lekan ogunmolu may 16, 2019opo140030/media/papers/thesis_slides.pdf · use of computers in...

Publications

Background

Conformal RTTreatment PlanningParameters

Robot-basedRadiation Therapy

Frameless andMaskless:Cyberknife system

BOO

Immobilization

Experiment Setup

Identification

3-DOF Control

AdaptiveNeuroControl

Deformation

Multi-DOFKinematics

BOO

Proposal Two-Player

Search

Fluence MapOptimization

Results

BOO II

References

A Multi-DOF Soft Robot Mechanism for Patient MotionCorrection and Beam Orientation Selection in Cancer Radiation

Therapy.

Lekan Ogunmolu

Department of Electrical Engineering

The University of Texas at Dallas, Richardson, TX

May 16, 2019

Lekan OgunmoluA Multi-DOF Soft Robot Mechanism for Patient Motion Correction and Beam Orientation Selection in Cancer Radiation Therapy.

Publications

Background




BOO

Immobilization

Experiment Setup

Identification

3-DOF Control


Deformation

Multi-DOFKinematics

BOO

Proposal Two-Player

Search


Results

BOO II

References

Publications

Olalekan Ogunmolu, A Multi-DOF Soft Robot Mechanism for Patient Motion Correction and BeamOrientation Selection in Cancer Radiation Therapy. PhD Thesis Manuscript. Hyperlink:utdallas.edu/ opo140030/media/Papers/thesis.pdf

Azar Sadeghnejad Barkousaraie, Olalekan Ogunmolu, Steve Jiang, and Dan Nguyen. A Fast DeepLearning Approach for Beam Orientation Selection Using Supervised Learning with ColumnGeneration on IMRT Prostate Cancer Patients. Under review at Medical Physics (Journal), April2019.

Olalekan Ogunmolu, Michael Folkerts, Dan Nguyen, Nicholas Gans, and Steve Jiang. Deep BOO!Automating Beam Orientation Selection in Intensity Modulated Radiation Therapy. AlgorithmicFoundations of Robotics XIII, International Workshop (WAFR), Merida, Mexico. December 2018.Published in Springer’s Proceedings in Advanced Robotics (SPAR) Book. 2019.

Olalekan Ogunmolu, Nicholas Gans, Tyler Summers. Minimax Iterative Dynamic Game: Applicationto Nonlinear Robot Control Tasks. IEEE/RSJ International Conference on Intelligent Robots andSystems (IROS), Madrid, Spain. October 2018.

Olalekan Ogunmolu, Adwait Kulkarn, Yonas Tadesse, Xuejun Gu, Steve Jiang, and Nicholas Gans.Soft-NeuroAdapt: A 3-DOF Neuro-Adaptive Pose Correction System For Frameless and MasklessCancer Radiotherapy. IEEE/RSJ International Conference on Intelligent Robots and Systems(IROS), Vancouver, BC, Canada. September 2017. DOI: 10.1109/IROS.2017.8206211.

Azar Sadeghnejad Barkousaraie, Olalekan Ogunmolu, Steve Jiang, and Dan Nguyen. Usingsupervised learning and guided Monte Carlo tree search for beam orientation optimization inradiation therapy. Under review at International Conference on Medical Image Computing andComputer Assisted Intervention, XXII (MICCAI), Shenzhen, China. October 2019.


https://ecs.utdallas.edu/~opo140030/media/Papers/thesis.pdf

http://algorithmic-robotics.org/wafr2018/

http://algorithmic-robotics.org/wafr2018/

https://arxiv.org/abs/1703.03821

https://arxiv.org/abs/1703.03821

http://iccr-mcma.org/iccr2019/


Publications

Background




BOO

Immobilization

Experiment Setup

Identification

3-DOF Control


Deformation

Multi-DOFKinematics

BOO

Proposal Two-Player

Search


Results

BOO II

References

Publications

Azar Sadeghnejad Barkousaraie, Olalekan Ogunmolu, Steve Jiang, and Dan Nguyen. Deep LearningNeural Network for Beam Orientation Optimization. To appear in International Conference on theuse of Computers in Radiation Therapy XVI (ICCR), Montreal, CA. June 2019.

Olalekan Ogunmolu, Dan Nguyen, Xun Jia, Weiguo Lu, Nicholas Gans, and Steve Jiang.Automating Beam Orientation Optimization for IMRT Treatment Planning: A Deep ReinforcementLearning Approach. Selected for Oral Presentation at the John R. Cameron Young InvestigatorsSymposium – 60th Annual Meeting of the American Association of Physicists in Medicine, Nashville,TN (AAPM). July 2018.

Yara Almubarak, Joshi Aniket, Olalekan Ogunmolu, Xuejun Gu, Steve Jiang, Nicholas Gans, andYonas Tadesse, Design and Development of Soft Robots for Head and Neck Cancer Radiotherapy.SPIE: Smart Structures + Nondestructive Evaluation, (SPIE), Denver, CO, U.S.A. March 2018.

Olalekan Ogunmolu, Xuejun Gu, Steve Jiang, and Nicholas Gans. Vision-based control of asoft-robot for Maskless Cancer Radiotherapy. IEEE Conference on Automation Science andEngineering (CASE), Fort-Worth, Texas, August 2016. DOI: 10.1109/CoASE.2016.7743378.

Olalekan Ogunmolu, Xuejun Gu, Steve Jiang, and Nicholas Gans. A Real-Time Soft-Robotic PatientPositioning System for Maskless Head-and-Neck Cancer Radiotherapy. IEEE Conference onAutomation Science and Engineering (CASE), Gothenburg, Sweden, August 2015. DOI:10.1109/CoASE.2015.7294318.




http://spie.org/conferences-and-exhibitions/smart-structures/nde?SSO=1

http://ecs.utdallas.edu/~opo140030/media/Papers/CASE2016.pdf

http://ecs.utdallas.edu/~opo140030/media/Papers/CASE2016.pdf

http://ecs.utdallas.edu/~opo140030/media/Papers/CASE2015/CASE2015.pdf

http://ecs.utdallas.edu/~opo140030/media/Papers/CASE2015/CASE2015.pdf

Publications

Background




BOO

Immobilization

Experiment Setup

Identification

3-DOF Control


Deformation

Multi-DOFKinematics

BOO

Proposal Two-Player

Search


Results

BOO II

References

Three Dimensional Conformal Radiation Therapy

Intensity Modulation: Control external beam’s physical delivery

Conform internally uniform fields with MLCs using a projection of target volumes [ Boyer et al.(1992)]

Improve tumor’s local control

L-R: Conventional radiotherapy. Conformal radiotherapy(CFRT) without intensity modulation. CFRT with intensity

modulation. Reprinted from Webb (2001).

A multi-leaf collimator for IMRT/3DCRT. c©VarianMedical Systems.


