Design and analysis of a novel hinged boom based on cable drive

Kijie DONG, Dunling LI,b,*, Qiuhong LIN, Hui QIU, Qing CONG,Xio LI

a School of Automation, Beijing University of Posts and Telecommunications, Beijing 100876, China

b College of Mechanical and Electrical Engineering, Shaanxi University of Science and Technology, Xi’an 712000, China

c Beijing Institute of Spacecraft System Engineering, China Academy of Space Technology, Beijing 100094, China

KEYWORDS Cable drive;Deployable structure;Dynamic simulation;Hinged boom;Stiffness;Type synthesis

Abstract Hinged booms are widely used in astrophysics missions; however, the trajectory and deployment velocity are difficult to control because they are usually driven by springs,which limits their application in narrow spaces.Thus,a novel hinged boom is highly required to achieve motion controllability.Through an equivalent substitution between the cable drive loop and the binary link in topology, a type synthesis method for the cable-driven single-degree-of-freedom chain is proposed based on the single-open-chain(SOC)adding method.According to the configuration design,a novel cable-driven hinged boom is proposed,aiming to achieve boom synchronism.Then,to preload easily, a method that preload is applied and measured at the cable ends is adopted and the relationship between the initial preload and the target preload is deduced. By analyzing the distribution of cable tension, a new stiffness model is proposed thus a stiffness equation is obtained.Finally, the dynamic simulation analysis and zero-gravity deployment experiment of the hinged boom is carried out to verify its reliability.This research provides a new way for the type synthesis of cable-driven single-degree-of-freedom chain and a new model for analyzing cable-driven stiffness. Moreover, the novel cable-driven hinged boom obtained in this study can be well-applied in the field of aerospace.

1. Introduction

In many space exploration missions,instruments of spacecrafts usually require a long-distance position and the support of deployable structures to meet functional requirements. While,considering the complexity, reliability and cost, small satellite platforms are often chosen,thus new requirements for deployable structures (e.g., simple, reliable, light weight, small stow to deployment ratio,large deployable length,appropriate basic frequency, etc.) are put forward. Among many of the deployable structures,the hinged boom is an appropriate scheme that can meet the requirements.It is widely used in space structures,such as rod-shaped antenna, magnetometer, gravity gradient boom and solar sail, etc.

A typical hinged boom is a linear deployable structure,which is composed of hinges, booms, hold down and release systems (HDRS) and accessories (such as payload interfaces,etc.). On the ground and during the launching phase, the hinged boom is in a stowed state to meet the envelope size requirement of vehicles, and is fixed to spacecrafts through the HDRS.After entering the orbit, the HDRS is unlocked and the boom with payloads deploys with the hinge driving force or the centrifugal force of the spacecraft,and the deployment mission succeeds after the structure is locked.The major difference between the hinged booms and other linear deployable structures is that the space trajectory of the booms during deployment is greater than the space occupied by the deployed booms, which is contrary to others.To date, typical examples of hinged booms are, for example, in 2003, a hinged boom, designed by the European Space Agency (ESA) and the Chinese Space Agency (CNSA), was successfully applied to the Geospace Double Star Project (DSP).The deployment of the boom was solely driven by centrifugal forces and boom hinges were equipped with a redundant self-locking mechanism,aiming to guarantee a play-free and accurate positioning after deployment.Moreover,in 2013,the hinged boom of the Jupiter Ice Moon Explorer successfully completed the ground deployment experiment.This boom was composed of six 1400 mm long and two 700 mm long Carbon Fiber Reinforced Plastics (CFRP) bars connected through hinges. The length after deployment could reach 10 m,and the average linear density was only 0.496 kg/m.

