Flux-free brazing of Mg-containing aluminium alloys by means of cold spraying

Abstract

In the present study, AlSi12 and AlSi10Cu4 were deposited onto Mg-containing aluminium alloys 6063 and 5754 by cold spraying. The influences of the two brazing alloys and spray parameters on coating formation were investigated. The microstructure of the coatings was characterized. Some coated samples were heat-treated at 590°C and 560°C in air to investigate the effect of the rupture of oxide scales on the diffusion of elements during heat-treatment. Some coated samples were brazed under argon atmosphere without any fluxes. The results show that AlSi12 had much better deposition behaviour than AlSi10Cu4. Due to the rupture of oxide scales, Cu and Si diffused into the substrate and a metallurgical bond formed between the brazing alloys and the substrates during heat-treatment. The coated samples could be brazed without any fluxes. Because the oxide scales prevented the formation of a metallurgical bond locally, the brazed samples had relatively low shear strengths of up to 43 MPa.

1 1 Introduction

Since the 1950s, mankind’s exploration of unknown territory has gradually shifted from Earth to extraterrestrial regions, such as the Moon and Mars. For extraterrestrial exploration, the technology of landing and walking is essential. Landing has two main methods: hard-landing and soft-landing. Hard-landing is a one-time detection method, such as the method used for Ranger4 [1] and Chang’e 1 [2]. It has no buffer mechanism; thus, the lander is destroyed when it touches the surface of the planet. For soft-landing, the landing speed is reduced through the buffer mechanism to achieve a controllable landing; hence, the lander can be protected from damage.

One effective method for soft-landing is to use a legged lander; according to the topological structure, a legged lander can be divided into two types: tripod type and cantilever type [3,4]. The tripod type has a simple structure and a small supporting capacity. Hence, it is suitable for landers with low mass, such as those used for the Mars Viking Program [5], Phoenix Program [6], Insight Program [7], Moon Surveyor Program [8–10], and Luna Program [11]. The cantilever type has good buffer capacity and large supporting stiffness. Hence, it has a good carrying capacity and is suitable for landers with heavy mass, such as those used for manned missions and the transport of cargos and scientific instruments. The typical application cases include the Apollo Program [12] and Chang’e 5 [13,14]. The legged lander cannot move, and it can only land on a relatively flat terrain. As a result, a legged lander has a very limited detection range.

Another effective method for soft-landing is to use a rover. It can use its visual system to walk and its equipment for sampling. The soft-landing method for the rover can be divided into two types: airbag type and aerial crane type. The airbag type has a small bearing capacity, so it is suitable for some small rovers, such as Spirit and Opportunity [15,16] on Mars. The aerial crane type is suitable for a planet with an atmosphere; thus, this method is used for the rovers Curiosity [17,18] and Perseverance to land on Mars. A highly efficient method is the cooperation of the legged lander and the rover. A lander is responsible for transporting the rover to the planet. It also acts as a transfer station for the rover to communicate with Earth. A lander employs the cantilever-type topological structure because it should have a high carrying capacity. The typical application cases include Apollo 15 and Lunar Rover Vehicle [19], Chang’e 3 and Yutu 1 [20], and Chang’e 4 and Yutu 2 [21]. However, the combination of a lander and a rover has some limitations. In particular, the separation of the lander and the rover results in heavy mass, large volume, and high launching costs. Moreover, the lander cannot move, and the rover needs to obtain supplies and send messages to Earth through the lander. These problems result in a very limited range of exploration.

Some researchers aimed to solve the above problems by proposing the mobile probe, which integrates the landing and walking functions. Legged robots have good adaptability on complex terrains, such as those with steps and gullies. Hence, some legged robots for detection have been designed, such as SpaceClimber [22] and SpaceBok [23,24]. For the methods of landing control, deep reinforcement learning (RL) was used on SpaceBok to realize repetitive controlled jumping and landing [25]. An integrated control method based on a model-free RL controller in conjunction with an auto-turned reward function was proposed to solve the landing control of quadruped robots on celestial bodies [26]. The dynamics of quadrupeds after touchdown were modeled as a Variable Height Springy Inverted Pendulum to solve the problem of landing on a flat horizontal surface for a quadruped robot [27]. A compliant landing control framework was proposed for quadruped robots; then, a model predictive control (MPC) algorithm was used to determine the joint torque and realize compliant landing [28]. A framework combining trajectory optimization and MPC was proposed to achieve continuous jumping and landing of the robot; this framework could also adapt to unknown height perturbations and uncertain dynamic models [29]. A new method using trajectory optimization, a high-frequency tracking controller, and a robust landing controller was proposed to stabilize a robot’s posture after impact [30]. An algorithm inspired by the falling cat was presented to reduce the impact at landing by using nonholonomic trajectory planning [31]. Based on the landing movements of cats, a new method combining trajectory optimization and machine learning was used to achieve a highly dynamic landing control [32]. A real-time and optimal landing controller was presented to determine the optimal touchdown postures; the controller helps Massachusetts Institute of Technology’s Mini Cheetah to recover from drops of up to 2 m [33]. Yin et al. [34,35] proposed a control method based on a state machine for soft-landing on the Moon to absorb the landing impact energy. The soft-landing process was analyzed based on a dynamic model through three numerical simulations [36]. These methods have significantly realized the landing control of legged robots. In this study, we proposed a six-legged mobile repetitive lander for Moon exploration. It can realize repetitive landing and walking by using compliance control of each drive motor. Compared with the separation of the lander and the rover, the mobile repetitive lander can reduce volume and launching costs. Solving the problem of landing is the most important thing to ensure the operation of exploration. In this study, a soft-landing method based on compliance control and optimal force control for the mobile repetitive lander is proposed to protect the whole system from damage, with the following features:

I. The soft-landing method can keep the lander’s body stable during the landing process.

II. The soft-landing method can limit the torque of the drive motors and weaken the peak torque during the landing process.

III. The soft-landing method can effectively adapt to complex terrains, such as step and slope.

The rest of this paper is organized as follows. In Section 2, the six-legged mobile repetitive lander with the serial–parallel leg mechanism, as well as its structure, is introduced. In Section 3, the soft-landing method based on compliance control and optimal force control for the mobile repetitive lander is proposed. In Section 4, experiments on complex terrains are conducted to validate the soft-landing method and its performances. In Section 5, the conclusions are presented.

2 2 System overview

2.1 2.1 Six-legged mobile repetitive lander

The six-legged mobile repetitive lander in this study is shown in Fig.1. It has six legs arranged around its body hexagonally. Each leg is a 3-degree-of-freedom serial–parallel mechanism composed of an abduction motor, a thigh motor, and a shank motor. All the motors are installed near the lander’s body to reduce leg inertia and improve the motion performance of the lander. The body coordinate frame (BCF) O_B-X_BY_BZ_B is located at the center of the body. X_B points forward, and Z_B points upward. The ground coordinate frame (GCF) O_G-X_GY_GZ_G coincides with the initial body frame, and it can be updated to the current BCF if necessary. The lander runs a real-time operating system, and the control frequency is 1000 Hz. Users send commands to the onboard computer via Wi-Fi. Then, the command can be sent to the drivers via EtherCAT.

