0
Research Papers

Ground Reaction Force Estimation in Prosthetic Legs With Nonlinear Kalman Filtering Methods

[+] Author and Article Information
Seyed Fakoorian

Department of Electrical Engineering
and Computer Science,
Cleveland State University,
Cleveland, OH 44115
e-mail: s.fakoorian@csuohio.edu

Vahid Azimi

School of Electrical and Computer Engineering,
Georgia Institute of Technology,
Atlanta, GA 30313

Mahmoud Moosavi

Department of Electrical Engineering
and Computer Science,
Cleveland State University,
Cleveland, OH 44115

Hanz Richter

Department of Mechanical Engineering,
Cleveland State University,
Cleveland, OH 44115
e-mail: h.richter@csuohio.edu

Dan Simon

Department of Electrical Engineering
and Computer Science,
Cleveland State University,
Cleveland, OH 44115
e-mail: d.j.simon@csuohio.edu

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received December 6, 2016; final manuscript received March 10, 2017; published online July 20, 2017. Assoc. Editor: Dumitru I. Caruntu.

J. Dyn. Sys., Meas., Control 139(11), 111004 (Jul 20, 2017) (11 pages) Paper No: DS-16-1583; doi: 10.1115/1.4036546 History: Received December 06, 2016; Revised March 10, 2017

A method to estimate ground reaction forces (GRFs) in a robot/prosthesis system is presented. The system includes a robot that emulates human hip and thigh motion, along with a powered (active) transfemoral prosthetic leg. We design a continuous-time extended Kalman filter (EKF) and a continuous-time unscented Kalman filter (UKF) to estimate not only the states of the robot/prosthesis system but also the GRFs that act on the foot. It is proven using stochastic Lyapunov functions that the estimation error of the EKF is exponentially bounded if the initial estimation errors and the disturbances are sufficiently small. The performance of the estimators in normal walk, fast walk, and slow walk is studied, when we use four sensors (hip displacement, thigh, knee, and ankle angles), three sensors (thigh, knee, and ankle angles), and two sensors (knee and ankle angles). Simulation results show that when using four sensors, the average root-mean-square (RMS) estimation error of the EKF is 0.0020 rad for the joint angles and 11.85 N for the GRFs. The respective numbers for the UKF are 0.0016 rad and 7.98 N, which are 20% and 33% lower than those of the EKF.

FIGURES IN THIS ARTICLE
<>
Copyright © 2017 by ASME
Your Session has timed out. Please sign back in to continue.

