Song Yimin,Qi Yang,Dong Gang,Sun Tao

Key Laboratory of Mechanism Theory and Equipment Design of Ministry of Education,Tianjin University,Tianjin 300350,China

Type synthesis of 2-DoF rotational parallel mechanisms actuating the inter-satellite link antenna

Song Yimin,Qi Yang,Dong Gang,Sun Tao*

Key Laboratory of Mechanism Theory and Equipment Design of Ministry of Education,Tianjin University,Tianjin 300350,China

This paper focuses on the type synthesis of two degree-of-freedom(2-DoF)rotational parallel mechanisms(RPMs)that would be applied as mechanisms actuating the inter-satellite link antenna.Based upon Lie group theory,two steps are necessary to synthesize 2-DoF RPMs except describing the continuous desired motions of the moving platform.They are respectively generation of required open-loop limbs between the fixed base and the moving platform and definition of assembly principles for these limbs.Firstly,all available displacement subgroups or submanifolds are obtained readily according to that the continuous motion of the moving platform is the intersection of those of all open-loop limbs.These subgroups or submanifolds are used to generate all the topology structures of limbs.By describing the characteristics of the displacement subgroups and submanifolds intuitively through employing simple geometrical symbols,their intersection and union operations can be carried out easily.Based on this,the assembly principles of two types are defined to synthesize all 2-DoF RPMs using obtained limbs.Finally,two novel categories of 2-DoF RPMs are provided by introducing a circular track and an articulated rotating platform,respectively.This work can lay the foundations for analysis and optimal design of 2-DoF RPMs that actuate the inter-satellite link antenna.

1.Introduction

With the rapid development of modern space technology,space flight vehicles including orbit satellites and manned spacecraft experience significant increments in both quantity and performance.This makes spacecraft tracking and surveying operations face great challenges,such as transmission velocity,coverage rate,and real-time capability,which are difficult to solve in terms of economics and politics.1,2To solve these problems,the tracking and data relay satellite(TDRS)has been invented and applied.3A TDRS can track nearly all flight vehicles in middle and low orbits and relay their data.This makes the performances of tracking and surveying operations improved dramatically.As a crucial component of a TDRS,the inter-satellite link antenna is required to track and capture inter-satellite links and then realize a higher rate of data transmission.The data transmission quality is directly related to the tracking accuracy and velocity of the intersatellite link antenna,which all depend on its actuated mechanism with two rotational degrees-of-freedom(DoFs).4

In recent decades,the actuated mechanism of the intersatellite link antenna has drawn more and more attentions from both academia and industry.5,6A common way to design the actuated mechanism is to employ a serial structure consisting of two shafts connected perpendicularly,such as the famous COMETS actuated mechanism.5With the increments of the aperture and weight of the antenna,it is inevitable to design the actuated mechanism with a serial structure toward a larger size and a heavier weight,which cannot be allowed due to the limitation of the satellite space.In addition,the accuracy and velocity performances are the disadvantages of this actuated mechanism with a serial structure.6Thus,inventing an actuated mechanism with advantages of good accuracy,high velocity,and a large ratio of stiffness to size is the potential direction.Bearing these in mind,the actuated mechanism with a parallel structure is one of the best options.In this paper,the actuated mechanism with a parallel structure is referred to as the 2-DoF rotational parallel mechanism(RPM).Literatures7–9show that the 5R mechanism may be the simplest structure of the 2-DoF RPMs,in which R denotes the revolute joint and the axes of these revolute joints intersect at the same point.Gosselin,7Lum,8and Liu9investigated the kinematics of the 5R mechanism and settled its kinematic optimization subjects to the prescribed workspace and the isotropy. By introducing the parallelogram mechanism,Baumann et al.10invented a 2-DoF RPM called after PantoS-cope,which has been applied to a micro invasive surgery.Driven by the requirements for a bionic wrist,Ross-Hime Designs,Inc.proposed two novel 2-DoF RPMs named as super seeker gimbal and Omni wrist,respectively.11The Omni wrist has been used to assemble a spatial parallel mechanism by NASA.12Yu and Kong et al.13,14investigated a type of 2-DoF RPM employing a graphic approach,in which the twist and wrench were denoted by different lines,and then the constraint of the mechanism could be obtained after giving the relationship among lines.In addition,Carricato and Parenti-Castelli15analyzed a 2-DoF RPM and claimed that the mechanism has a compact structure and good robustness.Gogu16designed a 2-DoF RPM with isotropy based on the condition number of Jacobian matrix.Huang et al.17carried out the kinematics of a 2-DoF RPM with its structure Uamp;RRUamp;SPU,in which U,P,and S denote the universal,prismatic,and spherical joints,respectively.Using the algebraic properties of displacement subsets,Herve′18synthesized a kind of nonoverconstrained orientation mechanism with two actuated limbs.

