A Convolutional Neural Network-Based Broad Incremental Learning Filter for Attenuating Physiological Tremors in Telerobot Systems

Abstract

While master-slave teleoperated robotic systems have extensive applications in practice, the physiological tremors can easily affect the control accuracy and even destroy the stability of the closed-loop control systems during operation. Hence, the development of some effective approaches for counteracting physiological tremors is of both theoretical and practical importance. In this paper, a broad learning network-based filter integrating a deep learning network and modified incremental learning algorithms is proposed to reconstruct and compensate for tremor signals. To strengthen the recognition of correlations between different moments, the lateral connectivity structure is adopted to obtain multi-scale feature maps. Each feature window is obtained from multi-scale feature maps generated by the convolutional neural network, which has an advantage that makes the feature nodes fuse the feature information of long time series and short time series by the lateral connection. The broad learning network is a unique construction, which only needs to obtain the input and the output to conveniently calculate the connection weights by the pseudo-inverse without involving backpropagation. It is known that the relation between the data X and the label Y can be represented as $XW=Y$, and the solution W can be obtained by the pseudo-inverse $W=X^{+}Y$. In addition, to guarantee the ill-posed problem, a ridge regression algorithm is used for the pseudo-inverse calculation. The effectiveness of our raised network architecture is illustrated by comparative simulation and experiment results.

1. Introduction

It is well known that teleoperation systems have gained widespread applications in many fields, such as medicine, space exploration, etc.; see [1,2,3] for examples. For such applications, it is generally required that the telerobot or its end-effector moves along the same trajectory as the operator hand to accomplish some specific tasks [4,5,6]. According to each human’s mechanism, slight tremors always exist in every human hand and are not pathological but natural [7,8]. During teleoperation, these physiological tremors can cause movement deflections of the telerobot from the objective. The physiological tremors have characteristics, i.e., a small amplitude range between 50 μm and 100 μm and a domain frequency range between 8 Hz and 12 Hz [9,10]. Hence, in some occasions requiring an accuracy of about 10 μm, such as microsurgery [11,12], the physiological tremor attenuation is a significant yet challenging problem.

To overcome the problem, numerous studies have been performed, with particular emphasis on cancelling out the physiological tremors, see [13,14], just to list a few. Generally, linear low-pass filters (LPFs) are employed for canceling tremors, but digital filters have a disadvantage that introduces amplitude and phase distortions [15]. Related studies have been proposed to overcome the limits of digital filters [16,17,18]. An adaptive band-limited multiple Fourier linear combiner with Kalman filter (BMFLC-KF) has been proposed to provide accurate time-frequency decomposition components in the band-limited signal for filtering and compensation [19]. Moreover, an autoregressive model with Kalman filter (AR-KF) has been proposed to estimate a real-time tremor based on the previous outputs [20]. These methods can reduce most phase distortion but still have a problem with a phase delay of 5 to 10 ms. Later, the multistep prediction-based algorithms were proposed to address the existence of phase delay in sensors and filter for accurate canceling of real-time tremors [21]. Because of the fusion of various algorithms, the computation costs are increased, which can be alleviated by the reduced-order Kalman-based enhanced band-limited multiple Fourier linear combiner (RKE-BMFLC) [22].

Although the algorithms mentioned above have a good performance in predicting tremors, they have some restrictions that need to be removed: the AR-KF method is a linear prediction model, which characterizes tremor signals as linear Gaussian distribution. It needs to consider the characteristic of tremors; when applying the BMFLC-KF method, a careful choice of proper parameters is required, since even a small frequency gap can make the method ineffective [19]. To overcome the problems, some small-scale sample machine learning algorithms are raised to compensate tremors; see typically the support vector machine (SVM)-based approaches [23,24], which exhibit a good generalization capability and are computationally attractive. Note that SVM may not be assigned exactly in the learning process and is not sensitive to small amplitude signals, and this could cause substantial losses in accuracy [25]. Such an issue can be well addressed by the deep-learning methodology. Moreover, the deep learning network has a high accuracy in predicting small signals due to its strong network depth [26]. In addition, to improve the accuracy of tremor prediction, a multi-step long-short-term memory (MS-LSTM) method has been used in [27].

