Conceptual design and analysis of legged landers with orientation capability

Rongfu LIN, Weizhong GUO, Changjie ZHAO, Mingda HE

State Key Laboratory of Mechanical System and Vibration, School of Mechanical Engineering, Shanghai Jiao Tong University, 200240, China

KEYWORDS Legged lander;Orientation analysis;Orientation capability;Parallel mechanism;Type synthesis

Abstract The lander with orientation capability is of fundamental importance for the‘‘Returning”mission because its body can be the base with a suitable launching angle for the ascender lifting off.This paper proposes the conceptual design and verification of Landers with Orientation Capability(LOCs).Firstly,the topological composition of the lander is analyzed and expressed in the form of Lie group,which provides the fundamental principle for the type synthesis of LOCs.Then,the type synthesis approach for designing LOCs is proposed, which includes designing the Equivalent Parallel Mechanism (EPM) and the auxiliary limbs.Numerous mechanisms of LOCs with twodimensional rotational motions are obtained based on the type synthesis approach by developing and combining the EPMs and auxiliary limbs.Afterward,kinematics and Jacobian matrix of a typical legged lander are developed,the workspace is analyzed,singularity configurations are analyzed based on the wrench graph method, and finally, the orientation capability is analyzed.The pitch and yaw angle reach 17.5°,considering the permissible range of joints.The proposed types of LOCs offer potential candidate for the‘‘Returning”mission of the exploration mission.

1.Introduction

Deep space exploration is in full swing in many nations and organizations.A lander plays a vital role in the deep exploration mission,1–3which mainly includes five stages:‘‘Circling,”‘‘Landing,”‘‘Exploring,”‘‘Returning,”and‘‘Base construction.”Up to now, it has been successfully carried out to the‘‘Returning”stage, whose main task is a samplereturn mission to collect and return samples from an extraterrestrial location to the Earth for analysis.During the‘‘Returning”stage,one of the most critical aspects is that the ascender(located on the top of the lander) needs to separate from the lander and lift off to the orbiter after samples have been collected.It is not difficult to find that the landers land on a relatively flat surface at the current stage, which means that the ascender can easily have a relatively good launching angle.Here, the launching angle refers to the angle between the top plane of the lander’s body and the surface.However, with the deepening of deep space exploration, the lander will also land on complex, uneven surfaces to explore complex areas.In this situation, to provide a good launching angle for the ascender, the lander’s orientation adjustment function is needed after landing on the surface.

Nomenclature LOC Lander with orientation capability RPM Rotational parallel mechanism 2D Two-Dimensional R Rotational joint P Prismatic joint U Universal joint S Spherical joint E A rigid body has no relative motion D A rigid body has the 3T3R motion ReAuL Reconfiguration auxiliary limb Mbody Motion of body MLegi Motion of the ith leg MUMaLi Motion of the upper part of the main limb MLMaLi Motion of the lower part of the main limb MULegi Motion of the upper part of the ith leg MLLegi Motion of the lower part of the ith leg MAuLij Motion of the jth auxiliary limb in the ith leg MAA_landing Motion of AA during landing phase MAA_orientation Motion of AA during orientation phase

Furthermore, the angle of the sun’s illumination changes during the day, and the angle directly affects the absorption efficiency of solar energy of the solar panels.Suppose that the body of the lander has orientation adjustment capability,and in that case,the solar panels’absorption efficiency of solar energy can be significantly improved by adjusting the angle of the solar panel attached to the body.However,the current lander is stationary and becomes a truss structure of landing gear with no orientation capability.Thus, it is urgent to design a Lander with Orientation Capability (LOC) for the coming mission.

