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1 Fuzzy-Control. 2 Control System Design model process Plant plant model Modeling control law controller design Controller Implementation Fuzzy- Control.

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Präsentation zum Thema: "1 Fuzzy-Control. 2 Control System Design model process Plant plant model Modeling control law controller design Controller Implementation Fuzzy- Control."—  Präsentation transkript:

1 1 Fuzzy-Control

2 2 Control System Design model process Plant plant model Modeling control law controller design Controller Implementation Fuzzy- Control

3 3 Fuzzy-Control versus classical Control Fuzzy controllers are nonlinear state space controllers with no internal dynamics Control loop contains additional dynamic transfer elements for integration and differentation (PID- fuzzy control) The dynamic behavior of a fuzzy controller is not different from a classical controller The main difference is the representation of the controller (parametrization) and therefore the design methodology Classic: model based Fuzzy: knowledge based

4 4 Pros and Cons Classical control Systematic design Stability and quality specifications are implictly met Model uncertainties and disturbance are rejected by means of robust control design Fuzzy control Heuristic design No guarantee of stability Controller is transparent and comprehensible State space controller requires observer, observer requires state space model, therefore why not directly classical control design

5 5 Fuzzy Control Fuzzy control makes sense if...  there is no system model in form of differential equations (e.g. behavior based robotics, due to environment)  the plant is highly nonlinear which complicates or prevents classical control design methods  control objectives are vague and imprecise, e.g. smooth switching of an automatic gear box  plant and control strategy are so simple that fuzzy control design requires less time and effort than classical control design

6 6 Types of Uncertainty Stochastic uncertainty example: rolling a dice Linguistic uncertainty examples : low price, tall people, young age Informational uncertainty example : credit worthiness, honesty

7 7 Classical Set young = { x  P | age(x)  20 } characteristic function:  young (x) = 1 : age(x)  20 0 : age(x) > 20 A=“young” x [years]  young (x) 1 0 {

8 8 Fuzzy Set Classical Logic Element x belongs to set A or it does not:  (x)  {0,1} A=“young” x [years]  A (x) 1 0 Fuzzy Logic Element x belongs to set A with a certain degree of membership:  (x)  [0,1] A=“young” x [years]  A (x) 1 0

9 9 Fuzzy Set Fuzzy Set A = {(x,  A (x)) : x  X,  A (x)  [0,1]} a universe of discourse X : 0  x  100 a membership function  A : X  [0,1] A=“young” x [years]  A (x) 1 0  =0.8 x=23 Definition :

10 10 Fuzzy set vs. Probabilty U ist the set of nontoxic liquids Bottle A belongs to U with probability p=0.9 Bottle B belongs to U with degree of membership  =0.9 A has the property „non toxic“ either completely with p=0.9 or not all (toxic) with p=0.1. B has the property „non toxic“ to a high degree (almost non-toxic) A B

11 11 Representation of Fuzzy Sets elementwise specifikation of membership degree discrete base set Fuzzy set definition via membership functions Base set X has values in a continuous range Fuzzy-set  is a real-valued scalar function with values in the unit interval

12 12 Types of Membership Functions x  (x) 1 0 abcd Trapezoid: x  (x) 1 0 Gaussian: N(m,s) m s x  (x) 1 0 ab Singleton: (a,1) and (b,0.5) x  (x) 1 0 abd Triangular:

13 13 Parametrization of Fuzzy-Sets Triangular functions Trapezoidal functions Bell-shaped functions

14 14 Classical truth values for conjunction, disjunction, implication and negation KonjunktionDisjunktionImplikation Negation

15 15 Truth value functions Restriction: The fuzzy truth value should coincide with the classical truth values for arguments restricted to the binary inputs 0 and 1. Example: truth value functions conjunction disjunktion negation Lukasiewicz-Implication Gödel-Implication

16 16 Definition t-Norm Definition: A function t : [0, 1] 2  [0, 1] is called t-Norm (triangular norm), if it complies with the axioms (T1) – (T4). Monotonic Kommutativity Associativity Truth value functions for conjunction

