# Optimisation of Energy Transfer in Reluctance Coil Guns: Application to Soccer Ball Launchers
## 1 Introduction
### Comparison of Existing Soccer Ball Launching Systems  
In this article, we focus on a case study having strong constraints: a real size soccer ball kicking system embedded in robots participating in the RoboCup (the autonomous robot soccer World Cup) in Middle Size League (Figure 1). This competition puts strong constraints on the robot size and weight, requiring to choose the launcher having the highest ratio of energy transmitted to launcher volume (Figure 2). The balls used for the competition are real soccer balls (diameter 22 cm) having a weight equal to 450 g. Size constraints on the robots are a maximum width and length \(L=W=52\) cm and a maximum height \(H=80\) cm. Considering these robots are all using omnidirectional propulsion with 3 or 4 wheels, the space for embedding the kicking system is very small and cannot exceed a length of 30 cm and a width of 20 cm, as shown in Figure 3. Moreover, its height must be limited because the mass centre of the robot must be as low as possible in order to allow high accelerations. A comparison of existing soccer ball launchers is first proposed in this section. Chemical propulsion has not been considered because they are dangerous and not reusable. Moreover, Ref. [1] shows that the size of a chemical launching system is equivalent to an electromagnetic one. The only advantage is that it does not need any power supply, but it is not a problem in our case considering that the robot has one. The reference for understanding ball kicking systems is the human. Professional soccer players shots can reach 130 km\(\cdot\)h\({}^{-1}\) = 36 m\(\cdot\)s\({}^{-1}\), corresponding to an kinetic energy equal to \(E_{K}=290\) J. As shown in Figure 4, the surface swept by the leg during a kick is important and approximately equal to one third of the surface of a circle having a radius \(R=80\) cm. The mass of a soccer player leg is approximately equal to \(m=20\) kg. Figure 1: RoboCup 2019—Sydney—Australia. Figure 3: Cut view of RCT robot 2020. Figure 2: RoboCup RCT robot 2020. Mechanical propulsion is also one of the most commonly used methods for propelling a soccer ball. A commercial system, shown in Figure 5, is able to launch soccer balls at a maximum speed of 105 km\(\cdot\)h\({}^{-1}\) = 29 m\(\cdot\)s\({}^{-1}\) corresponding to an kinetic energy equal to \(E_{K}=190\) J, using two 10 kg cylinders coated with rubber. The propulsion part (cylinders and motors) of the system weighs 15 kg and its dimensions are \(W=65\) cm and \(L=25\) cm and \(H=25\) cm. Another mechanical system that can be used for propelling a soccer ball is a robotic leg powered by a motor as shown in the kicking from Aiddas [9] or as described in [10]. These solutions, using a robotic arm, have multiple degrees of freedom [10] or a set of rotary and linear spring-loaded actuators [9]. The Aiddas solution shown in Figure 6 is composed of a 0.6 m robotic thigh rotating at 85 RPM and 0.6 m shank rotating at a maximum speed of 165 RPM, leading to a maximum ball speed of 21 m\(\cdot\)s\({}^{-1}\) corresponding to an kinetic energy equal to \(E_{K}=100\) J. However, the whole system is heavy (more than 50 kg) and its dimensions are important as it sweeps a 1.2 m radius cylinder. These mechanical systems are interesting for simulating a football player leg [9,10] and for training humans in real conditions. However, embedding them in a RoboCup robot is very difficult. This is probably the reason why the RoboCup research community is mainly focused on electromagnetic launchers, and more especially on variable reluctance coil guns. Authors in [11] introduced a design of a variable reluctance coil gun (Figure 7) that is used in the RoboCup robots of the 2019 World Champion team. This actuator's dimensions are \(L=30\) cm, \(W=9\) cm and \(H=9\) cm, its weight is 4.5 kg and ball maximum speed can reach 11.4 m\(\cdot\)s\({}^{-1}\), corresponding to a kinetic energy equal to \(E_{K}=29\) J. Output speed is smaller than mechanical design, but volume has been divided by factor 20 compared with an inertial rotating launcher.