An optimized proportional-derivative controller for the human upper extremity with gravity

Abstract

When Functional Electrical Stimulation (FES) is used to restore movement in subjects with spinal cord injury (SCI), muscle stimulation patterns should be selected to generate accurate and efficient movements. Ideally, the controller for such a neuroprosthesis will have the simplest architecture possible, to facilitate translation into a clinical setting. In this study, we used the simulated annealing algorithm to optimize two proportional-derivative (PD) feedback controller gain sets for a 3-dimensional arm model that includes musculoskeletal dynamics and has 5 degrees of freedom and 22 muscles, performing goal-oriented reaching movements. Controller gains were optimized by minimizing a weighted sum of position errors, orientation errors, and muscle activations. After optimization, gain performance was evaluated on the basis of accuracy and efficiency of reaching movements, along with three other benchmark gain sets not optimized for our system, on a large set of dynamic reaching movements for which the controllers had not been optimized, to test ability to generalize. Robustness in the presence of weakened muscles was also tested. The two optimized gain sets were found to have very similar performance to each other on all metrics, and to exhibit significantly better accuracy, compared with the three standard gain sets. All gain sets investigated used physiologically acceptable amounts of muscular activation. It was concluded that optimization can yield significant improvements in controller performance while still maintaining muscular efficiency, and that optimization should be considered as a strategy for future neuroprosthesis controller design.

Keywords: Feedback control; Functional electrical stimulation; Human; Musculoskeletal modeling and simulation; Optimization; Proportional-derivative; Upper extremity.

Fig. 1

Joint angle definitions for 3D arm model. The shoulder joint is modeled according to the Y-Z’-Y” convention.

Fig. 2

System block diagram of arm with proportional-derivative (PD) control. The PD Controller block is specified by the PD controller equation (Eq. (2)). u() are muscle stimulation values, θ(5) are joint angles, and θ̇(5) are angular velocities, with the subscript “targ” indicating target values.

Fig. 3

Boxplots showing (A) endpoint position errors and (B) endpoint orientation errors for the Generality Test. ‘3D 10-param’ is the set of 10 proportional-derivative (PD) gains optimized on the 3-dimensional arm model; ‘3D 2-param’ is the pair of 2 gains optimized on the 3-dimensional arm model; ‘2D 2-param’ is the pair of gains optimized on the planar arm model; ‘Hand-Tuned’ indicates the gain set manually tuned on the 3D arm model; and ‘Ziegler-Nichols’ denotes the gain set tuned on the 3D arm model using the Ziegler-Nichols method. Red ‘+’ symbols indicate individual outlier values.

Fig. 4

Boxplots showing (A) endpoint muscular effort and (B) whole-task mean muscular effort values for the Generality Test. ‘3D 10-param’ is the set of 10 proportional-derivative (PD) gains optimized on the 3-dimensional arm model; ‘3D 2-param’ is the pair of 2 gains optimized on the 3-dimensional arm model; ‘2D 2-param’ is the pair of gains optimized on the planar arm model; ‘Hand-Tuned’ indicates the gain set manually tuned on the 3D arm model; and ‘Ziegler-Nichols’ denotes the gain set tuned on the 3D arm model using the Ziegler-Nichols method. Red ‘+’ symbols indicate individual outlier values.

Fig. 5

Example task visualization: front (A) and side (B) views of the initial position (Position description: “Hand in front, in usable space #1.”), and front (C) and side (D) views of the target position (Position description: “Outstretched to object at right.”).

Fig. 6

(A) Endpoint position errors and (B) Endpoint orientation error for 5 proportional-derivative (PD) controller gain sets performing the example reaching task (Figure 5) from the Generality Test. ‘3D 10-param (opt)’ is the set of 10 PD gains optimized on the 3-dimensional arm model; ‘3D 2-param (opt)’ is the pair of 2 gains optimized on the 3-dimensional arm model; ‘2D 2-param (opt)’ is the pair of gains optimized on the planar arm model; ‘Hand-Tuned’ indicates the gain set manually tuned on the 3D arm model; and ‘Ziegler-Nichols’ denotes the gain set tuned on the 3D arm model using the Ziegler-Nichols method.

Fig. 7

Fatigue Robustness test results. Plots show performance metrics as a function of muscular fatigue over all muscles. Error bars show standard deviation of the estimated mean. (A) Endpoint position error in cm. (B) Endpoint orientation error in degrees. (C) Mean muscular effort over the entire movement.