However,it is concluded from the aforementioned literature review that:(1)the number of topology structures of 2-DoF RPMs is less than that of other parallel mechanisms apparently,which would not meet increasing demands in the fields of aerospace,bionic,rehabilitation,etc.and(2)the assembly principles for limbs between a moving platform and a fixed base have not been defined in systematical,effective,and distinct manners.

Since the introduction of Lie group theory by Herve′,19numerous literature studies20–24have demonstrated that Lie group theory is very effective for the type synthesis of parallel mechanisms.Lots of parallel mechanisms have been synthesized by means of this approach.For example,Li and Herve′20,21carried out the type synthesis of a class of parallel mechanism in terms of 3RM2TM,3RM1TM,and 2RM1TMtypes,in which RMand TMdenote rotation and translation.Meng et al.22employed the Lie algebra to calculate the intersection set of displacement subgroups and synthesized the type structures of limbs.Lee and Herve′23enumerated limb structures generating the product of two planar motions named G–G bond,and then synthesized a class of 3-DoF translational parallel mechanisms.Li and Herve′24analyzed the planarspherical bond and its generators,and then investigated the type synthesis of 1TM2RM-DoF parallel mechanisms without parasitic motions.

Based upon Lie group theory,this paper focuses on the type synthesis of 2-DoF RPMs that would be applied as mechanisms actuating the inter-satellite link antenna.Having outlined in Section 1 the significance and the existing problems in the type synthesis of 2-DoF RPMs,the remainder of this paper is organized as follows.In Section 2,all available displacement subgroups and submanifolds are obtained,which are used to generatethetopologystructuresoftheopen-looplimbsbetween a fixed base and a moving platform.In Section 3,the characteristics of the displacement subgroups and submanifolds are described intuitively.Bearing in mind how to assemble the selected limbs to a 2-DoF RPM with a fixed base and a moving platform,theassemblyprinciplesoftwotypesaredefinedinSection 4 to synthesize all 2-DoF RPMs.In Sections 5 and 6,two novel categories of 2-DoF RPMs are provided by introducing a circular track and an articulated rotating platform,respectively.The conclusions are drawn in Section 7.

2.Generation of open-loop limbs

Based on Herve′’s work,19,20the displacement submanifold of the moving platform of a 2-DoF RPM can be described as

where{MP}represents the displacement submanifold of the moving platform,P denotes the rotational center of the 2-DoF RPM,u and v represent the rotational axes of the 2-DoF RPM,and{R（P，u）}and{R（P，v）}denote the 1-D rotation about axis passing through point P and paralleling with unit vectors u and v,respectively.It is worth noticing that the 2-DoF RPMs synthesized in this paper are considered as realizing two rotations about the center of the moving platform.

Since all the limbs of the 2-DoF RPM share the same moving platform,the displacement submanifold{MP}can also be described as

where{Mi}represents the displacement subgroup or submanifold of the ith limb of the 2-DoF RPM,and can be further expressed as

Table 1 Potential limb motions for 2-DoF RPMs.

herein,{Ma，i}denotes the additional displacement subgroup or submanifold of the ith limb except the target submanifold,i.e.,{R（P，u）}{R（P，v）}in this paper.

In order to guarantee that the synthesized mechanism is capable of two rotations,Eq.(3)must satisfy the following condition:

where{E}denotes rigid connection without relative motions.