Fig.1 Model of the six-legged mobile repetitive lander: (a) top view and (b) axonometric drawing. Six legs are labeled by numbers for identification purpose.

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The six-legged mobile repetitive lander can realize the function of landing and walking, as shown in Fig.2. In particular, the lander can realize safe and stable landing based on the soft-landing control method, as shown in Fig.2(a)–Fig.2(d). Moreover, the lander can walk by the 3-3 gait, with three legs swinging and the three other legs supporting. The six legs are divided into two groups and walk in turn, as shown in Fig.2(e)–Fig.2(h).

Fig.2 (a–h) Working mode of the six-legged mobile repetitive lander.

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2.2 2.2 Leg mechanism and model analysis

2.2.1 2.2.1 Leg mechanism

The composition of the leg mechanism is shown in Fig.3(a). The abduction motor is fixed to the body and connects to the parallel four-bar mechanism. The thigh motor controls the thigh rod directly. The shank motor controls the shank rod by the rocker and the connecting rod of the parallel four-bar mechanism. Two passive linear springs are installed between the thigh rod and the shank rod, and they can reduce the torque of the motors during landing. The original length of the spring is L₀ = 0.21 m, and the stiffness is K_s = 4200 N/m. The leg coordinate frame (LCF) O_Li-X_LiY_LiZ_Li (i = 1,2,…,6) is located at the intersection of the rotational axis of the abduction motor, thigh motor, and shank motor. X_Li coincides with the rotational axis of the abduction motor, and Z_Li coincides with the rotational axis of the thigh motor. The dimensions of the leg mechanism are shown in Tab.1. The kinematic parameters at the input end are defined as q_i = (θ_ai,θ_ti,θ_si)^T, and they are limited to [−π/2,π/2], [π/2,3π/2], and [π/2,π], as shown in Fig.3(b). The coordinate of the foot tip in LCF is the output end. It is defined as ${\mathit{P}}_i^{\mathrm{L}}$ = ($x_i^{\mathrm{L}}$, $y_i^{\mathrm{L}}$, ${\mathit{\text{z}}}_i^{\mathrm{L}}$)^T, as shown in Fig.3(b).

Tab.1 Dimensions of the leg mechanism

Parameter	Dimension/m
lAD,lBC	0.10
lBE	0.50
lAB,lCD	0.50
lBF	0.21

Fig.3 Leg mechanism and kinematic parameter: (a) composition of the leg mechanism, (b) kinematic parameter, and (c) portrait view of the leg mechanism.

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2.2.2 2.2.2 Model analysis

The inverse kinematics in LCF is q_i = ${IK}_i^{\mathrm{L}}$(${\mathit{P}}_i^{\mathrm{L}}$), which entails deriving q_i from ${\mathit{P}}_i^{\mathrm{L}}$. As shown in Fig.3(c),

(1)${\theta }_{\mathrm{a}i}=\arctan \left({\displaystyle \frac{{\mathit{\text{z}}}_i^{\mathrm{L}}}{y_i^{\mathrm{L}}}}\right).$

Fig.3(c) shows that in the plane of the four-bar mechanism, the coordinate of ${\mathit{P}}_i^{\mathrm{L}}$ in O_Li-X_LiY'_LiZ'_Li is defined as ${\mathit{P}}_i^{{\mathrm{L}}^{{}^{\prime }}}$ = ($x_i^{{\mathrm{L}}^{{}^{\prime }}}$, $y_i^{{\mathrm{L}}^{{}^{\prime }}}$, ${\mathit{\text{z}}}_i^{{\mathrm{L}}^{{}^{\prime }}}$)^T. It can be calculated by

(2)$\{\begin{array}{rl}& x_i^{{\mathrm{L}}^{{}^{\prime }}}=x_i^{\mathrm{L}},\\ & y_i^{{\mathrm{L}}^{{}^{\prime }}}=y_i^{\mathrm{L}}\cos {\theta }_{\mathrm{a}i}+{\mathit{\text{z}}}_i^{\mathrm{L}}\sin {\theta }_{\mathrm{a}i},\\ & {\mathit{\text{z}}}_i^{{\mathrm{L}}^{{}^{\prime }}}=0.\end{array}$

Accordingly,

(3)$l_{\mathrm{A}\mathrm{E}}=\sqrt{{\left(x_i^{{\mathrm{L}}^{{}^{\prime }}}\right)}^2+{\left(y_i^{{\mathrm{L}}^{{}^{\prime }}}\right)}^2+{\left({\mathit{\text{z}}}_i^{{\mathrm{L}}^{{}^{\prime }}}\right)}^2}.$

In ΔABE, α_i = ∠BAE and β_i = ∠ABE can be derived from the cosine law of the triangle:

(4)$\{\begin{array}{l}{\alpha }_i=\arccos \left({\displaystyle \frac{l_{\mathrm{A}\mathrm{B}}^2+l_{\mathrm{A}\mathrm{E}}^2-l_{\mathrm{B}\mathrm{E}}^2}{2l_{\mathrm{A}\mathrm{B}}{l}_{\mathrm{A}\mathrm{E}}}}\right),\\ {\beta }_i=\arccos \left({\displaystyle \frac{l_{\mathrm{A}\mathrm{B}}^2+l_{\mathrm{B}\mathrm{E}}^2-l_{\mathrm{A}\mathrm{E}}^2}{2l_{\mathrm{A}\mathrm{B}}{l}_{\mathrm{B}\mathrm{E}}}}\right).\end{array}$

The phase angle of ${\mathit{P}}_i^{{\mathrm{L}}^{{}^{\prime }}}$ can be calculated by

(5)${\gamma }_i=\arctan \left(\frac{x_i^{{\mathrm{L}}^{{}^{\prime }}}}{y_i^{{\mathrm{L}}^{{}^{\prime }}}}\right).$

As a result,

(6)$\{\begin{array}{rl}& {\theta }_{\mathrm{a}i}=\arctan \left(\frac{{\mathit{\text{z}}}_i^{\mathrm{L}}}{y_i^{\mathrm{L}}}\right),\\ & {\theta }_{\mathrm{t}i}=\frac{3}{2}\text{π}-{\gamma }_i-{\alpha }_i,\\ & {\theta }_{\mathrm{s}i}=\frac{3}{2}\text{π}-{\gamma }_i-{\alpha }_i-{\beta }_i.\end{array}$

The forward kinematics in LCF is expressed by ${\mathit{P}}_i^{\mathrm{L}}$ = ${FK}_i^{\mathrm{L}}$(q_i), which entails deriving ${\mathit{P}}_i^{\mathrm{L}}$ from q_i. It can be calculated by

