"Body-In-The-Loop": Optimizing Device Parameters Using Measures of Instantaneous Energetic Cost

This paper demonstrates methods for the online optimization of assistive robotic devices such as powered prostheses, orthoses and exoskeletons. Our algorithms estimate the value of a physiological objective in real-time (with a body “in-the-loop”) and use this information to identify optimal device parameters. To handle sensor data that are noisy and dynamically delayed, we rely on a combination of dynamic estimation and response surface identification. We evaluated three algorithms (Steady-State Cost Mapping, Instantaneous Cost Mapping, and Instantaneous Cost Gradient Search) with eight healthy human subjects. Steady-State Cost Mapping is an established technique that fits a cubic polynomial to averages of steady-state measures at different parameter settings. The optimal parameter value is determined from the polynomial fit. Using a continuous sweep over a range of parameters and taking into account measurement dynamics, Instantaneous Cost Mapping identifies a cubic polynomial more quickly. Instantaneous Cost Gradient Search uses a similar technique to iteratively approach the optimal parameter value using estimates of the local gradient. To evaluate these methods in a simple and repeatable way, we prescribed step frequency via a metronome and optimized this frequency to minimize metabolic energetic cost. This use of step frequency allows a comparison of our results to established techniques and enables others to replicate our methods. Our results show that all three methods achieve similar accuracy in estimating optimal step frequency. For all methods, the average error between the predicted minima and the subjects’ preferred step frequencies was less than 1% with a standard deviation between 4% and 5%. Using Instantaneous Cost Mapping, we were able to reduce subject walking-time from over an hour to less than 10 minutes. While, for a single parameter, the Instantaneous Cost Gradient Search is not much faster than Steady-State Cost Mapping, the Instantaneous Cost Gradient Search extends favorably to multi-dimensional parameter spaces

Examples of Response Surface Estimates of the Three Methods.

<p>Data are shown for Subject 1 and Subject 8 (poor and good performance of Instantaneous Cost methods respectively). This figure illustrates the third-order polynomial fit to the means (± SE) in the Steady-State Cost Mapping method, the third order polynomial response surface that best predicts the measures from the Instantaneous Cost Mapping, and the series of linear response surfaces used to update the parameter guess in the Instantaneous Cost Gradient Search (the lightening of shades indicates the sequence of the fits). The linear response surfaces of the Instantaneous Cost Gradient Search for Subject 1 show a much higher degree of variability than Subject 8. This led the algorithm to converge at a value somewhat below the subject’s preferred step frequency (worse than any other subject). Interestingly, the Instantaneous Cost Mapping for Subject 1 estimated a similarly low minimum. For subject 8, the linear response surfaces are more regular, leading the gradient search algorithm to outperform both of the mapping methods.</p

Convergence of the Instantaneous Cost Gradient Search Method.

<p>The lines indicate the prescribed step frequencies over the course of the experiment. The experiment begins with an incorrect guess of the optimal step frequency that is randomly selected to be either 20% above or below the subject’s preferred step frequency. After a warm-up period, the step frequency is perturbed above and below the guess. The measurements of oxygen consumption during this perturbation process are used to estimate the slope of the underlying relationship between energetic cost and step frequency. This slope, or gradient, is used to update the guess of the optimal step frequency. Though the performance of the algorithm varied between subjects, the algorithm was always able to approach the energetic minimum. The average total walking-time was 56.5 minutes and the mean relative error of the final iteration value to the subject’s preferred step frequency was −0.29% ± 4.77% (mean ± SD). The lightening of shades indicates the progress of time.</p

Subject-Specific Data.

<p>The subject’s preferred cadence (step frequency) was evaluated while on a treadmill without a metronome. “Time constant” refers to the time constant of the metabolic response used in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0135342#pone.0135342.e008" target="_blank">Eq (1)</a>. This was characterized with a step-change in energetic requirements during the first six minutes of the Instantaneous Cost Mapping trial.</p

Qualitative Comparison of the Three Methods.

<p>SSCM refers to the traditional method of Steady-State Cost Mapping. Our methods rely on the estimation of Instantaneous Cost. ICM refers to the Instantaneous Cost Mapping and ICGS to the Instantaneous Cost Gradient Search.</p

Graphical Representation of the Response Surface Identification.

<p>The relationship <i>x</i>(<i>p</i>) between a controller parameter <i>p</i> (in this work, step-frequency) and the physiological objective <i>x</i> (in this work, metabolic energetic cost) was replicated by a response surface </p><p></p><p></p><p></p><p><mi>x</mi><mo>‾</mo></p><p><mo stretchy="true">(</mo><mi>p</mi><mo>,</mo><mi>λ</mi><mo stretchy="true">)</mo></p><p></p><p></p><p></p> defined by a set of shape parameters <i><b>λ</b></i>. The shape parameters were identified to minimize the error between predicted respiratory response <p></p><p></p><p></p><p><mi>y</mi><mo>‾</mo></p><p></p><p></p><p></p> (based on a model of the measurement dynamics [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0135342#pone.0135342.ref020" target="_blank">20</a>]) and actual measures <p></p><p></p><p></p><p><mi>y</mi><mo>^</mo></p><p></p><p></p><p></p>. This dynamic estimation process enabled us to use non-steady-state breath-by-breath measures to approximate the relationship between the control parameter <i>p</i> and energetic cost <i>x</i>. Optimization was performed with respect to the response surface, <p></p><p></p><p></p><p><mi>x</mi><mo>‾</mo></p><p><mo stretchy="true">(</mo><mi>p</mi><mo>,</mo><mi>λ</mi><mo stretchy="true">)</mo></p><p></p><p></p><p></p>, that approximates the energy-parameter relationship.<p></p

Optimization Results.

<p>Listed are estimates of the energetic minima resulting from the different algorithms. Values are in percent-error with respect to the subject’s preferred step frequency. SSCM refers to a Steady-State Cost Mapping with a best-fit third-order polynomial. ICM refers to the minima of the Instantaneous Cost Mapping. ICGS refers to terminal values of the Instantaneous Cost Gradient Search.</p