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Adaptive RBF neural network-computed torque control for a pediatric gait exoskeleton system: an experimental study

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Abstract

For pediatric rehabilitation, obtaining accurate coupled human-exoskeleton system models is challenging due to unknown model parameters caused by children’s dynamic growth and development. These factors make it difficult to establish precise and standardized models for exoskeleton control. Additionally, external disturbances, such as unpredictable movements or involuntary muscle contractions, further complicate the control process that must be addressed. This work presents the computed torque control (CTC) scheme compensated by a radial basis function neural network (RBFNN) for an uncertain lower-limb exoskeleton system. Primarily, the design, hardware architecture, and experimental procedure of a pediatric exoskeleton are briefly demonstrated. Thereafter, the proposed adaptive RBFNN-CTC (ARBFNN-CTC) is highlighted, where the adaptation of network weights depends on the Gaussian function and the Lyapunov equation. The adaptive RBFNN estimates the unknown model dynamics and compensates the CTC for the effective gait tracking of the coupled system in passive-assist mode. A Lyapunov stability is presented to ensure the convergence of error states into a significantly small domain. Finally, an experimental study with a pediatric subject (12 years) is carried out to investigate the effectiveness of the proposed control scheme. The gait tracking results show that the ARBFNN-CTC outperforms the traditional CTC by nearly 40\(\%\) over three gait cycles. Furthermore, the proposed approach’s generalizability is validated across various gait cycles, especially at 3,  10,  20, and 30 cycles. The high correlation coefficients of 0.996,  0.997, and 0.999 for the hip, knee, and ankle joints, respectively, at thirty gait cycles, highlight the potential of the ARBFNN-CTC scheme in achieving effective and consistent gait training outcomes over extended periods.

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Abbreviations

\(\tau _{\textrm{j,h}}\) :

Joint torque vector for human leg

\(\tau _{{ eth}}\) :

Interaction torque from exoskeleton to human

\(\ddot{q}_{\textrm{j,h}}\) :

Joint acceleration vector of the human leg

\(\dot{q}_{\textrm{j,h}}\) :

Joint speed vector of the human leg

\(q_{\textrm{j,h}}\) :

Joint state vector of the human leg

\(\ddot{q}_{\textrm{j,e}}\) :

Joint acceleration vector of the exoskeleton system

\(\dot{q}_{\textrm{j,e}}\) :

Joint speed vector of the exoskeleton system

\(q_{\textrm{j,e}}\) :

Joint state vector of the exoskeleton system

\(M_{{ h}}(q_{\textrm{j,h}})\) :

Positive definite inertia matrix for the human leg

\(C_{{ h}}(q_{\textrm{j,h}}, \dot{q}_{\textrm{j,h}})\) :

Coriolis/centrifugal matrix for the human leg

\(G_{{ h}}(q_{\textrm{j,h}})\) :

Gravitational vector for the human leg

\(\tau _{\textrm{j,e}}\) :

Joint torque vector for the exoskeleton system

\(\tau _{{ hte}}\) :

Interaction torque from human to exoskeleton

\(M_{{ e}}(q_{\textrm{j,e}})\) :

Positive definite inertia matrix for the exoskeleton system

\(C_{{ e}}(q_{\textrm{j,e}}, \dot{q}_{\textrm{j,e}})\) :

Coriolis/centrifugal matrix for the exoskeleton system

\(G_{{ e}}(q_{\textrm{j,e}})\) :

Gravitational vector for the exoskeleton system

\(M_{{ h,e}}(q_{\textrm{j,e}})\) :

Positive definite inertia matrix for the coupled human-exoskeleton system

\(C_{{ h,e}}(q_{\textrm{j,e}}, \dot{q}_{\textrm{j,e}})\) :

Coriolis/centrifugal matrix for the coupled human-exoskeleton system

\(G_{{ h,e}}(q_{\textrm{j,e}})\) :

Gravitational vector for the coupled human-exoskeleton system

\(\tau _D\) :

Disturbance vector for the coupled human-exoskeleton system

\(\mathbb {M}_{{ h,e}}(q_{\textrm{j,e}})\) :

