This pap er presen ts inno v ativ e mec hanisms capable of adv an tageously pro viding attitude con trol for spacecrafts. These new mec hanisms, whic h w e ha v e named exur e whe els , are the dynamic equiv alen t of a rotating wheel and can b e en tirely implemen ted with exures. A r e action whe el is a w ell kno wn device for con trolling the orien tation of spacecrafts [5]. It consists in a motorised y-wheel whic h is placed within the spacecraft. T o set the wheel in to angular rotation, a torque is applied to the wheel whic h in resp onse applies the opp osite torque bac k to the spacecraft, according to Newton's third la w. This reaction torque is ho w the spacecraft rotates in order to con trol its orien tation. In order to enable this wheel to rotate around a xed axis, sev eral metho ds ha v e b een implemen ted suc h as ball b earings, whic h suer from frictional losses and imp erfections whic h lead to vibrations and failure, as w ell as magnetic b earings whic h do not suer from these issues but ha v e an increased p o w er consumption and complexit y [5]. The sub ject of this pap er is to in tro duce alternativ e mec hanisms that are able to pro duce the same constan t angular momen tum as a rotating wheel, but whic h do not suer from the ab o v e defects. In order to reac h this goal, our in v en tions use exure mec hanisms to pro duce the required con- stan t angular momen tum. Note that the term exur e me chanism is exactly equiv alen t to c ompliant me chanism [18] [19]. The di cult y in this task is that exures only ha v e a limited strok e making it virtually imp ossible for a exure b earing wheel to rotate around a xed axis with constan t angular momen tum. W e therefore found alternate metho ds for generating angular momen tum b y using exure mec hanisms. T w o metho ds are presen ted in this pap er. The rst consists of a rigid b o dy whose cen tre of mass has a circular tra jectory around a xed p oin t, but the b o dy do es not rotate around its cen tre of mass. The b o dy mo v es in translation and acts dynamically as a p oin t mass, and thereb y generates angular momen tum in a constan t direction. The second consists of t w o b o dies rotating around their cen tres of mass, but whose total angular momen tum lies in a xed direction. The rst metho d w as successfully exploited in the IsoSpring pro ject whose goal w as to in tro duce new t w o degree of freedom oscillators in mec hanical clo c ks and w atc hes, in order to remo v e their traditional escap emen t mec hanism [16]. The second mec hanism is also inspired from the IsoSpring pro ject where a sphere oscillating around its cen tre of mass pro vided a t w o degree of freedom oscillator less sensitiv e to the direction of gra vit y [15]. The pap er presen ts exure wheel designs along with their implemen tations. Moreo v er, metho ds to con trol the uniform circular motion are presen ted, among whic h a no v el exure b earing whic h restricts the motion of a b o dy to translation on a circular orbit. T w o protot yp es w ere successfully built and tested. Finally , qualitativ e results from this pro of of concept are presen ted. 2 Instan t Lab | EPFL CONTENTS Con ten ts 1 In tro duction and statemen t of results 5 1.1 A ttitude con trol for spacecrafts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2 State of the art for reaction wheels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3 Inno v ativ e designs of exure wheels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.4 Description of demonstrators and results . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2 Flexure Wheels for spacecraft attitude con trol 8 2.1 A ttitude con trol for spacecrafts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.2 State of the art for reaction wheels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.2.1 Bearing t yp es . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.2.2 Issues with ball-b earing reaction wheels . . . . . . . . . . . . . . . . . . . . . . . 10 2.2.2.1 F ailures and mission impact . . . . . . . . . . . . . . . . . . . . . . . . 10 2.2.2.2 Micro-vibrations in reaction wheels . . . . . . . . . . . . . . . . . . . . 10 2.2.2.2.1 Rotor un balance . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2.2.2.2 Ball b earing vibration . . . . . . . . . . . . . . . . . . . . . . 12 2.2.2.3 Magnetic cleanliness in reaction wheels . . . . . . . . . . . . . . . . . . 12 2.3 Inno v ativ e designs of exure wheels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.3.1 Abstract designs of new mec hanisms . . . . . . . . . . . . . . . . . . . . . . . . 15 2.3.1.1 Planar mec hanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.3.1.1.1 In tro ductory example . . . . . . . . . . . . . . . . . . . . . . 15 2.3.1.1.2 F ormalisation of planar mec hanisms . . . . . . . . . . . . . . . 16 2.3.1.2 Spherical mec hanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.3.2 Metho ds of use . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.3.2.1 Obtaining constan t angular momen tum . . . . . . . . . . . . . . . . . . 23 2.3.2.2 Exp orting torque . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.3.2.2.1 Exp orting a torque b y c hanging the angular momen tum's am- plitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.3.2.2.2 Exp orting a torque b y c hanging the angular momen tum's di- rection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.3.3 Detailed designs of new mec hanisms . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.3.3.1 Planar Mec hanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.3.3.1.1 In v erter pan tographs . . . . . . . . . . . . . . . . . . . . . . . 27 3 Instan t Lab | EPFL CONTENTS 2.3.3.1.2 Com bining masses on circular and linear motions . . . . . . . 31 2.3.3.2 Spherical Mec hanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.4 Demonstrator protot yp es and results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.4.1 Obtaining the circular motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.4.2 Protot yp e 1: t w o serial in v erting pan tographs . . . . . . . . . . . . . . . . . . . 36 2.4.3 Protot yp e 2: one mass on circular orbit and t w o masses to stabilise the cen tre of mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.4.4 Exp erimen tal setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2.4.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3 Conclusion and future w ork 40 4 App endices 42 4.1 App endix: Angular momen tum of a mass on uniform circular motion . . . . . . . . . . 42 4.2 App endix: Angular momen tum of t w o masses on uniform circular motion . . . . . . . . 43 4.3 App endix: Angular v elo cit y of a CV-join t on circular orbits . . . . . . . . . . . . . . . . 