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.

References

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.pdf
10.
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
.
You do not currently have access to this content.