Robo3x-1.3 1Property of Penn Engineering, Vijay Kumar and Ani Hsieh

Video 3.1Vijay Kumar and Ani Hsieh

Robo3x-1.3 2Property of Penn Engineering, Vijay Kumar and Ani Hsieh

Vijay Kumar and Ani HsiehUniversity of Pennsylvania

Dynamics of Robot Arms

Robo3x-1.3 3Property of Penn Engineering, Vijay Kumar and Ani Hsieh

Lagrange’s Equation of Motion

Lagrangian

Kinetic Energy

Potential Energy

1-DOF n-DOF

Generalized Coordinates

Generalized Forces

Robo3x-1.3 4Property of Penn Engineering, Vijay Kumar and Ani Hsieh

Motion of Systems of Particles

• Center of Mass

Oa1

a3

mia2

rOPi

Pi

Newton’s 2nd Law

fi

Robo3x-1.3 5Property of Penn Engineering, Vijay Kumar and Ani Hsieh

Rigid Body as a System of Particles

• Constraints

• Holonomic Constraints

• Constraints on position

Fi

Pi

FjPj

O

rOPi rOPj

P

Robo3x-1.3 6Property of Penn Engineering, Vijay Kumar and Ani Hsieh

Holonomic Constraints

• Given a system with k particles and lholonomic constraints

Ø DOF = k – l

Ø n = k – l generalized coordinates

Ø

Ø are independent

Robo3x-1.3 7Property of Penn Engineering, Vijay Kumar and Ani Hsieh

Types of Displacements

• Actual

• Possible

• Virtual (or Admissible)

fi

Pi

fjPj

O

rOPi rOPj

P

Robo3x-1.3 8Property of Penn Engineering, Vijay Kumar and Ani Hsieh

Video 3.2Vijay Kumar and Ani Hsieh

Robo3x-1.3 9Property of Penn Engineering, Vijay Kumar and Ani Hsieh

Classification of Forces

Lagrangian

Constraint vs Applied

Applied Forces:Any forces that are notconstraint forces

Newtonian

Internal vs External

Robo3x-1.3 10Property of Penn Engineering, Vijay Kumar and Ani Hsieh

D’Alembert’s Principle

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

Robo3x-1.3 11Property of Penn Engineering, Vijay Kumar and Ani Hsieh

D’Alembert’s & Virtual Displacements

• Ci – Constraint Surface

• TCi – Tangent space of Ci

• Virtual Displacements

satisfy:

1.

2. Eqn of Motion

Ci

qTCi

Robo3x-1.3 12Property of Penn Engineering, Vijay Kumar and Ani Hsieh

Intuition for D’Alembert’s (1)

From Newton’s 2nd Law

Robo3x-1.3 13Property of Penn Engineering, Vijay Kumar and Ani Hsieh

Intuition for D’Alembert’s (2)

By definition

And,

and

b/c motion is constrained

and

Robo3x-1.3 14Property 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/

Robo3x-1.3 15Property of Penn Engineering, Vijay Kumar and Ani Hsieh

Video 3.3Vijay Kumar and Ani Hsieh

Robo3x-1.3 16Property of Penn Engineering, Vijay Kumar and Ani Hsieh

Principle of Virtual Work

The totality of the constraint forces does no virtual work.

Virtual Work

By D’Alembert’s Principle

Robo3x-1.3 17Property of Penn Engineering, Vijay Kumar and Ani Hsieh

Lagrange’s EOM for Systems of Particles (1)

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

Virtual Work

jth generalized force

Robo3x-1.3 18Property of Penn Engineering, Vijay Kumar and Ani Hsieh

Lagrange’s EOM for Systems of Particles (2)

Note: 1)

2)

Robo3x-1.3 19Property of Penn Engineering, Vijay Kumar and Ani Hsieh

Lagrange’s EOM for Systems of Particles (3)

Kinetic Energy

Robo3x-1.3 20Property of Penn Engineering, Vijay Kumar and Ani Hsieh

Lagrange’s EOM for Systems of Particles (4)