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BOO

Immobilization

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3-DOF Control


Deformation

Multi-DOFKinematics

BOO

Proposal Two-Player

Search


Results

BOO II

References

Conformal RT Treatment Planning Parameters

Optimal treatment parameters B good treatment outcome

dose-limiting structures

OARs within a target volume

doctor’s dose prescription

dose fractionation

patient positioning

dose distribution


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BOO

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3-DOF Control


Deformation

Multi-DOFKinematics

BOO

Proposal Two-Player

Search


Results

BOO II

References

Frame-based Radiotherapy Treatment

Accurately irradiate a moving target and a moving patient with the aid of robots[Schweikard et al.(1995); Webb (1999)]


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BOO

Immobilization

Experiment Setup

Identification

3-DOF Control


Deformation

Multi-DOFKinematics

BOO

Proposal Two-Player

Search


Results

BOO II

References

Frameless and Maskless Radiotherapy

c© Wiersma et al. (2009)


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BOO

Immobilization

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Identification

3-DOF Control


Deformation

Multi-DOFKinematics

BOO

Proposal Two-Player

Search


Results

BOO II

References

HexaPOD

Reprinted from Herrmann et al. (2011)


Publications

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BOO

Immobilization

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3-DOF Control


Deformation

Multi-DOFKinematics

BOO

Proposal Two-Player

Search


Results

BOO II

References

Cyberknife/Novalis systems


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BOO

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3-DOF Control


Deformation

Multi-DOFKinematics

BOO

Proposal Two-Player

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Results

BOO II

References

The Novalis ExacTrac Module

c©Novalis


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Deformation

Multi-DOFKinematics

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BOO II

References

The Case for Soft Robots

Frame-based immobilization

LINAC misalignments =⇒ negative dosimetry effects

× Fractionated treatments

Frameless RT

Incompatible with most conventional LINACs

Cyberknife/Novalis Systems

Reliance on pre-treatment images

Rigid motion compensation issues

Involuntary patient motion requires adaptive positioning

0Morphological computation, Cephalopods, Adaptive Controller for changing head dynamics: shape,

weight etc


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BOO

Immobilization

Experiment Setup

Identification

3-DOF Control


Deformation

Multi-DOFKinematics

BOO

Proposal Two-Player

Search


Results

BOO II

References

Beam Orientation Optimization

During treatment planning, a beam orientationoptimization problem (BOO) is separately solved

Radiation is delivered from ≈ (5− 15) different beamorientations during IMRT

BOO determines the best beam angle combinations fordelivering radiation

Process of determining beamlets’ intensities is termedfluence map optimization (FMO)


Publications

Background




BOO

Immobilization

Experiment Setup

Identification

3-DOF Control


Deformation

Multi-DOFKinematics

BOO

Proposal Two-Player

Search


Results

BOO II

References

Vision-based 1-DOF Control


Publications

Background




BOO

Immobilization

Experiment Setup

Identification

3-DOF Control


Deformation

Multi-DOFKinematics

BOO

Proposal Two-Player

Search


Results

BOO II

References

Sensors’ Noise Floor

0 500 1000 1500 2000

Time(sec)

680

700

720

De

pth

(m

m)

Kinect Xbox Tracking Noise on a Static Target at 684mm

Ground Truth = 680mm; Covariance = 21.5022

100 200 300 400 500 600 700 800 900 1000

Time(sec)

995

1000

1005

1010

De

pth

(m

m)

Kinect v1 Position Estimation of a Static Object

Ground Truth = 960mm; Covariance = 11.4707


Publications

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BOO

Immobilization

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3-DOF Control


Deformation

Multi-DOFKinematics

BOO

Proposal Two-Player

Search


Results

BOO II

References

Kalman Filter Model

x(k) = F(k)x(k − 1) + B(k)uk + Gkwk

zs = Hs(k)x(k) + vs(k) s = 1, 2

F =

[1 ∆T0 1

]; ak ∼ N (0, σa); Gk = I2x2;

w(k) ∼ N (0,Q(k)), W(k) =

(∆T 2

2∆T

)

Q = WWTσa2 =

∆T 4

4

∆T 3

2∆T 3

2∆T 2

σa2. (1)


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BOO

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3-DOF Control


Deformation

Multi-DOFKinematics

BOO

Proposal Two-Player

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Results

BOO II

References

Filtering Results

200 400 600 800 1000 1200 1400 1600 1800 2000 2200

Samples

670

680

690

700

710

Ob

se

rva

tio

n in

m

m

Kinect Xbox Tracking Noise on a Static Target at 684mm

Observation

200 400 600 800 1000 1200 1400 1600 1800 2000 2200

Samples

682

684

686

688

Ka

lma

n u

pd

ate

s in

m

m

Kalman Filter Estimates with Kinect Xbox. [Truth: 684mm, R: 70mm2

]

Estimates

100 150 200 250 300 350 400

Time(Secs

645

650

655

De

pth

(m

m)

Kinect v1 Tracking Noise on a Static Target at 651mm

Observation

100 150 200 250 300 350 400

Time(secs)

649

650

651

652

653

KF

E

stim

ate

s

(m

m)

Recursive KF Results of Raw Observation in Real Time

Priori

Posteriori

Xbox vs. Kinect v1

Fusion:

x(F )(k |k) = P(F )(k |k)N∑

s=1

[P(s)−1(k|k)x(s)(k|k)

]


Publications

Background




BOO

Immobilization

Experiment Setup

Identification

3-DOF Control


Deformation

Multi-DOFKinematics

BOO

Proposal Two-Player

Search


Results

BOO II

References

Fusion Results

Fusion of local state estimates.


Publications

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BOO

Immobilization

Experiment Setup

Identification

3-DOF Control


Deformation

Multi-DOFKinematics

BOO

Proposal Two-Player

Search


Results

BOO II

References

System Model and LQG Control

Obtain optimal model parameters from I/O data through,

G (t) = arg minθ

VN(θ, φN) (4)

From (4), we obtained

x(k + Ts) = Ax(k) + Bu(k) + Ke(k)

y(k) = Cx(k) + Du(k) + e(k) (5)

LQG cost:

J =K∑

k=0

xT (k)Q x(k) + u(k)T R u(k) + 2x(k)T N u(k)

Find u from ∆u = arg min∆u

J


Publications

Background




BOO

Immobilization

Experiment Setup

Identification

3-DOF Control


Deformation

Multi-DOFKinematics

BOO

Proposal Two-Player

Search


Results

BOO II

References

1-DOF Control Results

20 30 40 50 60 70 80 90 100

Time (seconds)

685

690

695

700

705

710H

ead P

ositio

n(m

m)

Experiment I Reference

Kinect Fused yhat(kT)

LQG Estimated yhat(kT)

10 20 30 40 50 60 70 80 90 100

Time (seconds)

680

690

700

710

Head P

ositio

n (

mm

)

Experiment IIReference

Kinect Fused yhat(kT)

LQG Estimated yhat(kT)

LQG Controller on mannequine head.


