# What Is an Artificial Muscle? A Systemic Approach.

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## Abstract

**:**

## 1. Introduction

## 2. The Actuator as a Stable System

**Figure 1.**Typical force/position and force/velocity characteristics of an acceleration-type actuator (

**a**) and (

**b**) and of a velocity-type actuator (

**c**) and (

**d**).

_{0}’, ‘r

_{0}’, ‘t

_{0}’, and current length, radius, and thickness are respectively noted, ‘l’, ‘r’, and ‘t’, the expansion of the pressurized cylindrical tube can be derived from classical stress-strain Treloar’s model [15]:

_{1}= (r/r

_{0}), λ

_{2}= (l/l

_{0}), λ

_{3}= (t/t

_{0}) and σ

_{1}, σ

_{2}, σ

_{3}are the corresponding stresses and G is the shear modulus of the rubber. By combining these equations, we derive the following expression linking the inflation pressure P to λ

_{2}:

_{0}/r

_{0}) is all the more low than the ratio (t

_{0}/r

_{0}) is low. As highlighted by Johnson and Soden [16], this phenomenon clearly corresponds to an instability phenomenon for the rubber tube considered as a system whose input is a control pressure bigger than the ‘ballooning pressure’ and the output is the tube length, as its radius.

_{1}= sin(α)/sin(α

_{0}) and λ

_{2}= cos(α)/cos(α

_{0}) where α

_{0}and α are, respectively, the initial and current braid angle. We deduce the new relationship:

_{0}= 20°. It is now clear that the system whose input is control pressure and output the muscle contraction length is stable for any pressure, however, limited by the rubber tensile range.

**Figure 2.**Interpretation of the braided sleeve movement of the McKibben artificial muscle as the internal process giving to the actuator its artificial muscle nature; (

**a**) combination of the pressurized inner tube and the braided sleeve inside the McKibben structure; and (

**b**) a comparison between the static behavior of an unrestrained thin rubber tube becoming unstable beyond the ballooning pressure and the stable behavior of the McKibben muscle inner tube (see text).

## 3. Concept of Linear Artificial Muscle

_{0}, b

_{1}are constants not equal to zero. Let us consider a normalized u-control between 0 and 1 where u = 1 corresponds to a maximum positive force F

_{max}. It can also be assumed that the x-position varies between 0 and a positive x

_{max}-value. If no load is considered, the equilibrium position of the artificial muscle, for a given u-value, is equal to Ku/b

_{0}. Consequently, we deduce: x

_{max}= F

_{max}/b

_{0}. It is then possible to rewrite Equation (4) in the following form:

_{0}= 0 and to the definition of a linear acceleration-type actuator if b

_{0}= b

_{1}= 0, in correspondence with Figure 1. According to the proposed definition, the linear artificial muscle behaves like an “active” spring whose stiffness is constant equal to (F

_{max}/x

_{max}). It is worthy to note that the linear artificial muscle considered in this section, as the biomimetic artificial muscle model considered in next section, are rough approximations of the real behavior of artificial or natural muscles due to the “linear” character—in the meaning of a straight line—of their static characteristics. For example, in the case of the McKibben artificial muscle that has already been mentioned, the isometric force versus muscle x-displacement exhibits a typical convex shape, complicated by a complex hysteresis phenomenon as described by Minh et al. [18], not considered in our study. In a different way, springs made of shape memory alloy, like those fabricated by Kim et al. [19], can exhibit a static characteristic for which a linear approximation of the static force versus deflection at constant temperature appears to be particularly relevant (see in [19], figure in page 81). We did not try, in the framework of this paper, to classify artificial muscles according to the concave or convex shape of their static characteristic, with or without hysteresis phenomenon. We consider, in the spirit of Hogan’s work on the skeletal muscle (see further) that “linear” static characteristics are a simple but powerful way for developing a general model of artificial muscle. More especially in this section, our proposed concept of linear artificial muscle can be considered as an ideal artificial muscle.

**Figure 3.**Use of the Hill-curve concept for defining the force-velocity characteristic of an artificial muscle, (

**a**) Hill’s quick-release test considered as a “release trial” of the artificial muscle from zero position and velocity, with a constant u-control and given load, and (

**b**) characterization of the corresponding velocity as the maximum velocity reached during contraction (redrawn from a typical skeletal muscle recording).

