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An admittance shaping controller for exoskeleton assistance of the lower extremities

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Abstract

We present a method for lower-limb exoskeleton control that defines assistance as a desired dynamic response for the human leg. Wearing the exoskeleton can be seen as replacing the leg’s natural admittance with the equivalent admittance of the coupled system. The control goal is to make the leg obey an admittance model defined by target values of natural frequency, peak magnitude and zero-frequency response. No estimation of muscle torques or motion intent is necessary. Instead, the controller scales up the coupled system’s sensitivity transfer function by means of a compensator employing positive feedback. This approach increases the leg’s mobility and makes the exoskeleton an active device capable of performing net positive work on the limb. Although positive feedback is usually considered destabilizing, here performance and robust stability are successfully achieved through a constrained optimization that maximizes the system’s gain margins while ensuring the desired location of its dominant poles.

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Notes

  1. For stiffness perturbations the area under \(\ln |S_h(j\omega )|\) remains constant, meaning that if the sensitivity increases in one frequency range, it will be attenuated in the same proportion elsewhere.

  2. Stiffness reduction (equivalently, increase in DC gain) stands apart because it obeys the zero sensitivity integral property.

Abbreviations

\(Z_\_(s)\) :

Mechanical impedance

\(Y_\_(s)\) :

Mechanical admittance

\(X_\_(s)\) :

Integral of the mechanical admittance (\(X_\_(s) = Y_\_(s)/s\))

\(S_\_(s)\) :

Sensitivity transfer function. This is a closed-loop transfer function that evaluates to 1 at all frequencies when the feedback gain is zero

\(T_\_(s)\) :

Complementary sensitivity transfer function, given by \(T_\_(s) = 1 - S_\_(s)\)

\(L_\_(s)\) :

Loop transfer function for root-locus analysis

\(N_\_(s)\) :

Numerator of a rational transfer function

\(D_\_(s)\) :

Denominator of a rational transfer function

\(W_\_(s)\) :

Loop transfer function for robustness analysis (Sect. 4)

h :

Human leg

e :

Exoskeleton mechanism, consisting of the actuator and arm

c :

Compliant coupling between the human leg and the exoskeleton mechanism, modeled as a spring and damper

f :

Feedback compensator for the exoskeleton

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Acknowledgments

This research was supported by a Grant from the Honda Research Institute, Mountain View, CA, USA.

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Authors and Affiliations

  1. Centre for Autonomous Systems, University of Technology, Broadway, Sydney, 2007, Australia

    Gabriel Aguirre-Ollinger

  2. Honda Research Institute, Mountain View, CA, 94043, USA

    Umashankar Nagarajan & Ambarish Goswami

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  1. Gabriel Aguirre-Ollinger
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  3. Ambarish Goswami
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Corresponding author

Correspondence to Gabriel Aguirre-Ollinger.

Appendices

Appendix 1: Target values for the dynamic response parameters of the assisted leg: computation

From (14) we have

$$\begin{aligned} \omega _{nh}^d = R_\omega \omega _{nh} \end{aligned}$$
(78)

We shall now define an intermediate target integral admittance \(X_{h,DC}(s)\) that differs from \(X_h(s)\) only in the trailing coefficient of the denominator:

$$\begin{aligned} X_{h,DC}(s) = \frac{1}{I_h s^2 + 2I_h \zeta _h \omega _{nh} s + I_h \omega _{nh,DC}^2} \end{aligned}$$
(79)

We choose \(\omega _{nh,DC}\) such that \(X_{h,DC}(s)\) meets the DC gain specification \(R_{DC}\):

$$\begin{aligned} \frac{X_{h,DC}(0)}{X_h(0)} = \frac{\omega _{nh}^2}{\omega _{nh,DC}^2} = R_{DC} \end{aligned}$$
(80)

yielding

$$\begin{aligned} \omega _{nh,DC} = \omega _{nh} \sqrt{R_{DC}^{-1}} \end{aligned}$$
(81)

Because the target integral admittance \(X_h^d(s)\) and the intermediate target \(X_{h,DC}(s)\) have the same DC gains (although in general they have different natural frequencies and different damping ratios), we can write

$$\begin{aligned}&X_h^d(0) = X_{h,DC}(0) \,\,\,\, \text {or} \nonumber \\&\frac{1}{I_h^d \omega _{nh}^{d\,\,2}} = \frac{1}{I_h \omega _{nh,DC}^2} \end{aligned}$$
(82)

