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

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

Nov 25, 2022 286 likes •767 views

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

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Visual Servoing, Intro Optimal Control

Lecture 12

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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

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Camera-Robot Configurations

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

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Image-based visual servoing

Current Image Goal Image

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Camera Motion to Image Motion

vx vy vz ωz ωx ωy

Slides adapted from Peter Corke

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The Image Jacobian

ω v ˆ f = f ρ ( ˙ u, ˙ v)T (X, Y, Z)T

✓ ˙ u ˙ v ◆ = ✓ − ˆ f/Z u/Z uv/ ˆ f −( ˆ f + u2/ ˆ f) v − ˆ f/Z v/Z ˆ f + u2/ ˆ f −uv/ ˆ f −u ◆ B B B B B B @ vx vy vz ωx ωy ωz 1 C C C C C C A Slides adapted from Peter Corke

The Image Jacobian

ω v ˆ f = f ρ ( ˙ u, ˙ v)T (X, Y, Z)T

✓ ˙ u ˙ v ◆ = ✓ − ˆ f/Z u/Z uv/ ˆ f −( ˆ f + u2/ ˆ f) v − ˆ f/Z v/Z ˆ f + u2/ ˆ f −uv/ ˆ f −u ◆ B B B B B B @ vx vy vz ωx ωy ωz 1 C C C C C C A Slides adapted from Peter Corke

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Optical flow Patterns

Slides adapted from Peter Corke

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Image-based visual servoing

Getting a camera velocity to minimize the error between the current and goal image

Current Image Goal Image

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Image-based visual servoing

Current Image Goal Image

✓ ˙ u ˙ v ◆ = ✓ − ˆ f/Z u/Z uv/ ˆ f −( ˆ f + u2/ ˆ f) v − ˆ f/Z v/Z ˆ f + u2/ ˆ f −uv/ ˆ f −u ◆ B B B B B B @ vx vy vz ωx ωy ωz 1 C C C C C C A

J(u, v, Z)

Slides adapted from Peter Corke

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Image-based visual servoing

Current Image Goal Image

        ˙ u1 ˙ v1 ˙ u2 ˙ v2 ˙ u3 ˙ v3         =   J(u1, v1, Z1) J(u2, v2, Z2) J(u3, v3, Z3)           vx vy vz ωx ωy ωz        

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Image-based visual servoing

        ˙ u1 ˙ v1 ˙ u2 ˙ v2 ˙ u3 ˙ v3         =   J(u1, v1, Z1) J(u2, v2, Z2) J(u3, v3, Z3)           vx vy vz ωx ωy ωz        

        vx vy vz ωx ωy ωz         =   J(u1, v1, Z1) J(u2, v2, Z2) J(u3, v3, Z3)  

−1

        ˙ u1 ˙ v1 ˙ u2 ˙ v2 ˙ u3 ˙ v3        

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Desired Pixel Velocity

Slides adapted from Peter Corke

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Simulation

Slides adapted from Peter Corke

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Point Correspondences

How to find them? Features, Markers

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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

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Training Deep Neural Networks for Visual Servoing

Bateux et al. ICRA 2018

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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
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Training Deep Neural Networks for Visual Servoing

Bateux et al. ICRA 2018

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Training Deep Neural Networks for Visual Servoing

Bateux et al. ICRA 2018

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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

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So far on Control

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Optimal Control and Reinforcement Learning from a unified point of view

Optimal Control Problem

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Trajectory Optimization and Reinforcement Learning

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Trajectory Optimization: find a optimal trajectory given non-linear dynamics and cost

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Reinforcement Learning: finding an optimal policy under unknown dynamics and given a reward = -cost

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Principle of Optimality – Example: Graph Search problem

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Forward Search

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Forward Search

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Backward Search

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Backward Search

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Principle of Optimality

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Bellman Equation

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Problem setup

System Dynamics Cost function

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Goal

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Formalize Cost-to-Go / Value function

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Optimal Value function = V with lowest cost

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Deriving the Bellman Equation

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Optimal Bellman Equation

Optimal Value function Optimal Policy

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Comparing Optimal Bellman and Value Function

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Infinite time horizon, deterministic system

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Infinite time horizon, deterministic system

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Infinite time horizon, deterministic system

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Finite Horizon, Stochastic system

Stochastic System Dynamics Cost function

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Finite Horizon, Stochastic system

Value function Optimal Value function Optimal Policy

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Finite Horizon, Stochastic system

Bellman Equation Optimal Bellman Equation

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Infinite Horizon, Stochastic system

Combining formulation from infinite horizon - discrete system with stochastic system derivation

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Continuous time systems

Hamilton-Jacobi-Bellman Equation

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How do you solve these equations?

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Linear Dynamical Systems, Quadratic cost – Linear Quadratic Regulator (LQR)

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Linear Dynamical Systems, Quadratic cost – Linear Quadratic Regulator (LQR)