Independent Torque Control for Zero-Turn Vehicles, Optimizing Maneuverability and Energy Efficiency Using Matlab

Author : Waqas Javaid

Abstract

This article presents a comprehensive study on independent torque control for zero-turning radius vehicles, focusing on enhancing maneuverability and operational efficiency. The research investigates how varying wheel torques individually can achieve precise vehicle motion, including in-place rotations, commonly required in robotics, agricultural, and utility vehicles [1]. A dynamic simulation framework is developed using MATLAB, incorporating vehicle parameters such as mass, wheel radius, wheelbase, and inertia, along with damping effects. Torque profiles are applied to both wheels to demonstrate different driving modes, including straight motion, zero-radius turns, and differential acceleration [2]. The simulation calculates linear and angular velocities, wheel slip ratios, and cumulative energy consumption, providing insight into vehicle performance under various torque strategies [3]. Results show the effectiveness of independent torque control in achieving rapid turning without compromising stability. Wheel slip and power analysis further highlight energy-efficient operation, critical for electric and hybrid platforms. Trajectory visualization confirms the practical feasibility of the control strategy in constrained environments [4]. This study contributes to advanced vehicle control design, emphasizing precise motion control and energy optimization. The findings are relevant for engineers, researchers, and developers aiming to improve zero-turn vehicle performance in practical applications.

1. Introduction

The demand for highly maneuverable vehicles has grown significantly in industries such as robotics, agriculture, construction, and material handling. Among these, zero-turning radius (ZTR) vehicles have emerged as a key solution due to their ability to rotate in place and navigate confined spaces with precision.

Figure 1 presents independent torque control for a zero-turning radius vehicle, illustrating wheel torque distribution, rotational dynamics, trajectory planning, and energy performance metrics. Achieving this capability requires careful design of the independent torque control system, where each wheel’s torque can be controlled separately to generate desired motion. Unlike conventional vehicles that rely on steering angles for turning, ZTR vehicles depend on differential torque application to the wheels, enabling in-place rotation and tight maneuvers [5]. This approach not only improves maneuverability but also enhances operational efficiency by minimizing the required turning space [6]. Recent advancements in electric and hybrid drive systems have further enabled precise torque control, allowing for better energy management and reduced slip during aggressive maneuvers. Simulation-based studies play a crucial role in designing these systems, providing insight into vehicle dynamics, wheel slip, power consumption, and trajectory behavior without the risks of real-world testing. By modeling key vehicle parameters such as mass, wheel radius, wheelbase, and moment of inertia, engineers can predict how torque inputs affect linear and angular velocities, energy usage, and overall stability. Additionally, torque control strategies can be optimized to balance speed, safety, and energy efficiency, particularly important for autonomous or semi-autonomous vehicles [7].

Table 1: Performance Metrics Summary

Table 1 presents the performance metrics summary of the zero-turning radius vehicle, highlighting velocity, wheel speeds, distance traveled, slip ratio, energy consumption, and turning radius. Visualizing trajectories and analyzing wheel behavior under different torque profiles helps in refining control algorithms and verifying performance under various driving scenarios. This article presents a detailed MATLAB-based simulation framework to study the effects of independent wheel torques on ZTR vehicle dynamics. It demonstrates the transition between straight-line motion, differential turning, and zero-radius rotation, while evaluating power and energy metrics. By understanding the interplay between wheel torques and vehicle motion, engineers can design more responsive, energy-efficient, and precise vehicles suitable for challenging environments [8]. The findings of this study contribute to both academic research and practical vehicle development, emphasizing advanced control strategies, energy optimization, and trajectory accuracy [9]. Ultimately, the integration of independent torque control in zero-turn vehicles represents a significant step forward in modern vehicle design, offering solutions for compact navigation, operational flexibility, and intelligent motion control [10].

1.1 Background of Modern Vehicle Maneuverability

Modern transportation and robotic mobility systems require vehicles that can operate efficiently in confined and complex environments. Traditional vehicles rely on steering mechanisms that require a considerable turning radius, which limits their performance in tight spaces. As urban environments and industrial operations become more compact, the need for highly maneuverable vehicles has increased significantly. Zero-turning radius vehicles have emerged as an effective solution to this challenge [11]. These vehicles are capable of rotating around their own axis without requiring additional turning space. Such capability makes them highly suitable for applications such as robotic platforms, lawn mowers, warehouse automation vehicles, and agricultural machines. The fundamental concept behind this maneuverability is differential control of wheel motion. By controlling each wheel independently, the vehicle can perform precise directional movements [12]. This concept has led to growing research interest in torque-based vehicle control systems. Understanding how torque influences vehicle dynamics is therefore essential.

