CC BY
7TECHNICAL SCIENCES
DESIGN OF NONLINEAR EXOSKELETON CONTROL SYSTEM BY GAIN SCHEDULING
METHOD
Ulikyan A.,
scientific supervisor, Candidate of Technical Sciences (Ph.D), associate professor of the Department of Control Systems, National Polytechnic University of Armenia.
Armenia, Yerevan Mkhitaryan A., Candidate of Technical Sciences (Ph.D), associate professor of the Department of Control Systems, National Polytechnic University of Armenia.
Armenia, Yerevan Khanamiryan Z.
Undergraduate master student of the Department of Control Systems, National Polytechnic University of Armenia.
Armenia, Yerevan DOI: 10.5281/zenodo.6532724
Abstract
The interest in exoskeletons is increasing year by year. There are designed to protect the joints and spine of people, and increases mobility, reduces the load of the musculoskeletal system during long hikes to transport load (backpacks, special equipment, etc.). The technology was used to reduce muscle fatigue and increase human performance. Exoskeletons, as well as any mobile artificial limbs, have control complexities due to nonlinearity and inter-influences between channels.
One of the advanced technologies in this field is an adaptive control, which is an important and interesting area of scientific and industrial researches. The need of adaptive control systems for exoskeletons used in rehabilitation medicine is because exoskeleton's control systems always have some parametric or dynamic uncertainties. There are used to adapt information about the dynamics of the processes to the parameters of PID controllers to ensure stability at all operating points.
The nonlinear control system of the exoskeleton's leg was considered in the work. Using the gain scheduling method, a group of linear PID controllers are designed to ensure the stability of the system at various operating points.
Keywords: exoskeleton, gain scheduling, MATLAB, modeling, multivariable control system, nonlinear systems, PID controller, Simulink.
I. INTRODUCTION To develop a control system, first we need to get a
The exoskeletons are designed to expand human mathematical model of it, do its investigation and then
functionality, in particular, to improve the life quality design an appropriate controller. of people with disabilities. The exoskeletons are de- This work will help to control the exoskeleton's
signed to protect the joints and spine of a human. They leg of the non-linear system through the gain schedul-
help to increase the mobility of the human, as well as ing method, which will be more effective for exoskele-
reduce the load of the musculoskeletal system when ton control. transporting long-term heavy load. II. Materials and method
The design and production of such devices are re- The Lagrange mechanics are used for modeling of
lated with election of solution of optimal parameters' the dynamic systems. Each mechanical system is char-
syntheses issues, design of control systems, transmis- acterized by a specific function called the Lagrange
sion mechanisms and software. Our main task is to de- function and has the following appearance: sign an exoskeleton's leg control system, which will allow the most accurate control of the nonlinear model.
L(q±, q2,... qs, fa, fa, ■■■ fa, t) = L(q, q, t), (1)
It contains of q and q parameters but does not depend on types of derivatives of q.
It is shown that mechanical condition of the system is characterized only by coordinates and velocities. Besides, the Lagrange function has very important feature, when t = ti and t = t2 moments the system is occupies a position in the space, which is described by qi and q2 coordinates, then its integral is:
rt'¿
= I L(q,q,t),
■>ti
which is approaches the lowest possible value. From (2) the equation of the motion is received:
S
d dL di
-----— = 0. (3)
at oq oq
In the case of a mechanical system with several degrees (s) of freedom, the equation will be: d dL dL
—— ■ —--— =0,where i = (1,2,...,s). (4)
dt dqt dqt ( )
If the Lagrange function is known, then the connection among acceleration, velocity, and coordinates can be found out from the equation (4) [1, 2].
To obtain a mathematical model of the system, let's consider a single-legged exoskeleton kinematic scheme:
Fig. 1. This is a sample of a figure caption.
Using the kinematic scheme shown in the Figure 1, it is possible to obtain the coordinates of the points A(x1,y1) and B(x2,y2) as following:
x1 = l1 sin d1,
Уl = -llcos6l,
x2 = l1 sin 81 + l2 sin 62, )
y2 = -l1 cos 81 — l2 cos d2, where x1,y1 and x2,y2 accordingly knee and ankle coordinates; 91 and Q2 are the angles of the leg joints with the vertical axis; l1 and l2 are the lengths of the leg joints [3,4].
