* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download slides
Laplace–Runge–Lenz vector wikipedia , lookup
Angular momentum operator wikipedia , lookup
Classical mechanics wikipedia , lookup
Photon polarization wikipedia , lookup
Center of mass wikipedia , lookup
Newton's theorem of revolving orbits wikipedia , lookup
Symmetry in quantum mechanics wikipedia , lookup
Velocity-addition formula wikipedia , lookup
Modified Newtonian dynamics wikipedia , lookup
Hunting oscillation wikipedia , lookup
Hamiltonian mechanics wikipedia , lookup
Dynamic substructuring wikipedia , lookup
Virtual work wikipedia , lookup
Relativistic mechanics wikipedia , lookup
Four-vector wikipedia , lookup
Derivations of the Lorentz transformations wikipedia , lookup
Seismometer wikipedia , lookup
Lagrangian mechanics wikipedia , lookup
Frame of reference wikipedia , lookup
Centrifugal force wikipedia , lookup
Fictitious force wikipedia , lookup
Work (physics) wikipedia , lookup
Centripetal force wikipedia , lookup
Inertial frame of reference wikipedia , lookup
Mechanics of planar particle motion wikipedia , lookup
Theoretical and experimental justification for the Schrödinger equation wikipedia , lookup
Relativistic angular momentum wikipedia , lookup
Computational electromagnetics wikipedia , lookup
Classical central-force problem wikipedia , lookup
Analytical mechanics wikipedia , lookup
Routhian mechanics wikipedia , lookup
Newton's laws of motion wikipedia , lookup
Equations of motion wikipedia , lookup
Ch. 7: Dynamics 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 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 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. Terminology • Definitions – Generalized coordinates: – Vector norm: measure of the magnitude of a vector • 2-norm: – Inner product: Euler-Lagrange Equations • We can derive the equations of motion for any nDOF system by using energy methods 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: 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: d L L i dt q i qi Ex: 1DOF system Ex: 1DOF system • Let the total inertia, J, be defined by: J r 2Jm Jl • : 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: I xx I I ij I yx I zx I xy I yy I zy I xz I yz I zz – 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: Iii r 2dm r 2 x, y, z dV r 2 x, y, z dxdydz V V V • r is the distance from the axis of rotation to the particle Inertia • The elements are defined by: Center of gravity x x x, y, z dxdydz x, y, z dxdydz I xx y 2 z 2 x, y , z dxdydz I yy (x,y,z) is the density I zz 2 z 2 2 y 2 principal moments of inertia I xy I yx xy x, y , z dxdydz cross products of inertia I xz I zx xz x, y , z dxdydz I yz I zy yz x, y , z dxdydz The i thpoint p ri pi 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 z d w x h y 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 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 dmv f ma dt – The rate change of the angular momentum equals the total torque applied to the body. dI OO O dt Newton-Euler Formulation • Euler equation Rii1 Force and Torque Equilibrium • Force equilibrium • Torque equilibrium Angular Velocity and Acceleration Initial and terminal conditions F L2 L1 • Moment of Inertia Next class… F L2 L1 JTF ( L1sin 1 L2 sin( 1 2)) L2 sin( 1 2) J L2 cos( 1 2) L1cos 1 L2 cos( 1 2) ( L1sin 1 L2 sin( 1 2)) L1cos 1 L2 cos( 1 2) J T L2 sin( 1 2) L2 cos( 1 2) det( J T ) L1L2 sin( 2) f 1 F f 2 f 2 f 1 tan 1