17 17 Examples for t-Norm Truth value function t( ,  ) = min{ ,  } for conjunction additional t-norms

18 18 Extension Principle For arbitrary functions f:  f(A) (y) = max{  A (x) | y=f(x)} f x  A (x) y  f(A) (y) Assume a fuzzy set A and a function f: How does the fuzzy set f(A) look like? f x  A (x) y  f(A) (y) max

19 19 Operators on Fuzzy Sets Union x 1 0  A  B (x)=min{  A (x),  B (x)}  A (x)  B (x) x 1 0  A  B (x)=max{  A (x),  B (x)}  A (x)  B (x) Intersection x 1 0  A  B (x)=  A (x)   B (x)  A (x)  B (x) x 1 0  A  B (x)=min{1,  A (x)+  B (x)}  A (x)  B (x)

20 20 Complement Negation:   A (x)= 1 -  A (x)   A  A (x)  1   A  A (x)  0 Classical law does not always hold: Example :  A (x) = 0.6   A (x) = 1 -  A (x) = 0.4   A  A (x) = max(0.6,0.4) = 0.6  1   A  A (x) = min(0.6,0.4) = 0.4  0

21 21 Fuzzy Relations classical relation R : X x Y defined by  R (x,y) = 1 if (x,y)  R 0 if (x,y)  R | { fuzzy relation R : X x Y defined by  R (x,y)  [0,1]  R (x,y) describes to which degree x and y are related It can also be interpreted as the truth value of the proposition x R y

22 22 Fuzzy Relations X = { rainy, cloudy, sunny } Y = { swimming, bicycling, camping, reading } X/Y swimming bicycling camping reading rainy cloudy sunny Example:

23 23 Fuzzy Sets & Linguistic Variables A linguistic variable combines several fuzzy sets. linguistic variable : temperature linguistics terms (fuzzy sets) : { cold, warm, hot } x [C]  (x) 1 0  cold  warm  hot 6020

24 24 Fuzzy Rules causal dependencies can be expressed in form of if-then- rules general form: if then example: if temperature is cold and oil is cheap then heating is high

25 25 if temperature is cold and oil price is low then heating is high if temperature is hot and oil price is normal then heating is low Fuzzy Rule Base Temperature : coldwarmhot Oil price: cheap normal expensive highhighmedium highmediumlow mediumlowlow Heating

26 26 fuzzy knowledge base Fuzzy Data-Base: Definition of linguistic input and output variables Definition of fuzzy membership functions Fuzzy Knowledge Base Fuzzy Rule-Base: if temperature is cold and oil price is cheap then heating is high …. x [C]  (x) 1 0  cold  warm  hot 6020

27 27 Schema of a Fuzzy Decision FuzzificationInferenceDefuzzification t  cold  warm  hot measured temperature if temp is cold then valve is open if temp is warm then valve is half if temp is hot then valve is close rule-base  cold =0.7  warm =0.2  hot =0.0 v  open  half  close crisp output for valve-setting

28 28 Fuzzification t 1 0  cold (t)=0.5 If temperature is cold... 15C p 1 0  cheap (p)=0.3 and oil is cheap... $13/barrel Determine degree of membership for each term of an input variable : temperature : t=15 C oilprice : p=$13/barrel 1. Fuzzification

29 29 Fuzzy Combination 2. Combine the terms in one degree of fulfillment for the entire antecedent by fuzzy AND: min-operator  ante = min{  cold (t),  cheap (p)} = min{0.5,0.3} = 0.3 t 1 0  cold (t)=0.5 if temperatur is cold... 15C p 1 0  cheap (p)=0.3 and oil is cheap... $13/barrel

30 30 Fuzzy Inference 3. Inference step: Apply the degree of membership of the antecedent to the consequent of the rule  high (h)... then heating is high  consequent (h) h 1 0  ante = h 1 0  high (h)  ante =  consequent (h) min-inference:  cons. = min{  ante,  high } prod-inference:  cons. =  ante  high

