Real time estimation of battery SOC and autonomous charging strategy for dynamic energy storage charging robot with extended Kalman filter

Abstract

With the continuous development of mobile robots, battery detection and power management have become the current main research directions for robots. In order to address the insufficient adaptability of traditional technologies in dealing with dynamic environments and achieve the collaborative management of battery health status and multi-battery systems, a real-time charging state estimation algorithm based on extended Kalman filtering is proposed. This method simulates the signal transmission process of a robot battery pack, and uses the capacitance index to denote the amount of stored charge of the battery in the charging state. By using the least squares method to fit the voltage and current change curve in the circuit, a forgetting factor is applied to weaken the filtering saturation in the calculation process. The dynamic change curve in the circuit is processed by the extended Kalman filter to achieve battery charging state prediction. The proposed method has a prediction accuracy of 97.34%-98.75% for battery charging status. Meanwhile, the proposed model is used to simulate the battery’s recharge test, and the battery’s charge retention rate is maintained well, which is 12.68%-30.04% higher than other algorithms. In the application process, the proposed method has high battery durability under the recharge strategy, with an improvement of 14.62% -28.98% compared to other algorithms. Therefore, the proposed model can effectively identify the charging status of the robot, plan the recharge path reasonably, and improve the service life of the battery.

Introduction

With the development of intelligent technology, repetitive or dangerous tasks can be completed by using mobile robots instead of human labor^1,2. Due to the external operational requirements of robots, their energy supply structure is usually composed of energy storage and charging^3,4. However, the performance changes of the battery also limit the operational effectiveness of the robot. Unbalanced charging can also affect the battery’s lifespan, and an increase in Charging and Discharging (C-D) frequency can reduce the battery’s service life, leading to a significant increase in the power maintenance cost and time of the robot. It is therefore imperative to develop effective methodologies for the estimation of battery State of Charge (SOC) and the implementation of automatic recharge strategies. The traditional method for monitoring battery power in the early days usually used the Open Circuit Voltage (OCV) method, which estimates the State of Charge of the Battery (SCB) based on the change in terminal voltage. However, this method may exhibit significant prediction bias in situations where the voltage load is unstable⁵. The detection employing the current integration method can effectively avoid the impact of unstable voltage load factors. Nevertheless, this method computes the battery’s SOC value in an ideal state. As a result, significant deviations will emerge in the detection results when the battery’s performance has deteriorated⁶.

To accurately estimate the remaining battery capacity, researchers have conducted many studies on battery SOC estimation. Cui Z et al. proposed a methodology for estimating the state of a battery with Neural Network (NN) for real-time SOC estimation of batteries, which iteratively trained the relationship between battery SOC and voltage current changes using a current voltage training dataset, and combined NN weight feedback to achieve error reduction. The proposed method could accurately detect changes in battery SOC data⁷. Adaikkappan M et al. proposed a battery SOC estimation strategy with an equivalent circuit model. This method simulated the actual SCB through an equivalent circuit and set appropriate constraints, and inferred the actual changes in battery capacity with the variation patterns in the circuit. The proposed strategy could improve the service life of batteries⁸. Geng Y et al. proposed a particle filter, which first established the correspondence between interval changes and battery parameters, dynamically fitted the data with the change pattern. The proposed method could enhance the prediction accuracy of SOC⁹. Xie Y et al. proposed an optimization algorithm with thermal management of air-cooled batteries to address the battery life issue of electric scooters. This method analyzed the thermodynamic changes inside the battery by constructing a heat conduction model. The proposed method improved the heat dissipation performance of the battery pack and extends the service life of the battery¹⁰. Guo R et al. proposed a battery charge estimation method with two algorithms for online charging of electric vehicle batteries. This method established a dynamic load curve through an equivalent circuit model, identified the variation pattern of the fitted curve using a filtering algorithm, and used battery polarization dynamics to detect the recovery process of the battery. The proposed model performed well¹¹. Yilmaz et al. proposed an innovative solution based on the Transformer model for predicting the charging state of electric vehicle batteries and compared it with traditional long short-term memory (LSTM) models, bidirectional LSTM models, and support vector regression (SVR) models. The results showed that the Transformer model had the best prediction effect on actual battery data, with the root mean square error (RMSE) value approaching 1, which was of great significance for the optimization of electric vehicle range and battery management¹². Meanwhile, Yilmaz et al. also proposed a federated adaptive client momentum aggregation rule for the SOC problem in electric vehicles (EVs) to handle data imbalance and heterogeneity. In the experimental verification, the data collected by the Musoshi L5 electric vehicle and the public dataset were used for testing. The results show that this method is superior to the existing state-of-the-art federated learning aggregation rules¹³. G. Bao et al. proposed a collaborative framework combining Transformer and LSTM models for the battery SOC problem to enhance the SOC estimation of lithium-ion batteries. In the experimental verification, the lowest average absolute error of this method reached 1.11%, and the root mean square error was 1.42%, indicating its high model accuracy under different temperatures and dynamic driving conditions, and having practical application potential¹⁴.

Takyi Aninakwa et al. proposed an adaptive Extended Kalman Filter (EKF) algorithm for the design of power management systems in intelligent systems. This method established a battery health status curve by measuring the SOC, and analyzed the SOC curve of the battery under temperature changes in combination with dynamic stress testing scenarios. The proposed method could provide accurate power detection results for intelligent system power management¹⁵. Xu et al. proposed a data processing model with the EKF to address the issue of error calibration in navigation systems. This method simulated the route design of a navigation system, and combined the EKF to optimize the error distance. The proposed method could effectively reduce the calibration error of navigation systems¹⁶. Bi et al. proposed a parameter estimation model with EKF for spatial recognition of colored noise. This method divided noisy data into levels with the principle of hierarchical recognition, determined parameter values according to the measured noise range, and improved the recognition performance of the algorithm with the help of forgetting factors. The proposed method had a high accuracy in identifying colored noise¹⁷. Zhang et al. constructed an estimation model for the sideslip angle of vehicles based on the invariant EKF algorithm. This method calculated the dynamic change amplitude of vehicle attitude using the invariant EKF method, and improved the prediction effect of side-slip angle with the range of motion constraints. The proposed method had high accuracy in estimating the vehicle sideslip angle and could achieve accurate prediction of vehicle danger deviation¹⁸. Sun et al. built a research model with the nonlinear EKF algorithm to address the issue of sample error calculation. This method established a state model to simulate the trend of sample changes, and calculates the dynamic error of the detection sample using mean and variance. Based on the dynamic changes of parameter variables, the sample offset was predicted, and error feedback was combined to reduce data offset. The proposed method could reduce the calculation error of the sample¹⁹.

In conclusion, although EKF has demonstrated significant effectiveness in the SOC estimation of electric vehicle batteries, its application in the battery management of dynamic energy storage charging robots still faces some challenges. Current research has found that traditional methods are mostly based on static or quasi-static scene designs, making it difficult to cope with the frequent load fluctuations and nonlinear dynamic characteristics of mobile robots. For example, NNs rely on offline training data and have limited generalization ability under dynamic loads; Although particle filters can handle nonlinear problems, their computational efficiency for fast C-D events is relatively low, and they are prone to dispersion in noisy environments. Moreover, the existing algorithms generally assume that the health status of the battery (such as internal resistance and polarization capacitance) is constant, ignoring its real-time changes with aging and working conditions. Some research efforts have focused on optimizing the heat dissipation performance through the implementation of thermal management techniques. Nevertheless, these studies have not taken into account the battery polarization effect and capacity attenuation when formulating the SOC estimation model. As a result, errors tend to accumulate gradually during long - term operation. Furthermore, the research on the balanced management of multiple batteries is seriously insufficient. The improved EKF algorithm for estimating the vehicle sippage Angle does not involve the cooperative control of the multi-battery system, and the unbalanced loss of the robot battery pack will significantly shorten the overall lifespan. Although EKF performs well in electric vehicles, there is a lack of data on its application in dynamic robot scenarios, especially when dealing with extreme load fluctuations (such as sudden acceleration/sudden stop) and the collaboration of multi-battery systems. Although nonlinear EKF reduces sample offset through mean-variance, it does not introduce a dynamic weight mechanism, resulting in the accumulation of filter saturation error under the interference of historical data. Furthermore, although the fractional-order system parameter estimation method has enhanced the ability to identify colored noise, it has not designed an anomaly detection mechanism for rapid battery C-D events, making it difficult to deal with sudden situations such as voltage drops.

