# Comparison of regression models for estimation of isometric wrist joint torques using surface electromyography

- Amirreza Ziai
^{1}and - Carlo Menon
^{1}Email author

**8**:56

https://doi.org/10.1186/1743-0003-8-56

© Ziai and Menon; licensee BioMed Central Ltd. 2011

**Received: **27 April 2011

**Accepted: **26 September 2011

**Published: **26 September 2011

## Abstract

### Background

Several regression models have been proposed for estimation of isometric joint torque using surface electromyography (SEMG) signals. Common issues related to torque estimation models are degradation of model accuracy with passage of time, electrode displacement, and alteration of limb posture. This work compares the performance of the most commonly used regression models under these circumstances, in order to assist researchers with identifying the most appropriate model for a specific biomedical application.

### Methods

Eleven healthy volunteers participated in this study. A custom-built rig, equipped with a torque sensor, was used to measure isometric torque as each volunteer flexed and extended his wrist. SEMG signals from eight forearm muscles, in addition to wrist joint torque data were gathered during the experiment. Additional data were gathered one hour and twenty-four hours following the completion of the first data gathering session, for the purpose of evaluating the effects of passage of time and electrode displacement on accuracy of models. Acquired SEMG signals were filtered, rectified, normalized and then fed to models for training.

### Results

It was shown that mean adjusted coefficient of determination $\left({\mathsf{\text{R}}}_{\mathsf{\text{a}}}^{2}\right)$ values decrease between 20%-35% for different models after one hour while altering arm posture decreased mean ${\mathsf{\text{R}}}_{\mathsf{\text{a}}}^{2}$ values between 64% to 74% for different models.

### Conclusions

Model estimation accuracy drops significantly with passage of time, electrode displacement, and alteration of limb posture. Therefore model retraining is crucial for preserving estimation accuracy. Data resampling can significantly reduce model training time without losing estimation accuracy. Among the models compared, ordinary least squares linear regression model (OLS) was shown to have high isometric torque estimation accuracy combined with very short training times.

## Background

SEMG is a well-established technique to non-invasively record the electrical activity produced by muscles. Signals recorded at the surface of the skin are picked up from all the active motor units in the vicinity of the electrode [1]. Due to the convenience of signal acquisition from the surface of the skin, SEMG signals have been used for controlling prosthetics and assistive devices [2–7], speech recognition systems [8], and also as a diagnostic tool for neuromuscular diseases [9].

However, analysis of SEMG signals is complicated due to nonlinear behaviour of muscles [10], as well as several other factors. First, cross talk between the adjacent muscles complicates recording signals from a muscle in isolation [11]. Second, signal behaviour is very sensitive to the position of electrodes [12]. Moreover, even with a fixed electrode position, altering limb positions have been shown to have substantial impact on SEMG signals [13]. Other issues, such as inherent noise in signal acquisition equipment, ambient noise, skin temperature, and motion artefact can potentially deteriorate signal quality [14, 15].

The aforementioned issues necessitate utilization of signal processing and statistical modeling for estimation of muscle forces and joint torques based on SEMG signals. Classification [16] and regression techniques [17, 18], as well as physiological models [19, 20], have been considered by the research community extensively. Machine learning classification methods in aggregate have proven to be viable methods for classifying limb postures [21] and joint torque levels [22]. Park et al. [23] compared the performance of a Hill-based muscle model, linear regression and artificial neural networks for estimation of thumb-tip forces under four different configurations. In another study, performance of a Hill-based physiological muscle model was compared to a neural network for estimation of forearm flexion and extension joint torques [24]. Both groups showed that neural network predictions are superior to Hill-based predictions, but neural network predictions are task specific and require ample training before usage. Castellini et al. [22] and Yang et al. [25], in two distinct studies, estimated grasping forces using artificial neural networks (ANN), support vectors machines (SVM) and locally weighted projection regression (LWPR). Yang concluded that SVM outperforms ANN and LWPR.

