This paper introduces a simple and effective method for selecting the maximum feedback gains in PD-type controllers applied to actuators where feedback delay and derivative signal filtering are present. The method provides the maximum feedback parameters that satisfy a phase margin criteria, producing a closed-loop system with high stability and a dynamic response with near-minimum settling time. Our approach is unique in that it simultaneously possesses: (1) a model of real-world performance-limiting factors (i.e., filtering and delay), (2) the ability to meet performance and stability criteria, and (3) the simplicity of a single closed-form expression. A central focus of our approach is the characterization of system stability through exhaustive searches of the feedback parameter space. Using this search-based method, we locate a set of maximum feedback parameters based on a phase margin criteria. We then fit continuous equations to this data and obtain a closed-form expression which matches the sampled data to within 2–4% error for the majority of the parameter space. We apply our feedback parameter selection method to two real-world actuators with widely differing system properties and show that our method successfully produces the maximum achievable nonoscillating impedance response.

References

References
1.
Kawamura
,
S.
,
Miyazaki
,
F.
, and
Arimoto
,
S.
,
1988
, “
Is a Local Linear PD Feedback Control Law Effective for Trajectory Tracking of Robot Motion?
,”
Proceedings of IEEE International Conference on Robotics and Automation
, Philadelphia, PA, Apr. 24–29, Vol. 3, pp.
1335
1340
.
2.
Colgate
,
J.
, and
Brown
,
J.
,
1994
, “
Factors Affecting the Z-Width of a Haptic Display
,” Proceedings of
IEEE
International Conference on Robotics and Automation
, San Diego, CA, May 8–13, Vol.
4
, pp.
3205
3210
.10.1109/ROBOT.1994.351077
3.
Hogan
,
N.
,
1985
, “
Impedance Control: An Approach to Manipulation: Part I—Theory
,”
ASME J. Dyn. Sys. Meas. Control
,
107
(
1
), pp.
1
7
.10.1115/1.3140702
4.
Ziegler
,
J. G.
, and
Nichols
,
N. B.
,
1942
, “
Optimum Settings for Automatic Controllers
,”
Trans. ASME
,
140
(
3
), pp.
759
768
.10.1115/1.2899060
5.
Åström
,
K. J.
,
1993
, “
Automatic Tuning and Adaptation for PID Controllers - A Survey
,”
Control Eng. Pract.
,
1
(
4
), pp.
699
714
.10.1016/0967-0661(93)91394-C
6.
Lee
,
C.-H.
,
2004
, “
Phase Margins
,”
Int. J. Comput. Cognit.
,
2
, pp.
63
100
.
7.
Poulin
,
E.
,
Pomerleau
,
A.
,
Desbiens
,
A.
, and
Hodouin
,
D.
,
1996
, “
Development and Evaluation of an Auto-Tuning and Adaptive PID Controller
,”
Automatica
,
32
(
1
), pp.
71
82
.10.1016/0005-1098(95)00105-0
8.
Ho
,
W. K.
,
Lim
,
K. W.
, and
Xu
,
W.
,
1998
, “
Optimal Gain and Phase Margin Tuning for PID Controllers
,”
Automatica
,
34
(
8
), pp.
1009
1014
.10.1016/S0005-1098(98)00032-6
9.
Lawrence
,
D.
,
1989
, “
Actuator Limitations on Achievable Manipulator Impedance
,” Proceedings of
IEEE
International Conference on Robotics and Automation
, Scottsdale, AZ, May 14–19, Vol.
1
, pp.
560
565
.10.1109/ROBOT.1989.100044
10.
Sourlas
,
D.
,
Choi
,
J.
, and
Manousiouthakis
,
V.
,
1994
, “
Best Achievable Control System Performance: The Saturation Paradox
,” Proceedings of the 33rd
IEEE
Conference on Decision and Control
, Lake Buena Vista, FL, Dec. 14–16, Vol.
4
, pp.
3816
3818
.10.1109/CDC.1994.411754
11.
Goldfarb
,
M.
, and
Sirithanapipat
,
T.
,
1999
, “
The Effect of Actuator Saturation on the Performance of PD-Controlled Servo Systems
,”
Mechatronics
,
9
(
5
), pp.
497
511
.10.1016/S0957-4158(99)00013-6
12.
Yaniv
,
O.
, and
Nagurka
,
M.
,
2004
, “
Design of PID Controllers Satisfying Gain Margin and Sensitivity Constraints on a Set of Plants
,”
Automatica
,
40
(
1
), pp.
111
116
.10.1016/j.automatica.2003.08.005
13.
Åström
,
K. J.
,
Panagopoulos
,
H.
, and
Hägglund
,
T.
,
1998
, “
Design of PI Controllers Based on Non-Convex Optimization
,”
Automatica
,
34
(
5
), pp.
585
601
.10.1016/S0005-1098(98)00011-9
14.
Li
,
D.
,
Gao
,
F.
,
Xue
,
Y.
, and
Lu
,
C.
,
2007
, “
Optimization of Decentralized PI/PID Controllers Based on Genetic Algorithm
,”
Asian J. Control
,
9
(
3
), pp.
306
316
.10.1111/j.1934-6093.2007.tb00416.x
15.
