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Computer Software Technologies for Intelligent Robot
VLADIMIR PAVLOVSKYKIAM RAS
Department of MechatronicsMiusskaya sq., 4 Moscow
ANTON ALISEYCHIKKIAM RAS
Department of MechatronicsMiusskaya sq., 4 Moscow
IGOR ORLOVKIAM RAS
Department of MechatronicsMiusskaya sq., 4 Moscow
Abstract: Nowadays new computer techniques for intelligent robots are the important point of investigations.Technologies for the intelligent manipulators capable to make decisions automatically is convenient to developusing laboratory tasks, like games with the objects manipulation. As examples of such tasks a) the game “Gomoku”and b) the game “Go” are considered. A set of problem-oriented technologies were investigated from the point ofview of the manipulator ManGo robot, namely its dynamic model and control system in Matlab Simulink. Thelogical move choice program in the game “Gomoku” is designed in Prolog and C languages. Vision system assupporting technology will be demonstrated as important part of the presentation. It is important to note, differentalgorithms can be implemented as manipulator software and hardware technologies. Besides classical algorithmsutilization, considerable attention has been given to the application of neural network paradigms for manipulatorscontrol. Addressing the problem the neural software for ManGo robot will be presented as well. The created robotspassed experimental working off which allowed to draw conclusions about the created software and hardwareeffectiveness in the implementation of various manipulators control algorithms.
Key–Words: Robotics, computer vision, neural networks
1 IntroductionNowadays one of the most important tasks in roboticsis creating intellectual robots. Among them creationof intellectual manipulator robots, able to automaticallymake decisions during their work, is being widely ex-plored. It is convenient to research such systems on labtasks such as different games, requiring to manipulatedifferent objects. In this work, hardware and softwaremeans of intellectual manipulator robots realization areconsidered, besides, two tasks and systems solving themare described. The first one is ManGo robot, developedfor solving some intellectual gaming tasks for robots,such as logical table games. Initially ManGo was cre-ated to play “Go” game with a human, but as the firstand easier application “Tic-Tac-Toe” game on a big, po-tentially unlimited field, is being considered. On thisgame, controlling the manipulator is being developed.
2 Logic Games Playing RobotIn an ancient Chinese game with a Japanese name“Gomoku” also known as Tic-Tac-Toe or “five-in-a-
row”, two player place symbols on an infinite desk, or adesk of m×n size by turns , until any of them places fiveof his symbols in a row vertically, horizontally, diago-nally, or until the desk is fully covered by symbols, if itis finite. In this case, neither of the players win. In caseof infinite desk the number of turns is limited. Eachplayer uses symbols of only one type: either “cross”,or “zero”. Each field of the desk can contain only onesymbol. Instead of “crosses” and “zeroes” in the origi-nal version stones of different colors (white and black)are used. In “Gomoku” game players place symbolson line intersections instead of fields. The desk size isusually limited by the size of cross-section paper sheet.For “Gomoku” game 15× 15 or 19× 19 (like in “Go”game) desk size is used. In the robot turn realizing pro-gram 20×20 desk is used. It is easy to prove that if thefirst player plays well enough, he can play draw at least.It is harder to prove that he can win but it is still obvi-ous. The first player has an advantage, for this reasonhe is given a handicap, restricting some moves. Partic-ularly it happens in “Renju” — one of the analoguesof “Gomoku”. It is known that professional “Renju”players, who move first gain an advantage at 10th and
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win the game “Gomoku” at 15th move. [8] Presents aprogram Victoria that always wins the game on desk15×15 if it moves first.
In computer applications, realizing the game “Fivein a Row” the move of computer player is often madewith the help of estimation function. For each emptyfield with indexes (i, j) its estimation (a real number) iscalculated, considering both profit of the player, and theprofit of an opponent that he could gain after makingthis move on field.
As the estimation function, for instance, the func-tion like ev(i, j) = evx(i, j) + aev0(i, j) Is used, whereev(i, j) — is an estimation of the players move in thisfield, and ev0(i, j) — is an estimation of his opponentsmove. Coefficient a is inverted ratio to aggressive-ness of the player. With its high value the strategy ofthe player has a defensive character and with its lowvalue — offensive. The program for the robot is writ-ten in Prolog language. The steps of a computer playerin games, written in Prolog are often realized with pro-duction rules. In the system both rules and estimatingfunction are used.
