A new dimension design method for the cantilever-type legged lander based on truss-mechanism transformation

Mechanism and Machine Theory 142 (2019) 103611

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Mechanism and Machine Theory journal homepage: www.elsevier.com/locate/mechmachtheory

Research paper

A new dimension design method for the cantilever-type legged lander based on truss-mechanism transformation Youcheng Han a, Weizhong Guo a,∗, Feng Gao a, Jianzhong Yang b a

State Key Laboratory of Mechanical Systems and Vibration, School of Mechanical Engineering, Shanghai Jiao Tong University, Shanghai, 200240, China b Institute of Spacecraft System Engineering, Beijing, 100094, China

a r t i c l e

i n f o

Article history: Received 28 April 2019 Revised 5 September 2019 Accepted 5 September 2019 Available online 20 September 2019 Keywords: Legged lander Dimension design Optimization design Performance atlas Truss-mechanism transformation

a b s t r a c t The cantilever-type legged lander (CTLL) is widely used in many extraterrestrial exploration activities, owing to the good buffering stability, large support stiffness, and excellent payload capacity. The paper will focus on the following two important and challenging aspects for its optimum dimension design: (1) One is to build a reliable and comprehensive optimization design model for the CTLL, including the optimization parameters of the topological structure, the performance indices of the landing behavior, and the constraint conditions of both landing stability and workspace; (2) The other aspect is to develop a systematic strategy to optimize both landing leg and overall lander respectively based on the optimization design model above. The paper proposes the trussmechanism transformation method (TMTM) to optimize the truss structure of the CTLL in the view of mechanisms and robotics with kinematics and dynamics characteristics. Furthermore, the design space dimensionality reduction method (DSDRM) is proposed for the multi-parameter optimization problem and develops the performance-chart based design methodology (PCbDM) as well. The paper can help one understand the relationship between different dimensions and the corresponding performances, and obtain the global optimum dimensions for both landing leg and overall lander. © 2019 Elsevier Ltd. All rights reserved.

1. Introduction Over the past 50 years, human sent a lot of legged landers for the scientific exploration of the extraterrestrial body, which conducted many practical experiments and achieved successful accomplishments. It’s undeniable that the landing mechanism of the legged lander plays a crucial role in the buffering process, and influences the success or failure of the mission directly. There are three main functions of the landing mechanism, including: (1) absorbing the instantaneous large landing impact energy, and protecting the exploration equipment; (2) guaranteeing the landing stability of the lander, to be specific, promising the engine tail nozzle not to get broken by touching the extraterrestrial surface, meanwhile, promising

Abbreviations: CTLL, cantilever-type legged lander; TMTM, truss-mechanism transformation method; DSDRM, design space dimensionality reduction method; PCbDM, performance-chart based design methodology; DOF, degree of freedom; COG, center of gravity; WVI, workspace volume index; GCI, global conditioning index; GTI, global transmission index; GPI, global payload index; GSI, global stiffness index; GLSI, global landing stability index; FASM, forceangle stability measure. ∗ Corresponding author. E-mail addresses: [email protected] (Y. Han), [email protected] (W. Guo), [email protected] (F. Gao), [email protected] (J. Yang). https://doi.org/10.1016/j.mechmachtheory.2019.103611 0094-114X/© 2019 Elsevier Ltd. All rights reserved.

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Fig. 1. Chang’e 3 Lunar Lander [1,4,28–30].

the lander not to tip over; (3) supporting the lander for a long time, and keeping the equipment on it carrying out experiments successfully [1]. Generally speaking, the legged lander contains two types according to the topological structure of the landing mechanism, namely the cantilever type and the tripod type. The former the cantilever type is widely used in many exploration activities among different countries, owing to the better vertical and horizontal buffering stabilities, larger support stiffness, and more excellent payload ability. Especially, it’s suitable for carrying heavy mass, including detecting instruments, cargos and human beings, typical application cases such as American Apollo Project (1969–1972) [2], Soviet/Russian LK Lunar Craft Program (1970–1971) [3], Chinese Chang’e 3 & 4 Program (2013-now, shown in Fig. 1) [4], etc. The latter the tripod type has a simpler structure relatively, smaller support stiffness, and smaller angle between the primary strut and the ground. Hence, it is more suitable for landing with low mass, typical application cases such as American Surveyor Program (1966–1968) [5], Viking Program (1975) [6], Mars Surveyor Program (1999) [7], Phoenix Program (2007) [8], Insight Program (2018) [9], Soviet/Russian Luna Program (1969–1976) [3], etc. Given that the cantilever-type legged lander (CTLL) has better comprehensive functions and performances, the paper will take this type of topological structure for the following investigation. Nevertheless, there still exist many challenges on the optimum dimension design problem for the CTLL (including both landing leg and overall lander). Three aspects need to be taken into account. For one thing, the conventional dimension design method for the past legged landers [1,10–12] is a structural design and analysis method based on iteration, trial and error, which takes the lander as truss structure with buffering characteristics, then investigates the structural strength, reliability, finite element analysis, etc. All that matters is there is not a cogent approach to determine the preliminary dimensions for the truss structure, which just depends on the experience and intuition of designers with the basic landing stability constraints. Once the preliminary dimensions are given tentatively, the iterative parametric modeling, simulation and experimental verification are used to derive the optimum structural dimensions. Therefore, the conventional method is time-consuming and inefficient: in Apollo Project [11,12], the researchers consider the lunar module as a six-DOF rigid body and present its touchdown dynamics mathematical procedure to simulate the landing process, then use the Monte Carlo approach to estimate the buffering probability with different structural dimensions and landing conditions; in Chang’e 3 Program [1], the researchers propose the parameterized modeling method of the landing mechanism, touchdown ground, initial conditions, dynamic payload, etc. The simulation analysis results are extracted automatically, including the landing stability, probability, and cushion stroke of the damper, etc. For another thing, it’s essential to establish reasonable and feasible evaluation indices for main performances related to the landing behavior. One should be careful to distinguish different performance indices if they can be used in the design process or just in the analysis or test process. The Apollo Project [13] proposes four statistical landing performance criteria of the landing gear, including the landing stability, primary-strut stroke, secondary-strut tension stroke, and secondarystrut compression stroke; the Chang’e 3 Program [1] presents five comprehensive evaluation indices, including the energy absorption capacity of unit mass, buffering force smoothness, material cushion rate, maximum impact acceleration, and tipover stability. However, all of these performance indices are mainly applied in the analysis or test process rather than the design process. Moreover, Liu et al. [14] put forward the axial crushing characteristic index of aluminum honeycomb and foam to characterize the cushion performance of the HIT legged lunar lander, but they don’t take other landing performances into account; Wu et al. [15] also propose three performance indices for the legged lander, including the distance from the COG (center of gravity) to the stability plane, the minimum distance from base to ground, and the impact force on the main body. However, these indices are still not very sufficient in the design process. What’s more, some other indices are widely recognized and applied in the fields of mechanisms and robotics, which provides referential significance for the performance criteria of the CTLL, such as the workspace volume index (WVI) [16], the global conditioning index (GCI) [17,18], the global transmission index (GTI) [19,20], the global payload index (GPI) [21], the global stiffness index (GSI) [21] for general robotic mechanisms, and the tipover stability index of force-angle stability measure (FASM) [22,23] for mobile manipulators, etc.

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Last but not least, the calculation and expression method for the optimum dimension design problem is of vital importance as well. For a long time, the cost function approach is widely used, which applies the multi-object numerical algorithm to solve the optimization problem, for instance, Wu et al. [15] utilize the second generation of non-inferior sorting genetic algorithm to solve the multi-objective optimization problem of the legged lander, and obtain the dimension results of the landing mechanism in different landing conditions. However, the cost function approach is easy to get a local optimum solution, or else it requires more computing time and effort to get the global optimum solution. Furthermore, it’s not perceptual to compare different solutions by visualization, and the designers still don’t know the optimal level of optimization results. On the contrary, in recent twenty years, the performance-chart based design methodology (PCbDM) [24-27] is popular in the fields of mechanisms and robotics. This approach is based on the physical model theory, which transforms the infinite dimensional design space into the finite non-dimensional design space. After plotting performance atlases in the finite non-dimensional design space, the designers can master the distribution characteristics of different performance indices intuitively and make perceptual comparations easily. Hence, the approach can provide with a good global optimization and visualization for the optimum dimension design process. It can make up for the shortcomings of the cost function approach and is appropriate for industrial application. However, it has a strict requirement that the quantity of the optimization parameters should be less than four, which constrains its application and development a lot. Moreover, the PCbDM is just used for one time in one optimization of the previous researches. Challenges will be generated for the optimization of complex mechanism with more than one multi-parameter dimension design chain. In conclusion, there is almost no directly feasible method for the optimum dimension design of the CTLL at present. Hence, the paper will focus on the following two important and challenging aspects to solve this problem: One is to build a reliable and comprehensive optimization design model of the CTLL, including both landing leg and overall lander: (1) the proposition of the truss-mechanism transformation method as the basis to build the optimization design model, which transforms the truss structure of the CTLL in landing process into robotic mechanisms with kinematics and dynamics characteristics; (2) the selection of optimization parameters for the topological structure; (3) the derivation of the performance evaluation indices for the landing behavior; (4) the identification of the constraint conditions for both landing stability and workspace. The other aspect is to develop a systematic strategy to optimize both landing leg and overall lander respectively based on the optimization design model above, including: (1) the proposition of the design space dimensionality reduction method as the basis of the performance-chart based design methodology; (2) the non-dimensional optimum design of both landing leg and overall lander respectively; (3) the identification of the normalization factor and the dimensional parameters. The strategy will help one to understand the relationship between different possible dimensions and the corresponding performances, and obtain the global optimum dimensions of both landing leg and overall lander. The paper is organized as follows: first of all, Section 2 gives a detailed introduction about the procedure of the proposed optimum dimension design strategy of the CTLL; next, Section 3 analyzes the structure, mobility, motion characteristics and kinematics model of both landing leg and overall lander respectively, based on the proposed truss-mechanism transformation method; afterward, Section 4 establishes the optimization design models of both landing leg and overall lander; thereafter, Sections 5 and 6 implement the optimization design processes of both landing leg and overall lander respectively; then, Section 7 obtains the optimum dimensional parameters of the CTLL; in the end, Section 8 concludes the paper and summarizes the main findings of this work. 2. Optimum dimension design strategy Inspired by the performance-chart based design methodology (PCbDM) [24], the paper will develop this method to get the global optimum dimensions and the perceptual contrasts with different performance indices. Nevertheless, it’s also very challenging to build the optimization design model and develop the systematic optimum dimension design strategy for the CTLL as mentioned above. As the introduction of the total, Fig. 2 explains the design strategy flowchart at length, which can be divided into four parts as follows: • Part 1: Optimization design model for the CTLL, including both landing leg and overall lander. (1) Step 1: Analyze the structures, functions and landing behaviors of both landing leg and overall lander, investigate their mobilities and motion characteristics by the truss-mechanism transformation method (TMTM); (2) Step 2: Establish the kinematics model as the basis of Step 3 & 4; (3) Step 3: Ensure optimization parameters, namely the design variables of the optimization design model, which should be constant structure parameters included in Jacobian matrices; (4) Step 4: Derive performance indices, namely the objective functions of the optimization design model. There are five types of indices considering different landing performances, including the workspace volume index (WVI), the global transmission index (GTI), the global payload index (GPI), the global stiffness index (GSI), the global landing stability index (GLSI), etc.; (5) Step 5: Identify constraint conditions of the optimization design model, including the landing stability constraints and the workspace constraints. • Part 2: Non-dimensional optimization design of the landing leg. Notably, the non-dimensional optimization design means there are no units and physical meaning for the obtained optimum parameters, which just represents the optimum proportional relationship between different parameters. (6) Step 6: Establish the physical model of the landing leg, to express the design space of the dimension parameters geometrically and visually; (7) Step 7: Plot performance atlases to represent the relationship between design parameters and performance indices. For the landing leg, the performance

