Russian Journal of Nonlinear Dynamics, 2023, vol. 19, no. 4, pp. 613-631. Full-texts are available at http://nd.ics.org.ru DOI: 10.20537/nd231208

NONLINEAR ENGINEERING AND ROBOTICS

MSC 2010: 70E60

Cable-Driven Parallel Robot: Distribution of Tension Forces, the Problem of Game Theory

E. A. Marchuk, A. Al Badr, Ya. V. Kalinin, A. V. Maloletov

This paper highlights the role of game theory in specific control tasks of cable-driven parallel robots. One of the challenges in the modeling of cable systems is the structural nonlinearity of cables, rather long cables can only be pulled but not pushed. Therefore, the vector of forces in configuration space must consist of only nonnegative components. Technically, the problem of distribution of tension forces can be turned into the problem of nonnegative least squares. Nevertheless, in the current work the game interpretation of the problem of distribution of tension forces is given. According to the proposed approach, the cables become actors and two examples of cooperative games are shown, linear production game and voting game. For the linear production game the resources are the forces in configuration space and the product is the wrench vector in the operational space of a robot. For the voting game the actors can form coalitions to reach the most effective composition of the vector of forces in configuration space. The problem of distribution of forces in the cable system of a robot is divided into two problems: that of preloading and that of counteraction. The problem of preloading is set as a problem of null-space of the Jacobian matrix. The problem of counteraction is set as a problem of cooperative game. Then the sets of optimal solutions obtained are approximated with a fuzzy

Received November 17, 2023 Accepted December 25, 2023

The work was supported by the RSF (Grant No. 22-29-01618).

Eugene A. Marchuk eugene.graduate@gmail.com

Amer Al Badr

a.albadr@innopolis.university

Yaroslav V. Kalinin claymor.vlg@gmail.com

Innopolis University

ul. Universitetskaya 1, Innopolis, 420500 Russia

Alexander V. Maloletov a.maloletov@innopolis.ru

Innopolis University

ul. Universitetskaya 1, Innopolis, 420500 Russia Volgograd State Technical University pr. Lenina 28, Volgograd, 400005 Russia

control surface for the problem of preloading, and game solutions are ready to use as is for the problem of counteraction. The methods have been applied to solve problems of large-sized cable-driven parallel robot, and the results are shown in examples with numerical simulation.

Keywords: cable-driven robot, parallel robot, distribution, null space, cooperative game, fuzzy logic, structural nonlinearity

1. Introduction

Cable-driven parallel robots are a relatively new branch in robotics whose concepts appeared in the 1980s. The well-known principles of cable-driven mechanisms such as winches and hoists have been used for centuries in different kinds of cranes, but they did not have automated control systems [1, 2]. Specific cranes driven and run by men were known in medieval China and Korea, they had absolutely the same structure as modern cable-driven parallel robots [3]. The only differences are an electrical drive and automated control in modern cable-driven machines. The first cable-driven parallel robots were the Stewart platforms, which stabilized marine cranes [4, 5]. Anyway, cable-driven parallel robots took their place in specific practical tasks at the beginning of the 21th century, but still are not very popular in industrial applications. The usual tasks for cable-driven parallel robots are: warehouse transportation, positioning cameras at stadiums, supporting exoskeletons, etc. However, the future for cable-driven parallel robots seems to be promising in large-scale additive manufacturing, e.g., concrete printing [6]. The advantages of cable-driven parallel robots are their scalability and simplicity of mechanisms, and their disadvantages are, in general, hard-to-solve control and design problems. Several scientific groups in the world are working on problems of cable-driven parallel robots nowadays, and the methodology of such robots is being developed too [7, 8].

The most exhaustive reviews and analyses on the relevant field of cable-driven parallel robot applications are given in [9-11].

Fig. 1. Large-sized cable-driven parallel robot "Tezuche" for concrete printing (project of Innopolis University)

In this paper, methods of artificial intelligence are proposed to solve the problems of force distribution in the group of lower cables of a large-sized cable-driven parallel robot, Fig. 1.

The method of game theory applied to the problem of a cable-driven parallel robot may be thought of as a scientific innovation in the field of cable-driven robotics. In such a way, each cable is assumed to be an intelligent agent and the agents can form coalitions to provide the most effective cooperative work.

2. State of the art

The paper is devoted to two fields that can be considered as belonging to the area of artificial intelligence: one is game theory, and the other is fuzzy logic. Game theory has been known since the 1920s and was primarily concerned with solving the problems of economics, then it became a more common thing [12]. Fuzzy logic has been known since the 1960s, and fuzzy systems that can be applied to a wide range of problems are based on the concepts of fuzzy logic [13]. Both are proposed to solve the tasks of force distribution in the specific problem of the cable system of a cable-driven parallel robot.

