Video 3.1 Vijay Kumar and Ani Hsieh Robo3x-1.3 1 Property of Penn Engineering, Vijay Kumar and Ani Hsieh
Dynamics of Robot Arms Vijay Kumar and Ani Hsieh University of Pennsylvania Robo3x-1.3 2 Property of Penn Engineering, Vijay Kumar and Ani Hsieh
Lagrange’s Equation of Motion Lagrangian Kinetic Potential Energy Energy 1-DOF n-DOF Generalized Coordinates Generalized Forces Robo3x-1.3 3 Property of Penn Engineering, Vijay Kumar and Ani Hsieh
Motion of Systems of Particles • Center of Mass f i a 2 P i m i r OPi a 1 O a 3 Newton’s 2 nd Law Robo3x-1.3 4 Property of Penn Engineering, Vijay Kumar and Ani Hsieh
Rigid Body as a System of Particles • Constraints P F i F j P j P i • Holonomic Constraints • Constraints on position r OPi r OPj O Robo3x-1.3 5 Property of Penn Engineering, Vijay Kumar and Ani Hsieh
Holonomic Constraints • Given a system with k particles and l holonomic constraints Ø DOF = k – l Ø n = k – l generalized coordinates Ø are independent Ø Robo3x-1.3 6 Property of Penn Engineering, Vijay Kumar and Ani Hsieh
Types of Displacements • Actual P f i f j P j P i • Possible r OPi r OPj • Virtual (or Admissible) O Robo3x-1.3 7 Property of Penn Engineering, Vijay Kumar and Ani Hsieh
Video 3.2 Vijay Kumar and Ani Hsieh Robo3x-1.3 8 Property of Penn Engineering, Vijay Kumar and Ani Hsieh
Classification of Forces Newtonian Lagrangian Internal vs External Constraint vs Applied Applied Forces: Any forces that are not constraint forces Robo3x-1.3 9 Property of Penn Engineering, Vijay Kumar and Ani Hsieh
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 Property of Penn Engineering, Vijay Kumar and Ani Hsieh
D’Alembert’s & Virtual Displacements • C i – Constraint Surface q TC i • TC i – Tangent space of C i C i • Virtual Displacements satisfy: 1. 2. Eqn of Motion Robo3x-1.3 11 Property of Penn Engineering, Vijay Kumar and Ani Hsieh
Intuition for D’Alembert’s (1) From Newton’s 2 nd Law Robo3x-1.3 12 Property of Penn Engineering, Vijay Kumar and Ani Hsieh
Intuition for D’Alembert’s (2) By definition and b/c motion is constrained And, and Robo3x-1.3 13 Property of Penn Engineering, Vijay Kumar and Ani Hsieh
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 Property of Penn Engineering, Vijay Kumar and Ani Hsieh
Video 3.3 Vijay Kumar and Ani Hsieh Robo3x-1.3 15 Property of Penn Engineering, Vijay Kumar and Ani Hsieh
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 Property of Penn Engineering, Vijay Kumar and Ani Hsieh
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 Property of Penn Engineering, Vijay Kumar and Ani Hsieh
Lagrange’s EOM for Systems of Particles (2) Note: 1) 2) Robo3x-1.3 18 Property of Penn Engineering, Vijay Kumar and Ani Hsieh
Lagrange’s EOM for Systems of Particles (3) Kinetic Energy Robo3x-1.3 19 Property of Penn Engineering, Vijay Kumar and Ani Hsieh
Lagrange’s EOM for Systems of Particles (4) - Potential Energy And if - Generalized Applied Forces Robo3x-1.3 20 Property of Penn Engineering, Vijay Kumar and Ani Hsieh
Summary • - vector in 3D • Virtual work • - component in the direction of DO virtual work vs. DO NOT Robo3x-1.3 21 Property of Penn Engineering, Vijay Kumar and Ani Hsieh
Video 3.4 Vijay Kumar and Ani Hsieh Robo3x-1.3 22 Property of Penn Engineering, Vijay Kumar and Ani Hsieh
Potential Energy Robo3x-1.3 23 Property of Penn Engineering, Vijay Kumar and Ani Hsieh
Kinetic Energy Kinetic energy of a rigid body consists of two parts Translational Rotational Inertia Tensor Robo3x-1.3 24 Property of Penn Engineering, Vijay Kumar and Ani Hsieh
Inertia Tensor Cross Principal Products of Moments of Inertia Inertia • 3x3 matrix • Symmetric matrix Robo3x-1.3 25 Property of Penn Engineering, Vijay Kumar and Ani Hsieh
Let denote the mass density Principal Moments of Inertia Cross Products of Inertia Robo3x-1.3 26 Property of Penn Engineering, Vijay Kumar and Ani Hsieh
Remarks Inertia tensor depends on • reference point • coordinate frame VS Robo3x-1.3 27 Property of Penn Engineering, Vijay Kumar and Ani Hsieh
