1 Introduction

With the increasing number of hemiplegic patients and demands for machine-assisted rehabilitation, varieties of rehabilitation exoskeleton robots have been developed for assistance [1,2,3,4]. In order to efficiently control robots, the first challenge is to recognize human motion intention in a rapid and accurate way. Recognition of human motion intention based on biological signals [5,6,7] has become increasingly popular due to its promising prospect in human–computer interaction. For the different expression forms of motion intention, existing recognition methods mainly include the forms of physiological signals, such as surface electromyography (sEMG) signals, neural signals, including electroencephalogram (EEG) signals, and such biological force signals, as human–computer interaction force signals.

Compared with other signals, sEMG signals are widely used in the robotic systems to obtain human motion intention due to its effectiveness, safety, portability and non-invasive, non-delay features [8, 9]. In order to improve patients' muscle coordination and exercise ability, the sEMG signals have been introduced as driving or feedback signals into the rehabilitation process for the elderly, stroke patients or others in need [10,11,12]. Previous researchers show that estimating joint torque is one of the effective approaches to realizing intention recognition [13]. Hence, it is necessary to propose a joint torque estimation method based on sEMG signals to realize intention recognition and then continuously controlling the robots.

Existing joint torque estimation methods can be listed as follows [14,15,16,17]. Cai [14] estimated the knee torque using Support Vector Regression (SVM) which required a large amount of matrix storage and computation, thus a large amount of machine memory and operation time were consumed. Besides, it was difficult to implement this algorithm for large-scale training samples. Researchers [15, 17] established various non-linear relationships between sEMG and joint torque, which didn’t take human muscle models into account. Tyler et al. [16] put forward a complex muscle activation model, including neural activation dynamics, muscle activation dynamics, muscle contraction dynamics, muscle moment arms and skeletal motion model. Obviously, too many human parameters needed to be identified, and complex functional relationships between variables needed to be established, which complicated the operation process and limited the application to robot controlling.

In prior research, the joint angle was often utilized to estimate joint torque. Angle was measured by different sensors in traditional methods [17, 18]. However, the application of the angle sensors not only brought the installation problems for exoskeleton robots, but also often needed complicated algorithms to figure out the angle. Aiming at the problems mentioned above, researchers have done lots of research on the angle estimation [19,20,21]. Zhang [19] used a feedforward artificial neural network model to make the mapping from sEMG to the elbow joint angle. Principle component analysis (PCA) or independent component analysis (ICA) was used in the process of feature extraction. Researchers [20,21,22] used Back-Propagation neural network (BPNN) and RBFNN to acquire the relationship between the sEMG and joint angle. The input, output and other parameters of the neural network needed to be set before the training which complicated the estimation process. Therefore, it is necessary to develop a relatively simple and accurate angle estimation strategy to ensure the estimation accuracy and simplify the operation.

In this paper, a sEMG-based joint torque estimation strategy combining with Hill-type muscle model (HMM) by using RBFNN and system identification is proposed to make the results of muscular movement digitized. Elbow joint angle is estimated through a transfer function model which interpreted a non-linear relationship. RBFNN is used to get the optimal factor of muscle activation, making the estimated joint torque more accurate. Compared to the existing torque estimation strategy, this method not only improves the estimation accuracy but also replaces the angle sensors. Finally, static and dynamic experiments based on these three different methods are conducted to prove that the improvements of this new strategy in terms of torque estimation accuracy.

2 Method

The joint torque estimation strategy will be carried out in several steps, which involve the processing of sEMG signals, system recognition and neural networks. This method will be introduced in two parts as follows. One is the joint angle estimation strategy, and the other is the muscle activation and muscle model.

2.1 Joint Angle Estimation Strategy

In order to replace the angle transducer and optimize the process of angle estimation, a new joint estimation angle method that establishes a relationship between sEMG and the joint angle is presented in this paper. A transfer function model is built up to relate sEMG signals from the biceps brachii to the joint angle. Furthermore, system identification is applied to estimate the parameters of the transfer function model, which only focuses on the input and output values. In this way, only a few calibration tests need to be done and simplify the hardware in practical application.

