There are enormous asteroids formed at the beginning of the solar system. Most of them are rich in minerals, and some of them are dangerous to the Earth [1,2]. Thus, exploring the asteroids is of great significance, with the purpose of obtaining great economic effectiveness, and protecting the Earth. Landing on an asteroid is a great step towards exploring these asteroids, and its merits include: 1) learning about asteroids in situ; 2) changing dangerous asteroid orbits; 3) using asteroids as platforms to observe other celestial bodies; 4) using asteroids as carrying devices; 5) establishing communication stations on asteroids, and 6) capturing asteroids with minerals and bringing them back to earth [3,4]. However, these merits can only be carried out after a lander is able to land safely on an asteroid. Landing safely or not is represented by the landing performance, and the landing performance is usually evaluated by both the overloading acceleration and stability time, which are included in the landing dynamic model. Thus, it is of great significance to estimate the key parameters that will induce a good landing performance, by building a landing dynamic model. Besides, the values of these key parameters can guide the design of the landing mechanism.

Presently, most of the landing dynamic models are designed for lunar landing mechanisms, but are not suited to small body landing mechanisms, because of different landing environments, with different landing strategies [5-11]. The Europe Space Agency (ESA), National Aeronautics and Space Administration (NASA), and Japan Aerospace Exploration Agency (JAXA) have developed small body landing mechanisms, but the landing dynamic model is rarely used to evaluate the landing performance, and to guide the design of the landing mechanism [12-16]. They just evaluate the landing performance by simulation with the given value of the parameters, but don’t show how these values are educed [17].

In this paper, an Asteroid Landing and In situ Exploring (“ALISE”) landing mechanism for asteroid with soft surface is presented. The landing dynamic model in the first turning stage is built by Lagrange equation. It can be found that there are three key parameters (retro-rocket thrust T, damping element damping c₁ and cardan element damping c₂) that will influence the landing performance. Thus, the values of these three parameters must be estimated, to achieve excellent landing performance. The paper firstly estimates the value of T, by considering that the retro-rocket will prevent the landing mechanism from overturning, by counteracting the turning energy. Secondly, the value of c₂ is estimated by a simplified dynamic model. Thirdly, after defining the values of T and c₁, the value of c₂ is calculated by the landing dynamic model, with the objective function of having the shortest stability time, and the constraint condition that the overloading acceleration is less than 10g. Lastly, the validities of these three values are tested, by simulating the landing performance in Adams software.

The ALISE landing mechanism aims at asteroids with soft surface(especially C type asteroid), which are softer than other asteroids, because of containing organic materials and amino acids [18,19], and the design is inspired by the Rosetta lander and the ST4/Champollion lander [20-22]. The ALISE landing mechanism includes a three-leg landing gear, and an anchoring system that is designed to avoid the flying away of the lander under low gravity. The landing gear contains landing foot, landing legs, cardan element, damping element, and equipment base. The damping element is realized by electromagnetic damping, which is a new technology in deep space exploration. This damping mode has the merits of high efficiency, easy control, adjustable damping, and so on, and has been used in the Rosetta landing mechanism [20]. The anchoring system contains anchoring element, propulsion element, rewinding element, and cushion element. The schematics and the performance parameters are shown

separately in Fig. 1 and Table 1. Besides, the retro-rocket is fixed on the upper surface of the equipment base, to supply thrust towards the landing slope, to prevent the rebound of the landing mechanism, when landing. There is no sign of the retro-rocket in the schematics, because it is a part of the control system, but not a part of the mechanical structure.