Different methods are known to solve problems of tension forces in the cable system of a robot, such as Dykstra's method, the puncture method, etc. [7, 8]. But all these methods are mainly given in the context of mathematical analysis, and have never been discussed in the context of artificial intelligence in numerous publications related to cable-driven parallel robots. Therefore, we can conclude that none of them have set the problem of forces distribution in terms of game theory.

Nevertheless, the game theory control approach has already been applied to several problems in different fields, i.e., aerospace, automotive systems, networks, etc. [14-16]. This concept originates from the 1960s and now it continues its heading [17-19]. Also, the same methods of game theory design based on the concept of using both cooperative and noncooperative games for distributing the loads were proposed to solve the static problems of structural mechanics in the 1980s [20].

Nowadays, fuzzy systems are used in a wide range of control tasks despite of the popularity of artificial neural networks in the field of artificial intelligence control. Moreover, fuzzy systems and artificial neural networks can be considered as similar things [21]. Fuzzy systems can be used as regulators in the field of artificial intelligence control, as regulators close to the field of classic control, e.g., fuzzy PID-type regulators [22, 23]. Different types of fuzzy control systems are listed and discussed in [24, 25]. Fuzzy systems are known as universal approximators, and several papers are devoted to fuzzy approximations of models of game theory, especially genetic fuzzy controllers for noncooperative games [26, 27]. Samples of fuzzy regulation for cable-driven systems can be found in [28-30].

Therefore, fuzzy control and game theory control appear to be relevant topics to examine them in the context of problems of cable-driven parallel robots.

3. Cable-driven parallel robots

The most common way to set the kinematics of a cable-driven parallel robot is a vector-and-matrix representation [7, 8]. Then, the segment of each cable between proximal and distal anchor points can be represented as

li = a - Rbi -r, (3.1)

where a, is the vector of the fth proximal anchor point in the world frame, b,i is the vector of the fth distal anchor point in the tool frame, r is the radius vector, and R is the rotation matrix.

The matrix of the geometric Jacobian provides relations between configuration and operational spaces:

, 6x8

ll_ . . . '8

•:iii2 ny2

The dynamics of the robot is described by the following typical and novel models. The first is the typical model of a viscoelastic body, namely, the Voigt model:

where f * is the tension force in the fth cable, Al is the deformation of the fth cable, E is the Young modulus, n is the dynamic viscosity of cable material, S is the cross section of the cable, and l0 is the length of the unloaded cable.

The second is a novel model of structural nonlinearity for cable-driven systems. According to the proposed method, the condition of structural nonlinearity can be incorporated into the model of cable with an activation function:

g(Al) = i tanh(fcAZ) • (1 + tanh(n • kAl)), (3.4)

where k and n are some coefficients.

The proposed type of activation function has been developed by the authors especially for the problems of a cable-driven parallel robot and differs from the conventional GELU functions, which are typical of artificial intelligence models [31, 32].

Then, the forces of tension in cables can be expressed in smooth and continuous form as

ft = f* • g(Al,) (3-5)

and the equation of dynamics becomes

Mq + Dq + gc = wp + Jf, (3.6)

where M is the mass matrix, D is the damping matrix, gc is the factor of centripetal force and angular momenta, wp is the outer wrench, J is the Jacobian matrix, f is the vector of forces in the cables, and q is the vector of generalized coordinates. Numerical computations for solving (3.6) are based on the Runge-Kutta or Adams-Bashforth methods [33].

Appropriate models of towers can be given using finite elements, but these methods have a very high computational cost. We can give a rough approximation of towers as Bernoulli beams, specifically, vertical uniform cantilever beams [34]. The deflection of the top of a beam can be expressed as

FL^

AX - 3EV (3'7)

where L is the length of a beam, E is Young's modulus, IA is the moment of area which depends on the shape of the cross-sectional area of a beam, and F is the force.

The model of a feeding hose is given according to the theory of catenary:

, ! (f - h \ l - 5

cosh"1 --+ 1--= 0,

\ a J a ,

; f N y5 (3-8)

cosh"1 ( - + 1 ) - - = 0 aa

where a and 5 are the special parameters of the catenary, f is the sag, h is the difference in heights between the ends of the catenary, and l is the length of span.

By solving Eq. (3.8), we can find parameters of the catenary of a feeding hose at any point of the trajectory of the mobile platform of the robot [35, 36].

It should be noted that, in the case of external disturbances such as wind pulsations or impact from the feeding hose, we deal with uncertainty in the position of proximal anchor points of upper cables. Taking into account the mechanical properties of the Bernoulli beam, we may assume that the proximal anchor points of lower cables always meet their given places [37]. This is the most important assumption for the control strategies described in the next chapter.

4. The problem of distribution: the group of lower cables

The relevant question is the problem of tension distribution in the group of lower cables, and two main problems of tension distribution are counteractions and preloading. The preliminary results obtained by the authors were reported in [38].

