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We analyze the possibility of taking advantage of artificial muscle’s own stiffness and damping, and substituting it for a classic proportional-integral-derivative controller (PID) controller an I controller. The advantages are that there would only be one parameter to tune and no need for a dynamic model. A stability analysis is proposed from a simple phenomenological artificial muscle model. Step and sinus-wave tracking responses performed with pneumatic McKibben muscles are reported showing the practical efficiency of the method to combine accuracy and load robustness. In the particular case of the McKibben artificial muscle technology, we suggest that the dynamic performances in stability and load robustness would result from the textile nature of its braided sleeve and its internal friction which do not obey Coulomb’s third law, as verified by preliminary reported original friction experiments. Comparisons are reported between three kinds of braided sleeves made of rayon yarns, plastic, and thin metal wires, whose similar closed-loop dynamic performances are highlighted. It is also experimentally shown that a sleeve braided with thin metal wires can give high accuracy performance, in step as in tracking response. This would be due to a low static friction coefficient combined with a kinetic friction exponentially increasing with speed in accordance with hydrodynamic lubrication theory applied to textile physics.

Artificial muscles are a special class of soft actuators thus named because they behave in a phenomenological—as opposed to anatomical—manner and, to some extent, behave like skeletal muscle. This means that, independently of its anatomical structure, the artificial muscle is entirely characterized by its so-called tension-length and tension-velocity curves which play the role of reference models for defining a rectilinear artificial muscle. We will consider in this article the case of rectilinear artificial muscles and, especially, one of its best-known realizations, the pneumatic McKibben artificial muscle. In a general manner, rectilinear fluidic artificial muscles are very promising for actuating soft robots or exoskeletons, but their highly nonlinear character makes their control particularly difficult. From the time the “rubber actuator” appeared in its renewed version designed by the Japanese tire manufacturer Bridgestone, multiple attempts have been made to control robots actuated by McKibben artificial muscles, by means of neural network control [

N. Hogan proposed in his seminal 1984 paper [_{0} and its current length _{max} at which the contraction force is equal to zero, we can consider, in Hogan’s spirit, the following relationship between the static force _{stat} and length _{max}_{damp}_{dyn}

We assume that the artificial muscle drives a given load _{max}[1 − (_{max})] − _{damp}

This equation can be normalized by dividing all terms by _{max}_{m}_{max}_{M}_{max} and _{damp}_{damp}/F_{max}_{max})] − _{damp}_{M}_{m}_{M}

In order to simulate the dynamic behavior of this model we must specify the damping force. This can be particularly complex due to the soft character of the artificial muscle and to shape changing of materials during muscle contraction. In the framework of our study, we will consider two simplified models. The first one corresponds to a linear viscous friction force whose constant coefficient will be denoted as _{v}_{damp}_{v}ẋ

The second considered friction model is much more original; it was inspired by our own work about McKibben muscle modeling [_{damp}_{s}_{k}^{−ẋ / ẋk} + _{k}_{s}_{k}_{k}

However linear or nonlinear the damping component may be, this dynamic model is essentially nonlinear due to the presence of the term in “_{I}_{d}

Let us derive this equation with respect to time; we get:

Let us assume that the closed-loop system is stable and that, during contraction, _{max}: when _{∞} —with all derivatives equal to zero—given by the following equation:

In order to check the possibility of converging towards the desired _{d}_{d}_{max} = 10 cm, and a viscous ratio (_{v}_{max}^{−t/T}_{M}_{m}_{I}_{I}^{−1}·s^{−1} appears to be a good compromise. In _{I}_{s}_{max}_{k}_{max}_{k}_{I}^{−1}·s^{−1} with a time scale corresponding to the best obtained result for a reasonable computing time. The result reported in

Simulation of the closed-loop control of the muscle model with a constant gain _{I}_{I}_{I}^{−1}·s^{−1}; (_{I}^{−1}·s^{−1} (see text).

