3.1. The resistance torque in the knee joint

Due to the different structures that make up the knee, such as bones, ligaments, muscles, and menisci, the knee joint ends up offering resistance to movement. Together with the friction between the bones, these forces generate a torque against the movement performed, called the resistance torque. It can be influenced by several factors such as the individual’s stretching, muscle tone, and frequency of physical activity, among others [Reference Neumann and Fapta17].

In ref. [Reference Stam18], the isokinetic strength test method, which performs dynamic contractions during a movement with the body velocity kept constant (isokinetic) without the influence of the force exerted by the subject, was used to find the knee torque curves. The author used a dynamometer that performs isometric, isokinetic, and isotonic tests. The results of the isokinetic tests showed a substantial improvement in the operated leg through the 8 weeks of treatment. The author indicates that this improvement in a short period may be partially due to the increase in muscle mass and neuromuscular coordination that follows the surgery, in addition to the reduction of pain and swelling in the joints. Isokinetic dynamometry is considered the gold standard for objectifying forces/moments in the knee [Reference Tittelboom, Alemdaroglu-Gürbüz, Hanssen, Heyrman, Feys, Desloovere, Calders and Van den Broeck19].

In this paper, it is hypothesized that is possible to find a model that reliably predicts knee resistance for a healthy population; the improvement in strength (increase in knee torque) may also imply the follow-up of the patient’s improvement during the rehabilitation treatment of the knee.

Table I.

Data of lower limb used in this paper [Reference Plagenhoef, Evans and Abdelnour15].

Table II.

CM of segments (%), Dempster [Reference Dempster16].

The possibility of modeling knee force and moment for healthy people may imply the creation of a force and moment reference for diagnoses in patients in the future using the procedure proposed in this paper.

4.1. Theoretical mechanical model

The cable-driven robot for lower limb movements, Section 2, will be used to collect cable force data for different knee angles. According to Section 3, the human knee is compared to a pulley segment. For the development of the theoretical model, the knee will be considered only as ideal support (friction-free) with free rotation around the transverse axis XX’, in the sagittal plane, Fig. 2(a). Another important assumption is that the test is static; that is, there is no motion involved and the acceleration is zero, Eq. (1). The assumption of static tests is made in function of low speeds and accelerations present in the kinematics of human gait rehabilitation [Reference Gonçalves and Krebs20].

(1) \begin{equation}\sum \vec {F}=0;\; \sum \vec {M}=0\end{equation}

From the free body diagram, Fig. 3, and the angles shown in Fig. 4, one can find the values of cable force and knee moments for each knee angle from the forces and moments balance.

Figure 3.

Free body diagram considering leg + foot.

Figure 4.

Detailing of the angles and cable coupling in the leg.

where

Rx – Knee support reaction on the x-axis;

Ry – Knee support reaction on the y-axis;

ß – Angle of the leg with the vertical, obtained through A_knee;

Af – Angle of the cable with the vertical, obtained through Beta;

P – Weight of the leg plus the foot, in Newtons;

T – Cable traction force, in Newtons;

t – Total leg dimension;

d – Distance from the place where the cable is connected to the splint to the sole of the foot;

d _CG – Distance from the center of the mass of the leg plus foot;

A _thigh – Angle of the thigh;

A _Knee – Angle of the leg with the thigh, reproduces the angle of the knee;

L – Initial cable length.

From the initial values – represented by the “_ini” – of thigh angle (which remains constant during the test), knee angle, and cable angle, it is possible to calculate the cable angle for any leg position by Eq. (2).

(2) \begin{equation}Af=tg^{-1}\left[\frac{\begin{array}{c} \cos \!\left(A\text{thigh}-A\text{Knee}_{\mathrm{ini}}+180^{\circ}\right)\left(t-d\right)+\\[5pt] tg\!\left(Af_{\mathrm{ini}}\right)L\cos \!\left(Af_{\mathrm{ini}}\right)-\sin\!(\beta )(t-d) \end{array}}{\begin{array}{c} L\cos \!\left(Af_{\mathrm{ini}}\right)-\sin \!\left(A\text{thigh}-A\text{Knee}_{\mathrm{ini}}+180^{\circ}\right)\\[5pt] \left(t-d\right)+\cos\!(\beta )(t-d) \end{array}}\right]\end{equation}

where $\beta$ is given by Eq. (3).

(3) \begin{equation}\beta =A_{\text{Knee}}-A_{\text{thigh}}-90^{\circ}\end{equation}

Solving the static equilibrium given in Eq. (1) for the x and y axes is possible to obtain:

(4) \begin{equation}\sum Fx=0\rightarrow Rx-Tx=0\rightarrow Rx=Tx=T \sin\!(Af)\end{equation}

(5) \begin{equation}\sum Fy=0\rightarrow Ry+Ty-P=0\rightarrow Ry=P-Ty=P-T \cos\!(Af)\end{equation}

The point where the foot meets the leg is used to calculate the equilibrium of moments in the system, as in Eq. (6).

