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Artificial Muscles: Actuators for Biorobotic Systems Glenn K Klute, Joseph M Czerniecki and Blake Hannaford The International Journal of Robotics Research 2002; 21; 295 DOI: 10.1177/027836402320556331 The online version of this article can be found at: http://ijr.sagepub.com/cgi/content/abstract/21/4/295

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Artificial Muscles: Actuators for Biorobotic Systems

Glenn K. Klute Dept. of Veterans Affairs Puget Sound Health Care System Seattle, WA 98108, USA Dept. of Electrical Engineering University of Washington Seattle, WA 98195, USA

Joseph M. Czerniecki Dept. of Veterans Affairs Puget Sound Health Care System Seattle, WA 98108, USA Dept. of Rehabilitation Medicine University of Washington Seattle, WA 98195, USA

Blake Hannaford Dept. of Electrical Engineering University of Washington Seattle, WA 98195, USA

Abstract

KEY WORDS—robotics, biomechanics, muscles, tendons

Biorobotic research seeks to develop new robotic technologies modeled after the performance of human and animal neuromuscular systems. The development of one component of a biorobotic system, an artificial muscle and tendon, is reported here. The device is based on known static and dynamic properties of biological muscle and tendon that were extracted from the literature and used to mathematically describe their force, length, and velocity relationships. A flexible pneumatic actuator is proposed as the contractile element of the artificial muscle and experimental results are presented that show the force-length properties of the actuator are muscle-like, but the force-velocity properties are not. The addition of a hydraulic damper is put forward to improve the actuator’s velocity-dependent properties. Further, an artificial tendon is set forth whose function is to serve as connective tissue between the artificial muscle and a skeletal structure. A complete model of the artificial muscle-tendon system is then presented which predicts the expected force-lengthvelocity performance of the artificial system. Experimental results of the constructed device indicate muscle-like performance in general: higher activation pressures yielded higher output forces, faster concentric contractions resulted in lower force outputs, faster eccentric contractions produced higher force outputs, and output forces were higher at longer muscle lengths than shorter lengths. The International Journal of Robotics Research Vol. 21, No. 4, April 2002, pp. 295-309, ©2002 Sage Publications

1. Introduction Over the past two decades, few revolutionary technological developments have occurred in the field of robotics. Expansion into new industries, such as agriculture, construction, environmental management (hazardous waste), human welfare, and associated medical services, and entertainment, will demand significant advances in the robotic state-of-the-art. In particular, these new robots will need to perform well in an uncontrolled, dynamic environment where a premium is placed on contact stability. Clearly, the requirements for a surgical robot interacting with both surgeon and patient or a team of robots picking fruit in an orchard are far different from those for a robot that assembles cell phones or hard drives in a clean room. Expansion into new industries will require robotic systems that extend or augment human performance while sharing the work environment with their human operators. The “biorobotic” approach to advancing the state-of-theart is to design robotic components whose systems emulate the very properties that allow humans and animals to operate successfully in ever-changing environments. Each component of a biorobotic system incorporates many of the known aspects of such diverse areas as: neuromuscular physiology, biomechanics, and cognitive sciences, into the design of sensors, actuators, circuits, processors, and control algorithms. 295

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THE INTERNATIONAL JOURNAL OF ROBOTICS RESEARCH / April 2002

The overall aim of the research presented here is to develop one component of a biorobotic system: a musculotendon-like actuator. An important step in the development is identifying the properties and performance that the artificial system is expected to achieve. The known static and dynamic properties of skeletal muscle and tendon are identified from the extensive physiological literature and mathematical models are used to describe their performance. These models serve as a means for comparison with artificial systems. In the development of an actual artificial muscle, our approach uses a flexible pneumatic actuator as the system’s contractile element. This actuator was patented by R. H. Gaylord and applied to orthotic appliances by J. L. McKibben in the late 1950s (Gaylord 1958; Schulte 1961; Nickel, Perry, and Garrett 1963). The experimental evidence presented demonstrates that the device has force-length properties similar to skeletal muscle. The addition of a passive hydraulic damper in parallel with the flexible pneumatic actuator provides a firstorder approximation of the desired force-velocity properties. As with biological systems, muscles attach to bone through tendons and the artificial system must be no different. One advantage of incorporating elastic tendon elements is to allow for energy storage in locomotor systems. Energy stored early in the stance phase of the gait cycle can be recovered late in the gait cycle, thereby minimizing metabolic energy expenditure during propulsive force generation. To achieve this capacity in our artificial system, we propose the use of a simple but effective artificial tendon based on the properties of the mammalian tendon. Lastly, we present the experimentally measured performance of our artificial muscle-tendon system and compare it with model predictions. The results document the ability of the artificial system to approximate the performance of the biological system in terms of its force, length, and velocity properties.

2. Biological Muscle and Tendon The first step in the development of an artificial muscle-tendon system is to identify the properties to be emulated. For most vertebrates under physiological conditions, the skeletal system is sufficiently light and rigid that its material properties do not affect static or dynamic performance. Muscle-tendon properties and their origin and insertion sites, however, are particularly important and can dramatically affect movement performance. While the origin and insertion locations can be obtained post-humously from cadaveric specimens, the properties of muscle and tendon are more difficult to obtain. As will be shown, muscle and tendon properties vary widely, not only between species but also within a species. The approach used here will be to present models and data of various species from the literature and then use this information as a guideline to specify a range appropriate for an artificial system.

