2.1. Elastic actuators

Elastic actuators are typically classified in two main types, depending on the placement of the elastic element [Reference Verstraten, Flynn, Geeroms, Vanderborght and Lefeber19]. A schematic of these two architectures can be found in Fig. 1. In SEA, the elastic element is placed between the drivetrain and the load, that is, in a series configuration. Another option is to place the elastic element in parallel with the drivetrain, resulting in a parallel elastic actuator (PEA). It is widely known that both architectures can generate energy-efficient cyclic motion if a (roughly) linear relationship exists between the required output torque and the output angle. This, for example, applies to resonant motion. Both architectures take advantage of the possibility of recirculating power between the motor, the spring and the load, an option which is unavailable in rigid actuators. In the latter, power often travels back and forth through the actuator, resulting in the accumulation of losses in its components [Reference Hitt, Sugar, Holgate, Bellman and Hollander31]. Recirculating power between the spring and the load has two advantages. First, power losses – which are more or less proportional to the power flowing through the actuator – are reduced. Second, the reduced power demand can often be met with a smaller electric motor and gearbox, which drives down the size, weight, and cost of the actuation system [Reference Verstraten, Flynn, Geeroms, Vanderborght and Lefeber19].

Figure 1.

Schematic representation of the two main elastic actuator architectures: (a) SEA, (b) PEA.

The mechanism by which the required motor power is reduced differs between the two architectures. In a well-designed PEA, the parallel spring will take over part of the load torque, resulting in a decrease of the torque required from the motor [Reference Haeufle, Taylor, Schmitt and Geyer18]. The motor will, however, rotate at the same speed as the output, since it is directly coupled to the latter. In contrast, a well-designed SEA will allow to reduce the required motor speed below that of the output. A SEA can, however, not compensate for a static load [Reference Beckerle, Verstraten, Mathijssen, Furnémont, Vanderborght and Lefeber15], although reductions in motor torque can be achieved in dynamic motions because of the reduced acceleration of the drivetrain, which is associated with a decrease in inertial torque. As we will discuss extensively in this manuscript, these characteristic differences between SEA and PEA will result in different efficiencies for resonant motions.

2.2. Energetic characterization

Several papers analyzed the energetic advantages of elastic actuators. Rather than being based on an analytical approach, most of these analyses are data driven. They typically consider a specific system and experimentally measure the forces and velocities needed for a specific task. Then, they simulate and optimize the properties of elastic elements so that the actuators produce the same force/velocity profiles with (hopefully) reduced power consumption. The advantage of this methodology is that it can be applied to a great variety of systems and tasks. On the other hand, the results are task specific and sensitive to experimental conditions, in the sense that the analysis must be repeated from scratch when changes are made to the task or to the new system. This approach is followed in several papers on exoskeletons [Reference Wang, van Dijk and van der Kooij24, Reference van den Bogert32–Reference Bacek, Moltedo, Rodriguez-Guerrero, Geeroms, Vanderborght and Lefeber34] and powered prostheses [Reference Verstraten, Flynn, Geeroms, Vanderborght and Lefeber19, Reference Grimmer, Eslamy and Seyfarth35–Reference Kamadan, Kiziltas and Patoglu38], where authors optimize elastic components in series or parallel to different human joints, taking gait data as a starting point for the kinematics. These results are task specific, since they depend on the locomotion speed. Several other examples of system-specific and task-specific analyses can be found, for example, on walking robots [Reference Moro, Tsagarakis and Caldwell25, Reference Gonzalez-Rodriguez, Gonzalez-Rodriguez and Castillo-Garcia39–Reference Reher, Cousineau, Hereid, Hubicki and Ames41], hopping robots [Reference Vu, Yu, Iida and Pfeifer42–Reference Liu, Rossi and Poulakakis44], manipulators [Reference Scheint, Sobotka and Buss45–Reference Shushtari, Nasiri, Yazdanpanah and Ahmadabadi48] and even wheeled robots [Reference Valero, Rubio and Llopis-Albert49]. In search of generalized rules for elastic actuator design, some authors have studied the motion of a pendulum at different frequencies. In [Reference Vanderborght, VanHam, Lefeber, Sugar and Hollander50, Reference Teitler and Proodian51] the efficiency of several variable stiffness actuators is compared in simulation and in [Reference Beckerle, Wojtusch, Rinderknecht and von Stryk13, Reference Vanderborght, VanHam, Lefeber, Sugar and Hollander50] the efficiency of series and parallel elastic actuators is compared in simulation and real-world experiments.

