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Modelling and analysis of coupling dynamics of swinging a lower limb exoskeleton

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Abstract

Exoskeleton is a promising technology to enhance the mobility of aged and disabled people. With comfortable human–exoskeleton interaction, mechanical and control designers aim at accurately tracking a desired trajectory, while saving the wearer’s energy expenditure is the preference from the viewpoint of biomechanics. Given all of these effects, we propose a new model, consisting of the swing dynamics of both human and robot’s lower extremities coupled by damped springs representing elastic and viscous properties of band and human tissue. With the coupling coefficients identified with an experimental platform, the analytical model yields consistent results compared with an experimental exoskeleton, especially in the prediction of the interactive forces. Further analyses are then performed based on this validated model, revealing the influences of desired trajectory, mass ratio, misalignment, coupling points, health condition and band tightness on the human–exoskeleton coupling dynamics. It is found that introducing gravity compensation and tuning the feedback gain improve the tracking accuracy, but hardly change the interactive force. The most comfortable interaction requires a healthy wearer coupled with a lightweight exoskeleton without any misalignment, but properly changing the trajectory, coupling points and tightness can partly reduce the interactive forces if the ideal condition is unachievable.

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Abbreviations

\(\dot{x}\) :

Derivative of x with respect to time t

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

Sign function of x

\(\text {sat}(x,y)\) :

Saturation function of x between the values \([-y,y]\)

\(\varvec{}{\theta }_\text {d}\) :

Vector of desired joint angles    [rad]

\(\varvec{}{\theta }_{i_1}\) :

Vector of joint angles: human (\(i_1=\text {h}\)) or robot (\(i_1=\text {r}\))             [rad]

\(\varvec{}{\tau }_\text {h}^\text {a}\) :

Vector of human active joint torques    [N m]

\(\varvec{}{\tau }_\text {h}^\text {ida}\) :

Vector of human active joint torques obtained by inverse dynamics analysis             [N m]

\(\varvec{}{\tau }_{i_1}\) :

Vector of joint torques: human (\(i_1=\text {h}\)) or robot (\(i_1=\text {r}\))             [N m]

\(\varvec{}{\tau }_{i_1}^\text {i}\) :

Vector of coupling joint torques: human (\(i_1=\text {h}\)) or robot (\(i_1=\text {r}\))    [N m]

\(\varvec{}{\tau }_\text {r}^\text {c}\) :

Vector of robot’s controlling joint torques    [N m]

\(\varvec{}{\tau }_\text {r}^\text {f}\) :

Vector of robot’s frictional joint torques    [N m]

\(\mathbf {C}_{i_1}(\varvec{}{\theta }_{i_1},\dot{\varvec{}{\theta }}_{i_1})\) :

Coriolis and centripetal matrix: human (\(i_1=\text {h}\)) or robot (\(i_1=\text {r}\))       [\(\text {N m s}\text { rad}^{-1}\)]

\(\mathbf {e}_{i_1}\) :

Vector of tracking errors: human (\(i_1=\text {h}\)) or robot (\(i_1=\text {r}\))    [rad]

\(\mathbf {F}_{i_2}\) :

Vector of coupling forces: thigh (\(i_2=\text {t}\)) or shank (\(i_2=\text {s}\))    [N]

\(\mathbf {G}_{i_1}(\varvec{}{\theta }_{i_1})\) :

Gravitational vector: human (\(i_1=\text {h}\)) or robot (\(i_1=\text {r}\))   [N m]

\(\mathbf {J}_{i_1,i_2}\) :

Jacobian matrix: human (\(i_1=\text {h}\)) or robot (\(i_1=\text {r}\)); thigh (\(i_2=\text {t}\)) or shank (\(i_2=\text {s}\))   [m]

\(\mathbf {M}_{i_1}(\varvec{}{\theta }_{i_1})\) :

