**3. Results**

This section presents the kinematic structures generated by the presented algorithms for the three poses defined earlier (Figure 2). The results are discussed and later compared by manipulability measure and manipulator length. A table with calculated DH parameters is also included. The visualisation of the final kinematic structures is shown in Figures 8–10.

The types of estimation A and B are not able to reach a pose in terms of orientation in general; however, in some cases (as shown in Figure 10b) the real solution was found for the type B. In addition, if compared with a similar D result (Figure 10d) for *Pose*(3), solution B provides shorter links. Furthermore, the A and B types are generated in a way where no collision of joints should occur.

**Figure 8.** Initial estimation results for Pose (1): (**a**) type A estimation; (**b**) type B estimation; (**c**) type C estimation; and (**d**) type D estimation.

The type C may perform very well if the *Z* axes of the base and pose are parallel, as shown in Figure 10c; on the other hand, if the axes are very close to each other (the perpendicular distance is short), the joints are in collision, as shown in Figure 9c.

Placing joints on an approximated spline provides the most general result of the provided algorithms, as shown in Figures 8d and 9d, but it is struggling with parallel axes—see Figure 10d. This could be mitigated by tuning the Bézier curve driven point related to the pose and placing the joints not in a plane that is defined by the two parallel *Z* axes.

**Figure 9.** Initial estimation results for Pose (2): (**a**) type A estimation; (**b**) type B estimation; (**c**) type C estimation; and (**d**) type D estimation.

**Figure 10.** Initial estimation results for Pose (3): (**a**) type A estimation; (**b**) type B estimation; (**c**) type C estimation; and (**d**) type D estimation.

*3.1. Resulting DH Parameters*

For a better evaluation, we include Table 1 with the generated DH parameters for *Pose*(1). The angles *θ* are considered as variables with zero offset.


**Table 1.** The DH parameters of manipulators for *Pose*(1).

#### *3.2. Manipulator Length Comparison*

In general, longer links of a manipulator demand more powerful motors because of higher torques. In Figure 11, there is a bar plot comparing the lengths of the resulting manipulators. The length was determined using Equation (48), where *ai* and *di* are DH parameters of *ith* link. *n* is the number of joints:

*n* (48)

**Figure 11.** Comparison of the length of generated manipulators for every pose.

#### *3.3. Manipulability Comparison*

To compare the results, we decided to evaluate the manipulability of the calculated kinematic structures. This scalar measure was obtained using the Yoshikawa algorithm [1], which describes how spherical the end-effector velocity ellipsoid is. It differs between 0 and 1, where the value 1 shows the best manipulability in all axes. If the value is close to 0, the mechanism might be dealing with singularities.

The results are shown for both translational and rotational motions in Figure 12 as a logarithmic graph, because the measure differs significantly between particular kinematic structures. It should be kept in mind that the presented manipulators have less than six DoFs, which is one of the reasons why the manipulability measure by Yoshikawa evaluates them with low numbers.

As expected, the type A kinematic structure has to deal with singularities and the manipulability tends to be the lowest for the given poses. Type B has almost the same translational manipulability as type D, but the last two joints are in a singular position, so the rotational manipulability drops in the case of type B kinematic structure. Type C performs better than types A and B. In the case of general poses *Pose*(1, <sup>2</sup>), type D provides the highest manipulability measures. However, for *Pose*(3) when *Z* axes are parallel, the manoeuvrability of the type D algorithm drops under the values of type C.

**Figure 12.** Logaritmic graph comparing the manipulability measures for given poses (1–3): (**a**) for translational motion; and (**b**) for rotational motion.