At the input of the program the array, consistingof numbers 0,1,2, that stand for empty and occupiedwith stones of dark and light colors respectively. Theprogram returns the number of a row and the number ofa column of a field that robot moves to.
The application in Visual Prolog 7.4, one of themost developed realizations of Prolog language was cre-ated for debugging the strategy of the computer player.A fast complier enables to ignore the details of algo-rithm and to formulate rules by declarative way, so itincreases the speed of coding the program. The usercan choose the player that makes the first move. If it isa computer, the window is opened with a move alreadymade. The first step is made randomly on a field that isless than 3 rows away from the center of the board. Theusers move is made by a click of a mouse on the field.
The players move rules are represented in the fol-lowing way:
• to put own sign fifth in a row;
• not to allow the opponent to put his sign fifth in arow (to perform “block”);
• not to allow the opponent to put his fourth sign ina row;
• to put own sign fourth in a row;
• to create a “fork”;
• not to allow the enemy to create a “fork”;
• to put a sign in the field that is located near themost of opponents three-sign rows and at least oneempty field;
• to put a sign in the field, the estimation of whichhas the highest value.
Among the fields with the maximum value of the es-timation function the fields that border with the max-imum number of cells, occupied by the opponent arecollected. The field among the collected ones is chosenat random. For the move search the estimation func-tion, described above is used. As the row estimationthe function ev(s,k) = 4k+1 is taken, where s is a sign,k — number of signs s in a row, parameter a is assumedto be equal to 1. For position storage, the program useslists; in order to lower the searching, only occupied slotsare stored. List in Prolog language is a persistent datatype, it allows backtrack. Unlike it in modifiable datatypes, in arrays of C language for instance, the changesare made in computer memory, so backtrack is unablefor them. The usage of lists allows the simplest situationmodelling, for instance, checking if the critical situationexists after placing on one field own and the opponentssign.
3 Manipulator and Control SystemMango manipulator (Fig. 1) has a SCARA-like kine-matic, that mostly suits object manipulation tasks on aplane, including desk games [6]. The first steps in cin-ematic analysis were carried out during the creation ofrobots design in CAR soft complex, the pneumatics ofItalian company “Pneumax” was used as the executingmotor. Optimal lengths of parts and attachment pointsof pneumatics were calculated to cover workspace of500 x 500 cm size, which is enough to work with al-most every knowledge-based logical desk game.
Due to simple two-section cinematic scheme the so-lution of inverse cinematic problem for robot control istrivial. Therefore, the most reliable in this case trajec-tory cinematic control with inverse link in the angles ofjoint rotation. The usage of pneumatics involves bothad-vantages (cheapness, higher speeds of motion, greatefforts, etc.) and drawbacks, the main of which is thedifficulty of accurate control. In order to solve this prob-lem the riveted PWM-control of the cylinders was real-ized. The control system of the manipulator ManGo for
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Figure 1: ManGo robot
the “Gomoky” game was realized on the pair microcon-troller Stm32F4 – PC with Windows OS, and the controlsystem – on the pair microcontroller Stm32F4 – smart-phone with Android OS.
3.1 Machine visionDesk recognition on a picture can be divided into sev-eral steps:
• Picture preparation (converting into binary matrix)
• Searching for intersections and edges of the board
• Searching for neighbors and beginning of the gridbuilding
• Completing the grid
The main steps of desk recognition algorithm on acertain example is described further.
3.1.1 Pattern RecognitionSince lines on the board are thin and therefore poorlydiscernible (especially because of desk bent relativelyto camera, natural non monotonous wood texture of theboard surface and occasional glints), high quality of the
picture is required. Using of camera with resolution ofno less than 3 megapixels is recommended.
Figure 2: The original picture (on the left), picture afteradaptive threshold (on the right)
To get a binary picture with sharp outlines, with-out noise, Adaptive Threshold [7]–[10] is applied to theoriginal picture(Fig. 2). Operations of dilatation anderosion follow next to smooth and clean the image.
