Artificial Muscles: Actuators for Biorobotic...

Klute, Czerniecki, and Hannaford - 1

Artificial Muscles: Actuators for Biorobotic Systems

Glenn K. Klute1,2, Joseph M. Czerniecki1,3, and Blake Hannaford2

1Dept. of Veterans Affairs, Puget Sound Health Care System, Seattle, WA 98108

2Dept. of Electrical Engineering, University of Washington, Seattle, WA 98195

3Dept. of Rehabilitation Medicine, University of Washington, Seattle, WA 98195

Keywords: Robotics, Biomechanics, Muscles, Tendons.

ABSTRACT

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 proposed to improve the actuator’s velocity-dependent

properties. Further, an artificial tendon is proposed 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-length-velocity 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.



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 (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 than 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-the-art 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.

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 1950’s (Gaylord, 1958; Schulte, 1961; Nickel et al., 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 first-order 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, 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 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

mimic 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. In contrast,

muscle-tendon properties and their origin and insertion sites 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 cross-bridge 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). Criticisms 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. Bahler, 1968; Wilkie, 1956; McCrorey et al., 1966; Ralston et

al., 1949). A Kelvin or “standard element” model can accommodate the parallel damping, but rarely is

the complexity of such a model worth the computational effort. Furthermore, experimentally derived

parameters for parallel element models are essentially non-existent. Examination of the existing

animal data reveals the magnitude of the force across the parallel element is negligible when the

muscle is at resting lengths or less. At muscle lengths greater than resting, this force component

rapidly increases.

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 also will ignore the

small contributions of a parallel element in our model.

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 (Bahler, 1967; Wilkie, 1956; McCrorey et al.,

1966). This element is usually neglected in skeletal muscle models when in series with a tendon

whose length is significantly longer than the muscle. In this architecture, the effect of the series

element elongation is negligible (e.g. the 10:1 ratio of the Achilles tendon to the gastrocnemius and

soleus muscles in the human lower limb). 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:



32

2

1,

kLLk

LLk

FF

oooL

L +

+

= (1)

where oLF , is the maximum isometric force generated by the contractile element when it is at its

“resting” length ( oL =1.0) and LF is the isometric force generated by the contractile element at muscle

length L . Values for the empirical constants 1k , 2k , and 3k 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 (figure 2). The human data has the

narrowest “operating” range demonstrated by 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) fit the data most

closely, while the other models (Bobbert and Ingen Schenau, 1990; Hoy et al., 1990) over estimate 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:

[ ][ ] [ ] baFbVaF Lm +=++ (2)



where mF is the muscle force, V is the muscle contraction velocity, and LF is the isometric muscle

force at the instantaneous muscle length from equation (1). The constants a and b are empirically

determined with experiments that report muscle velocity when the muscle is at resting length ( oLV , ).

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; assigned a positive value for velocity by convention).

Obviously, both biological and artificial muscles perform eccentric contractions (i.e. lengthening under

load; assigned a negative value for velocity by convention), but unfortunately, there exists significant

variation in output force for biological muscle under these conditions ( 0/ , ≤oLVV ).

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 oLF ,3.1 ⋅ (e.g. Hatze, 1981; Winters and Stark, 1985); however, values ranging

from as low as oLF ,2.1 ⋅ (Hof and van den Berg, 1981a) to as high as oLF ,8.1 ⋅ (Lehman, 1990) have

been used. Limited experimental evidence from the cat soleus muscle shows oLF ,3.1 ⋅ as the

asymptotic value (Joyce et al., 1969).

Values from the muscle physiology literature for the parameters in equation (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 equations (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:

5.135.4 1 −≤≤− k (3a)

2.280.9 2 ≤≤ k (3b)

0.145.3 3 −≤≤− k (3c)

41.012.0,

≤≤oLF

a (3d)

41.012.0,

≤≤oLV

b (3e)



for muscle lengths of 2.17.0 ≤≤ oLL and velocities of 0.1, ≤oLVV . For eccentric conditions, we

will use an “inverted Hill” model that asymptotes to oLF ,3.1 ⋅ but recognize further experimental work

in this area of muscle physiology is necessary.

2.2. Biological Tendon

There is a wide body of literature describing the behavior of tendon under load. Some investigators

have removed structural characteristics by reporting 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 stress increases slowly with strain at first. 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. Lewis and Shaw, 1997; Ker

et al., 1988).