This study was intended to compare performance of commonly utilized regression models for isometric torque estimation and identify their merits and shortcomings under circumstances where the accuracy of predictive models has been reported to be compromised. Wrist joint was chosen as its functionality is frequently impaired due to aging [26] or stroke [7], and robots (controlled by SEMG signals) are developed to train and assist affected patients [2, 3]. Performance of five different models for estimation of isometric wrist flexion and extension torques are compared: a physiological based model (PBM), an ordinary least squares linear regression model (OLS), a regularized least squares linear regression model (RLS), and three machine learning techniques, namely SVM, ANN, and LWPR.

### Physiological Based Model

where e_{j} is the processed SEMG signal of muscle j at time t, d is the electromechanical delay, α is the gain coefficient, u_{j}(t) is the post-processed SEMG signal at time t, and β_{1} and β_{2} the recursive coefficients for muscle j.

_{j}(t) for muscle j into post-processed values u

_{j}(t). Stability of this equation is ensured by satisfying the following constraints [32]:

_{j}(t) values to oscillate or even go to infinity. To ensure stability of this filter and restrict the maximum neural activation values to one, another constraint is imposed:

Neural activation values are conventionally restricted to values between zero and one, where zero implies no activation and one translates to full voluntary activation of the muscle.

where A is called the non-linear shape factor. A = -3 corresponds to highly exponential behaviour of the muscle and A = 0 corresponds to a linear one.

where F_{max,j} is the maximum voluntary force produced by muscle j.

where MA_{j} is moment arm at neutral wrist position for muscle j and τ_{j}(t) is the torque generated by muscle j at time t. Moment arms for flexors and extensors were assigned positive and negative signs respectively to maintain consistency with measured values.

where M is the number of muscles used in the model, and ΣPCSA_{j} is the summation of PCSA of the muscle represents by muscle j and PCSA of muscle j itself.

_{max}to one standard deviation of the reported values. Initial values for moment arms were fixed to the mean values in [43], and constrained to one standard deviation of the values reported in the same reference. Since these parameters are constrained within their physiologically acceptable values, calibrated models can potentially provide physiological insight [24]. Activation parameters A, C

_{1}, C

_{2}, and d were assumed to be constant for all muscles a model with too many parameters loses its predictive power due to overfitting [44]. Parameters x = {A, C

_{1}, C

_{2}, d, F

_{max,1}, ..., F

_{max,M}, MA

_{1}, MA

_{2}, ..., MA

_{M}} were tuned by optimizing the following objective function while constraining parameters to values mentioned beforehand:

Models were optimized by Genetic Algorithms (GA) using MATLAB Global Optimization Toolbox (details of GA implementation are available in [45]). GA has previously been used for tuning muscle models [20]. Default MATLAB GA parameters were used and models were tuned in less than 100 generations (MATLAB default value for the number of optimization iterations) [46].

### Ordinary Least Squares Linear Regression Model

where N is the number of samples considered (observations), M is the number of muscles, τ_{m} is a vector of measured torque values, SEMG is a matrix of processed SEMG signals, β is a vector of regression coefficients to be estimated, and ε is a vector of independent random variables.

_{e}) as shown:

### Regularized Least Squares Linear Regression Model

_{1}-regularized least squares (RLS) method for estimation of regression coefficients is known to overcome some of the common issues associated with the ordinary least squares method [48]. Estimated vector of regression coefficients using ℓ

_{1}-regularized least squares method $\left(\widehat{\beta}\right)$ is computed through the following optimization:

where *λ* ≥ 0 is the regularization parameter which is usually set equal to 0.01 [49, 50].

We used the Matlab implementation of the ℓ_{1}-regularized least squares method [51].

### Support Vector Machines

Support vectors machines (SVM) are machine learning methods used for classification and regression. Support vector regression (SVR) maps input data using a non-linear mapping to a higher-dimensional feature space where linear regression can be applied. Unlike neural networks, SVR does not suffer from the local minima problem since model parameter estimation involves solving a convex optimization problem [52].

