Determination and optimization of joint torques and joint reaction forces in therapeutic exercises with elastic resistance

(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.

(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)).

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).

(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).