In this paper, a novel analytical coupled trajectory optimization of a seven degrees-of-freedom (7DOF) Baxter manipulator utilizing extremum seeking (ES) approach is presented. The robotic manipulators are used in network-based industrial units, and even homes, by expending a significant lumped amount of energy, and therefore, optimal trajectories need to be generated to address efficiency issues. These robots are typically operated for thousands of cycles resulting in a considerable cost of operation. First, coupled dynamic equations are derived using the Lagrangian method and experimentally validated to examine the accuracy of the model. Then, global design sensitivity analysis is performed to investigate the effects of changes of optimization variables on the cost function leading to select the most effective ones. We examine a discrete-time multivariable gradient-based ES scheme enforcing operational time and torque saturation constraints in order to minimize the lumped amount of energy consumed in a path given; therefore, time-energy optimization would not be the immediate focus of this research effort. The results are compared with those of a global heuristic genetic algorithm (GA) to discuss the locality/globality of optimal solutions. Finally, the optimal trajectory is experimentally implemented to be thoroughly compared with the inefficient one. The results reveal that the proposed scheme yields the minimum energy consumption in addition to overcoming the robot's jerky motion observed in an inefficient path.
Issue Section:
Research Papers
References
1.
Meike
, D.
, and Ribickis
, L.
, 2011
, “Energy Efficient Use of Robotics in the Automobile Industry
,” 15th International Conference on Advanced Robotics
(ICAR
), Tallinn, Estonia, June 20–23, pp. 507
–511
.2.
Park
, J.-H.
, and Asada
, H.
, 1994
, “Concurrent Design Optimization of Mechanical Structure and Control for High Speed Robots
,” ASME J. Dyn. Syst. Meas. Control
, 116
(3
), pp. 344
–356
.3.
Bagheri
, M.
, Ajoudani
, A.
, Lee
, J.
, Caldwell
, D. G.
, and Tsagarakis
, N. G.
, 2015
, “Kinematic Analysis and Design Considerations for Optimal Base Frame Arrangement of Humanoid Shoulders
,” IEEE International Conference on Robotics and Automation
(ICRA
), Seattle, WA, May 26–30, pp. 2710
–2715
.4.
Red
, W.
, Troung-Cao
, H.-V.
, and Kim
, K.
, 1987
, “Robot Path Planning in Three-Dimensions Using the Direct Subspace
,” ASME J. Dyn. Syst. Meas. Control
, 109
(3
), pp. 238
–244
.5.
Luh
, J. Y.
, and Lin
, C. S.
, 1981
, “Optimum Path Planning for Mechanical Manipulators
,” ASME J. Dyn. Syst. Meas. Control
, 103
(2
), pp. 142
–151
.6.
Wiens
, G.
, and Berggren
, M.
, 1991
, “Suboptimal Path Planning of Robots: Minimal Nonlinear Forces and Energy
,” ASME J. Dyn. Syst. Meas. Control
, 113
(4
), pp. 748
–752
.7.
Jha
, D. K.
, Li
, Y.
, Wettergren
, T. A.
, and Ray
, A.
, 2015
, “Robot Path Planning in Uncertain Environments: A Language-Measure-Theoretic Approach
,” ASME J. Dyn. Syst. Meas. Control
, 137
(3
), p. 034501
.8.
Yun
, W. S.
, Cho
, D. W.
, and Baek
, Y. S.
, 1997
, “Dynamic Path Planning for Robot Navigation Using Sonor Mapping and Neural Networks
,” ASME J. Dyn. Syst. Meas. Control
, 119
(1
), pp. 19
–26
.9.
Huang
, P.
, Xu
, Y.
, and Liang
, B.
, 2006
, “Global Minimum-Jerk Trajectory Planning of Space Manipulator
,” Int. J. Control, Autom., Syst.
, 4
(4
), pp. 405
–413
.http://www.ijcas.com/admin/paper/files/IJCAS_v4_n4_pp.405-413.pdf10.
Hirakawa
, A. R.
, and Kawamura
, A.
, 1996
, “Proposal of Trajectory Generation for Redundant Manipulators Using Variational Approach Applied to Minimization of Consumed Electrical Energy
,” Fourth International Workshop on Advanced Motion Control
(AMC'96
), Mie, Japan, Mar. 18–21.11.
