Video 3.1 Vijay Kumar and Ani Hsieh Robo3x-1.3 1
Dynamics of Robot Arms Vijay Kumar and Ani Hsieh University of Pennsylvania Robo3x-1.3 2
Lagrange s Equation of Motion Lagrangian Kinetic Energy Potential Energy 1-DOF n-dof Generalized Coordinates Generalized Forces Robo3x-1.3 3
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
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
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
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
Video 3.2 Vijay Kumar and Ani Hsieh Robo3x-1.3 8
Classification of Forces Newtonian Lagrangian Internal vs External Constraint vs Applied Applied Forces: Any forces that are not constraint forces Robo3x-1.3 9
D Alembert s Principle The totality of the constraint forces may be disregarded in the dynamics problem for a system of particles Robo3x-1.3 10
D Alembert s & Virtual Displacements C i Constraint Surface TC i q TC i Tangent space of C i C i Virtual Displacements satisfy: 1. 2. Eqn of Motion Robo3x-1.3 11
Intuition for D Alembert s (1) From Newton s 2 nd Law Robo3x-1.3 12
Intuition for D Alembert s (2) By definition and And, b/c motion is constrained and Robo3x-1.3 13
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-1.3 14
Video 3.3 Vijay Kumar and Ani Hsieh Robo3x-1.3 15
Principle of Virtual Work The totality of the constraint forces does no virtual work. Virtual Work By D Alembert s Principle Robo3x-1.3 16
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-1.3 17
Lagrange s EOM for Systems of Particles (2) Note: 1) 2) Robo3x-1.3 18
Lagrange s EOM for Systems of Particles (3) Kinetic Energy Robo3x-1.3 19
Lagrange s EOM for Systems of Particles (4) And if - Potential Energy - Generalized Applied Forces Robo3x-1.3 20
Summary - vector in 3D Virtual work - component in the direction of DO virtual work vs. DO NOT Robo3x-1.3 21
Video 3.4 Vijay Kumar and Ani Hsieh Robo3x-1.3 22
Potential Energy Robo3x-1.3 23
Kinetic Energy Kinetic energy of a rigid body consists of two parts Translational Rotational Inertia Tensor Robo3x-1.3 24
Inertia Tensor Principal Cross Moments Products of Inertia 3x3 matrix Symmetric matrix Robo3x-1.3 25
Let denote the mass density Principal Moments of Inertia Cross Products of Inertia Robo3x-1.3 26
Remarks Inertia tensor depends on reference point coordinate frame VS Robo3x-1.3 27
Example Compute the inertia tensor of the block with the given dimensions. Assume is constant. Robo3x-1.3 28
Video 3.5 Vijay Kumar and Ani Hsieh Robo3x-1.3 29
Potential Energy for n-link Robot 1-Link Robot n-link Robot Robo3x-1.3 30
Kinetic Energy for n-link Robot (1) 1-Link Robot n-link Robot Robo3x-1.3 31
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-1.3 32
Kinetic Energy of n-link Robot (2) Robo3x-1.3 33
Euler-Lagrange EOM for n-link Robot (1) Assumptions: is quadratic function of and independent of Robo3x-1.3 34
Euler-Lagrange EOM for n-link Robot (2) Robo3x-1.3 35
Euler-Lagrange EOM for n-link Robot (3) Robo3x-1.3 36
Euler-Lagrange EOM for n-link Robot (4) Christoffel Symbols In matrix form Robo3x-1.3 37
Skew Symmetry Robo3x-1.3 38
Passivity Power = Force x Velocity Energy dissipated over finite time is bounded Important for Controls Robo3x-1.3 39
Bounds on D(q) - eigenvalue of Robo3x-1.3 40
Linearity in the Parameters System Parameters: Mass, moments of inertia, lengths, etc. Robo3x-1.3 41
2-Link Cartesian Manipulator (1) q 2 q 1 y 0 x 0 Robo3x-1.3 42
2-Link Cartesian Manipulator (2) q 2 q 1 y 0 x 0 Robo3x-1.3 43
2-Link Cartesian Manipulator (3) q 2 q 1 x 0 y 0 Robo3x-1.3 44
2-Link Cartesian Manipulator (4) q 2 q 1 y 0 x 0 Robo3x-1.3 45
Video 3.6 Vijay Kumar and Ani Hsieh Robo3x-1.3 46
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-1.3 47
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-1.3 48
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-1.3 49
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-1.3 50
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-1.3 51
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-1.3 52
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-1.3 53
Video 3.7 Vijay Kumar and Ani Hsieh Robo3x-1.3 54
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-1.3 55
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-1.3 56
2-Link Planar Manipulator (7) Christoffel Symbols Robo3x-1.3 57
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-1.3 58
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-1.3 59
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-1.3 60
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-1.3 61