Robots Extension Control using Fuzzy Smoothing - FUSION

Proceedings of the 2013 International Conference on Advanced Mechatronic Systems, Luoyang, China, September 25-27, 2013

Robots Extension Control using Fuzzy Smoothing Victor Vladareanu1, Paul Schiopu3

Mingcong Deng2

Faculty of Electronics, II, 313 SplaiulIndependenţei, 060042 “Politehnica” University of Bucharest, Bucharest, ROMANIA [email protected]

Dept. of Electrical and Electronic Eng., Koganei,, Tokyo University of Agriculture and Technology Tokyo 184-8588, JAPAN [email protected] ntry

Abstract— The paper presents a new control method for the robot extension control, using the Extenics concepts and Extension Theory techniques. The controller uses the Dependent Function to measure the degree of compatibility of the process variable and then takes the appropriate action to force the system into convergence around a desired set point. The output of the Dependent Function classifies the process variable value into one of four categories, concurrent with the nested intervals used in Extenics, which are then equated to fuzzy inputs sets of a linguistic variable for a simple fuzzy controller. This acts as a fuzzy smoothing of the controller output. The rationality and validity of the proposed model are demonstrated through simulation in the Matlab/Simulink environment. The results show that the proposed new controller architecture obtains remarkable results, while having the advantage of increased simplicity in design and setting of parameters. Throughout the paper, opportunities for further improvement and research are highlighted and discussed.

convenient to test the operation of the Extenics controller in a relatively simple task. Extensions to 2- and n- dimensional spaces can then be made using the advances [4] of Extension Theory as a whole. While Direct Current motors have been around for a long time, they continue to enjoy broad usage in industry and everyday applications alike. They are easy to use and simulate and representations of DC motors are readily available for virtually all programming languages. In spite of this, the performance of a given controller in regulating a DC motor is a good indication of how it will perform in increasingly complex tasks, such as robot actuators. They therefore provide a good benchmark for controller comparison and indeed have often been used as such in academia. Due to space constraints and the rather involved nature of some of the concepts present in Extenics, this paper will proceed to give a brief outline of such concepts, as are needed for the subject matter at hand, in the next chapter. Chapter III will then discuss the features and key concepts involved in the development and design of the Extension controller. A detailed description of the construction of the controller as well as the concepts needed for the simulation is presented in Chapter IV, while the final chapter shows the obtained results, discusses their significance and draws conclusion and inferences from these. Suggestions for further research are also comprised in the last chapter and throughout the paper, as appropriate.

Keywords—robot control; extension theory; extenics; fuzzy net

I.

INTRODUCTION (Heading 1)

This paper proposes a new type of controller for robot actuators using elements of Extenics, as well as fuzzy control. There are a number of key differences from the usual controller paradigm, which are discussed throughout the paper. Extenics is a relatively new science which deals with the condition and solving of contradictory problems. At its core lie a new formalism for the description of problem elements and a theory of transformations which seek to turn incompatible problems into compatible problems. It researches to what extent innovative ideas can be generated using algorithms and computers. It is classified as being part of Artificial Intelligence, but is frequently referred to as being a mixture of Mathematics, Engineering and Philosophy [1]. Extenics has been used extensively over the past decade in fields such as Data Mining, Marketing, Operations Research, Control and Detection [2]. Applications involving Extension control, however, have generally been limited to extending the range of controllability of a given process [3]. As a first test, this new controller is then used to regulate the speed of a DC motor. This is roughly equivalent to simulating a one-dimensional robot actuator and is very

II.

EXTENICS

Extension Set Theory is a new set theory which aims to describe the change of the nature of matters, thus taking both qualitative, as well as quantitative aspects into account. The theoretical definition for an extension set is as follows: supposingU to be an universe of discourse, u is any one element in U, k is a mapping of U to the realfield I, T=(TU, Tk, Tu) is given transformation, we call:

an extension set on the universe of discourse U, y=k(u) the Dependent Function of E (T ), and y’= Tkk(Tu u) the extension

Manuscript received August 10th, 2013. This work was supported in part by the Romanian Academy, the FP7IRSES RABOT project no. 318902/20122016 and the Romanian Scientific Research National Authority under Grant PN-II-PT-PCCA-2011-3.1-0190 Contract 149/2012(sponsor and financial support acknowledgment goes here).