Ker and others (1988) observed that there are two primary purposes for tendon: one is for transmitting

muscular action into forces for fine manipulation and the other is for 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 (Bennett et al., 1986; Herrick et al., 1978; Riemershma and Schamhardt, 1985), human

(Harris et al., 1966; Lewis and Shaw, 1997), pig (Bennett et al., 1986; Woo et al., 1981), rat (Minns



and Muckle, 1982), sheep (Bennett et al., 1986; Ker, 1987), and wallaby (Bennett et al., 1986; Ker at

al., 1986).

Compilation of this data (Klute et al., 2000b) reveals a monotonically increasing stress (σ ) v. strain

(ε ) relationship given by:

nEεσ = (4)

where the modulus of elasticity ( E ) and strain exponent ( n ) must be empirically determined.

In the biomechanics literature, a number of investigators have published models describing the

relationship between stress and strain or force and elongation in tendons (see figure 4: Bobbert et al.,

1986; Zajac, 1989; Bobbert and van Ingen Schenau, 1990; Voigt et al., 1995a and b; Hof, 1998).

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 complaint 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:

133335773 ≤≤ E (5a)

2≈n (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. Wu et al., 1997;

Yoda and Shita, 1994; Cocatre-Zilgien et al., 1996), 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 elements (Robinson et al., 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 et al., 1963),

flexible pneumatic actuators have seen more recent use in a variety of robotics applications (Inoue,

1987; Klute and Hannaford, 1998; Klute et al., 2000a; van der Smagt et al.,1996; Caldwell et al.,

1993a; b; 1994; 1995; 1997; van der Linde, 1998; Cai and Yamaura, 1996; Paynter, 1996; Noritsugu et

al., 1999; Festo Pneumatics, 2000; Birch et al., 2000). 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, and these models fall into several categories: (a)

empirical models (Chou and Hannaford, 1996; Gavrilovic and Maric, 1969; Medrano-Cerda et al.,



1995), (b) models based on geometry (Chou and Hannaford, 1996; Schulte, 1961; Tondu et al., 1994;

Inoue, 1987; Cai and Yamaura, 1996), and (c) models that include material properties (Chou and

Hannaford, 1996; Schulte, 1961). 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

( oo LLL ⋅≤≤⋅ 00.165.0 where oL refers to the resting length of the actuator) to assess the accuracy of

model predictions (Klute et al., 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. To minimize this effect, all our

actuators were constructed such that their resting length to diameter ratio was at least 14.

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). The force at each pressure (only P=5 bar is shown for clarity) 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 is somewhat similar to biological

muscle; however, there are a few significant differences between the artificial and natural actuators

(figure 6). The biological data has a parabolic force-length relationship whose slope monotonically

decreases with increases in length. In contrast, the output force of the flexible pneumatic actuator is



not parabolic and its slope monotonically increases with length. Furthermore, unlike biological

muscle, the flexible pneumatic actuator cannot be stretched beyond its resting length. In spite of these

differences, the force-length relationship of the flexible pneumatic actuator is close to the bio-mimetic

limits defined by equation (3 a-c).

Unfortunately, the force-velocity properties of the flexible pneumatic actuator are nowhere near the

bio-mimetic limits defined by equation (3 d-e). The results show the actuator has only a small amount

of natural damping as output force decreases only 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).

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 bio-mimetic 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

improving the force-velocity properties. 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 with a

corresponding loss in potential performance.

For a damper in parallel with the flexible pneumatic actuator, the desired damping force ( hydF ) is

simply:

mfpahyd FFF −= (6)

where fpaF is the flexible pneumatic actuator output force whose value can either be taken from

experiment or from a model published elsewhere (Klute and Hannaford, 2000). mF is the force

produced by biological muscle, modeled by equations (1) through (3).

Approaching the problem of calculating the desired orifice size using conservation of energy,

Bernoulli’s equation can be written as:

gP

gP

2v

2v 2

222

11 +=+ρρ

(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.

Furthermore, conservation of mass ( m ) demands:

21 mm = (8)

such that:

2211 vAvA ρρ = (9)

where A is the cross-sectional area. An empirical enhancement to this equation is the multiplication

of a discharge coefficient ( dC ) to the right side of equation (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:

221

41

2hyd

61

214

2v8

vdd CCF ρπφ

ρφπφ+

= (10)

where 2φ is the orifice diameter and 1φ 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:

−= 1

8v

42

2

41

21

21

hydφ

φπρφdC

F (11)

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 et al., 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, knowledge of the peak muscle forces, lengths,

and velocities are required and are application dependent.