While the deep learning network is a powerful tool, due to the need for many hyperparameters, and the use of complicated structures, a time-consuming training process is generally required. In addition, the deep structures are the black box that is difficult to analyze. Recently, a novel network structure broad learning system (BLS) has been proposed in [28]. Unlike the deep network, BLS has a simple structure and trains without hyper-parameters. It updates the weights of network by the pseudo-inverse calculation. In [29], it shows that BLS has the generalization capability in function approximation and proposes variant structures of original BLS. However, combined with our recent study [30], we find that the original BLS has some limitations in extracting the features’ process, as the original BLS cannot extract the potential information of time-frequency features for non-stationary signals. Thus, we point out that the original BLS must be redesigned as a novel variant to accommodate tremor signals.

Motivated by the above, in this study, we propose a novel convolutional neural network-based broad learning system filter (CNN-BLSF) combined with broad expansion: incremental learning to predict and compensate for tremor signals. In summary, the work of this study has the following novelties and contributions:

• For the original BLS, the task of feature extraction is generally achieved by the sparse autoencoder (SAE), which could not reflect the underlying relationship of the adjacent time sequence. To overcome the problem, a convolutional neural network is introduced to extract features, based on which a novel network structure, called the convolutional neural network-based BLS (CNN-BLS), is established.
• With our raised convolutional neural network, various feature maps, including feature information in long and short time series regions, can be well obtained. A lateral connection structure, such as the residual network, is employed in the network so that the feature information of the time sequence can be extracted. Our purpose of design is to construct multi-scale feature maps and fuse them.
• With the unique construction of BLS, an incremental learning algorithm, which adapts to BLS and can improve the performance of systems without retraining, is raised to remodel the network. Combining such an algorithm with our CNN-BLS, a convolutional neural network-based broad learning system with incremental learning (CNN-BLS-IL) is developed. It is worth mentioning that the raised incremental learning algorithm is better than the original incremental learning algorithm and thus could be more effective, as illustrated in Section 4.1.3.

The rest of this paper is organized as follows. The problem to be investigated is presented in Section 2. Some control strategies are given in Section 3. The proposed CNN-BLS-based approach is described in Section 4. The simulation and experiment results are shown in Section 5 to demonstrates the effectiveness of our approach. The paper is concluded in Section 6.

2. Problem Description

A teleoperated robotic system is composed of three parts: (master, channel, and slave). The effects of tremors are concentrated in the master part (haptic device), which leads the slave part (slave robot) to generate noisy signals.

In Figure 1, the haptic device (Touch X) has 6 degrees of freedom (DOFs). Its homogeneous transformation matrix between the i-th joint and the (i-1)-th joint generates deviation variables when the operator’s hand with tremor touches the 6-th joint of Touch X. Firstly, the original joint angle ${\theta }_o$ and the original link offset $d_o$ of each joint change as ${\theta }^{*}$ and $d^{*}$, which can be described as

$$\begin{array}{cc}\hfill & {\theta }_i^{*}={\theta }_{oi}+{\theta }_{mi}+\Delta {\theta }_i,i=1,2,\dots ,6\hfill \\ \hfill & d_i^{*}=d_{oi}+d_{mi}+\Delta d_i,i=1,2,\dots ,6,\hfill \end{array}$$

(1)

where ${\theta }_{mi}$ and $d_{mi}$ are the joint angle and the link offset generated by the operator, and $\Delta {\theta }_i$ and $\Delta d_i$ are the deviation joint angle and the deviation link offset caused by tremors. i represents i-th joint of the haptic device. With these disturbance variables, the homogeneous transformation matrix between adjacent joints ${}_{i-1}^{i}T$, according to the modified D-H notation, can be rewritten as

$$\begin{array}{cc}\hfill & {}_{i-1}^i{T}^{*}=\left[\begin{array}{cccc}c{\theta }_i^{*}& -s{\theta }_i^{*}& 0& a_{i-1}\\ s{\theta }_i^{*}c{\alpha }_{i-1}& c{\theta }_i^{*}c{\alpha }_{i-1}& -s{\alpha }_{i-1}& -s{\alpha }_{i-1}{d}_i^{*}\\ s{\theta }_i^{*}s{\alpha }_{i-1}& c{\theta }_i^{*}s{\alpha }_{i-1}& c{\alpha }_{i-1}& c{\alpha }_{i-1}{d}_i^{*}\\ 0& 0& 0& 1\end{array}\right],\hfill \\ & s=sin,c=cos,i=1,2,\dots ,6,\hfill \end{array}$$