Rubio
, F.
, Llopis-Albert
, C.
, Valero
, F.
, and Suñer
, J. L.
, 2016
, “Industrial Robot Efficient Trajectory Generation Without Collision Through the Evolution of the Optimal Trajectory
,” Rob. Auton. Syst.
, 86
, pp. 106
–112
.12.
Pham
, Q.-C.
, and Nakamura
, Y.
, 2015
, “A New Trajectory Deformation Algorithm Based on Affine Transformations
,” IEEE Trans. Rob.
, 31
(4
), pp. 1054
–1063
.13.
Jiang
, Q.
, and Gosselin
, C. M.
, 2010
, “Dynamic Optimization of Reactionless Four-Bar Linkages
,” ASME J. Dyn. Syst. Meas. Control
, 132
(4
), p. 041006
.14.
Bessonnet
, G.
, and Lallemand
, J.
, 1994
, “On the Optimization of Robotic Manipulator Trajectories With Bounded Joint Actuators or Joint Kinetic Loads Considered as Control Variables
,” ASME J. Dyn. Syst. Meas. Control
, 116
(4
), p. 819
.15.
Huang
, P.
, Xu
, Y.
, and Liang
, B.
, 2006
, “Minimum-Torque Path Planning of Space Robots Using Genetic Algorithms
,” Int. J. Rob. Autom.
, 21
(3
), p. 229
.16.
Garg
, D. P.
, and Kumar
, M.
, 2002
, “Optimization Techniques Applied to Multiple Manipulators for Path Planning and Torque Minimization
,” Eng. Appl. Artif. Intell.
, 15
(3–4
), pp. 241
–252
.17.
Dong
, J.
, and Stori
, J.
, 2006
, “A Generalized Time-Optimal Bidirectional Scan Algorithm for Constrained Feed-Rate Optimization
,” ASME J. Dyn. Syst. Meas. Control
, 128
(2
), pp. 379
–390
.18.
Mann
, M. P.
, Zion
, B.
, Rubinstein
, D.
, Linker
, R.
, and Shmulevich
, I.
, 2014
, “Minimum Time Kinematic Motions of a Cartesian Mobile Manipulator for a Fruit Harvesting Robot
,” ASME J. Dyn. Syst. Meas. Control
, 136
(5
), p. 051009
.19.
Shiller
, Z.
, 1996
, “Time-Energy Optimal Control of Articulated Systems With Geometric Path Constraints
,” ASME J. Dyn. Syst. Meas. Control
, 118
(1
), pp. 139
–143
.20.
Barnett
, E.
, and Gosselin
, C.
, 2015
, “Time-Optimal Trajectory Planning of Cable-Driven Parallel Mechanisms for Fully Specified Paths With g1-Discontinuities
,” ASME J. Dyn. Syst. Meas. Control
, 137
(7
), p. 071007
.21.
Mattmüller
, J.
, and Gisler
, D.
, 2009
, “Calculating a Near Time-Optimal Jerk-Constrained Trajectory along a Specified Smooth Path
,” Int. J. Adv. Manuf. Technol.
, 45
(9–10
), pp. 1007
–1016
.22.
Costantinescu
, D.
, and Croft
, E. A.
, 2000
, “Smooth and Time-Optimal Trajectory Planning for Industrial Manipulators along Specified Paths
,” J. Rob. Syst.
, 17
(5
), pp. 233
–249
.23.
Ariyur
, K. B.
, and Krstić
, M.
, 2003
, Real-Time Optimization by Extremum-Seeking Control
, Wiley
, Hoboken, NJ.24.
Krstić
, M.
, and Wang
, H.-H.
, 2000
, “Stability of Extremum Seeking Feedback for General Nonlinear Dynamic Systems
,” Automatica
, 36
(4
), pp. 595
–601
.25.
Krstić
, M.
, 2000
, “Performance Improvement and Limitations in Extremum Seeking Control
,” Syst. Control Lett.
, 39
(5
), pp. 313
–326
.26.
Wang
, H.-H.
, Yeung
, S.
, and Krstić
, M.