The hinge, as the most important part of hinged booms, is crucial for the folding and unfolding function. Hinges can be divided into two categories according to different principles:the rigid hinge and the flexure hinge. The rigid hinge is traditional, which constricts the degree of freedom (DOF) through the cooperation of the shafts and holes of the rigid parts; it generally has only one rotational degree of freedom. A rigid hinge can be further divided into a passive drive and an active drive according to the power source. The passive drive is generally a form of spring self-storage drive, a flat spiral spring and cylindrical helical torsional springs, which are commonly used ones.A rigid hinge generally has a self-locking mechanism, and an appropriate design of a self-locking mechanism and shafting contribute to high stiffness and precision.Unlike rigid hinges, flexure hinges do not usually have independent driving or locking mechanisms, and they do not have rotation or sliding pairs for constricting DOF. Through the buckling performance of its elastic shell, flexure hinges can be driven and supported by self-storage energy. Also, they have the advantages of simple composition, high stiffness to mass ratio, no gap passive expansion and high repeatable pointing accuracy.However, due to the geometric shape and flexibility of the flexure hinge, the stiffness is limited and cannot achieve high levels. Therefore, a hinged boom with rigid hinges is still the mainstream of linear deployable mechanism.

The spring-driven rigid hinge is widely employed in onetime deployments due to its simple structure,light weight,easy installation and high reliability.However,the uncontrollability of the spring drive may lead to some difficulties in controlling the trajectory and speed of the hinged boom during the deployment process.Considering that for the deployment of a space deployable structure,a multi-rigid body system in particular, the interference between links or between the structure and the spacecraft must be avoided, otherwise it may result in the failure of the whole structure or may affect the stability of spacecraft gestures. In this case, the sequential deployment control of the boom becomes a key and difficult point.Li et al.adopted two methods—torque control method and sequential trigger method;however,neither realize synchronous deployment and sequential expansion: it was difficult to realize synchronous deployment accurately for torque control, while trigger control failed to change the shortcomings of the large sweep trajectory.

Therefore, to achieve synchronous deployment, a hinged boom based on a cable drive is proposed in this paper.In addition to this introductory chapter that delineates the background and the research problem, this paper consists of a further four chapters, summarized as follows. Section 2 introduces the design of the cable drive configuration,including the type synthesis of cable drive and the design of multimodal motion. Section 3 introduces a detailed design of the hinged boom,including hinges design and cable drive design.Section 4 analyzes the preload and stiffness of the hinged boom. Section 5 illustrates the dynamic simulation and experiment. Section 6 concludes this paper.

2. Design of Cable-driven configuration

2.1. Cable-driven type synthesis

A cable drive is a typical flexure drive, in which flexure links can only sustain tension, thus the flexure link only has one motion constriction.In a flexible drive, if a rigid link connects with another rigid link by at least two flexible links, and the direction or the position of the force between the flexible links and the rigid link is different, the rigid link will have two constraints,which is similar to the case of being connected by a binary link. So, the two flexible links, which appear as a flexible loop, can be equal to a binary link. Simply put, the flexible loop can be expressed as‘‘◇”,and the two constraints of the flexible loop on the rigid link are represented by the thin double lines ‘‘═”.

Fig. 1 4-bar linkage and a flexure drive.

Fig. 2 Equivalent of a 4-bar linkage.

The comparison between the topology graph of the 4-bar linkage(Fig.1(a))and the flexible mechanism(Fig.1(b))indicates that these two mechanisms are topologically identical.In a rigid mechanism, a binary link, except a fixed link, can be replaced by two flexible links or a flexible loop. As shown in Fig. 2, a binary link in a parallelogram link is replaced by two cables,and the new mechanism is equal to a parallelogram link. In the typical 4-bar link, binary links, whose speeds to revolute pairs are different, can be replaced by flexible links according to the design profile curve of the wheel(like CAMs).

Among the developable structures, the developable scissor structure,one of the commonest,is widely used across various fields. Fig. 3 (a) shows a kind of scissor elevator schematic.However,due to the large error accumulation caused by many hinges, the large stow to deployment ratio and the heavy relative weight, scissor structure is restricted to its application in spacecraft. By replacing some constraints in the scissor structure with the cable drive, the number of links and hinges are reduced, and some new developable structure configuration may be available. A scissor structure is a kind of symmetric structure,and if one half is removed,a configuration that bars are in series is obtained, as shown in Fig. 3 (b). The obtained 5R link has multiple degrees of freedom,and its topology diagram is shown in Fig. 3. According to the Gruebler’s equation

Fig. 3 Schematic diagrams of mechanism and topology graphs.

