Method to build an initial adaptive Neuro-Fuzzy controller for joints control of a legged robot

Transcription

1 Method to bild an initial adaptive Nero-Fzzy controller for joints control of a legged robot J-C Habmremyi, P. ool and Y. Badoin Royal Military Academy-Free University of Brssels 08 Hobbema str, box:mrm, 000, Brssels, Belgim Jean-Clade.Habmremyi@rma.ac.be;Pkool@vb.ac.be; Yvan.Badoin@rma.ac.be Abstract De to the complexity of walking robots which has in general a great nmber of degrees of freedom, cognitive modelling controller sch as Fzzy Logic, Neral Networks seems to be reasonable. Fzzy Logic Controller is more sed becase it lets yo describe desired system behavior with simple if-then relations. Bt it has a major limitation becase in many applications, the designer has to derive if-then rles manally by trial and error. In this paper, we show an original method to fix initial parameters of a Sgeno fzzy controller apparently to Ziegler-Nichols rles. Simlations and application to a six-legged robot named AMRU5 has proved the effectiveness of this method. hen, we make the fzzy controller obtained adaptive by combining it to Neral Networks technologies. Introdction Many robots (maniplator and mobile robots, ntil now, are controlled sing linear controllers sch as PD, PID which are independent for each joint. It can be proven that those controllers are fairly effective. he two main reasons are []: - he large redction ratios between the actators and the link mechanism (non-linearities and copling terms become less important - he large feedback gains in the control loops (they enlarge the domain where the complete robot dynamics is locally eqivalent to a linear model. hese controllers limit the se of robots to slow motion applications and they operate over a small closely controlled range. However, the normal operational range of a robot is large and its payload also can vary. o have a controller which works on different operational range and take into accont the change of the payload, the environment and the ncertainties (friction, flexibility, necessitate a sort of on-line parameter estimation scheme in it (an adaptive controller. Most of classical adaptive controllers are based on the wellknown dynamic properties of robots which stiplate that parameters models (mass, moment inertia, link lengths are linear. hey were applied sccessflly in simple cases (sch as inverted pendlm system, planar two-link maniplator, five bar linkage maniplator where mathematical model can be dedced. hey were also applied in complex cases bt simplified by limiting for example, the normal operational range of the robot de to the difficlty of estimating ncertainties of the model. he problem become more complex for a walking robot which has in general a large nmber of degrees of freedom (we have 8 jst for the robot to walk and which reqires changing internal parameters depending on the environment that it explores. Also, it seems practically difficlt to bild a representative model of a walking robot de to the problem of having accrate internal parameters (distance between joints, moment inertia and to accrately model some complex phenomena sch as backslash, friction In this case, cognitive modelling sch as Fzzy Control and Neral Networks seems to be reasonable. hat s why, we have investigate the way to control (joints and gait control a six-legged robot named AMRU5 shown in Figre sing ANFIS (Adaptive Nero-Fzzy Inference System. In this paper, we are more focsed on the joints control of the robot and we show an original method on how to fix initial parameters of a Sgeno fzzy controller. Figre : Walking Robot AMRU5

