An optimized proportional-derivative controller for the human upper extremity with gravity
Introduction
Spinal cord injury (SCI) impairs movement and sensation below the level of injury. High-level SCI, which affects the cervical C1–C4 levels, compromises voluntary motor function below the neck. Although communication between the brain and peripheral neuromuscular system is impaired, muscle function remains intact. Functional Electrical Stimulation (FES) is a technology that uses electrical current to activate peripheral nerves that otherwise would be inactive due to injury (Crago et al., 1996) to restore useful muscular movement. FES neuroprostheses have been applied to numerous physiological systems, including upper extremity function, which is addressed in the present study.
Feedforward control is the form most commonly used for clinical FES applications (Peckham and Knutson, 2005, Lynch and Popovic, 2008). It entails calculating and applying muscle stimulation patterns using available information about the system, without the use of feedback signals. It is simple to implement and does not require sensors; however, this absence of sensors also makes the success of the movements generated heavily dependent on accurate models of the controlled system and environment.
Feedback control requires the use of sensors, which detect arm properties and allow the controller to correct its actions if they deviate from the desired behavior. Upper extremity (UE) FES applications of feedback control have included shoulder function (Yu et al., 2001), elbow extension (Giuffrida and Crago, 2001, Memberg et al., 2003), hand grasp (Kilgore et al., 1989), and wrist stabilization (Lemay and Crago, 1997).
Additionally, more advanced upper extremity FES controllers have been investigated. These can involve the combination of feedforward and feedback control (Abbas and Chizeck, 1995, Blana et al., 2009), reinforcement learning (Izawa et al., 2004, reinforcement learning controller for Functional Electrical Stimulation of a human arm (Masters of Science thesis) 2009 Case Western Reserve University Cleveland, OH, USA., Jagodnik, 2014), and artificial neural networks (Giuffrida and Crago, 2005, Hincapie and Kirsch, 2009).
Many projects that develop advanced controllers compare their new control method to more basic feedback control, e.g. proportional-derivative (PD) or proportional-integral-derivative (PID), to demonstrate the superiority of the newly-developed advanced controller. However, there is often minimal effort invested in adequately tuning the feedback controllers intended for comparison, and we hypothesize that these feedback controllers may often perform worse than they would have, had they been properly tuned. PID controller tuning algorithms include the Ziegler–Nichols method (Ziegler and Nichols, 1942, Astrom and Hagglund, 2004) and the Chien, Hrones, and Reswick method (Chien et al., 1952). However, these tuning methods can often result in poor performance (Astrom and Hagglund, 2001), particularly for nonlinear systems such as FES control. For example, when using Ziegler–Nichols tuning, overshoot is common for nonlinear systems (Dey and Mudi, 2009). Such tuning algorithms cannot be considered optimized. Because these simpler feedback controllers have not been given the same care in tuning as the more advanced controllers to which they are being compared, it is likely that inaccurate conclusions may be drawn when comparing these two classes of control algorithms.
For this reason, we propose to mathematically optimize a proportional-derivative (PD) controller gain set for a 3-dimensional human shoulder and arm system, and to compare its performance on dynamic reaching tasks to PD controller gain sets tuned using standard algorithms. We hypothesize that optimization will yield significantly improved performance when compared with standard, non-optimal, tuning methods. PD control was selected because it represents a basic feedback control architecture, and because goal-directed reaching movements with a single endpoint specified per task are being performed (Heaviside step function with no explicit trajectory specified); such a task specification could result in compromised performance should an integral control component be added, as in a PID controller. Additionally, PD control is consistent with the Equilibrium Point hypothesis, which effectively explains certain features of motor control (Bizzi et al., 1992, Feldman et al., 1998).
We have previously determined for a planar arm system that using simulated annealing to optimize PD control can yield excellent performance (Jagodnik and van den Bogert, 2010). To extend our previous work, we optimize a PD controller to perform goal-oriented reaching movements, using a 3-dimensional biomechanical model of a human arm that has 5 degrees of freedom (DOF). We explore two PD controller architectures: one with 2 gains, and another with 10. The optimized controller gain sets are applied to a large variety of point-to-point reaching tasks, and tested for their ability to generalize to tasks for which they had not been optimized, and for their ability to withstand muscular fatigue. The performance of our optimized controller gain sets is compared with that of three other PD controller gain sets that have not been optimized for this system, and conclusions are drawn about the utility of optimization for neuroprosthesis controller development.
Section snippets
Biomechanical model
For all experiments described, a 3-dimensional (3D) computational musculoskeletal model of the human arm was used that has 5 degrees of freedom (DOF) (Table 1) and 102 muscle elements grouped into 22 muscles (Table 2) (Chadwick et al., 2009). The model includes gravity and uses a fixed scapula as the base of the model. All joints are modeled as hinges, with the glenohumeral joint consisting of three such hinges (Chadwick et al., 2009); this joint is modeled according to the Y–Z′–Y″ convention (
Results
The gains resulting from the optimizations performed in this study are presented (Table 3). This table also lists the additional three PD controller gain sets used for comparison analyses.
Discussion
In this study, we optimized a PD controller for a 3D model of the human arm using simulated annealing. Two different PD control architectures were investigated, with 2 and 10 gain parameters. When PD controllers using the optimized gain sets were applied to a large set of tasks on which the gains had not been optimized, to test ability to generalize, both gain sets achieved excellent accuracy, with the 10-parameter gain set slightly, but not significantly, outperforming the 2-parameter gain set
Conclusion
We have optimized two proportional-derivative (PD) controller gain sets on a 3-dimensional biomechanical arm model performing goal-oriented reaching movements, and have demonstrated that optimization can yield significant improvements in controller accuracy over a wide range of dynamic reaching tasks, when compared with three other PD controller gain sets that had not been optimized for this system. The optimized controllers used physiologically reasonable levels of muscular effort to perform
Conflict of interest statement
None of the authors has a conflict of interest to disclose.
Acknowledgments
This project was funded by National Institutes of Health (NIH) fellowship #TRN030167, NIH Training Grant #T32-EB004314, and Ardiem Medical Arm Control Device Grant #W81XWH0720044. The authors thank Joris Lambrecht for his 3D arm visualization software, Dr. Peter Cooman for his input on project planning, Dr. Steven Sidik for statistical analysis guidance, and the CWRU High Performance Computing Cluster group for assistance with running simulations.
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