(7)$x_i^{{\mathrm{L}}^{{}^{\prime }}}=l_{\mathrm{A}\mathrm{B}}\cos {\theta }_{\mathrm{t}i}-l_{\mathrm{B}\mathrm{E}}\cos {\theta }_{\mathrm{s}i},$

(8)$y_i^{{\mathrm{L}}^{{}^{\prime }}}=l_{\mathrm{A}\mathrm{B}}\sin {\theta }_{\mathrm{t}i}-l_{\mathrm{B}\mathrm{E}}\sin {\theta }_{\mathrm{s}i},$

(9)$\{\begin{array}{rl}& x_i^{\mathrm{L}}=x_i^{{\mathrm{L}}^{{}^{\prime }}},\\ & y_i^{\mathrm{L}}=y_i^{{\mathrm{L}}^{{}^{\prime }}}\cos {\theta }_{\mathrm{a}i},\\ & {\mathit{\text{z}}}_i^{\mathrm{L}}=y_i^{{\mathrm{L}}^{{}^{\prime }}}\sin {\theta }_{\mathrm{a}i}.\end{array}$

The velocity at the output end of the leg in LCF can be expressed by

(10)${\dot{\mathit{P}}}_i^{\mathrm{L}}={\mathit{J}}_{\mathrm{v}i}^{\mathrm{L}}{\dot{\mathit{q}}}_i,$

where ${\mathit{J}}_{\mathrm{v}i}^{\mathrm{L}}$ is the velocity Jacobian matrix of the leg mechanism. It can be calculated by

(11)${\mathit{J}}_{\mathrm{v}i}^{\mathrm{L}}=\frac{\delta \left(F{K}_i^{\mathrm{L}}\left({\mathit{q}}_i\right)\right)}{\delta {\mathit{q}}_i}=\left[\begin{array}{ccc}0& -l_{\mathrm{A}\mathrm{B}}\sin {\theta }_{\mathrm{t}i}& l_{\mathrm{B}\mathrm{E}}\sin {\theta }_{\mathrm{s}i}\\ -\sin {\theta }_{\mathrm{a}i}\left(l_{\mathrm{A}\mathrm{B}}\sin {\theta }_{\mathrm{t}i}-l_{\mathrm{B}\mathrm{E}}\sin {\theta }_{\mathrm{s}i}\right)& l_{\mathrm{A}\mathrm{B}}\cos {\theta }_{\mathrm{a}i}\cos {\theta }_{\mathrm{t}i}& -l_{\mathrm{B}\mathrm{E}}\cos {\theta }_{\mathrm{a}i}\cos {\theta }_{\mathrm{s}i}\\ \cos {\theta }_{\mathrm{a}i}\left(l_{\mathrm{A}\mathrm{B}}\sin {\theta }_{\mathrm{t}i}-l_{\mathrm{B}\mathrm{E}}\sin {\theta }_{\mathrm{s}i}\right)& l_{\mathrm{A}\mathrm{B}}\sin {\theta }_{\mathrm{a}i}\cos {\theta }_{\mathrm{t}i}& -l_{\mathrm{B}\mathrm{E}}\sin {\theta }_{\mathrm{a}i}\cos {\theta }_{\mathrm{s}i}\end{array}\right].$

Based on the principle of virtual work, the force Jacobian matrix can be expressed by

(12)${\mathit{J}}_{\mathrm{F}i}^{\mathrm{L}}={\left({\mathit{J}}_{\mathrm{v}i}^{\mathrm{L}}\right)}^{-\mathrm{T}}.$

The action of the passive linear spring can influence the torque of the thigh motor and the shank motor. As shown in Fig.4, the force generated by the two passive springs in each leg can be expressed by

Fig.4 Analysis of the passive spring.

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(13)$F_{\mathrm{k}i}=2K_{\mathrm{s}}\left(l_{\mathrm{C}\mathrm{F}}-L_0\right),$

where l_CF is the current length of the spring. It can be calculated by

(14)$l_{\mathrm{C}\mathrm{F}}=\sqrt{l_{\mathrm{B}\mathrm{C}}^2+l_{\mathrm{B}\mathrm{F}}^2-2l_{\mathrm{B}\mathrm{C}}{l}_{\mathrm{B}\mathrm{F}}\cos \left(\mathrm{\angle }\mathrm{C}\mathrm{B}\mathrm{F}\right)}.$

The toque generated by the springs to the shank motor is

(15)$\begin{array}{rl}\mathrm{\Delta }{\tau }_{\mathrm{s}i}=& -F_1{l}_{\mathrm{A}\mathrm{D}}\sin \left(\mathrm{\angle }\mathrm{C}\mathrm{D}\mathrm{A}\right)=-F_1{l}_{\mathrm{B}\mathrm{C}}\sin \left(\mathrm{\angle }\mathrm{C}\mathrm{B}\mathrm{F}\right)\\ =& -F_{\mathrm{k}i}{l}_{\mathrm{B}\mathrm{C}}\sin \left(\mathrm{\angle }\mathrm{B}\mathrm{C}\mathrm{F}\right),\end{array}$

where F₁ is the component of F_ki along line CD.

Based on the force analysis performed on point B, the torque generated by the springs to the thigh motor is

(16)$\begin{array}{rl}\mathrm{\Delta }{\tau }_{\mathrm{t}i}=& F_{\mathrm{k}i}{l}_{\mathrm{A}\mathrm{B}}\sin \left(\mathrm{\angle }\mathrm{C}\mathrm{F}\mathrm{B}\right)-F_{\mathrm{k}i}\left(l_{\mathrm{A}\mathrm{B}}-l_{\mathrm{B}\mathrm{F}}\right)\sin \left(\mathrm{\angle }\mathrm{C}\mathrm{F}\mathrm{B}\right)\\ =& F_{\mathrm{k}i}{l}_{\mathrm{B}\mathrm{F}}\sin \left(\mathrm{\angle }\mathrm{C}\mathrm{F}\mathrm{B}\right).\end{array}$

In ΔBCF,

(17)$\{\begin{array}{l}\mathrm{\angle }\mathrm{C}\mathrm{B}\mathrm{F}=\text{π}+{\theta }_{\mathrm{s}i}-{\theta }_{\mathrm{t}i},\\ {\displaystyle \frac{l_{\mathrm{B}\mathrm{F}}}{\sin \left(\mathrm{\angle }\mathrm{B}\mathrm{C}\mathrm{F}\right)}}={\displaystyle \frac{l_{\mathrm{B}\mathrm{C}}}{\sin \left(\mathrm{\angle }\mathrm{C}\mathrm{F}\mathrm{B}\right)}}={\displaystyle \frac{l_{\mathrm{C}\mathrm{F}}}{\sin \left(\mathrm{\angle }\mathrm{C}\mathrm{B}\mathrm{F}\right)}}.\end{array}$

Therefore,

(18)$\mathrm{\Delta }{\tau }_{\mathrm{t}i}=-\mathrm{\Delta }{\tau }_{\mathrm{s}i}=F_{\mathrm{k}i}\frac{l_{\mathrm{B}\mathrm{F}}{l}_{\mathrm{B}\mathrm{C}}\sin \left(\text{π}+{\theta }_{\mathrm{s}i}-{\theta }_{\mathrm{t}i}\right)}{l_{\mathrm{C}\mathrm{F}}}.$

Hence, the torques generated by the springs to the input end can be defined as τ_spring,i = (0,Δτ_ti,Δτ_si)^T. The force at the output end of the leg in LCF can be expressed by

(19)${\mathit{F}}_i^{\mathrm{L}}={\mathit{J}}_{\mathrm{F}i}^{\mathrm{L}}\left[{\mathit{\tau }}_i+{\mathit{\tau }}_{\mathrm{s}\mathrm{p}\mathrm{r}\mathrm{i}\mathrm{n}\mathrm{g},i}\right],$

where τ_i is the torque of the input end. It can be expressed as τ_i = (τ_ai,τ_ti,τ_si)^T.

