# Multi-Objective PSO with Variable Number of Dimensions for Space Robot Path Optimization
## 1 Introduction  
Algorithms for finding the optimal path are used in our lives on a daily basis. Fast and accurate methods for finding the optimal route can make our lives more pleasant, e.g., when searching the optimal route using public transport [1], saving a huge amount of resources in cases of a proper delivery path planning [2], or even saving lives in cases of emergency vehicle routing [3]. The minimal distance of the path connecting the starting and target points is usually the first choice for the objective function used by the algorithms for optimal planning on the surface of the earth. Other objectives such as the shortest traveling time, the least power consumption during the travel, or the minimal risks are used by specific applications. Nevertheless, all the mentioned objectives are valid at once for the space robot path planning. Consider, for example, the surface of the moon, where various robot rovers must avoid surface irregularities, whether craters or, conversely, outcrops [4]. Power consumption and route reliability have at least the same importance as the length of the route itself in the case of the space robot path planning. This is due to the unavailability of replacement sources and the very limited (and costly) or complete impossibility of repairing the device in the event of a collision with an obstacle outside the Earth. The conventional graph-based algorithms including the famous [PERSON]'s [5] and A-star algorithms [6] follow only one objective during the optimal path search. All decisions are made based on weights of graph node connections that express one of the considered path parameters (the shortest distance, the least power consumption, etc.) or their linear combination where the weights of individual components have to be set a priori. Nevertheless, the space robot pathfinding problem (SRPP) optimization has to be considered as a multi-objective optimization problem (MOOP) [7] to prevent the loss ofgenerality of the solution. The set of compromise solutions (the so-called Pareto front) as a result of the multi-objective optimization problem gives much more insight into the conflicting objectives of the solved problem. The final solution can be then selected from the Pareto front by an expert or an automated system considering the gained knowledge about the problem under solution. Several authors have considered the SRPP as a multi-objective problem. For example, [PERSON] et al. consider two objectives during the path planning: (1) the total path's distance, and (2) the path's smoothness in [8]. Nevertheless, they combine the two objectives into a single weighted function, with weights being selected a priori. Another drawback of their approach is that they select the total number of the turning (or way) points (TP) between the start and target position a priori. This drawback can be overcome by employing the so-called grid-based solution, as in [9], where the solution domain is first divided into a set of regular subdomains (usually of a rectangular shape). Then, the subdomains occupied by obstacles cannot become the TPs while the empty subdomains can serve as the TPs. The decision space vector for optimization purposes is then a stream of subdomain addresses and its length has to be either fixed [10; 11] or some additional algorithm [9; 12] has to control the addition or deletion of TPs. The grid-based methods are relatively easy to implement. Nevertheless, they suffer from several drawbacks: the obstacles are not represented precisely in cases of a very coarse subdomain tessellation, or the number of subdomains reaches very high values, which leads to an unwanted explosion of the dimensions of the decision space vector [13]. The drawbacks of the aforementioned SRPP formulations as either single- or multi-objective problems with fixed dimensions turn our attention to the class of so-called variable number of dimensions (VND) algorithms. Some authors refer to these algorithms as metameric.