Whole-body dynamic motion planning with centroidal dynamics and full kinematics

Size: px
Start display at page:

Download "Whole-body dynamic motion planning with centroidal dynamics and full kinematics"

Transcription

1 Sep Introduction Linear Inverted Pendulum compute ZMP with point-mass model linear system, analytical solution co-planar contact solve kinematics separately Our approach dynamic constraint for 6 under-actuated DoF Full body trajectory optimization dynamic constraint for EVERY degrees of freedom Accuracy Complexity Overview task constraint nonlinear robot state trajectory whole body robot model optimization solver contact wrench profile inverse dynamics robot actuator input admissible contact regions Code can be downloaded from Drake https://drake.mit.edu Results Atlas playing monkey bars Atlas running Little Dog running

2 Dynamic constraint Consider a robot interacting with the environment, with foot and hand contact. The red dot r is the Center of Mass location.

3 Dynamic constraint The robot is in contact with the envinronment at point c, subject to contact wrench [F, τ ] and the i i i gravitational force mg at the CoM.

4 Dynamic constraint The rate of centroidal linear and angular momentum should equal to the total wrench at the CoM. The rate of centroidal linear momentum is m r. The centroidal angular momentum can be computed from the robot posture and velocity. [D.E.Orin et al] m r = j F j + mg Newton s law on CoM acceleration L = j (c j r) F j + τ j rate of centroidal angular momentum equals to external torque r = com(q) L = A(q)v compute CoM from posture compute angular momentum from robot state

5 Kinematic Constraint Accommodate a variety of kinematic constraints Position of an end-effector Orientation of an end-effector Gaze at a point Collision avoidance Quasi-static Figure: Solving inverse kinematics problem with different types of kinematic constraints.

6 Unscheduled contact sequence Hard to specify the contact sequence when mulitple contact points can be active. Exploit the complementarity constraint on normal contact force F n and distance to contact φ(c). [M.Posa] < φ j (c j ), F n j >= 0 φ j (c j ) 0 F n j 0 Figure: Illustration of the contact point c j, its distance φ j to the contact surface S j, and the local coordinate frame on the tangential surface, with unit vector t x, t y. The complementarity condition holds between contact distance φ j and the normal contact force F n j. r_foot_l_toe r_foot_l_toe Contact sequence BEFORE optimizing with complementarity constraints active contact inactive contact Contact sequence AFTER optimizing with complementarity constraints r_foot_r_heel r_foot_l_heel l_foot_r_toe l_foot_l_toe l_foot_r_heel l_foot_l_heel knot knot

7 Trajectory optimization 1 Sample the whole trajectory into N knot points.

8 Trajectory optimization Sample the whole trajectory into N knot points. 1 In k th point, assign the posture q[k], velocity v[k], contact wrench F[k], τ[k] and time duration h[k] as 2 decision variables.

9 Trajectory optimization Sample the whole trajectory into N knot points. 1 In k th point, assign the posture q[k], velocity v[k], contact wrench F[k], τ[k] and time duration h[k] as 2 decision variables. Solve a nonlinear optimization problem. 3

10 Trajectory optimization Sample the whole trajectory into N knot points. 1 In k th point, assign the posture q[k], velocity v[k], contact wrench F[k], τ[k] and time duration h[k] as 2 decision variables. Solve a nonlinear optimization problem. 3 N min q[k],v[k],h[k] r[k],ṙ[k], r[k] c j [k],f j [k],τ j [k] k=1 q[k] q nom [k] 2 Q q + v[k] 2 Q v + r[k] 2 + j (c Fj 1 [k] 2 + c 2 τ [k] 2 ) h[k] j L[k], L[k]

11 Trajectory optimization Sample the whole trajectory into N knot points. 1 In k th point, assign the posture q[k], velocity v[k], contact wrench F[k], τ[k] and time duration h[k] as 2 decision variables. Solve a nonlinear optimization problem. 3 N min q[k],v[k],h[k] r[k],ṙ[k], r[k] c j [k],f j [k],τ j [k] k=1 q[k] q nom [k] 2 Q q + v[k] 2 Q v + r[k] 2 + j (c Fj 1 [k] 2 + c 2 τ [k] 2 ) h[k] j L[k], L[k] m r[k] = j F j[k] + mg L[k] = j (c j[k] r[k]) F j [k] + τ j [k] s.t Dynamic constraint L[k] = A(q[k])v[k] r[k] = com(q[k])

12 Trajectory optimization Sample the whole trajectory into N knot points. 1 In k th point, assign the posture q[k], velocity v[k], contact wrench F[k], τ[k] and time duration h[k] as 2 decision variables. Solve a nonlinear optimization problem. 3 N min q[k],v[k],h[k] r[k],ṙ[k], r[k] c j [k],f j [k],τ j [k] k=1 q[k] q nom [k] 2 Q q + v[k] 2 Q v + r[k] 2 + j (c Fj 1 [k] 2 + c 2 τ [k] 2 ) h[k] j L[k], L[k] m r[k] = j F j[k] + mg L[k] = j (c j[k] r[k]) F j [k] + τ j [k] s.t Dynamic constraint L[k] = A(q[k])v[k] r[k] = com(q[k]) { q[k] q[k 1] = v[k]h[k] Backward-Euler integration L[k] L[k 1] = L[k]h[k]

