Optimal Sensor Placement for Motion Tracking of Soft Wearables Using Bayesian Sampling

Abstract

Soft sensors integrated or attached to robots or human bodies enable rapid and accurate estimation of the physical states of the target systems, including position, orientation, and force. While the use of a number of sensors enhances precision and reliability in estimation, it may constrain the movement of the target system or make the entire system complex and bulky. This article proposes a rapid, efficient framework for determining where to place the sensors on the system given the limited number of available sensors. In particular, given $m$ candidates in location for sensor placement, the algorithm recommends $m_{0}$ locations that guarantee the maximal estimation performance, based on Bayesian sampling. The sampling and optimization method aims to maximize the log-likelihood in nonparametric regression between the measured values of the selected sensors and the target references. The proposed approach for the optimal sensor placement is validated through two scenarios: full-body motion sensing with a soft wearable sensor suit and fingertip position tracking with a motion-capture system. The proposed algorithm successfully determines the sensor locations close to the optimum within 20 min of learning for both cases.

Introduction

Wearable strain sensors in soft robotics take advantage of physical compliance of the materials or the structures that mainly compose the robots,^1–4 for safe and interactive applications with humans.^5–9 While various types of soft sensing mechanisms exist for essentially monitoring and estimating the shape or the pose of host systems, certain characteristics of soft robots, such as dexterity in manipulation, adaptability to environments, and user comfort, should be seriously considered in design to address the issue of practicality.^10–13

There are two representative ways of pose estimation of soft wearables: a vision-based pose tracking method and direct measurement of strain using attachable sensors. The vision-based method uses external cameras to track the position of markers attached to the wearables.^14,15 State-of-art motion capture systems show fast and accurate tracking performance. However, due to the use of multiple cameras, these systems are not portable and cannot be used for outdoor activities.

While the direct-strain sensing method makes the system more mobile and portable, a limited number of sensors should be used to minimize physical interference.^16–18 In this case, attaching sensors to the body is necessary for estimating the pose of the wearer. The more sensors in general ensure the higher accuracy in state estimation.^19–21 However, attaching a large number of sensors may constrain the movement of the wearer with increased number of signal wires and bulky circuits and consequently limit the range of motion. This presents a trade-off between accuracy and mobility.

In this study, we are interested in enhancing the performance of pose estimation in soft wearables with a limited number of attachable sensors. If the number of sensors that are allowed to use is limited, we must consider where to locate these sensors on the system, to maximize the estimation performance by collecting the meaningful output signals that are highly correlated to the task states.^22–24 For instance, to analyze the full-body motions of the person using a wearable sensing suit, the location of each soft sensor that measures the joint angles or the limb motions becomes one of the critical factors that determine the sensing performance of the system.²⁵

Problem definition

The mathematical formulation of this motivation can be written as follows: given $m$ possible sensor placement locations, the goal is to find the best $m_{0} (\leq m)$ locations. Indeed, this problem has the optimal solution given the fixed $m_{0}$ locations, since the number of possible combinations for selecting $m_{0}$ sensors from $m$ locations is fixed. For all possible sensor placement combinations, a nonlinear regression between the output signals from the selected sensors to the task objective (e.g., position trajectory) is available with its fitting performance (i.e., reconstruction error or log likelihood explaining the fitting). Therefore, the estimation accuracy can be evaluated for every possible case. However, as the numbers $m$ and $m_{0}$ increase, evaluating all possible cases becomes practically infeasible.

