Nonlinear System Identification of Tremors Dynamics: A Data-driven Approximation Using Koopman Operator Theory

Abstract

People who suffer from tremors have difficulty performing activities of daily living. Efforts in developing a model of a limb with tremors can pave the way for non-surgical tremor suppression techniques. However, due to the nonlinearity, developing an accurate model of tremors is challenging. This paper implements a data-driven method for approximating the Koopman operator, which is capable of presenting nonlinear dynamics in a linear framework and is promising for predicting the nonlinear system. A dynamic model of tremors is developed with ultrasound (US) image data collected from a patient with essential tremor as they grasp objects. The method is applied to predict the patient’s tremor dynamics and is compared with the nonlinear Hammerstein-Wiener system identification technique.

I. INTRODUCTION

In pathological terms, a tremor is a movement of a body part that occurs involuntarily, rhythmically, and oscillatory [1]. Various diseases, such as Parkinson’s disease (PD), essential tremor (ET), and multiple sclerosis, can cause pathological tremors [2]. Over 11 million people in the United States live with tremors caused by neurodegenerative diseases such as Parkinson’s disease and Essential tremor [3]. Individuals who suffer from tremors in their hands and arms have difficulty performing daily activities.

In tremor research, a tremor model could be used to determine the type of tremor present [4], the tremor phase and frequency [5], and the distribution of tremorogenic activity among the major superficial muscles [6]. Accurate modeling of tremor disorders also facilitates control of deep brain stimulation (DBS) [7], functional electrical stimulation (FES) of tremoring muscles [8], electrical stimulation of afferent pathways [8], and wearable tremor suppression system [9]. The majority of the techniques discussed above are based on tremor models. Therefore, the effectiveness of the controller is closely related to the accuracy of the model.

Models of tremor dynamics can be categorized into two types: physics-based models and data-driven models. Physics-based models are built based on the first principles, empirical expressions, or highly sophisticated numerical/visualization tools. In contrast, data-driven models utilize information from prior data to identify the characteristics of the system. Some related physics-based tremor models have been studied by researchers in [10]–[12]. However, the nonlinear and stochastic behavior make it difficult to construct accurate physics-based models of tremor.

Due to the wide availability of miniature or wearable sensing technology, it has been possible to generate a large amount of data during the tremoring process. As a result, data-driven modeling can be used to identify unknown dynamics associated with tremors. The model-free methods in data-driven models, such as recurrent neural network models, are capable to predict the tremor voluntary motion [13]. However, due to the fact that the models obtained from the neural networks are packaged as black boxes, which simply map inputs to outputs, these models provide little insight into the design of tremor suppression control techniques. Meanwhile, the model-based methods, are capable of identifying unknown dynamics from data and learning data-driven representations that enable the representation of nonlinear systems in a linear fashion [14]. When compared to model-free methods, model-based methods produce more physically meaningful models and do not suffer from issues associated with tuning and computation load. In recent years, data-driven methods have been applied to approximate the Koopman operator [15], including DMD [16] and EDMD [17]. With the Koopman operator, a nonlinear system can be approximated by a linear system. The advantage of the Koopman operator theory is that the Koopman representation can be solved using standard linear operator theory [18].

The main contribution of this work is to present a data-driven approach of Koopman operator in modeling real tremors with ultrasound image data of muscle fascicle velocity. A preliminary study was recently performed to show that ultrasound can be used as a novel method to characterize tremor during grasping motions [19]. US has significant advantages over accelerometry and surface electromyography (sEMG), the two most common methods of tremor suppression. As opposed to accelerometry, which only measures a limb’s oscillatory displacements, ultrasound is designed to detect the tremor activity directly within the muscle. In contrast to sEMG, which measures only the superficial muscles’ tremors, ultrasound is not affected by artifacts caused by stimulation and muscle cross-talk. Therefore, it allows direct visualization of superficial as well as deep muscles.

In this paper, we applied the data-driven approximation of the Koopman operator with EDMD described in [17] and extend the technique by involving the best-fit linear regression to avoid introducing a weight matrix that needs to be tuned. We then collected US images from a patient with tremors as the patient grasped objects. Finally, we demonstrate that the proposed approach represents the system’s dynamics more accurately than nonlinear Hammerstein-Wiener model. To the best of the author’s knowledge, ultrasound imaging data has never been used to identify a dynamic model of a tremor system.

II. Koopman operator theory and Extended Dynamic Mode Decomposition

The objective of this section is to approximate the Koopman operator with EDMD [17] and to identify the nonlinear dynamics of tremor, which will be validated on an actual tremor system.