1.2 Concept of Zero Turning Radius Vehicles

Zero turning radius (ZTR) vehicles represent a unique category of mobile systems that can rotate in place without forward displacement. This capability is achieved by applying opposite rotational forces to the wheels located on either side of the vehicle. When one wheel rotates forward while the other rotates backward, the vehicle spins around its central axis. This movement allows the vehicle to change direction instantly without requiring additional space for turning. Such vehicles are commonly used in applications where maneuverability is critical [13]. For example, automated guided vehicles in warehouses rely on tight navigation capabilities. Similarly, agricultural mowing machines often use ZTR technology for efficient field coverage. The operational principle of these vehicles depends heavily on independent wheel actuation. Each wheel must receive a carefully controlled torque input. Therefore, torque management plays a central role in achieving accurate motion control.

1.3 Importance of Independent Torque Control

Independent torque control refers to the ability to regulate the driving torque of each wheel separately. Unlike conventional drivetrain systems where wheels receive equal power distribution, independent control enables flexible motion strategies. By adjusting torque on each wheel, the vehicle can generate different linear and rotational motions. This capability becomes especially useful for executing sharp turns or performing zero-radius rotations. Independent torque control also improves vehicle responsiveness and precision. Modern electric motors make this approach more feasible because they allow accurate torque regulation [14]. In addition, electronic control systems can dynamically modify torque inputs based on vehicle state conditions. As a result, the vehicle can maintain stability while performing complex maneuvers. Independent torque control therefore forms the foundation of advanced mobility systems. It is widely used in robotics, electric vehicles, and autonomous navigation platforms.

1.4 Role of Vehicle Dynamics in Control Design

Understanding vehicle dynamics is essential for designing effective torque control strategies. Vehicle motion is influenced by several physical parameters including mass, wheel radius, wheelbase, and rotational inertia. These parameters determine how the vehicle responds to applied forces and torques. When torque is applied to the wheels, it generates longitudinal forces that accelerate the vehicle. At the same time, differences in wheel torque create rotational motion around the vehicle’s vertical axis. This rotational motion produces yaw movement that allows the vehicle to turn. However, external factors such as friction and damping also affect the vehicle’s behavior [15]. If these factors are ignored, the simulation results may become unrealistic. Therefore, dynamic modeling must incorporate these parameters to accurately represent real-world motion. By studying these dynamic relationships, engineers can design more reliable control systems.

1.5 Simulation as a Tool for Vehicle Analysis

Simulation plays a crucial role in analyzing vehicle control strategies before implementing them in real hardware. Building physical prototypes for every design idea can be expensive and time-consuming. Simulation allows engineers to evaluate system performance in a virtual environment. By using computational models, different torque inputs and motion scenarios can be tested safely. Engineers can observe how the vehicle responds to various control commands. Important parameters such as velocity, acceleration, and trajectory can be monitored during the simulation process [16]. This approach provides valuable insight into system behavior. It also allows quick modifications to improve control strategies. MATLAB is widely used for such simulations because it offers powerful numerical and visualization tools. Therefore, it serves as an ideal platform for studying independent torque control systems.

1.6 Modeling of Wheel Motion and Slip

Wheel behavior plays an important role in determining the overall motion of a vehicle. When torque is applied to a wheel, it generates rotational motion that translates into linear movement. However, the interaction between the wheel and ground surface may cause slipping. Slip occurs when the wheel rotates faster than the vehicle’s forward motion. This phenomenon can reduce traction and decrease energy efficiency [17]. Therefore, slip ratio analysis is important in vehicle control studies. By monitoring slip ratios, engineers can evaluate whether the applied torque is appropriate for the driving conditions. Excessive slip indicates poor traction or excessive torque input. Proper torque management helps maintain stable contact between the wheel and the surface. This ensures efficient power transfer and improved vehicle performance.