The critical aspect of LOC design is designing the corresponding topology structure.After touching the surface, the lander and the surface are seen as a complex robotic parallel-parallel mechanism system.The first parallel one refers to each leg, whose limbs are two auxiliary backbones and one main backbone.The second parallel one refers to the lander,the four legs are regarded as the equivalent limbs, the surface is the base,and the body is the moving platform.Furthermore,the lander’s topological structures are different during performing landing and orientation capabilities.Thus, it is necessary to analyze the topological composition of the lander,which benefits the design of the topological structure of the LOC.Accordingly,the design issue of LOC becomes the problem of how to perform type synthesis of parallel-parallel mechanism with two- or three-Dimensional (3D) rotational motion according to the orientation task of the LOC.Due to the space limitations and considering the less dimension of the motion,this paper only focuses on the design of LOC with 2D rotational motion, also called LOC with two-Rotational (2R)motion.

Since the LOC with 2R motion is essentially a parallelparallel mechanism with 2R motion, the innovative significant achievements of the 2D Rotational Parallel Mechanisms(RPM) are introduced.The 5R mechanisms4(R denotes the revolute joint) may be the simplest structure of a 2D RPM,in which the axes of these R joints intersect at the same point.Wu et al.5investigated the kinematics of the 5R mechanism and restricted its kinematic optimization objectives to the prescribed workspace and the isotropy.Baumann et al.6invented a 2D RPM by introducing the parallelogram mechanism,which has been applied to a micro-invasive surgery.Sofka et al.7carried out the Omni-Wrist III robotic manipulator with a competitive alternative to traditional gimbals systems, used to assemble a spatial parallel mechanism by NASA.Yu et al.8investigated a type of 2D RPM employing a graphic approach,in which different lines denoted the twist and wrench.Then the constraint of the mechanism could be obtained after giving the relationship among lines.In addition, Carricato and Parenti9analyzed a 2D RPM that has a compact structure and good robustness.Gogu10designed a 2D RPM with isotropy based on the condition number of the Jacobian matrix.Zeng et al.11carried out the kinematics of a 2D RPM whose structure is U&RRU&SPU, where U, P, and S denote the universal, prismatic, and spherical joints respectively, and & stands for the parallel relationship.Herve12synthesized a kind of orientation mechanism with two actuated limbs and non-overconstraint by using the algebraic properties of displacement subsets.Song et al.13proposed some 2D RPMs that would be applied as mechanisms actuating the inter-satellite link antenna.A family of 2D RPMs with an equal-diameter spherical pure rotation was presented.The theoretical models of both kinematics and constraints inherited in the manipulators are analyzed through a graphical approach.However, the existing 2D RPMs cannot be directly applied to the LOC design because the lander is a complex mechanical system with different configuration requirements during landing and orientation capabilities.Nevertheless, the abundant research results of 2D RPMs are achieved and provide design references for LOCs.Therefore,this paper focuses on designing and verifying LOCs with 2R motion.The contributions include:(A)The concept of landers with orientation adjustment capability after landing is proposed.(B) The topological composition of the lander is analyzed and expressed in the form of Lie group.(C)The type synthesis procedure for designing LOCs is proposed, and numerous mechanisms of LOCs are provided.(D)One specific 4-SU equivalent mechanism for LOCs is taken as an example to illustrate the performance of the LOC by checking the kinematics, singularity analysis, workspace, and orientation capability.

The rest of this paper is organized as follows.The combinational principle of the lander, design concept and type synthesis procedure for the LOC is proposed in Section 2.Section 3 presents the type procedure for the Equivalent Parallel Mechanism (EPM), and numerous EPMs are obtained.The auxiliary limbs design, actuation schemes, and some typical LOCs are proposed in Section 4.Section 5 and Section 6 analyze the kinematics, singularity analysis, workspace, and orientation capability.The conclusions are drawn in Section 7.

2.Combinational principle, design concept and procedure for LOCs with 2R motion

2.1.Combinational principle of Chang’e landers

The existing Chang’e 4/5 lander sketch is shown in Fig.1,consisting of the body and four legs.Because of the topological structure, it can be seen as a parallel-parallel mechanism.The structure of each leg is divided into two parts, the lower and upper parts.The lower part contains the terrain adaptability mechanism connected to a footpad.The upper part includes a parallel mechanism with three limbs, including one main limb and two auxiliary limbs.In other words, the main limb is also divided into the upper and lower parts.Energy absorbers, such as a metal honeycomb, are installed in the limbs of the auxiliary limbs and the linkage connected to the footpad and middle platform.

