Joint torques estimation in human gait based on Gaussian process

Abstract

BACKGROUND:

Human gait involves activities in nervous and musculoskeletal dynamics to modulate joint torques with time continuously for adapting to varieties of walking conditions.

OBJECTIVE:

The goal of this paper is to estimate the joint torques of lower limbs in human gait based on Gaussian process.

METHOD:

The potential uses of this study include optimization of exoskeleton assistance, control of the active prostheses, and modulating the joint torque for human-like robots. To achieve this, Gaussian process (GP) based data fusion algorithm is established with joint angles as the inputs.

RESULTS:

The statistic nature of the proposed model can explore the correlations between joint angles and joint torques, and enable accurate joint-torque estimations. Experiments were conducted for 5 subjects at three walking speed (0.8 m/s, 1.2 m/s, 1.6 m/s).

CONCLUSION:

The results show that it is possible to estimate the joint torques at different scenarios.

Keywords

GP human gait joint torque prostheses robots mechanics

1. Introduction

Lower limb prostheses, human-like robots, and leg exoskeletons desire to replicate human gait mechanics using mechanical components [1]. Challenges both in software and hardware should be overcome for their wide applications [2]. Although various hardware mimicking biology have been made such as series elastic actuators [3] and Bidirectional Antagonistic Floating Spring Actuator [4], the performances are primarily limited [5], since the joint torques in human gait are modulated continuously and subconsciously during natural locomotion [6]. To put the enormous potentials of those devices into reality, one of the main challenges focuses on the high-level controller, which is in charge of estimating the joint torque [2].

Joint torque estimation known as motion intent [7] is of great significance for achieving optimization of exoskeleton assistance [8, 9] and lower limb prosthesis stance control [10] in the bipedal locomotion. For instance, Zhang et al. set the desired torque as a function of time to control the ankle exoskeleton [11] during walking. A control strategy for powered prostheses was designed by Thatte et al. [10] using the Gaussian process (GP), which was performed to specify the desired feed-forward torques for the knee and ankle joints. To tackle the control challenge, there are many kinds of literature estimating joint torques during walking, which involve activities in nervous and musculoskeletal dynamics [2, 12]. In summary, existing strategies are mainly classified as discrete techniques and continuous methods. The normal gait cycle was divided into several gait phases known as state machine in the discrete methods [13]. Depending on the gait phase, the joint torque was estimated to design the high-level controller synchronized to the gait cycle [14]. However, the human joint torques are modulated continuously for naturally adjusting to biomechanical demands. This calls for a robust model continuously estimating the joint torque. Some researchers attempted to establish the biomechanical model of the musculoskeletal system aiming to study human gait [15]. Surface electromyography (sEMG) signals were always used as the inputs to the model. Massimo et al. established a multi-DOF EMG-driven forward-dynamic model of human lower extremity [16]. Ao et al. developed an EMG-driven Hill-type musculoskeletal model to estimate the real-time ankle joint torque and control a power-assist ankle exoskeleton [17]. Nevertheless, the performance of this Hill-type model is limited mainly by the time-varying parameters. Also, the workload of model calibration is heavy. These drawbacks hinder its widespread use. With the development of machine learning, black-box models are usually applied in engineering applications using the biomedical signals measured form the user [18]. There are some measurable state variables in each of the human subsystems and segments manifesting human gait [7]. Therefore, human joint torques can be described by many different state variables, including sEMG, joint angles, ground reaction forces, and attitude angles of the feet. The complex cooperation between joints and segments in the lower extremity was investigated for potential use in developing high-level controllers for biped devices [2, 10]. In the study by Eslamy et al. [2], GP was adopted to estimate the ankle joint torque using the shank angular velocity and angle as the inputs of the model. Ankle joint torques were estimated based on the artificial neural networks using a low-cost pressure insole and tendon sensor [19]. A central difficulty in high precision estimations is in determining a model that can capture the dynamic information behind the biological signals.

The joint angles of lower limbs are the direct description of human gait [20], which can be realized by modulating the joint torques. This paper proposed a Gaussian process (GP) based data fusion method [21] for joint torque estimation in human gait. The motivation of this paper is to develop a GP based data fusion model for human lower limb torque learning. The model devotes to figure out the natural relationships between the angle and torque of a joint. Thus, it promises to offer superior performance. The statistical nature of the GP is capable of giving credibility for risk-based control. Experimental works are also presented to demonstrate the superior performance in learning joint torque.

