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Elastic-bound conditions for energetically optimal elasticity and their implications for biomimetic propulsion systems


Minimising the energy consumption associated with periodic motion is a priority common to a wide range of technologies and organisms. These include many forms of biological and biomimetic propulsion system, such as flying insects. Linear and nonlinear elasticity can play an important role in optimising the energetic behaviour of these systems, via linear or nonlinear resonance. However, existing methods for computing energetically optimal nonlinear elasticities struggle when actuator energy regeneration is imperfect: when the system cannot reuse work performed on the actuator, as occurs in many realistic systems. Here, we develop a new analytical method that overcomes these limitations. Our method provides exact nonlinear elasticities minimising the mechanical power consumption required to generate a target periodic response, under conditions of imperfect energy regeneration. We demonstrate how, in general parallel- and series-elastic actuation systems, imperfect regeneration can lead to a set of non-unique optimal nonlinear elasticities. This solution space generalises the energetic properties of linear resonance, and is described completely via bounds on the system work loop: the elastic-bound conditions. The choice of nonlinear elasticities from within these bounds leads to new tools for systems design, with particular relevance to biomimetic propulsion systems: tools for controlling the trade-off between actuator peak power and duty cycle; for using unidirectional actuators to generate energetically optimal oscillations; and further. More broadly, these results lead to new perspectives on the role of nonlinear elasticity in biological organisms, and new insights into the fundamental relationship between nonlinear resonance, nonlinear elasticity, and energetic optimality.

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Data availability

No datasets were generated or utilised in this study.


  1. The limit is a state of zero load over 50% of the work loop (Fig. 5Aii). If different quarter-cycles of the work loop are kinematically faster than others, then the limit of the time-domain duty cycle can be < 50%.

  2. 6 If the kinematic waveform, \(x\left(t\right)\), is composed of two symmetric half-cycles, i.e., \(x\left(t\right)=x\left(T-t\right),\forall t\), then both one-way drive elasticities will generate load waveforms of duty cycle 50% (Fig. 5Ciii). If this is not the case, then one will generate a waveform of duty cycle < 50%, and the other, a waveform of duty cycle > 50%.


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This work was supported by the Azrieli Foundation Faculty Fellowship the Israel Ministry of Science and Technology, and the Israel Science Foundation (Grant No. 1851/17). AP was additionally supported by the Jerusalem Brain Community Post-Doctoral Fellowship.

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1.1 A.1 Example constructions in PEA systems

For illustration, we derive specific forms of \(G^{ \pm } \left( x \right)\), and the elastic-bound condition, for two different PEA systems. First, consider a linear 1DOF PEA system, with time variable \(t\), displacement \(x\), inertia \(m\), and dissipation coefficient \(c\) (Fig. 2, main text). The system is driven by an actuator, generating simple harmonic oscillation, amplitude \(\hat{x}\):

$$ x\left( t \right) = \hat{x}\cos \left( {\omega t} \right), \dot{x}\left( t \right) = - \hat{x}\omega \sin \left( {\omega t} \right), \ddot{x}\left( t \right) = - \hat{x}\omega^{2} \cos \left( {\omega t} \right), $$

The required actuator load for this oscillation, inelastic (\(G\)) and elastic (\(F\)), is given by:

$$ \begin{aligned} G\left( t \right) & = m\ddot{x}\left( t \right) + c\dot{x}\left( t \right), \\ F\left( t \right) & = m\ddot{x}\left( t \right) + c\dot{x}\left( t \right) + F_{s} \left( {x\left( t \right)} \right), \\ \end{aligned} $$


$$ \begin{aligned} G\left( t \right) & = - m\hat{x}\omega^{2} \cos \left( {\omega t} \right) - c\hat{x}\omega \sin \left( {\omega t} \right), \\ F\left( t \right) & = - m\hat{x}\omega^{2} \cos \left( {\omega t} \right) - c\hat{x}\omega \sin \left( {\omega t} \right) + F_{s} \left( {x\left( t \right)} \right). \\ \end{aligned} $$

Parameterising \(G\left( t \right)\) and \(F\left( t \right)\) in terms of \(x\left( t \right) = \hat{x}\cos \left( {\omega t} \right)\), and splitting into two solutions (arcs) we have:

$$ \begin{aligned} G^{ \pm } \left( x \right) & = - m\omega^{2} x \pm c\omega \sqrt {\hat{x}^{2} - x^{2} } . \\ F^{ \pm } \left( x \right) & = F_{s} \left( x \right) - m\omega^{2} x \pm c\omega \sqrt {\hat{x}^{2} - x^{2} } . \\ \end{aligned} $$

The elastic-bound conditions then read: for \(x \in \left[ { - \hat{x},\hat{x}} \right]\),

$$ - m\omega^{2} x - c\omega \sqrt {\hat{x}^{2} - x^{2} } \le - F_{s} \left( x \right) \le - m\omega^{2} x + c\omega \sqrt {\hat{x}^{2} - x^{2} } . $$

Optimal linear, cubic, and quintic elasticities—if they exist, which is dependent on the system parameters—are thus:

$$ \begin{aligned} F_{s,1} \left( x \right) & = m\omega^{2} x, \\ F_{s,3} \left( x \right) & = m\omega^{2} x^{3} /\hat{x}^{2} , \\ F_{s,5} \left( x \right) & = m\omega^{2} x^{5} /\hat{x}^{4} . \\ \end{aligned} $$

Note that there is no more than one of each such optimal elasticity in this, or any other, system: one pure-linear, one cubic, etc. More broadly, we may define two families of optimal polynomial elasticities, interpolated linear-cubic and linear-quintic profiles, respectively:

$$ \begin{aligned} F_{s,1,3} \left( {x,\alpha } \right) & = \left( {1 - \alpha } \right)m\omega^{2} x + \alpha m\omega^{2} x^{3} /\hat{x}^{2} , \alpha \in \left[ {0,1} \right], \\ F_{s,1,5} \left( {x,\beta } \right) & = \left( {1 - \beta } \right)m\omega^{2} x + \beta m\omega^{2} x^{5} /\hat{x}^{4} , \beta \in \left[ {0,1} \right]. \\ \end{aligned} $$

The states \(\alpha = 0\) and \(\beta = 0\) represent linear profiles; and \(\alpha = 1\) and \(\beta = 1\), purely cubic and quintic profiles, respectively.