13 Trajectory optimization Sample the whole trajectory into N knot points. 1 In k th point, assign the posture q[k], velocity v[k], contact wrench F[k], τ[k] and time duration h[k] as 2 decision variables. Solve a nonlinear optimization problem. 3 N min q[k],v[k],h[k] r[k],ṙ[k], r[k] c j [k],f j [k],τ j [k] k=1 q[k] q nom [k] 2 Q q + v[k] 2 Q v + r[k] 2 + j (c Fj 1 [k] 2 + c 2 τ [k] 2 ) h[k] j L[k], L[k] m r[k] = j F j[k] + mg L[k] = j (c j[k] r[k]) F j [k] + τ j [k] s.t Dynamic constraint L[k] = A(q[k])v[k] r[k] = com(q[k]) { q[k] q[k 1] = v[k]h[k] Backward-Euler integration Quadratic interpolation on CoM L[k] L[k 1] = L[k]h[k] { r[k] r[k 1] = ṙ[k]+ṙ[k 1] h[k] 2 ṙ[k] ṙ[k 1] = r[k]h[k]

14 Trajectory optimization Sample the whole trajectory into N knot points. 1 In k th point, assign the posture q[k], velocity v[k], contact wrench F[k], τ[k] and time duration h[k] as 2 decision variables. Solve a nonlinear optimization problem. 3 N min q[k],v[k],h[k] r[k],ṙ[k], r[k] c j [k],f j [k],τ j [k] k=1 q[k] q nom [k] 2 Q q + v[k] 2 Q v + r[k] 2 + j (c Fj 1 [k] 2 + c 2 τ [k] 2 ) h[k] j L[k], L[k] m r[k] = j F j[k] + mg L[k] = j (c j[k] r[k]) F j [k] + τ j [k] s.t Dynamic constraint L[k] = A(q[k])v[k] r[k] = com(q[k]) { q[k] q[k 1] = v[k]h[k] Backward-Euler integration Quadratic interpolation on CoM L[k] L[k 1] = L[k]h[k] { r[k] r[k 1] = ṙ[k]+ṙ[k 1] h[k] 2 ṙ[k] ṙ[k 1] = r[k]h[k] { c j [k] = p j (q[k]) Kinematic constraint for contact c j [k] S j [k]

15 Monkey bars

16 Atlas running

17 Monkey bars

18 Little dog running Playback in 1/8x speed Playback in real speed.

19 Little dog walking Side view Front view

20 Little dog bounding Playback in 1/4x speed Play back in real speed.

21 Little dog rotary Playback in 1/4x speed Play back in real speed.

Single and Double plane pendulum

Single and Double plane pendulum Single and Double plane pendulum Gabriela González 1 Introduction We will write down equations of motion for a single and a double plane pendulum, following Newton s equations, and using Lagrange s equations.

More information

INSTRUCTOR WORKBOOK Quanser Robotics Package for Education for MATLAB /Simulink Users

INSTRUCTOR WORKBOOK Quanser Robotics Package for Education for MATLAB /Simulink Users INSTRUCTOR WORKBOOK for MATLAB /Simulink Users Developed by: Amir Haddadi, Ph.D., Quanser Peter Martin, M.A.SC., Quanser Quanser educational solutions are powered by: CAPTIVATE. MOTIVATE. GRADUATE. PREFACE

More information

Online Courses for High School Students 1-888-972-6237

Online Courses for High School Students 1-888-972-6237 Online Courses for High School Students 1-888-972-6237 PHYSICS Course Description: This course provides a comprehensive survey of all key areas: physical systems, measurement, kinematics, dynamics, momentum,

More information

Computer Animation. Lecture 2. Basics of Character Animation

Computer Animation. Lecture 2. Basics of Character Animation Computer Animation Lecture 2. Basics of Character Animation Taku Komura Overview Character Animation Posture representation Hierarchical structure of the body Joint types Translational, hinge, universal,

More information

Chapter 5 Newton s Laws of Motion

Chapter 5 Newton s Laws of Motion Chapter 5 Newton s Laws of Motion Force and Mass Units of Chapter 5 Newton s First Law of Motion Newton s Second Law of Motion Newton s Third Law of Motion The Vector Nature of Forces: Forces in Two Dimensions

More information

Linear and Rotational Kinematics

Linear and Rotational Kinematics Linear and Rotational Kinematics Starting from rest, a disk takes 10 revolutions to reach an angular velocity. If the angular acceleration is constant throughout, how many additional revolutions are required

More information

ACTUATOR DESIGN FOR ARC WELDING ROBOT

ACTUATOR DESIGN FOR ARC WELDING ROBOT ACTUATOR DESIGN FOR ARC WELDING ROBOT 1 Anurag Verma, 2 M. M. Gor* 1 G.H Patel College of Engineering & Technology, V.V.Nagar-388120, Gujarat, India 2 Parul Institute of Engineering & Technology, Limda-391760,

More information

Design-Simulation-Optimization Package for a Generic 6-DOF Manipulator with a Spherical Wrist

Design-Simulation-Optimization Package for a Generic 6-DOF Manipulator with a Spherical Wrist Design-Simulation-Optimization Package for a Generic 6-DOF Manipulator with a Spherical Wrist MHER GRIGORIAN, TAREK SOBH Department of Computer Science and Engineering, U. of Bridgeport, USA ABSTRACT Robot

More information

Simulation of Trajectories and Comparison of Joint Variables for Robotic Manipulator Using Multibody Dynamics (MBD)

Simulation of Trajectories and Comparison of Joint Variables for Robotic Manipulator Using Multibody Dynamics (MBD) Simulation of Trajectories and Comparison of Joint Variables for Robotic Manipulator Using Multibody Dynamics (MBD) Jatin Dave Assistant Professor Nirma University Mechanical Engineering Department, Institute

More information

EL5223. Basic Concepts of Robot Sensors, Actuators, Localization, Navigation, and1 Mappin / 12

EL5223. Basic Concepts of Robot Sensors, Actuators, Localization, Navigation, and1 Mappin / 12 Basic Concepts of Robot Sensors, Actuators, Localization, Navigation, and Mapping Basic Concepts of Robot Sensors, Actuators, Localization, Navigation, and1 Mappin / 12 Sensors and Actuators Robotic systems