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Fig. 2. Optimum dimension design strategy flowchart of the CTLL.

indices are chosen as WVI, GTI, GPI, GSI; (8) Step 8: Select optimum parameters with the best comprehensive performances in atlases. On account that there doesn’t exist the unique optimum solution for the multi-objective optimization problem, the determination of the optimum solution should balance different performance requirements. Thus, it’s appropriate to choose different combinations of possible non-dimensional parameters; (9) Step 9: Make comparations between different non-dimensional combinations and the corresponding performances, finally verify the applicability of the selected optimum parameters. • Part 3: Non-dimensional optimization design of the overall lander, this part is similar to Part 2 in general. (10) Step 10: Establish the physical model of the overall lander; (11) Step 11: Plot performance atlases for the overall lander, and the performance indices are chosen as WVI, GPI, GSI, GLSI; (12) Step 12: Select optimum parameters with the best comprehensive performances in atlases. For the same reason, it’s appropriate to choose different combinations of possible non-dimensional parameters; (13) Step 13: Make comparations between different non-dimensional combinations and the corresponding performances, finally verify the applicability of the selected optimum parameters. • Part 4: Dimensional solution for both landing leg and overall lander. Notably, the dimensional solution means the optimum parameters possess units, which has physical meaning and represents the real lengths of the structure. (14) Step 14: Identify the normalization factor of the overall lander by special application demand, to be specific, the CTLL in the stowed state should meet the envelope requirement of the carrying vehicle; (15) Step 15: Calculate dimensional parameters of the overall lander, by using the normalization factor solved in Step 14. Then, according to the common parameter appeared in both physical models of landing leg and overall lander, the normalization factor of the landing leg can be derived uniquely. Thus, the dimensional parameters of the landing leg can be derived. The detailed implementation of the proposed design strategy will be introduced in the following sections. 3. Structure and kinematics model 3.1. Structure and truss-mechanism transformation method The CTLL consists of one main body and four landing legs, and each landing leg contains one primary damper and two secondary dampers. Both primary damper and secondary damper are made by the energy-absorption material or device, such as the aluminum honeycomb damper, hydraulic damper, electromagnetic damper, etc. The primary damper aims at absorbing the vertical landing impact energy and supporting main mass of the CTLL after landing. Its outer sleeve is connected with main body by universal hinge, and inner piston is connected with footpad by spherical hinge. The secondary damper aims at absorbing the horizontal landing impact energy and assisting the primary damper to support the CTLL together. Its

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Fig. 3. Structure diagram of the CTLL in different states.

Fig. 4. Truss structure and its transformed mechanism of the landing leg.

outer sleeve is connected with main body by universal hinge, and inner piston is connected with the outer sleeve of primary damper by spherical hinge. The footpad can be seen as the end-effector of the landing leg, it aims at increasing the contact area between the CTLL and the ground in case of sinking into the extraterrestrial surface excessively, preventing the lateral slippage of the CTLL, and absorbing a little impact energy as well. Fig. 3 shows two different states for the CTLL before landing and after landing, which are called the pre-landing state and the post-landing state. The CTLL has different structural dimensions between the pre-landing state, the landing process, and the post-landing state. Firstly, in the pre-landing state, the CTLL reaches the predetermined orbit, all the landing legs are deployed and locked to provide a stable and reliable landing configuration. Then via a series of deceleration control and braking effect caused by the thrust engine, the CTLL makes preparation for landing on the extraterrestrial surface soon after. At this moment, the body axis of the CTLL is vertical to the landing surface, and all the four primary dampers and all the eight secondary dampers hold the same length respectively, as is shown in Fig. 3(a). Next, in the landing process, the dampers installed on the landing leg take effect and absorb large landing impact energy, the primary damper is shortened while the secondary damper is both possible to be shortened and stretched under different landing conditions. So in the post-landing state, the body axis of the CTLL is inclined to the landing surface, and the lengths of primary dampers and secondary dampers are different respectively, as is shown in Fig. 3(b). Thereafter, all dampers in the landing process will show the motion characteristics of prismatic pair, so that both landing leg and overall lander can be seen as mechanisms or robots with kinematics and dynamics characteristics, in contrast that they are just truss structures in the pre-landing and post-landing state. In this point of view, the landing process that the dampers absorb the landing impact energy and change their lengths is just the process that both landing leg and overall lander perform motion characteristics. For the sake of this, we present the truss-mechanism transformation method (TMTM) here, which transforms the truss structures of both landing leg and overall lander into movable robotic mechanisms, and further investigate their topologies, mobilities and motion characteristics. Fig. 4 illustrates the truss-mechanism transformation method for the landing leg, (a) denotes the truss structure of the landing leg, and (b) denotes the corresponding transformed mechanism. In this figure, U1 ,U2 ,U3 denote three universal

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hinges, S1 ,S2 ,S3 denote three spherical hinges. l1 and l3 represent the inner piston and the outer sleeve of the primary damper, l2 and l4 represent two secondary dampers, and l2 = l4 when in the pre-landing state. Also marked as: l1 = |AS1 |, l2 = |U2 S2 |, l3 = |AU1 |, l4 = |U3 S3 |. Using the theory of spatial mechanisms, the landing leg is studied as a multimode mechanism with specific mobilities and motion characteristics. (1) In the landing process, all dampers possess one-dimensional translational characteristic with buffering effect, so l1 ,l2 ,l4 are considered as the prismatic pairs P1 ,P2 ,P3 . As a result, each landing leg is equivalent to a hybrid mechanism composed of a parallel mechanism in series connection with a single open-chain mechanism. Fig. 4(b) shows the transformed mechanism of the landing leg: the parallel portion U&2UPS is also called the hip mechanism, consists of one primary limb U and two secondary limbs UPS; in addition, the series portion PS, consists of one primary damper P1 and one spherical hinge S in the ankle joint connected with the footpad. According to the modified Grübler-Kutzbach formula [31], the mobility of the landing leg (U&2UPS)-PS can be obtained by Eq. (1). Notably, the footpad is recognized as the end effector, and the spherical hinge is passive or local mobility to adapt to the extraterrestrial terrain.

MLEG = d (n − g − 1 ) +

g 

fi + ν − ξ = 6(8 − 9 − 1 ) + 18 + 0 − 3 = 3

(1)

i=1

We can deduce that each landing leg has 2R1T motion characteristics. Given the effect of the spherical hinge, then it has 6 DOF with 3R3T motion characteristics, explained by the screw theory in Appendix A. Hence, the overall lander has 3R3T full mobility, MCT LL = 6, its moving platform is the main body, while its static platform is the ground surface. (2) In the pre-landing state and the post-landing state, all dampers are considered as rigid linkages with fixed lengths. Hence, both landing leg and overall lander are considered as truss structures, of which the hip mechanism is represented as U&2US, including one primary limb U and two secondary limbs US. Moreover, the series portion is represented as S. Hence, the mobility of the landing leg (U&2US)-S can be obtained by Eq. (2), MLEG = MCT LL = 0. Using the screw theory we can also deduce that both landing leg and overall lander have 0R0T motion characteristics. Accordingly, in the prelanding state, both landing leg and overall lander hold the truss structure to fulfill the energy absorption capacity of the dampers, while in the post-landing state, the CTLL keeps a steady truss configuration for a long time to release the rover to the ground, and further carrying out the extraterrestrial rock sampling and fixed-point detection.

MLEG = d (n − g − 1 ) +

g 

fi + ν − ξ = 6(5 − 6 − 1 ) + 15 + 0 − 3 = 0

(2)

i=1

Above all, both landing leg and overall lander are investigated by TMTM. It’s a reasonable and practicable method as the basis to establish the kinematics model and implement the optimum dimension design. 3.2. Parameters and kinematics model Fig. 5 shows the parameters diagram for both overall lander and landing leg. There are three coordinate frames in the overall lander, all follow the right-hand rule: (1) the world coordinate frame {G}, in which we discuss the position and velocity analysis of the overall lander; (2) the body coordinate frame {B}: the point B is located in the geometry center of main body, xB and yB are perpendicular to the side surface of main body; (3) the waist coordinate frame {Ci } (i = 1, 2, 3, 4), in which we discuss the position and velocity analysis of the landing leg, the point C is the midpoint of |U2 U3 |, and xC is perpendicular to the side surface of main body, zC is parallel with the zB . In Figs. 4(b) and 5(b), the point A is the vertex of the triangle AS2 S3 . Moreover, Fi denotes the center point of S1 in the ith landing leg. For the hip mechanism of the landing leg, the static platform is the isosceles triangle U1 U2 U3 and the moving platform is the isosceles triangle AS2 S3 , their bottom edges are |U2 U3 | = 2a1 and |S2 S3 | = 2a2 , heights are b1 and b2 ; moreover, θ in represents the installation angle between the landing leg and the main body; the main body is a 2d × 2d × d cuboid, where d is restricted by the envelope diameter of the carrying vehicle; Rsp represents the radius of the supporting circle formed by the four footpads of the landing legs, which can also be considered as the radius of the static platform of the overall lander; Rms represents the distance between zB and the main strut universal hinge, which can also be considered as the radius of the moving platform of the overall lander; L0 is the distance from the COG (recognized as the geometric center of main body) to the engine nozzle skirt (the lowest point of main body); Lv is the distance from the engine nozzle skirt to the extraterrestrial surface at the pre-landing state; H0 is the distance from the COG to the ground at the pre-landing state, √ and there exists H0 = Lv + L0 ; Lh is the distance between zB and the connecting line of two adjacent footpads, and Rsp = 2Lh . Consequently, all structure parameters of the landing leg are listed as: l3 ,a1 ,b1 ,a2 ,b2 . Beyond that, all structure parameters of the overall lander are listed as: θ in ,Rms ,Rsp ,d,L0 ,Lh . Thus, the parameters mentioned above contain all the dimensions of the topological structure for the CTLL, although some of these aren’t independent. Moreover, based on the TMTM, the parameters l1 ,l2 ,l4 are taken as the actuation parameters for the CTLL rather than structure parameters, on account that their lengths will be changed by the landing impact energy. So the CTLL possesses twelve actuation parameters. In this point of view, the inverse kinematics model of the CTLL is more meaningful than the forward one.

