Abstract

This paper explores new ways to use energy shaping and regulation methods in walking systems to generate new passive-like gaits and dynamically transition between them. We recapitulate a control framework for Lagrangian hybrid systems, and show that regulating a state varying energy function is equivalent to applying energy shaping and regulating the system to a constant energy value. We then consider a simple one-dimensional hopping robot and show how energy shaping and regulation control can be used to generate and transition between nearly globally stable hopping limit cycles. The principles from this example are then applied on two canonical walking models, the spring loaded inverted pendulum (SLIP) and compass gait biped, to generate and transition between locomotive gaits. These examples show that piecewise jumps in control parameters can be used to achieve stable changes in desired gait characteristics dynamically/online.

References

1.
Westervelt
,
E. R.
,
Grizzle
,
J. W.
, and
Koditschek
,
D. E.
,
2003
, “
Hybrid Zero Dynamics of Planar Biped Walkers
,”
IEEE Trans. Autom. Control
,
48
(
1
), pp.
42
56
.10.1109/TAC.2002.806653
2.
Westervelt
,
E. R.
,
Grizzle
,
J. W.
,
Chevallereau
,
C.
,
Choi
,
J. H.
, and
Morris
,
B.
,
2007
,
Feedback Control of Dynamic Bipedal Robot Locomotion
,
CRC Press
, Boca Raton, FL, p.
528
.
3.
Powell
,
M. J.
,
Hereid
,
A.
, and
Ames
,
A. D.
,
2013
, “
Speed Regulation in 3D Robotic Walking Through Motion Transitions Between Human-Inspired Partial Hybrid Zero Dynamics
,”
2013 IEEE International Conference on Robotics and Automation
, Karlsruhe, Germany, May 6–10,
pp.
4803
4810
.10.1109/ICRA.2013.6631262
4.
Saglam
,
C. O.
, and
Byl
,
K.
,
2015
, “
Meshing Hybrid Zero Dynamics for Rough Terrain Walking
,” 2015 IEEE International Conference on Robotics and Automation (
ICRA
), Seattle, WA,
May 26–30
, pp.
5718
5725
.10.1109/ICRA.2015.7140000
5.
Ma
,
W.-L.
,
Hamed
,
K. A.
, and
Ames
,
A. D.
,
2019
, “
First Steps Towards Full Model Based Motion Planning and Control of Quadrupeds: A Hybrid Zero Dynamics Approach
,”
IEEE/RSJ International Conference on Intelligent Robots and Systems
, Macau, China, pp. 5498–5503.http://ames.caltech.edu/ma2019first.pdf
6.
Ortega
,
R.
,
Spong
,
M. W.
,
Gomez-Estern
,
F.
, and
Blankenstein
,
G.
,
2002
, “
Stabilization of a Class of Undercactuated Mechanical Systems Via Interconnection and Damping Assignment
,”
IEEE Trans. Autom. Control
,
47
(
8
), pp.
1218
1233
.10.1109/TAC.2002.800770
7.
Yi
,
B.
,
Ortega
,
R.
,
Wu
,
D.
, and
Zhang
,
W.
,
2020
, “
Orbital Stabilization of Nonlinear Systems Via Mexican Sombrero Energy Shaping and Pumping-and-Damping Injection
,”
Automatica
,
112
, p.
108661
.10.1016/j.automatica.2019.108661
8.
Bloch
,
A. M.
,
Leonard
,
N. E.
, and
Marsden
,
J. E.
,
2001
, “
Controlled Lagrangians and the Stabilization of Euler-Poincare Mechanical Systems
,”
Int. J. Robust Nonlinear Control
,
11
(
3
), pp.
191
214
.10.1002/rnc.572
9.
Blankenstein
,
G.
,
Ortega
,
R.
, and
Van Der Schaft
,
A. J.
,
2002
, “
The Matching Conditions of Controlled Lagrangians and IDA-Passivity Based Control
,”
Int. J. Control
,
75
(
9
), pp.
645
665
.10.1080/00207170210135939
10.
Chang
,
D. E.
,
Bloch
,
A. M.
,
Leonard
,
N. E.
,
Marsden
,
J. E.
, and
Woolsey
,
C. A.
,
2002
, “
The Equivalence of Controlled Lagrangian and Controlled Hamiltonian Systems
,”
ESAIM: Control, Optim. Calculus Var.
,
8
, pp.