We used epsilon support vector regression (ε-SVR) available in the LibSVM tool for Matlab [53]. Details of ε-SVR problem formulation are available in [54]. ε-SVR has previously been utilized for estimation of grasp forces [22, 25]. The Gaussian kernel was used as it enables nonlinear mapping of samples and has a low number of hyperparameters, which reduces complexity of model selection [55]. Eight-fold cross-validation to generalize error values and grid-search for finding the optimal values of hyperparameters C, γ and ε were carried out for each model.

### Artificial Neural Networks

_{e}is the estimated torque value.

ANN models were trained using Matlab Neural Network Toolbox. Hyperbolic tangent sigmoid activation functions were used to capture the nonlinearities of SEMG signals. For each model, the training phase was repeated ten times and the best model was picked out of those repetitions in order to overcome the local minima problem [52]. We also used early stopping and regularization in order to improve generalization and reduce the likelihood of overfitting [61].

### Locally Weighted Projection Regression

Locally Weighted Projection Regression (LWPR) is a nonlinear regression method for high-dimensional spaces with redundant and irrelevant input dimensions [62]. LWPR employs nonparametric regression with locally linear models based on the assumption that high dimensional data sets have locally low dimensional distributions. However piecewise linear modeling utilized in this method is computationally expensive with high dimensional data.

We used Radial Basis Function (RBF) kernel and meta-learning and then performed an eight-fold cross validation to avoid overfitting. Finally we used grid search to find the initial values of the distance metric for receptive fields, as it is customary in the literature [22, 25]. Models were trained using a Matlab version of LWPR [63].

## Methods

A Transducer Techniques TRX-100 torque sensor, with an axis of rotation corresponding to that of the volunteer's wrist joint, was used to measure torques applied about the wrist axis of rotation. Volunteer's forearm was secured to the rig using two Velcro straps. This design allowed restriction of arm movements. Volunteer placed their elbow on the rig and assumed an upright position.

### Protocol

Actions and repetitions for protocols.

Repetition | Action |
---|---|

1 | Wrist flexion with maximum torque |

1 | Wrist extension with maximum torque |

3 | Gradual wrist flexion until 50% MVC and gradual decrease to zero |

3 | Gradual wrist extension until 50% MVC and gradual decrease to zero |

3 | Gradual wrist flexion until 25% MVC and gradual decrease to zero |

3 | Gradual wrist extension until 25% MVC and gradual decrease to zero |

In order to capture the effects of passage of time on model accuracy, volunteers were asked to repeat the same session after one hour. This session was named session two. Electrodes were not detached in between the two sessions. After completion of session two, electrodes were removed from the volunteer's skin. The volunteer was asked to repeat another session in twenty four hours following session two while attaching new electrodes. This was intended to capture the effects of electrode displacement and further time passage.

Each volunteer was asked to supinate her/his forearm and exert isometric torques on the rig following the same protocol used before after completion of session 1. This was intended to study the effects of arm posture on model accuracy.

### SEMG Acquisition

Muscles monitored using SEMG.

Channel | Muscle | Action |
---|---|---|

1 | Extensor Carpi Radialis Longus (ECRL) | Wrist extension Radial deviation |

2 | Extensor Digitorum Communis (EDC) | Wrist extension Four fingers extension |

3 | Extensor Carpi Ulnaris (ECU) | Wrist extension Ulnar deviation |

4 | Extensor Carpi Radialis Brevis (ECRB) | Wrist extension Wrist abductor |

5 | Flexor Carpi Radialis (FCR) | Wrist flexion Radial deviation |

6 | Palmaris Longus (PL) | Wrist flexion |

7 | Flexor Digitorum Superficialis (FDS) | Wrist flexion |

8 | Flexor Carpi Ulnaris (FCU) | Wrist flexion Ulnar deviation |

SEMG signals were acquired at 1 kHz using a National Instruments (NI-USB-6289) data acquisition card. An application was developed using LabVIEW software that stored data on a computer and provided visual feedback for volunteers. Visual feedback consisted of a bar chart that visualized the magnitude of exerted torques, which aided volunteers to follow the protocol more accurately.

