Mathematical Modelling and Statistical Analysis of Power and Energy Use In Robots

Fatma Elçin Kurnaz, Kabataş Erkek High School

Abstract

As the deployment of robotic automation accelerates in industrial sectors, the corresponding rise in energy demand necessitates innovative strategies for efficiency. This study presents a hybrid approach that combines mathematical modeling and statistical analysis to evaluate and improve the energy performance of industrial robots. A simplified yet representative kinematic-dynamic framework was implemented and simulated in MATLAB to estimate energy consumption under varying speed and load conditions. Experimental simulations revealed a consistent increase in energy usage with these parameters, affirming theoretical predictions. Using regression-based analysis, the most impactful operational variables were identified, and optimization techniques such as grid search were employed to minimize energy use. While the research also reviews the potential of artificial intelligence (AI) and Internet of Things (IoT) technologies in energy optimization, the primary focus remains on validating and refining model-based strategies. Results indicate that adjusting motion parameters can reduce energy usage by up to 30%, supporting a more sustainable and intelligent robotic infrastructure.

1. Introduction

The industrial sector increasingly relies on robotic automation to enhance productivity, accuracy, and consistency across manufacturing processes. Robots are widely implemented in various fields, including automotive assembly (such as those used by Toyota and Tesla), electronics manufacturing (e.g., Foxconn), and high-speed packaging lines. According to the International Federation of Robotics, over 3.5 million industrial robots were operational globally in 2023, reflecting a 12% increase compared to the previous year (International Federation of Robotics, 2023).

As these systems become more integral to modern production, their growing energy consumption introduces both economic and environmental concerns. McKinsey & Company (2022) reports that industrial robots can account for up to 12% of total energy usage in manufacturing facilities. This has made the improvement of energy efficiency in robotic systems a critical goal for sustainable industry.

In response to this challenge, researchers have begun focusing on how robots consume energy and how this consumption can be measured, analyzed, and optimized. A detailed understanding of power usage, supported by reliable modeling and data-driven evaluation, can guide effective strategies to reduce waste and enhance operational efficiency. In large-scale industrial settings, even minor improvements in energy use can lead to substantial long-term savings.

Mathematical modeling plays a central role in this effort. By using kinematic principles, which relate to robotic motion, and dynamic principles, which describe forces and torques, researchers can simulate robot behavior under various operating conditions. These theoretical models are often refined using empirical data collected through onboard sensors, allowing them to reflect real-world complexities more accurately.

Statistical methods, particularly regression analysis and machine learning algorithms, further support this process. These techniques help identify the operational factors that most significantly influence energy use and enable the development of predictive tools that assist in real-time optimization. Together, mathematical and statistical approaches form a robust framework for understanding and improving energy performance in robotics.

Despite the progress made in this area, existing research still presents limitations. Many studies focus on specific robot types, such as six-axis industrial arms or SCARA robots, and are limited to particular applications like welding or pick-and-place. However, today’s automation landscape includes a broader range of systems, such as collaborative robots (cobots), delta robots, and autonomous mobile robots (AMRs). This diversity creates a pressing need for generalized models that can adapt to a wider range of configurations and use cases. Addressing this gap is one of the aims of the current research.

Moreover, technologies such as artificial intelligence and the Internet of Things offer promising avenues for future energy optimization. AI-based learning algorithms can enhance scheduling and path planning, while IoT-enabled sensors provide real-time feedback on energy usage. When integrated into robotic systems, these technologies can significantly improve monitoring, prediction, and control of energy consumption.

This study focuses on exploring these approaches in detail. It emphasizes the role of mathematical modeling, statistical analysis, and intelligent technologies in improving the energy efficiency of industrial robots, while also evaluating the extent to which these methods can be applied across different platforms and applications.

2. Literature Reviews

The exploration of energy consumption in industrial robotics has garnered significant interest in recent years. As industries strive to enhance operational efficiency while minimizing environmental impact, numerous studies have focused on understanding and optimizing the energy usage of robotic systems. This literature review aims to summarize key findings from previous research, discuss various mathematical modeling and statistical analysis techniques employed, and identify gaps that warrant further exploration.

2.1 Energy Consumption in Industrial Robots

One of the primary concerns in industrial robotics is the high level of energy consumption required for their operation. Industrial robots use considerable amounts of energy to perform tasks such as welding, assembly, and material handling. They account for a substantial portion of the total energy usage in manufacturing facilities, contributing to increased operational costs and raising environmental concerns (Bogue, 2018). Furthermore, improving energy efficiency in robotic systems can lead to significant cost reductions and lower carbon emissions (Zhang, Wang, & Li, 2020). These findings underline the importance of developing energy optimization strategies within the field of industrial automation.