Fig. 4 SOC adding method of cable-driven single DOF chain.

Table 1 Cable drive design.

where F is degree of freedom;n is number of links;p is number of joints.

The DOF of the 5R link is 5. For each additional binary link, the mechanism will add a link and two kinematic pairs.With a binary link being equivalized by a cable loop, it is assumed that the minimum increase of the cable loop is n,which can make the DOF of the mechanism become 1. Then,according to Eq.(1),ncan be obtained as 4,which means that by adding at least 4 cable loops, the 5R link can have a single DOF.

In cable-drive type synthesis,the cable loop cannot connect the two adjacent rigid links,otherwise,according to the Gruebler’s equation, the DOF of the three connected links is 0,which is equivalent to a rigid structure. For the synthesis of cable drive chain, a method based on Chu’s single-openchain (SOC) adding method, is proposed as follows(Fig. 4). To be specific, a cable loop is connected to 3R links to form a basic loop that is used as a kinematic chain M.Next, connecting a cable loop with a binary ling to form a single-open-chain O;then,adding a SOC to Mwhile guaranteeing only one connection between the rigid link in O,which will obtain a 2-loop kinematic chain M. Similarly, adding another SOC Oto Mwill obtain a 3-loop kinematic chain M3.

By using the cable-driven type synthesis method, all the four-loop single-DOF chains with cable drive can be obtained.After the isomorphism analysis,the corresponding mechanism atlases can be obtained (detailed analysis will not be covered here). Table 1 provides three typical kinds of the mechanism types in mechanism atlases, including double-color topology graphs and kinematic diagrams. In the kinematic diagrams,if the colors of the wheels and connected links are the same,it means the wheels are fixed with the links, while if not, it means they are connected by revolute pairs. The kinematic loops in case 1 are connected in series with each other, and each kinematic loop has four links, which is similar to the developable scissor structure.In case 2,an idler wheel is added to change the direction of the cable Cmotion,so that the link 6 can be directly connected to the fixed link 1. In case 3, each link is connected to the fixed link 1,and some idler wheels are also needed. If these cables to motors are connected, a lightweight multi-freedom manipulator configuration can be obtained. Case 1 is simpler and more reliable than other configurations because it has no idler wheels,and it is often used in solar wing synchronous deployable mechanisms, thus more suitable for a hinged boom configuration.

According to the type synthesis and considering the functional requirements, the design of a single synchronous cable-driven loop is shown in Fig. 5. In a cable drive, wheel rotation speeds are inversely proportional to their diameters,so the diameters of the two wheels in a synchronous cable drive are equal.The two wheels are connected by two cables,respectively,and they are fixed to avoid slipping.Guided by the idler wheels, the distance between the cables is narrowed down by changing the cables’directions,and this can make the envelope size of the drive smaller and the structure more compact, contributing in preventing cables from being hooked to other parts. The deployment process of the cable drive is shown in Fig. 6.

Fig. 5 Cable drive of boom.

2.2. Multi-mode deployment design based on metamorphic mechanism

At the end of the deployment, the DOF of the deployable structure needs to become 0 from 1 so that the deployable structure can become rigid to position and support instruments. The cable drive, on the one hand, benefits boom synchronization and controllability, and on the other hand,decreases the reliability of the hinges’locking due to hinge constraints. Due to the error of preloading, manufacturing and assembly, the initial position of each hinge has some relative errors. Accordingly, under the cable drive, there are some errors between the locking positions of hinges,which may lead to the failure of synchronous locking. If one hinge is locked,the constraint of the cable will become an obstacle to other hinges’ locking, which may lead to a lock failure. Therefore,to increase the reliability, it is necessary to remove the constraints of the cable drive on the links, in this case, the DOF of the deployable boom will change from single-degree-offreedom to multi-degree-of-freedom, and each hinge can be independently locked by its coiled spring.In the whole deployment process, the boom has multi-mode movement where its DOF changes from one-degree-of-freedom to multi-degreeof-freedom, and to zero-degree-of-freedom in the end, and the topological relationship between links also changes, which conforms to the definition of the metamorphic mechanism.As shown in Fig. 7, the topological relationship between links also changes in the process.The cable-driven multi-looped planar linkage becomes an open-chain after the cable drive negates,and after the hinges lock,an open chain conformation forms a rigid component.The multi-mode deployment design,based on the principle of metamorphic mechanism, can meet requirements whereby in one deployment, the mechanism needs a multi-mode motion for different developing stages of system, which is beneficial to reliability.