Publications

Background




BOO

Immobilization

Experiment Setup

Identification

3-DOF Control


Deformation

Multi-DOFKinematics

BOO

Proposal Two-Player

Search


Results

BOO II

References

Vision-based 3-DOF Control

Hardware Description


Publications

Background




BOO

Immobilization

Experiment Setup

Identification

3-DOF Control


Deformation

Multi-DOFKinematics

BOO

Proposal Two-Player

Search


Results

BOO II

References

Point Cloud Pre-Processing


Publications

Background




BOO

Immobilization

Experiment Setup

Identification

3-DOF Control


Deformation

Multi-DOFKinematics

BOO

Proposal Two-Player

Search


Results

BOO II

References

Head Pose Estimation

Set cloud’s centroid as measured point set P = −→p i

Get covariance matrix Σpx of measured and model pointsets: P and X

Set cyclic components of anti-symmetric matrix as ∆

Set Q(Σpx) =

[tr(Σpx) ∆T

∆ Σpx + ΣTpx − tr(Σpx)I3

]qR = max

eig(Q(Σpx)); qT = µx − R(qR)µp

xh = (qT , qR)


Publications

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BOO

Immobilization

Experiment Setup

Identification

3-DOF Control


Deformation

Multi-DOFKinematics

BOO

Proposal Two-Player

Search


Results

BOO II

References

Model Reference Adaptive Control

Head and IAB System Model

y = Ay + BΛ (u− f (y,u)) + w(k)

Realize f (y, u) with an RNN ≡ ΘTΦ(y)

In a ball BR ⊂ D

an ideal neural network (NN) approximation f (·) : Rn → Rm,can be realized to a sufficient degree of accuracy, εf > 0;

Outside BR :

‖ε(y)‖ ≤ kmax(y), ∀ y ∈ BR ;


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BOO

Immobilization

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Identification

3-DOF Control


Deformation

Multi-DOFKinematics

BOO

Proposal Two-Player

Search


Results

BOO II

References

Adaptive Neuro-Control Scheme

Choose ym = Amym + Bmr

Knowns: Bm and an Hurwitz Am

u = KTy y︸︷︷︸

state feedback

+ KTr r︸︷︷︸

optimal regulator

+ f (y, u)︸︷︷︸approximator

Ky and Kr are adaptive gains to be designed. NB:Kx = Kx − Kx

Assume ideal model matching conditionsKy = Ky , and Kr = Kr


Publications

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BOO

Immobilization

Experiment Setup

Identification

3-DOF Control


Deformation

Multi-DOFKinematics

BOO

Proposal Two-Player

Search


Results

BOO II

References

Lyapunov Analysis

Theorem: Given correct choice of adaptive gains Ky and Kr ,the error state vector, e(k) with closed loop time derivative e, isuniformly ultimately bounded, and the state y will convergeto a neighborhood of r (proof in (Ogunmolu et al., 2017,§V.A)).

Choose

V(e, Ky , KT

r ) = eTPe + tr(KT

y Γ−1y K

T

y |Λ|) + tr(KT

r Γ−1r Kr |Λ|)

Neural network model f (y) = ΘTΦ(y) + εf (y)


Publications

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BOO

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3-DOF Control


Deformation

Multi-DOFKinematics

BOO

Proposal Two-Player

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Results

BOO II

References

Proof Main Matter

V(e, Ky , KTr ) = −eTQe− 2eTPBΛεf

V(e, Ky , KTr ) ≤ −λlow‖e‖2 + 2‖e‖‖PB‖λhigh(Λ)εmax

Term Contributions

KTy y keeps y ∈ BR stable; K

Tr r reference tracking

f (y,u) ensures states starting outside set y ∈ BR

converge to BR in finite time


Publications

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BOO

Immobilization

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3-DOF Control


Deformation

Multi-DOFKinematics

BOO

Proposal Two-Player

Search


Results

BOO II

References

Proof Main Matter

λlow , λhigh ≡ minimum and maximum characteristic rootsof Q and Λ respectively

V(·) is thus negative definite outside the compact set:

χ =(

e : ‖e‖ ≤ 2‖PB‖λhigh(Λ)εmax (y)λlow (Q)

)Therefore, the error e is uniformly ultimately boundedy(t)→ 0 as t →∞


Publications

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BOO

Immobilization

Experiment Setup

Identification

3-DOF Control


Deformation

Multi-DOFKinematics

BOO

Proposal Two-Player

Search


Results

BOO II

References

Results

Choose

P =

−1705002668 0 00 −170500

2668 00 0 −170500

2668

and set

B =

1 1 0 0 0 00 0 1 1 0 00 0 0 0 1 1


Publications

Background




BOO

Immobilization

Experiment Setup

Identification

3-DOF Control


Deformation

Multi-DOFKinematics

BOO

Proposal Two-Player

Search


Results

BOO II

References

Results

10 20 30 40 50 60 70 80time (secs)

5

10

15

20

heig

ht (m

m) z-motion correction

z-setpoint

z

10 20 30 40 50 60 70 80time (secs)

0.5

1

1.5

2

pitch angle

(deg)

pitch motion correction

pitch set-point

pitch motion20 40 60 80 100 120 140 160 180

time (secs)

70

72

74

76

78

80

ro

ll (

de

g)

roll motion correction

roll setpoint

roll motion

[Left]: Goal command: (z , θ, φ) = (2.5mm, 0.25, 35) to(14mm, 1.6, 45)T . [Right]: Head roll tracking.


Publications

Background




BOO

Immobilization

Experiment Setup

Identification

3-DOF Control


Deformation

Multi-DOFKinematics

BOO

Proposal Two-Player

Search


Results

BOO II

References

Model of a 6-DOF SoRo Mechanism

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BOO

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3-DOF Control


Deformation

Multi-DOFKinematics

BOO

Proposal Two-Player

Search


Results

BOO II

References

Existing Modeling Approaches

Finite element modeling: Nesme et al. (2005, 2006); Bern et al. (2017); Gent (2012)

Constant curvature approaches: Hannan and Walker (2003, 2000); Jones and Walker (2006)

Piecewise constant curvature model: Jones and Walker (2006)

Cosserat brothers’ beam theory: Renda et al. (2014); Trivedi et al. (2008)

Non-constant curvature approaches

Continuum approximation of hyper-redundant sytems e.g. Mochiyama (2005); Chirikjianand Burdick (1995); Chirikjian (1994),

Spring-mass models for semi-rigid robots: Yekutieli et al. (2005); Zheng et al. (2012),

Geometric continuum models: Boyer et al. (2006); Gent (2012); Ogden (1997); Sedal et al.(2018); Holzapfel et al. (2000); Rucker et al. (2010); Demirkoparan and Pence (2007)


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BOO

Immobilization

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3-DOF Control


Deformation

Multi-DOFKinematics

BOO

Proposal Two-Player

Search


Results

BOO II

References

A Continuum Mechanics Model for IABDeformation

Context: Model-based approaches generally give bettermaterial responses

Contributions

Finite elastic (geometric continuum) model for IABdeformation

Component stresses, internal pressurization, and particlepositions/velocities

Synthesis of IAB contact velocities and head motion:manipulation dynamics, selection maps, and contact forces


Publications

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BOO

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3-DOF Control


Deformation

Multi-DOFKinematics

BOO

Proposal Two-Player

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Results

BOO II

References

IAB Kinematics


Publications

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BOO

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3-DOF Control


Deformation

Multi-DOFKinematics

BOO

Proposal Two-Player

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Results

BOO II

References

Fibers and deformation

The rate of deformation from a configuration B0 to acurrent configuration B in component form

dxi =∂xi∂Xα

dXα, with invariant form dx = FdX

for an observer O in e.g. basis Eα

A material line element (a fiber) dX at a point X areparticles lying along dX at a point X of a soft body

dX 6= 0 =⇒ FdX 6= 0 for all dX 6= 0. Therefore, F mustbe a non-singular tensor, imposing the restriction,det F 6= 0