Second, consider a nonlinear 1DOF PEA system, undergoing the same simple-harmonic motion, but with quadratic damping—a simplified model of an insect flight motor:

$$ \begin{aligned} G\left( t \right) & = m\ddot{x}\left( t \right) + {\text{sign}}\left( {\dot{x}\left( t \right)} \right)c\dot{x}\left( t \right)^{2} , \\ F\left( t \right) & = m\ddot{x}\left( t \right) + {\text{sign}}\left( {\dot{x}\left( t \right)} \right)c\dot{x}\left( t \right)^{2} + F_{s} \left( {x\left( t \right)} \right), \\ \end{aligned} $$


$$ \begin{aligned} G\left( t \right) & = - m\hat{x}\omega^{2} \cos \left( {\omega t} \right) - {\text{sign}}\left( {\sin \left( {\omega t} \right)} \right)c\hat{x}^{2} \omega^{2} \sin^{2} \left( {\omega t} \right), \\ F\left( t \right) & = - m\hat{x}\omega^{2} \cos \left( {\omega t} \right) - {\text{sign}}\left( {\sin \left( {\omega t} \right)} \right)c\hat{x}^{2} \omega^{2} \sin^{2} \left( {\omega t} \right) + F_{s} \left( {x\left( t \right)} \right). \\ \end{aligned} $$

Parameterising \(G\left( t \right)\) and \(F\left( t \right)\) in terms of \(x\left( t \right) = \hat{x}\cos \left( {\omega t} \right)\) and splitting into two solutions (arcs), we have:

$$ \begin{aligned} G^{ \pm } \left( x \right) & = - m\omega^{2} x \pm c\omega^{2} \left( {x^{2} - \hat{x}^{2} } \right), \\ F^{ \pm } \left( x \right) & = - m\omega^{2} x \pm c\omega^{2} \left( {x^{2} - \hat{x}^{2} } \right) + F_{s} \left( {x\left( t \right)} \right). \\ \end{aligned} $$

The elastic-bound conditions then read: for \(x \in \left[ { - \hat{x},\hat{x}} \right]\),

$$ - m\omega^{2} x - c\omega^{2} \left( {x^{2} - \hat{x}^{2} } \right) \le - F_{s} \left( x \right) \le - m\omega^{2} x + c\omega^{2} \left( {x^{2} - \hat{x}^{2} } \right). $$

Note that, because linear inertia and harmonic motion is common to both systems considered in this section, the optimal elastic profiles, if they exist, are the identical (identical optimal cubic elasticity, etc., as per Eqs. A.1.6, A.1.7).

1.2 A.2 Example constructions in SEA systems

For illustration, we derive specific forms of \(X^{ \pm } \left( x \right)\) and the associated elastic optimality condition, for an SEA system. Consider a linear, 1DOF, SEA, system, with time variable \(t\), displacement \(x\), inertia \(m\), and dissipation coefficient \(c\) (per Fig. 4, main text). The system is driven by an actuator, generating simple harmonic oscillation, amplitude \(\hat{x}\):

$$ x\left( t \right) = \hat{x}\cos \left( {\omega t} \right), \dot{x}\left( t \right) = - \hat{x}\omega \sin \left( {\omega t} \right), \ddot{x}\left( t \right) = - \hat{x}\omega^{2} \cos \left( {\omega t} \right), $$

The equation of motion for this system is:

$$ m\ddot{x}\left( t \right) + c\dot{x}\left( t \right) = F_{s} \left( {u\left( t \right) - x\left( t \right)} \right), $$

and thus provided that \(F_{s} \left( \cdot \right)\) is invertible, the required actuator displacement, \(u\left( t \right)\), and actuator load, are given by apply \(F_{s}^{ - 1} \left( \cdot \right)\) to Eq. A.2.2:

$$ \begin{aligned} u\left( t \right) & = x\left( t \right) + F_{s}^{ - 1} \left( {m\ddot{x}\left( t \right) + c\dot{x}\left( t \right)} \right), \\ F\left( t \right) & = F_{s} \left( {u\left( t \right) - x\left( t \right)} \right) = m\ddot{x}\left( t \right) + c\dot{x}\left( t \right). \\ \end{aligned} $$

It follows that the actuator displacement requirement may be alternately cast as:

$$ u\left( t \right) = x\left( t \right) + F_{s}^{ - 1} \left( {F\left( t \right)} \right), $$

To reformulate this requirement as \(u\left( F \right)\), we must reformulate \(x\left( t \right)\) in terms of \(F\). From the definition of the actuator load, we have:

$$ F\left( t \right) = - m\hat{x}\omega^{2} \cos \left( {\omega t} \right) - c\hat{x}\omega \sin \left( {\omega t} \right), $$

and thus, as with the PEA, a work loop representation, \(F\)-\(x\):

$$ F^{ \pm } \left( x \right) = - m\omega^{2} x \pm c\omega \sqrt {\hat{x}^{2} - x^{2} } . $$

To invert this into a loop \(x\)-\(F\), we go through the following process. Manipulating the root term in both \(F^{ + } \left( x \right)\) and \(F^{ - } \left( x \right)\) yields a single multivariable polynomial, describing the elliptical shape of the work loop, which can be solved for either \(F\) or \(x\):

$$ \left( {m^{2} \omega^{4} + c^{2} \omega^{2} } \right)x^{2} + 2Fm\omega^{2} x + \left( {F^{2} - c^{2} \omega^{2} \hat{x}^{2} } \right) = 0. $$

Note that \(F\) has replaced \(F^{ \pm }\) now that these branches are unified. The solution to this polynomial, in terms of \(x\), are the functions we denote \(X^{ \pm } \left( F \right)\):

$$ X^{ \pm } \left( F \right) = \frac{{ - mF \pm \frac{c}{\omega }\sqrt {\hat{x}^{2} \omega^{2} \left( {m^{2} \omega^{2} + c^{2} } \right) - F^{2} } }}{{m^{2} \omega^{2} + c^{2} }}. $$

The appropriate \(F\)-range for \(X^{ \pm } \left( F \right)\), \(\left[ { - \hat{F},\hat{F}} \right]\), is given by the extrema of Eq. A.2.6:

$$ \hat{F} = \omega \hat{x}\sqrt {m^{2} \omega^{2} + c^{2} } . $$

The derivatives \(X^{\prime \pm } \left( F \right)\) can be computed directly from Eq. A.2.8, as:

$$ X^{\prime \pm } \left( F \right) = \frac{1}{{m^{2} \omega^{2} + c^{2} }}\left( { - m \mp \frac{cF}{{\omega \sqrt {\hat{x}^{2} \omega^{2} \left( {m^{2} \omega^{2} + c^{2} } \right) - F^{2} } }}} \right), $$

and the elastic bound-conditions read:

$$ \begin{aligned} X^{\prime - } \left( F \right) & \le - \left( {F_{s}^{ - 1} } \right)^{^{\prime}} \left( F \right) \le X^{\prime + } \left( F \right), \forall F \in \left[ {F_{1} ,0} \right], \\ X^{\prime + } \left( F \right) & \le - \left( {F_{s}^{ - 1} } \right)^{^{\prime}} \left( F \right) \le X^{\prime - } \left( F \right), \forall F \in \left[ {0,F_{2} } \right]. \\ \end{aligned} $$

We note that this state condition can be fulfilled by some \(F_{s}^{ - 1} \left( F \right)\): the optimal state is accessible (Sect. 2.3.4). More specifically:

$$ X^{\prime - } \left( 0 \right) = X^{\prime + } \left( 0 \right) = \frac{ - m}{{m^{2} \omega^{2} + c^{2} }} \therefore \left( {F_{s,lin}^{ - 1} } \right)^{^{\prime}} \left( 0 \right) = \frac{m}{{m^{2} \omega^{2} + c^{2} }}, $$

where \(F_{s,1} \left( \cdot \right)\) is the system optimal linear elasticity, alternately expressible as:

$$ F_{s,lin} \left( {u - x} \right) = \frac{{m^{2} \omega^{2} + c^{2} }}{m}\left( {u - x} \right). $$

1.3 A.3 Proof of mechanical power optimality in PEA systems

In this section, we prove the PEA elastic optimality results that we have utilised in the main text, prior to this point in the appendix. We begin at the formulation of the system as per the work loop of Eq. 7:

$$ \begin{aligned} G^{ \pm } \left( x \right) & = G_{{{\text{mid}}}} \left( x \right) \pm G_{{{\text{arc}}}} \left( x \right), \\ F^{ \pm } \left( x \right) & = G_{{{\text{mid}}}} \left( x \right) \pm G_{{{\text{arc}}}} \left( x \right) + F_{s} \left( x \right), \\ \end{aligned} $$

over \(x \in \left[ {x_{1} ,x_{2} } \right]\), with \(G_{{{\text{arc}}}} \left( x \right) > 0\). If a system can be formulated as a work loop of this form, respecting the condition that the loop must be a simple closed curve, no more than bivalued at any \(x\), and showing net power dissipation, then the system is admissible under this analysis.

