PNEUMATIC ARTIFICAL MUSCLE FORCE FUNCTION APPROXIMATION
USING ANFIS
*Alexander HOŠOVSKÝ, **Milan BALARA *Department of Mathematics, Informatics and Cybenetics, Faculty of Manufacturing Technologies, Technical University of Košice
Bayerova 1, 08001 Prešov, Slovakia
Phone: + (421)(517)721360-165 Email: [email protected] **Department of Mathematics, Informatics and Cybenetics, Faculty of Manufacturing Technologies, Technical University of Košice
Bayerova 1, 08001 Prešov, Slovakia
Phone: + (421)(517)721360-165 Email: [email protected]
In order to model systems actuated by pneumatic artificial muscles, it is necessary to approximate a nonlinear function of generated force (static modeling part) which depends (basically) on two variables : pressure within the muscle and its contraction. In this paper an approximation of the mentioned function is carried out using Adaptive Neuro-Fuzzy Inference System (ANFIS) with data points extracted from the characteristic supplied by manufacturer. The powerful approximating capabilities of ANFIS are clearly evident when compared to the approximation using traditional least squares method with function finder. Matlab's Fuzzy Logic Toolbox with ANFIS environment was used for the fuzzy approximation while an online function finder with LSM was used for conventional surface approximation.
Keywords : ANFIS; generated force; contraction; pressure; approximation
1 INTRODUCTION
Pneumatic artificial muscles belong to the group of nonconventional actuators basically intended for low-cost industrial manipulators or robots, anthropomorphic hands or prosthetic limbs. Despite their simple construction (and hence low cost) and principle of operation, the mechanisms based on them remain quite complex to control due to the inherently nonlinear behavior associated with the material elasticity and friction. The modeling process of the PAM-based systems is divided into two parts : static modeling part deals with the modeling of generated force function which is nonlinear and hysteretic and dynamic modeling which incorporates Newtonian differential equations for motion and equations describing the dynamic processes of flowing compressed air [1]. This paper deals with the first part of this modeling process which necessitates the generated force function approximation in order to correctly map the current values of contraction and muscle pressure from the input space into the value of generated force so that a moment (the muscles are part of 1 DOF actuator with one rotation axis) causing the movement of arm can be determined.
2 PROBLEM DESCRIPTION
The modeled pneumatic artificial muscle is FESTO MAS-20. The version used for experiments was equipped with a force compensator restricting the value of generated force to 1600N. The contraction was in the range <-3,20> [%]. The problem formulation is that we are given a function [2]
g :
X
Y
(1)where
X ⊂R
n andY ⊂R
. This is the generated force function we want to approximate. We would liketo construct a system that is capable of performing static mapping
f : X Y
(2)where
X ⊂R
n andY ⊂R
by manipulating some adjustable parameter vector θ so that the error between g and f is as small as possible.for all
x=[ x
1, x
2, x
3, ... , x
n]
T∈
X
. In the case of ANFIS, the parameter vector consisted of linear and nonlinear parameters found in rules consequents and membership functions of Takagi-Sugeno fuzzy system. In case of online function finder these parameters represented the coefficients of determined function found using least squares [3].The data points used for ANFIS training (or finding out the coefficients using a function finder) were gathered from characteristics supplied by the manufacturer (figure 1). In this figure, a relationship between the generated force and muscle contraction for seven discrete values of muscle pressure is depicted.
Figure 1: Force-contraction-pressure relationship for FESTO MAS-20
In order to assure good generalization capabilities of the trained fuzzy systems with higher number of membership functions (the number of data points should be higher than the number of adjustable parameters), 329 data points were chosen to represent the training data set. The whole data set is depicted in three dimensions in figure 2. In this case the hysteresis of respective curves was neglected (it is specified to be max. 3% of the nominal muscle length)[4]. It is clearly visible that the curves are more nonlinear within the interval of shortest contractions. It was decided to use moderate number of membership functions distributed over the input universes of discourse (7) with grid partitioning of the training data set (grid partitioning proved to be more effective with smaller approximation error achieved through the use of lower number of fuzzy rules).
3 RESULTS
The total number of fuzzy rules was 49. The shape of membership functions was chosen to be generalized bell with linear consequents in fuzzy rules. The total number of parameters was 189 (147 linear and 42 nonlinear), i.e. well below the number of data points allowing for good generalization capabilities of the approximator. In figure 3, the course of error during the training of adaptive network is shown. It is evident that starting from epoch 220 on the change in error is insignificant, i.e. increasing the number of epochs of ANFIS training would not have led to further error reduction. It was possible to decrease it more be using a higher number of membership functions, yet the resulting average error (RSME = 2.36767) was considered acceptable.
