Dynamics of Serial Manipulators
Professor Nicola Ferrier
ME Room 2246, 265-8793
ferrier@engr.wisc.edu
ME 439
Professor N. J. Ferrier
Dynamic Modeling
• For manipulator arms:
– Relate forces/torques at joints to the motion
of manipulator + load
• External forces usually only considered at the
end-effector
• Gravity (lift arms) is a major consideration
ME 439
Professor N. J. Ferrier
Dynamic Modeling
• Need to derive the equations of motion
– Relate forces/torque to motion
• Must consider distribution of mass
• Need to model external forces
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Professor N. J. Ferrier
Manipulator Link Mass
• Consider link as a system of particles
– Each particle has mass, dm
– Position of each particle can be expressed
using forward kinematics
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Professor N. J. Ferrier
Manipulator Link Mass
• The density at a position x is r(x),
– usually r is assumed constant
• The mass of a body is given by
– where
is the set of material points
that comprise the body
• The center of mass is
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Professor N. J. Ferrier
Inertia
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Professor N. J. Ferrier
Equations of Motion
• Newton-Euler approach
–
–
–
–
P is absolute linear momentum
F is resultant external force
Mo is resultant external moment wrt point o
Ho is moment of momentum wrt point o
• Lagrangian (energy methods)
ME 439
Professor N. J. Ferrier
Equations of Motion
• Lagrangian using generalized coordinates:
• The equations of motion for a mechanical
system with generalized coordinates are:
– External force vector
– ti is the external force acting on the ith general
coordinate
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Professor N. J. Ferrier
Equations of Motion
• Lagrangian Dynamics, continued
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Professor N. J. Ferrier
Equations of Motions
• Robotics texts will use either method to
derive equations of motion
– In “ME 739: Advanced Robotics and
Automation” we use a Lagrangian approach
using computational tools from kinematics
to derive the equations of motion
• For simple robots (planar two link arm),
Newton-Euler approach is straight
forward
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Professor N. J. Ferrier
Manipulator Dynamics
• Isolate each link
– Neighboring links apply external forces and
torques
• Mass of neighboring links
• External force inherited from contact between tip
and an object
• D’Alembert force (if neighboring link is
accelerating)
– Actuator applies either pure torque or pure
force (by DH convention along the z-axis)
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Professor N. J. Ferrier
Notation
The following are w.r.t. reference frame R:
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Professor N. J. Ferrier
Force on Isolated Link
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Professor N. J. Ferrier
Torque on Isolated Link
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Professor N. J. Ferrier
Force-torque balance on manipulator
Applied by
actuators in
z direction
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external
Professor N. J. Ferrier
Newton’s Law
• A net force acting on body produces a
rate of change of momentum in
accordance with Newton’s Law
• The time rate of change of the total
angular momentum of a body about the
origin of an inertial reference frame is
equal to the torque acting on the body
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Professor N. J. Ferrier
Force/Torque on link n
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Professor N. J. Ferrier
Newton’s Law
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Professor N. J. Ferrier
Newton-Euler Algorithm
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Professor N. J. Ferrier
Newton-Euler Algorithm
1. Compute the inertia tensors,
2. Working from the base to the endeffector, calculate the positions,
velocities, and accelerations of the
centroids of the manipulator links with
respect to the link coordinates
(kinematics)
3. Working from the end-effector to the
base of the robot, recursively calculate
the forces and torques at the actuators
with respect to link coordinates
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Professor N. J. Ferrier
“Change of coordinates” for
force/torque
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Professor N. J. Ferrier
Recursive Newton-Euler
Algorithm
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Professor N. J. Ferrier
Two-link manipulator
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Professor N. J. Ferrier
Two link planar arm
DH table for two link arm
L2
L1
x0
Z0 1
Link
1
2
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x1
2
Z1
Var
1
2
x2
Z2

d

a
1
2
0
0
0
0
L1
L2
Professor N. J. Ferrier
Forward Kinematics: planar 2-link arm
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Professor N. J. Ferrier
Forward Kinematics: planar 2-link manipulator
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Professor N. J. Ferrier
Forward Kinematics: planar 2-link manipulator
w.r.t. base
frame {0}
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Professor N. J. Ferrier
Forward Kinematics: planar 2-link manipulator
position vector from origin of
frame 0 to c.o.m. of link 1
expressed in frame 0
position vector from origin
of frame 1 to c.o.m. of link 2
expressed in frame 0
position vector from origin of
frame 0 to origin of frame 1
expressed in frame 0
position vector from origin
of frame 1 to origin of frame
2 expressed in frame 0
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Professor N. J. Ferrier
Forward Kinematics: planar 2-link manipulator
w.r.t. base frame {0}
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Professor N. J. Ferrier
Point Mass model for two link planar arm
DH table for two link arm
m1
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Professor N. J. Ferrier
m2
Dynamic Model of Two Link Arm
w/point mass
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Professor N. J. Ferrier
General Form
Joint
torques
Inertia
(mass)
Coriolis &
centripetal
terms
Joint
accelerations
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Professor N. J. Ferrier
Gravity
terms
General Form: No motion
No motion so
Gravity terms
Joint torques
required to hold
manipulator in a
static position (i.e.
counter
gravitational forces)
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Professor N. J. Ferrier
Independent Joint Control revisited
• Called “Computed Torque Feedforward”
in text
• Use dynamic model + setpoints
(desired position, velocity and
acceleration from kinematics/trajectory
planning) as a feedforward term
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Professor N. J. Ferrier
Manipulator motion from input torques
Integrate to get
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Professor N. J. Ferrier
Dynamic Model of Two Link Arm
w/point mass
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Professor N. J. Ferrier
Dynamics of 2-link – point mass
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Professor N. J. Ferrier
Dynamics in block diagram of 2-link (point mass)
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Professor N. J. Ferrier
Dynamics of 2-link – slender rod
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Professor N. J. Ferrier