Video 3.1 Vijay Kumar and Ani Hsieh Robo3x-1.3 1 Property of - - PowerPoint PPT Presentation

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Video 3.1 Vijay Kumar and Ani Hsieh Robo3x-1.3 1 Property of - - PowerPoint PPT Presentation

Oct 09, 2023 174 likes •801 views

Video 3.1 Vijay Kumar and Ani Hsieh Robo3x-1.3 1 Property of Penn Engineering, Vijay Kumar and Ani Hsieh Dynamics of Robot Arms Vijay Kumar and Ani Hsieh University of Pennsylvania Robo3x-1.3 2 Property of Penn Engineering, Vijay

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Property of Penn Engineering, Vijay Kumar and Ani Hsieh

Video 3.1 Vijay Kumar and Ani Hsieh

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Property of Penn Engineering, Vijay Kumar and Ani Hsieh

Vijay Kumar and Ani Hsieh University of Pennsylvania

Dynamics of Robot Arms

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Lagrange’s Equation of Motion

Lagrangian Kinetic Energy Potential Energy 1-DOF n-DOF

Generalized Coordinates Generalized Forces

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Motion of Systems of Particles

  • Center of Mass

O

a1 a3

mi

a2

rOPi

Pi

Newton’s 2nd Law

fi

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Rigid Body as a System of Particles

  • Constraints
  • Holonomic Constraints
  • Constraints on position

Fi

Pi

Fj

Pj O

rOPi

rOPj P

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Holonomic Constraints

  • Given a system with k particles and l

holonomic constraints

Ø DOF = k – l Ø n = k – l generalized coordinates Ø Ø are independent

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Types of Displacements

  • Actual
  • Possible
  • Virtual (or Admissible)

fi

Pi

fj

Pj O

rOPi

rOPj P

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Video 3.2 Vijay Kumar and Ani Hsieh

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Classification of Forces

Lagrangian Constraint vs Applied Applied Forces:

Any forces that are not constraint forces

Newtonian Internal vs External

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D’Alembert’s Principle

The totality of the constraint forces may be disregarded in the dynamics problem for a system

  • f particles
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D’Alembert’s & Virtual Displacements

  • Ci – Constraint Surface
  • TCi – Tangent space of Ci
  • Virtual Displacements

satisfy:

1. 2. Eqn of Motion

Ci

q

TCi

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Intuition for D’Alembert’s (1)

From Newton’s 2nd Law

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Intuition for D’Alembert’s (2)

By definition And, and b/c motion is constrained and

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Property of Penn Engineering, Vijay Kumar and Ani Hsieh

D’Alembert’s Principle

Alternative Form:

  • 1. Tangent component of are the only ones to

contribute to the particle’s acceleration

  • 2. Normal components of are in equilibrium

w/

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Video 3.3 Vijay Kumar and Ani Hsieh

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Principle of Virtual Work

The totality of the constraint forces does no virtual work.

Virtual Work

By D’Alembert’s Principle

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Lagrange’s EOM for Systems of Particles (1)

System w/ k particles, l constraints, n = k-l DOF

Virtual Work jth generalized force

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Lagrange’s EOM for Systems of Particles (2)

Note: 1) 2)

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Lagrange’s EOM for Systems of Particles (3)

Kinetic Energy

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Lagrange’s EOM for Systems of Particles (4)

And if

  • Potential Energy
  • Generalized Applied

Forces

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Summary

  • vector in 3D
  • Virtual work
  • component in the direction of

DO virtual work vs. DO NOT

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Video 3.4 Vijay Kumar and Ani Hsieh

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Potential Energy

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Kinetic Energy

Kinetic energy of a rigid body consists of two parts Inertia Tensor

Translational Rotational

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Inertia Tensor

  • 3x3 matrix
  • Symmetric matrix

Principal Moments of Inertia Cross Products of Inertia

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Property of Penn Engineering, Vijay Kumar and Ani Hsieh

Let denote the mass density

Cross Products

  • f Inertia

Principal Moments of Inertia

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Remarks

Inertia tensor depends on

  • reference point
  • coordinate frame

VS

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Example

Compute the inertia tensor of the block with the given dimensions. Assume is constant.

