A design concept of parallel elasticity extracted from biological muscles for engineered actuators

Series elastic actuation that takes inspiration from biological muscle-tendon units has been extensively studied and used to address the challenges (e.g. energy efficiency, robustness) existing in purely stiff robots. However, there also exists another form of passive property in biological actuation, parallel elasticity within muscles themselves, and our knowledge of it is limited: for example, there is still no general design strategy for the elasticity profile. When we look at nature, on the other hand, there seems a universal agreement in biological systems: experimental evidence has suggested that a concave-upward elasticity behaviour is exhibited within the muscles of animals. Seeking to draw possible design clues for elasticity in parallel with actuators, we use a simplified joint model to investigate the mechanisms behind this biologically universal preference of muscles. Actuation of the model is identified from general biological joints and further reduced with a specific focus on muscle elasticity aspects, for the sake of easy implementation. By examining various elasticity scenarios, one without elasticity and three with elasticity of different profiles, we find that parallel elasticity generally exerts contradictory influences on energy efficiency and disturbance rejection, due to the mechanical impedance shift thus caused. The trade-off analysis between them also reveals that concave parallel elasticity is able to achieve a more advantageous balance than linear and convex ones. It is expected that the results could contribute to our further understanding of muscle elasticity and provide a theoretical guideline on how to properly design parallel elasticity behaviours for engineering systems such as artificial actuators and robotic joints.


Introduction
Articulated systems are commonly existing and used in both natural world (e.g. human beings and mammals) and engineering domain (e.g. robotic arms/hands [1] and legged robots [2]). Typically these systems use joints, which are driven by biological or artificial actuators (e.g. muscles, electric motors), as fundamental 'units' to achieve desirable movements such as reaching, grasping and walking. As a result, the performance of the whole system is highly dependent on that of joints.
For biological joints, they use muscles as the source of force for move [3] and accommodate for external disturbances through two different ways: passive adaptation by intrinsic properties of biological components and active control by the Central Nervous System (CNS). The former is regarded as an effective supplement and solution to simplify and cope with the challenges (e.g. time delay, energy use and robustness) existing in the latter one. From this perspective, muscles being the motor of biological joints, their inherent elasticity should be extremely crucial for the remarkable dynamic performances of animals. It appears that nature has found the effective way how to make the best use of this property of muscles after millions of years of evolution. As exemplified in figure 1, biologists have found, by measuring the passive force-length (elasticity) relationships in different frog muscles using laser diffraction techniques, a concave-upward or even exponential tendency is always presented [4][5][6].
Likewise, the similar elasticity behaviours are also experimentally identified in the muscles of rabbits [7,8], humans [9], etc. All these experimental evidences essentially reveal a universal preference in biological muscles. Furthermore, biomechanists have also pointed out that the output force for each muscle is the sum of an active component and a passive component [10], implying that the muscle-like elasticity could be technically recreated in the form of parallel elasticity. from the lateral gastrocnemius muscle of human. All these experimental evidences suggest a universally concaveupward elasticity behaviour in biological muscles. Data is taken from Reference [5,8,9].
In the line of engineering applications, roboticists have been attempting to integrate elastic property into artificial actuators for the purpose of enabling engineered systems comparable to biological ones.
Following this idea, series elastic structures that take inspiration from biological tendons have been extensively studied and used in today's robots. A representative physical realization is to combine mechanical springs in series with electric motors, namely, Series Elastic Actuators (SEAs) [11,12]. Also, many considerations go into the approach of achieving this elasticity through cable-driven actuations [13,14]. Apart from those, we also noticed several other studies that have looked at the utilization of parallel elasticity, which can be thought to mimic the elastic property within muscles. For example, [15] investigated the effects of parallel elasticity using a 2-DOF robotic finger with extension springs. They found that the integration of parallel elasticity leads to improved stability and robustness in robotic hands during grasping and manipulation. [16] compared a linear and a physiologically based parallel elasticity profile with a two-segment swinging robotic leg, and found that both can reduce the energetic cost of leg swinging through wisely choosing parameter values. A final example is [17], in which several cases of without parallel spring and with parallel springs of different stiffnesses and preloads were studied with a planar bipedal robot. The authors found that the addition of springs in parallel with the knee actuators can improve the energetic efficiency of walking, with higher stiffness providing greater benefit at higher speeds and lower stiffnesses providing benefit at lower speeds. All these attempts have demonstrated the effectiveness of parallel elasticity to achieve robust and efficient movements.
Despite impressive achievements, it is still a common and somewhat intractable problem to emulate the elastic property of biological muscles, even for a single robotic joint. A significant challenge lies in determining the profile of elasticity. Obviously, the straightforward implementation of muscle-like elasticity profile requires costly mechatronic design and, from our point of view, is not necessarily suitable due to the limits of current state-of-the-art technologies and the differences between biological and engineering systems. With these considerations in mind, the current paradigm is to use either a simple-to-design but not optimal-performance method of selecting a plausible elastic element (e.g. mechanical spring) from those commercially available off the shelf, or a better-performance but not versatile elasticity profile tailored to fulfill a specific task. In addition, there seems to be a consensus that nonlinear profiles of elasticity have a more positive effect than linear ones [18][19][20], however, at present it is still open what type of nonlinearity is a good fit. In fact, to the best of our knowledge, there are very few studies in the literature to address how to reasonably copy the elastic property of biological muscles into artificial systems.
As a step towards further understanding the functions of muscle elasticity and more effectively applying this property in engineering domain, this paper seeks to determine, through examination of a conceptual joint model, the potential functional benefits behind the preferred elasticity profiles in biological muscles and to distill a general guideline for the engineeringly applicable design of elasticity profiles. Firstly, a single-degree-of-freedom rotary joint model implementing periodic motions is developed, on the basis of analyzing the agonist-antagonist muscle configuration in biological joints. The elastic property of the muscle pair is seriously considered in this model. Then the effects of various elasticity profiles on two crucial but contradictory demands for artificial systems, energy efficiency and disturbance rejection, are compared and contrasted in detail, followed by an explanation of the mechanisms underlying the observed behaviours. Finally, several lessons that can be learned from this study are discussed.