Inertial matrix: human (\(i_1=\text {h}\)) or robot (\(i_1=\text {r}\))[\(\text {N m s}^2\text { rad}^{-1}\)]

\(\mathbf {s}_{i_1,i_2}\) :

Coordinate of coupling point: human (\(i_1=\text {h}\)) or robot (\(i_1=\text {r}\)); thigh (\(i_2=\text {t}\)) or shank (\(i_2=\text {s}\))   [m]

\(\Delta \mathbf {s}_{i_2}\) :

Vector of human–exoskeleton relative displacement: thigh (\(i_2=\text {t}\)) or shank (\(i_2=\text {s}\))   [m]

\(\alpha \) :

Angle between human and robot’s thighs due to hip misalignment    [rad]

\(\beta \) :

Angle of hip misalignment[rad]

\(\gamma \) :

Angle between human and robot’s shanks due to hip misalignment    [rad]

\(\theta _{i_1,i_3}\) :

Joint angle: human (\(i_1=\text {h}\)) or robot (\(i_1=\text {r}\)); hip (\(i_3=\text {h}\)) or knee (\(i_3=\text {k}\))   [rad]

\(\tau _{\text {h},i_3}^\text {a}\) :

Human active joint torque: hip (\(i_3=\text {h}\)) or knee (\(i_3=\text {k}\))   [N m]

\(\tau _{\text {h},i_3}^\text {ida}\) :

Human active joint torque based on inverse dynamics analysis: hip (\(i_3=\text {h}\)) or knee (\(i_3=\text {k}\))   [N m]

\(\tau _{\text {h},i_3}^\text {max}\) :

Maximum active joint torque of healthy people: hip (\(i_3=\text {h}\)) or knee (\(i_3=\text {k}\))   [N m]

\(\tau _{\text {r},i_3}^\text {sf}\) :

Static frictional torque in robot’s joint: hip (\(i_3=\text {h}\)) or knee (\(i_3=\text {k}\))   [N m]

\(\tau _{i_1,i_3}\) :

Joint torque: human (\(i_1=\text {h}\)) or robot (\(i_1=\text {r}\)); hip (\(i_3=\text {h}\)) or knee (\(i_3=\text {k}\))   [N m]

\(\tau _{i_1,i_3}^\text {i}\) :

Coupling joint torque: human (\(i_1=\text {h}\)) or robot (\(i_1=\text {r}\)); hip (\(i_3=\text {h}\)) or knee (\(i_3=\text {k}\))   [N m]

\(\omega \) :

Swing frequency      [\(\text {rad s}^{-1}\)]

a :

Magnitude of hip misalignment    [m]

\(b_{i_2}\) :

Coupling damping: thigh (\(i_2=\text {t}\)) or shank (\(i_2=\text {s}\))      [\(\text {N s m}^{-1}\)]

\(c_{i_3}\) :

Coefficient of viscous friction in robot joint: hip (\(i_3=\text {h}\)) or knee (\(i_3=\text {k}\))   [\(\text {N m s rad}^{-1}\)]

\(c_{i_1,i_2}\) :

Centre of mass: human (\(i_1=\text {h}\)) or robot (\(i_1=\text {r}\)); thigh (\(i_2=\text {t}\)) or shank (\(i_2=\text {s}\))   [m]

\(d_\text {a}\) :

Vertical distance from the coupling point to the laser sensor above it   [m]

\(d_\text {b}\) :

Vertical distance from the coupling point to the laser sensor below it   [m]

D :

Horizontal displacement at the coupling point      [m]

\(D_\text {a}\) :

Horizontal displacement measured by the laser sensor above the coupling point   [m]

\(D_\text {b}\) :

Horizontal displacement measured by the laser sensor below the coupling point   [m]

\(e_{i_1,i_3}\) :

Tracking error in each joint: human (\(i_1=\text {h}\)) or robot (\(i_1=\text {r}\)); hip (\(i_3=\text {h}\)) or knee (\(i_3=\text {k}\))   [rad]