44 4 Instan t Lab | EPFL 1 INTR ODUCTION AND ST A TEMENT OF RESUL TS 1 In tro duction and statemen t of results 1.1 A ttitude con trol for spacecrafts A k ey asp ect of sending a satellite in to space is b eing able to con trol its orien tation. This capabilit y is de ned as attitude c ontr ol [2]. A n um b er of tec hniques ha v e already b een dev elop ed in this goal. These include:  Magnetic torquers whic h tak e adv an tage of Earth's magnetic eld to generate Lorenz forces [2]  Thrusters whic h exp el matter and use Newton's third la w of motion [2]  Reaction wheels whic h also use the same action-reaction principle, but in rotation b y means of accelerating or decelerating a set of wheels. [2] The latter are the fo cus of this pap er as they are based on mec hanical engineering, do not exp el an y matter and do not require an y external in uence. 1.2 State of the art for reaction wheels A reaction wheel consists in a y-wheel whose angular rotation is motorised and whic h is placed within the spacecraft [3]. T o set the wheel in to angular rotation, the spacecraft applies a torque to the wheel whic h in resp onse applies the opp osite torque bac k to the spacecraft, according to Newton's third la w. This reaction torque rotates the spacecraft in order to con trol its orien tation (Fig. 1). The principle underlying the reaction wheel is that the spacecraft is an isolated system ha ving constan t total angular momen tum and when torque is applied to the reaction wheel, this increases its angular momen tum so the rest of the spacecraft is forced to rotate in the opp osite direction in order to k eep the total angular momen tum constan t. This alternativ e view of a reaction wheel highligh ts one of its fundamen tal requiremen ts: that it m ust b e able to pro vide constan t direction angular momen tum in the frame of the spacecraft. motor (a) Satellite and reaction wheel at rest Rota�on of reac�on wheel Opposite rota�onof satellitemotor (b) Satellite with its reaction wheel set in to motion Figure 1: The principle of a reaction wheel. During its use, the wheel naturally p erforms complete turns around its cen tre of mass and it remains in the same qualitativ e state. This prop ert y of b eing able to turn inde nitely in the same direction while retaining the same qualitativ e state is de ned as unlimite d r otation . This unlimited angular 5 Instan t Lab | EPFL 1 INTR ODUCTION AND ST A TEMENT OF RESUL TS rotation requires b earings at the in terface b et w een the rotating mass and its cen tral axis. Curren tly , the most common b earing implemen tation is ball b earings [13] [4]. Ho w ev er, these suer from a n um b er of issues, suc h as w ear, failure and the need for lubrication. Due to these issues, w e in v estigated the use of exur es, since these do not suer from the ab o v e defects [18]. The application of exures, also kno wn as c ompliant me chanisms [19], to reaction wheels has, to our kno wledge, not previously b een explored. 1.3 Inno v ativ e designs of exure wheels Flexures are a w ell studied category of mec hanical comp onen ts whic h rely on elastic deformation to enable motion [18] [19]. They are analogous to standard mec hanical building blo c ks suc h as a pin or slider (Fig. 2), but are free of friction (since no parts are sliding or rolling on eac h other). Their dra wbac ks include their limited range of motion and the fact that they act as springs and therefore require force to b e k ept a w a y from their neutral p osition. (a) XY translation stage made of t w o parallel leaf spring stages moun ted in series [16] (b) Butter y hinge whic h acts as a piv ot [17] Figure 2: Examples of exures Since the goal of this pap er is to nd a dynamical equiv alen t of a reaction wheel using exures, w e ha v e named our new mec hanisms exur e whe els. A k ey feature of reaction wheels is unlimited angular rotation. This asp ect can eviden tly not b e directly translated in to exures as these ha v e a limited strok e.1The conceptual con tribution of this article is the design of exure mec hanisms whic h b eha v e lik e y wheels ev en though eac h individual join t has a limited strok e. Generating angular momen tum using join ts with limited strok e As unlimited angular rota- tion seems out of reac h with exures, other t yp es of motion need to b e considered in order to obtain constan t angular momen tum. T w o t yp es of motion ha v e b een found to satisfy this condition. The rst is a rigid b o dy in translation on a circular tra jectory (Fig. 3). As the b o dy do es not rotate around its cen tre of mass, i.e. k eeps the same orien tation, it can b e considered as a p oin t mass. This t yp e of motion is of in terest since, when ev aluated at the cen tre of the orbit, the angular momen tum is constan t. This circular motion is implemen table with exures b y using planar t w o-degree-of-freedom (2-DOF) mec hanisms suc h as the XY-stage sho wn in Fig.2a. The issue with this single p oin t mass is that the cen tre of mass is not xed, whic h results in shaking forces exp orted to the spacecraft. The solution is to com bine p oin t masses in order to k eep the cen tre of mass xed while still conserving the o v erall angular momen tum (Fig. 4). 1Some progress in this direction seems to ha v e b een made using the Dirac b elt tric k [20] [21]. 6 Instan t Lab | EPFL 1 INTR ODUCTION AND ST A TEMENT OF RESUL TS Figure 3: The dieren t steps of a rigid b o dy in translation on a circular orbit �� �� Figure 4: Flexure Wheel concept consisting in com bining circular motions whic h results in an o v erall constan t angular momen tum. The second t yp e of motion that can also replace unlimited angular rotation is a rigid b o dy rotating in circles around its cen tre of mass. An example of suc h a motion is a h uman ey e trac king the edge of a circle: the ey e is alw a ys in motion ev en though its strok e is limited (Fig. 5). Note that ey e mo v emen ts are essen tially 2-DOF spherical rotations, see [15]. In this case, sev eral of the requiremen ts are already satis ed: the cen tre of mass remains xed and the angular momen tum has a constan t amplitude. Ho w ev er, the angular momen tum p oin ts in dieren t directions throughout the mo v emen t. Similarly as to the previously discussed planar mec hanisms, com bining these spherical mec hanisms prop erly enables angular momen tum to b e constan t (Fig. 6). eyeeye's line of sight tracked circle Figure 5: The dieren t steps of an ey e trac king the edge of a circle. 