The conceptual joint model with parallel elasticity
To explore the behaviour of parallel elasticity in detail, we consider a conceptual model with only one rotary degree-of-freedom, as illustrated in figure 2a, which is the basic component of articulated systems.
It consists of an upper segment and a lower segment (with mass m and inertia J ) that is able to rotate around the upper one. For the actuation of this model, an antagonistic pair of muscles, similar to animal muscle actuation, is included. Both the extensor and flexor muscle are described by a Hill-type muscle model [21] which has been used as the conceptual foundation in biomechanical modelling.
Despite only a single-degree-of-freedom joint, it is still challenging to implement a systematic investigation with the developed model. Because unlike existing studies that just choose muscle-tendon parameters based on biological data or empirical estimates, the primary objective of this study is to investigate the effects of various parallel elasticity behaviours by performing a parameter scan and to clarify possible design implications for parallel elasticity. With the actuation model in figure 2a, however, a large number of parameters of muscle activation and tendon-like (series) elasticity would have to be estimated, in addition to those describing the muscle-like elasticity. This causes the analysis to be computationally difficult. On a deeper level, the muscle-tendon units of biological organisms are dynamically coupled and have already been well tuned through evolution to provide the desired superiority like high efficiency and robustness. It is therefore very likely that the respective effects of As a second approximation, tendon-like series elasticity is ignored in this work, taking account of 1) tendons are another equally important elastic elements in biological joints, their elasticity profiles are also unknown, a full consideration of which would greatly complicate the whole investigation. A better approach is to study one element while keeping others in a suitably simple setting; 2) the stiffness of a muscle is normally considerably less than the tendon stiffness [22][23][24], therefore tendons can be viewed to be stiff if no large perturbations (e.g. ground impact at touchdown) are involved during simulation; 3) researchers have argued that this simplification greatly facilitates the analytic process at the expense of a relatively small accuracy [25,26]. Simplifying by reducing the model dimensionality, we thus arrive at a simplified joint model  (figure 2b) that allows a systemic investigation and comparison of muscle-like parallel elasticity. () e f l r     (2) here r is the radius of the virtual pulley, lr     is the deflection of the elastic element, () fl  is the elastic force generated by parallel elasticity 1 figure 2), the function f could be linear or nonlinear. In this paper, 0   is set as the position of optimal length 0 l of the two muscles at which there is no passive restoring force, without consideration of the asymmetric structure existing in biological joints. With this setup, the elastic force always acts with gravity. We can derive from equation (1) that the profile of elasticity has a direct impact on the performance of joint.
 The antagonistic configuration is reserved for a more straightforward correspondence with biological joints, even though it can be further reduced by replacing the unidirectional motors and elastic springs with bidirectional ones during engineering realization.

