Estimation of elbow angle using surface electromyographic signals

Abstract

Disability badly affects the performance of daily activities of a person thereby degrading the quality of his life. Physical rehabilitation can help restoration of the physical functions of such persons to a great extent. Numerous research are being performed for developing muscle models capable of estimating torque, joint angles, force, velocity etc. especially for upper and lower limb kinematics. This paper proposes three models for estimating elbow angle namely, Average Value model, a feed forward back-propagation neural network model and a Non-linear Auto-Regressive with eXogenous input (NARX) neural network model. All these models are studied with the sEMG signals acquired from the biceps brachii muscles of ten healthy subjects. The linear envelope of the signal is utilized for constructing two time domain features which serve as inputs to the models. The output of the models is the estimated elbow angles corresponding to the human intention identified from the sEMG signals. Regression coefficient values and Root mean square error values are considered for evaluating the accuracies of the models. The results obtained show that the models can be effectively used for implementation in control of human limb prosthetics and assistive devices.

Keywords

Surface electromyogram (sEMG)elbow kinematics estimation average value model non-linear auto-regressive with eXogenous input (NARX) neural network rehabilitation

1 Introduction

As per Persons with Disabilities Act 1995, disability has been defined as blindness; low vision; leprosy-cured; hearing impairment; locomotor disability; mental retardation and mental illness. Among these, locomotor disabilities include, those people without both arms or both legs; or are paralyzed and not able to move but crawl; or can move only using walking aids; or have acute and permanent problems of joints/muscles that have resulted in limited movement; or with no fingers or toes; or unable to move or pick up at least small things; or have stiffness or tightness in movement; or difficulty in balancing and coordinating body movements; or sensation loss in the body due to any reason; or any deformity of the body parts. In India, out of the total 121 crore population, about 2.68 crore persons are disabled which is 2.21% of the total population, as per Census 2011 [2]. Due to such disabilities, the performance of a person’s daily activities gets affected, leading to poor quality of life. Rehabilitation and assistive technologies can help them to recover from the lost physical functions to a great extent. Numerous researchers have been working on the development of rehabilitation systems for providing restoration of the lost physical functions, which will help to improve the quality of their lives [11 , 19–23]. Assistive devices can compensate for the physical limitations experienced by the disabled persons in their routine activities and can help them to reduce their dependence on others. Further, it can help the afflicted to enhance and upgrade their physical capabilities. Apart from these, they can provide assistance to aged persons to overcome the physical limitations that arise naturally in the old age. These systems become more effective only when they are associated with biological signals [6]. Electromyogram (EMG), which is the record of the electrical activity of the muscles during contraction and relaxation, is one of the most important biological signals which helps to imitate human intended motion. Hence, EMG can be used to identify the movement intention of the user by estimating torque, force, angle, acceleration etc. These are accomplished by developing models of human muscles with processed EMG as input and the quantity to be estimated as the output. This paper proposes three such muscle models for estimating human elbow kinematics by utilizing processed sEMG signals. The work ultimately aims at the development of an assistive device for supporting human elbow motion with its operational dynamics similar to that of natural elbow characteristics. Here, the proposed muscle models uses the EMG data from biceps brachii for identifying the movement intention of the user. The muscle model gives estimated elbow angle as the output which can be further used to drive the elbow assistive device.

Rosen et al. [20] used the Hill-based (HB) muscle skeleton model and the EMG signals to model the human arm status, to control a powered exoskeleton. Hill-based muscle processors were developed for the flexor and the extensor muscle groups separately for predicting the moment applied on the elbow joint by each one of these muscle groups. The sum of moment components, with (+) sign for agonist muscles and (–) sign for antagonist muscles, were taken to calculate the muscles’ net moment. Another model called dynamic human body model (DHBM) and the direct force control (DFC) were proposed by Fleischer et al. [3] to accomplish exoskeleton control for lower limb. The force produced by the muscle was also estimated using the muscle model.

Continuous motion estimation of human limb was proposed by Han et al. using a state-space electromyography (EMG) model [5] in which the forward dynamics of human joint movement was integrated into the Hill-based muscle model (HMM) to estimate the joint motion states directly. Features were extracted from EMG to construct measurement equations for the extended Hill-based muscle model to form the state-space model. With the state-space HMM, Kalman-type algorithm was used to estimate the joint kinematics from EMG signals. Gaussian Mixture Model (GMM) are also being used for estimation. Michieletto et al. developed a Gaussian Mixture Model to estimate the bending angle of knee joint of a single human joint using EMG from the leg muscles [14].

