Selection of Artiﬁcial Muscle Actuators for a Continuum Manipulator

2 Artiﬁcial muscle actuators have become a popular choice as actuation units for robotic 3 applications, particularly in the growing area of soft robotics. The precise speciﬁcation of 4 an artiﬁcial muscle actuator for a particular application requires the consideration of several 5 parameters that work together to achieve the performance characteristics of the actuator. This 6 paper explores the speciﬁcation of artiﬁcial muscle actuator parameters by presenting and 7 applying the analytical description of the actuator, simulation by ﬁnite element method for 8 investigating material stresses under a wide variety of conﬁgurations, and a speciﬁc parameter 9 selection process. This is followed by an experimental validation using an example actuator to 10 compare against the predicted actuator performance. Some discussion of appropriateness of 11 this type of actuator as a candidate solution for use in the example application of a dexterous 12 continuum manipulator is included.

Although hydraulic actuation offers high power density, mechanical rigidity and high dynamic response, 29 the force capability, F , for a given pressure, P , of a conventional linear hydraulic actuator is limited by 30 the piston area, A, given by: F = P · A. In view of the stringent diametric requirement of the present 31 application, the use of hydraulic artificial muscle actuators (AMA) (see Fig. 1), which will be shown to 32 have better peak force capability than conventional linear actuators for the same pressure and diameter, 33 is of interest Schulte (1961). While the concept of artificial muscle actuators and their use in robotics is 34 not new, the majority of application use pneumatic power to drive them Cardona (2012); Trivedi et al.

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(2008a); Klute and Hannaford (2000); Tsagarakis and Caldwell (2000). However, through the use of a 36 hydraulic medium it may be possible to mitigate some of the issues with responsiveness and rigidity that 37 have been encountered previously. There are two important advantages to a hydraulic approach: 1) a higher 38 pressure (by a factor of 10 compared with pneumatic) can be applied so that force and power density can 39 be further increased and the actuator diameter can be decreased; 2) liquid has a much lower compressibility 40 and therefore better rigidity than compressed gas. 41 Artificial muscle actuators consist of a contained internal bladder, surrounded by a flexible, braided outer 42 sheath. It is the geometry of this outer sheath that transmits the radial expansion of the internal bladder 43 due to applied pressure to contractile force along the longitudinal axis of the muscle actuator Davis et al. 44 (2003). As the radius of the bladder, and thus the outer mesh, increases, the individual strands of the mesh 45 which are woven in a over-under crossing pattern rotate relative to each other and to the long axis of the 46 actuator and shorten the longitudinal distance from one end of the strand to the other. The load capacity of the artificial muscle actuator is then a function of the geometry and orientation of the outer sheath and the 48 pressure applied to the internal bladder Chou and Hannaford (1996).

ANALYTICAL DESCRIPTION OF ARTIFICIAL MUSCLE ACTUATORS
There are two methods presented in the literature for modeling the transmission of internal pressure to 50 contractile force of an AMA. The first is a theoretical approach based upon energy conservation Schulte 51 (1961), while the second is an examination of the force profile of the surface pressure Tondu et al. (1996).

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The first approach is based on the principle that energy supplied to the actuator by the pressurized fluid must 53 leave the actuator through the application of a load over some distance. The second approach is based on 54 an examination of the distortion of the internal bladder under isobaric conditions. However, ultimately each 55 of these methods arrives at the base model Tsagarakis and Caldwell (2000). A summary of the first method 56 is presented here. It should be noted that this model does not account for possible effects of compressibility between the strands and the longitudinal axis of the actuator, γ(t), which is assumed to be uniform for all 63 strands within the braided sheath.

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The overall length of the actuator, L a (t), and the actuator diameter, D(t), can then be represented in 65 terms of the constants, n and b, and as functions of the variable γ(t), as seen in Eq. (1) and Eq. (2).

