OPtimum Take-Off Techniques and Muscle Design for Long Jump

Fast saltatory movements such as human running or jumping are characterised by alternating flight and contact phases. The general dynamics of the body during ground contact, represented by the centre of mass trajectory, is expressed by the pattern of ground reaction forces and determines the subsequent flight phase.

Several experimental investigations of the long jump can be found in the literature (e.g. Luhtanen and Komi, 1979; Lees et al., 1994; Stefanyshyn and Nigg, 1998). The limiting factors for attaining greater jumping distances are the ability to increase running speed (Hay, 1993) and the ability to build up large muscle forces (Alexander, 1990). A causal understanding of the contribution of different variables to jumping distance requires a mechanical model of the dynamics during the final support phase before take-off. Obviously, the long jump is a combination of a spring-like elastic operation of the leg and a hammer-like landing shock (Witters et al., 1992). An inverse dynamics approach to the running long jump (Stefanyshyn and Nigg, 1998) used only low run-up speeds, without special attention to impact dynamics. A variety of forward dynamic mechanical models have been developed for human jumps: a mass supported by a simple spring (Alexander et al., 1986; Blickhan, 1989), a two-segment model with a muscle-like torque generator (Alexander, 1990), a four-segment model with a number of muscle-like actuators (e.g. Pandy et al., 1990; Bobbert et al., 1996) and even more detailed models with many more segments and actuators (Hatze, 1981).

Our goal was to test a series of models whose components are important for describing the dynamics and optimal techniques for the long jump. In the first step, a close fit between measured ground reaction forces and the force predicted by a model was achieved (Seyfarth et al., 1999). This model, an extension of the spring-mass model of Blickhan (1989), was used to describe the spring-like operation of the supporting leg. It predicted an optimal angle of attack of approximately 45–55 ° for a leg stiffness of approximately 20 kN m⁻¹ and an approach speed of 10 m s⁻¹. The optimal angle decreased for a higher approach speed and increased for a higher leg stiffness, but was not much influenced by the representation of the distal mass. This model, however, did not take into account muscle properties which, to an unknown extent, contribute to the time course of the force and may shift optimum strategies.

To investigate the role of the muscles in the long jump, the simple model of Alexander (1990) was taken as a starting point. Using a Hill-like torque generator and massless legs, his model predicted the maximum possible approach speed for the long jump to result in optimal performance. The predicted optimal angle of attack for maximum jumping performance was approximately 70–75 °. To describe the first peak in the ground reaction force, the knee extensor muscle was characterised by prestretch without eccentric force enhancement and an extremely high isometric force (approximately 25 kN). This resulted in an unrealistic sudden rise in the ground reaction force. In contrast to experimental findings, the passive and active peak were not separated by a local minimum.

In the present study, we focus on four questions. (i) To what extent is the spring-like operation of the leg due to an interaction between leg geometry and muscle–tendon properties? (ii) To what extent can a highly activated knee extensor contribute to the first force peak? (iii) What is the role of elastic components and eccentric force enhancement in jumping performance? (iv) How does jumping performance depend on muscle architecture?

To investigate these issues, a two-segment model was constructed with a Hill-type knee extensor. The extensor muscle was characterised by eccentric force enhancement and included nonlinear serial and parallel elastic components. Distal masses can easily mask the contribution of the musculature under eccentric loading and were therefore neglected in our model. Moreover, only a minor influence of a distal mass representation on the jumping technique had been found in our earlier long jump model (Seyfarth et al., 1999).

Ground reaction forces and kinematic data were recorded in an experimental analysis of 30 long jumps by 18 sports students (Seyfarth et al., 1999). The jumping distance was measured as the shortest distance between the take-off position of the foot tip and the landing point.

The upper and lower leg were considered as a chain of two rigid segments, each of length l and zero mass. Ground contact was assumed to occur at the distal end of the lower segment. The mass of the jumper was represented as a point mass m at the hip. The distance between the point of ground contact and the hip was the leg length l_LEG. The leg angle α was defined as the orientation of the leg (l_LEG) relative to the horizontal plane (Fig. 1).

To describe the operation of the stance leg, at least one knee extensor muscle must be introduced (Fig. 1). The model of the muscle–tendon complex (MTC) consisted of three elements: (1) the contractile element (CE), described by a Hill-like force–velocity relationship with an eccentric force enhancement (see equations A1 and A2 in Appendix 1); (2) a parallel elastic component (PE); and (3) a serial elastic component (SE). Both elastic elements were characterised by a nonlinear stress–strain relationship (equation A4, Appendix 2). Muscle belly force F_CE+F_PE was equal to the serial tendon force F_SE.