Publications

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BOO

Immobilization

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Identification

3-DOF Control


Deformation

Multi-DOFKinematics

BOO

Proposal Two-Player

Search


Results

BOO II

References

Deformation Analysis of a Soft Continuum Robot

with

Ri ≤ R ≤ Ro , 0 ≤ Θ ≤ 2π, 0 ≤ Φ ≤ πri ≤ r ≤ ro , 0 ≤ θ ≤ 2π, 0 ≤ φ ≤ π (6)


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BOO

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3-DOF Control


Deformation

Multi-DOFKinematics

BOO

Proposal Two-Player

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Results

BOO II

References

Deformation Analysis

An isochoric homogeneous deformation implies

4

3π(R3 − R3

i

)=

4

3π(r3 − r3

i

), θ = Θ φ = Φ

r3 = R3 + r3i − R3

i , θ = Θ, φ = Φ (7)


Publications

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BOO

Immobilization

Experiment Setup

Identification

3-DOF Control


Deformation

Multi-DOFKinematics

BOO

Proposal Two-Player

Search


Results

BOO II

References

Stored Energy Invariants and Principal Ratios

Spherically symmetric deformation implies coincidence ofthe Lagrangean and Eulerian axes

Thus, principal ratios along azimuthal and zenith axes isλθ = λφ = r/R

We have λrλφλθ = 1 from the incompressibility of the IAB

material. Thus, λr = R2

r2 so that

I1 = λ2r + λ2

φ + λ2θ, and I2 = λ−2

r + λ−2φ + λ−2

θ . (8)


Publications

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BOO

Immobilization

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Identification

3-DOF Control


Deformation

Multi-DOFKinematics

BOO

Proposal Two-Player

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Results

BOO II

References

Strain Tensor and Deformation Gradient

Mooney-Rivlin strain energy form for small deformations:

W =1

2C1(I1 − 3) +

1

2C2(I2 − 3). (9)

Deformation gradient in spherical polar coordinates

F = λrer ⊗ eR + λφeφ ⊗ eΦ + λθeθ ⊗ eΘ

F =R2

r2er ⊗ eR +

r

Reφ ⊗ eΦ +

r

Reθ ⊗ eΘ. (10)

B = F FT := Left Cauchy-Green deformation tensor

C = FTF := Right Cauchy-Green deformation tensor


Publications

Background




BOO

Immobilization

Experiment Setup

Identification

3-DOF Control


Deformation

Multi-DOFKinematics

BOO

Proposal Two-Player

Search


Results

BOO II

References

Stress Laws and Constitutive Equations

Stress distribution on the internal continuum’s differential surface, dS .


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BOO

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3-DOF Control


Deformation

Multi-DOFKinematics

BOO

Proposal Two-Player

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Results

BOO II

References

Invariants of Deformation

I1 = tr(C) =R4

r4+

2 r2

R2; I2 = tr

(C−1

)=

r4

R4+

2R2

r2. (11)

For a constrained elastic material, we have the followingconstitutive relation

σ = G(F) + qF∂Λ

∂F(F)

= G(F)− pFF−Tdet(F)

= G(F)− pI (12)


Publications

Background




BOO

Immobilization

Experiment Setup

Identification

3-DOF Control


Deformation

Multi-DOFKinematics

BOO

Proposal Two-Player

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Results

BOO II

References

Cauchy stress and hydrostatic pressure

In terms of the stored strain energy, we have the stress tensorfield as

σ =

σrr σrφ σrθσφr σφφ σφθσθr σθφ σθθ

=∂W

∂FFT − pI, (13)

or

σ = C1B− C2C−2 − pI (14)

where C1,C2 are appropriate choices of the IAB materialmoduli;

σrr = −p + C1R4

r4− C2

r8

R8(15a)

σθθ = σφφ = −p + C1r2

R2− C2

R8

r8(15b)


Publications

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Contact-Free BVP for IAB Deformation

Consider an IAB with boundary conditions,

σrr |R=Ro = −Patm, σrr |R=Ri = −Patm − P (16)

If the stress components, σij , satisfy hydrostatic equilibrium,equilibrium equations for the body force b′s physical componentvectors, br , bθ, bφ are

−br =1

r2

∂

∂r(r2σrr ) +

1

r sinφ

∂

∂φ(sinφσrφ) +

1

r sinφ

∂

∂θ(σrθ)−

1

r(σθθ + σφφ) (17a)

−bφ =1

r3

∂

∂r(r3σrφ) +

1

r sinφ

∂

∂φ(sinφσφφ) +

1

r sinφ

∂

∂θ(σθφ)−

cotφ

r(σθθ) (17b)

−bθ =1

r3

∂

∂r(r3σθr ) +

1

r sin2 φ

∂

∂φ(sin2

φσθφ) +1

r sinφ

∂

∂θ(σθθ) (17c)

Cauchy’s 1st law of motion

div σT + ρb = ρv (18)


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Stress at Hydrostatic Equilibrium

Equilibrium therefore implies that

div σ = 0

1

r

∂

∂r(r2σrr ) = (σθθ + σφφ)

Whereupon,

P = 2C1

∫ R

Ri

(1

r− R6

r7

)dR + 2C2

∫ R

Ri

(r5

R6− R10

r11

)dR

≡∫ r

ri

[2C1

(r

R2− R4

r5

)+ 2C2

(r7

R8− R8

r9

)]dr .

(19)

(19) completely determines the deformation kinematics ofthe IAB material at rest.


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Example I: IAB Deformation (Extension)

Deformation Parameters

C1 = 11, 000 C2 = 22, 000 Ri = 10cm, ri = 13cmR = 15cm r = 16.60cm P = 14.52psi


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Example I: Deformation (Extension) Results

Mesh Time: 0.8838s Total Time: 4.7782sν = 0.45 ρ = 9.8446× 10−4kG/m3


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Example II: IAB Deformation (Extension)


C1 = 500, 000 C2 = 1, 000, 000 Ri = 7.5cm, ri = 12cmR = 10cm r = 13.21cm P = 14.5193psi


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Example II: Deformation Results (Extension)

Mesh Time: .9143s Total Time: 4.1445sν = 0.4995 ρ = 10−4 kg/m3


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Example III: IAB Deformation (Compression)


C1 = 500, 000 C2 = 1, 200, 000 Ri = 12cm, ri = 10cmR = 15cm r = 13.83cm P = −27.3631 psi


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Example III: Deformation Results (Compression)

Mesh Time: .8625s Total Time: 4.5338sν = 0.45 ρ = 12× 10−4 kg/m3


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Example IV: IAB Deformation (Compression)


C1 = 1.1e12 C2 = 2.2e10 Ri = 10cm, ri = 8cmR = 19cm r = 18.54cm P = −27.3631psi


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Example IV: Deformation Results (Compression)

Mesh Time: 0.823576s Total Time: 4.5098sν = 0.495 ρ = 2.0× 10−5 kg/m3


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Multi-DOF IAB Kinematics & Dynamics

OutlineSolve boundary-value problem for IAB kinematics withhead contact

Relate Deformation Kinematics to Contact Dynamics

Derive head velocity and orientation in terms of contactvelocities

Derive Newton-Euler’s Dynamics of the Head-IAB systemfor control


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Contact Kinematics

All friction cones lie within a softcontact type model: Nguyen (1988)

