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GMM-based Single-joint Angle Estimation usingEMG signals

Stefano Michieletto1, Luca Tonin1 Mauro Antonello1, Roberto Bortoletto1,Fabiola Spolaor2, Enrico Pagello1, and Emanuele Menegatti1

1 Intelligent Autonomous Systems Lab (IAS-Lab)2 Movement Analysis Laboratory

Department of Information Engineering (DEI)University of Padova

{michieletto,luca.tonin,antonelm,bortolet,fspolaor,epv,emg}@dei.unipd.it

http://robotics.dei.unipd.it

Abstract. This paper aims to explore the possibility to use Electromyography(EMG) to train a Gaussian Mixture Model (GMM) in order to estimatethe bending angle of a single human joint. In particular, EMG signalsfrom eight leg muscles and the knee joint angle are acquired during akick task from three different subjects. GMM is validated on new unseendata and the classification performances are compared with respect tothe number of EMG channels and the number of collected trials usedduring the training phase. Achieved results show that our framework isable to obtain high performances even using few EMG channels and witha small training dataset(Normalized Mean Square Error: 0.96, 0.98, 0.98for the three subjects, respectively), opening new and interesting per-spectives for the hybrid control of humanoid robots and exoskeletons.

Keywords: EMG signals, Gaussian Mixture Model, Gaussian MixtureRegression, Single-joint Angle Estimation

1 Introduction

The past two decades have seen growing interest in controlling robotic devicesby means of physiological human signals (i.e., Electroencephalogram (EEG) [22,23,30] and Electromyography (EMG) [2,3,16,17]). The main motivations for thishuman-inspired control modality are twofold: on one hand to catch the humanintention and translate them in actual action of external actuators (i.e., exoskele-ton, hand/arm rehabilitation devices, car assistive breaking); on the other hand,to extract specific information and patterns from the human muscular-skeletonstructure and movements to make the behavior of humanoid autonomous robotsmore natural, robust and efficient.

In this scenario, several studies have been conducted in order to model andcontrol movements of the upper limb [2, 3, 12–14, 17]. These works grounded onstandard signal processing techniques to extract useful features from EMG sig-nals (i.e., time-frequency analysis, wavelet transform) and on different machine

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learning algorithms (i.e., Neural Networks, Gaussian Mixture Model) to predictand replicate complex human movements [7, 10, 15, 24]. The results achieved bydifferent groups have shown high classification and prediction accuracy bothin simulated environments [3, 13] and in real-time control of prosthetic de-vices [2, 12, 14, 17]. Nevertheless, only few studies have focused on modelingand predicting lower limb related tasks by means of EMG signals [19,27] and onusing them as direct control input for robotic actuators [16].

This paper aims to explore the use of a Gaussian Mixture Model (GMM)for estimation of single-joint angle, and, in particular the angular aperture ofthe knee. A Gaussian Mixture Regression (GMR) technique is then used to re-trieve the data from the trained model. This approach enables an autonomousextraction of the task-related information encoded in EMG signals, without lossof generality. Moreover, such a probabilistic framework based on Mixture ofGaussians (MoG) distributions only require a reduced number of parameters tobe kept, resulting in lightweight models. In the past, several groups exploitedMoG—and in particular GMM—as a method to encode EMG information.Chu et al. [8] were able to recognize EMG patterns related to hand motion inten different subjects by means of a GMM classifier. Suresh et al. [29] exploiteda combination of EMG and GMM as a signature for people identification.

Furthermore, a GMM/GMR probabilistic framework requires less trainingdata to achieve good results and provides faster regression with respect to othertechniques usually used in the field of EMG pattern recognition (e.g., NeuralNetworks (NN) [32]). In particular, Fukuda et al. [11] proposed a FeedforwardNeural Network (FNN) that achieves performances comparable to GMM, butwith a large manual tuning effort and with the drawback of high sensitivity tothe specific case study. For these reasons, a GMM probabilistic framework hasbeen already widely adopted in robotic applications [21], Robot Learning fromDemonstration (RLfD) [5,6], sound categorization [25], grasp modeling [26] andaction recognition [18].