References

Sub, F. , Varol, H. A. , and Goldfarb, M. , 2010, “ Upslope Walking With a Powered Knee and Ankle Prosthesis: Initial Results With an Amputee Subject,” IEEE Trans. Neural Syst. Rehabil. Eng., 19(1), pp. 71–78. [PubMed]
Harvey, Z. , Potter, B. K. , Vandersea, J. , and Wolf, E. , 2011, “ Prosthetic Advances,” J. Surg. Orthop. Adv., 21(1), pp. 58–64 http://www.armdynamics.com/caffeine/uploads/files/21-1-9.pdf.
Martinez-Villalpando, E. C. , Weber, J. , Elliott, G. , and Herr, H. , 2008, “ Design of an Agonist-Antagonist Active Knee Prosthesis,” IEEE/RAS-EMBS International Conference on Biomedical Robotics and Biomechatronics, Scottsdale, AZ, Oct. 19–22, pp. 529–534.
Au, S. K. , Herr, H. , Weber, J. , and Martinez-Villalpando, E. C. , 2007, “ Powered Ankle-Foot Prosthesis for the Improvement of Amputee Ambulation,” 29th Annual IEEE Conference on Engineering in Medicine and Biology Society (EMBS), Lyon, France, Aug. 22–26, pp. 3020–3026.
Au, S. K. , Weber, J. , and Herr, H. , 2007, “ Biomechanical Design of a Powered Ankle Foot Prosthesis,” IEEE International Conference on Rehabilitation Robotics (ICORR), Noordwijk, The Netherlands, June 13–15, pp. 298–303.
Zhao, H. , Kolathaya, S. , and Ames, A. D. , 2014, “ Quadratic Programming and Impedance Control for Transfemoral Prosthesis,” International Conference on Robotics and Automation (ICRA), Hong Kong, China, May 31–June 7, pp. 1341–1347.
Gregg, R. D. , Lenzi, T. , Hargrove, L. J. , and Sensinger, J. W. , 2014, “ Virtual Constraint Control of a Powered Prosthetic Leg: From Simulation to Experiments With Transfemoral Amputees,” IEEE Trans. Rob., 30(6), pp. 1455–1471. [CrossRef]
Young, A. J. , Simon, A. M. , and Hargrove, L. J. , 2014, “ A Training Method for Locomotion Mode Prediction Using Powered Lower Limb Prostheses,” IEEE Trans. Neural Syst. Rehabil. Eng., 22(3), pp. 671–677. [CrossRef] [PubMed]
Gonzalez, M. , 2014, “ Biomechanical Analysis of Gait Kinetics Resulting From Use of a Vacuum Socket on a Transtibial Prosthesis,” Master's thesis, University of Nevada, Las Vegas, NV http://digitalscholarship.unlv.edu/cgi/viewcontent.cgi?article=1014amp;context=honors_theses.
Murakami, T. , Nakamura, R. , Yu, F. , and Ohnishi, K. , 1993, “ Force Sensorless Impedance Control by Disturbance Observer,” IEEE Power Conversion Conference, Yokohama, Japan, Apr. 19–21, pp. 352–357.
Simpson, J. W. L. , Cook, C. D. , and Li, Z. , 2002, “ Sensorless Force Estimation for Robots With Friction,” Australasian Conference on Robotics and Automation, Auckland, New Zealand, Nov. 27–29, pp. 94–99 https://pdfs.semanticscholar.org/17ad/b33f0e6f6ceaa05dbecbe5781232709967b0.pdf.
Phong, L. D. , Choi, J. , Lee, W. , and Kang, S. , 2015, “ A Novel Method for Estimating External Force: Simulation Study With a 4-DOF Robot Manipulator,” Int. J. Precis. Eng. Manuf., 16(4), pp. 755–766. [CrossRef]
Hacksel, P. J. , and Salcudean, S. E. , 1994, “ Estimation of Environment Forces and Rigid-Body Velocities Using Observers,” IEEE International Conference on Robotics and Automation (ICRA), San Diego, CA, May 8–13, pp. 931–936.
Rigatos, G. G. , 2012, “ A Derivative-Free Kalman Filtering Approach to State Estimation-Based Control of a Class of Nonlinear Systems,” IEEE Trans. Ind. Electron., 59(10), pp. 3987–3997. [CrossRef]
Sarkka, S. , 2007, “ On Unscented Kalman Filtering for State Estimation of Continuous-Time Nonlinear Systems,” IEEE Trans. Autom. Control, 52(9), pp. 1631–1641. [CrossRef]
Julier, S. J. , Uhlmann, J. K. , and Durrant-Whyte, H. F. , 1995, “ A New Approach for Filtering Nonlinear Systems,” American Control Conference (ACC), Seattle, WA, June 21–23, pp. 1628–1632.
Fakoorian, S. A. , Simon, D. , Richter, H. , and Azimi, V. , 2016, “ Ground Reaction Force Estimation in Prosthetic Legs With an Extended Kalman Filter,” IEEE International Systems Conference (SysCon), Orlando, FL, Apr. 18–21, pp. 338–343.
Richter, H. , and Simon, D. , 2014, “ Robust Tracking Control of a Prosthesis Test Robot,” ASME J. Dyn. Syst. Meas. Control, 136(3), p. 031011. [CrossRef]
van den Bogert, A. J. , Geijtenbeek, T. , Even-Zohar, O. , Steenbrink, F. , and Hardin, E. C. , 2013, “ A Real-Time System for Biomechanical Analysis of Human Movement and Muscle Function,” Med. Biol. Eng. Comput., 51(10), pp. 1069–1077. [CrossRef] [PubMed]
Azimi, V. , Simon, D. , and Richter, H. , 2015, “ Stable Robust Adaptive Impedance Control of a Prosthetic Leg,” ASME Paper No. DSCC2015-9794.
Richter, H. , Simon, D. , Smith, W. A. , and Samorezov, S. , 2015, “ Dynamic Modeling, Parameter Estimation and Control of a Leg Prosthesis Test Robot,” Appl. Math. Modell., 39(2), pp. 559–573. [CrossRef]
Mohammadi, H. , and Richter, H. , 2015, “ Robust Tracking/Impedance Control: Application to Prosthetics,” American Control Conference (ACC), Chicago, IL, July 1–3, pp. 2673–2678.
Warner, H. , 2015, “ Optimal Design and Control of a Lower-Limb Prosthesis With Energy Regeneration,” Master's thesis, Cleveland State University, Cleveland, OH http://engagedscholarship.csuohio.edu/cgi/viewcontent.cgi?article=1678&context=etdarchive.
Azimi, V. , Simon, D. , Richter, H. , and Fakoorian, S. A. , 2016, “ Robust Composite Adaptive Transfemoral Prosthesis Control With Non-Scalar Boundary Layer Trajectories,” American Control Conference (ACC), Boston, MA, July 6–8, pp. 3002–3007.
Warner, H. , Simon, D. , Mohammadi, H. , and Richter, H. , 2016, “ Switched Robust Tracking/Impedance Control for an Active Transfemoral Prosthesis,” American Control Conference (ACC), Boston, MA, July 6–8, pp. 2187–2192.
van der Merwe, R. , and Wan, E. A. , 2001, “ The Square-Root Unscented Kalman Filter for State and Parameter-Estimation,” IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP'01), Salt Lake City, UT, May 7–11, pp. 3461–3464.
Simon, D. , 2006, Optimal State Estimation: Kalman, H Infinity, and Nonlinear Approaches, Wiley, Hoboken, NJ. [CrossRef]
Safonov, M. , and Athans, M. , 1978, “ Robustness and Computational Aspects of Nonlinear Stochastic Estimators and Regulators,” IEEE Trans. Autom. Control, 23(4), pp. 717–725. [CrossRef]
Ljung, L. , 1979, “ Asymptotic Behavior of the Extended Kalman Filter as a Parameter Estimator for Linear Systems,” IEEE Trans. Autom. Control, 24(1), pp. 36–50. [CrossRef]
Ursin, B. , 1980, “ Asymptotic Convergence Properties of the Extended Kalman Filter Using Filtered State Estimates,” IEEE Trans. Autom. Control, 25(6), pp. 1207–1211. [CrossRef]
Reif, K. , Gunther, S. , Yaz, E. , and Unbehauen, R. , 2000, “ Stochastic Stability of the Continuous-Time Extended Kalman Filter,” IEEE Proc. Control Theory Appl., 147(1), pp. 45–72. [CrossRef]
Reif, K. , Gunther, S. , Yaz, E. , and Unbehauen, R. , 1999, “ Stochastic Stability of the Discrete-Time Extended Kalman Filter,” IEEE Trans. Autom. Control, 44(4), pp. 714–728. [CrossRef]
Fitts, J. M. , 1972, “ On the Observability of Nonlinear Systems With Applications to Nonlinear Regression Analysis,” Inf. Sci., 4(2), pp. 129–156. [CrossRef]
Sontag, E. D. , 1979, “ On the Observability of Polynomial Systems—I: Finite-Time Problems,” SIAM J. Control Optim., 17(1), pp. 139–151. [CrossRef]
Gard, T. C. , 1988, Introduction to Stochastic Differential Equations, M. Dekker, New York.
Zakai, M. , 1967, “ On the Ultimate Boundedness of Moments Associated With Solutions of Stochastic Differential Equations,” SIAM J. Control, 5(4), pp. 588–593. [CrossRef]
Shanying, Z. , Chen, C. , Li, W. , Yang, B. , and Guan, X. , 2013, “ Distributed Optimal Consensus Filter for Target Tracking in Heterogeneous Sensor Networks,” IEEE Trans. Cybern., 43(6), pp. 1963–1976. [CrossRef] [PubMed]
Tønne, K. K. , 2007, “ Stability Analysis of EKF-Based Attitude Determination and Control,” Master's thesis, Norwegian University of Science and Technology, Trondheim, Norway http://folk.ntnu.no/tomgra/Diplomer/Tonne.pdf.
Baras, J. S. , Bensoussan, A. , and James, M. R. , 1988, “ Dynamic Observers as Asymptotic Limits of Recursive Filters: Special Cases,” SIAM J. Appl. Math., 48(5), pp. 1147–1158. [CrossRef]
Gauthier, J. P. , Hammouri, H. , and Othman, S. , 1992, “ A Simple Observer for Nonlinear Systems Applications to Bioreactors,” IEEE Trans. Autom. Control, 37(6), pp. 875–880. [CrossRef]
Khademi, G. , Mohammadi, H. , Hardin, E. C. , and Simon, D. , 2015, “ Evolutionary Optimization of User Intent Recognition for Transfemoral Amputees,” IEEE Biomedical Circuits and Systems Conference (BioCAS), Atlanta, GA, Oct. 22–24, pp. 1–4.
Mao, X. , 2007, Stochastic Differential Equations and Applications, Elsevier, Cambridge, UK.