2 ANFIS Architectre. Fzzy Logic and Neral Networks Fzzy Logic Controller (FLC is more sed becase it lets yo describe desired system behavior with simple «if-then» relations. In many applications, this gets yo a simpler soltion in less design time. In addition, yo can se all available engineering know-how to optimise the system performance directly. While this is certainly the beaty of fzzy logic, at the same time it is a major limitation. In many applications, knowledge that describes desired system behavior is contained in data sets. he designer has to derive the «if-then» rles from the data sets manally, which reqires a major effort with large data sets. his is often done by trial and error. Withot adaptive capability, the performance of FLCs relies on two factors: the availability of hman experts, and knowledge acqisition techniqes to convert hman expertise into appropriate fzzy «if-then» rles and membership fnctions. hese two factors sbstantially restrict the application domain of FLCs. Changing shapes of membership fnctions can drastically inflence the qality of the FLC. hs methods for tning fzzy controllers are necessary. Artificial neral networks are highly parallel architectres consisting of simple processing elements, which commnicate throgh weighted connections. hey are able to approximate or to solve certain tasks by learning from examples. When data sets contain knowledge abot the system to be designed, a neral net promises a soltion becase it can train itself from the data sets. However, only few commercial applications of neral nets exist de to the lack of interpretation of the soltion, the prohibitive comptational effort and the difficlty to select the appropriate net model. It becomes obvios that a clever combination of the two technologies delivers the best of both. Nero-Fzzy [] is a combination of the explicit knowledge representation of the fzzy logic with the learning power of the neral nets.. Nero-Fzzy systems here are many approaches to combine fzzy logic and Neral Networks, known among them are: - Cooperative Nero-Fzzy systems where ANN learning mechanism determines the FIS membership fnctions or fzzy rles from the training data after ANN goes to the backgrond. - Concrrent Nero-Fzzy systems where ANN assists the FIS continosly to determine the reqired parameters. - Fsed Nero-Fzzy systems is the methods where FL and ANN share data strctres and Adaptive Nero-Fzzy Inference Systems knowledge representations. In the above methods FL and ANN are separated. In fsed systems, ANN learning algorithms are sed to determine the parameters of FIS. For that, Fzzy system is represented in a special ANN like architectre. Some of major works in this area are NEFCON, ANFIS[], FALCON, GARIC, FINES and many others.. ANFIS Architectre In or application, we se the ANFIS [] architectre. It keeps the strctre of the fzzy controller that is determined by the fzzy rles as depicted in Figre. Figre : ANFIS Architectre At layer, every node is adaptive (premise parameters with a node fnction which is the membership fnction. Node otpt at layer represents the firing strength of a rle. In or application, it is a prodct of all the incoming signals bt can be in general any -norm operators that performs fzzy AND. At layer, a normalised firing strengths is realised by making a ratio of rle s firing strength to the sm of all rle s firing strengths. Nodes at layer 4 are adaptive (conseqent parameters with node fnction which can be a first-order Sgeno, zero-order Sgeno, Mandani or skamoto fzzy model. Design of an initial zero-order Sgeno Fzzy Logic controller It is important to have an initial controller which works properly in a closed range. One of the reasons is that dring the on-line learning, optimization algorithm will be sed and the soltion cannot converge to the good one if parameters of the controller are set far away of the tre. We have to make a fzzy controller which work properly in a closed range then make it adaptive to take into accont ncertainties of the model. Many controller design avoid this problem by making first a classical controller (PD sally then add another controller (Fzzy or Neral Network to deal with ncertainties. his makes the controller more complex. In or method,