The entire process of angle estimation strategy is described in Fig. 1. In order to obtain high-quality required sEMG signals, a series of processing steps are adopted, part of which can refer to the paper [17, 23]. In this research, the biceps brachii is selected as the main responsible muscle of the elbow joint [24, 25]. Before collecting sEMG signals by the sEMG sensor (MyoWare Sensor), the muscle's surface must be cleaned with alcohol cotton to remove grease and stains. The fine hairs on the surface of the human body need to be removed with special cutting tools to ensure the high-quality original signals. Next, the sEMG sensor is put on the center of muscle and it should be mounted parallel to the direction of muscle fiber [26]. In addition, due to the influences of noise and other interference factors, the original sEMG signals are not suitable for direct application, so it must go through a series of filtering and rectification operations.

Fig. 1
figure 1

The procedure of the angle estimation strategy

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Besides, data amplitude is susceptible to several factors, such as electrodes location, subjects or even mental condition and so on. Normalizing sEMG data to 0–1 can solve some of these problems. Hence, after 1 Hz low-pass filter, the sEMG signals should be normalized with regard to the amplitude of sEMG.

Furthermore, in order to make the low amplitude increase of sEMG more obvious during the slow motion, the input signals of the transfer function model can be computed from the normalized sEMG signals, and the corresponding relationship can be described as shown below:

$$v\left( t \right) = - \frac{1}{{\ln \left( {u\left( t \right)} \right)}}$$
(1)

where

$$0 < u\left( t \right) < 1$$
(2)

Here, v(t) is the input signals of transfer function model and u(t) is the normalized sEMG signals. This research presents a relationship between the sEMG and the joint angle, which can be defined as follows:

$$G\left( s \right) = \frac{v\left( s \right)}{{\theta \left( s \right)}} = \frac{{k\left( {1 + T_{z} s} \right)}}{{\left( {1 + T_{p1} s} \right)\left( {1 + T_{p2} s} \right)\left( {1 + T_{p3} s} \right)}}$$
(3)

where G(s) is the transfer function; v(s) is the input signals of this model and it is the Laplace transform of the processed sEMG v(t); θ(s) is the output signals of this model and it is the Laplace transform of the elbow joint angle θ(t); s is a complex variable; k, Tz, Tp1, Tp2, Tp3 are all coefficients in this transfer function model. Moreover, the parameters of the transfer function are different under different load conditions. Also, all the coefficients can be obtained through system identification module in the Host personal computer (PC).

Usually, the signals obtained after these processing steps also contain gaussian white noise. Therefore, Kalman filtering is used to remove gaussian white noise and smooth the signals.

2.2 The Muscle Activation and Muscle Model

In this part, sEMG signals could be transformed into muscle activation through a series of processing, and then combined with the muscle model and neural network to obtain the estimated torque of the elbow joint.

2.2.1 From the sEMG Signals to the Muscle Activation

Some of the preprocess of sEMG has been described above and the following operations will be discussed in the following part. The operation procedure of torque estimation strategy is illustrated in Fig. 2. In this research, a model is proposed to characterize the non-linear relationship between sEMG level and muscle activation on the basis of the theory reported in the article [27], which can be expressed as:

$$a\left( t \right) = \frac{1}{q} \cdot \frac{{e^{A \cdot u\left( t \right)} - 1}}{{e^{A} - 1}}$$
(4)
Fig. 2
figure 2

The procedure of elbow joint torque estimation strategy

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where u(t) is the normalized sEMG signals; a(t) is muscle activation; A is a constant determining the degree of nonlinearity; q is a variable parameter that will be trained in the RBFNN, also its specific meaning and expression method will be explained in formula (17).

2.2.2 From Muscle Activation to the Estimated Torque

The muscle activation could be obtained through Eq. (4) and then used to compute the muscle force. This model, described in Fig. 3, has gone through many times of evolution and simplification. The picture illustrates that a muscle–tendon unit contains a contractile element (CE), a passive element (PE) and tendons. The force Fmu produced by the muscle–tendon unit can be given as [28]:

$$F_{mu} = \left( {F_{c} + F_{p} } \right) \cdot \cos \left( {\theta_{p} } \right)$$
(5)
Fig. 3
figure 3