In deployment process, the change of DOF is largely realized by the tension and looseness of the cable. As shown in Fig. 8, the blue cable is tangent to the wheel and the guide wheel at B and C,and the cable end is fixed at A.At the initial position, when the deployment angle is 0°, the angle between BOand OA is φ. When the boom rotates counterclockwise,the tangent point B comes closer to A, and when the deployment angle reaches 180°-φ, the tangent point B and cable end A overlap.As the boom continues rotating,the blue cable is slack and cable drive fails, then the hinges are locked when the booms deploy independently to 180°. Thus, when the deployment angle interval is (180°-φ, 180°], the deployable boom is a multi-DOF mechanism, and its hinges can rotate freely until they lock. In this way, the reliability of the hinge lock increases and the risk that many hinges cannot lock simultaneously is greatly reduced. In addition, cable relaxation can be realized by changing the wheel curvature radius, as shown in Fig. 9, but for this method, its machining process is more complex and the cost is higher than the method of changing the fixed points.

3. Design of cable-driven deployable boom

According to the configuration design illustrated in the previous section, the design of a deployable hinged boom based on a cable drive in this paper is mainly composed of a 90° root hinge, 180°mid-booms hinges, a 90° top hinge, HDRS and cable drive, as shown in Fig. 10.

The hinge is not essential but the key to ensuring the performance of boom deployment. The assembly of the hinge designed in this paper is shown in Fig. 11. For the hinge, the deployment energy is provided by a couple of tape springs and the driving torque of the spring is adjustable for the deployment need. To reduce radial clearance and increase the structure stiffness, the hinge has a couple of angular contact balls, in addition, the locking method, lock-pin and groove,also aims to ensure the stiffness and reliability of hinge locking. Also, the position precision of hinge deployment is achieved by adjusting the locking angle through the screws.In addition, during launching, the stowed hinges may collide.To avoid this, silicone pads are fixed on hinge bracket B for isolation.Through the above methods,the hinge and the boom are guaranteed to have high deployment precision and locking stiffness. Moreover, to control the deployment speed of the boom and reduce the locking impact,the root hinge can selectively connect with a viscous damper.

The hinge is also an important part of the cable drive. The cable wheels are arranged on the hinge brackets.For example,as shown in Fig. 12, a mid-boom hinge consists of a medium wheel and a small wheel,and the middle wheel,used to be driven with the next hinge, is fixed with hinge bracket A. The small wheel that drives the large wheel of the root hinge, is fixed on hinge bracket B,and the radius ratio of the two wheels is 1:2, so the deployment angle ratio is 2:1. In terms of fixing cable ends, taking the middle wheel as an example, it has two fixing points whereby positions are determined according to the design illustrated in section 2.2, aiming to ensure that the cable is slack before the hinge is locked. At the fixing points, there are two kinds of cable spikes, one is long with a screw to adjust the tension of the cable and the other is a short spike which is small for the compact space.

Fig.13 shows the cable drives in the hinged boom.Wire or Kevlar can be used to ensure the strength and stiffness of the cable. The connecting relation of components and other parts of the cable drives of the boom is shown in Table 2.

Fig.14 illustrates the unfolding process of the boom.In the initial state,the boom is stowed and fixed on the spacecraft by HDRS, while it will be deployed driven by coil springs when HDRS is unlocked.Because the moment of inertia of the components driven by the coil springs is different, the rotation velocity and acceleration of each hinge differ. In cable drives,the moment of cable on wheels is shown as the driving moment on the hinges with slower rotation speed and the resistance moment on the hinges with faster rotation speed.The moment functionally relates to the unfolding angle of the hinges.When the angles are equal, the moment becomes 0, thus, hinges of theboomtendtobeapproximatelyequal,θ=θ=θ=θ.In the final stage of the unfolding process,each hinge will be locked by the locking mechanism so that the boom can maintain a fixed unfolding shape.