In all the aforementioned works, the energy efficiency is expressed in terms of energy or power costs, in a form similar to

(1) \begin{equation}\eta_{1}=\frac{1}{input\;power}\end{equation}

which is task specific. The literature usually considers nonlinear and multi degree of freedom systems for which an analytical efficiency analysis is extremely challenging. The only analytical analysis found in literature [Reference Albert12] considers a simpler system (a SEA) and uses the traditional efficiency index

(2) \begin{equation}\eta_{2}=\frac{output\;power}{input\;power}.\end{equation}

This index assumes that work is done on the environment; if not, the efficiency goes to zero. Thus, experiments in ref. [Reference Albert12] are carried out by coupling the actuator with a dissipative environment and measuring the output work. Quite surprisingly, the outcome of their work is that the maximum efficiency increases with environmental damping. This can be explained by the fact that the energy is mainly dissipated by something that is considered external to the system while a minor amount of energy is dissipated internally.

Another common efficiency index in locomotion is the cost of transport (COT), which is defined as

(3) \begin{equation}COT=\frac{input\;power}{weight\times velocity}.\end{equation}

The COT is a nondimensional quantity and represents a ratio between the cost (input power) and the benefit (transported weight at a given speed) of a transportation mode [Reference Teitler and Proodian51, Reference Shi, Stapersma and Grimmelius52]. Similarly to other approaches, the COT is task-specific as it depends on the specific transportation velocity.

In conclusion, the existing efficiency analyses are empirical, that is, driven by simulation or experimental data rather than by analytical computations. Moreover, most of the indices found in the literature are task specific as they evaluate the efficiency of a system given a certain task: $\eta_{1}$ can only be used to compare efficiency within identical tasks while the COT depends on the transportation velocity. The only exception is the index $\eta_{2}$ which, however, implies that the environment dissipation improves efficiency. These indices are not representative of the concept of efficiency as it is intended in this work.

2.3. Proposed approach

In general, energy efficiency is related to the amount of input power that is wasted when completing a certain job or task. In this paper, we are interested in the task of maintaining a resonant motion. Here, efficiency regards the input power needed to support such a motion. To evaluate this scenario, we introduce a novel index $\eta$ which indicates “how fast” the internal energy is dissipated during a resonant motion. In particular, $\eta$ relates the system’s internal energy to the mean input power needed to support the resonant motion:

(4) \begin{equation}\eta=\frac{internal\:energy}{mean\:input\:power}.\end{equation}

The index $\eta$ is expressed in time units and characterizes the tendency of a passive system to perpetuate its natural motion. For the moment we can assume that the input power is equal to the dissipated power.Footnote 3 The index $\eta$ can roughly be interpreted as the duration of the natural motion when no external power is supplied. A low value for $\eta$ means that little motion can be observed before the system dissipates all its internal energy. Vice-versa, a high $\eta$ means long-lasting motion. The index $\eta$ ranges between $[0,+\infty)$ , where 0 implies no observable natural motion (no internal energy storage) while $+\infty$ corresponds to perpetual natural motion (no dissipation). Alternatively, the index $\eta$ can be interpreted as a measure of power consumption (denominator) with a normalization factor (numerator), eliminating the effect of the natural motion “magnitude.” In fact, different motion “magnitudes” involve different levels of internal energy, which in turn involve different dissipation levels.

5.1. Motor damping

Similarly to the case of PEA, motor damping can be compensated by positing $F(t)=d_{m}\dot{x}_{m}(t)$ which leads to the following system dynamics

(19a) \begin{align} m_{m}\ddot{x}_{m}(t) =-k(x_{m}(t)-x_{l}(t)) \end{align}

(19b) \begin{align} m_{l}\ddot{x}_{l}(t) = k(x_{m}(t)-x_{l}(t)) \end{align}

which exhibits a resonant frequency in

(20) \begin{equation}\omega_{S}=\sqrt{\frac{k}{m_{m}}+\frac{k}{m_{l}}.}\end{equation}

In order to observe the system during a resonant motion, we consider the initial conditions $x_{m}(0)=x_{l}(0)=0,\:\dot{x}_{m}(0)=\alpha\omega_{S}\mu$ and $\dot{x}_{l}(0)=-\alpha\omega_{S}$ leading to the following motion equations

(21) \begin{align} x_{m}(t) & =-\mu\alpha\sin(\omega_{S}t) \end{align}

(22) \begin{align} \dot{x}_{m}(t) & =-\mu\alpha\omega_{S}\cos(\omega_{S}t)\end{align}

(23) \begin{align} x_{l}(t) & =\alpha\sin(\omega_{S}t)\end{align}

(24) \begin{align} \dot{x}_{l}(t) & =\alpha\omega_{S}\cos(\omega_{S}t)\end{align}

where $\mu=m_{l}/m_{m}$ . The internal energy can be computed by considering

(25) \begin{align} K & =\frac{1}{2}m_{m}\dot{x}_{m}^{2}+\frac{1}{2}m_{l}\dot{x}_{l}^{2}=\frac{1}{2}\alpha^{2}\omega_{S}^{2}m_{l}(1+\mu)\cos^{2}\omega_{S}t \end{align}

