Optimization over Manifolds: Applications to Robotics and Channel Layout Design, Slides of Robotics

Download Optimization over Manifolds: Applications to Robotics and Channel Layout Design and more Slides Robotics in PDF only on Docsity!

Optimization over Manifolds with applications to

Robotic Needle Steering and Channel Layout Design

Trajectory Optimization

Optimization over vector spaces n

Nonholonomy Examples

Car pulling trailers: 2  1  1 ^1 Bicycle:^2  1 ^1

Rolling Ball:? (^2)  SO (3)

C-Spaces as Manifolds

Manifold: Topological space that near each point resembles Euclidean space

Other examples:

Optimization over Manifolds

n

Optimization over Manifolds

n

Define projection operator from tangent space to manifold

: Incremental rotation to reference rotation defined in terms of axis-angle

Parameterization: Incremental

Rotations

 Why not directly optimize over rotation matrix entries?

 Over-constrained (orthonormality)  Larger number of optimization variables

 Define local parameterization in terms of incremental rotation

r

r

Projection Operator

r

[ ]

e

r

: Point on SO(3) that can be reached by traveling along the geodesic in direction

e [ ] r

r

0 0 0

[ ]

z y z x y x

  

r r r r r r

where^ r

0

X^1 k

k

e X

k





and   is the

matrix exponential operator

Steerable Needle

Steerable needle

Target (^) Bladder

Prostate

Pelvis

Skin

Cowper’s gland

Steerable needles inside phantom tissue

Steerable needles navigate around sensitive structures (simulated)

Steerable Needle

[Webster, Okamura, Cowan, Chirikjian, Goldberg, AlterovitzUnited States Patent 7,822,458. 2010]

Bevel-tip

Highly flexible

Reaction forces from tissue

Follows constant curvature paths

State (needle tip)

• Position: 3D
• Orientation: 3D

SE (3) : 3  SO (3)

Steerable Needle Plans

Results

Why is minimizing twist important?

Channel Layout: Opt Formulation

Results