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Ch. 7: Dynamics
Example: three link cylindrical robot
• Up to this point, we have developed a
systematic method to determine the forward
and inverse kinematics and the Jacobian for
any arbitrary serial manipulator
– Forward kinematics: mapping from joint
variables to position and orientation of the
end effector
– Inverse kinematics: finding joint variables that
satisfy a given position and orientation of the
end effector
– Jacobian: mapping from the joint velocities to
the end effector linear and angular velocities
• Example: three link cylindrical robot
Why are we studying inertial dynamics and
control?
Kinematic vs dynamic models:
• What we’re really doing is modeling the manipulator
• Kinematic models
• Simple control schemes
• Good approximation for manipulators at low velocities and
accelerations when inertial coupling between links is small
• Not so good at higher velocities or accelerations
• Dynamic models
• More complex controllers
• More accurate
Methods to Analyze Dynamics
• Two methods:
– Energy of the system: Euler-Lagrange method
– Iterative Link analysis: Euler-Newton method
• Each has its own ads and disads.
• In general, they are the same and the results are the same.
Terminology
• Definitions
– Generalized coordinates:
– Vector norm: measure of the magnitude of a vector
• 2-norm:
– Inner product:
Euler-Lagrange Equations
• We can derive the equations of motion for any nDOF system by
using energy methods
Ex: 1DOF system
• To illustrate, we derive the equations of motion for a 1DOF
system
– Consider a particle of mass m
– Using Newton’s second law:
Euler-Lagrange Equations
• If we represent the variables of the system as generalized
coordinates, then we can write the equations of motion for an
nDOF system as:
d L L

 i
dt q i qi
Ex: 1DOF system
Ex: 1DOF system
• Let the total inertia, J, be defined by:
J  r 2Jm  Jl
• :
Inertia
• Inertia, in the body attached frame, is an intrinsic property of a
rigid body
– In the body frame, it is a constant 3x3 matrix:
I xx

I  I ij  I yx
I zx


I xy
I yy
I zy
I xz 

I yz 
I zz 
– The diagonal elements are called the principal moments of inertia
and are a representation of the mass distribution of a body with
respect to an axis of rotation:
Iii   r 2dm   r 2  x, y, z dV   r 2  x, y, z dxdydz
V
V
V
• r is the distance from the axis of rotation to the particle
Inertia
• The elements are defined by:
Center of gravity

  x
  x

 x, y, z dxdydz
 x, y, z dxdydz
I xx   y 2  z 2  x, y , z dxdydz
I yy
(x,y,z) is
the density
I zz
2
z
2
2
y
2
principal
moments
of inertia
I xy  I yx    xy  x, y , z dxdydz
cross
products
of inertia
I xz  I zx    xz x, y , z dxdydz
I yz  I zy    yz x, y , z dxdydz
The
i thpoint
p
ri
pi
The Inertia Matrix
Calculate the moment of inertia of a cuboid
about its centroid:
Since the object is symmetrical about the CG,
all cross products of inertia are zero
z
d
w
x
h
y
Inertia
• First, we need to express the inertia in the body-attached frame
– Note that the rotation between the inertial frame and the body
attached frame is just R
Newton-Euler Formulation
• Rules:
– Every action has an equal reaction
– The rate of change of the linear momentum equals
the total forces applied to the body
dmv
f 
 ma
dt
– The rate change of the angular momentum equals
the total torque applied to the body.
dI OO
O 
dt
Newton-Euler Formulation
• Euler equation
Rii1
Force and Torque Equilibrium
• Force equilibrium
• Torque equilibrium
Angular Velocity and Acceleration
Initial and terminal conditions
F
L2
L1
• Moment of Inertia
Next class…
F
L2
L1
  JTF
  ( L1sin  1  L2 sin(  1   2))  L2 sin(  1   2) 

J  
L2 cos( 1   2) 
 L1cos  1  L2 cos( 1   2)
  ( L1sin 1  L2 sin(  1   2)) L1cos  1  L2 cos( 1   2) 

J T  
 L2 sin(  1   2)
L2 cos( 1   2)


det( J T )  L1L2 sin(  2)
 f 1
F   
 f 2
f 2  f 1 tan  1