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Fig. 5. Parameters diagram for both overall lander and landing leg.

In the following, the paper will introduce the recapitulative expressions of the kinematic models for the sake of brevity. For the ith landing leg, the position model and velocity model can be expressed by Eqs. (3) and (4) respectively in frame {Ci }, where LegIPM denotes the inverse position model of the landing leg, i denotes the length vector of the dampers, i = [li1 li2 li4 ]T , Ci Fi and Ci VFi are the position vector and the velocity vector of the point Fi , and JLEG is the inverse Jacobian of the landing leg.

  i = LegIPM Ci Fi

(3)

˙ i = JLEG ·Ci VF  i

(4)

For the overall lander, the position model and velocity model can be expressed by Eqs. (5) and (6) respectively in frame {G}, where CTLLIPM denotes the inverse position model of the overall lander,  denotes the length vector of all the dampers,  = [1 2 3 4 ]T , G B and G VB are the position vector and the velocity vector of the COG, and JCTLL is the inverse Jacobian of the overall lander.

   = CT LLIPM G B

(5)

˙ = JCT LL ·GVB 

(6)

4. Optimization design model 4.1. Optimization parameters and dimensionality reduction method Optimization parameters, namely design variables, are the kernel dimension parameters that influence the kinematics and dynamics performances of the CTLL. For the problem of optimum dimension design, the optimization parameters must be constant values included in the Jacobian matrix and irrelevant to the landing process.

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Fig. 6. Flowchart of the DSDRM and its implementation.

Via careful investigation of Fig. 5, there are a total of eight optimization parameters ensured for the CTLL. Among them, five parameters are for the landing leg, including a1 ,b1 ,a2 ,b2 ,l3 , meanwhile, four parameters are for the overall lander, including θ in ,d,Rsp ,l3 . Notably, the parameter l3 exists in both landing leg and overall lander, so it can make a transfer effect on the normalization factors for both landing leg and overall lander. That is to say, after obtaining their optimum non-dimensional parameters respectively, we can firstly identify the normalization factor of the overall lander, and then the normalization factor of the landing leg can be determined uniquely. Because the value of l3 is obtained in the dimensional solution process of the overall lander. The PCbDM method has the advantages of multi-object global optimization and perceptual comparations with different solutions by visualization. However, it still has a strict demand that the quantity of the optimization parameters should be less than four [24], which gives a challenge to the optimum dimension design of the CTLL with eight parameters. To make up for this shortage, the paper proposes the design space dimensionality reduction method (DSDRM) illustrated in Fig. 6, which contains three key steps: firstly, converting the optimization design model for the CTLL with eight parameters into two separate models — the landing leg with five parameters and the overall lander with four parameters. So the PCbDM method can be implemented with the parameter quantity less than four in each part. And the optimization of the landing leg is followed by the overall lander; then, taking the similarity operation by a1 /b1 = a2 /b2 = χ for the landing leg, according to the low DOF parallel mechanism usually owns the geometrically similar static platform and moving platform to get better performances and easier forward kinematics solutions. χ is further assigned to 1.5 given the landing stability and supporting stiffness; finally, substituting θ in and d into the geometrical relationship equation Rms = d + b1 sinθin for the CTLL. This study further let θin = π /6 on account that the maximum rotation angle of the universal hinge is π /3, so their sum value can be equal to π /2. The detailed implementation of the DSDRM based PCbDM will be introduced in Sections 5 and 6. 4.2. Constraint conditions 4.2.1. Landing stability constraints According to the researches by Yang [1], there are mainly two types of landing stability constraints for the CTLL: one is the vertical stability constraint, which requires the engine nozzle skirt at the bottom of the main body cannot touch the rocks or the extraterrestrial ground; the other is the horizontal stability constraint, which demands the CTLL cannot tip over under the effect of the horizontal landing speed, so a large span of the legs are needed, as is shown in Fig. 5(a). Eq. (7) illustrates the criterion for the vertical stability Lv and horizontal stability Lh :



Lv − [Lv ] ≥ 0 Lh − [Lh ] ≥ 0

(7)

where [Lv ] is the minimum value of Lv that can ensure the CTLL safe in vertical direction, and [Lh ] is the minimum value of Lh that can ensure the CTLL safe in the horizontal direction. 4.2.2. Workspace constraints Based on the TMTM, the workspace here also means the buffering space of both landing leg and overall lander, which is restricted by the length range of the damper, the rotation angle range of the universal hinge and the spherical hinge. We use lmax and lmin to denote the maximum and the minimum values of the damper, β max and β min to denote the maximum and the minimum values of the first axis angle of the universal hinge, γ max and γ min to denote the maximum and the minimum values of the second axis angle of the universal hinge, and ξ max and ξ min to denote the maximum and the minimum values of the pivot angle of the spherical angle.

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Furthermore, considering all the four landing legs share common parameters, we use j (j ∈ {1, 2, 3, 4}) to denote the jth landing leg, and i to denote ith limb (i ∈ {1, 2, 3}). Therefore, we can obtain that there are a total of twelve workspace constraints for each landing leg, and a total of forty-eight workspace constraints for the overall lander, given by:

⎧ ⎪ ⎨ limin ≤ l ji ≤ limax βmin ≤ β ji ≤ βmax ⎪ γmin ≤ γ ji ≤ γmax ⎩ ξmin ≤ ξ ji ≤ ξmax

(8)

4.3. Performance indices In this paper, there are five performance indices for evaluating the landing behavior and touchdown performance of the CTLL. Among them, the first four indices are based on the works by Gosselin, Liu, and Gao, etc., which are the workspace volume index [16], global transmission index [19], global payload index [21], and global stiffness index [21]; while the fifth index, the global landing stability index, is put forward for the first time based on the work by Papadopoulos [22]. 4.3.1. Workspace volume index (WVI) Based on the TMTM, both landing leg and overall lander are taken as robotic mechanisms with workspace. In the landing process, they change their configurations from the initial pose at the pre-landing state to any possible pose in the workspace. The workspace volume index (WVI) [16] is used to represent how big the workspace is, actually it stands for the buffering extent and ability of both landing leg and overall lander. For the following four global indices, they are actually the average values of four local indices with respect to the workspace. So the WVI is also the basis for establishing all other performances. Both landing leg and overall lander should have a bigger workspace to adapt to the worst buffering condition. The WVI can be derived by the triple integral in the Cartesian coordinate system:

ηW =



V



dW =



xmax

xmin



ymax

dx

zmax

dy ymin

dz

(9)

zmin

where x, y, z are the three integral variables, and their ranges are: x ∈ [xmin , xmax ], y ∈ [ymin , ymax ], z ∈ [zmin , zmax ]. 4.3.2. Global transmission index (GTI) To improve the buffering efficiency, the CTLL should have a better motion/force transmission performance in the landing process, which means the landing impact energy can be absorbed more easily and quickly. For the landing leg, its function aims at transmitting the buffering motion/force from its input members (dampers) to its output member (end-effector of footpad). The local transmission index (LTI) and global transmission index (GTI) [19] are used to evaluate the transmission efficiency in the workspace and dimension design space respectively. As Eq. (10) indicates, the input transmission index μIi represents the power coefficient between the input twist screw $Ii (along with the motion direction of the damper) and the transmission wrench screw $Ti (offered by the damper to the moving platform) by reciprocal product operation; the output transmission index μOi represents the power coefficient between $Ti and the output twist screw $Oi (the buffering motion of the footpad); μT represents the LTI, is the minimum of μIi and μOi ; ηT represents the GTI, is the average value of μT with respect to the workspace. And we hope the transmission index of the CTLL is as large as possible, which satisfies μT , ηT ∈ [0, 1].

 i| μIi = |$|Ii$◦$Ii ◦$Ti |Tmax , μT = min {μIi , μOi },

Ti ◦$Oi | μOi = |$|T$i ◦$ Oi |max μ T ηT = V dWdW

(10)

V

It’s notable to say that the workspace for all performance indices including WVI is characterized by LTI > 0.50 to have a good motion/force transmission performance. 4.3.3. Global payload index (GPI) In the landing process, the landing leg will be subjected to the landing impact force at footpad. Moreover, the overall lander will be subjected to both landing impact force and moment at COG simultaneously. To obtain the CTLL with better impact-resistance ability considering both landing impact force and moment, the global payload index (GPI) [21] is used to evaluate the extremum of the landing impact force or moment that both landing leg and overall lander can bear in the landing process. Particularly, we hope the payload indices of the landing leg and the main body are as large as possible. T ]T , where [FT MT ]T is the output general force while τ is the The force equation can be given as: [F T M T ]T = Gτ , G = [GFT GM input joint general force, and G is the force Jacobian composed of the submatrices GF and GM . The relationship between the velocity Jacobian and the force Jacobian is: G = J T . The four GPIs are ηFmax , ηFmin , ηMmax and ηMmin , given by:

 ηF max =

ηMmax =



√

max(|λF i | )dW , dW √ V max λ dW | | ( ) Mi V , V dW V

ηF min = ηMmin =

√

V

V

min(|λF i | )dW V dW min(|λMi | )dW V dW

√

T G respectively. where λFi and λMi are the eigenvalues of the matrices GFT GF and GM M

(11)

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Fig. 7. Parameter diagram of the landing stability index.