2.1. Skeletal Muscle During the last century, two muscle modeling approaches have become well established: (1) a phenomenological approach originating with the thermodynamic experiments of Fenn and Marsh (1935) and Hill (1938) and (2) a biophysical crossbridge model originating with Huxley (1957). Both models have received numerous contributions from contemporary researchers who continue to refine and evolve these models. Hill-based models are founded upon experiments yielding parameters for visco-elastic series and parallel elements coupled with a contractile element (Figure 1). The results of such models are often used to predict force, length, and velocity relationships describing muscle behavior. Huxley-based models are built upon biochemical, thermodynamic, and mechanical experiments that describe muscle at the molecular level. These models are used to understand the properties of the microscopic contractile elements. Biomechanists, engineers, and motor control scientists using models to describe muscle behavior in applications ranging from isolated whole muscle to bi-articulate, multi-muscle systems have almost exclusively used Hill-based models (Zahalak 1990). Critics of this approach have noted that Hill models may be too simple and fail to capture the essential elements of real muscle. However, the Hill approach has endured because accurate data for model parameters exist in the literature, motor function described by Hill models has often been verified experimentally, and simple models often provide insights hidden by overly complex models (Winters 1995). The modeling task here is to provide a description of whole muscle performance to design an artificial system. As a result, a Hill-based model is an appropriate beginning. Several variations of the Hill model exist, but there are essentially three components: a parallel element, a series element, and a contractile element (Figure 1). Experimental evidence in the muscle physiology literature shows the parallel element to be a lightly damped, non-linear elastic element (e.g., Ralston et al. 1949; Wilkie 1956; McCrorey, Gale, and Alpert 1966; Bahler 1968). The parallel element is often ignored in skeletal muscle models (e.g., Woittiez et al. 1984; Bobbert and van Ingen Schenau 1990) because the force across the element is insignificant until the muscle is stretched beyond its physiological range (∼1.2 times the muscle’s resting length). As will be demonstrated, our choice of an artificial contractile element (a flexible pneumatic actuator) precludes operations at greater than resting lengths due to physical limitations, as such we will also ignore the small contributions of a parallel element in our model. There are flexible pneumatic actuators that can be stretched beyond their resting length; however, their contractile range (minimum to maximum length) is typically unchanged. As a result, the additional capacity beyond resting length comes at a cost of reduced capacity at lengths shorter than resting. For applications that warrant this capacity, inclusion of a parallel element would be appropriate. Such an element would have

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Klute, Czerniecki, and Hannaford / Artificial Muscles

297

Muscle 1

CE

Tendon PE

Modified from Hill, 1938

Fig. 1. Schematic of the three-element muscle model (Hill 1938) and tendon. Skeletal muscle is represented by a contractile element (CE), a series elastic element (SE), and parallel elastic element (PE).

only a small stiffness contribution at lengths less than 1.2 times resting, but would rapidly increase at longer lengths. The series elastic element is the stiffer of the two “springs” in the Hill model. Like the parallel element, it is also generally accepted to be a lightly damped element; thus the damping can be neglected without loss of accuracy. Animal data reveal the total series element elongation under load is only a small percentage of the muscle resting length (e.g., Wilkie 1956; McCrorey, Gale, and Alpert 1966; Bahler 1967). This element is usually neglected in skeletal muscle models when in series with a tendon whose length is significantly longer than the muscle (e.g., the 10:1 ratio of the Achilles tendon to the gastrocnemius and soleus muscles in the human lower limb) as its contribution to energy storage is negligible. As one of our primary applications of this research is a bio-mimetic, below-knee prosthesis for amputees, our model will also neglect the contributions of a series elastic element. 2.1.1. Isometric Properties of Muscle The contractile element is the dominant component of the Hill muscle model. Cook and Stark (1968) showed that this element could be treated as a force generator in parallel with a damping element. Under isometric conditions (zero contractile velocity), the output force can be described as a parabolic function of muscle length (Bahler 1968). Neglecting the series and parallel elements, this relationship is given by  2   FL L L + k3 , = k1 + k2 (1) FL,o Lo Lo where FL,o is the maximum isometric force generated by the contractile element when it is at its “resting” length (Lo = 1.0) and FL is the isometric force generated by the contractile element at muscle length L. Values for the empirical constants k1 , k2 , and k3 can be extracted from the physiology literature for the frog, rat, cat, and human and compared with values from several skeletal muscle models in the biomechanic literature (Table 1). Plots predicting the force as a function of length reveal the expected parabolic form but, importantly, there is considerable variability between results from the various species as well as between the skeletal muscle models

FL / FL,o − dimensionless

SE 0.8

0.6

0.4

0.2

Rat Cat Frog Human Hoy, Zajac, Gordon 1990 Bobbert & Ingen Schenau 1990 Woittiez et al. 1984

0 0.5

1

1.5

L / Lo − dimensionless

Fig. 2. Dimensionless relationship between force and length under isometric conditions at maximal activation for rat, cat, frog, and human compared with predictive skeletal muscle models (see text for citations).

(Figure 2). The human data has the narrowest “operating” range, with large changes in output force over a small range of contractile element lengths. The selected frog and rat muscles generated contractile forces over a larger range of lengths compared to the human, but the cat muscle surpassed all with the widest operating range of contractile element lengths. The model of Woittiez et al. (1984) fits the data most closely, while the other models (Bobbert and van Ingen Schenau 1990; Hoy, Zajac, and Gordon 1990) overestimate the output force at short contractile element lengths. 2.1.2. Non-Isometric Properties of Muscle Under non-isometric conditions, the output force of muscle is a function of both length and velocity for a given level of activation. It is well known that the output force of biological muscle drops significantly as the contraction velocities increase during shortening (Hill 1938). The general form of this relationship is given by (Fm + a)(V + b) = (FL + a)b,

(2)

where Fm is the muscle force, V is the muscle contraction velocity, and FL is the isometric muscle force at the instantaneous muscle length from eq (1). The constants a and b are empirically determined with experiments that report muscle velocity when the muscle is at resting length (VL,o ). Their values depend not only on the species of interest but also on the type of muscle fiber within a species. The majority of research conducted on the contractile element of muscle has involved concentric contractions (i.e., shortening under load and assignment of a positive value for

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298

THE INTERNATIONAL JOURNAL OF ROBOTICS RESEARCH / April 2002 Table 1. Isometric Force v. Length Data from the Literature k1 k2 k3 Lo 1

