## 5 Results and Discussion  
Most robotic structures that replicate the gait of human legs consist of rigid elements the are equivalent to biological bones, which are connected by revolute joints. The revolute joints act in a similar fashion to anatomically correct knee and hip joints; however, they have one main caveat: revolute joints have only one axis of rotation [8]. Anatomically inspired joints consist of rigid and flexible elements suspended in a network of muscles, tendons, and ligaments that work in unison to allow the joint to flex, move and contort [23; 35]. The current work proposes an alternative approach: A design pipeline that develops flexible-rigid structures that recreate the kinematics of human legs.  
Figure 8 shows the comparison between the hip and knee rotation angles during a full gait event. The OpenSim 3.0 prediction is shown in green, the flexible-rigid manipulator is shown in red and the human leg testing in blue. Figure 8a shows the hip rotation angles, and Figure 8c the knee rotation angles along the sagittal plane. The convention used here to define the rotation angles for the hip and for the knee is shown in Figure 7d--step 5. To compute the angles we associated a vector with the direction of each component of the manipulator (\(V_{hip}\) and \(V_{knee}\)) and compute the angle between that component and the unit vector \(k\) associated with the \(z\) axes of the Cartesian coordinate system, which is assumed to be the resting position Figure 7a--step 1 and Figure 7d--step 6:  
\[\theta_{i}=\arccos(\frac{\vec{V}_{i}\cdot\vec{k}}{|\vec{V}_{i}|*|\vec{k}|}), \qquad i\in\{hip,knee\} \tag{7}\]  
All three systems demonstrated similar kinematics during the four stages of gait. Figure 8b show the absolute errors between the average measured and predicted hip angles and Figure 8d show the absolute errors between the average measured and predicted knee angles. The error between the human leg and the robotic prototype is shown in blue and the error between the human leg and the OpenSim 3.0 simulation is shown in red. It should be noted that the errors are very small. Due to the pelvic-inspired component suspended and confined to a mounted position, there is limited variance in the range of motion resulting in a lower absolute error compared to the knee. Both the kinematics of the femur-inspired element and the large angular displacement shown in each stage of gait (Figure 7) effect the absolute error of the knee (Figure 8d). Table 2 indicated that the movements for both Stage 1 and Stage 3 were both the most dramatic causing the largest  
Figure 8: Simulated and measured angles of the hip (**a**) and knee (**b**) joints during the four stages of gait, and (**c**,**d**) the absolute errors, between angles across multiple trials (see Table 2). The shaded range represents the lower and upper limits of motion through simulated trials with the chosen configuration in a solid line.  
error for both joints. The results between the _OpenSim 3.0 model_ vs. _Prototype_ and _Human_ vs. _Prototype_ show the accuracy of replicating the recorded motions with this system. Figure 9 tracks the trajectory of the end effector of the femur and tibia along the sagittal axis. The Cartesian coordinates (\(\vec{v}\) and \(\vec{u}\)) are computed using Equation (7) for each joint angle (\(\theta\)) from the start and end of the gait experiment. This demonstrates the kinematic similarities of the human, simulation, and prototype gaits.