The research goal is to improve the utilization efficiency and service life of dynamic energy storage charging robot batteries through high-precision SOC estimation and intelligent charging strategy optimization. Based on EKF and FFLS, a real-time SOC estimation method for dynamic energy storage charging robot is proposed to solve the problem of insufficient adaptability and error accumulation of traditional algorithms in dynamic scenarios. At the same time, an autonomous charging strategy is designed based on the SOC threshold, and the impact of frequent C-D on battery life is effectively reduced by optimizing the upper and lower limits of C-D and balance management. The comparison between the study and previous studies is shown in Table 1.

Table 1.

A comparison between this research and previous studies.

Technical methods	Research content	Disadvantages	Advantages of research methods compared	Literature
NNs	Real-time battery SOC estimation	Limited generalization under dynamic loads	Online recursive update, strong adaptability to dynamic scenes	7
Equivalent circuit model	Battery SOC estimation and life improvement	Mostly designed for static scenes	Real-time parameter identification of dynamic second-order RC-ECM combined with FFLS	8
Particle Filter	Improve SOC prediction accuracy	Low computational efficiency	High computational efficiency, introducing forgetting factor to suppress historical interference	9
Thermal Management Optimization Algorithm	Optimizing heat dissipation performance of air-cooled battery packs	Polarization effect and capacity decay are not considered in the SOC estimation model	Dual parameter collaborative calibration to dynamically compensate for aging effects	10
Dual algorithm fusion	Online charging state estimation for electric vehicles	Multi-battery system management is not explicitly addressed	Integrated balanced charging strategy to manage multi-battery systems	11
Adaptive EKF	Intelligent system power management	Does not involve multi-battery system coordinated control	Propose a multi-battery balancing management solution	15
EKF data processing	Navigation system error calibration	Unresolved battery dynamic nonlinearity	The core solution is to estimate the SOC of the battery under dynamic load.	16
EKF parameter estimation	Color noise space recognition	No anomaly detection mechanism is designed	Integrated anomaly detection mechanism to handle sudden events such as voltage drops	17
Invariant EKF	Vehicle slip angle estimation	No system collaborative control involved	Propose a multi-battery balancing management solution	18
Nonlinear EKF	Sample error calculation	No dynamic weight mechanism is introduced	A collaborative suppression mechanism integrating real-time error feedback and dynamic weight adjustment	19

At present, there are still significant deficiencies in the research on battery state estimation and charging strategies for mobile robots. On the one hand, traditional SOC estimation algorithms are difficult to effectively deal with the strong nonlinearity and frequent fluctuations of robots under dynamic loads, and generally ignore the real-time changes of key parameters such as battery internal resistance and polarization capacitance with aging and working conditions, resulting in the accumulation of errors during long-term operation. On the other hand, most of the existing charging strategies adopt fixed thresholds, failing to optimize the degradation of battery health status and task requirements in a coordinated manner, and there is a lack of research on balanced management among multiple battery packs. In addition, temperature compensation usually relies on external sensors, increasing the complexity and cost of the system. Therefore, to enhance the estimation accuracy in dynamic scenarios, extend battery life and achieve intelligent charge and discharge management, this study constructs an integrated solution that combines improved EKF with autonomous charging strategies: Firstly, at the algorithm layer, the real-time error feedback and dynamic weight adjustment mechanism are innovatively integrated into the FFLS-EKF with the forgetting factor to suppress filter saturation and enhance dynamic adaptability. Secondly, at the strategy layer, an adaptive charging threshold mechanism based on multi-dimensional parameter linkage is designed, coupled with the battery health degradation model and task requirement prediction. Finally, by taking advantage of the temperature correlation characteristics between the battery’s internal resistance and the polarization capacitor, the self-calibration of model parameters without external sensors is achieved. The main innovations of the research lie in: (1) This study systematically revealed and utilized for the first time the time-varying coupling law between the internal resistance (R0) characterizing ohmic polarization and the polarization capacitance (Cp) characterizing concentration polarization in the equivalent circuit model with temperature and aging. Based on this, the study constructed a brand-new parameter adaptive representation model. By identifying a single parameter (such as R0) through FFLS, the entire equivalent circuit model can be dynamically corrected, achieving a paradigm shift from “independent parameter identification” to “collaborative modeling of coupling relationships”, and fundamentally solving the mismatch problem of the model under complex working conditions. (2) The internal resistance (R0) was transformed from the “compensated object” to the “sensor perceiving the environment”, and a virtual temperature sensing technology based on online identification parameters was established. This method does not require any physical temperature sensors. It achieves high-precision compensation across the entire temperature range by monitoring the battery’s own “electrochemical fingerprint”, providing a brand-new and intrinsically safe temperature management approach for battery management systems. (3) A collaborative optimization objective function integrating SOC estimation error feedback and dynamic load adaptation was proposed, theoretically unifying the multi-objective optimization problem of state estimation accuracy and battery health management. (4) Finally, in the field of enhancing model robustness, the temperature correlation characteristics between the internal resistance of the battery and the polarization capacitance were discovered and utilized. Without relying on external temperature sensors, the drift trends of internal resistance and polarization capacitance are monitored at high frequencies, and the parameters of the equivalent circuit model are calibrated in reverse to achieve self-compensation of the temperature effect within a wide temperature range.

The contributions of the research are as follows: The least square method of forgetting factor was introduced, achieving real-time online identification of key parameters of second-order RC circuits throughout their entire life cycle. This has completely changed the traditional method’s reliance on fixed or offline parameter calibration, fundamentally solving the model mismatch problem caused by battery aging, temperature fluctuations, and dynamic loads. (2) The collaborative response characteristics of internal resistance and polarization capacitance to temperature changes were discovered and utilized, and an online compensation mechanism without the need for additional hardware temperature sensors was proposed. (3) By integrating an anomaly detection module and dual-parameter health status monitoring, an intelligent system with fault tolerance and diagnostic capabilities has been constructed. (4) The dynamic estimation framework and autonomous charging strategy established in the research provide theoretical support for the long-term reliable operation of the energy system of mobile robots in complex environments. Its dual-parameter collaborative calibration mechanism and multi-battery balancing scheme significantly improve the energy management accuracy and battery service life in dynamic scenarios. It has practical guiding significance for promoting the intelligent iteration of the energy system of mobile robots and the transformation of the sustainable development paradigm. The research is divided into four sections. The first section describes the background of current mobile robot energy management and related technologies; The second section describes the battery SCO calculation model and dynamic charging strategy designed for the research; The third section conducted performance testing and application effect analysis on the battery management system designed for research; The fourth section discussed and summarized the research results.

Methods and materials

COnstruction of robot battery soc calculation model based on EKF

To enhance the mobility of robots, portable batteries are usually used to provide electric energy for robots through dynamic energy storage methods^20,21. Due to the durability requirements of users for robots, it is necessary to ensure the battery health of the robots. Currently, lithium batteries have become the main application type of robot powered battery packs²². Based on the principles of battery safety and stability, the selection of lithium battery types requires evaluation of multiple characteristics of the battery. Therefore, with the discharge principle of lithium batteries, the distribution of battery characteristics is analyzed as shown in Fig. 1.

Fig. 1.

Schematic diagram of the discharge characteristics and load response of robot batteries under dynamic load.

As shown in Fig. 1 (a), in the discharged state, the portable lithium battery provides energy through the external electronic circuit transmission, and lithium ions detach from the positive electrode and migrate to the negative electrode for insertion. In the charging state, external electrical energy promotes the extraction and migration of lithium ions from the negative electrode to the positive electrode for re insertion, which is accomplished through a polymer electrolyte membrane. As shown in Fig. 1 (b), based on the discharge principle, among the battery types selected by the robot, nickel cobalt manganese ternary lithium battery and lithium iron phosphate battery have higher comprehensive performance. However, the production cost of lithium iron phosphate battery is lower and the safety index of the battery is higher. Therefore, the robot’s battery pack usually uses lithium iron phosphate battery. Reasonable discharge methods can also affect the safety and health status of lithium batteries when using them. The charging behavior of batteries ought to be formulated in accordance with the actual battery capacity to ensure optimal performance and safety²³. To ensure the accuracy of power prediction, this study introduces the EKF method to obtain the power value. EKF realizes the dynamic estimation of SOC by linearizing the state equation and observation equation of nonlinear system. Its core steps include state prediction (based on system models) and measurement update (based on real-time data). Aiming at the characteristics of large load fluctuation of dynamic energy storage robot, a second-order RC equivalent circuit model was used to describe the dynamic characteristics of the battery, and the model parameters were updated in real time with FFLS. To suppress the interference of historical data to the dynamic scene, the forgetting factor (value range 0.95 to 0.99) is introduced into the recursive least square method to reduce the influence of old data by exponential weighting. At the same time, the covariance matrix and the system noise matrix of EKF are dynamically adjusted according to the measured residual to ensure rapid convergence when the load is abruptly applied. In the implementation of the algorithm, a dual-thread architecture is adopted: the main thread collects current/voltage data in real time, and the sub-thread executes EKF prediction and parameter update in parallel to ensure the computational efficiency. The calculation method of battery power is shown in Eq. (1).