2.2.
Modelling the parallel elasticity The goal of this work is to draw possible design clues for parallel elasticity by performing an exhaustive search of its profiles. For a systematic search, four scenarios with different elasticity configurations are compared and contrasted. Scenario 1 is the simplest case, in which simulation is executed without elasticity, and with which other configurations can be weighed. In scenarios 2 through 4, three families of elasticity with distinct profiles are explored, respectively, as illustrated in figure 3. At a more detailed level, for a start, a family of linear elasticity with the force-deflection relationship and also the four scenarios can be specified as follows:

Performance metrics and Simulation procedure
Being both robust to disturbances and energetically efficient is desirable for artificial machines, particularly for those robots expected to manipulate or locomote in complex environments such as disaster areas, dangerous regions and highly unstructured terrains. These two aspects are usually stated as the main concerns about using parallel elasticity in robotic joints: the parallel elasticity is able to share part of the actuator torques and thus helps reduce the active energy consumption, but it may also counteract the actuator torques sometimes; in turn, such counteraction can be beneficial to disturbance rejection, and the rapid mechanical feedback loop formed by elastic elements may further provide selfrighting after perturbation [27,28]. From this perspective, two subsequent stages are included in our simulation study, with the aim of examining the disturbance rejection and behaviours energy efficiency,

Disturbance rejection
In the first stage, the joint was commanded to execute a periodic motion between position A and B ( 60  , see figure 2) along a sinusoidal trajectory, during which an impulsive perturbation was applied to the joint after it converged to its steady state. Settling area is employed as a measure of disturbance rejection in this work, thanks to the simplification of our model [29]. Specifically, as depicted in figure 4, it indicates the total of the areas enclosed between two steady-state boundary lines and actual trajectory outside the steady-state region. This measure provides an intuitive identification regarding how much area is in departure from the undisturbed route due to the perturbation. Of particular note is that, because the parallel elasticity modelled in this paper is defined position-dependant, and naturally the settling area values are expected to vary as the perturbed positions. To deal with this issue, we sample the settling area data at10 , turning points calculated by  and returns to steady state at time t 2 . So, settling time is (t 2 -t 1 ), settling area is the area enclosed between two boundary lines and actual trajectory outside the steady-state region.

Energy efficiency
In the second stage, the joint was commanded to execute the same motion. In order to maintain this motion, the two unidirectional motors have to offer a certain contribution, in collaboration with the passive elastic elements. The second key performance criterion in this work is therefore energy efficiency.
Inspired by the criterion of Cost of transport (CoT) [30], which has widely used to quantitate the energy cost of humans, animals and robots of different sizes, here we adopt angular cost of transport (A-CoT) to describe the energy dissipated by the joint per unit radian, defined as where m , L and g denote the mass, length of the lower joint link and gravitational acceleration. M  is the active torque fed into the joint to the maintain the motion  , while d  represents the total angular displacement of  . Such definition characterizes the average energy cost of the joint during, for example, one motion period. Seeking to a good correspondence with the disturbance rejection defined above, we further refine the A-CoT to a specific position level by modifying the definition as where M  is the corresponding active torque when the joint goes through within a motion period (e.g. from ()   to ()   and back from ()   to ()   ),   is infinitely small (set to1  in this paper). Such definition not only helps understand the position-dependant elasticity profile, but, more importantly, enables the subsequent trade-off analysis between disturbance rejection and energy efficiency. Similar to the disturbance rejection behaviour, the energy efficiency levels at10 , turning points and 50 are sampled. Note that the A-CoT here is actually a different measure and not comparable with the commonly used CoT.