4. Analysis of preload and stiffness

4.1. Preload analysis

The design that places the cable in the boom,has some advantage:cables are rarely hooked to other parts,the envelope size adopted, and the initial preload is determined by a target preload through analysis.

Fig. 6 Deployment of cable drive.

Fig. 7 Topological relationship change.

Fig. 8 Cable drive in deployment.

Fig. 9 Changing wheel curvature radius.

Fig. 10 Stowed hinged boom.

Fig. 11 Composition of hinge.

Fig. 12 Cable wheels in a mid-boom hinge.

Fig. 13 Cable drives in hinged boom.

Table 2 Connecting relation of cable drives.

Fig. 14 Deployment of boom.

To simplify the cable-driven model,the drive system retains a single linkage cable, and the guide wheels are not taken into consideration. Thus the cable drive in Fig. 7 is equal to the model in Fig. 15, where the radiuses of wheels Oand Oare Rand R, A and F represent the position of the adjustable preload cable ends, and C and D are the tangent points of the cable and wheels. The wrap angles, which are the distance in degrees that the cable contacts the wheel,are αand β.The total length after preloading is l, which can be expressed as

where, L is the free length of cable without contacting either wheel.

At the initial state, preload Fis applied at point A of the cable wheel. Due to the friction between cable and wheels, the relation of the tensions in different sections of the cable is of the boom is small and the structure is compact. Despite these advantages, there are also some problems, for example,the hinge mechanism is complex, the operating space is small,etc.Among them,the problem of cable preload is particularly relevant.Due to the fact that the whole cable,except the cable ends, is in the carbon fiber tubes, it is difficult to measure the tension of the cable using a tensiometer and other measuring equipment, and it brings difficulties for cable preload. which is a crucial parameter in stiffness analysis.Thus,a new method based on applying initial preload at the ends of the cable is F> F> F. During the deployment of the boom, the tension Fin section AC decreases,and the tensions Fand Fin section CD and DF increase slightly.When the wheel Orotates until point C coincides with point A, if the boom continues unfolding,the cable will absolutely relax and the tension will be 0 in each section.After cable relaxation,with the boom re-stowed,the cable will be re-preloaded and the tension distribution will change. Since the boom material is carbon fiber,which has a high compressive modulus, we can assume that the length deformation of the boom before and after two preloads is ignored.Based on the same length of cable before and after the first unfolding, the change of tension is analyzed in detail as follows.

Fig. 15 Cable drive model.

At the initial state, the cable is preloaded at point A.Assuming the initial preload is F, according Euler equation of the flexible body friction transmission,

Similarly,the elongation of the cable in section CD and DF being influenced by preload at the initial stage is

The tension distribution changes are shown in Fig. 17, the dotted lines in the figure represents the initial preload distribution. We find that the tension in the free length region increases, while in contact regions, the tension decreases.

The total elongation of the cable after the boom is restowed is obtained as

k is a preload coefficient which is used to express the ratio of the initial preload Fto the target preload F.

Fig. 16 Tension distribution at initial state.

Fig. 17 Tension distribution at work state.

Taking the preload of the cable drive between the root hinge and the mid-boom hinge as an example, and relevant parameters are shown in Table 3. The initial preload is 40 N,and we can get the distribution curve of tension of the cable,as shown in Fig. 18. The red curve expresses the tension of the cable before the first unfolding of the boom, the blue expresses tension after the boom is re-stowed, and the xcoordinate expresses the length from point A to each section of cable. By calculation, the tension of cable in section CD before the first unfolding of the boom is 12.31 N, the tension in section CD after the boom is re-stowed is 13.54 N, and k is 2.95.For k,the smaller the preload coefficient k is,the smaller the difference between the initial preload and the target preload is, which is conducive to the lightness of the hinge and reduces the difficulty of preload.