Mamikoglu et al. has used integrated EMG signal to find the EMG-joint angle relation and the joint angles for elbow flexion-extension movement were estimated by using an auto-regressive integrated moving average with exogenous input (ARIMAX) model, which takes integrated EMG measurements as input [12]. Raj et al. extracted two time-domain parameters namely, Integrated EMG (IEMG) and Zero Crossing (ZC) from surface Electromyography (sEMG) signal from biceps and triceps and derived a multiple input-multiple output Nonlinear Auto Regressive with eXogenous inputs (NARX) structure based Multiple Layer Perceptron Neural Network (MLPNN) model, for identifying the elbow kinematics [18].

Zhang et al. has estimated 4-DoF kinematics at shoulder and elbow during coordinated arm movements [29] with different muscle activity decomposition and learning strategies. Principle component analysis (PCA) and independent component analysis (ICA) were employed for EMG mode decomposition with artificial neural network (ANN) for the estimation of multi-joint kinematics in variant application conditions. A back propagation neural network (BPNN) along with a Virtual Human Model (VHM) was proposed by Aung et al. to estimate the shoulder and elbow joint angles from the recorded EMG signals [1].

The recent research works described in the literature [3 , 20] require a number of parameters. For example, Hill-based method model [20] requires a large number of physiological parameters which are difficult to be identified accurately. Apart from these, a knowledge on the muscle’s length and velocity are required, in addition to the EMG signal, which defines the muscle activation level for predicting the force developed by the physiological model. To control a robotic device assisting the movement of human limb, angular velocity and position need to be further calculated from the EMG-recognized force indirectly which accumulates errors and thereby degrades the estimation accuracy. Also, HB is a complex model and introduces extra computational encumbrance. The proposed methods have the following advantages (i) They can be used directly for practical implementation of the assistive device/exoskeleton structure in its operational dynamics mimicking the natural elbow characteristics using less number of parameters (ii) provide development of database for operational dynamics of the exoskeleton for the handicapped as well as the aged people (iii) the method used in the proposed models is much easy and simple.

The organization of rest of the paper is as follows. The next section explains about the methodology of the work which includes data collection and its processing, and the proposed muscle models. Section III describes the experimental results followed by conclusion in the last section.

2 Methodology

An overview of the proposed system is shown in Fig. 1. The complete system shall be divided into two parts. Part I: EMG acquisition and processing, and Part II: Finding a relationship between EMG and elbow angle.

Fig.1

An outline of the proposed system [12].

2.1 EMG acquisition and processing

Surface EMG (sEMG) signals are utilized for the study, as the acquisition of these does not require expertise as in the case of intramuscular EMG, which are acquired using invasive type needle electrodes. The sEMG signal is acquired from ten subjects with an average age of 25 years, average weight of 66 kg and average height of 167 cm, using three Ag-AgCl electrodes, with two of them placed on the biceps brachii muscle of their right arm and the third kept as reference electrode placed on their wrist. ADXL335 accelerometer is used to acquire the kinematics of elbow movements. The subjects are instructed to perform forearm flexion and extension movements continuously at different average angular velocities of approximately 30 deg/s (slow movement), 75 deg/s (medium speed movement) and 120 deg/s (fast movement) starting from the extended arm position while standing. The signals are acquired at a sampling frequency of 10 kHz and is filtered using a fourth order IIR Butterworth filter with a lower cutoff frequency of 10 Hz and upper cutoff frequency of 400 Hz to remove motions artifacts and high frequency components to avoid signal aliasing respectively [4].

The filtered signal is then rectified and low pass filtered at a frequency of 20 Hz to extract the envelope of the signal. The combination of rectification and low pass filtering of the signal yields the linear envelope of the signal. Since the random changes of the EMG signal makes it difficult for analysis, it is suitable to consider the linear envelope of the signal; which results in a smoother wave as shown in Fig. 2.

Fig.2

Raw sEMG and linear envelope of rectified sEMG.

2.2 Proposed average value model

The obtained linear envelope of the sEMG signal is divided into several sectors called primary sectors; which is further divided into smaller sectors called secondary sectors and further into tertiary sectors and so on as shown in Fig. 3. In each primary sector, the sEMG values and corresponding elbow angles recorded using the accelerometer are considered. In Fig. 3 is shown three primary sectors (PS) and, one primary sector divided into four secondary sectors (SS). Tertiary sectors and higher level divisions are avoided for clear understanding.