The paper is structured as follows: Section 2 will describe the procedure toextract task-related features from EMG signals, the algorithm exploited for theknee angle estimation, and the modelization technique. Results of the applicationof the proposed approach will be provided in Section 3, as well as a posteriorianalysis on the minimum number of EMG channels and repetition required tohave high estimation performances. Finally, in Section 4 we will summarize anddiscuss the results achieved in light of possible future applications of this ap-proach.

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2 Methods

2.1 Experimental setup

Three healthy volunteers (S1–S3; age 30± 4; one female) participated in the ex-periment. No motor related problems have been reported. The study was carriedout in accordance with the principle of the Declaration of Helsinki3.

Subjects were asked to naturally kick a ball (diameter 17 cm) from a sittingposition. The ball was positioned on the floor at a fixed distance (2 cm) fromthe foot. After each kick, an operator was in charged of the ball reposition. Asadditional behavioral and motivational task, we asked to the subjects to try toshot the ball in a goal in front of them (distance 350 cm, width 122 cm, height76.2 cm). It is worth to notice that the task was fully self-paced.

Each participant performed several repetitions of the aforementioned taskover a single recording session (day). Table 1 illustrates the number of trialsperformed and the duration of the experiment for each subject.

Table 1. Number of kicks performed by each subject and the corresponding experimentduration

Subject Kicks Duration

S1 70 ∼8.5 mS2 67 ∼6.9 mS3 72 ∼6.5 m

2.2 Data collection

Electromyography (EMG) signals were acquired with an active 8-channel wirelessEMG system (BTS FREEEMG 3004) at 1000 Hz. The eight EMG electrodeswere placed on the left leg of each subject in order to cover the principal musculargroups active during the kick task. In more detail, the following muscle wererecorded: Rectus femoris, Vastus lateralis, Vastus medialis, Tibialis anterior,Gastrocnemius lateralis, Gastrocnemius medialis, Biceps femoris caput longus,Peroneus longus. An example of the position of EMG electrodes is illustrated inFigure 1.

Synchronously to the EMG signals, we recorded the kinematics of the leftleg by means of an optoelectronic system (BTS SMART-DX 1004). Six retroreflecting markers were placed on the subject’s leg as illustrated in Figure 1 (akaLGT, LLE, LHF, LLM, LVMH, LVTH ). Six infrared digital cameras recordedthe marker positions (namely, the leg kinematic) at 60 Hz during the whole

3 WMA Declaration of Helsinki - Ethical Principles for Medical Research InvolvingHuman Subjects http://www.wma.net/en/30publications/10policies/b3/

4 BTS S.p.A., Milan, Italy

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vknee-hip

vknee-ankle

alpha

LGT

LLE

LLM

Fig. 1. Example of recording setup for subject S2. EMG channels and motion capturemarkers are highlighted in green and red, respectively. Notice that not all channels andmarkers are visible. Vectors exploited for computing the angle alpha are reported.

recording session. Kinematic data was visually inspected and manually trackedwith BTS software suite.

2.3 Processing and feature extraction

First, EMG data was processed by means of signal rectification and smooth-ing in order to highlight the muscular activation during the kick tasks. Thismethod is widely exploited in literature to denoise EMG signals and extractuseful information and features for classification purposes [19, 20, 27]. In brief:the raw 8-channel EMG signals were high-pass filtered (zero-phase digital filter,4th order Butterworth, 30 Hz cut-off frequency) to remove artifacts; then, signalswere full-wave rectified and low-pass filtered (4th order Butterworth, 6 Hz cut-offfrequency). Signal processing was performed in MATLAB 8.05.