3 3 Soft-landing method

As shown in Fig.5, the process of soft-landing can be divided into three steps based on the state of the body: landing state, retracting state, and extending state. The landing state is the process of dropping while the lander is in the sky. When all the legs of the lander touch the ground, the process switches to the retracting state, and the body starts to retract to absorb the impact energy. Then, the process switches to the extending state, and the lander starts to lift the body to adjust the posture when the speed of the body decelerates to zero.

Fig.5 Process of soft-landing.

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During the landing state, the lander descends under the gravity. The compliance control is used to maintain the position of the active joints and deal with the unknown landing terrain. Then, the lander absorbs the impact energy and slows down when it is in the retracting state. The impact results in changes in the actual position and velocity of the active joints. The compliance control produces large joint torques and exceeds the capacity of the joints. An optimal force control method is proposed to control the peak torque of the joint and maximize the capacity of buffering absorption. When the lander is in the extending state, compliance control is also used to control the stability of the body and prepare for walking.

3.1 3.1 Landing state: compliance control

The compliance control shown in Eq. (20) is used to control the joint torques in the landing state process. The actual joint angle q_act,i and actual joint velocity ${\dot{\mathit{q}}}_{\mathrm{a}\mathrm{c}\mathrm{t},i}$ of each leg can be obtained from the encoders. q_init,i is the initial joint angle that should remain constant during the landing state. It remains unchanged and is measured when the lander starts landing. τ_req,i is the required joint torque of each leg. The active stiffness K and damping B must be low coefficients to avoid a large impact force. They cannot be zero because the leg would move. This movement is due to the gravity if the joint torques are zero. Hence, K and B are set to 26 N·m/rad and 2.5 N·m·s/rad, respectively.

(20)${\mathit{\tau }}_{\mathrm{r}\mathrm{e}\mathrm{q},\mathrm{i}}={\mathit{\tau }}_{\mathrm{c}\mathrm{o}\mathrm{m},i}=K\left({\mathit{q}}_{\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t},i}-{\mathit{q}}_{\mathrm{a}\mathrm{c}\mathrm{t},i}\right)+B\left(\mathbf{0}-{\dot{\mathit{q}}}_{\mathrm{a}\mathrm{c}\mathrm{t},i}\right)-{\mathit{\tau }}_{\mathrm{s}\mathrm{p}\mathrm{r}\mathrm{i}\mathrm{n}\mathrm{g},i}.$

A virtual compliance control shown in Eq. (21) is used to judge the state of each leg. As shown in Fig.6(a), all legs are in the aerial phase when the lander is in the landing state. Fig.6(b) shows that each leg has a different time to touch the ground because of the terrain’s irregularity. A virtual foot-tip force F_vi in GCF calculated by Eq. (22) is generated to judge the touchdown of each leg. ${}_{\mathrm{B}}^{\mathrm{G}}\mathit{R}$ is the rotation matrix from BCF to GCF, and ${}_{\mathrm{L}i}^{\mathrm{G}}\mathit{R}$ is the rotation matrix from LCF to BCF. When the leg starts to touch to ground, the actual angle and actual joint velocity generate great changes in a very short time because of the low active stiffness K and damping B. The virtual stiffness K_virtual and damping B_virtual must be large to make F_vi large enough to judge the touchdown. Hence, K_virtual and B_virtual are set to 130 N·m/rad and 12.5 N·m·s/rad, respectively.

Fig.6 (a–c) Change process from the landing state to the retracting state.

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(21)${\mathit{\tau }}_{\mathrm{v}i}=K_{\mathrm{v}\mathrm{i}\mathrm{r}\mathrm{t}\mathrm{u}\mathrm{a}\mathrm{l}}\left({\mathit{q}}_{\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t},i}-{\mathit{q}}_{\mathrm{a}\mathrm{c}\mathrm{t},i}\right)+B_{\mathrm{v}\mathrm{i}\mathrm{r}\mathrm{t}\mathrm{u}\mathrm{a}\mathrm{l}}\left(\mathbf{0}-{\dot{\mathit{q}}}_{\mathrm{a}\mathrm{c}\mathrm{t},i}\right).$

(22)${\mathit{F}}_{\mathrm{v}i}={}_{\mathrm{B}}^{\mathrm{G}}\mathit{R}{}_{\mathrm{L}i}^B{\mathit{R}\mathit{J}}_{\mathrm{F}i}^{\mathrm{L}}{\mathit{\tau }}_{\mathrm{v}i}.$

When the virtual vertical value F_{vi_z} shown in Fig.6(b) is greater than the threshold value, the leg state changes to the grounded phase. In the grounded phase, any contact force would cause the body to tilt. Hence, the stiffness K and damping B in Eq. (20) are set to zero.

(23)$\mathrm{L}\mathrm{e}\mathrm{g}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{e}=\{\begin{array}{ll}\mathrm{A}\mathrm{e}\mathrm{r}\mathrm{i}\mathrm{a}\mathrm{l}\mathrm{p}\mathrm{h}\mathrm{a}\mathrm{s}\mathrm{e},& \mathrm{i}\mathrm{f}{F}_{\mathrm{v}i\mathrm{\_}\mathit{\text{z}}}<50\mathrm{N},\\ \mathrm{G}\mathrm{r}\mathrm{o}\mathrm{u}\mathrm{n}\mathrm{d}\mathrm{e}\mathrm{d}\mathrm{p}\mathrm{h}\mathrm{a}\mathrm{s}\mathrm{e},& \mathrm{i}\mathrm{f}{F}_{\mathrm{v}i\mathrm{\_}\mathit{\text{z}}}⩾50\mathrm{N}.\end{array}$

The lander changes to the retracting state when all leg states switch to the grounded phase, as shown in Fig.6(c).