Comparing Eq.(3)with the 6-D rigid body motion displacement subgroup{D},it’s informed that{Ma，i}could be the following displacement subgroup or the displacement submanifold constituted by the combination of the following displacement subgroups:{E},{R（P，g）},{T（m）},{T（Plu）},{T（Plv）},{T（Plc）},and{T},where g represents unit vector that is not correlated with unit vectors u and v,m represents unit vector of the arbitrary translational axis,Pluand Plvdenote the translational plane that is perpendicular to vectors u and v,respectively,Plcrepresents the translational plane that is not perpendicular to vectors u and v,{T（m）}describes 1-D translation parallel to unit vector m,{T（Pl）}denotes 2-D translations parallel to plane Pl,and{T}represents 3-D translationsin space.Consequently,thefollowingxRMyTM(x=2,3 and y=0,1,2,3,respectively)motions shown in Table 1 could be the potential limb motions for 2-DoF RPMs,where RMand TMrepresent the rotational and translational motions,respectively,and x and y denote the numbers of the rotational and translational motions,respectively.

Through further analysis on those potential limb motions,it’s revealed that some xRMyTMmotions contain an inactive portion that does not contribute to the function of the whole unity.Namely,if the inactive portion being deleted from the xRMyTMmotions,then the remainders still own the capability to realize the desired 2-DoF rotational motion.

Since the 1-D translation lacks of capability to influence the property of the rotational motion and the 2-D translations could not change the rotational axis that is not perpendicular to its translational plane,the 1-D translational subgroup{T（m）}of the 2RM1TMmotion{R（P，u）}{R（P，v）}{T（m）},the 2-D translational subgroup{T（Plc）}of the 2RM2TMmotion{R（P，u）}{R（P，v）}{T（Plc）},and the 1-D translational subgroup{T（m）}of the 3R1T motion{S（P）}{T（m）}are inactive portions.Besides,it should be pointed out that the desired motion in this paper is 2-DoF rotations about the center point of the moving platform,so the u axis of the 2RM2TMmotion{R（P，u）}{R（P，v）}{T（Plv）}must pass through the center point P,which leads to the function of the 2RM2TMmotion{R（P，u）}{R（P，v）}{T（Plv）}equaling to the 2RMmotion{R（P，u）}{R（P，v）}.In consequence,the portion{T（Plv）}is notactivein the2RM2TMmotion{R（P，u）}{R（P，v）}{T（Plv）}.The inactive portions in the limb motions not only have no practical functions in assembling the target mechanism,but also increase the manufacture and assembly difficulties.Therefore,these motions including inactive portions are eliminated from the available limb motions for the 2-DoF RPMs.

In addition,the 6-D rigid body motion displacement subgroup{D}contains all the possible xRMyTMmotions,so the intersection of the displacement subgroup{D}and other xRMyTMmotions is still the xRMyTMmotion.Hence,in order to generate the desired 2-DoF rotational motion,the displacement subgroup{D}only can be utilized with the 2RMmotion{R（P，u）}{R（P，v）}.Furthermore,no assembly principle is needed for the displacement subgroup{D}.Therefore,the analysis of the displacement subgroup{D}is omitted in the following sections for simplicity.

With the help of the following subgroup equivalent properties:{R（P，u）}{T（Plu）}={G（u）},{T}={T（Plu）}{T（Plv）},and{R（P，u）}{R（P，v）}{R（P，g）}={S（P）},the available limb motions discussed in this paper can be classified and transformed into the following forms:2RMsubmanifold{R（P，u）}{R（P，v）},3RMsubgroup{S（P）},2RM2TMsubmanifold{G（u）}{R（P，v）},2RM3TMsubmanifold{G（u）}{G（v）},and 3RM2TMsubmanifold{G（u）}{S（P）},where{G（u）}represents the 3-D planar gliding motions perpendicular to vector u and{S（P）}denotes the 3-D rotations about point P.

