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Research Papers

Simulation Analysis of Linear Quadratic Regulator Control of Sagittal-Plane Human Walking—Implications for Exoskeletons

[+] Author and Article Information
Raviraj Nataraj

Department of Biomedical Engineering,
Chemistry, and Biological Sciences,
Stevens Institute of Technology,
1 Castle Point Terrace,
Hoboken, NJ 07030
e-mail: rnataraj@stevens.edu

Antonie J. van den Bogert

Professor
Department of Mechanical Engineering,
Cleveland State University,
2121 Euclid Avenue,
Cleveland, OH 44115
e-mail: a.vandenbogert@csuohio.edu

1Corresponding author.

Manuscript received February 11, 2017; final manuscript received August 1, 2017; published online August 22, 2017. Assoc. Editor: Kenneth Fischer.

J Biomech Eng 139(10), 101009 (Aug 22, 2017) (11 pages) Paper No: BIO-17-1059; doi: 10.1115/1.4037560 History: Received February 11, 2017; Revised August 01, 2017

The linear quadratic regulator (LQR) is a classical optimal control approach that can regulate gait dynamics about target kinematic trajectories. Exoskeletons to restore gait function have conventionally utilized time-varying proportional-derivative (PD) control of leg joints. But, these PD parameters are not uniquely optimized for whole-body (full-state) performance. The objective of this study was to investigate the effectiveness of LQR full-state feedback compared to PD control to maintain bipedal walking of a sagittal-plane computational model against force disturbances. Several LQR controllers were uniquely solved with feedback gains optimized for different levels of tracking capability versus control effort. The main implications to future exoskeleton control systems include (1) which LQR controllers out-perform PD controllers in walking maintenance and effort, (2) verifying that LQR desirably produces joint torques that oppose rapidly growing joint state errors, and (3) potentially equipping accurate sensing systems for nonjoint states such as hip-position and torso orientation. The LQR controllers capable of longer walk times than respective PD controllers also required less control effort. During sudden leg collapse, LQR desirably behaved like PD by generating feedback torques that opposed the direction of leg-joint errors. Feedback from nonjoint states contributed to over 50% of the LQR joint torques and appear critical for whole-body LQR control. While LQR control poses implementation challenges, such as more sensors for full-state feedback and operation near the desired trajectories, it offers significant performance advantages over PD control.

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Figures

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Fig. 1

Full gait-cycle phases shown against forward progression (+X) distance of hip-position for sagittal-plane walking model optimally tracking desired gait kinematics (Winter 1990) at forward walking speed of 1.325 m/s

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Fig. 2

The random perturbation types applied to hip of walking model for controller performance evaluation

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Fig. 3

Example (Q/R = 1) of optimal position (Kp) and velocity (Kv) feedback gain values (y-axis) of LQR state feedback controller over gait cycle (x-axis). Controllers have right/left symmetry so only controller gains for right-side joints are shown. The zero value is denoted for each feedback state y-axis. Note: Different scale on vertical axes according to the range (maximum–minimum) in gain values, which is given above each feedback gain profile. Kp, Kv units: N/m, N s/m for hip-X/Y and N/rad, N s/rad for positive angle.

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Fig. 4

Feedback torques from LQR control (Q/R = 1) in the stance leg plotted against vertical drop of the hip during collapse of single-leg stance. Stick figure of walking model (Top) depicts select single-leg support instance in gait cycle and respective changes in leg posture with increasing collapse. The LQR torques generated (Bottom) with hip drop during leg collapse and concurrently applying range of collapse velocities (associated with 0–100 cm/s hip drop velocity) are shown.

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Fig. 5

Time-to-fall against growing perturbation magnitude for LQR controllers with gains optimized according to relative weighting of better tracking (Q) versus less controller effort (R)

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Fig. 6

The RMS of the torque signal generated at each joint and then summed across all six joints during walking against a fixed bound perturbation. RMS results are shown for the total joint torque generated for all LQR controllers tested (top, left) and those controllers with Q/R ratio that did not result in a fall, i.e., “successful” (top, right). For successful controllers, RMS results are also shown for the feedback portion of the joint torques generated (bottom).

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Fig. 7

Plots showing “time-to-fall” against a growing perturbation (type#1) versus “feedback torque RMS” against a fixed bound perturbation (type#2) for tested LQR and PD controllers. Results are shown for all tested controllers (top) and for those controllers that did not produce a fall, i.e., “successful,” against the fixed bound perturbation (bottom). Note: controllers with higher gains denoted by thicker standard deviation lines.

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