4.3.4. Global stiffness index (GSI) Mechanism stiffness is another important index for the landing behavior, which can influence the rigidity and position accuracy of the mechanism topological structure of the CTLL. Both landing leg and overall lander should possess better mechanism stiffness and position accuracy, to decline the linear or angular deformation of the CTLL mechanism. Only under this condition, the dampers will bring out the energy-absorbing ability more sufficiently. Hence, the global stiffness index (GSI) [21] is used to evaluate the linear displacement deformation or the angular displacement deformation for both landing leg and overall lander in the landing process. Particularly, we hope the stiffness indices of both landing leg and overall lander are as little as possible, so the deformations of them are as little as possible. The deformation equation can be given as: [P T OT ]T = K −1 [F T M T ]T , K = [KPT KOT ]T = J T J, where [PT OT ]T denotes the linear deformation P and the angular deformation O. K is the stiffness matrix composed of the submatrices KP and KO with respect to P and O. So the four GSIs are ηPmax , ηPmin , ηOmax and ηOmin , given by:

 ηPmax =

ηOmax =



√

max(|λPi | )dW , dW √ V max λ dW | | ( ) Oi V , V dW V

ηPmin = ηOmin =

√

V

V

min(|λPi | )dW V dW min(|λOi | )dW V dW

√

(12)

where λPi and λOi are the eigenvalues of the matrices (KP−1 )T KP−1 and (KO−1 )T KO−1 respectively. 4.3.5. Global landing stability index (GLSI) Landing stability is one of the most important performance criteria for the dimension design of the CTLL, which influences the success or failure of the exploration mission. In the landing process, the CTLL is easy to tip over across the axis formed by any two adjacent footpads under the effect of the horizontal velocity. Hence, a good landing stability index should indicate the relationship between different dimensions and landing stability of the overall lander. The local landing stability index (LLSI) and global landing stability index (GLSI) are used to evaluate both horizontal and vertical stabilities to make the landing process secure, which take the effect of landing speed, net effective tipover force, support polygon, position of COG, etc. into consideration simultaneously. Particularly, we hope the GLSI for the overall lander is as big as possible. The indices can be derived based on the force-angle stability measure (FASM) [22] illustrated in Fig. 7, where aˆi denotes the ith tipover axis, Fi∗ denotes the magnitude of the effective tipover net force with respect to aˆi , di denotes the tipover radius which is also the distance vector from aˆi to Fi∗ , θ i denotes the tipover angle and θ i < 0 means the CTLL will tip over directly. The LLSI is configuration-dependent and given as Sk = min(θi · di  · Fi∗ ) for the four legs, so each configuration in the workspace corresponds with a specific Sk . The GLSI is configuration-independent and dimension-dependent, and given as the average value of Sk in the workspace, defined by Eq. (13). So it’s just affected by the given dimensions of the CTLL, can represent the comprehensive landing stabilities in both horizontal direction and vertical direction.



ηS = V

Sk dW dW

(13)

V

As a whole, all the performance indices are listed in Table 1, in which they are marked according to if they are appropriate for the landing leg or the overall lander. In the table, “ ” denotes the index is appropriate; on the contrary, the blank denotes the index is inappropriate. To be specific: (1) For the landing leg, it has three DOF with 2R1T motion characteristics, the 2R motion characteristics can reproduce 2T parasitic motions which composes the three-dimensional translational characteristics together with the original 1T characteristic. On the other hand, the passive spherical hinge in the ankle joint of

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Table 1 Performance indices for both landing leg and overall lander. Performance index

WVI

GTI

GPI

ηW

ηT

ηFmax , ηFmin

landing leg overall lander







GSI

GLSI

ηMmax , ηMmin

ηPmax , ηPmin

ηOmax , ηOmin

ηS









the landing leg can eliminate the payload moment and the angular deformation in the landing process. Therefore, the paper will focus on the three-dimensional translational motion, payload force, and the linear deformation for the performance investigation of the landing leg. As a result, ηT is chosen as the GTI, ηFmax and ηFmin are chosen as the GPI, ηPmax and ηPmin are chosen as the GSI; (2) For the overall lander, it has full mobility which means it has three translation and three rotation abilities. Hence, the paper will investigate the entire performances for the overall lander, including the three-dimensional translations, three-dimensional rotations, payload force, payload moment, linear deformation, angular deformation and the landing stability. As a result, ηFmax , ηFmin , ηMmax , ηMmin are chosen as the GPI, ηPmax , ηPmin , ηOmax , ηOmin are chosen as the GSI. Furthermore, the global landing stability index ηS of the overall lander is also taken into account. 5. Optimization design of the landing leg 5.1. Physical model of the design space For the landing leg, there are five optimization parameters in its physical model, including a1 , b1 , a2 , b2 , l3 . The mean R of these parameters are obtained by:

1 ( l3 + b1 + b2 + a1 + a2 ) = R 5

(14)

Dividing both sides of Eq. (14) by R, we can get the normalized equation:

r1 + r2 + r3 + r4 + r5 = 5

(15)

where ri (i = 1, 2, 3, 4, 5) are the normalized parameters, and r1 = l3 /R, r2 = b1 /R, r3 = b2 /R, r4 = a1 /R, r5 = a2 /R. Using the DSDRM strategy, the parameters r2 ,r4 of the static platform can be similar to the parameters r3 ,r5 of the moving platform, considering that the low DOF parallel mechanism usually owns the geometrically similar static platform and moving platform and symmetrically arranged limbs, to get better performances and easier kinematics solutions, such as the HALF manipulator, the Delta manipulator and the Star-like manipulator [24]. So the quantity of the optimization parameters of the landing leg can be reduced by the similarity operation, that is, U1 U2 U3 is similar to AS2 S3 , given by:

r4 r5 = =χ r2 r3

(16)

Substituting Eq. (16) into Eq. (15), we can get:

r1 + ( 1 + χ ) ( r2 + r3 ) = 5

(17)

So r1 , r2 , r3 are the three independent dimension parameters, and χ is a variable parameter. On account that χ is not allowed to be too small or too big, we here let 0.5 ≤ χ ≤ 2.0. Thus, constraint inequations can be derived as:



0.5 ≤ χ ≤ 2.0 0 < r1 < 5 0 < r2 , r3 < 5/ ( 1 + χ )

(18)

Eqs. (17)–(18) indicate the parameter design space of the landing leg, which is the rectangular pyramid ABCDE in frame O − r1 r2 r3 . When χ is given as a constant number, the design space will become an isosceles triangle plane, for instance, the plane ABC corresponds with χ = 0.5, and the plane ADE corresponds with χ = 2.0. For any χ ∈ [0.5, 2.0], the parameter design space can be illustrated in Fig. 8. The mapping function of the variables between frame O − r1 r2 r3 and frame A − xyz can be given by:



r1 = −cosδ1 x − sinδ1 cosδ2 y + 5 r2 = sinδ1 x − cosδ1 cosδ2 y r3 = sinδ2 y

or



x= y=

r2 sinδ1 r3 sinδ2

+

cosδ1 cosδ2 sinδ2 sinδ1 r3

(19)

(20)

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Fig. 8. Parameter design space of the landing leg.

Fig. 9. The WVI performance atlases with different χ .

where

δ1 = arccos

1+χ 1 + (1 + χ )

2

,

δ2 = arccos

1+χ 1 + 2 ( 1 + χ )2

(21)

5.2. Performance atlases of the landing leg Fig. 9 illustrates the WVI performance atlases and the parameter design space with different χ , defined by Eq. (16). We can infer that the design space becomes smaller and smaller as χ changing from 0.5 to 2.0, but all the WVI performance values belong to the same variation range of [0, 48]. According to the references [24,32], for the PCbDM with four parameters, the design space will become a cube and there is no better way to optimize collaboratively all these four parameters,

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Fig. 10. The GTI performance atlas of the landing leg.

Fig. 11. The GPI performance atlases of the landing leg.

one practicable way is to let one of the four parameters be a constant value. Consequently, given the supporting stiffness and landing stability, r2 < r4 and r3 < r5 , so χ > 1 is required. Herein, we further let χ = 1.5 for the following analysis. In Fig. 9(c), one can see that: (1) ηW is almost proportional to r1 if r2 or r3 is specified; (2) ηW doesn’t vary a lot with r2 or r3 if r1 is specified, so the length of the outer sleeve of the primary damper r1 is the main factor that influences the workspace volume; (3) as a whole, the relatively bigger ηW is necessary for the landing leg to improve the buffering capacity and the complex environment adaptability. The solution region should cover the good area of ηW to contrast with other performance indices collaboratively, which can be given by W V I = {(r1 , r2 , r3 )|1.5 < r1 < 4.0, 0.1 < r2 < 1.2}. Fig. 10 shows the GTI performance atlas of the landing leg. One can see that: (1) ηT is almost inversely proportional to r1 if r2 is specified; (2) the performance value in the right part is bigger than that in the left part, which means the length of the outer sleeve of the primary damper r1 should not be very big; (3) ηT has the variation range of [0.55, 1], and it is greater than or equal to 0.80 in most areas of the atlas with good GTI performance; (4) as a whole, the relatively bigger ηT is necessary to improve the motion/force transmission performance of the landing leg, to absorb the landing impact energy more easily and efficiently. The solution region can be given by GCI = {(r1 , r2 , r3 )|1.0 < r1 < 2.5}. Fig. 11 shows the GPI performance atlases of the landing leg, in which (a) is the atlas of ηFmax , (b) is the atlas of ηFmin . One can see that: (1) the performance distributions of ηFmax and ηFmin in the design space are almost similar, that is to say, the area that ηFmax achieves the maximum value is also near to the area that ηFmin achieves the maximum value, the area that ηFmax achieves the minimum value is also near to the area that ηFmin achieves the minimum value; (2) both ηFmax and ηFmin are inversely proportional to r3 when r1 is specified, which means the dimensions of the moving platform of the hip mechanism should be small to hold a better payload performance. In fact, we pay more attention to ηFmin than ηFmax to resist the landing impact force in case of the worst condition; (3) as a whole, a relatively bigger ηFmax , ηFmin are necessary to empower the landing leg capacities to resist larger landing impact force. The solution region can be given by GPI = {(r1 , r2 , r3 )|1.5 < r1 < 3.5, 0.8 < r2 < 1.2, 0 < r3 < 0.4}. Fig. 12 shows the GSI performance atlases of the landing leg, in which (a) is the atlas of ηPmax , (b) is the atlas of ηPmin . One can see that: (1) the distributions of the GSI performances are almost opposite to the GPI performances, that is to say,

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Fig. 12. The GSI performance atlases of the landing leg.

Fig. 13. Global optimum region of the landing leg.

when ηFmax achieves the maximum and the minimum value, respectively, ηPmin achieves the minimum and the maximum value; when ηFmin achieves the maximum and the minimum value, respectively, ηPmax achieves the minimum and the maximum value as well; (2) the performance distributions of ηPmax and ηPmin in the design space are almost similar, that is to say, the area that ηPmax achieves the maximum value is also near to the area that ηPmin achieves the maximum value, the area that ηPmax achieves the minimum value is also near to the area that ηPmin achieves the minimum value; (3) both ηPmax and ηPmin are proportional to r3 , which means the dimension r3 of the moving platform of the hip mechanism should be small to hold a better stiffness performance. In fact, we pay more attention to ηPmax than ηPmin to get a strict stiffness performance to decline the linear deformation when suffering from the landing impact force; (4) as a whole, the relatively smaller ηFmax , ηFmin are necessary to obtain a better mechanism stiffness of the landing leg. The solution region can be given by GSI = {(r1 , r2 , r3 )|1.5 < r1 < 3.5, 0.6 < r2 < 1.2, 0 < r3 < 0.4}. 5.3. Optimum parameters selection and applicability verification Figs. 9–12 illustrate the relationship between non-dimensional parameters and different performance indices for the landing leg, which offers credible evidence to compare multiple parameters and select the optimum non-dimensional combination for global optimization. According to the analysis above, the optimum solution region is restricted by WVI , GTI , GPI and GSI , here we take the intersection operation:

LEG = W V I ∩ GT I ∩ GPI ∩ GSI

(22)

where LEG denotes the global optimum region for the landing leg, which is given by Eq. (23) and shown in Fig. 13.