In order to generate topological structures of limbs corresponding to available displacement subgroup and submanifolds for 2-DoF RPMs,three steps need to be carried out.20,21,23The primary step is to utilize the revolute joint(R joint),spherical joint(S joint),and planar joint to replace the corresponding displacement subgroups {R（P，u）},{S（P）},and{G（u）}.Based on this,the primary structures of open-loop limbs are derived.The second step is to expand the primary structures by introducing the equivalent generators of the R joint,S joint,and planar joint.The third step is to simplify the redundant generators.Based on the properties of the displacement subgroups and submanifolds,some generators of certain displacement submanifolds can be simplified.

It is noted that the topology structures of limbs correspond to the displacement subgroups and submanifolds;however,the kind and number of the former are more than those of the latter.For brevity,the assembly principles would be defined for the displacement subgroups and submanifolds.

3.Subgroup and submanifold characteristics

Although the displacement subgroups or submanifolds can describe the continuous motions of the rigid body,the intersection operations of the displacement subgroups or submanifolds mainly depend on the mathematical inclusive concept.This method is inconvenient and inexplicit when being used to be applied to mechanisms with multiple limbs.For this reason,the authors of this paper25defined the subgroup characteristic using simple geometrical symbols and provided the intersection operation principles for spherical parallel mechanisms.For a reader’s convenience,the subgroup characteristic is recalled in this paper as shown in Fig.1 briefly,where Q is an arbitrary point within plane Pl.It should be noticed that the line in the characteristic symbol directly represents the rotational or translational axis,while the boundaries of the plane,cubic or sphere do not demonstrate the actual axes of the corresponding motion.More specifically,the translational plane denotes 2-D translations along any two axes parallel with the plane;the cubic means 3-D general translations along any three axes in space;and the sphere represents 3-D rotations about any three axes passing through the spherical center.

As the extension of the displacement subgroup characteristic,the displacement submanifold characteristic may be described by the combination of the displacement subgroup characteristics.However,part of certain displacement subgroup characteristics may merge with others when multiple displacement subgroups coexist in one displacement submanifold.

In order to describe the explicit characteristics of the feasible displacement submanifold for the 2-DoF RPM,the specific displacement submanifold satisfying the characteristic fusion conditions is equivalent to the product of displacement subgroups in a more concise and obvious manner,which is much more suitable for the synthesis of the 2-DoF RPM with multiple limbs.The equivalent results are expressed in Eqs.(5)and(6),and the characteristics of the available limbs for the 2-DoF RPM are shown in Fig.2.

where F is a fixed point;M and N are different arbitrary points in space.

4.Assembly principles

As shown in Fig.2,the displacement subgroup or submanifolds of available limbs constituting a 2-DoF RPM are provided,from which at least two subgroups or submanifolds can be selected to form an intersection set in terms of 2-D rotational submanifold under prescribed assembly principles.The selected subgroups or submanifolds correspond to the limbs with different type structures,which can be employed to constitute a 2-DoF RPM with a fixed base and a moving platform.Herein,the assembly principles of the displacement subgroups or submanifolds are the key and challenging issue to synthesize 2-DoF RPMs.

Fig.1 Displacement subgroup characteristics.

Fig.2 Displacement subgroup or submanifolds of available limbs.

As shown in Fig.3,a set of n displacement subgroups or submanifolds are selected from Fig.2 to be assembled toward a target submanifold,i.e.,a 2-D rotational submanifold.It is noted that the assemblage of the n displacement subgroups or submanifolds is carried out by unit of the intersection operation of two displacement subgroups or submanifolds successively.Bearing in mind the target submanifold,the assembly principles should be obeyed during the intersection operation of any two displacement subgroups or submanifolds.

The results of the intersection operation of any two displacement subgroups or submanifolds selected from Fig.2 can be classified into two types.The first type is referred to as Type A,which means that the intersection set of any two displacement subgroups or submanifolds directly equals to a 2-DoF rotational submanifold.The corresponding assembly principles and intersection set characteristics are given in Table 2 and Fig.4,in which Piand Qidenote the points in the ith displacement subgroup or submanifold,uiand virepresent the rotational axes in the ith displacement submanifold,and Plimeans the translational plane in the ith displacement submanifold.The other type is referred to as Type B,which represents that the intersection set of two displacement subgroups or submanifolds is a high-dimensional displacement subgroup or submanifold including a 2-D rotational submanifold,and the assembly principles and intersection set characteristics of Type B are given in Table 3 and Fig.5.

