NN :~:, 8 Nov., Adaptive H Control for Linear Slider with Friction Compensation positioning mechanism moving table stand manipulator Point to Point Control [G] Continuous Path Control ground Fig. Positoining Mechanism using Linear Slider
(# & Guide Table Motor -/, Position sensor Fig. Linear Slider mẍ + F = u m : mass of table, x: table position u: control input, F : Friction force & "# "# *+,-./ Fig. Experimental result without friction comp. mẍ = u, u = c r + ˆmẍ r ˆm = η m ẍ r r, r ė + λe, ẋ r = ẋ d λe c =, η m =., λ = sresearches PID... I [µm] PID... F = µn µ... PID...... dead-zone......
LuGre F ż α(x, ẋ = LuGre = σ z + σ ż + σ ẋ = α(x, ẋ ẋ z + ẋ σ f c + (f s f c exp{ (ẋ/ẋ s Fig. : Friction interface between two surfaces σ : stiffness, σ : damping coefficient, σ : viscous friction f c : Coulomb friction level f s : the level of the stiction force z : the average deflection of the bristles. α(x, ẋ : unknown nonlinear function F ż α(x, ẋ = LuGre = σ z + σ ż + σ ẋ = α(x, ẋ ẋ z + ẋ σ f c + (f s f c exp{ (ẋ/ẋ s z x F LuGre : x d,, e = x x d ẋ r = ẋ d λe r ė + λe mṙ = u F mẍ r
LuGre F = σ + σ ẋ + σ z σ α(x, ẋ ẋ z = θẋ + F z (x, ẋ, z θ = σ + σ, F z = σ z σ α(x, ẋ ẋ z θẋ : viscous friction force F z (x, ẋ, z : dynamic friction force which depend on z mṙ = u θẋ F z (x, ẋ, z mẍ r LuGre... mṙ = u θẋ F z (x, ẋ, z mẍ r F z (x, ẋ, z F z (x, ẋ, z F z (x, ẋ, z Neural Network(: NN NN F z (x, ẋ, z NN NN NN F z (x, ẋ, z F (x, ẋ = F nn (x, ẋ + ε(x, ẋ = W T S(x, ẋ + ε(x, ẋ W R l : optimal weight vector ε(x, ẋ : NN approximation error S(x, ẋ = [s, s,, s l ] T R l : known basis function vector s i (x, ẋ = exp ( (x µ i + (ẋ µ i σ i =,, l x, ẋ : input variables, σ : variance, µ i, µ i : centres Neural network F z (x, ẋ, z Upper bound for dynamic friction force F z (x, ẋ, z F z (x, ẋ, z σ + σ α(x, ẋ α min F zm (x, ẋ NN approximation for F zm (x, ẋ F zm (x, ẋ = W T S(x, ẋ + ε W : unknown optimal weight vector ε : NN approximation error
NN F z (x, ẋ, z... F zm (x, ẋ NN W NN ε NN F zm (x, ẋ F z (x, ẋ, z ε F z (x, ẋ, z + F zm (x, ẋ ε NN ε ε Ge:,, Wang: u = c r + ˆθẋ + ˆmẍ r Ŵ T S(x, ẋ sgn(r k sgn(r, ˆθ = η θ ẋr, η θ >, ˆm = η m ẍ r r, η m >, Ŵ = ΓS(x, ẋ r, Γ = Γ T >. c : positive constant k : positive design constant which satisfy ε d < k ε εd sgn(r ε k ε Ŵ NN NN NN r L γ... H mṙ = u θẋ F z (x, ẋ, z mẍ r u = c r + ˆθẋ + ˆmẍ r Ŵ T S(x, ẋ + v, ˆθ = η θ ẋr, η θ >, ˆm = η m ẍ r r, η m >, Ŵ = ΓS(x, ẋr, Γ = Γ T >. c : positive constant, ˆθ : estimates of θ ˆm : estimates of m Ŵ T S(x, ẋ: NN approximation of F zm (x, ẋ Ŵ : estimates of W v
v v = δ g Ṽ x = δ r v is sub optimal control input for the cost function J [ t { J = hr + δv γ (ε + ε } ] dτ + V (t sup ε,ε L h : positive constant or function γ : positive constant ε ε H V (t = mr + η θ θ + η m m + W T Γ W Experimental equipments Guide Table Motor Position sensor Fig. Linear slider The resolution of linear scale is [nm] Fig. 6 Reference for position x d [m]....... c =, η m =., λ =, η θ =, Γ = I 9, σ =, δ = disturbance attenuation level: γ =. design parameters for S(x, ẋ l =, µ =., µ =, µ =., µ =., µ =, µ =....6.8...6.8 Time t[s] Fig. 6 Reference Signal
"# &# +-*/ "# "# "# "# "# (*+,-./ Fig. 7 Experimental result using Wang s method /*-*,,.+6/-6 &( "# "# *+,-./, /*-*,,.+6#-76 &( Fig. 8 Error Signals: e = x d x "# "# *+,-./, H... ˆθ, ˆm, Ŵ ˆθ = κˆθ η θ ẋr, η θ > ˆm = κ ˆm η m ẍ r r, η m > Ŵ = κŵ + ΓS(x, ẋr, Γ = Γ T > κ...