First, consider evaluating the four integrals of mechanical power, Eq. 8, for \(F^{ \pm } \left( x \right)\). For the net power, \({\overline{\text{P}}}_{{\left( {\text{a}} \right)}}\), we have the result:

$$ {\overline{\text{P}}}_{{\left( {\text{a}} \right)}} = \frac{1}{T}\mathop \int \limits_{{x_{1} }}^{{x_{2} }} \left( {F^{ + } \left( x \right) - F^{ - } \left( x \right)} \right)dx = \frac{2}{T}\mathop \int \limits_{{x_{1} }}^{{x_{2} }} G_{{{\text{arc}}}} \left( x \right)dx; $$

that is, the net power is unaffected by elasticity, \(F_{s} \left( x \right)\), exactly as we would expect. For the other metrics, \({\overline{\text{P}}}_{{\left( {\text{b}} \right)}}\), \({\overline{\text{P}}}_{{\left( {\text{c}} \right)}}\), and \({\overline{\text{P}}}_{{\left( {\text{d}} \right)}}\), however, no self-cancellation occurs, and a general dependency on elasticity, \(F_{s} \left( x \right)\), remains. To gain further insight into the behaviour of these latter metrics, consider an alternative work loop parameterisation. Split \(F^{ + } \left( x \right)\) and \(F^{ - } \left( x \right)\) into four components, representing regions of positive (\(A\)) and negative (\(B\)) load:

$$ \begin{gathered} F^{ + } \left( x \right) = A^{ + } \left( x \right) - B^{ + } \left( x \right),\quad F^{ - } \left( x \right) = A^{ - } \left( x \right) - B^{ - } \left( x \right), \hfill \\ A^{ + } \left( x \right) = F^{ + } \left( x \right)\left[ {F^{ + } \left( x \right) \ge 0} \right]_{{\mathbb{I}}} ,\quad A^{ - } \left( x \right) = F^{ - } \left( x \right)\left[ {F^{ - } \left( x \right) \ge 0} \right]_{{\mathbb{I}}} , \hfill \\ B^{ + } \left( x \right) = - F^{ + } \left( x \right)\left[ {F^{ + } \left( x \right) \le 0} \right]_{{\mathbb{I}}} ,\quad B^{ - } \left( x \right) = - F^{ - } \left( x \right)\left[ {F^{ - } \left( x \right) \le 0} \right]_{{\mathbb{I}}} . \hfill \\ \end{gathered} $$

Thus, \(A^{ \pm } \left( x \right), B^{ \pm } \left( x \right) \ge 0, \forall x\), and individual power integrals for each component are necessarily positive:

$$ W\left( I \right) = \frac{1}{T}\mathop \int \limits_{{x_{1} }}^{{x_{2} }} I\left( x \right)dx \ge 0,{ }I\left( x \right) \in \left\{ {A^{ + } \left( x \right),B^{ + } \left( x \right),A^{ - } \left( x \right),B^{ - } \left( x \right)} \right\}. $$

This allows evaluation of the power metrics, Eq. 8, in terms of these \(W\left(I\right)\). For the net power, \({\overline{\mathrm{P}} }_{\left({\text{a}}\right)}\), we have the result:

$$ {\overline{\text{P}}}_{{\left( {\text{a}} \right)}} = W\left( {A^{ + } } \right) + W\left( {B^{ - } } \right) - W\left( {A^{ - } } \right) - W\left( {B^{ + } } \right), $$

where the terms \(W\left( {A^{ - } } \right)\) and \(W\left( {B^{ + } } \right)\) serve to reduce the actuator power consumption (via storage and release of negative power). We also have explicit results for \({\overline{\text{P}}}_{{\left( {\text{b}} \right)}}\), \({\overline{\text{P}}}_{{\left( {\text{c}} \right)}}\) and \({\overline{\text{P}}}_{{\left( {\text{d}} \right)}}\):

$$ \begin{aligned} {\overline{\text{P}}}_{{\left( {\text{b}} \right)}} & = W\left( {A^{ + } } \right) + W\left( {B^{ - } } \right) + W\left( {A^{ - } } \right) + W\left( {B^{ + } } \right), \\ {\overline{\text{P}}}_{{\left( {\text{c}} \right)}} & = W\left( {A^{ + } } \right) + W\left( {B^{ - } } \right), \\ {\overline{\text{P}}}_{{\left( {\text{d}} \right)}} & = W\left( {A^{ + } } \right) + W\left( {B^{ - } } \right) - W\left( {A^{ - } } \right) - W\left( {B^{ + } } \right) + W_{Q} , \\ \end{aligned} $$

with the special penalty integral for \({\overline{\text{P}}}_{{\left( {\text{d}} \right)}}\):

$$ W_{Q} = \frac{1}{T}\mathop \int \limits_{{x_{1} }}^{{x_{2} }} Q^{ + } F^{ + } \left[ {F^{ + } \le 0} \right]_{{\mathbb{I}}} dx + \frac{1}{T}\mathop \int \limits_{{x_{1} }}^{{x_{2} }} Q^{ - } F^{ - } \left[ {F^{ - } \ge 0} \right]_{{\mathbb{I}}} dx. $$

These explicit forms have several implications, representable in the form of the following inequalities, valid \(\forall F^{ \pm } \left( x \right)\):

$$ \begin{gathered} {\overline{\text{P}}}_{{\left( {\text{b}} \right)}} \ge {\overline{\text{P}}}_{{\left( {\text{a}} \right)}} , \hfill \\ {\overline{\text{P}}}_{{\left( {\text{c}} \right)}} \ge {\overline{\text{P}}}_{{\left( {\text{a}} \right)}} , \hfill \\ {\overline{\text{P}}}_{{\left( {\text{b}} \right)}} \ge {\overline{\text{P}}}_{{\left( {\text{c}} \right)}} , \hfill \\ {\overline{\text{P}}}_{{\left( {\text{d}} \right)}} \ge {\overline{\text{P}}}_{{\left( {\text{a}} \right)}} , \hfill \\ \end{gathered} $$

these results can be seen from the fact that \(W\left( I \right) \ge 0,\forall I\left( x \right)\). In the case of \({\overline{\text{P}}}_{{\left( {\text{d}} \right)}} \ge {\overline{\text{P}}}_{{\left( {\text{a}} \right)}}\), from the fact that \(Q^{ \pm } \left( x \right) > 0\) necessitates \(W_{Q} \ge 0\).

