# Model-Based Design and Evaluation of a Brachiating Monkey Robot with an Active Waist
## 2 The Reduced-Order Two-Link Dynamic Model of the Monkey Robot  
A monkey swinging in a tree, as shown in Figure 1a, is a spatial and complex dynamic motion. To effectively understand the essential dynamic characteristics of the system, we construct a reduced-order model of the system by considering the following issues: (i) The motion of the monkey is only considered in the sagittal plane, and any dynamics outside this plane are ignored; (ii) The \"simplest actuation scheme\" is adopted, where a fore/aft swing motion is generated by only one active and rotational degree-of-freedom (DOF) with torque \(\tau\). This DOF is located around the geometric center of the body, so the relative configuration of the \"upper body\" and the \"lower body\" can be actively controlled. This joint is referred to as \"joint 2\"; (iii) The upper and lower bodies are assumed to be rigid bodies with mass (\(m_{1}\) and \(m_{2}\)) and inertia (\(I_{1}\) and \(I_{2}\)); (iv) The mass of the rope is ignored. In this case, during the monkey swing motion, the rope can be treated as a massless rigid rod that connects to the fixed end by a passive revolute joint (referred to as \"joint 1\"), as shown in Figure 1b. The viscous damper with the damping coefficient \(c\) is added at this joint to represent the energy loss of the model; (v) During the monkey swing motion, the relative configuration between the hand of the monkey and the rope is fixed. Together with the issue described in (iv), the \"rope\" and the upper body of the monkey can be modeled together as a rigid-body link with length \(l_{1}\) and its center-of-mass (COM) is located close to the active joint (\(l_{1c}\) from the revolute joint); (vi) The motion and dynamic effects of the other arm are not considered because we intend to keep the reduced-ordermodel (or template) simple to extract the essential dynamics of the system. On the robot side, the robot is designed to have less inertia in the arms to fulfill the assumption of the template. After taking these considerations into account, the original complex system can be simplified as a two-link model, as shown in Figure 1b. The equations of motion (EOMs) of the reduced-order two-link model shown in Figure 1b was developed based on Lagrangian mechanics. The origin of the Cartesian coordinate system is located at joint 1. The angles \(\theta_{1}\) and \(\theta_{2}\) represent the configuration of the upper and lower links, respectively. The model can be regarded as a compound pendulum, and its behavior can be parameterized by two variables, \(\theta_{1}\) and \(\theta_{2}\). The quantitative formulation of the model is described as follows. First, the positions (\(\vec{P_{1}}\) and \(\vec{P_{2}}\)) of the masses (\(m_{1}\) and \(m_{2}\)) can be expressed as  
\[ \begin{array}{l}\vec{P_{1}}=l_{1c}cos\theta_{1}\hat{i}+l_{1c}sin\theta_{1} \hat{j}\\ \vec{P_{2}}=[l_{1}cos\theta_{1}+l_{2c}cos(\theta_{1}+\theta_{2})]\hat{i}+[l_{ 1}sin\theta_{1}+l_{2c}sin(\theta_{1}+\theta_{2})]\hat{j},\end{array}  \tag{1}\]  
and their velocities (\(\vec{V_{1}}\) and \(\vec{V_{2}}\)) can be derived as  
\[ \begin{array}{l}\vec{V_{1}}=l_{1c}\dot{\theta_{1}}(-sin\theta_{1}\hat{i}+cos \theta_{1}\hat{j})\\ \vec{V_{2}}=l_{1}\dot{\theta_{1}}(-sin\theta_{1}\hat{i}+cos\theta_{1}\hat{j}) +l_{2c}(\dot{\theta_{1}}+\dot{\theta_{2}})[(-sin(\theta_{1}+\theta_{2})\hat{i} +cos(\theta_{1}+\theta_{2})\hat{j}),\end{array}  \tag{2}\]  
The kinetic energy (\(T\)) and potential energy (\(V\)) of the model can be expressed as  
\[ \begin{array}{l}T=\frac{1}{2}m_{1}\dot{\vec{V_{1}}}\cdot\dot{\vec{V_{1}}}+ \frac{1}{2}{I_{1}\theta_{1}}^{2}+\frac{1}{2}m_{2}\dot{\vec{V_{2}}}\cdot\dot{ \vec{V_{2}}}+\frac{1}{2}{I_{2}(\dot{\theta_{1}}+\dot{\theta_{2}})}^{2}\\ V=-m_{1}g{l_{1}}cos\theta_{1}-m_{2}g[l_{1}sin\theta_{1}+l_{2c}sin(\theta_{1}+ \theta_{2})],\end{array}  \tag{3}\]  
Figure 1: (a) A monkey swings in a tree and (b) the simplified two-link model. where \(g\) represents the gravity constant.