Taking into account both the problems of deformations and external disturbance, we have to choose a control strategy. Since the tops of the towers mismatch their positions, the coordinates of proximal anchor points cannot be defined accurately. Also, the cables themselves are deformed under payload. Therefore, no geometrical model can be used to define the position and orientation of the mobile platform using configuration properties of the group of upper cables. This problem can be solved by decoupling the cable system into two groups. The group of four upper cables is responsible for three degrees of freedom which are pitch and roll angles, and heave in terms of motion of a ship. Thereafter, the group of four lower cables is responsible for surge and sway, and yaw angle. Under the given conditions without external disturbance and for a relatively smooth and slow law of motion, the errors in general coordinates with respect to surge and sway, and the yaw angle can be assumed to be small. Therefore, we do not need to compensate for these errors by running the group of lower cables, and running the group of upper cables is enough. In other words, the group of upper cables is responsible for gravity compensation tasks. The method for the case described above has been proposed and discussed by the authors in [37, 39].

The robot must be equipped with the following types of sensors:

• electronic levels attached to lower cables at distal anchor points;

• tension sensors under the pulleys at proximal anchor points of lower cables;

• accelerometer and gyroscope (IMU) attached to the mobile platform of the robot

to use the proposed control strategies.

In the next sections, the methods to run the group of lower cables are proposed. These cases can take place when the feeding hose is attached to the extruder which is mounted on the mobile platform of a large-sized cable-driven parallel robot. The cables of the lower group have to be

preloaded to prevent any sag, and the forces of counteractions have to be distributed taking into account the specific properties of cables that can only pull but not push.

As has been noted earlier, the system of cables of the robot can be decoupled, and now we are dealing with the group of lower cables. It should be noted that the given configuration for the group of four lower cables is such that they lie in the same horizontal plane. Thus, we can assume the object of the mobile platform to be a plate and reformulate the model as a planar problem. We obtain the system with three degrees of freedom which are surge and sway, and the yaw angle, and four joints, which are cables [40]. The example shows the calculation of counteractions against forces of inertia of the mobile platform described by (4.1). Thus, the system is overactuated, and different ways to solve its mechanical problems can be used according to the chosen criteria of optimization:

J

3x4p4x1 b

'-c

Mf3 q3x1.

(4.1)

Then we have to solve (4.1) as a nonhomogeneous system of linear algebraic equations, and the solution is

fc = fp + f„ (4.2)

where fp is a particular solution that depends on the right-hand side of the equation, and fs is a special solution that belongs to the null-space of Jb.

Therefore, the vectors of forces fp and fs define counteracting and preloading in the group of lower cables, respectively. The following sections are devoted to finding the optimal solutions for the problem of preloading and counteractions in different ways.

5. The problem of preloading: the null space approach

We propose to give the preloading for the lower group of cables via the algorithm of distributing the vectors of tension forces by exploring the null-space of the geometric Jacobian. The example given below shows the calculations for the specific case of the 3 x 4 matrix, and its rank is assumed to be r = 2:

u11 u12 u13 0 0 0

u21 u22 u23 0 0 0

u31 u32 u33 0 0 0 0

U

11 v12 v13 v14

21 v22 v23 v24

31 v32 v33 4 3

41 v42 v43 V44

V VT

Using the singular value decomposition of the matrix, we can find the basis of its null-space. In popular mathematical packages, svd and rank are conventional functions. The next task is to find a vector that belongs to null-space and does not have negative components, if it exists. Therefore, we have to solve an optimization problem and find the coefficients for the set of vectors of null-space to compose the vector of forces:

m

v13 v14 f. v13 v14

v23 + n • v24 • f to ^ fs = v23 v24

=

v33 v34 fs3 v33 v34

v43 v44 A. v43 v44

*i*Не можете найти то, что вам нужно? Попробуйте сервис подбора литературы.

v

v

J

b

v

v

Then we consider the last expression to be a problem of linear programming Ax ^ b, and in popular mathematical packages linprog is a conventional function.

Algorithm 1. % singular value decomposition returns three matrices

[U, S, V] = svd(Jb) % after eliminating the first r columns of V the last % columns of V form the basis of the null-space of matrix Jb for k = 1 : r

V:>fc = [ ]

end

V

null

V

% solver linprog provides a solution with the Dual-% Simplex Algorithm ('dual-simplex') or the % Interior-Point Algorithm ('interior-point')

% for some objective function f and lower bound x ^ 0 x = solver(f, Vnull, b)

V

null

x

The proposed algorithm is a kind of solver that solves a specific problem of null-space. The algorithm has a relatively low computational cost, but for the set of points of the trajectory it obviously provides polygonal chains in the space of forces. Thus, we face a problem in realtime execution, the less the discretization time step, the less the biases of polygonal chains, and vice versa. It has a great impact on the energy efficiency of the robot and particularly on the smoothness of the motion of the end-effector.