More information

Animations in Creo 3.0

Animations in Creo 3.0 Animations in Creo 3.0 ME170 Part I. Introduction & Outline Animations provide useful demonstrations and analyses of a mechanism's motion. This document will present two ways to create a motion animation

More information

Practical Work DELMIA V5 R20 Lecture 1. D. Chablat / S. Caro Damien.Chablat@irccyn.ec-nantes.fr Stephane.Caro@irccyn.ec-nantes.fr

Practical Work DELMIA V5 R20 Lecture 1. D. Chablat / S. Caro Damien.Chablat@irccyn.ec-nantes.fr Stephane.Caro@irccyn.ec-nantes.fr Practical Work DELMIA V5 R20 Lecture 1 D. Chablat / S. Caro Damien.Chablat@irccyn.ec-nantes.fr Stephane.Caro@irccyn.ec-nantes.fr Native languages Definition of the language for the user interface English,

More information

School of Biotechnology

School of Biotechnology Physics reference slides Donatello Dolce Università di Camerino a.y. 2014/2015 mail: donatello.dolce@unicam.it School of Biotechnology Program and Aim Introduction to Physics Kinematics and Dynamics; Position

More information

PHYSICS AND MATH LAB ON GRAVITY NAME

PHYSICS AND MATH LAB ON GRAVITY NAME RIVERDALE HIGH SCHOOL PHYSICS AND MATH LAB ON GRAVITY NAME Purpose: To investigate two of the levels of gravity experienced by Florida teachers during their recent ZeroG flight. Background: NASA supplied

More information

Reavis High School Physics Honors Curriculum Snapshot

Reavis High School Physics Honors Curriculum Snapshot Reavis High School Physics Honors Curriculum Snapshot Unit 1: Mathematical Toolkit Students will be able to: state definition for physics; measure length using a meter stick; measure the time with a stopwatch

More information

Presentation of problem T1 (9 points): The Maribo Meteorite

Presentation of problem T1 (9 points): The Maribo Meteorite Presentation of problem T1 (9 points): The Maribo Meteorite Definitions Meteoroid. A small particle (typically smaller than 1 m) from a comet or an asteroid. Meteorite: A meteoroid that impacts the ground

More information

SIMULATION OF WALKING HUMANOID ROBOT BASED ON MATLAB/SIMMECHANICS. Sébastien Corner

SIMULATION OF WALKING HUMANOID ROBOT BASED ON MATLAB/SIMMECHANICS. Sébastien Corner SIMULATION OF WALKING HUMANOID ROBOT BASED ON MATLAB/SIMMECHANICS Sébastien Corner scorner@vt.edu The Robotics & Mechanisms Laboratory, RoMeLa Department of Mechanical Engineering of the University of

More information

Constraint satisfaction and global optimization in robotics

Constraint satisfaction and global optimization in robotics Constraint satisfaction and global optimization in robotics Arnold Neumaier Universität Wien and Jean-Pierre Merlet INRIA Sophia Antipolis 1 The design, validation, and use of robots poses a number of

More information

NEWTON S LAWS OF MOTION

NEWTON S LAWS OF MOTION NEWTON S LAWS OF MOTION Background: Aristotle believed that the natural state of motion for objects on the earth was one of rest. In other words, objects needed a force to be kept in motion. Galileo studied

More information

Page 1. Klyde. V Klyde. ConcepTest 9.3a Angular Displacement I. ConcepTest 9.2 Truck Speedometer

Page 1. Klyde. V Klyde. ConcepTest 9.3a Angular Displacement I. ConcepTest 9.2 Truck Speedometer ConcepTest 9.a Bonnie and Klyde I Bonnie sits on the outer rim of a merry-go-round, and Klyde sits midway between the center and the rim. The merry-go-round makes one complete revolution every two seconds.

More information

Geometric Constraints

Geometric Constraints Simulation in Computer Graphics Geometric Constraints Matthias Teschner Computer Science Department University of Freiburg Outline introduction penalty method Lagrange multipliers local constraints University

More information

Physics 9e/Cutnell. correlated to the. College Board AP Physics 1 Course Objectives

Physics 9e/Cutnell. correlated to the. College Board AP Physics 1 Course Objectives Physics 9e/Cutnell correlated to the College Board AP Physics 1 Course Objectives Big Idea 1: Objects and systems have properties such as mass and charge. Systems may have internal structure. Enduring

More information

5. Universal Laws of Motion

5. Universal Laws of Motion 5. Universal Laws of Motion If I have seen farther than others, it is because I have stood on the shoulders of giants. Sir Isaac Newton (1642 1727) Physicist 5.1 Describing Motion: Examples from Daily

More information

Pre-requisites 2012-2013

Pre-requisites 2012-2013 Pre-requisites 2012-2013 Engineering Computation The student should be familiar with basic tools in Mathematics and Physics as learned at the High School level and in the first year of Engineering Schools.