7 Instan t Lab | EPFL 2 FLEXURE WHEELS F OR SP A CECRAFT A TTITUDE CONTR OL Figure 6: Flexure Wheel concept consisting in com bining spherical mec hanisms whic h results in an o v erall constan t angular momen tum. 1.4 Description of demonstrators and results T w o demonstrators w ere fabricated. They b oth use masses translating on circular orbits and rely on dieren t com binations of mass tra jectories. The rst demonstrator uses exclusiv ely masses on circular orbits whic h all con tribute to the angular momen tum while their com bination k eeps the cen tre of mass xed. The second demonstrator mixes t w o classes of motion: a mass on a circular orbit whic h generates the angular momen tum and a pair of masses oscillating linearly . The latter do not con tribute to the angular momen tum, but correct the cen tre of mass so that it remains xed for the en tire device. The protot yp es w ere tested in a pro of-of-concept exp erimen t whic h mimics the absence of gra vit y of outer space. The setup consists in susp ending the exure wheel on the end of a string and the frame holding the exure wheel can b e seen as the satellite whic h the exure wheel m ust set in to rotation (Fig. 7). The system w as initially at rest and as the exure wheel w as put in to motion, the en tire system rotated successfully . (a) Demonstrator protot yp e 1 (b) Demonstrator protot yp e 2 Figure 7: Ov erview of the t w o demonstrators 2 Flexure Wheels for spacecraft attitude con trol 2.1 A ttitude con trol for spacecrafts A ttitude con trol can b e de ned as the pro cess of con trolling the orien tation of an aerospace v ehicule in space [2]. This is an essen tial asp ect of a space mission as it is required for exmaple for correct 8 Instan t Lab | EPFL 2 FLEXURE WHEELS F OR SP A CECRAFT A TTITUDE CONTR OL orien tation of instrumen ts suc h as a telescop e, an tennas and solar panels as w ell as mano euv ering t w o spacecrafts for do c king. A ttitude con trol is separated in to t w o main categories: spin stabilisation and three-axis stabilisation. Spin stabilisation consists in setting the spacecraft spinning, using the gyroscopic action of the rotating spacecraft mass as the stabilizing mec hanism. Three-axis stabilisation consists in holding the spacecraft in the desired orien tation without an y rotation. [2] W e will fo cus on the three-axis stabilisation as it is more relev an t for the presen ted mec hanisms. T yp es of actuators The main t yp es of actuators for attitude con trol are:  Thrusters: basically small ro c k ets whic h exp el matter and use the principle of action/reaction to orien t the spacecraft. This is the same principle as in a jet engine or in Hero's aeolipile (a radial steam turbine that dates bac k to appro x. 100BC). If w e consider the system consisting of the spacecraft and the matter to b e exp elled as isolated (i.e no external forces act up on the system), the momen tum is conserv ed. Once the matter is exp elled in one direction, to conserv e momen tum, the spacecraft mo v es in the opp osite direction. A single actuator therefore deliv ers a force. T o generate a pure torque, a pair of thrusters m ust b e used. These actuators can deliv er a large amoun t of torque but cannot w ork inde nitely as fuel runs out and they ha v e limited precision of stabilit y .  Reaction wheels/momen tum wheels: these electrically p o w ered wheels whic h are put in to rotation at a con trolled sp eed also use the principle of action/reaction, but for rotation. Indeed, b y again considering the spacecraft and reaction wheel system to b e isolated, the angular momen tum is conserv ed. Therefore if the reaction wheel is accelerated (with resp ect to the spacecraft) in one direction, the spacecraft will rotate in the opp osite direction. [2]  Magnetic torquers: Coils or (on v ery small satellites) p ermanen t magnets exert a torque against [Earth's] lo cal magnetic eld [2]  Con trol momen t gyroscop es: These are similar to the reaction wheel, but instead of v arying the sp eed of the wheel, the wheel is spun at a constan t sp eed and direction of its axis is c hanged. This pro vides a gyroscopic eect. The maxim um torque is higher than for reaction wheels, but the design is more complex, more exp ensiv e and hea vier. These device are therefore usually used on larger satellites, suc h as the In ternational Space Station. [2]  Solar sails: pro duce thrust b y the reaction force of re ecting inciden t ligh t. [2]  Gra vit y-gradien t stabilisation: for a long enough satellite, the gra vitational eld is not assumed to b e constan t, but a gradien t. This allo ws the satellite to stabilise itself with its long axis p oin ting to w ards Earth. Indeed, its lo w er end feels a stronger attraction than the upp er end. [2] The spacecraft is not an en tirely closed system: it receiv es some external forces due to solar radiation pressure, gra vit y gradien ts, geo-magnetic torque and aero dynamic drag [9] [6]. Excess momen tum can build up in the spacecraft. In the case of reaction wheels, the consequence is that the wheels will ev en tually reac h their maxim um sp eed and will no longer b e able to con trol the attitude. This calls for a moment desatur ation mano euvre in whic h thrusters are turned on to bring the reaction wheels bac k to a normal sp eed [2]. Numerical v alues for these external torques are giv en in [9] and [6] for a nano-satellite resp. pico- satellite. Their total con tribution are in the order of 105[Nm]. 9 Instan t Lab | EPFL 2 FLEXURE WHEELS F OR SP A CECRAFT A TTITUDE CONTR OL 2.2 State of the art for reaction wheels 2.2.1 Bearing t yp es The curren t reaction wheels use an actually wheel (i.e. a mass in angular rotation around one of its principal axes of inertia). This unlimited angular rotation requires b earings for whic h t w o main solutions curren tly exist: the standard ball b earings and the more elab orate and complex magnetic b earings. Ball b earing reaction wheels The great ma jorit y of to da y's reaction wheels use ball b earings. This solution is relativ ely simple and c heap, but has the dra wbac ks of friction, lubrication and w ear whic h limit its lifetime and reliabilit y . Moreo v er, if the wheel sp eed passes b y zero, stiction aects the spacecraft stabilit y [4] [5]. As further explained in Sect. 2.2.2.2, ball b earing reaction wheels are also resp onsible for micro vi- bration in satellites, either due to an im balance of the wheel (i.e. the axis of rotation is not aligned with the cen tre of mass) or due to unev en racew a ys [4] [5]. Magnetic b earing reaction wheels The alternativ e solution is to use magnetic b earings in whic h the wheel axis is k ept in place with magnetic levitation. F or this reason, they ha v e no matter-on-matter friction and o v ercome the issue of stiction when sp eed passes b y zero. They are also capable of reac hing m uc h higher sp eeds than the ball b earing solution. [5] These t yp es of b earing normally ha v e activ e con trol to main tain the wheel axis in the correct p osition. This requires sensing and con tin uous p o w er input. This activ e con trol do es ha v e its adv an tages, it can for example adjust the axis of rotation to eliminate the im balance. This in v olv es ensuring that the axis of rotation passes b y the wheel's cen tre of gra vit y and that it is parallel to the wheel's principal axis [5]. Moreo v er, gim balling of the wheel is ev en p ossible whic h leads to gyroscopic forces. This means the all three spacecraft axes ma y b e con trolled b y just one wheel [11]. 