(24)$\begin{array}{rl}& \mathrm{L}\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{e}\mathrm{r}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{e}\\ =& \{\begin{array}{ll}\mathrm{L}\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{i}\mathrm{n}\mathrm{g},& \mathrm{i}\mathrm{f}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}\mathrm{e}\mathrm{x}\mathrm{i}\mathrm{s}\mathrm{t}\mathrm{s}\mathrm{a}\mathrm{n}\mathrm{a}\mathrm{e}\mathrm{r}\mathrm{i}\mathrm{a}\mathrm{l}\mathrm{l}\mathrm{e}\mathrm{g}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{e},\\ \mathrm{R}\mathrm{e}\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{n}\mathrm{g},& \mathrm{i}\mathrm{f}\mathrm{a}\mathrm{l}\mathrm{l}\mathrm{l}\mathrm{e}\mathrm{g}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{s}\mathrm{a}\mathrm{r}\mathrm{e}\mathrm{g}\mathrm{r}\mathrm{o}\mathrm{u}\mathrm{n}\mathrm{d}\mathrm{e}\mathrm{d}.\end{array}\end{array}$

3.2 3.2 Retracting state: optimal force control

In the retracting state process, the lander absorbs the impact energy. An intuitive idea for obtaining a good deceleration involves maximizing the sum of the vertical foot-tip force of all six legs. The object can be modeled as

(25)$max\sum _{i=1}^6{F}_{\mathit{\text{z}}i},$

where F_zi is shown in Fig.7. The peak torque of the joint should be controlled; thus, for each leg, the joint torque generated by the supporting force F_i = (0,0,F_zi)^T should satisfy

Fig.7 Supporting force in the retracting process.

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(26)$-{\mathit{\tau }}_{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{t}}⩽{\mathit{\tau }}_i=-{\left[{\mathit{J}}_{\mathrm{F}i}^{\mathrm{L}}\right]}^{-1}{}_{\mathrm{L}i}^{\mathrm{B}}{\mathit{R}}^{\mathrm{T}}{}_{\mathrm{B}}^{\mathrm{G}}{\mathit{R}}^{\mathrm{T}}{\mathit{F}}_i-{\mathit{\tau }}_{\mathrm{s}\mathrm{p}\mathrm{r}\mathrm{i}\mathrm{n}\mathrm{g},i}⩽{\mathit{\tau }}_{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{t}},$

where the $⩽$ sign means that the elements of left matrix are all less than or euqal to the corresponding elements of right matrix. τ_limit is the limited torque of the drive motor. Equation (26) can be then expressed as

(27)${\mathit{A}}_i{F}_{\mathit{\text{z}}i}⩽{\mathit{b}}_i,$

where A_i and b_i are 6 × 1 matrixes. They can be expressed by

(28)${\mathit{A}}_i=\left[\begin{array}{r}{\left[{\mathit{J}}_{\mathrm{F}i}^L\right]}^{-1}{}_{\mathrm{L}i}^{\mathrm{B}}{\mathit{R}}^{\mathrm{T}}{}_{\mathrm{B}}^{\mathrm{G}}{\mathit{R}}^{\mathrm{T}}{\left[\begin{array}{ccc}0& 0& 1\end{array}\right]}^{\mathrm{T}}\\ -{\left[{\mathit{J}}_{\mathrm{F}i}^L\right]}^{-1}{}_{\mathrm{L}i}^{\mathrm{B}}{\mathit{R}}^{\mathrm{T}}{}_{\mathrm{B}}^{\mathrm{G}}{\mathit{R}}^{\mathrm{T}}{\left[\begin{array}{ccc}0& 0& 1\end{array}\right]}^{\mathrm{T}}\end{array}\right],{\mathit{b}}_i=\left[\begin{array}{c}{\mathit{\tau }}_{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{t}}-{\mathit{\tau }}_{\mathrm{s}\mathrm{p}\mathrm{r}\mathrm{i}\mathrm{n}\mathrm{g},i}\\ {\mathit{\tau }}_{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{t}}+{\mathit{\tau }}_{\mathrm{s}\mathrm{p}\mathrm{r}\mathrm{i}\mathrm{n}\mathrm{g},i}\end{array}\right].$

Merging all six legs yields

(29)$\mathit{A}\cdot {\mathit{F}}_{\mathit{\text{z}}}⩽\mathit{b},$

where A is a 36 × 6 matrix, F is a 6 × 1 matrix, and b is a 36 × 1 matrix. Note that

(30)$\mathit{A}=\left[\begin{array}{rrrrrr}{\mathit{A}}_1& & & & & \\ & {\mathit{A}}_2& & & & \\ & & {\mathit{A}}_3& & & \\ & & & {\mathit{A}}_4& & \\ & & & & {\mathit{A}}_5& \\ & & & & & {\mathit{A}}_6\end{array}\right],{\mathit{F}}_{\mathit{\text{z}}}=\left[\begin{array}{r}{F}_{\mathit{\text{z}}1}\\ F_{\mathit{\text{z}}2}\\ F_{\mathit{\text{z}}3}\\ F_{\mathit{\text{z}}4}\\ F_{\mathit{\text{z}}5}\\ F_{\mathit{\text{z}}6}\end{array}\right],\mathit{b}=\left[\begin{array}{r}{\mathit{b}}_1\\ {\mathit{b}}_2\\ {\mathit{b}}_3\\ {\mathit{b}}_4\\ {\mathit{b}}_5\\ {\mathit{b}}_6\end{array}\right].$

The roll angle $w_{x\mathrm{B}}^{\mathrm{G}}$ and pitch angle $w_{y\mathrm{B}}^{\mathrm{G}}$ of the body shown in Fig.7 should be equal to zero to keep the stability of the body. Therefore, a torque τ_body is needed to control the body, and it can be generated by the PID controller as follows:

(31)${\mathit{\tau }}_{\mathrm{b}\mathrm{o}\mathrm{d}\mathrm{y}}=\left[\begin{array}{c}{k}_{\mathrm{p}}\left(0-w_{x\mathrm{B}}^{\mathrm{G}}\right)+k_{\mathrm{d}}\left(0-{\dot{w}}_{x\mathrm{B}}^{\mathrm{G}}\right)+k_{\mathrm{i}}\int \left(0-w_{x\mathrm{B}}^{\mathrm{G}}\right)\mathrm{d}t\\ k_{\mathrm{p}}\left(0-w_{y\mathrm{B}}^{\mathrm{G}}\right)+k_{\mathrm{d}}\left(0-{\dot{w}}_{y\mathrm{B}}^{\mathrm{G}}\right)+k_{\mathrm{i}}\int \left(0-w_{y\mathrm{B}}^{\mathrm{G}}\right)\mathrm{d}t\end{array}\right],$

where k_p, k_d, and k_i are the control parameters of the proportional gain, derivative gain, and integral gain. $w_{x\mathrm{B}}^{\mathrm{G}}$, ${\dot{w}}_{x\mathrm{B}}^{\mathrm{G}}$, $w_{y\mathrm{B}}^{\mathrm{G}}$, and ${\dot{w}}_{y\mathrm{B}}^{\mathrm{G}}$ can be obtained from the inertial measurement unit.

