UTIAS C. J. Damaren University of Toronto Institute for - - PowerPoint PPT Presentation
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
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Cooperating Flexible Manipulators
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)
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Rigid Manipulators
adaptive feedforward + PD feedback
⇒ passivity property due to collocation ⇒ [problem is “square”] ⇒ dynamics are linear in mass properties
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✈ ✲ ✻
F0
✬ ✫ ✩ ✪
✈
✈ ✈ r r r
BM BM+1
✬ ✫ ✩ ✪ ✈ ✈ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ r r r ✈
B1
✬ ✫ ✩ ✪
BN Closed-Loop Multibody System
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Characteristics
⇒ rigid body nonlinearities “plus vibration modes”
are noncollocated
⇒ Nonminimum phase system ⇒ Nonpassive system
may be uncertain
⇒ robust and/or adaptive control
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✲
input
u(t) G
✲ y(t)
G is a general input/output map G is passive if
τ yT(t)u(t) dt ≥ 0 , ∀τ > 0
G is strictly passive if
τ yT(t)u(t) dt ≥ ε τ uT(t)u(t) dt, ε > 0 , ∀ τ > 0
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✲ ❥ ✲
ud(t) G
+ −
y(t)
❄ ❥ ✛
yd(t)
+ −
H u(t)
✻
disturbance/ feedforward plant reference signal controller
If G is passive and H is strictly passive with finite gain, then the closed-loop system is L2-stable:
{ud, yd} ∈ L2 ⇒ {y, u} ∈ L2
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payload position:
ρ = F1(θ1, qe1) = F2(θ2, qe2)
payload velocity:
˙ ρ = J1θ(θ1, q1e) ˙ θ1 + J1e(θ1, q1e) ˙ q1e = J2θ(θ2, q2e) ˙ θ2 + J2e(θ2, q2e) ˙ q2e
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The joint torques are determined from
τ: τ = τ 1 τ 2
C1JT
1θ
C2JT
2θ
C1 and C2 with 0 < Ci < 1 and C1 + C2 = 1 are
load-sharing parameters.
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µ-tip rate: ˙ ρµ = µ ˙ ρ + (1 − µ)[C1J1θ ˙ θ1 + C2J2θ ˙ θ2] µ-tip position: ρµ(t) . = µρ(t) + (1 − µ)[C1F1(θ1, 0) + C2F2(θ2, 0)]
For µ = 1, ρµ = ρ For µ = 0, ρµ .
= C1F1(θ1, 0) + C2F2(θ2, 0)
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✲
input
G
✲ ˙
ρµ(t)
This system is passive for µ < 1 when the payload is large, i.e.,
τ ˙ ρT
µ(t)
τ(t) dt ≥ 0 , ∀τ > 0
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Rigid task-space equations:
M ρρ¨ ρ + Cρ(ρ, ˙ ρ) ˙ ρ = τ
PLUS Elastic equations consistent with a cantilevered payload.
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Including only the payload mass properties:
M ˙ ν + ν⊗Mν
τ W ( ˙ ν, ν, ν)a
where
M = m1 −c× c× J
v ω
ν⊗ = ω× O v× ω×
CM0(ρ) O O SM0(ρ)
a is a column of mass properties.
Note: ν = P (ρ) ˙
ρ
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desired trajectory: {ρd, ˙
ρd, ¨ ρd}
tracking error:
ρµd . = ρd
filtered error:
sµ = ˙
ρµ, Λ = ΛT > O
If sµ ∈ L2, then
ρµ → 0 as t → 0.
body-frame ‘desired’ trajectory:
νd = P (ρ) ˙ ρd
body-frame ‘reference’ trajectory:
νr = νd − P (ρ)Λ ρµ
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control law:
νr, νr, ν) a(t) − Kdsµ = P T[ M ˙ νr + ν⊗
r
Mν] − Kd[ ˙
ρµ]
adaptation law:
˙
νr, νr, ν)P (ρ)sµ, Γ = ΓT > O
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✲ ✍✌ ✎☞ ✲ ✍✌ ✎☞ ✲
−P TW a −P TW a G
τ d + − + −
✲ ① ✛
sµ
① ✛
Kd
✻
Γ1
s
✖✕ ✗✔
❅ ✛ ✖✕ ✗✔
❅ ✛ ✻ ✻ ✻
P TW WTP
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y-position (m) vs. x-position (m) PD des. 0.5 0.6 0.7 0.8 0.9 1.0
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time (sec) x-position (m) vs. time time (sec) y-position (m) vs. time y-position (m) vs. x-position (m)
5 10 15 20 25 0.5 0.6 0.7 0.8 0.9 1.0 5 10 15 20 25 des. fixed pars. 0.5 0.6 0.7 0.8 0.9 1.0
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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
0.00 0.01 0.02 5 10 15 20 25 C1 = 0.25
0.01 0.03 0.05 5 10 15 20 25
0.0 0.1 5 10 15 20 25
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time (sec) mass (kg) vs. time time (sec) c_y (kg-m) vs. time time (sec) J_zz (kg-m ) vs. time 2 estimate payload value 0. 10. 20. 30. 40. 5 10 15 20 25
0.0 1.0 2.0 3.0 4.0 5.0 5 10 15 20 25 0. 5. 10. 15. 20. 5 10 15 20 25
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µ-tip rates + load-sharing