Synthesising the information from Eqs. A.3.2 and A.3.9 yields the following conclusions. From Eq. A.3.2, \({\overline{\text{P}}}_{{\left( {\text{a}} \right)}}\) is independent of \(F_{s} \left( x \right)\), whereas \({\overline{\text{P}}}_{{\left( {\text{b}} \right)}}\), \({\overline{\text{P}}}_{{\left( {\text{c}} \right)}}\), and \({\overline{\text{P}}}_{{\left( {\text{d}} \right)}}\) are (in general) dependent. From Eq. A.3.9, the minimum possible value that \({\overline{\text{P}}}_{{\left( {\text{b}} \right)}}\), \({\overline{\text{P}}}_{{\left( {\text{c}} \right)}}\) and \({\overline{\text{P}}}_{{\left( {\text{d}} \right)}}\) can take, under any \(F_{s} \left( x \right)\), is this independent value of \({\overline{\text{P}}}_{{\left( {\text{a}} \right)}}\). There is no immediate guarantee that this minimum is attainable, but we may explore the conditions for it to be attained. Given \(W\left( I \right) \ge 0,\forall I\left( x \right)\):

$$ \begin{aligned} {\overline{\text{P}}}_{{\left( {\text{b}} \right)}} & = {\overline{\text{P}}}_{{\left( {\text{a}} \right)}} \Rightarrow W\left( {A^{ - } } \right) + W\left( {B^{ + } } \right) = 0 \Rightarrow W\left( {A^{ - } } \right) = W\left( {B^{ + } } \right) = 0, \\ {\overline{\text{P}}}_{{\left( {\text{c}} \right)}} & = {\overline{\text{P}}}_{{\left( {\text{a}} \right)}} \Rightarrow W\left( {A^{ - } } \right) + W\left( {B^{ + } } \right) = 0 \Rightarrow W\left( {A^{ - } } \right) = W\left( {B^{ + } } \right) = 0, \\ {\overline{\text{P}}}_{{\left( {\text{d}} \right)}} & = {\overline{\text{P}}}_{{\left( {\text{a}} \right)}} \Rightarrow W_{Q} = 0. \\ \end{aligned} $$

Consider these two sets of conditions separately. For \({\overline{\text{P}}}_{{\left( {\text{b}} \right)}}\) and \({\overline{\text{P}}}_{{\left( {\text{c}} \right)}}\), given that \(A^{ - } \left( x \right)\) and \(B^{ + } \left( x \right)\) cannot change sign (Eq. A.3.4), the integral condition \(W\left( {A^{ - } } \right) = W\left( {B^{ + } } \right) = 0\) implies the functional condition \(A^{ - } \left( x \right) = B^{ + } \left( x \right) = 0,\forall x\); that is\(,\forall x\):

$$ \begin{gathered} F^{ + } \left( x \right)\left[ {F^{ + } \left( x \right) \le 0} \right]_{{\mathbb{I}}} = 0, \hfill \\ F^{ - } \left( x \right)\left[ {F^{ - } \left( x \right) \ge 0} \right]_{{\mathbb{I}}} = 0. \hfill \\ \end{gathered} $$

For \({\overline{\text{P}}}_{{\left( {\text{d}} \right)}}\), the condition \(W_{Q} = 0\) implies directly the identical condition, that, \(\forall x\):

$$ \begin{gathered} F^{ + } \left( x \right)\left[ {F^{ + } \left( x \right) \le 0} \right]_{{\mathbb{I}}} = 0, \hfill \\ F^{ - } \left( x \right)\left[ {F^{ - } \left( x \right) \ge 0} \right]_{{\mathbb{I}}} = 0. \hfill \\ \end{gathered} $$

It follows that this single pair of conditions, Eqs. A.3.11 or A.3.12, ensure optimality in mechanical power both necessarily and sufficiently: they are the sole conditions to ensure that the inequalities of Eq. A.3.9 become equalities, and thus, the mechanical power metrics take their minimum possible values.

These conditions are interpretable in terms of \(F_{s} \left( x \right)\), in the following way. To ensure that the expressions in Eqs. A.3.11 or A.3.12 are zero-valued \(\forall x\), the following must be true:

$$ \begin{gathered} \forall x\;{\text{either}}\;F^{ + } \left( x \right) = 0\;{\text{or}}\;\left[ {F^{ + } \left( x \right) \le 0} \right]_{{\mathbb{I}}} = 0, \hfill \\ \forall x\quad {\text{either}}\;F^{ - } \left( x \right) = 0\;{\text{or}}\;\left[ {F^{ - } \left( x \right) \ge 0} \right]_{{\mathbb{I}}} = 0. \hfill \\ \end{gathered} $$

Computing the Iverson bracket, these conditions become:

$$ \begin{gathered} \forall x\;{\text{either}}\;F^{ + } \left( x \right) = 0\;{\text{or}}\;F^{ + } \left( x \right) > 0, \hfill \\ \forall x\;{\text{either}}\;F^{ - } \left( x \right) = 0\;{\text{or}}\;F^{ - } \left( x \right) < 0, \hfill \\ \end{gathered} $$

or, as two simple inequalities:

$$ \begin{gathered} F^{ + } \left( x \right) \ge 0, \hfill \\ F^{ - } \left( x \right) \le 0. \hfill \\ \end{gathered} $$

This is the intuitive result of Sect. 2.2: the presence of negative power in the system, i.e. \(F\dot{x} < 0\), necessarily implies that the system is suboptimal in terms of mechanical power.

What are the conditions on \(F_{s} \left( x \right)\) to ensure Eq. A.3.15 is satisfied? This is simple to compute. As \(F^{ \pm } \left( x \right) = G^{ \pm } \left( x \right) + F_{s} \left( x \right)\), by the system definition, Eq. A.3.1, it follows that:

$$ \begin{gathered} G^{ + } \left( x \right) + F_{s} \left( x \right) \ge 0, \hfill \\ G^{ - } \left( x \right) + F_{s} \left( x \right) \le 0, \hfill \\ \end{gathered} $$

and thus:

$$ G^{ - } \left( x \right) \le - F_{s} \left( x \right) \le G^{ + } \left( x \right), $$

which is the elastic-bound condition, Eq. 10 (Sect. 2.2). We have demonstrated that this condition is necessary and sufficient to ensure that any of the mechanical power metrics, \({\overline{\text{P}}}_{{\left( {\text{b}} \right)}}\), \({\overline{\text{P}}}_{{\left( {\text{c}} \right)}}\) and \({\overline{\text{P}}}_{{\left( {\text{d}} \right)}}\), are minimised. Note that (i) we have computed this optimality condition analytically, not numerically, in contrast to most existing treatments of absolute-power minimisation [58, 59, 63]; (ii) we required no calculus of optimisation to do so: optimality can be deduced simply from the form of the power integrals in the \(F\)-\(x\) domain; and (iii) no differentiability requirements have been imposed on any system parameters. Work-loop analysis is a powerful tool for analysing the optimality of system parameters with respect to mechanical power consumption.