, 2000
, “Experimental Application of Extremum Seeking on an Axial-Flow Compressor
,” IEEE Trans. Control Syst. Technol.
, 8
(2
), pp. 300
–309
.27.
Binetti
, P.
, Ariyur
, K. B.
, Krstić
, M.
, and Bernelli
, F.
, 2003
, “Formation Flight Optimization Using Extremum Seeking Feedback
,” J. Guid. Control Dyn.
, 26
(1
), pp. 132
–142
.28.
Cochran
, J.
, Kanso
, E.
, Kelly
, S. D.
, Xiong
, H.
, and Krstić
, M.
, 2009
, “Source Seeking for Two Nonholonomic Models of Fish Locomotion
,” IEEE Trans. Rob.
, 25
(5
), pp. 1166
–1176
.29.
Cochran
, J.
, Siranosian
, A.
, Ghods
, N.
, and Krstić
, M.
, 2009
, “3-d Source Seeking for Underactuated Vehicles Without Position Measurement
,” IEEE Trans. Rob.
, 25
(1
), pp. 117
–129
.30.
Ghaffari
, A.
, Krstić
, M.
, and Seshagiri
, S.
, 2014
, “Power Optimization and Control in Wind Energy Conversion Systems Using Extremum Seeking
,” IEEE Trans. Control Syst. Technol.
, 22
(5
), pp. 1684
–1695
.31.
Frihauf
, P.
, Krstić
, M.
, and Başar
, T.
, 2013
, “Finite-Horizon Lq Control for Unknown Discrete-Time Linear Systems Via Extremum Seeking
,” Eur. J. Control
, 19
(5
), pp. 399
–407
.32.
Manzie
, C.
, and Krstić
, M.
, 2009
, “Extremum Seeking With Stochastic Perturbations
,” IEEE Trans. Autom. Control
, 54
(3
), pp. 580
–585
.33.
Stanković
, M. S.
, and Stipanović
, D. M.
, 2009
, “Discrete Time Extremum Seeking by Autonomous Vehicles in a Stochastic Environment
,” 48th IEEE Conference in Decision and Control, 2009 Held Jointly With the 2009 28th Chinese Control Conference
(CDC/CCC
), Shanghai, China, Dec. 15–18, pp. 4541
–4546
.34.
Liu
, S.-J.
, and Krstić
, M.
, 2014
, “Discrete-Time Stochastic Extremum Seeking
,” IFAC Proc. Volumes
, 47
(3
), pp. 3274
–3279
.35.
Choi
, J.-Y.
, Krstić
, M.
, Ariyur
, K. B.
, and Lee
, J. S.
, 2002
, “Extremum Seeking Control for Discrete-Time Systems
,” IEEE Trans. Autom. Control
, 47
(2
), pp. 318
–323
.36.
Rotea
, M. A.
, 2000
, “Analysis of Multivariable Extremum Seeking Algorithms
,” American Control Conference
(ACC
), Chicago, IL, June 28–30, pp. 433–437.37.
Walsh
, G. C.
, 2000
, “On the Application of Multi-Parameter Extremum Seeking Control
,” American Control Conference
(ACC
), Chicago, IL, June 28–30, pp. 411–415.38.
Ariyur
, K. B.
, and Krstić
, M.
, 2002
, “Multivariable Extremum Seeking Feedback: Analysis and Design
,” 15th International Symposium on Mathematical Theory of Networks and Systems, Notre Dame, IN, August 12–16, pp. 1–15.39.
Li
, Y.
, Rotea
, M. A.
, Chiu
, G.-C.
, Mongeau
, L. G.
, and Paek
, I.-S.
, 2005
, “Extremum Seeking Control of a Tunable Thermoacoustic Cooler
,” IEEE Trans. Control Syst. Technol.
, 13
(4
), pp. 527
–536
.40.
Zhang
, Y.
, Rotea
, M.
, and Gans
, N.
, 2011
, “Sensors Searching for Interesting Things: Extremum Seeking Control on Entropy Maps
,” 50th IEEE Conference on Decision and Control and European Control Conference
(CDC-ECC
), Orlando, FL, Dec. 12–15, pp. 4985
–4991
.41.
