2 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 manipulatorForward kinematics: mapping from joint variables to position and orientation of the end effectorInverse kinematics: finding joint variables that satisfy a given position and orientation of the end effectorJacobian: mapping from the joint velocities to the end effector linear and angular velocitiesExample: three link cylindrical robotOr any other coordinate frame on the manipulator
3 Why are we studying inertial dynamics and control? Kinematic vs dynamic models:What we’re really doing is modeling the manipulatorKinematic modelsSimple control schemesGood approximation for manipulators at low velocities and accelerations when inertial coupling between links is smallNot so good at higher velocities or accelerationsDynamic modelsMore complex controllersMore accurate
4 Methods to Analyze Dynamics Two methods:Energy of the system: Euler-Lagrange methodIterative Link analysis: Euler-Newton methodEach has its own ads and disads.In general, they are the same and the results are the same.
5 Terminology Definitions Generalized coordinates: Vector norm: measure of the magnitude of a vector2-norm:Inner product:Definitions of other norms
6 Euler-Lagrange Equations We can derive the equations of motion for any nDOF system by using energy methods
7 Ex: 1DOF systemTo illustrate, we derive the equations of motion for a 1DOF systemConsider a particle of mass mUsing Newton’s second law:Mass is constrained to move vertically. Note that the potential energy is defined with respect to a ‘zero energy state’
8 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:
10 Ex: 1DOF system Let the total inertia, J, be defined by: : Note that this result gives a nonlinear equation: we can linearize using the small angle assumption
11 InertiaInertia, in the body attached frame, is an intrinsic property of a rigid bodyIn the body frame, it is a constant 3x3 matrix: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:r is the distance from the axis of rotation to the particle
12 Inertia The elements are defined by: Center of gravity The point principal moments of inertiar(x,y,z) is the densitycross products of inertiaThe point
13 The Inertia MatrixCalculate 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
14 InertiaFirst, we need to express the inertia in the body-attached frameNote that the rotation between the inertial frame and the body attached frame is just R
15 Newton-Euler Formulation Rules:Every action has an equal reactionThe rate of change of the linear momentum equals the total forces applied to the bodyThe rate change of the angular momentum equals the total torque applied to the body.