In (82) we substitute \(\omega _{nh}^d\) with (78) and \(\omega _{nh,DC}\) with (81), thereby obtaining the required value for \(I_h^d\):

$$\begin{aligned} I_h^d = \frac{I_h}{R_{DC} R_\omega ^2} \end{aligned}$$
(83)

In order to obtain \(\zeta _h^d\), we compute the values of the resonant peaks for \(X_h(j\omega )\) using Eq. (12) and \(X_h^d(j\omega )\) using Eq. (13):

$$\begin{aligned} M_h = \frac{1}{2 \, I_h \, \omega _{nh}^2 \zeta _h \sqrt{1 - \zeta _h^2}} \,\,\,\, \text {for }X_h(j\omega ) \end{aligned}$$
(84)

and

$$\begin{aligned} M_h^d = \frac{1}{2 \, I_h^d \, \omega _{nh}^{d\,\,2} \, \zeta _h^d \sqrt{1 - \zeta _h^{d\,\,2}}} \,\,\,\, \text {for }X_h^d(j\omega ) \end{aligned}$$
(85)

Computing the ratio \(M_h^d/M_h\) and applying (83) yields

$$\begin{aligned} \frac{M_h^d}{M_h} = \frac{R_{DC} \zeta _h \sqrt{1 - \zeta _h^2}}{\zeta _h^d \sqrt{1 - \zeta _h^{d\,\,2}}} \end{aligned}$$
(86)

Equating the left-hand side of (86) to \(R_M\) (Definition (15)) yields

$$\begin{aligned} \zeta _h^d \sqrt{1 - \zeta _h^{d\,\,2}} = \frac{R_{DC}}{R_M} \zeta _h \sqrt{1 - \zeta _h^2} \end{aligned}$$
(87)

Now we define the right-hand side of (87) as

$$\begin{aligned} \rho = \frac{R_{DC}}{R_M} \zeta _h \sqrt{1 - \zeta _h^2} \end{aligned}$$
(88)

yielding

$$\begin{aligned} \zeta _h^{d\,\,4} - \zeta _h^{d\,\,2} + \rho ^2 = 0 \end{aligned}$$
(89)

for which the solution that ensures the existence of a resonant peak is

$$\begin{aligned} \zeta _h^d = \sqrt{ \frac{1 - \sqrt{1 - 4 \rho ^2}}{2} } \end{aligned}$$
(90)

Appendix 2: Pole placement solution for the feedback compensator based on the properties of the positive-feedback root locus

Given a target dominant pole \(p_h^d\), the phase of \(L_{hecf}(p_h^d)\) is computed as

$$\begin{aligned} \varPhi (\sigma _f, \, \omega _{d,f}, \, p_h^d) = \sum _{i=1}^{N_z} \psi _i - \sum _{i=1}^{N_p} \phi _i - \phi _f - \overline{\phi }_f \end{aligned}$$
(91)

where

$$\begin{aligned} \psi _i&= \arctan \left( {\frac{{\text {Im}}\{p_h^d - z_{hec,i}\} }{{\text {Re}}\{p_h^d - z_{hec,i}\} }} \right) \nonumber \\ \phi _i&= \arctan \left( {\frac{{\text {Im}}\{p_h^d - p_{hec,i}\} }{{\text {Re}}\{p_h^d - p_{hec,i}\} }} \right) \nonumber \\ \phi _f&= \arctan \left( {\frac{{\text {Im}}\{p_h^d\} - \omega _{d,f}}{{\text {Re}}\{p_h^d\} + \sigma _f}} \right) \nonumber \\ \overline{\phi }_f&= \arctan \left( {\frac{{\text {Im}}\{p_h^d\} + \omega _{d,f}}{{\text {Re}}\{p_h^d\} + \sigma _f}} \right) \end{aligned}$$
(92)

Here \(z_{hec,i}\) are the zeros of \(L_{hecf}(s)\) and \(p_{hec,i}\) are the poles of \(L_{hecf}(s)\) excepting those at \(s = -\sigma _f \pm j\omega _{d,f}\). Thus \(N_p = N_z = \) 4. A valid solution for \(\sigma _f\) and \(\omega _{d,f}\) satisfies \(\varPhi (\sigma _f, \, \omega _{d,f}, \, p_h^d) =\) 0 for positive feedback.