The motion of the legged lander is analyzed as follows.The motion of the body Mbodyis the intersection of the motion of each leg MLegi,

where i denotes the sequence number of the leg,and j presents the sequence number of the auxiliary limb.Since the motion of the lower part of each leg MLLegiis the motion of the lower part of the main limb MLMaLi, i.e., MLLegi= MLMaLi, the motion of each leg MLegiis the union of the motions of the leg’s upper part MULegiand lower part MULegior MLMaLi,

where MLegi_landingand MLegi_orientationmean the ith leg’s motion during landing and orientation phases, respectively.

2.2.Combinational principle of LOCs

The motions of a LOC during landing and orientation phases are analyzed: during landing, the function is the same as that of the existing Chang’e 4/5 lander, which needs to bear large impact force,and become a truss without any relative motion;during the orientation phase, after landing and staying on the surface, the body of the lander needs to adjust its orientation,i.e., the body’s motion is the orientation motion.Thus, the motions during these two phases are expressed as

The combinational motion principle of LOCs can be analyzed during the orientation and landing phases.

Fig.1 Combination principle of a legged lander.

During orientation phase: According to the successful experience and structure of the Chang’e 4/5 landers, the structure and function of a novel LOC are analyzed.Each leg is regarded as a composite limb that consists of one main limb and two auxiliary limbs.Since the main limb needs to absorb and bear large impact force during the landing phase,no actuated joint is designed into this limb.Furthermore,the auxiliary limbs are regarded as transmission limbs,which do not provide any additional constraints to the upper part of the legs.The motion of the upper part of legs (i.e., Eq.(3)) is expressed as

Thus, the main limbs are designed to provide all the constraints for the lander.The body’s motion is determined by the intersection of the four main limbs,and Eq.(1)can be written as

Thus, during landing, the motion of each leg needs to satisfy Eqs.(10) and (11), which means that the motion of the upper part of the main leg is stationary, and the intersection motion of the four lower parts of the main limbs is E.

According to motion relationships during the orientation phase (Eq.(7)), and during the landing phase (Eqs.(10) and(11)), the motion transformation between these two phases is analyzed as follows.Since the structure of the lower part of the main limbs during these two phases keep the same, the motions are equal to each other, which is expressed as

2.3.Design concept and type synthesis procedure of LOCs

According to the combinational principle of LOCs,during the orientation phase, the main limbs provide all the constraints for the leg while auxiliary limbs provide no constraints.However, the auxiliary limbs transform their configurations and motions during the landing phase.So, one design concept based on main-ReAuxiliary limbs is proposed: Firstly, we design the structure for the lander, which is regarded as an Equivalent Parallel Mechanism (EPM).The EPM consists of the four main limbs and the body.Secondly, we design the auxiliary limbs to transform the configurations between landing and orientation phases.The detailed type synthesis procedure of LOCs is described as follows.

(1) Motion requirement of the task.The motion of the LOC is 2R motion.

(2) Type synthesis of EPMs for the LOCs.

(3) Design of ReAuxiliary limbs and actuation scheme.

(4) Assembly of EPMs and auxiliary limbs to obtain the mechanisms for LOCs.

3.Type synthesis of EPMs with 2R motion

3.1.Type synthesis procedure of EPMs

By combining the screw theory14and Lie group theory, the type synthesis procedure for the EPMs can be introduced as follows:

Step 1.Express the constraints of EPMs that have the desired mobility using wrench system W.

Step 2.Determine the number of limbs (typically the number of limbs is not less than that of DOFs of the mechanism to be designed), and then decompose the wrench system of each limb following the condition that the union of limb wrench systems should equal W.

Step 3.Identify the corresponding twist system and their Lie subgroup or submanifold expressions for each limb from the constraint systems derived in the previous step using the reciprocal condition.

Step 4.Configure the kinematic joints for each limb based on the derived Lie subgroup or submanifold expressions and generate the kinematic chains of all limbs.