Related works

Previous approaches to select a limited number of sensors from possible locations are based on robot kinematics and model reduction. Ahn et al. used proper orthogonal decomposition with a linearized model to estimate the shape of an elastic chain.²⁶ Kim et al. used a piecewise constant curvature model together with a basis function to track the shape of continuum robots.²⁷ While these approaches benefit from model-based methods that eliminate the risk of nonexplainable optimization results, the models assume a local linear behavior²⁶ or combinations of linear basis functions.²⁷ Alternatively, Spielberg et al. used a point cloud-based feature extractor to learn sparsification of sensor distribution and considered a binary sensor probability weighting. This approach is purely data driven but requires heavy calculation cost of point cloud.²⁸

Contribution

We propose a data-driven method for optimal sensor placement, only relying on nonparametric model evaluation (i.e., Gaussian process regression [GPR]²⁹). Since the proposed method does not use a neural network-based learning model, it facilitates fast convergence to the suboptimal solution, ensuring robustness with a small amount of dataset.³⁰ A binary random variable is assigned to each candidate location for sensor placement, and the objective function then becomes to make only $m_{0}$ selected locations to be valued 1 that best explains the target output. To overcome sparsity problem and guarantee sufficient exploration where the case of $m_{0}$ is much smaller than $m$ , Thompson sampling with the beta distribution prior is used to substitute direct sampling from the Bernoulli distribution, representing the binary random variable assigned to each sensor.³¹ Moreover, a binary reward-based empirical loss and update rule is designed for fair sampling and weight update.

For validation, a sensor placement problem for capturing the full-body motions with a soft wearable sensing suit is presented.²⁵ A total of 20 soft strain sensors were strategically placed on various regions across the wearer’s body, and we were able to identify the optimal sensor locations, focusing only on 8 out of the 20 available candidates by employing the proposed algorithm.

For application, identifying the optimal locations of strain sensors to track 3D hand motions is presented. Hand motions were precisely measured using the changes in position of 29 markers affixed to the back of the subject’s hand. By measuring distances between each of two adjacent markers, a total of 36 distance information can be used as strain information on the back of the hand. Finally, we identified the optimal four locations for accurately estimating the specified hand motions.

Although previous approaches have shown that both model-based and model-free methods for sensor placement are available,^27,28 our study provides a solution to the sparsity problem using both nonparametric regression and Thompson sampling, and it enables a theoretical approach to the optimization process when compared with supervised learning that is trained as a complete black-box model. Finally, the validation results demonstrate that a rapid optimization process finds a combination of a limited number of sensor locations close to the optimum from a large number of possible candidates.

Materials and Methods

Problem definition

If the output signals from multiple sensors are collected to estimate the task state (e.g., position or orientation), a total of $m$ sensor signals at a single timestep are defined as a vector $x \in R^{m}$ , and the task state is given as $y \in R^{n}$ . When an ample amount of data is collected, a general calibration method, such as curve fitting or machine learning, can be applied to make an estimation function $f : R^{m} \to R^{n}$ . The datasets of the sensors and the task state can now be expressed with matrices $X \in R^{N \times m}$ and $Y \in R^{N \times n}$ , respectively, where $N$ is the size of the dataset. A likelihood $p_{f} (Y | X)$ explains how accurately the selected (or optimized) function approximation works. If the hyperparameters of the model are optimized, the quality of the data $X$ will determine the overall likelihood. For simplicity, $X$ and $Y$ are called the input and the output matrices, respectively, in our problem.

For the next step, $m_{0}$ dimensions are selected from $m$ dimensions of the input, which is equivalent to selecting $m_{0}$ sensor locations from possible $m$ candidates. The selection of the input is realized by defining a masking vector $z \in {0, 1}^{m}$ with an element-wise product $x_{0} = x \cdot z \in R^{m}$ . The likelihood for the selected input is given as $p_{f} (Y | z, X)$ with a constraint ${∥ z ∥}_{1} = m_{0}$ .

