Draft:Zero Reference Model

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:

1. The joint number i and its type (R for rotational and P for prismatic)
2. The joint direction as a vector ${\textstyle u_{0}^{i}}$
3. The joint center as a position ${\textstyle Q_{0}^{i}}$
4. The Hand Data as a position ${\textstyle Q_{0}^{h}}$ and two orthogonal vectors ${\textstyle u_{0}^{a}}$ (direction of gripping) and ${\textstyle u_{0}^{t}}$ (direction of the gripper tongs)

Example of a simple serial manipulator for illustrating the Zero Reference Model

The ZRM table for a simple serial manipulator with 6 joints is shown below:

Joint nb ${\textstyle i}$, joint type	Joint dir ${\textstyle u_{0}^{i}}$	Joint pos ${\textstyle Q_{0}^{i}}$	Hand data
1, R	(0, 0, 1)	(0, 0, 0)	${\textstyle Q_{0}^{h}}$ = (b + c + d + e + f, 0, a) ${\textstyle u_{0}^{a}}$ = (1, 0, 0) ${\textstyle u_{0}^{t}}$ = (0, 0, 1)
2, R	(0, 1, 0)	(0, 0, a)
3, R	(0, 1, 0)	(b, 0, a)
4, R	(1, 0, 0)	(b + c, 0, a)
5, R	(0, 1, 0)	(b + c + d, 0, a)
6, R	(1, 0, 0)	(b + c + d + e, 0, a)

The transformation of the hand with respect to the base is calculated as follows:

$[T]=[T_{0}^{1}].[T_{1}^{2}]...[T_{n-1}^{n}].[T_{n}^{h}]$

Each individual transformation ${\textstyle T_{i-1}^{i}}$ is the combination of the translation ${\textstyle Q_{i-1}^{i}}$ and the rotation ${\textstyle R_{i-1}^{i}}$ with:

• For a rotative joint: ${\textstyle Q_{i-1}^{i}=Q_{0}^{i}-Q_{0}^{i-1}}$ and ${\textstyle R_{i-1}^{i}=Rot(q_{i},u_{0}^{i})}$
• For a prismatic joint: ${\textstyle Q_{i-1}^{i}=(Q_{0}^{i}-Q_{0}^{i-1})+q_{i}.u_{0}^{i}}$ and ${\textstyle R_{i-1}^{i}=Id}$

The transformation ${\textstyle [T_{n}^{h}]}$ is given as:

$[T_{n}^{h}]={\begin{pmatrix}u_{0}^{t}\times u_{0}^{a}&u_{0}^{t}&u_{0}^{a}&Q_{0}^{h}-Q_{0}^{n}\\0&0&0&1\end{pmatrix}}$

Discussion

While the ZRM representation is not as compact as the DH parameters, it has a number of advantages:

• It is really straightforward and less error-prone, e.g. when describing the kinematics of a manipulator from the vendor documentation.
• It leads to simple calculations, especially when using unitary quaternions for the rotations, as ${\textstyle R_{i}^{j}=cos({\frac {q_{j}}{2}})+sin({\frac {q_{j}}{2}}).(u_{i0_{x}}i+u_{i0_{y}}j+u_{i0_{z}}k)}$

A more compact form can be obtained by using ${\textstyle Q_{i-1}^{i}}$ instead of ${\textstyle Q_{0}^{i}}$ in the third column, and a position and quaternion in the fourth. The table above becomes:

Joint nb ${\textstyle i}$, joint type	Joint dir ${\textstyle u_{i-1}^{i}}$	Joint pos ${\textstyle Q_{i-1}^{i}}$	Hand data
1, R	(0, 0, 1)	(0, 0, 0)	${\textstyle Q_{6}^{h}}$ = (f, 0, 0) ${\textstyle R_{6}^{h}}$ = ${\textstyle Id}$
2, R	(0, 1, 0)	(0, 0, a)
3, R	(0, 1, 0)	(b, 0, 0)
4, R	(1, 0, 0)	(c, 0, 0)
5, R	(0, 1, 0)	(d, 0, 0)
6, R	(1, 0, 0)	(e, 0, 0)

It is important to note that the ZRM of a serial manipulator can be described even if the manipulator cannot reach the zero value on all joints because of mechanical limitations.