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 practical realization of such a system suffers from a penalty in terms of the space to

accommodate numerous springs as well as their associated weight. Our approach was to utilize two

springs to minimize this penalty, albeit with the commensurate loss of performance (Klute et al.,

2000b). A model of this two-spring system is given by:

xKF eq2spring = (12)

where 2springF is the force of the two offset springs and x is the deformation. The equivalent spring

constant ( eqK ) is given by:

≥<

=aba

aaeq

LxKKLxK

K (13)

where aL is the offset distance between springs aK and bK and



ba

baba

KKKKKK

+= (14).

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 (equation 4) to

structural properties, the force versus elongation relationship is given by:

n

ot

ottEST L

LLEAF

−=

,

, (15)

where ESTF is the modeled tendon force, A is the tendon’s cross-sectional area, ot,L is the tendon’s

resting length, and tL the tendon length under load. Resting lengths and areas for a wide variety of

tendons are available in the literature.

The two spring constants ( aK and bK ) and the offset distance between the two springs ( aL ) can be

selected by minimizing the difference in stored energy between the two-spring implementation and the

EST model:

( )

−∫Lt

EST2springmin FF (16).

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 development of an

artificial muscle powered by a flexible pneumatic actuator and a mechanical 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 (Winter, 1990;

Gitter et al., 1991; Spoor et al., 1990). 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 cross-sectional 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

others (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

equations 4, 5, and 15 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 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 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 aK and bK , found by minimizing equation 16, 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 axial-torsional Bionix (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 muscle-tendon



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 muscle-tendon was first tested at an activation pressure

of 2 bar for each velocity specified, and then again at each activation pressure of 3, 4, and 5 bar. To

ensure the actuator pressure remained constant during changes in length, a pneumatic ballast tank

(tank:actuator volume ratio of 4000:1) was attached in close proximity to the flexible pneumatic

actuators.

5.2. Results

To predict the expected performance of the artificial muscle-tendon constructed as identified above,

we generated model results using two flexible pneumatic actuator 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, predicted force outputs are higher at longer muscle lengths than shorter lengths.

Concentric contractions yield lower predicted force outputs with a convex force-velocity profile while

eccentric contractions 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. Also, predicted forces were proportional to



changes in activation pressure, however only the maximum pressure case is shown for clarity (P=5

bar).

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, while the peak force of 2000

N occurred under eccentric conditions at 300 mm/sec. 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 on the order of 25 N.

5.3. Discussion

The force-length-velocity relationship of the artificial muscle-tendon 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, un-like 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 force-

velocity 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 et al., 2000b). 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 also 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 ( oL ) is selected at construction, however the

contraction ratio ( oLL ) 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 muscle- and 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 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.

ACKNOWLEDGEMENTS

The Department of Veterans Affairs, Veterans Health Administration, Rehabilitation Research and

Development Service, project number A0806C, supported the research reported here. Glenn K. Klute,

Ph. D. is a Research Health Scientist at the VA Puget Sound Health Care Systems, Seattle, WA.

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Figure 1: Schematic of the three-element muscle model (Hill, 1938) and tendon: contractile element

(CE), series elastic element (SE), and parallel elastic element (PE).

��

CE SE

TendonPE

Muscle

Hill, 1938

��

CE SE

TendonPE

Muscle

Hill, 1938



Figure 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).

0

0.2

0.4

0.6

0.8

1

0.5 0.7 0.9 1.1 1.3 1.5L / Lo - dimensionless

FL /

FL,o

- di

men

sion

less

Rat

Cat

Frog

Human

Bobbert et al

Woittiez et al.1984

Bobbert & Ingen Schenau1990

Hoy, Zajac, Gordon1990



Figure 3: Output force at maximal activation over a muscle’s eccentric (lengthening under load,

negative velocity by convention) and concentric (shortening under load, positive velocity by

convention) velocity range for rat, cat, frog, and human compared with predictive skeletal muscle

models (see text for citations).

��

��

0

0.25

0.5

0.75

1

1.25

1.5

1.75

-0.5 -0.25 0 0.25 0.5 0.75 1V / VL,o - dimensionless

Fm /

F L,o

- di

men

sion

less

Rat

Cat

Frog

Human

Hof & van den Berg1981

Woittiez et al.1984

Bobbert et al. 1986and

Bobbert & Ingen Schenau 1990

Lehman, 1990

Hatze, 1981



Figure 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).

0

20

40

60

80

100

120

0 2 4 6 8 10 12Strain - percent

Stre

ss -

MPa

Bobbert et al.1986

Bobbert & Ingen Schenau1990 Voigt et al.

