In Robotics engineering, Zero Reference Model (ZRM) is a method for describing the kinematics of a serial manipulator, alternative to the Denavit-Hartenberg parameters (DH). It was introduced by Gupta in 1981. with the name Zero Reference Position Description. Some authors claim that ZRM is superior to DH, for example in the case of geometric parameters calibration
Zero Reference Model The basic principle of ZRM consists in describing the location of the robot joints and their rotation/translation axes, in the base coordinate system, when all joint values are equal to zero. Gupta presents the ZRM for a serial manipulator as a table with four columns containing:
The joint number i and its type (R for rotational and P for prismatic) The joint direction as a vector u 0 i {\textstyle u_{0}^{i}}
The joint center as a position Q 0 i {\textstyle Q_{0}^{i}}
The Hand Data as a position Q 0 h {\textstyle Q_{0}^{h}} and two orthogonal vectors u 0 a {\textstyle u_{0}^{a}} (direction of gripping) and u 0 t {\textstyle u_{0}^{t}} (direction of the gripper tongs)
The ZRM table for a simple serial manipulator with 6 joints is shown below:
The transformation of the hand with respect to the base is calculated as follows:
[ T ] = [ T 0 1 ] . [ T 1 2 ] . . . [ T n − 1 n ] . [ T n h ] {\displaystyle [T]=[T_{0}^{1}].[T_{1}^{2}]...[T_{n-1}^{n}].[T_{n}^{h}]}
Each individual transformation T i − 1 i {\textstyle T_{i-1}^{i}} is the combination of the translation Q i − 1 i {\textstyle Q_{i-1}^{i}} and the rotation R i − 1 i {\textstyle R_{i-1}^{i}} with:
For a rotative joint: Q i − 1 i = Q 0 i − Q 0 i − 1 {\textstyle Q_{i-1}^{i}=Q_{0}^{i}-Q_{0}^{i-1}} and R i − 1 i = R o t ( q i , u 0 i ) {\textstyle R_{i-1}^{i}=Rot(q_{i},u_{0}^{i})}
For a prismatic joint: Q i − 1 i = ( Q 0 i − Q 0 i − 1 ) + q i . u 0 i {\textstyle Q_{i-1}^{i}=(Q_{0}^{i}-Q_{0}^{i-1})+q_{i}.u_{0}^{i}} and R i − 1 i = I d {\textstyle R_{i-1}^{i}=Id}
The transformation [ T n h ] {\textstyle [T_{n}^{h}]} is given as:
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