During the contact phase, the knee extensor muscle was assumed to be fully activated and had a constant moment arm r at the knee joint. The resulting torque was in equilibrium with the moment generated by the ground reaction force, F_G, with the corresponding variable moment arm d (Fig. 1, equation A5). As distal masses and the foot were neglected, the orientation of the force generated by the leg, i.e. the ground reaction force, was always identical to the leg angle α.

The position of the point mass m at touch-down was described by the angle of attack α₀ and the knee angle φ₀ (α₀≈60–65 °, φ₀≈165–170 °; Lees et al., 1994). For elite long jumpers, the initial velocity v₀ is approximately 10–11 m s⁻¹ and has only a small vertical component (less than 1 m s⁻¹; Hay et al., 1986). For an average male athlete, we assumed a body mass of 80 kg (Luhtanen and Komi, 1979).

The height of the centre of mass is approximately 1 m for an upright standing male human 1.8 m tall (Alexander, 1990). We assumed a slightly longer leg (segment length l=0.6 m) because (1) the centre of rotation during ground contact shifts from the heel to the ball of the foot and (2) the leg (including the foot) is almost stretched at touch-down and take-off.

The muscle–tendon complex was represented by the contractile element and two elastic components, a serial elastic component and a parallel elastic component (Fig. 1). Its mass m_MTC was not included in the calculations of the dynamics. The reference length of the muscle–tendon complex (l_MTC) was approximately the length of the real leg segment (0.5 m). The muscle design was described by (i) the ratio R_-ℓ of the fibre length l_f to the total muscle–tendon complex length l_MTC, R_-ℓ=l_f/l_MTC, and (ii) the ratio R_A of the physiological cross-sectional area of the muscle fibres A_f to the serial tendon cross-sectional area A_t, R_A=A_f/A_t.

Assuming an average muscle–tendon complex density ρ_MTC of approximately 1100 kg m⁻³ (Spector et al., 1980; Winters and Woo, 1990), the geometry of the fibres and tendons (cross-sectional areas and lengths) could be expressed in terms of m_MTC, R_A and R_-ℓ (equations A8 and A9, Appendix 4). The cross-sectional area of the parallel element A_PE accounted for a small part (1 %) of the muscle fibre area A_f, the remaining area belonging to the fibres themselves: A_CE=0.99A_f.

The maximum isometric force of the muscle fibres F_Max was defined assuming a muscle fibre maximum isometric stress σ_Max of 300 kPa (Alexander and Vernon, 1975; Close, 1972; equation A3). The maximum shortening velocity v_Max is the product of the fibre length l_f and the maximum fibre strain rate σ?_Max and was assumed to be 14 l_f s⁻¹ (Spector et al., 1980).

In reality, the moment arm d of the ground reaction force is much shorter than described in a two-segment model (Fig. 1). After the first 40 ms of ground contact, the centre of pressure shifts from the ankle joint to the ball of the foot. Therefore, to generate realistic values of the ground reaction force, an unrealistically high maximum isometric muscle force F_Max of approximately 13 kN had to be assumed (see also Alexander, 1990).

The trajectory of the point mass, given by the leg force and the gravitational force, was obtained by integrating the equations of motion (equations A6 and A7) numerically with a constant time step (0.01 ms). The calculation was terminated at the instant of take-off, i.e. when the ground reaction force became zero. The resulting jumping distance was calculated assuming a frictionless parabolic flight trajectory and landing at the intersection of the centre of mass trajectory with the ground (equation A10).

The optimal angle of attack α_opt leading to a maximum jumping distance was investigated with respect to (i) the running speed at touch-down v₀, (ii) the total muscle–tendon complex mass m_MTC, (iii) the ratio of muscle fibre to serial tendon cross-sectional areas R_A, and (iv) the relative fibre length R_-ℓ. During the take-off phase, maximum activation of the muscle was assumed because this resulted in the longest jumps and agrees with experimental observations (Kyröläinen et al., 1987).

