Dynamics of Serial Manipulators

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Dynamics of Serial Manipulators

• Professor Nicola Ferrier
• ME Room 2246, 265-8793
• ferrier_at_engr.wisc.edu

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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

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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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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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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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Inertia
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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)

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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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Equations of Motion

• Lagrangian Dynamics, continued

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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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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
• DAlembert 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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Notation
The following are w.r.t. reference frame R
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Force on Isolated Link
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Torque on Isolated Link
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Force-torque balance on manipulator
Applied by actuators in z direction
external
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Newtons Law

• A net force acting on body produces a rate of
  change of momentum in accordance with Newtons
  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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Force/Torque on link n
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Newtons Law
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Newton-Euler Algorithm
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Newton-Euler Algorithm

• Compute the inertia tensors,
• Working from the base to the end-effector,
  calculate the positions, velocities, and
  accelerations of the centroids of the manipulator
  links with respect to the link coordinates
  (kinematics)
• 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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Change of coordinates for force/torque
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Recursive Newton-Euler Algorithm
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Two-link manipulator
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Two link planar arm
DH table for two link arm
L2
L1
x0
x1
x2
?2
?1
Z2
Z0
Z1
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Forward Kinematics planar 2-link arm
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Forward Kinematics planar 2-link manipulator
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Forward Kinematics planar 2-link manipulator
w.r.t. base frame 0
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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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Forward Kinematics planar 2-link manipulator
w.r.t. base frame 0
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Point Mass model for two link planar arm
DH table for two link arm
m1
m2
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Dynamic Model of Two Link Arm w/point mass
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General Form
Coriolis centripetal terms
Inertia (mass)
Joint torques
Gravity terms
Joint accelerations
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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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Independent Joint Control revisited

• Called Computed Torque Feedforward in text
• Use dynamic model setpoints (desired position,
  velocity and acceleration from kinematics/trajecto
  ry planning) as a feedforward term

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Manipulator motion from input torques
Integrate to get
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Dynamic Model of Two Link Arm w/point mass
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Dynamics of 2-link point mass
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Dynamics in block diagram of 2-link (point mass)
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Dynamics of 2-link slender rod