31 31 Fuzzy Aggregation h then heating is high... then heating is medium... then heating is low 4. Aggregation: Aggregate all the rules consequents using the max-operator for union

32 32 Defuzzification 5. Determine crisp value from output membership function for example using “Center of Gravity”-method: h 1 0  consequent (h) COG 73

33 33 Mamdani-Controler Mamdani-Regler is based on a finite set  of If-Then-Rules R  of the form inputs: x 1,...,x n outputs: y The fuzzy sets  R (i) are often associated with linguistic terms, e.g. vague concepts such as „roughly zero“, „medium large“ „negative small “. Fuzzy sets and linguistic terms are often synomously.

34 34 Mamdani-Controller Rules R = {R 1,...,R r } can be interpreted as piecewise, defined, imprecise function

35 35 Mamdani-Controller For a crisp input vector (a 1,...,a n ) over the input variables (x 1,...,x n ) the resulting output fuzzy set is Fuzzy set  R can be interpreted as a fuzzy relation across the product space X 1 x... x X n and Y Fuzzy set  R,a1,...,an corresponds to the image of the one-element set {(a 1,...,a n )} under the fuzzy relation  R.

36 36 Mamdani-Controller projection of the crisp input x 1 onto output y Rule base R = {R 1,R 2,R 3 }, R i : if x is  Ai then y is  Bi

37 37 Defuzzifizication Center of gravity (COG) Center of output fuzzy sets Center of singletons y 1 0 y 1 0 COG ysys y 1 0 y1y1 y2y2 y3y3 ysys ysys

38 38 Defuzzification Mean of maxima (MOM) Centroid of largest (COL) y 1 0 MOM ysys y 1 0 COL ysys COG

39 39 Defuzzification If X is A then Y is A´ If X is B then Y is B´ x 1 0 A B y 1 0 A´B´ Mean of maximaCenter of gravity

40 40 Fuzzy Toolbox

41 41 Fuzzy Toolbox

42 42 Fuzzy Toolbox

43 43 Fuzzy Toolbox

44 44 Mamdani-Fuzzy Controler Tank u o h valve tank tank: Saturation valve: Saturation w C0C0

45 45 Fuzzy Toolbox: Tank Level Control Inputs Level error Change of error Output Change of valve position Nonlinear system

46 46 Fuzzy-PD-Regler

47 47 Fuzzy-PD-Regler : Linguistische Variablen Wasserstand h Änderung des Wasserstand dh/dt Ventilstellung u

48 48 Fuzzy-PD-Regler : Regelbasis If (level is okay) then (valve is no change) If (level is low) then (valve is open fast) If (level is high) then (valve is close fast) If (level is okay) and (rate is positive) then (valve is close slow) If (level is okay) and (rate is negative) then (valve is open slow)

49 49 Takagi-Sugeno-Kang-Controler Takagi-Sugeno-Kang-Controler are based on rules of the form rule premise is identical to Mamdani controlers rule conclusion is a crisp (linear) function of the inputs The rule conclusion is a local model of the input-output relationship in the region defined by the rule premise in intermediate regions the output results from interpolation between the individual regions (gain scheduling)

50 50 Takagi-Sugeno-Kang-Regler TSK-controler often employ linear functions in the conclusion Gain-Scheduling An verschiedenen Arbeitspunkten wird ein lineares Modell der Strecke gebildet, bsw. durch Linearisierung um den Arbeitspunkt. Für jeden Arbeitspunkt wird aufgrund des dort gültigen lokalen linearen Modells ein linearer Regler entworfen. Im laufenden Betrieb wird dann je nach Arbeitspunkt zwischen den verschiedenen Reglern bzw. Reglerparametern hin- und hergeschaltet. Um sprungförmige Änderungen der Eingangsgröße zu vermeiden, erfolgt der Übergang vom alten zum neuen Regler möglichst stetig.