2.2.2 Issues with ball-b earing reaction wheels 2.2.2.1 F ailures and mission impact In the case of ball b earings, reaction wheels ha v e a limited mec hanical lifetime. Indeed, they are sub ject to w ear and to electrostatic disc harge (due to space w eather) [7]. In the past, sev eral reaction wheel failures ha v e b een rep orted in missions among whic h Ha y abusa, Kepler and Da wn. These can signi can tly compromise the mission as the satellite loses its abilit y to orien t itself [2]. 2.2.2.2 Micro-vibrations in reaction wheels The satellite p oin ting stabilit y is a critical asp ect of a mission for earth observ ation, space astronom y and adv anced scien ti c missions as it directly reduces the capabilities of optical and imaging systems [4]. One of the main dra wbac ks of ball-b earing reaction wheels is the induced micro-vibrations due to man ufacturing and assem bly tolerances. Indeed, reaction wheels are a ma jor source of disturbance and ha v e b een studied and describ ed in detail b y Le [4]. Le lists three approac hes to reduce these micro-vibrations: deal with the initial ph ysical cause, add a damp er and use activ e con trol. (The latter is more adapted for activ ely con trolled magnetic b earings). 10 Instan t Lab | EPFL 2 FLEXURE WHEELS F OR SP A CECRAFT A TTITUDE CONTR OL The ph ysical causes of the micro-vibrations are separated in three categories: rotor un balances, ball b earing vibration and torque ripple. Rotor un balances are caused b y the misalignmen t b et w een the wheel's axis of rotation and its principal axis of inertia. Ball b earing vibration is caused b y imp erfections of the b earings' balls, racew a ys or cages. T orque ripple is caused b y the switc hing of the motor's coils (w e will no longer consider this eect as it is smaller than the others). One last issue with reaction wheels is the need for magnetic cleanliness. 2.2.2.2.1 Rotor un balance Rotor un balances is kno wn as one of the ma jor disturbing factors of micro-vibrations. The rotor un balances come under t w o forms:  Static un balance is due to the axis of rotation not passing b y the wheel's cen tre of mass. This induces a cen trifugal force on the b earing, Fs=mrg!2, wherem is the wheel's mass, rg the distance b et w een the cen tre of mass and the axis of rotation and ! the wheel's angular v elo cit y . W e de ne as static unb alanc e Us=mrg whic h dep ends only on the wheel's dimensions on not on its sp eed. The units of the static un balance are therefore[kgm], but can also b e expressed in [gcm]. X Y�Principal axis of iner�aaxis of rota�on Figure 8: Static un balance  Dynamic un balance is due to a misalignmen t b et w een the rotation axis and the principal axis of inertia. This results in gyroscopic forces describ ed b y Eq. 5. In this case, ev en if the wheel rotates at a constan t rate, the system is con tin uously exp orting a torque whic h is p erp endicular to the eectiv e axis of rotation and whose direction turns with the wheel. 11 Instan t Lab | EPFL 2 FLEXURE WHEELS F OR SP A CECRAFT A TTITUDE CONTR OL � X YPrincipal axis of iner�aaxis of rota�on Figure 9: Dynamic un balance 2.2.2.2.2 Ball b earing vibration Ball b earing vibrations are caused b y imp erfections in the b earings' balls, cages and racew a ys. This is a w ell-studied eld as it is a general mec hanics question. As explained in [14], ball-b earing vibrations can b e split in to three groups dep ending on their origin: man ufacturing, structure and handling. Structure imp erfections consist in imp erfections and w a vi- ness whic h excite the resonances of the b earing and surrounding structures. The result is structural vibrations [4].  Man ufacturing: the parts constituting the ball b earings are nev er p erfectly man ufactured. This means that neither the balls nor the racew a ys are p erfectly round and smo oth, but presen t defects describ ed b y w a viness.  Structure: imp erfections and w a viness excite the resonances of the b earing and surrounding structures. The result is structural vibrations [4].  Handling: con taminations and imp erfections created during the assem bly pro cess or v eri cation tests are vibration sources due to handling . F or example, mec hanical impact during v eri cation tests suc h as random vibration and sho c k tests can create suc h imp erfections. [4] 2.2.2.3 Magnetic cleanliness in reaction wheels Satellites can b e equipp ed with highly sensitiv e measuremen t instrumen ts whic h are disturb ed b y the magnetic elds of the satellite itself. Reaction wheels, often accommo dated to meet p oin ting requiremen ts, are a kno wn source of disturbance to b oth the DC and A C magnetic elds [12]. The strategies to mitigate these disturbances include k eeping a distance b et w een reaction wheels and instrumen ts, adding m u-metal shielding and selecting the most similar magnets to build the motor. 2.3 Inno v ativ e designs of exure wheels Wh y wheels? The new mec hanisms that will b e presen ted ha v e a similar use and prop erties as reaction wheels and con trol momen t gyroscop es (CMGs) for the fact that they rely on rigid mec hanical parts in mo v emen t, do not exp el an y matter remain xed to the spacecraft and do not require an y forces external to the spacecraft. Indeed, the mec hanisms that will b e presen ted dier from magnetic torquers as these rely on 12 Instan t Lab | EPFL 2 FLEXURE WHEELS F OR SP A CECRAFT A TTITUDE CONTR OL electro-magnetic forces. They also dier from thrusters as they do not exp el matter. Last but not least, they dier from solar sails as they function without external forces (suc h as solar radiation pressure). W e therefore fo cus on reaction wheels and CMGs in the rest of this w ork. Moreo v er, for simplicit y w e will limit ourselv es to reaction wheels for the rest of the state of the art as CMGs also con tain rotating wheels and therefore suer from similar problems. The issues discussed for reaction wheels can th us b e extended to CMGs. W e ha v e ho w ev er not y et adressed a cen tral asp ect of reaction wheels and CMGs: wh y are wheels used in particular? The answ er comes from the fact that the spacecraft's orien tation m ust b e stable. This means that the spacecraft m ust not receiv e an y exp orted torque or forces from the attitude con trol system. If a constan t torque is exp orted from the attitude con trol system, the spacecraft will accelerate in rotation. If an alternating torque is exp orted, the spacecraft will shak e in rotation. In the case of forces, w e only consider alternating forces whic h will induce a shaking in translation as w ell as in rotation of the spacecraft (if the spacecraft's attitude con trol system's cen tres of gra vit y are not coinciden t). Moreo v er, the mec hanism m ust b e able to pro vide a constan t angular momen tum to comp ensate for the built up angular momen tum due to external forces suc h as solar wind or gra vit y gradien t. In order to translate these requiremen ts in to ph ysical terms, a brief in tro duction to systems me- c hanics is required. In tro duction to systems mec hanics: W e wish analyse the forces and torque that an attitude con trol mec hanism exp