Step 5.Assemble EPMs using the synthesized limb structures, in which some assembly conditions are required to ensure that the union of the limb wrench systems is W.

Step 6.Check the synthesized mechanism’s movement continuity to ensure that the EPMs have full-cycle mobility throughout their workspaces, except singular configurations.The constraints are denoted by the symbols, as shown in Table 1.

3.2.Type synthesis of EPMs

According to the procedure, the type synthesis of EPMs with 2R motion can be processed as follows.

According to Step 1, for the EPMs with 2R motion, there are 2D rotational Degree of Freedoms(DOFs,) and based on the screw theory, it is a 4D constraint set.Then, according to the reciprocal condition, two kinds of equivalent wrench systems are obtained, which is proven by the screw theory.Note that other equivalent wrench systems can be seen in Ref.15.The equivalent wrench systems of Case I are presented in Fig.2(a),and it consists of three independent 1D forces and one 1D couple.Because two 1D forces can derive the 1D couple, the equivalent wrench systems of Case II are presented in Fig.2(b), consisting of four independent 1D forces.

According to Step 2, considering that the lander has four legs, the four equivalent wrench systems can be decomposed and assigned to the limbs.In order to make the structure more symmetrical, 4D constraints could be distributed evenly into four limbs.Then,there are four limbs with a 1D force for Case I,while there are three limbs with 1D force and one limb with a 1D couple for Case II.Thus, there are two kinds of limbs for the two cases:one is the limb with a 1D force,and the other is the limb with a 1D couple.

According to Steps 3 and 4, as for the limb with 1D force,the twist system is composed of 3D rotations and 2D translations, and it can be realized by one corresponding Lie group,which is expressed as S(N)T(u,v).S(N) denotes 3D rotations about point N, and T(u,v) presents 2D translations within a plane formed by two linearly independent unit vectors v and w.Based on the Lie group, the kinematic structures of S(N)T(u,v) can be SuPvP, SuvU, SuPuR limbs (uP: prismatic joint,which translates along u;uR: revolute joint which rotates around u; S: spherical joint;uvU: universal joint which rotates around u and v).On the limbs with the 1D couple, the twistsystem is composed of 3D translations and 2D rotations.Therefore it can be realized by one corresponding Lie group which can be expressed as R(A,u)R(A,v)T,where R(A,u)means 1D rotation about an axis parallel to unit vector u and passing through point A,and T means 3D translations in space.R(A,u)R(A,v)is also equivalent to U(A,u,v),in which U(A,u,v)means the 2D R motions around the unit vector u and v.Therefore,the kinematic structures of R(A,u)R(A,v)T can beuPvPwPuvU,uvUwPuRvR,uvUwPuvU, and PR(Pa)RR, where Pa means the parallelogram linkage.Thus, the typical sketches of the limbs are presented in Table 2.

Fig.2 Two special cases of equivalent wrench systems.

According to Step 5, the EPMs can be obtained by assembling the four limbs for the two cases listed in Table 3.The 3-SU&UPU EPM is obtained for Case I of equivalent wrench systems by assembling three SU limbs and one UPU limb, as shown in Fig.3(a).By the same approach, the 4-SU EPM is obtained for Case II of equivalent wrench systems, as shown in Fig.3(b).Each limb is selected as SU for the EPM to make the mechanism have symmetric performance.Since the lower part of the leg of the existing lander is S-joint, the S-joint is arranged on the lower part, and U-joint is arranged on the upper part of the EPM.Furthermore, the upper part is Ujoint which has 2D motion, and it meets the condition of Eq.(15).Thus, the 4-SU EPM has potential application for legged landers.

According to Step 6, the 4-SU EPM keeps the geometric relationship of the constraint system during performing 2R motion except for the singularity configuration.Therefore,the rank of the constraint system is 4.So,there is no instantaneity in the motion process.

