Dynamics of Serial Manipulators

ME 439 Professor N. J. Ferrier

Dynamics of Serial Manipulators

Professor Nicola FerrierME Room 2246, 265-8793

[email protected]


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


Dynamic Modeling

• Need to derive the equations of motion– Relate forces/torque to motion

• Must consider distribution of mass• Need to model external forces


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


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


Inertia


• 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


• 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


Equations of Motion

• Lagrangian Dynamics, continued


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


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)


Notation

The following are w.r.t. reference frame R:


Force on Isolated Link


Torque on Isolated Link


external

Applied by actuators in z direction

Force-torque balance on manipulator


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


Force/Torque on link n


Newton’s Law


Newton-Euler Algorithm


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


“Change of coordinates” for force/torque


Recursive Newton-Euler Algorithm


Two-link manipulator


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


Forward Kinematics: planar 2-link arm


Forward Kinematics: planar 2-link manipulator



w.r.t. base frame {0}



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



w.r.t. base frame {0}


Point Mass model for two link planar armDH table for two link arm

m1 m2


Dynamic Model of Two Link Arm w/point mass


General Form

Joint torques

Inertia (mass)

Joint accelerations

Gravity terms

Coriolis & centripetal terms


General Form: No motion

Joint torques required to hold manipulator in a

static position (i.e. counter

gravitational forces)

Gravity terms

No motion so


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


Manipulator motion from input torques

Integrate to get


Dynamic Model of Two Link Arm w/point mass


Dynamics of 2-link – point mass


Dynamics in block diagram of 2-link (point mass)


Dynamics of 2-link – slender rod

Documents

Dynamics of Serial Manipulators