Determination and optimization of joint torques and joint reaction forces in therapeutic exercises with elastic resistance
Article Outline
- Abstract
- 1. Introduction
- 2. Methods
- 3. Results
- 4. Discussion
- 5. Conclusions
- Conflict of interest
- Appendix A. Determination of shear and axial components of the joint reaction force ϕ→
- Appendix B. Nomenclature
- References
- Copyright
Abstract
A model has been developed to definitively characterize the resistance properties and the joint loading (i.e., shear and compressive components of the joint reaction force) in single-joint exercises with ideal elastic bands. The model accounts for the relevant geometric and elastic properties of the band, the band pre-stretching, and the relative positioning among the joint center of rotation and the fixation points of the band. All the possible elastic torque profiles of ascending–descending, descending, or ascending type were disclosed in relation to the different ranges of joint angles. From these results the elastic resistance setting that best reproduces the average-user's knee extensor torque in maximal isometric/isokinetic efforts was determined. In this optimized setting, the shear tibiofemoral reaction force corresponding to an anterior (posterior) tibial displacement was 65% smaller than (nearly the same as) that obtained in a cam-equipped leg-extension equipment for equal values of resistance torque peak, whereas the compressive tibiofemoral reaction force was 22% higher. Compared to a weight-stack leg-extension equipment, an elastic resistance optimized setting has the potential to give a more effective quadriceps activation across the range of motion, and greatly reduces the anterior cruciate ligament strain force, which represents the main drawback of existing open kinetic-chain knee-extension exercises.
Keywords: Elastic resistance, Torque, Joint loading, Knee extension, Tibiofemoral joint
1. Introduction
Elastic resistance exercises have gained increasing popularity in recent decades and are widely used in today's conditioning and rehabilitation programs. Since the pioneering research of Aniansson et al. [1], many clinical randomized controlled trials assessed the effectiveness of elastic resistance exercises in improving strength, balance, proprioception, and functionality, as well as in the treatment of chronic pain and injury prevention [2]. Several electromyographic studies investigated muscle activity patterns during knee and shoulder rehabilitation exercises with elastic resistances [3], [4], [5]. In contrast, little attention has been devoted in the literature to understand the specific resistive properties provided by elastic bands or tubing [2], [6], [7], [8]. An accurate and rational knowledge of joint loading during elastic resistance exercises is still completely lacking. The determination of the axial and shear joint reaction forces, and of the forces carried by the ligaments, represents a crucial step to plan effective therapeutic exercises and understand all the clinical implications.
The magnitude of the force exerted by an homogeneous and ideal elastic band is given by 
, where E is the Young's modulus, which characterizes the stiffness of the material, S0 and l0 are the resting cross-sectional area and length of the undeformed band, l is the actual length of the band after its elongation strain, and the factor 
represents the relative elongation of the band. Color-coded bands with different stiffness and/or cross-sectional area are typically marketed in bundle, to provide a range of resistances that meets the different user's needs and demands. The two ends of an elastic band are generally fixed to a point C of a stable support and to a point P of the exercising limb; the change in l during the exercises gives a modulation of the elastic resistance within the joint range of motion (ROM). A point within the length of the elastic band may be fixed to C to shorten the effective resting length of the band, and increase both the mean value of 
and its overall increase 
during the exercise 
. Of course, elastic bands can be connected in series (in parallel) to increase the effective value of l0 (S0) and produce the opposite (the same) effects. Both 
and 
can also be modulated by changing the distance of the user from C or the level of the band pre-stretching.
Nevertheless, the linear increase of 
with l would appear to establish a severe limitation in the management of the resistance force. In fact, the force of the muscle–tendon unit typically decreases during a shortening contraction, due to the muscle force–length relationship. However, it is well known that the muscle force is optimized during a single-joint strengthening exercise when the torque of the resistance about the joint axis reproduces the distinctive variable joint torque produced by healthy users within the joint ROM in maximal isometric/isokinetic efforts. With changing joint angle, joint torques display either a ascending, descending, or ascending–descending type trend, depending on the specific joint and joint movement [9]. Hughes et al. [6] highlighted that an ascending–descending resistance torque profile may be obtained in shoulder abduction exercises with elastic bands, due to the change in direction of the elastic force and the concurrent change in the resistance moment arm. However, the shoulder abduction torque-angle curve is of descending type [10] and the implications of the selected elastic resistance setting on the shoulder joint loading were not discussed.