3.1.2 Searching for Line IntersectionsNow it is time to get to the next step, which is pat-terns usage. Before anything else, the edges of the deskmust be found. It is very important to find the bottomedge (the top is hard to find due to desk bent, it is of-ten the same color as the surroundings), and at leastone of the side edges, to define the position of the deskon the picture. Patterns describe how line intersectionsshould look like in binary picture. All that is need to bedone is to apply these patterns during hit-or-miss trans-form [16]. However, because of image flaws (noise anddiscontinuous lines), some edge points are missed andsome detected points don’t belong to the edge. Nextstep is to detect an line. OpenCV library offers methodsof fitting line in array of points, but their accuracy is notsufficient. The only option is to run through all near-est to each other pairs of points, chose those with moreprobable angle (since possible range is known for eachedge). Associated with pair of points line is added tovector with one “vote” or, if a line with similar param-eters has already been added, count of votes of latterincrements. Line with biggest count is the one.
Intersections inside the grid are found in the sameway, through patterns.
3.1.3 Grid BuildingFLANN (Fast Library for Approximate Nearest Neigh-bors) [11] is a library, that implement nearest neigh-bor search method (interface for the library is con-tained in OpenCV). It uses point array in matrix form
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as input, and the result is k-dimensional tree, datastructure, dividing the space for sorting points in k-dimensional space. In this case, the space is obviouslytwo-dimensional. Each point of the desk has no morethan four direct neighbors. In process of searching offour nearest for each point, restrictions are placed ac-cording to situation:
• Little distance between points, the distances be-tween the point and each one of the neighborsshouldn’t differ too much from each other;
• Right and left neighbors must lie on the same hor-izontal line;
• Top and bottom neighbors must lie on the sameline, which is bent from the main vertical to nomore than 30 degrees.
Point becomes a part of the grid if it has no less thantwo neighbors.
During this part of the algorithm, vector of 6-slotarrays is created, each array associated with the singlepoint. The first four slots contain neighbor indexes (or−1, if there is no neighbor in a certain direction), andthe other two contain horizontal and vertical line num-bers, to which the point belongs. Fig. 3 shows how thepoints and their relations were found. The points on thetop werent found either because the distances betweenthem were too small or detected points in the area wereto scarce to find any relations. Next picture shows thefound lines as well. Next step is completing the grid
Figure 3: Lines and relations found
to a bounding rectangle. It can be done by calculatingintersections of found lines, if there is any. Thats whyfurther actions take place:
1. If the point doesnt have one neighbor, it is calcu-lated due to the coordinates of the point and the oppositeneighbor. Afterwards, the new point is added to a vec-tor of new possible points, and the counter of current isassigned 1. 2. Before adding the coordinate to the vec-tor, it’s important to check whether there is a point with
close coordinates. If so, the counter increments, and thecoordinates are changed to the average between the oldand the new one, if not, a new point is added. 3. In theend, the point is added to the grid, if the counter valueis at least two.
The process continues until there is no more pointsto add. As a result is supposed to be a matrix N (width)x M (height) size.
3.1.4 Finishing the Grid
As can be seen on Fig. 4, points lies on intersections offound lines and edges.
Figure 4: Intersection points are found
Number of missing points between the edges andrectangle can be calculated due to the grid that makes itpossible to find average distance between points on eachhorizontal line. This way grid is expanded to zones L,R and B (Fig. 4, left).
To find the rest of the points the following actionsare performed: 1. Diagonals, and their intersectionswith vertical lines are found, as shown on Fig. 5, left.2. Now we can find the rest of horizontal lines. 3. Findthe rest of the points as intersections of vertical and hor-izontal lines. Point search completed (Fig. 5, right).
Figure 5: Looking for diagonals, horizontal lines, inter-sections of diagonals and horizontal lines, points foundon the desk (respectively)
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3.2 Black and White Stones RecognitionThe idea of the algorithm is based upon the fact that il-lumination doesnt change during the game. The initialpicture of empty desk is saved. The new picture (withstones on the desk) and the old one (without) are con-verted to HSV (Hue, Saturation, Value) format. Oncethe matrix of desk points on the picture is already found,the average value of brightness (Value), and intensity(Saturation) in the point surroundings are compared be-tween two images. The less the Saturation is, the more“grayer” is the color. It means that black and white(stone colors) will have the lowest Saturation value. Thepart of the image, where the white stones are, will havethe higher Value, and the black stones will have lowerValue. To remove the flecks off the stones, the pictureis corrected by Gaussian blur.
