ME5751�Robotics Motion Planning

Yizhe Chang (chang@cpp.edu)�Lecture Note Set #1

Outline

• Course Overview
• Review: Coordinate Rotation
• 2D Rotation
• 3D Rotation

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Course Introduction

• What is motion planning?
• Piano moving problem: (Given an open subset U in n-dimensional space and two compact subsets C0 and C1 of U, where C1 is derived from C0 by a continuous motion, is it possible to move C0 to C1 while remaining entirely inside U? wolfram alpha)
• Basically, it is a search problem

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Course Introduction

• What is motion planning?
• Motion planning is a term used in robotics for the process of breaking down a desired movement task into discrete motions that satisfy movement constraints and possibly optimize some aspect of the movement.

(umich robotics)

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Course Introduction

• 4 Major Modules:

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• Proportional motion control
• Potential field generation (cost map)
• A* search
• Probabilistic road map (PRM)

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Proportional Control for point tracking

• One out of many point tracking method
• “Plan” robot velocity by given position road map

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Potential Field Generation

• Brushfire (Breadth first search) algorithm

A* path planning

• The most difficult part of our course

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https://fab.cba.mit.edu/classes/865.21/topics/path_planning/robotic.html

Probabilistic Road Map

• It is random, but faster
• We will only implement a “lumped version”

Minor Topics and Gap Topics

• Minor Topics

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• Mobility
• Localization
• Probabilistic Mapping
• Other Planning Methods

Outline

• Course Overview
• Review: Coordinate Transformation and Forward Kinematics
• Review: Homogeneous Transformation
• Differential Kinematics

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Review: Coordinate Transformation

• Degree of Freedoms

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Review: Coordinate Transformation

• DOF Analysis

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Review: DoF for 2 wheel differential robot

• How many DoF does this differential driving robot has?
• Hint: our robot doesn’t fly

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Review: DoF for 2 wheel differential robot

• Intuitively, 2, x translation and rotation, no y translation
• Or we can think, two wheel can independently move
• L = 3, J1 =2, J2 =0
• DOF=3x(3-1)-2x2=2

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Forward Kinematics

• Forward Kinematics refers to the use of the kinematic equations of a robot to compute the position of the end-effector from specified values for the joint parameters

Where is the end effector?

Forward Kinematics

Review: Coordinate Transformation

• Starting Problem:
• We have a point, in O₂ coordinate: P²(x2,y2)
• What is its coordinate in O₁: P¹(x1,y1)
• What is its coordinate in O₀: P¹(x0,y0)

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P² (x,y)

x₂

y₂

φ

Review: Coordinate Transformation

• Starting with 2D

Review: Rotation Matrix

Review: Rotation Matrix

P¹ (l_x, l_y)

Review: Rotation Matrix

P¹ (1, 0)

Review: Rotation Matrix

P² (l_x, l_y)

x₂

y₂

φ

Review: Rotation Matrix

P² (l_x, l_y)

x₂

y₂

φ

Mini-summary

P² (l_x, l_y)

x₂

y₂

φ

O₂

Rotation in 3D

z₀

y₀

y₁

z₁

θ

Rotation in 3D

z₀

x₀

y₀

x₁

y₁

z₁

θ

Rotation in 3D

z₀

x₀

y₀

x₁

y₁

z₁

θ

Rotation in 3D

z₀

x₀

y₀

x₁

y₁

z₁

θ

Basic 3D rotation

Question: composite 3D rotation

z₀

x₀

y₀

x₁

y₁

z₁

θ₁

z₀

y₀

y₁

z₁

θ₂

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z₂

y₂

x₁

x₂

x₀

Question: composite 3D rotation

z₀

y₀

y₁

z₁

θ₂

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z₂

y₂

x₂

Question: Composite Rotation 2

z₀

y₀

y₁

z₁

θ₂

x₁

x₀

z₀

y₁

z₁

x₀

z₂

y₂

x₂

θ₁

Question: Composite Rotation 2

Question: Composite Rotation 2

Mini Summary:

Additional Reading: Euler Angle

• How many fixed-body rotation at MOST we need to rotate from original frame to an ARBITRARY frame?
• A) 1
• B) 2
• C) 3
• D) 4

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Additional Reading: Euler Angle

• How many fixed-body rotation at MOST we need to rotate from original frame to an ARBITRARY frame?
• A) 1
• B) 2
• C) 3
• D) 4

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Additional Reading: Euler Angle

• Define a set of intermediate angles:
• 1. Rotate along z₀ by φ from x₀y₀z₀=> x₁y₁z₁
• 2. Rotate along y₁ by θ from x₁y₁z₁=> x₂y₂z₂
• 3. Rotate along z₂ by ψ from x₂y₂z₂=> x₃y₃z₃

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Euler Angle

• Step 1: Rotate along z₀ by φ from x₀y₀z₀=> x₁y₁z₁

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Euler Angle

• Step 2: Rotate along y₁ by θ from x₁y₁z₁=> x₂y₂z₂

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Euler Angle

• Step 3: Rotate along z₂ by ψ from x₂y₂z₂=> x₃y₃z₃

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Euler Angle

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Euler Angle

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Mini summary

• Euler angle is one of many ways to represent 3d rotation�(e.g. rotate along z₀, then y₁, then z₂)
• Other methods include pitch-roll-yaw, axis/angle representation
• For Euler angle:
• 1. Rotate along z₀ by φ from x₀y₀z₀=> x₁y₁z₁
• 2. Rotate along y₁ by θ from x₁y₁z₁=> x₂y₂z₂
• 3. Rotate along z₂ by ψ from x₂y₂z₂=> x₃y₃z₃

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Summary

• Rotation can be represented by rotation matrix
• For 3d rotation, body-fixed coordinate system is commonly used
• For 3d rotation, the sequence of rotation matters!