Given a solution for \(\sigma _f\) and \(\omega _{d,f}\), the loop gain (54) is computed as

$$\begin{aligned} K_L(\sigma _f, \, \omega _{d,f}, \, p_h^d) = - \mathcal {K}_{L,f} \, \overline{\mathcal {K}}_{L,f} \, \mathcal {K}_{L,hec} \end{aligned}$$
(93)

where

$$\begin{aligned}&\mathcal {K}_{L,f} = [({\text {Re}}\{p_h^d\} + \sigma _f)^2 + ({\text {Im}}\{p_h^d\} - \omega _{d,f})^2]^\frac{1}{2} \nonumber \\&\overline{\mathcal {K}}_{L,f} = [({\text {Re}}\{p_h^d\} + \sigma _f)^2 + ({\text {Im}}\{p_h^d\} + \omega _{d,f})^2]^\frac{1}{2} \nonumber \\&\mathcal {K}_{L,hec} = \displaystyle \frac{\displaystyle \prod _{i=1}^{N_p}[{\text {Re}}\{p_h^d - p_{hec,i}\}^2 + {\text {Im}}\{p_h^d - p_{hec,i}\}^2]^\frac{1}{2}}{\displaystyle \prod _{i=1}^{N_z}[{\text {Re}}\{p_h^d - z_{hec,i}\}^2 + {\text {Im}}\{p_h^d - z_{hec,i}\}^2]^\frac{1}{2}} \end{aligned}$$
(94)

Appendix 3: Target values for the dynamic response parameters of the assisted leg: computation

In the dynamic walker (DW) of Fig. 16, described in Aguirre-Ollinger (2014), a combination of adaptive frequency oscillator (AFO) and adaptive Fourier analysis (Petric et al. 2011) generates a cyclic phase signal \(\varphi (t)\). We use \(\varphi \) to generate bell-shaped torques on the hip joints in lieu of hip muscle torques. The hip torque profiles for the stance and the swing phases are given, respectively, by

$$\begin{aligned}&\tau _{st}(\varphi ) = \frac{\tau _{st,max}}{K_{st}} \, \varphi \, \exp {\left( {1-\frac{\varphi }{K_{st}}} \right) } \nonumber \\&\tau _{sw}(\varphi ) = \frac{\tau _{sw,max}}{K_{sw}} \, \varphi \, \exp {\left( {1-\frac{\varphi }{K_{sw}}} \right) } \end{aligned}$$
(95)

The knee joint possesses torsional stiffness \(\kappa _{kn}\) and damping \(\nu _{kn}\). The knee becomes locked when the leading leg reaches full extension and is released again at toe-off. In order to ensure a stable gait, the hip joint of the leading leg is stiffened during the interval from knee-lock to ground collision. The torsional stiffness and damping for this interval are, respectively, \(\kappa _{sw}\) and \(\nu _{sw}\). The virtual spring \(\kappa _{sw}\) has an equilibrium defined by an inter-leg angle \(\beta =\beta _{eq}\) (Fig. 16b).

The numerical parameters employed in the simulation of Sect. 5 are given below.

  • Multibody system: \(M=\) 19.62 kg, \(m=\) 15.38 kg, \(l_2 = \) 0.4165 m, \(l_3 = \) 0.4845 m, foot radius = 0.1 m

  • Hip joint damping: \(b_h = \) 2.194 Nms/rad

  • Stance-phase hip torque: \(\tau _{st,max} = \) 39.7 Nm, \(K_{st} =\) 0.5

  • Swing-phase hip torque: \(\tau _{sw,max} = \) 35.3 Nm, \( K_{sw} = \) 0.5

  • Hip joint stiffness and damping at the end of the swing phase \(\kappa _{sw} = \) 96 Nm/rad, \(\nu _{sw} = \) 32 Nms/rad; equilibrium point: \(\beta _{eq} = \) 0.6 rad

  • Knee joint stiffness and damping: \(\kappa _{kn} = \) 8.43 Nm/rad, \(\nu _{kn} = \) 0.966 Nms/rad.

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Aguirre-Ollinger, G., Nagarajan, U. & Goswami, A. An admittance shaping controller for exoskeleton assistance of the lower extremities. Auton Robot 40, 701–728 (2016). https://doi.org/10.1007/s10514-015-9490-8

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