0
Research Papers

Exponential Stabilization of Fully Actuated Planar Bipedal Robotic Walking With Global Position Tracking Capabilities

[+] Author and Article Information
Yan Gu

School of Mechanical Engineering,
Purdue University,
West Lafayette, IN 47906
e-mail: gu49@purdue.edu

Bin Yao

Professor
School of Mechanical Engineering,
Purdue University,
West Lafayette, IN 47906;
The State Key Laboratory of Fluid Power
and Mechatronic Systems,
Zhejiang University,
Hangzhou, Zhejiang 310027, China
e-mail: byao@purdue.edu

C. S. George Lee

School of Electrical and Computer Engineering,
Purdue University,
West Lafayette, IN 47906
e-mail: csglee@purdue.edu

1Corresponding author.

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received February 19, 2017; final manuscript received October 19, 2017; published online December 19, 2017. Assoc. Editor: Hashem Ashrafiuon.

J. Dyn. Sys., Meas., Control 140(5), 051008 (Dec 19, 2017) (11 pages) Paper No: DS-17-1107; doi: 10.1115/1.4038268 History: Received February 19, 2017; Revised October 19, 2017

This paper focuses on the development of a model-based feedback controller to realize high versatility of fully actuated planar bipedal robotic walking. To conveniently define both symmetric and asymmetric walking patterns, we propose to use the left and the right legs for gait characterization. In addition to walking pattern tracking error, a biped's position tracking error in Cartesian space is included in the output function in order to enable high-level task planning and control such as multi-agent coordination. A feedback controller based on input–output linearization and proportional–derivative control is then synthesized to realize exponential tracking of the desired walking pattern as well as the desired global position trajectory. Sufficient stability conditions of the hybrid time-varying closed-loop system are developed based on the construction of multiple Lyapunov functions. In motion planning, a new method of walking pattern design is introduced, which decouples the planning of global motion and walking pattern. Finally, simulation results on a fully actuated planar biped show the effectiveness of the proposed walking strategy.

FIGURES IN THIS ARTICLE
<>
Copyright © 2018 by ASME
Your Session has timed out. Please sign back in to continue.