(2)

where $\alpha$ is the kinematic link twist. The six matrixes ${}_{i-1}^i{T}^{*}$ ($i=1,2,\dots ,6$) can be multiplied together as the pose matrix of the end effector ${}_0^6{T}^{*}$,

$${}_6^0{T}^{*}=\left[\begin{array}{cccc}{n}_{x1}^{*}& n_{x2}^{*}& n_{x3}^{*}& p_x^{*}\\ n_{y1}^{*}& n_{y2}^{*}& n_{y3}^{*}& p_y^{*}\\ n_{z1}^{*}& n_{z2}^{*}& n_{z3}^{*}& p_z^{*}\\ 0& 0& 0& 1\end{array}\right],$$

(3)

where $n_{xi}^{*}$, $n_{yi}^{*}$, $n_{zi}^{*}$ ($i=1,2,3$) are the rotational elements and $p_x^{*}$, $p_y^{*}$, $p_z^{*}$ are the position vector.

Influenced by tremors, the pose of the end effector change, which makes the slave robot deviates from its original track. The trajectory deviation over 100 μm results in failure on the occasion of requiring precise manual positioning [31,32,33]. The tremor filter needs to be designed in the teleoperation system for the tremor compensation to eliminate tremors.

3. Control Strategies

Considering the stability and precision of the teleoperation system, we integrate some control strategies into the system’s control mode. As shown in Figure 2, the operator gives a command to the haptic device to obtain an actual trajectory $S_{m1}$, and $S_m$ is a filtered trajectory. After receiving the operation signal, the robot follows the haptic device and generates a trajectory $S_s$. $S_e$ is the position error of $S_m$ and $S_s$. $J_m^{+}$ and $J_s^{+}$ are the pseudo-inverse of the Jacobian of the master device and the slave device, respectively. ${\dot{q}}_{md}$, $q_m$ are the master desired joint velocity, the master joint angle. ${\dot{q}}_{sd}$, $q_s$ are the slave desired joint velocity and the slave joint angle. ${\tau }_m$, ${\tau }_s$ and ${\tau }_o$ are the master torque and the slave torque generated by the PD controller, and the torque generated by the operator.

3.1. Haptic Force Feedback

Force feedback is integrated into the haptic device to improve tactile perceptual interaction between an operator and a slave robot, which can be written as [34,35,36].

$$\begin{array}{cc}\hfill & F_m=K_m(S_m-S_s)=K_m{S}_e\hfill \\ \hfill & F_s=K_s(S_m-S_s)=K_s{S}_e.\hfill \end{array}$$

(4)

Its control object is the tracking error $S_e$, i.e., $S_e=S_m-S_s$. The value of feedback force decreases with the tracking error $S_e$ reducing, and finally the whole control system converges.

Depending on the adjustment of the feedback force, $S_m$ and $S_s$ can be modified to the desired trajectories $S_{md}$ and $S_{sd}$. The pose information $S_{md}$ and $S_{sd}$ are translated into the joint velocity information ${\dot{q}}_{md}$ and ${\dot{q}}_{sd}$ based on the Jacobian matrix.

$$\begin{array}{cc}\hfill & {\dot{S}}_{md}=J_m{\dot{q}}_{md}\to {\dot{q}}_{md}=J_m^{+}{\dot{S}}_{md}\hfill \\ \hfill & {\dot{S}}_{sd}=J_s{\dot{q}}_{sd}\to {\dot{q}}_{sd}=J_s^{+}{\dot{S}}_{sd}.\hfill \end{array}$$

(5)