More information

Lab 8: Ballistic Pendulum

Lab 8: Ballistic Pendulum Lab 8: Ballistic Pendulum Equipment: Ballistic pendulum apparatus, 2 meter ruler, 30 cm ruler, blank paper, carbon paper, masking tape, scale. Caution In this experiment a steel ball is projected horizontally

More information

Discrete mechanics, optimal control and formation flying spacecraft

Discrete mechanics, optimal control and formation flying spacecraft Discrete mechanics, optimal control and formation flying spacecraft Oliver Junge Center for Mathematics Munich University of Technology joint work with Jerrold E. Marsden and Sina Ober-Blöbaum partially

More information

Prioritizing linear equality and inequality systems: application to local motion planning for redundant robots

Prioritizing linear equality and inequality systems: application to local motion planning for redundant robots Prioritizing linear equality and inequality systems: application to local motion planning for redundant robots Oussama Kanoun, Florent Lamiraux, Pierre-Brice Wieber, Fumio Kanehiro, Eiichi Yoshida and

More information

Operational Space Control for A Scara Robot

Operational Space Control for A Scara Robot Operational Space Control for A Scara Robot Francisco Franco Obando D., Pablo Eduardo Caicedo R., Oscar Andrés Vivas A. Universidad del Cauca, {fobando, pacaicedo, avivas }@unicauca.edu.co Abstract This

More information

A Simple Algorithm for Generating Stable Biped Walking Patterns

A Simple Algorithm for Generating Stable Biped Walking Patterns International Journal of Computer Applications (975 8887) A Simple Algorithm for Generating Stable Biped Walking Patterns Hayder F. N. Al- Shuka Baghdad University,Mech. Eng. Dep., Iraq Burkhard J. Corves

More information

Rotation, Angular Momentum

Rotation, Angular Momentum This test covers rotational motion, rotational kinematics, rotational energy, moments of inertia, torque, cross-products, angular momentum and conservation of angular momentum, with some problems requiring

More information

Orbital Mechanics. Angular Momentum

Orbital Mechanics. Angular Momentum Orbital Mechanics The objects that orbit earth have only a few forces acting on them, the largest being the gravitational pull from the earth. The trajectories that satellites or rockets follow are largely

More information

USING MS EXCEL FOR DATA ANALYSIS AND SIMULATION

USING MS EXCEL FOR DATA ANALYSIS AND SIMULATION USING MS EXCEL FOR DATA ANALYSIS AND SIMULATION Ian Cooper School of Physics The University of Sydney i.cooper@physics.usyd.edu.au Introduction The numerical calculations performed by scientists and engineers

More information

KERN COMMUNITY COLLEGE DISTRICT CERRO COSO COLLEGE PHYS C111 COURSE OUTLINE OF RECORD

KERN COMMUNITY COLLEGE DISTRICT CERRO COSO COLLEGE PHYS C111 COURSE OUTLINE OF RECORD KERN COMMUNITY COLLEGE DISTRICT CERRO COSO COLLEGE PHYS C111 COURSE OUTLINE OF RECORD 1. DISCIPLINE AND COURSE NUMBER: PHYS C111 2. COURSE TITLE: Mechanics 3. SHORT BANWEB TITLE: Mechanics 4. COURSE AUTHOR:

More information

C is a point of concurrency is at distance from End Effector frame & at distance from ref frame.

C is a point of concurrency is at distance from End Effector frame & at distance from ref frame. Module 6 : Robot manipulators kinematics Lecture 21 : Forward & inverse kinematics examples of 2R, 3R & 3P manipulators Objectives In this course you will learn the following Inverse position and orientation

More information

Version PREVIEW Practice 8 carroll (11108) 1

Version PREVIEW Practice 8 carroll (11108) 1 Version PREVIEW Practice 8 carroll 11108 1 This print-out should have 12 questions. Multiple-choice questions may continue on the net column or page find all choices before answering. Inertia of Solids

More information

Transcription Methods for Trajectory Optimization

Transcription Methods for Trajectory Optimization Transcription Methods for Trajectory Optimization A beginners tutorial Matthew P. Kelly Cornell University mpk7@cornell.edu February 8, 5 Abstract This report is an introduction to transcription methods

More information

MCE/EEC 647/747: Robot Dynamics and Control

MCE/EEC 647/747: Robot Dynamics and Control MCE/EEC 647/747: Robot Dynamics and Control Lecture 4: Velocity Kinematics Jacobian and Singularities Torque/Force Relationship Inverse Velocity Problem Reading: SHV Chapter 4 Mechanical Engineering Hanz

More information

Today. Laws of Motion Conservation Laws Gravity tides. What is the phase of the moon?

Today. Laws of Motion Conservation Laws Gravity tides. What is the phase of the moon? Today Laws of Motion Conservation Laws Gravity tides What is the phase of the moon? How is mass different from weight? Mass the amount of matter in an object Weight the force that acts upon an object You

More information

Inverse Kinematics. Ming Yao

Inverse Kinematics. Ming Yao Mathematics for Inverse Kinematics 15 464: Technical Animation Ming Yao Overview Kinematics Forward Kinematics and Inverse Kinematics Jabobian Pseudoinverse of the Jacobian Assignment 2 Vocabulary of Kinematics

More information

Introduction to Computer Graphics Marie-Paule Cani & Estelle Duveau

Introduction to Computer Graphics Marie-Paule Cani & Estelle Duveau Introduction to Computer Graphics Marie-Paule Cani & Estelle Duveau 04/02 Introduction & projective rendering 11/02 Prodedural modeling, Interactive modeling with parametric surfaces 25/02 Introduction

More information

THEORETICAL MECHANICS

THEORETICAL MECHANICS PROF. DR. ING. VASILE SZOLGA THEORETICAL MECHANICS LECTURE NOTES AND SAMPLE PROBLEMS PART ONE STATICS OF THE PARTICLE, OF THE RIGID BODY AND OF THE SYSTEMS OF BODIES KINEMATICS OF THE PARTICLE 2010 0 Contents

More information

3.6 Solving Problems Involving Projectile Motion

3.6 Solving Problems Involving Projectile Motion INTRODUCTION 1-2 Physics and its relation to other fields introduction of physics, its importance and scope 1-5 Units, standards, and the SI System description of the SI System description of base and

More information

Rotation: Moment of Inertia and Torque

Rotation: Moment of Inertia and Torque Rotation: Moment of Inertia and Torque Every time we push a door open or tighten a bolt using a wrench, we apply a force that results in a rotational motion about a fixed axis. Through experience we learn