Fig. 2A shows an example of predicted force patterns (modelled ground reaction force F_G,Mod and muscle–tendon complex force F_MTC,Mod) with a fully activated knee extensor muscle compared with a measured ground reaction force F_G,Exp for a 6.9 m jump at an approach speed of 9 m s⁻¹ from Seyfarth et al. (1999). This curve can be taken as an example for the other tracings. The simulated jump resulted in a jumping distance of 6.3 m, with the angle of the velocity vector at take-off being approximately 14 ° (measured angle 21 °).

No preactivation was assumed before the moment of touch-down. The general pattern of the predicted dynamics agreed with the experimental result. In both cases, a sharp rise in ground reaction force (first peak) was followed by a prolonged force at a lower level (second peak), but the two peaks were not separated by a local minimum in the simulation. In Fig. 2A, the measured force curve starts 10 ms earlier than the simulated curve because there was no segment representing the foot in the model that would be decelerated first (Seyfarth et al., 1999). The predicted muscle force had achieved its maximum value by 15 ms after touch-down, when the muscle fibres were at their maximum stretching velocity. The serial tendon had a positive strain during the whole ground contact period (SE in Fig. 2B). It was stretched rapidly immediately after touch-down, shortened very slightly over the next approximately 30 ms (plateau phase) and shortened continuously over the second half of ground contact. The muscle fibres shortened during the first few milliseconds and during the last 35 ms before take-off (CE in Fig. 2B), when the muscle force was less than the maximum isometric force F_Max (neglecting the force of the parallel elastic element). During most of the ground contact period (approximately 10–80 ms), the muscle fibres were lengthening and therefore absorbing energy.

Because of the large ground reaction force (up to approximately 5 kN) during the first 20 ms of ground contact, the leg showed a negative work loop (Fig. 3B). After maximum knee flexion, the path of the force–length curve was almost linear (Fig. 3B), in contrast to the nonlinear operation of the extensor muscle (MTC, Fig. 3A). The total energy lost (approximately 280 J, i.e. 44 % of the total muscle work or 9 % of the initial kinetic energy) was absorbed by the muscle fibres (Fig. 4).

The shape of the contractile element work loop was almost rectangular (Fig. 3A). Only 8 % of the energy transferred to the contractile element was returned to the point mass during the concentric phase (Fig. 4). In contrast, in the serial elastic component, approximately 80 % of the stored energy was reusable as mechanical energy. The remainder was absorbed by the contractile element.

In Fig. 5, the influence of the angle of attack α₀ and the running speed v₀ on the jumping distance, the percentage of mechanical energy of the point mass at take-off relative to that at the instant of touch-down, and the angle of the velocity vector at take-off is shown. For high approach speeds (greater than 6m s⁻¹), the optimal angle of attack (approximately 65–70 °) was insensitive to the approach speed (Fig. 5A). The change in mechanical energy during the take-off phase was approximately

±5 % while using an optimal angle of attack (Fig. 5B). For running speeds greater than 5 m s⁻¹, optimal performance required a net loss of energy. At lower approach velocities, the leg generated net positive work for optimal performance. This required optimum angles of attack greater than 70 ° (Fig. 5B). The angle of the velocity vector at take-off relative to the horizontal plane (take-off velocity angle, Fig. 5C) increased for decreased angles of attack if the running speed was less than 8 m s⁻¹. At higher speeds, the take-off velocity angle was insensitive to the angle of attack or even decreased slightly for small angles of attack.

Compared with the situation in Fig. 2 (initial knee angle 170 °), a smaller knee angle φ₀ led to a shorter jumping distance (for φ₀=160 ° a difference of −19 cm, and for φ₀=150 ° a difference of −38 cm) and a decreased take-off velocity angle (for φ₀=160 ° a difference of −2 °, and for φ₀=150 ° a difference of −4 °). Nevertheless, the percentage of the take-off to touch-down mechanical energy increased for smaller knee angles φ₀ (for φ₀=160 ° a difference of +1.7 %, and for φ₀=150 ° a difference of +3.2 %).

The influence of the angle of attack, the total muscle mass m_MTC, the ratio of muscle fibre to tendon cross-sectional area R_A and the relative fibre length R_-ℓ on the jumping distance is shown in Fig. 6.

Keeping the architectural variables (R_A, R_-ℓ) constant, the muscle mass determined the maximum force exerted by the leg and had a strong influence on the jumping distance (Fig. 6A). For masses greater than 4 kg, the optimal angle of attack (approximately 65 °) was fairly insensitive to changes in muscle mass. Note that the muscle mass is not part of the accelerated mass, it simply represents the volume of the musculature.