Fig.3

Three Primary sectors and first primary sector divided into four secondary sectors.

After determining sectors and subsectors, next step is to determine the variables namely, Base Average (BA), first variation term (ΔBA), second variation term (Δ²BA), and so on. The average value of each of the primary sectors is called Base Averages. The average of the secondary sectors are determined and are subtracted from the respective Base Averages. Then the average of all these differences are calculated which forms the first variation term, denoted as ΔBA. Similarly for tertiary sectors, second variation term Δ²BA and for quaternary sectors Δ³BA and so on. The size and the levels of division of these sectors are determined through different attempts which give maximum accuracy in estimation. Thus it is observed that two levels namely, BA and ΔBA are sufficient for the estimation of elbow angles with considerable accuracy. Hence these variables are used to form mathematical equations of Average value model, to represent a muscle model for the estimation of elbow angles at different angular velocities. The variation of these quantities with elbow angle is shown in Fig. 4.

Fig.4

Relationship of (a) BA with Elbow Angle (b) ΔBA with Elbow Angle.

As is evident from Fig. 4, BA and ΔBA do not have a purely linear relation with elbow angle. Even then, for the simplicity of the model, these quantities are assumed to be linear with elbow angle. Hence the muscle model is formed using these two quantities as shown in Equation (1). $\begin{matrix} ({BA}_{1}) P_{1} + (Δ {BA}_{1}) P_{2} = θ_{1} \\ ({BA}_{2}) P_{1} + (Δ {BA}_{2}) P_{2} = θ_{2} \end{matrix}$ (1) $\begin{matrix} ⋮ \\ ({BA}_{n}) P_{1} + (Δ {BA}_{n}) P_{2} = θ_{n} \end{matrix}$ where θ is the elbow angle in degrees and n is the number of equations which is equal to the number of primary sectors. As shown in Equation (1), n linear equations are formed using the BAs, ΔBAs and the corresponding elbow angles. In matrix form, Equation (1) can be written as $[\begin{matrix} {BA}_{1} & Δ {BA}_{1} \\ {BA}_{2} & Δ {BA}_{2} \\ ⋮ & ⋮ \\ {BA}_{n} & Δ {BA}_{n} \end{matrix}] [\begin{matrix} \begin{matrix} P_{1} \end{matrix} \\ P_{2} \end{matrix}] = [\begin{matrix} θ_{1} \\ θ_{2} \\ ⋮ \\ θ_{n} \end{matrix}]$ (2) which is of the form Ax = B where $A = [\begin{matrix} {BA}_{1} & Δ {BA}_{1} \\ {BA}_{2} & Δ {BA}_{2} \\ ⋮ & ⋮ \\ {BA}_{n} & Δ {BA}_{n} \end{matrix}], x = [\begin{matrix} P_{1} \\ P_{2} \end{matrix}], B = [\begin{matrix} θ_{1} \\ θ_{2} \\ ⋮ \\ θ_{n} \end{matrix}]$ (3)

Since the model is formed using average values of sEMG and elbow angles, the model is named as Average Value Model. As seen in Equation (2), it is an over-determined linear system, which implies that the number of equations is more than the number of unknowns. Such systems do not have an exact solution. In such cases, a solution is sought with the smallest amount of error. If the error for an over-determined linear system Ax = B is defined as e = Ax - B, then the error e is a vector whose size is measured using a norm. Hence the solution of over-determined system Ax = b is determined by minimizing ∥Ax - b∥ for some norm.

It is common to determine the solution x by minimizing the energy of the error, $J (x) = {∥ Ax - B ∥}^{2}$ (4)

Taking the derivative of J (x) with respect to x and setting it to zero, gives x as, $x = {(A^{T} A)}^{- 1} A^{T} B$ (5)

Equation (5) gives the least square solution whereby $x = {[\begin{matrix} P_{1} & P_{2} \end{matrix}]}^{T}$ is determined. Here, the values of P₁ and P₂ vary with subjects. Now, the model can be depicted more clearly as shown in Fig. 5.

Fig.5

A flowchart representing the proposed Average Value Model.