1995

EST ModelZajac1989

Hof1998



Figure 5: Flexible pneumatic actuator force (Fa) as a function of normalized length and velocity 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.



Figure 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).

0

0.2

0.4

0.6

0.8

1

0.6 0.7 0.8 0.9 1 1.1 1.2 1.3L / Lo - dimensionless

FL /

FL,o

- di

men

sion

less

Rat

Cat

Frog

Human

Flexible PneumaticActuator



Figure 7: Schematic of three-element artificial muscle-tendon actuator. Flexible pneumatic actuator

serves as contractile element, hydraulic damper yields an approximate force-velocity response, and

mechanical tendon acts as energy storage element.

Flexible Pneumatic Actuator

Hydraulic Damper

Mechanical Tendon

��

Artificial Muscle

Eccentric Orifice

Concentric Orifice


Hydraulic Damper

Mechanical Tendon

��

Artificial Muscle

Eccentric Orifice

Concentric Orifice



Figure 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).


Artificial Tendon

Pneumatic Ballast Tank

Hydraulic Cylinder

MTS Hydraulic Ram

Hydraulic Ballast Chamber

Linear Bearing Race

Artificial Tendon

Linear Bearing Pillow Block

Load Cell




Figure 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.



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



Figure 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).

L=35

0 m

m

L=33

0 m

m

L=31

0 m

m



Table 1: Isometric force v. length data from the literature. 1k 2k 3k oL oLF , Rat1 -4.50 8.95 -3.45 27.0 mm 0.29 N p≤0.001 Frog2 -6.79 14.69 -6.88 31.0 mm 0.67 N r2=0.96 Cat3 -5.71 11.52 -4.81 31.9 mm 0.18 N r2=0.99 Human4 -13.43 28.23 -13.96 215.9 mm 193.1 N r2=0.75 Skeletal Muscle Model5

-6.25 12.50 -5.25 60.0 mm 3000 N

1 Bahler (1968) equation 16. Rat gracilis anticus muscle at 17.5 degrees C. Bahler’s equation is valid over the range oo LLL ⋅≤≤⋅ 2.17.0 . 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 degrees C. For Wilkie’s data, oo LLL ⋅≤≤⋅ 34.168.0 is valid.

3 McCrorey et al. (1966) data from figures 6 (controlled-release data) and 10. Cat tenuissimus muscle at 37 degrees C. For McCrorey’s data, oo LLL ⋅≤≤⋅ 35.173.0 is valid. 4 Ralston et al. (1949) data from figure 5. Human pectoralis major, sternal portion at 37 degrees 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 et al., 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, oo LLL ⋅≤≤⋅ 28.182.0 is valid. 5 Woittiez et al. (1984) equation A-23. Constants for Woittiez model of skeletal muscle were chosen based on frog experiments by Gordon et al., 1966; Hill, 1953; and ter Keurs et al., 1978. Other published skeletal muscle models include Bobbert et al. (1986), Bobbert and Ingen Schenau (1990), and Hoy et al. (1990). The model of Bobbert et al. (1986) uses the same values as Woittiez and co-workers. The models of Bobbert and Ingen Schenau, (1990) and Hoy et al. (1990) both conform to the general parabolic shape, but neither provides a mathematical equation.



Table 2: Force v. velocity data from the literature. oLFa , - none oLF , - N oLVb ,/ - none oLV , - mm/s Rat1 0.356 4.30 0.38 144 Frog2 0.27 0.67 0.28 42 Cat3 0.27 0.18 0.30 191 Human4 0.81 200 0.81 1115 Skeletal Muscle Model5

0.224 0.224

Human6 0.41 3000 0.39 756 Human7 0.41 2430 0.41 780 Human8 0.12 0.12

1Wells (1965) table 1. Rat tibialis anterior muscle at 38 degrees C. 2Abbott and Wilkie (1953) their figure 5. Frog sartorius muscle at 0 degrees C. 3McCrorey et al. (1966) their figure 1. Cat tenuissimus muscle at 37 degrees C. 4Ralston et al., (1949) their figure 1; but see also Wilkie (1950). Human pectoralis major in-vivo, sternal portion at 37 degrees C. 5Woittiez et al. (1984) their equations A-22 and B-8. Model of generic skeletal muscle. Woittiez’s dimensionless model did not require the specification of oLF , or oLV , . 6Bobbert et al. (1986). Model of human triceps surae. 7Bobbert and van Ingen Schenau (1990). Model of human triceps surae. 8Hof and van den Berg (1981). Model of human triceps surae. Hof and ven den Berg’s model also did not require the specification o f oLF , or oLV , .

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