For a realistic value of the muscle fibre to tendon cross-sectional area ratio of approximately 100 (Winters and Woo, 1990; Pierrynowski, 1985), the optimal angle of attack was approximately 65–70 ° (Fig. 6B). At high ratios (R_A>140) and angles of attack less than 65 °, slightly higher distances can be achieved. The muscle can produce higher forces and the tendon becomes more compliant. In contrast, for thick tendons (R_A<30), this optimum shifted to steeper angles (more than 70 °). However, the jumping performance was not much influenced by R_A.

The influence of the relative fibre length R_-ℓ proved to be more complex (Fig. 6C). Very short muscle fibres (R_-ℓ=0.01) led to unrealistically large jumping distances (for humans more than 9 m at an approach speed of 9 m s⁻¹) at realistic optimal angles of attack α_opt of 60–65 °. A relative fibre length of approximately 0.05 predicted the smallest α_opt of approximately 60 ° and a jumping distance of approximately 9 m. For longer fibres, α_opt increased. For example, at R_-ℓ=0.15, a realistic jumping distance (6.7 m) with an α_opt of 65–70 ° was calculated.

Compared with the situation in Fig. 2, the predicted jumping distance was quite insensitive to changes in maximum shortening speed of the muscle fibres v_Max (less than 1 % change in distance for twice or half the original speed). Halving or doubling the tendon compliance also led to a surprisingly small change in jumping distance (less than 3 %). Thus, jumping distance is insensitive to muscle speed and to the serial tendon compliance.

In contrast, changes in the total muscle–tendon complex mass m_MTC (to 50 % or 200 %), of the maximum isometric muscle fibre stress σ_Max (to 50 % or 200 %) and of eccentric force enhancement N_ECC (defined as the ratio between maximum eccentric force F_ECC and maximum isometric force F_Max; from 1.8 to 1.4 or 2.2, respectively) led to a considerable change in jumping distance (m_MTC, −24 % or +25 %, σ_Max, −22 % or +35 %, N_ECC, −8 % or +8 %). Thus, jumping distance is sensitive to muscle strength and to eccentric force enhancement.

The model presented here predicted the angle of attack used in the long jump to be insensitive to running speed (>6 m s⁻¹) and muscle design. The angles agreed with the experimental observations of Seyfarth et al. (1999) within a range of approximately 5 °. Furthermore, the model gave a reasonable explanation of why long jumpers use relatively low take-off velocity angles and of the extent to which they take advantage of energy losses in leg muscle. Although the model was very simple, it provided insights into muscular dynamics at high stretching velocities and into the contribution of a fully activated knee extensor muscle to the shape of the ground reaction force over time in running and jumping.

Because of relatively short muscle fibres (R_-ℓ=0.16) and high muscle activation (shown experimentally by Kyröläinen et al., 1987), lengthening of the muscle–tendon complex takes place largely within the serial tendon. Therefore, the muscle–tendon complex work loop was strongly influenced by the J-shaped nonlinear stress–strain curve of the tendon (Fig. 3A). The geometry function, given by the ratio of ground reaction force to muscle–tendon complex force, was high for an almost stretched knee joint exactly at the instant at which tendon stiffness was low (for small deflections). Furthermore, during muscle shortening before take-off, the maximum force decreased continuously with increasing shortening velocity. This resulted in an almost linear region of the work loop during unloading (Fig. 3B). The region of linear behaviour could be extended by elastic components in other leg muscles with more compliant tendons (e.g. the Achilles tendon) or by reducing the muscle activation just before the active peak, as observed experimentally (Kyröläinen et al., 1987), and it may simplify the control of jumps.

In contrast to a spring-mass system with a constant leg stiffness (Seyfarth et al., 1999), a realistic muscle automatically adapts its stiffness at different leg shortening velocities, which keeps the optimal jumping techniques robust (Fig. 7). Characterisation of this effect requires a proper description of the muscle eccentric force enhancement together with suitable compliance characteristics of the serial elastic tendon.

Taking the force–length relationship of the knee extensor into account (Fig. 7), the spring-like behaviour of the leg would be slightly enhanced. Depending on the initial fibre length l_f with respect to the optimal fibre length l_opt, this leads to a slightly increased dependency of the predicted optimal angle on running speed (l_f=l_opt/1.4 in Fig. 7). Nevertheless, for maximum performance, we assume that the muscle works at near the maximum l_f>l_opt/1.2 within the force–length curve.