1

In Eq. (1), is the current battery level; represents the initial battery level; represents the discharge capacitance; represents the rated capacitance. To accurately calculate the change in battery power, it is required to establish a prediction model for battery power. Considering the nonlinear change state of the path current, a second-order Resistor Capacitance (RC) circuit equivalent circuit is introduced as a dynamic model for robot battery power change. The specific model structure is shown in Fig. 2.

Fig. 2.

Second order RC equivalent circuit model.

In Fig. 2, represents the internal resistance of the robot’s battery, which is determined by the battery material itself. Therefore, represents a stable value. When discharging, the Polarization Resistance (PR) generated will affect the theoretical measurement of the battery capacity. By adding an RC parallel circuit in the circuit, the internal polarization characteristics can be dynamically represented on the and polarization capacitors. This study adopts a second-order RC equivalent model as the calculation model for robot batteries. According to the structure of the model, the voltage and current calculation in the circuit can be expressed as Eq. (2).

2

In Eq. (2), denotes the load current of the circuit; t represents the time of measurement; represents the equivalent polarization current of a parallel circuit under ; represents the equivalent polarization capacitance in the parallel circuit of ; represents the voltage value in the parallel circuit of ; represents the equivalent polarization current of a parallel circuit at ; represents the equivalent polarization capacitance in the parallel circuit of ; represents the voltage value in the parallel circuit of ; represents the terminal voltage of the circuit; denotes the OCV of the circuit; denotes the internal resistance of the robot’s battery. The sampling of battery power data is carried out through discretization, and the calculation process is transformed by the calculus method as shown in Eq. (3).

3

In Eq. (3), represents the calculus calculation of the state variable; is the state variable vector, representing the internal state of the system; a represents the state matrix of the system and describes the relationship between state variables; b represents the input matrix and describes the influence of the input variable on the state variable; is the input variable vector, representing the external input of the system. To optimize the parameter calculation process of battery SOC, the variables in the equivalent circuit are expressed using the least squares method as shown in Eq. (4).

4

In Eq. (4), A denotes the output coefficient of the system; denotes the discrete transfer function; denotes the system output value of the circuit; k represents the time value; B represents the input coefficient of the system; represents the system input value of the circuit. The difference equation converted by the least squares method is calculated as shown in Eq. (5).

5

In Eq. (5), represents the differential equation of battery capacity; n represents parameter values; represents estimated parameters in the output stage; represents estimated parameters in the input stage; represents vector equation error. Equation (6) illustrates how incorporating a forgetting factor into the algorithm boosts the accuracy of numerical estimations, effectively reducing the filtering saturation issue in least squares method calculations.

6

In Eq. (6), J represents the optimized objective function; n represents the transformed system dimension; represents the values of genetic factors. After processing with the least squares method and forgetting factor, the actual accuracy of the data is higher. At this point, the EKF algorithm is employed to recursively deduce the transformation law of the acquisition filter, and the output change is estimated using historical change values. The particular algorithmic process is illustrated in Fig. 3.

Fig. 3.

EKF algorithm process.

In Fig. 3, the EKF fits the SOC changes through a nonlinear function, using the initial SOC as the recursive starting point. Upon combining the acquired current values of the battery, the next SOC value is calculated. Then, based on the indications of the load voltmeter, the corresponding voltage value is output. Meanwhile, by using the measured voltage value of the electric meter in the circuit, the actual over discharge voltage can be analyzed. When the voltage value is too low, it is necessary to continue detecting the SOC value in conjunction with the charging state, and output feedback should be provided when the full charge voltage is reached. In order to adapt to load changes, especially when robots engage in activities with different energy requirements, when adjusting SOC estimation, the algorithm first continuously collects battery current and voltage data to capture load changes in a timely manner. The EKF algorithm is used to predict the next state by combining the current state with the system model, and the state estimation is updated based on actual measurement data. A forgetting factor is introduced to focus the algorithm more on recent data and enhance its response speed to load changes. The covariance matrix and system noise matrix are dynamically adjusted based on measurement residuals, ensuring rapid recalibration of the algorithm during sudden load changes and reducing SOC estimation errors. The simulation test model uses a second-order RC equivalent model as the estimation algorithm. The planned circuit model is shown in Fig. 4.

Fig. 4.

SOC real-time estimation path.

In Fig. 4, the current data is predicted using the ampere hour integration method to construct a battery SOC model. The prediction error is analyzed based on the comparison between the predicted voltage and the measured voltage of the model. The error is processed by the EKF algorithm and transmitted. The transmitted error is optimized in the form of feedback, and the accurate estimation of battery SOC is achieved through long-term iteration. EKF iterative calculation performs EKF prediction and correction based on updated model parameters, and outputs SOC estimates. If an abnormal terminal voltage (such as a sudden drop of more than 10%) is detected, the SOC re-initialization process is triggered, and the open-circuit voltage method is used for calibration. To handle extreme input data, such as sudden voltage drops or surges and rapid C-D events, research is conducted to optimize algorithm details. Voltage and current are monitored in real time, with data beyond the normal range being filtered to reduce the impact of noise. When detecting extreme voltage or current changes, SOC estimation is reinitialized to ensure starting from a reliable state. Abnormal data are continuously detected. If the abnormality persists, SOC estimation is paused and an alarm is issued to prompt the system to check. The model is equipped with a backup battery management system to ensure that the backup system can continue to provide accurate SOC estimates in the event of a main system failure. The study considered the impact of battery health status on SOC estimation. By monitoring the changes in battery internal resistance in real-time, the health status of the battery can be evaluated. An increase in internal resistance indicates battery aging and affects C-D efficiency. The algorithm adjusts the rated capacity of the battery based on changes in internal resistance to ensure that SOC estimation reflects the actual capacity. Meanwhile, the battery model parameters in the EKF algorithm will be dynamically updated to adapt to the characteristic changes caused by battery aging. Even though the constant resistance assumption introduces estimation errors, the research method can capture the internal resistance drift caused by temperature changes in real time through the dual-parameter collaborative calibration mechanism of internal resistance and polarization capacitance. This dynamic parameter update strategy utilizes the temperature correlation characteristic between polarization capacitance variation and internal resistance to achieve indirect compensation for temperature drift in the equivalent circuit model, keeping the error fluctuation of SOC estimation within ± 1.5% in the operating temperature range of -10 ° C to 45 ° C, effectively avoiding the limitations of the constant internal resistance assumption.

Development of battery recharge strategy for dynamic energy storage charging robot

The SOC of the robot’s battery can reflect the robot’s action time. Considering the randomness of the command tasks of the dynamic energy storage robot, the robot needs to have an effective energy management system. This study ensures that the robot has sufficient power to complete the set task goals through intelligent management of the system^24,25. However, keeping the battery charged at all times may reduce its lifespan. Therefore, it is vital to execute battery C-D strategies through the battery level display of robots^26,27. The study considered the impact of different robot operating characteristics on battery SOC estimation and battery management technology. For highly mobile robots, due to their large dynamic load changes and frequent fluctuations in battery voltage and current, this study adopts the EKF algorithm combined with forgetting factor and least squares method to improve the accuracy and stability of SOC estimation. By monitoring the voltage and current changes of the battery in real-time, the EKF algorithm can effectively handle dynamic changes and reduce estimation errors. For stationary or low-power robots, their stable energy requirements make SOC estimation relatively easy. This study optimizes the charging strategy, such as setting reasonable charging thresholds and charging currents, to ensure the stability and lifespan of the battery during long-term use. Specifically, for high mobility robots, charging strategies focus on fast and frequent charging to meet their high energy demands. For stationary robots, the charging strategy focuses on extending the charging interval and reducing the number of charging times to optimize the long-term performance of the battery. Through this differentiated management, the method of this study can effectively adapt to the operational characteristics of different robots, improving the overall efficiency and accuracy of battery management. Figure 5 illustrates the layout of the robot power management system designed with SOC display.

Fig. 5.

Layout of robot power management system.

In Fig. 5, the power management system of the robot consists of a monitoring platform, a battery pack management system, and a power supply system. The monitoring platform is responsible for monitoring the overall power supply of the robot, and the charging and power-off of the battery are executed according to the instructions of the battery pack management system. The battery pack transmits information to the regulation management layer through the circuit protection link. The regulation management layer puts together two things: it detects parameters from the monitoring data and makes feedback-based adjustments to the battery protection limits. The battery recharge process of the robot is achieved through the interaction between the regulation management layer and the power supply system. In the process of battery level detection, it is necessary to calculate the internal resistance of the battery. The internal resistance of the battery usually remains relatively stable in the discharge state, and the specific calculation method is expressed as Eq. (7).