3.2. PD Controller

The joint information is obtained as the control variables from Equation (5). The PD controller is designed for the master part and the slave part, which can be described as follows:  

$$\begin{array}{cc}\hfill & {\tau }_m=P_m(q_m-q_{md})+D_m({\dot{q}}_m-{\dot{q}}_{md})=P_m\Delta q_m+D_m\Delta {\dot{q}}_m\hfill \\ \hfill & {\tau }_s=P_s(q_s-q_{sd})+D_s({\dot{q}}_s-{\dot{q}}_{sd})=P_s\Delta q_s+D_s\Delta {\dot{q}}_s,\hfill \end{array}$$

(6)

where ${\tau }_m$ and ${\tau }_s$ are the master torque and the slave torque. $P_m$, $P_s$ are proportion gain of the controller and $D_m$, $D_s$ are differential gain of the controller. The control objects of the controller are the joint angle $\Delta q_m$, $\Delta q_s$ and the joint velocity $\Delta {\dot{q}}_m$, $\Delta {\dot{q}}_s$.

To describe the controller module clearly, a block diagram of the designed controller is given in Figure 3, where P is the proportion control unit, and D is the differential control unit. $\delta$ denotes m (master device) or s (slave device). The master device (Touch X) and the slave device (Telerobot) are both 6 DOFs, where each joint is driven by a motor and is controlled by a power steering gear. The micro-controller unit (MCU) controls the power steering gear to move devices by torque.

3.3. Model Structure of CNN-BLSF

3.3.1. Physical Model Structure of CNN-BLSF

In order to address the effects caused by physiological tremors, a novel CNN-BLS-based filter is proposed in this work. The physical model of CNN-BLSF contains: (i) sampling unit; (ii) tremor filtering unit; (iii) control unit. In Figure 4, each unit of the whole CNN-BLSF system is shown.

• As the central part of the sampling module, the internal measurement unit (IMU) is used to sample the operator’s real-time hand movements. It can obtain a three-axis position acceleration and a three-axis joint angular velocity from a human.
• In the tremor filtering unit module, the CNN-BLS network algorithm is integrated to forecast the tremor signals. The compensation tremor signals $x_p$, $y_p$, $z_p$ and ${\theta }_{xp}$, ${\theta }_{yp}$, ${\theta }_{zp}$ have the same magnitude but opposite phase compared with tremor signals, which can neutralize the tremor signals in the actual signals x, y, and z.
• The control module integrates the calculation of inverse kinematics, the driver of a single joint, and the motion feedback of deflection sensors, which converts inverse kinematics into motion control variables for the robot manipulator.

Figure 4. Block diagram of the tremor filter.

Figure 4. Block diagram of the tremor filter.

3.3.2. Mathematical Model Structure of CNN-BLSF

To better analyze the relationship between the input and output of the tremor filter, the mathematical model is shown in Figure 5, where $d\left(k\right)$, $n\left(k\right)$ and $n_p\left(k\right)$ are a desired signal, a faint signal caused by tremors and a predicted tremor signal, respectively. An actual signal $S_1\left(k\right)$ with the tremor can be expressed as

$$S_1\left(k\right)=d\left(k\right)+n\left(k\right).$$

(7)

The input data of the CNN-BLSF is ($S_1\left(k\right)$, $n_p\left(k\right)$), and the output of the CNN-BLSF $S\left(k\right)$ is

$$S\left(k\right)=S_1\left(k\right)-n_p\left(k\right)=d\left(k\right)+n\left(k\right)-n_p\left(k\right)=d\left(k\right)+\Delta n.$$

(8)

According to the Equation (8), when $n\left(k\right)=n_p\left(k\right)$, the end item $\Delta n=n\left(k\right)-n_p\left(k\right)$ is zero. We expect the tremor attenuation filter to be designed so that the predicted error $\Delta n$ is closer to zero.

Figure 5. Mathematical model of the tremor filter.

Figure 5. Mathematical model of the tremor filter.

4. Design of CNN-BLS Tremor Filter

To guarantee the predicted error $\Delta n$ is closer to zero, a novel variant of the broad learning network is proposed in this paper, which integrates the advantage of the broad learning network (fast) and convolutional neural network (accurate). In addition, an incremental learning algorithm can be applied to specific networks, such as the broad learning network. To better complete the build of the tremor filter, the increment learning is combined with our designed network.