1.4 A.4 Proof of mechanical power optimality in SEA systems

We begin with a general SEA system defined as per Eq. 11. We then place the following restrictions, defining the admissibility of the system for our analysis. We require that there exist time-domain functions \(x\left( t \right)\), \(\dot{x}\left( t \right)\), \(F\left( t \right)\), and \(\dot{F}\left( t \right)\), all continuous and real-valued over all real-valued \(t\), and periodic with period \(T\). Consider the function \(\dot{F}\left( t \right)\) in more detail. We define two sets of times, \(T^{ + }\) and \(T^{ - }\), in relation to this function: \(T^{ + } = \left\{ {t : \dot{F}\left( t \right) < 0} \right\}\), and \(T^{ - } = \left\{ {t : \dot{F}\left( t \right) \ge 0} \right\}\). Note the association of “\(+\)” with “\(< 0\)” and “\({-}\)” with “\(\ge 0\)” is intentional. We require that these sets of times exist. If they do, they are necessarily unique, and together span all \(t\), i.e. \(T^{ + } \cup T^{ - } = {\mathbb{R}}\). We can then use these time windows to segment \(\dot{F}\left( t \right)\) into two functions: \(\dot{f}^{ + } \left( t \right) = \dot{F}\left( t \right)\), defined for \(t \in T^{ + }\), and \(\dot{f}^{ - } \left( t \right) = \dot{F}\left( t \right)\), defined for \(t \in T^{ - }\). It follows that \(\dot{f}^{ + } \left( t \right) < 0\), \(\forall t \in T^{ + }\), and \(\dot{f}^{ + } \left( t \right) \ge 0\), \(\forall t \in T^{ + }\), and we may reconstruct \(\dot{F}\left( t \right)\) as:

$$ \dot{F}\left( t \right) = \left\{ {\begin{array}{*{20}c} {\dot{f}^{ + } \left( t \right)} & {t \in T^{ + } ,} \\ {\dot{f}^{ - } \left( t \right)} & {t \in T^{ - } .} \\ \end{array} } \right. $$

We now apply the same process to \(x\left( t \right)\), \(\dot{x}\left( t \right)\) and \(F\left( t \right)\):

$$ x\left( t \right) = \left\{ {\begin{array}{*{20}c} {x^{ + } \left( t \right)} & {t \in T^{ + } ,} \\ {x^{ - } \left( t \right)} & {t \in T^{ - } ,} \\ \end{array} } \right. \dot{x}\left( t \right) = \left\{ {\begin{array}{*{20}c} {\dot{x}^{ + } \left( t \right)} & {t \in T^{ + } ,} \\ {\dot{x}^{ - } \left( t \right)} & {t \in T^{ - } ,} \\ \end{array} } \right. $$
$$ F\left( t \right) = \left\{ {\begin{array}{*{20}c} {f^{ + } \left( t \right)} & {t \in T^{ + } ,} \\ {f^{ - } \left( t \right)} & {t \in T^{ - } ,} \\ \end{array} } \right. $$

defining \(x^{ \pm } \left( t \right)\), \(\dot{x}^{ \pm } \left( t \right)\) and \(F^{ \pm } \left( t \right)\). We require the following condition be satisfied by these functions: that the value of \(x^{ + } \left( t \right)\), \(x^{ - } \left( t \right)\), \(\dot{x}^{ + } \left( t \right)\) and \(\dot{x}^{ - } \left( t \right)\), at all \(t\), must be uniquely defined by the value \(F\left( t \right)\). In equivalent terms: \(f^{ + } \left( t \right)\) and \(f^{ - } \left( t \right)\) must be invertible: the periodic waveform \(F\left( t \right)\) must be composed to two monotonic half cycles.

If this condition is satisfied, we may parameterise \(x^{ \pm } \left( t \right)\) and \(\dot{x}^{ \pm } \left( t \right)\) in terms of \(F\):

$$ \begin{gathered} x\left( t \right) = \left\{ {\begin{array}{*{20}c} {X^{ + } \left( {F\left( t \right)} \right)} & {t \in T^{ + } ,} \\ {X^{ - } \left( {F\left( t \right)} \right)} & {t \in T^{ - } ,} \\ \end{array} } \right. \dot{x}\left( t \right) = \left\{ {\begin{array}{*{20}c} {\dot{X}^{ + } \left( {F\left( t \right)} \right)} & {t \in T^{ + } ,} \\ {\dot{X}^{ - } \left( {F\left( t \right)} \right)} & {t \in T^{ - } ,} \\ \end{array} } \right. \hfill \\ \dot{F}\left( t \right) = \left\{ {\begin{array}{*{20}c} {\dot{F}^{ + } \left( {F\left( t \right)} \right)} & {t \in T^{ + } ,} \\ {\dot{F}^{ - } \left( {F\left( t \right)} \right)} & {t \in T^{ - } ,} \\ \end{array} } \right. \hfill \\ \end{gathered} $$

where \(x^{ \pm } \left( t \right)\) and \(\dot{x}^{ \pm } \left( t \right)\) have now been reformulated into the functions \(X^{ \pm } \left( F \right)\), \(\dot{X}^{ \pm } \left( F \right)\) and \(F^{ \pm } \left( F \right)\), defined over an interval in force, \(F \in \left[ {F_{1} ,F_{2} } \right]\), where \(F_{1} = \min F\left( t \right)\) and \(F_{2} = \max F\left( t \right)\). The function \(X^{ \pm } \left( F \right)\) we will recognise as a work loop: \(dx \cdot dF\) is the differential of work. The final condition which we require to be satisfied is: \(X^{ + } \left( F \right) \ge X^{ - } \left( F \right),\forall F \in \left[ {F_{1} ,F_{2} } \right]\) (NB: the motivation behind the allocation of the superscript symbols, \(\pm\)). We note that the conditions \(X^{ + } \left( {F_{1} } \right) = X^{ - } \left( {F_{1} } \right)\) and \(X^{ + } \left( {F_{2} } \right) = X^{ - } \left( {F_{2} } \right)\) are already necessarily satisfied by nature of the continuity of \(x\left( t \right)\).

If these conditions are satisfied, then we consider the system admissible for our analysis, and we may proceed. Consider an elastic element in the system, with elastic load profile \({F}_{s}\left(\delta \right)\), continuous, differentiable, and real-valued over all real-valued \(\delta \). We require that \({F}_{s}\left(\delta \right)\) be invertible. We may then express the SEA dynamics and power requirements (cf. Equation 12) as:

$$ \begin{aligned} u\left( t \right) & = x\left( t \right) + F_{s}^{ - 1} \left( {F\left( t \right)} \right), \\ \dot{u}\left( t \right) & = \dot{x}\left( t \right) + \dot{F}\left( t \right)\left( {F_{s}^{ - 1} } \right)^{^{\prime}} \left( {F\left( t \right)} \right), \\ P\left( t \right) & = F\left( t \right)\dot{u}\left( t \right), \\ \end{aligned} $$

where \(\left( {F_{s}^{ - 1} } \right)^{^{\prime}} \left( F \right)\) denotes \(\frac{d}{dF}F_{s}^{ - 1} \left( F \right)\). Note that we have used the differential relation.