LEG = {(r1 , r2 , r3 )|1.5 < r1 < 2.5, 0.8 < r2 < 1.2, 0 < r3 < 0.4}

(23)

So in the global optimum region LEG , we can calculate the performance values with different possible non-dimensional combinations (r1 , r2 , r3 ) and select one combination as the optimum non-dimensional parameters. Using the numerical

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calculation method, the iteration step size should satisfy Eq. (24), so that the dimensional parameters can meet the basic requirements for design, manufacture and assembly:

ε 1 < r · D ≤ ε 2

(24)

where r denotes the step size of r1 , r2 , r3 , furthermore, D is the normalization factor characterized by the order of magnitude 100 m, ɛ1 is the manufacture and assembly accuracy characterized by the order of magnitude 10−4 m, and ɛ2 is the maximum allowable variation of dimension parameters characterized by the order of magnitude 10−2 m. Thus, we can obR tain 10−4 < r ≤ 10−2 , here let r = 10−2 to execute the numerical calculation by the commercial software MATLAB

, and finally get 2851 non-dimensional parameters (r1 , r2 , r3 ) ∈ LEG . Some typical results are listed in Appendix B. For the reason that there is not an absolute or unique optimum solution for the multi-objective optimization problem, and it relies on the designer to decide the importance degree of different performance indices. For the landing leg, the more ideal the parameter combination is, the better the comprehensive performances are, as long as meeting the demands: (1) a bigger ηW , so the landing leg has a bigger buffering capacity and can adapt to the complex environment; (2) a bigger ηT , so the landing leg has a better force and motion transformation capacity; (3) a bigger ηFmax and ηFmin , which empower the landing leg capacities to resist larger landing impact force, and the index ηFmin is more important than ηFmax ; (4) a smaller ηPmax and ηPmin , so the linear deformation of the landing leg will be smaller, and the index ηPmax is more important than ηPmin . Here, we contrast different combinations in Appendix B and draw the conclusions: (1) for No. 80, it has the biggest ηW = 18.0319 in the list, but ηT is a little small and ηPmax is a little big; (2) for No. 22, it has the biggest ηT = 0.9230 in the list, but it also has the smallest ηFmin and the biggest ηPmax ; (3) for No. 77, it has the biggest ηF max = 1.9073 and the smallest ηPmin = 0.5153 in the list, but ηT is a little small; (4) for No. 69, it has the biggest ηF min = 0.8173 and the smallest ηPmax = 1.7125 in the list, but ηT is still a little small. As a result, to develop the landing leg with the best comprehensive capacities, we choose No. 57 as the optimum non-dimensional parameters, so r1 = 2.20, r2 = 1.05, and r3 = 0.07, the corresponding performance indices are: ηW = 9.5438, ηT = 0.8608, ηF max = 1.7557, ηF min = 0.8108, ηPmax = 1.7223, ηPmin = 0.5600. 6. Optimization design of the overall lander 6.1. Physical model of the design space For the overall lander, there are four optimization parameters in its physical model, including θ in , d, Rsp , l3 . We let θin = π /6 for the reason that the maximum rotation angle of the universal hinge is π /3. Moreover, θ in and d can be replaced by Rms according to Rms = d + b1 sinθin in the geometrical relationship, and b1 can be obtained in the optimization design process of the landing leg. Thus, using the DSDRM strategy, the quantity of the optimization parameters for the overall lander can be reduced to three, including l3 , Rms , Rsp . The mean T of these parameters are obtained by:

1 (l3 + Rms + Rsp ) = T 3

(25)

Dividing both sides of Eq. (25) by T, we can get the normalized equation:

t1 + t2 + t3 = 3

(26)

where ti (i = 1, 2, 3) are the normalized parameters, and t1 = l3 /T , t2 = Rms /T , t3 = Rsp /T . Considering the structural characteristic of the overall lander that l3 + Rms > Rsp , its corresponding non-dimensional condition is t1 + t2 > t3 , so constraint equations can be derived as:



t2 < t3 0 < t1 < 3 0 < t2 , t3 < 1.5

(27)

Eqs. (26)∼(27) indicate the parameter design space of the overall lander, which is illustrated in Fig. 14 as well. The mapping function of the variables between frame O − t1 t2 t3 and frame U − xyz can be given by:

⎧ √ 6 x +√ 3 ⎨t1 = − √ 3 t = t3 =

⎩2 or



x= y=

6 x √6 6 x 6

√ 3 6 2 √ 3 2 2

− +

− −

2 y √2 2 y 2

(28)

√

6 t √2 1 2 t 2 1

−

√ 2t2

(29)

6.2. Performance atlases of the overall lander Fig. 15 shows the WVI performance atlas of the overall lander. One can see that: (1) ηW is almost proportional to t3 if t1 or t2 is specified, further to say, when t3 changes from 1.2 to 1.5, ηW will gradually approach to the peak value;

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Fig. 14. Parameter design space of the overall lander.

Fig. 15. The WVI performance atlas of the overall lander.

(2) ηW doesn’t vary a lot with t1 or t2 if t3 is specified, which means the support radius t3 of the overall lander is the main factor that influences the workspace volume; (3) as a whole, the relatively bigger ηW is necessary for the overall lander to improve the buffering capacity and the complex environment adaptability. The solution region can be given by W V I = {(t1 , t2 , t3 )|0.5 < t1 < 1.5, 0.1 < t2 < 1.1, 1.2 < t3 < 1.5}. Fig. 16 shows the GPI performance atlases of the overall lander, in which (a) is the atlas of ηFmax , (b) is the atlas of ηFmin , (c) is the atlas of ηMmax , (d) is the atlas of ηMmin . One can see that: (1) the performance distributions of ηFmax and ηFmin in the design space are almost opposite, that is to say, the area that ηFmax achieves the maximum value is also near to the area that ηFmin achieves the minimum value, the area that ηFmax achieves the minimum value is also near to the area that ηFmin achieves the maximum value; (2) the performance distributions of ηMmax and ηMmin in the design space are almost similar, that is to say, the area that ηMmax achieves the maximum value is also near to the area that ηMmin achieves the maximum value, the area that ηMmax achieves the minimum value is also near to the area that ηMmin achieves the minimum value; (3) for the ηFmax , the performance value in the upper left part is bigger than that in the lower right part, and the lower right part still possesses big ηFmax ; (4) for the ηFmin , the performance value in the lower right part is bigger than that in the upper left part, but the upper left part possesses very small ηFmin ; (5) for the ηMmax and ηMmin , the performances are almost proportional to t3 if t1 or t2 is specified, but they don’t vary a lot with t1 or t2 if t3 is specified, which means the support radius t3 is the main factor for the overall lander to resist the landing impact moment; (6) actually, we pay more attention to ηFmin , ηMmin than ηFmax , ηMmax to get better payload abilities to resist the landing impact force and moment on the main body of the overall lander; (7) as a whole, the relatively bigger ηFmax , ηFmin , ηMmax , ηMmin are necessary. The solution region can be given by GPI = {(t1 , t2 , t3 )|0.5 < t1 < 1.1, 0.6 < t2 < 1.1, 1.2 < t3 < 1.5}. Fig. 17 shows the GSI performance atlases of the overall lander, in which (a) is the atlas of ηPmax , (b) is the atlas of ηPmin , (c) is the atlas of ηOmax , (d) is the atlas of ηOmin . One can see that: (1) the distributions of ηPmax and ηPmin are almost opposite to ηFmin and ηFmax , that is to say, when ηFmax achieves the maximum and the minimum value, respectively, ηPmin achieves the minimum and the maximum value; when ηFmin achieves the maximum and the minimum value, respectively, ηPmax achieves the minimum and the maximum value as well; (2) the distributions of ηOmax and ηOmin are almost opposite to ηMmin and ηMmax , that is to say, when ηMmax achieves the maximum and the minimum value, respectively, ηOmax achieves the minimum and the maximum value; when ηMmin achieves the maximum and the minimum value, respectively, ηOmin achieves the minimum and the maximum value; (3) the distributions of ηPmax and ηPmin are almost similar, while the distributions of ηOmax and ηOmin are almost opposite; (4) for the ηPmax , the performance value in the

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Fig. 16. The GPI performance atlases of the overall lander.

Fig. 17. The GSI performance atlases of the overall lander.

upper left part is bigger than that in the lower right part, and the lower right part possesses small ηPmax ; (5) for the ηPmin , the performance value in the lower right part is bigger than that in the upper left part, but the lower right part still possesses very small ηPmin ; (6) for the ηOmax and ηOmin , the performances in the lower right part are bigger than the else region; (7) actually, we pay more attention to ηPmax , ηOmax than ηPmin , ηOmin to get better stiffness abilities to decline the linear and angular deformations when the main body is suffering from the landing impact force and the impact

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Fig. 18. The GLSI performance atlas of the overall lander.

Fig. 19. Global optimum region of the overall lander.

moment; (8) as a whole, the relatively smaller ηPmax , ηPmin , ηOmax , ηOmin are necessary. The solution region can be given by GSI = {(t1 , t2 , t3 )|0.3 < t1 < 1.1, 0.6 < t2 < 1.2, 1.2 < t3 < 1.5}. Fig. 18 shows the GLSI performance atlas of the overall lander. One can see that: (1) ηS is almost proportional to t3 if t1 or t2 is specified; (2) ηS doesn’t vary a lot with t1 or t2 if t3 is specified, which means the support radius t3 of the overall lander is the main factor that influences the landing tipover stability; (3) the peak value of ηS can be reached when t2 = 0.0 and t3 = 1.5; (3) for the overall lander with the best GLSI, the solution region is GLSI = {(t1 , t2 , t3 )|0.3 < t1 < 1.8, 0 < t2 < 1.2, 1.2 < t3 < 1.5}. 6.3. Optimum parameters selection and applicability verification Figs. 15–18 illustrate the relationship between non-dimensional parameters and different performance indices for the overall lander, which offers credible evidence to compare multiple parameters and select the optimum non-dimensional parameters for global optimization. According to the analysis above, the optimum region of the solution is restricted by WVI , GPI , GSI and GLSI , here we take the intersection operation:

CT LL = W V I ∩ GPI ∩ GSI ∩ GLSI

(30)

where CTLL denotes the global optimum region for the overall lander, which is given by Eq. (31) and shown in Fig. 19:

CT LL = {(t1 , t2 , t3 )|0.5 < t1 < 1.1, 0.6 < t2 < 1.1, 1.2 < t3 < 1.5}

(31)

So in the global optimum region CTLL , we can calculate the performance values with different possible non-dimensional combinations (t1 , t2 , t3 ) and select one combination as the optimum non-dimensional parameters. According to Eq. (24), R here we take the iteration step size t = 10−2 to execute the numerical calculation by the commercial software MATLAB