A. The Koopman Operator Theory

Consider a discrete-time nonlinear dynamical system $(\mathcal{M},n,\mathbf{F})$:

$$x_{n+1}=\mathbf{F}\left(x_n\right).$$ (1)

where $x\in \mathcal{M}\subseteq {ℝ}^N$ is the state, $n\in \mathbb{Z}$ is the discrete time, $N$ is the number of states, and $\mathbf{F}$ is the evolution operator.

The Koopman operator $\mathcal{K}$ is an infinite-dimensional linear operator that operates on the all measurement functions $g:\mathcal{M}\to ℝ$ of a system. The action of the Koopman operator is:

$$\mathcal{K}g=g\circ \mathbf{F}$$ (2)

where $\circ$ denotes the composition of $g$ with $\mathbf{F}$. In a discrete-time system with a timestep $\text{Δ}t$, Koopman operator $\mathcal{K}$ acts on measurement function g as follows:

$${\mathcal{K}}_{\text{Δ}t}g\left(x_n\right)=g\left({\mathbf{F}}_{\text{Δ}t}\left(x_n\right)\right)=g\left(x_{n+1}\right)$$ (3)

This means that, for any measurement $g$ and state $x_n$, the Koopman operator defines an infinite-dimensional linear dynamical system that propagates the observation of the state $g\left(x_n\right)$ to the next time step $g\left(x_{n+1}\right)$. Therefore, the state of the system $x_n$ at a given time can either be derived directly by nonlinear dynamics $\mathbf{F}$ or by updating all the measurement functions $g$ through the Koopman operator $\mathcal{K}$. Consider the eigenvalue problem

$$\mathcal{K}{\phi }_k={\lambda }_k{\phi }_k$$ (4)

where ${\lambda }_k$ is $k$th Koopman eigenvalues associated with the $k$th ${\phi }_k$ are the Koopman eigenfunctions. The original nonlinear dynamical system (1) and Koopman operator dynamical system (3) are linked by full-state observable $\mathbf{g}(\mathbf{x})=\mathbf{x}$, which can be be represented in terms of Koopman eigenfunctions $\phi$ as

$$\mathbf{x}=\mathbf{g}(\mathbf{x})=\sum _{k=1}^{\infty }{\mathbf{v}}_k{\phi }_k(\mathbf{x})$$ (5)

where ${\mathbf{v}}_k$ and ${\phi }_k$ are the $k$th Koopman mode and eigenfunction, respectively. Therefore, the representation of $\mathbf{F}(\mathbf{x})$ can be evaluated through Koopman operator as:

$$\mathbf{F}(\mathbf{x})=(\mathcal{K}\mathbf{g})(\mathbf{x})=\mathcal{K}\sum _{k=1}^{\infty }{\mathbf{v}}_k\left({\phi }_k\right)(\mathbf{x})=\sum _{k=1}^{\infty }{\lambda }_k{\mathbf{v}}_k{\phi }_k(\mathbf{x})$$ (6)

From (6), once ${\lambda }_k$,${\phi }_k$,${\mathbf{v}}_k$ are known, we can predict the future state. Next we will apply EDMD algorithm to approximate ${\lambda }_k$,${\phi }_k$,${\mathbf{v}}_k$ directly from data with a best-fit linear model.

B. Approximating the Koopman Operator with Extended Dynamic Mode Decomposition

The purpose of this section is to present EDMD [17] to approximate ${\lambda }_k$,${\phi }_k$,${\mathbf{v}}_k$ defined in (6). Suppose we have a dynamical system (1), a data set of snapshot pairs, ${\left\{\left(x_m,y_m\right)\right\}}_{m=1}^M$ and a dictionary of observables $\mathcal{D}$ as

$$\mathbf{X}=\left[{\mathbf{x}}_{\mathbf{1}},\cdots ,{\mathbf{x}}_{\mathbf{M}}\right],\mathbf{Y}=\left[{\mathbf{y}}_{\mathbf{1}},\cdots ,{\mathbf{y}}_{\mathbf{M}}\right],\mathcal{D}=\left\{{\psi }_1,\cdots ,{\psi }_{N_K}\right\}$$ (7)

where ${\mathbf{x}}_i\in \mathcal{M}$ and ${\mathbf{y}}_i\in \mathcal{M}$ are snapshots of the system state with ${\mathbf{y}}_i=\mathbf{F}\left({\mathbf{x}}_{\mathbf{i}}\right)$, $M$ is the number of snapshots and $\mathbf{\Psi }(\mathbf{x})=\left[{\psi }_1(\mathbf{x}),{\psi }_2(\mathbf{x}),\cdots ,{\psi }_{N_K}(\mathbf{x})\right]$, $N_k$ is number of the observables functions. Then define $\mathrm{\Upsilon }$ as

$$\mathrm{\Upsilon }(\mathbf{x})=\sum _{k=1}^{N_K}{a}_k{\psi }_k(\mathbf{x})=\Psi (\mathbf{x})\mathbf{a}$$ (8)

where $\mathbf{a}$ is the weight of elements in $\Psi$.