\(dF = \dot{F}dt\) to reformulate \(\frac{d}{dt}F_{s}^{ - 1} \left( F \right)\) and \(\dot{F}\left( t \right)\frac{d}{dF}F_{s}^{ - 1} \left( F \right)\). Using the sets of times, \(T^{ + }\) and \(T^{ - }\), we may segment these additional functions in the manner of Eq. A.4.2:

$$ \begin{gathered} u\left( t \right) = \left\{ {\begin{array}{*{20}c} {u^{ + } \left( t \right)} & {t \in T^{ + } ,} \\ {u^{ - } \left( t \right)} & {t \in T^{ - } ,} \\ \end{array} } \right. \dot{u}\left( t \right) = \left\{ {\begin{array}{*{20}c} {\dot{u}^{ + } \left( t \right)} & {t \in T^{ + } ,} \\ {\dot{u}^{ - } \left( t \right)} & {t \in T^{ - } ,} \\ \end{array} } \right. \hfill \\ P\left( t \right) = F\left( t \right)\dot{u}\left( t \right), \hfill \\ \end{gathered} $$

and parameterise them in terms of \(F\), in the manner of Eq. A.4.3:

$$ \begin{gathered} u\left( t \right) = \left\{ {\begin{array}{*{20}c} {U^{ + } \left( {F\left( t \right)} \right)} & {t \in T^{ + } ,} \\ {U^{ - } \left( {F\left( t \right)} \right)} & {t \in T^{ - } ,} \\ \end{array} } \right. \dot{u}\left( t \right) = \left\{ {\begin{array}{*{20}c} {\dot{U}^{ + } \left( {F\left( t \right)} \right)} & {t \in T^{ + } ,} \\ {\dot{U}^{ - } \left( {F\left( t \right)} \right)} & {t \in T^{ - } ,} \\ \end{array} } \right. \hfill \\ P\left( t \right) = \left\{ {\begin{array}{*{20}c} {P^{ + } \left( {F\left( t \right)} \right)} & {t \in T^{ + } ,} \\ {P^{ - } \left( {F\left( t \right)} \right)} & {t \in T^{ - } ,} \\ \end{array} } \right. \hfill \\ \end{gathered} $$

Now, we can relate these segmented and parameterised functions to known system functions (Eqs. A.4.2 and A.4.3). Then, over \(F \in \left[ {F_{1} ,F_{2} } \right]\):

$$ \begin{aligned} u^{ \pm } \left( t \right) & = x^{ \pm } \left( t \right) + F_{s}^{ - 1} \left( {f^{ \pm } \left( t \right)} \right), \\ \dot{u}^{ \pm } \left( t \right) & = \dot{x}^{ \pm } \left( t \right) + \dot{f}^{ \pm } \left( t \right) \cdot \left( {F_{s}^{ - 1} } \right)^{^{\prime}} \left( {f^{ \pm } \left( t \right)} \right), \\ p^{ \pm } \left( t \right) & = f^{ \pm } \left( t \right) \cdot \left( {\dot{x}^{ \pm } \left( t \right) + \dot{f}^{ \pm } \left( t \right)\left( {F_{s}^{ - 1} } \right)^{^{\prime}} \left( {f^{ \pm } \left( t \right)} \right)} \right), \\ U^{ \pm } \left( F \right) & = X^{ \pm } \left( F \right) + F_{s}^{ - 1} \left( F \right), \\ \dot{U}^{ \pm } \left( F \right) & = \dot{X}^{ \pm } \left( F \right) + \dot{F}^{ \pm } \left( F \right) \cdot \left( {F_{s}^{ - 1} } \right)^{^{\prime}} \left( F \right), \\ P^{ \pm } \left( F \right) & = F \cdot \left( {\dot{X}^{ \pm } \left( F \right) + \dot{F}^{ \pm } \left( F \right) \cdot \left( {F_{s}^{ - 1} } \right)^{^{\prime}} \left( F \right)} \right). \\ \end{aligned} $$

In addition, we can define the work-loop tangent functions, \(X^{\prime \pm } \left( F \right) = \frac{d}{dF}X^{ \pm } \left( F \right)\) and \(U^{\prime \pm } \left( F \right) = \frac{d}{dF}U^{ \pm } \left( F \right)\), where:

$$ X^{\prime \pm } \left( F \right) = \frac{{\dot{X}^{ + } \left( F \right)}}{{\dot{F}^{ + } \left( F \right)}}, U^{\prime \pm } \left( F \right) = X^{\prime \pm } \left( F \right) + \left( {F_{s}^{ - 1} } \right)^{^{\prime}} \left( F \right), $$

and thus:

$$ P^{ \pm } \left( F \right) = F \cdot \dot{F}^{ \pm } \left( F \right) \cdot \left( {X^{\prime \pm } \left( F \right) + \left( {F_{s}^{ - 1} } \right)^{^{\prime}} \left( F \right)} \right). $$

We are now able to evaluate the actuator power consumption metrics (a)–(d) (Sect. 2.1) in terms of \(P^{ \pm } \left( F \right)\). For the net power, metric (a):

$$ {\overline{\text{P}}}_{{\left( {\text{a}} \right)}} = \frac{1}{T}\mathop \int \limits_{0}^{T} P\left( t \right)dt = \frac{1}{T}\left( {\mathop \int \limits_{{t \in T^{ + } }}^{ } p^{ + } \left( t \right)dt + \mathop \int \limits_{{t \in T^{ - } }}^{ } p^{ - } \left( t \right)dt} \right) = \frac{1}{T}\left( {\mathop \int \limits_{{F_{2} }}^{{F_{1} }} \frac{{P^{ + } \left( F \right)}}{{\dot{F}^{ + } \left( F \right)}}dF + \mathop \int \limits_{{F_{1} }}^{{F_{2} }} \frac{{P^{ - } \left( F \right)}}{{\dot{F}^{ - } \left( F \right)}}dF} \right), $$

where, flipping limits; \(\left[ {F_{2} ,F_{1} } \right]\) to \(\left[ {F_{1} ,F_{2} } \right]\), we obtain:

$$ {\overline{\text{P}}}_{{\left( {\text{a}} \right)}} = \frac{1}{T}\left( {\mathop \int \limits_{{F_{1} }}^{{F_{2} }} F \cdot \left( {X^{\prime - } \left( F \right) - X^{\prime + } \left( F \right)} \right)dF} \right). $$

We require that the sets of time \(T^{ \pm }\) permit an adequate definition of integration (i.e. they are composed of continuous intervals). We observe that \({\overline{\text{P}}}_{{\left( {\text{a}} \right)}}\) is independent of \(F_{s} \left( \delta \right)\): the net power is unaltered by elasticity. We can perform the same process for \({\overline{\text{P}}}_{{\left( {\text{d}} \right)}}\) and thereby recover both \({\overline{\text{P}}}_{{\left( {\text{b}} \right)}}\) and \({\overline{\text{P}}}_{{\left( {\text{c}} \right)}}\), as we already know that they are special cases of \({\overline{\text{P}}}_{{\left( {\text{d}} \right)}}\). Evaluating \({\overline{\text{P}}}_{{\left( {\text{d}} \right)}}\), we obtain:

$$ \begin{aligned} {\overline{\text{P}}}_{{\left( {\text{d}} \right)}} & = \frac{1}{T}\mathop \int \limits_{0}^{T} \left( {P\left( t \right) + Q\left( t \right)\left| {P\left( t \right)} \right|\left[ {P\left( t \right) \le 0} \right]_{{\mathbb{I}}} } \right)dt = = \frac{1}{T}\mathop \int \limits_{{t \in T^{ + } }}^{ } \left( {p^{ + } \left( t \right) + q^{ + } \left( t \right)\left| {p^{ + } \left( t \right)} \right|\left[ {p^{ + } \left( t \right) \le 0} \right]_{{\mathbb{I}}} } \right)dt \\ & \quad + \frac{1}{T}\mathop \int \limits_{{t \in T^{ - } }}^{ } \left( {p^{ - } \left( t \right) + q^{ - } \left( t \right)\left| {p^{ - } \left( t \right)} \right|\left[ {p^{ - } \left( t \right) \le 0} \right]_{{\mathbb{I}}} } \right)dt, \\ \end{aligned} $$

