Abstract
The minimum jerk principle is commonly used for trajectory planning of robotic manipulators. However, since this principle is stated in terms of the robot’s kinematics, there is no guarantee that the joint controllers will actually track the planned acceleration and jerk profiles because the tuning of the controllers’ gains is decoupled from the trajectory planning. Bearing this in mind, in this paper we introduce a comprehensive framework for optimal estimation of the gains of PID-like controllers for tracking minimum-jerk (MJ) robot trajectories. The proposed methodology relies mainly on a novel variant of error-based performance indices (ISE, ITSE, IAE and ITAE) which are adapted to the tracking of MJ trajectories. Furthermore, the particle swarm optimization (PSO) algorithm is used to search for optimal values for the gains of the controllers of all joints simultaneously. The resulting approach is much simpler than recent developments based on more complex performance indices, in which joint controllers were individually optimized. The proposed approach is general enough to easily encompass the tuning of fractional PID controllers and a comprehensive set of experiments are reported comparing the performances of standard and fractional PID controllers for the task of interest.
Similar content being viewed by others
References
Aghababa, M. P.: Optimal design of fractional-order PID controller for five bar linkage robot using a new particle swarm optimization algorithm. Soft. Comput. 20(10), 4055–4067 (2016)
Ahn, K.K., Truong, D.Q.: Online tuning fuzzy PID controller using robust extended Kalman filter. J. Process Control 19(6), 1011–1023 (2009). Usa derivada do erro. Descreve bem o modelo tipo Mandani e todas suas etapas.
Aloulou, A., Boubaker, O.: Minimum Jerk-Based Control for a Three Dimensional Bipedal Robot. In: International Conference on Intelligent Robotics and Applications (ICIRA’2011), pp. 251–262 (2011)
Angel, L., Viola, J.: Fractional order PID for tracking control of a parallel robotic manipulator type delta. ISA Trans. 79, 172–188 (2018). https://doi.org/10.1016/j.isatra.2018.04.010
Anwaar, H., Yixin, Y., Ijaz, S., Ashraf, M. A., Anwaar, W.: Fractional order based computed torque control of 2-link robotic arm. Adv. Sci. Technol. Res. J. 12(1), 273–284 (2018)
Aström, K. J., Hägglund, T.: Revisiting the Ziegler-Nichols step response method for PID control. J. Process. Control. 14(6), 635–650 (2004)
Bezanson, J., Edelman, A., Karpinski, S., Shah, V. B.: Julia: a fresh approach to numerical computing. SIAM Rev. 59(1), 65–98 (2017)
Boudjehem, B., Boudjehem, D.: Fractional PID controller design based on minimizing performance indices. IFAC-PapersOnLine 49(9), 164–168 (2016). https://doi.org/10.1016/j.ifacol.2016.07.522
Breteler, M. D. K., Meulenbroek, R. G., Gielen, S. C.: An evaluation of the minimum-jerk and minimum torque-change principles at the path, trajectory, and movement-cost levels. Mot. Control. 6(1), 69–83 (2002)
Chopade, A. S., Khubalkar, S. W., Junghare, A., Aware, M., Das, S.: Design and implementation of digital fractional order PID controller using optimal pole-zero approximation method for magnetic levitation system. IEEE/CAA J. Autom. Sin. 5(5), 977–989 (2018)
Das, S.: Springer (2008)
Dounis, A.I., Kofinas, P., Alafodimos, C., Tseles, D.: Adaptive fuzzy gain scheduling PID controller for maximum power point tracking of photovoltaic system. Renew. Energy 60, 202–214 (2013). Fuzzy PID gain scheduling
Dulǎu, M., Gligor, A., Dulǎu, T. M.: Fractional order controllers versus integer order controllers. Procedia Eng. 181, 538–545 (2017). https://doi.org/10.1016/j.proeng.2017.02.431
Eberhart, R., Kennedy, J.: Particle swarm optimization. In: Proceedings of the IEEE International Conference on Neural Networks, vol. 4, pp. 1942–1948 (1995)
