## 5 Experimental Results  
Experiments were carried out with the prototype described in the Section 2.2 in order to demonstrate the efficiency and robustness of the proposed technique. It is important to highlight that the proposed control strategy is valid for a system with \(n\) links, \(n>2\). Our prototype has three links and two actuators, with one passive joint, which is the ankle joint, and two active joints actuated by micro motors, which are the knee and hip joints (see Figure 3). The parameters of the mathematical model for our prototype are in Table 3, where the components of the matrices can be found in [24]. Notice that the masses \(m_{i}\) are much higher than the \(m_{bi}\). The controller design parameters of (12) were specified to place the poles of the system in the real axis and sufficiently far from the imaginary axis with \(\lambda_{1}=784\) and \(\lambda_{2}=56\) (\(p_{i}=-28\)). For the experiment, the passive joint angle was fixed to \(q_{1}=65^{\circ}\). The third joint of the prototype acts like the beak (see Figure 3), which has to follow a desired path with measurements taken at different target points. A grasping torque with the value \(\tau_{c,l}=0.015\) Nm (position of the screw \(\Delta x=0.5\) mm from Table 4) was exerted at the base. A large trajectory was designed to cover a wide range of the workspace, assuring that \(p_{2}\
eq 0\), which means that the matrix \(\mathbf{P}_{n-1}^{-1}\) is full rank and the control law (12) can be implemented. The desired trajectory of the end effector of the manipulator is defined by \(q_{10}=65^{\circ}\) and, the desired trajectory \(\bar{q}_{2}\) imposed by the control objective \(\{\bar{q}_{2}\in\mathbb{R}:y_{1}^{*}=g_{1}(q_{10},\bar{q}_{2},y_{2}^{*})=0\}\), with \(y_{2}^{*}=q_{3}^{*}\). The desired trajectory, \(q_{3}^{*}\) is defined by Bezier curves, which are smooth curves that can be differentiated indefinitely where the position and time of the trajectory is defined in Table 5.  
Figure 8 shows the angular coordinates of the passive joint, of the second joint (active) to maintain the equilibrium at the passive joint and of the third joint (active) together with its reference. Notice that the controller achieves an almost perfect tracking positioning in the third joint while maintaining the equilibrium at the first one. In essence, the controller balances the resultant torque at the second joint. In Figure 9 we show the torque of the motors during the movement. The saturation torque of the motors is \(0.5\) Nm, and therefore, they are far from saturating along the experiment, which is essential in this bio-inspired application. Recall that the resultant torque at the passive joint was defined as \(\tau_{1}:=\mathbf{M_{c}\dot{q}_{2n}}+\mathbf{C_{c1}\dot{q}_{2n}}+g_{1}\). Let us define the inertial and centripetal terms in this joint as \(\epsilon:=\mathbf{M_{c}\ddot{q}_{2n}}+\mathbf{C_{c1}\dot{q}_{2n}}\), so that \(\tau_{1}=\epsilon+g_{1}\). Thus, in Figure 9(a) we show a new experiment following the trajectory \(q_{3}^{\ast}\) defined in Table 5 but with the second joint locked instead of using the proposed controller. The points where the passive joint would have collapsed and the system would have fallen down if the grasping torque was not increased are depicted with circles. This experiment demonstrates that: (1) inertial and centripetal terms are very small in comparison with the gravity ones, and hence corroborating our approximation (5) of \(|\epsilon|<<|g_{1}|\) implying that \(|\tau_{1}|\simeq|g_{1}|\); (2) without control we have to increase the grasping torque to be able of maintaining the equilibrium at the passive joint (at least 0.03 Nm which correspond to \(\Delta x=1\) mm). On the other hand, in Figure 9(b), we show the performance of the proposed controller along the same trajectory of the initial experiment. It is clear that the controller is able to minimize the gravity term \(|g_{1}|\) up to the same size of \(|\epsilon|\), and therefore, guaranteeing \(|\tau_{1}|\leq\tau_{c,l}\) while the posture is maintained.