Propelling projectiles with a controlled speed and trajectory is a technological challenge having lots of applications ranging from kicking soccer balls to launching rockets or satellites, including testing military ammunitions. In this introduction, we focus on propulsion techniques having strong accelerations, so that an important speed can be obtained in a short distance. Common propulsion techniques of this type are among the following ones:

#### 1.1. 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·h

${}^{-1}$ = 36 m·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.

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·h

${}^{-1}$ = 29 m·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 Adidas [

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 Adidas 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·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·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.

Table 1 shows a comparison between existing ball launchers, including humans. These solutions are very different. This comparison is done considering the weight and size for each system, as it is a strong constraint in our case study. It is important to note that the energy transferred to the ball is close to the maximum value for all systems, except for the robot leg. This one is based on an industrial actuator able to carry heavy loads, and largely oversized for launching a soccer balls in terms of torque and power. Because it is not used at full power, its ratio of energy transmitted to launcher volume is very low compared to other solutions. However, this solution takes too much space due to the rotation of the leg and is not relevant for a small size launcher.

In conclusion of this section, the most relevant launching systems in terms of energy transferred to the ball for a given actuator volume are reluctance coil guns, with a ratio of energy transmitted to launcher volume better than rotating inertial launchers by a factor $2.5$, and better than most humans by a factor 5.

Since these electromagnetic coil guns seem to be the most promising solution for launching balls, this paper will only focus on improving that solution in order to maximize this energy transfer without changing the volume and the weight of the actuator.

#### 1.2. Reluctance Coil Guns: A Ball Launcher That Can Be Optimized

Even if reluctance coil guns are a relevant solution for kicking soccer balls efficiently, it is important to note that they are not very efficient in terms of energy conversion. In [

11], electrical energy for the coil gun is stored in a capacitor having a capacitance value

C = 4700

$\mathsf{\mu}$F under 425 V. Stored electrical energy is equal to:

Consequently, the ratio of ball kinetic energy to the input electrical one is only $7\%$, and the ratio of the overall mechanical transmitted energy (including the kinetic energies of the iron rod, the lever and the ball as explained later) to the input electrical one is $12\%$. However, the energy necessary for kicking like human soccer players is already stored in the capacitor. This means that if a robot’s kick is 10 times less powerful than a human one, it is not an issue related to available energy, but it is a problem of inefficiency of energy transfer in reluctance coil guns.

Optimizing this energy transfer can be done in two main ways without changing the size and the weight of the launcher. The first one is to adjust the initial position of the plunger, and the length of its non-magnetic extension. [

12] shows that energy transmission can be increased by

$70\%$ using this technique compared to the reference case presented in [

11]. This optimization is interesting because nothing is changed on the coil gun structure and size; it is only an optimization of initial conditions and a plunger parameter adjustment.

A second way of improving the energy transfer of a coil gun is to modify its inner structure by splitting the coil and the energy storage capacitor into several ones [

7,

13], without changing the overall quantity of coil copper and the overall capacitance value. Instead of sending an energy pulse to a single coil, a sequence of smaller energy pulses will be sent to the different coils propagating the magnetic force along the coil as the plunger enters it. The number of coils and the triggering sequence are the parameters to be optimized.

This paper focuses on this second method for optimizing the energy transfer in a reluctance coil gun. It is divided into three sections:

Section 2 recalls the principles of coil guns.

Section 3 describes 4 mechatronic coupled models of reluctance coil guns. All these coil guns use the same coil copper quantity and have the same overall electrical energy storage capacitance, but they have, respectively, one, two, three and four coils. The electromagnetic part of each model has been implemented using

FEMM 4.1, a finite element electromagnetic simulation tool, and Matlab Simulink is used for modelling the electrical and mechanical parts.

Section 4 presents results, which are discussed in order to conclude on the most relevant coil structure for maximizing the ball speed and the energy transfer of the reluctance coil gun, while maintaining a high level of robustness.