orts to the rest of the spacecraft. In order to ac hiev e this, w e assume the spacecraft to b e an inertial frame. This means that the spacecraft's orien tation and p osition are not aected b y the attitude con trol system. W e consider the attitude con trol mec hanism to b e a system of rigid b o dies. It consists in a set of N massesmi . The system's total mass is M and its cen tre of mass is lo cated b y v ector X . The v alues w e are in terested in are the exp orted force Fexp and torque Mexp that the system applies to the spacecraft. In an arbitrary reference frame, the angular momen tum ~Lsys the en tire system is giv en b y: L=MX_X+Lg . (1) This equation has t w o terms. The rst is dep enden t on the frame's p osition with resp ect to the solid's cen tre of mass X , its time deriv ativ e _X and the solid's mass M . The second term of the equation is Lg , the angular momen tum ev aluated at the solid's cen tre of mass X . It is de ned in terms of  whic h is the p osition of eac h mass elemen t dm in the frame of the cen tre of gra vit y: Lg=Z _dm . (2) W e do not wish a reaction wheel to exp ort forces and therefore its cen tre of mass m ust remain xed. Indeed, the exp orted force Fexp is giv en b y: Fexp=MX . (3) T o ha v e no exp orted force, w e therefore ha v e X= 0 . Moreo v er, it w ould mak e no sense that X= 0 and _X6= 0 as this w ould lead to the wheel forev er drifting a w a y from the spacecraft at a constan t v elo cit y . W e therefore also ha v e _X= 0 , whic h pro v es that the cen tre of mass m ust remain xed in order to exp ort no forces to the spacecraft. 13 Instan t Lab | EPFL 2 FLEXURE WHEELS F OR SP A CECRAFT A TTITUDE CONTR OL In the case w e are in terested in where _X= 0 , w e ha v e L=Lg . (4) In the case of a standard reaction wheel, the system consists in a single solid part in angular rotation around its cen tre of mass. This enables Lg to b e written as Lg= ! , where  is the three b y three inertia tensor. As explained in [1] the di cult y in this equation is that, in the inertial frame,  c hanges as the solid rotates. F or this reason it is easier to place ourselv es in the non-inertial frame of the solid itself, in whic h  is constan t. The consequence is that the exp orted torque Mexp is sligh t y more elab orate: Mexp=_L=_Lg=  _!+!! (5) The last step is in tro ducing the fact that the wheel rotates around one of its principal axes of inertia. This means that ! is an eigen v ector of  and therefore !!=!!=0 . This leads to the simple and w ell kno wn equation: Mg=_Lg=  _! . (6) If the rotation w ere not along a principal axis of inertia, the wheel w ould exp ort an additional torque. Moreo v er, this torque lies in the plane p erp endicular to the axis of rotation and turns with the solid. Stabilit y and accuracy of orien tation as w ell as lifetime are k ey asp ects of an attitude con trol system. The stabilit y and accuracy of orien tation are impacted b y the micro-vibrations and misalignmen t in ball b earings. The lifetime is also determined b y the ball b earings. Moreo v er, assem bling a reaction wheel is elab orate as the system m ust b e encapsulated in a lo w pressure atmosphere prev en t degassing of lubrican t. It can therefore b e understo o d that these b earings are a critical and limiting factor in reaction wheels. F or these reasons, the goal of this w ork is to pro vide new exure mec hanism designs capable of pro viding attitude con trol. The main c hallenge lies in the fact that the essence of a wheel is unlimited angular rotation. This b eha viour ob viously cannot b e directly translated in to exures as they ha v e a limited range of motion. W e therefore need an articulated structure in whic h eac h join t has a limited range of motion and whic h, for an external observ er, b eha v es lik e a wheel. T o b e able to pro vide attitude con trol, the mec hanism cannot exp ort an y forces to the spacecraft and m ust b e able to exp ort a con trolled torque to the spacecraft. Moreo v er, the mec hanism m ust b e able to ha v e a constan t angular momen tum in order to k eep the spacecraft in a xed orien tation ev en in presence of external forces. Indeed, these external forces cause a build-up of angular momen tum to the whole spacecraft whic h m ust b e comp ensated b y the attitude con trol system. T w o exp ort torque to the rest of the spacecraft, the mec hanism's angular momen tum m ust v ary . This can ac hiev ed b y c hanging its amplitude, its direction, or b oth. The actuation m ust also b e frictionless in order to a v oid w ear, the release of particles and the need for lubrication. The use of a standard rotating motor is therefore prohibited, as suc h a motor is in itself a wheel (with ball b earings) and therefore has the issues w e are in fact trying to solv e. W e can no w summarize the mec hanisms' sp eci cations. Summary of mec hanism sp eci cations 1. No exp orted forces 2. Capable of pro viding angular momen tum that is constan t in direction and in amplitude 3. Capable of mo difying the amplitude of angular momen tum, either in amplitude or direction 4. The actuation m ust b e frictionless 14 Instan t Lab | EPFL 2 FLEXURE WHEELS F OR SP A CECRAFT A TTITUDE CONTR OL Outline and o v erview of found mec hanisms and metho ds of use: W e start b y examining mec hanisms that satisfy conditions 1 and 2 (whic h is equiv alen t to a wheel rotating at constan t sp eed). W e then discuss condition 3 with dieren t strategies to mo dify the angular momen tum. Detailed designs are then presen ted whic h allo ws the actuation of condition 4 to b e dealt with. T w o main classes of mec hanisms ha v e b een in v en ted: planar and spherical mec hanisms. The planar mec hanisms consist in a set of p oin t masses eac h that collectiv ely ha v e a con trollable angular momen tum. The spherical mec hanisms are based on a set of solids that rotate around their cen tre of gra vit y in an alternate manner. By com bining these solids, a total angular momen tum with a constan t direction is obtainable. The in v en ted mec hanisms ha v e t w o metho ds of use. A rst in whic h the mec hanism is considered as a t w o-degree-of-freedom (2-DOF) oscillator. In this case the mec hanism mo v es at its natural frequency and the angular momen tum is c hanged b y adjusting the oscillation amplitude (whic h is equiv alen t to mo difying the momen t of inertia of the system). The second metho d of use consists in constraining the mec hanism to circular orbits of xed amplitude. This reduces its degrees of freedom to one. In this case, the mec hanism do es not oscillate at a xed frequency , but is free to follo w its orbit at an y arbitrary sp eed. 2.3.1 Abstract designs of new mec hanisms 2.3.1.1 Planar mec hanisms 2.3.1.1.1 In tro ductory example T o in tro duce the principle of these mec hanisms, w e will rst giv e a sp eci c design as an example. W e will start from a usual wheel in rotation and gradually bring mo di cations that do not c hange its b eha viour from a external p oin t of view (i.e. that do not c hange its exp orted forces and torque). The starting p oin t is a standard rotating wheel as sho wn in Fig. 10. XY XY �� Figure 10: F ron t and isometric views of a wheel in rotation. W e no w consider the mass of the wheel to b e concen trated at t w o opp osite sides of the wheel, represen ted b y the dark circles in Fig. 11. 