dynamics. relationship between the joint actuator torques and the motion of the structure ...
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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