51 51 Takagi-Sugeno-Kang-Regler Input X : {very low, low, high, very high} Non-overlapping input sets Rules: If X is very low then y=x If X is low then y=1 If X is high then y=x-2 If X is very high then y=3

52 52 Takagi-Sugeno-Kang-Regler Non overlapping input fuzzy sets Exact representation of local models

53 53 Takagi-Sugeno-Kang-Regler input X : {very low, low, high, very high} Overlapping input fuzzy sets rules: If X is very low then y=x If X is low then y=1 If X is high then y=x-2 If X is very high then y=3

54 54 Takagi-Sugeno-Kang-Regler Interpolation between local linear models y=x y=1 y=3 y=x-2

55 55 Takagi-Sugeno-Kang-Regler Lineare TSK-Regler k i (z(t)) sei der Wahrheitswert der Prämisse der i-ten Regel für die Eingangsgröße z(t) Normierung der Wahrheitswerte Die Ausgangsgröße u des TSK-Reglers ergibt sich aus der mit k i gewichteten Überlagerung der Ausgangsgrößen F i x der einzelnen Regler.

56 56 Takagi-Sugeno-Kang-Regler TSK-Modelle lassen sich auch zur Modellierung einer Strecke heranziehen. Für das Zustandsmodell ergibt sich Für einen geschlossenen Kreis ohne äußere Anregung ergibt sich als TSK- Modell Reduktion auf einen Index l mit Hilfe von ergibt für den geschlossenen Kreis

57 57 Positiv und negativ definite Matrizen Eine Matrix A ist positiv definit wenn für alle x gilt x T A x > 0 Eine Matrix A ist negativ definit wenn für alle x gilt x T A x < 0

58 58 Stabilitätssatz für kontinuierliche TSK-Systeme Gegeben sei ein kontinuierliches System in der Form Dieses System besitzt eine globale, asymptotisch stabile Ruhelage x=0, wenn eine gemeinsame, positiv definite Matrix P für alle Teilsysteme A existiert so dass die Matrix für alle l negativ definit (M l <0) ist. Die Frage nach der Existenz einer solchen Matrix P lässt sich in ein LMI-Problem (Lineare Matrix Ungleichung) überführen, für welches effiziente Lösungsalgorithmen bereitstehen, so dass die Frage der Stabilität des Systems auf einfachem Wege beantwortet werden kann.

59 59 Takagi-Sugeno-Kang-Regler-Entwurf Linearisiere das nichtlinearen Modell an den Arbeitspunkten  =-0.7,  =-0.3,  =0.0,  =0.3,  =0.7

60 60 Takagi-Sugeno-Kang-Regler-Entwurf Entwerfe mit der Methode der Polplatzierung Zustandsregler für die linearisierten Modelle Partitioniere Eingangsvariable  durch dreieckige Fuzzy-Zugehörigkeitsfunktion, deren Schwerpunkte gerade den Arbeitspunkten entsprechen. Für jeden Arbeitspunkt beschreibe die Zustandsrückführung als TSK-Regel. Bilde den gesamten TSK-Fuzzy-Regler aus der Überlagerung (Gain-Scheduling) der Regeln an den verschiedenen Arbeitspunkten.

61 61 Takagi-Sugeno-Kang-Regler-Entwurf Linearisierung um  =0 >> [A,B,C,D]=linmod('cart_pole'); >> k=acker(A,B,[-2, -2, -1+i, -1-i]) k = TSK-Regel

62 62 Takagi-Sugeno-Kang-Regler-Entwurf Gleichgewichtzustand für  =0.4 >> x=[0.4; 0; 0; 0]; >> ix=[1]; >> dx=[0; 0; 0; 0]; >> idx=[1; 2]; >> [x, u, y, dx]=trim('cart_pole',x,[],[],ix,[],[],dx,idx) x= [ ] u= dx=[ ]

63 63 Takagi-Sugeno-Kang-Regler-Entwurf Linearisierung um Gleichgewichtszustand für  =0.4 >> [A,B,C,D]=linmod('cart_pole‚x,u); >> k=acker(A,B,[-2, -2, -1+i, -1-i]) k = TSK-Regeln

