Configuration: A specificaton of the position of all points of a mechanism
Degrees of Freedom (DoF): Number of independent variables required to define a system. Ex: robot system
Configuration space (C-space): DOF-dimension space of all configurations
DOF of a planar body: m = 3
DOF of a spatial body: m = 6
DOF = summation of body freedoms - summation of independent constraints from joints.
There are six type of joints and degrees of freedom.
Revolute, 1
Helical, 1
Prismatic, 1
Universal, 2
Cylindrical, 2
Spherical, 3
GRUBLER's formula for degrees of freedom:
DOF = m(N-1-J) + Summation(f_i)
m = 3 or 6
N - number of bodies
J - Number of joints
f_i - number of freedoms provided by joints
manipulating a rigid object? 6 DOF
operating a laser pointer? 5 DOF
carrying a tray of glasses to keep them vertical? 4 DOF: planar body + height
Language of 3D robot modeling: 1. UDRF: Unifies Robot Description Format.
The description of UDRF consists of Link and Joint
2. Xacro
Co-ordinate system
cartesian: Rotational
x-axis: Roll
y-axis: Yaw
z-axis: Pitch
Calculation of inertia matrix
[Ixx Ixy Ixz]
[Iyx Iyy Iyz]
[Izx Iyz Izz]
Box Inertia
Ixx = 1/2m(b^2 + c^2)
Iyy = 1/2m(a^2 + c^2)
Izz = 1/2m(b^2 + a^2)
Here we have considered the side are which are not the axis for which we are calculating inertia for.
Solid sphere Inertia
Izz = 1/2mR^2
Ixx = Iyy = 1/12m(3R^2 + h^2)
Cylindrical inertial matrix
Ixx == Iyy == Izz == 2/5mR^2