The forces on the foot tips need to satisfy the following to realize the torque:

(32)$\sum _{i=1}^6{\mathit{r}}_i\times {\mathit{F}}_i={\mathit{\tau }}_{\mathrm{b}\mathrm{o}\mathrm{d}\mathrm{y}},$

where r_i is the vector from the body to the foot tip, as shown in Fig.7. Equation (32) can be rearranged as

(33)$\mathit{A}\mathit{e}\mathit{q}\cdot {\mathit{F}}_{\mathit{\text{z}}}=\mathit{b}\mathit{e}\mathit{q},$

where Aeq is a 2 × 6 matrix, and beq is a 2 × 1 matrix. They can be expressed by

(34)$\begin{array}{c}\mathit{A}\mathit{e}\mathit{q}=\left[\begin{array}{rrrrrr}{r}_{1y}& r_{2y}& r_{3y}& r_{4y}& r_{5y}& r_{6y}\\ -r_{1x}& -r_{2x}& -r_{3x}& -r_{4x}& -r_{5x}& -r_{6x}\end{array}\right],\\ \mathit{b}\mathit{e}\mathit{q}={\mathit{\tau }}_{\mathrm{b}\mathrm{o}\mathrm{d}\mathrm{y}}.\end{array}$

As a result, the optimization model can be summarized as follows:

(35)$\mathrm{M}\mathrm{a}\mathrm{x}\sum _{i=1}^6{F}_{\mathit{\text{z}}i}\mathrm{s}.\mathrm{t}.\{\begin{array}{l}\mathit{A}\cdot {\mathit{F}}_{\mathit{\text{z}}}⩽\mathit{b}\\ \mathit{A}\mathit{e}\mathit{q}\cdot {\mathit{F}}_{\mathit{\text{z}}}=\mathit{b}\mathit{e}\mathit{q}\end{array}.$

This model can be considered as a linear programming problem and solved by the simplex algorithm.

Hence, the required joint torque of each leg in the retracting state process can be expressed by

(36)${\mathit{\tau }}_{\mathrm{r}\mathrm{e}\mathrm{q},i}=-{\left[{\mathit{J}}_{\mathrm{F}i}^{\mathrm{L}}\right]}^{-1}{}_{\mathrm{L}i}^{\mathrm{B}}{\mathit{R}}^{\mathrm{T}}{}_{\mathrm{B}}^{\mathrm{G}}{\mathit{R}}^{\mathrm{T}}{\left[\begin{array}{ccc}0& 0& 1\end{array}\right]}^{\mathrm{T}}{F}_{\mathit{\text{z}}i}-{\mathit{\tau }}_{\mathrm{s}\mathrm{p}\mathrm{r}\mathrm{i}\mathrm{n}\mathrm{g},i}.$

In the retracting state process, the height of the body drops, and the velocity eventually decelerates to zero. Accordingly, the velocity of the foot tip in BCF is zero. It can be calculated by

(37)${\dot{\mathit{P}}}_i^{\mathrm{B}}={}_{\mathrm{L}i}^{\mathrm{B}}{\mathit{R}\mathit{J}}_{\mathrm{v}i}^{\mathrm{L}}{\dot{\mathit{q}}}_i.$

The average of the vertical value ${\dot{P}}_{\mathit{\text{z}}i}^{\mathrm{B}}$ of all six legs can be expressed by

(38)${\overline{\dot{P}}}_{\mathit{\text{z}}}^{\mathrm{B}}=\frac{1}{6}\sum _{i=1}^6{\dot{P}}_{\mathit{\text{z}}i}^{\mathrm{B}}.$

When ${\overline{\dot{P}}}_{\mathit{\text{z}}}^{\mathrm{B}}$ is equal to zero, the lander changes to the extending state.

(39)$\mathrm{L}\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{e}\mathrm{r}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{e}=\{\begin{array}{ll}\mathrm{R}\mathrm{e}\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{n}\mathrm{g},& \mathrm{i}\mathrm{f}{\overline{\dot{P}}}_{\mathit{\text{z}}}^{\mathrm{B}}>0,\\ \mathrm{E}\mathrm{x}\mathrm{t}\mathrm{e}\mathrm{n}\mathrm{d}\mathrm{i}\mathrm{n}\mathrm{g},& \mathrm{i}\mathrm{f}{\overline{\dot{P}}}_{\mathit{\text{z}}}^{\mathrm{B}}⩽0.\end{array}$

3.3 3.3 Extending state: compliance control

In the extending state process, the lander stands on the ground and starts to lift the body to the desired height. The required joint torque can be calculated by compliance control and body stabilizer, as shown in Eq. (40).

(40)${\mathit{\tau }}_{\mathrm{r}\mathrm{e}\mathrm{q},i}={\mathit{\tau }}_{\mathrm{c}\mathrm{o}\mathrm{m},i}+{\mathit{\tau }}_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{b},i}.$

Compliance control can be expressed by

(41)${\mathit{\tau }}_{\mathrm{c}\mathrm{o}\mathrm{m},i}=K\left({\mathit{q}}_{\mathrm{r}\mathrm{e}\mathrm{f},\mathrm{i}}-{\mathit{q}}_{\mathrm{a}\mathrm{c}\mathrm{t},i}\right)+B\left({\dot{\mathit{q}}}_{\mathrm{r}\mathrm{e}\mathrm{f},i}-{\dot{\mathit{q}}}_{\mathrm{a}\mathrm{c}\mathrm{t},i}\right)-{\mathit{\tau }}_{\mathrm{s}\mathrm{p}\mathrm{r}\mathrm{i}\mathrm{n}\mathrm{g},i}.$

The reference joint angle q_ref,i and reference joint velocity ${\dot{\mathit{q}}}_{\mathrm{r}\mathrm{e}\mathrm{f},i}$ of each leg can be processed by trajectory planning. The active stiffness K and damping B should be large to improve the control for the trajectory planning, and they are set to 260 N·m/rad and 50 N·m·s/rad, respectively. Trajectory planning can be calculated by

(42)$\{\begin{array}{l}{\mathit{P}}_i^{\mathrm{L}}={}_{\mathrm{L}i}^{\mathrm{B}}{\mathit{R}}^{\mathrm{T}}\left[{}_{\mathrm{B}}^{\mathrm{G}}{\mathit{R}}^{\mathrm{T}}\left({\mathit{P}}_i^{\mathrm{G}}-{\mathit{P}}_{\mathrm{B}}^{\mathrm{G}}\right)-{\mathit{P}}_{\mathrm{L}i}^{\mathrm{B}}\right],\\ {\mathit{q}}_{\mathrm{r}\mathrm{e}\mathrm{f},i}=I{K}_i^{\mathrm{L}}\left({\mathit{P}}_i^{\mathrm{L}}\right),\\ {\dot{\mathit{q}}}_{\mathrm{r}\mathrm{e}\mathrm{f},i}=\left({\mathit{q}}_{\mathrm{r}\mathrm{e}\mathrm{f},i}-{\mathit{q}}_{\mathrm{r}\mathrm{e}\mathrm{f},i\mathrm{\_}\mathrm{l}\mathrm{a}\mathrm{s}\mathrm{t}}\right)\mathrm{d}t,\end{array}$

where ${\mathit{P}}_i^{\mathrm{G}}$ is the coordinate of the foot tip in GCF, ${\mathit{P}}_{\mathrm{B}}^{\mathrm{G}}$ is the coordinate of the body in GCF, ${\mathit{P}}_{\mathrm{L}i}^{\mathrm{B}}$ is the coordinate of LCF in BCF, and q_{ref,i_last} is the reference joint angle in the last control period. dt is the control cycle, and it is set to 0.001 s.