(6) \begin{equation}Ry \sin \!\left(\beta \right)\! t-Rx \cos \!\left(\beta \right)\! t-P \sin \!\left(\beta \right)\! d_{\mathrm{CG}}+T \sin \!\left(\beta +Af\right) d = 0\end{equation}

Isolating T in Eq. (6), one has:

(7) \begin{equation}T=\frac{P\sin\!(\beta )(t-d_{\mathrm{CG}})}{\sin\!(\beta +Af)(t-d)}\end{equation}

In this way, it is possible to calculate the theoretical force in the cable at each position of the leg, Eq. (7). The resulting theoretical moment is then the sum of the moments of Rx and Ry, as in Eq. (8), where MjT is the total moment at the theoretical knee.

(8) \begin{equation}MjT=\left(P-T \cos \!\left(Af\right)\right)\sin \!\left(\beta \right)\! t-T\sin\!(Af)\cos\!(\beta )t\end{equation}

Figure 5.

Flow diagram of the clinical tests with healthy subjects.

4.2. Biomechanical model

In the real case, it is known that the knee offers resistance to movement due to ligaments, muscles, and friction between bones, among others. This real knee resistance implies an experimental moment that will be called Me. In this way, the total moment in the experimental knee will be the sum of the theoretical moment previously calculated, Eq. (8), with the unknown experimental moment Me, where MjE is the total moment in the experimental knee.

(9) \begin{equation}MjE=MjT+Me\end{equation}

As Me is unknown, to calculate MjE, the moment balance around the foot-to-leg point contact is performed again, Eq. (10).

(10) \begin{equation}\sum \vec {M}=0\rightarrow MjE=P \sin\!(\beta )d_{\mathrm{CG}}-T \sin\!(\beta +Af)d\end{equation}

Figure 6.

Experimental setup.

With Eq. (10) and the cable force data for each leg angle, one can then calculate the experimental moment for each angle. The calculation of the experimental moment is done indirectly from the traction force measured by the load cell, Eq. (10).

4.3. Experimental methodology

The Research Ethics Committee of the UFU (Brazil) approved this study (CAAE N° 51133821.4.0000.5152), and written informed consent was obtained before any data collection. To participate in the tests, volunteers had to be between 15 and 30 years old, have a body mass of up to 100 kg, be sedentary (practice physical activity at most once a week), and not have had problems with arthrosis, dislocations, or knee and hip operations. Tests were performed on 6 men and 6 women.

The flow diagram of the clinical test with healthy subjects is shown in Fig. 5.

Table III.

Data statistics of the subjects.

Figure 7.

(a) Theoretical knee cable forces all tests; (b) Experimental knee cable forces all tests.

Figure 8.

(a) Theoretical knee moments; (b) Experimental knee moments.

Figure 9.

(a) Cable forces for up and down in two subjects; (b) Knee moments for up and down in two subjects.

Figure 10.

ANOVA test for all subjects and thigh angles.

Two procedures were created: the first was to test 10 individuals from the population, and the second was to test the two remaining subjects. The first protocol tested three thigh angles, Fig. 6, (the first between 0° and 5°, the second from 5° to 10°, and the last from 10° to 15°) only in the descent movement, that is, flexion of the knee. The second tested the same thigh angle (around 0°) in three phases of descent and ascent, working on knee flexion and extension.

The general testing protocol included the following:

• Completion of a questionnaire with age, gender, race, frequency of physical activity, and body mass (measured with a Plenna Ice scale with a minimum capacity of 10 kg and a maximum of 150 kg and graduation of 100 g);

• Test only on the dominant leg;

• Measurement of dominant leg size, the subject should be sitting with feet barefoot on the floor and knee at approximately 90°, the examiner measured the distance from the lateral epicondyle of the femur to the ground with a Cescorf 2 m anthropometric tape measure and 1 mm resolution;

• Measurement of leg circumference and amount of thigh and calf fat dominant using a Mitutoyo plicometer with a sensitivity of 0.1 mm and amplitude of 85 mm reading;

• The test should be performed barefoot, to avoid the extra weight of the shoe;

• Placement of the splint and the velcro connection with the cable approximately 10 cm from the lateral malleolus and measuring the distance from the velcro to the sole with the tape measure anthropometric;

• A position to sit in the chair in which the subject remained upright was standardized over the hip, extra support was used to keep the thigh at the same level during the whole experiment.

For each thigh angle, a test was performed:

• The thigh angle was then measured with an M-D SmartTool digital inclinometer Building Products with 1/10th of a degree resolution, and also the cable start angle;

• The initial length of the cable up to the fastening velcro was measured with the tape measure anthropometric;

• The knee angle was measured with a Mitutoyo goniometer with a range from 0 to 90° and a resolution of 1°, and, after about 1 min in position, the force in the cable was measured by the cable-driven robot, 10 measurements were obtained and an average was used for load calculations;

• The examiner operates the robot, the leg descent, and repeated the previous step until the maximum flexion, so the examiner changed the angle of the thigh and repeated the process.

In the case of the second procedure, after the complete descent, the examiner collected the cable to different angles of extension, also measuring the force in the cable at each of these.

Figure 11.

ANOVA for group 1, without test cut (a) and after the cut (b).

Figure 12.

ANOVA for group 2, without test cut (a) and after the cut (b).

Figure 13.

ANOVA for group 3 without test cut.

Figure 6 shows in detail the coupling of the goniometer on the leg for reading knee angles and the velcro connection on the splint to attach the cable to the individual’s leg, and the subjects were positioned in the same way using the thigh support, Fig. 6.