Projections of the linear velocities on the x and y axes can be obtained using the Euler-Lagrange equation:
= fi(q±),yi = дl(Цl),
*2= f2(R2),y2= 92(R2), (6)
qi = Bi,q2 = 02, . dfi , d(lisin9i) dBi Xi = dq1^qi= 39--~dt = i eicos(d1>
. dgi , d(—li cosSi) ddi (7)
yi=W^qi=-36--"dt == eisin(ei)
df2 d(l1sind1 + l2sind2) dd2 X2 = dq2'q2= 39 dt =
= l1 91cos(91) + l2 92cos(92) dg2 3(—l1cos91 — l2cos92) d92 (8)
y2=~dq2'q2= 39 dt =
= l1 91sin(91) + l2 92sin(92)
d91 . d92 ei= — =«i,92= — =«2, (9)
where and m2 are the angular velocities.
Thus, the linear velocity will be obtained by the square sum of the velocity projections:
xf + y1 =
= l(li 9iCos(9i))2 + (lJiSin(9i))2 =
= pA) = iA, ^2 =
x2 + y2 =
= ^(lA) + M2) + 2lil29i92cos (9i - e2).
The Lagrange function is obtained by the difference of kinetic-potential energies:
L = T-V
where T is the kinetic energy; V is the potential energy. The kinetic energy equation will be:
T=\^Ysmy2)=
2
1 • • 2 • 2 = -(m1l2^el2+m2((l1e1) +Q262) +
+ 2l1l26162 005(61-62))). The potential energy equation will be:
V = ^ mgh = w.1gj1 + m.2gy2 =
= -(m.1 + m2)gh1 cos^) - m.2gh2 cos(&2). So, the Lagrangian function will be:
L = ^ (m^lel +m2((lA)2 + (I282) +
+2I1I2ÖA 005(61-62))) + + (m.1 + m2)gh1 cos^) + m2gh2 cos(&2), where m is the mass of the finger and g is the gravitational constant. The equations of motion will take the following form:
(11)
(12)
(13)
(14)
(15)
M,
M,
_ d Í dL\ dL
dt\ddj d01 _ d ( dL\ dL
= dt\dé2) ds2
82) +
M1 = (m1 +m2)l1d1 + m2l2B2 cos(91 +m2l2d2 sin(81 — d2) +m1+ m2gsin(d1),
M2 = m2he2 + +m2hÖ1 cos(91 — 62) — —m.21181 sin(01 — O2) + m2g sin^) .
(16)
(17)
where M1 and M2 are the moments of joints.
The created model in MATLAB/Simulink environment of the nonlinear system of the exoskeleton's leg is represented in Figure 2. The Gain Scheduling method is used for the design of multidimensional automatic control systems (MACS) with similar non-linear interconnections [3, 4]. This method makes it possible to design controllers at different operating points, each of which ensures the stability of the system (Fig. 3) [5].
v.
Fig. 2. The simulation model of the nonlinear system of the mechanical part of the exoskeleton.
To apply the gain scheduling method, it is necessary to linearize the given nonlinear system in the operating points, taking into account the range of movements of the leg during human walking, and then analyze the obtained linear systems. After, PID controllers for each operating points are designed, obtain the dependence of the Kp, Ki, Kd parameters on the system observation parameters, then integrate them into the nonlinear multivariable system [6].
Fig. 3. Block diagram of control system with gain scheduler.
The 91 and d2 angles formed by the joints of the leg (hip and knee) with a vertical axis were chosen as observation parameters. Their ranges can be determined in case when during walking knee and hip angles vary accordingly [0: pi/3] and [-pi/6: pi/6] radians. Based on the obtained ranges was performed linearization of the nonlinear system at 441 operating points. The obtained liner systems were analyzed. The step response curves of the closed loop systems are observed in the Fig. 4, 5, which indicate that it is necessary to design controllers for system stabilization [7,8].
(Rii Ri2
H) ■ <i8>
MATLAB/Simulink environment is shown in Fig. 6 and in Fig.7, where box with dotted lines indicates the mechanical part of the exoskeleton leg [9], which is shown in Figure 2. In the block diagram q1±, q12, q2 \ and q22 are interrelationships which can be represented by the matrix Q:
time(sec)
Fig. 4. The step responses of 91 closed-loop control system.