(26) \begin{align} U & =\frac{1}{2}k(x_{m}-x_{l})^{2}=\frac{1}{2}\alpha^{2}k(1+\mu)^{2}\sin^{2}\omega_{S}t\end{align}

which leads to

(27) \begin{equation}E=U+K=\frac{1}{2}\alpha^{2}k(1+\mu)^{2}.\end{equation}

Thus, the total internal energy E is constant over time and is completely stored in the resonant mode. As a consequence, no temporary energy arises.Footnote 5 The mean dissipated power for damping compensation can be expressed as

(28) \begin{equation}\bar{P}_{d_{m}}=d_{m}\frac{1}{T}\int_{0}^{T}\dot{x}_{m}^{2}(t)dt\end{equation}

where $T=2\pi/\omega_{S}$ . Considering that $\frac{1}{T}\int_{0}^{T}\sin^{2}\omega_{S}t=\frac{1}{2}$ , we have

(29) \begin{equation}\bar{P}_{d_{m}}=\frac{1}{2}d_{m}\mu^{2}\alpha^{2}\omega_{S}^{2}=\frac{1}{2}\alpha^{2}k\frac{\mu(1+\mu)}{m_{m}}d_{m}\end{equation}

and we can finally compute the index

(30) \begin{equation}\eta_{m}=\frac{E_{r}}{\bar{P}_{d_{m}}}=\frac{m_{m}+m_{l}}{d_{m}}\frac{m_{m}}{m_{l}}.\end{equation}

5.2. Spring damping

When the damping is located in the spring, it is convenient to explicitly distinguish between the resonant and the translatory modes. To this aim, the SEA model (18a)-(18b) can be expressed in modal coordinates as

(31a) \begin{align} z_{1} = m^{-1}(\mu x_{l}+x_{m}) \end{align}

(31b) \begin{align} z_{2} = m^{-1}(x_{l}-x_{m})\end{align}

where $m=1+\mu$ , leading to the modal representation

(32) \begin{equation}M_{p}\left[\begin{array}{c}\ddot{z}_{1}\\\ddot{z}_{2}\end{array}\right]+D_{p}\left[\begin{array}{c}\dot{z}_{1}\\\dot{z}_{2}\end{array}\right]+K_{p}\left[\begin{array}{c}z_{1}\\z_{2}\end{array}\right]=\left[\begin{array}{c}F\\-\mu F\end{array}\right]\end{equation}

where

(33a) \begin{eqnarray}M_{p} & = & \left[\begin{array}{cc}mm_{m} & 0\\0 & mm_{l}\end{array}\right],\end{eqnarray}

(33b) \begin{eqnarray}D_{p} & = & \left[\begin{array}{cc}d_{m}+d_{l} & d_{l}-\mu d_{m}\\d_{l}-\mu d_{m} & m^{2}d_{s}+\mu^{2}d_{m}+d_{l}\end{array}\right],\end{eqnarray}

(33c) \begin{eqnarray}K_{p} & = & \left[\begin{array}{cc}0 & 0\\0 & m^{2}k\end{array}\right].\end{eqnarray}

Here the states $z_{1}$ and $\dot{z}_{1}$ describe the translatory mode while $z_{2}$ and $\dot{z}_{2}$ represent the resonant mode. The modal transformation (31a)-(31b) is energy preserving and allows to easily distinguish the resonant energy, stored in states $z_{2}$ and $\dot{z}_{2}$ , from the temporary energy, stored in states $z_{1}$ and $\dot{z}_{1}$ .

If we consider a single dissipation source given by $d_{s}$ , the damping matrix $D_{p}$ is diagonal and the Eqs. (32) become decoupled:

(34a) \begin{align} mm_{m}\ddot{z}_{1} =F \end{align}

(34b) \begin{align} mm_{l}\ddot{z}_{2}+m^{2}d_{s}\dot{z}_{2}+m^{2}kz =-\mu F. \end{align}

Focusing on the latter row we can impose

(35) \begin{equation}F=-\frac{m^{2}}{\mu}d_{s}\dot{z}_{2}=\frac{1+\mu}{\mu}(x_{m}-x_{l})\end{equation}

to have perfect damping compensation. Similarly as before, initial conditions can be chosen on order to observe the following resonant motion

(36) \begin{align} z_{2}(t) & =\alpha\sin(\omega_{S}t) \end{align}

(37) \begin{align} \dot{z}_{2}(t) & =\alpha\omega_{S}\cos(\omega_{S}t) \end{align}

which implies a constant level of resonant energy

(38) \begin{equation}E_{r}=K_{2}+U_{2}=\frac{1}{2}mm_{l}\dot{z}_{2}^{2}+\frac{1}{2}m^{2}kz^{2}=\frac{1}{2}\alpha^{2}m^{2}k.\end{equation}