, and finally get 1447 non-dimensional parameters (t1 , t2 , t3 ) ∈ CTLL . Some typical results are listed in Appendix C. For the reason that there is not an absolute or unique optimum solution for the multi-objective optimization problem, it relies on the designer to decide the importance degree of different performance indices. For the overall lander in landing process, the main body of the CTLL changes the pose in its workspace. Meanwhile, it suffers from the landing impact force/moment with the corresponding linear/angular deformation. A good landing stability is necessary in case of tipping

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over. Obviously, the more ideal the parameter combination is, the better the comprehensive performances are, as long as meeting the demands: (1) a bigger ηW , which means the overall lander has a bigger buffering capacity and can adapt to the complex landing environment; (2) a bigger ηFmax , ηFmin , ηMmax , ηMmin , which empower the overall lander capacities to resist larger landing impact force and moment, and the indices ηFmin , ηMmin are more important than ηFmax , ηMmax ; (3) a smaller ηPmax , ηPmin , ηOmax , ηOmin , so the linear and angular deformations of the overall lander will be smaller as well, and obviously, the indices ηPmax , ηOmax are also more important than ηPmin , ηOmin ; (4) a bigger ηS , so the overall lander will hold better horizontal and vertical stabilities. Here, we contrast different combinations in Appendix C and draw the conclusions: (1) for No. 5, it has the biggest ηW = 1.0406, ηF min = 0.9070, ηS = 0.6378, and the smallest ηPmax = 1.4011 in the list, but ηFmax is very small and ηPmin is very big; (2) for No. 1, it has the biggest ηF max = 2.1025 and the smallest ηPmin = 0.2278 in the list, but ηW , ηFmin , ηS are very small and ηPmax , ηOmax are very big; (3) for No. 9, it has the biggest ηMmax = 1.4506 and the smallest ηOmin = 0.7476 in the list, but ηOmax is a little big; (4) for No.24, it has the biggest ηMmin = 0.1941 and the smallest ηOmax = 28.6589 in the list, but ηFmax is very small and ηPmin is very big. As a result, to develop the overall lander with the best comprehensive capacities, we determine to choose No. 52 as the optimum non-dimensional parameters, so t1 = 0.6, t2 = 0.9, and t3 = 1.5, the corresponding performance indices are: ηW = 0.7179, ηF max = 1.7564, ηF min = 0.7854, ηMmax = 1.1033, ηMmin = 0.1841, ηPmax = 2.1808, ηPmin = 0.3454, ηOmax = 37.7583, ηOmin = 4.4700, ηS = 0.6189. 7. Dimensional parameter calculation Up to now, we have obtained the non-dimensional optimum parameters for both landing leg and overall lander. In the next, to derive the dimensional optimum parameters, we will identify the normalization factor of the overall lander firstly. For the sake that in the stowed state, the CTLL should meet the envelope demand of the carrying vehicle, that is to say, the diameter of the CTLL should be less than the envelope diameter φ e of the carrying vehicle, given by:

 √  max 2 2d, 2Rms < φe

(32)

According to the equation d = Rms − b1 sinθin and the proportional relationship of the non-dimensional parameters solved in Sections 5 and 6, we can deduce:

√ 2 2d < φe

(33)

Here we take φe = 3650mm in reference to the envelope demand of Chang’e 3 [28], and obtain Rms < 1534.61mm, so we let Rms = 1500mm for the following steps. Hence, in the light of r1 : r2 : r3 = 2.2 : 1.05 : 0.07 and t1 : t2 : t3 = 0.6 : 0.9 : 1.5, so the normalization factor of the overall lander can be calculated by: T = Rms /t2 = 1500/0.9 = 1666.67mm. Thus, the optimum dimension parameters of the overall lander can be obtained by: l3 = t1 · T = 10 0 0mm, Rsp = t3 · T = 250 0mm. In the next, the normalization factor of the landing leg can be ensured uniquely by: R = l3 /r1 = 454.55mm, so the optimum dimension parameters of the landing leg can be obtained by: b1 = r2 · R = 477.27 mm, b2 = r3 · R = 31.82mm, a1 = b1 · χ = 715.91mm, a2 = b2 · χ = 47.73mm. As a consequence, all the optimization parameters of the CTLL are optimized with the best comprehensive performances. 8. Conclusions The paper proposes the optimum dimension design method for the cantilever-type legged lander, and solves two important and challenging aspects: (1) One is to build a reliable and comprehensive optimization design model for the CTLL. Based on the proposed trussmechanism transformation method, both landing leg and overall lander in the landing process are seen as mechanisms or robots with kinematics and dynamics characteristics. Hence, the optimization problem for the truss structure of the CTLL in the landing process is converted into the kinematics and dynamics optimization for the mechanisms or robots with motion characteristics. Then, the optimization design model is established, including the optimization parameters of the topological structure, constraint conditions of both landing stability and workspace, and performance indices of WVI, GTI, GPI, GSI, and GLSI; (2) The other aspect is to develop a systematic strategy to optimize both landing leg and overall lander respectively based on the optimization design model. The design space dimensionality reduction method is proposed for the multiparameter optimization problem of the CTLL with eight optimization parameters, and develops the performance-chart based design methodology. Finally, the relationship between different possible dimensions and the corresponding performances is analyzed, and the global optimum dimensions for the CTLL are obtained. The results shown by the performance atlases are perceptual and credible. Acknowledgement The author thanks the partial financial supports under the projects from the National Natural Science Foundation of China (Grant No. 51735009) and the Research Fund of State Key Lab of MSV, China (Grant No. MSV-ZD-2016–08). Furthermore, Youcheng Han especially wishes to thank Weiqing Xu, for his silent support and the shared reading time.

20

Y. Han, W. Guo and F. Gao et al. / Mechanism and Machine Theory 142 (2019) 103611

Appendix A Appendix A aims at explaining the motion characteristics of the landing leg by screw theory illustrated by Fig. 20. First, the leg coordinate frame {L} is established to discuss this problem: the point L is located in the cross point of the universal hinge U1 , yL is always in line with the first rotational axis of U1 , zL is in line with the second rotational axis of U1 at the initial time. Let β 1 and γ 1 denote the rotation angles of U1 with respect to yL and zL . The unit vector of AS1 is:

 A S1 =

AS1

 AS1 

= (cβ1 cγ1 , sγ1 , −sβ1 cγ1 )

T

where c represents cosine and s represents sine. Considering that the secondary limb of the landing leg possesses 6 DOF, the motion screw system of the hip mechanism (the parallel portion of the leg, U&2UPS) is composed of two motion screws by U1 :



$1 = (0, 1, 0, 0, 0, 0)T T $2 = (−sβ1 , 0, −cβ1 , 0, 0, 0)

(34)

(1) For the ankle joint point of the landing leg without considering the spherical hinge S1 , the motion screw system is:

⎧ T ⎨$1 = (0, 1, 0, 0, 0, 0) T $2 = (−sβ1 , 0, −cβ1 , 0, 0, 0)   T ⎩ T  $3 = 01 × 3 , A S1 = (0, 0, 0, cβ1 cγ1 , sγ1 , −sβ1 cγ1 )

(35)

Hence, Eq. (35) indicates that $1 , $2 , $3 can constitute the three-order screw system with two linear vectors and one couple, so it represents the 2R1T motion characteristics. (1) For the ankle joint point of the landing leg considering the spherical hinge S1 : Three mutually orthogonal linear vectors at point S1 are used to substitute the equivalent screw system generated by the spherical hinge, which are represented as $4 , $5 , $6 , their unit direction vector are s4 , s5 , s6 .

⎧ ⎨s4 = (−cβ1 sγ1 , cγ1 , sβ1 sγ1 )T T s = (−sβ1 , 0, −cβ1 ) ⎩5 T s6 = (cβ1 cγ1 , sγ1 , −sβ1 cγ1 )

Let rS denote the position vector of the ankle joint point S1 in frame {L}, rS = (xS , yS , zS )T , and yields:



rS × s4 =

0 zS −yS

−zS 0 xS

yS −xS 0



−cβ1 sγ1 cγ1 sβ1 sγ1





=

−zS cγ1 + yS sβ1 sγ1 −zS cβ1 sγ1 − xS sβ1 sγ1 yS cβ1 sγ1 + xS cγ1



Fig. 20. Motion Screw System of the Landing Leg.

Y. Han, W. Guo and F. Gao et al. / Mechanism and Machine Theory 142 (2019) 103611

 rS × s5 =

 rS × s6 =

0 zS −yS

−zS 0 xS

yS −xS 0

0 zS −yS

−zS 0 xS

yS −xS 0



−sβ1 0 −cβ1





 =

cβ1 cγ1 sγ1 −sβ1 cγ1

−yS cβ1 −zS sβ1 + xS cβ1 yS sβ1





 =

21

−zS sγ1 − yS sβ1 cγ1 zS cβ1 cγ1 + xS sβ1 cγ1 −yS cβ1 cγ1 + xS sγ1



Hence, the screw system of the landing leg considering the spherical hinge in ankle joint can be obtained by:

⎧ $1 = (0, 1, 0, 0, 0, 0)T ⎪ ⎪ ⎪ ⎪$2 = (−sβ1 , 0, −cβ1 , 0, 0, 0)T ⎪ ⎨ T $3 = (0, 0, 0, cβ1 cγ1 , sγ1 , −sβ1 cγ1 ) T $4 = (−cβ1 sγ1 , cγ1 , sβ1 sγ1 , −zS cγ1 + yS sβ1 sγ1 , −zS cβ1 sγ1 − xS sβ1 sγ1 , yS cβ1 sγ1 + xS cγ1) ⎪ ⎪ ⎪ T ⎪ ⎪ ⎩$5 = (−sβ1 , 0, −cβ1 , −yS cβ1 , −zS sβ1 + xS cβ1 , yS sβ1 ) T $6 = (cβ1 cγ1 , sγ1 , −sβ1 cγ1 , −zS sγ1 − yS sβ1 cγ1 , zS cβ1 cγ1 + xS sβ1 cγ1 , −yS cβ1 cγ1 + xS sγ1 )

(36)

According to the linear combination of the motion screws in the above screw system by Eq. (36), we can obtain:

⎧ $1 ⇒ (0, 1, 0, 0, 0, 0)T ⎪ ⎪ ⎪ ⎪ $ ⇒ (0, 0, 1, 0, 0, 0)T ⎪ ⎨ 2 $3 ⇒ (0, 0, 0, 1, 0, 0)T $ ⇒ (1, 0, 0, 0, 0, 0)T ⎪ ⎪ ⎪$4 ⇒ 0, 0, 0, 0, 1, 0 T ⎪ ( ) ⎪ ⎩ 5 $6 ⇒ (0, 0, 0, 0, 0, 1)T

(37)

Eq. (37) indicates after linear combination and simplification, $ 4 , $ 1 , $ 2 can represent the three-order equivalent linear vectors, while $ 3 , $ 5 , $ 6 can represent the three-order equivalent couples. Therefore, the motion screw system of landing leg considering the spherical hinge is full rank and six order, so the landing leg has 3R3T motion characteristics. Appendix B For the landing leg, some typical results of the non-dimensional parameters in global optimum region and its corresponding performance values are listed in Table 2.

