## 4 Discussion
In our case, we were not interested in the investigation of motor control strategy, so our approach was a simple one, though relatively efficient [42], that just looks for the minimization of the maximum muscle force referred to the physiological cross-sectional area of the muscle. The results are consistent with the known electromyographic patterns [58], as for most of the published works [31; 51; 52; 56; 57]. We suppose that even if the estimation of the muscle forces had been slightly different, the effect of these forces on the tension of the ligaments would not have changed substantially. Concerning the biomechanical structure, while designing our model, we realized how difficult was to arrange the ligaments in the model to avoid the excessive resistance to movement and, at the opposite, to prevent joint dislocation. Beyond the generic anatomical description of the attachment areas [34], the specific data reported in the literature [6; 22; 27; 59] did not help very much: they showed that the attachment points can differ among subjects, and a great variability exists about the mechanical properties. The matching between attachment points, ligaments properties and bone surface geometry is very critical. For example, by using the force/deformation characteristics and stiffness values provided in [26], the anterior bundle of the anterior cruciate ligament (ACL-a) reaches a force of 45 N for a deformation \(\varepsilon=0.06\), which correspond to a lengthening of just 2.1 mm (rest length L\({}_{0}=35.2\) mm). A stiffer ligament, for example the deep posterior bundle of the medial collateral ligament (MCL-dp), goes from 0 N to 135 N for a length increase of just 2.4 mm and achieves 217 N for elongation of 3 mm. Therefore, the acceptable range of length changes is very small. Furthermore, the force distribution between all ligaments affects the knee joint kinematics. In our model, the only flexion/extension movement was imposed, and the remaining 5 d.o.f. were completely free. Obtaining the results reported in Figure 6 was an important achievement in our point of view. In fact, the rotations and the displacements are within a reasonable range and the screw home mechanism is reproduced both in the unloaded condition and also when the external forces and muscle forces were applied. About the muscle forces, our estimation takes into account the muscles arrangements, their lever arms in relation to the different functional axes and the physiological cross-sectional areas. A total of 23 muscles were considered because some of the knee muscles are double-joint muscles, acting also at the hip (hamstrings) and the ankle (gastrocnemius muscles), and so the optimization procedure has to take into account the whole limb. A comparison of our result to a reference pattern of muscle force generation is difficult, because, as mentioned above, there are differences between the data provided by the different authors. However, the phases of activity and the relative amplitudes obtained for our muscles (see Figure 5) are consistent with their role, and can easily be understood: the first phase of the activity of the vasti group (from 0 to 30% of the stride cycle) corresponds to the load acceptance phase, in which the quadriceps is required to control the knee flexion after the impact with the ground; the second phase of activity, starting in the second half of the stance phase, is required to sustain the knee flexion while the ground reaction is still present. The activity of the vasti is extended to the first part of the swing phase, since an extensor moment is required to prevent excessive knee flexion while the limb is thrown forward. Interestingly the rectus femoris (RF) becomes active in this phase, since its action is required at the hip joint to produce the forward acceleration of the thigh, and thus, it contributes to the extensor moment together with the vasti muscles.