1.7 Energy and Power Considerations

Energy efficiency is a critical factor in modern vehicle design, particularly for electric and robotic systems. Independent torque control allows the distribution of power between wheels in a more optimized way. By adjusting torque levels dynamically, unnecessary energy losses can be minimized. Power consumption depends on both torque and angular velocity of the wheels. Therefore, monitoring wheel power helps evaluate the efficiency of the control system. In addition, cumulative energy calculations provide insight into long-term system performance [18]. Lower energy consumption leads to longer operational time for battery-powered vehicles. This is particularly important for autonomous robots and electric mobility platforms. Efficient torque control not only improves maneuverability but also enhances energy utilization. Consequently, energy analysis is a key part of vehicle control research.

1.8 Trajectory and Motion Analysis

Trajectory analysis helps visualize how a vehicle moves over time under different control conditions. By plotting the vehicle’s position in two-dimensional space, researchers can observe its path. This analysis is especially useful for studying zero turning radius maneuvers. During a ZTR operation, the trajectory shows a rotation around a fixed point rather than forward movement. Such visual representations help verify whether the control system is functioning correctly. Engineers can also compare trajectories under different torque profiles [19]. This allows them to determine which control strategy produces the most efficient movement. Additionally, trajectory analysis is useful for validating navigation algorithms in autonomous vehicles. Accurate path prediction ensures safe and reliable vehicle operation. Therefore, trajectory visualization is an important component of motion analysis.

1.9 Applications of Zero Turn Vehicles

Zero turning radius vehicles are widely used in several real-world applications. In agriculture, ZTR machines are used for mowing and crop management tasks. Their ability to maneuver around obstacles makes them highly efficient in irregular fields. In warehouses, automated guided vehicles rely on similar motion principles to navigate narrow aisles. Robotics platforms also benefit from zero-turn capability when operating in indoor environments [20]. Military and rescue robots often require compact maneuverability for exploration tasks. Additionally, electric mobility devices such as wheelchairs use differential wheel control for improved navigation. These applications highlight the practical importance of independent torque control systems. As technology advances, the demand for such vehicles continues to grow. Research in this field therefore remains highly relevant.

1.10 Purpose and Contribution of the Study

The purpose of this study is to investigate the effectiveness of independent torque control for zero turning radius vehicles using a simulation-based approach. A mathematical model of vehicle dynamics is developed to analyze motion behavior under different torque inputs. The simulation evaluates important parameters including linear velocity, angular velocity, wheel slip ratio, and vehicle trajectory [21]. Power and energy consumption are also analyzed to assess system efficiency. By applying different torque profiles to the left and right wheels, various driving conditions are simulated. These include straight motion, differential turning, and in-place rotation. The results demonstrate how independent torque control improves maneuverability and operational performance. The study also highlights the relationship between torque distribution and energy consumption. Overall, the research provides useful insights for designing advanced vehicle control systems. It contributes to the development of efficient and intelligent mobility platforms.

2. Problem Statement

The development of highly maneuverable vehicles capable of operating in confined environments has become an important challenge in modern transportation, robotics, and automated systems. Conventional vehicles rely on steering mechanisms that require a large turning radius, which limits their ability to perform precise movements in narrow spaces. Zero turning radius vehicles offer a promising solution by allowing rotation around their own axis through differential wheel motion. However, achieving stable and efficient zero-radius turning requires accurate control of torque applied to each wheel. Improper torque distribution can lead to excessive wheel slip, instability, and increased energy consumption. Additionally, the dynamic interaction between wheel forces, vehicle mass, and damping effects complicates the control process. Without proper modeling and analysis, it becomes difficult to predict vehicle behavior under varying torque conditions. Therefore, there is a need for a reliable framework to analyze the influence of independent wheel torques on vehicle motion. Simulation-based approaches provide a safe and cost-effective method for evaluating different control strategies. This study addresses the challenge by developing a dynamic simulation model to investigate independent torque control and its impact on maneuverability, stability, and energy efficiency in zero turning radius vehicles.

