1. Manipulator Dynamics Lagrange approach Newton-Euler approach
Hamiltonian approach

2. Lagrange dynamics The Lagrangian, “L”, of any system is defined as:
(1)
where:

3. Lagrangian Dynamics for 2-Link Manipulator
The Lagrangian, “L”, of any system is defined as:

4. Lagrange dynamics
Furthermore, we have:

5. Lagrange Equation

6. Two-link manipulator

7. Kinetic energy of n-1 link

8. Potential energy of link-1
potential energy of link-1 can be presented as:

9. kinetic energy of link-2

10. Lagrangian dynamics of 2-link manipulator (cont)

11. Lagrangian dynamics of 2-link manipulator
Kinetic energy of link – 2 is:

12. Lagrangian dynamics of 2-link manupulator
Potential energy of link 2 is:
P2 = - m2gd1C1 – m2gd2C12
where C1 = cos (θ1)
C12 = cos(θ1 + θ2)

13. Total kinetic and potential energy
Kinetic energy: K= K1 + K2

14. Total potential energy
Potential energy: P1 + P2

15. Lagrangian of the system
L = K – P

16. Lagrange dynamics

17. Lagrange dynamics

18. Torque T1

19. Link 2

20. Torque T2

21. Torques - interpretation

22. Torques