Rat Frog2 Cat3 Human4 Skeletal Muscle Model5

–4.50 –6.79 –5.71 –13.43

8.95 14.69 11.52 28.23

–3.45 –6.88 –4.81 –13.96

–6.25

12.50

–5.25

FL,o

27.0 mm 31.0 mm 31.9 mm 215.9 mm

0.29 N 0.67 N 0.18 N 193.1 N

60.0 mm

3000 N

p ≤ 0.001 r 2 = 0.96 r 2 = 0.99 r 2 = 0.75





1

(Bahler 1968) eq (16). Rat gracilis anticus muscle at 17.5 C. Bahler’s equation is valid over the range 0.7 Lo ≤ L ≤ 1.2 Lo . Bahler fitted the data using a “least mean square fit (significant to p ≤ 0.001).” 2 (Wilkie 1956) data from Figures 2 and 4. Frog sartorius muscle at 0◦ C. For Wilkie’s data, 0.68 Lo ≤ L ≤ 1.34 Lo is valid. 3 (McCrorey, Gale and Alpert 1966) data from Figures 6 (controlled-release data) and 10. Cat tenuissimus muscle at 37◦ C. For McCrorey’s data, 0.73 Lo ≤ L ≤ 1.35 Lo is valid. 4 (Ralston et al. 1949) data from Figure 5. Human pectoralis major, sternal portion at 37◦ C. Caveats include: (1) results are not that of isolated muscle as the insertion tendon is necessarily included, (2) Ralston did not report muscle length so the results presented here used data from a single cadaver (Wood, Meek and Jacobsen 1989), and (3) performance of residual amputee muscles may not be the same as those of intact individuals. (Wilkie 1950) estimated the length change of Ralston’s data to be three times less than normal and the peak isometric force to be five times less than normal. For Ralston’s data, 0.82 Lo ≤ L ≤ 1.28 Lo is valid. 5 (Woittiez et al. 1984) eq (A23). Constants for Woittiez model of skeletal muscle were chosen based on frog experiments by Hill (1953), Gordon, Huxley and Julian (1966), ter Keurs, Iwazumi and Pollack (1978). Other published skeletal muscle models include Bobbert, Huijing and van Ingen Schenau (1986), Bobbert and van Ingen Schenau (1990), and Hoy, Zajac and Gordon (1990). The model of Bobbert, Huijing and van Ingen Schenau (1986) uses the same values as Woittiez et al. (1984). The models of Bobbert and van Ingen Schenau (1990) and Hoy, Zajac and Gordon (1990) both conform to the general parabolic shape, but neither provides a mathematical equation.

velocity by convention). Obviously, both biological and artificial muscles perform eccentric contractions (i.e., lengthening under load and assignment of a negative value for velocity by convention) but, unfortunately, there exists significant variation in output force for biological muscle under these conditions (V /VL,o ≤ 0). Most skeletal muscle models assume an asymptotic relationship between force and velocity under eccentric conditions. These models modify the “classic” Hill equation (eq (2)) to obtain an “inverted Hill” model to describe lengthening muscle behavior. Asymptotic values for these models are typically assumed to be 1.3 FL,o (e.g., Hatze 1981; Winters and Stark 1985); however, values ranging from as low as 1.2 FL,o (Hof and Van den Berg 1981) to as high as 1.8 FL,o (Lehman 1990) have been used. Limited experimental evidence from the cat soleus muscle shows 1.3 FL,o as the asymptotic value (Joyce, Rack, and Westbury 1969). Values from the muscle physiology literature for the parameters in eq (2) can again be extracted from the literature for the frog, rat, cat, and human along with values from several skeletal muscle models (Table 2). Examination of the parameters clearly indicates there is a wide range of possible values.

Plotting the force as a function of velocity when the muscle is at its resting length reveals a hyperbolic shape where the muscle force monotonically decreases under concentric conditions and increases under eccentric conditions (see Figure 3). Also evident is the significant variation across animals as well as across the published models purporting to portray the performance of human skeletal muscle. 2.1.3. Summary of Muscle Model Properties Due to the wide variance in model parameters that describe muscle performance, specifying exact parameters for the constants in eqs (1) and (2) would certainly be controversial. However, by specifying a range, we can identify performance that would be acceptable. For designing an artificial skeletal muscle for locomotion applications, the following constraints can be used to envelope the desired performance: −4.5 ≤ k1 ≤ −13.5

(3a)

9.0 ≤ k2 ≤ 28.2

(3b)

−3.5 ≤ k3 ≤ −14.0

(3c)

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Klute, Czerniecki, and Hannaford / Artificial Muscles Table 2. Force v. Velocity Data from the Literature a/FL,o FL,o b/VL,o dimensionless N dimensionless Rat1 Frog2 Cat3 Human4 Skeletal Muscle Model5 Human6 Human7 Human8

299

VL,o mm/s

0.356 0.27 0.27 0.81

4.30 0.67 0.18 200

0.38 0.28 0.30 0.81

144 42 191 1115

0.224 0.41 0.41 0.12

— 3000 2430 —

0.224 0.39 0.41 0.12

— 756 780 —

(Wells 1965) Table 1. Rat tibialis anterior muscle at 38◦ C. (Abbott and Wilkie 1953) their Figure 5. Frog sartorius muscle at 0◦ C. 3 (McCrorey, Gale and Alpert 1966) their Figure 1. Cat tenuissimus muscle at 37◦ C. 4 (Ralston et al. 1949) their Figure 1, but see also Wilkie (1950). Human pectoralis major in-vivo, sternal portion at 37◦ C. 5 (Woittiez et al. 1984) their eqs (A22) and (B8). Model of generic skeletal muscle. Woittiez’s dimensionless model did not require the specification of FL,o or VL,o . 6 (Bobbert, Huijing and van Ingen Schenau 1986). Model of human triceps surae. 7 (Bobbert and van Ingen Schenau 1990). Model of human triceps surae. 8 (Hof and Van den Berg 1981). Model of human triceps surae. Hof and ven den Berg’s model also did not require the specification of FL,o or VL,o . 1 2