3.3 User InterfaceAn application for Android is a user interface, where itis possible to set rules of the game and start it. Onceplay has began, it automatically tries to connect to robotvia Bluetooth, after succeeding, if you choose to start agame, it enables you to “calibrate” the desk, recognizeit with internal camera of the device. The internal toolsof the application include GNU Go engine [12], deskand stones recognition algorithms, and Bluetooth con-nection between ManGo robot and the device. GNU Gois a free Go playing program. It doesnt have a graphi-cal interface, but it supports two protocols to “commu-nicate” To other programs: Go Modem Protocol andGo Text Protocol (GTP). In this application GTP isused. The program is realized on C++ programminglanguage. Image processing is realized on C++ lan-guage with usage of OpenCV (Open Computer Vision)library. The application interface, Bluetooth connectionand camera treatment is realized on Java, with usage ofAndroid features for developers (Android SDK), whichenables us to control Android API.
4 Neural Network based ControlAll functions of manipulator control can be realized bymeans of neural network models. But before replacingcontrol elements it is worth considering advantages andshortcomings of neural network models in contrast withtraditional approaches.
The main convenience of neural network modelsis the capability of self-organization. Primary idea is
just to remember some basic interrelations between in-put and output vector signals Xi → Yi and make somekind of interpolations of Y for X 6= Xi ∈ {X} Neu-ral network shouldnt remember all input signals, but theset of remembered {Xi} ought to represent correspon-dence {X} → {Y}. The conventional structure of for-mal neuron ” j”: X , Y , M j and L j are vectors. X andY are representing activity of input and output neuronsand M j and L j are weights of input and output connec-tions. The formal neuron ” j” activity a j is calculatedlike some monotone increasing function of X ∗M j dotproduct. Model of formal neuron can be considered asa memory cell. Weights of connections are responsiblefor data storage. Input weights M j are the ”address”of the cell. When X is close enough to M j to makea j = f (X ∗M j) > 0, neuron ” j” plays back the data,stored by weights L j. The contribution of neuron ” j” tooutput layer activity Y is proportional to a j. As a ruleseveral neurons a j can add up to output neurons activityvector Y . All formal neurons and connections are con-similar. But functions of layers X , A and Y are different.Layer X represents input signal, A is mapping the inputsignal space of states and Y should represent Y in cor-respondence {X} → {Y}. Also the rules for M and Lconnections weights learning must be dissimilar. Inter-connected layers pairs XA and AY only look alike. PairXA has to remember the states of input layer X , whilepair AY must memorize desirable states of output layerY . Neuron ” j” from layer A should remember the meanactivity X while a j > 0 and neuron ”k” from layer Y —vary weights Lk to get suitable activity yk for differentXi (and corresponding Ai). Learning rules for vectorsM j and Lk should be like
∆M j = η1(X−M j)a j∆t (1)
∆Lk = η2(ysk− yk)A∆t (2)
where ysk — specified activity of yk and η1, η2 are coef-
ficients, and limt→∞ η = 0, limt→∞ ∑η = ∞.Both learning rules are designed to minimize the
distinction in parentheses, but the directions of vectorsM j and Lk changes (with reference to vectors Xi and Ai)are quite different. The task of converting {X} → {Y}allows to choose arbitrary activity level for hidden layerA. Both learning rules ( 1) and ( 2) are changing weightsonly for connections with active neurons of layer A. Themore layer A neurons are active simultaneously, the lessresolution of the input signal space of states we can get(in the sense of difference between vectors M j). So it isbetter to limit the number of active layer A neurons byfew units.