15 Instan t Lab | EPFL 2 FLEXURE WHEELS F OR SP A CECRAFT A TTITUDE CONTR OL XY XY �� Figure 11 Since the mass is only con tained the t w o dark circles, the rest of the wheel can b e remo v ed and replaced with a simple lev er linking the t w o masses, as in Fig. 12. XY XY �� Figure 12 W e can ev en remo v e this lev er and w e are left with t w o indep enden t masses at opp osite ends of a circular tra jectory , as in Fig. 13. XY XY �� Figure 13 In this con guration, w e remo v ed the requiremen t for con tin uous angular rotation. This is the k ey step that enables exures to b e in tro duced. A reaction wheel can therefore b e replaced b y a mec hanism in whic h t w o p oin t masses are at opp osite ends of a same circular tra jectory . 2.3.1.1.2 F ormalisation of planar mec hanisms This rst example in tuitiv ely brough t to ligh t ho w a rotating wheel can b e replaced b y a set of p oin t masses. T o pro ceed in a more systematic and analytical manner, it is necessary to tak e step bac k to understand ho w p oin t masses aect a system's cen tre of mass and angular momen tum through their resp ectiv e mo v emen ts. These consideration will ev en tually pro vide building blo c ks whic h, when put together, are able to pro vide mec hanisms suc h as the previous example whic h can replace a reaction wheel. In trinsic angular momen tum: Dep ending on the tra jectory that a p oin t mass has, its angular momen tum ma y or ma y not b e zero dep ending at the p oin t at whic h it is ev aluated. In other w ords, for 16 Instan t Lab | EPFL 2 FLEXURE WHEELS F OR SP A CECRAFT A TTITUDE CONTR OL some tra jectories it is p ossible to nd a p oin t of observ ation at whic h the angular momen tum is zero at all times while for other tra jectories it is imp ossible to nd suc h p oin ts of observ ation. In the case where it is imp ossible to observ e zero angular momen tum, w e de ne the p oin t mass as ha ving intrinsic angular momentum . A Omv r a L =mrv+mav cos( ) = constAvrt OL =mrv =const Figure 14: System with in trinsic angular momen tum consisting of a mass m on a uniform circular motion. An example of tra jectory that has in trinsic angular momen tum is a mass with a clo c kwise uniform circular motion as sho wn in Fig. 14. In this system, a mass m on a uniform circular motion of radius r at v elo cit y v . When ev aluating the angular momen tum LA from an arbitrary p oin t A at a distance a from the orbit's cen tre O ,LA is non-zero, though not constan t (see app endix for complete calculation). Ho w ev er, when ev aluating from O , the angular momen tum LO is constan t and non-zero. An imp ortan t remark is that this mass on a circular orbit cannot, on its o wn, replace a reaction wheel since its cen tre of mass (the mass itself ) is not xed; it follo ws the same circular orbit. An example of tra jectory that do es not ha v e in trinsic angular momen tum is that of a mass oscillating bac k and forth on a straigh t line as sho wn in Fig. 15. Indeed, ev en if the angular momen tum LA observ ed at p oin t A is equal to mrv , when observ ed from p oin ts B or C, the angular momen ta LB and LC are b oth zero at all times. mvC BA r L =mrv = const L =0L =0 AC B Figure 15: System without in trinsic angular momen tum consisting of a mass m mo ving in a straigh t line at v elo cit y v . A last example is t w o masses that are eac h oscillating on their resp ectiv e linear tra jectories whic h are at 90 ° as sho wn in Fig. 16. The t w o masses m are lo cated at p oin ts A andB and their oscillation amplitudes R are equal. Moreo v er these oscillations ha v e a relativ e phase shift of 90o. The cen tre of 17 Instan t Lab | EPFL 2 FLEXURE WHEELS F OR SP A CECRAFT A TTITUDE CONTR OL mass is therefore lo cated at p oin t C whic h follo ws a uniform circular motion of radius R=2 (an ticlo c kwise in this illustration). Concerning the angular momen ta, when observing at p oin ts D andE , their angular momen ta are non-zero due to masses A resp ectiv ely B . Ho w ev er, when observing from p oin t O , the angular momen tum is alw a ys zero and therefore the system has no in trinsic angular momen tum y xvB vAA=R cos(�t)xB=R sin(�t)y C= (cos(�t)x+sin(�t)y)R 2 O QP L =0O L =0QL =0P R/2 R� Figure 16: System without in trinsic angular momen tum consisting of t w o iden tical masses oscillating linearly around a same p oin t. W e ha v e no w pro vided the necessary building blo c ks to create sev eral equiv alen ts of a reaction wheel using p oin t masses. W e ha v e indeed sho wn the circular orbit in whic h the angular momen tum can b e constan t, but whic h suers from a mo ving cen tre of mass. W e ha v e also sho wn the last example whic h has a mo ving cen tre of mass but with no angular momen tum. It is then easy to understand that sev eral of these mec hanisms can b e com bined to reac h a xed cen tre of mass while still ha ving angular momen tum. Sev eral com binations exist. A rst example is sho wn in Fig. 17 in whic h the p oin t mass from Fig. 14 (with in trinsic angular momen tum) is com bined with the system of Fig. 16 (without in trinsic angular momen tum). The mass with uniform circular motion pro vides the angular momen tum but suers from a mo ving cen tre of mass. The t w o masses comp ensate the cen tre of mass displacemen t without adding angular momen tum. When these t w o subsystem are com bined, the total system therefore has a xed cen tre of mass and angular momen tum L that is constan t in b oth amplitude and direction. Another p ossible com bination is taking t wice the p oin t mass of Fig. 14 so that the displacemen ts of their resp ectiv e cen tres of mass comp ensate eac h other while their angular momen ta are added. This is in fact the underlying approac h that leads to the in tro ductory example (Fig.13). 18 Instan t Lab | EPFL 2 FLEXURE WHEELS F OR SP A CECRAFT A TTITUDE CONTR OL y xO R/2 R� y xO R/2 R�y xO R/2 R� �+ = Mass (and centre of mass) on circular orbitCentre of mass on circular orbit Centre of mass is fixed Figure 17: Equiv alen t of a reaction wheel (righ t) consisting in a mass on a circular orbit (left) com bined with a set of t w o masses oscillating around the cen tre of this orbit (cen tre). 