Fuzzy-Control.

Präsentation zum Thema: "Fuzzy-Control."—  Präsentation transkript:

1 Fuzzy-Control

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

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 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 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 Types of Uncertainty Stochastic uncertainty example: rolling a dice
Linguistic uncertainty examples : low price, tall people, young age Informational uncertainty example : credit worthiness, honesty Stochastic uncertainty an event occurs with a given probability lexical or linguistic uncertainty imprecise description of an object or concept 3. Informational uncertainty uncertainty caused by missing or incomplete information

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

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

9 Fuzzy Set Definition : 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(x) Membership function also called characteristic function A=“young” 1 =0.8 x [years] x=23

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 m=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) B A

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 m is a real-valued scalar function with values in the unit interval

12 Types of Membership Functions
Trapezoid: <a,b,c,d> Gaussian: N(m,s) (x) (x) 1 1 s a b c d x m x Triangular: <a,b,b,d> Singleton: (a,1) and (b,0.5) (x) (x) 1 1 a b d x a b x

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

14 Classical truth values for conjunction, disjunction, implication and negation
Konjunktion Disjunktion Implikation Negation

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

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

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

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

20 Complement Negation: A(x)= 1 - A(x)
Classical law does not always hold: AA(x)  1 AA(x)  0 Example : A(x) = 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 { Fuzzy Relations classical relation
R : X x Y defined by mR(x,y) = if (x,y)  R 0 if (x,y)  R { | fuzzy relation R : X x Y defined by mR(x,y)  [0,1] mR(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 Fuzzy Relations Example: X = { rainy, cloudy, sunny }
Y = { swimming, bicycling, camping, reading } X/Y swimming bicycling camping reading rainy cloudy sunny 0.0 0.2 0.0 1.0 0.0 0.8 0.3 0.3 1.0 0.2 0.7 0.0

23 Fuzzy Sets & Linguistic Variables
A linguistic variable combines several fuzzy sets. linguistic variable : temperature linguistics terms (fuzzy sets) : { cold, warm, hot } (x) cold warm hot 1 20 60 x [C]

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

25 Fuzzy Rule Base Heating Temperature : cold warm hot Oil price: cheap
normal expensive high high medium high medium low medium low low 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

26 Fuzzy Knowledge Base (x) cold warm hot 1 20 60 x [C]
Fuzzy Data-Base: Definition of linguistic input and output variables Definition of fuzzy membership functions (x) cold warm hot 1 20 60 x [C] Fuzzy Rule-Base: if temperature is cold and oil price is cheap then heating is high ….

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

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

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

30 Fuzzy Inference 3. Inference step: Apply the degree of membership of the antecedent to the consequent of the rule high(h) consequent(h) h 1 ante =0.3 ... min-inference: cons. = min{ante , high } ... then heating is high high(h) consequent(h) 1 ... ante =0.3 prod-inference: cons. = ante • high h

31 Fuzzy Aggregation 4. Aggregation: Aggregate all the rules consequents using the max-operator for union ... then heating is high ... then heating is medium ... then heating is low 1 h

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

33 Mamdani-Controler Mamdani-Regler is based on a finite set  of If-Then-Rules R  of the form inputs: x1,...,xn outputs: y The fuzzy sets mR(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 Mamdani-Controller Rules R = {R1,...,Rr} can be interpreted as piecewise, defined, imprecise function

35 Mamdani-Controller For a crisp input vector (a1,...,an) over the input variables (x1,...,xn) the resulting output fuzzy set is Fuzzy set mR can be interpreted as a fuzzy relation across the product space X1 x ... x Xn and Y Fuzzy set mR,a1,...,an corresponds to the image of the one-element set {(a1,...,an)} under the fuzzy relation mR.

36 Mamdani-Controller projection of the crisp input x1 onto output y
Rule base R = {R1,R2,R3}, Ri : if x is mAi then y is mBi

37 Defuzzifizication COG Center of gravity (COG)
Center of output fuzzy sets Center of singletons 1 ys y 1 y ys 1 y1 y2 ys y3 y

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

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

40 Fuzzy Toolbox

41 Fuzzy Toolbox

42 Fuzzy Toolbox

43 Fuzzy Toolbox

44 Mamdani-Fuzzy Controler
Tank u C0 valve h w o tank tank: valve: Saturation Saturation

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

46 Fuzzy-PD-Regler

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

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

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

55 Takagi-Sugeno-Kang-Regler
Lineare TSK-Regler ki(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 ki gewichteten Überlagerung der Ausgangsgrößen Fix der einzelnen Regler.