When an appropriate function approximation model is given (i.e., $p (Y | X)$ is explicitly given from the data), it is straightforward that an optimal $z$ can be always found among $(\begin{matrix} m \\ m_{0} \end{matrix})$ possible cases. However, such calculation contains factorial time complexity, which makes it impossible to solve the problem when $m$ becomes significantly large. Instead, an approximation model for selecting $z$ and attempting to optimize it can be a solution. To realize this, $z$ is assumed to be sampled from an $m$ -dimensional Bernoulli distribution $p (z | τ)$ as $p (z_{i} | τ_{i}) = τ_{i}^{z_{i}} {(1 - τ_{i})}^{1 - z_{i}},$ (1)where $z_{i}$ and $τ_{i}$ indicate the $i^{t h}$ components of $z$ and $τ$ , respectively. This means that the equality constraint ${∥ z ∥}_{1} = m_{0}$ is relaxed to the optimization constraint, in which the expectation of ${∥ z ∥}_{1} = m_{0}$ is equal to $m_{0}$ , given that $z$ is sampled from the posterior distribution of $z$ given $τ, X$ and $Y$ . To optimize the parameter $τ$ , we employ a beta distribution prior parameterized by $θ = [α, β]$ given as $p_{θ} (τ) = \frac{τ^{α - 1} {(1 - τ)}^{β - 1}}{\int_{0}^{1} x^{α - 1} {(1 - x)}^{β - 1} d x} .$ (2)

The overall selection process of $m_{0}$ inputs among the possible $m$ candidates is summarized in Figure 1. From the distribution variable $τ$ , the Bernoulli distribution $p (z | τ)$ is generated. $z$ is then sampled from the distribution, and the likelihood of $p_{f} (Y | z, X)$ can be calculated. As a result, the objective for the optimization to find the optimal distribution $p_{θ} (τ)$ is based on the maximum a posteriori (MAP) approach, in particular, finding the parameter $θ$ of the distribution $p_{θ} (τ)$ , which is given as $\begin{array}{l} θ^{*} = {argmax}_{θ} \log p (θ | Y, X), \\ s . t . E_{z \sim p (z | Y, X, τ)} (∥ z_{1} ∥) = m_{0} \end{array}$ (3)where the primary objective is to find the MAP solution, constrained by the masking parameter $z$ . The objective function can be reorganized into $\max_{θ} \log p (θ | Y, X) = \max_{θ} [\log p (Y | X, θ) + \log p (θ)]$ (4)using the Bayes’ rule, and the constraint in Equation (3) can be satisfied by setting the constraints on $z$ , which is sampled from the posterior distribution $p (z | Y, X, τ)$ . The marginal likelihood of $p (Y | X, θ)$ is given as $\begin{array}{l} \log p (Y | X, θ) = \log (\int ∫ p (Y, z, τ | X, θ) dzd τ) \\ = \log (\int ∫ p_{f} (Y | z, X) p (z | τ) p_{θ} (τ) dzd τ) . \end{array}$ (5)

Using Jensen’s inequality³² allows us to swap the log and the integral as $\log p (Y | X, θ) \geq \int ∫ q (z, τ) \log (\frac{p (Y, z, τ | X, θ)}{q (z, τ)}) d z d τ,$ (6)where a variational distribution $q (z, τ)$ substitutes $p (z | Y, X, τ)$ , satisfying the constraint given in Equation (3). The right-hand side of Equation (6) is the evidence lower bound of likelihood (ELBO) of the log likelihood, which can be decomposed as $ELBO = E_{z, τ \sim q (z, τ)} [log p_{f} (Y | z, X) + log p (z | τ) + log p_{θ} (τ) - log q (z, τ)] .$ (7)

FIG. 1.

An overview of the sensor placement framework. Sensor signal $X$ and reference trajectory $Y$ are the given dataset, and the objective is to find the sensor placement by determining binary variable $z$ , which is sampled from the Bernoulli distribution parameterized by $τ$ . The parameter $τ$ is determined from the prior distribution, which uses Thompson sampling.

Sampling process

The joint variational distribution $q (z, τ)$ is actually factorized into a product of two distributions: $q (z, τ) = q (τ) q (z | τ)$ (8)