References

Vukobratović, M. , and Stepenenko, J. , 1972, “ On the Stability of Anthropomorphic Systems,” Math. Biosci., 15(1), pp. 1–37. [CrossRef]
Kajita, S. , Kanehiro, F. , Kaneko, K. , Fujiwara, K. , Harada, K. , Yokoi, K. , and Hirukawa, H. , 2003, “ Biped Walking Pattern Generation by Using Preview Control of Zero-Moment Point,” IEEE International Conference on Robotics and Automation (ICRA), Taipei, Taiwan, Sept. 14–19, pp. 1620–1626.
Aubin, J. P. , 2009, Viability Theory, Springer, Berlin. [CrossRef]
Wieber, P. , 2002, “ On the Stability of Walking Systems,” International Workshop on Humanoid and Human Friendly Robotics, Tsukuba, Japan, Dec. 11–12, pp. 53–59. https://hal.archives-ouvertes.fr/inria-00390866/document
Pratt, J. E. , Carff, J. , Drakunov, S. , and Goswami, A. , 2006, “ Capture Point: A Step Toward Humanoid Push Recovery,” IEEE-RAS International Conference on Humanoid Robots (ICHR), Genova, Italy, Dec. 4–6, pp. 200–207.
Pratt, J. , and Tedrake, R. , 2006, “ Velocity-Based Stability Margins for Fast Bipedal Walking,” Fast Motions in Biomechanics and Robotics, Springer, Berlin, pp. 299–324. [CrossRef]
Koolen, T. , De Boer, T. , Rebula, J. , Goswami, A. , and Pratt, J. , 2012, “ Capturability-Based Analysis and Control of Legged Locomotion—Part 1: Theory and Application to Three Simple Gait Models,” Int. J. Rob. Res., 31(9), pp. 1094–1113. [CrossRef]
Pratt, J. , Koolen, T. , De Boer, T. , Rebula, J. , Cotton, S. , Carff, J. , Johnson, M. , and Neuhaus, P. , 2012, “ Capturability-Based Analysis and Control of Legged Locomotion—Part 2: Application to M2V2, a Lower-Body Humanoid,” Int. J. Rob. Res., 31(10), pp. 1117–1133. [CrossRef]
Grizzle, J. , Abba, G. , and Plestan, P. , 2001, “ Asymptotically Stable Walking for Biped Robots: Analysis Via Systems With Impulse Effects,” IEEE Trans. Autom. Control, 46(1), pp. 51–64. [CrossRef]
Grizzle, J. W. , Chevallereau, C. , Sinnet, R. W. , and Ames, A. D. , 2014, “ Models, Feedback Control, and Open Problems of 3D Bipedal Robotic Walking,” Automatica, 50(8), pp. 1955–1988. [CrossRef]
Ames, A. D. , Cousineau, E. A. , and Powell, M. J. , 2012, “ Dynamically Stable Bipedal Robotic Walking With NAO Via Human-Inspired Hybrid Zero Dynamics,” International Conference on Hybrid Systems: Computation and Control (HSCC), Beijing, China, Apr. 17–19, pp. 135–144.
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. [CrossRef]
Morris, B. , and Grizzle, J. W. , 2009, “ Hybrid Invariant Manifolds in Systems With Impulse Effects With Application to Periodic Locomotion in Bipedal Robots,” IEEE Trans. Autom. Control, 54(8), pp. 1751–1764. [CrossRef]
Morris, B. , and Grizzle, J. W. , 2005, “ A Restricted Poincaré Map for Determining Exponentially Stable Periodic Orbits in Systems With Impulse Effects: Application to Bipedal Robots,” IEEE International Conference on Decision and Control (CDC), Seville, Spain, Dec. 10–15, pp. 4199–4206.
Griffin, B. , and Grizzle, J. , 2015, “ Walking Gait Optimization for Accommodation of Unknown Terrain Height Variations,” American Control Conference (ACC), Chicago, IL, July 1–3, pp. 4810–4817.
Buss, B. G. , Ramezani, A. , Hamed, K. A. , Griffin, B. , Galloway, K. S. , and Grizzle, J. W. , 2014, “ Preliminary Walking Experiments With Underactuated 3D Bipedal Robot MARLO,” IEEE International Conference on Robots and Systems (IROS), Chicago, IL, Sept. 14–18, pp. 2529–2536.
Da, X. , Harib, O. , Hartley, R. , Griffin, B. , and Grizzle, J. W. , 2016, “ From 2D Design of Underactuated Bipedal Gaits to 3D Implementation: Walking With Speed Tracking,” IEEE Access, 4, pp. 3469–3478. [CrossRef]
Shiriaev, A. , Perram, J. W. , and Canudas-de Wit, C. , 2005, “ Constructive Tool for Orbital Stabilization of Underactuated Nonlinear Systems: Virtual Constraints Approach,” IEEE Trans. Autom. Control, 50(8), pp. 1164–1176. [CrossRef]
Shiriaev, A. S. , Freidovich, L. B. , and Manchester, I. R. , 2008, “ Can We Make a Robot Ballerina Perform a Pirouette? Orbital Stabilization of Periodic Motions of Underactuated Mechanical Systems,” Annu. Rev. Control, 32(2), pp. 200–211. [CrossRef]
Manchester, I. R. , Mettin, U. , Iida, F. , and Tedrake, R. , 2011, “ Stable Dynamic Walking Over Uneven Terrain,” Int. J. Rob. Res., 30(3), pp. 265–279. [CrossRef]
Ames, A. D. , Galloway, K. , Sreenath, K. , and Grizzle, J. W. , 2014, “ Rapidly Exponentially Stabilizing Control Lyapunov Functions and Hybrid Zero Dynamics,” IEEE Trans. Autom. Control, 59(4), pp. 876–891. [CrossRef]
Ames, A. D. , 2014, “ Human-Inspired Control of Bipedal Walking Robots,” IEEE Trans. Autom. Control, 59(5), pp. 1115–1130. [CrossRef]
Hamed, K. A. , and Grizzle, J. W. , 2014, “ Event-Based Stabilization of Periodic Orbits for Underactuated 3-D Bipedal Robots With Left-Right Symmetry,” IEEE Trans. Rob., 30(2), pp. 365–381. [CrossRef]
Branicky, M. S. , 1998, “ Multiple Lyapunov Functions and Other Analysis Tools for Switched and Hybrid Systems,” IEEE Trans. Autom. Control, 43(4), pp. 475–482. [CrossRef]
Gu, Y. , Yao, B. , and Lee, C. S. G. , 2016, “ Bipedal Gait Recharacterization and Walking Encoding Generalization for Stable Dynamic Walking,” IEEE International Conference on Robotics and Automation (ICRA), Stockholm, Sweden, May 16–21, pp. 1788–1793.
Ramezani, A. , Hurst, J. W. , Hamed, K. A. , and Grizzle, J. W. , 2013, “ Performance Analysis and Feedback Control of ATRIAS, a Three-Dimensional Bipedal Robot,” ASME J. Dyn. Syst. Meas. Control, 136(2), p. 021012. [CrossRef]
Hamed, K. A. , Buss, B. G. , and Grizzle, J. W. , 2016, “ Exponentially Stabilizing Continuous-Time Controllers for Periodic Orbits of Hybrid Systems: Application to Bipedal Locomotion With Ground Height Variations,” Int. J. Rob. Res., 35(8), pp. 977–999. [CrossRef]
Grizzle, J. W. , Abba, G. , and Plestan, F. , 1999, “ Proving Asymptotic Stability of a Walking Cycle for a Five DOF Biped Robot Model,” International Conference on Climbing and Walking Robots, Portsmouth, UK, Sept. 14–15, pp. 4048–4053.
Yao, B. , Hu, C. , and Wang, Q. , 2012, “ An Orthogonal Global Task Coordinate Frame for Contouring Control of Biaxial Systems,” IEEE Trans. Mechatron., 17(4), pp. 622–634. [CrossRef]
Khalil, H. K. , 1996, Nonlinear Control, Prentice Hall, Upper Saddle River, NJ.
Bainov, D. , and Simeonov, P. , 1993, Impulsive Differential Equations: Periodic Solutions and Applications, Vol. 66, CRC Press, Boca Raton, FL.