Results
It can be seen from equation (1) that the addition of parallel elasticity changes dynamics of the joint system. In this section, we show how and to what extent it affects the disturbance rejection and energy efficiency behaviours. To facilitate analysis, we restrict the turning locations of 0 NL C  to the range of [20 , 40 ]  .

Effect of parallel elasticity on disturbance rejection
In

Explanation of the behavioural changes: a frequency-domain view
We showed a series of changes in energy efficiency and disturbance rejection behaviours of the joint, resulting from the addition of various profiles of parallel elasticity. From an overall perspective, introducing elasticity in parallel to actuators would probably have a negative effect on energy efficiency, with the exception of the tiny area marked by dots in figure 5f. An intuitive explanation for this result is the increased difference in frequency between motion trajectory and the joint system. The frequency of motion trajectory used during simulation is 1Hz, lower than the resonance frequency of the joint itself which nonlinearly depends on joint position, from ~1.93Hz at10 down to ~1.80Hz at 60 . In this circumstance, if parallel elasticity is integrated, the resonance frequency of the joint system will be enhanced, thereby expanding the frequency difference. Consequently, different amount of active contribution is required to compensate such difference caused by parallel elasticity.
For a more detailed insight, we also examined the mechanical impedance of the synthetic joint system, a central property characterising the intrinsic dynamics of mechanical systems, which is defined as the ratio of the applied torque to the resultant angular velocity By changing the joint mechanical impedance, the relationship between the angular trajectory of the joint and the net muscle torque required to produce it is correspondingly changed. The effect is illustrated in impedance. This gives a more fundamental explanation for energy efficiency behaviours in figure 5, since, obviously, the higher the magnitude of mechanical impedance the more energy is required in order to achieve the same motion. For example, at 1Hz, compared to the case of no elasticity, there is a slight increase in the magnitude of mechanical impedance at the location of 10 o when parallel elasticity is To interpret the changes in disturbance rejection behaviour, we first look at what determines the settling area that we use as a measure. By the definition in section 2.3, two aspects are involved: (i) deviation magnitude, which denotes the degree of departure of the joint from undisturbed route due to perturbation, and (ii) response delay, describing how fast the subsequent motor torque would counteract the deviation in case it occurs. The first factor, deviation magnitude, is inversely proportional to the magnitude of mechanical impedance, since clearly, given a perturbation, the higher the magnitude of mechanical impedance, the less the joint would deviate. Together with the impedance magnitude shown in the left column of figure 6, we can validly infer the change in the deviation magnitude. In terms of response delay, we look at the phase plot of mechanical impedance for a quantitative analysis. A negative impedance phase means the active torque lags the joint velocity, thus a certain time is needed for the subsequent motor torque to react once derivation takes place. In contrast, a positive phase indicates that the active effort is in leading position. Such phase lead enables the subsequent motor torque to play a role when derivation is occurring, and is thus equivalent to enhancing the capability of resistance to perturbation.
Referring to figure 6, we can identify how various elasticity configurations affect the disturbance rejection.
So far we have revealed the individual contributions of deviation magnitude and response delay to disturbance rejection. However we still do not have a conclusive answer as to their collective effect, since contradictory trends always appear with them. In fact, it is difficult to establish a mathematical correlation between them due to the high nonlinearity of the problem. But the numerical results in figure 5a-c and 6 imply that the ability to resist deviation from the start of it occurring seems to be a key. Specifically, according to the above analysis, such resistance of deviation is directly reflected in increased magnitude and advanced phase of the mechanical impedance. To further test the above explanation and extrapolation, we investigate the behavioural changes by letting the joint move at the frequency of 3Hz. The magnitude and phase data of mechanical impedance can refer again to figure 6, and the observed results are demonstrated in figure 7. We found that the energy efficiency behaviours can be well explained by the corresponding impedance magnitude, exhibiting an inverse correlation. As for the disturbance rejection, the trend is not sensible with small C NL , because the amount of increase in phase lead is not significant and neutralized by the lowered impedance magnitude.
But with nonlinear coefficient C NL increasing, the improvement in phase lead becomes significant. As a result, visible enhancements are seen, supporting the aforementioned extrapolation.

