Comparative Study of Polynomial Trajectory Planning Methods (Cubic Vs Quintic) for Robotic Manipulators with Smoothness and Energy Considerations

Abstract

Trajectory planning has been an important part of motion control of the robotic manipulators, as it directly impacts their smoothness, accuracy, and energy consumption. Continuous, dynamically feasible, and energy-efficient joint trajectories are essential in such cases as pick-and-place, welding, and assembly where the robots are operating at a high speed. Polynomial trajectory planning, in particular quintic and cubic polynomials has evolved to be a popular technique of trajectory generation, owing to its simplicity and ability to address the boundary conditions. In this paper, a comparative analysis of cubic and quintic polynomial trajectories of robotic manipulators is given with special emphasis on motion smoothness, jerk reduction, and energy expenditure. In order to explain and make a comparison both analytically and by simulation, a simple two-link (2R) manipulator model is selected. Analysis is carried out on both trajectories by having the same conditions on the boundaries and then MATLAB simulations are used to show the changes of the position, velocity, acceleration, and jerk. The resultant performance of cubic trajectories reflecting low computation costs implies discontinuity in acceleration and jerk, hence reduced fluid motion and increased energy demands on short time intervals. Conversely, quintic curves also provide continuous acceleration and velocity with close to zero endpoint jerk, and consequently, allow even smoother and efficient motion. The major conclusions of the research are as follows: cubic polynomies are suitable at slow speed or simple operations, and quintic polynomials trajectories are selected when working with precision-based and high-speed robots, where the smoothness and energy-saving are essential. Keywords- Trajectory planning, robotic manipulators, cubic polynomial, quintic polynomial, motion.

Introduction

This study focuses on robotic trajectory planning for multi-degree-of-freedom manipulators, comparing cubic and quintic polynomial motion profiles using MATLAB simulations. As robotics has advanced, smooth, precise, and energy-efficient motion has become essential for industrial and high-precision applications. Trajectory planning connects kinematics and dynamics by generating joint paths that satisfy constraints on position, velocity, acceleration, and torque.

Cubic polynomial trajectories are simple and computationally efficient, ensuring smooth position and velocity but failing to maintain continuous acceleration. This causes jerk, vibrations, higher torque fluctuations, and reduced mechanical stability. They are therefore suitable mainly for low-speed or low-cost robotic systems.

Quintic polynomial trajectories extend cubic models by also ensuring acceleration continuity and reducing jerk. This results in smoother, more stable, and energy-efficient motion, making them ideal for high-speed, precise, and delicate applications such as surgery or advanced assembly. Smoother trajectories also improve controller performance, reduce wear on actuators, and lower energy consumption.

The literature review shows the evolution from early polynomial methods in the 1970s–80s to modern high-order trajectory planning techniques. Research consistently shows that smoother trajectories reduce vibration, improve accuracy, and enhance energy efficiency. Quintic and higher-order polynomials are widely recognized as superior for high-performance robotic systems, though cubic methods remain common due to their simplicity.

The theoretical background explains robot kinematics and dynamics, emphasizing the importance of continuous position, velocity, acceleration, and jerk minimization for stable motion. Polynomial trajectory planning defines joint motion using time-based equations, where higher-order polynomials allow more boundary conditions and smoother behavior.

Conclusion

This research showcased a comparative study of cubic and quintic polynomial trajectory planning for robotic arms, with the emphasis on motion smoothness, jerk minimization, and energy efficiency. A two-link planar manipulator was then created in MATLAB/Simulink and used to conduct the evaluation of both methods under similar boundary conditions. The analysis demonstrated that although the cubic trajectories presented a simple solution in computation, the discontinuous acceleration and ascendancy torque variations were its drawbacks. Coming to the other side, the quintic played very smooth and continuous position, velocity, and acceleration profiles with the help of reducing mechanical vibrations and torque spikes along the way. Moreover, quintic trajectories were jerk-minimizing and as they reached their near-zero values for the start and end points, the motion became smoother and more stable which was perfect for precision and human interaction applications. The energy comparison indicated that the quintic trajectory was able to consume energy close to 25–30% less than because of the smooth acceleration and dwindled torque peaks. However, even if the quintic methods are computationally expensive, a modern controller will manage it easily with the commensurate gain in stability and actuator life. Therefore, while the cubic ones can serve the simple or cost-sensitive systems, the quintic paths are the main character for advanced, high-performance industrial robotic systems which need precision, reliability, and energy efficiency. To put it briefly, the changing of the trajectory order from cubic to quintic brought along with it a significant increase in the properties of smoothness, energy efficiency, and dynamic stability.

References

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Copyright

Copyright © 2026 Kishore L, Jaipprakesh S A, Dr. Anbarasi M P. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.