Dynamics

DynamicsDynamics

relationship between the joint actuator torques relationship between the joint actuator torques and the motion of the structureand the motion of the structure

Derivation of dynamic model of a manipulatorDerivation of dynamic model of a manipulator

Simulation of motion

Design of control algorithms

Analysis of manipulator structures

Method based on Lagrange formulationMethod based on Lagrange formulation

Lagrange FormulationLagrange Formulation

Generalized coordinatesGeneralized coordinates n variables which describe the link positions of an n-

degree-of-mobility manipulator

The Lagrange of the mechanical system

Lagrange FormulationLagrange Formulation

The Lagrange of the mechanical system

Function of generalized coordinates

Kinetic energy

Potential energy

Lagrange FormulationLagrange Formulation

The Lagrange’s equations

Generalized force Given by the nonconservative force Joint actuator torques, joint friction torques, joint torqu

es induced by interaction with environment

Lagrange Formulation Example 4.1

Lagrange Formulation Example 4.1

Rotor inertia

Reduction gear ratio

Stator is fixed on the previous link

Actuation torque

Viscous friction

Initial position

Generalized coordinate?

Kinetic energy?

Potential energy?

Lagrange Formulation Example 4.1

Lagrange Formulation Example 4.1

Generalized coordinate: theta Kinetic energy

Potential energy

Lagrange Formulation Example 4.1

Lagrange Formulation Example 4.1

Lagrangian of the system

Lagrange Formulation Example 4.1

Lagrange Formulation Example 4.1

Contributions to the generalized force

Dynamic of the model

Relations between torque and joint position, velocity and acceleration

Mechanical StructureMechanical Structure Joint actuator torques are delivered by the motors

Mechanical transmission

Direct drive

Computation of Kinetic EnergyComputation 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 LinkKinetic Energy of Link

Kinetic energy of link i is given by

Kinetic Energy of LinkKinetic Energy of Link

Kinetic energy of a rigid body (appendix B.3)

ilTil

Tlll iiiii

IppmT 21

21

translational rotational

Kinetic Energy of LinkKinetic 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 LinkKinetic 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 MotorKinetic 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

According to the angular velocity composition rule

Angular position of the rotor

attention Kinetic energy of rotor

Kinetic Energy of ManipulatorKinetic Energy of Manipulator

Computation of Potential EnergyComputation of Potential Energy

Consider a manipulator with n rigid links

Equations of MotionEquations of Motion

Equations of MotionEquations of Motion

Equations of MotionEquations of Motion

For the acceleration terms For the quadratic velocity terms For the configuration-dependent terms

Joint Space Dynamic ModelJoint Space Dynamic Model

Viscous friction torques Coulomb

friction torques

Actuation torques

Force and moment exerted on the environment

Multi-input-multi-output; Strong coupling; NonlinearityMulti-input-multi-output; Strong coupling; Nonlinearity