Combination of parameters

0 100 200 300 400 500 600 700 800

t, sec

Fig. 2. Calculated parameters m and n

But we definitely do not need to calculate these optimal values in real-time. Therefore, we just use a preliminary calculated set of optimal solutions for the given workspace. A possible way is fuzzy approximation that provides a smooth and continuous control surface for the given data set. This approach is discussed in the next section.

The results for linear programming solutions are shown in Figs. 2, 3.

s

Forces in Null Space (Linear Programming)

50 100 150 200 250 300 350 400 450 500 550

t, sec

Fig. 3. Forces in null space (LP-solutions)

6. The problem of counteraction: the game theory approach

Now we propose to solve the problem of counteraction for the lower group of cables using the algorithm of distributing the vectors of tension forces by setting the problem as a cooperative game. The example given below shows an n-person game, which is a linear production game, and the number of actors is assumed to be n = 4. The players can form coalitions, cooperating with one another to reach the most effective configuration. The value of a coalition K is the maximum profit it can achieve with all the resources possessed by its members. It can be obtained by solving a corresponding linear programming problem as follows:

+ cnxn),

v(K ) = maxC xl + c2x2 +

x>0

ai,lxl + ai,2x2 +

+ ai,nxn ^ "y ^ b,

ies

S3

where ci is a given market price, xi is an imputation of i-th player, ai j is the amount of resource, bi is a vector of resources, and Vi = 1 : m, Vj = 1 : n.

Turning back to the problem of the robot, we assume that the components of the vector of forces in configuration space are assumed now to be n specific types of resources, and m types of products, by the number of the rows of the Jacobian matrix, can be produced out of them. Each component of a row of the Jacobian matrix gives some amount of these resources to make its product.

Note that for the right part of the given constraints we use the corresponding values from the right-hand side of (4.1). Hence, we instantly obtain the sum of bij for each ith case without calculating each bi j separately. We obtain the conditions for each discretization time step:

v(K) = m&x(clxl + c2x2 + c3 x3 + c4x4),

j1 ,1x1 + j1 , 2x2 + j1 , 3x3 + j1 , 4x4 ^ Fx, j2,1 x1 + j2,2x2 + j2,3x3 + j2, 4x4 ^ Fy, j3,1x1 + j3,2x2 + j3,3x3 + j3,4x4 ^ x1, x2, x3, x4

" 0.

Such problems of linear programming can be visualized using convex polytopes. In our case of cooperative game, we have a tetrahedron whose vertices correspond to the players, and its inner space is the imputation set, and after truncation of the tetrahedron with given planar constraints we obtain the core of a game. The solution of a game belongs to the core, which is the set of feasible allocations or imputations. We consider it to be a geometrical interpretation of a game, which is related to the inner space of the tetrahedron of the imputation set.

On the other hand, the problem of forces distribution can be associated with a voting game, where winning coalition gives sufficient voices for the passage of the decision [41]. To each winning coalition we assign v(K{¿}) = v(K), and to the others we assign v(K{¿}) = 0.

To solve the problem of a voting game, at first we have to define the number of possible combinations of players without repetitions:

= kl(n-k)l {6A)

where n is the total number of players and k is the number of players in coalition K.

It should be noted that the factorial grows very fast, i. e., 8! = 40,320 possible combinations. Nevertheless, it seems to be a relevant technique, because for real systems of cable-driven parallel robots the number of cables is usually no more than 8, and the number of possible combinations without repetitions even for n = 8 is not so large:

Cg1 + C82 + C83 + C84 + C85 + C86 + C87 + C88 = 255, (6.2)

and this number can be also reduced after excluding coalitions associated with singular configurations.

For the given number of players n = 4 the result is:

C41 + C42 + C43 + C44 = 4 + 6 + 4 + 1 = 15 (6.3)

and the coalitions are: 1 player:

K{1}, K{2}, K{3}, K{4},

d (2 (3 (4

2 players:

3 players:

4 players:

K{1, 2}, K{1, 3}, K{1, 4},

© © (!)

K{2, 3}, K{2, 4}, K{3, 4}, © © ©

K{12, 3}, K{12, 4}, K{13, 4}, K{2 3, 4},

© © © ©

K{1, 2, 3, 4}.

Also, we have to take into account singular configurations in the cable system. It becomes possible if two of two cables are tightened along the same straight line, e.g., for a rectangular base these are diagonal lines, Figs. 4, 5. Therefore, for the given configuration of the cable system

3 4

Fig. 4. Arbitrary nonsingular configuration of the group of lower cables 2 1

3 4

Fig. 5. Singular configuration of the group of lower cables (2nd and 4th cables)

the coalitions K{1, 3} and K{2, 4} must be excluded from the pool of possible combinations in calculations.

Then for each coalition we solve a system of linear algebraic equations. Each solution is found with minimum norm least-squares which is the conventional function lsqminnorm in popular mathematical packages. An example is given for K{1, 4} to check if it is a winning coalition:

j11 0 0 j 14 "A" /2 /3 /4 F x

j21 0 0 j 14 = Fy

j31 0 0 j 14 M4>.