Hill-based muscle model (HMM). It contains a number of elements, that is PE parallel elastic element; CE contractile element; Fmu force of the musculotendinous unit; Fm force of muscle fiber; Fc force of contractile element; Fp force of parallel elastic element; Lt length of tendon; Lm length of muscle fiber; Lmu length of the musculotendinous unit; θp pennation angle

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where Fmu represents the force generated by the muscle–tendon unit; Fc is the force generated by the contractile element; Fp denotes the force generated by the passive element; θp is the Pennation angle. Force generated by CE and PE can be given by the following equations:

$$F_{c} = f_{c} \left( {l_{m} } \right) \cdot f_{c} \left( {v_{m} } \right) \cdot a\left( t \right) \cdot F_{mo}$$
(6)
$$F_{p} = f_{p} \left( {l_{m} } \right) \cdot F_{{m{\text{o}}}}$$
(7)

Here, a(t) is muscle activation; Fmo represents the maximum isometric muscle force; fc (lm) and fc (vm) indicate the force–length and force–velocity relationship for the CE, respectively, while fp(lm) represents the force–length relationship for the PE. In order to calculate Fmu, these relationships can be defined by the following equations:

$$f_{c} \left( {l_{m} } \right) = \left\{ \begin{gathered} r_{0} \cdot \left( {\frac{{L_{m} }}{{L_{mo} }}} \right)^{2} + r_{1} \cdot \frac{{L_{m} }}{{L_{mo} }} + r_{2} ,0.5 \le \frac{{L_{m} }}{{L_{mo} }} \le 1.5 \hfill \\ 0,otherwise \hfill \\ \end{gathered} \right.$$
(8)
$$f_{c} \left( {v_{m} } \right) = 1$$
(9)
$$f_{p} \left( {l_{m} } \right) = e^{{10 \cdot \frac{{L_{m} }}{{L_{mo} }} - 15}}$$
(10)

where Lm is the muscle fiber length and Lmo represents the optimal fiber length; r0, r1, and r2 are set to be the constants.

The following formula can be derived from Fig. 3:

$$L_{mu} = L_{t} + L_{m} \cdot \cos \left( {\theta_{p} } \right)$$
(11)

Here, Lt is the length of the tendons and Lmu is the musculotendinous unit length. To simplify the calculation process, the following assumptions are made according to the article [27]:

$$L_{mu} = \eta + \mu \cdot \theta_{e}$$
(12)
$$\tau = F_{mu} \cdot \mu$$
(13)

where τ is the joint moment at time t; θe is the estimated joint angle; η and μ are constants. Many body parameters can be referred to the previous research [28,29,30,31]. The parameters mentioned above are shown in Table 1. In this research, the whole experiment will be conducted in two ways, i.e. the static and dynamic experiments, as shown in Figs. 4 and 5, respectively.

Table 1 Definition and values of parameters
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Fig. 4
figure 4

System layout in the static experiment

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Fig. 5
figure 5

System layout in the dynamic experiment

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Experimental setup for static experiment is shown in Fig. 4. In this case, the upper arm is always perpendicular to the lower arm. The hand pulls the wire rope and the force should be completely generated by the biceps brachii. A tension sensor (JLBS-MD-10KG) is utilized to measure the tension during the experiment. The actual torque can be calculated through this formula:

$$\tau_{a1} = F_{t} \cdot r_{a}$$
(14)

where τa1 represents the actual torque in the static case, Ft represents the force generated by the tension sensor, ra is the length of the lower arm.

Experimental setup for dynamic case is shown in Fig. 5. The upper arm should always be perpendicular to the ground throughout the experiment. The hand lifts the bar and the force is completely generated by the biceps brachii too. In the dynamic experiment, the actual torque can be calculated roughly according to the formula (15), and the calculation model is shown in Fig. 6.

$$\tau_{a2} = r_{a} \cdot F = \left( {G_{1} + \frac{{G_{2} }}{2}} \right) \cdot r_{a} \cdot \sin \theta$$
(15)
Fig. 6
figure 6

The actual torque calculation model in the dynamic experiment

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where τa2 represents the actual torque in the dynamic experiment; G1 is the mass of the barbell; G2 represents the mass of the lower arm; θ is the angle of the elbow joint; F is the force perpendicular to the ground. In both cases, muscle activation a1(t) can be deduced through the actual torque and HMM. Moreover, the muscle activation mentioned in the article [27] is described as:

$$a_{2} \left( t \right) = \frac{{e^{A \cdot u\left( t \right)} - 1}}{{e^{A} - 1}}$$
(16)

