## 2 Methods
### Hybrid Dynamic Model of Exoskeleton and Child's Body  
Since the exoskeleton is designed for children between five and eight years old [38], the kinematic chain formed by the child's body and the exoskeleton is computed to analyze the motion of a natural walking process, where the balance of the body must be kept and guided by the movement of the swing leg, while the support leg joints are locked by the electrical brakes. Figure 3 shows a kinematic model of the hybrid system, where the blue and red dots represent the limb center of masses and the joint positions, respectively. In order to numerically simulate the dynamic behavior of the hybrid system formed by the child and exoskeleton, two dynamic models are built to reproduce the walking during the single support phase in the sagittal plane. The first model uses the Simscape Multibody software from Matlab(r), which allows for modeling and simulating the multibody mechanical systems without extracting mathematical equations. This numerical model is used to test the control strategies in a relevant simulation environment. The second model is built using mathematical expressions that allow for designing a control strategy that ensures closed-loop stability. Figure 1: Exoskeleton mechanisms. Figure 2: Exoskeleton main design. Given the short time span of the double support in bipedal walking, the dynamics of this phase is neglected and considered as an instantaneous event [42]. Here, the same approach is used; therefore, the focus is on the single-support walking phase, where one leg is over the ground while the other leg is in forward motion. Thus, the mathematical model considers the swing leg as an open kinematic chain with three actuated degrees of freedom (DoF) and the support leg as an inverted pendulum with the joints locked by the electrical brakes and the contact between the support leg-end and the floor as a passive pivot. Even though the support leg has actuated joints, the idea of activating the brakes to impose motion restriction is proposed to reduce the torque requirements and energy consumption, leaving the support leg motion as a passive element. Since the motion is analyzed in the sagittal plane, the Lagrange method is used to find the governing equation of the planar mechanism. In this scenario, the angular position of the swing leg's hip, knee, and ankle are defined, as shown in Figure 4, with the generalized coordinates \(\mathbf{q}:=[\theta_{1}\ \theta_{2}\ \theta_{3}]^{T}\), respectively. Those coordinates are used to define the Euler-Lagrange equation as follows:  
\[D(\mathbf{q})\ddot{\mathbf{q}}+C(\mathbf{q},\dot{\mathbf{q}})\dot{\mathbf{q}}+G (\mathbf{q})=\tau, \tag{1}\]  
where \(D(\mathbf{q})\) is the inertial matrix; \(C(\mathbf{q},\dot{\mathbf{q}})\) is the matrix of Coriolis and centripetal forces; \(G(\mathbf{q})\) is the matrix of gravitational effects; \(\tau:=[\tau_{1}\ \tau_{2}\ \tau_{3}]^{T}\) is the vector of external torques; and \(\dot{\mathbf{q}}\) and \(\ddot{\mathbf{q}}\) are the vectors of angular velocities and accelerations, respectively. To set a validation test, the natural and forced responses of the model are simulated using the body parameter shown in Table 2. First, the natural response is obtained by letting the mechanism be in its initial condition, different from the rest position, and by releasing it without torque input. The mechanism shows its natural behavior when approaching an equilibrium point, as shown in Figure 5a. This figure allows us to identify the resting position of the mechanism and shows the damping effect on the joint's behavior. The second validation test looks for a forced response. In this case, the mechanism with similar initial conditions to the previous test is exposed to a constant input torque of 2 Nm in each joint. The results of this simulation are shown in Figure 5b, where how the mechanism approaches a final position is visible, which is different than the resting position of the  
Figure 3: Kinematic model system. natural response.