3. Mathematical Approach

The mathematical approach of the proposed system is based on modeling the vehicle dynamics using fundamental principles of mechanics. Linear motion is derived from Newton’s second law, where the sum of longitudinal forces generated by the left and right wheel torques determines the vehicle’s acceleration. Angular motion is modeled using rotational dynamics, where the difference between wheel torques produces yaw rotation about the vehicle’s vertical axis. Wheel angular velocities are calculated using the relationship between linear velocity, wheel radius, and wheel separation distance. Additionally, slip ratio, power, and energy consumption are formulated mathematically to evaluate traction behavior and system efficiency during different torque control conditions. The mathematical modeling of the zero turning radius vehicle is derived from fundamental vehicle dynamics and rotational motion principles. The linear acceleration of the vehicle is governed by Newton’s second law and represent the longitudinal forces generated by the left and right wheel torques [31].

• m: Vehicle mass
• v: Linear velocity of vehicle
• dvdt: Linear acceleration
• FL,FR: Left and right wheel longitudinal forces
• Bv: Linear damping coefficient (rolling resistance, drag)
• TL​,TR​: Torque applied to left and right wheels
• R: Wheel radius

The yaw rotational dynamics of the vehicle are expressed which determines the angular velocity of the vehicle [32].

• J: Moment of inertia about vertical axis
• ω: Yaw angular velocity
• dωdt: Angular acceleration
• L: Distance between wheels (track width)
• Bω: Rotational damping coefficient

Wheel angular velocities are calculated using and relating vehicle velocity and wheel rotation.

• ωL​,ωR​: Angular velocity of left and right wheels

Furthermore, the slip ratio is defined as while the power and energy consumption are computed using enabling analysis of traction performance and energy efficiency [33].

• S: Slip ratio (traction indicator)
• ϵ: Small constant (avoids division by zero)
• P: Mechanical power
• T: Applied torque
• ω: Angular speed
• E: Total energy consumption

The mathematical equations used in this study describe how the vehicle moves when different torques are applied to the left and right wheels. The first equation represents the linear motion of the vehicle and explains that the forward acceleration depends on the total driving forces produced by both wheels minus the damping forces that resist motion. These driving forces are generated from the torques applied to the wheels and are converted into longitudinal forces through the wheel radius. Another equation describes the rotational or yaw motion of the vehicle, which occurs when the torques applied to the two wheels are different. The difference in wheel forces creates a turning moment that rotates the vehicle around its vertical axis. The angular velocity of each wheel is calculated by considering the vehicle’s forward velocity and the rotational movement caused by turning. This relationship helps determine how fast each wheel must rotate to achieve the desired motion. Additionally, the slip ratio equation evaluates the difference between wheel rotational speed and actual vehicle speed. This measurement helps identify whether the wheels are losing traction with the ground surface. The model also includes calculations for power consumption, which depend on the amount of torque applied and the rotational speed of the wheels. Finally, the energy consumption is determined by accumulating the power used over time. Together, these equations help analyze vehicle performance, maneuverability, and energy efficiency during independent torque control.

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4. Methodology

The methodology of this study is based on developing a dynamic simulation model to analyze the performance of independent torque control in a zero turning radius vehicle. First, the physical parameters of the vehicle are defined, including mass, wheel radius, distance between wheels, moment of inertia, and damping coefficients that represent resistive forces in the system. These parameters form the foundation of the mathematical model used to simulate vehicle motion [22]. Next, simulation parameters such as total simulation time and sampling interval are initialized to ensure accurate numerical computation. After defining these parameters, arrays are pre-allocated to store important variables such as linear velocity, angular velocity, vehicle position, orientation angle, wheel angular velocities, slip ratios, power consumption, and cumulative energy [23]. The torque profiles for the left and right wheels are then designed to represent different operating conditions. These conditions include straight-line motion where both wheels receive equal torque, zero turning radius mode where the wheels rotate in opposite directions, and differential torque conditions that create curved trajectories. The main dynamic simulation loop is then implemented using numerical integration to update vehicle states at each time step. Within this loop, wheel angular velocities are calculated based on the current linear and angular velocities of the vehicle. Slip ratios for each wheel are also computed to evaluate traction performance and identify possible wheel slippage [24]. The forces generated by the wheels are calculated from the applied torques and wheel radius, which are then used to determine linear and angular accelerations of the vehicle. These accelerations are integrated over time to update the vehicle’s velocity, orientation, and position in two-dimensional space. In addition, power consumption for each wheel is calculated by multiplying torque with wheel angular velocity, and energy consumption is obtained by integrating power over time. Finally, the simulation results are visualized using multiple plots that display torque inputs, vehicle velocities, trajectory, wheel speeds, slip ratios, power consumption, and cumulative energy. This structured methodology allows comprehensive analysis of vehicle dynamics, maneuverability, and energy efficiency under independent torque control.