There are awls beneath the landing feet to preventing sliding, and contact switches inside the landing feet, to generate landing signals. The cardan element has the functions of both absorbing the horizontal impact when landing, and adjusting the attitude of the equipment base after landing, while the damping element is just used to absorb the vertical impact. The anchoring element, which connects to the rewinding element via a thread, will be pushed rapidly into the asteroid by the propulsion element. At the time of penetration of the anchoring element, the rewinding element will quickly rewind the thread. When the thread is instantaneously tense, the cushion element, which is composed of a compression spring, will absorb the impact, to protect the rewinding motor. The retro-rocket will be activated at the time of landing, and supply a constant force lasting about 5 seconds toward the equipment base, to prevent the landing mechanism from rebounding.

The surface of the asteroid with soft surface is soft. When landing, the awls will penetrate the asteroid some depth, to prevent sliding of the landing mechanism, and the retrorocket would be activated, to counteract the rebound of the landing mechanism. Then, the landing mechanism will turn clockwise or counterclockwise around the feet’s anchoring points. The interaction between the awls and the surface is three-dimensional and very complicated, and few reasonable models can express this interaction accurately. Furthermore, the dynamic parameters in the first turning stage are enough to express the landing performance. Thus, the paper only develops the landing dynamic of the first turning stage. There are two-dimensional and three-dimensional landing dynamic models. But considering that the two-dimensional model is successfully adopted by the lunar lander, and that it is simpler than the three-dimensional model, a twodimensional landing dynamic is built for the ALISE landing mechanism.

Some hypotheses are made in building the landing dynamic model: 1) the gravity on the asteroid is of the order of magnitude about 10^-4 m/s², therefore the gravity is ignored; 2) the friction between the landing feet and landing legs is ignored; 3) The stiffness of the landing gear is far greater than that of the damping element vertically, and of the cardan element horizontally. So the flexibility of the landing gear is ignored; 4) the impulse acting on the landing mechanism when shooting the anchoring system is ignored; 5) the overturning of the landing mechanism is over, before the anchoring system tenses the thread.

As shown in the right part of Fig. 2, the landing mechanism will turn around the point O when landing. The whole system has three degrees of freedom: rotational DOF of m₁; rotational DOF of m₂; and translational DOF of m₂. The Lagrange equation is introduced, to buildng dynamic model. The kinetic energy T, potential energy V, and Rayleigh’s Dissipation Function ψ_q are shown in equations (1) and (2) respectively. The meanings and values of the parameters in equations are shown in Fig. 2 and Table 2.

Then, the Lagrange dynamic equations are deduced, as shown in equations (3), (4) and (5).

The parameters

of landing dynamic characteristics can be solved by equations (3), (4) and (5) with the initial values. However, in a complicated landing impact, the initial values are difficult, or impossible to solve accurately - they can only be estimated. A schematic of the impact is shown in the left part of Fig. 2. Firstly, the impacting force acting on m1 at point O is far larger than the other external force, thus the impulse of m₂ acting on m₁ can be ignored. So the angular momentum of m1 around O is conservational. Secondly, ignoring the retro-rocket thrust T, the angular momentum of the system composed of m₁ and m₂ is conservational. Thirdly, there is a damping element between the m₁ and m₂ vertically, therefore the vertical velocity of the m₂ changes continuously. Therefore the following equation (6) can be deduced. Meanings and values of the parameters in equation (6) are shown in Table 2.

where, ω₁, ω₂ and V_21y can be calculated from equations (6). Thus, the initial values to solve the equations (3), (4) and (5) are obtained, as shown in equation (7).

From the landing dynamic model, as shown in equations (3), (4) and (5), it can be found that the landing performance parameters

are determined by T, c₁ and c₂. So to have a superior landing performance, it is necessary to determine the proper values of T, c₁, and c₂.

The retro-rocket supplies constant force T towards the upper surface of the equipment base, to prevent the landing mechanism from rebounding and overturning. In order to find the safe thrust, it is assumed that the landing mechanism is a rigid body, and its initial kinetic energy is

counteracted entirely by the retro-rocket thrust. There are counterclockwise overturning, and clockwise overturning. They need different values of thrust, to prevent overturning. Thus, it is necessary to estimate the thrust separately for the two types of overturning, and then the largest value is taken as the retro-rocket thrust value.