Figure 3: Training data set for ANFIS
Figure 4: Input variable membership function distribution after training
The resulting membership function distribution after 300-epoch ANFIS training can be seen in figure 4. This distribution reflects the extent of nonlinearity in different parts of the characteristics. The relationship between the force and the pressure (for constant contraction) is almost linear while the relationship between the force and the contraction (for constant pressure) is clearly nonlinear (especially for the smallest contractions) with uneven membership functions distribution. In figure 5, resulting fuzzy surface of trained approximator is depicted.
Figure 5: Resulting fuzzy surface of trained fuzzy approximator
The results from ANFIS were compared to a classic method of surface fitting using least squares. In this experiment a multitude of various functions (polynomial, power series, logarithmic, exponential growth and decay, roman surfaces etc.) was tested with a goal of finding function that would result in lowest sum of squared absolute errors. The number of coefficients was restricted to 6. The function that ranked first in this experiment was Roman surface (minus) scaled and with XY offset, the equation of which was
k cyd
2−
axb
2−
axb
2−
cyd
2
k
2−
axb
2−
cyd
2
2axb
22 cyd
2
f
(4) with coefficients k = 1.20966x103, a = 1.00944x10-5, b = 1.13258x10-4, c = -2.238x10-5, d = -2.8276x10-5,f = 1.11279x103. In this case average error was RSME = 54.35586, i.e. much higher compared to ANFIS.
A surface plot for this function is shown in figure 6.
Figure 7: Absolute error vs. Z data (calculated force value)
The plot surface for a function approximated by roman surface looks smoother, however RSME for this function is unacceptably high. The smoothness of fuzzy surface could be increased by using more membership functions for respective universes of discourse. This would have also further decreased the average approximation error. In figure 7 absolute error versus calculated force values for roman surface function approximation is depicted. The source of largest absolute errors became the boundaries of actual force function (force compensation at 1600N and zero force at high muscle contractions).
Table 1: Generated force comparison between fuzzy approximator and LS approximated function
p [bar] c [%] Fact Ffuzzy Froman efuzzy eroman
2 2.75 370 369.13 429.9 0.87 59.9 3 6.25 482 485.7 466.07 3.7 15.93 4 -1.25 1270 1273.15 1316.7 3.15 46.7 4 8.75 554 552.7 534.03 1.3 19.97 4 13.25 365 367.19 366.47 2.19 1.47 5 17.75 330 331.36 370.23 1.36 40.23
In table 1, the results for 6 randomly chosen test points are shown. These test points were selected to test the generalization properties of the approximator. It is obvious that fuzzy approximator performs much better in terms of approximation error with average error 2.095 (roman surface approximation had an average error 30.7). The points were chosen to be different from the points in training data set but only for one of the independent variables (contraction) since the actual force for a pressure between discrete steps of one bar could not be extracted from the manufacturer's characteristics.
4 CONCLUSION
In this paper a static modeling of PAM generated force function using two methods was carried out. This function is basically a 3D function (where muscle contraction and pressure represent independent variables). Using 329 data points gathered experimentally, at first an adaptive network was trained to perform function approximation. In a second part of experiment, function finder using a multitude of predefined functions was applied. The coefficients of respective functions were determined using traditional least squares method. ANFIS significantly outperformed a function finder even for moderate number of rules (49) while it retains a space for improvement. Since the number of data points was higher than the number of adjustable parameters, resulting fuzzy approximator showed good generalization capabilities. The decision
which method should be applied is problem-dependent since the fuzzy systems are basically more difficult to analyze while for this specific application they seem to be a better solution (due to their lower approximation error).
REFERENCES
[1] VERRELST, B.: A Dynamic Walking Biped Actuated By Pleated Artificial Muscles. PhD. Thesis. Vrije Universiteit Brussel, February 2005
[2] PASSINO, K.M., YURKOVICH, S.: Fuzzy Control. Addison-Wesley 1998. pp.235-236. ISBN 0-201-18074-X
[3] JANG, R.J.S. et al.: Neuro-Fuzzy and Soft Computing. Prentice Hall 1997. pp.340-341. ISBN 0-13-261066-3
[4] HESSE, S.,: The Fluidic Musle In Application. Blue Digest 2003. pp.32 [5] Fuzzy Logic Toolbox User's Guide. The MathWorks Inc. 2008
[6] ESPINOSA, J. et al.: Fuzzy Logic, Identification and Predictive Control. Springer 2005. ISBN 1-85233-828-8
ACKNOWLEDGEMENTS
This work was supported by the Science grant agency of Ministry of Education of Slovak Republic,VEGA, grant No. 1/4077/07 and the Institutional task of the Faculty of Manufacturing Engineering of Technical University in Košice, no. 1/2009.