4.1. Algorithm of CNN-BLS

4.1.1. Broad Learning Network

Professor Chen and Liu proposed a novel and simple network frame in [28]. Moreover, Chen and Liu further demonstrated the approximation capability of the BLS for generic functions and proposed several variants of the original BLS in [29]. Firstly, as shown in Figure 6, $G=[x,y,z]$ is denoted as the input data, where X, Y, and Z are the three-axis input signals, respectively. The input data $G=[x,y,z]$ are randomly mapped to n groups feature nodes, which can be expressed as the equation of the form

$$Z_i=\varphi \left(G{W}_{f_i}+{\beta }_{f_i}\right),i=1,\dots ,n,$$

(9)

where $W_{f_i}$ and ${\beta }_{f_i}$ are random weights and random biases. $\varphi (·)$ represents a random mapping function.

Remark 1. 

For the original BLS, the feature layer is obtained from a random mapping function, which regularizes the broad network. In addition, SAE is used to fine-tune the weights of feature extraction ($W_{fi}$, ${\beta }_{f_i}$), and it has a good performance in classification. However, the original BLS feature extraction method cannot obtain enough information for time-series features that adjacent sequences have connections. Thus, a new network structure for feature extraction is proposed to replace the SAE of the original BLS.

All generated feature nodes can be written as $Z^n=[Z_1,Z_2,\dots ,Z_n]$, and then, are activated to be m enhancement nodes by a activation function $\xi (·)$. The j-th enhancement node is represented as

$$H_j=\xi \left(Z^n{W}_{e_j}+{\beta }_{e_j}\right),j=1,\dots ,m,$$

(10)

where $W_{e_j}$ and ${\beta }_{e_j}$ are also random weights and random biases. Denote the input layer as

$$I_n^m=\left[Z_i\right|H_j],i=1,\dots ,n, j=1,\dots ,m.$$

(11)

Finally, the output of BLS can be represented as

$$Y=\left[Z_1,\dots ,Z_n\mid H_1,\dots ,H_m\right]W^{*}=I_n^m{W}^{*}.$$

(12)

According to the pseudo-inverse calculation equation, the Equation (12) can be rewritten as the following form

$$W^{*}={\left(I_n^m\right)}^{+}Y={\left({\left(I_n^m\right)}^T{I}_n^m\right)}^{-1}{\left(I_n^m\right)}^{T}Y.$$

(13)

A ridge regression algorithm is used to avoid the inexistence of pseudo-inverse of $I_n^m$ [37], and Equation (13) is rewritten as

$$W^{*}={(\lambda E+{\left(I_n^m\right)}^T{I}_n^m)}^{-1}{\left(I_n^m\right)}^{T}Y.$$

(14)

4.1.2. Structure of Our CNN-BLS Network

In Figure 7, the internal algorithm of the designed tremor filter is shown and the algorithm of our designed network is given in Algorithm 1, where $\gamma$ represents the number of feature maps, $\delta$ represents the number of enhancement nodes and $f^{\left(\gamma \right)}(·)$ represents the output of the $\gamma$-th pooling layer. We point out that the primary network framework is the BLS. Since the original BLS does not extract time feature sequences well, the network fuses a simple CNN with a lateral connection approach [38].

Algorithm 1: Convolutional Broad Learning Filter Algorithm: Increment of Feature Mapping Nodes and Enchancement Nodes.

The input data $G=[x,y,z]$ are sent to the convolutional network layer, which contains $\gamma$ convolution and pooling process and can yield some feature sequences concerning the input. In Figure 8, the convolution and pooling operation process is given. Firstly, the input time series can form the feature maps. The process uses auto-padding to generate a $3\times 3$ feature map of the same size as the convolution kernel. Then, the feature map can be mapped as feature series based on the max pooling. Finally, through the process of convolution and pooling, the i-th feature sequence $\overrightarrow{z_i}$ is represented as

$$\overrightarrow{z_i}=f^{\left(i\right)}\left(G{W}_{con{v}_i}\right),i=1,\dots ,\gamma ,$$