Wang
,
C.
, and
Li
,
D.
,
2011
, “
Decentralized PID Controllers Based on Probabilistic Robustness
,”
ASME J. Dyn. Sys. Meas. Control
,
133
(
6
), p.
061015
.10.1115/1.4004781
16.
Colgate
,
J.
, and
Schenkel
,
G.
,
1994
, “
Passivity of a Class of Sampled-Data Systems: Application to Haptic Interfaces
,”
American Control Conference
, June 29–July 1, Vol.
3
, pp.
3236
3240
.
17.
Lawrence
,
D.
,
1988
, “
Impedance Control Stability Properties in Common Implementations
,” Proceedings
IEEE
International Conference on Robotics and Automation
, Philadelphia, PA, Apr. 24–29, Vol.
2
, pp.
1185
1190
.10.1109/ROBOT.1988.12222
18.
An
,
J.
, and
Kwon
,
D.-S.
,
2004
, “
In Haptics, the Influence of the Controllable Physical Damping on Stability and Performance
,” Proceedings of
IEEE
/RSJ
International Conference on Intelligent Robots and Systems
, Sept. 28–Oct. 2, Vol.
2
, pp.
1204
1209
.10.1109/IROS.2004.1389560
19.
Mehling
,
J.
,
Colgate
,
J.
, and
Peshkin
,
M.
,
2005
, “
Increasing the Impedance Range of a Haptic Display by Adding Electrical Damping
,”
Eurohaptics Conference, Symposium on Haptic Interfaces for Virtual Environment and Teleoperator Systems
, Mar. 18–20, pp.
257
262
.10.1109/WHC.2005.79
20.
Hulin
,
T.
,
Preusche
,
C.
, and
Hirzinger
,
G.
,
2006
, “
Stability Boundary for Haptic Rendering: Influence of Physical Damping
,”
IEEE
/RSJ
International Conference on Intelligent Robots and Systems
, Beijing, China, pp.
1570
1575
.10.1109/IROS.2006.282043
21.
Franklin
,
G. F.
,
Powell
,
J. D.
, and
Baeini
,
A. E.
,
1986
,
Feedback Control of Dynamic Systems
,
Addison-Wesley
,
Reading, MA
.
22.
Ogata
,
K.
,
1990
,
Modern Control Engineering
,
Prentice-Hall
,
Englewood Cliffs, NJ
.
23.
Suchomski
,
P.
,
2001
, “
Robust PI and PID Controller Design in Delta Domain
,”
Control Theory Appl., IEE Proc.
,
148
(
5
), pp.
350
354
.10.1049/ip-cta:20010627
24.
Li
,
K.
,
2013
, “
PID Tuning for Optimal Closed-Loop Performance With Specified Gain and Phase Margins
,”
IEEE Trans. Control Syst. Technol.
,
21
(
3
), pp.
1024
1030
.10.1109/TCST.2012.2198479
25.
Lee
,
C.-H.
, and
Teng
,
C.-C.
,
2003
, “
Calculation of PID Controller Parameters by Using a Fuzzy Neural Network
,”
ISA Trans.
,
42
(
3
), pp.
391
400
.10.1016/S0019-0578(07)60142-6
26.
Diolaiti
,
N.
,
Niemeyer
,
G.
,
Barbagli
,
F.
, and
Salisbury
,
J.
,
2006
, “
Stability of Haptic Rendering: Discretization, Quantization, Time Delay, and Coulomb Effects
,”
IEEE Trans. Rob.
,
22
(
2
), pp.
256
268
.10.1109/TRO.2005.862487
27.
Pratt
,
G.
, and
Williamson
,
M.
,
1995
, “
Series Elastic Actuators
,”
Intelligent Robots and Systems 95, Proceedings of 1995 IEEE/RSJ International Conference on Human Robot Interaction and Cooperative Robots'
, Vol.
1
, pp.
399
406
.
28.
Weir
,
D.
,
Colgate
,
J.
, and
Peshkin
,
M.
,
2008
, “
Measuring and Increasing Z-Width With Active Electrical Damping
,”
Symposium on Haptic Interfaces for Virtual Environment and Teleoperator Systems
, Reno, NV, Mar. 13–14, pp.
169
175
.10.1109/HAPTICS.2008.4479938
29.
Rossa
,
C.
,
Lozada
,
J.
, and
Micaelli
,
A.
,
2013
, “
Stable Haptic Interaction Using Passive and Active Actuators
,”
IEEE
International Conference on Robotics and Automation
, Karlsruhe, Germany, May 6–10, pp.
2386
2392
.10.1109/ICRA.2013.6630901
30.
NASA-JSC DRC team valkyrie, accessed 2014-02-10, http://www.theroboticschallenge.org/node/59
31.
Bao
,
J.
, and
Lee
,
P. L.
,
2007
,
Process Control: The Passive Systems Approach
,
Springer, London, UK
.
32.
Paine
,
N.
,
Oh
,
S.
, and
Sentis
,
L.
,
2014
, “
Design and Control Considerations for High-Performance Series Elastic Actuators
,”
IEEE/ASME Trans. Mech.
,
19
(
3
), pp.
1080
1091
.10.1109/TMECH.2013.2270435
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