and thus \({\overline{\text{P}}}_{{\left( {\text{d}} \right)}} = {\overline{\text{P}}}_{{\left( {\text{a}} \right)}} + {{\Delta \overline{P}}}\), where:

$$ {{\Delta \overline{P}}} = \frac{1}{T}\mathop \int \limits_{{F_{1} }}^{{F_{2} }} \left( {\frac{{Q^{ - } \left( F \right)\left| {P^{ - } \left( F \right)} \right|\left[ {P^{ - } \left( F \right) \le 0} \right]_{{\mathbb{I}}} }}{{\dot{F}^{ - } \left( F \right)}} - \frac{{Q^{ + } \left( F \right)\left| {P^{ + } \left( F \right)} \right|\left[ {P^{ + } \left( F \right) \le 0} \right]_{{\mathbb{I}}} }}{{\dot{F}^{ + } \left( F \right)}}} \right)dF. $$

We can see that \({{\Delta \overline{P}}} \ge 0\) in any system that we have defined:

(i) For power metric (d), \(Q\left( t \right) > 0\), \(\forall t\), and so \(Q^{ \pm } \left( F \right) > 0\), \(\forall F\).

(ii) \(\left|{P}^{\pm }\left(F\right)\right|\ge 0\), \(\forall F\), and \({\left[{P}^{-}\left(F\right)\le 0\right]}_{\mathbb{I}}\ge 0\), \(\forall F\), necessarily.

(iii) \(\dot{F}^{ - } \left( F \right) \ge 0\) and \(\dot{F}^{ + } \left( F \right) < 0\), by our construction.

Thus \({{\Delta \overline{P}}} \ge 0\). Physically, this is a simple restatement of the principle that the actuator power consumption is always greater than or equal to the net work—as studied also in “Appendix A.3”. It follows that if we can use elasticity, \(F_{s} \left( \delta \right)\), to ensure that \({{\Delta \overline{P}}} = 0\) (i.e. \({\overline{\text{P}}}_{{\left( {\text{a}} \right)}} = {\overline{\text{P}}}_{{\left( {\text{d}} \right)}}\)) then we will have reached a state of minimum \({\overline{\text{P}}}_{{\left( {\text{d}} \right)}}\) w.r.t. elasticity. Based on the time-domain formulation of \({\overline{\text{P}}}_{{\left( {\text{a}} \right)}}\) and \({\overline{\text{P}}}_{{\left( {\text{d}} \right)}}\), we can see that \({\overline{\text{P}}}_{{\left( {\text{a}} \right)}} = {\overline{\text{P}}}_{{\left( {\text{d}} \right)}}\) when \(P\left( t \right) \ge 0\), \(\forall t\)—i.e. the absence of negative work; and the global resonance condition of Ma and Zhang [50,51,52,53]. What elasticity, \(F_{s} \left( \delta \right)\), will ensure that this condition is satisfied? If \(P\left( t \right) \ge 0\), \(\forall t\), then \(p^{ + } \left( t \right) \ge 0,\forall t \in T^{ + }\) and \(p^{ - } \left( t \right) \ge 0,\forall t \in T^{ - }\); and thus, \(P^{ \pm } \left( F \right) \ge 0\), \(\forall F \in \left[ {F_{1} ,F_{2} } \right]\). Hence:

$$ F \cdot \dot{F}^{ \pm } \left( F \right) \cdot \left( {X^{\prime \pm } \left( F \right) + \left( {F_{s}^{ - 1} } \right)^{^{\prime}} \left( F \right)} \right) \ge 0, \forall F \in \left[ {F_{1} ,F_{2} } \right]. $$

We may disassemble this condition according to the signs of \(F\) and \(\dot{F}^{ \pm } \left( F \right)\):

(A) Consider \(F > 0\), i.e. \(F \in \left( {0,F_{2} } \right]\). Then:

$$ \dot{F}^{ \pm } \left( F \right) \cdot \left( {X^{\prime \pm } \left( F \right) + \left( {F_{s}^{ - 1} } \right)^{^{\prime}} \left( F \right)} \right) \ge 0, \forall F \in \left( {0,F_{2} } \right], $$

(A.i) Take \(\dot{F}^{ + } \left( F \right) < 0\), over \(X^{\prime + } \left( F \right)\). Then:

$$ - \left( {F_{s}^{ - 1} } \right)^{^{\prime}} \left( F \right) \ge X^{\prime + } \left( F \right), \forall F \in \left( {0,F_{2} } \right], $$

(A.ii) Or, take instead \(\dot{F}^{ - } \left( F \right) > 0\), \(\dot{F}^{ - } \left( F \right) \ne 0\), over \(X^{\prime - } \left( F \right)\). Then:

$$ - \left( {F_{s}^{ - 1} } \right)^{^{\prime}} \left( F \right) \le X^{\prime - } \left( F \right), \forall F \in \left( {0,F_{2} } \right], \dot{F}^{ - } \left( F \right) \ne 0. $$

(B) Consider the case \(F < 0\), i.e. \(F \in \left[ {F_{1} ,0} \right)\). Then:

$$ \dot{F}^{ \pm } \left( F \right) \cdot \left( {X^{\prime \pm } \left( F \right) + \left( {F_{s}^{ - 1} } \right)^{^{\prime}} \left( F \right)} \right) \le 0, \forall F \in \left[ {F_{1} ,0} \right), $$

(B.i) Take \(\dot{F}^{ + } \left( F \right) < 0\), over \(X^{\prime + } \left( F \right)\). Then:

$$ - \left( {F_{s}^{ - 1} } \right)^{^{\prime}} \left( F \right) \le X^{\prime + } \left( F \right), \forall F \in \left( {0,F_{2} } \right], $$

(B.ii) Or, take instead \(\dot{F}^{ - } \left( F \right) > 0\), \(\dot{F}^{ - } \left( F \right) \ne 0\), over \(X^{\prime - } \left( F \right)\). Then:

$$ - \left( {F_{s}^{ - 1} } \right)^{^{\prime}} \left( F \right) \ge X^{\prime - } \left( F \right), \forall F \in \left( {0,F_{2} } \right], \dot{F}^{ - } \left( F \right) \ne 0. $$

(C) Consider \(F = 0\) and/or \(\dot{F}^{ - } \left( F \right) = 0\).

Then the elasticity is unbounded. However, note that as \(F = \delta F \to 0\), from both below and above zero, then we have the four limits:

$$ \begin{gathered} X^{\prime - } \left( {\delta F} \right) \le - \left( {F_{s}^{ - 1} } \right)^{^{\prime}} \left( {\delta F} \right) \le X^{\prime + } \left( {\delta F} \right), \delta F \to 0, \delta F < 0, \hfill \\ X^{\prime + } \left( {\delta F} \right) \le - \left( {F_{s}^{ - 1} } \right)^{^{\prime}} \left( {\delta F} \right) \le X^{\prime - } \left( {\delta F} \right), \delta F \to 0, \delta F > 0, \hfill \\ \end{gathered} $$

and thus, if the values \(X^{\prime \pm } \left( 0 \right)\) exist, then we require that \(- \left( {F_{s}^{ - 1} } \right)^{^{\prime}} \left( 0 \right) = X^{\prime \pm } \left( 0 \right)\). This contrasts to the behaviour of these conditions as \(\dot{F} = \delta \dot{F} \to 0\): there, \(X^{\prime} \to \infty\) (unless, \(\dot{x} \to 0\) more rapidly), and so, in general, the optimal compliance will be unbounded in this limit.

