) -Л)
-
да.
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
Fig. 6. Time evolution of the fifth link angular position and reference
3
0
0
Conclusions
We consider the trajectory tracking control problem for serial robotic manipulators equipped by revolute and prismatic joints. The generalized coordinates of such mechanical systems include the rotation angles of the manipulator links which makes it possible to consider the trajectory tracking control problem in a cylindrical phase space. The paper proposes and substantiates a technique for constructing a relay control law consisting of the sum of two components: the feedforward term and the sign function of a feedback component. The asymptotic stability property of the set of all equilibrium positions of the closed-loop system is obtained by constructing a Lyapunov function which is periodic in angular coordinates with a semi-definite time derivative and using the principle of quasi-invariance for non-autonomous systems of ordinary differential equations. The numerical simulation of a five-link robotic manipulator demonstrates the good applicability of the controller.
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Андреев Александр Сергеевич - родился в городе Шураб Таджикской ССР (1952). Окончил с отличием механико-математический факультет Ташкентского государственного университета по специальности «Общая механика» (1974). Защитил диссертацию на соискание учёной степени доктора физико-математических наук на тему «Развитие прямого метода Ляпунова в исследовании устойчивости движений неавтономных механических систем» по специальности «Теоретическая механика» (1990, МГУ им. М. В. Ломоносова). С 1975 по 1980 годы работал в Институте сейсмологии АН Узбекской ССР. С 1980 года по 1991 год работал в Ташкентском политехническом институте в должностях ассистента, старшего преподавателя, доцента, заведующего кафедрой. С 1991 года работает в Филиале Московского государственного университета в Ульяновске (с 1996 года Ульяновский государственный университет) вначале в должности профессора, затем с 1992 года по октябрь 2008 года -заведующим кафедрой механики и теории управления, с октября 2008 года и по настоящее время - заведующим кафедрой информационной безопасности и теории управления. Научные интересы - теория устойчивости и управления движением механических систем, математическое моделирование. Опубликовал свыше 200 научных статей по указанным направлениям.
Россия, 432017 Ульяновск, Льва Толстого, 42 Ульяновский государственный университет E-mail: asa5208@mail.ru ORCID: 0000-0002-9408-0392
Перегудова Ольга Алексеевна - родилась в Ульяновске (1975). Окончила механико-математический факультет МГУ им. М. В. Ломоносова (филиал в Ульяновске) по специальности «Механика» (1997). Защитила диссертацию на соискание учёной степени доктора физико-математических наук на тему «Решение задач об устойчивости и управлении движением неавтономных механических систем на принципах сравнения и декомпозиции» по специальности «Теоретическая механика» (2009, МГУ им. М. В. Ломоносова). С 2009 года работает на кафедре «Информационная безопасность и теория управления» Ульяновского государственного университета в должности профессора. Научные интересы - устойчивость и управление движением колесных мобильных роботов и манипуляторов. Опубликовала свыше 100 научных статей по указанным направлениям.
Россия, 432017 Ульяновск, Льва Толстого, 42 Ульяновский государственный университет E-mail: peregudovaoa@gmail.com ORCID: 0000-0003-2701-9054
Петровичева Юлия Владимировна - родилась в Ульяновске (1986). Окончила факультет математики и информационных технологий Ульяновского государственного университета по направлению «Механика» (2008). Защитила диссертацию на соискание учёной степени кандидата физико-математических наук на тему «Математическое моделирование нелинейных управляемых систем с непрерывным и разрывным управлением» по специальности «Математическое моделирование, численные методы и комплексы программ» (2013, УлГУ). С 2019 года работает на кафедре «Информационная безопасность и теория управления» Ульяновского государственного университета в должности доцента. Научные интересы -математическое моделирование, устойчивость и управление движением. Опубликовала свыше 20 научных статей по указанным направлениям.
Россия, 432017 Ульяновск, Льва Толстого, 42 Ульяновский государственный университет E-mail: petrovichevayulia@yandex.ru