2. Method

Generally speaking, joint torques in human gait are complex involving musculoskeletal dynamics, ligament forces, and bone-on-bone forces [20]. The problem of estimating joint torques is to develop an appropriate mapping function $y=f(x)$ between the input $x$ and the joint torque $y$ . In the study by Eslamy et al. [2], the shank angle and shank angular velocity were regarded as the inputs. In the present paper, GP is performed to learn the mapping function $f$ . GP is fully specified by its mean and covariance function [22] as given by Eq. (1)

$\displaystyle f(x)\sim\mathcal{GP}\left({m(x),\sum({x,x})}\right)$ (1)

where, $m(x)$ is the mean function and $\sum({x,x})$ is the covariance function which specifies the covariance between pairs of random variables. The covariance function evaluated at pairs $x_{1}=[x_{1}^{1}\cdots x_{1}^{n}]$ and $x_{2}=[{x_{2}^{1}}\cdots{x_{2}^{m}}]$ is defined as follows:

$\displaystyle\sum({x_{1},x_{2}})=\left({{\begin{array}[]{*{20}c}{\varsigma({x_% {1},x_{1}})}&{\varsigma({x_{1},x_{2}})}&\cdots&{\varsigma({x_{1},x_{m}})}\\ {\varsigma({x_{2},x_{1}})}&{\varsigma({x_{2},x_{2}})}&\cdots&{\varsigma({x_{2}% ,x_{m}})}\\ \vdots&\vdots&\cdots&\vdots\\ {\varsigma({x_{n},x_{1}})}&{\varsigma({x_{n},x_{2}})}&\cdots&{\varsigma({x_{n}% ,x_{m}})}\\ \end{array}}}\right)_{n\times m}$ (2)

The covariance specifies which structure the learned function is likely to be and in turn, determine the generalization ability of the model. Commonly used covariances in the field of machine learning include the squared exponential (SE), Matern class (MC), etc. The SE is suitable to model smooth dynamics and can convert global correlation into local correlation. MC is Suitable to model dynamics and kinematics with different roughness. The Matern class of covariance function given as follows

$\displaystyle\sum(r)=\frac{2^{1-\nu}}{\Gamma(\nu)}\left({\frac{\sqrt{2\nu}r}{l% }}\right)^{\nu}K_{\nu}\left({\frac{\sqrt{2\nu}r}{l}}\right)$ (3)

where, $r=|{x_{i}-x_{j}}|$ ; $\nu$ and $l$ are hyper-parameters which are positive. $K_{\nu}$ is a modified Bessel function, and $\Gamma$ is the gamma function.

In addition, we can combine existing base kernel functions to make a new one by virtue of the sum and product constructions to model different dynamic characteristics. Here, composite covariance function was designed with a combination of MC, and WN. The MC is used to regress nonlinear dynamics. WN was used to figure out the system noise. Hyper-parameters are optimized using the gradient optimization method to maximize the marginal likelihood. The partial derivatives of marginal likelihood function with respect to the hyper-parameters is acquired:

$\displaystyle\frac{\partial\ln p({y|z,\theta})}{\partial\theta}=-1/2\textit{% trace}\left(\sum^{-1}\frac{\partial\sum}{\partial\theta}\right)+1/2y^{T}\sum^{% -1}\frac{\partial\sum}{\partial\theta}\sum^{-1}y$ (4)

3. Experimental study

Five subjects without musculoskeletal or neurological dysfunctions gave written informed consent prior to participation in the experiments.

Optical motion capture system (from Vicon Inc.) and force plates were used for validating the proposed method. 16 markers (10 mm in diameter) were fixed on the subjects’ lower limbs. Joint angles of lower limbs can be acquired by the optical motion capture system capturing the markers. Ground reaction force/moment for each foot were measured by the force plates. Joint torques can be found by inverse dynamic embedded in the commercial software from Vicon Inc. The signals were acquired with a sampling rate of 100 Hz. After a practicing phase, all the subjects were required to walk on the treadmill at three walking speeds (0.8 m/s, 1.2 m/s, and 1.6 m/s). The data set was divided into training and testing groups. To be specific, 1 trial was used for training and the other three were used for validation. The joint angles in the left leg were regarded as the training inputs when the right joint torques were predicted. Similarly, when we predicted the left joint torques, the joint angles in the right leg were used as the model inputs. RMS and $r^{2}$ values were used to evaluate the estimation quality and give a comprehensive understanding of the estimation results. The criterion used by Boyle and Frean [23], that $r^{2}$ values were higher than 0.8, was regarded as the acceptable estimation.