The main purpose of this study is to provide a general view of all possible resistance torque curves that can be obtained in single-joint exercises by handling all the relevant elastic resistance parameters and the relative positions among the joint center of rotation and the fixation points of the elastic band. Moreover, this work is aimed at finding the optimal elastic resistance setting that reproduces the average-user's knee extensor torque curve. Finally, the compressive and shear tibiofemoral joint reaction forces are calculated for this elastic resistance setting and compared with those obtained in a leg-extension machine equipped with a variable-resistance cam.
2. Methods
A two-dimensional geometrical sketch of a single-joint exercise with a resistance provided by an ideal and homogeneous elastic band is shown in Fig. 1. The plane π containing the exercising limb and the elastic band may be inclined by an angle α with respect to a horizontal plane. The joint center of rotation O is fixed and is the origin of a Cartesian x–y reference for π. The position of the limb is individuated by the angle θ between the longitudinal limb axis and a reference orientation, for example the negative y direction, which coincides with the orientation of the downward-directed maximum-slope straight lines of π. The elastic band is fixed and in a point P of the exercising limb and in a point C of a stable support. The elastic resistance force acting on P is always directed towards C (referred to as the center of 
) and its magnitude is given by
(1)
is the elastic constant, 
, 
, 
and 
are the distances of 
and P from the origin O, and 
is the small angle between the longitudinal limb axis and OP. In this equation, the Carnot's theorem has been used to express the variable distance 
as a function of the joint angle θ and of the constant parameters 
. In a more general setting, the elastic band is fixed in a point A of a stable support and then forced to pass through point C prior to its fixation on P (Fig. 1b). The elastic resistance force acting on P is still directed towards the center C, whereas in this case 
and 
, the sign before 
depending whether P0 falls within the segment AC (+ sign) or CP (− sign). The magnitude of 
simply generalizes to
(2)
Fig. 1.
(a) Geometrical sketch of a single-joint exercise with a resistance provided by an ideal homogeneous elastic band. The joint center of rotation O is assumed to be fixed. The elastic band is fixed to a point C of a stable support and to a point P of the exercising limb. The distance |CP0| defines the resting length of the band. The elastic force
acting on P is always directed towards C (the center of 
) and its intensity is proportional to |PP0|. The moment arm 
of 
depends on the angle β between OP and CP. The point G is the center of mass of the exercising limb. The small angle 
between the longitudinal limb axis and OG is assumed to be negligible 
. (b) The band may be pre-stretched so that an elastic force is exerted on the limb even when P coincides with the center of 
. For example, the band may be fixed in a point A of a stable support and then forced to pass through point C (the center of 
) prior to its fixation on P.