More information

Force/position control of a robotic system for transcranial magnetic stimulation

Force/position control of a robotic system for transcranial magnetic stimulation Force/position control of a robotic system for transcranial magnetic stimulation W.N. Wan Zakaria School of Mechanical and System Engineering Newcastle University Abstract To develop a force control scheme

More information

Lecture 16. Newton s Second Law for Rotation. Moment of Inertia. Angular momentum. Cutnell+Johnson: 9.4, 9.6

Lecture 16. Newton s Second Law for Rotation. Moment of Inertia. Angular momentum. Cutnell+Johnson: 9.4, 9.6 Lecture 16 Newton s Second Law for Rotation Moment of Inertia Angular momentum Cutnell+Johnson: 9.4, 9.6 Newton s Second Law for Rotation Newton s second law says how a net force causes an acceleration.

More information

SOLID MECHANICS TUTORIAL MECHANISMS KINEMATICS - VELOCITY AND ACCELERATION DIAGRAMS

SOLID MECHANICS TUTORIAL MECHANISMS KINEMATICS - VELOCITY AND ACCELERATION DIAGRAMS SOLID MECHANICS TUTORIAL MECHANISMS KINEMATICS - VELOCITY AND ACCELERATION DIAGRAMS This work covers elements of the syllabus for the Engineering Council exams C105 Mechanical and Structural Engineering

More information

Here is a guide if you are looking for practice questions in the old Physics 111 tests. SUMMARY

Here is a guide if you are looking for practice questions in the old Physics 111 tests. SUMMARY 31 May 11 1 phys115_in_phys111_exams-new.docx PHYSICS 115 MATERIAL IN OLD PHYSICS 111 EXAMS Here is a guide if you are looking for practice questions in the old Physics 111 tests. SUMMARY Giambattista

More information

Physics 211 Week 12. Simple Harmonic Motion: Equation of Motion

Physics 211 Week 12. Simple Harmonic Motion: Equation of Motion Physics 11 Week 1 Simple Harmonic Motion: Equation of Motion A mass M rests on a frictionless table and is connected to a spring of spring constant k. The other end of the spring is fixed to a vertical

More information

Rotation, Rolling, Torque, Angular Momentum

Rotation, Rolling, Torque, Angular Momentum Halliday, Resnick & Walker Chapter 10 & 11 Rotation, Rolling, Torque, Angular Momentum Physics 1A PHYS1121 Professor Michael Burton Rotation 10-1 Rotational Variables! The motion of rotation! The same

More information

Chapter 10 Rotational Motion. Copyright 2009 Pearson Education, Inc.

Chapter 10 Rotational Motion. Copyright 2009 Pearson Education, Inc. Chapter 10 Rotational Motion Angular Quantities Units of Chapter 10 Vector Nature of Angular Quantities Constant Angular Acceleration Torque Rotational Dynamics; Torque and Rotational Inertia Solving Problems

More information

DESIGN, IMPLEMENTATION, AND COOPERATIVE COEVOLUTION OF AN AUTONOMOUS/TELEOPERATED CONTROL SYSTEM FOR A SERPENTINE ROBOTIC MANIPULATOR

DESIGN, IMPLEMENTATION, AND COOPERATIVE COEVOLUTION OF AN AUTONOMOUS/TELEOPERATED CONTROL SYSTEM FOR A SERPENTINE ROBOTIC MANIPULATOR Proceedings of the American Nuclear Society Ninth Topical Meeting on Robotics and Remote Systems, Seattle Washington, March 2001. DESIGN, IMPLEMENTATION, AND COOPERATIVE COEVOLUTION OF AN AUTONOMOUS/TELEOPERATED

More information

Center of Mass Estimator for Humanoids and its Application in Modelling Error Compensation, Fall Detection and Prevention

Center of Mass Estimator for Humanoids and its Application in Modelling Error Compensation, Fall Detection and Prevention Center of Mass Estimator for Humanoids and its Application in Modelling Error Compensation, Fall Detection and Prevention X Xinjilefu, Siyuan Feng, and Christopher G. Atkeson Abstract We introduce a center

More information

Physics 111: Lecture 4: Chapter 4 - Forces and Newton s Laws of Motion. Physics is about forces and how the world around us reacts to these forces.

Physics 111: Lecture 4: Chapter 4 - Forces and Newton s Laws of Motion. Physics is about forces and how the world around us reacts to these forces. Physics 111: Lecture 4: Chapter 4 - Forces and Newton s Laws of Motion Physics is about forces and how the world around us reacts to these forces. Whats a force? Contact and non-contact forces. Whats a

More information

A B = AB sin(θ) = A B = AB (2) For two vectors A and B the cross product A B is a vector. The magnitude of the cross product

A B = AB sin(θ) = A B = AB (2) For two vectors A and B the cross product A B is a vector. The magnitude of the cross product 1 Dot Product and Cross Products For two vectors, the dot product is a number A B = AB cos(θ) = A B = AB (1) For two vectors A and B the cross product A B is a vector. The magnitude of the cross product

More information

Kinematics & Dynamics

Kinematics & Dynamics Overview Kinematics & Dynamics Adam Finkelstein Princeton University COS 46, Spring 005 Kinematics Considers only motion Determined by positions, velocities, accelerations Dynamics Considers underlying

More information

Intersection of Objects with Linear and Angular Velocities using Oriented Bounding Boxes

Intersection of Objects with Linear and Angular Velocities using Oriented Bounding Boxes Intersection of Objects with Linear and Angular Velocities using Oriented Bounding Boxes David Eberly Geometric Tools, LLC http://www.geometrictools.com/ Copyright c 1998-2016. All Rights Reserved. Created:

More information

Phys 111 Fall P111 Syllabus

Phys 111 Fall P111 Syllabus Phys 111 Fall 2012 Course structure Five sections lecture time 150 minutes per week Textbook Physics by James S. Walker fourth edition (Pearson) Clickers recommended Coursework Complete assignments from

More information

Average Angular Velocity

Average Angular Velocity Average Angular Velocity Hanno Essén Department of Mechanics Royal Institute of Technology S-100 44 Stockholm, Sweden 199, December Abstract This paper addresses the problem of the separation of rotational

More information

SYLLABUS FORM WESTCHESTER COMMUNITY COLLEGE Valhalla, NY lo595. l. Course #: PHYSC 111 2. NAME OF ORIGINATOR /REVISOR: Dr.