Figure 5 shows the flowchart of the proposed Average Value Model. In this, the raw EMG signal taken from biceps brachii muscle serves as the input. This signal is rectified and low-pass filtered to extract the linear envelope of the signal. This extracted linear envelope is divided into primary sectors and the primary sectors are further divided into secondary sectors as shown in Fig. 3. Base Averages (BA) and first variation terms (ΔBA) are determined from the primary and secondary sectors. These variables are used as inputs to the Average Value Muscle Model for the estimation of elbow angle.

Regression and RMSE [9 , 27] values are used to determine the accuracy of the estimated values of elbow angles. Regression measures the correlation between estimated values and desired values. An R value of 1 means a close relationship and 0 means a random relationship. The Primary sectors which are used to obtain the coefficient values P₁ and P₂ shall be called Training set and the Primary sectors which are used for estimation of elbow angles with the obtained coefficient values shall be called Testing set. Table 1 shows the regression values corresponding to different lengths of Training/Testing sets. From the analysis, it is observed that the coefficients formed by Training sets of primary sectors of length 20000 samples (corresponding to 2 seconds), and Testing sets of primary sectors of length 5000 samples (corresponding to 0.5 seconds) give sufficiently accurate results. Similarly, the lengths of secondary sectors were also determined step by step through thorough analysis. Then with these values fixed, the coefficients of the linear equations, P₁ and P₂ are determined for each subject separately. These coefficients may be assumed to represent the physical and other psychological characteristics which affect the system during measurement of the signal. Hence the values of P₁ and P₂ are found to vary with subjects. The range of values obtained for P₁ and P₂ for 10 subjects is shown as boxplot in Fig. 6.

Fig.6

Range of coefficients P₁ and P₂ of the proposed mathematical model.

Table 1

Regression values obtained with different lengths of PS of Training/Testing sets for coefficient determination and estimation respectively

Subject #	Speed of movement	2500/2500	5000/2500	5000/5000	10000/2500	10000/5000	20000/2500	20000/5000
1	Fast	0.94622	0.94505	0.82149	0.93007	0.81397	0.93857	0.83696
	Medium	0.8927	0.89948	0.90228	0.90802	0.91226	0.91996	0.91408
	Slow	0.77526	0.79366	0.80367	0.81304	0.81773	0.81096	0.82388
2	Fast	0.68834	0.68855	0.84093	0.69528	0.84091	0.69296	0.84239
	Medium	0.66701	0.66635	0.75372	0.66303	0.76956	0.65586	0.78169
	Slow	0.92071	0.92604	0.9414	0.93111	0.94757	0.93344	0.9509
3	Fast	0.6561	0.67511	0.71615	0.71036	0.72811	0.61679	0.67843
	Medium	0.84412	0.86091	0.89025	0.66601	0.91065	0.88796	0.93133
	Slow	0.86048	0.87185	0.88267	0.86912	0.89716	0.92653	0.92971
4	Fast	0.74441	0.7446	0.84789	0.73106	0.84699	0.72673	0.83891
	Medium	0.66238	0.67009	0.68744	0.66822	0.70825	0.65678	0.71694
	Slow	0.78108	0.79867	0.81344	0.80847	0.82046	0.81438	0.82294
5	Fast	0.94565	0.95497	0.8324	0.95187	0.86528	0.95088	0.88276
	Medium	0.88561	0.90242	0.89446	0.91978	0.90341	0.9278	0.90831
	Slow	0.89322	0.90948	0.9191	0.92303	0.93567	0.93108	0.94668
6	Fast	0.83008	0.79652	0.81682	0.77593	0.82766	0.72873	0.80833
	Medium	0.85776	0.86579	0.80566	0.86701	0.81976	0.86447	0.82709
	Slow	0.74941	0.75589	0.79572	0.76453	0.79474	0.78768	0.80018
7	Fast	0.6251	0.62045	0.60246	0.62479	0.62515	0.62593	0.63354
	Medium	0.87297	0.87629	0.92008	0.88279	0.92752	0.89404	0.93797
	Slow	0.77559	0.79427	0.80927	0.80547	0.81352	0.81406	0.82075
8	Fast	0.61534	0.65283	0.57799	0.66471	0.58817	0.68759	0.59355
	Medium	0.6391	0.65452	0.67215	0.66816	0.66262	0.68952	0.71172
	Slow	0.87454	0.89711	0.89357	0.91782	0.91907	0.93041	0.93329
9	Fast	0.95116	0.95063	0.79981	0.9439	0.81264	0.9369	0.82647
	Medium	0.86949	0.87313	0.91929	0.88998	0.93288	0.90024	0.94417
	Slow	0.77404	0.79704	0.80642	0.80833	0.81878	0.80898	0.82355
10	Fast	0.88958	0.89676	0.91939	0.89961	0.9247	0.87986	0.90489
	Medium	0.84357	0.8518	0.90176	0.85705	0.9086	0.87951	0.9388
	Slow	0.69092	0.70087	0.80398	0.70736	0.80738	0.75435	0.85487
	Average	0.80073	0.80970	0.81972	0.80886	0.83004	0.81910	0.83884

Artificial Neural Networks have proved to be a good tool for estimation [1 , 29]. Hence, two Neural Network Models are also developed with the same inputs and output and are compared with the proposed Average Value Model.