FC = fc ∈ Rn : ‖f tcij ‖ ≤ µij‖fnci‖,

i = 1, . . . , k, j = 1, . . . ,mi(20)

f tcij= tangent component of jth element

of contact force

f nci= i th contact’s normal force, and µij

is fcij ’s coefficient of friction

Contact force within friction cone:

Fci =

[I 00 nci

] [fciτci

], (21) IAB-Head Soft Contact


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Contact Map

Soft contact type is the map

Gi (rci , ξh) =

[I 0

ω(rci ) I

]Bi (ξh, ξr ) (22)

Head never rolls out of convex sum of conic forces ofindividual IABs

For multiple IABs acting on the head, resultant head forceis a superposition of individual IAB forces

Fh =[G1, . . . ,G8

]( Fc1

Fc8

)= GFc , (23)

where Fh ∈ R6 and Fc ∈ Rm1 × Rm2 × . . .× Rm8


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BVP for IAB-Head Dynamics

Velocity constraint dual(vciωci

)=

[I ω(rci )0 I

](vchωch

). (24)

If vc is the conjugate velocity to fc , then forces exerted bythe fingers are (

vcωc

)= GT

(vhωh

)(25)

Equations of motion for IAB continuum

ρ+ ρdivv = 0, (26a)

σT = σ, (26b)

divσT + ρb = ρv (26c)


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IAB Forces under Entropy

Head forces (see derivation in (Ogunmolu, 2019, Appendix C.)) are in part the internal pressurization, Pi ,and the stress tensor components σφφ(ε), σθθ(ζ),

P =

∫ r

ri

[1

r

(−2p + 2C1

r2

R2− 2C2

R8

r8

)− ρbr + ρ cos θ

(2rφ cos θ + r cos θφ− 2r θφ sin θ

)−ρ sinφ

(cos θ(−r + r θ2 + rφ2) + sin θ(2r θ + r θ)

) ]dr (27a)

σφφ(ε) = −∫ πε

[rρ[

cosφ(

2r θφ cos θ + (2rφ + rφ) sin θ)

+

sin θ(

2r θ cos θ + r θ cos θ + (r − r θ2 − rφ2))

sinφ]− ρrbθ

]dφ, 0 ≤ ε ≤ π (27b)

σθθ(ζ) = −∫ 2π

ζ

[−rρbθ sinφ + rρ sinφ cosφ

(r − rφ2

)− rρ sin2

φ(

2rφ + rφ)]

dθ, 0 ≤ ζ ≤ 2π

(27c)

where 0 ≤ ε ≤ π, 0 ≤ ζ ≤ 2π and gravity forces.


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Piola-Kirchoff Stress Tensor & Contact Forces

For an infinitesimal vector element dA in B0, we havedA = NdA for a surface dA with outward normal map N

Similarly, in configuration B, da = n da for an outwardnormal map n

Therefore on the IAB boundary, volume preservationimplies that ∫

∂Bσ n da =

∫∂B

J σH N dA. (28)

Define the Piola-Kirchoff stress tensor field

S = J HT σ where H = F−T (29)

It follows thatσda = STdA.


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Contact force and stress fields

Contact Wrench

fci = STi dAi = JiσiHidAi = JiσiF

−1i dAi (30)

τci = fci × rci (31)

whereupon,

fci =

(R2i

r2i

Pi +Ri

riσφφi (ε) +

Ri

riσθθi (ζ)

)ncidAi (32)

and the contact force map is

Fci =

[I 00 nci

] [fci

fci × rci

]. (33)


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Contact coordinates and head motion

α1 = (u1, v1) ∈ U1, and αh = (uh, vh) ∈ Uh

fi (ui , vi ) : U → Si ⊂ R3|i = 1, h.


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Differential Geometry of Contact Coordinates

Define contact coordinates η = (α1, αh, ψ)

Let g ∈ Ω ⊂ SE(3) and let R ∈ SO(3) be g ’s rotatory component

Ω = S1in1i=1, Shi

nhi=1 for which the IAB and head remain in contact

At a point of contact, η must satisfy

g f1(α1) = fh(αh) (34a)

R n1(α1) = −nh(αh) (34b)

Additionally, the orientation of the tangent planes of α1 and αh imply that

R∂f1

∂α1

M−11 Rψ =

∂fh

∂αh

M−1h (35)

Hence, we have the contact equations (see derivation in Ogunmolu (2019))

αh = M−1h (Kh + K1)−1

(ωt − K1vt

)(36a)

α1 = M−11 Rψ(Kh + K1)−1 (ωt −Khvt ) (36b)

ψ = ωn + ThMhαh + T1M1α1 (36c)


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Contact Equations

where

Mi =

‖ ∂fi∂ui ‖ 0

0 ‖ ∂fh∂vi‖

, Rψ =

[cosψ − sinψ− sinψ − cosψ

](37a)

Th = yTh∂xh

∂αh

M−1h , T1 = yT1

∂x1

∂α1

M−11 , ωn = zTh ω (37b)

Kh =[xTh , yTh

]T ∂nTh

∂αh

M−1h , K1 = Rψ

[xT1 , yT1

]T ∂nT1

∂α1

M−11 Rψ (37c)

ωt =[xTh , yTh

]T [nh × ω

]T, vt =

[xTh , yTh

]T [(−fh × ω + v)

]T. (37d)


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Multi-IAB Kinematics

IAB configuration space with respect to the spatial frameat a certain time can then be described bygst(r) : r→ gst(r) ∈ SE (3)

Strain state of the IAB is characterized by the strain field

ξi (r) = g−1i

∂gi∂r∈ se(3) = g−1

i g ′i (38)

g ′i : tangent vector at gi such that g ′i ∈ Tgi (r)SE(3)

IAB’s strain field as an exponential map of SE (3)

gi (r) = exp‖r‖ξi = I + ξi ‖r‖+ω

‖ω‖2(1− cos(‖r‖‖ω‖)) ξ2

i

+ω3

‖ω‖3(‖r‖‖ω‖ − sin(‖r‖‖ω‖)) ξ3

i . (39)


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Forward Kinematics

FK Jacobian:(viabiωiabi

)=∂Kiabi

∂ri

dr

dtK−1iabi

= Ji (ri )ri (40)

Contact forces/velocities mapped by the contact Jacobian:

Jci (ξh, ξiabi ) =

[I ω(rci )0 I

]Jri , (41)

where Jci : ξri →[vTci , ωT

ci

]Tξr = (ξr1 , ξr2 , · · · , ξr8): positions and orientations for eachof the 8 IABs


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Manipulation Map

For a selection matrix BTi (ξh, ξiabi ) ∈ Rm

i for a particularmanipulation task

GTi (ξh, ξiabi )ξh = BT

i (ξh, ξiabi )Jci (ξh, ξri )ξiabi (42)

Manipulation constraint:GT

1

GT2...

GT8

(

vhωh

)=

BT

1 Jc1 0 · · · 00 BT

2 Jc2 · · · 0...

.... . .