This means that $z$ and $τ$ are hierarchically sampled from two separate distributions. Then, the variational distribution relevant to $τ$ can be substituted with the true distribution $p_{θ} (τ)$ . However, the distribution related to sampling of $z$ contains the constraint term. Therefore, the complex constraint term is removed from the optimization and replaced with a term that draws $z$ during the sampling process using $q (z | τ)$ . By assigning a value of 1 to $z$ corresponding to the top $m_{0}$ values in the sampled $τ$ values in order of magnitude, and assigning 0 to the rest, the norm ${∥ z ∥}_{1} = m_{0}$ is always fixed to $m_{0}$ . In other words, the variational distribution that samples $z$ is considered to be a deterministic distribution through the top $m_{0}$ sampling process. From this point of view, the ELBO relation can be written in a sampling form, in which a total of $N_{E}$ samples are used: $ELBO = \frac{1}{N_{E}} \sum_{i = 1}^{N_{E}} [log p_{f} (Y | z^{(i)}, X) + log p (z^{(i)} | τ^{(i)}) + log p_{θ} (τ^{(i)})],$ (9)where $τ^{(i)}$ and $z^{(i)}$ indicate $i^{th}$ sampled values from $τ^{(i)} \sim p_{θ} (τ) and z^{(i)} \sim Top m_{0} of τ^{(i)} .$ (10)

Optimization through Thompson sampling

Calculation of the ELBO consists of three trackable terms. The likelihood $p_{f} (Y | z, X)$ can be obtained from a dataset. The process of calculating the first term $p_{f} (Y | z, X)$ utilizes a dataset that was already prepared. When the masking vector $z$ is given, the values of the dimensions not selected in the top $m_{0}$ selection are all masked to be 0. Therefore, by using nonparametric regression, such as GPR, the likelihood can be easily calculated. The second term $p (z | τ)$ and the third term $p_{θ} (τ)$ can be calculated from Equations (1) and (2), respectively.

Updating both parameters through the gradient descent method will be difficult due to the constraints that each parameter possess. Therefore, adopting the updating method used in conjugate prior between $θ$ and $τ$ is considered here. From above, the ELBO is calculated using the mean value of $N_{E}$ samples. Each of these $N_{E}$ samples is compared with the mean value. The values that significantly deviate above or below the mean are then categorized as success or failure, respectively. This allows to transform the ELBO calculation result into a Bernoulli trial that satisfies the conjugate prior for the beta distribution. When examining the $z$ values for the samples categorized as either success or failure, $m_{0}$ dimensions are valued to be 1. For these dimensions, either $α$ or $β$ value is updated in the case of success and failure, respectively. The threshold for determining success or failure and the update values is heuristically determined. First, the statistical values for the calculated ELBO for each sampled $z$ are given as $\begin{array}{l} {ELBO}_{i} = \log p_{f} (Y | z^{(i)}, X) + \log p (z^{(i)} | τ^{(i)}) + \log p_{θ} (τ^{(i)}), \\ μ_{ELBO} = \frac{1}{N_{E}} \sum_{i = 1}^{N_{E}} {ELBO}_{i}, σ_{ELBO} = \sqrt{\frac{1}{N_{E}} \sum_{i = 1}^{N_{E}} {({ELBO}_{i} - μ_{ELBO})}^{2}}, \end{array}$ (11)and for each sampled $z$ , the labels can be assigned as $Class (i) = {\begin{matrix} 1, if {ELBO}_{i} \geq μ + 1.5 σ \\ - 1, if {ELBO}_{i} \leq μ - 1.5 σ \end{matrix}$ (12)where $μ$ and $σ$ are the values from Equation (11). Finally, the values for $α$ and $β$ are updated using the following rule: $\begin{array}{l} α \leftarrow γ α + \frac{λ}{N_{E}} \sum_{i = 1}^{N_{E}} z^{(i)} {ELBO}_{i} (Class (i) + 1), \\ β \leftarrow γ β + \frac{λ}{N_{E}} \sum_{i = 1}^{N_{E}} z^{(i)} {ELBO}_{i} (- Class (i) + 1) . \end{array}$ (13)where $γ < 1$ is the forgetting factor for avoiding overfitting or local minima, and $λ$ is the learning rate for adjusting the update speed.