At the beginning of the extending state, the coordinate of the foot tip in GCF for each leg can be calculated by

(43)${\mathit{P}}_i^{\mathrm{G}}={}_{\mathrm{B}}^{\mathrm{G}}\mathit{R}{}_{\mathrm{L}i}^B\mathit{R}F{K}_i^{\mathrm{L}}\left({\mathit{q}}_i\right).$

Hence, the average height of all six legs H₀ shown in Fig.8 can be expressed by

Fig.8 Extended height of the body in the process of extending.

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(44)$H_0=\frac{1}{6}\sum _{i=1}^6{H}_i=\frac{1}{6}\sum _{i=1}^6\left(-P_{\mathit{\text{z}}i}^{\mathrm{G}}\right).$

The extended height of the body is H_init−H₀, where H_init is the desired height. Therefore, the trajectory planning of the body in the extending state can be expressed by

(45)$\begin{array}{rl}& \{\begin{array}{l}{\mathit{P}}_{\mathrm{B}}^{\mathrm{G}}=\left(\begin{array}{c}{P}_{x\mathrm{B}}^{\mathrm{G}}\\ P_{y\mathrm{B}}^{\mathrm{G}}\\ P_{\mathit{\text{z}}\mathrm{B}}^{\mathrm{G}}\end{array}\right)=s\left(\begin{array}{c}0\\ 0\\ H_{\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}}-H_0\end{array}\right),\\ {\mathit{w}}_{\mathrm{B}}^{\mathrm{G}}=\left(\begin{array}{c}{w}_{x\mathrm{B}}^{\mathrm{G}}\\ w_{y\mathrm{B}}^{\mathrm{G}}\\ w_{\mathit{\text{z}}\mathrm{B}}^{\mathrm{G}}\end{array}\right)=\left(1-s\right)\left(\begin{array}{c}{w}_{x\mathrm{B}0}^{\mathrm{G}}\\ w_{y\mathrm{B}0}^{\mathrm{G}}\\ 0\end{array}\right),\end{array}\\ & s=\{\begin{array}{ll}{\displaystyle \frac{2t^2}{t_{\mathrm{e}\mathrm{x}\mathrm{t}\mathrm{e}\mathrm{n}\mathrm{d}}^2}},& \mathrm{i}\mathrm{f}t⩽{\displaystyle \frac{t_{\mathrm{e}\mathrm{x}\mathrm{t}\mathrm{e}\mathrm{n}\mathrm{d}}}{2}},\\ -{\displaystyle \frac{2t^2}{t_{\mathrm{e}\mathrm{x}\mathrm{t}\mathrm{e}\mathrm{n}\mathrm{d}}^2}}+{\displaystyle \frac{4t}{t_{\mathrm{e}\mathrm{x}\mathrm{t}\mathrm{e}\mathrm{n}\mathrm{d}}}},& \mathrm{i}\mathrm{f}t>{\displaystyle \frac{t_{\mathrm{e}\mathrm{x}\mathrm{t}\mathrm{e}\mathrm{n}\mathrm{d}}}{2}},\end{array}\end{array}$

where $w_{x\mathrm{B}0}^{\mathrm{G}}$ and $w_{y\mathrm{B}0}^{\mathrm{G}}$ are the roll angle and pitch angle at the beginning of the extending state, respectively, and t_extend is the total extending time.

For the body stabilizer, the resultant force in the vertical direction should be kept at zero to maintain a stable height during the extending state process. Moreover, torque is needed to make the roll angle and pitch angle remain zero and control the body attitude. We assume that the mass of the lander is concentrated at the center of the body to simplify the body stabilizer. Therefore, the body stabilizer can be expressed by

(46)$\{\begin{array}{rl}& \sum _{i=1}^6{F}_{\mathit{\text{z}}i}-mg=0,\\ & \sum _{i=1}^6{\mathit{r}}_i\times {\mathit{F}}_i={\mathit{\tau }}_{\mathrm{b}\mathrm{o}\mathrm{d}\mathrm{y}}.\end{array}$

Equation (46) can be rearranged as

(47)$\mathit{C}\cdot {\mathit{F}}_{\mathit{\text{z}}}=\mathit{D},$

where C is a 3 × 6 matrix, and D is a 3 × 1 matrix. They can be expressed by

(48)$\begin{array}{c}\mathit{C}=\left[\begin{array}{cccccc}1& 1& 1& 1& 1& 1\\ r_{1y}& r_{2y}& r_{3y}& r_{4y}& r_{5y}& r_{6y}\\ -r_{1x}& -r_{2x}& -r_{3x}& -r_{4x}& -r_{5x}& -r_{6x}\end{array}\right],\\ \mathit{D}=\left[\begin{array}{c}mg\\ {\mathit{\tau }}_{\mathrm{b}\mathrm{o}\mathrm{d}\mathrm{y}}\end{array}\right].\end{array}$

Equation (47) can be solved by a pseudoinverse, which turns out to be

(49)${\mathit{F}}_{\mathit{\text{z}}}={\mathit{C}}^{†}\mathit{D}.$

Therefore, the joint torque generated by the body stabilizer can be calculated by

(50)${\mathit{\tau }}_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{b},i}=-{\left[{\mathit{J}}_{\mathrm{F}i}^{\mathrm{L}}\right]}^{-1}{}_{\mathrm{L}i}^{\mathrm{B}}{\mathit{R}}^{\mathrm{T}}{}_{\mathrm{B}}^{\mathrm{G}}{\mathit{R}}^{\mathrm{T}}{\left[\begin{array}{ccc}0& 0& 1\end{array}\right]}^{\mathrm{T}}{F}_{\mathit{\text{z}}i}.$

The control procedure for the entire soft-landing process is concluded in Fig.9.

Fig.9 Control procedure for the entire soft-landing process.

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4 4 Experiments and results

4.1 4.1 Experiment scene

As shown in Fig.10, an experimental platform is built to simulate the gravity of the Moon and verify the effectiveness of the soft-landing method. The platform consists of a six-legged mobile repetitive lander, a balancing weight, a rope, a cantilever, two pulleys, and soils. The mass of the balancing weight can be calculated by

Fig.10 Experiment scene for the lander.

