## 1 Introduction  
Lithium ion batteries are widely used in spacecraft, aircraft, and electric vehicles. Hence, an accurate prognosis for the remaining useful life (RUL) and time until end of discharge voltage (EODV) is essential for these applications [1]. Significant research has been conducted in order to get higher energy density, reduced weight and cost, longer life, or shorter recharge times [2]. New materials and technologies are continually being explored with the goals of improving the technology and satisfying consumer requirements. The ability to precisely monitor and manage battery health will increase the performance and avoid catastrophic failures, mainly in aerospace systems. The failure of a battery can lead to loss of operation, reduced competency, stoppage, and even catastrophic failure [3]. There are two main prognostic and health management approaches: physics of failure and data driven. Physics of failure prognostic methods utilize knowledge of a product's life cycle, loading conditions, geometry, material properties, and failure mechanisms to predict its RUL [4]. Data-driven techniques extract features from performance data such as current, voltage, time, and impedance using statistical and machine learning techniques to track the product's degradation and estimate its RUL [5]. The current state of a battery is usually indicated by two common indices, namely state-of-charge (SOC) and state of health (SOH). SOC is quantified as the percentage of charge remaining in a battery before recharging during the current cycle. SOH indicates the RUL of a battery [6]. The particle filter (PF) has been extensively used in estimating the RUL of lithium ion batteries [6; 7; 8; 9; 10; 11; 12]. Particle filters are sequential Monte Carlo methods that track and predict the state of the system [13]. [PERSON] et al. [7] proposed a novel approach to predict the state of health (SOH) and RUL using the Dempster-Shafer theory (DST) and the Bayesian Monte Carlo (BMC) method. An empirical model based on the physical degradation behaviour of lithium ion batteries was developed. DST was used to initialize the model parameters by combining sets of training data, and BMC was used to update the model parameters and evaluate the RUL. [PERSON] and [PERSON] [14] presented an empirical model to describe battery behavior during individual discharge cycles as well as over its life cycle. Consequently, the model has been used in a particle filter framework to predict the RUL for individual discharge cycles as well as for cycle life. An extension to [PERSON] et al. 's [7] work was proposed by [PERSON] et al. [6] by testing an empirical second-order polynomial model for RUL predictions and compared it with [PERSON] el al. 's particle filter model. Their results showed that the particle filter predicted the RUL more accurately than the polynomial model. [PERSON] et al. [15] used a lumped parameter battery model to perform diagnostics on lithium ion batteries. Their work was similar to [PERSON] and [PERSON] [14] in regards to the state transition model in EOD predictions. The particle filter has been widely used in prognostics of other applications. [PERSON] et al. [16] used a physics-based model of a centrifugal pump combined with PF in predicting the failure of a pump. This research involves the development of an improved prognostic model to determine End of Life (EOL) and End of Discharge (EOD), thus increasing the accuracy of the RUL prediction of the lithium ion batteries. [PERSON] et al. [17] performed prognostics on lithium ion batteries based on battery performance analysis and flexible support vector regression (F-SVR). In their research, energy efficiency and working temperature were used as the input data and they discovered that the non-iterative prediction model based on F-SVR is suitable for long-term prediction, whereas an iterative multi-step prediction model based on SVR is suitable for short-term predictions. [PERSON] et al.