Validation: Sensor placement for body motion sensing

Experimental setup

The performance of our sensor placement framework is validated through two application scenarios for soft strain sensor placement for motion sensing. First, we introduce the selection of sensor placements for a soft wearable suit, using the sensor signal data and the reference data obtained from the motion capture system (MOCAP), as shown in Figure 2.²⁵ The goal is to determine the optimal set of sensor locations from a wearable suit with 20 potential positions that can effectively track specific motions. A resistive-type soft strain sensor embedded with liquid-metal microchannels was placed at each candidate location.³³ In addition, motion capture markers (OptiTrack) were placed on several body locations to track the reference motions of the wearer. Both the resistance signals from the soft sensors and the location data of the human body from the MOCAP were collected simultaneously. The signals from the sensors became the input $X$ , and the actual motion coordinates obtained through the MOCAP were set to be the output $Y$ . We clarify that the input and the output data used in this validation experiment are from our previous work.²⁵ In this experiment, we selected three full-body motions for estimation: squat, bend and reach, and windmill.

FIG. 2.

Experimental setting of data collection process in.²⁰ A total of 20 soft strain sensors are placed at the whole-body sensing suit, which is marked red. Together, a total of 13 motion capture trackers are placed at the suit, which are marked blue.

At each timestep, the data from both the soft sensor and the MOCAP were retrieved and stored. The data were collected over T = 22,680 timesteps. With the 20 sensors used and the 3D position data obtained from 13 MOCAP trackers, the dimensions of the input and the output were $X \in R^{T \times 20}$ and $Y \in R^{T \times 39}$ , respectively.

When the value of $m_{0} = 8$ , which determines the masking vector $z$ selecting only a subset of the sensor signals, is assumed to be predefined, the input $X$ used for calibration was masked by calculating $X_{0} ≔ X diag (z)$ . The model calibrating $Y$ from the masked input $X_{0}$ utilized GPR with the squared-exponential kernel being employed. The hyperparameters of the kernel function and the noise level were tuned together during the optimization process.

Results

The result of the sensor selection process is shown in Figure 3. Among the 20 candidates, eight locations were selected by comparing the expected value of $τ$ after 200 optimization steps. Figure 3A shows the evolution of expectation of $τ$ from the initial step to 200 optimization steps. The expected value is based on the beta distribution parameters $α$ and $β$ , and the snapshot of the beta distribution based on these parameters at the $200^{t h}$ optimization step is given in Figure 3B. Overall, the selected eight sensor locations were {2,4,9,16,17,18,19,20}, as shown in Figure 3C, indicating the expectation of $τ$ after the optimization. Figure 4 shows the average values of the log likelihood of the training dataset for each optimization step, and the values gradually increased, meaning that the optimization process improved the selection strategy. This can be proved by the reconstruction error of the validation dataset, as shown in Figure 4. The actual tracking performance was visualized in Figure 5A, showing the position tracking result of three motion capture markers #6, #9, and #13. We also evaluated the tracking performance in the case of using all 20 available sensors and using the linear covariance prediction given in.²⁶ The performances of all 13 motion capture markers are listed in Table 1 and visualized in Figure 5B. A normalized root-mean-square error was calculated together with a t-test result between our method and using a linear method. The result indicates that our method performs better than the previous sensor placement approach²⁶ and a similar error level to that in the case of using all 20 available sensors.

FIG. 3.

Experimental result of sensor placement for tracking 13 motion capture markers. (A) Expected value of $τ$ over optimization iteration is given. At the initial phase, the expectation is initialized to 0.5, which guarantees fair exploration. (B) Beta distribution prior of each sensor location and $τ$ is sampled from this distribution. (C) Visualization of $E (τ)$ at the final iteration, and the colored bar indicates the selected eight sensors.

FIG. 4.

Average values of log-likelihood and normalized reconstruction error using the GPR after each optimization step. The log-likelihood is calculated from the training dataset, and the reconstruction error is calculated from the validation dataset. The bold lines indicate the average values of five trials. GPR, Gaussian process regression.

FIG. 5.