Zhang
, Y.
, Shen
, J.
, Rotea
, M.
, and Gans
, N.
, 2011
, “Robots Looking for Interesting Things: Extremum Seeking Control on Saliency Maps
,” IEEE/RSJ International Conference on Intelligent Robots and Systems
(IROS
), San Francisco, CA, Sept. 25–30, pp. 1180
–1186
.42.
Ghaffari
, A.
, Seshagiri
, S.
, and Krstić
, M.
, 2012
, “Power Optimization for Photovoltaic Micro-Converters Using Multivariable Gradient-Based Extremum-Seeking
,” American Control Conference
(ACC
), Montreal, QC, June 27–29, pp. 3383
–3388
.43.
Bagheri
, M.
, and Naseradinmousavi
, P.
, 2017
, “Novel Analytical and Experimental Trajectory Optimization of a 7-Dof Baxter Robot: Global Design Sensitivity and Step Size Analyses
,” Int. J. Adv. Manuf. Technol.
, 93
(9–12
), pp. 4153
–4167
.44.
Bagheri
, M.
, Naseradinmousavi
, P.
, and Morsi
, R.
, 2017
, “Experimental and Novel Analytical Trajectory Optimization of a 7-Dof Baxter Robot: Global Design Sensitivity and Step Size Analyses
,” ASME
Paper No. DSCC2017-5004.45.
Naseradinmousavi
, P.
, Machiani
, S. G.
, Ayoubi
, M. A.
, and Nataraj
, C.
, 2017
, “Coupled Operational Optimization of Smart Valve System Subject to Different Approach Angles of a Pipe Contraction
,” J. Struct. Multidiscip. Optim.
, 55
(3
), pp. 1001
–1015
.46.
Naseradinmousavi
, P.
, Bagheri
, M.
, and Nataraj
, C.
, 2016
, “Coupled Operational Optimization of Smart Valve System Subject to Different Approach Angles of a Pipe Contraction
,” ASME
Paper No. DSCC2016-9627.47.
Naseradinmousavi
, P.
, Segala
, D. B.
, and Nataraj
, C.
, 2016
, “Chaotic and Hyperchaotic Dynamics of Smart Valves System Subject to a Sudden Contraction
,” ASME J. Comput. Nonlinear Dyn.
, 11
(5
), p. 051025
.48.
Van Dijk
, N.
, Van de Wouw
, N.
, Nijmeijer
, H.
, and Pancras
, W.
, 2007
, “Path-Constrained Motion Planning for Robotics Based on Kinematic Constraints
,” ASME
Paper No. DETC2007-34780.49.
Davidor
, Y.
, 1991
, Genetic Algorithms and Robotics: A Heuristic Strategy for Optimization
, Vol. 1
, World Scientific
, Singapore.50.
Ghaffari
, A.
, Seshagiri
, S.
, and Krstić
, M.
, 2015
, “Multivariable Maximum Power Point Tracking for Photovoltaic Micro-Converters Using Extremum Seeking
,” Control Eng. Pract.
, 35
, pp. 83
–91
.51.
Ghaffari
, A.
, Krstić
, M.
, and Seshagiri
, S.
, 2014
, “Power Optimization for Photovoltaic Microconverters Using Multivariable Newton-Based Extremum Seeking
,” IEEE Trans. Control Syst. Technol.
, 22
(6
), pp. 2141
–2149
.52.
Ghaffari
, A.
, Krstić
, M.
, and Nešić
, D.
, 2012
, “Multivariable Newton-Based Extremum Seeking
,” Automatica
, 48
(8
), pp. 1759
–1767
.53.
Ariyur
, K. B.
, and Krstić
, M.
, 2002
, “Analysis and Design of Multivariable Extremum Seeking
,” American Control Conference
(ACC
), Anchorage, AK, May 8–10, pp. 2903
–2908
.54.
Bai
, E.-W.
, Fu
, L.-C.
, and Sastry
, S. S.
, 1988
, “Averaging Analysis for Discrete Time and Sampled Data Adaptive Systems
,” IEEE Trans. Circuits Syst.
, 35
(2
), pp. 137
–148
.Copyright © 2018 by ASME
You do not currently have access to this content.