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function of E(T), wherein, TU, Tk and Tu are transformations of the respective universe of discourse U. Dependent Function k and element u. If T≠e, that is to say the transformation is not identical, four more concepts can be outlined, as follows: - positive extensible field (or positive qualitative change field) of E (T): -

negative extensible field (or negative qualitative change field) of E (T ):

-

positive stable field (or positive quantitative change field) of E (T ):

-

negative stable field (or negative quantitative change field)of E (T ):

-

extension boundary of E (T ):

basis of this can quantitatively describe the objective reality of “differentiation among the same classification” and further describe the process of qualitative change and quantitative change [2]. Suppose x is any point in real axis, and X=is any interval in real field, then

is the Extension Distance between point x and interval, where can be an open interval, a closed interval, or a half-open and a half-closed interval X. This is, in effect, the distance between the point considered and the closest border of the interval. It can be noticed that when the point is on the border of the interval (i.e. x=a or x=b), the Extension Distance will be null, while the minimum possible value for the Extension Distance is the negative of the half of the interval length. This of course applies to the definition of Extension Distance as considered here, taking into account the centre of the interval as the point of interest and in keeping with the one-dimensional aspect of the formulation. Further generalizations to „side distance” and „n-dimensional distance” are put forth in papers such as [4, 5], which merit consideration but are beyond the scope of this application. With the aid of this new take on the distance between a point and an interval, a new concept can be introduced. Place value is an indicator of the relative position of a point with relation to two nested intervals. Suppose , then the specified place value of point x about the nest of intervals composed of intervals X0and X is

This is further illustrated in Figure 1 [2].

This describes the locational relation between point x and the nest of intervals composed of X0 and X [2]. Further use is made of this new definition of distance in order to define a new indicator for the measurement of compatibility within an Extension Set. This indicator is called Dependent Function and is defined as follows. Constitute a nest of three intervals by standard positive field X0, positive field X and interval, i.e. , then for any , the elementary Dependent Function k(x) of optimal point at the midpoint of interval X0 is

Fig. 1. Universe of Discourse in an Extenics Transformation

The tool for solving contradictory problems is extension transformation. Through certain extension transformations, unknowable problems can be transformed to knowable problems and unfeasible problems can be transformed to feasible problems. In any real environment, change is a constant, which is to say at any given moment any number of transformations is taking place. It is therefore convenient, in attempting to solve a contradictory problem, or indeed any problem, to consider which transformation may effect a change in the qualitative aspects of the problem formulation, so as to arrive at the desired result. One of the main research areas of Extenics is investigating and formalizing the effects of transformation upon the objects, elements, conditions and universe of discourse of a given problem. With the aim of measuring the degree of compatibility or incompatibility in a given problem set, Extension Theory has introduced the notion of Extension Distance. New concepts of “distance” and “side distance” which describe distance are established, to break the classical mathematics rule that the distance between points and intervals is zero if the point is within the interval. The Dependent Function established on the

This provides an indicator for the degree of compatibility of a given problem which has been expressed numerically, much in the same way as a membership function determines the degree of membership in a fuzzy set. In Extenics, however, the Dependent Function is generalized to the entire real domain, so that it also takes into account qualitative changes, as well as

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quantitative ones. It should be noted that this is the simplest, earliest definition of the Dependent Function. Subsequent papers both by the original authors [2, 6] as well as new researchers [5,7] have led to generalized and, in some cases, very involved expressions for dependents functions. For the purposes of this paper, however, the elementary Dependent Function will suffice. III.

being made depending on the class the current output is assigned to (sort of like a switch of controllers). The input variable for the fuzzy controller falls within one of four classes, as discussed in the theoretical approach: “Not – Controlled” (NC) for values of the Dependent Function (k) smaller than -1; “Transformable” (Tr) for k between -1 and 0; “Acceptable” (Acc) for k between 0 and 1 and “Compatible” (OK) for k greater than 1. These are the Dependent Function values of the limits of the nested intervals and are the same in any Extenics application, being a property of how the Dependent Function is calculated rather than the problem itself. In our case, “Not – Controlled” corresponds to the process variable value being outside the last nested interval, where an Extenics problem would be considered completely non – compatible. This makes the controller act vigorously, in order to force the system into compatibility. “Transformable” is the label assigned to the largest nested interval. Since it is past the rise time mark (0.7 of the reference value), the controller is clearly having an appropriate effect, however controller action must still be maintained in order to reach the goal. The name of the label comes from this being the region, in Extension Theory, where a generic transformation can most likely lead to compatibility. For a Dependent Function value that is positive, but less the one, the process value is classified as “Acceptable”. This is very close to the desired settling band (±2% of the reference value), so the controller action is minimal. Once the signal passes into the settling band, controller output is zero and the system is classified as being “Compatible”. Figure 2 shows these categories for the Dependent Function value in relation to the process variable, assuming a reference value of 1.

