ROBOT DYNAMICS
T. Bajd and M. Mihelj
T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010
Robot dynamics
• In contrast to kinematics, dynamics represents the part of
mechanics, which is interested into the forces and torques
which are producing the motion of a mechanism.
• The analysis of robot dynamics enables us to consider
– the torques necessary to compensate the gravity forces of robot
segments,
– the differences in moments of inertia occurring during the robot
motion,
– dynamic couplings caused by simultaneous movements of all
robot segments.
T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010
Forward and inverse dynamics
Applied torques
T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010
Joint motions
Two-segment robot mechanism
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The dynamic analysis of a robot is based on a twosegment robot mechanism.
The motion of the robot manipulator with two
rotational joints occurs in the vertical plane.
Both segments are of equal length.
The dynamic model is simplified by assuming that
the whole mass of each segment is concentrated in
its center of mass.
Such a pair of segments appears both in the
anthropomorphic and in the SCARA robot structures.
The robot trajectory is denoted by the two joint
angles.
The robot is placed into the fixed reference frame
with z axis aligned with the axis of the first joint.
T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010
Torque in the second joint
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T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010
Position, velocity and acceleration
of the center of mass of the
second segment
Torque in the second joint
•
The motion of the second segment
mass is given by Newton’s law
•
In addition to the force of gravity,
the mass
is acted upon by the
force , transmitted by the
massless segment
T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010
Center of mass acceleration
• Robot segments and are rigid, thus
Centripetal acceleration
T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010
Tangential acceleration
Torque in the second joint
• The torque in the second joint is
• or
T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010
Torque in the second joint
• Considering
• the torque in the second joint is
• With
Inertial coupling
Inertial
T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010
Centrifugal
Gravitational
Torque in the first joint
• Relation between the total torque of external forces and
the time derivative of the angular momentum
• The sum of the torques produced by the external forces
T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010
Angular momentum
• The angular momentum of the mass
equals
• with
• The angular momentum of the mass
• with
T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010
equals
Torque in the first joint
• With
Inertial
Inertial coupling
Coriolis
Centrifugal
Gravitational
T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010
Dynamic model in matrix form
• The torques in the robot joints can be written in the
following general form
• where
T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010
Inertial matrix
b11
b21
b12
b22
• Inertial matrix
T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010
Coriolis and centrifugal terms
c11
c12
c21
• Coriolis and centrifugal terms
T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010
Gravitational terms
g1
g2
• Gravitational terms
T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010
Forward and inverse dynamic model
• Inverse dynamic model with friction (diagonal matrix of
the joint friction coefficients )
• Forward dynamic model with friction
T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010
Forward dynamic model block scheme
T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010