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(51)$\{\begin{array}{l}\left(m_1+m_2\right)g_{\mathrm{m}\mathrm{o}\mathrm{o}\mathrm{n}}=\left(m_2-m_1\right)g_{\mathrm{e}\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{h}},\\ g_{\mathrm{m}\mathrm{o}\mathrm{o}\mathrm{n}}={\displaystyle \frac{1}{6}}{g}_{\mathrm{e}\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{h}},\end{array}$

where m₂ = 95 kg is the mass of the lander; g_moon and g_earth are the gravitational acceleration of the Moon and Earth. Therefore, the mass of the balancing weight should be m₁ = 67.8 kg. The lander lands from a height of 2.35 m to simulate the soft-landing process. Two terrains are constructed to prove the validity of the soft-landing method. As shown in Fig.11, the height of the step is 0.13 m, and the degree of the slope is approximately 8.92°.

Fig.11 Parameters of the terrains: (a) height of the step and (b) degree of the slope.

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4.2 4.2 Experiment analysis

The experiment on the terrain with a step is shown in Fig.12. At the beginning of the process, the lander is in the landing state. It drops from the sky, as shown in Fig.12(b). At t = 1.510 s, the legs start to touch the ground, and the state of each leg changes from the aerial phase to the grounded phase successively. Leg 3 touches the ground first because of the step, as shown in Fig.12(c). The grounded time of each leg is shown in Tab.2. The lander changes to the retracting state when all legs touch the ground at t = 1.581 s, as shown in Fig.12(d). Then, the height of the body descends until the body speed drops to zero at t = 1.863 s, as shown in Fig.12(e) and Fig.12(f). Finally, the lander changes to the extending state, and the height of the body extends to the desired height at t = 4.063 s, as shown in Fig.12(h). The extended height of the body is approximately 0.229 m.

Tab.2 Grounded time of each leg on the terrain with a step

Leg number	Grounded time/s
1	1.565
2	1.580
3	1.510
4	1.581
5	1.578
6	1.572

Fig.12 (a–h) Snapshots of the experiment on the terrain with a step.

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The joint torques of all six legs for the soft-landing process on the terrain with a step are shown in Fig.13(a). The limited torque τ_limit of each joint is set to 50 N·m. When t < 1.581 s, the lander is in the landing state (line A). All six legs touch the ground in succession. The torques of the thigh motor and shank motor in the process of touching the ground are shown in Fig.13(b). With leg 3 as the example, the leg is in the aerial phase (line D) when t < 1.510 s. It changes to the grounded phase at t = 1.510 s when the leg touches the ground (line E). Then, the stiffness K and damping B of the compliance control are set to zero. The joint torque is only affected by the two passive springs. Accordingly, the other legs are controlled in the same way.

Fig.13 Joint torques of the legs on the terrain with a step: (a) the torques of all six legs in the whole process and (b) the torques of the thigh motor and shank motor in the process of touching the ground.

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When the lander is in the retracting state (line B), the optimal force control helps the body decelerate as quickly as possible. The average height of all six legs reduces, as shown in Fig.14. Fig.14(a) shows that when t = 1.863 s, the average vertical value of the velocity of all six legs in BCF drops to zero. The average height of all six legs is 0.521 m, as shown in Fig.14(b). Then, the lander changes to the extending state (line C). The height of the body rises to approximately 0.75 m at t = 4.063 s and is maintained. Fig.14(c) illustrates the roll angle and pitch angle of the experiment. They have oscillations in the process of touching the ground. They are close to zero when the body is in the stable state.

Fig.14 Results of the experiment on the terrain with a step: (a) the average vertical value of the velocities of all six legs in BCF, (b) the average height of all six legs, and (c) the roll angle and pitch angle of the body.

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The soft-landing experiment on the terrain with a slope is shown in Fig.15. The lander changes to the retracting state when all legs touch the ground at t = 1.502 s. The order of the legs is shown in Fig.16. Legs 4 and 5 touch the ground first because of the slope. The grounded time of each leg is shown in Tab.3.

Tab.3 Grounded time of each leg on the terrain with a slope

Leg number	Grounded time/s
1	1.496
2	1.502
3	1.470
4	1.420
5	1.421
6	1.477

Fig.15 (a–h) Snapshots of the experiment on the terrain with a slope.

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Fig.16 Order of the legs.

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The joint torques of all six legs for the soft-landing process on the terrain with a slope are shown in Fig.17(a). The limited torque of each joint is also set to 50 N·m. Fig.18(a) shows that when t = 1.731 s, the average vertical value of the velocity of all six legs in BCF drops to zero. The body of the lander rises to approximately 0.182 m during the extending state process, as shown in Fig.18(b).

Fig.17 Joint torques of the legs on the terrain with a slope: (a) the torques of all six legs in the whole process and (b) the torques of the thigh motor and shank motor in the process of touching the ground.

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Fig.18 Results of the experiment on the terrain with a slope: (a) the average vertical value of the velocities of all six legs in BCF, (b) the average height of all six legs, and (c) the roll angle and pitch angle of the body.

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A comparative experiment is conducted to illustrate the effectiveness of the soft-landing method proposed in this study. The control strategy of the comparative experiment is to change the active stiffness K and damping B continuously when the lander is in the retracting and extending states [34]. The experiment on the terrain with a step is shown in Fig.19. At t = 1.520 s, the legs start to touch the ground successively. The grounded time of each leg is shown in Tab.4. The total time of the retracting state is 0.299 s.

Tab.4 Grounded time of each leg on the terrain with a step for comparative experiment

Leg number	Grounded time/s
1	1.589
2	1.520
3	1.576
4	1.579
5	1.578
6	1.591

Fig.19 (a–h) Snapshots for the comparative experiment.

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The joint torques of all six legs for the comparative experiment on the terrain with a step are shown in Fig.20. When t < 1.591 s, the lander is in the landing state (line A). Moreover, the lander changes to the extending state when t > 1.89 s (line C). The maximum torque of the active joints is 130.5 N·m in the soft-landing process. The result of the comparative experiment shows that the optimal force control we proposed can effectively control the joint torques and reduce the impact energy.

Fig.20 Joint torques of the legs for the comparative experiment.

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5 5 Conclusions

This study proposes a soft-landing method for the six-legged mobile repetitive lander. The method based on compliance control and optimal force control can make responses to different states during the landing process. It can also realize soft-landing safely in rough terrains, such as those with steps and slopes. Moreover, an experimental platform is built to simulate the gravity of the Moon. The results show that the proposed method can realize a stable and safe soft-landing on complex terrains under the gravity of the Moon. In future work, the fault-tolerant soft-landing method will be studied to solve the landing problem, such as the failure of several drive motors. Based on the form of the drive motors’ failure, the two kinds of failure are the uncontrollable failure and the locked failure. For uncontrollable failure, the failed joints lose the supporting capacity, and the landing problem can be solved by dealing with the conditions of losing several legs. For locked failure, the failed joints cannot move and can be used for support. The body of the lander tilts when the lander touches the ground. The potential strategy of this condition is to plan the trajectory of the body and make the lander adjust to a stable state.

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