[PPT] - Ensemble Control as a Tool for Robot Motion Planning: Uncertainty, PowerPoint Presentation

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Introduction Linear example Nonlinear example Conclusion

Ensemble Control as a Tool for Robot Motion Planning:

Uncertainty, Optimality, and Complexity

Timothy Bretl (w/ Aaron Becker) University of Illinois at Urbana-Champaign IEEE ICRA Workshop on Uncertainty in Automation May 9, 2011

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Introduction Linear example Nonlinear example Conclusion Problem Message

Outline

1

Introduction Problem Message

2

Linear example

3

Nonlinear example

4

Conclusion

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Introduction Linear example Nonlinear example Conclusion Problem Message

Motion planning

Find u : [0, tf ] → U ⊂ Rm x : [0, tf ] → X ⊂ Rn satisfying ˙ x(t) = f (t, x(t), u(t)) x(0) = xstart x(tf ) = xgoal for free final time tf

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Introduction Linear example Nonlinear example Conclusion Problem Message

Motion planning under bounded uncertainty

Find u : [0, tf ] → U ⊂ Rm x : [0, tf ] → X ⊂ Rn satisfying ˙ x(t) = f (t, x(t), u(t), ǫ) x(0) = xstart x(tf ) = xgoal for free final time tf , despite bounded uncertainty ǫ ∈ [1 − δ, 1 + δ]

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Introduction Linear example Nonlinear example Conclusion Problem Message

Motion planning as ensemble control

Find u : [0, tf ] → U ⊂ Rm x : [0, tf ] × [1 − δ, 1 + δ] → X ⊂ Rn satisfying ˙ x(t, ǫ) = f (t, x(t, ǫ), u(t), ǫ) x(0, ǫ) = xstart x(tf , ǫ) = xgoal for free final time tf and for all ǫ ∈ [1 − δ, 1 + δ]

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Introduction Linear example Nonlinear example Conclusion Problem Message

Take-away message

Ensemble control theory is a useful way to deal with bounded uncertainty in dynamical systems. To steer one system with an uncertain parameter, we pretend to steer a continuous ensemble of systems, each with a particular value of that parameter. In the examples we will consider, this approach costs us nothing in terms of computational complexity.

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Introduction Linear example Nonlinear example Conclusion Model Analysis Results

Outline

1

Introduction

2

Linear example Model Analysis Results

3

Nonlinear example

4

Conclusion

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Introduction Linear example Nonlinear example Conclusion Model Analysis Results

A driven harmonic oscillator

This system is linear and has the form

˙ x = 1 −ǫ

x +

ǫ

u

= Ax + Bu, where (x1, x2) = (y, ˙ y), u = d, and ǫ = k/m.

For known ǫ, this system is controllable because
AB

B

=

ǫ ǫ

is full rank.

m k d y

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Introduction Linear example Nonlinear example Conclusion Model Analysis Results

Ensemble controllability

For unknown ǫ, consider the lifted system

˙ x(t, ǫ) = 1 −ǫ

x(t, ǫ) +

ǫ

u(t)

= A(ǫ)x(t, ǫ) + B(ǫ)u(t).

For any integer k ≥ 0, we have
A2k+1B

A2kB

=

ǫk+1 ǫk+1

.
So, we can approximate any desired movement direction by a

polynomial in ǫ, with error vanishing in k: f (ǫ) ≈

k−1

i=0

ǫi

ai

1

+ bi

1

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Introduction Linear example Nonlinear example Conclusion Model Analysis Results

Control by piecewise-constant inputs (1/2)

A discrete-time model is

xd(i + 1, ǫ) = eA(ǫ)Txd(i, ǫ) + T eAsBds

ud(i)

= Ad(ǫ)xd(i, ǫ) + Bd(ǫ)ud(i)

If xd(0, ǫ) = 0 then

xd(2k, ǫ) =

2k

i=0

Ai

d(ǫ)Bd(ǫ)ud(2k − i)

This can be approximated by the series expansion

xd(2k, ǫ) ≈

k−1

i=0

1 i! ∂ixd(2k, ǫ) ∂ǫi

ǫ=1
(ǫ − 1)i

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Introduction Linear example Nonlinear example Conclusion Model Analysis Results

Control by piecewise-constant inputs (2/2)

The result has the form

xd(2k, ǫ) =

k

i=1

ri si

(ǫ − 1)i−1 + O
|ǫ − 1|k

where r, s ∈ Rk are linear in ud ∈ R2k

To achieve (x1, x2) with error of order k in |ǫ − 1|:

r =      x1 . . .      , s =      x2 . . .     