SYLLABUS FORM WESTCHESTER COMMUNITY COLLEGE Valhalla, NY lo595. l. Course #: PHYSC 111 2. NAME OF ORIGINATOR /REVISOR: Dr. SYLLABUS FORM WESTCHESTER COMMUNITY COLLEGE Valhalla, NY lo595 l. Course #: PHYSC 111 2. NAME OF ORIGINATOR /REVISOR: Dr. Neil Basescu NAME OF COURSE: College Physics 1 with Lab 3. CURRENT DATE: 4/24/13

More information

9.1 Rotational Kinematics: Angular Velocity and Angular Acceleration

9.1 Rotational Kinematics: Angular Velocity and Angular Acceleration Ch 9 Rotation 9.1 Rotational Kinematics: Angular Velocity and Angular Acceleration Q: What is angular velocity? Angular speed? What symbols are used to denote each? What units are used? Q: What is linear

More information

CONTRIBUTIONS TO THE AUTOMATIC CONTROL OF AERIAL VEHICLES

CONTRIBUTIONS TO THE AUTOMATIC CONTROL OF AERIAL VEHICLES 1 / 23 CONTRIBUTIONS TO THE AUTOMATIC CONTROL OF AERIAL VEHICLES MINH DUC HUA 1 1 INRIA Sophia Antipolis, AROBAS team I3S-CNRS Sophia Antipolis, CONDOR team Project ANR SCUAV Supervisors: Pascal MORIN,

More information

Salem Community College Course Syllabus. Course Title: Physics I. Course Code: PHY 101. Lecture Hours: 2 Laboratory Hours: 4 Credits: 4

Salem Community College Course Syllabus. Course Title: Physics I. Course Code: PHY 101. Lecture Hours: 2 Laboratory Hours: 4 Credits: 4 Salem Community College Course Syllabus Course Title: Physics I Course Code: PHY 101 Lecture Hours: 2 Laboratory Hours: 4 Credits: 4 Course Description: The basic principles of classical physics are explored

More information

CALIBRATION OF A ROBUST 2 DOF PATH MONITORING TOOL FOR INDUSTRIAL ROBOTS AND MACHINE TOOLS BASED ON PARALLEL KINEMATICS

CALIBRATION OF A ROBUST 2 DOF PATH MONITORING TOOL FOR INDUSTRIAL ROBOTS AND MACHINE TOOLS BASED ON PARALLEL KINEMATICS CALIBRATION OF A ROBUST 2 DOF PATH MONITORING TOOL FOR INDUSTRIAL ROBOTS AND MACHINE TOOLS BASED ON PARALLEL KINEMATICS E. Batzies 1, M. Kreutzer 1, D. Leucht 2, V. Welker 2, O. Zirn 1 1 Mechatronics Research

More information

Intelligent Robotics Lab.

Intelligent Robotics Lab. 1 Variable Stiffness Actuation based on Dual Actuators Connected in Series and Parallel Prof. Jae-Bok Song (jbsong@korea.ac.kr ). (http://robotics.korea.ac.kr) ti k Depart. of Mechanical Engineering, Korea

More information

Chapter 10: Linear Kinematics of Human Movement

Chapter 10: Linear Kinematics of Human Movement Chapter 10: Linear Kinematics of Human Movement Basic Biomechanics, 4 th edition Susan J. Hall Presentation Created by TK Koesterer, Ph.D., ATC Humboldt State University Objectives Discuss the interrelationship

More information

Chapter 21 Rigid Body Dynamics: Rotation and Translation about a Fixed Axis

Chapter 21 Rigid Body Dynamics: Rotation and Translation about a Fixed Axis Chapter 21 Rigid Body Dynamics: Rotation and Translation about a Fixed Axis 21.1 Introduction... 1 21.2 Translational Equation of Motion... 1 21.3 Translational and Rotational Equations of Motion... 1

More information

Principles and Laws of Motion

Principles and Laws of Motion 2009 19 minutes Teacher Notes: Ian Walter DipAppChem; TTTC; GDipEdAdmin; MEdAdmin (part) Program Synopsis This program begins by looking at the different types of motion all around us. Forces that cause

More information

Chapter 3: Force and Motion

Chapter 3: Force and Motion Force and Motion Cause and Effect Chapter 3 Chapter 3: Force and Motion Homework: All questions on the Multiple- Choice and the odd-numbered questions on Exercises sections at the end of the chapter. In

More information

s r or equivalently sr linear velocity vr Rotation its description and what causes it? Consider a disk rotating at constant angular velocity.

s r or equivalently sr linear velocity vr Rotation its description and what causes it? Consider a disk rotating at constant angular velocity. Rotation its description and what causes it? Consider a disk rotating at constant angular velocity. Rotation involves turning. Turning implies change of angle. Turning is about an axis of rotation. All

More information

Kinematics and Dynamics of Mechatronic Systems. Wojciech Lisowski. 1 An Introduction