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The question about remembering output signals Yis not so obvious. If there are N neurons in layer A andthe activity limit is 1 neuron, there are N different statesof layer A activity. If limit is 2 there are N(N−1)/2 dif-ferent states of activity, if 3 there are N(N−1)(N−2)/6states and so on. One can think, that high activity is bet-ter. But if we want to memorize more than one indepen-dent signal Y , we should be able to solve the system ofp equations (p pairs Xi → Yi with corresponding innerlayer activity Ai = {aTi
1 j}):
N
∑j=1
l jka1j = y1
k ,N
∑j=1
l jkapj = yp
k . (3)
The system is solvable for p ≤ N and activity limitdoesnt change the number of independent output sig-nals Y , that can be memorized on N elements of hiddenlayer A. It means that there is no reason for choosinghigh activity level. Self-organizing maps (SOM) elab-orated by T. Kohonen [13], [14] can solve the mappingtask for the input signal space of states. Originally SOManswer is the activity of only one neuron of inner layer,”winner takes all (WTA)”. For the Y interpolation pur-poses better to have several active neurons. SOM withWTA can be easily expanded to SOM with activity cen-ter (AC). It is worth mentioning, that SOM learning ruleis exactly (1). Learning rule (2) is used in backpropaga-tion paradigm [15], [16]. Backpropagation technic wassuccessfully implemented for solving different tasks.But the tendency to use purely rule (2) (without rule(1)) resulted in various problems while learning and insome measure useful attempts to solve these problemsby means of randomization, normalization, orthogonal-ization, using multiple layers (deep learning) and so on.Both rules (1) and (2) are used in counterpropagationnetwork [17]. It was developed in 1986 by R. Hecht-Nielsen. It is guaranteed to find the correct weights, un-like regular back propagation networks that can becometrapped in local minimums during training. Amazinglycounterpropagation is less popular, then backpropaga-tion model. Perhaps its easier to implement pure rule(2), then think about cooperation of rules (1) and (2).Also, original counterpropagation use WTA competi-tive network (like original SOM), while for purposes ofsmooth converting {X} → {Y} its better to apply CA.Different counterpropagation network extensions wereused for manipulation tasks control. The general ideawas to convert some data about actual and specified ma-nipulator states X into control signal Y , as shown onFig. 6. Two-link manipulator arm is moving in horizon-
Figure 6: Neural network manipulator control
tal plane. Two pneumocylinders operate the arm move-ments. Each pneumocylinder position is controlled bysending pulse-width modulation (PWM) to valves V3and V4 (Fig. 7). Valve 4 should be opened to move thepiston rod to the right. If valves V1 and V2 are switchedto the opposite position, short PWM impulses to V3 willcause the slow motion of piston rod leftwards.
Figure 7: Pneumocontrol structure. 1 — air pressuresource; 2 — pneumocylinder ; V1 – V4 —electricallycontrolled valves; C1 — air flow constrictor
Unpretentiousness of manipulator structure, tasksand control functions was very helpful for detail studyand main ideas understanding of neural network manip-ulator control. It gave possibility to start the work fromvery simple tasks for neural net and then gradually in-crease the complexity of the tasks.
First the quasistatic tasks were examined. The sim-plest task is to control only one manipulator arm linkand to be able to stop at specified φ s starting from ar-
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bitrary φ . This task can be easily solved by the instru-mentality of standard calculations. The φ2 angle is con-trolled by y2 position. The traditional control formulafor ∆y2 to reach specified φ s
2 starting from arbitrary φ2is (see Fig. 6 for designations):
∆y2 =√
a2 +b2−2abcosφ s2−
√a2 +b2−2abcosφ2
(4)And ∆y2 is proportional to valves V3 or V4 openingtime, T = k∆y2, which can be transformed to PWM pa-rameters (frequency, width and number of pulses andvalve number (V3 or V4)). In experiments the mean mis-match of single movement was not very high, with min-imum about 5% of ∆φ2 at φ s
2 = π/2. But the displace-ment could be repeated from closer position and 2−−3movements usually allowed to reach desirable error oflink end position less than 2 mm. Can the results of tasksolution be improved by means of neural net implemen-tation? Equation (4) gives ideal solution for ideal armlink control model. Real arm link and its control canslightly differ from ideal model. During traditional con-trol installation and tuning only a, b, φ and multipliersof transformation ∆φ2 to PWM parameters could be ad-justed, while the function (1) and direct proportionalityT = k∆y2 to PWM parameters are unalterable. Neuralnets are representing transformation function in tabularform and are capable for functions form fine tuning dur-ing adjustments to real control conditions.
Only one net unit was used in the first experiment.Two angles of the arm link position (actual π/2 andspecified φ s
2) were forming X for neural net and Y wascomposed of four PWM parameters. The initial valuesfor vectors M j and Lk were calculated by (4) and lawof transformation ∆y2 to PWM parameters. 1000 in-put signals X (formed of random φ s
2 and previous φ2)and learning rules (1) and (2) application resulted in im-provement of minimum mean mismatch to 3,6% for 80neurons of inner layer. 40 neurons net reached 4,7% af-ter 350 steps of learning. 20 neurons started from 9%of minimum mean mismatch and reached reduction toonly 7,6%. So 20 neurons arent enough for good tabu-lar representation of function (4).