4. Results

GP has been developed in order to provide the generalized mapping from inputs and the joint torque. The inputs were related to three walking speeds (0.8 m/s, 1.2 m/s, and 1.6 m/s). The torques were scaled to the percentage of body mass to assist in comparing different subjects. Figure 1 demonstrates a typical set of experimental results of a single step presented as percentages of the gait cycle from heel – contact, where the 95% confidence interval is shown in grey.

Figure 1.

Estimation results of joint torques at walking speed of 0.8 m/s, 1.2 m/s and 1.6 m/s respectively where the 95% confidence interval is shown in grey.

As shown in Fig. 1, the estimation results are quite acceptable, with most of the expected values falling in the confidence interval. In addition, the torques tend to change magnitude with the walking speeds.

Figure 2.

$r^{2}$ values relating to the three walking speeds. a) $r^{2}$ values at walking speed of 0.8 m/s. b) $r^{2}$ values at walking speed of 1.2 m/s. c) $r^{2}$ values at walking speed of 1.6 m/s.

Figure 3.

RMS errors. a) RMS errors at walking speed of 0.8 m/s. b) RMS errors at walking speed of 1.2 m/s. c) RMS errors at walking speed of 1.6 m/s.

To better display the effectiveness of the proposed GP, $r^{2}$ values of for the three walking speeds (0.8 m/s, 1.2 m/s, and 1.6 m/s) were computed and demonstrated in Fig. 2. The squares show the mean of the $r^{2}$ values at every specific speed. The upper and lower bounds are the maximum and minimum $r^{2}$ values, respectively. 4 values out of 135 values are unacceptable. About 30% of the $r^{2}$ values are equal or greater than 0.95, which shows the superiority of the algorithm to some extent.

For each trail, the RMS errors are demonstrated in Fig. 3. The squares show the mean of the RMS error at every specific speed. The upper and lower bounds are the maximum and minimum RMS errors, respectively. The maximum of the RMS error is about 0.25 Nm/kg. And the minimum of the RMS error is below 0.05 Nm/kg. The distribution of the RMS errors is basically irregular. The RMS results are also acceptable.

It should be noted that a balance between model precision and complexity must be reconciled. The dilemma can be solved by the Bayesian information criterion. Another question that should be considered is that any information may contain noise. Filtering is required in some special cases. A fusion of velocities or sEMG may improve the learning results. This requires more investigations in the future.

5. Discussion

Human gait is definitely complex, involving co – operation between different joints and segments in the lower extremities. There are many measurable states that can be used to estimate joint torques in lower limbs. In the study by Eslamy et al. [2], shank angles and angular velocities were treated as the inputs of a GP model to estimate the ankle torque. Experiments were conducted to verify the effectiveness of the GP model. However, the performance of the GP model for estimating knee and hip torques remains unknown. In reality, the knee torque and hip torque are somewhat hard to estimate compared with ankle torques, as ankle torques are the least variable due to the constraint by the ground. Thus, torque estimation of the three joints in lower limbs should be investigated extensively. In our former study, long short – term memory with a convolutional autoencoder was proposed to learn the joint torques in human gait [24]. A smart shoe was designed to acquire information concerning ground reaction force and foot motion. Then, these information were regarded as the inputs of the proposed model. Experimental results show that ankle torque, knee torque and hip torque can be estimated accurately with 98% of the $r^{2}$ values higher than 0.8. In order to exploit the temporal connections among the measured signals, joint torque learning based on GP was also developed [25]. Joint torque estimation under different trained and untrained speed levels was verified by experiments. Smart shoes should be used to measure the ground reaction force and foot motion, when joint torques were estimated using the methods by Yang and Yin [24, 25]. Joint torque estimation based on GP model using other measurable states have not been studied. To extend the application scope, GP based data fusion algorithm is established with joint angles as the inputs in this paper. Joint torques may be learned when designing the high – level controller of prosthetics. In this situation, smart shoes for measuring foot motion may be impossible or inaccurate. Thus, the contra – lateral leg angles used as the inputs may be a better choice.

6. Conclusions

This paper presents a GP – based algorithm for joint torque learning in human gait which can figure out the natural relationships among the correlated biomedical signals. Experimental results show that the proposed data fusion model has an excellent performance in the estimation of joint torques. 97% of the $r^{2}$ values are higher than 0.8. The estimation results are quite good with most of the expected values falling in the confidence interval. It is of great significance in constructing a high – level controller for human – like robots.

Footnotes

Acknowledgments

This work is supported in part by the National Key Research and Development Project (No. 2019YFB1312500) and in part by the National Natural Science Foundation of China (Nos. 62073156, 62103280).

Conflict of interest

None to report.

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