\(F_0\) :

Force constant in the coupling coefficient identification   [N]

\(F_x\) :

Horizontal force measured by the force sensor   [N]

g :

Gravitational acceleration[\(\text {m s}^{-2}\)]

H :

Wearer’s height    [m]

\(h_\text {c}\) :

Health condition

\(i_1\) :

Index of human (\(i_1=\text {h}\)) and robot (\(i_1=\text {r}\))

\(i_2\) :

Index of thigh (\(i_2=\text {t}\)) and shank (\(i_2=\text {s}\))

\(i_3\) :

Index of hip (\(i_3=\text {h}\)) and knee (\(i_3=\text {k}\))

\(I_{i_1,i_2}\) :

Moment of inertia: human (\(i_1=\text {h}\)) or robot (\(i_1=\text {r}\)); thigh (\(i_2=\text {t}\)) or shank (\(i_2=\text {s}\))   [\(\text {kg m}^2\)]

\(k_\text {d}\) :

Derivative feedback gain in robot control   [\(\text {N m s rad}^{-1}\)]

\(k_{i_2}\) :

Coupling stiffness: thigh (\(i_2=\text {t}\)) or shank (\(i_2=\text {s}\))   [\(\text {N m}^{-1}\)]

\(k_\text {p}\) :

Proportional feedback gain in robot control   [\(\text {N m rad}^{-1}\)]

\(l_{i_1,i_2}\) :

Length: human (\(i_1=\text {h}\)) or robot (\(i_1=\text {r}\)); thigh (\(i_2=\text {t}\)) or shank (\(i_2=\text {s}\))   [m]

\(m_{i_1,i_2}\) :

Mass: human (\(i_1=\text {h}\)) or robot (\(i_1=\text {r}\)); thigh (\(i_2=\text {t}\)) or shank (\(i_2=\text {s}\))   [kg]

M :

Wearer’s body mass    [kg]

\(\overline{P}_{\text {h},i_3}^\text {a}\) :

Time-averaged human active joint power: hip (\(i_3=\text {h}\)) or knee (\(i_3=\text {k}\))   [W]

\(\overline{P}_{\text {r},i_3}^\text {c}\) :

Time-averaged robot’s controlling joint power: hip (\(i_3=\text {h}\)) or knee (\(i_3=\text {k}\))   [W]

\(r_{i_2}\) :

Coupling tightness: thigh (\(i_2=\text {t}\)) or shank (\(i_2=\text {s}\))

\(r_\text {m}\) :

Mass ratio

\(s_{i_1,i_2}\) :

Distance from segment proximal end to coupling point: human (\(i_1=\text {h}\)) or robot (\(i_1=\text {r}\)); thigh (\(i_2=\text {t}\)) or shank (\(i_2=\text {s}\))   [m]

t :

Time   [s]

T:

Swing period   [s]

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Acknowledgements

This research is supported by National Natural Science Foundation of China (Grants No. 12072068, 11932015, 52175046, and 11872147), and Sichuan Science and Technology Program (Grant No. 2022JDRC0018). The authors thank Mr. Gan Liu and Mr. Li Chen for operating the exoskeleton and data collection.

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Authors and Affiliations

  1. School of Aeronautics and Astronautics, University of Electronic Science and Technology of China, Chengdu, 611731, China

    Yao Yan, Zhenlei Chen, Cheng Huang & Qing Guo

  2. Zhejiang Grand Orient Chair industry Co., Ltd, Zhejiang, 313300, China

    Yao Yan

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  1. Yao Yan
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Correspondence to Qing Guo.

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Yan, Y., Chen, Z., Huang, C. et al. Modelling and analysis of coupling dynamics of swinging a lower limb exoskeleton. Nonlinear Dyn 111, 1213–1234 (2023). https://doi.org/10.1007/s11071-022-07876-8

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