UTIAS 1/26 An Adaptive Controller for Two Cooperating Flexible Manipulators UTIAS C. J. Damaren University of Toronto � Institute for Aerospace Studies 4925 Dufferin Street � Toronto, ON M3H 5T6 � Canada � Presented at the International Symposium on Adaptive Motion of Animals and � Machines (AMAM) � �
Outline of Presentation UTIAS 2/26 • Cooperating Flexible Manipulators • Passivity Ideas • Large Payload Dynamics UTIAS • The Adaptive Controller • Experimental Apparatus and Results � � • Conclusions � � � � �
Adaptive Control of UTIAS Rigid Manipulators 3/26 • Motivation: Mass property uncertainty • Typical Controller Structure: adaptive feedforward + PD feedback UTIAS • Stability established using: ⇒ passivity property due to collocation ⇒ [problem is “square”] � ⇒ dynamics are linear in mass properties � � � � � �
UTIAS 4/26 UTIAS � � � � � � �
Cooperating Flexible Manipulators UTIAS 5/26 ✬ ✩ ✈ ✈ r r r B M B 1 B 1 � � ✈ ✈ ❅ F 0 � ❅ � ✫ ✪ ❅ � B M +1 ❅ � ❅ � ✬ ✩ ✬ ✩ ✻ r r r ❅ � ❅ � ❅ ❅ � ❅ ✈ ✈ ✲ ✈ B N � � ✠ � B 0 UTIAS ✫ ✪ ✫ ✪ � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � Closed-Loop Multibody System � � � � � �
Cooperating Flexible Manipulator Systems: UTIAS Characteristics 6/26 • Nonlinear system ⇒ rigid body nonlinearities “plus vibration modes” • Input actuation and controlled output are noncollocated UTIAS ⇒ Nonminimum phase system ⇒ Nonpassive system � • System is “rigidly” overactuated � � • Vibration frequencies and/or mass properties � may be uncertain � ⇒ robust and/or adaptive control � �
Passivity Definitions UTIAS 7/26 ✲ y (t) u (t) ✲ G input output G is a general input/output map G is passive if � τ UTIAS y T ( t ) u ( t ) dt ≥ 0 , ∀ τ > 0 0 � G is strictly passive if � � � τ � τ � y T ( t ) u ( t ) dt ≥ ε u T ( t ) u ( t ) dt, � 0 0 ε > 0 , ∀ τ > 0 � �
Passivity Theorem UTIAS 8/26 disturbance/ feedforward plant ✲ ❥ + u d (t) y (t) ✲ G − ✻ ❥ ❄ − + u (t) ✛ y d (t) H controller reference signal UTIAS If G is passive and H is strictly passive with finite � gain, then the closed-loop system is L 2 -stable: � � � { u d , y d } ∈ L 2 ⇒ { y , u } ∈ L 2 � � �
Kinematics UTIAS 9/26 payload position: ρ = F 1 ( θ 1 , q e 1 ) = F 2 ( θ 2 , q e 2 ) payload velocity: UTIAS ρ = J 1 θ ( θ 1 , q 1 e ) ˙ ˙ θ 1 + J 1 e ( θ 1 , q 1 e ) ˙ q 1 e = J 2 θ ( θ 2 , q 2 e ) ˙ θ 2 + J 2 e ( θ 2 , q 2 e ) ˙ q 2 e � � � � � � �
Modified Input UTIAS 10/26 The joint torques are determined from � τ : � τ 1 � � C 1 J T � 1 θ τ = = � τ C 2 J T τ 2 2 θ UTIAS C 1 and C 2 with 0 < C i < 1 and C 1 + C 2 = 1 are load-sharing parameters. � � � � � � �
Modified Output UTIAS 11/26 µ -tip rate: ρ + (1 − µ )[ C 1 J 1 θ ˙ θ 1 + C 2 J 2 θ ˙ ρ µ = µ ˙ ˙ θ 2 ] µ -tip position: UTIAS ρ µ ( t ) . = µ ρ ( t ) + (1 − µ )[ C 1 F 1 ( θ 1 , 0 ) + C 2 F 2 ( θ 2 , 0 )] � � For µ = 1 , ρ µ = ρ For µ = 0 , ρ µ . � = C 1 F 1 ( θ 1 , 0 ) + C 2 F 2 ( θ 2 , 0 ) � � � �
Passivity Results UTIAS 12/26 ✲ ˙ τ (t) � ✲ ρ µ (t) G input output This system is passive for µ < 1 when the payload is large, i.e., UTIAS � τ ρ T ˙ τ ( t ) dt ≥ 0 , ∀ τ > 0 µ ( t ) � � 0 � � � � � �
Large Payload Motion Equations I UTIAS 13/26 Rigid task-space equations: M ρρ ¨ ρ + C ρ ( ρ , ˙ ρ ) ˙ ρ = � τ PLUS UTIAS Elastic equations consistent with a cantilevered payload. � � � � � � �
Large Payload Motion Equations II UTIAS 14/26 Including only the payload mass properties: ν + ν ⊗ Mν � = P − T ( ρ ) � M ˙ τ � �� W ( ˙ ν , ν , ν ) a where UTIAS � m 1 − c × � � v � M = , ν = , c × J ω � � ω × � � C M 0 ( ρ ) � O O ν ⊗ = � , P = v × ω × S M 0 ( ρ ) O � W is the regressor. � a is a column of mass properties. � Note: ν = P ( ρ ) ˙ ρ � �
Key Definitions UTIAS 15/26 desired trajectory: { ρ d , ˙ ρ d , ¨ ρ d } tracking error: ρ µd . ρ µ = ρ µ − ρ µd , � = ρ d filtered error: ρ µ , Λ = Λ T > O s µ = ˙ ρ µ + Λ � � UTIAS If s µ ∈ L 2 , then � ρ µ → 0 as t → 0 . body-frame ‘desired’ trajectory: � ν d = P ( ρ ) ˙ ρ d � body-frame ‘reference’ trajectory: � � ν r = ν d − P ( ρ )Λ � ρ µ � � �
The Adaptive Controller I UTIAS 16/26 control law: τ = P T W ( ˙ � ν r , ν r , ν ) � a ( t ) − K d s µ Mν ] − K d [ ˙ = P T [ � r � ν r + ν ⊗ M ˙ ρ µ + Λ � � ρ µ ] UTIAS adaptation law: ˙ a = − Γ W T ( ˙ � ν r , ν r , ν ) P ( ρ ) s µ , � Γ = Γ T > O � � � � � �
The Adaptive Controller II UTIAS 17/26 ✎☞ ✎☞ + + τ − � � τ d ① ① ✍✌ ✍✌ − P T W a ✲ ✲ ✲ ✲ s µ G − − ✻ ✻ ✛ K d − P T W � a UTIAS ✗✔ ✗✔ ✛ ✛ ✛ ✖✕ ❅ � ✖✕ ❅ � Γ 1 � ❅ � ❅ s ✻ ✻ � P T W W T P � � � � � �
Experimental Apparatus UTIAS 18/26 UTIAS � � � � � � �
Closed-Loop Configuration UTIAS 19/26 UTIAS � � � � � � �
Payload UTIAS 20/26 UTIAS � � � � � � �
Mode Shapes UTIAS 21/26 UTIAS � � � � � � �
PD Feedback Alone ( C 1 = C 2 = 0 . 5 , µ = 0 . 8 ) UTIAS 22/26 y-position (m) vs. x-position (m) 1.0 0.9 UTIAS 0.8 des. PD 0.7 � � 0.6 � � 0.5 � -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 -0.0 � �
Nonadaptive Results ( C 1 = C 2 = 0 . 5 ) UTIAS 23/26 x-position (m) vs. time y-position (m) vs. time y-position (m) vs. x-position (m) -0.1 1.0 1.0 0.9 0.9 -0.2 0.8 0.8 des. -0.3 fixed 0.7 0.7 pars. -0.4 0.6 0.6 UTIAS -0.5 0.5 0.5 0 5 10 15 20 25 0 5 10 15 20 25 -0.5 -0.4 -0.3 -0.2 -0.1 time (sec) time (sec) � � � � � � �
Adaptive Results UTIAS 24/26 x-pos (m) error vs. time y-pos (m) error vs. time z-orientation (rad) vs. time 0.02 0.05 0.1 0.01 0.03 0.00 0.01 -0.01 0.0 -0.01 -0.02 fixed par. UTIAS C1 = 0.25 -0.03 -0.03 C1 = 0.75 -0.04 -0.05 -0.1 0 5 10 15 20 25 0 5 10 15 20 25 0 5 10 15 20 25 time (sec) time (sec) time (sec) � � � � � � �
Parameter Estimates UTIAS 25/26 2 mass (kg) vs. time c_y (kg-m) vs. time J_zz (kg-m ) vs. time 40. 5.0 20. 4.0 30. 15. 3.0 20. 2.0 10. 1.0 UTIAS 10. estimate 5. 0.0 payload value 0. -1.0 0. 0 5 10 15 20 25 0 5 10 15 20 25 0 5 10 15 20 25 time (sec) time (sec) time (sec) � � � � � � �
Summary of Presentation UTIAS 26/26 • Passivity-based adaptive control: µ -tip rates + load-sharing • Adaptive feedforward depends only on “payload equations” UTIAS • Robust since passivity depends only on a large payload � • Results exhibit good tracking � � � � � �
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