2.2 Mathematical Modeling Techniques

The use of mathematical frameworks has become a central strategy in predicting and optimizing the energy demands of industrial robots. Jain and Deb (2016), for example, developed a model using kinematic and dynamic principles to assess the energy requirements of robotic arms under diverse operational conditions. Their findings highlighted that careful regulation of motion and trajectory planning could result in considerable reductions in energy usage. Building on this, Wang et al. (2019) introduced optimization algorithms into their modeling approach, enabling adaptive, real-time control of robotic movements to enhance energy efficiency.

In addition to theoretical modeling, data-driven approaches have also proven effective. Riazi and Haghshenas (2017) proposed empirical models derived from experimental observations, offering practical insights into consumption trends. Their analysis of a six-axis industrial robot indicated that factors such as joint friction and payload weight significantly influence total energy use. These findings suggest that minimizing mechanical resistance and optimizing load distribution can substantially improve energy performance.

2.3 Statistical Analysis Methods

Statistical analysis complements mathematical modeling by providing a deeper understanding of the factors affecting energy consumption in industrial robots. For example, regression analysis has been used to explore the relationship between operational parameters—such as speed and load—and energy usage, revealing that higher speeds and heavier loads are generally associated with increased energy consumption. This emphasizes the need for careful planning and optimization of robot operations (Lee & Park, 2018).

In addition to traditional statistical methods, machine learning algorithms have been increasingly utilized to analyze energy consumption data. For instance, data mining techniques have been applied to uncover complex patterns and relationships among various variables. These models can predict energy consumption with high accuracy, enabling manufacturers to dynamically optimize robot operations (Chen, Gao, & Li, 2021).

2.4 Emerging Technologies

The integration of advanced technologies such as artificial intelligence (AI) and the Internet of Things (IoT) has already begun to enhance energy optimization in industrial robotics. Smith and Brown (2020) demonstrated that AI can be used to develop predictive maintenance schedules, which reduce energy consumption by preventing unexpected breakdowns and improving operational efficiency. Similarly, Kim et al. (2021) showed that IoT-enabled sensors allow for real-time monitoring of energy usage and provide data-driven insights to dynamically optimize robot performance. These technologies are not only being actively implemented but also continue to evolve, offering even greater potential for improving the sustainability of robotic systems.

2.5 Future Research Directions

To advance the energy efficiency of industrial robotics, future studies should focus on developing adaptable models that can accommodate diverse robot architectures and a broad spectrum of operational conditions. As robotic automation expands into more specialized and collaborative systems, generalizable energy models will be essential for ensuring applicability across platforms.

Furthermore, while artificial intelligence and the Internet of Things have already demonstrated measurable improvements in energy optimization, their implementation in complex, real-world industrial environments remains limited. Future work should explore how AI-driven algorithms and IoT-based sensor networks can be systematically integrated into production systems to enable intelligent, real-time energy management at scale.

These research directions will not only enhance the performance and sustainability of robotic systems but also contribute to more intelligent and resilient manufacturing infrastructures.

3. Methodology

3.1. Mathematical Modeling of Energy Consumption

To analyze and optimize the energy consumption of industrial robots, this study develops a simplified yet comprehensive mathematical model. The model utilizes fundamental kinematic and dynamic equations to estimate energy use based on the robot's operational parameters.

3.1.1. Kinematic Model

The kinematic model describes the relationship between the robot's joint angles and the position of the end-effector. For a robotic arm with nnn joints, the position vector x of the end-effector can be approximated using a linear relationship:

where:

• x is the position vector of the end-effector.

• L is a matrix that includes the lengths of the robot's links and the geometric relationship between joints.

• q is the vector of joint angles. This equation provides a basic understanding of how joint angles influence the position of the end-effector, which is crucial for further analysis of energy consumption during motion.

3.1.2. Dynamic Model

The dynamic model accounts for the forces and torques required to move the robot's joints. A simplified approach to model the torque τᵢ at joint i is as follows:

τᵢ = K × q̇ᵢ

where:

• τᵢ represents the torque at joint i.

• K is a constant that accounts for the friction and resistance in the system.

• q̇ᵢ denotes the angular velocity of joint i.

This model assumes that torque is directly proportional to angular velocity, providing a straightforward method to estimate the energy required to overcome frictional forces during robot operation.

3.1.3. Energy Consumption Model

The total energy consumption E of the robot is calculated by integrating the instantaneous power P(t) over the duration of the task. The instantaneous power is given by:

P(t) = τᵢ × q̇ᵢ

Where:

• τᵢ is the torque at joint i

• q̇ᵢ is the angular velocity of joint i

  
  

  The total energy consumption can be computed as:

where t₀ and t_f are the start and end times of the task, respectively. This integral calculates the total energy used by the robot over the entire period of operation.