### Signal Processing

## Results and Discussion

where τ_{e}(i) is the estimated and τ_{m}(i) is the measured torque value for sample i, n corresponds to the total number of samples tested, and τ_{m,flex} and τ_{m,ext} are the maximum flexion and extension torques exerted by each volunteer. The absolute value of τ_{m,ext} is considered because of the negative sign assigned to extension torque values during signal acquisition.

^{2}is a measure of the percentage of variation in the dependant variable (torque) collectively explained by the independent variables (SEMG signals):

where $\overline{{\tau}_{\mathsf{\text{m}}}}$ is the mean measured torque.

^{2}has a tendency to overestimate the regression as more independent variables are added to the model. For this reason, many researchers recommend adjusting R

^{2}for the number of independent variables:

where ${\mathsf{\text{R}}}_{\mathsf{\text{a}}}^{2}$ is the adjusted R^{2}, n is the number of samples and k is the number of SEMG channels.

^{®}Core™2 Duo 2.5 GHz processor and 6 GB of RAM. Table 3 compares mean training times for models trained using the original and resampled data sets.

Model training times for original and resampled data sets.

Time (s) | PBM | OLS | RLS | SVM | ANN | LWPR |
---|---|---|---|---|---|---|

Original | 1,080.07 | 0.01 | 1.98 | 19,125.31 | 166.73 | 5,195.03 |

Resampled | 10.96 | 0.00 | 0.03 | 15.32 | 9.40 | 18.63 |

One-way Analysis of Variance (ANOVA) failed to reject the null hypothesis that NRMSE and ${\mathsf{\text{R}}}_{\mathsf{\text{a}}}^{2}$ have different mean values for each model, meaning that the difference between means is not significant (with minimum P-value of 0.95). We used reduced data sets with data resampled every 100 samples for the rest of the study.

### Number of Muscles

Comparison of joint torque estimation for models trained with two, five, and eight SEMG channels.

Model | 8 channels | 5 channels | 2 channels | ||||
---|---|---|---|---|---|---|---|

NRMSE | ${R}_{a}^{2}$ | NRMSE | ${R}_{a}^{2}$ | NRMSE | ${R}_{a}^{2}$ | ||

PBM | Mean | 2.73% | 0.85 | 3.07% | 0.86 | 4.59% | 0.77 |

STD | 0.97% | 0.13 | 1.03% | 0.11 | 1.32% | 0.19 | |

OLS | Mean | 2.88% | 0.84 | 3.17% | 0.77 | 4.82% | 0.63 |

STD | 0.94% | 0.11 | 1.06% | 0.13 | 1.81% | 0.23 | |

RLS | Mean | 2.83% | 0.82 | 3.11% | 0.79 | 4.73% | 0.69 |

STD | 0.93% | 0.10 | 1.01 | 0.11 | 1.31% | 0.18 | |

SVM | Mean | 2.85% | 0.82 | 3.00% | 0.80 | 4.77% | 0.73 |

STD | 1.00% | 0.09 | 1.04% | 0.10 | 1.02% | 0.14 | |

ANN | Mean | 2.82% | 0.82 | 3.03% | 0.81 | 4.74% | 0.69 |

STD | 0.95% | 0.09 | 1.05% | 0.12 | 1.17% | 0.18 | |

LWPR | Mean | 3.03% | 0.75 | 3.19% | 0.78 | 4.97% | 0.69 |

STD | 1.14% | 0.21 | 1.19% | 0.13 | 1.31% | 0.21 |

This result appears to be in contrast to the results obtained by Delp et al. [66] where extrinsic muscles of the hand are expected to contribute substantially to torque generation. However, due to the design of our testing rig, volunteers only generated torque by pushing their palms against the torque-sensing plate and their fingers did not contribute to torque generation. Therefore the addition of SEMG signals of extrinsic muscles to the model did not result in a significant increase in accuracy.