Second, we computed the kinematic of the knee-joint angle from the posi-tion of the markers placed on the leg. We considered two vectors: vknee−hip andvknee−ankle, as illustrated in Figure 1. Hip, knee and ankle positions were com-

5 MATLAB, The MathWorks, Inc., Natick, Massachusetts, United States

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puted from markers {LGT}, {LLE} and {LLM }, respectively. The knee anglewas computed at each time instant t according to the following formula:

α = arctan

(‖vknee−hip × vknee−ankle‖vknee−hip · vknee−ankle

)(1)

Then, angle data was resampled to match the sampling rate of EMG signals.Finally, we extracted 1000 ms of EMG and angle data before and after the

maximum angular aperture of the knee during each kick. We defined the afore-mentioned period (2000 ms) as trial and we used it as input for the classificationalgorithm.

2.4 Modelization

For modelization purposes, we exploited a stochastic approach in order to ad-dress the high variability of the input EMG signals. Information extracted fromEMG was used as input of a Gaussian Mixture Model (GMM) to estimate itscorrelation with the knee bending angle α.

The aim of GMM is to obtain the weighted sum of K Gaussian componentswhich best approximates the input dataset representing the set of kick trialsused for the training. In this particular case, the total number of data sampleswas N = nT , where n is the number of trials used to train the system, andT = 2000 is the number of observations acquired during each trial. A single datain input at the framework is described in Equation 2.

ζj = {t, ξ, α} ∈ RD ξ = {ξc}c=1,...,C (2)

where:

– t ∈ R is the time elapsed from the beginning of the trial (ms);– ξc ∈ R is the cth EMG channel;– ξ ∈ RC is the set of considered channels, 1 ≤ |ξ| ≤ 8;– α ∈ R is the knee bending angle;– 3 ≤ D ≤ 10 is the dimensionality of the problem.

The GMM was trained through the Expectation-Maximization (EM) algo-rithm [9], resulting in a probability distribution of the train dataset later used toperform the regression of the knee angle. The Expectation-Maximization algo-rithm iteratively estimates the optimal parameters θ = (πk, µk, Σk) that charac-terizes the K mixtures composing the GMM. The algorithm can be separated intwo cyclic phases: Expectation and Maximization. The EM loop stops when theincrement of the log-likelihood L =

∑Nj=1 log (p (ζj |θ)) at each iteration becomes

smaller than a defined threshold ε, given by L(t+1)L(t) < ε.

The algorithm optimizes the parameters of the K Gaussian components bymaintaining a monotone increasing likelihood during the local search of the max-imum. This approach enables an autonomous extraction of the kick characteristicEMG signal while still maintaining an appropriate generalization.

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Finally, the resulting probability density function is computed:

p (ζj) =K∑k=1

πkN (ζj ;µk, Σk) (3)

where:

– πk are priors probabilities;– N (ζj ;µk, Σk) are Gaussian distribution defined by µk and Σk, respectively

mean vector and covariance matrix of the k-th distribution.

The main drawback in the learning process lies in the EM requirement of aprior specification for the model complexity (i.e., the number of components K).On one hand, an overestimation of this parameter might lead to over-fitting and,consequently, to a poor generalization; on the other hand, an underestimationwill result to poor regression performances. To deal with this issue we introducedan entropy based selection of the best number of components, K, in the GMM.

Several entropy based model selection techniques has been proposed in litera-ture (e.g., Bayesian Information Criterion (BIC) [28], Akaike Information Crite-rion (AIC) [1], Minimum Description Length (MDL) [4], and Minimum MessageLength (MML) [31]). Although, in [8] the authors proposed a specific criteria toestimate the value of the K parameter in the case of EMG signals, in this workwe preferred a more standard approach based on BIC. In our experiments thewhole learning process has been repeated with different GMM complexities byusing BIC (Equation 4) as index of model quality with respect to the number ofcomponents K.

SBIC = −2L+ np logN (4)

where:

– L =∑Nj=1 log (p (ζj |θ)) is the log-likelihood for the considered model θ;

– np = (K−1)+K(D+ 12D(D+1)) is the number of free parameters required

for a mixture of K components with full covariance matrix.

