dynamics of serial manipulators
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ME 439 Professor N. J. Ferrier
Dynamics of Serial Manipulators
Professor Nicola FerrierME 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
ME 439 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
ME 439 Professor N. J. Ferrier
Manipulator Link Mass• The density at a position x is (x),
– usually 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
ME 439 Professor N. J. Ferrier
Inertia
ME 439 Professor N. J. Ferrier
• 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)
Equations of Motion
ME 439 Professor N. J. Ferrier
• Lagrangian using generalized coordinates:
• The equations of motion for a mechanical system with generalized coordinates are:
– External force vector
– i is the external force acting on the ith general coordinate
Equations of Motion
ME 439 Professor N. J. Ferrier
Equations of Motion
• Lagrangian Dynamics, continued
ME 439 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
ME 439 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)
ME 439 Professor N. J. Ferrier
Notation
The following are w.r.t. reference frame R:
ME 439 Professor N. J. Ferrier
Force on Isolated Link
ME 439 Professor N. J. Ferrier
Torque on Isolated Link
ME 439 Professor N. J. Ferrier
external
Applied by actuators in z direction
Force-torque balance on manipulator
ME 439 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
ME 439 Professor N. J. Ferrier
Force/Torque on link n
ME 439 Professor N. J. Ferrier
Newton’s Law
ME 439 Professor N. J. Ferrier
Newton-Euler Algorithm
ME 439 Professor N. J. Ferrier
Newton-Euler Algorithm
1. Compute the inertia tensors, 2. 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)
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
ME 439 Professor N. J. Ferrier
“Change of coordinates” for force/torque
ME 439 Professor N. J. Ferrier
Recursive Newton-Euler Algorithm
ME 439 Professor N. J. Ferrier
Two-link manipulator
ME 439 Professor N. J. Ferrier
Two link planar arm
Z0 1
DH table for two link arm
x0 x2
Z2
Link Var d a
1 1 1 0 0 L1
2 2 2 0 0 L2
L1L2
x1
Z1
2
ME 439 Professor N. J. Ferrier
Forward Kinematics: planar 2-link arm
ME 439 Professor N. J. Ferrier
Forward Kinematics: planar 2-link manipulator
ME 439 Professor N. J. Ferrier
Forward Kinematics: planar 2-link manipulator
w.r.t. base frame {0}
ME 439 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 1 to origin of frame 2 expressed in frame 0
position vector from origin of frame 0 to origin of frame 1 expressed in frame 0
ME 439 Professor N. J. Ferrier
Forward Kinematics: planar 2-link manipulator
w.r.t. base frame {0}
ME 439 Professor N. J. Ferrier
Point Mass model for two link planar armDH table for two link arm
m1 m2
ME 439 Professor N. J. Ferrier
Dynamic Model of Two Link Arm w/point mass
ME 439 Professor N. J. Ferrier
General Form
Joint torques
Inertia (mass)
Joint accelerations
Gravity terms
Coriolis & centripetal terms
ME 439 Professor N. J. Ferrier
General Form: No motion
Joint torques required to hold manipulator in a
static position (i.e. counter
gravitational forces)
Gravity terms
No motion so
ME 439 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
ME 439 Professor N. J. Ferrier
Manipulator motion from input torques
Integrate to get
ME 439 Professor N. J. Ferrier
Dynamic Model of Two Link Arm w/point mass
ME 439 Professor N. J. Ferrier
Dynamics of 2-link – point mass
ME 439 Professor N. J. Ferrier
Dynamics in block diagram of 2-link (point mass)
ME 439 Professor N. J. Ferrier
Dynamics of 2-link – slender rod
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