Video 3.1 Vijay Kumar and Ani Hsieh

Transcription

1 Video 3.1 Vijay Kumar and Ani Hsieh Robo3x-1.3 1

2 Dynamics of Robot Arms Vijay Kumar and Ani Hsieh University of Pennsylvania Robo3x-1.3 2

3 Lagrange s Equation of Motion Lagrangian Kinetic Energy Potential Energy 1-DOF n-dof Generalized Coordinates Generalized Forces Robo3x-1.3 3

4 Motion of Systems of Particles Center of Mass a 2 O P i m i r OPi a 1 f i a 3 Newton s 2 nd Law Robo3x-1.3 4

5 Rigid Body as a System of Particles F i P Constraints P i Pj F j r OPi r OPj Holonomic Constraints Constraints on position O Robo3x-1.3 5

6 Holonomic Constraints Given a system with k particles and l holonomic constraints Ø DOF = k l Ø n = k l generalized coordinates Ø Ø are independent Robo3x-1.3 6

7 Types of Displacements f i P Actual P i Pj f j Possible r OPi O r OPj Virtual (or Admissible) Robo3x-1.3 7

8 Video 3.2 Vijay Kumar and Ani Hsieh Robo3x-1.3 8

9 Classification of Forces Newtonian Lagrangian Internal vs External Constraint vs Applied Applied Forces: Any forces that are not constraint forces Robo3x-1.3 9

10 D Alembert s Principle The totality of the constraint forces may be disregarded in the dynamics problem for a system of particles Robo3x

11 D Alembert s & Virtual Displacements C i Constraint Surface TC i q TC i Tangent space of C i C i Virtual Displacements satisfy: Eqn of Motion Robo3x

12 Intuition for D Alembert s (1) From Newton s 2 nd Law Robo3x

13 Intuition for D Alembert s (2) By definition and And, b/c motion is constrained and Robo3x

14 D Alembert s Principle Alternative Form: 1. Tangent component of are the only ones to contribute to the particle s acceleration 2. Normal components of are in equilibrium w/ Robo3x

15 Video 3.3 Vijay Kumar and Ani Hsieh Robo3x

16 Principle of Virtual Work The totality of the constraint forces does no virtual work. Virtual Work By D Alembert s Principle Robo3x

17 Lagrange s EOM for Systems of Particles (1) System w/ k particles, l constraints, n = k-l DOF Virtual Work j th generalized force Robo3x