The average power needed for damping compensation can be computed from (35) as

(39) \begin{equation}\bar{P}_{d_{s},r}=m^{2}d_{s}\frac{1}{T}\int_{0}^{T}\dot{z}_{2}^{2}(t)dt=\frac{1}{2}\alpha^{2}m^{3}k\frac{d_{s}}{m_{l}}\end{equation}

and finally we have

(40) \begin{equation}\eta_{s}=\frac{E_{r}}{\bar{P}_{d_{s}}}=\frac{m_{l}}{md_{s}}=\frac{m_{m}m_{l}}{m_{m}+m_{l}}\frac{1}{d_{s}}.\end{equation}

5.2.1. Accounting for Temporary Energy

By substituting (35) in (34a) we find that, in order to compensate for $d_{s}$ , some energy is introduced in the mode $z_{1}$ leading the two masses to oscillate in-phase. Unfortunately, the system cannot permanently store this energy because, as reported in (34a) the mode $z_{1}$ exhibits pure integrator dynamics. The temporary energy is just temporarily stored in the state $\dot{z}_{1}$ as kinetic energy and then exits the system as negative input power. In particular, given the input (35) and a proper choice of initial conditions, we can solve Eq. (34a) and compute

(41) \begin{equation}\dot{z}_{1}=-\alpha\frac{m}{m_{l}}d_{s}\sin\omega_{S}t.\end{equation}

The associated kinetic energy is

(42) \begin{equation}K_{1}=\frac{1}{2}mm_{m}\dot{z}_{1}^{2}=\frac{1}{2}\alpha^{2}\frac{m^{3}}{\mu m_{l}}d_{s}^{2}\sin^{2}\omega_{S}t\end{equation}

which increases in the first quarter of the period and decreases in the second quarter. In a half period ( $T/2$ ) such temporary energy is released as negative input power which may be dissipated by the motor driver. Such dissipated power can be computed by relating the released energy with the releasing period $T/2$ leading to

(43) \begin{equation}\bar{P}_{d_{s},tA}^{WC}=\frac{\max(K_{1})}{T/2}=f\alpha^{2}\frac{m^{3}}{\mu m_{l}}d_{s}^{2}\end{equation}

where $f=1/T=\omega_{S}/2\pi$ is the resonance frequency of the SEA and $\omega_{S}$ is defined in (20). It is worth highlighting that such dissipated power is related to worst case condition. Indeed, the amount of temporary energy is reduced when the load inertia $m_{l}$ is constrained by some downstream dynamics. For example, in the limit case of fixed load inertia there will be no temporary energy because no in-phase oscillation can occur. In contrast, when the load mass is completely unconstrained, we have the highest temporary energy and higher associated dissipated power which represents the worst case condition (WC) for efficiency. The worst case index can be computed considering such worst case dissipated power as

(44) \begin{equation}\eta_{s}^{WC}=\frac{E_{r}}{\bar{P}_{d_{s},tA}^{WC}}=\frac{\eta_{s}}{1+\frac{fd_{s}}{\mu k}}\end{equation}

or equivalently as

(45) \begin{equation}\eta_{s}^{WC}=\frac{\eta_{s}}{1+\sigma d_{s}},\quad\sigma=\frac{m_{m}}{m_{l}\sqrt{k\cdot m_{m}||m_{l}}}\end{equation}

where the notation $m_{m}\parallel m_{l}$ indicates the parallel of motor and load inertias $m_{m}\parallel m_{l}=\frac{m_{m}m_{l}}{m_{m}+m_{l}}$ . This index accounts for dissipation of type A, while dissipation of type B is not observed because no dissipation term related to state $\dot{z}_{1}$ exists in Eqs. (34a)-(34b). By looking at the coefficient $\sigma$ it seems that such dissipation (type A) increases when the motor inertia is high and at the same time the load inertia is low. Also, it decreases with the actuator stiffness. Thus, rigid systems which avoid the combination of high motor inertia and lightweight load seem to be more efficient in the sense that they are less prone to dissipation due to the temporary energy.