4.Design of auxiliary limbs and actuation scheme

4.1.Design of ReAuxiliary limbs

The process to design the ReAuLs includes two stages: one is to design a parallel mechanism with given motion during performing orientation capability, and the other is to design the transform approach to switch the configuration of the auxiliary limb for performing landing and orientation capabilities.

During performing orientation capability:the body’s motion is the intersection of main limbs.The main limb consists of the upper part and the lower part.The auxiliary limbs are connected to the upper part, so the critical aspect becomes the design of the two ReAuxiliary limbs for the parallel mechanism with the motion of the main limb’s upper part.The motion of the upper part limb may be R-motion, U-motion,and UP-motion.Based on the process for designing PMs stated in Section 4.1, the kinematic structures of auxiliary limbs for these three motions are illustrated.For the R-motion, themotion of the auxiliary limb can be 2T1R (T means Translational motion, and R Rotational moiton), 2T2R, 2T3R, and 3T3R.If we introduce 2R1T,2R2T,2T3R,and 3T3R motions to the two auxiliary limbs, the overconstraints of the parallel mechanisms for the upper part are 6, 4, 2, 0.For the Umotion and UP-motion, the motion of the auxiliary limb can be 2T2R, 2T3R, and 3T3R.If we introduce 2R2T, 2T3R,and 3T3R motions to the two auxiliary limbs, the overconstraints of the parallel mechanisms for the upper part are 4,2, 0.The construction of parallel mechanisms with overconstraints turns out to be a challenging task because the required geometric conditions are challenging to meet.Consequently,the fabricated mechanism may not exhibit the expected mobility.Thus, to reduce the overconstraints of the parallel mechanism for the upper part, we design the auxiliary limb as the 3T3R motion, which is expressed as D.The kinematic structures of the D motion can bewPuvUS,wPuvUS,wPSS,uRuvUS,vRUS anduRSS.

Table 2 Typical limbs for EPMs.

Table 3 Typical EPMs with 2R motion.

Fig.3 4-SU EPM for landers.

During performing landing capability: the auxiliary limbs need to be transformed to another configuration to maintain the upper part of a leg to be stationary (i.e., be a truss structure).Two singularity configurations for maintaining a truss mechanism stationary are introduced.(A) One is boundary singularity.This singularity configuration can be obtained when some links reach their limiting position or fix one joint to the fixed body.Taking two links connected by the P joints as an example, the links cannot translate along one direction if reaching the extreme position because of the range of the physical P joint; taking two links connected by the R joint as another example,one link cannot rotate if it is locked to a fixed link by means of a locking device.(B) The other is the dead point.This singularity configuration can be obtained when the axes of some links are coincident.Considering the planar four-link mechanism as an example, when the connecting rod and the passive crank are coincident, the crank cannot move no matter how significant the moment is attached on the driving link on this stationary position.This singularity position is called a dead point of the mechanism.

The two stated approaches can obtain the two kinds of singularity configurations for the auxiliary limbs.By the approach of boundary singularity, if the RUS limb is used as the auxiliary limb,when the RU link is fixed on the body,this limb becomes US limb, as shown in Fig.4(a); if the PUS limb is used as the auxiliary limb, when the P joint is fixed, the PU link will be locked on the body,and then this limb becomes US limb, as shown in Fig.4(b).The motions of the limbs satisfy

4.2.Actuation scheme

The actuated joints could not be selected arbitrarily.16,17The selection of the actuated joints must ensure that, in a general configuration,the DOF of the mechanism with all the blocked actuated joints is zero.In other words, any load on the endeffector can be balanced by the wrenches of the actuator inputs.Accordingly, the actuation scheme for the LOCs is listed as follows:

Fig.4 Typical configurations of auxiliary limbs.

(1) For the main limb: if the upper part of the main limb is UP chain,the P joint is fixed at the boundary singularity during the landing phase.After landing, the P joint will be a passive joint so that the LOC performs the orientation capability.If the upper part of the main limb is U or R chain, they are passive joints during the landing and orientation phases.