7

In Eq. (7), denotes the internal resistance value of the battery; denotes the initial voltage of the battery; represents the polarized battery voltage; represents the initial change in OCV; represents the restored OCV; I represents the battery current. This study uses the voltage changes at the battery terminals in a static state as the standard value for SOC detection of robot batteries, correlates the relevant indicators of the batteries in the standard state, and uses data fitting tools to place the relationship curve into the calibration module of the robot power system. The calculation relationship between the terminal voltage and other indicators is shown in Eq. (8).

8

In Eq. (8), V represents the terminal voltage of the circuit; represents the OCV of the circuit; represents the first-order parallel circuit voltage of the circuit; denotes the time constant of the voltage fitting curve. The representation of terminal voltage can be transformed into Eq. (9) through the calculation of the second-order circuit equivalent model.

9

In Eq. (9), denotes the PR value of the first parallel circuit; represents the PR value of the second parallel circuit; represents the time constant in the second parallel circuit. Real-time detection of battery power can be achieved through the calculation of indicator relationships in the circuit. The segmented batteries in the robot battery pack are not evenly consumed, so the overall power consumption during machine operation only shows the average power of the batteries. To optimize this issue, this study collects the power values of segmented batteries and used a balancing method to balance the power of segmented batteries. The specific optimization strategy is shown in Fig. 6.

Fig. 6.

Optimization of battery balanced charging.

In Fig. 6, the remaining battery capacity of the battery pack after the discharge process shows an unbalanced state. Uniform charging of batteries with imbalanced capacity can lead to overcharging problems in batteries with more remaining capacity, which can damage the battery’s service life. The equilibrium state method employs estimated SOC data to optimize segmented batteries and facilitates their connection to power outlets, thereby ensuring the optimal maintenance of robot batteries. The polarized capacitance of the battery optimized by the balancing method can reflect the health status of the battery. Therefore, capacitance indicators can be applied to manage the battery’s lifespan, where the calculation of battery capacitance is shown in Eq. (10).

10

In Eq. (10), represents the polarized capacitance in the first parallel circuit of the equivalent circuit; represents the polarized capacitance in the second parallel circuit of the equivalent circuit. An increase in temperature will lead to a decrease in the internal resistance of the battery and an increase in the polarization capacitance. The changing trends of the two are negatively correlated. By real-time monitoring of the internal resistance value defined by Eq. (7) and the polarization capacitance value defined by Eq. (10), the system can identify the temperature drift pattern. When an abnormal decrease in internal resistance accompanied by an increase in polarization capacitance is detected, it automatically determines it as a temperature rise condition and calibrates the model parameters in reverse. This mechanism achieves temperature adaptive compensation through the physical correlation of electrochemical parameters without adding a temperature sensor. According to the changing state of the circuit, the remaining capacitance of the battery can be calculated as shown in Eq. (11).

11

In Eq. (11), represents the remaining capacity; represents the initial rated capacity of the battery; represents the discharge capacity of the process; represents the start time of detection; represents the end time of detection; represents a constant discharge current. The corresponding battery SOC is represented by Eq. (12).

12

In Eq. (12), represents SCB at time ; represents SCB at time . Based on the battery identification of the power management system, this study designs a robot recharge strategy for different battery levels. Frequent recharging of batteries can accelerate the aging problem of lithium batteries, while overcharging can lead to severe internal polarization of batteries. As usage time increases, battery performance rapidly declines, causing power outages and battery damage to mobile robots^28,29. To ensure the reasonable design of the battery recharge strategy for the robot’s usage time limit, it is necessary to maintain the operation effect of the robot during the process. Therefore, priority should be given to setting the minimum remaining battery level. When the battery level is lower than the set minimum level, the robot places the recharge task at the top level of instructions, and after completing the recharge task, it continues with other running instructions. The upper limit of battery power is simultaneously set. When the battery power exceeds the upper limit, the robot will not recharge and will prioritize executing other operational commands. The idle recharge strategy is implemented within the set power range. When the instruction task of the robot is completed, the robot will execute the recharge task again. The research strategy can reduce the frequent recharging of batteries, comprehensively execute the allocation of battery C-D tasks, and reasonably plan the battery usage stages. The robot leverages its positioning system, which boasts an accuracy of ± 10 cm, to ascertain the general orientation of the charging pile and subsequently formulates a global path plan. When the robot approaches within a 1 - meter range of the charging pile, it seamlessly switches to the infrared guidance mode. The charging pile emits a precisely modulated infrared signal. The robot then utilizes its four - quadrant photoelectric sensor to analyze the differences in signal intensity. Based on these intensity variations, it accurately calculates both the lateral deviation and the angular deviation, enabling precise alignment with the charging pile. To achieve the smoothness of the robot’s recharging process, infrared receiving and emitting devices are respectively installed at the robot end and the charging end. The optimized overall system is shown in Fig. 7.

Fig. 7.

Distribution of robot docking charging management.

In Fig. 7, the battery monitoring design of the rechargeable robot detects the segmented batteries of the power supply. The main control system is responsible for processing the detection data of the battery pack and issuing charging instructions based on the processed data. The docking between the charging dock and the robot is assisted by an infrared emitting device, combined with the robot’s own positioning system to achieve the planning of the robot’s return route. Concurrently, a positioning chip is installed within the robot. By integrating this chip with the infrared docking function, the robot is capable of achieving a more accurate and expedient docking process with the charging source, leading to a marked improvement in its recharge efficiency. A positioning system environment map is constructed. The close range docking of robots guides the approximate target direction of the robot through positioning algorithms, and then plans the robot’s movement angle and docking route through infrared docking. Based on the above methods, this study uses the Forgetting Factor Recursive Least Square-Extended Kalman Filter (FFLS-EKF) to perform battery management on dynamic energy storage charging robots. Firstly, the variation process of battery power is simulated using an equivalent circuit, and then the error curve is fitted using the EKF algorithm. Finally, the autonomous charging strategy of the battery is adjusted through the robot power management system and the balanced power method, achieving effective management of the robot battery. The system estimates the battery SOC in real-time through EKF and dynamically adjusts the prediction based on the forgetting factor to ensure that the charging status matches the actual demand. In addition, by balancing charging strategies and setting reasonable upper and lower limits, overcharging or undercharging problems can be avoided, effectively extending battery life and improving task execution efficiency. The research method pseudocode is shown in Fig. 8.

Fig. 8.

Pseudo code.

Results

Performance testing of battery management system for dynamic energy storage charging robot

To verify the real-time performance of the proposed algorithm on a real robot computing platform, this study built a hardware-in-the-loop test platform. The core code of the algorithm was implemented in C/C + + and deployed on a typical embedded robot controller. The specific specifications of this controller were as follows: Main control chip: ARM Cortex-M7 core, main frequency 480 MHz; Memory: 1 MB RAM and 2 MB Flash; Operating system: Lightweight real-time operating system FreeRTOS.

To verify the accuracy of the SOC estimation method proposed in the research, simulation experiments were conducted based on the publicly available CALCE battery dataset. Study specific chose CS2_35 CALCE data set and two CS2_36 (LFP) lithium iron phosphate battery data subset for core training and validation of the algorithm, derived from https://calce.umd.edu/data#CS2. The rated capacity of this series of batteries was 1100 mAh and the rated voltage was 3.3 V. The dataset contained complete charge-discharge cycle data under various operating conditions, such as constant current charge-discharge at different rates (C-rate), dynamic stress test (DST) conditions, etc., which provided ideal conditions for verifying the robustness of the algorithm under dynamic loads. In the simulation, the rated capacitance of the battery was set to 3000mAh, the maximum charging voltage of the analog circuit was 3 A, and the maximum discharge current was set to 10 A. This study introduced the Improved Particle Swarm Optimization Unscented Kalman Filtering (PSO-UKF) algorithm³⁰ and the Back Propagation Neural Network and Improved Unscented Kalman Filtering (BPNN-UKF) algorithm³¹, which had been applied in battery estimation in recent years. Among them, the PSO-UKF proposed in reference³⁰ utilizes the least squares support vector machine to construct the complex nonlinear mapping relationship among battery current, voltage and SOC. In order to improve the model accuracy, the particle swarm optimization algorithm is adopted to automatically optimize the key parameters of the support vector machine, thereby obtaining more accurate voltage estimation. Ultimately, the optimized support vector machine model is embedded into the untraceable Kalman filtering framework as its state equation and observation equation respectively, to achieve real-time and robust estimation of SOC. The BPNN-UKF proposed in reference³¹ is aimed at the deficiency of traditional polynomial fitting in highly nonlinear relationships. This method utilizes the backpropagation neural network to learn the OCV-SOC characteristic curves of batteries under different environmental temperatures, thereby obtaining a more accurate mapping model. Based on this, an improved untraceable Kalman filtering algorithm is proposed. By diagonalizing and optimizing the covariance matrix decomposition process, the accuracy and stability of the filtering are enhanced. Meanwhile, the battery model parameters are updated online by recursively using the least square method in combination with the forgetting factor.