EXTENSION CONTROL

The dominant view with respect to the use of Extenics for control applications relates to improving the range of controllability, rather than the quality of control parameters. In this regard, papers such as [3, 8] merit consideration. Excellent results have also been obtained by [9]. This paper presents a new method, which aims to improve control quality by designing a controller that uses some of the innovative concepts and the mathematical apparatus of Extenics. The basic idea is using the Dependent Function as a means of judging whether the controlled parameter is within an acceptable range and altering the output of the controller commensurately with the degree of incompatibility shown by the Dependent Function. Thus it provides a powerful indicator of reduced complexity for the controller status of the system. This means that the controller output should be higher in cases where compatibility is low to non – existent and will decrease as the system nears convergence. This, in Extenics terms, is the transformation occurring in the universe of discourse which should transform the problem from an incompatible form (lack of convergence) to compatibility. In this sense, the aim is simply to equate the property of converge with the positive transitive field pertaining to this transformation. It should be noted that, as transformations go, increasing or decreasing the controller output is one of the most rudimentary options available. Therefore, as seen from the point of view of Extenics, ample opportunities for further research should be present in investigating the effect of different, more complex transformations on the universe of discourse. The aim of the Extension controller is bring the desired controlled parameter (the speed of the DC motor) to converge to the reference value set by the operator. The nested intervals in the Extenics representation of the problem are symmetrical ranges around the optimum (i.e. the reference value). These can be set at will, provided the configuration of the nested intervals remains the same (as in, they still include each other as in the definition of the Dependent Function). In fact, further research may experiment with unsymmetrical intervals for one or more of the extension sets, different and more complex formulations for the Dependent Function in order to achieve different behaviour. As such, a classification of sorts can be made, with each nested interval acting as a class. These can be equated with fuzzy sets or linguistic variables for a fuzzy smoothing of the output (as is seen later) or can be useful in a number of different implementations. It is even possible to provide different controllers to act on the process, with the selection

Fig. 2. Dependent Function Classification

This is the same graph that is being shown in all subplots, each of them has been zoomed and panned so as to better illustrate each of the four categories. As can be seen from last subplot, the range for the “Compatible” classification is quite small. For an added measure of accuracy, it has been reduced to ±1% of the reference value.

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IV.

implementation they were chosen to be concentric, symmetrical and roughly coinciding with established indicators such as the marks for response time, rise time and settling time. This is explained visually in Figure 4. The Dependent Function value is obtained in the simulation by using an Interpreted Matlab Function block, after having defined the elementary Dependent Function in a Matlab m-file. This provides a simple, instantaneous calculation for the degree of compatibility of the process variable. Within the simulation, this classification is done using the fuzzy controller. Each of the four classes is equated with a set of the input fuzzy linguistic variable. Minor adjustments can be made to account for the way a fuzzy controller processes inputs: the sets do not need to be mutually exclusive and the process is usually smoother when there is some overlapping. The exception to this is in the third set, which does not pass its right outer boundary, so as to prevent the controller from having a residual output once the “Compatible” stage is reached. Figure 5 below shows the arrangements of the input variable and output variable sets.

EXTENION CONTROL METHOD WITH FUZZY SMOOTHING

After the text edit has been completed, the paper is ready for the template. As mentioned in the previous chapter, the degree of incompatibility, measured using the Dependent Function, will provide a scale of the output of the controller. The controller simulations and tests are performed using a simple DC motor configuration in the Matlab / Simulink programming environment. An overall representation of the process, as seen in a Simulink simulation, is given in Figure 3. The motor model, while not at all involved, takes into account the existence of load (although this is not used in this application) and provides a scope to the armature current, which needs to be monitored, as very high currents may cause permanent damage to a motor in a real-life setting. The Dependent Function is calculated with respect to the process variable and not the error, as is the most common case with controllers. It shows the degree of compatibility between the current value of the process variable and the desired state (convergence on the reference value). It cannot, however, offer information on whether the error is positive or negative. Therefore, the simulation must account for the sign of the error, which is extracted and multiplied to the end result of the controller output, as can be seen in the overall diagram in Figure 3.