Fareh, R., Bettayeb, M., Rahman, M. H.: Control of serial link manipulator using a fractional order controller. Int. Rev. Autom. Control 11(1), 1–6 (2018). https://doi.org/10.15866/ireaco.v11i1.13275
Flash, T., Hogan, N., Richardson, M. J. E.: Optimization Principles in Motor Control. In: M. Arbib (Ed.) the Handbook of Brain Theory and Neural Networks, 2Nd Edn., pp. 827–830. MIT Press (2003)
Flash, T., Meirovitch, Y., Barliya, A.: Models of human movement: Trajectory planning and inverse kinematics studies. Robot. Auton. Syst. 61(4), 330–339 (2013)
Gaing, Z. L.: A particle swarm optimization approach for optimum design of PID controller in AVR system. IEEE Trans. Energy Conver. 19(2), 384–391 (2004). https://doi.org/10.1109/TEC.2003.821821
Grimholt, C., Skogestad, S.: Optimal PID control of double integrating processes. IFAC-PapersOnLine 49 (7), 127–132 (2016). https://doi.org/10.1016/j.ifacol.2016.07.228
Hogan, N.: An organizing principle for a class of voluntary movements. J. Neurosci. 4(11), 2745–2754 (1984)
An, J., Zhao, S. L., Jiang, L., Liu, H.: Solving the time-jerk optimal trajectory planning problem of a robot using augmented Lagrange constrained particle swarm optimization. Math. Probl. Eng. 2017(ID-1921479), 1–10 (2017). https://doi.org/10.1155/2017/1921479
Kaserer, D., Gattringer, H., Mueller, A.: On-line robot-object synchronization with geometric constraints and limits on velocity, acceleration and jerk. IEEE Robot. Autom. Lett., 1–7 (2018). https://doi.org/10.1109/LRA.2018.2849827
Kathuria, T., Kumar, V., Rana, K. P. S., Azar, A. T.: Control of a Three-Link Manipulator Using Fractional-Order PID Controller. In: Azar, A.T., Radwan, A.G., Vaidyanathan, S. (eds.) Fractional Order Systems: Optimization, Control, Circuit Realizations and Applications, chap. 16, pp 477–510. Academic Press (2018)
Kelly, R., Carelli, R.: A class of nonlinear PD-type controllers for robot manipulators. J. Robot. Syst. 13 (12), 793–802 (1996)
Kelly, R., Davila, V. S., Perez, J. A. L.: Control of robot manipulators in joint space. Springer Science & Business Media (2006)
Kumar, A., Gaidhane, P. J., Kumar, V.: A nonlinear fractional order PID controller applied to redundant robot manipulator. In: Proceedings of the 6th International Conference on Computer Applications In Electrical Engineering-Recent Advances (CERA’2017), pp. 527–532 (2017)
Kyriakopoulos, K. J., Saridis, G.: Minimum jerk for trajectory planning and control. Robotica 12(2), 109–113 (1994)
Kyriakopoulos, K. J., Saridis, G. N.: Minimum Jerk Trajectory Planning for Robotic Manipulators. In: Cooperative Intelligent Robotics in Space, vol. 1387, pp. 159–165. International Society for Optics and Photonics (1991)
Li, M., Zhou, P., Zhao, Z., Zhang, J.: Two-degree-of-freedom fractional order-PID controllers design for fractional order processes with dead-time. ISA Trans. 61, 147–154 (2016)
Liu, G., Daley, S.: Optimal-tuning PID control for industrial systems. Control. Eng. Pract. 9, 1185–1194 (2001)
Llama, M. A., Kelly, R., Santibañez, V.: A stable motion control system for manipulators via fuzzy self-tuning. Fuzzy Sets Syst. 124(2), 133–154 (2001)
Lozano, R., Valera, A., Albertos, P., Arimoto, S., Nakayama, T.: PD control of robot manipulators with joint flexibility, actuators dynamics and friction. Automatica 35(10), 1697–1700 (1999)
Mattos, C. L. C., Barreto, G. A., Cavalcanti, F. R. P.: An improved hybrid particle swarm optimization algorithm applied to economic modeling of radio resource allocation. Electron. Commer. Res. 14(1), 51–70 (2014)
Meza, J. L., Santibáñez, V., Soto, R., Llama, M. A.: Fuzzy self-tuning PID semiglobal regulator for robot manipulators. IEEE Trans. Ind. Electron. 59(6), 2709–2717 (2012)
O’Brien, R.T. Jr, Howe, J.M.: Optimal PID controller design using standard optimal control techniques. In: Proceedings of the 2008 American Control Conference (ACC’2008), pp. 4733–4738 (2008)