4.1.1 Counterclockwise overturning

When landing with the initial velocity V_x=-0.5m/s and V_y=0m/s in 2-1 mode, the landing mechanism will turn counterclockwise, and have the most possibility to overturn counterclockwise. This schematic is shown in the left part of the Fig. 3. The meanings and values of the parameters in the schematic are shown in Table 3.

The landing mechanism turns around the O point counterclockwise after impact, and the initial overturn angular velocity ω_L can be estimated from the angular momentum conservation of m around O. Thus, the following equation is obtained:

The solution of equation (8) is:

The largest allowable turning range of the landing mechanism without overturning is from P₁ to P₂. So the initial kinetic energy of the landing mechanism after impact should be counteracted totally by the retro-rocket thrust T₁ during the angle ∠P₁OP₂, which equals π/2. Therefore, the following equation is obtained:

Obtaining:

where, T₁ is the retro-rocket thrust preventing the landing mechanism from counterclockwise overturning.

4.1.2 Clockwise overturning

When landing with the initial velocity V_x=0.5m/s and V_y=-1.5m/s in 1-2 mode, the landing mechanism will turn clockwise, and have the most possibility to overturn clockwise. This schematic is shown in the right part of Fig. 3. The meanings and values of the parameters in the schematic are shown in Table 2 and Table 3.

The landing mechanism turns around the O point clockwise after impact, and the initial turning angular velocity ω_R1 can be estimated from the angular momentum conservation of m around O. Thus, the following equation is obtained:

Obtaining:

When the landing mechanism turns to the P₂ position with the retro-rocket thrust T₂, the angular velocity ω_R2 can

be calculated by the energy conversion.

The tangential velocity of m in the P₂ position is:

Then the landing mechanism will turn continuously around point B. It is assumed that the recovery coefficient e equals 0.6 between the landing mechanism and the landing surface (e equals 0.5 between wood and wood; e equals 0.56 between steel and steel). Thus, the rebound velocity V₂₃ can be expressed as:

Then the initial angular velocity ω_R3 around the point B is obtained:

The largest allowable turning range of the landing mechanism, without overturning, is from P₂ to P₃. So the kinetic energy of the landing mechanism after P₂ position should be counteracted totally by the retro-rocket T₂ during the angle ∠P₂BP₃, which equals π/2-θ. Thus, the following equation is obtained:

Combining equations (13-18) yields the retro-rocket thrust T₂ :

Substituting e with 0.6, we obtain:

The final retro-rocket thrust T should be no less than T₁ or T₂. In the paper, the retro-rocket thrust T is set to 65 N.

The dynamic model of the landing mechanism in the vertical direction could be simply expressed as Fig. 4 and equation (21), in which k is the stiffness of the landing mechanism.

The initial conditions of equation (21) can be written as follows:

The overloading accelerations of m₂ are different, on account of different c₁. When the damping c₁ equals 900Ns/m, the numerical solution of the equation (21) is shown in Fig. 5.

It can be found that the overloading acceleration of m₂ is 30m/s², and the stroke of the damping is 0.13m. These overloading acceleration and stroke are feasible for the landing mechanism. Thus, the parameters of the damping element are set to c₁=900 Ns/m and S=0.13m, respectively.

The c₂ is a rotational damping produced by two motors in the cardan element, and it is used to absorb the horizontal impact when landing. Therefore, the value of c₂ can be changed, by controlling the motors. The landing performance in changing c₂ that is varied depending on different initial landing velocities, will be better compared with that in constant c₂.

The value of c₂ can be calculated through the landing dynamic model shown in equations (3), (4), (5) with proper objective function and constraint conditions, after determining the values of T and c₁.