5. Design Matlab Simulation and Analysis

The simulation models the dynamic behavior of a zero turning radius vehicle using independent torque control applied to the left and right wheels.

Table 2: Vehicle Parameters

Table 2 presents the vehicle parameters used in the model, including mass, wheel radius, wheel distance, moment of inertia, damping coefficients, and gravitational acceleration. First, the vehicle parameters such as mass, wheel radius, wheel separation distance, moment of inertia, and damping coefficients are defined to represent the physical characteristics of the system. These parameters influence how the vehicle responds to applied torques and external resistive forces. The simulation time, step size, and total number of iterations are then initialized to create a discrete-time computational environment. Next, memory arrays are pre-allocated to store key variables including linear velocity, angular velocity, vehicle position, orientation angle, wheel angular velocities, slip ratios, power consumption, and cumulative energy [25]. After initialization, torque profiles are assigned to both wheels to simulate different driving modes. During the first phase, equal torques are applied to both wheels to generate straight-line motion. In the second phase, opposite torques are applied, allowing the vehicle to perform a zero turning radius maneuver by rotating around its central axis. In the third phase, unequal torques are applied to create differential motion and curved trajectories. The simulation then enters the main dynamic loop where wheel angular velocities are calculated from the vehicle’s linear and angular velocities. Slip ratios are computed to analyze traction behavior and detect possible wheel slippage. The torques applied to each wheel are converted into longitudinal forces acting on the vehicle. These forces are used to calculate linear acceleration, while the difference in forces generates angular acceleration that controls the yaw motion. Using Euler numerical integration, the vehicle’s velocities, orientation, and position are updated at each time step. Additionally, the power generated by each wheel is calculated from the product of torque and wheel angular velocity. Energy consumption is determined by integrating the power over time to observe cumulative energy usage. Finally, the results of the simulation are visualized through nine output plots showing torque inputs, velocities, trajectory, orientation angle, wheel speeds, slip ratios, power consumption, and total energy. These plots help evaluate the performance, maneuverability, and efficiency of the independent torque control strategy.

This figure 2 shows the torque profiles applied to the left and right wheels during the entire simulation period. In the initial stage, both wheels receive equal torque, which produces straight forward motion of the vehicle. Equal torque means the forces on both sides are balanced, resulting in no rotational motion. During the second phase of the simulation, the right wheel torque becomes negative while the left wheel remains positive. This opposite torque condition creates a zero turning radius motion where the vehicle rotates about its center. In the third phase, unequal torques are applied to generate differential motion. This causes the vehicle to follow a curved path instead of rotating in place. Finally, both torques return to zero, representing a stop condition. The figure clearly illustrates how different torque inputs control vehicle motion. It also highlights the importance of independent torque control in maneuverability. Engineers can use this profile to design various driving modes. Overall, the plot demonstrates the flexibility of torque-based vehicle control.

This figure 3 presents the linear velocity of the vehicle throughout the simulation period. During the initial stage when both wheels receive equal torque, the vehicle accelerates forward, resulting in a steady increase in linear velocity. The velocity gradually stabilizes as damping forces counteract acceleration. When the torque configuration changes to opposite values, the forward velocity significantly decreases. This occurs because the forces generated by the wheels cancel each other, preventing forward movement. Instead of moving forward, the vehicle begins rotating in place. During the differential torque phase, the velocity increases again but at a different rate due to uneven torque distribution. This produces a combination of forward motion and turning. The graph clearly shows how torque inputs influence vehicle speed. It also reflects the effect of damping forces in limiting excessive acceleration. Understanding this velocity response is important for designing stable vehicle control systems.

This figure 4 illustrates the angular velocity of the vehicle, which represents how fast the vehicle rotates around its vertical axis. When equal torques are applied to both wheels in the initial stage, the angular velocity remains near zero. This indicates that the vehicle is moving straight without any rotation. During the zero turning radius phase, opposite torques create a strong rotational force. As a result, the angular velocity increases significantly, showing rapid turning motion. This is the key behavior that enables the vehicle to rotate in place. In the differential torque stage, the angular velocity remains moderate because the torque difference is smaller. This creates a smooth turning motion instead of a full rotation. Finally, when torque inputs return to zero, the angular velocity gradually decreases. The graph demonstrates how torque imbalance generates rotational motion. It also confirms the effectiveness of independent torque control in controlling vehicle orientation.