64 64 Takagi-Sugeno-Kang-Regler-Entwurf Gleichgewichtzustand für  =0.7 >> x=[0.7; 0; 0; 0]; >> ix=[1]; >> dx=[0; 0; 0; 0]; >> idx=[1; 2]; >> [x, u, y, dx]=trim('cart_pole',x,[],[],ix,[],[],dx,idx) x= [ ] u= dx=[ ]

65 65 Takagi-Sugeno-Kang-Regler-Entwurf Linearisierung um Gleichgewichtszustand für  =0.7 >> [A,B,C,D]=linmod('cart_pole‚x,u); >> k=acker(A,B,[-2, -2, -1+i, -1-i]) k = TSK-Regeln

66 66 TSK-Regelbasis

67 67 TSK-Kennfeld

68 68 Logikbasierte Regler  Mamdani-Regler interpretieren Fuzzy-Regeln als stückweise definierte unscharfe Funktionen.  Logikbasierte Regler interpretieren Fuzzy-Regeln im Sinne von logischen Implikationen.  Betrachte Regeln mit nur einer Eingangs-Fuzzy-Menge if x is  then y is  Wie beim Mamdani-Regler, ergibt sich als Ausgabe-Fuzzy- Menge exakt die Fuzzy-Menge, wenn der Eingangswert x einen Zugehörigkeitsgrad von Eins zur Fuzzy-Menge  aufweist.  Im Gegensatz zum Mamdani-Regler wird die Ausgabe-Fuzzy- Menge jedoch umso größer, je schlechter die Prämisse zutrifft, d. h. je geringer der Wert  (x) wird.

69 69 Logikbasierte Regler  Im Extremfall  (x) = 0 erhalten wir als Ausgabe die Fuzzy- Menge, die konstant Eins ist.  Der Mamdani-Regler würde hier die Fuzzy-Menge, die konstant Null ist, liefern.  Bei einem logikbasierten Regler sollte die Ausgabe-Fuzzy-Menge daher als Menge der noch möglichen Werte interpretiert werden.  Wenn die Prämisse überhaupt nicht zutrifft (  (x) = 0), kann auf- grund der Regel nichts geschlossen werden und alle Ausgabe- werte sind möglich.  Trifft die Regel zu 100 % zu (  (x) = 1), so sind nur noch die Werte aus der (unscharfen) Menge zulässig.

70 70 Logikbasierte Regler  Eine einzelne Regel liefert daher jeweils eine Einschränkung aller noch möglichen Werte.  Da alle Regeln als korrekt (wahr) angesehen werden, müssen alle durch die Regeln vorgegebenen Einschränkungen erfüllt sein, d. h. die resultierenden Fuzzy-Mengen aus den Einzelregeln müssen im Gegensatz zum Mamdani-Regler miteinander geschnitten werden.

71 71 Logikbasierte Regler Sind r Regeln der Form vorgegeben, ist die Ausgabe-Fuzzy-Menge bei einem logikbasierten Regler daher bei der Eingabe x = a Hierbei muss noch die Wahrheitswertfunktion der Implikation  festgelegt werden. Mit der Gödel-Implikation erhalten wir während die Lukasiewicz-Implikation zu führt.

72 72 Fuzzy Gütemasse zur Regleroptimierung Optimierungs- verfahren Optimierungs- verfahren Fuzzy- güte- anforderungen Fuzzy- güte- anforderungen Strecke Linearer dynamischer Regler Linearer dynamischer Regler Stellgröße : u skalares Gütemass Regelgröße : y(t) Führungsgröße : w(t) Regler-Parameter {K,T n, T v } Regelgröße : y w - + e

73 73 Genetisches Fuzzy System Evolutionärer Algorithmus Evolutionärer Algorithmus Bewertungs- Schema Bewertungs- Schema Strecke Fuzzy Regler F: e  u Fuzzy Regler F: e  u Stellgröße : u Gütemass Regelgröße : y Führungsgröße : w Fuzzy-Regeln Fuzzy-Mengen Regelgröße : y w - + e