4.2. Analysis of cable-driven stiffness

In the stiffness analysis of the boom, due to ductility of the cable, the stiffness of the cable drive is much lower than that of the hinges. Therefore, the hinges can be regarded as rigid bodies, and the transmission stiffness of the cable drive is equivalent to that of the boom.The cable-driven stiffness can be obtained by solving the amount of elastic deformation of the cable under the linkage load (the working load).

As shown in Fig.19,if assuming Oas the driven wheel,Ois the driving wheel, and the driving and resistance moments are Tand T (linkage load). Taking the driven wheel as an example, when the linkage load T is applied, it is equivalent to increasing the cable tension F(F=T/2R)at the tangential point between the driven wheel and the cable’s tight edge.

In traditional stiffness analysis of the cable drive,the driven wheel wrap angle is divided into a loose elastic slip region, a tight edge elastic slip region and a non-slip region.However, in the tight side of this cable drive, according to the preload distribution illustrated in the previous section,the friction force is in the same direction as F, so all of the contact region is elastic slip region. After the load is applied,the tension distribution of the tight side is shown in Fig. 20,and the dotted line represents the tension distribution before loading. The tension at a point on the contact region can be expressed by the wrap angle of the arc from the tangent point.According to Euler equation,the tension at one point can be expressed as (F+ F) e.

Fig. 18 Distribution of tension.

Firstly, taking the cable micro-element in the slip region of the tight side as an example, the change of length under the effect of tension F is established as The actual length change dδof the cable micro-element is the sum of length change caused by torque dδ and the length change caused by the pre-load force dδ. That is,

Then,the cable elongation generated by torque on the tight side of the driven wheel is

Table 3 Design parameters of cable drive.

Fig. 19 Cable drive in load condition.

Fig. 20 Tension distribution in tight side.

Three sections of cable on each side are in a series relationship,and the two sides act in parallel,as shown in Fig.22,thus the total longitudinal stiffness of the cable is the addition of these two stiffnesses in parallel.

Fig. 21 Tension distribution in slack side.

According to Eqs.(33)and(37),the main factors that affect stiffness are the elastic modulus E, the equivalent crosssectional area of the cable A, the friction coefficient between the cable and the wheel μ, the wheel radiuses Rand R, the free length L, preload F, load Fand the wrap angles α,β, α’and β’. It can be found that E and A are directly proportional to the linear stiffness K. When the spacecraft enters the orbit and the boom deploys under low temperature, the preload and load torque are assumed to be 500 N and 12 Nm. According to the cable drive parameters as shown in Table 3, the torsional stiffness of the two wheels Oand Oare 2.23 × 10N˙sm/(°) and 5.58 × 10N˙sm/(°) respectively,and the linear stiffness is 4.33 × 10N/m. Fig. 22 shows the influence rule of design parameters on the stiffness. Fig. 23(a) shows that longitudinal stiffness K increases with the increase of Fand F, and the growth rate decreases slowly;F> F, K is a constant and decreases dramatically. Thus,if structural strength and other conditions allow, the preload Fshould be appropriately increased to make F

Fig. 22 Simplification of stiffness model of cable drive.

It can be seen that longitudinal stiffness K is affected by the ratio of Fto F. Fig. 23(b) shows that K increases with the increase of μ,and tends to be constant when close to the point of F=F.Fig.23(c)shows that K decreases with the increase of α(β, α’or β’), but the effect of αis slight. Fig. 23(d)shows that, where the transmission ratio remains unchanged,increasing the radius of wheels can significantly improve the rotational stiffness K, but also reduces longitudinal stiffness K.

The major difference between the stiffness model in this paper and the ones proposed by Werkmeister and Slocumis that, the preload is considered to be a constant and its distribution is overlooked in previous stiffness analyses, which can cause the deviation of stiffness analysis. In this section, a new stiffness model is established by analyzing the distribution of preload, so the stiffness calculation is more reasonable,accurate, and has universality.

5. Dynamic simulation and development experiment

In this paper,MSC.Adams is used to simulate the dynamics of the hinged boom.Firstly,the dynamic calculation model of the mechanism is established. It is assumed that the frictional moment of each hinge is constant during the deployment process, and this constant is the maximum frictional moment.Also, the driving moment of the deployment drive coil spring is simplified as a linear function of the deployment angle.The locking process of hinges is simulated by bistop twosided collision function.

