# Model-Based Design and Evaluation of a Brachiating Monkey Robot with an Active Waist
## 4 Swing Dynamics of the Model and the MonkeyBot  
The reduced-order two-link model described in Section 2 is designed to simulate and predict the dynamic behavior of the MonkeyBot described in Section 3, which helps to ease the process of developing the control law and investigating the effect of the parameters of the model/robot. The behavior of the empirical robot is much more complex than that of the model; for example, energy loss is one of the important issues. We do understand a discrepancy between the model and the robot definitely exists, but this is a typical synergy of \"template vs. anchor\" [37] and it is tolerable as long as the model (i.e., template) is good enough to provide the correct behavioral trend of the robot (i.e., anchor). On the model side, in addition to the use of the active joint \(\theta_{2}\) to represent the actuator effect of the robot, the model has to include the energy loss term to simulate the energy loss of the robot. Here, to make the reduced-order model as simple as possible, we intend to use just one viscous damping dashpot mounted at the swing joint (i.e., at joint \(\theta_{1}\)) to represent the overall energy loss of the robot. While the physical specifications of the MonkeyBot, such as the mass, inertia, and lengths, can be directly mapped to the model parameters, the only parameter that needs to be determined is the damping coefficient \(c\) of the \"resultant\" viscous damping at the swing joint. The free swing test of the model/robot was conducted to determine the damping coefficient \(c\) of the model. The experimental environment was set up to record the robot's motion for quantitative analysis. Five markers were mounted in the setup: two on the top of two rods (i.e., two joint 1s), one on the COM, one on the arm, and one on the lower body. When the robot swung or brachiated on the rod, its sagittal-plane motion was recorded by a stationary high-definition (HD) camcorder (HDR-SR11, Sony, Tokyo, Japan). While the qualitative motion of the robot can be easily observed by the video, the quantitative motion of the robot can be computed based on the markers' positions in the sequential snapshots taken by the camcorder. In the experiments, the MonkeyBot was posed at two different initial heights (\(\theta_{1}=100^{\circ},115^{\circ}\), corresponding to \(90^{\circ}\pm 10^{\circ}\) and \(90^{\circ}\pm 25^{\circ}\) swings) and released. The robot started to swing, and the swing amplitude gradually decreased, owing to the energy loss. The profile \(\theta_{1}\) versus time \(t\) was recorded for data analysis. On the model side, models with various damping coefficients \(c\) were simulated to generate the corresponding \(\theta_{1}\) versus \(t\) profiles. Next, the root mean squared (RMS) errors between the peak value of trajectories in the first five periods of the robot's swing motion, and the models with various \(c\) values were computed. The damping coefficient \(c=0.0185\) was selected because this model has the smallest RMS error. Figure 4 plots the trajectories \(\theta_{1}\) versus \(t\) of the robot and the model with this damping coefficient, and the subplots (a) and (b) correspond to the initial condition \(\theta_{1}=100^{\circ}\) and \(\theta_{1}=115^{\circ}\), respectively. Both plots show a decent trajectory match between the robot and the model, and the RMS errors are \(0.42^{\circ}\) and \(0.44^{\circ}\), accordingly. The frequencies shown in Figure 4 represent the natural frequency of the model and the robot. According to a paper [38], the stride frequencies of different-sized animals have a certain relation to their body mass, and those frequencies also match the natural frequencies if the animals are modeled as the spring-mass systems (i.e., resonance region) because the brachiating motion of the robot is basically a pendulum motion that has a certain natural frequency. To actuate the waist of the robot/model at a frequency similar to the natural frequency is similar to driving the legs of the animals at a frequency similar to the natural frequency.