2.3 Proposed neural network model

The neural network model [7 , 25] is a multi-input-single-output (MISO) neural network, developed with the same inputs and outputs used in Average Value model. It is a two layer feed-forward neural network (NN) with one input layer, one hidden layer and one output layer. In the input layer of NN, no computations are being performed; it just transmits the inputs from the external environment to the network. Hence input layers are not usually considered as a layer. The hidden layer consists of three hidden neurons with sigmoid activation function and one linear output neuron. Figure 7 shows the feed-forward NN model for the estimation of elbow angle. Thus NN model gives a non-linear mathematical model because of the non-linear sigmoid activation function in the hidden layer.

Fig.7

Feed-forward NN model for the estimation of elbow angle.

Output of the hidden layer is given by, $y_{H} = tansig (\sum_{j = 1}^{2} w_{kj} x_{j} + b_{k})$ (6) where x_j denotes the jth input, w_kj denotes the synaptic weights connecting the kth neuron in the hidden layer and jth input, and b_k denotes the bias of kth neuron in the hidden layer.

Output of the neural network is given by, $y_{O} = purelin (\sum_{i = 1}^{3} w_{1 i} y_{i} + b)$ (7) where y_i denotes the ith input from the hidden layer, w_1i denotes the synaptic weights connecting the single neuron in the output layer and ith input from the hidden layer, and b denotes the bias of single output neuron in the output layer.

During the development of the network, the samples presented to it is divided into 3 parts – training, validation and testing. 80% of the input data samples are used for the training of the network, 10% for validation and 10% for testing. The Levenberg-Marquardt Backpropagation algorithm [28] is used for training of the feed-forward network. This algorithm typically takes more memory, but less time. During training, the network is adapted according to its error. Training automatically stops when generalization stops improving, as indicated by an increase in the mean square error of validation samples. Validation measures the generalization of the neural network model and helps to terminate training when generalization stops improving. Testing provides an independent measure of network performance during and after training.

2.4 Proposed nonlinear autoregressive with eXogenous input (NARX) neural network model

NARX Neural Network Model has been found to be a good tool for accurate prediction of angular displacements and angular velocities [18]. It is a recurrent dynamic network which helps to predict future values of a time series with its own past values and past values of an external input. It is mathematically expressed as

$\begin{matrix} y (t) & = & f (y (t - 1), \dots, y (t - d), \\ x (t - 1), \dots, x (t - d)) \end{matrix}$ (8) where x (t) and y (t) are the input and output respectively at time t, d is the delay and f is a nonlinear function. The NARX model is based on linear ARX model. As seen in Equation (6), the next value of the dependent output signal y (t) is regressed on previous values of the output signal and previous values of an independent (exogenous) input signal. Here, a two-layered NARX neural network model is proposed which takes the same two inputs namely, BA and ΔBA in the input layer, and gives the estimated elbow angle at the output. It consists of 10 hidden neurons with sigmoid activation function in the hidden layer and a linear output neuron in the output layer. The value of d in Equation (6) is taken as 2. The structure of the proposed network is shown in Fig. 8.

Fig.9

Regression values obtained with training, validation and testing of the NARX model.

Output of the NARX neural network is given by, $y_{O} = purelin (\sum_{i = 1}^{m} w_{1 i} (tansig (\sum_{j = 1}^{n} w_{kj} x_{j} + b_{k})) + b)$ (9)

where m is the number of hidden neurons which is equal to 10 in this case, n is the number of inputs, x_j denotes the jth input, w_kj denotes the synaptic weights connecting the kth neuron in the hidden layer and jth input, and b_k denotes the bias of kth neuron in the hidden layer, y_i denotes the ith input from the hidden layer, w_1i denotes the synaptic weights connecting the single neuron in the output layer and ith input from the hidden layer, and b denotes the bias of single output neuron in the output layer.

