# Uniform vs. Lognormal Kinematics in Robots: Perceptual Preferences for Robotic Movements
## 3 Robotic Arm Modelling
### Pose of an Object  
To spatially define the tool located at the end of the robot, it is necessary to know its pose. Meanwhile, for the position of a point in space, its definition requires a coordinate frame (CF). This position is given in Cartesian coordinates by the projections of the point on the three coordinate axes. In the case of a robotic arm, this CF is usually located at the base of the robot.  
To define the orientation of an object, two CFs are necessary, with one (the fixed CF) taken as a base and another (a mobile CF) rigidly attached to the object whose orientation is to be defined and both must be placed with the same origin of coordinates. The position and orientation of the object are independent magnitudes, meaning that modifying either one does not imply a modification of the other and that both can be calculated separately.  
The rotation matrix is one of the most used methods to define orientations in robotics. The CF \(OXYZ\) is defined as fixed and the \(OUVW\) as mobile. The rotation matrix is a \(3\times 3\) matrix in which the first column represents the unit vector \(\mathbf{\hat{u}}\), the second column the unit vector \(\mathbf{\hat{v}}\) and the third the unit vector \(\mathbf{\hat{w}}\), all related to the fixed CF.  
Figure 2 shows the mobile and fixed CFs and both placed with the same origin.  
The rotation matrix that relates both CFs is given by,  
\[R=\begin{pmatrix}n_{x}&o_{x}&a_{x}\\ n_{y}&o_{y}&a_{y}\\ n_{z}&o_{z}&a_{z}\end{pmatrix}=\begin{pmatrix}-1&0&0\\ 0&1&0\\ 0&0&-1\end{pmatrix} \tag{1}\]  
It is noted that the first column indicates that the \(\mathbf{\hat{u}}\) vector is oriented in the negative direction of the \(x\)-\(axis\)\((\mathbf{n}=(-1,0,0))\), the second column indicates that the \(\mathbf{\hat{v}}\) vector is in the same direction as the \(y\)-\(axis\) and the third column indicates that the \(\mathbf{\hat{w}}\) vector points in the opposite direction to the \(z\)-\(axis\)\((\mathbf{a}=(0,0,-1))\).  
If both mobile and fixed CFs have the same origin of coordinates, then if the coordinates of a point in the mobile CF are known, they can be obtained in the fixed system as follows:  
\[\begin{pmatrix}r_{x}\\ r_{y}\\ r_{z}\end{pmatrix}=R\begin{pmatrix}r_{u}\\ r_{v}\\ r_{w}\end{pmatrix} \tag{2}\]  
As discussed above, the pose of an object is defined by the position and orientation of its CF with respect to the fixed CF. The mathematical tool used to define the pose of an object is the homogeneous transformation matrix, defined as:  
\[T=\begin{pmatrix}R&\mathbf{P}\\ \mathbf{f}&w\end{pmatrix}=\begin{pmatrix}n_{x}&o_{x}&a_{x}&p_{x}\\ n_{y}&o_{y}&a_{y}&p_{y}\\ n_{z}&o_{z}&a_{z}&p_{z}\\ 0&0&0&1\end{pmatrix} \tag{3}\]  
where the matrix \(R\) of \(3\times 3\) is the rotation matrix described above, the vector \(\mathbf{p}\) indicates the position of the mobile coordinate frame referring to the fixed CF, the vector \(\mathbf{f}\) represents the perspective vector, which is usually a null vector in robotics and finally, the scalar \(w\) represents the scaling factor, which always has a unit value in robotics.  
Figure 2: Example of CFs to define the orientation of a mobile object.  
As an example, the matrix that relates the mobile CF to the fixed one shown in Figure 3 is given by the homogeneous transformation matrix:  
\[fT_{m}=\left(\begin{array}{cccc}0&0&-1&L_{1}\\ -1&0&0&-L_{2}\\ 0&1&0&0\\ 0&0&0&1\end{array}\right) \tag{4}\]  
The nomenclature \(fT_{m}\) indicates the homogeneous transformation matrix that relates the pose of the mobile CF to the fixed one.