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

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 (Ml<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 Takagi-Sugeno-Kang-Regler-Entwurf
Linearisiere das nichtlinearen Modell an den Arbeitspunkten q=-0.7, q=-0.3, q=0.0, q=0.3, q=0.7

60 Takagi-Sugeno-Kang-Regler-Entwurf
Entwerfe mit der Methode der Polplatzierung Zustandsregler für die linearisierten Modelle Partitioniere Eingangsvariable q 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 Takagi-Sugeno-Kang-Regler-Entwurf
Linearisierung um q=0 >> [A,B,C,D]=linmod('cart_pole'); >> k=acker(A,B,[-2, -2, -1+i, -1-i]) k = TSK-Regel

62 Takagi-Sugeno-Kang-Regler-Entwurf
Gleichgewichtzustand für q=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=4.5557 dx=[ ]

63 Takagi-Sugeno-Kang-Regler-Entwurf
Linearisierung um Gleichgewichtszustand für q=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 Takagi-Sugeno-Kang-Regler-Entwurf
Gleichgewichtzustand für q=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=9.0799 dx=[ ]

65 Takagi-Sugeno-Kang-Regler-Entwurf
Linearisierung um Gleichgewichtszustand für q=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 TSK-Regelbasis

67 TSK-Kennfeld

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 m then y is n 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 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 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 Logikbasierte Regler während die Lukasiewicz-Implikation zu führt.
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 Fuzzy Gütemasse zur Regleroptimierung
skalares Gütemass Optimierungs- verfahren Fuzzy- güte- anforderungen Regler-Parameter {K,Tn, Tv} Regelgröße : y(t) Führungsgröße : w(t) Linearer dynamischer Regler Stellgröße : u Strecke w e + - Regelgröße : y

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

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

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

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 Präferenzen m1(a) kombiniere Handlungen m1(a) a1 a  m2(a) m2(a) a1+2 a1  2 a2

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

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

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

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. mB(r) denotes the desirability of behavior B for executing action r. preference for a response crisp response mB(y) mB(y) heading y heading y

87 Fuzzy Behavior Coordination
aggregate preferences for actions of active behaviors m1(a) fuse actions fuse preferences m1(a) a1 a m2(a) m2(a) a1+2 a1+2 a1  2 a2

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

90 Fuzzy Behavior for Obstacle Avoidance
total response

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 s1 s2 s3 s4 activation 1.0 w3 w2 w1 actual strength

92 Priority Based Behavior Coordination of Preferences
w1=0.5 w2=0.25 w3=0.25

93 Priority based behavior coordination

94 Priority based behavior coordination

95 Priority based behavior coordination

96 Learning Behavioral Responses from Demonstration
parameters demonstration optimization scheme (EA, PSO) fitness training set <st,rt> match preference response with examples behaviors and behavior coordination mi(s,r) s,r1 s,r2

97 Criteria for Matching Preferences with Training Examples
Consistency: demonstrated actions rt associated with a perceptual state st should have a high preference mi(st,rt) if the behavior is active wi(st)>0 in the context st. Preferences should be as general as necessary. mi(s,r) s,r1 s,r2

98 Criteria for Matching Preferences with Training Examples
Specificity: the preference mi(st,r) for actions r that have not been demonstrated in a perceptual state st should be low. Preferences should be as specific as possible. mi(s,r) s,r1 s,r2

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 mi(s,r) s,r1 s,r2

100 Example 200 state-action pairs generated from random distribution of obstacles based on three behaviors identification of parameters dmin, dmax for obstacle avoidance original: dmin=1, dmax=3 , fitness=265 PSO optimum: dmin=0.92, dmax=3.76 , fitness=267 0.5 1 1.5 2 2.5 3 3.5 4 210 220 230 240 250 260 270 d min max fitness 4 x 3.8 3.6 3.4 3.2 x max 3 d 2.8 2.6 2.4 2.2 2 0.5 1 1.5 2 d min

101 Simulation Original trajectory dmin=10, dmax=20
trajectory with adapted parameters dmin=12.5, dmax=16

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

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

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

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

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

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

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

110 Fuzzy Behavior Coordination
fuse preferences fuse actions

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 fails COL ok COL fails