**4. Discussion**

The outcomes of this paper are aimed to be utilised in the synthesis of kinematic structures of robotic manipulators. Other applications are also possible, for example, in rapid kinematic analysis and simulation to evaluate kinematic options if only one position or pose on a trajectory is desired to be reached. However, only topics related to the synthesis problem are going to be discussed here.

This paper provided four algorithms to estimate the initial value of a kinematic structure for later implementation in optimisation algorithms. As mentioned previously, especially in algorithms searching the local minimum of an objective function, every initial guess can provide different results. Therefore, we decided to present the types (A and B) of estimation even though they do not fulfil the given pose in terms of orientation. The reason is that an optimisation algorithm may overcome this issue later and the solution could achieve the pose on a given trajectory with a better final value of the objective function than with the other presented types (C and D). This always depends on the type and properties of the trajectory. Therefore, we sugges<sup>t</sup> implementing all four estimation types into an optimisation algorithm and compare the results afterwards.

If one would like to know which one of the four structures is the best, this question is not easy to answer. We chose three poses to demonstrate the advantages and disadvantages of particular algorithms and compared the final structures for these poses by manoeuvrability and length. The type D provides the most general structure that can reach any pose, but one must be careful in cases when some axes of the base and the pose are parallel, for example. We sugges<sup>t</sup> to always visualise all initial structures for better evaluation.

The presented algorithms are based on the Denavit–Hartenberg convention, generating DH parameters. During this work, a question arose, of how suitable the DH convention

is for general structures and especially for their synthesis for given trajectory. As mentioned previously, it is possible to find DH parameters between two coordinate frames using the common perpendicular approach, but it is required to "borrow" another two DH parameters from the next joint and to add them to the already existing four parameters. This is obvious since we need six parameters (three translations and three rotations) in general to describe the motion of a frame in Euclidean space. Therefore, using an optimisation algorithm to synthesise a manipulator that fulfils the DH convention may be limiting. The frames representing joints cannot freely rotate and translate wherever the algorithm tends to, but they have to follow the XZ planes in which the common perpendiculars between neighbouring joints lie. On top of that, the algorithm may not find the global or local minimum because of this limit at all. The reason is that traditionally DH parameters serve for the description of existing robots and once they are obtained, some local coordinates of the joints may be located outside of the rigid body along the Z axis, and although they are still closely tied to the particular joint and representing its kinematics, they are not representing the real (physical) position of the joint. On the other hand, in terms of synthesis when a rigid body does not exist yet, the location (transformation) of the coordinate frames of joints is the only known and crucial parameter and its variability should not be limited during the synthesis process anyhow.

The comparison of the synthesis of kinematic structure using DH convention and other standard approaches, such as screw theory [34] for instance, will be an interesting topic for future research. However, this matter has no impact on the work presented in this paper. The kinematic structures obtained using the four algorithms may be translated into any other standard description of the structure of a manipulator.