dead-zone { η θ ẋr, if e > ρ ˆθ =, if e ρ { η m ẍ r r, if e > ρ ˆm =, if e ρ { ΓS(x, ẋr, if e > ρ Ŵ =, if e ρ ρ ±" η θ ẋr, if ˆθ < B ˆθ and η ˆθ θẋr ˆθ = ˆθ η θ ẋr + η θ ˆθη θ ˆθ η θẋr, otherwise η m ẍ r r, if ˆm < B ˆm ˆm and η mẍ r r ˆm = ˆm η m ẍ r r + η m ˆmη m ˆm η mẍ r r, otherwise ΓS(x, ẋr, if Ŵ T Ŵ < B Ŵ Ŵ= and (ΓS(x, ẋr T Ŵ ΓS(x, ẋr Γ Ŵ Ŵ T ΓS(x, ẋr, otherwise Ŵ T ΓŴ θ B ˆθ, m B ˆm, W B ˆθ ˆθ( B ˆθ, ˆm( B ˆm, Ŵ ( B ˆθ +.+(-666(((-/,(7.7 & ( & "# "# *+,-(((./- +.+(-666(((-/,(7#.87 & ( & "# "# *+,-(((./- thetahat ~st~.. thetahat ~th~.. +.+(-666(((-/,(7.87 & ( & "# "# *+,-(((./- +.+(-666(((-/,(7#.87 & ( & "# "# *+,-(((./- thetahat ~th~.. thetahat ~th~.. Fig. 9 Error Signals: e = x d x κ =. Fig. 9: ˆθ Signals κ =.
mhat ~st~ mhat ~th~.... mhat ~th~ mhat ~th~....,+.,,+, & & "# "# (*+++,-.*/ & & "# "# (*+++,-.*/,+#,,+, & & "# "# (*+++,-.*/ "& +6 "# "# "# "# "# (*+++,-.*/ Fig. : ˆm Signals κ =. Fig. : Ŵ Signals κ =. dead-zone dead-zone /*-*,,.+6/-6 /*-*,,.+6-76 &( "# "# &( *+,-./, "# "# *+,-./, "# "# Fig. Error Signals: e = x d x /*-*,,.+6#-76 /*-*,,.+6#-76 &( &( *+,-./, "# "# *+,-./, ρ = 8 thetahat ~st~ thetahat ~th~.... thetahat ~th~ thetahat ~th~ Fig. : ˆθ Signals.... ρ = 8
dead-zone dead-zone mhat ~st~ mhat ~th~.... Fig. : mhat ~th~ mhat ~th~.... ˆm Signals ρ = 8,+.,,+, & & "# "# (*+++,-.*/ & & "# "# (*+++,-.*/ Fig. :,+#,,+#, & Ŵ Signals & "# "# (*+++,-.*/ & & "# "# (*+++,-.*/ ρ = 8 /*-*,,.+ 6/-6 /*-*,,.+ 6-76 /*-*,,.+ 6-76 "# "# &( *+,-./, &( "# "# *+,-./, &( "# "# *+,-./, /*-*,,.+ 6#-76 /*-*,,.+ 6#-76 /*-*,,.+ 68-76 "# "# &( *+,-./, &( "# "# &( *+,-./, "# "# *+,-./, Fig. 6 Error Signals: e = x d x +.+-666-/, 7#.87 +.+-666-/, 7#.87 +.+-666-/, 7.87 (#& # # "# " " "& *+,-./- (#& # # "# " " "& *+,-./- (#& # # "# " " "& *+,-./- +.+-666-/, 7#.7 +.+-666-/, 7#.87 +.+-666-/, 7.87 (#& # # "9 # #" #": *+,-./- (#& # # "9 # #" #": *+,-./- (#& # # "9 # #" #": *+,-./- Fig. 7 Error Signals: e = x d x errors within.[µm]
thetahat ~st~ thetahat ~th~ thetahat ~th~...... thetahat ~th~ thetahat ~th~ thetahat ~th~...... Fig. 8 ˆθ Signal(B ˆθ = mhat ~st~ mhat ~th~ mhat ~th~...... mhat ~th~ mhat ~th~ mhat ~th~...... Fig. 9 ˆm Signal(B ˆm =,+.,,+,,+, & & "# "# (*+++,-.*/ & & "# "# (*+++,-.*/ & & "# "# (*+++,-.*/ & & "# "# (*+++,-.*/ & & "# "# (*+++,-.*/ & & "# "# (*+++,-.*/,+#,,+#,,+&, Fig. Ŵ Signal(B Ŵ = /*-*,,.+6/-6 /*-*,,.+6#/-6 /*-*,,.+6-76 &( "# "# *+,-./, &( "# "# *+,-./, &( "# *+,-./, "# Fig. : Error signals (
