# Quadrupedal Robots Whole-Body Motion Control Based on Centroidal Momentum Dynamics
## 2 Methods
### Application to Quadrupedal Robot Dynamic Motion
#### 2.3.1 Walking Pattern based on Capture Point Dynamics  
The walking pattern is defined as a set of time series of joint angles for the desired walking [25]. To ensure stable walking, a reduced model of the robot is indispensable. Recently, the linear inverted pendulum model (LIPM) required for the control of bipedal walking robots rise in popularity [26; 27; 28; 29; 30]. Once the walking pattern is using simple models, such as the LIPM, the motion of the robot can be equivalent to the motion of the centre of mass of the robot and the motion of the supporting foot. In general, the walking pattern was generated with a predetermined footstep position. If a set of time series of robot footstep positions is given, the set of time series of the CoM trajectory of the robot is the general walking pattern [31]. In the simplified model, when a quadruped robot is supporting its body on two legs, the behaviour of a pair of legs acting in unison can be represented by an equivalent virtual leg [32]. In this paper, we extend the concept of virtual leg. No matter how many legs in contact with the ground, it can be equivalent to a virtual leg, as depicted in Figure 4. GRFs \(F_{i}\) at the contact point can be mapped to the CoM though the matrix \(B\). \[\underbrace{ \begin{bmatrix}I_{3}&\cdots&I_{3}\\ \left[P_{1}-r\right]\times&\cdots&\left[P_{i}-r\right]\times\end{bmatrix} }_{B} \begin{bmatrix}F_{1}\\ \vdots\\ F_{i}\end{bmatrix} = \begin{bmatrix}F\\ \tau\end{bmatrix}  \tag{15}\]  
where \(F= \begin{bmatrix}f_{x}&f_{y}&f_{z}\end{bmatrix} ^{T}\), \(\tau= \begin{bmatrix}\tau_{x}&\tau_{y}&\tau_{z}\end{bmatrix} ^{T}\). We assume that the virtual foot is located at the midpoint between the two supporting foot, which strike the ground recnetly. Its dominant dynamics can be represented by a single inverted pendulum, which is comprised of a point mass and a massless telescopic leg in contact with the ground. According to LIPM, the relationship between the CoM and \(P_{vl}\) can be expressed as follows:  
\[ \begin{bmatrix}p_{x}\\ p_{y}\end{bmatrix} =\frac{1}{\bar{z}+g} \begin{bmatrix}x(\bar{z}+g)+z\bar{x}\\ y(\bar{z}+g)+z\bar{y}\end{bmatrix} -\frac{1}{m\left(\bar{z}+g\right)} \begin{bmatrix} \tau_{y}\\ \tau_{x}\end{bmatrix}  \tag{16}\]  
where \(x\), \(y\) and \(z\) denote the horizontal, lateral and vertical CoM position, \(p_{x}\) and \(p_{y}\) denote the horizontal and lateral position of the virtual leg, \(\tau_{y}\) and \(\tau_{x}\) denote the torque about the \(y\) axis and \(x\) axis of CoM, \(m\) is the total mass of the quadruped robot and \(g\) is the acceleration due to gravity. The Linear Inverted Pendulum Model, described by [PERSON] et al. [26; 27] assumes that the height of the CoM remains constant and the change in the rotational momentum about the CoM is zero. Figure 4: The simplification of the quadruped robot to the Linear Inverted Pendulum model. Based on these assumptions, the dynamic equations of the robot are linear and decoupled, and the relationship between the CoM and \(P\) can be written:  
\[ \begin{bmatrix}\dot{x}\\ \dot{y}\end{bmatrix} =w^{2}\left( \begin{bmatrix}x\\ y\end{bmatrix} - \begin{bmatrix}p_{x}\\ p_{y}\end{bmatrix} \right) \tag{17}\]  
where \(w{=}\sqrt{g/z_{0}}\)  is the natural frequency of the pendulum. [PERSON] et al. derive the unstable part of LIPM from its orbital energy and named it the Capture Point [33]. This is the point on the floor where the robot (modelled as an LIPM) has to step to come to a complete rest. Equivalent values of the Capture Point have been analysed by [PERSON] as the 'extrapolated center of mass' in [34] and DCM by [PERSON] et al. in [35]. The Capture Point is comprised of the horizontal CoM position, velocity, and the natural frequency of the pendulum. It can be expressed as:  
\[ \begin{bmatrix}\xi_{x}\\ \dot{\xi}_{y}\end{bmatrix} = \begin{bmatrix}x\\ y\end{bmatrix} +\frac{1}{w} \begin{bmatrix}\dot{x}\\ \dot{y}\end{bmatrix}  \tag{18}\]  
The equation of the Capture Point can be rewritten as  
\[ \begin{bmatrix}\dot{x}\\ \dot{y}\end{bmatrix} {=}-w\left(