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 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 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 ME 439 Professor N. J. Ferrier

Inertia ME 439 Professor N. J. Ferrier

Equations of Motion Newton-Euler approach Lagrangian (energy methods) 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 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

Force-torque balance on manipulator Applied by actuators in z direction external 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 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 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 DH table for two link arm L1 L2 x0 x1 x2 2 Z0 1 Var  d  a 1 1 L1 2 2 L2 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 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 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 arm DH 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 Coriolis & centripetal terms Inertia (mass) Joint torques Gravity terms Joint accelerations ME 439 Professor N. J. Ferrier

General Form: No motion No motion so Joint torques required to hold manipulator in a static position (i.e. counter gravitational forces) Gravity terms 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 ME 439 Professor N. J. Ferrier