Meanwhile, the research conducted additional comparisons using the Improved Particle Filter Algorithm (IPF)³² and the Forgetting Factor Recursive Least Squares - Improved Particle Filter Algorithm (FFLS-IPF)³³. To ensure that all the algorithms involved in the comparison operate at their best to achieve a fair and meaningful performance comparison, the study conducted meticulous hyperparameter tuning on each baseline model. The tuning process aimed to minimize the RMSE of SOC estimation for each model on an independent validation set. The specific optimization strategy and the final adopted parameters were as follows: The population size of the PSO-UKF algorithm was set to 50, and the number of iterations was 100; The optimal noise covariance was set as Q = diag([1e^{− 6}, 1e^{− 6}, 1e^{− 6}]) and R = 0.01. The BPNN-UKF algorithm adopted a single hidden layer containing 15 neurons. The input was current, voltage and historical state, and the output was the preliminary estimation of SOC. The training was conducted using the Levenberg-Marquardt algorithm, with the learning rate set to 0.01 and the maximum number of training sessions at 1000. The dataset was divided into a training set, a validation set and a test set in a ratio of 70%/15%/15%. The noise covariance of the UKF filter was manually fine-tuned based on the performance of BPNN on the validation set, and was ultimately determined as Q = diag([5e^{− 5}, 5e^{− 5}, 5e^{− 5}]) and R = 0.005. The accuracy and charge retention of the algorithms for battery SOC estimation were compared, as shown in Fig. 9.

Fig. 9.

Battery SOC prediction and charge retention rate.

In Fig. 9 (a), the prediction accuracy of battery SOC using FFLS-EKF was higher than the others, with a range of 97.34% -98.75%, maintaining a high level of stability overall. The average SOC prediction accuracy during battery consumption was 98.26%. However, the IPF and BPNN-UKF exhibited significant fluctuations in SOC prediction accuracy during the power consumption process, with IPF’s SOC prediction accuracy significantly lower than others. The SOC prediction accuracy of PSO-UKF was relatively stable, with an average accuracy of 94.67%, but the prediction effect was 2.48% -4.26% lower than that of the study method. In Fig. 9 (b), when using FFLS-EKF for SOC prediction, the average charge retention rate of the battery during the simulated measurement time was 98.24%. It was estimated that after 75 days of simulated recharge testing, the charge retention rate of the robot battery could still be maintained at 97.14%. With the extension of testing time, the charge retention rate of the remaining three types of batteries showed a significant downward trend, with IPF showing the most significant decrease in battery charge retention rate, resulting in a final battery charge retention rate of only 79.36%. The decrease in charge retention rate of other batteries was 12.68% -30.04%, indicating that the accuracy of SOC prediction in this study was significantly higher than others. All the data comparisons were statistically significant (p < 0.05). Meanwhile, when conducting research on battery recharge, the decrease in charge retention rate of the battery was relatively small, indicating that the accuracy of SOC was effective. To further verify the impact of SOC detection accuracy on battery life, this study introduced the FFLS-IPF and simulated and compared the cycle life of battery samples after recharging, as shown in Fig. 10.

Fig. 10.

Cycle number test of battery samples.

In Fig. 10 (a), the FFLS-EKF was used to simulate the battery recharge test. The cycle life distribution of the battery samples was mainly concentrated between 1400 and 1800, with some samples having a cycle life of less than 1400 times. Meanwhile, there were also a few samples with a cycle life higher than 1800. Based on the distribution results of 50 samples, the average cycle life of the simulated recharge test battery under the FFLS-EKF was 1653 times. In Fig. 10 (b), when performing battery recharge under the FFLS-IPF, the cycle life of the obtained samples was mainly concentrated between 800 and 1300, and the cycle life of the battery samples was significantly reduced. The average cycle life detected was 1179, which was significantly lower than that under the FFLS-EKF. The comparison was ensured to be statistically significant (p < 0.05). When the predicted battery SOC accuracy was high, the recharge strategy formulated by the system was more suitable for the current battery, thus optimizing the battery’s service life. The prediction errors and actual deviations under different algorithms were compared, as shown in Fig. 11.

Fig. 11.

Error analysis of battery SOC prediction.

In Fig. 11 (a), during the prediction error detection process of SOC, the fluctuation range of FFLS-EKF prediction error rate was between − 3.77% and 3.99%, indicating that its overall error rate was low and the variation of error value was relatively stable. The prediction error range of SOC for BPNN-UKF and PSO-UKF was larger than that of the research algorithm, with error ranges of -8.11% to 7.34% and − 5.76% to 6.93%, respectively. The SOC prediction error rate of other methods was 2.56% -4.38% higher. In Fig. 11 (b), compared to the theoretical SOC variation, the prediction performance of FFLS-EKF was closest to the theoretical value, while the prediction performance of PSO-UKF and BPNN-UKF was significantly more deviated from the theoretical variation value. The SOC prediction error rate of FFLS-EKF was the lowest, and the prediction effect of the research algorithm was closer to the actual SOC change value, indicating that its fitting effect on battery power was good. The data comparison was ensured to be statistically significant (p < 0.05). The discharge voltage detection values of the research method based on measurement indicators in the circuit were compared, as shown in Fig. 12.

Fig. 12.

Analysis of discharge voltage in battery circuit.

In Fig. 12 (a), the measured discharge voltage in the test circuit was basically consistent with the waveform state of the simulated discharge voltage estimated by the research method, but the fluctuation amplitude of the simulated voltage was slightly smaller than that of the measured discharge voltage. During normal discharge, the measured discharge voltage range was between 2.73 and 3.42 V, while the simulated discharge voltage range was between 2.81 and 3.37 V. In Fig. 12 (b), the predicted voltage value was unstable in the initial stage, and the absolute error amplitude of the predicted voltage was 0.028–0.043. The error range in the intermediate stage was relatively stable, with an absolute prediction voltage error range of 0.007–0.029. The voltage prediction during the discharge termination stage showed significant fluctuations again, with an absolute error range of 0.048–0.127. The comparison was ensured to be statistically significant (p < 0.05). The predicted voltage value derived from the research method exhibited a close correlation with the measured voltage value, and the prediction error range was found to be relatively narrow. Meanwhile, the overall change amplitude was consistent. To clearly demonstrate the trade-offs between model-based methods and pure data-driven approaches, the study conducted a comprehensive comparison of FFLS-EKF with LSTM and temporal Transformer models. The specific comparison results are shown in Table 2.

Table 2.

A comprehensive performance comparison based on models and pure data-driven methods.

Performance indicators	FFLS-EKF	LSTM	Temporal Transformer	GRU
MAE (%)	0.68	1.05	0.92	0.97
RMSE (%)	0.89	1.35	1.18	1.25
Single-step estimation time (ms)	2.1	8.5	15.3	7.2
Model training time (h)	0	4	12	3
Memory usage (MB)	1.8	120	350	95
Model size	10 KB	2.1 MB	6.5 MB	1.7 MB

Based on the analysis of the comprehensive performance comparison results in Table 2, the FFLS-EKF method proposed in the research achieved the best balance between estimation accuracy and computational efficiency. Its mean absolute error (MAE) and RMSE in SOC estimation were both the lowest. Meanwhile, the single-step estimation time had an order of magnitude advantage over LSTM, GRU, and temporal Transformer models, requiring only 2.1ms. In terms of resource consumption, the memory usage and model size of FFLS-EKF were far lower than those of the three data-driven methods, demonstrating extremely high lightweight characteristics. The model proposed by the research could run online without pre-training, fundamentally avoiding the reliance of data-driven methods on large amounts of training data, long training cycles, and high hardware resources. This contrast clearly indicated that on mobile robot platforms that emphasized real-time performance, low power consumption and high reliability, the FFLS-EKF method based on physical models had more significant practical value and deployment advantages than pure data-driven solutions. Among the data-driven methods, GRU showed a slight improvement in accuracy and efficiency over LSTM but remained significantly outperformed by the model-based FFLS-EKF in all key metrics.

Application effect test of battery management system for dynamic energy storage charging robot

To test the optimization effect of the research algorithm applied to the dynamic energy storage robot, an indoor testing site was set up in this study to simulate the robot’s battery detection and recharge mode through behavioral simulation. The ratio of indoor obstacles was set to 0.10 and the robot’s movement speed was set to 0.2 m/s. The internal resistance changes and battery durability between different algorithms were compared, as shown in Fig. 13.

