Energy-Efficient Trajectory Tracking of a 2-DOF Planar Manipulator Using Optimized Computed Torque Control

Abstract

This study presents an optimisation-based control strategy for improving the energy efficiency of robotic manipulators without compromising motion accuracy. A two-degree-of-freedom (2-DOF) planar manipulator is modeled using the Euler–Lagrange formulation to capture its nonlinear dynamic behavior [1]. The control architecture employs the standard Computed Torque Control (CTC) method, which linearizes the nonlinear dynamics through model-based compensation. Unlike conventional fixed or high-gain tuning approaches that often result in excessive torque demand and increased energy consumption, the proposed approach adopts an optimal CTC framework. Within this framework, the proportional and derivative gains are determined through a constrained nonlinear optimization process that minimizes the integral of squared joint torques—defined as the energy cost function—while ensuring the trajectory tracking error remains within a predefined tolerance. The nonlinear optimization is implemented in MATLAB using a simulation-based iterative procedure, where the system dynamics are embedded within each optimization cycle. The resulting optimized controller demonstrates a significant reduction in actuator effort and overall energy expenditure compared to the conventional CTC method. Although not universally optimal, gain-optimized controllers present a practical and effective means for achieving energy-efficient robotic motion control.

Introduction

The study addresses the energy efficiency of robotic manipulators in industrial and service applications, highlighting that traditional control methods focus on accuracy and high-speed tracking but often ignore energy consumption. Excessive torque generation and high-frequency actuation increase energy use, actuator heating, and mechanical wear, creating a trade-off between tracking performance and energy efficiency.

Computed Torque Control (CTC):
CTC is a model-based approach that linearizes and decouples the nonlinear dynamics of manipulators by compensating for inertia, Coriolis, centrifugal, and gravitational forces. Standard proportional-derivative (PD) gains are tuned for rapid, precise tracking, but high-gain control increases energy expenditure and mechanical stress.

Proposed Contribution:
The study develops an optimization-based CTC framework for a 2-DOF planar manipulator that:

1. Minimizes total actuation energy (integral of squared joint torques).

2. Maintains trajectory tracking errors within allowable tolerances.

3. Uses constrained nonlinear optimization (MATLAB’s fmincon) to systematically select optimal Kp and Kd gains for energy-efficient control.

Dynamic Modeling:

• The manipulator’s dynamics are derived using the Euler-Lagrange formulation, producing the standard model:

M(q)q¨+C(q,q?)q?=τM(q) \ddot{q} + C(q, \dot{q})\dot{q} = \tauM(q)q¨?+C(q,q??)q??=τ

• The mass matrix M(q) captures configuration-dependent inertia, while the Coriolis/centrifugal matrix C(q, q?) accounts for joint coupling effects. Gravitational forces are neglected for horizontal planar motion.

Trajectory Planning:

• Quintic polynomial trajectories are used to ensure smooth motion with continuous position, velocity, acceleration, and zero jerk at endpoints.

• Smooth trajectories prevent torque spikes, supporting both energy efficiency and accurate tracking.

Energy Optimization Approach:

• Focuses on controller gains rather than trajectory redesign, offering a generalizable method for fixed desired paths.

• Reduces energy use without sacrificing tracking precision, as confirmed by simulation studies.

Significance:
The framework demonstrates that optimization-based CTC can substantially improve energy efficiency in robotic manipulators while preserving high-accuracy motion tracking. This methodology supports sustainable and intelligent robotic system design, balancing operational performance with reduced energy costs and mechanical wear.

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Conclusion

This work presented a comprehensive framework for achieving energy-efficient trajectory tracking in robotic manipulators by optimizing computed torque controller gains. Starting with a complete Euler–Lagrange dynamic model for a 2-DOF planar manipulator, the study established the foundation for accurate inverse dynamics control. The conventional computed torque control (CTC) scheme was extended to an optimization-based variant (O-CTC), formulated as a constrained nonlinear problem minimizing the integral of squared joint torque while maintaining strict tracking accuracy [10]. Simulation results showed that the optimized controller achieved a 67% reduction in total energy cost, significantly lower torque peaks, and tracking errors below 0.005 radians, confirming improved efficiency without loss of precision. These findings demonstrate that energy-aware gain tuning can effectively balance performance and sustainability in robotic control systems. Beyond the numerical results, the study underscores the importance of optimization-driven control design. By incorporating energy considerations into the control synthesis stage, it offers a scalable and adaptable approach for developing efficient, low-wear robotic manipulators. Future work may explore real-time adaptive optimization, machine learning–based parameter tuning, and experimental validation on physical robotic platforms.

References

[1] M. W. Spong, S. Hutchinson, and M. Vidyasagar, Robot Modeling and Control. New York, NY, USA: John Wiley & Sons, 2006. [2] J. J. E. Slotine and W. Li, Applied Nonlinear Control. Englewood Cliffs, NJ, USA: Prentice Hall, 1991. [3] J. J. Craig, Introduction to Robotics: Mechanics and Control, 3rd ed. Upper Saddle River, NJ, USA: Pearson Prentice Hall, 2005. [4] C. Canudas de Wit, B. Siciliano, and G. Bastin, Theory of Robot Control. London, U.K.: Springer, 1996. [5] L. Sciavicco and B. Siciliano, Modelling and Control of Robot Manipulators. New York, NY, USA: McGraw-Hill, 2000. [6] M. Gopal, Control Systems: Principles and Design. New Delhi, India: McGraw Hill Education, 2003. [7] R. V. Dukkipati, Robotics. New Delhi, India: New Age International Publishers, 2007. [8] D. Goldfarb and M. J. Todd, “Nonlinear programming,” in Handbooks in Operations Research and Management Science: Optimization. Amsterdam, The Netherlands: Elsevier, 1991. [9] M. W. Spong, “Computed torque control of robot manipulator,” IEEE Control Systems Magazine, vol. 6, no. 4, pp. 13–24, 1986. [10] C. S. G. Lee and M. J. Chung, “An adaptive control strategy for mechanical manipulators,” IEEE Transactions on Automatic Control, vol. 29, no. 9, pp. 837–840, 1984.

Copyright

Copyright © 2026 R. Vishnu Varthan , Sanjay Ganesh S., 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.