Robot Dynamics (1)

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ROBOT DYNAMICS
DYNAMIC ANALYSIS AND FORCES
Prof. Charlton S. Inao Defence University Debre
Zeit , Ethiopia
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Introduction

• Analysis of forces, torques, inertias, loads, and
  accelerations.
• Derivation of dynamic equations of motion.
• Allows determination of important loads for
  design.
• Assists in the selection of actuators.

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Newtonian Mechanics

• Easier for simpler systems.
• Familiar to users

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Lagrangian Mechanics

• Easier for more complicated systems
• Based on systems energies
• Systematic
• Lagrangian is the difference between kinetic and
  potential energies of the system

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Lagrangian Relationships

• Lagrangian relationships are

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Review of Basics

• Potential energy is energy stored in an object
  due to its position or arrangement.
• Kinetic energy is energy of an object due to its
  movement - its motion

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Illustration
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Lagrangian Mechanics

• Lagrangian field theory is a formalism in
  classical field theory. It is the field-theoretic
  analogue of Lagrangian mechanics
• Lagrangian mechanics is used for discrete
  particles each with a finite number of degrees of
  freedom. Lagrangian field theory applies to
  continua and fields, which have an infinite
  number of degrees of freedom.
• Deals with energy.. The difference between
  Kinetic and Potential energy.

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• In this chapter, were going to learn about a
  whole new way of looking at things.
• Consider the system of a mass on the end of a
  spring. We can analyze this, of course, by using
  F ma to write downm mx -kx.
• The solutions to this equation are sinusoidal
  functions, as we well know.
• We can, however, ?gure things out by using
  another method which doesnt explicitly use
• F ma.
• In many (in fact, probably most) physical
  situations, this new method is far superior to
  using F ma.

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Lagrangian mechanics

• Please see examples in the book for the
  application of Lagrangian mechanics. For the
  following system we can derive equations of
  motion as

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Example 1
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Kinematics Solution
Wangular velocity of the wheel 2rdiameter X
dot linear velocity
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Dynamics Solution
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Notes
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Notes
velocity
distance
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Example 2
Newtonian mechanics VS Lagrangian mechanics
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Solution
a) Lagrangian Mechanics
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b) Newtonian Mechanics
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Example 3
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Solution 3
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a) KEKE cart KE pendulum
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b) PEPE spring PE pendulum
Pendulum
Spring
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b.1) PE pendulum
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LK-P
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Solution 3 continued
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Review/Reference
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Solution 3 continued
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Example 4 (For home study or work out exercise)
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Solution 4
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Solution 4 contd..
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Solution 4 contd..
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Solution 4 contd..
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Solution 4 contd..
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Taking the derivatives of the Langrangian and
substituting into Equation 4.4 yields to the ffg.
Equation of Motion
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Equation of motion in Matrix Form
Solution 4 contd..
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Example 5 (Class Discussion)
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Solution 5
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Solution 5 .. In matrix form
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DETAILED SOLUTION No. 5
Solve for Kinetic Energy for each link
Solve for K1
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RECALL
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Reference Moment of InertiaThin rod about axis
through end perpendicular to length
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Solve for the X and Y component of M2
Horizontal component up to point m2
vertical component up to point m2
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Comcenter of mass
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Differentiate XD and YD to get the velocity and
the velocity of the arm distance up to point m2
U r
Vcos?
U V

Recall
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Calculate the VD ,up to Mass2and square it, for
linear an d angular component of momentum
expression, V2D
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Solve for K2
K1
K2
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Solve for K2
Angular momentum
linear momentum
Recall
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Solve for Lagrangian,L K-P
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Differentiation with respect to time, in terms of
both variables, i. e. r,?
1st
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Differentiation of L with respect to ?
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Solve for Lagrangian, K-P
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Detailed calculation for VD2
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• THE END