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Then, calculating the volume of a cylinder and substituting in the functions for L a (t) and D(t), (3) The first derivatives of L a (t) and V (t) with respect to γ(t) are calculated as From Eq. (4) and Eq. (5), the first derivative of V (t) with respect to L a (t) is given as From the principle of virtual work we have: Figure 3. Ratio of force capacity for artificial muscle actuator to force capacity of a piston and cylinder when the maximum cross-sectional area of each actuator is equivalent (F = P · A max ) as a function of braid angle. and solving Eq. (7) for the force output with Eq. (1 and 6) results in where F a (t) is the contractile force and P a (t) is the pressure differential across the bladder wall. Because 73 the term within the brackets in Eq. (9) can be greater than 1 (lim γ(t)→0 = 2), the force capability of a 74 artificial muscle can be greater than that of a hydraulic piston actuator for the same area and pressure.

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It is assumed here that the force from the hydraulic piston actuator is F = P · A where the diameter is 76 equivalent to the maximum diameter of the AMA. In reality the available area may be reduced by the piston 77 rod cross-sectional area. Notice, however, that this advantage comes at the cost of having the force/pressure 78 relationship vary with γ(t) or actuator length L a (t), illustrated in Fig. 3. When the contraction angle is 79 35.3 • , the two actuators are equivalent. Further, it can be seen (Fig. 3)

BRAID PARAMETER SELECTION
When designing an artificial muscle actuator for a given application, it is likely that the available supply 89 pressure is known as well as the desired length, L max , and maximum diameter, D max , of the actuator 90 which occurs when the braid angle, γ = 54.7 • . Therefore, the optimal design is the one that maximizes is presented as where D o = b πn (obtained by evaluating Eq. 2 at γ = 90 • )is the theoretical maximum muscle diameter.
which is formulated by analyzing the fractional component of the hoop stress realized within an individual 109 strand of the braided mesh. Each of the N strands encircles the bladder n times. This result can then 110 be compared against the tensile limit of the mesh material, which is typically either a nylon polymer

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The use of the artificial muscle actuator as a means of power input to the system carries with it several 114 advantages previously discussed. However, it is also necessary to understand the failure limits for each 115 component. An expression for determining stress in the braided mesh was discussed in the previous section; 116 however, it is necessary to understand the failure limits of the internal bladder due to applied pressure 117 as well as the bladder is likely to be the weaker of the two components that make up the actuator. The strands. The value for the edge length can be calculated as where b, n, and N are the strand length, number of turns per strand, and strand number, respectively, as 128 before. The braid angle is a measure of AMA contraction and thus changes as the AMA is pressurized.

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This relationship is described in Section 2. In addition to the parameters EL and γ, it is also necessary to 130 consider the wall thickness, t, of the bladder material when determining a proper design to prevent failure.

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Further, the pressure within the bladder, P a is an important consideration for evaluating the stress within  of a bladder segment for a given set of parameters. As expected, the greatest deformation occurs at the 155 center of the segment while the deformation at the edges is zero as specified by the boundary conditions.

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Here it is shown that the maximum calculated deformation is on the order of 0.5 mm for this example. where the spacing between strands becomes large. Thus, for this application, the bladder stress can be used 196 to set the lower limit for the strand number. It is desirable to approach this lower limit as fewer strands Finally, the effect of bladder wall thickness were examined. Figure 11 shows that the wall stress increases 199 quickly when the wall thickness is smaller than approximately 0.15 mm and changes more gradually above   Figure 10. Plot of the bladder stress calculated using FEA versus strand number. Braid angle is set to 45 • for all cases. Figure 11. Plot of the bladder stress calculated using FEA versus bladder wall thickness. Braid angle is set to 45 • for all cases.

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The previous text provided a means of designing the AMA by analyzing the constraints in terms of strand 204 stress and bladder stress for a given set of input conditions. Equation 12, when combined with the results 205 presented in Section 3.1, allows for a determination of the optimal combination of strand diameter, D s , and 206 strand number, N , within the braid for which the strand stress does not exceed the tensile strength of the 207 braid material and the stress in the bladder wall does not exceed the limits of the bladder material. The 208 thickness of the bladder wall is also an independent design parameter that can be minimized in order to 209 allow the actuator to reach full elongation and therefore the wall thickness can be included as a design 210 variable.