In order to better understand closed-loop stability, we propose applying to our nonlinear system a classic linearization around the desired position considered a system’s equilibrium point. We will limit, in a first step, our analysis to the linear viscous friction case. From Equation (3) we derive:

Let us define the state variables as follows: _{1} = _{2} = _{3} = _{d}_{1d} = _{2d} = _{3d} = 0 in such a way that _{1} = _{1d} + _{1}, _{2} = _{2d} + _{2}, _{3} = _{3d} + _{3}. Let us define the following integral term _{3}):

From Equation (10), the following state representation is deduced in which the term in

In a classic way, we can deduce the matrix relationship _{1} _{2} _{3}]^{T} and the characteristic polynomial corresponding to det(_{3}_{3} is the 3 × 3 unit matrix and

Analyzing the system stability from this equation has, however, a meaning only if all coefficients are constant. This can be made from a simple physical interpretation of _{3}): let us assume that the system transitory state takes a finite amount of time to put the system from its initial position to the neighborhood of _{d}_{3}) ≈ _{d}_{I}_{d}_{d}

By reporting Equations (14) and (15) into (13), we get:

It is now possible to apply the classic Routh–Hurwitz stability criterion peculiar to a third order system in the form ^{3} + ^{2} +

Two important facts can be deduced from this stability condition: the higher the load _{max}_{I} value of about 363 m^{−1}·s^{−1} is deduced; in fact, this parameter induces some intermediate _{I}_{max} − _{d}_{max} and, on the other hand, we can consider that _{I}

Its value, in the case of our simulations, is equal to about 24.5 m^{−1}·s^{−1} and is still greater than the chosen value of 10 associated to a load

Is it possible to apply such analysis to our nonlinear kinetic friction model? We believe it is if we assume that, when the system is removed from its equilibrium position, its restoring velocity can be considered to be constant and equal to the initial slope of the kinetic friction _{d}

By using this new expression of _{d}_{v}_{I}

As stated earlier, increasing load or desired position can determine stability, but also make a large difference (_{k}_{s}_{max} − _{d}_{max} and _{I}^{−1}·s^{−1} in the case of numerical values used for our simulation, very close to the empirically chosen 10 m^{−1}·s^{−1}. Although questionable due to its very simplifying assumptions, this stability analysis suggests that a closed-loop stable control by means of a single linear integral action is feasible for the McKibben muscle without adding any supplementary damping devices, as experimentally confirmed in further reported experiments.

We consider three prototypes of McKibben pneumatic artificial muscle, hand-made at the laboratory, with braided sleeves made of three different materials: the first one, inspired by the historical version of the McKibben muscle, the sleeve is braided with cotton (

Various braided sleeves of the pneumatic McKibben artificial muscle, (

We give in

Main characteristics of the three considered McKibben artificial muscle prototypes.

Initial Length (cm) | Initial Radius (mm) | Initial Braid Angle (deg) | Max. Force (daN) (at 5 Bar) | Max. % Contraction (at 5 Bar) | |
---|---|---|---|---|---|

Rayon yarns | 36 | 8 | 23 | 105 | 0.35 |

Plastic wires | 32 | 6 | 30 | 30 | 0.35 |

Metal wires | 32 | 6 | 30 | 30 | 0.35 |

We discussed in

Experimental studies in friction coefficient for fibers and yarns are generally performed with specific apparatus [

The following principle was considered: if during contraction it is possible to exhibit almost constant velocity portions, the recorded artificial muscle force is a direct estimate of the kinetic friction force corresponding to this velocity, since the inertia force is then negligible. Because the load is accurately known, the kinetic friction coefficient can be estimated for different speed values. We give in _{k}

Experimental apparatus used to estimate the kinetic friction of a flattened sample of our rayon braided sleeve sliding over itself.

Experimental estimation of the kinetic friction coefficient of our flattened rayon sheath sliding over itself (

The relevance and efficiency of our proposed controller was experimentally tested on the setup shown in

Experimental setup for testing the closed-loop positioning controller of the three considered McKibben pneumatic artificial muscles driving various loads (

This is the reason why, in further reported step responses of experimental curves (see

Two kinds of signals were considered for testing static and dynamic accuracy of our McKibben muscle prototypes: step signals and sinus-wave signals. In both cases, the desired position is defined as an _{max}], where 0 corresponds to the initial muscle state _{max} corresponds to the fully contracted state of the muscle. In the case of a step signal, the desired step position was defined between a lower bound and an upper bound. Because the first prototype made of rayon yarns is relatively long and can generate a relatively high power, we decided to distinguish a “short” step with _{max}_{M}_{I}^{−1}·s^{−1} was considered in all reported experiments. Moreover, we give in _{I}_{I}^{−1}·s^{−1}), we again highlight the load robustness with a response time of about 1 s, albeit in a tighter load range, due to the weaker artificial muscle power. However, the important point to be emphasized is the impressive accuracy obtained in the case of the artificial metal muscle: an accuracy of 1/10 mm can be read on