How does jumping performance depend on muscle architecture? In our model, prestretch was excluded because no antagonist was present. An antagonist muscle would reduce the net knee torque. Assuming a maximum isometric force of the extensor muscle of 10 kN, a realistic preactivation of 60 % did not reproduce the high force peak observed experimentally of approximately 7–8 kN (see Fig. 2A). Only the unrealistic assumption of a completely preactivated extensor without an antagonist produced an appropriate initial peak in amplitude. The shape of this first force peak was characterised by a rapid rise followed by a steady descent similar to that described by Alexander (1990). However, the shape of the curve in Alexander’s (1990) model was not due to force enhancement by eccentric operation of the muscle but was instead a direct consequence of the geometric conditions and an assumed prestretch of the series elastic component (geometry function of the leg, see equation A5; Van Ingen Schenau, 1989), which resulted in a rapid rise in ground reaction force (force increases from zero to maximum within one time step).

So, including realistic tendon properties and eccentric muscle loading in a model is not sufficient to describe the initial peak in force. To achieve this, it is necessary to include a combination of passive properties due to the distributed mass of the system (Seyfarth et al., 1999) and muscle properties.

Jumping performance was not very dependent on tendon compliance. At high approach speeds, high eccentric forces are an important feature and increase the vertical momentum at the expense of mechanical energy loss. This requires stiff tendons to profit from high stretching velocities immediately after touch-down. In contrast, low tendon stiffness allows a larger amount of energy to be stored elastically and extends the period during which the activated muscle fibres are stretched. As shown in vertical jumping (Zajac, 1993), energy storage and muscle work output are compensating effects. Thinner tendons could improve mechanical efficiency but prevent muscle fibres from making use of high eccentric forces.

Jumping distance was more sensitive to changes in relative fibre length R_-ℓ than to the ratio of tendon to muscle fibre cross-sectional areas R_A (Fig. 6). Good jumpers benefit from short muscle fibres and long tendons. This is a well-known architectural feature of animals specialised for jumping (Alexander, 1988; Ettema, 1996). Nevertheless, the optimal angle of attack was not very dependent on the actual muscle design (Fig. 6). Exactly this observation was found in an experimental analysis of long jumpers with a variety of abilities (Fig. 8).

Furthermore, the maximum muscle velocity v_Max did not play a crucial role in the performance of the take-off phase because the muscle produces very little positive work (Fig. 4). Therefore, maximum performance was observed for the shortest fibres. Nevertheless, intrinsic maximum muscle velocity could have a positive influence on running speed.

We cannot reproduce accurately the magnitude of the first force peak because no distributed masses corresponding to those of the supporting leg were included in the present model (Seyfarth et al., 1999). This results in shorter jumps (−9 %) and somewhat flatter take-off velocity angles (−7 °) than those observed experimentally.

Furthermore, a foot is not included, and thus the sequence of impacts generated by the contact of the heel pad and the ball of the foot was not represented in our model (see Hatze, 1981). The lack of the foot also resulted in unrealistically high muscle forces and a shorter contact time (Stefanyshyn and Nigg, 1998). Furthermore, plantarflexion, swinging movements of the leg and arm lifting raise the take-off position of the body centre of mass (Hay and Nohara, 1990; Seyfarth et al., 1999). Our model cannot simulate the details of prestretch because no antagonists are present and the segments have zero mass. In reality, preactivation before touch-down does occur (Kyröläinen et al., 1987). We compensated for this to some degree by introducing an instantaneous increase in activation state.

Very short muscle fibres require a specific muscle architecture (e.g. characterised by large pennation angles) and reduce the range of muscle lengths over which high forces can be produced (as a result of the force–length relationship). This was not included in our model.