5.3. Load damping

In the case of load damping we can consider the following modal equations

(46a) \begin{align} mm_{m}\ddot{z}_{1} +d_{l}\dot{z}_{1}+d_{l}\dot{z}_{2}=F \end{align}

(46b) \begin{align} mm_{l}\ddot{z}_{2} +d_{l}\dot{z}_{1}+d_{l}\dot{z}_{2}+m^{2}kz_{2}=-\mu F \end{align}

with states $\dot{z}_{2},\:z_{2}$ representing the oscillatory mode and $\dot{z}_{1},\:z_{1}$ the translatory mode. In this case both system modes are involved in energy dissipation, leading to type B dissipation. Focusing on (46b), damping terms can be compensated by imposing $F=-\frac{d_{l}}{\mu}(\dot{z}_{1}+\dot{z}_{2})=-\frac{d_{l}}{\mu}\dot{x}_{l}$ which lets the system oscillate at its natural frequency. The system dynamics then becomes

(47a) \begin{align} m_{m}\ddot{z}_{1} +d_{l}\dot{z}_{1}+d_{l}\dot{z}_{2}=0 \end{align}

(47b) \begin{align} m_{l}\ddot{z}_{2} +mkz_{2}=0. \end{align}

Similarly to previous sections, initial conditions can be chosen on order to observe the following resonant motion

(48a) \begin{align} z_{2}(t) =\alpha\sin(\omega_{S}t)\end{align}

(48b) \begin{align} \dot{z}_{2}(t) =\alpha\omega_{S}\cos(\omega_{S}t). \end{align}

The resonant energy is stored in the states $\dot{z}_{2},\:z_{2}$ and can be computed as in the previous section

(49) \begin{equation}E_{r}=\frac{1}{2}\alpha^{2}m^{2}k.\end{equation}

If we neglect the temporary energy by considering $z_{1}=\dot{z}_{1}=0$ , the average dissipated power is

(50) \begin{equation}\bar{P}_{d_{l},r}=d_{l}\frac{1}{T}\int_{0}^{T}\dot{z}_{2}^{2}(t)dt=\frac{1}{2}\alpha^{2}\omega_{S}^{2}d_{L}\end{equation}

and the index $\eta_{l}$ can be expressed as

(51) \begin{equation}\eta_{l}=\frac{E_{r}}{\bar{P}_{d_{l},r}}=\frac{m_{m}+m_{l}}{d_{l}}\frac{m_{m}}{m_{l}}.\end{equation}

5.3.1. Accounting for Temporary Energy

When compensating for load damping, we have dissipation of both type A and B. The first is due to the temporary energy exiting the system as negative input power (and then being dissipated by the motor driver). The second is due to the dissipative terms $d_{l}\dot{z}_{1}$ in (46a) and (46b). As before, to characterize the worst case dissipation, the trajectory of $z_{1}$ is computed by considering an unconstrained load. In particular, Eq. (47a) can be solved in the Laplace domain by replacing $z_{2}$ with its transform leading to

(52) \begin{equation}z_{1}(s)=\frac{-\alpha\gamma\omega_{S}}{(s+\gamma)(s^{2}+\omega_{S}^{2})}\end{equation}

where $\gamma=\frac{d_{l}}{m_{l}}$ . Then by anti-transformation and proper choice of initial conditions it is possible to express $z_{1}$ as

(53) \begin{equation}z_{1}(t)=\frac{-\alpha}{\gamma^{2}+\omega_{S}^{2}}(\gamma\omega_{S}e^{-\gamma t}+\gamma\omega_{S}\cos\omega_{S}t+\gamma^{2}\sin\omega_{S}t).\end{equation}

The exponential term just represents a transient and is neglected from here on. The power within system modes can be computed as $P_{d_{l}}+P_{d_{l}}^{WC-B}=d_{l}(\dot{z}_{1}+\dot{z}_{2})^{2}$ . It also accounts for type B dissipation due to the worst case temporary energy. The trajectory $\dot{z}_{1}$ can be derived from (53) as

(54) \begin{equation}\dot{z}_{1}(t)=\frac{\alpha}{\gamma^{2}+\omega_{S}^{2}}(\gamma^{2}\omega_{S}\cos\omega_{S}t-\gamma\omega_{S}^{2}\sin\omega_{S}t)\end{equation}

while $\dot{z}_{2}$ is expressed in (48b). By defining $h=\frac{\omega_{S}^{2}}{\gamma^{2}}=\frac{kmm_{l}}{d_{l}^{2}}$ , the dissipated power can be computed as

(55) \begin{equation}P_{d_{l},r}+P_{d_{l},tB}^{WC}=d_{_{l}}\alpha^{2}\left(\frac{h+2}{h+1}\omega_{S}\cos\omega_{S}t-\frac{h}{h+1}\gamma\sin\omega_{S}t\right)^{2}\end{equation}

and on average we have

(56) \begin{equation}P_{d_{l},r}+P_{d_{l},tB}^{WC}=\frac{1}{2}\alpha^{2}\omega_{S}^{2}(1+3\xi)d_{l}\end{equation}

where $\xi=\frac{1}{h+1}$ . The efficiency index accounting for worst case type B dissipations can be computed as