Example Compute the inertia tensor of the block with the given dimensions. Assume is constant. Robo3x-1.3 28 Property of Penn Engineering, Vijay Kumar and Ani Hsieh
Video 3.5 Vijay Kumar and Ani Hsieh Robo3x-1.3 29 Property of Penn Engineering, Vijay Kumar and Ani Hsieh
Potential Energy for n-Link Robot • 1-Link Robot • n-Link Robot Robo3x-1.3 30 Property of Penn Engineering, Vijay Kumar and Ani Hsieh
Kinetic Energy for n-Link Robot (1) • 1-Link Robot • n-Link Robot Robo3x-1.3 31 Property of Penn Engineering, Vijay Kumar and Ani Hsieh
Review of the Jacobian ' ∈ ℝ $ !: ℝ $ → ℝ & !(') ∈ ℝ & ,1 ,1 . . ⋯ ,- . ,- $ ,! δ! = … = ⋮ ⋱ ⋮ J ,- . ,- $ ,1 ,1 & & ⋯ ,- . ,- $ = ,1 5 J ij ,- 6 Robo3x-1.3 32 Property of Penn Engineering, Vijay Kumar and Ani Hsieh
Kinetic Energy of n-Link Robot (2) Robo3x-1.3 33 Property of Penn Engineering, Vijay Kumar and Ani Hsieh
Euler-Lagrange EOM for n-Link Robot (1) Assumptions: • is quadratic function of • and independent of Robo3x-1.3 34 Property of Penn Engineering, Vijay Kumar and Ani Hsieh
Euler-Lagrange EOM for n-Link Robot (2) Robo3x-1.3 35 Property of Penn Engineering, Vijay Kumar and Ani Hsieh
Euler-Lagrange EOM for n-Link Robot (3) Robo3x-1.3 36 Property of Penn Engineering, Vijay Kumar and Ani Hsieh
Euler-Lagrange EOM for n-Link Robot (4) Christoffel Symbols In matrix form Robo3x-1.3 37 Property of Penn Engineering, Vijay Kumar and Ani Hsieh
Skew Symmetry Robo3x-1.3 38 Property of Penn Engineering, Vijay Kumar and Ani Hsieh
Passivity • Power = Force x Velocity • Energy dissipated over finite time is bounded • Important for Controls Robo3x-1.3 39 Property of Penn Engineering, Vijay Kumar and Ani Hsieh
Bounds on D(q) • - eigenvalue of • Robo3x-1.3 40 Property of Penn Engineering, Vijay Kumar and Ani Hsieh
Linearity in the Parameters System Parameters: • Mass, moments of inertia, lengths, etc. Robo3x-1.3 41 Property of Penn Engineering, Vijay Kumar and Ani Hsieh
2-Link Cartesian Manipulator (1) q 2 q 1 y 0 x 0 Robo3x-1.3 42 Property of Penn Engineering, Vijay Kumar and Ani Hsieh
2-Link Cartesian Manipulator (2) q 2 q 1 y 0 x 0 Robo3x-1.3 43 Property of Penn Engineering, Vijay Kumar and Ani Hsieh
2-Link Cartesian Manipulator (3) q 2 q 1 x 0 y 0 Robo3x-1.3 44 Property of Penn Engineering, Vijay Kumar and Ani Hsieh
2-Link Cartesian Manipulator (4) q 2 q 1 y 0 x 0 Robo3x-1.3 45 Property of Penn Engineering, Vijay Kumar and Ani Hsieh
Video 3.6 Vijay Kumar and Ani Hsieh Robo3x-1.3 46 Property of Penn Engineering, Vijay Kumar and Ani Hsieh
2-Link Planar Manipulator (1) System parameters: • Link lengths • Link center of mass location y 1 x 1 • Link masses q 2 y 2 P Q y 0 x 2 q 1 O x 0 Robo3x-1.3 47 Property of Penn Engineering, Vijay Kumar and Ani Hsieh
2-Link Planar Manipulator (2) Recall y 1 x 1 q 2 y 2 P Q y 0 x 2 q 1 O x 0 Robo3x-1.3 48 Property of Penn Engineering, Vijay Kumar and Ani Hsieh
2-Link Planar Manipulator (3) y 1 x 1 q 2 y 2 P Q y 0 x 2 q 1 O x 0 Robo3x-1.3 49 Property of Penn Engineering, Vijay Kumar and Ani Hsieh
2-Link Planar Manipulator (4) Kinetic Energy = Translational + Rotational Translational y 1 x 1 q 2 y 2 P Q y 0 x 2 q 1 O x 0 Robo3x-1.3 50 Property of Penn Engineering, Vijay Kumar and Ani Hsieh
2-Link Planar Manipulator (5) Kinetic Energy = Translational + Rotational Rotational y 1 x 1 q 2 y 2 P Q y 0 x 2 q 1 O x 0 Robo3x-1.3 51 Property of Penn Engineering, Vijay Kumar and Ani Hsieh
2-Link Planar Manipulator (6) Kinetic Energy = Translational + Rotational Rotational y 1 x 1 q 2 y 2 P Q y 0 x 2 q 1 O x 0 Robo3x-1.3 52 Property of Penn Engineering, Vijay Kumar and Ani Hsieh
2-Link Planar Manipulator (7) Kinetic Energy = Translational + Rotational Rotational y 1 x 1 q 2 y 2 P Q y 0 x 2 q 1 O x 0 Robo3x-1.3 53 Property of Penn Engineering, Vijay Kumar and Ani Hsieh
Video 3.7 Vijay Kumar and Ani Hsieh Robo3x-1.3 54 Property of Penn Engineering, Vijay Kumar and Ani Hsieh
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