Concatenating all these conditions, we obtain the elastic-bound conditions, Eq. 20:

$$ \begin{gathered} X^{\prime - } \left( F \right) \le - \left( {F_{s}^{ - 1} } \right)^{^{\prime}} \left( F \right) \le X^{\prime + } \left( F \right), \forall F \in \left[ {F_{1} ,0} \right]; \hfill \\ X^{\prime + } \left( F \right) \le - \left( {F_{s}^{ - 1} } \right)^{^{\prime}} \left( F \right) \le X^{\prime - } \left( F \right), \forall F \in \left[ {0,F_{2} } \right]. \hfill \\ \end{gathered} $$

In this proof, we have demonstrated that these conditions are sufficient and necessary to ensure that any of the mechanical power metrics, \({\overline{\text{P}}}_{{\left( {\text{b}} \right)}}\), \({\overline{\text{P}}}_{{\left( {\text{c}} \right)}}\), and \({\overline{\text{P}}}_{{\left( {\text{d}} \right)}}\), take the value of the net power, \({\overline{\text{P}}}_{{\left( {\text{a}} \right)}}\)—a value less than or equal to their minimum possible value over the space of \(F_{s} \left( \cdot \right)\). However, as discussed in Sect. 2.3, it is possible that Eq. 20 cannot be satisfied by any linear or nonlinear elasticity, in which case the value of the net power, \({\overline{\text{P}}}_{{\left( {\text{a}} \right)}}\), is always less than the minimum value(s) of \({\overline{\text{P}}}_{{\left( {\text{b}} \right)}}\), \({\overline{\text{P}}}_{{\left( {\text{c}} \right)}}\), and \({\overline{\text{P}}}_{{\left( {\text{d}} \right)}}\) over the space of \(F_{s} \left( \cdot \right)\). In this scenario, these minimum value(s) must be computed via some other method. The condition on \(X^{\prime \pm } \left( F \right)\) that ensures that Eq. A.4.20 admits some solution for \(\left( {F_{s}^{ - 1} } \right)^{^{\prime}} \left( F \right)\) is simply that:

$$ \begin{gathered} X^{\prime + } \left( F \right) \ge X^{\prime - } \left( F \right), \forall F \in \left[ {F_{1} ,0} \right], \hfill \\ X^{\prime - } \left( F \right) \ge X^{\prime + } \left( F \right), \forall F \in \left[ {0,F_{2} } \right], \hfill \\ \end{gathered} $$

implying, at \(F = 0\), that \(X^{\prime + } \left( 0 \right) = X^{\prime - } \left( 0 \right)\).

1.5 A.5 Proof of conditional absolute-load invariance in PEA systems.

The PEA elastic-bound conditions (Eq. 10) also define a region of invariance in the absolute load integral, \(\overline{P}_{\left| F \right|}\), as per Sect. 2.4. Here we provide proof. Consider a general PEA system, as per Eq. 7, with its attendant conditions for admissibility. We may evaluate \({\overline{\text{P}}}_{\left| F \right|}\) by segmenting \(F\left( t \right)\) into two time-domain profiles, \(F^{ + } \left( t \right)\) following \(F^{ - } \left( t \right)\), without loss of generality, representing \(F^{ \pm } \left( x \right)\) in the time domain. The time \(T^{*}\) represents the transition between \(F^{ \pm } \left( t \right)\). Under an elasticity that is optimal in terms of mechanical power consumption, \(F^{ + } \left( t \right) \ge 0\) and \(F^{ - } \left( t \right) \le 0\), and thus we have:

$$ {\overline{\text{P}}}_{\left| F \right|} \propto \frac{1}{T}\mathop \int \limits_{0}^{T} \left| {F\left( t \right)} \right|dt = \frac{1}{T}\mathop \int \limits_{{T^{*} }}^{T} F^{ + } \left( t \right)dt - \frac{1}{T}\mathop \int \limits_{0}^{{T^{*} }} F^{ - } \left( t \right)dt. $$

Via the differential relation, \(dt = 1/\dot{x}dx\), we may recast this integral into an integral over \(x\), with the velocity functions \(\dot{x}^{ + } \left( x \right)\) and \(\dot{x}^{ - } \left( x \right)\) representing the velocities associated with time windows \(\left[ {T^{*} ,T} \right]\) and \(\left[ {0,T^{*} } \right]\), respectively, matching \(F^{ \pm } \left( x \right)\). This yields:

$$ {\overline{\text{P}}}_{\left| F \right|} \propto \mathop \int \limits_{{x_{1} }}^{{x_{2} }} \frac{{F^{ + } \left( x \right)}}{{\dot{x}^{ + } \left( x \right)}}dx + \mathop \int \limits_{{x_{1} }}^{{x_{2} }} \frac{{F^{ - } \left( x \right)}}{{\dot{x}^{ - } \left( x \right)}}dx, $$

where, note, the negative sign arises because of the need to switch integration over the window \(\left[ {x_{2} ,x_{1} } \right]\) to \(\left[ {x_{1} ,x_{2} } \right]\). If, then, we have a symmetric waveform, \(x\left( t \right) = x\left( {T - t} \right)\), \(\forall t\), in the time-domain, or \(\dot{x}^{ + } \left( x \right) = - \dot{x}^{ - } \left( x \right) = \dot{x}_{{{\text{ref}}}} \left( x \right) > 0\), \(x \in \left[ {x_{1} ,x_{2} } \right]\) in the displacement domain, then we have:

$$ {\overline{\text{P}}}_{\left| F \right|} \propto \mathop \int \limits_{{x_{1} }}^{{x_{2} }} \frac{{F^{ + } \left( x \right) - F^{ - } \left( x \right)}}{{\dot{x}_{{{\text{ref}}}} \left( x \right)}}dx. $$

Under the definition of \(F^{ \pm } \left( x \right)\), this reduces to:

$$ {\overline{\text{P}}}_{\left| F \right|} \propto \mathop \int \limits_{{x_{1} }}^{{x_{2} }} \frac{{G_{{{\text{arc}}}} \left( x \right)}}{{\dot{x}_{{{\text{ref}}}} \left( x \right)}}dx, $$

which is independent of \(F_{s} \left( x \right)\). That is, under the waveform symmetry condition, \(x\left( t \right) = x\left( {T - t} \right)\), \(\forall t\), the metric \({\overline{\text{P}}}_{\left| F \right|}\) is invariant under any elasticity satisfying Eq. 10.

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Pons, A., Beatus, T. Elastic-bound conditions for energetically optimal elasticity and their implications for biomimetic propulsion systems. Nonlinear Dyn 108, 2045–2074 (2022).

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  • Energetic optimality
  • Biomimetic propulsion
  • Nonlinear elasticity
  • Inverse problem
  • Global resonance
  • Flapping-wing flight