(57) \begin{equation}\eta_{l}^{WC-B}=\frac{E_{r}}{P_{d_{l},r}+P_{d_{l},tB}^{WC}}=\frac{m_{m}+m_{l}}{d_{l}}\frac{m_{m}}{m_{l}}\frac{1}{(1+3\xi)}.\end{equation}

Again, this index is worst case because the load mass is unconstrained, leading to the maximum temporary energy flowing into the system (maximum in-phase oscillations). As we are not yet accounting for type A dissipations, $\eta_{l}^{WC-B}$ describes the case of a system capable of recovering the negative input power. We can say that a SEA with an energy recovery circuit can exhibit an efficiency index with a value between $\eta_{l}$ in (51), where the temporary energy was entirely neglected, and $\eta_{l}^{WC-B}$ in (57), depending on the load/environment dynamics. Interestingly, the ratio between $\eta_{l}$ and $\eta_{l}^{WC-B}$ is given by the coefficient $\frac{1}{(1+3\xi)}$ which can reduce the worst case efficiency by no more than a quarter. Indeed, since $h>0$ , one has that $0\leq\xi\leq1$ . When k is high the quantity $\xi$ is small and we again find that a series elastic system with high stiffness leads to higher efficiency in resonant motions, because the temporary energy flow is minimized. A similar effect can be observed for high load and low motor inertia.

Finally, to compute type A dissipation we need to consider the negative input power dissipated by the motor driver in absence of an energy recovery circuit. As in previous section $\bar{P}_{d_{l},tA}^{WC}$ is given by the kinetic energy which is temporarily stored in the state $\dot{z}_{1}$ and then released

(58a) \begin{align} K_{1} =\frac{1}{2}mm_{m}\dot{z}_{1}^{2} \end{align}

(58b) \begin{align} =\frac{1}{2}mm_{m}\alpha\omega^{2}\left[\frac{1}{h+1}\cos\omega_{S}t-\frac{h}{h+1}\sin\omega_{S}t)\right]^{2}.\end{align}

The negative input power can be computed as before considering the amount of released energy $\max(K_{1})$ within the releasing period $T/2$ . After some small computationsFootnote 6 this leads to

(59) \begin{equation}\bar{P}_{d_{l},tA}^{WC}=\frac{\max(K_{1})}{T/2}=\frac{1}{2}\alpha^{2}f\frac{mm_{m}}{m_{^{2}}}d_{l}^{2}(1-\xi)\end{equation}

and the worst case efficiency index in absence of an energy recovery circuit is

(60a) \begin{align} \eta_{l}^{WC} =\frac{E_{r}}{P_{d_{l},r}+P_{d_{l},tB}^{WC}+P_{d_{l},tA}^{WC}} \end{align}

(60b) \begin{align} =\frac{\eta_{l}}{1+\sigma d_{l}(1-\xi)+3\xi}. \end{align}

Interestingly, when $\xi=0$ Eq. (60a) reduces to the following expression

(61) \begin{equation}\eta_{l}^{WC}=\frac{\eta_{l}}{1+\sigma d_{l}}\end{equation}

which is similar to the case of spring damping (45).

7.1. Simulation Validation

In our simulations, we consider a scenario where the load is unconstrained (maximum temporary energy) and no energy recovery circuit is available. This can be considered as a worst case scenario for the SEA, because in several applications the load is somehow constrained. For example, if we consider a legged robot with the robot’s foot as a load, we find that the foot is constrained to the ground during the stance phase, and it is constrained by gravity forces during other phases. Similarly, if we consider pick-and-place applications, dynamic constraints due to gravity apply. A set of inertial ( $m_{m},\:m_{l}$ ) and damping ( $d_{m},\:d_{s},\:d_{l}$ ) parameters are considered while the spring stiffness (k) is fixed. The set of inertial parameters ranges from $10^{-4}\;Kgm^{2}$ to $1\;Kgm^{2}$ and includes typical values of rotor reflected inertias, representative of a wide set of electromagnetic gear-motors, from direct-drive to high gear-ratio. Similarly, we impose realistic damping values in the range of $(10^{-3},10^{-2})\;Nm/rad$ . The stiffness parameter is set to $k=100\;Nm/rad$ and represents a typical choice for SEAs whose stiffness ranges from some tens to some hundreds of $Nm/rad$ . We chose to fix this value as the efficiency index $\eta$ do not depend on k, as reported in Table II

The following damping compensation laws are implemented

(64) \begin{align} F_{PEA} & =\left(d_{m}+d_{s}+d_{l}\right)\dot{x}_{l}(t) \end{align}

(65) \begin{align} F_{SEA} & =\left(d_{m}+\frac{1+\mu}{\mu}d_{l}\right)\dot{x}_{m}(t)-\left(\frac{1+\mu}{\mu}d_{s}+\frac{1}{\mu}d_{l}\right)\dot{x}_{l}(t)\end{align}

for the case of PEA and SEA, as derived in the analysis in sections 4 and 5.