Here, u(t) is the normalized sEMG signals. Lots of experiment results show that there is a deviation between a1(t) and a2(t). Therefore, q is used to describe this relationship, which can be described as:

$$q = \frac{{a_{2} \left( t \right)}}{{a_{1} \left( t \right)}}$$
(17)

According to the paper [32], RBFNN is a kind of local approaching neural networks, which is often used to handle non-linear control. What's more, compared with other neural networks, RBFNN has some sound characteristics, such as high approximation accuracy and fast training speed. Therefore, RBFNN will be used to get the real-time parameter q in this paper.

Generally speaking, RBFNN has three layers [33], namely input layer, hidden layer and output layer. The input layer is used to receive all elements which have strong correlations with the output results, and passes them to the hidden layer. The hidden layer performs multivariate nonlinear transformation on the input vector for feature extraction. The output layer is trained by the model to determine the output weights of neurons in each hidden layer, then outputs the weights after linear combination. The relationships between the three layers can be expressed by the following two formulas:

$$y = \sum\limits_{i = 1}^{n} {w_{i} } h_{i}$$
(18)
$$h_{i} = \exp \left\{ {\left. { - \frac{{\parallel x - c_{i} \parallel^{2} }}{{2 \cdot b_{i}^{2} }}} \right\}} \right.b_{i} i_{th}$$
(19)

where x and y are the input and output, respectively; wi represents the weights for the node i from the hidden layer to the output layer, i = 1,2,3,…n; hi is the activation function for hidden layer. ci and bi are the parameters of basis function and its width for ith node in the hidden layer.

The training process of RBFNN can be roughly divided into two stages. Firstly, the center of the activation function for the hidden layer is selected. In RBFNN, self-organized center selection is the most widely used learning algorithm and orthogonal least square method is the most commonly utilized self-organized center selection method. Therefore, the orthogonal least square method is chosen to select the center. Furthermore, the Gram–Schmidt algorithm is used to select and update the center. Next, determine the output weights of neurons in each hidden layer. Adaptive gradient descent is used to adapt the weights. All the values of RBFNN can be acquired when the output errors meet the requirements. In this paper, muscle activation a2(t), Normalized sEMG u(t), elbow joint angle θ(t) and raw sEMG r(t) are selected as the inputs of the neural network [34], while q is the output of the neural network. The inputs of neural network are in conformity with the following principles. One is that all inputs have certain impacts on the output. The other is that the numbers of inputs should not be too small and the comprehensive effects could make the output more accurate. The structure of the RBFNN is shown in Fig. 7.

Fig.7
figure 7

Schematic diagram of neural network

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3 Experimental Setup

In this research, a healthy male adult, aged 23, height 180.5 cm, weight 68 kg, has participated in the experiments. The employed experimental methods of this paper have been approved by the Institutional Review Board of Nanjing University of Aeronautics and Astronautics.

As stated above, this paper will verify the feasibility of the torque estimation strategy in static and dynamic cases. In the static case, just as shown in Fig. 4, the experimenter stands by the experimental platform, keeping the upper and lower arms perpendicular, and pulls the tension sensor by the biceps brachii. The force signals are collected by the tension sensor (JLBS-MD-10KG). In the dynamic case, just as demonstrated in Fig. 5, the experimenter holds barbells of different mass, raising and laying down them at a constant rate with different frequencies. The experiments are conducted at different frequencies (1/2 Hz, 1/3 Hz and 1/4 Hz) and loads (0 kg, 3 kg and 5 kg). Elbow angle is provided by the encoder (MINI-1024 ATI Industrial Automation) and angle calculation algorithm is carried out in the microcontroller (ALIENTEK STM32F103C8T6). The angle signals are sent to the development board (ALIENTEK STM32F407) through the WIFI module (ATK-ESP8266) and then transmitted to the Target PC. Finally, the relationship between the sEMG and the joint angle will be established in System identification module in the Host PC. The experimenter takes a 3-min break to relieve muscle fatigue after each test. Moreover, to avoid electromagnetic interference, the unnecessary motors and machines are removed.