74 74 Traditionelle KI Robotik Beschränkungen des SMPA-Ansatzes Unsicherheiten in der Sensorik und den Aktuatoren Unvollständiges Wissen über die Umgebung Fehlende Rückkopplung während der Ausführung Wahr- nehmen Modell- ieren Planen Handeln WahrnehmungKartierungWegplanungAusführung

75 75 Verhaltensbasierte Robotik Keine explizite Modellierung der Umgebung Körperlichkeit und Situiertheit Enge Kopplung zwischen Wahrnehmung und Handeln Ein primitives Verhalten implementiert eine Regelung für eine spezifische Teilaufgabe Das Gesamtverhalten erwächst aus der Integration und Wechselwirkung der primitiven Verhalten Wahrnehmung Handeln

76 76 Verhaltensbasierte Robotik mit Fuzzy-Reglern Hindernisvermeidung If obstacle is close and obstacle is left then turn right If obstacle is close and obstacle is right then turn left If obstacle is close and obstacle is ahead then turn sharp left If obstacle is far then turn zero Zielpunktansteuerung If goal is far and goal is left turn left If goal is ahead turn zero If goal is far and goal is right turn right If goal is near turn zero

77 77 Hierarchische Verhaltensbasierte Architekturen Wand folgen Tür passieren Zielpunkt ansteuern Hindernis- vermeidung Navigation Manipulation reichen greifen abstellen … primitive Verhalten koordinierende Verhalten

78 78 Kontextabhängiges Überlagern von Verhaltensweisen schlichtende Fuzzy-Regeln aktivieren oder deaktivieren Verhalten anhand des aktuellen Kontext if obstacle-close then obstacle-avoidance if obstacle-far then go-to-target fusionierende Fuzzy-Regeln kombinieren die Präferenzen für Handlungen der aktiven Verhalten  kombiniere Handlungen kombiniere Präferenzen  1 (a) a  2 (a) a1a1 a2a2 a 1+2  1 (a)  2 (a) a 1  2

79 79 Situationsbedingte Fuzzy-Regeln zur Planung Fuzzy- Plan zur sequentiellen Aktivierung von Verhalten If obstacle then avoid If not obstacle and in(Corr1) and not in(Corr2) then follow(Corr1) If not obstacle and in(Corr1) and not near(Door2) then follow(Corr2) If not obstacle and near(Dorr5) and not in(Room5) then cross(Dorr5) If in(Room5) then still

80 80 Fuzzy Behavior Coordination for Robot Learning from Demonstration Frank Hoffmann University of Dortmund, Germany

81 81 Outline robotic behaviors based on fuzzy preferences fuzzy obstacle avoidance behavior coordination of multiple behaviors learning behaviors from demonstration

82 82 Learning Robotic Behaviors from Demonstration human teacher drives robot manually while perception action pairs are recorded learning task: adapt a collection of robotic behaviors to mimic the demonstrated behavior problems partially observable decision process credit assignment: Which behavior is responsible for the observed action? ambiguity of training examples due to ambigious control actions perceptual aliasing

83 83 Behavior Based Robotics robotic behavior: direct mapping from perceptions to actions the overall behavior emerges from the cooperation and coordination of concurrently active primitive behaviors perception s response r

84 84 Behavior Coordination and Command Fusion Behavior coordination is concerned with how to decide which behavior to activate at each moment. Command fusion is concerned with how to combine the desired actions of multiple behaviors into one command to be sent to the robot‘s actuators.

