Ch. 7: Dynamics

1. Ch. 7: Dynamics

2. Example: three link cylindrical robot • Up to this point, we have developed a systematic method to determine the forward and inverse kinematics and the Jacobian for any arbitrary serial manipulator • Forward kinematics: mapping from joint variables to position and orientation of the end effector • Inverse kinematics: finding joint variables that satisfy a given position and orientation of the end effector • Jacobian: mapping from the joint velocities to the end effector linear and angular velocities • Example: three link cylindrical robot

3. Why are we studying inertial dynamics and control? • Kinematic vs dynamic models: • What we’re really doing is modeling the manipulator • Kinematic models • Simple control schemes • Good approximation for manipulators at low velocities and accelerations when inertial coupling between links is small • Not so good at higher velocities or accelerations • Dynamic models • More complex controllers • More accurate

4. Methods to Analyze Dynamics • Two methods: • Energy of the system: Euler-Lagrange method • Iterative Link analysis: Euler-Newton method • Each has its own ads and disads. • In general, they are the same and the results are the same.

5. Terminology • Definitions • Generalized coordinates: • Vector norm: measure of the magnitude of a vector • 2-norm: • Inner product:

6. Euler-Lagrange Equations • We can derive the equations of motion for any nDOF system by using energy methods

7. Ex: 1DOF system • To illustrate, we derive the equations of motion for a 1DOF system • Consider a particle of mass m • Using Newton’s second law:

8. Euler-Lagrange Equations • If we represent the variables of the system as generalized coordinates, then we can write the equations of motion for an nDOF system as:

9. Ex: 1DOF system

10. Ex: 1DOF system • Let the total inertia, J, be defined by: • :

11. Inertia • Inertia, in the body attached frame, is an intrinsic property of a rigid body • In the body frame, it is a constant 3x3 matrix: • The diagonal elements are called the principal moments of inertia and are a representation of the mass distribution of a body with respect to an axis of rotation: • r is the distance from the axis of rotation to the particle

12. Inertia The point • The elements are defined by: Center of gravity principal moments of inertia r(x,y,z) is the density cross products of inertia

13. The Inertia Matrix Calculate the moment of inertia of a cuboid about its centroid: Since the object is symmetrical about the CG, all cross products of inertia are zero

14. Inertia • First, we need to express the inertia in the body-attached frame • Note that the rotation between the inertial frame and the body attached frame is just R

15. Newton-Euler Formulation • Rules: • Every action has an equal reaction • The rate of change of the linear momentum equals the total forces applied to the body • The rate change of the angular momentum equals the total torque applied to the body.

16. Newton-Euler Formulation • Euler equation

19. Force and Torque Equilibrium • Force equilibrium • Torque equilibrium

20. Angular Velocity and Acceleration

21. Initial and terminal conditions

22. Next class… F L2 L1 • Moment of Inertia

23. F L2 L1