Rat Cat Frog Human Hof & van den Berg 1981 Bobbert et al. 1986, and Bobbert & Ingen Schenau 1990 Woittiez et al. 1984 Lehman 1990 Hatze 1981

1.6

Fm / FL,o − dimensionless

1.4 1.2 1

0.12 ≤

a ≤ 0.41 FL,o

(3d)

0.12 ≤

b ≤ 0.41 VL,o

(3e)

for muscle lengths of 0.7 ≤ L/Lo ≤ 1.2 and velocities of |V /VL,o | ≤ 1.0. For eccentric conditions, we will use an “inverted Hill” model that asymptotes to 1.3 FL,o but recognize further experimental work in this area of muscle physiology is necessary.

0.8 0.6 0.4

2.2. Biological Tendon

0.2

There is a wide body of literature describing the behavior of tendon under load. Some investigators have reported material properties using stress versus strain, while others have reported the form in which the data was recorded, namely force versus elongation. The stress v. strain curve of tendon exhibits a characteristic “toe” region where initially stress increases slowly with strain. Further strain results in more rapid increases in stress, followed by a region where stress increases linearly with strain until failure. Confounding problems include the freshness of the specimen (some investigators have used embalmed tissue), grip failures, and appropriate strain rates (e.g., Ker, Alexander, and Bennett 1988; Lewis and Shaw 1997).

0 −0.5

0

0.5

1

V / VL,o − dimensionless

Fig. 3. Output force at maximal activation over a muscle’s eccentric (i.e., lengthening under load and is assigned a negative value for velocity by convention) and concentric (i.e., shortening under load and is assigned a positive value for velocity by convention) velocity range for rat, cat, frog, and human compared with predictive skeletal muscle models (see text for citations).

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THE INTERNATIONAL JOURNAL OF ROBOTICS RESEARCH / April 2002

Ker, Alexander, and Bennett (1988) observed that there are two primary purposes for tendon: (1) transmitting muscular action into forces for fine manipulation, and (2) energy storage during locomotion. The focus of our efforts is on energy storing tendons due to our interest in lower limb prosthetic applications. While data on human lower limbs is sparse, data for a variety of energy storing animal tendons is abundant. Experimental data is available in the literature for the camel (Alexander et al. 1982), deer (Bennett et al. 1986), dog (Webster and Werner 1983), donkey (Bennett et al. 1986), horse (Herrick, Kingsbury, and Lou 1978; Riemersma and Schamhardt 1985; Bennett et al. 1986), human (Harris, Walker, and Bass 1966; Lewis and Shaw 1997), pig (Woo et al. 1981; Bennett et al. 1986), rat (Minns and Muckle 1982), sheep (Bennett et al. 1986; Ker et al. 1987), and wallaby (Bennett et al. 1986; Ker, Dimery, and Alexander 1986). Compilation of this data (Klute, Czerniecki, and Hannaford 2000a) reveals a monotonically increasing stress (σ ) v. strain (ε) relationship given by: σ = Eε n ,

(4)

where the modulus of elasticity (E) and strain exponent (n) must be empirically determined. In the biomechanics literature, a number of investigators (Bobbert, Huijing and van Ingen Schenau 1986; Zajac 1989; Bobbert and van Ingen Schenau 1990; Voigt et al. 1995a, b; Hof 1998) have published models describing the relationship between stress and strain or force and elongation in tendons (Figure 4). Nearly all of the experimental evidence is bounded within the extremes described by the models of Bobbert and van Ingen Schenau (1990) and Hof (1998). The data from Hof, on the lower bound, is based on an in vivo human experiment and may indicate that in vivo tendon is more compliant than in vitro or post-mortem tendon. For design applications, we interpret this data as defining a model of tendon whose relationship between stress and strain is described within 5773 ≤ E ≤ 13333 n ≈ 2,

(5a) (5b)

where the units of E are MPa and n is dimensionless.

3. Artificial Muscle and Tendon The stated objective of this research is the development of a muscle- and tendon-like actuator. While some investigators have proposed the use of commercially available actuators (e.g., Yoda and Shiota 1994; Cocatre-Zilgien, Delcomyn and Hart 1996; Wu, Hwang and Chang 1997), others have suggested novel actuators such as shape-memory alloys (Mills 1993), electro-reactive gels (Mitwalli et al. 1997), ionic polymer-metal composites (Mojarrad and Shahinpoor 1997), and stepper motors with ball screw drives connected to elastic

120

100

Bobbert & Ingen Schenau 1990 Zajac 1989 Voigt et al. 1995 EST Model Bobbert et al. 1986 Hof 1998

80

Stress − MPa

300

60

40

20

0 0

2

4

6

8

10

12

Strain − percent

Fig. 4. Published models describing the relationship between stress and strain (see text for citations). Nearly all experimental evidence is bounded within extremes described by models of Bobbert and van Ingen Schenau (1990) and Hof (1998).

elements (Robinson, Pratt and Paluska 1999). Our approach has used a flexible pneumatic actuator made from an inflatable inner bladder sheathed with a double helical weave which contracts lengthwise when expanded radially. 3.1. Artificial Contractile Element First popularized by J. L. McKibben as an orthotic appliance for polio patients (Nickel, Perry and Garrett 1963), flexible pneumatic actuators have seen more recent use in a variety of robotics applications (Inoue 1987; Caldwell, Razak and Goodwin 1993a, b, 1994, 1995; Cai and Yamaura 1996; Paynter 1996; vanderSmagt, Groen and Schulten 1996; Caldwell, Medrano-Cerda and Bowler 1997; Pack, Christopher Jr. and Kawamura 1997; van der Linde 1998; Noritsugu, Tsuji and Ito 1999; Birch et al. 2000; Klute, Czerniecki and Hannaford 2000b; Hannaford, Jaax and Klute 2001; Northrup, Sarkar and Kawamura 2001). This actuator serves as the contractile element of our artificial muscle. There are a number of parameters that significantly affect the performance of the flexible pneumatic actuator. The contraction stroke length, force generated, and air volume consumption are all dependent on the geometry and material of the inner bladder and exterior braided shell. A number of investigators have attempted to model the actuator; these models fall into several categories: (a) empirical models (Gavrilovic and Maric 1969; Medrano-Cerda, Bowler and Caldwell 1995; Chou and Hannaford 1996), (b) models based on geometry (Schulte 1961; Inoue 1987; Tondu, Boitier and Lopez 1994; Cai and Yamaura 1996; Chou and Hannaford