(2) For the auxiliary limb: the joint located on the base is enacted as an actuated joint, which is preferable to reduce the inertia and realize the reconfiguration of LOC for the landing and orientation capabilities.Therefore, if the auxiliary limb is RUS, the R joint is selected as the actuated joint.If the auxiliary limb is PUS, the P joint is chosen as the actuated joint.

The symbols of the actuated joints are represented with an underscore, such as Por R.

4.3.Typical LOCs with 2R motion

After designing the EPMs and the auxiliary limbs, the structures of landers could be obtained by assembling an EPM(as listed in Table 3) and two auxiliary limbs (Fig.4).The 4-SU is selected as the EPM,and RUS is chosen as the auxiliary limbs.Thus,one typical LOC is obtained,which has four symmetrical kinematic structures of legs and is presented as 4-(U&RUS)-S, as shown in Fig.5(a).The two configurations of one leg during performing landing and orientation capabilities are shown in Fig.5(b) and Fig.5(c), respectively.Therefore, to maintain stability and singularity, two R joints of the auxiliary limbs in each leg can be selected as the actuated joints according to the actuation scheme, which is underlined in Fig.5.Thus, the whole mechanism of this LOC is redundant.

5.Kinematics modeling, Jacobian matrix, and singularity analysis during performing orientation capability

In this section,the typical LOC(see Fig.5)during performing the orientation ability is taken as an example,the inverse kinematics and Jacobian matrix are established, and singularity is analyzed.When unlocking the actuated joints,each leg’s mechanism becomes SU.Moreover, this LOC’s mechanism can be presented as an equivalent, 4-SU, combining the body and four limbs AiBior Li(i = 1–4) with the length li.This mechanism has two DOFs, which could be verified based on the DOF formula

where d is the order of the mechanism,n is the number of links,g denotes the number of kinematic pairs, fipresents the freedom of the ith pair, and v means the number of redundant constraints.

5.1.Inverse kinematics

To evaluate the kinematic performance of the 2D RPM, one should first formulate its kinematic equations for the inverse position analysis.Then, according to Ref.9, a group of suitable main parameters is selected for the typical LOC configuration, as shown in Fig.6(a) and Fig.6(b).

Without loss of generality, one can define the original pose of the moving platform as the center C being located on the z-axis, and the plane uCv being parallel to the plane xOy.Then, the mapping operator between the global coordinate frame and the moving coordinate frame under the original pose can be formulated as

Fig.6 Typical configuration of LOC.

5.2.Jacobian matrix

As shown in Fig.6(a), the infinitesimal twist19of the moving platform can be written as the linear combination of the instantaneous twist screw of each limb

where $Pdenotes the infinitesimal twist of the moving platform, $j,irepresents the unit screw of the jth 1-DoF joint of the ith limb, and ˙θj,irepresents the angular rate of the jth 1-DoF joint of the ith limb.Note that a U-joint can be seen as a kinematic equivalent with two revolute joints with perpendicular axes that pass through the center of the U joint.Likewise,an S joint can be regarded as a kinematic equivalent with three orthogonal revolute joints whose axes pass through the S joint’s center point.

Constraint analysis: According to the reciprocal screw theory, one constraint wrench screw $Ci,can be obtained in each SU limb, which satisfies

and B1x,B1y,and B1zare the location of B1with respect to the global coordinate frame O-xyz.

Combine Eqs.(34) and (37) into a matrix form, and then the complete velocity mapping between input and output can be obtained by

5.3.Singularity analysis based on wrench graph method

It is noteworthy that more than two active joints can be added for the LOC, and the LOC will be a redundant mechanism to avoid this singularity and ensure mobility.

6.Workspace and rotation capability

6.1.Workspace analysis

Once the inverse kinematic model is established,the reachable workspace of the mechanism can be determined by considering some constraints imposed by joints and actuators.A group of suitable main dimensional parameters is selected, as shown in Fig.7.Considering the mechanical constraint, the set of permissible joint configurations is determined as follows:

Fig.7 Wrench graph of 4-SU mechanism.

where Bizmeans the z-value of Bipoint; θi1, θi2are the angle ranges of the U joint in each limb.The corresponding reachable workspace with the set of permissible joints when zC>40-mm is determined in Fig.8.Each point in the workspace is point C(xC, yC, zC).