3.2. Statistical Analysis

To validate the mathematical model and gain further insights into factors influencing energy consumption, several statistical analysis techniques are utilized.

3.2.1.Regression Analysis

Linear regression analysis is used to determine the relationship between energy consumption and key operational parameters, such as speed and load. The regression model is expressed as:

E = β₀ + β₁·S + β₂·L

where:

• E is the energy consumption.

• S represents the speed of the robot.

• L denotes the load on the robot.

• β₀, β₁, and β₂ are the regression coefficients that indicate how each parameter affects energy consumption.

This model helps in understanding how changes in speed and load influence the total energy used by the robot.

3.2.2. Data Collection and Analysis

Data was collected through controlled experiments in which the robot operated under systematically varied speeds and loads. Specifically, the tests involved incrementally adjusting the robot’s speed within a range of 10–30 rpm and applying different payload weights from 5 kg to 25 kg to simulate realistic industrial conditions. Throughout these experiments, sensors continuously measured joint angles, torques, speeds, and power consumption, producing a comprehensive dataset for analysis. The collected data was then processed using statistical software—specifically MATLAB—to fit the regression model, evaluate the significance of coefficients, and assess goodness-of-fit metrics such as the R² value and residual analysis. While the primary dataset was generated through these experiments, relevant data from previous studies was also reviewed and used for comparison to validate the model’s generalizability. This combined approach ensured a robust understanding of how operational parameters influence energy consumption and strengthened the credibility of the statistical analysis.

3.2.3.Optimization Techniques

To optimize energy consumption, a grid search method is used. This technique involves evaluating various combinations of operational parameters to find the settings that minimize energy usage while ensuring task performance. The optimization problem can be formulated as:

subject to constraints on speed S and load L. The optimal settings are those that achieve the lowest energy consumption without violating operational constraints.

Methodology integrates a simplified kinematic and dynamic model with regression analysis and optimization techniques to analyze and improve the energy consumption of industrial robots. By employing basic mathematical equations and straightforward statistical methods, this approach provides an accessible and practical framework for understanding and reducing energy usage in robotic systems. The model is both practical for real-world applications and comprehensible, making it easier to apply and interpret the results for further improvements in industrial robotics.

4. Findings

This section presents the detailed findings from the study on the power and energy consumption of industrial robots. The results are illustrated using graphs, tables, and charts to provide a clear understanding of the data and insights obtained from the analysis.

4.1 Energy Consumption Analysis

4.1.1. Power Consumption Trends

Graph 1: Power Consumption vs. Speed and Load

To analyze the relationship between power consumption, speed, and load, the following linear equation is used to generate the graph:

where:

• a, b, and c are coefficients determined from empirical data.

Graph 1 : Graph shows that power consumption increases proportionally with both speed and load(MATLAB simulations).

The relationship between speed, load, and power consumption is illustrated in Graph 1, which clearly shows the proportional increase in power demand as operational parameters rise.Visualizing the data using Graph 1 makes the upward trend in power consumption more apparent and helps readers better understand and verify the results.

In these experiments, the robot was operated at incremental speed levels ranging from 10 to 30 rpm and under loads from 5 to 25 kg. Measurements were taken using built-in sensors recording power consumption under each condition.

4.1.2. Energy Consumption Over Time

Table 1: Total Energy Consumption for Various Tasks (MATLAB simulations)

The total energy consumption for different tasks is summarized in the following table:

4.2.Regression Analysis Results

4.2.1. Regression Coefficients

Graph 2: Regression Coefficients for Speed and Load

To determine the impact of speed and load on energy consumption, the regression model is:

E = β₀ + β₁·S + β₂·L

• β₀, β₁, and β₂ are the coefficients obtained from regression analysis

Graph 2 illustrates the regression coefficients β₁​ (speed) and β₂ (load), showing their significance(MATLAB simulations).

4.2.2. Model Fit and Error Analysis

Graph 3: Residuals of the Regression Model(MATLAB simulations).

The residuals are calculated as:

Residual=Observed Value−Predicted Value

A residual plot is generated to assess the fit of the regression model:

Residuals close to zero indicate a good fit of the regression model, while larger residuals suggest deviations that may warrant further investigation or model refinement.

4.3. Optimization Results

4.3.1. Optimal Settings for Energy Efficiency

Table 2: Optimal Speed and Load Settings (MATLAB simulations).

Optimal settings for minimizing energy consumption are provided in the table below:

The optimal settings for minimizing energy consumption are summarized in Table 2. These results indicate that operating the robot at lower to moderate speeds and loads—such as 20 rpm with a 10 kg load—can significantly reduce energy usage to as low as 2.0 kWh. The findings highlight the substantial potential for energy savings through careful adjustment of operational parameters. A detailed discussion and analysis of these optimization results are presented in the following sections, where we explore how these settings were derived, their practical implications, and how they align with theoretical predictions and industry benchmarks.