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Klute, Czerniecki, and Hannaford / Artificial Muscles

1000 800

Fa − N

1996), and (c) models that include material properties (Schulte 1961; Chou and Hannaford 1996). Our most recent efforts to improve model predictions of performance have incorporated non-linear material properties; however, further work is necessary to achieve satisfactory results (Klute and Hannaford 2000). We conducted a series of experiments on actuators of different sizes over a range of operating pressures (0 to 5 bar), velocities (–150 mm/sec eccentric to +300 mm/sec concentric), and lengths (0.65 Lo ≤ L ≤ 1.00 Lo , where Lo refers to the resting length of the actuator) to assess the accuracy of model predictions (Klute, Czerniecki and Hannaford 1999; Klute and Hannaford 2000). Because actuator output force is proportional to radial expansion, actuators with short resting lengths perform poorly because radial expansion is constrained at the actuator ends by its construction. Consistent with our application requirements, all the actuators tested here had a resting length to diameter ratio of at least 14 and no end constraint performance issues were observed. Measuring the output force under constant velocity conditions while maintaining constant pressure over the actuator’s working length, we found the output force to be a function of length but not a function of velocity (Figure 5; P = 5 bar only for clarity). The force at each pressure (P = 2 to 5 bar) increases from zero at the actuator’s minimum length to the maximum value obtained when the actuator is at its longest length (equal to the unpressurized “resting” length). Comparing the force-length data to the biological muscle of the frog, rat, cat, and human reveals the actuator performance is quite different from biological muscle (Figure 6). The biological data has a parabolic force-length relationship whose peak output is at the resting length. In contrast, the output force of the flexible pneumatic actuator is nearly linear (slight slope increase with length). Furthermore, unlike biological muscle, the flexible pneumatic actuator used here cannot be stretched beyond its resting length. For alternative actuator designs that can be stretched beyond resting length, the biological model would likely necessitate the inclusion of a parallel element. Under these circumstances, the artificial actuator would be expected to mimic a similar parabolic shape, except at ∼1.2 times resting length or greater the output force would be expected to rapidly increase with increasing length. However, over the restricted operating range of the artificial actuator tested here (lengths equal to or shorter than the resting length), the performance is close to the bio-mimetic limits defined by eqs (3a)-(3c). Unfortunately, the force-velocity properties of the flexible pneumatic actuator are nowhere near the bio-mimetic limits defined by eqs (3d)-(3e). The results show the actuator has only a small amount of natural damping as output force decreases slightly with increases in velocity. At maximum contraction velocity, the output force decreased by only 6 percent while biological muscle decreases to zero output force (Figures 3 and 5).

301

600 400 200 0 −200 1

0

0.9 0.8

200 0.7 400

0.6

V − mm/sec

L / Lo − dimensionless

Fig. 5. Flexible pneumatic actuator force (Fa) as a function of velocity and normalized length at an activation pressure of 5 bar. Actuator presented here was constructed with natural latex rubber bladder (interior diameter of 12.8 mm, wall thickness of 1.6 mm) and polyester exterior braid (minimum interior diameter 12.8 mm when stretched taught). Lower activation pressures exhibited proportionally lower forces but are not shown for clarity. The ripples along the velocity profiles are artifacts of hydraulic pump used by the testing instrument.

In conclusion, these tests on the flexible pneumatic actuator demonstrate that, under isometric conditions, the actuator is a reasonable approximation to biological muscle. However, under dynamic conditions, the performance must be improved. 3.2. Damping Element To create an artificial muscle whose properties are biomimetic under both isometric and dynamic conditions, a hydraulic damper with a passive, fixed orifice was added in parallel to the flexible pneumatic actuator (Figure 7). This design preserves the muscle-like force-length characteristics while attempting to mimic the desired force-velocity properties of biological muscle. An actively controlled orifice with a fast response might be able to perfectly mimic biological muscle; however, such an orifice would add a significant level of complexity to the system. Adopting a fixed orifice approach results in a simpler system, albeit one with a corresponding loss in potential performance. For a damper in parallel with the flexible pneumatic actuator, the desired damping force (Fhyd ) is simply Fhyd = Ffpa − Fm ,

(6)

where Ffpa is the flexible pneumatic actuator output force whose value can either be taken from experiment or from

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302

THE INTERNATIONAL JOURNAL OF ROBOTICS RESEARCH / April 2002 Furthermore, conservation of mass (m) demands

FL / FL,o − dimensionless

1

0.6

0.4 Rat Cat Frog Human Flexible Pneumatic Actuator

0.2

0.7

0.8

0.9

1

1.1

1.2

1.3

L / Lo − dimensionless

Fig. 6. Dimensionless relationship between force and length under isometric conditions at maximal activation for rat, cat, frog, and human compared with a flexible pneumatic actuator at 5 bar (see text for citations).

