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Swift augmented human–robot dynamics modeling for rehabilitation planning analyses

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Abstract

With the widespread implementation of robotics exoskeletons in rehabilitation, modeling and dynamics analysis of such highly nonlinear coupled systems has become significantly important. In this paper, a swift numerical human–robot dynamics modeling has been developed to achieve accurate and realistic interpretation. This takes into consideration the separation and impact between multiple bodies for rehabilitation planning. To this end, first, a novel parallel algorithm combined with sequential interaction conditions is proposed based on the numerical recursive Newton–Euler method. The approach begins by deriving separated numerical models for the complicated system: i.e. both the human and the robot. These models are then augmented, with a primary focus on reducing the error of the interaction conditions, including forces and positions. The accuracy of the proposed model, with a computational complexity of O(n), is assessed by comparing to a previously validated nonrecursive analytical model with a higher computational complexity of O(n^4). Additionally, the quality of the connection between the human and the robot is assessed to establish a suitable control objective and an effective interaction strategy for rehabilitation planning. The study employs a lower-limb walking assistive robot developed in the ARAS lab (RoboWalk) to validate the proposed method. The algorithm is empirically implemented on the RoboWalk test stand, ensuring the integrity of the proposed dynamics modeling. The human–robot interaction forces are estimated with an accuracy of 2 N, in the presence of friction and measurement noise. Finally, the effectiveness of the model-based controller is assessed by using the proposed method, providing valuable tools for the enhancement of overall performance of such a complex dynamics system.

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Notes

  1. Plücker coordinates are a system of coordinates for representing lines in space. They are named after Julius Plücker, who introduced them in the 1860s. This system utilizes a pair of vectors to capture the essence of a line. The first three Plücker coordinates represent the direction of the line, while the last three Plücker coordinates represent the moment vector that lies perpendicular to the line and intersects it at a specific point. Together, the six components of these vectors, referred to as Plücker coordinates, provide a comprehensive representation of the line. Plücker coordinates offer several advantages over traditional methods of representing lines. They are compact, invariant under translation and rotation, and possess valuable geometric properties. These characteristics make them particularly well suited for applications in rigid body dynamics, where they facilitate the analysis of motion, forces, and torques acting on rigid bodies. Whether in robotics, physics, or automotive engineering, Plücker coordinates stand as a powerful tool for understanding and manipulating the motion of objects in three-dimensional space [45].

  2. Plücker coordinates.

Abbreviations

\(\boldsymbol{a}_{i}\) :

Spatial acceleration of body \(i\)

\(\boldsymbol{C}\) :

A 10 × 1 vector represents Coriolis, centrifugal, and gyration effects of the user multibody dynamics systems

\(\boldsymbol{F}_{ext}\) :

External forces applied to the model in workspace

\({}^{stance} \boldsymbol{F}_{ext}^{foot}\) :

Stance foot external forces in workspace

\({}^{swing} \boldsymbol{F}_{ext}^{foot}\) :

Swing foot external forces in workspace

\(\boldsymbol{F}_{ext}^{HJ}\) :

External forces on the hypothetical joint of the model in workspace

\(\boldsymbol{F}_{i}^{B}\) :

The net spatial force applied to body \(i\)

\(\boldsymbol{F}_{i}^{ext}\) :

External spatial force applied to body \(i\)

\(\boldsymbol{F}_{int}\) :

Robot–user interaction forces at feet and saddle in workspace

\(\boldsymbol{F}_{int}^{A_{L}}\), \(\boldsymbol{F}_{int}^{A_{R}}\) :

Known forces applied to the RoboWalk ankle joint

\(\boldsymbol{F}_{int}^{seat}\) :

Assistive force applied to the user

\({}^{v} \boldsymbol{F}_{int}^{seat}\) :

Robot saddle vertical interaction force in workspace

\(\boldsymbol{F}_{j}\) :

The spatial force applied by body \(i\) to its successor body \(j\)

\(\boldsymbol{G}\) :

A 10 × 1 vector represents the gravity effects of the user multibody dynamics systems

\(\boldsymbol{I}_{i}\) :

Spatial moment of inertia of body \(i\)

\(\boldsymbol{J}_{A_{L}}\), \(\boldsymbol{J}_{A_{R}}\) :

Robot’s left and right ankle Jacobian

\(\boldsymbol{J}_{FRF}\) :

Jacobian of the contact point

\(\boldsymbol{J}_{s}\) :

Jacobian of saddle force applying point

\(l_{1}\) :

The radius of the saddle of the robot

\(l_{3}\), \(l_{2}\) :