Figures

Grahic Jump Location
Fig. 1

The robotic model of the hip robot/prosthesis system. It is desired to eliminate the load cells on the foot and instead estimate forces with an EKF.

Grahic Jump Location
Fig. 2

State estimation of joint displacements of the robot with the use of EKF and four measurements during normal gait speed. Note that the initial estimate quickly converges to the true state. (a) The estimate of the hip displacement, (b) the estimate of the thigh angle, (c) the estimate of the knee angle, and (d) the estimate of the ankle angle.

Grahic Jump Location
Fig. 3

State estimation of the robot velocities with the use of EKF and four measurements during normal gait speed

Grahic Jump Location
Fig. 4

Horizontal and vertical GRFs estimations with the use of EKF and four measurements during normal gait speed

Grahic Jump Location
Fig. 5

The simulation results show the boundedness of the estimation error with small initial error and small noise terms. The sixth state is used here for illustration purposes, but similar results hold for all of the other states as well. (a) The estimate of x6(t) and (b) the estimation error ξ6(t).

Grahic Jump Location
Fig. 6

The simulation results show the divergence of the estimation error with small initial error and large noise terms. The sixth state is used here for illustration purposes, but similar results hold for all of the other states as well. (a) The estimate of x6(t) and (b) the estimation error ξ6(t).

Grahic Jump Location
Fig. 7

The simulation results show the divergence of the estimation error with large initial error and small noise terms. The sixth state is used here for illustration purposes, but similar results hold for all of the other states as well. (a) The estimate of x6(t) and (b) the estimation error ξ6(t).

Tables

Errata

Discussions

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