Linear versus concave versus convex profiles
The profile of parallel elasticity plays a significant role on the behavioural changes. Linear profiles are the most commonly used type thanks to its ease of modelling and physical implementation. Similarly, consistent distribution trends are observed when the joint moves at 3Hz (figure 8d-f), indicating that concave profiles are also superior to other two types in the high-frequency mode. At a closer look, we can find that concave profiles are particularly beneficial for those robots concerning energy cost. Since at the joint location of 50 o , the energetic level is significantly worsened (approximately 0.1 at 10 compared to 0.7 at 50 ), and unfortunately the enhancement in terms of energy efficiency thus gained by exploiting convex elasticity is extremely limited, no more than 4% (from 0.77 to 0.74, see figure 8f). Conversely, if concave elasticity is employed, energy efficiency would be expected to improve by up to roughly 13% (from 0.75 to 0.65), without sacrificing much of the disturbance rejection property.
Moreover, we note that Mahdi Khoramshahi et al. [18] found that spinal joint with piecewise linear elasticity, essentially a simple variation of the concave elasticity, is able to achieve a better balance between energy efficiency and speed than purely linear ones in quadrupedal locomotion. Such trade-off analysis above also provides a likely explanation as to why biological systems shape the similar muscular elasticity characteristic (see figure 1). Figure

Reducing the downside of parallel elasticity
We showed that employing parallel elasticity would inevitably introduce unwanted side effects. These negative consequences can be reduced or even avoided by proper designs. To facilitate analysis, the equilibrium point of parallel elasticity in our model is assumed at 0 , making the elastic torque always in the same direction with gravitational torque. Hence, the first possible way is to include an offset in the equilibrium position of parallel elasticity [16]. By choosing the correct offset, it will be possible to recycle the negative work done by gravity, thereby improving the energy efficiency of active efforts.
Another common design technique is to add (active or passive) clutches into the joint structure, which will allow decoupling of the parallel elasticity when not desired [31]. Note that this method may require additional mechatronic designs that may complicate the structure. However, we do not discuss the details here.

Application to artificial systems
As identified in the previous sections, our main finding is the benefit of the muscle-like concave elasticity characteristics for overcoming the trade-off between energy efficiency and disturbance rejection during joint rotating. The result is able to contribute to the design of rotary actuator units or robotic joints, thereby helping to enhance the performance of articulated systems such as robotic limbs and prostheses.
This improvement in hardware design is particularly useful for those systems concerning both energy cost and robustness. For example, this would be the case for a legged robot needing to locomote in highly rugged terrains with power autonomy [2,32]. Such scenarios typically occur in applications such as search and rescue and planetary exploration. Another example would be robotic hands that need to achieve energy-efficient and robust grasping and manipulation [15]. By reasonably integrating identified elasticity characteristics into their joints, the performance of these systems is expected to be enhanced. In addition, since the improvement is obtained purely from the self-adjustment of joint dynamics, it may also help reduce the dependence and burden of active control.
In order to implement identified parallel elasticity properties in a robotic joint, many design proposals for nonlinear elasticity can be utilized. For example, Kim and Deshpande [33] presented a design methodology for nonlinear stiffness by using a noncircular pulley-spring mechanism. With the synthesis procedure they reported, the shape of the noncircular pulley can be obtained for a desired elasticity profile.
Schmit and Okada [34] presented a non-circular cable spool mechanism to synthesize a nonlinear rotational spring. Depending on the shape of the spool, various torque-angle relationships can be realized.
Endo et al. [35] also proposed a design of non-circular pulley-spring mechanism to produce the desired nonlinear torque profile. In addition, the linear springs in these mechanisms can be also replaced by other compliant materials like rubber and viscoelastic polyurethane, which could make the mechanisms easier to be integrated into lightweight and small robots. Representative examples of such designs include those proposed in [36] and [37]. With these clever mechanisms, it is possible to produce various types of elasticity profiles. And as a theoretical reference for hardware designers, the result of this study reveals that concave ones may be more reasonable when designing elasticity in parallel with actuators.