Ultimately, in comparison to the first setting (Fig. 1a), a pre-stretching of the elastic band is allowed by the latter setting (Fig. 1b). The moment arm of 
about the joint axis is
(3)
β has been expressed as a function of θ with the use of Euler's theorem. The axial moment (or torque) of 
about the joint axis is the product of the magnitude of 
and its moment arm:
(4)Given the position 
of the joint center of rotation, the elastic torque 
depends on the elastic constant k, the joint angle θ, the distances 
and 
between O and the elastic band fixation points C and P, the angle 
, the ratio 
between the coordinates of C, and the resting length or pre-stretching level 
. However, the normalized torque
(5)
, and 
. The rotational dynamic equation of the exercising limb turns out to be
(6)
is the overall joint net torque produced by the forces 
of the muscles (both agonist and antagonist) crossing the joint, and by the joint reaction forces due to ligament tension and bone-to-bone contacts; 
is the distance of the center of mass (G) of the limb from the joint center of rotation (O). Eq. (6) determines the joint torque 
, given the kinematics of the exercise.The general analytical expressions of the tangent (shear), 
, and normal (axial), 
, components of the joint reaction force 
are derived in Appendix A, and are given by
(7)
(8)
. Actually, 
and 
can only be calculated when the single muscle forces 
are known. To this end, electromyography measurements and muscle-architecture data are employed in conjunction with appropriate optimization procedures [11], [12], [13]. In fact, Eq. (6) only gives the overall joint torque 
, which depends on both the muscle and the joint reaction forces. However, the moment arms of the joint reaction forces are typically negligible and the joint torque is mainly determined by the muscle forces, i.e., 
where 
is the moment arm of 
. When only the main agonist muscle force 
is taken into account, and the other synergistic and antagonist muscle forces are neglected, than one has
(9)
.In the following, with the use of Eq. (5), we will derive and discuss all the possible types of elastic resistance torque profiles in single-joint exercises. We will also design the elastic resistance setting that best reproduces the average-user's knee extensor torque in maximal isometric/isokinetic efforts. To this end, the isometric/isokinetic experimental points of Knapik et al. [14] were fitted with Eq. (5), handling 
, and 
as fit parameters. The convergence of the fitting procedure requires a proper initialization of such parameters. This can be easily achieved by comparing qualitatively the experimental isometric/isokinetic torque trend with the families of all the possible elastic resistance torque curves (this point is addressed in detail in Section 4).
For a leg-extension exercise with this optimized elastic resistance setting, the compressive and shear tibiofemoral joint reaction forces will then be derived from the general Eqs. (7), (8), under the limitation established by Eq. (9). With this assumption, 
represents the quadriceps force, 
and 
the moment arm and the traction angle of the patellar tendon. The dependences of 
and 
on the knee angle θ were taken from Herzog and Read data [15]. For this exercise, α is 90° in Eqs. (6), (7), (8) (the exercising limb moves in a vertical plane), and 
(the lower leg is vertical) when the knee is 90° flexed (the knee flexion angle 
is given by 
).
3. Results
Fig. 2a–c displays the dependences of 
(i.e., 
normalized to unity) on 
, for different values of 
and 
. It can be deduced by Eq. (5) that either a pair of values of 
and 
, or the new pair of values defined by the transformations
(10)
. 
vanishes whenever the band is stretched and O, P, and C are aligned 
, and whenever the elastic band loses its tension 
. Thus, the elastic band provides a useful resistive torque 
only inside a range limited by the values 
and 
where 
. 
is always equal to 180° and is defined by the condition that OC and OP have opposite orientations (Fig. 3a and b). When the band is pre-stretched to A, 
is equal to 0° and is defined by the condition that OC and OP have the same orientation (Fig. 3c). When the band is fixed in C, 
is again defined by the condition that OC and OP have the same orientation, and again 
, provided that 
in this configuration (Fig. 3d); whereas, if 
in such configuration, then 
becomes greater than 0° 
and corresponds to the value of 
at which 
equals 
(Fig. 3e). This important effect allows the modulation of both the width 
of the resistive range 
, and the slope 
. With change in θ*, 
displays an ascending–descending behavior in 
, the location of the maximum depending on the values of 
and 
. For 
and 
, 
is discontinuous in 
and, in the limit 
, takes positive values which increase up to unity with increasing 
(Fig. 2a). Therefore, 
may assume a descending behavior inside the whole 180° range of joint angles 
. On the contrary, 
may assume an ascending behavior only in sub-ranges of 
of amplitude smaller than 120° (see for example the curve in Fig. 2c obtained for 
and 
).
Fig. 2.