18 Lagrange s EOM for Systems of Particles (2) Note: 1) 2) Robo3x

19 Lagrange s EOM for Systems of Particles (3) Kinetic Energy Robo3x

20 Lagrange s EOM for Systems of Particles (4) And if - Potential Energy - Generalized Applied Forces Robo3x

21 Summary - vector in 3D Virtual work - component in the direction of DO virtual work vs. DO NOT Robo3x

22 Video 3.4 Vijay Kumar and Ani Hsieh Robo3x

23 Potential Energy Robo3x

24 Kinetic Energy Kinetic energy of a rigid body consists of two parts Translational Rotational Inertia Tensor Robo3x

25 Inertia Tensor Principal Cross Moments Products of Inertia 3x3 matrix Symmetric matrix Robo3x

26 Let denote the mass density Principal Moments of Inertia Cross Products of Inertia Robo3x

27 Remarks Inertia tensor depends on reference point coordinate frame VS Robo3x

28 Example Compute the inertia tensor of the block with the given dimensions. Assume is constant. Robo3x

29 Video 3.5 Vijay Kumar and Ani Hsieh Robo3x

30 Potential Energy for n-link Robot 1-Link Robot n-link Robot Robo3x

31 Kinetic Energy for n-link Robot (1) 1-Link Robot n-link Robot Robo3x

32 Review of the Jacobian q R $ f(q) R & f: R $ R & J = f q. δf q $ = f. f. q. q $ f & f & q. q $ J ij = f 5 q 6 Robo3x

33 Kinetic Energy of n-link Robot (2) Robo3x

34 Euler-Lagrange EOM for n-link Robot (1) Assumptions: is quadratic function of and independent of Robo3x

35 Euler-Lagrange EOM for n-link Robot (2) Robo3x

36 Euler-Lagrange EOM for n-link Robot (3) Robo3x

37 Euler-Lagrange EOM for n-link Robot (4) Christoffel Symbols In matrix form Robo3x

38 Skew Symmetry Robo3x

39 Passivity Power = Force x Velocity Energy dissipated over finite time is bounded Important for Controls Robo3x

40 Bounds on D(q) - eigenvalue of Robo3x

41 Linearity in the Parameters System Parameters: Mass, moments of inertia, lengths, etc. Robo3x

42 2-Link Cartesian Manipulator (1) q 2 q 1 y 0 x 0 Robo3x

43 2-Link Cartesian Manipulator (2) q 2 q 1 y 0 x 0 Robo3x

44 2-Link Cartesian Manipulator (3) q 2 q 1 x 0 y 0 Robo3x

45 2-Link Cartesian Manipulator (4) q 2 q 1 y 0 x 0 Robo3x

46 Video 3.6 Vijay Kumar and Ani Hsieh Robo3x

47 2-Link Planar Manipulator (1) System parameters: Link lengths Link center of mass location Link masses y 0 q 2 y 1 P x 1 y 2 Q O q 1 x 0 x 2 Robo3x

48 2-Link Planar Manipulator (2) Recall y 0 q 2 y 1 P x 1 y 2 Q O q 1 x 0 x 2 Robo3x

49 2-Link Planar Manipulator (3) y 1 x 1 y 0 q 2 P y 2 Q O q 1 x 0 x 2 Robo3x

50 2-Link Planar Manipulator (4) Kinetic Energy = Translational + Rotational Translational y 1 x 1 y 0 q 2 P y 2 Q O q 1 x 0 x 2 Robo3x

51 2-Link Planar Manipulator (5) Kinetic Energy = Translational + Rotational Rotational y 1 x 1 y 0 q 2 P y 2 Q O q 1 x 0 x 2 Robo3x

52 2-Link Planar Manipulator (6) Kinetic Energy = Translational + Rotational Rotational y 1 x 1 y 0 q 2 P y 2 Q O q 1 x 0 x 2 Robo3x

53 2-Link Planar Manipulator (7) Kinetic Energy = Translational + Rotational Rotational y 1 x 1 y 0 q 2 P y 2 Q O q 1 x 0 x 2 Robo3x

54 Video 3.7 Vijay Kumar and Ani Hsieh Robo3x

55 2-Link Planar Manipulator (8) Kinetic Energy = Translational + Rotational Rotational y 1 x 1 y 0 q 2 P y 2 Q O q 1 x 0 x 2 Robo3x

56 2-Link Planar Manipulator (6) y 1 x 1 y 0 q 2 P y 2 Q O q 1 x 0 x 2 Robo3x

57 2-Link Planar Manipulator (7) Christoffel Symbols Robo3x

58 2-Link Planar Manipulator (8) Potential Energy y 1 x 1 y 0 q 2 P y 2 Q O q 1 x 0 x 2 Robo3x

59 2-Link Planar Manipulator (9) Putting it all together y 1 x 1 1 y 0 q 2 P y 2 Q O q 1 x 0 x 2 Robo3x

60 Newton-Euler vs. Euler-Lagrange Ø N-E: Newton s Laws of Motion Ø N-E: Explicit accounting for constraints Ø N-E: Explicit accounting of the reference frame Ø E-L: D Alembert s Principle + Principle of Virtual Work Ø E-L: Invariant under point transformations Robo3x

61 Summary Lagrangian D Alembert s Principle + Principle of Virtual Work Euler-Lagrange EOM Properties of the E-L EOM Examples: 2 Link Planar Manipulators Robo3x