The relative position and orientation of the axes of two successive joints can be specified by two link parameters, link length and link twist angle.
The two link parameters are always constant and are specified as part of the mechanical design.
The orientation of a tool can be represented in Joint Coordinates by YPR (Yaw, Pitch and Roll) convention. In rectangular or Cartesian coordinates, the same can be represented by a rotation matrix R, where the three columns of R correspond to the normal, sliding and approach vectors resply.
The approach vector is aligned with the roll axis and points away from the wrist. Consequently, it represents the direction in which the tool is pointing.
The sliding vector is orthogonal to the approach vector and aligned with the open-close axis of the tool.
Yaw, Pich and Roll motions are rotations about normal, sliding and approach vectors.
In order to manipulate objects in space, it is required to control both the position and orientation of the tool / end effector in three-dimensional space.
A relationship between the joint variables and the position and orientation of the tool is to be formulated.
The direct kinematics problem is the following: given the vector of joint variables, of a robotic manipulator, how do we determine the position and orientation of the tool with respect to a co-ordinate frame attached to the robot base.
It is necessary to have a concise formulation of a general solution to the direct kinematic problem.