(a) Dependence of the normalized elastic torque
on the relative joint angle 
, for 
, and for different values of 
(δ=1000, 100, 4, 1, 0, −0.5, −1, −1.414). For δ<−1.414, 
inside a range 
of width 
smaller than 90°. (b) Dependence of the normalized elastic torque 
on the relative joint angle 
, for 
and for different values of 
(δ=1000, 100, 4, 1, 0, −0.5, −0.75, −1.118). For δ<−1.118, 
inside a range 
of width 
smaller than 90°. The same set of curves are obtained for 
and δ=2000, 200, 8, 2, 0, −1, −1.5, −2.236 (see transformations defined by Eq. (10)). (c) Dependence of the normalized elastic torque 
on the relative joint angle 
, for 
and for different values of 
(δ=100, 10, 0, −0.75, −0.85, −0.9, −0.95, −1.005). For δ<−1.005, 
inside a range 
of width 
smaller than 90°. The same set of curves are obtained for 
and δ=1000, 100, 0, −7.5, −8.5, −9, −9.5, −10.05 (see transformations defined by Eq. (10)).
Fig. 3.
Resistance settings (relative positioning of O, P, P0, C) that define the extremes
(a and b) and 
(c–e) of the joint range 
wherein the elastic band provides a useful resistive torque 
. In (a, d, and e) the band is directly fixed in C, whereas in (b and c) the elastic band is pre-stretched (the band is fixed in a point A of a stable support and then forced to pass through point C prior to its fixation on P).
The optimal elastic resistance setting that reproduces the average-user's knee extensor torque 
(normalized to unity) in maximal isometric/isokinetic efforts for knee flexion angle 
smaller than 90° (
) is given in Fig. 4a. This optimal 
curve was obtained with the following values of the fit parameters: 
, 
, and 
(Fig. 4b). The curve also reproduces the normalized isokinetic average-users’ knee extensor curves, because the isometric and isokinetic trends closely overlap when normalized to unity [16].
Fig. 4.
(a) Resistance setting (relative positioning of O, P, P0, C) that reproduces the average-user's knee extensor torque (normalized to unity) in maximal isometric/isokinetic efforts for knee flexion angle
smaller than 90°. The knee flexion angle 
and the unspecific general joint angle θ are related by the equation 
(
for 90° knee flexion, 
at full knee-extension). (b) 
curve obtained by fitting the isometric experimental points of Knapik et al. [14] with Eq. (5) and the following values of the fit parameters: 
, 
, and 
. (c) Dependence of the shear, 
, and compressive, 
, components of the tibiofemoral joint reaction force on the knee flexion angle 
, for different values of the elastic torque peak 
(
), in a single-joint knee-extension exercise with an elastic resistance applied distally on the lower leg 
. The elastic resistance setting provides an optimized resistive torque profile that reproduces the average-user's knee extensor torque in maximal isometric/isokinetic efforts (
, 
, and 
). A positive (negative) shear force 
constrains the tibial plateau posterior (anterior) translation with respect to the femur, reflecting a load on the PCL (ACL).
As represented in Fig. 4c, in this optimized setting, and for 
(elastic resistance applied distally on the lower leg), the shear tibiofemoral force 
takes negative (positive) values for 
smaller (greater) than 40°, thus producing in this range a strain force on the anterior (posterior) cruciate ligament, ACL (PCL). The peak value of the ACL-loading (PCL-loading) component of 
occurs for 
within 15–19° (85–88°) and increases nearly linearly with the peak value 
of the elastic resistance torque. The axial component 
of the tibiofemoral force is of compressive type 
, peaks for 
within 60–66°, and reaches values as high as 6000N for 
.