In the landing dynamic model, it can be found that during the turning, c₂ will influence the angular momentum of m₁. The smaller the angular momentum of m₁, the harder the landing mechanism is to overturn. Thus, the smaller angular momentum of m₁ is set to be the objective function to solve c₂. The constraint conditions to solve c₂ are: 1) the horizontal overloading acceleration of m₂ is less than 10g; 2) the turning angle of m₂ relative to m₁ is less than 10°. The flow chart to calculate c₂ is shown in Fig. 6.

The values of c₂ solved with different landing velocities are shown in Fig. 7. The data in Fig. 7 can be expressed as counter in Fig. 8. The landing mechanism will have a more stable landing when c₂ varies with Vx and Vy according to the

relationship shown in Fig. 7 or Fig. 8. The varying range of c₂ is between 17 Nm.s/rad and 111 Nm.s/rad.

The validities of the landing dynamic model and the parameters T, c₁, c₂ need to be verified, by testing the landing performance. In the paper, the famous Adams software is used to simulate the landing performance. The landing mechanism has three classic landing modes, called 1-2 mode, 2-1 mode and 1-1-1 mode, respectively. Thus, the landing performances are tested in these three landing modes. The simulation parameters are shown in Table 4.

The landing performances in extreme landing conditions (V_x=-0.5 m/s, V_y=-1.5 m/s) are tested with the estimated parameters T, c₁ and c₂. The overloading acceleration of the equipment base reflects the damping performance, and the angular velocity of the landing legs reflects the stability time. Their values in the three classic landing modes are shown in Fig. 9. The left graph shows that the largest overloading accelerations of the equipment base are: about 80 m/s² in 1-2 mode; about 30 m/s² in 2-1 mode, and about 90 m/s² in 1-1-1 mode. The right graph shows that the stability times are: about 3.6 second in 1-2 mode; about 1.5 second in 2-1 mode and about 3.7 second in 1-1-1 mode. It can be found that the overloading accelerations are all less than 10g, and the landing stability times are all less than 4 second. Thus, the estimations of the parameters T, c₁ and c₂ are valid.

There are two modes of c₂ to select when landing: one is constant c₂ (c₂ remains constant at about 111 Nms/rad in all

landing velocities), the other is optimal c₂ (c₂ varies according to different initial landing velocities, as the rules shown in Fig. 7 or Fig. 8). Constant c₂ has the merits of easy control, whereas optimal c₂ can induce a better landing performance. The landing performances between the two modes of c₂ are compared in three classic landing modes, in the conditions that V_x=0.1 m/s and V_y =-0.5 m/s, respectively. The optimal c₂ in relation to V_x=0.1 m/s and V_y =-0.5 m/s is 55 Nms/rad,

which is shown in Fig. 7 and Fig. 8. The landing performance simulation results are shown in Fig. 10, Fig. 11 and Fig. 12, respectively, and the maximum overloading accelerations and stability times shown in these figures are summarized in Table 5. It can be found that both the maximum overloading accelerations and stability times with constant c₂ are all larger, than that with optimal c₂. Thus, optimal c₂ would lead to better landing performance.

A landing mechanism for asteroid with soft surface is presented. Reasonable values of the three key parameters (retro-rocket thrust T, damping element damping c₁ and cardan element c₂) that influence the landing performance of the landing mechanism presented in the paper are: T=65N; c₁=900Ns/m; and c₂ changes between 17 Nms/rad and 111Nms/rad, in relation to the initial landing velocities. The overloading accelerations of the equipment base are less than 10g, and the stability times are less than 5s, when landing with the estimated values of T, c₁, and c₂. Furthermore, optimal c₂ in relation to the landing velocities will lead to a good landing performance, with comparison to constant c₂. The simulation results show that the landing dynamic model, and the methods to estimate key parameters, are reasonable. The estimations of key parameters can be used to guide the design of the landing mechanism.

In future, the landing mechanism will be manufactured, and then the validities of the landing mechanism and its parameters will be tested physically, under microgravity.