The log-likelihood measures how well the model fits the data, while the secondterm is introduced to avoid data overfitting and maintain the model generalenough. In our experiments the best BIC value was obtained with K = 15components.

The Gaussian Mixture Regression (GMR) has been used to retrieve a smoothgeneralized version of the signal encoded in the associated GMM. So that, theconditional expectation of knee joint angle α is calculated from the consecutivetemporal value t and the EMG signals ξ known a priori. As we already said, thek-th Gaussian component is defined by the parameters (πk, µk, Σk), where:

µk = {µp,k µα,k} Σk =

[Σp,k Σpα,kΣαp,k Σα,k

](5)

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with µp and Σp respectively the mean and the covariance of the known a pri-ori information. The conditional expectation and its covariance can be estimatedrespectively using Equation 6 and 7.

α = E [α |t, ξ ] =K∑k=1

βkαk (6)

Σs = Cov [α |t, ξ ] =K∑k=1

β2kΣα,k (7)

where:

– βk =πkN (t,ξc|µp,k,Σp,k )∑Kj=1N (t,ξc|µp,j ,Σp,j )

is the weight of the k-th Gaussian component

through the mixture;– αk = E [αk |t, ξ ] = µα,k + Σαp,k (Σp,k)

−1({t, ξ} − µp,k) is the conditional

expectation of αk given {t, ξ};– Σα,k = Cov [αk |t, ξ ] = Σα,k + Σαp,k (Σp,k)

−1Σpα,k is the conditional co-

variance of αk given {t, ξ}.

Thus, the generalized form of the motions ζ = {t, ξ, α} required only theweight, mean and covariance of the Gaussian components calculated throughthe EM algorithm.

Finally, we exploited the Normalized Mean Square Error (NMSE) in order toevaluate the effectiveness of the GMM-based system. This function measures thegoodness of fit between test and reference data, in our case α (the data estimatedthrough the GMR) and α (the angle calculated by means of the motion capturesystem):

NMSE(t) = 1−∥∥∥∥ α(t)− α(t)

α(t)− µt(α)

∥∥∥∥2 (8)

where:

– t is the temporal instant from the beginning of the trial (ms);– α(t) is the estimated angle at the instant t;– α(t) is the angle calculated through the motion capture at the instant t;– µt(α) is the mean along the time of the angles given by the motion capture.

NMSE costs vary between −∞ (bad fit) to 1 (perfect fit). Zero is the valuereached from a straight line in fitting the reference.

3 Results

The trained GMM was tested on unseen data from the three different subjects.In this first step, we adopted a totally data-driven approach by using all eight

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EMG channels and a fixed amount of training data (60 repetitions from eachsubject) as input for the GMM.

Figure 2 shows the comparison between the average actual (blue line) andestimated (red line) angular aperture of the knee joint for subjects S1, S2, S3,respectively. Confidence bounds correspond to the standard deviation computedalong the different repetitions of the kick.

3.2

3

2.8

2.6

2.4

2.2

2

1.8

1.6

1.40 0.5 1 1.5 2

An

gle

[ra

d]

Time [s]

Subject S1

0 0.5 1 1.5 2 0 0.5 1 1.5 2

Subject S2 Subject S3

Actual

Estimated

Time [s] Time [s]

Fig. 2. Actual (in blue) and estimated (in red) angular aperture for the three subjects.Mean and standard deviation are reported (solid line and bounds, respectively). Verti-cal black line corresponds to the moment with maximum angular aperture during eachkick.

For all subjects, the angular variation estimated by the GMM follows theactual one with high accuracy (NMSE: 0.91, 0.97, 0.96, for S1–3, respectively).Subject S1 reports slight perturbations in the estimated angle around 0.5 s,probably due to muscular contractions before starting the kick.