ρA1 v1 = ρA2 v2 ,

(9)

Flexible Pneumatic Actuator

Mechanical Tendon

Hydraulic Damper

Concentric Orifice

Fig. 7. Schematic of three-element artificial muscle-tendon actuator. Flexible pneumatic actuator serves as contractile element, hydraulic damper yields an approximate forcevelocity response, and mechanical tendon acts as energy storage element.

a model published elsewhere (Klute and Hannaford 2000). Fm is the force produced by biological muscle, modeled by eqs (1) through (3). Approaching the problem of calculating the desired orifice size using conservation of energy, Bernoulli’s equation can be written as P1 v2 P2 v2 + 1 = + 2, ρ 2g ρ 2g

where A is the cross-sectional area. An empirical enhancement to this equation is the multiplication of a discharge coefficient (Cd ) to the right side of eq (9). This coefficient accounts for vena contracture at the orifice, an effective reduction in cross-sectional area at high Reynold’s numbers. The value of this coefficient is 1.0 at low Reynold’s numbers, but can be as low as 0.61 for very high Reynold’s numbers. We have set this parameter to 1.0 for our simulations but plan future experimental measurements. Substitution and simplification yields an equation for determining the ideal orifice size, φ24 =

π v12 φ16 ρ , 8Fhyd Cd2 + ρπ φ14 v12 Cd2

(10)

where φ2 is the orifice diameter and φ1 is the cylinder bore diameter. Equation (10) gives the instantaneous orifice diameter required to obtain the precise amount of damping desired. However, as previously noted, such an adjustable orifice would add a significant level of complexity to the system. The damping force generated by a fixed orifice damper can be calculated from  4  φ1 πρφ12 v12 Fhyd = −1 . (11) 8 Cd2 φ24

Artificial Muscle

Eccentric Orifice

(8)

such that

0.8

0 0.6

˙ 2, m ˙1 = m

(7)

where P is pressure, v is fluid velocity, ρ is the fluid density, and g is gravity. The subscripts 1 and 2 refer to cylinder and orifice locations.

Since the hydraulic damping force is a function of the square of the velocity, the shape of the force v. velocity curve will be convex instead of the desired concavity exhibited by biological muscle. However, if the fixed orifice size is selected based on the muscle length as well as velocity, the effect of this design can be minimized. Other caveats for this simple hydraulic damping model include omission of piston rod lipseal friction, frictional losses within the hydraulic lines, and the effects of vena contracture at the orifice. Furthermore, many of these losses are a function of the Reynold’s number. In this system, the Reynold’s number varies widely depending on location and actuator contraction velocity. In order to optimize the damper design, the orifice diameter is selected by minimizing the difference between a flexible pneumatic actuator in parallel with perfect, Hill-like damping and a flexible pneumatic actuator with fixed orifice hydraulic damping (Klute, Czerniecki and Hannaford 1999). This design approach results in an actuator system that mimics biological muscle to the greatest degree possible while maintaining simplicity. To complete the design process, application dependent knowledge of the peak muscle forces, lengths, and velocities are required.

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Klute, Czerniecki, and Hannaford / Artificial Muscles 3.3. Artificial Tendons As with the biological system, the artificial system must also have a tendon in series with muscle. One approach to tendon design uses a number of offset linear springs arranged in parallel to achieve the desired parabolic force-length relationship (Frisen et al. 1969). With more springs comes better performance, but in practice such a system would be difficult to implement due to the necessary space and associated weight. Our compromise was to utilize two springs, mindful of the commensurate loss of performance (Klute, Czerniecki and Hannaford 2000a). A model of this two-spring system is given by F2spring = Ka x

x < La

(12)

F2spring = Ka x + Kb (x − La )

x ≥ La

(13)

where F2spring is the force of the two offset springs, Ka and Kb are the spring constants of the two springs, La is the offset distance between them, and x is the deformation. Based on the existing animal data (see Section 2.2), we defined a generic, energy storing tendon (EST) using a least squares fit, such that the resulting modulus of elasticity and strain exponent are given by E = 10289 MPa and n = 1.91. It is appropriate to be cautious with this analysis because different methods were used to collect the data and some data points represent many specimens, while others represent only a single specimen. Nevertheless, the EST model provides a simple design requirement to select the spring constants of the two-spring system. To convert from material (eq (4)) to structural properties, the force versus elongation relationship is given by n  Lt − Lt,o , (14) FEST = EA Lt,o where FEST is the modeled tendon force, A is the tendon’s cross-sectional area, Lt,o is the tendon’s resting length, and Lt the tendon length under load. Resting lengths and areas for a wide variety of tendons are available in the literature. The two spring constants (Ka and Kb ) and the offset distance between the two springs (La ) can be selected by minimizing the difference in stored energy between the two-spring implementation and the EST model  min (F2spring − FEST ). (15) Lt

4. Artificial Triceps Surae and Achilles Tendon Design For the purposes of describing their performance, the properties of skeletal muscle and biological tendon have been presented in a dimensionless format. Likewise, the theoretical