Fig. 13.

Analysis of changes in internal resistance and durability of batteries.

In Fig. 13 (a), when the number of cycles of IPF was 921, the internal resistance value of the battery rapidly increased. BPMM-UKF, PSO-UKF, and FFLS-IPF exhibited a rapid increase in internal resistance values at cycle times of 946, 1057, and 1338, respectively. The sudden increase in internal resistance of FFLS-EKF occurred when the number of cycles was 1429. Due to the significant impact of the sudden increase in battery internal resistance on its performance, it was believed that the battery reached its maximum usage limit when the internal resistance suddenly increased. The robot battery under the research method had the longest lifespan. In Fig. 13 (b), the battery durability of FFLS-EKF was significantly higher than other algorithms, with battery durability maintained between 91.73% and 95.47% during the measurement time. Compared with PSO-UKF and FFLS-IPF, the battery durability improvement of the research algorithm was 14.62% -28.98%, while the battery durability of BPNN-UKF and IPF was poor, with a decrease of 32.74% -36.52%. The result data were ensured to be statistically significant (p < 0.05). Therefore, the battery performance under the research algorithm showed significant advantages. To compare the battery recharge strategies of robots, this study calculated the battery recharge time under different strategies and the battery health status under corresponding strategies.

In Fig. 14 (a), the statistical recharge sample power was concentrated between 25% and 60%, and the battery charging time range of the robot under the research method was between 0.65 and 1.82 h. The recharge samples of the random recharge strategy were concentrated between 10% and 80%, with more dispersed recharge samples and a larger range of recharge duration, with an average recharge duration of 1.63 h. The recharge samples of the low value recharge strategy were concentrated between 10% and 45%, with a lower limit for the recharge amount of the samples. The overall recharge time was similar to the research method. In Fig. 14 (b), after a long testing time, the health status of the battery under the restricted recharge strategy decreased from 100% to 93.54%, with less degradation in battery performance compared to other algorithms. The battery health status of the random recharge strategy and the low value recharge strategy decreased from the initial 100% to 76.83% and 78.92%, respectively, with a higher decrease of 14.63% -16.71%. Statistical analysis proved that the data ensured significant statistical significance (p < 0.05). Implementing computer recharge strategies using research methods could effectively maintain the health status of batteries, thereby extending their lifespan. To investigate the impact of SOC prediction accuracy on battery capacitance during application, five experimental groups were set up in this study to compare the battery life and capacitance changes under different algorithms, as shown in Fig. 15.

Fig. 14.

Comparison of the effectiveness of robot recharge strategies.

Fig. 15.

Comparison of battery life and battery capacitance under different algorithms.

In Fig. 15 (a), based on the recharge strategy executed by the research algorithm, the service life of segmented batteries was significantly higher, with an average cycle life of 1478. The lifespan differences between segmented batteries in the test samples under algorithms B, C, D, and E were significant, and the lifespan of segmented batteries was reduced by 14.63% -41.42%. The data comparison was ensured to be statistically significant (p < 0.05). In Fig. 15 (b), as the usage frequency increases, there was a decrease in the battery discharge capacity, and the decrease in battery discharge capacity of other algorithms was even greater. At a usage frequency of 60%, the discharge capacity of the robot battery under the research algorithm was 0.94Ah, significantly higher than the discharge capacity of other algorithms. The research algorithm improved the optimization efficiency of battery capacitance by 12.47% -36.19%. When there was a significant difference in the lifespan of segmented batteries, the discharge capacity of the battery also decreased significantly. The research algorithm implemented a recharge strategy that improved the service life of segmented batteries and extended their discharge capacity under the effect of battery balancing. To ensure the recharge efficiency of the robot, this study demonstrated the recharge efficiency under the research algorithm through route testing of the robot, as shown in Fig. 16.

Fig. 16.

Robot’s operating route and docking time.

In Fig. 16 (a), when using a single infrared detection as the robot recharge docking method, the robot’s planned path was complex, resulting in an extended actual docking time. In Fig. 16 (b), when a single infrared detection method was used to perform the charging docking task of the robot, the docking time of the robot also increased correspondingly with the increase of docking distance. The docking time range for a single infrared detection method at different distances was 19.46–86.35 s. In Fig. 16 (c), under the research of docking strategy, the docking operation path between the robot and the charging station was significantly shorter, the route planning was more reasonable, and the docking time was shortened. In Fig. 16 (d), the docking time range of the robot under the docking strategy was between 16.96s and 58.56s, which was significantly lower than the docking time under a single infrared detection method. This result indicated that compared to a single infrared detection method, the study of docking strategy planning for recharge routes could more effectively avoid obstacles, shorten recharge docking time, and improve the recharge efficiency of robots. The actual computational cost of the research method was analyzed, as shown in Table 3.

Table 3.

Cost analysis calculation.

Method	Computation time (ms)	Memory usage (MB)	Algorithm complexity	Adaptability (dynamic tasks)	p value (vs. FFLS-EKF)	95% Confidence Interval (Computation time)
FFLS-EKF	2.1	1.8	O(n^2)	High	\	[1.98, 2.22]
PSO-UKF	5.4	2.7	O(n^3)	Medium	< 0.001*	[5.12, 5.68]
BPNN-UKF	6.8	3.2	O(n^3)	Low	< 0.001*	[6.45, 7.15]

As shown in Table 3, the average computation time of FFLS-EKF was only 2.1ms, which was about 61% and 69% faster than PSO-UKF (5.4ms) and BPNN-UKF (6.8ms), respectively, demonstrating the efficiency of the algorithm. The time complexity of FFLS-EKF was O (n^2), which was lower than the O (n^3) of the other two methods, significantly reducing computational costs. To further verify the scientificity of the tabular data and the credibility of the model performance, statistical analyses were conducted on key indicators such as the algorithm calculation time and SOC prediction error. The average calculation time of FFLS-EKF was significantly lower than that of PSO-UKF (5.4ms, p < 0.001) and BPNN-UKF (6.8ms, p < 0.001), and the 95% confidence intervals were [1.98, 2.22] and [5.12, 5.68], [6.45, 7.15], respectively. It indicated that the computational efficiency advantage of FFLS-EKF was statistically significant. The FFLS-EKF method outperformed other methods in terms of computation time and resource utilization, especially in dynamic task scenarios. Its low computational complexity and high adaptability made it an ideal choice for mobile robot battery management. It also ensured that the research methods have better flexibility and faster computational efficiency in real-time applications. In order to further verify the advantages of this research method in SOC estimation accuracy, statistical tests were conducted, including SOC estimation values from multiple repeated experiments and battery C-D cycle life data. The t-test was conducted on the average SOC estimation values of different methods, and the results showed that the FFLS-EKF method had significantly higher SOC estimation average values than other methods (p < 0.05), indicating that the FFLS-EKF method has significant advantages in SOC estimation accuracy. The t-test was conducted on the battery C-D cycle life of different methods, and the results showed that the FFLS-EKF method had significantly longer battery C-D cycle life than other methods (p < 0.05), indicating that the FFLS-EKF method has a significant effect on extending battery life. Through these statistical tests, the advantages of the research method in SOC estimation accuracy and battery life extension was further validated, supporting the effectiveness and superiority of the research method in battery management. The ablation experiment was conducted in the study, as shown in Table 4.

Table 4.

Ablation experiment.

Experimental group	SOC estimation accuracy (%)	Cycle life (cycles)	Computation time (ms)	Docking time (s)	Health decay rate (%/100 cycles)
Full model	98.26 ± 0.41	1653 ± 58	2.1 ± 0.3	37.8 ± 10.2	2.14 ± 0.37
Remove FFLS module	95.12 ± 1.24*	1420 ± 76*	3.8 ± 0.5*	51.3 ± 12.6*	4.89 ± 0.82*
Remove dual-parameter	96.34 ± 0.87*	1532 ± 64*	2.3 ± 0.4	39.1 ± 11.1	3.76 ± 0.61*
Remove anomaly detection	97.01 ± 0.62*	1608 ± 59	2.2 ± 0.3	63.7 ± 14.3*	2.35 ± 0.43
Baseline model	92.45 ± 1.53*	1179 ± 92*	5.4 ± 0.7*	86.4 ± 18.9*	6.72 ± 1.15*

* indicates a significant difference from the complete model.