...0 0 · · · BT

8 Jc8

riab1

riab2

...riab8

,

(43)


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Planar Manipulation Example

G2 =

I 0

ω

−rc2 sinφ2

rc2 cosφ2

0

I

1 00 10 00 00 00 0

G1 =

[I 0

ω

−rc1sinφ1

−rc1cosφ10

I

] 1 00 10 00 00 00 0


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Head Planar Manipulation Map

G3 =

I 0

ω

rc3 sinφ3

rc3 cosφ3

0

I

1 00 10 00 00 00 0

G(x, y, φ) =

1 0 rc1cosφ1

0 1 −rc1sinφ1

1 0 −rc2cosφ2

0 1 −rc2sinφ2

1 0 −rc3cosφ3

0 1 rc3sinφ3

1 0 rc4cosφ4

0 1 rc4sinφ4

T

where G (·) is the manipulation map for all forces with respectto xy coordinates.


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Multi-IAB Lagrangian Dynamics

L(r, r) = T (r, r)− V (r). (44)

Pneumatic system equation

d

dt

∂L

∂ri− ∂L

∂ri= τi , i = 1, . . . ,m (45)

Define the Eulerian strain rate tensor

Γ = grad v(r, t). (46)

Dropping explicit time-dependence, we have fromCauchy’s first law

div (σTv)− tr(σΓ) + ρb · v = ρv · v. (47)


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Balance of mechanical energy

∫Bρb · v dv +

∫∂B

fρ · v da =d

dt

∫B

1

2ρv · v dv +

∫B

tr(σΓ) dv

(48)

Taking cognizance that Σ = 12 (Γ + ΓT ), we have

T (r, r) =1

2ρv · v, V (r) = tr(σΣ). (49)

whereupon, we find that

τ =

[ρ 0 0

0 ρ r2 0

0 0 ρ r2 sin2 φ

] [r

φ

θ

]+ diag

[2 ρ r

(θ sin2 φ + φ

)ρr(r θ sin 2φ− φ

)−ρr θ sinφ (r cosφ + sinφ)

][r

φ

θ

](50)

Compactly, we write the IAB actuator dynamics as

Miabj (rj , φj )rj + Ciabj (rj , φj , θj , φj)rj = τj (51)Lekan Ogunmolu

A Multi-DOF Soft Robot Mechanism for Patient Motion Correction and Beam Orientation Selection in Cancer Radiation Therapy.

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Newton-Euler Equations for IAB-Head System

No actuator torques:

Mh(ζ)ζ + Ch(ζ, ζ)ζ + Nh(ζ, ζ) = 0 (52)

Manipulation constraint:

GT (ζ, r)ζ = J(ζ, r)r. (53)

Wherefore, we find that(d

dt

∂L

∂ζ− ∂L

∂ζ

)δζ + GJ−T

(d

dt

∂L

∂r− ∂L

∂r

)= GJ−Tτ

(54)

Equations (54) and (53) completely describe the system.


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BOO, FMO

Aim high-precision, geometrically-shaped photons to tumors

Reprinted from David Shepard’s AAPM Slides

Beamlets from photons; optimal beam angles; FMO process Bintensity modulation


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Existing Approaches

Stochastic optimization approaches (SA, GA): Bortfeld andSchlegel (1993); Sodertrom and Brahme (1993); Pugachevet al. (2000); Pugachev and Xing (2002); Aleman et al. (2008);Bertsimas et al. (2013)

Gradient search: Stein et al. (1997); Craft (2007); Bertsimaset al. (2013)

Feature-based machine learning: Lu et al. (2006); Li and Lei(2010)

Mixed-integer LP, branch and cut, beam angle eliminationalgorithms: Wang et al. (2003); D D’Souza et al. (2004); Limet al. (2007); Jia et al. (2011)


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An ADP + Monte-Carlo Evaluation Proposal

Context: For 180 discretized angles in a 5 beam plan,there are 188, 956, 800, 000 possible search directions

Proposal

Monte-Carlo game planning strategy

Deep neural network policy: map patients geometry tobeam angles

Refine accuracy of beams selection policy with fictitiousself-play [Heinrich et al. (2015)]


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State Representation: Network Input Plane

0Net policy: produces a subjective probability distribution about a rational decision-making agent’s

preference for a lottery (or value) in an uncertain environment. With new information, decision-maker’ssubjective probability distribution gets revised. Repeatedly sampling from this probability distribution enablesthe transition between episode contexts, i.e., xk → xk+1.


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Two-Player Framework

Players base their decisions on a random event’s outcome

Guided by a nonstationary Markovian policy setΠpi = Πp1 ,Πp2 such that

πp1 ∈ πp1

0 , πp1

1 , . . . , πp1

T ⊆ Πp1

πp2 ∈ πp2

0 , πp2

1 , . . . , πp2

T ⊆ Πp2

Stochastic action selection strategyπ(u|x) := πp1 ,πp2 contain control sequencesup1

t 0≤t≤T and up2t 0≤t≤T


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Cost-to-go

Optimal cost-to-go value function for state

V ∗t (x) = infπp1∈Πp1

supπp2∈Πp2

E

[T−1∑t=i

Vt(x0, f (xt , πp1 , πp2))

],

x ∈ X; V ∗T (x) = 0, ∀ x ∈ X

Each player generates a mixed strategy determined byaveraging the outcome of individual plays

Find optimal saddle point control pair up∗1t , u

p∗2t such that

V ?p1≤ V ?

t ≤ V ?p2∀ πp1

t , πp2t 0≤t≤T .


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Game Tree Simulation

Network roll-out policy guides a tree’s game, Γ, toward abest-first set of beam angle candidates

Essentially, a sampling-based lookout algorithm

Focus on state space regions with least FMO score forbeam angle combinations

Lookout simulation steps: Selection; Expansion;Simulation; Back-up

A ‘best move’ for current beam block selected, after eachiteration

0Selection: from root node, recursively apply child selection policy to navigate tree branches until an

expandable node is encountered. Expansion: iteratively add one or more children to the current node, basedon the available move probabilities.


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Fluence Map Optimization (FMO)

Suppose X is the total discretized VOI ’s in a targetvolume

Suppose B1 ∪ B2 ∪ . . . ∪ Bn ⊆ B represents the partitionsubset of a beam B

Suppose further that Dij(θk) is the matrix that describeseach dose influence, di .

Dij(θk) is computed for each dose to voxel i occupying abixel, j , incident from a beam angle, θk at every 360/ϕ.NB: j ∈ θk


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The FMO Problem

Pre-calculated dose term:Ax =

∑swsvsDs

ijxs | Dsij ∈ Rn×l , n l

Find decision variable xj that maximizes dose to tumor,and minimizes dose to critical structures and body tissuesfor all k ∈ 1, . . . , n

min1

vs

∑s∈OARs

‖(bs − w sDsijxs)+‖2

2 +1

vs

∑s∈PTVs

‖(wsDsijxs − bs)+‖2

2

subject to x ≥ 0.

Restated as

min1

2‖Ax− b‖2

2 subject to x ≥ 0.


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FMO Lagrangian

Lagrangian:

L(x,λ) =1

2‖Ax− b‖2

2 − λTx.

