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An Adaptive Controller for TwoCooperating Flexible Manipulators

C. J. Damaren

University of TorontoInstitute for Aerospace Studies4925 Dufferin StreetToronto, ON M3H 5T6CanadaPresented at the International Symposium on Adaptive Motion of Animals andMachines (AMAM)

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Outline of Presentation

• Cooperating Flexible Manipulators

• Passivity Ideas

• Large Payload Dynamics

• The Adaptive Controller

• Experimental Apparatus and Results

• Conclusions

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Adaptive Control ofRigid Manipulators

• Motivation: Mass property uncertainty

• Typical Controller Structure:adaptive feedforward + PD feedback

• Stability established using:⇒ passivity property due to collocation⇒ [problem is “square”]⇒ dynamics are linear in mass properties

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Cooperating Flexible Manipulators

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Closed-Loop Multibody System

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Cooperating Flexible Manipulator Systems:Characteristics

• Nonlinear system⇒ rigid body nonlinearities “plus vibration modes”

• Input actuation and controlled outputare noncollocated

⇒ Nonminimum phase system⇒ Nonpassive system

• System is “rigidly” overactuated

• Vibration frequencies and/or mass propertiesmay be uncertain

⇒ robust and/or adaptive control

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Passivity Definitions-

inputu(t) G - y(t)

output

G is a general input/output mapG is passive if∫ τ

0

yT (t)u(t) dt ≥ 0 , ∀τ > 0

G is strictly passive if∫ τ0

yT (t)u(t) dt ≥ ε∫ τ

0

uT (t)u(t) dt,

ε > 0 , ∀ τ > 0

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Passivity Theorem

- j -ud(t) G+− y(t)

?j� yd(t)+ −Hu(t)6

disturbance/feedforward plant

reference signalcontroller

If G is passive and H is strictly passive with finitegain, then the closed-loop system is L2-stable:

{ud,yd} ∈ L2⇒ {y,u} ∈ L2

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Kinematics

payload position:

ρ = F1(θ1,qe1) = F2(θ2,qe2)

payload velocity:

ρ̇ = J1θ(θ1,q1e)θ̇1 + J1e(θ1,q1e)q̇1e

= J2θ(θ2,q2e)θ̇2 + J2e(θ2,q2e)q̇2e

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Modified Input

The joint torques are determined from τ̂ :

τ =

[τ 1τ 2

]=

[C1J

T1θ

C2JT2θ

]τ̂

C1 and C2 with 0 < Ci < 1 and C1 + C2 = 1 areload-sharing parameters.

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Modified Output

µ-tip rate:

ρ̇µ = µρ̇ + (1− µ)[C1J1θθ̇1 + C2J2θθ̇2]

µ-tip position:

ρµ(t).= µρ(t) + (1− µ)[C1F1(θ1,0) + C2F2(θ2,0)]

For µ = 1, ρµ = ρFor µ = 0, ρµ

.= C1F1(θ1,0) + C2F2(θ2,0)

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Passivity Results

-

inputτ̂ (t) G - ρ̇µ(t)

output

This system is passive for µ < 1when the payload is large, i.e.,∫ τ

0

ρ̇Tµ (t)τ̂ (t) dt ≥ 0 , ∀τ > 0

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Large Payload Motion Equations I

Rigid task-space equations:

M ρρρ̈ +Cρ(ρ, ρ̇)ρ̇ = τ̂

PLUS

Elastic equations consistent with acantilevered payload.

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Large Payload Motion Equations II

Including only the payload mass properties:

Mν̇ + ν⊗Mν︸ ︷︷ ︸ = P−T (ρ)τ̂W (ν̇,ν,ν)a

where

M =

[m1 −c×c× J

], ν =

[v

ω

],

ν⊗ =

[ω× O

v× ω×

], P =

[CM0(ρ) O

O SM0(ρ)

]W is the regressor.a is a column of mass properties.Note: ν = P (ρ)ρ̇

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Key Definitions

desired trajectory: {ρd, ρ̇d, ρ̈d}tracking error:

ρ̃µ = ρµ − ρµd, ρµd.= ρd

filtered error:

sµ = ˙̃ρµ + Λρ̃µ, Λ = ΛT > O

If sµ ∈ L2, then ρ̃µ → 0 as t→ 0.body-frame ‘desired’ trajectory:

νd = P (ρ)ρ̇d

body-frame ‘reference’ trajectory:

νr = νd − P (ρ)Λρ̃µ

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The Adaptive Controller I

control law:

τ̂ = P TW (ν̇r,νr,ν)â(t)−Kdsµ= P T [M̂ν̇r + ν

⊗r M̂ν]−Kd[ ˙̃ρµ + Λρ̃µ]

adaptation law:˙̂a = −ΓW T (ν̇r,νr,ν)P (ρ)sµ,

Γ = ΓT > O

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The Adaptive Controller II

-���-��� -

−P TWâ

−P TWa Gτ̂ − τ̂ d+

−+

−-x

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P TW WTP

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Experimental Apparatus

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Closed-Loop Configuration

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Payload

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Mode Shapes

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PD Feedback Alone (C1 = C2 = 0.5, µ = 0.8)

y-position (m) vs. x-position (m)

PDdes.

0.5

0.6

0.7

0.8

0.9

1.0

-0.6 -0.5 -0.4 -0.3 -0.2 -0.1 -0.0

time (sec)

z-orientation (rad) vs. time

-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

0 5 10 15 20 25

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Nonadaptive Results (C1 = C2 = 0.5)

time (sec)

x-position (m) vs. time

time (sec)

y-position (m) vs. time y-position (m) vs. x-position (m)

-0.5

-0.4

-0.3

-0.2

-0.1

0 5 10 15 20 250.5

0.6

0.7

0.8

0.9

1.0

0 5 10 15 20 25

des.fixedpars.

0.5

0.6

0.7

0.8

0.9

1.0

-0.5 -0.4 -0.3 -0.2 -0.1

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Adaptive Results

time (sec)

x-pos (m) error vs. time

time (sec)

y-pos (m) error vs. time

time (sec)

z-orientation (rad) vs. time

fixed par.

C1 = 0.75

-0.04

-0.03

-0.02

-0.01

0.00

0.01

0.02

0 5 10 15 20 25

C1 = 0.25

-0.05

-0.03

-0.01

0.01

0.03

0.05

0 5 10 15 20 25-0.1

0.0

0.1

0 5 10 15 20 25

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Parameter Estimates

time (sec)

mass (kg) vs. time

time (sec)

c_y (kg-m) vs. time

time (sec)

J_zz (kg-m ) vs. time2

estimatepayload value

0.

10.

20.

30.

40.

0 5 10 15 20 25-1.0

0.0

1.0

2.0

3.0

4.0

5.0

0 5 10 15 20 250.

5.

10.

15.

20.

0 5 10 15 20 25

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Summary of Presentation

• Passivity-based adaptive control:µ-tip rates + load-sharing

• Adaptive feedforward dependsonly on “payload equations”

• Robust since passivity dependsonly on a large payload

• Results exhibit good tracking