Figures

Grahic Jump Location
Fig. 1

A bipedal robot walking in the Xw-Zw plane: (a) left leg in support and (b) right leg in support

Grahic Jump Location
Fig. 2

Encoding the swing-leg pattern using the support-leg angle qst and the swing-leg angle qsw

Grahic Jump Location
Fig. 5

Simulation results of previous work based on orbital stabilization and support-swing gait characterization. Green (blue) dashed: desired swing-leg (trunk) trajectory determined by the desired walking pattern.

Grahic Jump Location
Fig. 3

A planar biped with lumped masses, massless thin feet,and identical legs. (m = 10 kg, MH = 5 kg, MT = 5 kg, and l=(r/2)=0.5 m.)

Grahic Jump Location
Fig. 6

Simulation results of proposed walking strategy underthe left-right gait characterization. Dashed lines: desired joint trajectories generated by gi(s¯,qsw,q3)=0 (i∈{L,R}) and sd(t).

Grahic Jump Location
Fig. 4

Desired walking patterns of swing-leg angle qsw with respect to support-leg angle qst: (a) symmetric and (b) asymmetric

Grahic Jump Location
Fig. 7

Symmetric walking with sd(t) = 0.6t − 0.1 m, KPi = diag [28, 28, 28], and KDi = diag[11, 11, 11]. Dashed lines: desired joint trajectories generated by gi(s¯,qsw,q3)=0 (i∈{L,R}) and sd(t).

Grahic Jump Location
Fig. 8

Symmetric walking with sd(t)=2.3e−0.3(t+0.5)+0.6t−2.1(m), KPi=diag[28,28,28], and KDi = diag[11, 11, 11]. Dashed lines: desired joint trajectories generated by gi(s¯,qsw,q3)=0 (i∈{L,R}) and sd(t).

Grahic Jump Location
Fig. 9

Asymmetric walking with sd(t) = 0.6t − 0.1 m, KPi = diag [12, 12, 12], and KDi = diag[7, 7, 7]. Dashed lines: desired joint trajectories generated by gi(s¯,qsw,q3)=0(i∈{L,R}) and sd(t).

Grahic Jump Location
Fig. 10

Asymmetric walking with sd(t) = 0.6t − 0.1 m, KPL = KPR = diag[28, 28, 28], and KDL = KDR = diag[11, 11, 11]. Dashed lines: desired joint trajectories generated by gi(s¯,qsw,q3)=0 (i∈{L,R}) and sd(t).

Grahic Jump Location
Fig. 11

Asymmetric walking with sd(t) = 0.6t − 0.1 m, KPi = diag[6, 6, 6], and KDi = diag[5, 5, 5]. Dashed lines: desired joint trajectories generated by gi(s¯,qsw,q3)=0 (i∈{L,R}) and sd(t).

Tables

Errata

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In