2.3.1.2 Spherical mec hanisms A critical design criterion in spatial applications is optimising the amoun t of angular momen tum can b e stored with resp ect to the mec hanism's w eigh t and v olume. In the planar mec hanisms, this requires the circular orbits of the masses to b e as large as p ossible or the mass sp eed to b e increased. There are no w a ys of optimising the shap e or arrangemen t of these p oin t masses as their eect relies directly on their eectiv e mass. T o o v ercome this limitation of using p oin t masses in translation, solid masses are considered. In this case, a rigid b o dy whose mo v emen t is not restricted to translation within a plane, but to rotation around its cen tre of gra vit y . The adv an tage of this approac h is that it is not the solid's mass that is resp onsible for the angular momen tum, but its momen t of inertia. By using a solid whose mass is distributed as far as p ossible from the cen tre of rotation, it is p ossible to obtain a large momen t of inertia (and therefore a large angular momen tum) for a ligh t mec hanism. Circular orbits for spherical mec hanisms Since these mec hanisms will b e implemen ted with exures, the solid in rotation has a limited range of motion. It is ho w ev er p ossible for suc h an ob ject to come close to con tin uous motion b y follo wing a closed circular orbit. W e will use the analogy of a planet to describ e the orien tation of the solid. W e assume that, at rest, the 'north p ole' p oin ts up w ards along thez -axis. The circular orbit previously men tioned in v olv es the north p ole mo ving on a circular orbit around the z -axis (the planet's axis b et w een its north and south p oles therefore cuts the z -axis with a constan t angle). The dieren t steps of suc h a circular motion are sho wn in Fig. 18 in whic h a cardan join t pro vides the motion of the solid part. In this gure, one end of the cardan join t is the ground and the other, whic h corresp onds to the north p ole, follo ws the said circular motion. 19 Instan t Lab | EPFL 2 FLEXURE WHEELS F OR SP A CECRAFT A TTITUDE CONTR OL (a) (b) (c) (d) (e) (f ) Figure 18: Cardan join t follo wing a circular orbit. Angular momen tum of a spherical mec hanism on a circular orbit A t an y momen t of the circular orbit, the solid's instan taneous angular v elo cit y is a v ector that lies within the plane describ ed b y thez -axis and the north p ole. Indeed, since the tilt angle of the planet from the z -axis do es not c hange, there is no comp onen t of the rotation that exits the said plane. Moreo v er, as sho wn in Sect. 4.3, if the solid's kinematics are go v erned b y a constan t-v elo cit y join t (CV join t) , the angular v elo cit y will remain the same throughout the orbit (it will naturally rotate with the solid). The trace of the angular v elo cit y will therefore describ e a cone whose ap ex is the solid's cen tre of rotation. Indeed, the angular momen tum v ector, when departing from the solid's cen tre of rotation, p oin ts on to a circle that lies in a plane parallel to the xy -plane and cen tred on the z -axis. Also, since the angular momen tum is prop ortional (and colinear) to the angular v elo cit y , the angular momen tum also describ es a similar cone. Conclusion: The angular momen tum of a spherical mec hanism (with CV join t) on circular motion has a constan t amplitude, a constan t z -comp onen t, but c hanges direction in the xy -plane. The ev olution of angular momen tum with resp ect to time is sho wn in Fig. 19 whic h sho ws the cardan in dieren t con gurations and the corresp onding angular momen tum. The plane con taining the angular momen tum, the north p ole and the z -axis is also sho wn. Note: a cardan join t is not a CV-join t, but it has nev ertheless b een used for illustrativ e purp oses as it is more visually understandable than other join ts. 20 Instan t Lab | EPFL 2 FLEXURE WHEELS F OR SP A CECRAFT A TTITUDE CONTR OL (a) (b) (c) (d) (e) (f ) Figure 19: Same cardan as Fig. 18 follo wing a circular orbit. The angular momen tum is sho wn along with the circle on to whic h it p oin ts. The plane con taining the angular momen tum and the solid's north p ole is also sho wn. Com bining spherical mec hanisms to obtain a constan t angular momen tum W e will no w restrict ourselv es to the case of CV join ts. T o mak e the dra wings clearer, w e also only represen t the dieren t solids in rotation b y their resp ectiv e angular momen ta. An example of this represen tation is Fig. 20 whic h sho ws the angular momen tum for eac h step as w ell as the cone that is sw ept. xyz (a) xyz (b) xyz (c) xyz (d) xyz (e) xyz (f ) Figure 20: The dieren t steps of angular momen tum for a spherical mec hanism on a coun ter-clo c kwise circular orbit. The angular momen tum is dotted when at the rear side of the cone. The goal of these exure wheel b eing to generate a constan t angular v elo cit y , w e ask ourselv es the follo wing question: Ho w can w e isolate the angular momen tum's constan t z -comp onen t while remo ving 21 Instan t Lab | EPFL 2 FLEXURE WHEELS F OR SP A CECRAFT A TTITUDE CONTR OL the alternating xy -comp onen ts? A solution is to com bine spherical mec hanisms so that their resp ectiv e angular momen ta will (v ectorially) summed. The simplest com bination consists in t w o iden tical spherical mec hanisms that are at opp osite ends of their resp ectiv e orbits. Since these t w o mec hanisms are in an ti-phase, their angular momen ta ha v e the samez -comp onen t, but opp osite x - andy -comp onen ts. Since these individual angular momen ta are v ectorially summed, the total angular is therefore constan t in direction and amplitude along the z -axis and a exure wheel has th us b een created. Fig. 21 illustrates this case in whic h the rst spherical mec hanism's angular momen tum and cone is sho wn along with angular momen tum of the second spherical mec hanism. It can b e seen that throughout the dieren t steps the v ectorial sum of the angular momen ta remains constan t along the z -axis. xyz (a) xyz (b) xyz (c) xyz (d) xyz (e) xyz (f ) Figure 21: The dieren t steps of angular momen ta of t w o spherical mec hanisms com bined in order to ac hiev e an o v erall constan t angular momen tum along the z -axis. Conclusion: A exure wheel can b e created with t w o iden tical spherical mec hanisms at opp osite ends of their circular orbit Some additional considerations ma y b e noted concerning the com bination of spherical mec hanisms:  It is p ossible to com bine an y n um b er of spherical mec hanisms, as long as the x - andy -comp onen ts of their angular momen ta cancel eac h other out. F or example, three spherical mec hanisms whic h ha v e a 120owith resp ect to eac h other  It is not strictly necessary to ha v e iden tical spherical mec hanisms on circular orbits of iden tical amplitude. The only requiremen t is that summed xy -comp onen ts of the angular momen tum is zero. It then b ecomes imaginable to use spherical mec hanisms with dieren t momen ts of inertia (whic h c hanges the cone amplitude) and dieren t orbit amplitudes (whic h c hanges the cone angle).  The spherical mec hanisms can b e arbitrarily placed with resp ect to one another. Indeed, as a spherical mec hanism's cen tre of mass is xed, the exp orted forces (whic h are zero) and exp orted torque do not dep end on its lo cation. This is similar to a standard reaction wheel whic h can b e placed an ywhere on the spacecraft. 