P{1, 4}

S{1, 4}

v(K )

S{1, 4} = lsqminnorm(P{1, 4}, v(K)),

fn < 0 ^ fn = 0

P{1, 4} • S{1, 4} = v(K{1, 4}), P {i}^S {i} = v(K {i}), S{i} = argmin ||v(K{i}) — v(K)||.

Algorithm 2.

% solver_1 finds all the possible combinations without % repetition for a given number of players n that are % the coalitions K

K{i} = solver_1(n) % solver_2 calculates v(K{i}) which is the worth of % coalition K{i}

v(k{i}) = solver_2(Jb, F, K{i}) % solver_3 finds the appropriate coalition K{i} and cor-% responding solution f according to the given criteria m fs = solver_3(v(K{i}), m)

The proposed algorithm is a kind of a solver that finds an optimal solution in the sense of game theory. The algorithm uses combinatorial interpretation of a game, which is related to the problem of possible combinations without repetitions.

It should be recommended for real tasks due to satisfactory small execution time of the algorithm and smoothness of the solution as is. Thereafter, it becomes suitable for the real-time control problems of a cable-driven parallel robot.

The problem could definitely be easily set and solved in the sense of linear programming with conventional solvers, e.g., linprog. But now we want to set and solve the problem of distribution, which is definitely the problem of game theory, in the sense of game theory. In other words, we assign specific behavior to the elements of the mechanical system of the robot. Hence, the cables are assumed to be the actors and become able to follow appropriate strategies according to the given conditions: compete or cooperate. According to the paradigm of artificial intelligence, an intelligent agent is an agent acting in an intelligent manner, implying the ability to react rationally to some events [42]. Thus, each participant of the game becomes an intelligent agent. Hence, forming a coalition or group of tightened cables belongs to the field of artificial intelligence and means a kind of optimal distribution based on the concepts of game theory.

The results for game solutions are shown in Figs. 6, 10 and 11.

Technically, a similar method of special nonnegative least squares, but without game interpretation and with added condition to check nonsingular solutions is reported by the authors in [43].

7. Fuzzy approximation

From a mathematical standpoint, we have to approximate a vector field that corresponds to the set of solutions for the given task of the preloading problem. The control surface is desired to be smooth and continuous, and it seems to be a fuzzy approximation. Obviously, we do not need to use all the power of deep learning and deep neural networks, and the relatively small

Winning coalitions

0 50 100 150 200 250 300 350 400 450 500 550

t, sec

Fig. 6. The winning coalitions

fuzzy system could suffice for the given problem. According to Kosko's theorem, fuzzy systems can be assumed to be universal approximators [44].

*i*Не можете найти то, что вам нужно? Попробуйте сервис подбора литературы.

According to Hornik's theorem, multilayer feedforward networks can be also assumed to be universal approximators [45]. So, the ANFIS approximation seems to be an appropriate way for the case being discussed in this paper, although other techniques can be used too. Such an approximation has shown good results in numerical simulations for the robot model with an attached feeding hose [36, 47].

To train the ANFIS for each of the control tasks shown above, we use the method proposed by the authors in [47]. The trajectory law given by a parametric curve, namely, Lissajous lines, is used to form a quasi-regular grid, then the set of optimal solutions has to be processed with a Takagi-Sugeno-Kang fuzzy system.

input inputmf rule outputmf output

Fig. 7. ANFIS as a regulator of tension distribution in the group of lower cables

Fig. 8. Fuzzy control surface for the regulator of tension distribution in the group of lower cables

Experimental verification has shown that a fuzzy system with two inputs and seven triangular membership functions is sufficient to approximate the set of solutions obtained with numerical modeling of the preloading problem, Figs. 7 and 8. The data set with approximately 105 rows is enough to train the ANFIS as a regulator of tension distribution in the group of lower cables. It only requires tens of iterations and takes on a average no more than hundreds of seconds for modern laptops. For example, the PC with CPU AMD Ryzen 7 5700X 4.6 GHz and RAM 48 Gb 3200 MHz solves the problem of training such an ANFIS in less than 2 minutes.

After training, the quality of the control system is checked using the trajectory law of Frantz's squircle, and the quality criteria are the errors in the position and orientation of the mobile platform of the robot [46, 47]. The feeding hose is assumed to be attached to the extruder, which is mounted on the mobile platform of the robot [36, 38].

The results for fuzzy approximation of LP-solutions are shown in Fig. 9.