Step responses performed by our rayon yarns McKibben muscle driving various loads: “long” (

Step responses performed by our plastic wire McKibben muscle driving various loads (

Step responses performed by our metal McKibben muscle driving various loads (

If we now consider the case of a sinus-wave tracking, with the value of _{I}_{I}^{−1}·s^{−1} for all loads), as shown in

Sinus-wave tracking at a 0.1 Hz operating frequency in the case of the rayon yarn-braided McKibben muscle (

In the case of plastic and metal wire-braided prototypes, we imposed a higher frequency equal to 0.2 Hz. It is noteworthy that the range of working frequencies of a given McKibben pneumatic artificial muscle depends on the dynamic performances of the pneumatic device feeding it as on the artificial muscle volume to be filled. That is the reason why a lower frequency was used for the first muscle whose initial internal volume is about twice this of the two other prototypes. _{I}^{−1}·s^{−1}.

Sinus-wave tracking performed at a 0.2 Hz operating frequency in the case of the plastic and metal wire-braided McKibben muscle, (

We report in

We also tested the ability of the controller to reject a manual perturbation: during the sinus-wave tracking, the operator manually caught the load and released it after some seconds. It can be checked in

The observed similarities in step and sinus-wave responses, as the reported differences in steady state accuracy and oscillations during perturbation rejection suggest that the three considered McKibben artificial muscles follow the same damping model with different parameters. Due to the good accordance between simulated and experimental step responses in all cases, we are led to consider that the assumed nonlinear damping model would be common to the three prototypes. This is, however, a surprising conclusion because if it is expected in the case of the sheath made of rayon yarns, it is more questionable in the non-fibrous case of plastic and metal-wire sheaths. A classic explanation given in textile technology is that the increase of kinetic friction with speed emphasizes the fundamental role of the lubrication agents during sliding. At high speeds and low loads, lubrication is partly hydrodynamic and, as written by Hansen and Tabor in their analysis of hydrodynamic factors in the friction of fibers and yarns [

We would like to finally highlight a double conclusion: first general a one and a then one more specifically devoted to McKibben artificial muscle technology. In a general way, we considered in this paper any rectilinear artificial muscle as being an original actuator characterized by its own stiffness proportional to control variable and an associated damping component which gives to the artificial muscle its open-loop stability. As a consequence, we suggested that a simple I controller would be relevant to control its positioning in closed-loop instead of the classic PID, with the following advantages: (1) there is no need for a dynamic model of the artificial muscle to look for high static and dynamic accuracies which are often problematic in soft actuator control; and (2) there is only a single parameter to tune which can eventually be adapted to tasks with specific compliance requirements. Although our stability analysis has to be improved, it appeared in good accordance with our experimental results. We believe that this I controller could be efficiently applied to a large class of artificial muscles if a damping component, whatever its nature, exists.

Our I controller has proved to be particularly efficient in the case of the tested pneumatic McKibben muscles. During our experiments made with three prototypes respectively braided with rayon yarns, plastic wires and metal wires, a surprising result occurred: by using thin metal wires, a static accuracy of about 1/10 mm was obtained without losing load robustness and stable perturbation rejection. Good tracking performance is also obtained with this prototype. Further experiments in friction will try to better analyze how a non-fibrous material used to braid a typical McKibben muscle sheath can behave like a textile material with damping performances apparently well more adapted to an accurate, quick and robust closed-loop control.

The simulation as experimental reported results are, however, limited to a closed-loop control of the artificial muscle contraction. A further step would be to apply this approach to a pair of antagonistic artificial muscles in order to check if similar accuracy and load robustness performances can be obtained with a “double effect”—positive and negative rotation—artificial muscle actuator.

I thank sincerely the three reviewers for their very pertinent remarks that helped this article become clearer and more precise.

The author declare no conflict of interest.