Figure 4.

Comparison between measured (left) and analytical (right) efficiency indices for PEAs.

Figure 5.

Comparison between measured (left column) and theoretic indices for SEAs.

Efficiency is measured according to (9) as

(66) \begin{equation}\hat{\eta}=E_{r}/mean(P_{in}^{+}(t))\end{equation}

where $P_{in}^{+}$ represents the power flowing from the driver to the motor (which is dissipated in the mechanical domain) plus the power flowing back from the motor to the driver (which is dissipated within the motor driver, as a recovery circuit is not included). More specifically, $P_{in}^{+}(t)$ is computed as the absolute value $P_{in}^{+}(t) = |\dot{x}_{m}(t)F(t)|$ where $\dot{x}_{m}$ and F represent the motor velocity and the motor input force or torque, proportional to the current.

Figure 4 shows, for the case of PEA, the measured index $\hat{\eta}$ against the index $\eta$ analytically computed as in Table II. One can observe that measured and theoretic plots present the same shape. The maximum relative error between the two is quantified as 0.264%. Logarithmic scales on the X and Y axes are chosen in order to span very different values of $m_m$ and $m_l$ . Similarly, we choose logarithmic scales on the Z axis in order to highlight differences between measured and theoretic indices, as the efficiency indices span very different magnitudes. Figure 5 shows the same results for the case of SEA. Here the measured values are compared with the proposed indices, either accounting or not accounting for the effects of temporary energy. As expected, the measured efficiency presents the same shape of the worst case index $\eta^{WC}$ . This is not because the index $\eta^{WC}$ is more accurate, but because we simulated worst case scenarios (maximum temporary energy). We highlight that on a linear scale the discrepancy of indices $\eta$ and $\eta^{WC}$ would be not significant, as we show here after. Nevertheless, the proposed indices are able to capture even small differences and show consistency with measured efficiency even on a logarithmic scale. We can conclude that, both for PEA and SEA, the measured efficiency agrees with our theoretical expectations, thus validating our theoretical analysis.

Before proceeding, we highlight some particularities regarding the case of SEA. First, the difference between indices $\eta$ and $\eta^{WC}$ on a linear scale is extremely small. To make it noticeable, we needed to apply an extremely low value for the stiffness $k=1\: Nm/rad$ – outside of the typical stiffness value ranges for SEA designs – and high distal friction values. These results are reported in Figure 6, where a linear scale is used for the Y axis. In these plots, we avoid showing the measured efficiency as it would overlap with $\eta^{WC}$ (as reported in Figure 5). Three different values are considered for $m_m$ and two for k. Even if $\eta$ and $\eta^{WC}$ are very close to each other, some differences can be observed in case of high distal friction. This agrees with the equations reported in Table III. Also, by comparing the first three rows A1, A2, and A3 – where we considered a stiffness $k=100\: Nm/rad$ – with the last three rows B1, B2 and B3 – where we considered a very low stiffness $k=1\: Nm/rad$ – one can observe that $\eta^{WC}$ decreases, leading to more noticeable differences. This is because $\eta^{WC}$ depends on $\sigma$ and $\xi$ which in turns depends on k. Instead, $\eta$ does not change with k as reported in Table II. Thus, if we neglect the (small) temporary energy dissipation, we can state that k affects the resonance frequency location, but not the efficiency magnitude. Finally, we highlight that even if the plots reported in Figure 5 for the SEA show a maximum discrepancy between $\eta$ and $\eta^{WC}$ for high $m_m$ and low $m_l$ , this is not a general result. This is just due to the combination of two choices: the considered parameter set and the logarithmic scale, as can be observed by comparing Figures 5 and 6.

Figure 6.

Comparison between indices $\eta$ and $\eta^{WC}$ on linear scales, for SEAs with increasing values of parameters $d_s$ and $d_l$ and for different values of $m_m$ .

7.2. Experimental Validation

In the previous section, an extensive validation has been provided in simulation, considering a wide set of inertia parameters. Since it is not practically feasible to test many mechanical arrangements in experiments, we choose to experimentally validate the few scenarios representing the more complex conditions highlighted by the proposed methodology. In particular, this section proposes an experimental validation considering the case of a SEA with different lightweight loads. The experimental setup consists of the elastic actuator prototype in figure 7. It is composed of a direct drive torque motor M connected in series to a spring S (used in torsional mode) and to an exchangeable load L. System parameters have been estimated using a procedure similar to the one described in ref. [Reference Calanca, Capisani, Ferrara and Magnani53] and are reported in Table IV, where $m_{li}$ for $i=1...4$ refers to different load inertias. As the inherent mechanical damping is low, we used a soft spring in order to obtain noticeably differences between indices $\eta$ and $\eta^{WC}$ as discussed in the previous section. For each load, Table III reports the parameters $\xi$ and $\sigma$ defined in (56) and (45). The motor Coulomb friction is compensated using a $\mathrm{sign}(\dot{x}_{m})$ function and neglected because its nonlinear nature is not considered in our analysis. Static and stick-slip friction are not compensated because of their complex dynamics.