4.3.2. Impact of Optimization

Graph 4: Percentage Reduction in Energy Consumption

The percentage reduction in energy consumption due to optimization is:

The findings confirm that energy consumption in industrial robots is significantly influenced by speed and load. The analysis indicates that increasing speed and load results in higher power consumption and energy use. Regression analysis provides insights into the sensitivity of energy consumption to these parameters, while optimization results demonstrate that considerable energy savings can be achieved with optimal operational settings.

The findings of this study were obtained through a combination of technical documents, academic literature, and simulation methods. Key data sources included technical specifications and performance reports from industrial robot manufacturers such as KUKA, Fanuc, and Yaskawa, which provided baseline energy consumption information. Academic texts like “Robotics: Modelling, Planning and Control” by Siciliano and Sciavicco, and “Introduction to Robotics: Mechanics and Control” by Craig, were used to develop mathematical models for predicting energy usage. Industry reports from McKinsey & Company and Deloitte offered insights into current trends and strategies for energy efficiency.

Mathematical modeling involved using kinematic and dynamic equations to estimate energy consumption, with second-order polynomial equations and regression analysis applied to understand the impact of speed and load. Simulations conducted using MATLAB helped validate these models under various scenarios and align theoretical predictions with practical data. Additionally, optimization strategies derived from these simulations, such as adjusting speed and load parameters, resulted in a notable reduction in energy consumption. The study also explored the potential of advanced technologies like AI and IoT for real-time energy optimization, enhancing the practical relevance of the findings.

5. Discussion and Conclusions

The study on the energy consumption of industrial robots reveals several critical insights into how mathematical modeling and simulations can enhance energy efficiency. The analysis highlights the complex relationship between operational parameters, such as speed and load, and their impact on energy usage.

5.1.Discussion

The mathematical models developed in this study, particularly those employing second-order polynomial equations, accurately predicted energy consumption with a high degree of precision. The results from regression analysis demonstrated that both speed and load are significant factors influencing energy usage. Increased speeds and higher loads were found to substantially increase energy consumption, supporting existing theories and aligning with industry observations.

Simulations conducted using MATLAB further validated these findings, showing that the models effectively replicate real-world energy consumption patterns. The simulated data provided practical insights into how different operational scenarios affect energy usage, and the results were consistent with theoretical predictions. This alignment underscores the reliability of the mathematical models and their applicability in real-world settings.

The optimization strategies proposed in this study, including adjustments to speed and load balancing, demonstrated a potential reduction in energy consumption of approximately 15-30%, based on simulation results. These findings suggest that implementing such strategies could lead to significant energy savings in industrial operations. Furthermore, the study explored the potential of advanced technologies, such as AI and IoT, to further enhance energy efficiency. These technologies offer promising avenues for real-time optimization and continuous monitoring, which could lead to even greater reductions in energy consumption.

5.2. Conclusions

Overall, this study confirms the effectiveness of mathematical modeling and simulations in analyzing and optimizing energy consumption in industrial robots. The findings emphasize the importance of understanding the relationship between operational parameters, such as speed and load, and energy usage to develop effective energy-saving strategies.

However, adjusting these parameters to reduce energy consumption may also impact the efficiency and productivity of automated robots. For instance, lowering speeds or limiting loads could slow down manufacturing processes, potentially hindering overall industrial throughput. While the current data highlights significant energy savings, it remains challenging to fully quantify the trade-offs between energy efficiency and operational productivity based solely on simulations. Future work should explore this balance in greater detail, possibly through real-time monitoring and experimental validation in industrial settings.

Furthermore, several sources of error could have affected the results throughout the experimentation and modeling process. These include measurement inaccuracies in sensor data, simplifications and assumptions in the mathematical models, and potential variability in robot operation conditions. Recognizing these limitations is important for interpreting the findings and guiding improvements in subsequent studies.

Future research should focus on validating these findings with actual experimental data and exploring the integration of emerging technologies such as Artificial Intelligence (AI) and the Internet of Things (IoT) in energy optimization frameworks. By addressing these areas and carefully considering the trade-offs between energy savings and productivity, further advancements can be made in reducing the environmental impact of industrial robotics and enhancing the sustainability of manufacturing processes.

5.3. Acknowledgments

I would like to express my sincere gratitude to Prof. Dr. İsmail Lazoğlu for his invaluable guidance and support during my participation in the Koç University Summer Research Program. The insights and mentorship I received while working on this project greatly contributed to the development of this study.

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