393
422
.10.1051/cocv:2002045
11.
Kotyczka
,
P.
, and
Sarras
,
I.
,
2012
, “
Equivalence of Immersion and Invariance and IDA-PBC for the Acrobot
,”
IFAC Proc. Vol.
,
45
(
19
), pp.
36
41
.10.3182/20120829-3-IT-4022.00013
12.
McCourt
,
M. J.
, and
Antsaklis
,
P. J.
,
2009
, “
Connection Between the Passivity Index and Conic Systems
,” University of Notre Dame, Notre Dame, IN, Report No. ISIS-9-009.
13.
Spong
,
M.
,
Holm
,
J.
, and
Lee
,
D.
,
2007
, “
Passivity-Based Control of Bipedal Locomotion
,”
IEEE Rob. Autom. Mag.
,
14
(
2
), pp.
30
40
.10.1109/MRA.2007.380638
14.
Gregg
,
R. D.
, and
Spong
,
M. W.
,
2010
, “
Reduction-Based Control of Three-Dimensional Bipedal Walking Robots
,”
Int. J. Rob. Res.
,
29
(
6
), pp.
680
702
.10.1177/0278364909104296
15.
Garofalo
,
G.
,
Ott
,
C.
, and
Albu-Schäffer
,
A.
,
2012
, “
Walking Control of Fully Actuated Robots Based on the Bipedal SLIP Model
,”
2012 IEEE International Conference on Robotics and Automation
, St. Paul, MN,
May 14–18
, pp.
1456
1463
.10.1109/ICRA.2012.6225272
16.
Sinnet
,
R. W.
, and
Ames
,
A. D.
,
2015
, “
Energy Shaping of Hybrid Systems Via Control Lyapunov Functions
,”
American Controls Conference
, Chicago, IL, July 1–3, pp.
5992
5997
.10.1109/ACC.2015.7172280
17.
Goswami
,
A.
,
Thuilot
,
B.
, and
Espiau
,
B.
,
1996
, “
Compass-Like Biped Robot Part I: Stability and Bifurcation of Passive Gaits
,”
Ph.D. thesis
,
INRIA
, Rocquencourt, France.https://hal.inria.fr/inria-00073701/document
18.
Gregg
,
R. D.
,
Tilton
,
A. K.
,
Candido
,
S.
,
Bretl
,
T.
, and
Spong
,
M. W.
,
2012
, “
Control and Planning of 3-D Dynamic Walking With Asymptotically Stable Gait Primitives
,”
IEEE Trans. Rob.
,
28
(
6
), pp.
1415
1423
.10.1109/TRO.2012.2210484
19.
Garofalo
,
G.
, and
Ott
,
C.
,
2019
, “
Repetitive Jumping Control for Biped Robots Via Force Distribution and Energy Regulation
,”
Human Friendly Robotics
,
Springer
, Berlin, pp.
29
45
.
20.
Geyer
,
H.
,
Seyfarth
,
A.
, and
Blickhan
,
R.
,
2006
, “
Compliant Leg Behaviour Explains Basic Dynamics of Walking and running
,”
Proc. R. Soc. B: Biol. Sci.
,
273
(
1603
), pp.
2861
2867
.10.1098/rspb.2006.3637
21.
Garcia
,
M.
,
Chatterjee
,
A.
,
Ruina
,
A.
, and
Coleman
,
M.
,
1998
, “
The Simplest Walking Model: Stability, Complexity, and Scaling
,”
ASME J. Biomech. Eng.
,
120
(
2
), pp.
281
288
.10.1115/1.2798313
22.
Ouakad
,
H.
,
Nayfeh
,
A.
,
Choura
,
S.
,
Abdel-Rahman
,
E.
,
Najar
,
F.
, and
Hammad
,
B.
,
2008
, “
Nonlinear Feedback Control and Dynamics of an Electrostatically Actuated Microbeam Filter
,”
ASME
Paper No. IMECE2008-68965.10.1115/IMECE2008-68965
23.
Yeatman
,
M.
,
Lv
,
G.
, and
Gregg
,
R. D.
,
2019
, “
Decentralized Passivity-Based Control With a Generalized Energy Storage Function for Robust Biped Locomotion
,”
ASME J. Dyn. Syst., Meas., Control
,
141
(
10
), p.
101007
.10.1115/1.4043801
24.
Garofalo
,
G.
, and
Ott
,
C.