Considering the correlation coefficient between estimated joint torque and measured joint torque, the validity of estimated joint torque can be measured as follows:

$$\rho_{{T_{a} ,T_{e} }} = \frac{{Cov\left( {T_{a} ,T_{e} } \right)}}{{\sigma_{{T_{a} }} \cdot \sigma_{{T_{e} }} }}$$
(20)

where Ta and Te represent the actual joint torque and estimated joint torque; Cov defines the covariance. In addition, σTa and σTe indicate the standard deviation of actual and estimated joint torque. Apart from that, in order to calculate the error between the actual joint torque and the estimated joint torque, root mean square error (RMSE) and normalized root mean square error (NRMSE) are taken into account [17], which are shown as:

$$RMSE = \sqrt {\frac{{\sum\nolimits_{t = 1}^{N} {\left( {T_{at} - T_{et} } \right)} }}{N}}$$
(21)
$$NRMSE = \frac{RMSE}{{T_{et\max } - T_{et\min } }}$$
(22)

where N is the number of joint torque data; Tat, Tet are the actual joint torque and estimated joint torque at time t. Tetmax, Tetmax represent the maximum and minimum of Tet.

4 Experiments and Analysis

Results of the static case are shown in Fig. 8. Considering the contingency in a single experiment, three experiments are conducted under the static condition in search of a more general conclusion. The measuring standards ρ, RMSE and NRMSE in each experiment are listed in Table 2. The results of experimental methods (Method 1–3) proposed in this research and the articles [17, 28] are shown successively. In the Method 2, the researcher used the IMU to get the joint angle and did not consider the optimal factor in the process of getting muscle activation while the method just established a non-linear relationship between the sEMG and joint torque in the Method 3.

Fig. 8
figure 8

Above are relevant experimental data in the first experiment. T1, T2, T3 represent the estimated torque obtained by Methods 1, 2 and 3, respectively. E1, E2, E3 represent errors between actual torque and estimated torque obtained by Methods 1, 2 and 3, respectively. a Raw sEMG signals b Actual torque c Estimated torque obtained by three estimation strategies d Errors between actual torque and three different estimated torques

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Table 2 Comparisons of the experimental results obtained by methods 1, 2, 3 (M1, M2, M3) in the three experiments
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In the static experiments, the average correlation coefficient between actual joint torque and estimated joint torque obtained by Methods 1, 2 and 3 are 97.88%, 95.90% and 94.86%, respectively. The results show that the estimated joint torque calculated by Method 1 has better correlations with the actual joint torque than that by Methods 2 and 3. In addition, according to the pictures (c) in three trials, the estimated torque obtained by Method 1 is highly in accordance with the trend of actual torque, demonstrating that Method 1 can identify the variation like muscle changes, which cannot be accomplished by Methods 2 and 3. The average RMSE between actual joint torque and estimated joint torque obtained by Methods 1, 2 and 3 are 1.447 Nm, 3.908 Nm and 5.714 Nm, respectively. The average NRMSE between actual joint torque and estimated joint torque obtained by Methods 1, 2 and 3 are 5.446%, 9.200% and 14.574%, respectively. Besides, according to the pictures (d) in three trials, errors obtained by Method 1 are mainly focus on the range of ± 4 Nm, which are acceptable. Plus, errors obtained by Method 1 are smaller than that by Methods 2 and 3, which indicates that at most time, estimated joint torque obtained by Method 1 is closer to actual joint torque than that by Methods 2 and 3. In conclusion, in the static experiment, Method 1 has a better performance than Methods 2 and 3.

In the dynamic case, experiments have been conducted three times at each frequency and load. The results of the dynamic experimental method under 0 kg, 3 kg and 5 kg are shown in Figs. 9, 10 and 11, respectively. Moreover, the average values of ρ, RMSE and NRMSE at different frequencies (1/2 Hz, 1/3 Hz and 1/4 Hz) and loads (0 kg, 3 kg and 5 kg) are recorded in Tables 3, 4 and 5.

