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Trajectory planning for energy minimization of industry robotic manipulators using the Lagrange interpolation method


We propose to use Lagrange interpolation method to express each joint trajectory function to realize trajectory planning for energy minimization of industrial robotic manipulators. We give the position constraints to the industrial robotic manipulators and the stability of the industrial robotic manipulators should be satisfied. In order to avoid Runge’s phenomenon of Lagrange interpolation method, we use the Chebyshev interpolation points for our approach. Through derivation, the angular velocity functions and angular acceleration functions can be obtained. Lagrange interpolation method can satisfy position constraints, and ensure the smoothness of joint angular positions, velocities, accelerations, and joint torques. By taking these functions into the Performance Index (PI) of energy minimization, as well as the direct iteration method used for optimization of energy consumption, we can obtain the optimal trajectory for industrial robotic manipulators.

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θ :

vector of joint angles of the robotic manipulators

\(\dot \theta \) :

vectors of joint angular velocities of the robotic manipulators

\(\ddot \theta \) :

vectors of joint angular accelerations of the robotic manipulators


symmetric positive definite inertial matrix

V(θ, \(V(\theta ,\ddot \theta )\)):

component of the torque depending on centrifugal force and the Coriolis forces


component depending on gravity forces

τ :

vectors of joint torques of the robotic manipulators

n :

number of the joints of the robotic manipulators

j :

j th joint of the robotic manipulators

N :

number of time interpolation points

l i :

basis polynomials of the j th segment of time for Lagrange interpolation method

ω :

basis polynomial for error function of the Lagrange interpolation method

a j_k :

coefficients of the polynomial function


  1. Alavandar, S. and Nigam, M., “Neuro-Fuzzy based Approach for Inverse Kinematics Solution of Industrial Robot Manipulators,” International Journal of Computers, Communications & Control, Vol. 3, No. 3, pp. 224–234, 2008.

    MathSciNet  Google Scholar 

  2. Kumar, N., Borm, J. H., Panwar, V., and Chai, J., “Tracking Control of Redundant Robot Manipulators using RBF Neural Network and an Adaptive Bound on Disturbances,” Int. J. Precis. Eng. Manuf., Vol. 13, No. 8, pp. 1377–1386, 2012.

    Article  Google Scholar 

  3. Kim, I. M., Kim, H. S., and Song, J. B., “Design of Joint Torque Sensor with Reduced Torque Ripple for a Robot Manipulator,” Int. J. Precis. Eng. Manuf., Vol. 13, No. 10, pp. 1773–1779, 2012.

    Article  Google Scholar 

  4. Van, M., Kang, H. J., and Suh, Y. S., “A Novel Neural Second- Order Sliding Mode Observer for Robust Fault Diagnosis in Robot Manipulators,” Int. J. Precis. Eng. Manuf., Vol. 14, No. 3, pp. 397- 406, 2013.

    Article  Google Scholar 

  5. Ghalia, M. B. and Alouani, A. T., “A Robust Trajectory Tracking Control of Industrial Robot Manipulators using Fuzzy Logic,” Proc. of IEEE Computer Society Southeastern Symposium on System Theory, pp. 268–268, 1995.

    Chapter  Google Scholar 

  6. Cosner, C., Anwar, G., and Tomizuka, M., “Plug in Repetitive Control for Industrial Robotic Manipulators,” Proc. of IEEE International Conference on Robotics and Automation, pp. 1970–1975, 1990.

    Chapter  Google Scholar 

  7. Munasinghe, S. R., Nakamura, M., Goto, S., and Kyura, N., “Trajectory Planning for Industrial Robot Manipulators Considering Assigned Velocity and Allowance under Joint Acceleration Limit,” International Journal of Control, Automation and Systems, Vol. 1, No. 1, pp. 68–75, 2003.

    Google Scholar 

  8. Mendes, M. F., Kraus Jr, W., and De Pieri, E. R., “Variable Structure Position Control of an Industrial Robotic Manipulator,” Journal of the Brazilian Society of Mechanical Sciences, Vol. 24, No. 3, pp. 169–176, 2002.

    Article  Google Scholar 

  9. Ata, A. A., Faris, W. F., and Sa'adeh, M. Y., “Optimal Trajectory Selection for a Three DOF Manipulator with Minimum Energy Consumption,” International Journal of Applied Engineering Research, Vol. 2, No. 1, pp. 45–63, 2007.