Apply Koopman operator $\mathcal{K}$ on (8), we have

$$(\mathcal{K}\mathrm{\Upsilon })(\mathbf{x})=(\mathbf{\Psi }\circ \mathbf{F}(\mathbf{x}))\mathbf{a}=\mathbf{\Psi }(\mathbf{x})(\mathbf{K}\mathbf{a})+r_1(\mathbf{x})$$ (9)

where $r_1(\mathbf{x})$ is the residual term.

$\mathbf{K}$ can be obtained by minimizing $r_1(\mathbf{x})$ as:

$$\mathbf{K}\triangleq {\mathbf{G}}^{†}\mathbf{A}$$ (10)

where $†$ denotes the pseudoinverse and

$$\mathbf{G}=\frac{1}{M}\sum _{m=1}^M\mathbf{\Psi }{\left({\mathbf{x}}_{\mathbf{m}}\right)}^{*}\mathbf{\Psi }\left({\mathbf{x}}_{\mathbf{m}}\right);\mathbf{A}=\frac{1}{M}\sum _{m=1}^M\mathbf{\Psi }{\left({\mathbf{x}}_{\mathbf{m}}\right)}^{*}\mathbf{\Psi }\left({\mathbf{y}}_{\mathbf{m}}\right)$$ (11)

Once $\mathbf{K}$ is obtained by (10), the eigenfunction $\phi$ can be approximated by:

$${\phi }_j(\mathbf{x})=\mathbf{\Psi }(\mathbf{x}){\xi }_j$$ (12)

Where ${\xi }_j$ is the $j$th eigenvector of $\mathbf{K}$ associated with the eigenvalue ${\lambda }_j$. Then define the vector-valued function $\mathbf{\Phi }$ as

$$\mathbf{\Phi }(\mathbf{x})=\left[{\phi }_1(\mathbf{x}),{\phi }_2(\mathbf{x}),\cdots ,{\phi }_{N_K}(\mathbf{x})\right]=\mathbf{\Psi }(\mathbf{x})\mathbf{\Xi }$$ (13)

where $\mathbf{\Xi }=\left[{\mathbf{\xi }}_1,{\mathbf{\xi }}_2,\cdots ,{\mathbf{\xi }}_{N_K}\right]$. From the full state observable (5), the entire vector-valued observable can be expressed as:

$$\mathbf{g}{(\mathbf{x})}^T={\mathbf{x}}^T=\sum _{k=1}^{N_k}{\phi }_k(\mathbf{x}){\mathbf{v}}_k^T=\mathbf{\Phi }(\mathbf{x})V^T+r_2(\mathbf{x})$$ (14)

where $\mathbf{V}=\left[{\mathbf{v}}_1,{\mathbf{v}}_2,\cdots ,{\mathbf{v}}_{N_K}\right]$ and $r_2(\mathbf{x})$ is the residual term. Similar to (10), to determine $\mathbf{v}$ is also a least-squares problem. By minimizing $r_2(\mathbf{x})$, $\mathbf{v}$ is obtained as:

$${\mathbf{V}}^{\mathbf{T}}\triangleq {\mathbf{P}}^{†}\mathbf{D}$$ (15)

where $†$ denotes the pseudoinverse and

$$\mathbf{P}=\frac{1}{M}\sum _{m=1}^M{\left(\mathbf{\Psi }\left({\mathbf{x}}_{\mathbf{m}}\right)\mathbf{\Xi }\right)}^{*}\mathbf{\Psi }\left({\mathbf{x}}_{\mathbf{m}}\right)\mathbf{\Xi };\mathbf{D}=\frac{1}{M}\sum _{m=1}^M{\left(\mathbf{\Psi }\left({\mathbf{x}}_{\mathbf{m}}\right)\mathbf{\Xi }\right)}^{*}{\mathbf{x}}_{\mathbf{m}}^T$$ (16)

As opposed to the technique presented in [17], which requires tuning weight matrix $B$, we extend the method by involving the best-fit linear regression to approximate Koopman modes v with (16).

