Design and Control of a Passive Noise Rejecting Variable Stiffness Actuator

Abstract

Inspired by the biomechanical and passive properties of human muscles, we present a novel actuator named passive noise rejecting Variable Stiffness Actuator (pnrVSA). For a single actuated joint, the proposed design adopts two motor-gear groups in an agonist-antagonist configuration coupled to the joint via serial non-linear springs. From a mechanical standpoint, the introduced novelty resides in two parallel non-linear springs connecting the internal motor-gear groups to the actuator frame. These additional elastic elements create a closed force path that mechanically attenuates the effects of external noise. We further explore the properties of this novel actuator by modeling the effect of gears static frictions on the output joint equilibrium position during the co-contraction of the agonist and antagonist side of the actuator. As a result, we found an analytical condition on the spring potential energies to guarantee that co-activation reduces the effect of friction on the joint equilibrium position. The design of an optimized set of springs respecting this condition leads to the construction of a prototype of our actuator. To conclude the work, we also present two control solutions that exploit the mechanical design of the actuator allowing to control both the joint stiffness and the joint equilibrium position.

Appendix

1.1 Sensitivity Matrix Computation

Let’s represent (2) in a compact way, with the following definition:

$$\begin{aligned} {\left\{ \begin{array}{ll} -U_1' (\hat{\vartheta }) - U_2' (\hat{\vartheta } + \hat{q}) = {\hat{\tau }}_{\vartheta } \\ U_4' ({\hat{\vartheta }}^a) + U_3' (\hat{\vartheta }^a -\hat{q}) = {\hat{\tau }}_{\vartheta ^a}\\ - U_2' (\hat{\vartheta } + \hat{q}) + U_3' ({\hat{\vartheta }}^a -\hat{q}) = {\hat{\tau }}_{q} \end{array}\right. } \iff f (\alpha \text {,} \tau ) = 0. \end{aligned}$$

By resourcing to the implicit function theorem, the equation $f (\alpha \text {,} \tau ) = 0$ locally defines a function $\alpha (\tau )$ (equilibrium configuration) with sensitivity:

$$\begin{aligned} \frac{\partial \alpha }{\partial \tau } = - {\left[ \frac{\partial f}{\partial \alpha }\right] }^{-1} \frac{\partial f}{\partial \tau }. \end{aligned}$$

as easily follows by numerical derivation of the constrain equation $f (\alpha (\tau ) \text {,} \tau ) = 0$:

$$\begin{aligned} \frac{\partial f}{\partial \alpha } \frac{\partial \alpha }{\partial \tau } + \frac{\partial f}{\partial \tau } = 0 \rightarrow \frac{\partial \alpha }{\partial \tau } = - {\left[ \frac{\partial f}{\partial \alpha }\right] }^{-1} \frac{\partial f}{\partial \tau }. \end{aligned}$$

Using the analytical expression of f given by (2), we obtain:

$$ \frac{\partial f}{\partial \alpha } = \begin{bmatrix} \frac{\partial f}{\partial \hat{\vartheta }} \frac{\partial f}{\partial {\hat{\vartheta }}^a} \frac{\partial f}{\partial \hat{q}} \end{bmatrix} = \begin{bmatrix} -U_1'' - U_2''&0&-U_2'' \\ 0&U_4'' + U_3''&- U_3'' \\ -U_2''&U_3''&- U_2'' - U_3'' \\ \end{bmatrix}, $$

and:

$$ \frac{\partial f}{\partial \tau } = \begin{bmatrix} -1&0&0 \\ 0&-1&0 \\ 0&0&-1 \\ \end{bmatrix} $$

which eventually results in the following expression:

$$ \frac{\partial \alpha }{\partial \tau } = {\begin{bmatrix} -U_1'' - U_2''&0&-U_2'' \\ 0&U_4'' + U_3''&- U_3'' \\ -U_2''&U_3''&- U_2'' - U_3'' \\ \end{bmatrix}}^{-1} $$

$$ \frac{\partial \alpha }{\partial \tau } = \begin{bmatrix} \frac{\partial \hat{\vartheta }}{\partial {\hat{\tau }}_{\vartheta }}&\frac{\partial \hat{\vartheta }}{\partial {{\hat{\tau }}_{\vartheta }}^a}&\frac{\partial \hat{\vartheta }}{\partial {\hat{\tau }}_q}\\ \frac{\partial {\hat{\vartheta }}^a}{\partial {\hat{\tau }}_{\vartheta }}&\frac{\partial {\hat{\vartheta }}^a}{\partial {{\hat{\tau }}_{\vartheta }}^a}&\frac{\partial {\hat{\vartheta }}^a}{\partial {\hat{\tau }}_q}\\ \frac{\partial \hat{q}}{\partial {\hat{\tau }}_{\vartheta }}&\frac{\partial \hat{q}}{\partial {{\hat{\tau }}_{\vartheta }}^a}&\frac{\partial \hat{q}}{\partial {\hat{\tau }}_q} \end{bmatrix} = $$

$$ \left[ {\begin{matrix} -(U_2'' U_3'' + U_2'' U_4'' + U_3'' U_4'') &{} - U_2'' U_3'' &{} U_2'' (U_3'' + U_4'') \\ U_2'' U_3'' &{} U_1'' U_2'' + U_1'' U_3'' + U_2'' U_3'' &{} -U_3'' (U_1'' + U_2'')\\ U_2'' (U_3'' + U_4'') &{} U_3'' (U_1'' + U_2'') &{} -(U_1'' + U_2'')(U_3'' + U_4'') \end{matrix}} \right] $$

$$\begin{aligned} \cdot \frac{1}{U_1'' U_2'' U_3'' + U_1'' U_2'' U_4'' + U_1'' U_3'' U_4'' + U_2'' U_3'' U_4'' } \end{aligned}$$

1.2 Actuator Specifications

The specifications of the actuator are recapped in Fig. 13.

Fig. 13

Source [14]

The VIACTORS VSA datasheet of the pnrVSA. In the plots on the right hand side we report the pnrVSA characteristic curves for different internal motor pretensions. This pretension has to be interpreted as the applied torque at motor capstan, ranging from 15 to 90% of the stall torque. The VIACTORS Variable Stiffness Joint Datasheet was developed within the VIACTORS project, which is a part of the EU 7th Framework Programme.