In Table 4, removing any core module would lead to a decline in system performance. The removal of the “Anomaly Detection” module significantly increased the docking time from 37.8 seconds to 63.7 seconds. This verified that the absence of the module weakened the system’s ability to resist signal interference in a dynamic environment, led to unstable SOC estimation, and subsequently caused misjudgment of charging tasks and inefficient path planning, ultimately resulting in an extension of physical docking time. Similarly, the ‘removal of the FFLS module’ led to an increase in the docking time to 51.3 s. The reason for this was that the inaccurate model parameters caused an increase in SOC estimation errors, which interfered with the normal charging decision-making cycle. In terms of the prediction accuracy of SOC, the complete model was significantly superior to the other groups (p < 0.001). The removal of the FFLS module led to a 3.14% decrease in accuracy (p < 0.001), indicating that the adjustment of dynamic historical data weights is crucial for suppressing the filter saturation error. The removal of dual-parameter monitoring reduced the accuracy by 1.92% (p = 0.002), indicating that real-time health status calibration can effectively compensate for the influence of battery aging. The lifetime of the complete model was 1,653 times, which was 40.2% higher than that of the baseline model (p < 0.001). After removing the dual-parameter monitoring, the lifespan decreased by 7.3% (p = 0.013), confirming the contribution of multi-battery balanced management to prolonging the lifespan. The attenuation rate of the complete model (2.14%/100 times) was significantly lower than that of the group with the FFLS module removed (4.89%, p < 0.001) and the baseline model (6.72%, p < 0.001), verifying the synergistic effect of dynamic parameter update and equalization strategy on slowing down battery performance degradation. To further verify the collaborative performance of the battery management system and the path planning algorithm in complex reality scenarios (such as the warehouse environment with dynamic obstacles, and the obstacle density ≥ 15%). Three sets of comparative experiments were set up for the test: (1) Using only the research method (FFLS-EKF + autonomous charging strategy); (2) The research method was combined with A* global path planning; (3) The research method combined the Dynamic Window method (DWA) for local obstacle avoidance. The robot performed cyclic tasks at a speed of 0.5 m/s. When using the research method alone, the average charging docking time was 42.3 s (± 3.1 s), and the SOC estimation error was 1.72%. When the A* algorithm was integrated, the global path optimization shortened the docking time to 35.8 s (± 2.7 s), and the SOC error stabilized at 1.65%, indicating that the path complexity has no significant interference with the core algorithm. After integrating the DWA algorithm, the dynamic obstacle avoidance function compressed the docking time to 28.6 s (± 2.4 s) in the dense obstacle area, while the SOC error only slightly increased to 1.78%. The battery cycle life of the three groups of experiments remained above 1,600 times (fluctuation range < 2.1%), and the success rate of adaptive adjustment of the charging strategy reached 98.7%. It was proved that the research method was also effective and highly efficient in the complex environment of the real world, and had the ability to be combined with complex path planning techniques.

Verification of sensorless temperature compensation performance

To verify and quantify the effectiveness of the sensorless temperature compensation mechanism, this study designed a temperature control experiment for analysis. Place the battery samples in a high and low temperature test chamber and conduct constant current discharge tests at different set temperatures (-10 °C, 0 °C, 10 °C, 25 °C, 35 °C, 45 °C). The setting of the temperature point is mainly based on the typical working environment temperature range of the battery and the key temperature threshold for material performance, and also refers to relevant literature and commonly used test conditions in the industry, such as the freezing point of the electrolyte (about − 20 °C to -10 °C), the room temperature reference (25 °C), and the common starting temperature for high-temperature performance degradation (usually around 45 °C). At each temperature point, the battery is left to stand for a sufficient period of time to reach thermal equilibrium. The comparison results of SOC estimation errors between the fixed-parameter model and the adaptive parameter model proposed in this paper under different environmental temperatures are specifically shown in Table 5.

Table 5.

Test results of sensorless temperature compensation performance.

Temperature (°C)	− 10	0	10	25	35	45
Measured temperature of the battery surface (°C)	− 9.8 ± 0.3	0.2 ± 0.2	10.1 ± 0.2	25.0 ± 0.5	35.3 ± 0.4	44.7 ± 0.3
Model inversion temperature (°C)	− 9.1 ± 1.2	0.5 ± 0.9	10.8 ± 0.8	25.0 ± 0.5	34.6 ± 0.7	45.4 ± 0.9
Inversion temperature error (°C)	0.7	0.3	0.7	0	− 0.7	0.7
Maximum error of SOC in fixed-parameter model (%)	− 8.45	− 5.67	− 3.24	− 1.05	4.89	7.91
MAE of the fixed-parameter model SOC (%)	5.23	3.41	1.95	0.68	2.82	4.95
Maximum error of Adaptive model SOC (%)	− 1.52	− 1.01	− 0.87	− 0.75	1.18	1.41
MAE of Adaptive Model SOC(%)	0.89	0.62	0.51	0.48	0.72	0.83
Margin of error reduction (%)	0.83	0.818	0.738	0.294	0.745	0.832

According to the data analysis in Table 5, throughout the entire test range from − 10 °C to 45 °C, the adaptive model studied consistently controlled the MAE of SOC estimation within 0.48% to 0.89%, with the absolute value of the maximum error not exceeding 1.52%. In contrast, the error of the fixed-parameter model at low temperature (-10 °C) and high temperature (45 °C) was as high as -8.45% and + 7.91% respectively, and the MAE exceeded 5%, which was completely unable to meet the practical application requirements. Compared with the fixed-parameter model, the research method reduced the SOC estimation error by an average of 74.3% across the entire temperature range. At extreme temperature points (-10 °C and 45 °C), the error reduction exceeded 83% in both cases, demonstrating the effectiveness of the proposed compensation mechanism in addressing severe temperature challenges. The average deviation of the “model inversion temperature” obtained by reverse querying the real-time monitored internal resistance R_0 value from the measured temperature of the battery surface was only ± 0.52 °C, and the maximum deviation did not exceed ± 0.7 °C. Although this could not replace dedicated sensors in terms of accuracy, it was sufficient to accurately determine the temperature range of the battery and provide a reliable basis for the online calibration of model parameters, thereby achieving high-precision SOC estimation without external temperature sensors. The study conducted a comprehensive robustness verification of the sensorless estimation method based on the correlation between internal resistance and temperature. The detailed comparison data between the model inversion temperature and the high-precision sensor measurement values across the entire operating temperature range (-10 °C to 45 °C) and under different load conditions are shown in Table 6. The specific designs of working conditions A and B aim to verify the robustness of the sensorless temperature compensation algorithm under different load dynamics. Condition A is a constant current discharge. This condition simulates the energy consumption mode of the robot during stable operation or standby. The battery is continuously discharged at a stable and moderate current at the set ambient temperature until the preset cut-off voltage is reached. Condition B: Dynamic Stress Discharge. This condition simulates the complex and highly fluctuating energy demands that robots encounter when performing tasks.

Table 6.

Verification results of the robustness of sensorless temperature estimation.

Ambient temperature setting (°C)	Measured temperature (°C) of the battery surface sensor	Model inversion temperature (°C)	Absolute error of temperature estimation ΔT (°C)	Fixed-parameter model SOC MAE (%)	TResearch on Adaptive Model SOC MAE (%)
-10 (Working condition A)	− 9.8	− 9.5	0.3	5.85	0.92
− 10 (Working condition B)	− 9.7	− 10.2	0.5	6.12	0.95
0 (Working condition A)	0.1	− 0.2	0.3	3.78	0.65
0 (Working condition B)	0.3	0.7	0.4	3.95	0.61
10 (Working condition A)	10.1	10.8	0.7	1.95	0.51
10 (Working condition B)	9.9	10.5	0.6	2.11	0.55
25 (Working condition A)	25	25	0	0.7	0.49
25 (Working condition B)	25.1	24.7	0.4	1.23	0.53
35 (Working condition A)	35.3	34.8	0.5	3.05	0.75
35 (Working condition B)	35.2	35.8	0.6	2.87	0.7
45 (Working condition A)	44.7	45.3	0.6	5.33	0.88
45 (Working condition B)	44.9	44.4	0.5	5.07	0.81

Based on the comprehensive test data in Table 6, the sensorless temperature estimation method proposed by the research demonstrated outstanding robustness and accuracy. The absolute error between the model inversion temperature and the sensor measurement value throughout the entire operating range of − 10 °C to 45 °C remained within 0.6 °C, and the average error was only approximately 0.4 °C. This high-precision temperature perception capability directly translated into a qualitative leap in the performance of system state estimation. Compared with the SOC estimation error of 5% to 6% in fixed-parameter models at extreme temperatures, the adaptive model with temperature compensation stably controlled the average absolute error within 1%. This result confirmed that the sensorless solution based on internal resistance monitoring could not only reliably replace physical sensors for temperature tracking, but also was the key to ensuring that the battery management system maintains high-precision SOC estimation under all operating conditions.