The second step consisted in the preselection of the EMG input channelsmodeled by the GMM in order to reduce the dimensionality of the system and theamount of uninformative signals that might affect the estimation performances.First, we trained eight different GMM by using as input one single EMG channelfor each model. Then, we computed the NMSE between the actual angle andthe one estimated by each GMM. Thus, we selected the channel with the bestNMSE and we used it in combination with the other channels to train seven newGMMs. The group with the best NMSE was selected. This iterative procedurewas performed—step by step—until including all the channels.

Figure 3 depicts the NMSE for each channel group and subject. It is worthto notice that x-axis corresponds to the different groups (and, for construction,to the number of channel in each group) and they are not including the samechannels for each subject. For subject S1 the best NMSE is achieved with onlyone channel (0.967), while for subjects S2 and S3 two channels are needed (0.989,0.987, respectively). After reaching the maximum NMSE the trend is negativefor all subjects, meaning that additional channels decrease the performances ofthe system.

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1.04

1.02

1

0.98

0.96

0.94

0.92

NM

SE

1 2 3 4 5 6 7 8Channel group

Subject S1

Subject S2

Subject S3

Fig. 3. NMSE computed by the iterative channel selection procedure for each subject.

Table 2 illustrates the muscles corresponding to the EMG channels in thegroups with the best NMSE. As expected from physiology, they are all locatedin the upper part of the leg. It is interesting to notice that a comparison betweenthe classification performances and the location of the muscles shows that onlyfor subject S1 (the one with the worst performances) the selected muscle is onthe back of the leg.

Table 2. Muscles combinations for best estimated performances

Subject Muscles Location #

S1 Biceps femoris caput longus Upper leg - back (1)S2 Rectus femoris & Vastus lateralis Upper leg - front (2)S3 Vastus medialis & Vastus lateralis Upper leg - front (2)

The last step was to study the variation of estimation performances accordingto the number of trial used to train the GMM. On this purpose, we preselectedthe channels that led to the best performances (as reported in the previous step)and we modified the number of trials exploited during the training phase. InFigure 4, we report the NMSE values for different size of the training set andfor each subject.

For all the considered subjects, 10 trials are sufficient to reach a NMSEgreater than 0.86, while 30 repetitions of the kicks are enough to achieve aneven higher estimation (NMSE: 0.954). Subjects S2 and S3 showed an increasingtrend, while subject 1 needs the whole training set to overcame the accuracy

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NM

SE

10 20 30 40 50 60

Size training set

Subject S1

Subject S2

Subject S3

1

0.98

0.96

0.94

0.92

0.9

0.88

0.86

Fig. 4. Estimation accuracy with different size of the training set.

reached by using only 10 trials. This might be due to the use of a single channelwith a small number of trials, or to different behaviors of subject S1 during therepetitions of the kick, or to the sliding in a different position of an EMG sensorduring the acquisition of the testing data.

4 Conclusions

This paper described a probabilistic framework to estimate a single-joint angleby means of EMG signals. Physiological information has been encoded througha Gaussian Mixture Model, while the joint angle related to a new sequence of un-seen EMG data has been estimated from the model by using Gaussian MixtureRegression. The framework has been validated by comparing the EMG based es-timated angle with respect to kinematics data acquired from the motion capturesystem. Furthermore, we performed additional analysis in order to minimize theexperimental parameters (such as the number of EMG channels and trainingtrials) with a minimum loss of performances.

Results highlighted that this approach ensures high accuracy in the estima-tion of knee-angle for all subjects (NMSE> 0.956) when trained with 30 trialsby using at worst only two EMG channels. More interestingly, we showed thateven data from only 10 trials seems to guarantee high accuracy (NMSE>0.86),confirming the robustness of this approach.

The next challenge is to extend the framework to an online approach openingnew and intriguing perspectives for a robust and bio-inspired control of humanoidrobots and exoskeleton.

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Acknowledgements

This research has been supported by “Consorzio Ethics” through a grant forresearch activity on the project “Rehabilitation Robotics”.

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