,
2017
, “
Energy Based Limit Cycle Control of Elastically Actuated Robots
,”
IEEE Trans. Autom. Control
,
62
(
5
), pp.
2490
2497
.10.1109/TAC.2016.2599781
25.
Goswami
,
A.
,
Espiau
,
B.
, and
Keramane
,
A.
,
1997
, “
Limit Cycles in a Passive Compass Gait Biped and Passivity-Mimicking Control Laws
,”
Auton. Rob.
,
4
(
3
), pp.
273
286
.10.1023/A:1008844026298
26.
Sinnet
,
R. W.
,
2015
, “
Energy Shaping of Mechanical Systems Via Control Lyapunov Functions With Applications to Bipedal Locomotion
,”
Ph.D. thesis
,
Texas A&M University
, College Station, TX.https://oaktrust.library.tamu.edu/handle/1969.1/154978
27.
Spong
,
M. W.
, and
Bullo
,
F.
,
2005
, “
Controlled Symmetries and Passive Walking
,”
IEEE Trans. Autom. Control
,
50
(
7
), pp.
1025
1031
.10.1109/TAC.2005.851449
28.
McGeer
,
T.
,
1990
, “
Passive Dynamic Walking
,”
Int. J. Rob. Res.
,
9
(
2
), pp.
62
82
.10.1177/027836499000900206
29.
Naldi
,
R.
, and
Sanfelice
,
R. G.
,
2013
, “
Passivity-Based Control for Hybrid Systems With Applications to Mechanical Systems Exhibiting Impacts
,”
Automatica
,
49
(
5
), pp.
1104
1116
.10.1016/j.automatica.2013.01.018
30.
Coleman
,
M. J.
,
Chatterjee
,
A.
, and
Ruina
,
A.
,
1997
, “
Motions of a Rimless Spoked Wheel: A Simple Three-Dimensional System With Impacts
,”
Dyn. Stab. Syst.
,
12
(
3
), pp.
139
159
.10.1080/02681119708806242
31.
Nathan
,
A.
,
1977
, “
The Rayleigh-Van Der Pol Harmonic Oscillator
,”
Int. J. Electron. Theor. Exp.
,
43
(
6
), pp.
609
614
.10.1080/00207217708900770
32.
Pillai
,
S. U.
,
Suel
,
T.
, and
Cha
,
S.
,
2005
, “
The Perron-Frobenius Theorem: Some of Its Applications
,”
IEEE Signal Process. Mag.
,
22
(
2
), pp.
62
75
.10.1109/MSP.2005.1406483
33.
Heim
,
S.
, and
Spröwitz
,
A.
,
2019
, “
Beyond Basins of Attraction: Quantifying Robustness of Natural Dynamics
,”
IEEE Trans. Rob.
,
35
(
4
), pp.
939
952
.10.1109/TRO.2019.2910739
34.
Quintero
,
D.
,
Villarreal
,
D. J.
,
Lambert
,
D. J.
,
Kapp
,
S.
, and
Gregg
,
R. D.
,
2018
, “
Continuous-Phase Control of a Powered Knee–Ankle Prosthesis: Amputee Experiments Across Speeds and Inclines
,”
IEEE Trans. Rob.
,
34
(
3
), pp.
686
701
.10.1109/TRO.2018.2794536
35.
Lv
,
G.
,
Zhu
,
H.
, and
Gregg
,
R. D.
,
2018
, “
On the Design and Control of Highly Backdrivable Lower-Limb Exoskeletons: A Discussion of Past and Ongoing Work
,”
IEEE Control Syst. Mag.
,
38
(
6
), pp.
88
113
.10.1109/MCS.2018.2866605
36.
Xu
,
C.
,
Ming
,
A.
, and
Chen
,
Q.
,
2014
, “
Characteristic Equations and Gravity Effects on Virtual Passive Bipedal Walking
,”
2014 IEEE International Conference on Robotics and Biomimetics (ROBIO 2014)
, Bali, Indonesia,
Dec. 5–10
, pp.
1296
1301
.10.1109/ROBIO.2014.7090512
37.
Mohammadi
,
A.
, and
Gregg
,
R. D.
,
2019
, “
Variable Impedance Control of Powered Knee Prostheses Using Human-Inspired Algebraic Curves
,”
ASME J. Comput. Nonlinear Dyn.
,
14
(
10
), p.
101007
.10.1115/1.4043002
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