The model could be improved by introducing other leg muscles. For example, the bi-articular gastrocnemius muscle couples knee extension to plantarflexion (Van Ingen Schenau, 1989) and improves the utilisation of the work capacity of the extensor muscle. However, the net knee torque can be increased through the simultaneous activation of other muscles not spanning the knee (e.g. hip extensors and plantar flexors) (Zajac, 1993). Inclusion of these muscles and their effects on the operation of the knee extensors awaits for further investigation. To keep the tendon strain below the critical value for rupture, a high stiffness of the serial tendon had to be assumed. This meant that, with a fully activated muscle, spring-like sinusoidal patterns of the ground reaction force were not possible. To represent the spring-like behaviour during ground contact, additional elastic structures (deformation of the foot, Achilles tendon, ligaments) should be introduced into the model. With regard to the spring-mass model (Blickhan, 1989; Seyfarth et al., 1999), the leg can produce only a certain maximum force and therefore has a limited capacity to operate like a spring. Thus, accurate prediction of the limited take-off velocity angle of human long jumps can only be achieved by taking the muscle properties into account.

Appendix 1: the force–velocity relationship of the muscle fibres

Appendix 3: the ground reaction force

Appendix 4: the muscle design variables

Appendix 5: jumping distance

 
• A_CE

  physiological cross-sectional area of muscle fibres

 
• A_f

  physiological cross-sectional area of muscle belly (contractile element and parallel elastic component)

 
• A_PE

  physiological cross-sectional area of parallel elastic component (PE)

 
• A_t

  physiological cross-sectional area of serial tendon element (SE)

 
• c₁–c₄

  constants in the stress–strain relationship of the elastic elements (serial elastic component and parallel elastic component)

 
• CE

  contractile element

 
• d

  moment arm of the ground reaction force

 
• F_CE

  muscle fibre force

 
• F_ECC

  maximum eccentric force of muscle fibres

 
• F_G

  ground reaction force

 
• F_G,Exp

  experimentally measured ground reaction force (from Seyfarth et al., 1999)

 
• F_G,Mod

  modelled ground reaction force

 
• F_Max

  maximum isometric force of muscle fibres

 
• F_MTC

  muscle force

 
• F_MTC,Mod

  modelled internal muscle force

 
• F_PE

  force of parallel elastic element

 
• F_SE

  force of serial elastic element

 
• G

  constant in the Hill equation (equation A1)

 
• g

  gravitational acceleration

 
• l

  length of each leg segment

 
• l_f

  length of the muscle belly (contractile element and parallel elastic component)

 
• l_LEG

  leg length, distance between point of ground contact and hip

 
• l_MTC

  reference length of the muscle–tendon complex

 
• l_opt

  optimum fibre length

 
• l_t

  length of serial tendon (serial elastic component)

 
• m

  point mass at the hip

 
• m_MTC

  mass of muscle–tendon complex MTC muscle–tendon complex

 
• N_ECC

  eccentric force enhancement (F_ECC/F_Max)

 
• PE

  parallel elastic element

 
• r

  moment arm of the patella tendon

 
• R_A

  ratio of cross-sectional areas (A_f/A_t) of the muscle

 
• R_-ℓ

  relative fibre length (l_f/l_MTC)

 
• SE

  serial elastic component

 
• v₀

  approach velocity

 
• v_0,x

  horizontal component of the touch-down velocity

 
• v_0,y

  vertical component of the touch-down velocity

 
• v_CE

  muscle fibre lengthening velocity

 
• v_E,x

  horizontal component of the take-off velocity

 
• v_E,y

  vertical component of the take-off velocity

 
• v_Max

  maximum shortening velocity of muscle fibres

 
• V_MTC

  volume of the muscle–tendon complex

 
• (x,y)

  horizontal and vertical coordinates of the point mass

 
• (⁠⁠, ÿ)

  second time derivative of the coordinates of the point mass

 
• (x_E,y_E)

  coordinates at take-off

 
• x_JD

  jumping distance

 
• α

  leg angle

 
• α₀

  angle of attack (leg angle at touch-down)

 
• α_opt

  optimal angle of attack resulting in maximum predicted jumping distance

 
• σ

  strain

 
• σ_Max

  maximum muscle fibre strain rate

 
• σ_C

  critical strain dividing the non-linear and linear behaviour of the tendon

 
• φ

  knee angle

 
• φ₀

  initial knee angle at touch-down

 
• ρ_MTC

  density of the muscle–tendon complex

 
• σ

  stress

 
• σ_Max

  maximum isometric stress of muscle fibres

We thank Professor R. F. Full (Berkeley) for useful comments on a draft of this paper. This work was supported by a DAAD Doktorandenstipendium im Rahmen des gemeinsamen Hochschulsonderprogramms III von Bund und Länder to A.S. and by DFG grant Bl 236/7.