4. Discussion
The universal resistance torque curves displayed in Fig. 2a–c reveal the rich variety of torque profiles that can be generated with the use of a simple elastic band. In general, the elastic force produces a useful resistance torque 
with an ascending–descending bell-shaped behavior inside a range of joint angles delimited by the values 
and 
where 
vanishes 
. With changing the parameters 
and 
, one can control both the width 
of the range, and the position of the 
peak within the range, i.e., the torque curve asymmetry. Indeed, 
can be reduced from 180° to 90° or less, enabling a fast change of 
with the change in the joint angle, that is, a rapid increase/decrease of the slope 
. This overcomes a major limitation of gravitational resistance strength-training equipment, where the cam or the pulley system only allows relatively small variations of the resistance torque with the joint angle. For example, in commercial leg-extension equipment the cam moment arm's percentage of variation does not exceed 20% throughout the ROM [16]. Actually, these cam profiles do not properly reproduce the real isometric/isokinetic torque-angle curves for the knee extensor muscles, where the torque near complete knee-extension (at 10° knee flexion) is nearly 25% of the peak torque occurring around 60° of knee flexion [14]. The elastic torque peak can be gradually shifted within the range 
towards the lower values of 
, down to 
, thus realizing a descending torque profile on a whole 180° range of joint angles. On the other hand, the peak value of 
cannot be shifted up to 
, and ascending torque profiles can only be obtained in ranges of joint angles of amplitude not exceeding 120°. On the whole, these elastic torque profiles well reproduce the typical ascending, descending, and ascending–descending human torque curves corresponding to the different joint and joint movements [9].
Fig. 2a–c constitutes an effective operative tool in the management and optimization of the elastic resistance. Trainers and therapists can easily select the proper values of 
and 
that individuate a desired torque profile in a sub-range of 
of amplitude corresponding to the joint ROM of a specific exercise. To establish the exact correspondence between the value of 
and those of the real joint angles it is sufficient to set the ratio 
according to the relation 
. For example, the normalized average-user's knee extensor torque in maximal isometric/isokinetic efforts, for knee flexion angle smaller than 90°, is well reproduced by the elastic torque curve 
obtained for 
and 
in the range 
(Fig. 2a). Thus, setting the values of 
and 
such that 
, while maintaining 
, one shifts the optimal portion of the torque curve to the range 
of knee joint angles 
. The best fit to the isometric data of Knapik et al. [14] gives nearly these values for such parameters (
, 
, and 
) and the agreement is excellent (Fig. 4b). The same procedure is applicable to many other joint and joint movements, taking into account the effects of the weight of the exercising limb whenever the exercising plane is not horizontal 
. Once the optimized profile of the normalized resistance torque 
has been established, one can freely adjust the absolute value of the peak resistance torque 
by changing the cross-sectional area or the Young's modulus of the elastic band, with no change in the optimized 
profile.
In traditional strength-training equipment, the inertial effects associated with accelerations of the weight stack (and of any other mobile part of the equipment) may considerably affect the nominal change of resistance torque established by the cam geometry. Typically, the torque is increased (decreased) at the beginning (end) of the concentric phase and at the end (beginning) of the eccentric phase [17]. In elastic resistance exercises, the equipment inertial effects are negligible, enabling a more accurate control of the resistance administration.
The optimization of the resistance does not only consist of equalizing the resistance torque to the average-user's maximal isometric/isokinetic joint torque; it is frequently more important to minimize the overall joint load or the stress on specific joint structures while maintaining a substantive muscle activation. For this reason, the shear and compressive components of tibiofemoral joint reaction force (
and 
) obtained with the elastic resistance setting that optimizes the knee extensor force (Fig. 4c) were compared with those obtained in a leg-extension machine equipped with a variable-resistance cam (see Fig. 4b in Ref. [18]) on equal values of resistance torque peak 
. Surprisingly, for 
, the ratio of the values of 
or 
obtained using these two different pieces of equipment is nearly independent of 
: with the use of the elastic resistance the ACL-loading component of 
turns out to be 65% smaller, the PCL-loading component of 
is nearly the same (±1%), whereas the compressive tibiofemoral force 
comes to be 22% higher. With a progressive decrease of 
below 60Nm, the above percentage differences decrease progressively, as the torque exerted by the weight of the limb (skank and foot) becomes comparable with that of the external resistance. Notably, with elastic resistance the ACL strain force is decreased without increasing the PCL strain force. The increased compressive tibiofemoral force enhances the loading on the contact joint surfaces, but at the same time stabilizes the joint reducing the anterior/posterior tibial translation and consequently the ACL/PCL loading. These results clearly indicate that the elastic resistances can be effectively used in controlling the anterior tibial displacement in open kinetic-chain knee-extension exercises.