thetahat ~st~ thetahat ~st~ thetahat ~th~...... Fig. : ˆθ signals ( mhat ~st~ mhat ~st~ mhat ~th~............ Fig. : ˆm signals (,+.,,+#.,,+, & # # & "# (*+++,-.*/ "# & # # & "# "# & # # & "# (*+++,-.*/ "# Fig. : Ŵ signals ( LuGre NN H
+.+-666-/,7(.7 +.+-666-/,7(.87 +.+-666-/,7(.7 +.+-666-/,7(.87 # &( # "# " " # &( *+,-./- # "# " " # &( *+,-./- # "# " " # &( *+,-./- # "# " " *+,-./- +.+-666-/,7.87 +.+-666-/,7(.87 +.+-666-/,7.87 +.+-666-/,7(.87 # &( # "# " " # &( *+,-./- # "# " " # &( *+,-./- "# " " # &( *+,-./- # "# " " *+,-./- +.+-666-/, 7(.7 +.+-666-/, 7(.87 +.+-666-/, 7.87 # &( # "# " " # &( *+,-./- # "# " " # &( *+,-./- # "# " " *+,-./- +.+-666-/, 7.87 +.+-666-/, 7(.87 +.+-666-/, 79.87 dead-zone # "# " " # &( *+,-./- # &( # "# " " # &( *+,-./- # "# " " *+,-./-,/,6.777.-8/8,/,6.777.-8/98,/,6.777.-8/8,/,6.777.-8/:8 & (* & "# " & (* +,-./. & "# " & (* +,-./. & "# " & (* +,-./. & "# " +,-./.,/,6.777.-8/98,/,6.777.-8/98,/,6.777.-8/:8,/,6.777.-8/:8 & (* & "# " & (* +,-./. & "# " (* +,-./. 9 "# " & (* +,-./. & "# " +,-./.,/,6.777.- 8/8,/,6.777.- 8/98,/,6.777.- 8/98 & (* & "# " & (* +,-./. & "# " & (* +,-./. & "# " +,-./.,/,6.777.- 8/98,/,6.777.- 8/98,/,6.777.- 8:/98 dead-zone & "# " & (* +,-./. & (* & "# " & (* +,-./. & "# " +,-./. 6--67+/8868+++/.+99 6--67+/8868+++/.+9;9 6--67+/8868+++/.+99 6--67+/8868+++/.+9;9 ( *+& ( "# "# "#& "## ( *+&,-./+++/ ( "# "# "#& "## ( *+&,-./+++/ ( "# "# "#& "## ( *+&,-./+++/ ( "# "# "#& "##,-./+++/ 6--67+/8868+++/.+9:;9 6--67+/8868+++/.+9;9 6--67+/8868+++/.+9:;9 6--67+/8868+++/.+9:;9 ( *+& ( "# "# "#& "## ( *+&,-./+++/ ( "# "# "#& "## ( *+&,-./+++/ ( "# "# "#& "## ( *+&,-./+++/ ( "# "# "#& "##,-./+++/ 6--67+/8868+/. 99 6--67+/8868+/. 9;9 6--67+/8868+/. 9;9 ( *+& ( "# "# "#& "## ( *+&,-./+++/ ( "# "# "#& "## ( *+&,-./+++/ ( "# "# "#& "##,-./+++/ 6--67+/8868+/. 9:;9 6--67+/8868+/. 9:;9 6--67+/8868+/. 9<;9 dead-zone ( *+& ( "# "# "#& "## ( *+&,-./+++/ ( "# "# "#& "## ( *+&,-./+++/ ( "# "# "#& "##,-./+++/