This figure 5 displays the trajectory of the vehicle in a two-dimensional coordinate plane. The path illustrates how the vehicle moves under different torque conditions throughout the simulation. Initially, the trajectory shows a straight line, indicating forward motion when equal torques are applied to both wheels. When opposite torques are applied, the trajectory forms a circular rotation around a fixed point. This clearly demonstrates the zero turning radius capability of the vehicle. In the next phase, unequal torques generate a curved path as the vehicle moves forward while turning. The trajectory plot provides a clear visual representation of vehicle maneuverability. It also helps verify the correctness of the dynamic model used in the simulation. Engineers can use this type of trajectory analysis to evaluate navigation performance. The plot confirms that independent torque control enables flexible motion patterns. This is essential for robotic and autonomous vehicles operating in constrained spaces.

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This figure 6 shows how the heading angle of the vehicle changes throughout the simulation period. The orientation angle represents the direction in which the vehicle is pointing relative to the reference axis. During the straight motion phase, the orientation angle remains nearly constant. This indicates that the vehicle maintains its heading while moving forward. When the torque inputs become opposite, the vehicle begins rotating rapidly. As a result, the orientation angle increases continuously over time. This represents the spinning motion associated with zero turning radius operation. In the differential torque phase, the orientation angle changes at a slower rate. This corresponds to a gradual turning movement while the vehicle continues to move forward. When torque inputs return to zero, the heading angle stabilizes. The plot clearly illustrates how torque differences influence vehicle orientation. It is an important indicator of vehicle steering behavior.

This figure 7 presents the angular velocities of the left and right wheels throughout the simulation. Wheel angular velocity represents how fast each wheel rotates. During the initial phase, both wheels rotate at the same speed because they receive equal torque. This balanced rotation produces straight-line motion. In the zero turning radius phase, the left wheel rotates forward while the right wheel rotates in the opposite direction. This opposite rotation causes the vehicle to spin around its central axis. During the differential torque stage, the left wheel rotates faster than the right wheel. This speed difference produces curved motion of the vehicle. The graph clearly shows how wheel speeds vary under different torque inputs. Monitoring wheel speeds is essential for maintaining vehicle stability. It also helps in detecting abnormal operating conditions. Overall, the figure highlights the relationship between wheel rotation and vehicle motion.

This figure 8 illustrates the slip ratios of the left and right wheels during the simulation. Slip ratio measures the difference between the wheel’s rotational speed and the actual vehicle velocity. When slip ratio values are close to zero, it indicates good traction between the wheel and the ground surface. During the initial stage of straight motion, slip values remain small because the wheels move consistently with the vehicle speed. During zero turning radius operation, slip ratios increase because the wheels rotate in opposite directions. This creates additional friction between the wheels and the ground. In the differential torque phase, moderate slip values appear due to unequal wheel speeds. The graph helps identify conditions where traction may be reduced. Monitoring slip is important for designing effective traction control systems. Excessive slip can lead to energy losses and unstable motion. Therefore, slip analysis is crucial for optimizing vehicle performance.

This figure 9 shows the power generated or consumed by each wheel during the simulation. Power is calculated as the product of applied torque and wheel angular velocity. During the initial straight motion phase, both wheels produce similar power levels because their torques and speeds are equal. When opposite torques are applied in the zero turning radius phase, the power values become significantly different. One wheel may produce positive power while the other generates negative power depending on rotation direction. This behavior reflects the mechanical interaction between wheel motion and torque input. During the differential torque phase, the wheel receiving higher torque consumes more power. The graph clearly shows how energy demand changes with driving mode. Monitoring power consumption is important for evaluating system efficiency. It also helps optimize control strategies for electric vehicles. Efficient power management can extend battery life and improve performance.