(15)

where $W_{con{v}_i}$ is the kernel weight of the i-th convolution layer and $f^{\left(i\right)}(·)$ represents the output of the i-th pooling layer. Denote $\gamma$ feature sequences as network feature nodes, and they are written as

$$Z^{\gamma }=[\overrightarrow{z_1},\dots ,\overrightarrow{z_{\gamma }}]=[Z_1,\dots ,Z_{\gamma }].$$

(16)

Similar to the Equations (10)–(14), the $\delta$ enhancement nodes can be generated by $Z^{\gamma }$ and they can be written as

$$H_j=\xi \left(Z^{\gamma }{W}_{e_j}+{\beta }_{e_j}\right),j=1,\dots ,\delta .$$

(17)

Our designed network input layer can be written as

$$II=\left[Z_i\right|H_j],i=1,\dots ,\gamma ,j=1,\dots ,\delta .$$

(18)

We have

$$W={(\lambda E+{\left(II\right)}^{T}II)}^{-1}{\left(II\right)}^{T}Y.$$

(19)

Figure 8. Process of single convolution.

Figure 8. Process of single convolution.

Remark 2. 

We point out that the BLS filter shows the capacity to cancel tremor is unpowerful because the feature extraction part may not effectively obtain corresponding feature maps. Hence, the above work shows our employed CNN to replace the SAE of BLS.

To enrich the feature extraction process of the network, we propose an approach concerning multi-scale extraction. In other words, it can be defined as a feature fusion method. By obtaining these multi-scale feature maps, we solve the problem of BLS and obtain better performance.

4.1.3. Broad Expansion: Incremental Learning

For the specific broad network construction, the incremental learning algorithm can be applied to improve the performance of the network, which contains three incremental methods: (i) increment of additional enhancement nodes; (ii) increment of additional feature mapping nodes; (iii) increment of input. To improve the regression capacity of the proposed algorithm, methods (i) and (ii) are applied, which have a fast remodeling to form robust feature extraction and non-linear capability of the model.

Increment of feature nodes: Suppose the tracking performance of the model is terrible. In that case, the reason may be that the feature layer cannot obtain underlying input data information, leading to the corresponding enhancement nodes not being enough to learn. Here, we show the detail of method (i), as follows.

Assume the initial model structure comprises: $\gamma$ convolution layers and pooling layers, and $\delta$ enhancement nodes. The additional feature nodes are denoted as

$${\overrightarrow{z}}_{\gamma +1}=f^{(\gamma +1)}\left(G{W}_{con{v}_{\gamma +1}}\right),$$

(20)

The corresponding yeild enhancement nodes can be respresented as

$$\Delta H_{\gamma +1}=\left[\xi \left({\overrightarrow{z}}_{\gamma +1}\Delta W_1+\Delta {\beta }_1\right),\dots ,\xi \left({\overrightarrow{z}}_{\gamma +1}\Delta W_m+\Delta {\beta }_{\delta }\right)\right],$$

(21)

where $\Delta W_i$ and $\Delta {\beta }_i$ are incremental random parameters corresponding to the $(n+1)$-th feature node. Then, denote the new input layer as

$$I{I}_{\gamma +1}=\left[II\right|{\overrightarrow{z}}_{\gamma +1}|\Delta H_{\gamma +1}],$$

(22)

and its pseudo-inverse can be calculated by the following equations:

$${\left(I{I}_{\gamma +1}\right)}^{+}=\left[\begin{array}{c}{\left(II\right)}^{+}(E-D{B}^T)\\ B^T\end{array}\right],$$

(23)

where $D={\left(II\right)}^{+}\left[{\overrightarrow{z}}_{\gamma +1}\right|\Delta H_{\gamma +1}]$, $C=I{I}_{\gamma +1}-II·D$ and

$$B^T=\left\{\begin{array}{cc}{C}^{+}\hfill & \mathrm{if}C\ne 0\hfill \\ {\left(\overrightarrow{1}+D^{T}D\right)}^{-1}{D}^T{\left(II\right)}^{+}\hfill & \mathrm{if}C=0\hfill \end{array}\right..$$

(24)

Thus, the new weights can be obtained as

$$W_{\gamma +1}=\left[\begin{array}{c}W-D{B}^{T}Y\\ B^{T}Y\end{array}\right].$$

(25)

Remark 3. 