The limitations of the present study concern the calculation of tibiofemoral joint loads in leg extension exercise (Fig. 4c). Only the agonist muscle force 
, provided by the quadriceps, was included in Eqs. (7), (8), which give the general expressions of the shear and axial joint reaction forces in single-joint exercises with elastic resistances. However, this approximation is largely accepted in single-joint, open kinetic-chain knee-extension exercises, performed in non weight-bearing position, with tights and upper body stabilized in seated posture [19]. In these conditions, muscle activation occurs predominantly in the quadriceps, and the other synergistic muscle force contributions are completely negligible. The antagonistic activity of knee flexors may become considerable only near full knee extension [20], and this antagonistic activity usually decreases after a period of familiarization [21]. Hintermeister et al. [4] examined the electromyographic activity of eight lower extremity muscles during five elastic resistance knee rehabilitation exercises. They concluded that hamstring/quadricep cocontraction is insignificant even in the closed-chain exercises. Moreover, due to the common quadriceps insertion site, the quadriceps force acting on the lower-leg can be modeled as a single force vector 
, thus justifying the use of Eq. (9) for the calculation of the tibiofemoral reaction force in leg-extension exercise.
5. Conclusions
The following main conclusions can be drawn from the present study:
Conflict of interest
The author confirms that there is no conflict of interest in relation to this work.
Appendix A. Determination of shear and axial components of the joint reaction force 
The resultant joint reaction force 
is determined by the force equation applied to the exercising limb
(A1)
being the acceleration of G. The tangent (shear), 
, and normal (axial), 
, components of 
are obtained by projecting the above equation on the tangent (t) and principal normal (n) of the circular trajectory of G [22], [23]:
(A2)
(A3)
, defined as the angle of 
and the longitudinal limb axis (γi=0 for a pure axial compression), and 
is the joint angular velocity. With the use of Eq. (2) (to express 
as a function of the joint angle θ and of the constant parameters 
), trigonometric addition formulas (for 
and 
), and Euler's and Carnot's theorems (to express 
and 
as a function of the joint angle θ and of the constant parameters 
), Eqs. (A2), (A3) give the final expressions of 
and 
reported in Section 2 (Eqs. (7), (8)).Appendix B. Nomenclature
Points
- A
fixation point of the elastic band on a stable support. A may coincide with C
- C
center of the elastic force: the elastic force is always directed towards the center C
- G
center of mass of the exercising limb
- O
joint center of rotation
- P
fixation point of the elastic band on the exercising limb
- P0
point that defines the resting length
of the unreformed band: 
(
) when the band is fixed at the point C (point A).
Distances and coordinates
moment arm (about the joint axis) of the elastic force
moment arm (about the joint axis) of the ith muscular force
distance between C and O
distance between G and O
distance between P and O
- δ
- l
actual length of the elastic band after its elongation strain
resting length of the unreformed band
and 
coordinates of C
Angular quantities
- α
inclination angle of the plane contacting the exercising limb with respect to an horizontal plane
- β
angle between OP and CP
traction angle of the ith muscular force
angle between the longitudinal limb axis and OP
- θ,
, and 
joint angle, joint angular velocity, and joint angular acceleration.
- θ*
Dynamic quantities
elastic force
ith muscular force
joint reaction force
compressive (axial) component of
shear (tangent) component of
- k
elastic constant
moment of inertia (about the joint axis) of the exercising limb
- M
mass of the exercising limb
axial moment or torque (about the joint axis) of the elastic force
axial moment or torque (about the joint axis) of muscular and joint reaction forces
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PII: S1350-4533(11)00148-2
doi:10.1016/j.medengphy.2011.06.011
© 2011 IPEM. Published by Elsevier Inc. All rights reserved.