Introduce the auxiliary variable z, we have

minx

1

2‖Ax− b‖2

2, subject to z = x, z ≥ 0,


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FMO Primal and Dual Updated

Solving the x and z sub-problems, we have

x, z-updates

xk+1 =(

ATA + ρI)−1 (

ATb + ρzk − λk)

zk+1 = Sλ/ρ

(xk+1 + λk

)where Sλ/ρ(τ) = (x− λ/ρ)+ − (−τ − λ/ρ)+, and

λk+1 = λk − γ(zk+1 − xk+1),

with γ as the step length controlling parameter.


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Results: Dose Wash Plot


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Results: Dose Wash


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Previously Proposed Future Work (Last Fall)

Supervised Pretraining

For example, PlanIQ or Column generation to eliminateplan quality gap and save training process time

Kalman filtering of predictions from neural network policyto obtain stable probabilities

Robust policy improvement of pre-training anglepredictions

e.g. Monte-Carlo Tree Search or Graph ConvolutionalNetworks and guided tree search [Zhuwen et al. (2018)].


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Supervised Pre-Training of Deep BOO Policy

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Generative MDP Prior/State Representation

Generative model to find near-optimal beams in largestate-space MDP: Sadeghnejad Barkousaraie et al.(2019a,b)


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Generative MDP Prior

For a subset Blim of discretized beams set, B, find optimalbeams set B?lim via greedy linear programming strategy

How?Iteratively add beams to Blim until |Blim|, reaches athreshold, Blim

Or when the master problem’s optimality condition is met

Add beams only if it reduces the current objectivefunction’s value from a previous iteration


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Column Generation Procedure

A deep policy (base policy) parameterizes the columngeneration functional mapping

Beam orientation input is a zero-initialized vector whoseparticular indices are complemented over the course ofbackpropagation

Base policy minimizes MSE between CG prediction, xcgk+1,

and KKT targets xkktk+1

Policy outputs a probability distribution which informs thebeam index to which a new beamlet xk+1 is inserted atnext iteration


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Policy Recovery

Search task is to recover a policy p(uk |xk ;µ) or πµ(uk |xk)

Simulate state-action cost-to-go approximation of runningbeam trajectories by drawing samples from

Qk(xk , uk) = E [gk(xk , uk ,wk) + Jk+1(xk + 1)] (55)

Average Qk(xk , uk) costs of each control uk ∈ Uk(xk) toobtain an estimate Qk(xk , uk)

Find approximate rollout control µk(xk) from theminimization

µk(xk) = arg minuk∈Uk (xk )

Qk(xk , uk). (56)

0Policy distributed over controls, uk , conditioned on the states, xk , and parameterized by a rollout

(tree) policy, µ. Jk+1(xk + 1) denotes the cost of running the base policy starting from xk+1, andgk (xk , uk ,wk ) is the cost of running the current tree policy.


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Monte-Carlo Tree Search Stages

Simulate single agent games from x0 up to a fixed depthwith child selection policy

As search progresses towards convergence given simulationlength, bound worst possible bias by a quantity thatconverges to zero

Rewrite (55) as

µ?k(xk) = µk(xk)n ± µjK∑j=1

E[Tj(n)] (57)

UCT single player:

Q(xk , uk) = Qj(xk , uk) + cp · π(xk)

√ln Np

1 + Nc+ κ (58)

0Child selection policy drawn from µ0, . . . , µN−1.


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Results: Dose distribution on select prostate cases


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DVH curves comparison


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References

Arthur L Boyer, Timothy G Ochran, Carl E Nyerick, Timothy J Waldron, and Calvin J Huntzinger. Clinicaldosimetry for implementation of a multileaf collimator. Medical physics, 19(5):1255–1261, 1992. 4

Steve Webb. Intensity-Modulated Radiation Therapy. Institute of Physics Publishing Ltd, Bristol andPhiladelphia, 2001. 4

Achim Schweikard, Rhea Tombropoulos, and John R Adler. Robotic Radiosurgery with Beams of AdaptableShapes. In Nicholas Ayache, editor, Computer Vision, Virtual Reality and Robotics in Medicine, pages138–149, Berlin, Heidelberg, 1995. Springer Berlin Heidelberg. ISBN 978-3-540-49197-2. 6

Steve Webb. Conformal intensity-modulated radiotherapy (imrt) delivered by robotic linac-testing imrt to thelimit? Physics in Medicine & Biology, 44(7):1639, 1999. 6

Rodney D Wiersma, Zhifei Wen, Meredith Sadinski, Karl Farrey, and Kamil M Yenice. Development of aframeless stereotactic radiosurgery system based on real-time 6d position monitoring and adaptive headmotion compensation. Physics in Medicine & Biology, 55(2):389, 2009. 7

Christian Herrmann, Lei Ma, and Klaus Schilling. Model Predictive Control For Tumor Motion CompensationIn Robot Assisted Radiotherapy. IFAC Proceedings Volumes, 44(1):5968–5973, 2011. 8

Olalekan Ogunmolu, Adwait Kulkarni, Yonas Tadesse, Xuejun Gu, Steve Jiang, and Nicholas Gans.Soft-neuroadapt: A 3-dof neuro-adaptive patient pose correction system for frameless and masklesscancer radiotherapy. In IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS),Vancouver, BC, CA, pages 3661–3668. IEEE, 2017. 26

Matthieu Nesme, Maud Marchal, Emmanuel Promayon, Matthieu Chabanas, Yohan Payan, and FrancoisFaure. Physically Realistic Interactive Simulation For Biological Soft Tissues. Recent ResearchDevelopments in Biomechanics, 2:1–22, 2005. 32

Matthieu Nesme, Francois Faure, and Yohan Payan. Hierarchical Multi-Resolution Finite Element Model forSoft Body Simulation. Lecture Notes in Computer Science, 4072:40–47, 2006. 32

James M Bern, Grace Kumagai, and Stelian Coros. Fabrication, modeling, and control of plush robots. InIntelligent Robots and Systems (IROS), 2017 IEEE/RSJ International Conference on, pages 3739–3746.IEEE, 2017. 32


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References

Alan Gent. Engineering with Rubber. How to Design Rubber Components. Carl Hanser Verlag Publicationbs,Munich, 2012. 32

Michael W Hannan and Ian D Walker. Kinematics and the Implementation of an Elephant’s TrunkManipulator and Other Continuum Style Robots. Journal of Robotic Systems, 20(2):45–63, 2003. 32

Michael W Hannan and Ian D. Walker. Novel Kinematics for Continuum Robots. pages 227–238. Advancesin Robot Kinematics, 2000. 32

Bryan A Jones and Ian D Walker. Kinematics for Multisection Continuum Robots.pdf. 22(1):43–55, 2006. 32

Federico Renda, Michele Giorelli, Marcello Calisti, Matteo Cianchetti, and Cecilia Laschi. Dynamic model ofa multibending soft robot arm driven by cables. IEEE Transactions on Robotics, 30(5):1109–1122,2014. 32

Deepak Trivedi, Amir Lotfi, and Christopher D Rahn. Geometrically Exact Models For Soft RoboticManipulators. IEEE Transactions on Robotics, 24(4):773–780, 2008. 32

Hiromi Mochiyama. Hyper-flexible robotic manipulators. In IEEE International Symposium onMicro-NanoMechatronics and Human Science, 2005, pages 41–46. IEEE, 2005. 32