It is noted that the incremental learning algorithm is a practical and fast remodeling approach. It can reduce the computational complexity of numerous training data. Since the feature extraction part of the original BLS is replaced by our method, the incremental learning with added feature nodes should be different from the original incremental learning. According to the above contents, the novel incremental learning is combined with multi-scale feature maps, where the added feature nodes are given by other scale feature maps.

Increment of enhancement nodes: If the non-linear degrees of the fitting curves are not enough to fit the target curves, the reason may be that the number of enhancement nodes is insufficient and can not achieve a good performance. Similar to method (i), the network connection weight of method (ii) can be calculated and obtained. Denote the new input layer as

$$I{I}^{\delta +1}=\left[II\right|\xi (Z^{\gamma }{W}_{e_{\delta +1}}+{\beta }_{e_{\delta +1}})]=\left[II\right|H_{\delta +1}],$$

(26)

where $W_{e_{\delta +1}}$ and ${\beta }_{e_{\delta +1}}$ are randomly generated, and we have

$$W^{\delta +1}=\left[\begin{array}{c}W-D{B}^{T}Y\\ B^{T}Y\end{array}\right],$$

(27)

where $D={\left(II\right)}^{+}\left[H_{\delta +1}\right]$, $C=I{I}^{\delta +1}-II·D$ and

$$B^T=\left\{\begin{array}{cc}{C}^{+}\hfill & \mathrm{if}C\ne 0\hfill \\ {\left(\overrightarrow{1}+D^{T}D\right)}^{-1}{D}^T{\left(II\right)}^{+}\hfill & \mathrm{if}C=0\hfill \end{array}\right..$$

(28)

5. Experiments

To explain the performance of different algorithms, sum square error (SSE), root mean square error (RMSE) and R-square ($R^2$) are applied, which can be written as

$$SSE=\sum _{t=1}^T{(n\left(t\right)-n_p\left(t\right))}^2$$

(29)

$$RMSE=\sqrt{\frac{{\sum }_{t=1}^T{(n\left(t\right)-n_p\left(t\right))}^2}{N}}$$

(30)

$$R^2=1-\frac{{\sum }_{t=1}^{T}n\left(t\right)-n_p\left(t\right)}{{\sum }_{t=1}^{T}n\left(t\right)-\overline{n}\left(t\right)}$$

(31)

where T and N are the period and the number of samples. $n\left(t\right)$, $n_p\left(t\right)$ and $\overline{n}\left(t\right)$ are the real values, the predicted values and the average of the real values, respectively. The comparative simulation is processed in the MATLAB robotics toolbox and Libsvm toolbox. Figure 9 shows a simulated robot manipulator and the desired and actual motion and pose.

5.1. Parameters Setting

According to the standard D-H table construction rules, we set four parameters (d, a, alpha, offset) to build the simulation robot, respectively. See

Table 1. D-H table of the simulated robot manipulator.

i	Theta	d/mm	a/mm	Alpha/rad	Offset/rad
1	q1	105.03	0	1.571	0
2	q2	0	−174.42	0	−1.571
3	q3	0	−174.42	0	0
4	q4	75.66	0	1.571	−1.571
5	q5	80.09	0	−1.571	0
6	q6	44.36	0	0	0

for specific parameter information. For the different filters, the parameters of filters were shown in

Table 2. Parameters and initial conditions of various methods.

Various Methods	Parameters and Initial Conditions
SVM	$p=0.4$; $n=0.1$; Radial Basis Function (RBF)
BLS	$C=2^{-30}$; $s=0.8$; $N11=8$; $N2=10$; $N33=80$
BLS-IL	$C=2^{-30}$; $s=0.8$; $N11=8$; $N2=10$; $N33=10$; $m1=10$; $m2=10$; $m3=50$
CNN-BLS	$s=0.8$; $N11=3$; $N2=10$; $N33=8$; $InitialLearnRate=1e-3$; Stochastic Gradient Descent with Momentum (SGDM); $\gamma =3$
CNN-BLS-IL	$s=0.8$; $N11=3$; $N2=10$; $N33=8$; $InitialLearnRate=1e-3$; SGDM; $m1=3$; $m2=3$; $m3=5$; $\gamma =3$;

.

