# Model-Based Design and Evaluation of a Brachiating Monkey Robot with an Active Waist
## 4 Swing Dynamics of the Model and the MonkeyBot
The model has a natural swing frequency of around \(\omega_{\rm n}=0.714\) Hz when \(\theta_{1}=90^{\circ}\). If the model starts with the at rest configuration and the joint 2 is actuated with a swing signal at the same frequency, the model can gradually increase its swing magnitude (\(\theta_{1}\)), as shown in Figure 5a. This control scheme is hereafter referred to as the \"open-loop\" method. Figure 5a also reveals that the open-loop method can quickly build up energy to increase the swing amplitude \(\theta_{1}\) to \(90^{\circ}\pm 50^{\circ}\) within 15 s. Because the double pendulum is a nonlinear system and the natural frequency changes when the configuration changes, its swing motion does not gradually converge to a certain amplitude, but exhibits amplitude variations. In addition, the model has less capability for disturbance rejection. The closed-loop strategy is developed to compensate for the drawbacks of the open-loop strategy. When joint 2 is actuated with the same rotation direction as joint 1 (\(\theta_{1}\)), the energy generated from joint 2 can be injected into the model. Figure 6 demonstrates this phenomenon. It can be assumed that the model initially swings toward the left (\(\theta_{1}>0\)) and joint 2 is fixed at a positive angle (\(\theta_{2}>0\)), as shown in Figure 6a. After the model reaches its highest left position, as shown in Figure 6b, and starts to change motion direction (\(\theta_{1}<0\)), if joint 2 is actuated to have a negative angular velocity (\(\theta_{2}<0\)), as shown in Figure 6c, the swing motion of joint 1 (\(\theta_{1}<0\)) can be maintained to overcome the damping loss or even increased to enlarge the swing amplitude. When the model reaches its highest right position, as shown in Figure 6d, joint 2 has a negative value (\(\theta_{2}<0\)) and is ready for the next actuation (\(\theta_{2}>0\)) after the model starts to change motion direction (\(\theta_{1}>0\)). This configuration is identical to the one shown in Figure 6a; thus, the swing motion can be periodically generated. Figure 5b plots the trajectories of the joint 1 angles of the model versus time when the model is actuated with this control scheme. This scheme is hereafter referred to as the \"closed-loop\" method. The model can successfully build up the swing amplitude \(\theta_{1}\) to \(90^{\circ}\pm 10^{\circ}\) within 4 s; however, the amplitude seems not to increase until the end of simulation (15 s). Moreover, this method was not functional on the robot when it started at a rest condition \(\theta_{1}=90^{\circ}\) for two reasons: First, the joint angle measurement (\(\theta_{1}\)) of the robot has noise, and this may lead to a wrong actuation initiation, especially when \(\theta_{1}\) is close to \(90^{\circ}\). Second, empirically, the actuation of joint 2 (\(\theta_{2}\)) needs time to finish its configuration change. When \(\theta_{1}\) is close to \(90^{\circ}\), the control algorithm may quickly switch the motion direction of joint 2, and in this case, the robot cannot correctly initiate its periodic motion pattern. Therefore, the hybrid control scheme  
Figure 4: A simulation (blue solid curve) of the two-link model and the experimental result (red dashed curve) of the MonkeyBot with same damping coefficients and two different initial heights: (**a**) \(c=0.0185\), \(\theta_{1\_0}=100^{\circ}\) and (**b**) \(c=0.0185\), \(\theta_{1\_0}=115^{\circ}\). Figure 5: A simulation of the two-link model with (**a**) open-loop and (**b**) closed-loop control methods. is utilized: The model/robot firstly uses an open-loop control strategy. After the swing amplitude approaches the maximum value, the control algorithm switches to a closed-loop to increase the motion control stability. The overall control flow chart is shown in Figure 7.  
While the size and mass of the upper body is, in general, fixed and determined by the characteristics of the components, the size and mass of the lower body has some freedom to be adjusted for better dynamic performance of the robot itself.