3 we design an initial Nero-fzzy controller which works similarly as a classical one. After, we make it adaptive to deal with ncertainties. o find good parameters of a fzzy logic controller is not an easy task becase they have in general a lot of parameters. If we have n inpt, m trianglar membership fnctions ( parameters to adjst by membership fnctio and a zero-order Sgeno FIS is m n sed, we have to fix parameters. For illstration of the method, we will se a fzzy system with trianglar membership fnctions (N, P, inpt (, n- and n- and a zero-order Sgeno FIS as shown on Figre, bt this method can be generalised. In this case, we have to fix 4 parameters. Bt if we fix parameters of the membership fnction, we have only 8 parameters to fix. A typical rle of sch a system has the form: If i is N, i- is P and i- is N then the otpt z = p i + q i + r i + s ( where { p, q, r s is the parameter set of one node., } he eqation ( become Sgeno FIS. z = s ( for a zero-order Figre : Zero-Order Sgeno: two trianglar MF and three inpt he first method to fix parameters of a controller is simple trial and error. Unfortnately, intitive tning procedres can be difficlt to develop in the case of Sgeno FIS becase a change in one tning constant tends to affect the performance of others terms in the controller s otpt. Also, the great nmber of parameters makes this method practically impossible. he second method can be the analytical approach to the tning problem. It involves a mathematical model of the process. his method cannot be sed becase the advantage of fzzy logic is precisely the fact that it is sed on complex processes where the establishment of a reliable model is nimaginable. he third approach to the tning problem is something of a compromise between prely self-teaching trial and error techniqes and the more rigoros analytical techniqes. It was originally proposed by John G. Ziegler and Nathaniel B. Nichols [5] and remains poplar today becase of its simplicity and its applicability to process which can be describes by a gain, a time constant and a deadtime (which is the case of joints of robots actated by DC motor. Ziegler and Nichols came p with a practical method for estimating the proportional, the integral and the derivative parameters of a PID controller. In this paper, we show how these techniqes can be applied in the design of a fzzy controller.. How the method was developed? Many techniqes sed to trn a Mandani Fzzy Model (which has less parameter compare to Sgeno Fzzy Model are intitive. In many papers, books they show which parameters to increase or to decrease by considering the rise time, the overshoot and the steady state error [4]. hese techniqes seem more like the art than engineering and they are difficlt to apply them to Sgeno Fzzy Model. he best soltion to trn parameters of a Sgeno Fzzy Model is by fsing Neral Networks to Fzzy Logic Systems. Bt we need data to train the system. he first soltion can be to collect them from a classical controller implemented to the real robot by giving random trajectories to the actators. We noticed that we cannot cover all possible operating regions, the time to read data (on encoders, actators, and to write them on a stored device is too short (the microcontroller has no mch time sometime to write all the vale and there is noise on the data. he error obtained after training is still big by sing these data. Another original soltion cold be the se of Ziegler- Nichols rles originally applied to PID controllers. he analoge PID controller is expressed by the eqation: t = p t + i t dt + d de dt ( ( where e is the difference between the set point and the process otpt and the command signal., and d are controller parameters. p i wo practical methods can be sed to have a first estimate of the PID controller parameters: - the step-response method - and the freqency response method (only this method will be considered in this paper.. he step-response method his method is based on a registration of the open-loop response of the system, which is characterized by two parameters. he parameters (a and L are determined from a nit step response of the process, as shown in Figre. When those parameters are known, the controller parameters are obtained from able. Controller ype P PI PID p a 0.9 a. a i d 0. al aL al able : Parameters obtained from step-response method

4 Figre : Step-response method Figre 4: Freqency response method.. he freqency-response method he idea of this method is to determine the point where the Nyqist crve of the open-loop system intersects the negative real axis. his is done by connecting the controller to the process and setting the parameters so that pre proportional loop system is obtained. he gain of the controller is then increased ntil the closed-loop systems reaches the stability limit. When this occrs, the gain and the period of oscillation shown on Figre 4 are determined. he controller parameters are then given by the able. Controller ype P PI PID p i 0.54 d able : Parameters obtained from freqencyresponse method.. Method of trning an UFLC based on the freqency-response If the ltimate gain and the ltimate period of the process was determined by experiment or simlation, eqation can be write as follow:. de ( t = 0.6 t + e t dt ( ( dt here exist different methods to convert eqation ( into discrete form for digital implementation sch as stin approximations (or trapezoidal approximations, ramp invariance, rectanglar approximations When the sampling time is short, all these methods have nearly the same performance. We ll se rectanglar approximations. Eqation ( becomes: n (4. n ( = i i = n. n n ( n = 0.6 n + i i = (4-(5 gives ( = ( ( n = + n + n (6 Where , 4 and = = 40 0 = 40 Eqation (6 can now be sed to bild a FLC controller that we called a UFLC (Unit FLC. UFLC will be determined by the eqation: ( = e ( + e ( n + e ( n (7 where e ( (5 is between - and. If we define a step t (0.00 for example, we can define a set A of nmbers between - and as follow: A = {, + t, + t,, t, t,} hen we constitte all possible set { e (, e ( n, e ( n } with nmbers which belong to the set A. From each set, we calclate ( sing eqation(7. Finally, we can se the set { e (, e ( n, e ( n, ( } to train the Nero- Fzzy Controller. Using a hybrid learning paradigm (least sqare error algorithm for conseqent parameters which are linear and backpropagation for premise parameters, we noticed that the initial membership fnctions did not change (premise parameters remain the same, only conseqent parameters change With trianglar membership fnctions choose for or illstration, we have analytical expression shown in the able. Rle e ( e ( n e ( n n ( N N N N N P N P N 4 N P P 5 P N N 6 P N P 7 P P N 8 P P P 6 5 (8 0 ( + 0 ( ( + 0 (4 + 0 ( able : Rles of the system sed for illstration