And if - Potential Energy

- Generalized Applied Forces

Robo3x-1.3 21Property of Penn Engineering, Vijay Kumar and Ani Hsieh

Summary

• - vector in 3D

• Virtual work

• - component in the direction of

DO virtual work vs. DO NOT

Robo3x-1.3 22Property of Penn Engineering, Vijay Kumar and Ani Hsieh

Video 3.4Vijay Kumar and Ani Hsieh

Robo3x-1.3 23Property of Penn Engineering, Vijay Kumar and Ani Hsieh

Potential Energy

Robo3x-1.3 24Property of Penn Engineering, Vijay Kumar and Ani Hsieh

Kinetic Energy

Kinetic energy of a rigid body consists of two parts

Inertia Tensor

Translational Rotational

Robo3x-1.3 25Property of Penn Engineering, Vijay Kumar and Ani Hsieh

Inertia Tensor

• 3x3 matrix

• Symmetric matrix

Principal Moments of

Inertia

Cross Products of

Inertia

Robo3x-1.3 26Property of Penn Engineering, Vijay Kumar and Ani Hsieh

Let denote the mass density

Cross Products of Inertia

Principal Moments of

Inertia

Robo3x-1.3 27Property of Penn Engineering, Vijay Kumar and Ani Hsieh

Remarks

Inertia tensor depends on

• reference point

• coordinate frame

VS

Robo3x-1.3 28Property of Penn Engineering, Vijay Kumar and Ani Hsieh

Example

Compute the inertia tensor of the block with the given dimensions.

Assume is constant.

Robo3x-1.3 29Property of Penn Engineering, Vijay Kumar and Ani Hsieh

Video 3.5Vijay Kumar and Ani Hsieh

Robo3x-1.3 30Property of Penn Engineering, Vijay Kumar and Ani Hsieh

Potential Energy for n-Link Robot

• 1-Link Robot

• n-Link Robot

Robo3x-1.3 31Property of Penn Engineering, Vijay Kumar and Ani Hsieh

Kinetic Energy for n-Link Robot (1)

• 1-Link Robot

• n-Link Robot

Robo3x-1.3 32Property of Penn Engineering, Vijay Kumar and Ani Hsieh

Review of the Jacobian

!: ℝ$ → ℝ&' ∈ ℝ$!(') ∈ ℝ&

= ,!,-.

… δ!,-$

=

,1.,-.

⋯ ,1.,-$

⋮ ⋱ ⋮,1&,-.

⋯ ,1&,-$

J

Jij = ,15,-6

Robo3x-1.3 33Property of Penn Engineering, Vijay Kumar and Ani Hsieh

Kinetic Energy of n-Link Robot (2)

Robo3x-1.3 34Property of Penn Engineering, Vijay Kumar and Ani Hsieh

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

Assumptions:

• is quadratic function of

• and independent of

Robo3x-1.3 35Property of Penn Engineering, Vijay Kumar and Ani Hsieh

Euler-Lagrange EOM for n-Link Robot (2)

Robo3x-1.3 36Property of Penn Engineering, Vijay Kumar and Ani Hsieh

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

Robo3x-1.3 37Property of Penn Engineering, Vijay Kumar and Ani Hsieh

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

Christoffel Symbols

In matrix form

Robo3x-1.3 38Property of Penn Engineering, Vijay Kumar and Ani Hsieh

Skew Symmetry

Robo3x-1.3 39Property of Penn Engineering, Vijay Kumar and Ani Hsieh

Passivity

• Power = Force x Velocity

• Energy dissipated over finite time is bounded

• Important for Controls

Robo3x-1.3 40Property of Penn Engineering, Vijay Kumar and Ani Hsieh

Bounds on D(q)

• - eigenvalue of

•

Robo3x-1.3 41Property of Penn Engineering, Vijay Kumar and Ani Hsieh

Linearity in the Parameters

System Parameters:

• Mass, moments of inertia, lengths, etc.