Comparison of training data set size on joint torque estimation.

Model | 25% training | 90% training | |||
---|---|---|---|---|---|

NRMSE | ${R}_{a}^{2}$ | NRMSE | ${R}_{a}^{2}$ | ||

PBM | Mean | 4.41% | 0.81 | 2.32% | 0.96 |

STD | 2.49% | 0.09 | 0.59% | 0.04 | |

OLS | Mean | 4.19% | 0.80 | 2.19% | 0.97 |

STD | 2.19% | 0.10 | 0.58% | 0.04 | |

RLS | Mean | 4.14% | 0.82 | 2.07% | 0.97 |

STD | 2.13% | 0.08 | 0.51% | 0.03 | |

SVM | Mean | 4.39% | 0.85 | 2.02% | 0.97 |

STD | 2.46% | 0.09 | 0.92% | 0.03 | |

ANN | Mean | 5.87% | 0.73 | 2.34% | 0.96 |

STD | 2.20% | 0.20 | 0.61% | 0.03 | |

LWPR | Mean | 6.41% | 0.69 | 3.43% | 0.87 |

STD | 3.14% | 0.29 | 0.84% | 0.07 |

where ΣPCSA_{flexors} is the summation of PCSA of all flexor muscles, ΣPCSA_{extensors} is the summation of PCSA of all extensor muscles, PCSA_{flexor} is the PCSA of the flexor muscle used for training, PCSA_{extensor} is the PCSA of the extensor muscle used for training, τ_{flexor}(t) is the torque of the flexor muscle used for training at time t, and τ_{extensor}(t) is the torque of the flexor muscle used for training at time t.

Similarly PBM training with the five primary wrist muscles was carried out with modified ΣPCSA terms. Half of the summation of PCSA values for non-primary flexors was added to each of the two primary flexors while a third of the summation of PCSA values for non-primary extensors was added to the ΣPCSA term of each of the three primary extensors.

These modifications allowed tuned parameters to stay within their physiologically acceptable values, even though less SEMG channels were used for training models.

### Cross Session

Effects of passage of time and electrode displacement on joint torque estimation.

Model | After 1 hour | After 24 hours | |||
---|---|---|---|---|---|

NRMSE | ${R}_{a}^{2}$ | NRMSE | ${R}_{a}^{2}$ | ||

PBM | Mean | 5.28% | 0.56 | 5.54% | 0.47 |

STD | 2.68% | 0.24 | 2.95% | 0.26 | |

OLS | Mean | 4.84% | 0.59 | 5.29% | 0.51 |

STD | 2.98% | 0.27 | 3.04% | 0.25 | |

RLS | Mean | 4.81% | 0.63 | 5.19% | 0.54 |

STD | 2.91% | 0.23 | 2.98% | 0.27 | |

SVM | Mean | 5.35% | 0.54 | 6.76% | 0.46 |

STD | 2.22% | 0.21 | 2.95% | 0.28 | |

ANN | Mean | 5.40% | 0.53 | 6.44% | 0.51 |

STD | 2.15% | 0.28 | 3.09% | 0.31 | |

LWPR | Mean | 5.42% | 0.60 | 5.93% | 0.59 |

STD | 3.00% | 0.23 | 3.18% | 0.30 |

Mean ${\mathsf{\text{R}}}_{\mathsf{\text{a}}}^{2}$ values after one hour decreased 34%, 28%, 25%, 34%, 35%, and 20% while mean NRMSE decreased 93%, 68%, 70%, 88%, 91%, and 79% for PBM, OLS, RLS, SVM, ANN, and LWPR, respectively. After twenty four hours mean NRMSE values decreased further. High standard deviations of NRMSE and ${\mathsf{\text{R}}}_{\mathsf{\text{a}}}^{2}$ values suggest unreliability of model predictions with passage of time and electrode displacement. Therefore it is crucial for models trained using SEMG signals to be retrained frequently regardless of the model utilized.

### Arm Posture

Effects of forearm supination on joint torque estimation.