(A) Reconstruction result of motion tracking of three markers: markers 6, 9, and 13. A three-dimensional position is given with its ground truth, prediction using eight selected sensors, comparison for using linear method,²¹ and prediction using all of 20 sensors. Both ground truth and prediction values are normalized. (B) Normalized reconstruction error over total 13 markers using all sensors, our method, and the linear method (**p < 0.01, ***p < 0.001).

Table 1.

Normalized Reconstruction Error for Motion Tracking of Various Markers (Normalized Root-Mean-Square Error, %)

CASE Marker	Linear (21)	Ours	Use all sensors
#1	5.533	5.404	4.948
#2	6.385	5.479	5.088
#3	5.403	5.310	4.979
#4	6.528	5.244	4.814
#5	6.092	5.115	3.627
#6	6.158	5.486	4.302
#7	4.874	4.684	4.783
#8	4.826	4.751	4.747
#9	5.647	5.310	4.785
#10	3.718	3.092	3.314
#11	2.761	3.094	3.303
#12	3.938	2.860	3.652
#13	3.200	2.954	3.617

To verify the effectiveness of the proposed algorithm, we obtained 2000 samples of $z$ that contain randomly selected eight sensors, and we calculated the reconstruction error likewise in Figure 4. The results of sorting the calculated reconstruction errors of the 2000 samples in the ascending order are shown in Figure 6A. Additionally, the 2000 samples were divided into five groups of 400 samples each, and the frequency of sensor occurrence for the $z$ samples in each group is shown in the histogram in Figure 6B. Since eight sensors were selected for each sample and there are 400 samples in a single group, approximately 160 samples would be selected for each sensor if the sorting was not performed in the ascending order. However, looking at the histogram of the first group with the lowest reconstruction error, it can be seen that the eight sensors, selected by our algorithm, were indeed chosen with a high frequency. To understand the correlation between the frequency of sensor occurrences and $E [τ]$ shown in Figure 3C, the scatter plot, showing the frequency of sensor occurrences and the values of $E [τ]$ for each group, is presented in Figure 6C. Moreover, the Pearson correlation coefficient calculated for each sample is shown in Figure 6A. The coefficient shows a higher value when the reconstruction error was lower valued, proving that our algorithm effectively selected sensors that produced high-accuracy results.

FIG. 6.

(A) Reconstruction error of the randomly selected $z$ values, which is sorted by the ascending order (blue line). Pearson correlation coefficient between $E [τ]$ and the appearance rate is shown in purple line. (b) Five histograms show the frequency of occurrence of each of the 20 sensors in the $z$ values of the sorted samples in ascending order, displayed for every 400 samples. The ideal frequency of randomly selected case is 160, indicated by the dotted line. (c) Scatter plot and linear regression line of the $E [τ]$ and the appearance rate for each 400 samples are given.

Application: Sensor placement for finger tracking

Based on the validation of the framework using dataset from previous work, we show the practical usage of the sensor placement strategy for tracking 3D fingertip motions. In this scenario, we aim to find the optimal locations to which soft strain sensors can be attached to the back of the hand to predict the tip position of two fingers. While we used the data from the 20 soft sensors that were already placed in the candidate locations to collect the input data $X$ in the previous validation experiment, in this experiment, we demonstrate how the optimal locations can be selected using only MOCAP data.

A total of 29 MOCAP markers (Facial marker 3 mm, OptiTrack) were attached to the fingers and the back of the hand, as shown in Figure 7. The nonuniform spacing of the markers was intended to avoid the interference between the markers so that the tracking processor can easily separate their signals. The absolute coordinate positions of these markers correspond to the output $Y$ . Among these output data, the location of a specific marker was targeted for estimation. Unlike the previous method where the sensors were already placed at all possible locations and the data were collected, a pair of two adjacent markers were considered as if a virtual strain sensor was attached. The distance between these adjacent markers was defined as the input $X$ , and 36 virtual sensor positions were obtained, as indicated by the dotted lines in Figure 7.

FIG. 7.

Experimental setting of fingertip motion tracking. A total of 29 motion tracking markers are placed on the left hand, which is considered an output. The inputs are determined by the distance between two adjacent markers.