    Google Scholar 

  10. Hirakawa, A. R. and Kawamura, A., “Trajectory Planning of Redundant Manipulators for Minimum Energy Consumption without Matrix Inversion,” Proc. of IEEE International Conference on Robotics and Automation, Vol. 3, pp. 2415–2420, 1997.

    Article  Google Scholar 

  11. Sato, A., Sato, O., Takahashi, N., and Kono, M., “Trajectory for Saving Energy of a Direct-Drive Manipulator in Throwing Motion,” Artificial Life and Robotics, Vol. 11, No. 1, pp. 61–66, 2007.

    Article  Google Scholar 

  12. Izumi, T., Yokose, Y., and Tamai, R., “Minimum Energy Path Search for a Manipulator in Consideration of all Nonlinear Characteristics by Ga and Its Experiments,” Electrical Engineering in Japan, Vol. 157, No. 3, pp. 26–34, 2006.

    Article  Google Scholar 

  13. Gasparetto, A. and Zanotto, V., “A New Method for Smooth Trajectory Planning of Robot Manipulators,” Mechanism and Machine Theory, Vol. 42, No. 4, pp. 455–471, 2007.

    Article  MATH  MathSciNet  Google Scholar 

  14. Costantinescu, D. and Croft, E., “Smooth and Time-Optimal Trajectory Planning for Industrial Manipulators along Specified Paths,” Journal of Robotic Systems, Vol. 17, No. 5, pp. 233–249, 2000.

    Article  Google Scholar 

  15. Field, G. and Stepanenko, Y., “Iterative Dynamic Programming: An Approach to Minimum Energy Trajectory Planning for Robotic Manipulators,” Proc. of IEEE International Conference on Robotics and Automation, Vol. 3, pp. 2755–2760, 1996.

    Article  Google Scholar 

  16. Saramago, S. F. P. and Steffen, V., “Optimization of the Trajectory Planning of Robot Manipulators Taking into Account the Dynamics of the System,” Mechanism and Machine Theory, Vol. 33, No. 7, pp. 883–894, 1998.

    Article  MATH  MathSciNet  Google Scholar 

  17. Saramago, S. F. P. and Steffen, V., “Optimal Trajectory Planning of Robot Manipulators in the Presence of Moving Obstacles,” Mechanism and Machine Theory, Vol. 35, No. 8, pp. 1079–1094, 2000.

    Article  MATH  MathSciNet  Google Scholar 

  18. Craig, J. J., “Introduction to Robotics: Mechanics and Control,” Upper Saddle River: Pearson Prentice Hall, Vol. 3, pp. 165–185, 2005.

    Google Scholar 

  19. Garg, D. P. and Kumar, M., “Optimization Techniques Applied to Multiple Manipulators for Path Planning and Torque Minimization,” Engineering Applications of Artificial Intelligence, Vol. 15, No. 3, pp. 241–252, 2002.

    Article  MathSciNet  Google Scholar 

  20. Runge, C., “Über empirische Funktionen und die Interpolation zwischen äquidistanten Ordinaten,” Zeitschrift für Mathematik und Physik, Vol. 46, pp. 224–243, 1901.

    MATH  Google Scholar 

  21. Stewart, G. W., “Afternotes on Numerical Analysis,” Philadelphia, PA: Society for Industrial and Applied Mathematics, pp. 131–134, 1993.

    Google Scholar 

  22. Abramowitz, M. and Stegun, I., “Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables,” New York, pp. 878–879, 883, 1972.

    Google Scholar 

  23. Jennings, A., “A Direct Iteration Method of Obtaining Latent Roots and Vectors of a Symmetric Matrix,” Mathematical Proceedings of the Cambridge Philosophical Society, pp.755–765. 1967.

    Google Scholar 

  24. Amiri, I. S. and Afroozeh, A., “Ring Resonator Systems to Perform Optical Communication Enhancement Using Soliton,” Springer, p. 37, 2014.

    Google Scholar 

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Correspondence to Chang-Soo Han.

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Luo, LP., Yuan, C., Yan, RJ. et al. Trajectory planning for energy minimization of industry robotic manipulators using the Lagrange interpolation method. Int. J. Precis. Eng. Manuf. 16, 911–917 (2015).

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  • Energy minimization
  • Trajectory planning
  • Industrial robotic manipulators
  • Lagrange interpolation method
  • Performance Index (PI)
  • Iterative method