Kinematics and Dynamics of Mechatronic Systems. Wojciech Lisowski. 1 An Introduction Katedra Robotyki i Mechatroniki Akademia Górniczo-Hutnicza w Krakowie Kinematics and Dynamics of Mechatronic Systems Wojciech Lisowski 1 An Introduction KADOMS KRIM, WIMIR, AGH Kraków 1 The course contents:

More information

Dynamics. Basilio Bona. DAUIN-Politecnico di Torino. Basilio Bona (DAUIN-Politecnico di Torino) Dynamics 2009 1 / 30

Dynamics. Basilio Bona. DAUIN-Politecnico di Torino. Basilio Bona (DAUIN-Politecnico di Torino) Dynamics 2009 1 / 30 Dynamics Basilio Bona DAUIN-Politecnico di Torino 2009 Basilio Bona (DAUIN-Politecnico di Torino) Dynamics 2009 1 / 30 Dynamics - Introduction In order to determine the dynamics of a manipulator, it is

More information

Astro 110-01 Lecture 10 Newton s laws

Astro 110-01 Lecture 10 Newton s laws Astro 110-01 Lecture 10 Newton s laws Twin Sungrazing comets 9/02/09 Habbal Astro110-01 Lecture 10 1 http://umbra.nascom.nasa.gov/comets/movies/soho_lasco_c2.mpg What have we learned? How do we describe

More information

Kinematics is the study of motion. Generally, this involves describing the position, velocity, and acceleration of an object.

Kinematics is the study of motion. Generally, this involves describing the position, velocity, and acceleration of an object. Kinematics Kinematics is the study of motion. Generally, this involves describing the position, velocity, and acceleration of an object. Reference frame In order to describe movement, we need to set a

More information

Laws of Motion, Velocity, Displacement, and Acceleration

Laws of Motion, Velocity, Displacement, and Acceleration Physical Science, Quarter 1, Unit 1.1 Laws of Motion, Velocity, Displacement, and Acceleration Overview Number of instructional days: 13 (1 day = 53 minutes) Content to be learned Add distance and displacement

More information

Chapter 4. Kinematics - Velocity and Acceleration. 4.1 Purpose. 4.2 Introduction

Chapter 4. Kinematics - Velocity and Acceleration. 4.1 Purpose. 4.2 Introduction Chapter 4 Kinematics - Velocity and Acceleration 4.1 Purpose In this lab, the relationship between position, velocity and acceleration will be explored. In this experiment, friction will be neglected.

More information

Chapter 4: Newton s Laws of Motion

Chapter 4: Newton s Laws of Motion Chapter 4: Newton s Laws of Motion Dynamics: Study of motion and its causes. orces cause changes in the motion of an object. orce and Interactions Definition ( loose ): A force is a push or pull exerted

More information

First Semester Learning Targets

First Semester Learning Targets First Semester Learning Targets 1.1.Can define major components of the scientific method 1.2.Can accurately carry out conversions using dimensional analysis 1.3.Can utilize and convert metric prefixes

More information

Full- day Workshop on Online and offline optimization for humanoid robots. at IEEE IROS 2013 in Tokyo

Full- day Workshop on Online and offline optimization for humanoid robots. at IEEE IROS 2013 in Tokyo Full- day Workshop on Online and offline optimization for humanoid robots at IEEE IROS 2013 in Tokyo Organizers: Eiichi Yoshida, Katja Mombaur, Tom Erez, Yuval Tassa Nov 7, 2013 TALK ABSTRACTS Tamim Asfour

More information

3. Interpolation. Closing the Gaps of Discretization... Beyond Polynomials

3. Interpolation. Closing the Gaps of Discretization... Beyond Polynomials 3. Interpolation Closing the Gaps of Discretization... Beyond Polynomials Closing the Gaps of Discretization... Beyond Polynomials, December 19, 2012 1 3.3. Polynomial Splines Idea of Polynomial Splines

More information

Mechanism and Control of a Dynamic Lifting Robot

Mechanism and Control of a Dynamic Lifting Robot Mechanism and Control of a Dynamic Lifting Robot T. Uenoa, N. Sunagaa, K. Brownb and H. Asada' 'Institute of Technology, Shimizu Corporation, Etchujima 3-4-17, Koto-ku, Tokyo 135, Japan 'Department of

More information

Lecture L2 - Degrees of Freedom and Constraints, Rectilinear Motion

Lecture L2 - Degrees of Freedom and Constraints, Rectilinear Motion S. Widnall 6.07 Dynamics Fall 009 Version.0 Lecture L - Degrees of Freedom and Constraints, Rectilinear Motion Degrees of Freedom Degrees of freedom refers to the number of independent spatial coordinates

More information

Robot Kinematics and Dynamics. Herman Bruyninckx Katholieke Universiteit Leuven Department of Mechanical Engineering Leuven, Belgium

Robot Kinematics and Dynamics. Herman Bruyninckx Katholieke Universiteit Leuven Department of Mechanical Engineering Leuven, Belgium Robot Kinematics and Dynamics Herman Bruyninckx Katholieke Universiteit Leuven Department of Mechanical Engineering Leuven, Belgium c August 21, 2010 Contents 1 Introduction 9 1.1 Robot motion...............................................