Fig. 3. Overall Model of the Simulation

Considering the three nested interval to be bands around the desired set point value of the process variable, the value of the Dependent Function provides information as to where the current process value is located.

Fig. 5. Fuzzy Controller Sets

While it is certainly possible to assign direct mathematical meaning to the degree of compatibility, it seems more feasible to pass the controller output through a process of so-called “fuzzy smoothing”. This simply means that an additional fuzzy controller is attached to the end of the output path and is indeed an integral part of the controller as a whole. The output of the Dependent Function is passed to the fuzzy controller, which classifies it with a degree a membership and

Fig. 4. Nested Intervals of the Extension Controller

Again, the width and position of the bands around the set point value may be altered at will, but for this first

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reconstructs an output through defuzzification. While this is in no way necessary, fuzzy smoothing has shown markedly improved results in reducing jamming in an array of applications [10, 11] and is also convenient for the task at hand, since establishing a direct mathematical equivalence would require considerably more experimentation.

interval ranges and fuzzy linguistic variables as described earlier. The representation of the fuzzy rule base is given below in Figure 6. To illustrate the simplicity in designing the fuzzy controller, Figure 7 shows a comparison between the Rule Viewer used here and one used in an implementation of a simple fuzzy PD controller [11]. For this application, the effects of load, disturbance and noise were ignored within the simulation. The overall controller output was also limited, which would protect the motor in a real-life situation against damage to its components. RESULTS AND CONCLUSION Figure 8 below shows the result of the Extension controller action on the process variable. It has very small overshoot (8%) with good values for rise time (2.4s) and settling time (4.7s). The steady state error is negligible at 0.03% of the reference value.

Fig. 6. Fuzzy Controller Rule Base

The obvious interpretation of the nested Extenics intervals as fuzzy linguistic variables also makes the use of a fuzzy smoother an easy choice for implementation.

Fig. 8. Extension Controller Performance (Scope)

Observing the effect on the armature current values (Figure 9), it is easy to see that the controller performance is more than satisfactory, with maximum currents in the region of 4A not posing a danger to the motor.

Fig. 7. Extenics and Regular FLC RuleViewers (comparison)

Fig. 9. Armature Current (Scope)

As is quite the norm in implementations of fuzzy smoothing, neither the rule base, nor the structure of the fuzzy controller, are very complex. The controller has a 1 input to 1 output structure, making use of the equivalence between the nested

The advantage of the Extension controller is that these results were obtained with no need for added complexity, either in the design or the implementation of the simulation. The controller architecture is very straightforward, once the

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aim is precisely to formalize the process of innovation, there is virtually no end to the possibilities for further research. Also, as Extension Theory continues to grow and mature as a discipline in itself, the theoretical advances made are sure to have a favourable impact on this type of implementation.

function interpretation block is in place, and even the additional fuzzy controller could have been omitted and replaced with a PID – like structure, for example. This, however, is not necessary, as even the design of the fuzzy smoother is not at all involved, with a 1:1 equivalence between the classification sets and the fuzzy variable sets, and then later between the fuzzy input and output variables. As opposed to a normal, standalone fuzzy controller implementation, there are no complex rule bases to sift through, and there are no schemas or complex algorithms needed to determine the simulation parameters (such as the gains in a PID implementation). While the locations and ranges of the nested extended sets needs to be specified and there is some tweaking involved in this, their optimization is not vital and perfectly viable results can be obtained with simple, intuitive values (for example setting the “Acceptable” range the same as the settling band of ±2% of the reference value). The novelty presented in this paper is implementing Extension Theory methods in a controller structure, with the aim of improving control quality as regards classical control indicators (overshoot, settling time, expended current, etc.), as well as robustness and ease of implementation. These aims have been followed and highlighted throughout the paper. It is also important, in the authors’ opinion, to propose a controller structure using the innovations brought about by Extension. Theory, which can then be improved and perfected by subsequent research. With that in mind, using the Extension controller for the purpose of controlling the speed of a simple DC motor with no load, disturbance or noise present is only the first step. Further simulation in this regard is needed to test the controller behaviour for robot actuators with load, noise, and disturbance. The final goal to be reached is a physical implementation on an actual mechatronics system. Good results have also been achieved with virtual projection methods [12, 13, 14]. Throughout the paper, possibilities for further research have been outlined and discussed. Extension control, as discussed in this paper, benefits greatly from being a novelty approach to controller design. While this paper proves a working model can be established with basic parameters, the possibilities for tweaking and optimizing in the hopes of obtaining improved performance are virtually limitless. Changes, both subtle and large, will be brought about by further experimentation and the development of Extension Theory as a whole. There are of course parallels to be drawn to Fuzzy control, as well as other types of Artificial Intelligence control algorithms. However, Extension control is unique in a number of aspects. Perhaps most importantly, it represents a shift in the paradigm of controller structure. While the controllers themselves have evolved greatly over the years, changes in the way one looks at controllers and controller structures have not been frequent, save perhaps for the first implementation of fuzzy controllers and their acceptance in industry. By way of being an implementation of a more generalized theory, whose

