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