85 85 Behavior Coordination Mechanisms behavior coordination mechanisms arbitration command fusion priority- based state- based winner- take-all voting super- position fuzzy multiple objective

86 86 Fuzzy Behavioral Responses Each behavior states its preferences for all possible responses instead of a single crisp response r. Preferences for actions are described by fuzzy sets.  B (r) denotes the desirability of behavior B for executing action r. B()B() heading  preference for a response B()B() crisp response heading 

87 87 Fuzzy Behavior Coordination aggregate preferences for actions of active behaviors fuse actions fuse preferences  1 (a) a  2 (a) a1a1 a2a2 a 1+2  1 (a)  2 (a) a 1  2 a 1+2

88 88 Fuzzy Behavior for Obstacle Avoidance Mapping from perceptions to preferences are described by fuzzy rules. If distance is small then preference is low If distance is large then preference is high Preference as a function of proximity

89 89 Fuzzy Behavior for Obstacle Avoidance Preference for a heading  depends on distances to obstacles d i according to  D (d i ) similarity of heading  with obstacle direction  i local perceptual space

90 90 Fuzzy Behavior for Obstacle Avoidance total response

91 91 Priority Based Behavior Coordination of Preferences behaviors are characterized by activation s (dynamic) applicability: environmental context desirability: task context priority (static) behavior coordination mechanism s1s1 s2s2 s3s3 s4s4 activation w1w1 w2w2 w3w3 actual strength 1.0

92 92 Priority Based Behavior Coordination of Preferences w 1 =0.5 w 2 =0.25 w 3 =0.25

93 93 Priority based behavior coordination

94 94 Priority based behavior coordination

95 95 Priority based behavior coordination

96 96 Learning Behavioral Responses from Demonstration training set behaviors and behavior coordination match preference response with examples optimization scheme (EA, PSO)  i (s,r) s,r 1 s,r 2 fitness behavior parameters demonstration

97 97 Criteria for Matching Preferences with Training Examples Consistency: demonstrated actions r t associated with a perceptual state s t should have a high preference  i (s t,r t ) if the behavior is active w i (s t )>0 in the context s t. Preferences should be as general as necessary.  i (s,r) s,r 1 s,r 2

98 98 Criteria for Matching Preferences with Training Examples Specificity: the preference  i (s t,r) for actions r that have not been demonstrated in a perceptual state s t should be low. Preferences should be as specific as possible.  i (s,r) s,r 1 s,r 2

99 99 Criteria for Matching Preferences with Training Examples Smoothness: similar responses for the same state should have similar preferences Inherently achieved by fuzzy representation of behavioral preferences Optimal compromise among all criteria  i (s,r) s,r 1 s,r 2

100 100 Example d min d max fitness d min d max x 200 state-action pairs generated from random distribution of obstacles based on three behaviors identification of parameters d min, d max for obstacle avoidance original: d min =1, d max =3, fitness=265 PSO optimum: d min =0.92, d max =3.76, fitness=267 x

101 101 Simulation Original trajectory d min =10, d max =20 trajectory with adapted parameters d min =12.5, d max =16

102 102 Conclusions fuzzy framework for learning robotic behaviors from demonstration matching of ambigious and uncertain perception action pairs with behavioral responses evaluation on synthetic datasets

103 103 Preference Responses for... (1) (2)

104 104 Preference Responses for... (3) (4)

105 105 Preference Responses for... (5) (6) (7) (8)

106 106 Learning Behavioral Response... (9) (10) (11) (12)

107 107 Learning Behavioral Response... (13) (14)

108 108 Fuzzy Behavioral Responses Each behavior states its preferences for all possible responses instead of a crisp response R i. Preferences for actions are described by fuzzy sets, where the degree of membership  B (R) denotes the desirability of behavior B for executing action R. B()B()  obstacle avoidance B()B()  goal seeking

109 109 Defuzzifizication Center of gravity (COG) Mean of maxima (MOM) Centroid of largest (COL) y 1 0 COG ysys y 1 0 MOM ysys y 1 0 COL ysys COG

110 110 Fuzzy Behavior Coordination fuse preferences fuse actions

111 111 Fuzzy Command Fusion defuzzification of multimodal output sets in case of conflicting preferences might result in “undesirable” control actions centroid of largest defuzzification operator additional external constraint-based fusion mechanism segregate rules with mutually exclusive output recommendations and use winner-takes-all heuristic COG failsCOL okCOL fails

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