The distances between the adjacent markers and the absolute coordinate position of each marker were set as the input and the output, respectively, and the results of the experiment to determine the optimal sensor locations are summarized in Figure 8. Similar to the result of Figure 3, Figures 8A–D show the changes in the expected values of $τ$ for each location over the optimization timestep, the beta distribution in the final optimization step, the changes in the log likelihood over the timestep, and the results derived from the top five locations with the highest expected $τ$ values. These results pertain to the experiment for estimating the tip position of both the thumb and the index finger. The recommended sensor locations for estimating the thumb, the index finger, and both of them together are shown in Figure 8E. Figure 8F presents the verification results for all combinations of the locations 1 to 21 regarding the recommendation for two fingers (total of 20,349 cases). The combinations including the locations 22–36 were omitted due to the heavy time complexity of verifying all possible combinations, and they were far from the target finger positions.

FIG. 8.

Experimental result of sensor placement for fingertip motion tracking, in which (A) through (D) is the result of tracking both thumb and index finger. (A) Expected value of $τ$ over optimization iteration is given. At the initial phase, the expectation is initialized to 0.5. (B) Beta distribution prior of each sensor location, and $τ$ is sampled from this distribution. (C) An average value of log-likelihood of sampled $z$ for after each optimization step. (D) Visualization of $E (τ)$ at the final iteration. (E) Selected sensor locations of various fingertip tracking targets are given. For both thumb and index finger, {4, 10, 11, 12, 15} is chosen. (F) Validation for all possible cases was given.

The five-sensor locations recommended by the proposed algorithm were 4, 10, 11, 12, and 15. The selected combination by the algorithm, based on the log-likelihood during verification, was the fifth among all possible $2529^{t h}$ combinations, as shown in Figure 8E. The combination that ranked the first was the 13,727^th combination 4, 10, 12, 15, and 16, suggesting that the performance of selecting the location 16 was slightly better than the location 11. While the best-performing combination among all possible cases was not exactly identified, the ability to quickly find a combination showing a good enough performance without evaluating all possible cases was demonstrated.

Conclusion

In this study, we proposed an algorithm to find the optimal locations for placing a fixed number of sensors to maximize target estimation accuracy. Although evaluating all possible combinations would guarantee the optimal solution, it is impractical due to the significant computational resources required. Moreover, evaluating individual sensors fails to capture interactive effects, and direct optimization during the iterative process through sampling is both time-consuming and yields the results with a reduced performance, as demonstrated by our experimental results. To address these challenges, we introduced Thompson sampling, a method widely used solving solving Bandit problems. By adapting Thompson sampling to our specific domain, the proposed algorithm effectively enhanced the performance in terms of both speed and optimality.

Previous sensor optimization methods in soft robotics have often relied on assumptions of linearity or focused primarily on task-specific problems. In contrast, our approach finds a solution through mathematical modeling by selecting $m_{0}$ out of $m$ possible sensor attachment locations to estimate the $n$ -dimensional tracking objective. This advantage allows for a generalization to a wide range of soft sensor-related research, as it is not limited to specific applications. In future research related to sensor optimization, we anticipate that the proposed method can be further developed by taking into account the system’s dynamics or considering the observability. In addition, determining the optimal value of $m_{0}$ considering practicality and efficiency should be considered.

Footnotes

Authors’ Contributions

K.D. proposed the overall research, developed the sensor placement framework, and wrote the article. S.K. conducted the experiments and performed numerical analysis. Y.-L.P. directed the overall research and organized and wrote the article.

Data Availability

Source codes used in this study are available at github.com/mochacoco/osp. Detailed experimental results and theoretical approach for direct optimization and real-time demonstration videos are given in e-pdf:

Author Disclosure Statement

The authors declare no competing financial or nonfinancial interests.

Funding Information

This work was supported in part by the National Research Foundation of Korea (Grants RS-2021-NR059648 and RS-2023-00208052) and in part by the Institute of Information & Communications Technology Planning & Evaluation grant funded by the Korea Government (Grant 2021-0-00896).

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