Positive definite inertia matrix with nominal model parameters of coupled human-exoskeleton system

\(\mathbb {C}_{{ h,e}}(q_{\textrm{j,e}}, \dot{q}_{\textrm{j,e}})\) :

Coriolis/Centrifugal matrix with nominal model parameters of coupled human-exoskeleton system

\(\mathbb {G}_{{ h,e}}(q_{\textrm{j,e}})\) :

Gravitational vector with nominal model parameters of coupled human-exoskeleton system

\(\delta \) :

Parametric perturbator

\(F(\dot{q}_{\textrm{j,e}})\) :

Friction vector

Ç\(_\textrm{F}\) :

Coulomb friction

\(V_\textrm{F}\) :

Viscous friction

\(\sigma \) :

Angular speed parameter

\(\rho \) :

lumped uncertain function

\(\xi \) :

Angular tracking error

\(\dot{\xi }\) :

Angular tracking error derivative

\(k_\textrm{p}\) :

Proportional gain

\(k_\textrm{d}\) :

Derivative gain

\(\hat{\rho }\) :

Estimation of lumped uncertain function

\(\text {x}\) :

Input vector to the RBFNN

\(\textrm{z}\) :

Output vector to the RBFNN

\(\bar{\angle }\) :

Weight matrix of the RBFNN

\(h(\textrm{x})\) :

Output of the Gaussian activation function with n hidden nodes

\((h_i(\text {x}))\) :

Output of the Gaussian activation function for the i-th hidden node

\(c_i\) :

Centre distance from the origin in Gaussian function

\(\sigma _{\textrm{wi}}\) :

Curve width of Gaussian function

\(\varepsilon \) :

Small positive constant

\(\bar{\angle }^*\) :

Optimal weight matrix

\(\zeta \) :

Approximation error

\(\zeta _0\) :

Finite constant parameter

\(\hat{\bar{\angle }}\) :

Estimation of optimal weight matrix

\(\tilde{\bar{\angle }}\) :

Estimation error of weight matrix

\({\mathcal {P}}\) :

Symmetric and positive definite matrix

\(\beta \) :

Positive constant parameter

\({\mathcal {Q}}\) :

Symmetric and positive Hermitian matrix

\(\kappa _1\) :

Positive constant parameter

\(\Vert ~\Vert _{\textrm{F}}\) :

Frobenius norm

\(\lambda _{\textrm{min}}({\mathcal {Q}})\) :

Minimum eigenvalue of matrix \({\mathcal {Q}}\)

\(\lambda _{\textrm{max}}({\mathcal {P}})\) :

Maximum eigenvalue of matrix \({\mathcal {P}}\)

\({\angle _{\textrm{max}}}\) :

Maximum valued element in ideal weight matrix

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Acknowledgements

The authors are grateful to DSIR-PRISM, India, under which this research and development project (DSIR/PRISM/78/2016) is carried out. The second author would like to thank Al-Baath University, the Ministry of Higher Education, Syrian Arab Republic for their support during the higher studies. The authors deeply acknowledge the role of the physical therapist Mr. Kandarpa Jyoti Das, from the institute hospital during the motion capture experiment and motion assistance. Moreover, the authors also thank Mechatronics and Robotics Laboratory, IITG, for the research support.

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Authors and Affiliations

  1. Department of Mechanical Engineering, Indian Institute of Technology Guwahati, Guwahati, Assam, 781039, India

    Jyotindra Narayan, Mohamed Abbas, Bhavik Patel & Santosha K. Dwivedy

  2. Department of Design and Production, Al-Baath University, Homs, Syria

    Mohamed Abbas

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  1. Jyotindra Narayan
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Correspondence to Jyotindra Narayan.

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Narayan, J., Abbas, M., Patel, B. et al. Adaptive RBF neural network-computed torque control for a pediatric gait exoskeleton system: an experimental study. Intel Serv Robotics 16, 549–564 (2023). https://doi.org/10.1007/s11370-023-00477-3

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