0 20 40 60 80 100 120 140 160 180 200

t, sec

Fig. 9. Forces in null space: LP-solutions (solid lines) vs. fuzzy approximation (dashed lines)

8. Summary and discussion

Firstly, we should note that including specific techniques of artificial intelligence into the model is not a solution for all problems of cable-driven parallel robots. On the other hand, the proposed methods for distributing specific resources, such as forces of tension, can be useful not only for cable-driven robots, but also for any other types. The proposed mathematical models have a simple description in terms of linear algebra and combinatorics that can be convenient for most cases. The listed examples are not limited to applications with the game theory approach and fuzzy approximations. Other compatible mathematical models can be also used to solve such specific problems in the field of cable-driven robotics.

Checking the solutions 1500,-.-.-.-.—®-.-,. . . =n

1000

500

v / \ ' \ / \ / * ' \ £ \ / \ / y \ / v \

-500; [MX: X/vV /\ N

-1000

—1500-1-

0 50 100 150 200 250 300 350 400 450 500 550

t, sec

Fig. 10. Forces in operational space: input (action) and solution of the cooperative game (counteraction)

2000,-1-1-1-1-^^-, I I i-n

_tp fP

!800 _ \\_Z

J 2 J 4

1600 -'

1400

vm

0 50 100 150 200 250 300 350 400 450 500 550

t, sec

Fig. 11. Forces in configuration space: solution of the cooperative game

Checking the solutions

pi___pg X X pi pa V V

-Mlg Ml

/ -VyTN

* \ » / \ # \ / \ / / v X '

y V V \/ v V

* /\ A \ ' X L A x y\ i_______'^..-'X-/ A A A x / » / \ » / V \ > / t\ \ v y / \ \

Solution of cooperative game

/T ÎÏ

fP2~ — fl

wi

Nevertheless, we can see the results of numerical simulations, and the proposed methods should work well for cable-driven parallel robots. The appropriate aims and achieved results in the control tasks of a cable system of a large-sized cable-driven parallel robot are listed below:

• preloading: the null-space concept has been used to provide preloading for the lower group of cables,

• counteraction: the game theory approach has been used to solve the problem of counteraction for the lower group of cables,

• real-time: fuzzy approximation has been given to the obtained sets of solutions for the problem of preloading, and game solutions are ready to use as is for the problem of counteraction.

The results for the proposed techniques relative to reducing errors in the position and orientation of the mobile platform of a large-sized cable-driven parallel robot are shown in Figs. 12-14. Position and orientation errors in centimeters and centiradians are assumed to be satisfactory for a cable-driven robot with linear dimensions in tens of meters.

Position errors

t, sec Orientation errors

0 500 1000 1500 2000 2500

t, sec

Fig. 12. Errors in the position and orientation of a mobile platform: w/o feeding hose and w/o correction of tension distribution

9. Conclusions

The problem of tension distribution in the system of a cable-driven parallel robot has been solved by finding a solution to the underdetermined system of linear algebraic equations. The structural nonlinearity of cables must be taken into account, then the components of the vector of forces in the configuration space of the robot can be only nonnegative. Particular solution depends on the right side of the system and corresponds to counteraction with respect to external disturbance. Special solution belongs to the null-space of the Jacobian matrix and corresponds to preloading of the cable system of the robot. The problem of preloading has been formalized as

Position errors

0.1 0

-0.1

-A, Ay A,

"V

500 1000 1500 2000 2500 t, sec

Orientation errors

500 1000 1500 2000 2500 t, sec

Fig. 13. Errors in the position and orientation of a mobile platform: with feeding hose and w/o correction of tension distribution

Position errors

0.1 0 -0.1

500 1000 1500

t, sec

Orientation errors

2000 2500

TJ

0

500

1000 1500

t, sec

2000 2500

Fig. 14. Errors in the position and orientation of a mobile platform: with feeding hose and with correction of tension distribution: preloading (fuzzy approximation of LP-solution) and counteraction (game solution)

a problem of linear programming, and the solutions have been approximated with fuzzy systems. The problem of counteraction has been formalized as a problem of game theory and successfully solved in two different ways. The cables were considered to be the actors and two concepts of cooperative games were shown: the linear production game and the voting game. According to the proposed approach, the cables were participants in the games and became intelligent agents that formed coalitions to reach the most profitable configuration. The proposed methods have been applied to solve problems of a large-sized cable-driven parallel robot, and the promising

results are shown in examples with numerical simulation. Future work should focus on studying the ways to make the mathematical modeling of cable-driven parallel robots easier and on finding new simple mathematical descriptions for the models of this specific type of robotic systems.

Acknowledgments

The authors thank Leisan Vasilova for useful remarks and Andrew Mikhailov for his contribution in the development of software for cable-driven robots.

Conflict of interest

The authors declare that they have no conflict of interest.

References

[1] Samset, I., The History of Rope Transportation, in Winch and Cable Systems, Forestry Sci., vol. 18, Dordrecht: Springer, 1985, pp. 23-54.

[2] An Encyclopaedia of the History of Technology, I. McNeil (Ed.), London: Routledge, 1996.