III. Data-driven modeling of tremors

This study was approved by the Institutional Review Board (IRB) at the North Carolina State University (IRB approval number: 19247). One participant (62 year old, male) with essential tremor consented to participate. The inclusion criteria of the study is that participants exhibiting tremors who fall within the age range of 18 to 90 years old. Subjects will fall under the tremor subject category if they exhibit tremor with re-emergent properties during the grasping motion and exhibit no other health conditions other than the essential tremor that impact tremor severity and inhibits motion to ensure that clear ultrasound images are collected.

A. Experimental Procedure

This study aimed at collecting US imaging data as a patient with tremors grasps objects. To perform the experiments, the participants were asked to grasp and hold a mug placed in a vertical upright position right in front of them. A linear ultrasound transducer (L7.5SC Prodigy Probe, S-Sharp, Taiwan, size: 2 cm × 2 cm × 10 cm) with center frequency 7.5 MHz connected to a US imaging system (Prodigy Probe, S-Sharp, Taiwan) was used to collect the B-mode US signals and an IMU sensor (Yost Labs Inc., USA, size: 2 cm × 2.5 cm × 5 cm) were attached to their forearm. The US system detected the flexor carpi radialis (FCR) and the IMU captured the wrist motion during the whole grasping motion. The fascicle velocity information was processed offline using the speckle tracking algorithm described in our previous work [19]. A MATLAB detrend and integral function [20] was used to remove drift from the collected IMU signals and obtain wrist motion information in degrees.

The participant performed three grasping motion trials. During each trial, the participant was instructed to grasp and hold the object for 5 seconds for four times during which US images and IMU motions were captured. A sketch of the experimental protocol, one collected B-mode US image, and one of the experiment results are shown in Fig. 1.

Fig. 1.

(a) Experimental setup for experiments, a participant with tremor holding a mug cup while the US transducer and IMU are strapped to the forearm to capture images of the FCR and motion of the wrist. (b) B-mode image with location of FCR marked. (c) Tremor behavior happened after about 5 secs, fascicle velocity (shown in blue) and joint angle from IMU (shown in red) are correlated.

B. Model Comparison and results

The tremor state is defined as $x={\left[x_1,x_2\right]}^T$, where $x_1$ is the wrist joint angle (deg) and $x_2$ is the FCR fascicle velocity from ultrasound (mm/sec). We evaluated the proposed Koopman-based EDMD in predicting $x$ by the root mean square error (RMSE) and the normalized root mean square error (NRMSE). The performance of the proposed method was compared to nonlinear Hammerstein-Wiener models generated from MATLAB System Identification Toolbox [20]. Table. I presents the RMSE and NRMSE for each method averaged over 20 randomly sampled initial conditions with 500 samples for each trial. Fig. 2 reports the average NRMSE for all trials. The result for modeling the tremor (10 seconds duration) is shown in Fig. 3.

TABLE I.

Comparison of the averaged RMSE and NRMSE(%)

Model	States
	RMSE		NRMSE
	x1	x2	x1	x2
Koopman-EDMD	0.0037	0.3172	3.4619	0.8867
Nonlinear-Wiener	0.0254	7.2214	23.2653	19.7089

Fig. 2.

Average NRMSE for 20 random initial conditions, with the standard deviation indicated by the black bars.

Fig. 3.

Prediction of tremor with Koopman operator from one of the experiment results.

From Table I and Fig. 2, we observe, compared to nonlinear Hammerstein-Wiener models, the proposed models achieved minimal averaged RMSE, NRMSE, and standard deviation. As a consequence, the Koopman model more accurately reflects the real behavior of tremor dynamics state.

IV. CONCLUSIONS AND FUTURE WORKS

This paper addresses a data-driven Koopman operator theory based EDMD approach in the modeling of tremors in people who have Essential Tremor or Parkinson’s disease. With US image data collected from a patient with tremors, we demonstrated the feasibility of constructing a dynamic model of tremors based on Koopman operator theory. We validated its performance by comparing it with the Hammerstein-Wiener nonlinear model.

According to the current study, some limitations of using US for tremor modeling include the fact that US probes are rigid and may not conform to a limb. Therefore, they are not suitable for continuous monitoring of muscle activity in patients with tremor. In order to address this issue, our group has conducted several preliminary studies on the development of wearable US transducers for monitoring muscle activity. [21], [22]. Based on the proposed data-driven model, another line of future research may involve the development of a real-time FES closed-loop control that can be applied to individual-specific stimulation for the inhibition of wrist tremor activity.