Draft: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)

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

More information Joint nb ...

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)

Close

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

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

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:

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

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:

More information Joint nb ...

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)

Close

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.