Here, the sample time is 50 s, and the interval is set as 1 s. To achieve the construction of the SVM filter, we used the Libsvm toolbox, where p and n represent the loss parameter of epsilon-SVR, and RBF is the type of kernel function. For other filters, C is the regularization parameter for sparse regularization, and s is the shrinkage parameter for enhancement nodes. $N11$, $N2$, and $N33$ are the feature nodes of each window, the number of windows of feature nodes, and the number of enhancement nodes, respectively. $m1$, $m2$, and $m3$ are the number of feature nodes of each increment step, the number of enhancement nodes related to the incremental feature nodes of each increment step, and the number of enhancement nodes in each incremental learning, respectively. For the CNN, we used SGDM to update weights, where the initial learning rate is 0.001. The number of convolution layers and pooling layers is 3. As shown in Figure 10a,b, the track performance varies as the number of feature nodes increases. To guarantee the performance of track and track error, we proposed that the number of feature nodes is set as 30.

5.2. Simulated Tremor Signal

Considering the characteristics of tremors, we define the tremor signal as having a small amplitude and high-low frequency, which can be written as

$$\begin{array}{cc}\hfill & n\left(t\right)=0.04sin\left(0.055\pi t\right)+0.03sin\left(0.078\pi t\right)\hfill \\ & +0.05sin\left(0.083\pi t\right)+0.007sin\left(1000\pi t\right)+0.01cos\left(500\pi t\right)\hfill \end{array}$$

(32)

where the front part is the low-frequency signal and the back part is the high-frequency signal. Figure 11a shows the trajectory of our simulated tremor signal.

5.3. Tremor Forecast Results

Figure 11b shows the desired operation trajectory and the actual operation trajectory with tremor. Our actual operation trajectory deviates from the desired trajectory within 100 μm. The bias 100 μm leads to the failure of high-precision tasks. Figure 12a,b represent the track performance and the track errors between the different algorithms, respectively. From Figure 12a,b, we can observe that our designed algorithm has a good performance in regression and loss. The loss of CNN-BLS-IL converges to zero throughout, indicating that it almost overlaps with the actual tremor signal. In Figure 13b, it shows the recovery trajectory by the different filters. The specific evaluation metrics are given in

Table 3. Compare among evaluation results with different methods: SSE; RMSE; R2; Time.

Methods and Evaluations	Train		Test		Total Time	R2
	SSE	RMSE	SSE	RMSE
BLS	0.0184	0.0027	0.0184	0.0192	0.05	84.6%
BLS-IL	0.0161	0.0025	0.0191	0.0196	0.054	83.9%
CNN-BLS	0.0016	0.0008	0.0012	0.005	4.02	98.9%
CNN-BLS-IL(Our)	0.00073	0.00054	0.0005	0.0031	4.03	99.6%
SVM	0.0302	0.0246	0.0302	0.0246	0.1	76.1%

. In summary, the results verify the effectiveness of our proposed approach (CNN-BLS-IL) in the tremor attenuation.

6. Conclusions

In this paper, we propose a novel variant network CNN-BLSF for forecasting and compensating physiological tremors. At first, to simplify the calculation of model parameters, the structure of the original BLS is used as our rebuild basis. Then, the SAE of the original BLS is replaced by the CNN to extract the time series feature efficiently. Moreover, to better obtain in-depth feature information from the multi-feature layers, the structure of lateral connection is adopted. Moreover, the incremental learning algorithm is adopted to update the network in real-time, which can improve the accuracy by adding new nodes. Finally, compared with the different existing methods, our proposed filter is more efficient in forecasting tremors, and we believe our filter can be applied to the medical area requiring higher accuracy.

The work improves the correlation of series by using the lateral connection, which seems to be somewhat complex. In view of this, and the superiorities of long short term memory (LSTM) in solving sequence correlation problems, in future study, we will try to achieve the combination of LSTM with BLS.