# Investigation of Zero Moment Point in a Partially Filled Liquid Vessel Subjected to Roll Motion
## 2 Methodology
### Mathematical Modelling  
The mathematical modelling employed in this study had three aspects: the ZMP description of a dynamic liquid region, computationally capturing the dynamics of the air and liquid regions in a domain subjected to periodic motion, and extracting essential information from the computational model for the calculation of ZMP of the dynamic liquid region. The ZMP [14] methodology describes the stability of bipeds by defining a point where the gravitational force acting on the biped balances the resultant horizontal inertia, which results in no moment along the horizontal plane at the point of contact of the biped with the ground. The ZMP can be calculate using the approach presented by [PERSON] [14] along with the table cart method [15; 16]. Between these two techniques, the table cart method can be seamlessly integrated to the liquid domain due to its dependence on the Center of Mass. Figure 1: Computational domain employed in the study. Figure 2: Grid independence study comparing center of mass (COM), zero moment point (ZMP) and pressure P, for coarse and fine meshes. A basic outline of a liquid tank carrying robot is depicted in Figure 3A, highlighting the CoM and the free surface between the air and liquid regions that is subjected to extreme topological changes during the robot's motion. Figure 3B shows the cart table description of this robot where a massless table is arranged within a robot and a cart is shown moving on it. Here the cart containing mass \"m\" at position (x, y) on the table represents the liquid domain of the container, while the table is considered to have the same shape and dimensions of support polygon as that of a robot foot print [27]. The principle of ZMP was adapted from [PERSON] [15] and [PERSON] [27], who employ the following relationship to locate the ZMP with respect to the origin:  
\[ZMP_{x}=x-\frac{y_{c}}{g}\ddot{x} \tag{1}\]  
where \(g\) is the gravitational constant, the location of CoM of the cart along the horizontal axis measured from the origin is \(x\), while \(\ddot{x}\) is its acceleration and \(y_{c}\) is the height of CoM of the cart. For the computation  
Figure 3: Liquid carrying robot: (**A**) simplified model, and (**B**) table cart description. of _ZMP_ for a liquid-carrying robot, the value of \(y_{c}\) and \(x\) are extracted from CoM, while \(\ddot{x}\) is obtained from a second order time differencing scheme. The interDyMFoam (\"DyM\" stands for \"Dynamic Mesh\") solver of OpenFOAM (OF), generally employed to simulate problems involving multiphase flow coupled with moving domains, was modified for this research to calculate the ZMP while solving the three-dimensional Navier-Stokes equations over two fluid domains consisting of incompressible air and water by employing the Volume of Fluid (VOF) approach over a finite volume discretization. The VOF method describes each phase by the fraction \(\alpha_{i}\) of the volume of the \(i\)'th cell that is occupied by the dispersed phase. The InterDyMFoam solver uses a PIMPLE approach, which combines both the Pressure Implicit with Splitting of Operators (PISO) and the Semi-Implicit Method for Pressure Linked Equations (SIMPLE) algorithms. Its main structure is inherited from the original PISO, but it allows equation under-relaxation to ensure the convergence of all the equations at each time step, as outlined by [PERSON] [25]. The fluids (air and water) are treated as incompressible, immiscible, and isothermal, while the interface between them is captured using a VOF technique coupled with a moving mesh function. To resolve fluid behavior under dynamic conditions, the conservation of mass and momentum equations are first solved over the liquid and air regions.