Table 2 Some typical results in global optimum region of the landing leg. WVI

GTI

GPI

No.

r1

r2

r3

ηW

ηT

ηFmax

ηFmin

GSI

ηPmax

ηPmin

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

1.50 1.50 1.55 1.55 1.55 1.60 1.60 1.60 1.60 1.65 1.65 1.65 1.65 1.70 1.70 1.70 1.70 1.75 1.75 1.75 1.75 1.80

1.15 1.20 1.10 1.15 1.20 1.05 1.10 1.15 1.20 1.05 1.10 1.15 1.20 1.00 1.05 1.10 1.15 1.05 1.10 1.15 1.20 0.90

0.25 0.20 0.28 0.23 0.18 0.31 0.26 0.21 0.16 0.29 0.24 0.19 0.14 0.32 0.27 0.22 0.17 0.25 0.20 0.15 0.10 0.38

0.0881 0.3600 0.0450 0.2456 0.4931 0.0113 0.1256 0.4800 0.6788 0.0750 0.3300 0.8325 0.8888 0.0338 0.2044 0.6000 1.2919 0.4688 1.0256 1.8975 1.4681 0.0169

0.9140 0.9026 0.9191 0.9106 0.9012 0.9215 0.9166 0.9061 0.8994 0.9204 0.9134 0.9015 0.8976 0.9221 0.9180 0.9095 0.8961 0.9148 0.9053 0.8897 0.8939 0.9230

1.0252 1.0622 1.0152 1.0621 1.0927 1.0013 1.0483 1.1083 1.1203 1.0282 1.0912 1.1573 1.1546 1.0154 1.0812 1.1507 1.2120 1.1308 1.2005 1.2725 1.2332 1.0053

0.6347 0.6502 0.6218 0.6606 0.6739 0.6143 0.6632 0.6933 0.6918 0.6438 0.6888 0.7164 0.7096 0.6379 0.6901 0.7242 0.7365 0.7156 0.7402 0.7521 0.7413 0.6116

2.5392 2.4830 2.6362 2.4039 2.3441 2.6653 2.3486 2.2193 2.2446 2.4749 2.2371 2.1121 2.1525 2.4958 2.2093 2.0550 2.0176 2.1001 1.9914 1.9496 1.9972 2.7026

0.9586 0.9110 0.9742 0.9104 0.8771 0.9974 0.9253 0.8589 0.8480 0.9544 0.8762 0.8145 0.8171 0.9735 0.8870 0.8171 0.7720 0.8377 0.7804 0.7339 0.7577 0.9902

(continued on next page)

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Table 2 (continued) No.

23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80

r1

1.80 1.80 1.80 1.85 1.85 1.85 1.85 1.90 1.90 1.90 1.90 1.95 1.95 1.95 1.95 2.00 2.00 2.00 2.00 2.05 2.05 2.05 2.05 2.10 2.10 2.10 2.10 2.15 2.15 2.15 2.15 2.20 2.20 2.20 2.20 2.25 2.25 2.25 2.25 2.30 2.30 2.30 2.30 2.35 2.35 2.35 2.35 2.40 2.40 2.40 2.40 2.45 2.45 2.45 2.45 2.50 2.50 2.50

r2

0.95 1.00 1.05 1.05 1.10 1.15 1.20 0.85 0.90 0.95 1.00 1.05 1.10 1.15 1.20 0.90 0.95 1.00 1.05 0.90 0.95 1.00 1.05 0.90 0.95 1.00 1.05 0.90 0.95 1.00 1.05 0.90 0.95 1.00 1.05 0.85 0.90 0.95 1.00 0.85 0.90 0.95 1.00 0.90 0.95 1.00 1.05 0.85 0.90 0.95 1.00 0.85 0.90 0.95 1.00 0.85 0.90 0.95

r3

0.33 0.28 0.23 0.21 0.16 0.11 0.06 0.39 0.34 0.29 0.24 0.17 0.12 0.07 0.02 0.30 0.25 0.20 0.15 0.28 0.23 0.18 0.13 0.26 0.21 0.16 0.11 0.24 0.19 0.14 0.09 0.22 0.17 0.12 0.07 0.25 0.20 0.15 0.10 0.23 0.18 0.13 0.08 0.16 0.11 0.06 0.01 0.19 0.14 0.09 0.04 0.17 0.12 0.07 0.02 0.15 0.10 0.05

WVI

GTI

GPI

ηW

ηT

ηFmax

ηFmin

GSI

ηPmax

ηPmin

0.1163 0.3713 0.8175 1.3406 2.2650 3.4875 2.3269 0.1088 0.2981 0.6488 1.1850 2.8163 4.1231 5.7750 3.4106 1.1288 1.7700 2.6475 3.7988 1.7850 2.5838 3.6469 5.0063 2.6775 3.6131 4.8300 6.3450 3.7275 4.8150 6.1838 7.8900 5.0419 6.2531 7.7906 9.5438 5.4319 6.4819 7.8825 9.5438 7.0819 8.3006 9.7631 11.5294 10.2394 11.8256 13.4700 14.9213 11.1075 12.4819 13.9275 15.4388 13.4756 14.7000 16.0706 17.2725 15.8775 16.9669 18.0319

0.9220 0.9184 0.9115 0.9073 0.8947 0.8762 0.8882 0.9227 0.9216 0.9182 0.9123 0.8974 0.8821 0.8602 0.8832 0.9175 0.9120 0.9036 0.8916 0.9145 0.9080 0.8984 0.8850 0.9109 0.9035 0.8929 0.8775 0.9066 0.8981 0.8860 0.8695 0.9010 0.8919 0.8783 0.8608 0.9035 0.8957 0.8846 0.8695 0.8979 0.8885 0.8756 0.8589 0.8795 0.8652 0.8517 0.8417 0.8810 0.8686 0.8565 0.8458 0.8695 0.8591 0.8488 0.8408 0.8596 0.8512 0.8443

1.0475 1.1177 1.1984 1.2545 1.3356 1.4298 1.3319 1.0421 1.1001 1.1634 1.2526 1.4161 1.5049 1.5832 1.4484 1.2314 1.3244 1.4171 1.5108 1.3286 1.4106 1.5143 1.5740 1.4182 1.4925 1.5774 1.6369 1.4897 1.5663 1.6298 1.7055 1.5607 1.6170 1.6936 1.7557 1.5615 1.6117 1.6691 1.7472 1.5987 1.6557 1.7272 1.7981 1.7068 1.7801 1.8352 1.8836 1.6762 1.7492 1.8102 1.8785 1.7264 1.7984 1.8503 1.9073 1.7594 1.8246 1.8933

0.6616 0.7107 0.7457 0.7586 0.7715 0.7788 0.7658 0.6449 0.6945 0.7281 0.7592 0.7866 0.7921 0.7928 0.7867 0.7411 0.7670 0.7853 0.7963 0.7626 0.7814 0.7949 0.8013 0.7726 0.7888 0.8005 0.8061 0.7820 0.7957 0.8043 0.8087 0.7876 0.7995 0.8071 0.8108 0.7767 0.7922 0.8021 0.8085 0.7792 0.7934 0.8031 0.8092 0.7942 0.8038 0.8106 0.8173 0.7813 0.7939 0.8037 0.8129 0.7812 0.7952 0.8059 0.8152 0.7816 0.7959 0.8081

2.3778 2.1224 1.9582 1.9106 1.8582 1.8354 1.8875 2.5275 2.2354 2.0605 1.9104 1.7958 1.7761 1.7865 1.7940 2.0273 1.8951 1.8067 1.7569 1.9299 1.8340 1.7694 1.7411 1.8938 1.8072 1.7491 1.7263 1.8551 1.7808 1.7380 1.7221 1.8354 1.7702 1.7334 1.7223 1.9124 1.8191 1.7659 1.7365 1.9071 1.8238 1.7715 1.7465 1.8296 1.7807 1.7487 1.7125 1.9213 1.8487 1.7950 1.7413 1.9442 1.8556 1.7941 1.7380 1.9596 1.8644 1.7873

0.9280 0.8506 0.7816 0.7483 0.7033 0.6606 0.7026 0.9392 0.8714 0.8155 0.7527 0.6755 0.6398 0.6043 0.6509 0.7808 0.7295 0.6838 0.6443 0.7352 0.6937 0.6541 0.6216 0.7063 0.6689 0.6319 0.5989 0.6801 0.6449 0.6123 0.5790 0.6590 0.6266 0.5931 0.5600 0.6740 0.6409 0.6091 0.5754 0.6606 0.6267 0.5930 0.5575 0.6118 0.5755 0.5434 0.5178 0.6331 0.5968 0.5624 0.5298 0.6181 0.5811 0.5475 0.5153 0.6051 0.5694 0.5368

Y. Han, W. Guo and F. Gao et al. / Mechanism and Machine Theory 142 (2019) 103611

23

Appendix C For the overall lander, some typical results of the non-dimensional parameters in global optimum region and its corresponding performance values are listed in Table 3.

Table 3 Some typical results in global optimum region of the overall lander. WVI

GPI

No.

t1

t2

t3

ηW

ηFmax

ηFmin

ηMmax

ηMmin

GSI

ηPmax

ηPmin

ηOmax

ηOmin

GLSI

ηS

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57

1.02 0.99 0.96 0.93 0.90 0.99 0.96 0.93 0.90 0.87 0.96 0.93 0.90 0.87 0.84 0.93 0.90 0.87 0.84 0.81 0.87 0.84 0.81 0.78 0.84 0.81 0.78 0.75 0.81 0.78 0.75 0.72 0.81 0.78 0.75 0.72 0.78 0.75 0.72 0.69 0.66 0.78 0.75 0.72 0.69 0.66 0.75 0.72 0.69 0.66 0.63 0.60 0.72 0.69 0.66 0.63 0.60

0.60 0.60 0.60 0.60 0.60 0.63 0.63 0.63 0.63 0.63 0.66 0.66 0.66 0.66 0.66 0.69 0.69 0.69 0.69 0.69 0.72 0.72 0.72 0.72 0.75 0.75 0.75 0.75 0.78 0.78 0.78 0.78 0.81 0.81 0.81 0.81 0.84 0.84 0.84 0.84 0.84 0.87 0.87 0.87 0.87 0.87 0.90 0.90 0.90 0.90 0.90 0.90 0.93 0.93 0.93 0.93 0.93

1.38 1.41 1.44 1.47 1.50 1.38 1.41 1.44 1.47 1.50 1.38 1.41 1.44 1.47 1.50 1.38 1.41 1.44 1.47 1.50 1.41 1.44 1.47 1.50 1.41 1.44 1.47 1.50 1.41 1.44 1.47 1.50 1.38 1.41 1.44 1.47 1.38 1.41 1.44 1.47 1.50 1.35 1.38 1.41 1.44 1.47 1.35 1.38 1.41 1.44 1.47 1.50 1.35 1.38 1.41 1.44 1.47

