## 1 Introduction  
MRs are sophisticated machines that implement reprogramming to perceive, analyze, communicate, and carry out a wide range of activities, including warehouse operations, military and civilian surveillance, healthcare support, customer support, and many more [1]. Also, MRs have been used to acquire data in a wide range of applications, including the inspection of electricity grids, the gathering of traffic-related information, tracking of chemical and biological contamination in major waterways, combat search and rescue, maritime surveillance, agricultural assessments, and environmental assessment [2, 3]. Energy efficiency is an important challenge to the performance of MRs since their mobility is restricted by robust and costly batteries. The lifespan and operating duration of autonomous frameworks can be estimated by anticipating and regulating their consumption of energy and time duration, which are crucial. The importance of investigating how much energy MRs utilize cannot be underestimated [4]. To fulfill the objectives of minimizing energy usage that also helps to reduce the operating time, the energy issue of MRs has received greater prominence. The consumption of energy and time constraints-based modeling of MRs through computational equations might be highly rigorous for the assessment of the impact of operational situations on energy intake, which serves as a roadmap to support energy-efficient and time constraintstechniques [5]. Since the MR itself can comprehend the quantity of energy needed for motion along with the precise consumption of energy for every component, energy usage can be adjusted to fit various circumstances, and thus, the amount of available energy backup can be determined. Consumption of Energy must be taken into consideration when performing acquisition activities because of the inadequate power available to MRs, and it is essential to ensure time constraints for job completion to accomplish time-sensitive tasks. Figure 1 describes the complete scenarios of energy requirements by the different parts of the MR.  
Additionally, an MR might collide with dynamic obstacles during its movements. For instance, a driverless car might come across pedestrians and traffic signals while gathering traffic-related information. Therefore, while collecting data, an MR must be fully conscious of both energy and time and always be ready to re-plan its movement direction as needed. In relation to path re-planning, various researchers have conducted several investigations and studies. The path-planning techniques in unknown and dynamic environments are provided in [6; 7; 8; 9], and the path re-planning techniques in urgent scenarios have been researched in [10; 11]. In the last century, for optimization purposes, numerous evolutionary algorithm (EA)-based heuristic algorithms have been proposed [12], including ant colony optimization (ACO) [13], artificial bee colony (ABC) [14], differential evolution (DE) [15], fireworks algorithm (FWA) [16], genetic algorithm (GA) [17], and PSO [18]. The PSO technique is widely used for the navigation of a MR to optimize the energy and minimize the time required by them to accomplish predefined tasks with higher accuracy and efficiency. Initially, by the inspiration of fish schooling and bird flocking, [PERSON] and [PERSON] developed PSO dependent upon a population-based optimization method. Nowadays, PSO has emerged as one of the most well-liked optimization methods and can be efficiently used to address a variety of real-world issues due to its ease of usage and quick convergence [19; 20; 21]. Particles within a solution region are initially created arbitrarily for the very first PSO.