Fig. 3 Assemblage of n displacement subgroups or submanifolds.

Table 2 Assembly principles of Type A.

Fig.4 Intersection set characteristics of Type A.

It should be noticed that the intersection sets of certain types given in Table 3,for instance,Type B.5.2,Type B.7.2,and Type B.8.5,contain displacement submanifolds which are not included in Fig.2,such as{R（P2，u1）}{R（P2，v1）}{T（n）},{S（P1）}{T（n）},and{R（P2，u1）}{R（P2，v1）}{T（Pl2）}.Seemingly,these displacement submanifolds can achieve 2-D rotational motion;nevertheless,the translational subgroups in each aforementioned displacement submanifold practically make no contribution to the feature of the rotational ingredient,which means that these translational subgroups are not indispensable for the intersection operation.Therefore,such submanifolds including the inactive translational subgroups have been culled in Fig.2.Actually,the aforementioned displacement submanifolds including the inactive translational subgroups can be generated by special limb structures,such as closed-loop limbs or parallelogram structures.Regarding the displacement submanifolds containing the inactive translational subgroups,the translational subgroups and rotational ingredient should be treated separately when assembling these displacement submanifolds including the inactive translational subgroups with other displacement subgroups or submanifolds.More specifically,the rotational ingredient should obey the assembly principles given in Tables 2 and 3 in this paper.The translation subgroups should be eliminated by the translational subgroup intersection operation,whose assembly principles have been given in the authors’previous work.25The intersection operation ofthedisplacementsubmanifolds including the inactive translational subgroups will be demonstrated in detail by examples.

According to the assembly principles shown in Tables 2 and 3,the minimum limb number of synthesized 2-DoF RPMs complying the assembly principles of Type A is two,while it becomes three when applying the assembly principles of Type B.Furthermore,2-DoF RPMs with four,five,and more limbs can also be synthesized by reusing the Type A and Type B assembly principles.Moreover,the type structures of 2-DoF RPMs would be enriched with the help of limb structure equivalent replacement.By the aforementioned steps,all kinds of 2-DoF RPMs would be successfully synthesized.In order to illustrate the synthesis procedures,the type synthesis of 2-DoF RPMs with two and three limbs is selected as example to be carried out in details.

According to Lie group theory,lots of different limb structures correspond to the same displacement subgroup or submanifold,for instance,RRR,RU,UR,and S are all limbstructures for the displacement subgroup{S（P）}.For simplicity,only R joint is chosen in this paper to construct limb structures.Corresponding to thedisplacementsubgroup or submanifolds of available limbs shown in Fig.2,RPuRPv,（RRR）PS, （RRR）uRPv, （RRR）u（RRR）v,and （RRR）u（RRR）PSare selected as exemplary limbs to represent 2RMsubmanifold{R（P，u）}{R（P，v）},3RMsubgroup{S（P）},2RM2TMsubmanifold{G（u）}{R（P，v）},2RM3TMsubmanifold{G（u）}{G（v）},

and 3RM2TMsubmanifold{G（u）}{S（P）},respectively,where RPuindicates that the rotational axis of R joint parallels with unit vector u and passes through point P,（RRR）PSdenotes that all three R joints within the bracket pass through point P and their motions can equal to one spherical rotation,and （RRR）uimplies that the axes of three R joints within the bracket parallel to unit vector u and their motions equal to one planar motion.

Table 3 Assembly principles of Type B.

Fig.5 Intersection set characteristics of Type B.

Therefore,applying the assembly principles of Type A to assemble the exemplary limb structures constructed before,the type synthesis of 2-DoF RPMs with two limbs can be accomplished and the typical type structures are shown in Table 4.