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The strand diameter and number can then be used in the manufacture of an appropriate braided mesh. For 212 the braiding machines that manufacture this sort of braided mesh, the necessary input parameter is the pick 213 count or the number of times the strands cross the center line per unit length Omeroglu (2006). This input 214 setting can be calculated from the optimized strand number, N , as where the length, L c , is the actuator length at the diameter of the core, D c (independent parameter as long 216 as it is smaller than D o ), that the mesh is being braided onto such that Eqs. 1 and 2 become Here the strand length, b, and number of turns, n, are constants which can be calculated from the input 219 parameters L max and D max and the calculated value for γ min (Eq. 11) as Therefore, for a given combination of wire diameter, D s , and strand number, N , the braiding machine can 222 be configured using the appropriate pick per unit length setting calculated from Eq. 14.

TESTING OF AMA LOAD CAPACITY
An evaluation of the accuracy of the predicted beam load capacity as a function of AMA extension and 224 pressure, as formulated in Section 2, was carried out. It was not possible to produce a muscle actuator at a 225 scale appropriate for this application as an inner bladder material with the correct diameter was not found 226 to be available. Thus a larger version of the muscle actuator, with an 8.8 mm maximum outer diameter, was 227 produced (Fig. 12) using latex surgical tubing (OD 3.2 mm, ID 1.6 mm) as the internal bladder and nylon 228 expandable mesh (OD 4.4 mm, ID 3.2 mm) and the outer sleeve. The actuator from Fig. 12 was connected 229 to a rigid support at one end and to a calibrated spring scale (OHAUS, 4 kg capacity) on the other end 230 (Fig. 13). The internal bladder was inflated using an instrumented syringe (BARD Caliber Inflation Device) 231 which provided a measure of the inflation pressure.   (1996); Davis and Caldwell (2006). Deviations between the experimental 243 and theoretical actuator forces are likely due to frictional effects which become more observable at higher 244 braid angles.
245 Figure 14. Plot of force output from the artificial muscle actuator predicted analytically (solid line) and determined experimentally (dots).

DISCUSSION
The use of artificial muscle actuators for robotic applications has been expanding as they provide several 246 advantages including dexterous mobility and compliance when coming into contact with other surfaces 247 or objects Trivedi et al. (2008b). This is particularly important in the application of minimally invasive 248 surgery where the robot maneuvers in a unpredictable and sensitive environment. Further, the use of a 249 hydraulic artificial muscle actuator for this purpose provides the opportunity for greater force output for the 250 given size constraints, as shown in Fig. 3 where it is seen that the theoretical load capacity of the AMA is 251 twice that of a conventional hydraulic actuator for the same diameter. In Section 2, a method for modeling 252 the AMA and its load capacity was given showing that actuator force is a function of internal pressure and 253 actuator length. The procedure for defining this method includes several simplifying assumptions; however, 254 it was demonstrated that at the prototype scale for this application, the predicted output compared well 255 with the experimental results. As was shown, the maximum contraction of the AMA is set by the braid 256 geometry at a braid angle of 54.7 • . However, the maximum elongation of the AMA is something that can 257 be designed and thus allows for an optimization of the AMA characteristics in order to achieve the greatest 258 stroke length while avoiding failure.

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The methods presented here make it possible to identify the appropriate braid characteristics to achieve 260 maximum AMA performance. Maximum performance is obtained by minimizing the achievable braid 261 angle. The minimum braid angle is dependent on the number of strands within the braided mesh and 262 the diameter of those strands Davis and Caldwell (2006). The limiting conditions placed on these two 263 quantities are the yield stress of the strands and the stress limit of the inner bladder. was not in its fully extended condition when the limit of the spring scale was reached. The theoretical 277 prediction was found to be accurate at small braid angles. If the theoretical calculation for AMA load 278 capacity is extended towards smaller braid angles, then the load capacity of the prototype AMA would 279 approach 80 N as the braid angle approached 10 • . If we then extend this model to the design scale for the 280 example application Berg (2013c), the predicted load capacity of the AMA would be 25.9 N as the braid 281 angle approached 10 • for the same supply pressure.

CONFLICT OF INTEREST STATEMENT
The authors declare that the research was conducted in the absence of any commercial or financial 283 relationships that could be construed as a potential conflict of interest.

AUTHOR CONTRIBUTIONS
285 The Author Contributions section is mandatory for all articles, including articles by sole authors. If an 286 appropriate statement is not provided on submission, a standard one will be inserted during the production 287 process. The Author Contributions statement must describe the contributions of individual authors referred 288 to by their initials and, in doing so, all authors agree to be accountable for the content of the work.