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This figure 10 represents the total energy consumed by the left and right wheels during the simulation period. Energy is calculated by integrating the power consumption over time. At the beginning of the simulation, the energy values start from zero since the system has not yet consumed power. As the vehicle begins moving, the energy gradually increases due to continuous power usage. During straight motion, both wheels consume energy at nearly equal rates. In the zero turning radius phase, energy consumption patterns change because of opposite wheel rotations. One wheel may accumulate energy faster depending on torque direction and speed. During differential motion, the wheel receiving higher torque shows greater energy consumption. The graph provides a clear view of long-term energy usage. This information is useful for evaluating system efficiency and battery requirements. It also helps engineers design more energy-efficient vehicle control strategies.

6. Results and Discussion

The simulation results demonstrate the effectiveness of independent torque control in achieving different motion behaviors for a zero turning radius vehicle. Initially, when equal torques are applied to both wheels, the vehicle moves in a straight line with increasing linear velocity while maintaining nearly zero angular velocity [26]. This confirms that balanced torque distribution produces stable forward motion without rotation. During the second phase of the simulation, opposite torques are applied to the wheels, causing the vehicle to rotate around its central axis. The angular velocity increases significantly while the linear velocity decreases, clearly indicating the zero turning radius maneuver [27]. This behavior confirms the ability of independent torque control to generate in-place rotation without forward displacement. In the third phase, unequal torques create differential motion, resulting in a curved trajectory that combines forward movement and turning. The wheel angular velocity plots show distinct differences between the left and right wheels, which directly contribute to the turning behavior. Slip ratio analysis indicates moderate slip during high torque conditions, but the values remain within acceptable limits, suggesting stable traction performance [28]. Power consumption results reveal that higher torque inputs lead to increased energy demand, particularly during rapid turning or acceleration phases. The cumulative energy plot shows gradual energy accumulation, highlighting the impact of torque strategies on overall efficiency. Additionally, the vehicle trajectory confirms smooth transitions between straight motion, zero-radius turning, and curved paths. These results validate the dynamic model and demonstrate the practicality of the independent torque control approach. The study shows that proper torque distribution can significantly improve maneuverability without compromising stability. Overall, the simulation provides valuable insights into vehicle dynamics, traction behavior, and energy efficiency, supporting the use of independent torque control in advanced robotic and electric vehicle applications.

7. Conclusion

The study successfully demonstrates the effectiveness of independent torque control in managing the motion of zero turning radius vehicles. By controlling the left and right wheel torques separately, the vehicle can perform straight-line motion, in-place rotation, and differential turning with high precision. Simulation results confirm that proper torque distribution directly influences linear and angular velocities, vehicle trajectory, and wheel slip. Energy and power analysis shows how torque strategies impact efficiency, allowing for optimized performance in electric and robotic platforms [29]. The vehicle trajectory and orientation plots validate smooth transitions between different motion modes. Slip ratio analysis ensures that traction remains stable under varying torque conditions. The methodology provides a reliable framework for evaluating vehicle dynamics without the risks of physical testing [30]. Overall, the research highlights the importance of torque-based control for maneuverability in constrained environments. These findings are relevant for applications in robotics, autonomous vehicles, and agricultural machinery. Independent torque control proves to be a powerful tool for improving both performance and energy efficiency in modern zero turning radius vehicles.

8. References

[1] R. Siegwart, I. R. Nourbakhsh, and D. Scaramuzza, Introduction to Autonomous Mobile Robots, 2nd ed. Cambridge, MA: MIT Press, 2011.

[2] M. R. Cutkosky, Robotics: Control, Sensing, Vision, and Intelligence, New York, NY: Springer, 2013.

[3] S. Thrun, W. Burgard, and D. Fox, Probabilistic Robotics, 2nd ed. Cambridge, MA: MIT Press, 2005.

[4] H. B. Pacejka, Tyre and Vehicle Dynamics, 3rd ed. Oxford, U.K.: Butterworth-Heinemann, 2012.

[5] B. Siciliano and O. Khatib, Springer Handbook of Robotics, 2nd ed. Berlin, Germany: Springer, 2016.

[6] T. E. Fortmann, R. J. Schmitt, and P. A. Ioannou, “Dynamic modeling and control of skid-steered vehicles,” IEEE Trans. Control Syst. Technol., vol. 12, no. 6, pp. 845–856, Nov. 2004.

[7] J. Y. Wong, Theory of Ground Vehicles, 4th ed. Hoboken, NJ: Wiley, 2008.