Regarding 2-DoF RPMs with three limbs,there are two solutions.As shown in Fig.3,the intersection set of any two displacement subgroups or submanifolds could participate in subsequent intersection operation.Therefore,the first method to synthesize 2-DoF RPMs with three limbs is to utilize each limb combination of Type A as one unity and then rerun the assembly principles of Type A to find the suitable assembly principles for each limb combination to construct 2-DoF RPMs with three limbs.On the other hand,taking each intersection set of Type B as unity and determining the suitable assembly principles for applicable limb combinations is the other way to synthesize 2-DoF RPMs with three limbs.The typical type structures of 2-DoF RPMs with three limbs are shown in Table 5.

It should be pointed out that the type synthesis of 2-DoF RPMs with two limbs and three limbs is taken as example to illustrate the type synthesis method.Through this method,2-DoF RPMs with more limbs can also be synthesized.However,for the sake of simplicity,the type structures of 2-DoF RPMs with more limbs are not exhibited in this paper.

5.2-DoF RPM with circular track

As mentioned above,the 2-DoF RPM will be synthesized by selecting the required limbs corresponding to the displacement subgroup or submanifolds in Fig.2,whose intersection operations have to obey the assembly principles provided in Section 4.Compared with the parallel mechanisms with more than two degrees offreedom,the 2-DoF RPM has more constraints,which leads to the difficulty and complexity in the design of type structures.As an alternative form of a revolute pair,a circular track is made of one circular path and its cor-responding movable slide.By using the movable slide to translate along the circular path,the circular track can achieve 1-D rotation about its center line without locating any physical joint along the rotational axis.25By providing a circular path along its circumference,the circular track transfers the rotation about its center line to a translation along its circular path,which relieves the assembly difficulty associated with the intersecting rotational axes.

Table 4 Typical type structures of 2-DoF RPMs with two limbs.

As shown in Fig.6,a 2-DoF RPM with a circular track is proposed.The type structure can be represented bydenotes the circular track whose center line parallels with the unit vector w1and passes through the circular track’s center point O.More specifically,the axes of two R joints and the distal axis of U joint are parallel to each other in the 1st,2nd,and 3rd limbs,respectively;the proximal axes of U joint in the 1st,2nd,and 3rd limbs are passing through the center point of U joint in the 4th limb;the center line of the circular track which is shared by the 1st and 3rd limbs coincides with the axis of R joint in the 4th limb and passes through the center point of U joint in the 4th limb,which can be designated as the center of the moving platform.

It should be pointed out that,as shown in Fig.6,the circular track is capable of supporting the movable slides in the 1st and 3rd limbs to realize the translation along its circular path.Based on Lie group theory,this motion generated by the circular track should be described as{R（O，w1）},a rotational motion toward the center line of the circular track.However,as shown in Fig.6,the slide of the 2nd limb is fixed to the circular track.This deters the 2nd limb from achieving the aforementioned rotational motion.Therefore,the displacement submanifolds of the four limbs can be expressed as

Bearing in mind the target submanifold,employing the assembly principle of Type B.5.2 in the intersection operation of{M1}and{M2}leads to

The translational subgroup{T（w1）}in Eq.(11)is inactive for the following intersection operation with the 3rd and 4th displacement submanifolds and it should be treated separately from the rotational ingredient{R（P，u2）}{R（P，v2）}.

Employing the assembly principle of Type A.5,the intersection operation of{M1}∩{M2}and{M3}can be derived as

Table 5 Typical type structures of 2-DoF RPMs with three limbs.

Similarly,employing the assembly principle of Type A.2 in the intersection operation of{M1}∩{M2}∩{M3}and{M4}yields

Fig.6 Schematic diagram of a 2-DoF RPM with a circular track.

As shown in Fig.6,it is informed from Eq.(13)that the RPM shown in Fig.6 is capable of two rotations described by a 2-D rotational submanifold,whose rotational axes are determined by the 2nd limb and rotational center coincides with the center of the moving platform.