0.2079 0.4716 0.7080 0.9570 1.0406 0.1712 0.3996 0.6833 0.9643 1.0356 0.1395 0.3548 0.6574 0.9673 1.0348 0.1178 0.2998 0.6257 0.9579 1.0279 0.2636 0.5987 0.9234 1.0054 0.2454 0.5507 0.8844 0.9769 0.2274 0.5199 0.8334 0.9332 0.1277 0.2318 0.4824 0.7847 0.1359 0.2411 0.4718 0.7350 0.8306 0.1127 0.1502 0.2449 0.4410 0.6743 0.1262 0.1577 0.2464 0.4200 0.6210 0.7179 0.1329 0.1669 0.2381 0.3990 0.5723

2.1025 2.0401 1.9514 1.8218 1.6867 2.0811 2.0284 1.9377 1.8213 1.6860 2.0601 2.0201 1.9334 1.8067 1.6866 2.0080 1.9997 1.9284 1.7983 1.6899 1.9796 1.9169 1.7973 1.6951 1.9608 1.9086 1.7875 1.6967 1.9458 1.8981 1.7891 1.7014 1.9020 1.9383 1.8932 1.7886 1.8957 1.9273 1.8931 1.7895 1.7182 1.9033 1.9150 1.9329 1.8962 1.8020 1.9261 1.9299 1.9402 1.8987 1.8158 1.7564 1.9485 1.9503 1.9602 1.9077 1.8360

0.5656 0.5931 0.6749 0.7950 0.9070 0.6272 0.6242 0.6915 0.8006 0.9042 0.7000 0.6510 0.7067 0.8160 0.9032 0.8019 0.6957 0.7181 0.8262 0.9003 0.7461 0.7359 0.8290 0.8973 0.7775 0.7484 0.8349 0.8894 0.7891 0.7576 0.8301 0.8794 0.8519 0.7782 0.7605 0.8234 0.8223 0.7608 0.7514 0.8116 0.8450 0.7900 0.7656 0.7305 0.7342 0.7884 0.7367 0.7189 0.6959 0.7164 0.7635 0.7854 0.6850 0.6703 0.6519 0.6920 0.7297

1.2234 1.3260 1.3917 1.4496 1.4494 1.1787 1.3006 1.3902 1.4506 1.4404 1.1084 1.2791 1.3774 1.4440 1.4186 0.9814 1.2211 1.3684 1.4327 1.4033 1.1315 1.3381 1.4103 1.3790 1.0359 1.2985 1.3795 1.3507 0.9281 1.2391 1.3413 1.3230 0.5600 0.8554 1.1785 1.2984 0.4582 0.7544 1.0913 1.2457 1.2351 0.2721 0.3822 0.6744 0.9979 1.1657 0.2590 0.3328 0.5852 0.9164 1.0937 1.1033 0.2491 0.3127 0.5252 0.8437 1.0119

0.1338 0.1469 0.1596 0.1725 0.1880 0.1299 0.1451 0.1589 0.1735 0.1896 0.1243 0.1433 0.1580 0.1740 0.1907 0.1154 0.1396 0.1580 0.1749 0.1924 0.1339 0.1567 0.1753 0.1941 0.1280 0.1554 0.1745 0.1939 0.1219 0.1524 0.1738 0.1941 0.0899 0.1181 0.1496 0.1721 0.0840 0.1120 0.1443 0.1704 0.1912 0.0640 0.0792 0.1076 0.1390 0.1665 0.0617 0.0751 0.1008 0.1340 0.1625 0.1841 0.0596 0.0717 0.0956 0.1285 0.1567

4.5482 3.3427 2.4057 1.7803 1.4011 3.8655 3.0753 2.3219 1.7943 1.4343 2.9659 2.8454 2.2597 1.7609 1.4592 2.1068 2.5168 2.2100 1.7438 1.4794 2.1955 2.1113 1.7370 1.4838 1.9988 2.0539 1.6998 1.5025 1.8814 1.9961 1.7147 1.5420 1.5565 1.9232 1.9826 1.7457 1.6768 1.9834 2.0486 1.8141 1.7156 1.8544 1.9959 2.1850 2.1637 1.9655 2.2423 2.3481 2.4930 2.3924 2.1899 2.1808 2.6291 2.8414 2.9372 2.7018 2.5740

0.2278 0.2418 0.2660 0.3135 0.3732 0.2342 0.2457 0.2705 0.3126 0.3751 0.2404 0.2482 0.2721 0.3186 0.3746 0.2549 0.2539 0.2735 0.3226 0.3739 0.2599 0.2768 0.3215 0.3718 0.2657 0.2793 0.3256 0.3710 0.2693 0.2824 0.3248 0.3687 0.2805 0.2708 0.2839 0.3244 0.2813 0.2730 0.2836 0.3246 0.3607 0.2777 0.2755 0.2712 0.2823 0.3197 0.2713 0.2711 0.2691 0.2828 0.3149 0.3454 0.2648 0.2654 0.2634 0.2807 0.3097

59.4194 48.5248 40.8461 34.9949 29.6968 65.1449 50.5665 41.4459 34.5813 29.2920 72.6375 52.6135 42.3925 34.6113 29.1832 87.8595 56.9418 42.7216 34.4708 28.9026 63.5985 44.3350 34.7241 28.6589 71.2497 45.9097 35.6497 29.3251 80.3132 49.5497 36.7889 29.6590 144.3475 87.5916 52.7872 38.5475 159.6934 98.9527 59.0153 40.6513 32.2161 254.1373 179.1296 108.8225 65.5827 44.5426 279.4570 197.7543 125.4167 74.2358 49.0056 37.7583 304.8101 222.9942 142.9005 84.1917 56.3492

0.8747 0.8652 0.8005 0.7891 0.8441 1.3508 1.0916 0.9026 0.7476 0.9290 1.9312 1.4113 1.2758 1.0766 1.2187 3.9391 2.4571 1.5032 1.3208 1.4051 3.5298 2.1374 1.6101 1.7048 5.3782 2.7691 2.1051 2.0976 7.2934 3.8456 2.6007 2.3944 15.4403 9.2466 4.7871 3.2511 18.0364 11.3019 6.0871 3.9189 3.3041 25.8116 21.2348 13.1659 7.3108 4.8114 28.0277 22.5027 15.1308 8.6405 5.5796 4.4700 28.2663 23.4793 16.2960 9.5332 6.3264

0.3260 0.4168 0.4771 0.5519 0.6378 0.3169 0.4140 0.4807 0.5415 0.6322 0.3005 0.4037 0.4705 0.5441 0.6266 0.2977 0.4028 0.4660 0.5436 0.6206 0.3881 0.4674 0.5389 0.6145 0.3781 0.4679 0.5444 0.6158 0.3793 0.4692 0.5436 0.6153 0.3471 0.3875 0.4690 0.5445 0.3480 0.3908 0.4628 0.5477 0.6185 0.3162 0.3497 0.3946 0.4596 0.5452 0.3181 0.3525 0.3984 0.4647 0.5484 0.6189 0.3193 0.3548 0.4003 0.4694 0.5495

(continued on next page)

24

Y. Han, W. Guo and F. Gao et al. / Mechanism and Machine Theory 142 (2019) 103611 Table 3 (continued) No.

58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80

t1

0.57 0.69 0.66 0.63 0.60 0.57 0.54 0.66 0.63 0.60 0.57 0.54 0.51 0.63 0.60 0.57 0.54 0.51 0.57 0.54 0.51 0.54 0.51

t2

0.93 0.96 0.96 0.96 0.96 0.96 0.96 0.99 0.99 0.99 0.99 0.99 0.99 1.02 1.02 1.02 1.02 1.02 1.05 1.05 1.05 1.08 1.08

t3

1.50 1.35 1.38 1.41 1.44 1.47 1.50 1.35 1.38 1.41 1.44 1.47 1.50 1.35 1.38 1.41 1.44 1.47 1.38 1.41 1.44 1.38 1.41

WVI

GPI

ηW

ηFmax

ηFmin

ηMmax

ηMmin

GSI

ηPmax

ηPmin

ηOmax

ηOmin

GLSI

ηS

0.6465 0.1279 0.1631 0.2233 0.3773 0.5145 0.6053 0.1174 0.1506 0.2083 0.3443 0.4701 0.5580 0.1136 0.1393 0.1868 0.3128 0.4164 0.1341 0.1710 0.2655 0.1198 0.1590

1.7985 1.9732 1.9728 1.9749 1.9259 1.8663 1.8228 1.9932 1.9949 1.9965 1.9481 1.8950 1.8560 1.9991 2.0107 2.0137 1.9733 1.9264 2.0205 2.0256 1.9987 2.0368 2.0451

0.7369 0.6266 0.6214 0.6128 0.6578 0.6910 0.6998 0.5813 0.5730 0.5742 0.6199 0.6528 0.6605 0.5679 0.5439 0.5413 0.5819 0.6137 0.5272 0.5195 0.5484 0.5046 0.4991

1.0132 0.2243 0.2800 0.4409 0.7918 0.9293 0.9540 0.2073 0.2492 0.3837 0.7172 0.8656 0.9031 0.2297 0.2429 0.3449 0.6646 0.7904 0.2644 0.3371 0.5957 0.2590 0.3315

0.1792 0.0558 0.0679 0.0881 0.1234 0.1507 0.1719 0.0522 0.0629 0.0816 0.1163 0.1433 0.1650 0.0528 0.0592 0.0757 0.1090 0.1349 0.0565 0.0707 0.1012 0.0500 0.0665

2.6639 3.2475 3.3515 3.5053 3.1657 3.0538 3.2049 3.8614 4.0473 4.1237 3.7289 3.6609 3.8474 4.3185 4.6404 4.7815 4.3791 4.3306 5.0653 5.2467 4.9420 5.2617 5.5208

0.3289 0.2583 0.2589 0.2595 0.2756 0.2997 0.3220 0.2532 0.2533 0.2538 0.2694 0.2913 0.3117 0.2520 0.2498 0.2501 0.2628 0.2829 0.2473 0.2474 0.2564 0.2428 0.2422

42.5758 356.4978 251.0487 170.3410 98.1826 65.1353 48.8106 414.0993 298.0411 201.4731 116.7752 78.8320 57.6749 476.0408 347.2752 241.1547 143.4912 95.9616 412.0402 285.8423 176.0610 486.0235 348.0968

5.1062 29.2927 23.7382 17.4059 9.9653 6.8375 5.2584 29.7816 23.8991 17.6509 10.2962 7.0414 5.2698 28.1083 23.2155 17.2995 10.3415 7.1257 21.6509 16.3877 10.2633 20.0299 15.3333

0.6099 0.3249 0.3571 0.4017 0.4726 0.5470 0.6154 0.3302 0.3628 0.4056 0.4732 0.5477 0.6187 0.3379 0.3695 0.4108 0.4764 0.5480 0.3763 0.4163 0.4770 0.3842 0.4211

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