303

development of an artificial muscle and tendon intended to mimic their biological counterparts have also been formulated without regard to a specific muscle or tendon. The next step, construction of a physical muscle-tendon actuator, necessitates specification. To this end, we are currently developing a bio-mimetic limb for transtibial human amputees for which the properties of the triceps surae muscles and Achilles tendon have been extracted from the literature. 4.1. Specifications Simply to achieve walking, the ankle musculature must provide a plantar-flexion torque of approximately 110 N-m, a range of motion of 30 degrees, a velocity of 225 degrees per second, and a sustained energy output of 36 Joules per step across an ankle moment arm of 47 mm (Spoor et al. 1990; Winter 1990; Gitter, Czerniecki, and DeGroot 1991). Using this data, the peak force produced by the triceps surae can be calculated to be approximately 2340 N, however, this peak occurs during a concentric contraction (muscle shortening under load). Accounting for tendon lengthening using the EST model, the muscle velocity at peak force can be estimated at 35 mm/sec; indicating that under isometric conditions, this same muscle would need to produce slightly more force (∼2400 to 2500 N). Cross-sectional area and length data for the Achilles tendon can also be extracted from the literature. The mean crosssectional area from published reports is 65 mm2 [49 mm2 (Yamaguchi et al. 1990), 54 mm2 (Lewis and Shaw 1997), 55 mm2 (Woittiez et al. 1984), 61 mm2 (Abrahams 1967), 65 mm2 (Voigt et al. 1995a), 81 mm2 (Komi 1990), 89 mm2 (Ker et al. 1987)]. The length of the Achilles tendon is more difficult to ascertain due to the area across which it attaches to bone and muscle. Yamaguchi and co-authors (Yamaguchi et al. 1990) cite two references for tendon lengths of the gastrocnemius medialis, gastrocnemius lateralis, and soleus muscles that form the triceps surae. The average length of these tendons, collectively taken as the Achilles tendon, is 363.5 mm. This data, when used in conjunction with eqs (4), (5), and (14), provide the design specification for the artificial tendon. 4.2. Design To generate peak forces desired during walking, our design uses two flexible pneumatic actuators in parallel in lieu of a single, larger actuator. This design provides limited functionality in the event of an actuator failure and balances the force of a centrally located, fixed orifice hydraulic damper in parallel with the actuators. A pair of linear bearings is incorporated in the design to prevent binding between the hydraulic cylinder and its piston. The linear bearings also introduce a limitation in the maximum force produced by the system. In the event of an actuator failure, the off-axis load capacity of the linear bearings is 900 N, thus the maximum force of the two

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actuators in parallel is limited to 1800 N. This is less than the desired 2500 N identified above and will result in the generation of lower than desired propulsive forces. This limitation will be corrected in the next generation of artificial muscle and tendon. To accommodate differences in hydraulic volumes between the rod and bore sides of the hydraulic damper cylinder piston, a small hydraulic ballast chamber was incorporated as the hydraulic fluid is nearly incompressible. The flexible pneumatic actuators were constructed with a natural latex rubber bladder with an interior diameter of 12.8 mm and a wall thickness of 1.6 mm. The polyester exterior braid had a minimum diameter of 12.8 mm and the completed actuator assembly had a resting length of 250 mm. The hydraulic damper had a cylinder bore of 22.2 mm, the cylinder rod diameter was 6.4 mm, and the stroke length was 76 mm. The eccentric orifice diameter (on the bore side of the cylinder) was 1.5 mm and the concentric orifice diameter (on the rod side of the cylinder) was 1.2 mm. The density of the hydraulic fluid was 900 kg/m3 . The actuators and damper are then placed together in series with a mechanical, two-spring tendon (Figure 8). The spring constants Ka and Kb , found by minimizing eq (15), are 65 N/mm and 115 N/mm, respectively, with an offset of 8.9 mm.

5. Artificial Muscle-Tendon Experiments A series of experiments were conducted using an axialtorsional BionixTM (MTS Systems, Minnesota, USA) testing instrument to measure the force-length-velocity properties (see Figure 8). The experiments measured the force output over a range of constant activation pressures (2 to 5 bar), velocities (–300 mm/sec eccentric to +300 mm/sec concentric), and actuator-tendon lengths (290 ≤ L ≤ 370 mm). 5.1. Methods The MTS testing instrument was used to apply a series of constant velocity profiles during concentric and eccentric contractions by the artificial muscle-tendon system. The tested velocities included: 1, 10, 25, 50, 100, 150, 200, 250, and 300 mm/sec. For each specified velocity, the artificial muscletendon first contracted concentrically over its full displacement range, and then it contracted eccentrically to complete the cycle. Due to acceleration and deceleration at the ends of the contraction range, the MTS testing instrument was able to constrain the velocity to the specified value for only the middle 90 percent of the contraction range. The artificial muscletendon was first tested at an activation pressure of 2 bar for each velocity specified, and then again at activation pressures of 3, 4, and 5 bar. To ensure the actuator pressure remained constant during changes in length, an over-sized pneumatic ballast tank (tank:actuator volume ratio of 4000:1) was attached in close proximity to the flexible pneumatic actuators. While the size of the ballast tank may not be of concern for

fixed installation applications, portable systems will demand minimization of this and other system components. 5.2. Results To predict the expected performance of the artificial muscletendon constructed as identified above, we generated model results using two flexible pneumatic actuators as the contractile element in parallel with a fixed orifice hydraulic damper, and then in series with a mechanical, two spring tendon. The model allowed prediction of the force-length-velocity relationship prior to construction (Figure 9). In general, the model results indicate a first-order approximation of skeletal muscle and biological tendon. Isometrically (zero contractile velocity), predicted force outputs are higher at longer muscle lengths than shorter lengths. Concentric contractions (i.e., shortening under load and is assigned a positive value for velocity by convention) yield lower predicted force outputs with a convex force-velocity profile while eccentric contractions (i.e., lengthening under load and is assigned a negative value for velocity by convention) yield higher predicted force outputs with a concave force-velocity profile. For comparison, skeletal muscle exhibits a concave force-velocity profile during concentric contractions and a convex force-velocity profile during eccentric contractions. The peak isometric force predicted is approximately 1600 N at an activation pressure of 5 bar and this force is predicted to be proportional to changes in activation pressure (Figure 9 shows only P = 5 bar for clarity). The peak eccentric force is predicted to be approximately 2100 N at maximum actuator length and velocity. The model also reveals a region of zero force output at short muscle lengths and high concentric velocities. Under these conditions, the flexible pneumatic actuator (contractile element) is not capable of generating sufficient power to perform internal work on the parallel damping element while also performing work on the environment. The experimental results reveal the force-length-velocity relationship of the artificial muscle-tendon actuator (Figures 10 and 11). In general, faster concentric contractions resulted in lower force outputs, faster eccentric contractions produced higher force outputs, and output forces were higher at longer muscle lengths than shorter lengths. As expected from model predictions, the concentric force-velocity profile was convex while the eccentric force-velocity profile was concave. At the maximum activation pressure (P = 5 bar), the maximum isometric force was 1600 N as predicted. The peak eccentric force observed (∼2000 N) at maximum actuator length and velocity was slightly less than the predicted peak (2100 N). Lower pressures exhibited proportionally lower forces but are not shown for clarity. Lipseal friction of the hydraulic cylinder is evident during changes in contraction direction and is of the order of 25 N. The region of zero force output at short muscle lengths and high velocities is observable, but smaller than predicted.

