Download Kinetic Energy of Link

Dynamics
Dynamics



relationship between the joint actuator torques
and the motion of the structure
Derivation of dynamic model of a manipulator

Simulation of motion

Design of control algorithms

Analysis of manipulator structures
Method based on Lagrange formulation
Lagrange Formulation

Generalized coordinates


n variables which describe the link positions of an ndegree-of-mobility manipulator
The Lagrange of the mechanical system
Lagrange Formulation

The Lagrange of the mechanical system
Function of
generalized
coordinates
Kinetic
energy
Potential
energy
Lagrange Formulation

The Lagrange’s equations

Generalized force


Given by the nonconservative force
Joint actuator torques, joint friction torques, joint
torques induced by interaction with environment
Lagrange Formulation
Example 4.1
Actuation
torque
Reduction
gear ratio
Generalized coordinate?
Rotor
inertia
Kinetic energy?
Viscous
friction
Stator is fixed on the
previous link
Potential energy?
Initial position
Lagrange Formulation
Example 4.1

Generalized coordinate: theta
Kinetic energy

Potential energy

Lagrange Formulation
Example 4.1

Lagrangian of the system
Lagrange Formulation
Example 4.1

Contributions to the generalized force

Dynamic of the model

Relations between torque and joint position,
velocity and acceleration
Mechanical Structure
 Joint actuator torques are
delivered by the motors
Mechanical transmission
Direct drive
Computation of Kinetic Energy

Consider a manipulator with n rigid links
Kinetic energy
of link i
Kinetic energy of
the motor
actuating joint i.
The motor is
located on link i-1
Kinetic Energy of Link

Kinetic energy of link i is given by
Kinetic Energy of Link

Kinetic energy of a rigid body (appendix B.3)
Tli  12 mli p lTi p li  12 iT I li i
translational
rotational
Kinetic Energy of Link

Translational

Centre of mass

Rotational

Inertia tensor

Inertia tensor is constant when referred to the link
frame (frame parallel to the link frame with origin
at centre of mass)
Constant
inertia tensor
Rotation matrix from link i
frame to the base frame
Kinetic Energy of Link

Express the kinetic energy as a function of the
generalized coordinates of the system, that are the
joint variables

Apply the geometric method for Jacobian
computation to the intermediate link

The kinetic energy of link i is
Kinetic Energy of Motor


Assume that the contribution of the stator is
included in that of the link on which such motor is
located
The kinetic energy to rotor i

On the assumption of rigid transmission
Angular position of the rotor

According to the angular velocity composition rule

Kinetic energy of rotor
attention
Kinetic Energy of Manipulator
Computation of Potential Energy

Consider a manipulator with n rigid links
Equations of Motion
Equations of Motion
Equations of Motion



For the acceleration terms
For the quadratic velocity terms
For the configuration-dependent terms
Joint Space Dynamic Model
Viscous
friction
torques
Coulomb
friction
torques
Actuation
torques
Force and moment
exerted on the
environment
Multi-input-multi-output; Strong coupling; Nonlinearity