[3] Kim, P. W., Scientific Disciplines of Geojunggi (the Traditional Crane) in Korean Science, Technology and History Class, Eurasia J. Math. Sci. Technol. Educ, 2017, vol. 13, no. 9, pp. 6147-6163.

[4] Landsberger, S. E., Design and Construction of a Cable-Controlled, Parallel Link Manipulator, Master's Thesis, Cambridge, Mass., Massachusetts Institute of Technology, 1984, 123 pp.

[5] Albus, J.S., Cable Arrangement and Lifting Platform for Stabilized Load Lifting, Patent US No 4,883,184 (28 Nov 1989).

[6] Tho, T. P. and Thinh, N. T., Using a Cable-Driven Parallel Robot with Applications in 3D Concrete Printing, Appl. Sci., 2021, vol. 11, no. 2, Art. 563, 23 pp.

[7] Zi, B. and Qian, S., Design, Analysis and Control of Cable-Suspended Parallel Robots and Its Applications, Singapore: Springer, 2017.

[8] Pott, A., Cable-Driven Parallel Robots: Theory and Application, Cham: Springer, 2018.

[9] Morris, M. and Shoham, M., Applications and Theoretical Issues of Cable-Driven Robots, in 2009 Florida Conf. on Recent Advances in Robotics (Boca Raton, Fla, 2009), 29 pp.

[10] Tho, T. P. and Thinh, N. T., An Overview of Cable-Driven Parallel Robots: Workspace, Tension Distribution, and Cable Sagging, Math. Probl. Eng., 2022, vol. 2022, Art. ID 2199748, 15 pp.

[11] Zhang, Z., Shao, Z., You, Z., Tang, X., Zi, B., Yang, G., Gosselin, C., and Caro, S., State-of-the-Art on Theories and Applications of Cable-Driven Parallel Robots, Front. Mech. Eng., 2022, vol. 17, no. 3, Art. 37, 23 pp.

[12] von Neumann, J. and Morgenstern, O., Theory of Games and Economic Behavior, Princeton, N.J.: Princeton Univ. Press, 1944.

[13] Zadeh, L. A., Fuzzy Sets, Inform. Control, 1965, vol. 8, no. 3, pp. 338-353.

[14] Fedin, A. P., Kalinin, Ya. V., and Marchuk, E.A., ANN and Elements of Differentional Games in the Model of Regulated Wheel Slippage Process in a Braking Mode, in Proc. of the Internat. Conf. "Nonlinearity, Information and Robotics" (Innopolis, Russia, Dec 2020), 4 pp.

[15] Zhang, J.-F., Challenges and Opportunities Facing Game Theory and Control: An Interview with Tamer Ba§ar, Natl. Sci. Rev., 2020, vol. 7, no. 7, pp. 1142-1146.

*i*Не можете найти то, что вам нужно? Попробуйте сервис подбора литературы.

[16] Wu, J., Wu, D., Yan, Y., Zhang, N., Bao, C., Wang, F., Steering and Braking Game Control Architecture Based Minimax Robust Stability Control for Emergency Avoidance of Autonomous Vehicles, Sci. China Technol. Sci., 2022, vol. 65, no. 4, pp. 943-955.

[17] Isaacs, R., Differential Games: A Mathematical Theory with Applications to Warfare and Pursuit, Control and Optimization, New York: Wiley, 1965.

[18] Bauso, D., Game Theory with Engineering Applications, Philadelphia, Pa.: SIAM, 2016.

[19] Marden, J.R. and Shamma, J. S., Game-Theoretic Learning in Distributed Control, in Handbook of Dynamic Game Theory, T. Ba§ar, G. Zaccour (Eds.), Cham: Springer, 2018, pp. 511-546.

[20] Rao, S. S., Game Theory Approach for Multiobjective Structural Optimization, Comput. Struct., 1987, vol. 25, no. 1, pp. 119-127.

[21] Jang, J.-S. R., ANFIS: Adaptive-Network-Based Fuzzy Inference System, in IEEE Trans. Syst. Man Cybern, 1993, vol. 23, no. 3, pp. 665-685.

[22] Reznik, L., Fuzzy Controllers, Burlington, Mass.: Newnes/Elsevier, 1997.

[23] Ross, T. J., Fuzzy Logic with Engineering Applications, 3rd ed., New York: Wiley, 2010.

[24] Piegat, A., Fuzzy Modeling and Control, Stud. Fuzziness Soft Comput., vol. 69, Berlin: Springer, 2001.

[25] Haykin, S., Neural Networks and Learning Machines, 3rd ed., New York: Pearson, 2016.

[26] Desouky, S.F. and Schwartz, H. M., Self-Learning Fuzzy Logic Controllers for Pursuit-Evasion Differential Games, Robot. Auton. Syst., 2011, vol. 59, no. 1, pp. 22-33.