Model | NRMSE | ${R}_{a}^{2}$ | |
---|---|---|---|

PBM | Mean | 9.55% | 0.22 |

STD | 5.69% | 0.32 | |

OLS | Mean | 8.93% | 0.25 |

STD | 5.37% | 0.33 | |

RLS | Mean | 8.86% | 0.23 |

STD | 5.30% | 0.29 | |

SVM | Mean | 8.65% | 0.24 |

STD | 4.47% | 0.37 | |

ANN | Mean | 9.13% | 0.23 |

STD | 4.76% | 0.36 | |

LWPR | Mean | 10.05% | 0.25 |

STD | 5.49% | 0.30 |

Comparison of models investigated.

Criteria | PBM | OLS | RLS | SVM | ANN | LWPR |
---|---|---|---|---|---|---|

Least training time | * | |||||

Physiological insight | * | |||||

Does not require SEMG processing | * | * | * | |||

Supination sensitivity | * | * | * | * | * | * |

Time passage sensitivity | * | * | * | * | * | * |

Electrode placement sensitivity | * | * | * | * | * | * |

## Conclusions

Eleven volunteers participated in this study. During the first session, 33,520 samples from eight SEMG channels and a torque sensor were acquired while volunteers followed a protocol consisting of isometric flexion and extension of the wrist. We then processed SEMG signals and resampled every 100 samples to save model training time. Subsequently we trained models using identical training data sets. When using 90% of data as training data set and the rest of the data as testing data, we attained ${\mathsf{\text{R}}}_{\mathsf{\text{a}}}^{2}$ values of 0.96 ± 0.04, 0.97 ± 0.04, 0.97 ± 0.03, 0.97 ± 0.03, 0.96 ± 3, and 0.87 ± 0.07 for PBM, OLS, RLS, SVM, ANN, and LWPR respectively. All models performed in a very comparable fashion, except for LWPR that yielded lower ${\mathsf{\text{R}}}_{\mathsf{\text{a}}}^{2}$ values and higher NRMSE values.

Models trained using the data set from session one were tested using two separate data sets gathered one hour and twenty four hours following session one. We showed that Mean ${\mathsf{\text{R}}}_{\mathsf{\text{a}}}^{2}$ values after one hour decrease 34%, 28%, 25%, 34%, 35%, and 20% for PBM, OLS, RLS, SVM, ANN, and LWPR, respectively. Tests after twenty four hours showed even further performance deterioration. Therefore it was concluded that all models considered in this study are sensitive to passage of time and electrode displacement.

The effects of the number of SEMG channels used for training were explored. Models trained with SEMG channels from the five primary forearm muscles were shown to be of similar predictive ability compared to models trained with all eight SEMG channels. However, models trained with two SMEG channels resulted in predictions with lower ${\mathsf{\text{R}}}_{\mathsf{\text{a}}}^{2}$ and higher NRMSE values.

Finally models trained with forearm in a pronated position were tested with data gathered from forearm in the supinated position. Mean NRMSE values increased 2.50, 2.10, 2.13, 2.04, 2.24, and 2.32 times for PBM, OLS, RLS, SVM, ANN, and LWPR.

The substantial decrease in predictive ability of all models with passage of time, electrode displacement, and altering arm posture necessitates regular retraining of models in order to sustain estimation accuracy. We showed that resampling the data set substantially reduces the training time without sacrificing estimation accuracy of models. All models were trained in under 20 seconds while OLS was trained in under 10 ms. Low training times achieved in this work render regular retraining feasible.

## Declarations

### Acknowledgements

Authors would like to thank the reviewers for their valuable suggestions, Zeeshan O Khokhar and Zhen G Xiao for designing the test rig, and Mojgan Tavakolan and Farzad Khosrow-Khawar for providing insight on machine learning techniques. This work is supported by the Canadian Institutes of Health Research (CIHR), the BC Network for Aging Research (BCNAR), and the Natural Sciences and Engineering Research Council of Canada (NSERC).

## Authors’ Affiliations

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