ACKNOWLEDGMENT This work was supported in part by the Romanian Academy, the FP7 IRSES RABOT project no. 318902/2012-2016 and the Romanian Scientific Research National Authority under PN-IIPT-PCCA-2011-3.1-0190 Contract 149/2012 (sponsor and financial support acknowledgment goes here). REFERENCES Cai Wen, “Extension Set and Non-Compatible Problems”, Advances in Applied Mathematics and Mechanics in China, Peking: International Academic Publishers, 1990, 1-21. [2] Yang Chunyan, Cai Wen, “Extension Engineering”, Science Press, Beijing, 2002. [3] Wang Xingyu, Li Jian. „Extension Control” [ J]. Control theory & applications, 1994, 11（1）：125-128. [4] Smarandache, F., &Vlădăreanu, V. (2012). “Applications of Extenics to 2D-Space and 3D-Space”. Extenics in Higher Dimensions, 39 (ISBN 9781599732039). [5] O. I. Şandru, L. Vlǎdǎreanu, P. Şchiopu, V. Vlǎdǎreanu, A. Şandru, “Multidimensional Extenics Theory”, U.P.B. Sci. Bull., Series A, Vol. 75, Iss. 1, 2013, ISSN 1223-7027. [6] Cai Wen, Yang Chunyan, Lin Weichu. Extension Engineering Methods [M]. Beijing：Science Press，1997 [7] Vladareanu V., Sandru O.I., Schiopu P., Sandru A., Vladareanu L., „Extension Hybrid Force-Position Control of Mechatronics Systems”, accepted for publication in Communications in Cybernetics, Systems Science and Engineering, CRC Press, Beijing 2013 [8] Hu Chen, Wang Xingyu. The Design of Extension Language Controller [C]. Beijing: Matter-element Analysis to Extenics[C]. Beijing: Science and Technology Literature Press, 1995, 12:173-179 [9] Zhang Yong, Wu Xiaopei, etc. Extension Control Method in a Kind of Multivariable Self-correcting System[J]. Journal of Nanjing University of Science and Technology, 2002, 26（5）: 486-489, 498 [10] Vladareanu L, Tont G, Ion I, Vladareanu V, Mitroi D, “Modeling and Hybrid Position-Force Control of Walking Modular Robots”, ISI Proceedings, Recent Advances in Applied Mathematics, Harvard University, Cambridge, USA, 2010, pg. 510-518, ISBN 978-960-474150-2, ISSN 1790-2769. [11] Vladareanu V, “Control Structures for the Position Control of a DC Motor”, MSc. Thesis, University of Newcastle, Faculty of Science, Agriculture and Engineering, 2011 [12] Vladareanu, L., Tont, G., Ion, I., Munteanu, M. S., Mitroi, D., "Walking Robots Dynamic Control Systems on an Uneven Terrain", Advances in Electrical and Computer Engineering, ISSN 1582-7445, e-ISSN 18447600, vol. 10, no. 2, pp. 146-153, 2010, doi: 10.4316/AECE.2010.02026. [13] Vladareanu L, Velea LM,Munteanu RI, Curaj A, Cononovici S, Sireteanu T, Capitanu L, Munteanu MS, “Real time control method and device for robot in virtual projection”, patent no. EPO-09464001, 18.05.2009. [14] Vasile Alexandru, E. Carpus, P. Svasta, I. Ignat , Drumea Andrei, Tapus Adina, Dumitru Georgiana, “The Processing and Identification of the Mobile Information”, SIITME 2008, International Symposium for Design and Technology of Electronic Packages, 14th edition, Predeal, Romania, 286-290, ISSN 183-5123. [1]

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