Visual Servoing, Intro Optimal Control Lecture 12 What will you - PowerPoint PPT Presentation

Visual Servoing, Intro Optimal Control Lecture 12

What will you take home today? Visual Servoing Interaction Matrix Control Law Case-Study: Learning-based approach Introduction Optimal Control Principle of Optimality Bellman Equation Deriving LQR

Camera-Robot Configurations Image from: CHANG, W., WU, C.. Hand-Eye Coordination for Robotic Assembly Tasks. International Journal of Automation and Smart Technology ,

Image-based visual servoing Current Image Goal Image

Camera Motion to Image Motion ω x ω z v x v z ω y v y Slides adapted from Peter Corke

The Image Jacobian The Image Jacobian ω ω f = f f = f v ) T v ) T ( ˙ ( ˙ u, ˙ u, ˙ ˆ ˆ ρ ρ ( X, Y, Z ) T ( X, Y, Z ) T 0 0 1 1 v x v x v v v y v y ✓ ˙ ✓ ˙ B B C C ✓ − ˆ ✓ − ˆ uv/ ˆ uv/ ˆ − ( ˆ − ( ˆ f + u 2 / ˆ f + u 2 / ˆ ◆ ◆ ◆ ◆ B B C C u u f/Z f/Z 0 0 u/Z u/Z f f f ) f ) v v v z v z B B C C = = − ˆ − ˆ f + u 2 / ˆ f + u 2 / ˆ ˆ ˆ − uv/ ˆ − uv/ ˆ B B C C v v ˙ ˙ 0 0 f/Z f/Z v/Z v/Z f f f f − u − u ω x ω x B B C C B B C C ω y ω y @ @ A A ω z ω z Slides adapted from Peter Corke Slides adapted from Peter Corke

Optical flow Patterns Slides adapted from Peter Corke

Image-based visual servoing Getting a camera velocity to minimize the error between the current and goal image Current Image Goal Image

Image-based visual servoing Current Image Goal Image J ( u, v, Z ) 0 1 v x v y B C − ˆ uv/ ˆ − ( ˆ f + u 2 / ˆ ✓ ◆ ✓ ◆ B C u ˙ f/Z 0 u/Z f f ) v v z B C = − ˆ f + u 2 / ˆ ˆ − uv/ ˆ B C v ˙ 0 f/Z v/Z f f − u ω x B C B C ω y @ A ω z Slides adapted from Peter Corke

Image-based visual servoing     ˙ u 1 v x Current Image Goal Image ˙ v 1 v y       J ( u 1 , v 1 , Z 1 )     ˙ u 2 v z     J ( u 2 , v 2 , Z 2 ) =       ˙ v 2 ω x     J ( u 3 , v 3 , Z 3 )     ˙ u 3 ω y     ˙ v 3 ω z

Image-based visual servoing     ˙ u 1 v x ˙ v 1 v y       J ( u 1 , v 1 , Z 1 )     ˙ u 2 v z     = J ( u 2 , v 2 , Z 2 )       ˙ v 2 ω x     J ( u 3 , v 3 , Z 3 )     ˙ u 3 ω y     ˙ v 3 ω z     ˙ v x u 1 ˙ v y v 1 − 1       J ( u 1 , v 1 , Z 1 )     ˙ v z u 2     J ( u 2 , v 2 , Z 2 ) =       ˙ v 2 ω x     J ( u 3 , v 3 , Z 3 )     ˙ u 3 ω y     ˙ v 3 ω z

Desired Pixel Velocity Slides adapted from Peter Corke

Simulation Slides adapted from Peter Corke

Point Correspondences How to find them? Features, Markers

What will you take home today? Visual Servoing Interaction Matrix Control Law Case-Study: Learning-based approach Introduction Optimal Control Principle of Optimality Bellman Equation Deriving LQR

Training Deep Neural Networks for Visual Servoing Bateux et al. ICRA 2018 1. Instead of using features, use the whole image to compare to given goal image a. Challenge: Small convergence region due to non-linear cost function

Training Deep Neural Networks for Visual Servoing Bateux et al. ICRA 2018

Training Deep Neural Networks for Visual Servoing Bateux et al. ICRA 2018

What will you take home today? Visual Servoing Interaction Matrix Control Law Case-Study: Learning-based approach Introduction Optimal Control Principle of Optimality Bellman Equation Deriving LQR

So far on Control

Optimal Control and Reinforcement Learning from a unified point of view Optimal Control Problem

Trajectory Optimization and Reinforcement Learning 1. Trajectory Optimization: find a optimal trajectory given non-linear dynamics and cost 2. Reinforcement Learning: finding an optimal policy under unknown dynamics and given a reward = -cost

Principle of Optimality – Example: Graph Search problem

Forward Search

Forward Search

Backward Search

Backward Search

Principle of Optimality

Bellman Equation

Problem setup System Dynamics Cost function

Goal

Formalize Cost-to-Go / Value function

Optimal Value function = V with lowest cost

Deriving the Bellman Equation

Optimal Bellman Equation Optimal Value function Optimal Policy

Comparing Optimal Bellman and Value Function

Infinite time horizon, deterministic system

Infinite time horizon, deterministic system

Infinite time horizon, deterministic system

Finite Horizon, Stochastic system Cost function Stochastic System Dynamics

Finite Horizon, Stochastic system Value function Optimal Value function Optimal Policy

Finite Horizon, Stochastic system Bellman Equation Optimal Bellman Equation

Infinite Horizon, Stochastic system Combining formulation from infinite horizon - discrete system with stochastic system derivation

Continuous time systems Hamilton-Jacobi-Bellman Equation

How do you solve these equations?

Linear Dynamical Systems, Quadratic cost – L inear Q uadratic R egulator (LQR)

Linear Dynamical Systems, Quadratic cost – L inear Q uadratic R egulator (LQR)