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Video 3.5 Vijay Kumar and Ani Hsieh

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Potential Energy for n-Link Robot

  • 1-Link Robot
  • n-Link Robot
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Kinetic Energy for n-Link Robot (1)

  • 1-Link Robot
  • n-Link Robot
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Review of the Jacobian

!: ℝ$ → ℝ& ' ∈ ℝ$ !(') ∈ ℝ& = ,! ,-. … δ! ,-$ = ,1

.

,-. ⋯ ,1

.

,-$ ⋮ ⋱ ⋮ ,1

&

,-. ⋯ ,1

&

,-$ J

Jij

= ,1

5

,-6

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Kinetic Energy of n-Link Robot (2)

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Euler-Lagrange EOM for n-Link Robot (1)

Assumptions:

  • is quadratic function of
  • and independent of
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Euler-Lagrange EOM for n-Link Robot (2)

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Euler-Lagrange EOM for n-Link Robot (3)

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Euler-Lagrange EOM for n-Link Robot (4)

Christoffel Symbols In matrix form

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Skew Symmetry

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Passivity

  • Power = Force x Velocity
  • Energy dissipated over finite time is bounded
  • Important for Controls
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Bounds on D(q)

  • eigenvalue of
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Linearity in the Parameters

System Parameters:

  • Mass, moments of inertia, lengths, etc.
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2-Link Cartesian Manipulator (1)

x0 y0

q1 q2

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2-Link Cartesian Manipulator (2)

x0 y0

q1 q2

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2-Link Cartesian Manipulator (3)

y0 x0

q1 q2

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2-Link Cartesian Manipulator (4)

x0 y0

q1 q2

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Video 3.6 Vijay Kumar and Ani Hsieh

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2-Link Planar Manipulator (1)

System parameters:

  • Link lengths
  • Link center of mass location
  • Link masses

y0 x0 x1 y1 x2 y2

P Q O

q1 q2

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2-Link Planar Manipulator (2)

Recall

y0 x0 x1 y1 x2 y2

P Q O

q1 q2

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2-Link Planar Manipulator (3)

y0 x0 x1 y1 x2 y2

P Q O

q1 q2

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2-Link Planar Manipulator (4)

Kinetic Energy = Translational + Rotational Translational

y0 x0 x1 y1 x2 y2

P Q O

q1 q2

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2-Link Planar Manipulator (5)

Kinetic Energy = Translational + Rotational Rotational

y0 x0 x1 y1 x2 y2

P Q O

q1 q2

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2-Link Planar Manipulator (6)

Kinetic Energy = Translational + Rotational Rotational

y0 x0 x1 y1 x2 y2

P Q O

q1 q2

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2-Link Planar Manipulator (7)

Kinetic Energy = Translational + Rotational Rotational

y0 x0 x1 y1 x2 y2

P Q O

q1 q2

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Video 3.7 Vijay Kumar and Ani Hsieh

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2-Link Planar Manipulator (8)

Kinetic Energy = Translational + Rotational Rotational

y0 x0 x1 y1 x2 y2

P Q O

q1 q2

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2-Link Planar Manipulator (6)

y0 x0 x1 y1 x2 y2

P Q O

q1 q2

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2-Link Planar Manipulator (7)

Christoffel Symbols

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2-Link Planar Manipulator (8)

Potential Energy

y0 x0 x1 y1 x2 y2

P Q O

q1 q2

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2-Link Planar Manipulator (9)

Putting it all together

y0 x0 x1 y1 x2 y2

P Q O

q1 q2

1

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Newton-Euler vs. Euler-Lagrange

Ø N-E: Newton’s Laws of Motion Ø N-E: Explicit accounting for constraints Ø N-E: Explicit accounting of the reference

frame

Ø E-L: D’Alembert’s Principle + Principle of

Virtual Work

Ø E-L: Invariant under point transformations

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Summary

  • Lagrangian
  • D’Alembert’s Principle + Principle of Virtual

Work

  • Euler-Lagrange EOM
  • Properties of the E-L EOM
  • Examples: 2 Link Planar Manipulators