0.4 0.6
time(sec-)
Fig. 5. The step responses of 92 closed-loop control system.
The block diagram and model of a nonlinear system with PID controllers designed by using gain scheduling method in
Fig. 6. Exoskeleton leg Simulink model with PID controllers designed by using gain scheduling method.
Fig. 7. Block diagram of the exoskeleton leg control system.
Designed a PID controller for one operating point and used in 441 operating points and their step response curves were extracted (Fig. 8). From the received curves, it can be seen that the system exhibits different stabilization parameters at different operating points, which need to be corrected by designing separate PID regulators for each operating point [10,11]. As a result, the PID regulator was designed for each with the same stability parameters (8 = 2.34%, tr = 0. 0005sec., ts = 0.004sec.). The step response curves of which are shown in Fig. 9.
PID controller Kp, Ki, Kd parameters dependence from 9i and Q2 observation parameters derived in the Figure 10, 11, 12 - hip joint dependencies, Figure 13, 14, 15 - knee joint dependencies). This shows how the PID controller parameters should be changed at different working angles (operating points).
Step responses
0 0.2 0.4 0.6 D.B 1 1.2 1.4 1.6 1.8 time{sec.) ^ .3
Fig. 8. Linearized system step responses without gain scheduler.
timefsec.)
Fig. 9. Linearized system step responses with gain scheduler.
Fig. 10. Hip PID controller KP dependence from 81 and d2 observation parameters.
ThetBj 0 Thsta,
Fig. 11. Hip PID controller KI dependence from 91 and d2 observation parameters.
Theta, 0 -OS Theta]
Fig. 12. Hip PID controller KD dependence from 91 and d2 observation parameters.
Fig. 13. Knee PID controller Kp dependence from 91 and 62 observation parameters.
2.5
Th»t»2 0 •°-5 Theta,
Fig. 14. Knee PID controller KI dependence from 91 and 92 observation parameters.
T&eta, ® -о» Thetai
Fig. 15. Knee PID controller KD dependence from 9-i and 92 observation parameters. RESULTS
As a result of the human gait algorithm which used for the designed exoskeleton, the coincidence of the input-output signals becomes apparent (Fig. 16) . A human gait algorithm was given to the system input to test the obtained method. In the Figure 16, the desired angles of the hip and knee joints are shown in red and blue lines, respectively, and the output angles are shown in dotted lines. The results show that the input and output angles correspond.
0.3 0.2
¥
Щ* о СЛ
с
< -0.1 -0.2 -0.Э
0 0.2 0.4 0.6 0.8 1
time{sec.)
Fig. 16. Hip PID controller KD dependence from 9-i and 92 observation parameters.
CONCLUSION
This paper presents the mathematical model and design features of the control system of the exoskele-ton's leg. The nonlinear model of the exoskeleton's leg is developed and investigated. The open and close loop systems have been developed and analyzed. For the system stabilization, the PID controllers are developed by using the gain scheduling method in 441 operating points. It is concluded that the gain scheduling method results are satisfied the necessary requirements for use in the exoskeleton control system.
ACKNOWLEDGEMENTS
The work was supported by the Science Committee of RA, in the frames of the research project № 21DP-2B003.
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ТЕПЛОВОЙ РАСЧЕТ СОЛНЕЧНОГО КОЛЛЕКТОРА И ЕГО ТЕХНИКО-ЭКОНОМИЧЕСКОЕ ОБОСНОВАНИЕ
Салманова Ф.А.
Доктор философии по техническим наукам, доцент, Институт Радиационных Проблем, г. Баку Мустафаева Р.М. Кандидат технических наук, доцент, Институт Радиационных Проблем, г. Баку
Саламов О.М.
Кандидат физико-математических наук, доцент, Институт Радиационных Проблем, г. Баку
Махмудова Т.А.
Кандидат физико-математических наук, доцент, Институт Радиационных Проблем, г. Баку
Юсупов И. М. Инженер
Институт Радиационных Проблем, г. Баку
Велизаде И.Э. Инженер
Институт Радиационных Проблем, г. Баку