Fig. 9
figure 9

Above are experimental results under no load condition. T1, T2, T3 represent the estimated torque obtained by Methods 1, 2 and 3, respectively. E1, E2, E3 represent errors between actual torque and estimated torque obtained by Methods 1, 2 and 3, respectively. A1, A2 are the actual angle and the estimated angle; a Raw sEMG signals b Actual angle and estimated angle c Actual torque d Estimated torque obtained by three estimation strategies e Errors between actual torque and three different estimated torques

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Fig. 10
figure 10

Above are experimental results under 3 kg load condition. T1, T2, T3 represent the estimated torque obtained by Methods 1, 2 and 3, respectively. E1, E2, E3 represent errors between actual torque and estimated torque obtained by Methods 1, 2 and 3, respectively. A1, A2 are the actual angle and the estimated angle; a Raw sEMG signals b Actual angle and estimated angle c Actual torque d Estimated torque obtained by three estimation strategies e Errors between actual torque and three different estimated torques

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Fig. 11
figure 11

Above are experimental results under 5 kg load condition. T1, T2, T3 represent the estimated torque obtained by Methods 1, 2 and 3, respectively. E1, E2, E3 represent errors between actual torque and estimated torque obtained by Methods 1, 2 and 3, respectively. A1, A2 are the actual angle and the estimated angle; a Raw sEMG signals b Actual angle and estimated angle c Actual torque d Estimated torque obtained by three estimation strategies e Errors between actual torque and three different estimated torque

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Table 3 Comparisons of the experimental results obtained by methods 1, 2, 3(M1, M2, M3) AT 1/2 Hz
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Table 4 Comparisons of the experimental results obtained by methods 1, 2, 3(M1, M2, M3) AT 1/3 Hz
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Table 5 Comparisons of the experimental results obtained by methods 1, 2, 3(M1, M2, M3) AT 1/4 Hz
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Just as the pictures (b) shown in Figs. 9, 10 and 11, the estimated angle and actual angle have positive correlations, and the errors between the actual and estimated angle are smaller enough to be accepted. Tables 3, 4 and 5 show that the estimated torque obtained by Method 1 has a better correlation with the actual torque than that by Methods 2 and 3. Except that, RMSE and NRMSE calculated by Method 1 are smaller than that by Methods 2 and 3 at most of time, which means the accuracy of estimated torque obtained by Method 1 is higher compared to the other two methods. Results show that compared to the other two existing torque estimation strategies, this method is proved to make progress in the aspect of torque estimation accuracy under different experimental conditions. The correlation coefficient increases by 2–9%; root-mean-square error (RMSE) reduces by 0.2–2.5 Nm; normalized root-mean-square error (NRMSE) reduces by 0.5–9.5%. Moreover, under the same load condition, the performances at high frequency (1/2 Hz) are better than low frequency (1/4 Hz) while worse than moderate frequency (1/3 Hz); at the same frequency, the performances under 0 kg are slightly worse than moderate load (3 kg) condition and better than the big load (5 kg). Two main reasons can explain these two phenomena. Firstly, the degree of muscle activation is relatively low as the experimenter lifts the low-mass barbell at a low speed, resulting in relatively poor estimation accuracy. The other is that the bigger the load is, the longer the load time is, the more energy the experimenter consumes, and the greater possibility of muscle shaking and fatigue. It requires that the torque estimation strategy can accurately detect the muscle condition in a very short time and make the corresponding changes, which is no doubt a big challenge even if the new proposed method can detect muscle changes to a certain extent.

5 Conclusions and Future Works

In order to identify the motion intention, a new torque estimation strategy is proposed in this paper which integrates sEMG, tension sensor and angle signals to estimate real-time elbow joint torque. Firstly, system identification is proposed to estimate elbow angle, which can be used in the Hill-type muscle model, and replace the use of the angle transducer. In addition, the adaptive adjustment factor is presented to describe the non-linear relationship between sEMG signals and the muscle activation, enabling the estimated torque to be more accurate. Finally, static and dynamic experiments are conducted respectively to verify the improvements of this strategy in terms of torque estimation accuracy.

The purpose of intention recognition is controlling rehabilitation exoskeleton robots and helping patients with hemiplegia to carry out limb rehabilitation training. Thus, developing control strategies based on this torque estimation method to control the rehabilitation exoskeleton robots will be the focus of future works.