5 he same procedre can be applied to the step-response method and to derive rles of a controller which depends from the parameters a and L...4 Use of the UFLC on a real process In practice, error will not belong always between - and. We need some transformation to se the UFLC design on a real process. If the minimm error of the system is a and the maximm is b (a and b was determined by the limitation of each joint, a redced error between - and can be expressed as follow: b + a e ( = (8 b a and b a b + a = e ( + (9 Eqation (9 in (6 gives e ( b a b + a ( = ( en ( + en ( n + en ( n + ( + + Eqation (0 becomes after simplification: b a 6 b + a ( = ( + ( 5 Figre 5 shows how UFLC is sed on a real process. Figre 5: Use of the UFLC on a real process (error (0 G ( s = ( ( s + his process has the ltimate gain = 8 and the ltimate period = π.6. From the able and able, we can easily design a PID and an UFLC. Figre 6 shows the Matlab schematic sed to compare the two controllers with the step fnction as the inpt. Figre 6: Comparison between PID controller and UFLC he otpt of the two controllers and their reslt of the error (otpt of the PID controller sbtract to the otpt of the UFLC are shown on Figre 7 and 8. he error is less than How to determine? wo ways can be sed to determine the ltimate gain : - By sing the proportional classical controller and change ntil the system reach the p stability limit - By sing the method mentioned above, we can easily bild a UFLC which react exactly as a classical controller. For example, if we change to the vale of ( for rle,,, p p 4 and to for rle 5, 6, 7, 8; the UFLC will react as a proportional classical controller. Figre 7: Otpt of the PID controller and the UFLC..6 Application to a known fnction transfer o allow comparison between a classical PID controller and a UFLC, we have applied the method to the process with a transfer fnction Figre 8: Error between the PID controller and the UFLC

6 4 Conclsions We show in this paper original methods to trn an initial ANFIS controller based on Ziegler-Nichols rles. he methods were applied to a zero-order Sgeno with trianglar membership fnctions and inpts. Analytical expression of trning rles was dedced from these methods. hese methods can be easily extended to more than trianglar membership fnctions and to other types of fzy model: first-order Sgeno, Mandani, skamoto. Indeed, the strctre of membership fnctions can be fond in a strctre with more than membership fnctions. he zero-order Sgeno is a particlar case of a first-order Sgeno (jst the constant element is no nll, of a Mandani fzzy model (each rle s conseqent is specified by a fzzy singleto and of a skamoto (each rle s conseqent is specified by a step membership fnction center at the constant. We obtain with the methods mentioned above a first approximation of the controller which can be refining after with adaptive paradigm. 5 References []. Yoshikawa, Fondations of Robotics: Analysis and Control. Massachsetts Institte of echnology, USA, 990 [] D. Nack, F. lawonn and R. rse. Combining Neral Networks and Fzzy Controllers FLAI 9, Linz, Astria, Jn. 8-Jl., 99 [] J. S. R. Jang, C.. Sn and E. Miztani. Nero- Fzzy and Soft Compting. Prentice-Hall (U, 997 [4] B. Sbdhi, A. S. Morris, Fzzy and Nero-Fzzy approaches to control a flexible single-link maniplator IMechE 00, 9 May 00 [5] J.G. Ziegler and N.B. Nichols, Optimm settings for atomatic controllers, rans. ASME, 64, 759, 94