[27] Pickering, L. and Cohen, K., Genetic Fuzzy Controller for the Homicidal Chauffeur Differential Game, in Applications of Fuzzy Techniques: North American Fuzzy Information Processing Society Annual Conference (NAFIPS, Halifax, NS, Canada, 2022), S.Dick, V.Kreinovich, P.Lingras (Eds.), Lect. Notes Netw. Syst., vol. 500, Cham: Springer, 2023, pp. 196-204.

[28] Amiri, R., Moussavi, S.Z., and Hamedi, J., Neuro-Fuzzy Intelligent Control Algorithm for Cable-Driven Robots with Elastic Cables, in Proc. of the Internat. Symp. on Computer Science and Intelligent Controls (ISCSIC, Budapest, Hungary, 2017), pp. 57-62.

[29] Aguas, X. I., Cuaycal, A., Paredes, I., and Herrera, M., A Fuzzy Sliding Mode Controller for Planar 4-Cable Direct Driven Robot, Enfoque UTE, 2018, vol. 9, no. 4, pp. 99-109.

[30] Tho, T. P. and Thinh, N. T., Sagging Cable Analysis and Evaluation of 4-Degree-of-Freedom Cable Robot Using Adaptive Neural Fuzzy Algorithm, Int. J. Mech. Eng. Robot. Res., 2022, vol. 11, no. 2, pp. 73-78.

[31] Hendrycks, D. and Gimpel, K., Gaussian Error Linear Units (GELUs), arXiv:1606.08415 (2016).

[32] Marchuk, E. A. and Kalinin, Ya. V., Specifity of Including of Structural Nonlinearity in Model of Dynamics of Cable-Driven Robot, Mekhatronika, Avtomatizatsiya, Upravlenie, 2021, vol. 22, no. 10, pp. 547-552 (Russian).

[33] Mathews, J.H. and Fink, K.D., Numerical Methods Using MATLAB, 4th ed., New York: Pearson, 2003.

[34] Esfandiari, R. S. and Lu, B., Modeling and Analysis of Dynamic Systems, 3rd ed., Boca Raton, Fla.: CRC, 2018.

[35] Irvine, M., Cable Structures, New York: Dover, 1992.

[36] Marchuk, E. A., Kalinin, Ya. V., Ivanou, M.A., and Maloletov, A.V., On Mathematical Modeling of Sagging Elements of Large-Sized Parallel Cable-Driven Robot, Izv. Volgogradsk. Gos. Tekhn. Univ., 2022, no. 9 (268), pp. 56-62 (Russian).

[37] Marchuk, E. A., Kalinin, Ya. V., Sidorova, A. V., and Maloletov, A. V., On the Problem of Position and Orientation Errors of a Large-Sized Cable-Driven Parallel Robot, Russian J. Nonlinear Dyn., 2022, vol. 18, no. 5, pp. 745-760.

[38] Marchuk, E. A. and Sharonov, N. G., Problem of Tension Distribution in a System of Cable-Driven Parallel Robot: Game Theory Formalization, Izv. Volgogradsk. Gos. Tekhn. Univ., 2023, no. 9 (280), pp. 51-59 (Russian).

[39] Marchuk, E.A., Kalinin, Ya. V., and Maloletov, A.V., Error Compensation in Position and Orientation of Mobile Platform of Cable-Driven Robots via Tensile Forces Measurement, Mekhatronika, Avtomatizatsiya, Upravlenie, 2022, vol. 23, no. 10, pp. 515-522.

[40] De Luca, A., Robots with Kinematic Redundancy, Sapienza Universita di Roma, 2016, 48 p.

[41] McCain, R.A., Game Theory: A Nontechnical Introduction to the Analysis of Strategy, 3rd ed., River Edge, N.J.: World Sci., 2014.

[42] Russell, S.J. and Norvig, P., Artificial Intelligence: A Modern Approach, 3rd ed., Upper Saddle River, N.J.: Prentice Hall, 2009.

[43] Marchuk, E. A., Mikhailov, A. S., Kalinin, Ya. V., and Maloletov, A. V., On the Problem of Tension Forces Distribution in Cable System of Cable-Driven Parallel Robot, Mekhatronika, Avtomatizatsiya, Upravlenie, 2023, vol. 24, no. 12, pp. 643-651.

[44] Kosko, B., Fuzzy Systems As Universal Approximators, IEEE Trans. Comput., 1994, vol. 43, no. 11, pp. 1329-1333.

[45] Hornik, K., Stinchcombe, M., and White, H., Multilayer Feedforward Networks Are Universal Approximators, Neural Netw., 1989, vol. 2, no. 5, pp. 359-366.

[46] Frantz, M., Save This Family! An Instructive Sequence of Curves, Math. Mag., 2018, vol. 91, no. 4, pp. 267-273.

[47] Marchuk, E., Kalinin, Ya., and Maloletov, A., On Smooth Planar Curvilinear Motion of Cable-Driven Parallel Robot End-Effector, IFAC-PapersOnLine, 2022, vol. 55, no. 10, pp. 2475-2480.