Oliveira, P. W., Barreto, G. A., Thé, G. A. P.: A novel tuning method for PD control of robotic manipulators based on minimum jerk principle. In: Proceedings of the 2018 Latin American Robotic Symposium (LARS’2018), pp. 396–401 (2018)
Padula, F., Ionescu, C., Latronico, N., Paltenghi, M., Visioli, A., Vivacqua, G.: A gain-scheduled PID controller for propofol dosing in anesthesia. In: Proceedings of the 9th IFAC Symposium on Biological and Medical Systems (BMS’2015), pp. 545–550, Berlin, Germany (2015)
Piazzi, A., Visioli, A.: Global minimum-jerk trajectory planning of robot manipulators. IEEE Trans. Ind. Electron. 47(1), 140–149 (2000)
Podlubny, I.: vol. 198, 1st edn. Academic Press (1999)
Poli, R., Kennedy, J., Blackwell, T.: Particle swarm optimization. an overview. Swarm Intell. 1(1), 33–57 (2007)
Resende, C.Z., Carelli, R., Sarcinelli-Filho, M.: A nonlinear trajectory tracking controller for mobile robots with velocity limitation via fuzzy gains. Control Eng. Pract. 21(10), 1302–1309 (2013). Usa Takagi-Sugeno para computar os ganhos do controlador PID.
Rohrer, B., Fasoli, S., Krebs, H. I., Hughes, R., Volpe, B., Frontera, W. R., Stein, J., Hogan, N.: Movement smoothness changes during stroke recovery. J. Neurosci. 22(18), 8297–8304 (2002)
Sabir, M. M., Ali, T.: Optimal PID controller design through swarm intelligence algorithms for sun tracking system. Appl. Math. Comput. 274, 690–699 (2016)
Sharma, R., Gaur, P., Mittal, A. P.: Performance analysis of two-degree of freedom fractional order PID, controllers for robotic manipulator with payload. ISA Trans. 58, 279–291 (2015). https://doi.org/10.1016/j.isatra.2015.03.013
Skogestad, S.: Tuning for smooth PID control with acceptable disturbance rejection. Ind. Eng. Chem. Res. 45(23), 7817–7822 (2006). https://doi.org/10.1021/ie0602815
Suleiman, W.: On inverse kinematics with inequality constraints: new insights into minimum jerk trajectory generation. Adv. Robot. 30(17–18), 1164–1172 (2016)
Viola, J., Angel, L.: Tracking control for robotic manipulators using fractional order controllers with computed torque control. IEEE Lat. Am. Trans. 16(7), 1884–1891 (2018). https://doi.org/10.1109/TLA.2018.8447353
Visioli, A.: Optimal tuning of PID controllers for integral and unstable processes. Proc.-Control Theory Appl. 148(2), 180–184 (2001)
Visioli, A.: Practical PID Control, 4th edn. Springer (2006)
Visioli, A.: Research trends for PID controllers. Acta Polytechn. 52(5), 133–154 (2012)
Wolpert, D. M., Ghahramani, Z., Jordan, M. I.: Are arm trajectories planned in kinematic or dynamic coordinates? an adaptation study. Exper. Brain Res. 103(3), 460–470 (1995)
Zhuang, M., Atherton, D. P.: Automatic tuning of optimum PID controllers. IEEE Proc. D - Control Theory Appl. 140(3), 216–224 (1993)
Ziegler, J. G., Nichols, N. B.: Optimum settings for automatic controllers. Trans. ASME 64, 759–768 (1942)
Ziliani, G., Visioli, A., Legnani, G.: Gain scheduling for hybrid force/velocity control in contour tracking task. Int. J. Adv. Robot. Syst. 3(4), 367–374 (2006)
Acknowledgments
This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES - Finance Code 001) and by the Brazilian National Research Council (CNPq) via the grant 309451/2015-9. This work was also partially supported by the project INCT (National Institute of Science and Technology) under the grant CNPq 465755/2014-3, FAPESP (São Paulo Research Foundation) 2014/50851-0.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Oliveira, P.W., Barreto, G.A. & Thé, G.A.P. A General Framework for Optimal Tuning of PID-like Controllers for Minimum Jerk Robotic Trajectories. J Intell Robot Syst 99, 467–486 (2020). https://doi.org/10.1007/s10846-019-01121-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10846-019-01121-y