# Initial Estimation of Kinematic Structure of a Robotic Manipulator as an Input for Its Synthesis
## 1 Introduction
Another usage of NLP is trying to find an optimal design minimizing the path length in joint space [20]. The results of the already described papers [19; 20] seem promising in their application of nonlinear programming to solve task-based custom manipulator design, however, there is one unanswered question in their work. In general, nonlinear programming traditionally requires that a starting point is given as part of the problem data, and comparative numerical testing is done using these traditional starting points [21]. In the case of robotic manipulators, if there is a given random path for a robotic manipulator, how should the starting point (initial values, initial estimation) of its kinematic structure look like? The work [19] mentions a set of input values without any detailed explanation of how those values were determined. In [20], eight initial seeds are applied, but they are random values, which might be a cause of why no solution was found at all in some cases. In [17], they searched for a global optimum; however, the input values are also random, which may extend the time of solving the objective function. Even in a book by [PERSON] et al. [22] which serves as an introduction to the optimisation of kinematic structures, the initial values for all described methods are obtained randomly without any detailed discussion. Therefore, a possible answer to the previously stated question will be addressed in this paper using multiple approaches. In this paper, a geometric analysis to estimate kinematic structures and related calculations are proposed and discussed. The outcomes may avoid relying on randomness, which in the case of local-optimum-search algorithms may more frequently lead to convergence, making them reliable but still much faster than global-optimum-search methods. Moreover, the results can also be used in the previously mentioned algorithms using GAs, where they can serve as an optional first generation input, and in global-optimum-search algorithms, where they can reduce the time needed for the optimisation. In addition to that, the procedure may serve as an input for other custom manipulator design challenges such as collision avoidance [23]. A Denavit-Hartenberg notation [24] (standard DH parameters) was applied in the presented algorithms to obtain a general kinematic structure of a robot. The DH parameters were widely used, however, they also bring some disadvantages in the case of general structures. These are discussed later. Some algorithms behind the automatic placement of DH parameters have already been presented. In [25], a vector-algebra is applied to extract the parameters. [PERSON] [26] creates a string of elementary translations and rotations from the user-defined base coordinate to the end-effector and factorises the string afterwards. An approach by [PERSON] et al. [27] is using line geometry to obtain the parameters. In addition to that, some researchers proposed their algorithms and they also identified (and verified) the DH parameters with an external sensor device in simulation [28] or experiments [29]. However, all of these algorithms were applied to already existing robots or similar devices. On the other hand, the algorithms presented in this paper are here to determine and \"create\" a non-existing manipulator (its kinematic structure) that can achieve a freely given pose.