_{M}= Mg/F

_{max}, we get:

_{M}= 0, the system behaves like a first order for which maximum velocity noted V

_{max}is given by:

_{M}is not equal to zero, we can derive the maximum velocity noted V from the classic solution resulting from the second-order differential equation. It is easy to show that, for a given r

_{M}-ratio, the ratio (V/V

_{max}) only depends on u and the damping Z-factor. We get:

_{max}and x

_{max}parameters of the artificial muscle static characteristic and to the load-ratio r

_{M}by the following relation-ship:

_{0})-coefficient role in the Hill curve: on the one hand, it characterizes the concave curvature of the pseudo-Hill curve and, on the other hand, the lower is c, the more curved is the pseudo-Hill curve, as this is the case for the (a/F

_{0})-coefficient in the case of the Hill curve. In the bound-case where c tends to infinite, i.e., b

_{0}tends to 0, the pseudo-Hill curve tends to the straight line: (V/V

_{max}) = (1 − r

_{M}/u). In this case, as previously noted, the actuator lost its artificial muscle nature to be a velocity-type actuator. Moreover, the case c = 1, for which the concave curvature of the pseudo-Hill curve is still little marked, corresponds to the limit for which all “release trials” are performed without overshooting since the limit between underdamping and overdamping is given by Z = 1 and so by: r

_{M}= c

^{2}. For any c < 1, the time-response will exhibit an overshooting for any load greater than c

^{2}F

_{max}/g. Moreover, we would like to show that the linear form of the pseudo-Hill curve can also be considered in the case of a more biomimetic type of artificial muscle.

**Figure 4.**Linear artificial muscle characteristics; (

**a**) force-position characteristic; and (

**b**) force-velocity characteristic defined as a pseudo-Hill curve (for u = 1)—see text.

## 4. Biomimetic Artificial Muscle for Antagonist Actuators

#### 4.1. Biomimetic Artificial Muscle Combining a Non-Linear Static Characteristic with a Linear Viscosity

_{max}and x

_{max}are, respectively, the maximum force and maximum x-contraction corresponding to u = 1, as illustrated in Figure 5a. Although this actuator is clearly non-linear due to the product of u by x, it is always in accordance with our definition of the artificial muscle: whatever the external force generating, for a given u, an equilibrium position x

_{e}, it is obvious that a variation of δx around this equilibrium position produces a return force δF = −(uF

_{max}/x

_{max})δx and so the actuator stability results. Let us associate to this non-linear static model a linear damping term similar to the one considered in the previous section. During the quick-release process, the value of u-control is constant and it is interesting to remark that for u = 1, which corresponds to the usual physiological conditions of the experimental quick-release under maximal neural activation, the model’s equations become the same for the biomimetic muscle model and the linear one. As a consequence, the curves of Figure 4b can also be considered for the force-velocity characteristic of the biomimetic artificial muscle in the case u = 1. The Hill curve equation can take several forms. We will consider the following one:

_{M}= Mg/F

_{max}where V

_{max}is the maximum velocity for no load and F

_{max}is interpreted as the tension producing no displacement i.e., the tension P of Equation (12) which prevents any isotonic contraction of the artificial muscle because this tension is equal to the maximum muscle force F

_{max}. In their paper about biorobotic actuators, Klute, Czerniecki, and Hannaford [22] consider that the (a/F

_{max})-coefficient characterizing the curvature of the Hill curve belongs to a typical [0.12–0.41] range. In fact, if we take into account other studies, non-considered by the previous authors, this coefficient can reach higher values: a 0.81-value is, for example, considered by Ralston et al. for a human major pectoralis whose data are reproduced in Figure 5b [23]. These last authors also emphasize, at their time, that the (a/F

_{max})-coefficient is subject to wide variation for a given type of muscle. On Figure 5c, we tried to match the two extreme cases for (a/F

_{max}), 0.12 and 0.81, with a corresponding pseudo-Hill curve as defined in previous section: a c equal to 0.12 has been considered in the case of the Hill curve characterized by (a/F

_{max}) = 0.12 and a c equal to 0.5 has been considered in the case of the Hill curve with (a/F

_{max}) = 0.81. If a certain discrepancy is notable for the lower value in (a/F

_{max}), the concordance between the proposed model and the real data (represented by circles) is rather good for (a/F

_{max}) = 0.81.