Towards hardware realization of nonlinear parallel elasticity
While the theoretical results indicated that concave parallel elasticity provides robotic joints with better passive performances, several practical considerations with hardware realization have to be made. First of all, implementing nonlinear elasticity usually needs additional mechanical designs, which leads to high system complexity, e.g. manufacturing and operation. Notably, such design can add weight and inertia of the system, lowering, in turn, the energy efficiency. In addition, it is unsuitable for small and lightweight platforms to integrate these complex nonlinear mechanisms. On the side of control, nonlinear property as well as the additional structural components may complicate the modelling, analysis and parameter identification of the whole system. In summary, we have to think about whether or not the performance enhancement outweighs the complexity increase involved in doing so on the side of the overall system.

Limitations and future work
Although we have shown the superiority of concave parallel elasticity, which agrees well with biological evidence, the use of the presented model has several limitations. As mentioned previously, tendon-like series elasticity was not considered in our model. And because of this, no large impulsive forces (e.g. ground impact at touchdown in legged locomotion) were adopted during simulation, as series elasticity is predominant for impulse absorption. Force-length and force-velocity relations of the active muscle force were simplified as well [38,39]. Damping variations within muscles were not considered. Consequently, the incomplete mapping caused by such approximations between our conceptual model and biological joints may interfere with the interpretation of other biological principles. In addition to the simplifications inside muscles, the biarticular structure of muscles was also not taken into account. Such structures are argued to contribute to energy efficiency and disturbance rejection of a joint as well. To purely examine the utilization of parallel elasticity, the joint was open-loop controlled during simulations, which may also result in noise to the disturbance rejection metric.
Our future work, therefore, will improve the model accuracy by reasonably removing these approximations. We emphasize 'reasonably', because, despite without a faithful correspondence to biological reality, advantages in terms of ease of modelling and reduction of parameter dimensionality were indeed seen in our studies, simplifying data analysis thereby facilitating basic principles revealing.
More importantly, our studies are conducted in the context of robotic application, rather than pure biomechanics. We have to think about the transfer of the presented models to robotic prototypes, that is, practical constraints of state-of-the-art technologies. Obviously, a relatively complicated model is challenging for straightforward hardware realization, as discussed above. Apart from improving the joint model, suitable active control is also indispensable if the joint system is put in practical use. Thus, we will also work on developing appropriate control strategies to enhance joint stability and adaptivity.

Conclusions
In this work, the effects of parallel elasticity on energy efficiency and disturbance rejection and the reasons underlying these behavioural changes are explored with a conceptual joint model. The results may contribute to the design of articulated machines concerning both energy cost and robustness, such as robotic manipulators working in limited space or legged robots locomoting in highly rugged terrains that need to deliberately swing and place the legs. Besides, several lessons can be learned from this study.
First, not any nonlinear parallel elasticity is superior to linear ones. It depends on the type of nonlinearity.
In principle, a concave profile is advantageous. Second, natural frequency of a joint system can be altered by parallel elasticity, either increased or decreased depending on the profiles introduced, and matched to specific tasks, for example high speed moving. Third, the universal elasticity profile exhibited within biological muscles is probably evolved as the result of a trade-off between multiple objectives, among which energy efficiency and disturbance rejection might be two contributing factors.