## 1 Introduction  
The concept of robots performing domestic tasks is still under development [1; 2]. In particular, to transfer fluids between vessels by robotic manipulators is the general scope of this research. A full robotic fluid pouring task based on physical modelling in unconstrained settings is a complex task. Vessel identification, grasping, fluid dynamics and solid body kinematics are some of the modelling tasks to perform by robotics. Coupling fluid dynamics and solid kinematics for pouring liquids is the particular subject of this work. The purpose is to provide a robust control for pouring using a free surface model that can overcome spillage regardless of the boundary conditions in tested conditions. Fully comprehensive reasoning of pouring liquids can be found in [1]. There, the problem is described as a general ontology, where physical dynamics are set aside, and the logic of pouring is conducted so further approaches of pouring can follow the same logical principles. However, quantitative approaches are case-dependent. By applying physical governing equations and constraining them to a given geometry, space and the relevant computational limitations, any particular model will arise with new issues. The problem of a flow over a free overall in an open channel has been studied widely by assuming parallel streamlines of the flow and no compaction of the nappe [2; 3]. This assumption turns to be very efficient for tracking the flow trajectories at the expense of fine detail visualization. On the other hand, pure fluid simulation methods have been developed to leverage the potential of novel parallel computation using GPUs to perform fast fluid simulations like the smoothed particle hydrodynamics method (SPH) [4], or the volume-of-fluid method [5]. Both methods must deal with their own limitations (i.e., boundary conditions), but both have proven consistency in free surface modelling [6; 7]. Specially, in the case of SPH, fluid particles' motion properties are processed using the GPU to avoid CPU overheads [4]. SPH was implemented and discussed in the work described herein. Recent work has developed adaptive pouring of liquids method [8; 9] based on particle position instead of a free surface approach like the present work. Using a SPH flow, the method in [9] relies in the position of fluid particles. The approximation of the particle positions is supported by a calibration stage where an added discrepancy reduces the cost of spillage during pouring. The ability to classify collected versus spilled fluid is the main contribution here. Other training methods of pouring have been developed recently [10; 11]. A training-based approach using thermal imaging for feedback is used in [10]. It determines liquid volumes using neural networks and a proportional-integral-derivative (PID) controller for robot kinematics. Neural networks eventually have to be reinforced with new training when pouring phases switch between fluid regimes. Nonetheless, the usage of a PID controller is interesting for our work because it shares the idea of controlling the final position of the free fall. In our work, the PID controls the fluid free surface rather than only the vessel geometry, as in [10]. The work in [11] trains the pouring task by learning imitations. Fluid dynamics are not part of the method; rather, a trial and error method is used by the robot to learn from human demonstrations. Typically, rigid body and fluid motion models can be merged together in a closed loop controller to transfer liquid from one vessel to the other at constant outflow velocity with relative ease using free falling parabolic trajectories [12]. However, maintaining constant outflow requires a dedicated control, and moreover, initial and sometimes final phases of pouring are inherently non stationary. Another challenge to consider is the issue of liquid adhesion to the vessel crest. Usually, this is avoided by using a sharp spilling edge and/or guaranteeing a minimum velocity at the outlet edge.