# Modeling the Dynamics of Two Cooperating Robots
## 1 Introduction  
The A team of robots used to complete a difficult or complex task together is called a robot formation. An example of a task that is impossible for one mobile robot to implement is the transport of objects of large size or significant mass. This type of problem concerns both the living world and technology. In the natural environment, a number of solutions can be found, an example of which is the transport of objects by ants or the very similar problem of two people moving a table. The cooperation of robots is the subject of a number of works, concerning, e.g., the problems of pushing a box [1], pushing a large load with its own support in the form of wheels [2], tugboats moving a tanker [3], transporting an object above the robots [4], or moving of an object by robots next to it [5] or two robots transporting a beam [6]. The authors of such works most often use, as transport robots, two-wheeled robots [7], n-wheel robots [6], and tracked vehicles [8]. The tasks performed by systems or formations of transport robots concern not only the movement of large masses, but also the cooperation of manipulators placed on vehicles [9], or performing behavioral tasks in an unknown environment based on signals from cameras or scanners [2]. In the literature, we found solutions based on considerable simplifications, e.g., robots and the object being moved are modeled with concentrated masses [8], as well as works considering cooperation and interaction between the robot and the object [4]. An alternative approach that simplifies the description of the kinematics of a complex system is the introduction of one-dimensional systems (unidirectional interaction) in the form of ropes or chains. We have here a number of examples such as construction cranes on ropes transporting an object [10], a team of helicopters transporting a tank [11], and transporting an object using drones [12]. Another approach is to use omnidirectional wheels, which allows a vehicle without holonomic limitations to be obtained [13]. In the literature can also be found solutions consisting of connecting objects transported with the use of spring shock absorbers, explicitly allowing for possible displacement in a kinematic pair [8]. This approach allows the use of virtual structure algorithms or leader tracking [1, 14]. An interesting solution is a structure [7, 15] that allows a formationof robots to transport a large object thanks to the connection of the object with the center of rotation of the two-wheeled mobile robot by means of an element equipped with two swivel joints. The construction of the dynamics of models of cooperating robots transporting an object is a current topic, as evidenced by publications [16; 17]. The work [18] presents a model of a system of three mobile robots connected with the use of swivel joints with a large-size plate being transported. The most popular solution that allows the movement of large masses is a tractor-trailer type system. The tractor is an active unit containing drives and control, while the trailer is a passive element connected to the tractor by articulation. In modeling dynamics tasks of this type of nonholonomic systems, approaches are used both with the assumption of movement without slippage and with consideration of lateral and longitudinal slippage of wheels [16; 17]. The presented solution assumes no slippage of the driving wheels, similarly to the mathematical model of a single nonholonomic mobile robot. The principles of modeling with such assumptions can be found in article [14]. Modeling of mobile robots using the Lagrange and Newton-Euler equations can be found in [19], an approach using [PERSON]'s approach is found in [20], and generalized rules for modeling systems with holonomic and nonholonomic constraints are included in [21]. It should be noted that the assumption of rolling without slippage is used when the slippage is so small that it can be ignored. This happens when the velocities are low and there is sufficient friction between the wheels and the ground.