6.2-DoF RPM with articulated rotating platform

In general,the moving platform of a parallel mechanism is always connected with a fixed base by open-loop limbs as one rigid body.In order to achieve required motion,such as rotations with large angles and gripping motion for irregular shapes,an articulated traveling platform26,27and articulated gripping platform28have been introduced to parallel mechanisms to replace the rigid moving platform.Inspired by this idea,an articulated rotating platform is defined in this paper as that the platform is composed by more than one rigid part articulated by one or more joints to generate relatively rotational motion between its subparts.Without loss of generality,the articulated rotating platform can be divided into two parts in terms of sub-part I and sub-part II.The function of sub-part I is to support and generate relative rotational motion for subpart II,which is utilized to connect with the end-effector to export the desired motion.By introducing the articulated rotating platform to the 2-DoF RPM,the assembly difficulty can be reduced significantly.

As shown in Fig.7,a 2-DoF RPM with an articulated rotating platform is proposed.29The moving platform is composed of sub-part I and sub-part II,which are articulated by one revolute joint.Sub-part I and sub-part II are actuated by closed-loop limbs I and II,respectively.Closed-loop limbs I is constructed by the 1st and 3rd limbs,which are jointed to the 1st rod.Similarly,closed-loop limbs II is made up of the 2nd and 4th limbs,which are linked to the 2nd rod.As shown in Fig.7,the distal axes of U joints and revolute axes of R joints in closed-loop limbs I and closed-loop limbs II are parallel to each other,respectively;the proximal axes of U joints in each close-loop limbs coincide with each other and pass through the center of sub-part I,which can be designated as the center of the moving platform;the axes of the three R joints in each closed-loop limbs are arranged within one plane,respectively.

It is worth noticing that sub-part I and sub-part II are articulated through one R joint,whose axis passes through the center point of the articulated rotating platform.This articulated structure enables the articulated rotating platform to generate one relative rotation{R（P，w）}between its two sub-parts.Due to the function of sub-part I,the relative rotation generated by the structure of the articulated platform should be ascribed to the motion of sub-part I.Furthermore,since the 1st actuated joint passes through the midpoint of the 1st rod and the center point of the articulated platform locates at the midpoint of rod B1B3,which is the opposite side of the 1st rod in the parallelogram A1B1B3A3,the parallelogram structure ensures that sub-part I has the rotational motion{R（P，u1）}.Similarly,sub-part II is also guaranteed to have rotational motion{R（P，u2）}.Therefore,combining the motions generated by the proximal axes of U joints and the motions produced by the parallelograms,the displacement submanifolds of each closed-loop limbs can be described as

Fig.7 Schematic diagram of a 2-DoF RPM with an articulated rotating platform.

where g1and g2are unit vectors within the parallelogram plane in closed-loop limbs I and closed-loop limbs II,respectively.The displacement subgroups{T（g1）}and{T（g2）}are generated by the parallelogram and inactive for the following intersection operation.Therefore,they should be treated separately from the rotational ingredient {S（P）} and{R（P，u2）}{R（P，v2）}.

Bearing in mind the target submanifold,employing the assembly principle of Type A.2 in the intersection operation of{M1}and{M2}leads to

As show in Fig.7,it is informed from Eq.(16)that the proposed RPM is capable of 2-D rotations,whose rotational axes are determined by closed-loop limbs II and rotational center coincides with the center of the moving platform.

From what has been discussed in Sections 5 and 6,even though the mechanisms in both Figs.6 and 7 belong to the category of multiple closed-loop mechanisms,they still contain unique characteristics.Fig.6 shows a typical example of 2-DoF RPMs with a circular track,while Fig.7 displays a prototype of 2-DoF RPMs constructed of an articulated rotating platform.Therefore,the fundamental differences between these mechanisms come from the unique properties of these core components:the circular track and the articulated rotating platform.As discussed before,the circular track is able to provide a rotation without placing any physical joint along its rotational axis by transforming the rotational motion about its central axis to a translation along its circular path.This relieves the assembly difficulties regarding the intersectional axes.On the other hand,the articulated rotating platform reforms the traditional rigid moving platform into two subparts,which are rotationally connected by one R joint.Compared with a traditional rigid platform,the distinct construction of the articulated rotating platform supplies an extra rotation.From what has been discussed above,the core construction concepts of the mechanisms in Figs.6 and 7 are different.Besides,the layouts of limbs are different between these mechanisms.Even though both mechanisms in Figs.6 and 7 are composed offour limbs,the two opposite limbs of the mechanism in Fig.7 are demanded to form a parallelogram,which is the essential condition to ensure the 2-DoF rotation of the mechanism in Fig.7,while the mechanism in Fig.6 has a limb connected to the centers of the circular track and the moving platform.