Gregory S Chirikjian and Joel W Burdick. The kinematics of hyper-redundant robot locomotion. IEEEtransactions on robotics and automation, 11(6):781–793, 1995. 32

Gregory S Chirikjian. Hyper-redundant manipulator dynamics: A continuum approximation. AdvancedRobotics, 9(3):217–243, 1994. 32

Yoram Yekutieli, Roni Sagiv-Zohar, Ranit Aharonov, Yaakov Engel, Binyamin Hochner, and Tamar Flash. Adynamic model of the octopus arm. i. biomechanics of the octopus reaching movement. Journal ofneurophysiology, 2005. 32

Tianjiang Zheng, David T Branson, Rongjie Kang, Matteo Cianchetti, Emanuele Guglielmino, MaurizioFollador, Gustavo A Medrano-Cerda, Isuru S Godage, and Darwin G Caldwell. Dynamic continuum armmodel for use with underwater robotic manipulators inspired by octopus vulgaris. In 2012 IEEEInternational Conference on Robotics and Automation, pages 5289–5294. IEEE, 2012. 32


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Frederic Boyer, Mathieu Porez, and Wisama Khalil. Macro-continuous computed torque algorithm for athree-dimensional eel-like robot. IEEE Transactions on Robotics, 22(4):763–775, 2006. 32

R.W. Ogden. Non-linear Elastic Deformations. Dover Publicationbs, Inc., Mineola, New York, 1997. 32

Audrey Sedal, Daniel Bruder, Joshua Bishop-Moser, Ram Vasudevan, and Sridhar Kota. A continuum modelfor fiber-reinforced soft robot actuators. Journal of Mechanisms and Robotics, 10(2):024501, 2018. 32

Gerhard A Holzapfel, Thomas C Gasser, and Ray W Ogden. A new constitutive framework for arterial wallmechanics and a comparative study of material models. Journal of elasticity and the physical science ofsolids, 61(1-3):1–48, 2000. 32

D Caleb Rucker, Bryan A Jones, and Robert J Webster III. A geometrically exact model for externally loadedconcentric-tube continuum robots. IEEE transactions on robotics: a publication of the IEEE Roboticsand Automation Society, 26(5):769, 2010. 32

Hasan Demirkoparan and Thomas J Pence. Swelling of an Internally Pressurized Nonlinearly Elastic Tubewith Fiber Reinforcing. International journal of solids and structures, 44(11-12):4009–4029, 2007. 32

Van-Duc Nguyen. Constructing force-closure grasps. The International Journal of Robotics Research, 7(3):3–16, 1988. 54

Olalekan Patrick Ogunmolu. A Multi-DOF Soft Robot Mechanism for Patient Motion Correction and BeamOrientation Selection in Cancer Radiation Therapy. PhD thesis, The University of Texas at Dallas,2019. 57, 61

Thomas Bortfeld and Wolfgang Schlegel. Optimization of beam orientations in radiation therapy: Sometheoretical considerations. Physics in Medicine & Biology, 38(2):291, 1993. 73

Svante Sodertrom and Anders Brahme. Optimization of the Dose Delivery In A Few Field Techniques UsingRadiobiological Objective Functions. Medical physics, 20(4):1201–1210, 1993. 73

AB Pugachev, AL Boyer, and L Xing. Beam orientation optimization in intensity-modulated radiationtreatment planning. Medical Physics, 27(6):1238–1245, 2000. 73

Andrei Pugachev and Lei Xing. Incorporating prior knowledge into beam orientaton optimization in imrt.International Journal of Radiation Oncology* Biology* Physics, 54(5):1565–1574, 2002. 73


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Dionne M Aleman, Arvind Kumar, Ravindra K Ahuja, H Edwin Romeijn, and James F Dempsey.Neighborhood Search Approaches to Beam Orientation Optimization i Intensity Modulated RadiationTherapy Treatment Planning. Journal of Global Optimization, 42(4):587–607, 2008. 73

Dimitris Bertsimas, Valentina Cacchiani, David Craft, and Omid Nohadani. A Hybrid Approach To BeamAngle Optimization In Intensity-modulated Radiation Therapy. Computers & Operations Research, 40(9):2187–2197, 2013. 73

Jorg Stein, Radhe Mohan, Xiao-Hong Wang, Thomas Bortfeld, Qiuwen Wu, Konrad Preiser, C Clifton Ling,and Wolfgang Schlegel. Number and orientations of beams in intensity-modulated radiation treatments.Medical Physics, 24(2):149–160, 1997. 73

David Craft. Local beam angle optimization with linear programming and gradient search. Physics inMedicine & Biology, 52(7):N127, 2007. 73

Renzhi Lu, Richard J Radke, Linda Hong, Chen-Shou Chui, Jianping Xiong, Ellen Yorke, and AndrewJackson. Learning The Relationship Between Patient Geometry And Beam Intensity In BreastIntensity-modulated Radiotherapy. IEEE transactions on biomedical engineering, 53(5):908–920, 2006.73

Yongjie Li and Jie Lei. A Feasible Solution To The Beam-angle-optimization Problem In RadiotherapyPlanning With A Dna-based Genetic Algorithm. IEEE Transactions on biomedical engineering, 57(3):499–508, 2010. 73

Chuang Wang, Jianrong Dai, and Yimin Hu. Optimization Of Beam Orientations And Beam Weights ForConformal Radiotherapy Using Mixed Integer Programming. Physics in Medicine & Biology, 48(24):4065, 2003. 73

Warren D D’Souza, Robert R Meyer, and Leyuan Shi. Selection Of Beam Orientations InIntensity-modulated Radiation Therapy Using Single-beam Indices And Integer Programming. Physicsin Medicine & Biology, 49(15):3465, 2004. 73

Gino J Lim, Michael C Ferris, Stephen J Wright, David M Shepard, and Matthew A Earl. An optimizationframework for conformal radiation treatment planning. INFORMS Journal on Computing, 19(3):366–380, 2007. 73


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Xun Jia, Chunhua Men, Yifei Lou, and Steve B. Jiang. Beam Orientation Optimization For IntensityModulated Radiation Therapy Using Adaptive L2,1-minimization. Physics in Medicine and Biology, 56(19):6205–6222, 2011. 73

Johannes Heinrich, Marc Lanctot, and David Silver. Fictitious self-play in extensive-form games. InInternational Conference on Machine Learning, pages 805–813, 2015. 74

Li Zhuwen, Qifeng Chen, and Vladlen Koltun. Combinatorial optimization with graph convolutional networksand guided tree search. In Advances in Neural Information Processing Systems. NeurIPS, 2018. 85

Azar Sadeghnejad Barkousaraie, Olalekan Ogunmolu, Dan Nguyen, and Steve Jiang. Using SupervisedLearning and Guided Monte Carlo Tree Search for Beam Orientation Optimization in RadiationTherapy. In Under Review at International Conference on Medical Image Computing and ComputerAssisted Intervention, XXII, 2019a. 87

Azar Sadeghnejad Barkousaraie, Olalekan Ogunmolu, Steve Jiang, and Dan Nguyen. A Fast Deep LearningApproach for Beam Orientation Selection Using Supervised Learning with Column Generation on IMRTProstate Cancer Patients. Medical Physics, American Association of Physicists in Medicine, 2019b. 87


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