Table IV.

Parameters of the Experimental Setup

Figure 7.

The SEA prototype used as testbed.

Two optical encoders E1 and E2 are used to measure motor and load position, respectively. They are connected to an industrial rack that drives a motor current amplifier and provides an EtherCat connection to a standard PC on which the control algorithm runs. No energy recovery system is present in the motor driver.

In our analysis, we assume the resonant energy is maintained by perfect damping compensation. However, it is very challenging to obtain perfect damping compensation in the real world. Static and stick-slip friction dynamics are hard to model and estimation of system parameter may suffer some bias. Thus, in practice, friction and damping forces will always be either overestimated or underestimated, resulting in either an increase or a decrease in the initial system energy. To keep the internal energy constant, we therefore relied on an energy-based control approach. Energy control injects power into the system until a desired level $E_{d}$ of internal energy E is reached. This strategy has been used for example to control PDWs [Reference Spong and Bhatia54]. According to [Reference Spong and Bhatia54] energy control can be formulated as

(67) \begin{eqnarray}F(t) & = & k_{e}(E_{d}-E(t))\dot{x}_{m}(t)\end{eqnarray}

where $k_{e},$ is a control gain and E is the overall system energy. One can observe that the input power $P_{in}(t)=F(t)\dot{x}_{m}(t)$ is positive when the system’s energy E is below the energy target $E_{d}$ and negative otherwise. The stability of the control law (67) can be proven considering E(t) as a Lyapunov function candidate, as in ref. [Reference Spong and Bhatia54]. In the case of SEA, E(t) can be computed as

(68) \begin{equation}E=U+K=\frac{1}{2}m_{m}\dot{x}_{m}^{2}+\frac{1}{2}m_{l}\dot{x}_{l}^{2}+\frac{1}{2}k(x_{m}-x_{l})^{2}.\end{equation}

However, as the control law (67) may inject power in both the resonant and translatory modes, a slightly modified version is proposed here, to minimize the effect of temporary energy:

(69) \begin{equation}F(t)=k_{e}(E_{d}-E(t))\dot{z}_{2}(t)\end{equation}

The difference between (67) and (69) is that the latter injects energy directly on the state $\dot{z}_{2}$ which stores the resonant energy. Nothing prevents some energy from being injected into the state $\dot{z}_{1}$ as a side effect but in minor part in comparison to (67). It can be proven that the energy control law (69) implicitly acts like a damping compensation law and its stability can be proven considering $E_{r}(t)$ , as expressed in (38), as a candidate Lyapunov function (proof is beyond the scope of this paper). Typical energy trends can be observed in Figure 8. After transients, the resonant energy $E_{r}$ reaches an almost constant value, while the temporary energy $E_{t}$ is still oscillating, alternating between energy accumulation and energy release phases, as expected. Power trends are reported in the lower plot where $P_{in}$ represents the input power and $P_{dis}=P_{dis,r}+P_{dis,tB}$ the power dissipated within the system, in both resonant and nonresonant modes.

Figure 8.

Energy and power trends in SEA simulation using energy control as in (67).

Data have been collected in 24 experiments characterized by 4 different loads, with different inertia, and with 6 different oscillation amplitudes corresponding to 6 different values for $E_{d}$ in (67). Figure 9 reports the measured efficiency in comparison with the analytical expectations, including worst case expectations as we did for the simulation results (Figures 4 and 5). The main differences in comparison with the simulation are that (1) the motor inertia is not changed, leading to a two-dimensional plot similar to those in Figure 6 instead of a three-dimensional one; (2) some noise can be observed in the measured index $\hat{\eta}$ , probably due to stiction effects, shaft misalignment and cogging torques, which can influence the experimental outcome depending on the motion amplitude and on the initial experimental conditions. In particular, stiction effects may present a dynamic constraint on the load, possibly affecting the temporary energy level. In the case of higher load inertias, one can see an agreement between the measured efficiency $\hat{\eta}$ and the analytical index $\eta$ . For the lowest load inertia, the index $\eta$ – which neglects the temporary energy – overestimates the system efficiency which lies between $\eta$ and $\eta^{WC}$ , as expected. This is observed only for the lowest load inertia because this situation involves higher levels of temporary energy, in accordance with (44) and (60a).

Figure 9.

SEA experimental validation: measured $\hat{\eta}$ in comparison with theoretic expectations $\eta$ and $\eta^{WC}$ .