More information

CS 775: Advanced Computer Graphics. Lecture 11 : Motion Editing

CS 775: Advanced Computer Graphics. Lecture 11 : Motion Editing CS 775: Advanced Computer Graphics Lecture 11 : Motion Editing Working with motion data: Captured on a single perfomer How to use to motion to animate different characters? How to change the motion? How

More information

Gravitational Potential Energy

Gravitational Potential Energy Gravitational Potential Energy Consider a ball falling from a height of y 0 =h to the floor at height y=0. A net force of gravity has been acting on the ball as it drops. So the total work done on the

More information

AST 101 Lecture 7. Newton s Laws and the Nature of Matter

AST 101 Lecture 7. Newton s Laws and the Nature of Matter AST 101 Lecture 7 Newton s Laws and the Nature of Matter The Nature of Matter Democritus (c. 470-380 BCE) posited that matter was composed of atoms Atoms: particles that can not be further subdivided 4

More information

April 07, 2015. Force motion examples.notebook MOTION AND FORCES. GRAVITY: a force that makes any object pull toward another object.

April 07, 2015. Force motion examples.notebook MOTION AND FORCES. GRAVITY: a force that makes any object pull toward another object. Force motion examples.notebook April 07, 2015 MOTION AND FORCES GRAVITY: a force that makes any object pull toward another object Feb 15 12:00 PM 1 FRICTION: a force that acts to slow down moving objects

More information

Rigid body dynamics using Euler s equations, Runge-Kutta and quaternions.

Rigid body dynamics using Euler s equations, Runge-Kutta and quaternions. Rigid body dynamics using Euler s equations, Runge-Kutta and quaternions. Indrek Mandre http://www.mare.ee/indrek/ February 26, 2008 1 Motivation I became interested in the angular dynamics

More information

Lab 1: Measuring of the Acceleration Due to Gravity. Geology 202: Earth s Interior

Lab 1: Measuring of the Acceleration Due to Gravity. Geology 202: Earth s Interior Lab 1: Measuring of the Acceleration Due to Gravity Geology 202: Earth s Interior Introduction: In order to make constraints on Earth s average density o,weneed to know its mass M and volume V. The mass

More information

SOFA. Flexible Plugin. Benjamin Gilles, François Faure, Maxime Tournier, Matthieu Nesme

SOFA. Flexible Plugin. Benjamin Gilles, François Faure, Maxime Tournier, Matthieu Nesme SOFA Flexible Plugin Benjamin Gilles, François Faure, Maxime Tournier, Matthieu Nesme Training Days, Montpellier 22 OCTOBRE 2013 Objectives Deformable solid simulation Modularity Unification of mesh-based

More information

Time Domain and Frequency Domain Techniques For Multi Shaker Time Waveform Replication

Time Domain and Frequency Domain Techniques For Multi Shaker Time Waveform Replication Time Domain and Frequency Domain Techniques For Multi Shaker Time Waveform Replication Thomas Reilly Data Physics Corporation 1741 Technology Drive, Suite 260 San Jose, CA 95110 (408) 216-8440 This paper

More information

Modeling and Control of Multi-Contact Centers of Pressure and Internal Forces in Humanoid Robots

Modeling and Control of Multi-Contact Centers of Pressure and Internal Forces in Humanoid Robots The 2009 IEEE/RSJ International Conference on Intelligent Robots and Systems October 11-15, 2009 St Louis, USA Modeling and Control of Multi-Contact Centers of Pressure and Internal Forces in Humanoid

More information

Chapter 9 Rotation of Rigid Bodies

Chapter 9 Rotation of Rigid Bodies Chapter 9 Rotation of Rigid Bodies 1 Angular Velocity and Acceleration θ = s r (angular displacement) The natural units of θ is radians. Angular Velocity 1 rad = 360o 2π = 57.3o Usually we pick the z-axis

More information

Chapter. 4 Mechanism Design and Analysis

Chapter. 4 Mechanism Design and Analysis Chapter. 4 Mechanism Design and Analysis 1 All mechanical devices containing moving parts are composed of some type of mechanism. A mechanism is a group of links interacting with each other through joints

More information

PHYSICAL QUANTITIES AND UNITS

PHYSICAL QUANTITIES AND UNITS 1 PHYSICAL QUANTITIES AND UNITS Introduction Physics is the study of matter, its motion and the interaction between matter. Physics involves analysis of physical quantities, the interaction between them

More information

Animating reactive motion using momentum-based inverse kinematics

Animating reactive motion using momentum-based inverse kinematics COMPUTER ANIMATION AND VIRTUAL WORLDS Comp. Anim. Virtual Worlds 2005; 16: 213 223 Published online in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/cav.101 Motion Capture and Retrieval

More information

4 is harder than 6: Inverse kinematics for underactuated robots

4 is harder than 6: Inverse kinematics for underactuated robots 4 is harder than 6: Inverse kinematics for underactuated robots Peter Corke February 2014 Many low-cost hobby class robots, such as shown in Figure 1 have only 4 joints (degrees of freedom). This document

More information

Lecture-VIII. Work and Energy

Lecture-VIII. Work and Energy Lecture-VIII Work and Energy The problem: As first glance there seems to be no problem in finding the motion of a particle if we know the force; starting with Newton's second law, we obtain the acceleration,

More information

Epipolar Geometry and Visual Servoing

Epipolar Geometry and Visual Servoing Epipolar Geometry and Visual Servoing Domenico Prattichizzo joint with with Gian Luca Mariottini and Jacopo Piazzi www.dii.unisi.it/prattichizzo Robotics & Systems Lab University of Siena, Italy Scuoladi

More information

Aircraft Flight Dynamics!

Aircraft Flight Dynamics! Aircraft Flight Dynamics Robert Stengel, Princeton University, 2014 Course Overview Introduction to Flight Dynamics Math Preliminaries Copyright 2014 by Robert Stengel. All rights reserved. For educational

More information

A Python Project for Lagrangian Mechanics

A Python Project for Lagrangian Mechanics The 3rd International Symposium on Engineering, Energy and Environments 17-20 November 2013, Pullman King Power Hotel, Bangkok A Python Project for Lagrangian Mechanics Peter Slaets 1, Pauwel Goethals

More information