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Klute, Czerniecki, and Hannaford / Artificial Muscles

305

MTS Hydraulic Ram Linear Bearing Race

Hydraulic Ballast Chamber

Artificial Tendon

Flexible Pneumatic Actuator

Hydraulic Cylinder

Linear Bearing Pillow Block

Load Cell Pneumatic Ballast Tank

Fig. 8. Experimental set-up for measuring force-length-velocity properties of artificial muscle-tendon actuator at various pressures. The pneumatic ballast tank ensures pressure within actuator is independent of length (i.e., volume).

2000 2000 1500

Force − N

Force − N

1500 1000

500

1000

500 0 −400

0 −400

−200

360 0

340 320

200

300 400

Velocity − mm/sec

280

−200

360 0

340 320

200

Length − mm

Fig. 9. Theoretical predictions of force-length-velocity relationship for artificial triceps surae in series with mechanical Achilles tendon for an activation pressure of 5 bar.

300 400

Velocity − mm/sec

280

Length − mm

Fig. 10. Experimental force-length-velocity results for artificial muscle-tendon actuator at 5 bar.

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2000 1800 L=350 mm

1600

Force − N

1400 1200

L=330 mm

1000 800 600

L=310 mm

400 200 0 −400

−300

−200

−100 0 100 Velocity − mm/sec

200

300

400

Fig. 11. Force versus velocity experimental results (solid lines) plotted with model predictions (dotted lines) at 5 bar for three muscle-tendon lengths (310, 330, and 350 mm).

5.3. Discussion The force-length-velocity relationship of the artificial muscletendon actuator is qualitatively similar to the triceps surae and Achilles tendon model prediction and is a first-order approximation of skeletal muscle and biological tendon. The flexible pneumatic actuator serves as the contractile element whose force-length relationship is reasonably close to skeletal muscle. Like skeletal muscle, the force outputs are higher at longer muscle lengths than shorter lengths, and the force is proportional to activation pressure. However, unlike skeletal muscle, the flexible pneumatic actuator cannot be stretched beyond its resting length, and the force v. velocity curve has a concave shape rather than convex. These remain as significant differences when compared to the biological actuator. To approximate the desired force-velocity characteristics, a hydraulic damper with fixed, flow restricting orifices was incorporated into the system. However, since the hydraulic damping force is a function of the square of the velocity while skeletal muscle is hyperbolic, the shape of the forcevelocity curve is not biological. For concentric contractions, the force-velocity curves of the artificial system are convex instead of the desired concave, while for eccentric contractions the curves are concave rather than the desired convex profile. Refinements to the proposed system, such as stepper motor controlled needle valves or passive orifices constructed from materials that deform under pressure, can certainly be introduced. Another significant difference between the artificial and biological system is the lipseal friction inherent in the hydraulic damper. The effect of this friction is particularly evident at changes in contraction direction (concentric and eccentric). The current design uses a single acting cylinder

which minimizes lipseal friction as it uses only one rod seal; however, the differences in volume between the rod and bore sides of the cylinder necessitates the use of a small hydraulic ballast chamber. A double acting cylinder would eliminate the need for the ballast chamber, but it uses two rod seals and would increase the observed lipseal friction. The performance of energy storing tendons is emulated with two offset linear springs arranged in parallel. Previous tests demonstrate the tendon design meets performance requirements while minimizing the weight and volume penalties (Klute, Czerniecki and Hannaford 2000a). The performance of this design could be improved, but only at the cost of additional springs. Another approach, using lightweight polymer springs with non-linear elastic properties, might better mimic the biological stress-strain curve while reducing weight. The design provides a first-order approximation of biological muscle and tendon and is suitable for a number of applications. The design can be modified to meet the application requirements just as biological muscle and tendon come in different sizes. The force output can be scaled by using either a flexible pneumatic actuator with a larger diameter or by placing several smaller diameter actuators in parallel. The length of the actuator (Lo ) is selected at construction; however, the contraction ratio (L/Lo ) remains fixed at approximately 0.7 and is independent of actuator diameter. The components of the system are inexpensive and widely available.

6. Conclusions The design of the artificial musclo-tendon actuator proposed in this paper is based on known static and dynamic properties of vertebrate skeletal muscle and tendon that were extracted from the literature. These properties were used to mathematically describe the unique force, length, and velocity relationships and specify the performance requirements for an artificial muscle-tendon actuator. To meet these requirements, we constructed an artificial muscle, consisting of two flexible pneumatic actuators in parallel with a hydraulic damper, and placed it in series with a bi-linear, two-spring implementation of an artificial tendon intended to mimic the triceps surae and Achilles tendon. To verify the system’s performance, we experimentally measured the output force for various velocity and activation profiles enveloped by the maximum conditions expected during human locomotion. The experimental results show the actuator-damper-tendon system behaves in a muscleand tendon-like manner. The use of physical as opposed to virtual models imposes demanding constraints on the study of organism-scale systems. The number of assumptions that can be included in a musculo-skeletal or neuro-mechanical model is dramatically reduced when embedded controllers, sensors, and actuators are involved. This unique artificial muscle-tendon system can serve as the actuator in numerous applications yet to be explored. Prosthetic limbs, robotic systems that share the work

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Klute, Czerniecki, and Hannaford / Artificial Muscles space with humans, entertainment robots, service robots for the elderly, tele-operated devices, and systems that extend or augment human capabilities are but a few of the possibilities.

Acknowledgments The Department of Veterans Affairs, Veterans Health Administration, Rehabilitation Research and Development Service, project number A0806C and merit review A2448R, supported this research.

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