For the inter-satellite link antenna,the aforementioned different properties lead to differences in tracking velocity,working space,and stiffness performance for the mechanisms in Figs.6 and 7.Mainly based on type structure distinctions and not taking the optimal design into account,compared with the mechanism in Fig.7,the mechanism in Fig.6 has lower velocity and less working space because of the constriction of U joint located in the center of the moving platform.However,thanks to this U joint,the mechanism in Fig.6 should possess better stiffness performance than that in Fig.7.

However different these mechanisms are,the mechanisms shown in Figs.6 and 7 still retain the merits of multiple closed-loop mechanisms.Compared with serial structure mechanisms,the geometrical errors of joints in limbs of multiple closed-loop mechanisms have a higher possibility to be compensated by the closed-loop structure,which makes the mechanisms in Figs.6 and 7 to obtain advantage in accuracy.Besides,the high acceleration effect and exerted payload on the platform are more likely to be rationally distributed to multiple closed-loop limbs.Consequently,the high velocity and stiffness performance of these mechanisms are better than those of serial structure mechanisms.Based on these outstanding properties generated from the type structure,the mechanisms shown in Figs.6 and 7 are suitable for the inter-satellite link antenna.

7.Conclusions

Based upon Lie group theory,this paper focuses on the type synthesis of 2-DoF RPMs that would be applied as mechanisms actuating the inter-satellite link antenna.The following conclusions are drawn.

(1)According to that the continuous motion of the moving platform is the intersection of those of all open-loop limbs,all available displacement subgroup and submanifolds are obtained readily.These subgroup and submanifolds are used to generate topology structures of limbs.

(2)By describing the characteristics of the displacement subgroup and submanifolds intuitively through employing simple geometrical symbols,their intersection and union operations can be carried out easily.

(3)The assembly principles of two types are defined,which are used as the rules to assemble the selected limbs to a 2-DoF RPM with a fixed base and a moving platform.

(4)Two novel categories of 2-DoF RPMs are provided by introducing a circular track and an articulated rotating platform,respectively,which demonstrate the employment of the assembly principles in the topology synthesis of 2-DoF RPMs.

Acknowledgments

This research work was supported by the National Natural Science Foundation of China(No.51475321)and Tianjin Research Program of Application Foundation and Advanced Technology(No.15JCZDJC38900 and No.16JCYBJC19300).

Appendix A.Supplementary material

Supplementary data associated with this article can be found,in the online version,at http://dx.doi.org/10.1016/j.cja.2016.05.005.

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Song Yimin is a professor and Ph.D.supervisor in the School of Mechanical Engineering at Tianjin University.His main research interests are robotics and mechanisms,mechanical dynamics,and mechanical transmission.

Sun Tao is an associate professor in the School of Mechanical Engineering at Tianjin University.His main research interests include robotics and mechanisms.

25 November 2015;revised 6 January 2016;accepted 17 April 2016

Available online 22 October 2016

Articulated rotating platform;

Assembly principle;

Circular track;

Lie group theory;

Rotational parallel mechanism;

Tracking and data relay satellite;

Type synthesis

?2016 Chinese Society of Aeronautics and Astronautics.Production and hosting by Elsevier Ltd.This is anopenaccessarticleundertheCCBY-NC-NDlicense(http://creativecommons.org/licenses/by-nc-nd/4.0/).

*Corresponding author.Tel.:+86 22 27406327.

E-mail address:stao@tju.edu.cn(T.Sun).

Peer review under responsibility of Editorial Committee of CJA.

Production and hosting by Elsevier

http://dx.doi.org/10.1016/j.cja.2016.05.005

1000-9361?2016 Chinese Society of Aeronautics and Astronautics.Production and hosting by Elsevier Ltd.

This is an open access article under the CC BY-NC-ND license(http://creativecommons.org/licenses/by-nc-nd/4.0/).