In the process of development, the frictional resistance moment of a hinge is supposed to be M, other resistance moment (e.g., the resistance moment of high frequency cable)is M,the driving moment of the coil spring is T,and the additional moment is M. Then torque Tin the deployment process is expressed as,

The additional torque M of the cable drive in the unfolding process of the boom is caused by unsynchronized unfolding.Supposing that the actual rotation angles of wheels Oand Oare θand θshown in Fig. 19, the corresponding torque M can be simplified as a function of θand θ.When F>F,according to the series and parallel relation to the cable and its stiffness formula, the total elongation can be deduced as follows

Fig. 23 Stiffness analysis.

Fig. 24 Dynamic simulation result.

When M is positive, it means that the unfolding speed of wheel Ois greater than that of wheel O, and the additional torque is regarded as a driving moment for wheel O; while when M is negative,it means that the unfolding speed of wheel Ois less than that of wheel O, and the additional torque of wheel Ois regarded as a resistance moment for wheel O

The parameters of the hinged boom are imported into Adams to analyze the unfolding and locking process. The curve of the relationships between deployment angle and speed of the root hinge and time is shown in Fig. 24(a), and the displacement, velocity and acceleration of the top hinge are shown in Fig. 24(b). It can be found that the velocity and acceleration are fluctuant because of the flexibility of the cable and the change of cable tension in the deployment process.The boom is fully deployed and locked in the following 1.5 s since it beginning unfold.

The zero-gravity deployment experiment was carried out on the boom, and the unrolling process is shown in Fig. 25. The boom fully deployed at 1.62 s. There is a deviation between the experiment results and the simulation. This is mainly because the zero-gravity air buoyancy devices are connected to the booms, which increases the moment of inertia of the booms. Moreover, zero-gravity devices cannot uninstall all the friction of the hinges. However, given that the results are close to the simulation results, thus it proves the effectiveness of the simulation.

Fig. 25 Zero-gravity deployment experiment.

6. Conclusions

(1) Through building a topological connection between the cable drive and planar linkage mechanism and based on the SOC method, this paper proposes a type synthesis method for the single-degree-of-freedom chain with the cable drive, and design a cable-driven configuration of the hinged boom based on this method. In order to improve the reliability of boom locking, based on the principle of a metamorphic mechanism, two methods are adopted in this paper to change the degree of freedom and topological structure of the mechanism during the unlocking process,leading to the failure of the cable drive before the hinges are locked,and reducing the risk that all hinges are unlikely to lock synchronously due to over-constraint.

(2) According to the configuration design, a cable-driven hinged boom whose booms can be linked to synchronously unfold is designed, whereby the unfold process can be controlled.The cable drive is placed inside of the booms, which can make the boom structure more compact and avoid being hooked to other parts.In addition, this design can reduce the gaps of the hinges and enable the hinges to carry a greater preload.(3) This paper adopts a method whereby a preload is applied and measured at the cable ends, and deduces the relationship between the initial preload Fand the target preload F, aiming to solve the cable tension problem, which is difficult to be measured inside of booms.Based on the analysis of the distribution of cable tension, a new stiffness model is proposed, and some shortcomings of the previous cable-driven stiffness model are corrected to further improve analysis.According to the new stiffness model, the dynamic simulation analysis of hinged boom is carried out by MSC.Adams, and the reliability of simulation is verified by a zero-gravity deployment experiment.

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.

Acknowledgement

The authors are grateful to YU Chunyu for discussions and LI Biao and WANG Yufeng for providing data.They also thank the anonymous reviewers for their critical and constructive review of the manuscript. This study was co-supported by the National Natural Science Foundation of China (No.51775052), Beijing Natural Science Foundation (No.21C10109),Jiangsu Key Laboratory of Advanced Food Manufacturing Equipment & Technology (No. FMZ202022) and Beijing Municipal Key Laboratory of Space-ground Interconnection and Convergence of China.