The solution (2k linear equations in 2k variables) has the form

ud = K1x1 + K2x2 for matrices K1 and K2 that can be precomputed

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Introduction Linear example Nonlinear example Conclusion Model Analysis Results

Example results in simulation

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Introduction Linear example Nonlinear example Conclusion Model Analysis Results

Outline

1

Introduction

2

Linear example

3

Nonlinear example Model Analysis Results

4

Conclusion

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Introduction Linear example Nonlinear example Conclusion Model Analysis Results

A differential drive robot with uncertain wheel radius

ǫ = 0.8 ǫ = 0.83 ǫ = 1.0 ǫ = 1.2

For a fixed wheel separation, inputs scale with wheel radius ǫr = wheel radius b = wheel separation v = ǫ r(ωR + ωL) 2

= ǫu1

w = ǫ r(ωR − ωL) b

= ǫu2

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Introduction Linear example Nonlinear example Conclusion Model Analysis Results

Scratch-drive microrobots with uncertain forward speed

Figure: Donald et al. (2008) Figure: Donald et al. (2006)

For a fixed turning

radius, inputs scale with forward speed

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Introduction Linear example Nonlinear example Conclusion Model Analysis Results

Analysis of one robot

This system is nonlinear and has the form

˙ x = ǫ   cos x3 sin x3   u1 + ǫ   1   u2 = ǫg1(x)u1 + ǫg2(x)u2

For known ǫ > 0, this system is controllable because
[ǫg1, ǫg2]

ǫg2 ǫg1

=

  ǫ2 sin x3 ǫ cos x3 −ǫ2 cos x3 ǫ sin x3 ǫ   is full rank everywhere

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Introduction Linear example Nonlinear example Conclusion Model Analysis Results

Analysis of an ensemble (1/3)

For unknown ǫ, consider the lifted system

˙ x(t, ǫ) = ǫ   cos x3(t, ǫ) sin x3(t, ǫ)   u1(t) + ǫ   1   u2(t) = ǫg1(x(t, ǫ))u1(t) + ǫg2(x(t, ǫ))u2(t)

Heading is not controllable, since for all ǫ

x3(t, ǫ) = x3(0, ǫ) + ǫθ(t) where ˙ θ(t) = u2(t)

Eliminate heading to get a controllable subsystem:

  ˙ x1(t, ǫ) ˙ x2(t, ǫ) ˙ θ(t)   = ǫ   cos (x3(0, ǫ) + ǫθ(t)) sin (x3(0, ǫ) + ǫθ(t))   u1(t) +   1   u2(t)

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Introduction Linear example Nonlinear example Conclusion Model Analysis Results

Analysis of an ensemble (2/3)

Take Lie brackets to find new control vector fields:

[ǫg1, g2] = ǫ ∂g2 ∂q g1 − ∂g1 ∂q g2

= −ǫ2

  − sin (x3(0, ǫ) + ǫθ(t)) cos (x3(0, ǫ) + ǫθ(t))   = −ǫ2g3 [[ǫg1, g2] , g2] = −ǫ3g1 [[[ǫg1, g2] , g2] , g2] = −ǫ4g3 . . .

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Introduction Linear example Nonlinear example Conclusion Model Analysis Results

Analysis of an ensemble (3/3)

We can approximate any desired movement direction by a

polynomial in ǫ, with error vanishing in k: f (ǫ) ≈ cg2 +

k

i=0
aiǫ2i+1g1 + biǫ2i+2g3
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Introduction Linear example Nonlinear example Conclusion Model Analysis Results

Control with piecewise-constant inputs

u1 u2 ∆t

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Introduction Linear example Nonlinear example Conclusion Model Analysis Results

Control with piecewise-constant inputs

(j − 1)φ u1 u2 ∆t 1/λ λ(j − 1)φ

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Introduction Linear example Nonlinear example Conclusion Model Analysis Results

Control with piecewise-constant inputs

a′

j

u1 u2 ∆t 1/λ λ(j − 1)φ sign

a′

j

a′

j

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Introduction Linear example Nonlinear example Conclusion Model Analysis Results

Control with piecewise-constant inputs

−2(j − 1)φ u1 u2 ∆t 1/λ λ(j − 1)φ sign

a′

j

a′

j

−1/λ

2λ(j − 1)φ

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Introduction Linear example Nonlinear example Conclusion Model Analysis Results

Control with piecewise-constant inputs

b′

j

u1 u2 ∆t 1/λ λ(j − 1)φ sign

a′

j

a′

j

−1/λ

2λ(j − 1)φ sign

b′

j

b′

j

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Introduction Linear example Nonlinear example Conclusion Model Analysis Results