More than 80 neurons of inner layer make the learn-ing time longer (proportionally to N2), but do not leadto sufficient mismatch decrease. 200 and 400 neuronsmake minimum mean mismatch equal to 3,4% and 3,3%respectively. Its meaning, that 3,3% mismatch is basedon unpredicted random errors. Second experiment wasbased on dividing transformation {X}→ {Y} into threestages. On the first stage nonlinear transfers actual φ2
and specified φ s2 to y2 and ys
2 was performed separatelyon the same net with 15 neurons of inner layer. On thesecond stage ∆y2 = ys
2− y2 and abs(∆y2) were calcu-lated. And on the third stage ∆y2 was converted intofour PWM parameters (7 neurons for frequency, 10 forwidth, 10 for number of pulses and 4 for valve num-ber (V3 or V4). This configuration demonstrated min-imum mean mismatch 3,5%. The advantage of threestage approach is that all of three stages are simpler,than conversion in one stage. The single transformationwas two-dimensional and start working good from 80inner layers neurons, while three stages transformationsare one-dimensional and need less neurons. The sec-ond stage (of three stages) was not realized on neuralnet, but for these linear transformations 10 neurons aremore than enough. The main benefit of neurons num-ber reducing is acceleration of learning process. Forthree stage approach minimum mean mismatch 3,5%was reached after processing of 70 input signals. Forsmall simple tasks the difference between one and sev-eral stages transformation isnt sufficient. But for moredifficult tasks with dimensionality more than 10 goodtabular representation of functions becomes impracti-cal. Too many neurons are needed and the learning timegrows unreasonably. The way of solving this problem isto split the complex task into several simpler tasks withdimensionality less than 6–7, 2–3 is desirable and 1 isthe best. After solving the single link quasistatic con-trol task its time to consider two-link quasistatic controltask. The positions of links are defined by angles and,since the dynamic of the process is not analysed, canbe controlled independently. The only difference withtraining single link quasistatic control task is that thegoals arent angles φ s
1 , φ s2 , but coordinates xs
1, xs2. The
task of conversion coordinates to angles can be eas-ily solved geometrically, but also it can be solved bymeans of neural net. Anyway specified coordinates xs
1,xs
2 should be transferred to φ s1 , φ s
2 . Angles φ1 and φ2 aremeasured by angle sensors. So pairs φ s
1 , φ1 and φ s2 , φ2
can be transformed to operating pneumocylinders posi-tions PWM parameters. The traditional control formula(4) or one of two neural nets approaches can be usedfor fulfilling the conversions. Mismatches of all threemethods for two-link arm are proportional to results forone link with multiplier 1,41. Learning times for neu-ral nets approaches are approximately four times longer.The best results for two-link arm also showed the threestage transformations method. Minimum mean mis-match 4,8% of ∆X was reached after processing of 300input signals. 1–2 movements were necessary to reach
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desirable less than 2 mm error of two-link arm end po-sition in the center of desk and 2–3 movements at thedesk edge. There are some important questions left toinvestigate in the quasistatic manipulator control task:balancing of learning rules coefficients, emphasizing ofsufficient variables, quick tuning without changing theshape of nonlinear transformations and others. The nextstep is considering dynamic manipulator control tasksin order to reach the specified points by one movement,without additional correcting movements. The correc-tions must be made during the movement. These taskshave higher complexity, because except main variablestheir time derivatives should be taken in account. Also,in contrast to quasistatic control, high transformationsperformance is needed to have time to make several cor-rections during quick movement. But any solutions ofdynamic manipulator control tasks by neural net meanswill greatly extend the manipulators application area.
5 ConclusionThe performed experiments have proven the effec-tiveness of software and hardware tools of intelligentrobotics and their correspondence to the tasks. De-tailed development of these instruments will provideshorter (faster) worktime of the robots, and improvedlogical features of the robots. Future plans are devel-oping fully “assembled” intelligent robot-manipulatorwith planning and playing games abilities based on vi-sual system and neural “nervous” system. The computertechnologies that have been developed will be the uni-versal tools for this activity.
Acknowledgements: The research was supported bythe Keldysh Institute of Applied Mathematics and in thecase of the second author, it was also supported by theGrant Agency of RFBR (grant No. 15-08-08769).
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