_{max}= 9 cm, F

_{max}= 200.1 N) given by Ralston et al. (see table in Figure 5b) for simulating the corresponding time-response of the quick-release—interpreted as a step-response of value u = 1—with c = 0.5. Although real-time responses are not reported in the Ralston et al. article, the authors specify that ‘[the sigmoid character of curves] is always the case expect with very heavy loads, when the curve tends to be flat over a large portion of the excursion’, which is well checked in Figure 5d with a minor overshooting appearing for high loads. Moreover, rising times are in very good accordance with other available experimental results [27]. It is also worthy to note that, in the case of u = 1, the time responses shown in Figure 5d correspond both to the non-linear static characteristic combined with a linear viscous friction and to the full linear artificial muscle model. This suggests that a purely linear artificial muscle, as we defined it, can approach, in a very satisfying way, the dynamic behavior of a real muscle and, therefore, the question arises to know which could be the interest of considering the non-linear static characteristic of Figure 5a instead of the linear one. The answer to this question will be given by the consideration of the actuator made of two antagonist artificial muscles.

**Figure 5.**Biomimetic artificial muscle and its comparison with the skeletal muscle, (

**a**) force/position characteristic, (

**b**) real data used for the comparison between a human pectoralis major and the proposed biomimetic artificial muscle with a linear viscous damping (rewritten from [23]), (

**c**) comparison between the force/velocity characteristic with u = 1 in two extreme cases of Hill’s curve—the circles correspond to the real data given in (b)—(see text), and (

**d**) time-simulation of the quick-release of the biomimetic artificial muscle in load conditions given in (b).

#### 4.2. Constant Moment Arm Antagonist Artificial Muscle Actuator

_{b}(u

_{b},x

_{b}) and F

_{t}(u

_{t},x

_{t}) are the forces generated by the artificial biceps and the triceps for, respectively, a u

_{b}-control and a x

_{b}-muscle contraction length, and a u

_{t}-control and a x

_{t}-muscle contraction length. If x

_{b}

_{0}are x

_{t}

_{0}are, respectively, the initial x

_{b}and x

_{t}muscle contraction lengths, we have:

_{b}− u

_{t}) − (x

_{b0}− x

_{t0})/x

_{max}where u

_{b}and u

_{t}vary between 0 and 1 and, consequently, u varies between ‘−1 − (x

_{b0}− x

_{t0})/x

_{max}’ and ‘+1 − (x

_{b0}− x

_{t0})/x

_{max}’. In the case where x

_{b0}= x

_{t0}, the static characteristic of the actuator takes a particularly simple form, as follows:

^{2}F

_{max}/x

_{max}and joint angle varies in the range [−θ

_{max}, +θ

_{max}] where θ

_{max}= (x

_{max}/2r). We give in Figure 7a the corresponding torque/angle characteristic in the normalized form T/rF

_{max}versus θ/θ

_{max}, limited to the case of positive actuator angle and torque.

_{1}, we get:

_{b}’ and is ‘u

_{t}’ cannot be combined into a single control variable. In the particular symmetrical case in which x

_{b}

_{0}= x

_{t}

_{0}= x

_{0}, the static torque equation is simplified into the following one:

_{b}− u

_{t}) is responsible for the actuator torque while (u

_{b}+ u

_{t}) is responsible for the joint stiffness. If it is true that, whatever the choice for x

_{b}

_{0}and x

_{t}

_{0}, the actuator stiffness is given by ‘(u

_{b}+ u

_{t})(r

^{2}F

_{max}/x

_{max})’ and is so proportional to (u

_{b}+ u

_{t}), the other torque component is proportional to (u

_{b}− u

_{t}) only when x

_{b}

_{0}= x

_{t}

_{0}= x

_{0}. In this last symmetrical case, the joint angle varies in the range [−θ

_{max}, +θ

_{max}] where θ

_{max}= (x

_{max}− x

_{0})/r, and the corresponding static torque can then be rewritten as follows:

**Figure 7.**Torque-angle characteristics for the constant moment arm antagonist actuator, (

**a**) the case of two identical linear artificial muscles, and (

**b**) the case of two identical biomimetic artificial muscles showing the ability of the actuator to modify its angular stiffness (from slope 1 to 2, for example, by increasing the sum of agonist and antagonist control values) while keeping the same equilibrium position (for simplicity reasons we limited our graphs to positive angle and torque).