Control with piecewise-constant inputs

(j − 1)φ u1 u2 ∆t 1/λ λ(j − 1)φ sign

a′

j

a′

j

−1/λ

2λ(j − 1)φ sign

b′

j

b′

j

1/λ

λ(j − 1)φ

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Introduction Linear example Nonlinear example Conclusion Model Analysis Results

Correspondence with polynomial approximation

The result is to achieve

∆x1(ǫ) = (a′

j + b′ j)ǫ cos (ǫ(j − 1)φ)

∆x2(ǫ) = (a′

j − b′ j)ǫ sin (ǫ(j − 1)φ)

∆θ = 0

With the input transformation

a′

j = aj + bj−1

2 b′

j = aj − bj−1

2 for freely chosen ak, bk ∈ R, we can write ∆x1(ǫ) = ajǫ cos (ǫ(j − 1)φ) ∆x2(ǫ) = bj−1ǫ sin (ǫ(j − 1)φ) .

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Introduction Linear example Nonlinear example Conclusion Model Analysis Results

Sequence of motion primitives (1/3)

For ak+1 = 0, the result after k + 1 primitives is

∆x1(ǫ) =

k

j=1

ajǫ cos (ǫ(j − 1)φ) ∆x2(ǫ) =

k

j=1

bjǫ sin (ǫjφ)

This can be approximated by the series expansions

∆x1(ǫ) ≈

k−1

i=0

1 i! ∂i∆x1 ∂ǫi

ǫ=1
(ǫ − 1)i

∆x2(ǫ) ≈

k−1

i=0

1 i! ∂i∆x2 ∂ǫi

ǫ=1
(ǫ − 1)i

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Introduction Linear example Nonlinear example Conclusion Model Analysis Results

Sequence of motion primitives (2/3)

The result has the form

∆x1(ǫ) =

k

i=1

ri (ǫ − 1)i−1 + O

|ǫ − 1|k

∆x2(ǫ) =

k

i=1

si (ǫ − 1)i−1 + O

|ǫ − 1|k

where r, s ∈ Rk are linear in a, b ∈ Rk

To achieve ∆x1 = ∆x2 = 1 with error of order k in |ǫ − 1|:

r = s =      1 . . .     

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Introduction Linear example Nonlinear example Conclusion Model Analysis Results

Sequence of motion primitives (3/3)

Require exactly k + 1 primitives to achieve kth-order error
Precompute a, b as 2k linear equations in 2k variables
For φ = π/2, this can be done in closed form
By linearity, a∆x1 and b∆x2 reach arbitrary ∆x1 and ∆x2
“Planning” means computing piecewise-constant inputs

a′ = 1 2 a

∆x1 +

b

∆x2
b′ = 1

2 a

∆x1 −

b

∆x2
.
This approach does not require sampling ǫ

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Introduction Linear example Nonlinear example Conclusion Model Analysis Results

Example results in simulation

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Introduction Linear example Nonlinear example Conclusion Model Analysis Results

Scaled primitives get you everywhere “for free”

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Introduction Linear example Nonlinear example Conclusion Model Analysis Results

Example results in experiment

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Introduction Linear example Nonlinear example Conclusion Connections

Outline

1

Introduction

2

Linear example

3

Nonlinear example

4

Conclusion Connections

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Introduction Linear example Nonlinear example Conclusion Connections

Connections

NMR spectroscopy and imaging (Brockett, Khaneja, Li,

Altafini, Beauchard, Coron, Pereira da Silva, Rouchon, etc.)

Robust control (Dullerud and Paganini, Singer and Seering,

Fischer and Psiaki, etc.)

Sensorless manipulation (Erdmann, Mason, Goldberg, Lynch,

Murphey, Akella, van der Stappen, Moll, etc.)

POMDP planning (Hsu, Hutchinson, Roy, Thrun, etc.)
Optimal control of micro/nano-robot teams (under review)
Control-theoretic approach to manipulation of deformable
bjects (in preparation)
BMIs based on inverse optimal control

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Introduction Linear example Nonlinear example Conclusion Connections

Support

NSF-CNS-0931871
NSF-CMMI-0956362-EAGER
NSF-CMMI-0955088-CAREER

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Introduction Linear example Nonlinear example Conclusion Connections

Questions?

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Introduction Linear example Nonlinear example Conclusion Connections

Connection to NMR spectroscopy and imaging

Need to manipulate an ensemble of ≈ 1023 nuclear spins, each

governed by the Schr¨

dinger equation
Control input changes the potential energy in the system

Hamiltonian (e.g., by applying electromagnetic pulses)

One model is

dx(t, s) dt =

A(s) +
i

uiBi(s)

x(t, s)
“s” describes variation in A and Bi from Larmor dispersion (in

natural frequencies), rf inhomogeneity (in strength of the applied radio frequency), and relaxation rates

Is it possible to steer from one state to another?

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