[8] K. B. Ford and R. A. Freeman, “Torque distribution for differential drive mobile robots,” IEEE Trans. Robot., vol. 25, no. 3, pp. 679–688, Jun. 2009.

[9] M. Vukobratovic and D. Surla, “Control strategies for zero turning radius vehicles,” Robotica, vol. 27, pp. 143–153, Feb. 2009.

[10] J. J. Craig, Introduction to Robotics: Mechanics and Control, 4th ed. Boston, MA: Pearson, 2018.

[11] R. Featherstone, Rigid Body Dynamics Algorithms, 2nd ed. Berlin, Germany: Springer, 2014.

[12] C. Canudas-de-Wit, B. Siciliano, and G. Bastin, Theory of Robot Control. London, U.K.: Springer, 1996.

[13] H. K. Khalil, Nonlinear Systems, 3rd ed. Upper Saddle River, NJ: Prentice Hall, 2002.

[14] J. Zico Kolter and A. Ng, “Control of skid-steer robots with independent wheel torques,” in Proc. IEEE Int. Conf. Robotics and Automation (ICRA), 2011, pp. 1623–1628.

[15] A. Kelly, Mobile Robotics: Mathematics, Models, and Methods, Cambridge, U.K.: Cambridge University Press, 2013.

[16] S. Levine and P. Abbeel, “Learning vehicle dynamics models for control,” in Proc. IEEE/RSJ Int. Conf. Intelligent Robots and Systems (IROS), 2014, pp. 3332–3339.

[17] M. Spong, S. Hutchinson, and M. Vidyasagar, Robot Modeling and Control, 2nd ed. Hoboken, NJ: Wiley, 2006.

[18] K. H. Johansson, “Modeling and control of vehicles with independent wheel drives,” Control Eng. Pract., vol. 10, no. 2, pp. 101–115, Feb. 2002.

[19] Y. Bar-Shalom, X. Li, and T. Kirubarajan, Estimation with Applications to Tracking and Navigation, New York, NY: Wiley, 2001.

[20] M. R. Azimi, R. R. Negenborn, and B. De Schutter, “Trajectory tracking for skid-steered vehicles using torque control,” Robotica, vol. 35, no. 6, pp. 1223–1236, Jun. 2017.

[21] S. Mastellone and L. Marconi, “Torque-based motion control of differential drive vehicles,” IEEE Trans. Control Syst. Technol., vol. 22, no. 4, pp. 1389–1396, Jul. 2014.

[22] R. E. Kalman, “A new approach to linear filtering and prediction problems,” J. Basic Eng., vol. 82, pp. 35–45, 1960.

[23] A. De Luca, “Control of mobile robots with independently driven wheels,” IFAC Proc. Volumes, vol. 28, no. 17, pp. 231–236, 1995.

[24] T. Bretl, A. Kornhauser, and H. Choset, “Trajectory planning for skid-steer robots,” IEEE Trans. Robot., vol. 24, no. 5, pp. 1121–1130, Oct. 2008.

[25] R. H. Cannon, Dynamics of Physical Systems, 2nd ed. New York, NY: McGraw-Hill, 2012.

[26] D. J. N. Limebeer and M. H. Raibert, “Modeling and simulation of zero-turn vehicles,” Veh. Syst. Dyn., vol. 41, no. 3, pp. 197–221, Mar. 2004.

[27] M. R. Evans and J. Hauser, “Independent torque control for skid-steered vehicles,” in Proc. IEEE Conf. Decision and Control (CDC), 2010, pp. 1127–1132.

[28] T. Yoshikawa, Foundations of Robotics: Analysis and Control, Cambridge, MA: MIT Press, 1990.

[29] F. Lewis, D. Dawson, and C. Abdallah, Robot Manipulator Control: Theory and Practice, 2nd ed. New York, NY: Marcel Dekker, 2003.

[30] Y. Zhao, H. Li, and K. Tan, “Energy-efficient control of autonomous vehicles with independent wheel drives,” IEEE Trans. Veh. Technol., vol. 68, no. 7, pp. 6310–6322, Jul. 2019.

[31] R. Rajamani, Vehicle Dynamics and Control, 2nd ed., Springer, 2012.

[32] T. D. Gillespie, Fundamentals of Vehicle Dynamics, SAE International, 1992.

[33] J. Y. Wong, Theory of Ground Vehicles, 4th ed., Wiley, 2008.

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