#### 4.3. Elbow-Type Antagonist Artificial Muscle Actuator

**A’**on the fixed link (arm) and is inserted in

**B’**on the mobile link (forearm), as illustrated in Figure 8a.

**A**of Figure 8a and elbow joint center

**E**is noted A and the distance between point

**B**and

**E**is noted B. Two offsets are considered, under the form of angles α and β, to express the fact that the distances

**AA**’ and

**BB**’ are not equal to zero in the reality. On the other hand, it is supposed that the ‘tendon’ of the triceps is driven around the elbow joint center with a constant radius r.

**Figure 8.**Elbow-type artificial muscle actuator, (

**a**) actuator scheme, and (

**b**) corresponding torque-angle characteristic (see text).

_{b}and F

_{t}the forces produced by the biceps and the triceps, θ the elbow joint angle, and T the torque generated by the two muscles. From a simple geometric analysis of Figure 8a, we deduce the following relationships for L(θ) =

**AB**:

**E**to the line segment [

**AB**], now depends on the joint angle according to the following relationship:

_{max}-value lower than 180°. It is also assumed that, in the zero-joint angle position, the triceps is fully contracted—i.e., its x

_{b}-position is equal to x

_{max}—while the biceps is not contracted—i.e., its x

_{t}-position is equal to 0. In order to make easier the computation of the actuator torque, it is considered that the biceps can contract at least of a x

_{max}-length and so the following expression for the maximum joint value θ

_{max}is deduced: θ

_{max}= (x

_{max}/r). It is worthy to note that no passive tension is taken into account in our model. By applying the non-linear static model of Equation (11) to the artificial biceps and triceps force expressions, we get:

_{max}, we get the new expression for the biceps torque:

_{b}and T

_{t}-expressions are now proportional to rF

_{max}and we can then define the normalized corresponding total torque T/rF

_{max}= (T

_{b}− T

_{t})/(rF

_{max}) as we did it for the constant arm antagonist actuator. We give in Figure 8b the corresponding static torque-angle characteristic of the elbow-type actuator for α = β = 10°, θ

_{max}= 180° − α − β, (B/A) = 0.2 and by imposing (u

_{b}+ u

_{t}) = 1—the input u

_{b}is denoted u on the graph. By comparison with the torque-angle characteristic shown in Figure 7b, the new static characteristic no longer exhibits a uniformly decreasing slope in the θ-range corresponding to a given constant u-value. Let us consider, for example, the case u = 1 and let us assume that the actuator is submitted to a constant resistive torque: a double equilibrium position can result from such a situation corresponding in Figure 7b to the points

**P**and

**Q**. If the system is set in

**Q**and if it removed from this equilibrium position, for example, by a positive angular variation, its restoring torque is negative, as illustrated in Figure 7b, leading the actuator to asymptotically come back to

**Q**but, if the actuator is set in the equilibrium position

**P**, the restoring torque, corresponding to the same positive angular variation, is now positive with, as a consequence, to move the actuator to the point

**Q**. While

**Q**is an asymptotically stable equilibrium position,

**P**is only an astatic equilibrium position.

## 5. Conclusions

_{max}, F

_{max}and viscosity coefficient peculiar to the considered artificial muscle model. We also show that our linear artificial muscle model can relatively well approach the Hill curve of a real skeletal muscle by means of a judicious c-coefficient.

## Acknowledgments

## Conflicts of Interest

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Tondu, B. What Is an Artificial Muscle? A Systemic Approach. *Actuators* **2015**, *4*, 336-352.
https://doi.org/10.3390/act4040336

**AMA Style**

Tondu B. What Is an Artificial Muscle? A Systemic Approach. *Actuators*. 2015; 4(4):336-352.
https://doi.org/10.3390/act4040336

**Chicago/Turabian Style**

Tondu, Bertrand. 2015. "What Is an Artificial Muscle? A Systemic Approach." *Actuators* 4, no. 4: 336-352.
https://doi.org/10.3390/act4040336