The length of the shank and thigh of the robot

\(\boldsymbol{M}\) :

Represents a 10 × 10 inertia matrix of the user multibody dynamics systems

\({}^{\boldsymbol{i}} \boldsymbol{P}_{\boldsymbol{P}_{\boldsymbol{i}}}\) :

The translation vector from \(\boldsymbol{P}_{\boldsymbol{i}}\) to \(\boldsymbol{i}\)

\(\boldsymbol{q}\) :

Position of the model joints in joint space

\(\dot{\boldsymbol{q}}\) :

Velocity of the model joints in joint space

\(\ddot{\boldsymbol{q}}\) :

Acceleration of the model joints in joint space

\({}^{\boldsymbol{i}} \boldsymbol{R}_{\boldsymbol{P}_{\boldsymbol{i}}}\) :

The rotation matrix from \(\boldsymbol{P}_{\boldsymbol{i}}\) to \(\boldsymbol{i}\)

\(S_{robot}\) :

Robot model switch parameter

\(S_{user}\) :

User’s foot switch sensor

\(\boldsymbol{v}_{i}\) :

Spatial velocity of body \(i \)

\(\mathbf{X}\) :

A 6 × 6 transfer matrix from one coordinate system to another

\(X_{ankle}\) :

X coordinate of the robot ankle in Cartesian workspace

\(\boldsymbol{X}_{feet}\) :

Positions of the robot or user ankles in workspace

\({}^{stance\_foot} \boldsymbol{X}_{HJ}\) :

Force transfer matrices from the user hypothetical joint to stance foot

\(X_{saddle}\) :

X coordinate of the robot saddle in Cartesian workspace

\(\boldsymbol{X}_{seat}\) :

Positions of the robot saddle in workspace

\({}^{v} \ddot{\boldsymbol{X}}_{seat}\) :

Vertical part of the robot saddle acceleration in workspace

\({}^{HJ} \boldsymbol{X}_{stance\_foot}\) :

Force transfer matrices from stance foot to the user hypothetical joint

\(\boldsymbol{X}_{pelvis}\) :

Positions of the user pelvis in workspace

\({}^{h} \ddot{\boldsymbol{X}}_{pelvis}\) :

Horizontal part of the user pelvis acceleration in workspace

\(Y_{ankle}\) :

Y coordinate of the robot ankle in Cartesian workspace

\(Y_{saddle}\) :

Y coordinate of the robot saddle in Cartesian workspace

\(\varepsilon \) :

A small positive number

\(\theta _{1}\) :

The saddle angle of robot

\(\theta _{2}\) :

The angle between the robot thigh and saddle

\(\theta _{3}\) :

Robot knee angle

\(\theta _{4}\) :

The angle between shank of the robot and the normal to the ground

\(\mu \) :

The set of available bodies

\(\boldsymbol{\tau}_{measure}\) :

Measured robot knee torque

\(\boldsymbol{\tau}_{HL}\), \(\boldsymbol{\tau}_{KL}\), \(\boldsymbol{\tau}_{AL}\) :

Joint torques of the left hip, knee, and ankle in the user model

\(\boldsymbol{\tau}_{HR}\), \(\boldsymbol{\tau}_{KR}\), \(\boldsymbol{\tau}_{AR}\) :

Joint torques of the right hip, knee, and ankle in the user model

\(\boldsymbol{\tau}_{model}\) :

Model joint torques

\(\hat{\boldsymbol{\tau}}_{model}\) :

Model joint torques without floor reaction forces

\(\boldsymbol{\tau}_{T}\) :

Trunk joint torque in the user model

\(\times ^{*}\) :

Spatial force product sign

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Authors and Affiliations

  1. Department of Mechanical Engineering, K. N. Toosi University of Technology, Tehran, Iran

    Vahid Akbari, Omid Mahdizadeh, S. Ali A. Moosavian & Mahdi Nabipour

  2. Department of Biomechanical Engineering, University of Twente, Enschede, The Netherlands

    Mahdi Nabipour

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Contributions

V.A. and O.M. wrote and organized the manuscript for accuracy and clarity. M.N. provided a non-recursive method. S.M. and M.N. provided guidance and oversight throughout the project and valuable input to enhance quality.

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Correspondence to Vahid Akbari.

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Akbari, V., Mahdizadeh, O., Moosavian, S.A.A. et al. Swift augmented human–robot dynamics modeling for rehabilitation planning analyses. Multibody Syst Dyn (2024). https://doi.org/10.1007/s11044-024-09975-3

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