The NARX neural network is created in Matlab using Neural Net Time Series App. The samples obtained with 250 ms window size are presented to the network. Out of these samples, 453 samples (80%) are used for training, 57 samples (10%) for validation and 57 samples (10%) for testing (shown in Appendix). The Levenberg-Marquardt algorithm is used for training of the network. Figure 9 shows the regression values obtained with training, validation and testing of the NARX model.

Fig.8

Structure of proposed NARX model.

3 Experimental results and discussions

The Average Value model, NN model and NARX model are tested with the data of 10 subjects at different angular velocities. For evaluating the accuracy of estimation, regression values and Root mean square error (RMSE) values are considered. Regression values and RMSE values are found to be equally good with Average Value model and NN model and better in NARX model. The comparison of the regression values and RMSE values obtained with all the three models are shown in Tables 2 and 3 respectively.

Table 2
Comparison of regression values obtained with NARX, NN and AV models

Subject # Regression

NARX NN AV

1 0.9995 0.848 0.858

2 0.9995 0.874 0.858

3 0.9997 0.843 0.846

4 0.9994 0.730 0.793

5 0.9995 0.935 0.913

6 0.9995 0.742 0.812

7 0.9995 0.791 0.797

8 0.9994 0.887 0.746

9 0.9997 0.906 0.900

10 0.9995 0.873 0.865

Average 0.9995 0.843 0.839

Subject #	Regression
1	0.9995	0.848	0.858
2	0.9995	0.874	0.858
3	0.9997	0.843	0.846
4	0.9994	0.730	0.793
5	0.9995	0.935	0.913
6	0.9995	0.742	0.812
7	0.9995	0.791	0.797
8	0.9994	0.887	0.746
9	0.9997	0.906	0.900
10	0.9995	0.873	0.865
Average	0.9995	0.843	0.839

Table 3

Comparison of rmse values obtained with NARX, NN and AV models

Subject #	RMSE
	NARX	NN	AV
1	1.59	22.25	21.26
2	1.72	18.00	18.32
3	1.40	19.10	23.64
4	1.92	39.09	42.84
5	1.58	22.25	21.48
6	1.79	61.04	43.15
7	1.55	31.97	31.35
8	1.73	19.10	21.60
9	1.41	18.81	20.38
10	1.56	29.63	32.63
Average	1.62	28.13	27.66

From Tables 2 and 3, it can be seen that the average regression value is very close to 1 and RMSE value is low in NARX model whereas they are almost the same with NN and Average Value models. Hence NARX model can be considered as an approximate non-linear mathematical model for human elbow which takes Base Average and ΔBA of sEMG as inputs and gives elbow angle as the output. The other two models can also be considered as mathematical models of human elbow, with Average value model as a linear model and NN model as a non-linear model, for the estimation of elbow kinematics with BAs and ΔBAs as inputs, but with a lower accuracy.

The graphs indicating a comparison between estimated values obtained with Average Value model and NN model at different angular velocities are shown in Fig. 10. The comparison plot of NARX is shown separately in Fig. 11 for more clarity. The advantage of NN and NARX models is that the same model gives meaningful results with all the 10 subjects whereas in Average Value method, the model is subject dependent.

Fig.10

Comparison of estimated elbow angles obtained with average value model and NN model for (a) fast movement (b) medium speed movement (c) slow movement.

Fig.11

Comparison of estimated elbow angles obtained with NARX model for (a) fast movement (b) medium speed movement (c) slow movement.

The relation between estimated values and desired values of elbow angle obtained with NARX model is shown in Fig. 12.

Fig.12

Relation between estimated elbow angles obtained with NARX model and desired elbow angle.

4 Conclusion

In this paper, a linear and two nonlinear models have been proposed for forming a mathematical model for the estimation of elbow kinematics, with Base average and its first variation term as inputs. The results obtained emphasize that out of the three proposed models, NARX neural network shall be considered to drive a motor connected on an assistive device for human elbow support in the next phase of the work. The dynamics of the movements of assistive limb shall be studied and compared with the estimated value. The other models called Average Value model and NN model have also been found to give satisfactory results. The Average Value model is the simplest to develop and implement. However, it is found to be subject dependent. As a future scope, the methodology can be extended for a full body exoskeleton including knee and hip of leg for human locomotion.

Footnotes

Appendix

A portion of the samples presented to the NARX network is shown in Table 4.

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