6.2.Section cutting plane in workspace

The section cutting plane,when z=760 mm,of the workspace is shown in Fig.9.Only one singularity configuration occurs on the plane’s center in this plane.Four motion trajectories are represented as different colored curves, as shown in Fig.9.P1,P2,P3,and P4are the intersection point of different colored curves.It is shown that the four motion trajectories cannot switch to each other on the intersection points P1,P2, P3, and P4.The legged lander has different configurations on each intersection point.The configuration corresponding to point P1of the blue curve is represented as configuration c,and the configuration on the P1of the red curve is represented as configuration b.Ii(i = 1–8) is the connecting point among different colored curves.It is shown that the four motion trajectories can switch to each other on these connecting points Ii(i = 1–8).The configurations corresponding to connecting points on the different colored curve are the same,for example,the configuration corresponding to the point I1is shown in configuration a.

6.3.Orientation capability

In order to illustrate the orientation capability and the configuration intuitively, a reachable orientation is represented by a united vector whose direction is perpendicular to the moving platform(body).This vector can be expressed as a corresponding vector shown in Fig.10(a),whose direction passes through a united sphere’s center and the point on the united sphere.Then, the overall reachable orientation can be represented on the unit spherical surface.

Fig.8 Reachable workspace with the set of constraints.

Fig.9 Section cutting plane z = 760 mm.

Fig.10 Orientation space with constraints of joints.

Furthermore,the orientation angle is the angle between the current pose and the initial pose,and it can be presented by the distance between the point above the surface and the sphere surface, as shown in Fig.10(b).The color bar represents the z value of the center of body C.Finally, the pitch angle of the body can reach 17.5°, whose orientation vector is(0.213,0.213,0.953), and the corresponding configuration is shown in Fig.11.Because of the legs’symmetric structures,the yaw range of the body is the same as that of the pitch.Therefore, it is concluded that the orientation capability of the two rotations of the lander satisfies the rotational requirement of landers.3It is noteworthy that the maximum orientation angles of the body with respect to the base plane formed by the centers of the footpads of the lander are equal when landing on a slope or even terrains.

Fig.11 An orientation configuration of LOC.

7.Conclusions

This paper firstly proposes the concept of landers with orientation capability after landing on the surface of the planetary bodies.This kind of LOC has two additional functions compared with the current Chang’e 4/5 lander: (A) It can provide a suitable launching angle for the ascender lifting off while performing the‘‘Returning”mission, even landing on the uneven surface.(B)It can enhance the efficiency of absorbing the solar energy for the lander’s solar panels(assembled on the lander’s body).

This paper discovers the combinational principles of the current landers and the designed LOCs, which are parallelparallel mechanisms.Then, the design concept and procedure of LOCs are proposed: designing equivalent parallel mechanisms and the auxiliary limbs respectively and assembling them to form the topological structure of LOCs.Two specific equivalent wrench systems for the EPMs with 2R motion are proposed.Numerous EPMs can be obtained according to these two equivalent wrench systems.The auxiliary limbs are designed based on the boundary and dead point singularity.A family of LOCs is obtained by assembling the structures of EPMs and the ReAuxiliary limbs.Finally, one specific 4-SU equivalent mechanism for the LOC is taken as an example to illustrate the performance of the LOC by calculating the kinematics, singularity analysis, workspace, and orientation capability.The result shows that pitch and yaw angle reach 17.5° considering the permissible range of joints, which satisfies the‘‘Returning”mission.Thus, the proposed types of legged landers with orientation capability offer potentially suitable choices for the‘‘Returning”mission of the exploration mission.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

The authors acknowledge the partial financial support under the projects from the National Natural Science Foundation of China(Nos.51735009 and 51905338),the State Key Laboratory of Mechanical System and Vibration,China (No.MSVZD-2016-08) and the China Postdoctoral Science Foundation(No.2019M651487).