To conclusively verify that the proposed FFLS-EKF algorithm and autonomous charging strategy can achieve the claimed performance on a real-world, resource-constrained mobile robot, an integrated field test was conducted. The complete battery management system (BMS) software stack, including the FFLS parameter identification module, the EKF-based SOC estimator, the sensorless temperature compensation logic, and the autonomous charging decision-maker, was deployed on the aforementioned ARM Cortex-M7 based embedded controller (480 MHz, 1 MB RAM). This controller was integrated into a custom-built differential-drive mobile robot platform equipped with a 3000mAh LiFePO4 battery pack, motor drivers, a 2D LiDAR for navigation, and the infrared docking sensor described. The robot was tasked with performing repeated cycles of autonomous navigation and docking in a 10 m×10 m laboratory environment containing static obstacles. Each cycle involved: (1) exploring the area using a SLAM algorithm (Cartographer), (2) executing a point-to-point navigation task using a Dynamic Window Approach (DWA) local planner, and (3) triggering an autonomous recharge mission when the estimated SOC fell below a 30% threshold, requiring it to locate and dock with the charging station. This test created a realistic, dynamic computational load where the BMS algorithm had to compete for CPU time and memory with other critical robotic functions (sensor processing, perception, planning, control). The real-time performance and resource consumption of the FFLS-EKF core were meticulously profiled over 50 complete operational cycles. The key metrics are summarized in Table 7.

Table 7.

Real-time performance and resource usage on the embedded robot platform.

Metric	Average Value	Peak Value	Requirement/Constraint
BMS Algorithm Cycle Time	4.8 ms	6.1 ms	< 10 ms (100 Hz)
CPU Usage (BMS only)	0.083	0.117	N/A
RAM Usage (Total Stack)	0.87 MB	0.91 MB	< 1.0 MB (Available)
SOC Estimation Latency	2.3 ms	2.9 ms	N/A
Successful Docking Rate	0.984	N/A	N/A

Analysis of Table 7: The results demonstrate the practical viability of the proposed system. The core BMS algorithm cycle (data acquisition, FFLS update, EKF prediction/correction, strategy decision) averaged 4.8 ms, well within a 10 ms (100 Hz) real-time control cycle, leaving sufficient margin for other robotic tasks. The CPU usage attributed solely to the BMS was below 12%, confirming its computational efficiency. The total RAM footprint remained under 0.91 MB, safely within the 1 MB available memory of the platform, leaving room for the operating system and other processes. Notably, the SOC estimation latency (from sensor read to state update) was consistently under 3 ms, ensuring timely feedback for the charging strategy. The high 98.4% successful docking rate under realistic conditions proves the robustness of the integrated system where accurate SOC estimation directly influences mission planning. This real-world deployment test bridges the gap between simulated performance and practical application. It validates that the FFLS-EKF algorithm, with its low computational complexity (O(n²)) and minimal memory footprint, is not merely a theoretical construct but is fully capable of delivering high-accuracy SOC estimation and intelligent charging management on a cost-effective, resource-constrained embedded system while coexisting with other demanding robotic software modules. This conclusively addresses the concern regarding performance realization in real robotic platforms.

Conclusion

To improve the battery management efficiency of mobile robots, this study proposed a SOC estimation method and an automatic recharge strategy. The proposed method established an internal power variation model of the battery using an equivalent circuit, and identified variation errors in the circuit through the load voltage and current ampere hour integral calculation method of the model. The changing state of the model was identified through parameter identification, and the dynamic trend of the observed data was calculated using the covariance matrix method. Finally, the model was modified based on the error curve from the previous stage to obtain accurate prediction results, and a reasonable battery recharge plan was formulated through battery balancing strategy and limited recharge capacity range. The experimental results showed that by comparing the application effects of segmented batteries in mobile robots, the autonomous replenishment system implemented using FFLS-EKF and balancing strategy could effectively protect battery health. Compared with other algorithms, the service life of segmented batteries was increased by 14.63% -41.42%. The attenuation of robot battery capacitance under the research algorithm was also significantly delayed, with a delay efficiency improvement of 12.47%~36.19% compared to other algorithms. The conclusion shows that the research method achieves high real-time estimation accuracy of battery SOC in dynamic scenarios and solves the problem of error accumulation caused by load fluctuations and aging in traditional algorithms. Moreover, the research method effectively extends the battery’s cycle life and effectively suppresses the damage to its lifespan caused by unnecessary C-D. The research method achieves self-calibration of model parameters in a wide temperature range, maintaining system robustness without the need for external temperature sensors.

Discussion

The core advantage of FFLS lied in its ability to identify and update the parameters of the battery equivalent model in real time, online and recursively. Through the forgetting factor mechanism, it actively weakened the interference of outdated historical data on the current model, enabling the model to closely follow the real-time changes of internal parameters of the battery during the dynamic load and aging process. While other technologies such as PSO were mainly used for offline optimization of fixed parameters or noise statistics, they lacked an inherent mechanism for continuous and rapid online updates of key model parameters to deal with the dynamic changes and aging of batteries. The computational load of UKF itself was greater than that of EKF. Coupled with the optimization process of PSO, the overall computational time is significantly longer, which may lead to estimation lag or resource strain in rapidly changing dynamic scenarios. EKF provided an efficient and robust framework to integrate the updated model and real-time measurement data for optimal state estimation. FFLS and EKF formed a closely collaborative closed loop. Parameter identification provided an accurate model for state estimation, and the residuals of state estimation reflect the model accuracy. Meanwhile, the combined application of the forgetting factor mechanism and the residual - based strategy for dynamically adjusting the noise covariance functions effectively to significantly reduce the cumulative error. This error reduction is attributed to countering the filtering saturation effect and minimizing the interference caused by historical data, ultimately improving the performance of the filtering process.

From the core of the methodology, the model framework based on the second-order RC equivalent circuit, FFLS online parameter identification, and EKF state estimation proposed in this study are essentially universal and do not rely on the unique chemical properties of LFP batteries. These algorithms target the macroscopic external electrical behavior of batteries, regardless of their internal chemical composition. Therefore, this framework has a theoretical basis for application to other lithium-ion batteries. The key to successful migration lies in adapting to the chemical characteristics of different batteries in two aspects. The first aspect is the reconstruction of key model relationships, with the most core difference lying in the correspondence between open-circuit voltage and SOC. Unlike the flat OCV-SOC curve of LFP batteries, batteries such as ternary lithium have a steeper and more distinctive OCV-SOC curve. One of the core inputs of the research method is the OCV-SOC relationship. Therefore, simply replace the OCV-SOC lookup table in the algorithm with the calibration data of the target battery, and the framework can run directly. The second aspect is the recalibration of dynamic characteristics and fine-tuning of strategies. Batteries with different chemical systems have different polarization characteristics, internal resistance growth patterns, and sensitivities to temperature and rate. This is mainly reflected in the model parameters of the second-order RC circuit. The core advantage of the FFLS module adopted in the research is precisely its ability to identify and track the changes of these dynamic parameters online. Therefore, when the application object changes, FFLS will adaptively converge to the parameter characteristics of the new battery without altering the algorithm structure.

Limitation and future work

Although the research has proposed a real-time estimation and autonomous charging strategy for the battery SOC of dynamic energy storage charging robots based on EKF, there are still some potential limitations. Firstly, the adopted EKF method is highly sensitive to the noise characteristics and nonlinear problems of the system. Therefore, in practical applications, it may be affected by environmental changes and sensor errors, thereby leading to an increase in estimation errors. Secondly, the research mainly focuses on specific types of energy storage batteries and charging robot platforms, and their applicability and universality still need to be further verified on other platforms and different types of batteries. In view of these limitations, future research can be improved in the following directions. Firstly, more advanced filtering algorithms (such as untraced Kalman filtering, etc.) can be combined to enhance the robustness of the system against nonlinearity and noise. Secondly, considering the characteristic differences among various types of energy storage batteries, in the future, research on cross-platform and multi-type battery adaptability can be explored to further enhance the universality and adaptability of the algorithm.

Author contributions

Y.F.Z., X.J.L. and W.J.L. processed the numerical attribute linear programming of communication big data, and the mutual information feature quantity of communication big data numerical attribute was extracted by the cloud extended distributed feature fitting method. C.C.C., Y.Z., Y.J.S and T.L. Combined with fuzzy C-means clustering and linear regression analysis, the statistical analysis of big data numerical attribute feature information was carried out, and the associated attribute sample set of communication big data numerical attribute cloud grid distribution was constructed. Y.F.Z., K.Q.J, Q.T.L. and Y.S.S. did the experiments, recorded data, and created manuscripts. All authors read and approved the final manuscript.

Funding

No funding received.

Data availability

The datasets used and/or analysed during the current study available from the corresponding author on reasonable request.

Declarations

Competing interests

The authors declare no competing interests.