Robo3x-1.3 42Property of Penn Engineering, Vijay Kumar and Ani Hsieh

2-Link Cartesian Manipulator (1)

x0

y0q1

q2

Robo3x-1.3 43Property of Penn Engineering, Vijay Kumar and Ani Hsieh

2-Link Cartesian Manipulator (2)

x0

y0q1

q2

Robo3x-1.3 44Property of Penn Engineering, Vijay Kumar and Ani Hsieh

2-Link Cartesian Manipulator (3)

y0

x0q1

q2

Robo3x-1.3 45Property of Penn Engineering, Vijay Kumar and Ani Hsieh

2-Link Cartesian Manipulator (4)

x0

y0q1

q2

Robo3x-1.3 46Property of Penn Engineering, Vijay Kumar and Ani Hsieh

Video 3.6Vijay Kumar and Ani Hsieh

Robo3x-1.3 47Property of Penn Engineering, Vijay Kumar and Ani Hsieh

2-Link Planar Manipulator (1)

System parameters:

• Link lengths

• Link center of mass location

• Link masses

y0

x0

x1y1

x2

y2P Q

Oq1

q2

Robo3x-1.3 48Property of Penn Engineering, Vijay Kumar and Ani Hsieh

2-Link Planar Manipulator (2)

Recall

y0

x0

x1y1

x2

y2P Q

Oq1

q2

Robo3x-1.3 49Property of Penn Engineering, Vijay Kumar and Ani Hsieh

2-Link Planar Manipulator (3)

y0

x0

x1y1

x2

y2P Q

Oq1

q2

Robo3x-1.3 50Property of Penn Engineering, Vijay Kumar and Ani Hsieh

2-Link Planar Manipulator (4)

Kinetic Energy = Translational + Rotational

Translational

y0

x0

x1y1

x2

y2P Q

Oq1

q2

Robo3x-1.3 51Property of Penn Engineering, Vijay Kumar and Ani Hsieh

2-Link Planar Manipulator (5)

Kinetic Energy = Translational + Rotational

Rotational

y0

x0

x1y1

x2

y2P Q

Oq1

q2

Robo3x-1.3 52Property of Penn Engineering, Vijay Kumar and Ani Hsieh

2-Link Planar Manipulator (6)

Kinetic Energy = Translational + Rotational

Rotational

y0

x0

x1y1

x2

y2P Q

Oq1

q2

Robo3x-1.3 53Property of Penn Engineering, Vijay Kumar and Ani Hsieh

2-Link Planar Manipulator (7)

Kinetic Energy = Translational + Rotational

Rotational

y0

x0

x1y1

x2

y2P Q

Oq1

q2

Robo3x-1.3 54Property of Penn Engineering, Vijay Kumar and Ani Hsieh

Video 3.7Vijay Kumar and Ani Hsieh

Robo3x-1.3 55Property of Penn Engineering, Vijay Kumar and Ani Hsieh

2-Link Planar Manipulator (8)

Kinetic Energy = Translational + Rotational

Rotational

y0

x0

x1y1

x2

y2P Q

Oq1

q2

Robo3x-1.3 56Property of Penn Engineering, Vijay Kumar and Ani Hsieh

2-Link Planar Manipulator (6)

y0

x0

x1y1

x2

y2P Q

Oq1

q2

Robo3x-1.3 57Property of Penn Engineering, Vijay Kumar and Ani Hsieh

2-Link Planar Manipulator (7)

Christoffel Symbols

Robo3x-1.3 58Property of Penn Engineering, Vijay Kumar and Ani Hsieh

2-Link Planar Manipulator (8)

Potential Energy

y0

x0

x1y1

x2

y2P Q

Oq1

q2

Robo3x-1.3 59Property of Penn Engineering, Vijay Kumar and Ani Hsieh

2-Link Planar Manipulator (9)

Putting it all together

y0

x0

x1y1

x2

y2P Q

Oq1

q2

1

Robo3x-1.3 60Property of Penn Engineering, Vijay Kumar and Ani Hsieh

Newton-Euler vs. Euler-Lagrange

Ø N-E: Newton’s Laws of MotionØ N-E: Explicit accounting for constraintsØ 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

Robo3x-1.3 61Property of Penn Engineering, Vijay Kumar and Ani Hsieh

Summary

• Lagrangian

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

• Euler-Lagrange EOM

• Properties of the E-L EOM

• Examples: 2 Link Planar Manipulators