Research Papers

Fractional PD-IλDμ Error Manifolds for Robust Tracking Control of Robotic Manipulators

[+] Author and Article Information
Aldo Jonathan Muñoz-Vázquez

School of Engineering,
Autonomous University of Chihuahua,
Campus II,
Chihuahua 31100, Chihuahua, Mexico
e-mail: aldo.munoz.vazquez@gmail.com

Vicente Parra-Vega

Robotics and Advanced Manufacturing,
Center for Research and Advanced Studies,
Saltillo 25900, Coahuila, Mexico
e-mail: vparra@cinvestav.mx

Anand Sánchez-Orta

Robotics and Advanced Manufacturing,
Center for Research and Advanced Studies,
Saltillo 25900, Coahuila, Mexico
e-mail: anand.sanchez@cinvestav.mx

Gerardo Romero-Galván

Electrical and Electronic Engineering
Autonomous University of Tamaulipas
Reynosa-Rodhe 88779, Tamaulipas, Mexico
e-mail: gromero@docentes.uat.edu.mx

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT,AND CONTROL. Manuscript received September 25, 2017; final manuscript received September 23, 2018; published online November 8, 2018. Assoc. Editor: Srinivasa M. Salapaka.

J. Dyn. Sys., Meas., Control 141(3), 031006 (Nov 08, 2018) (6 pages) Paper No: DS-17-1485; doi: 10.1115/1.4041605 History: Received September 25, 2017; Revised September 23, 2018

Linear proportional-integral-derivative (PID) controller stands for the most widespread technique in industrial applications due to its simple structure and easy tuning rules. Recently, considering fractional orders λ and μ, there has been studied the fractional-order PIλDμ (FPID) controller to provide salient advantages in comparison to the conventional integer-order PID, such as, a more flexible structure and a preciser performance. In addition, proportional and derivative (PD) and PID error manifolds have been classically proposed; however, there remains the question on how FPID-like error manifolds perform for the control of nonlinear plants, such as robots. In this paper, this problem is addressed by proposing a PD-IλDμ error manifold for novel vector saturated control. The stability analysis shows convergence into a small vicinity of the origin, wherein, such hybrid combination of integer- and fractional-order error manifolds provides further insights into the closed-loop response of the nonlinear plant. Simulations studies are carried out to illustrate the feasibility of the proposed scheme.

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Grahic Jump Location
Fig. 1

Bode's diagrams of transfer function Gi(s). Notice that in this particular example, the fractional response is in between of phase and gain to the integer order cases.

Grahic Jump Location
Fig. 2

Regulation: bended tracking errors q̃(t). Joint 1 in solid line, joint 2 in dot dashed line: (a) t versus q̃(t): λ = 0, μ = 0, (b) t versus q̃(t): λ = 0.5, μ = 0.25, and (c) t versus q̃(t): λ = 1, μ = 1.

Grahic Jump Location
Fig. 3

Regulation: error variable S(t) and control torque τ(t): (a) t versus S(t): λ = 0.5, μ = 0.25 and (b) t versus τ(t): λ = 0.5, μ = 0.25

Grahic Jump Location
Fig. 4

Tracking: bended tracking errors q̃(t). Joint 1 in solid line, joint 2 in dot dashed line: (a) t versus q̃(t): λ = 0, μ = 0, (b) t versus q̃(t): λ = 0.6, μ = 0.5, and (c) t versus q̃(t): λ = 1, μ = 1.

Grahic Jump Location
Fig. 5

Tracking: error manifold S(t) and control torque τ(t): (a) t versus S(t): λ = 0.6 μ = 0.5 and (b) t versus τ(t): λ = 0.6, μ = 0.5



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