2.3.2 Metho ds of use It will no w b e discussed ho w these abstract designs can b e used to pro vide attitude con trol. This will b e done in t w o steps: a rst whic h presen ts ho w a constan t angular momen tum can b e obtained and a 22 Instan t Lab | EPFL 2 FLEXURE WHEELS F OR SP A CECRAFT A TTITUDE CONTR OL second step whic h discusses ho w this angular momen tum can b e c hanged in order to exp ort the desired torque. 2.3.2.1 Obtaining constan t angular momen tum A k ey asp ect of all presen ted mec hanisms is that they follo w circular motion, whic h is not normally ac hiev ed with exures. Indeed, the t yp e of motion obtained with exures is usually bac k-and-forth, whether it b e rotational or rectilinear. T w o main metho ds to enable suc h circular motion ha v e b een found. The rst consists in creating a new t yp e of exures that restrict motion to a circle, whic h are de ned as Cir cular Motion Flexur es (CMF s). The second option consists in using mec hanisms that ha v e t w o degrees-of-freedom, but only using them as one-degree-of-freedom. Mec hanically restricting the motion to circular orbits with Circular Motion Flexures This metho d consists in design a no v el t yp e of exures that restrict the motion to a circle. Suc h a exure is therefore de ned as Cir cular Motion Flexur e (CMF). The design is based on the buc kling of a b eam with a circular cross-section, as sho wn in Fig. 22. Due to the circularit y of the cross-section, the buc kling can tak e place in an y direction. The cen tre of the b eam, whic h de ects to the side, is therefore restricted to a circle. Middle of beam restricted to circular mo�onBuckling of beamBeam with circular cross sec�on Figure 22 By observing this system in more detail, it can b e seen that if a mass is at the middle of the buc kled b eam, this mass has an additional three degrees-of-freedom in rotation since the t w o half-b eams linking the mass to the ground ha v e in ternal degrees of freedom. Moreo v er, this system lac ks radial stiness. Indeed, if a radial force is applied, suc h as the cen trifugal force if the mass is in circular motion, the size of the orbit increases. T o solv e this issue, a design that uses rigid parts and lo calized exible elemen ts is prop osed. The design is sho wn in Fig. 23 in whic h t w o rigid segmen ts are link ed together b y a spherical join t (three degrees-of-freedom)and are link ed to the ground b y CV-join ts (t w o degrees-of-freedom). Note: Other com binations of join ts are p ossible, as long as no h yp erstatisms are induced. 23 Instan t Lab | EPFL 2 FLEXURE WHEELS F OR SP A CECRAFT A TTITUDE CONTR OL BucklingCV-joint (2DOF) CV-joint (2DOF)Spherical joint (3DOF) Figure 23 Using the mec hanisms as isotropic 2-DOF oscillators A mass needn't b e ph ysically re- stricted to a xed circular orbit in order to follo w a circular motion. Indeed, the mec hanism has t w o degrees-of-freedom and is implemen ted with exures whic h pro vide a restoring force. If this restoring force is isotropic, the mec hanism can b e seen as an isotropic 2-DOF harmonic oscillator. In this case, when oscillating freely , the mass's tra jectory is an ellipse and its p erio d is indep endan t of its amplitude (Fig. 24). A circular motion is then nothing more than an ellipse with the righ t prop ortions. Figure 24: Illustration of an isotropic 2-DOF harmonic oscillator. [16] These prop erties ha v e b een adv an tageously used for their time-k eeping applications in the IsoSpring pro ject (see [16]). Indeed, as the mass follo ws a circular motion at a constan t rate, it can b e directly link ed to the rest of the gear train b y means of a crank in a mec hanical clo c k or w atc h (Fig. 25). 24 Instan t Lab | EPFL 2 FLEXURE WHEELS F OR SP A CECRAFT A TTITUDE CONTR OL Figure 25: Illustration of the crank mec hanism used to driv e the 2-DOF oscillator. [16] In this case, increasing the angular momen tum it is not done b y mo difying the angular frequency (as it is giv en b y the oscillator's natural frequency), but only the orbit's amplitude. This is equiv alen t to increasing the system's momen t of inertia. Sev eral practical issues arise when using the mec hanisms as 2-DOF oscillators due to asymmetries in the mec hanism. Indeed, if the restoring force is not p erfectly isotropic (inheren tly presen t in man ufac- turing and assem bly), the orbit will not b e p erfectly circular nor elliptic. If left to itself, the tra jectory will degenerate in to patterns suc h as Lissa jous gures. Moreo v er, the restoring force is no longer cen- tral and therefore creates a torque (when ev aluated at the neutral p osition) whic h is exp orted to the spacecraft. F orcing circular orbits with an activ e con trol This approac h is similar to the previous metho d, but b y ha ving the mass go at a rate dieren t from its natural frequency b y means of actuation. This case can also b e seen as a harmonic oscillator, but where the actuation pro vides additional stiness whic h c hanges the frequency . This means that throughout the motion the actuation is alw a ys applying force or torque. This is a serious disadv an tage as it means that the system is con tin uously consuming p o w er, whic h is scarce is spatial applications. Ho w ev er, if the actuation guides the mass at its natural frequency , it will only ha v e to mak e up for the small departures from the ideal circular tra jectory due the mec hanism's asymmetries, whic h w ould not requires a large consumption. 2.3.2.2 Exp orting torque T o remain as general as p ossible, a v ector, in this case the angular momen tum, can see itself c hange in amplitude, in direction, or a com bination of b oth. Both scenarios will b e separately discussed with a fo cus on the c hange of amplitude, for whic h t w o sub-metho ds exist. 2.3.2.2.1 Exp orting a torque b y c hanging the angular momen tum's amplitude T w o mec hanism categories capable of pro viding constan t angular momen tum ha v e b een sho wn: planar and 25 Instan t Lab | EPFL 2 FLEXURE WHEELS F OR SP A CECRAFT A TTITUDE CONTR OL spherical mec hanisms. In b oth cases, the angular momen tum dep ends on the orbit amplitude and v elo cit y . There are therefore t w o lev ers with whic h to adjust the angular momen tum. In the case of planar mec hanisms, w e can consider a single mass on a circular orbit without losing generalit y . The angular momen tum L is giv en b y: L=mrv (7) =mr2! (8) =I! , withI=mr2, (9) wherem is the mass, r its orbit radius, v its v elo cit y and ! its angular v elo cit y . In the case of spherical mec hanisms, w e assume the solid's mass distribution is spherically symmetric (i.e. that the momen t of inertia is the same for all rotation axes and can therefore b e giv en b y a scalar I ). Moreo v er, since spherical masses are com bined to remo v e the angular momen tum's alternating xy -comp onen ts, only the v ertical comp onen t Lz is considered and is giv en b y: Lz=Isin(L)! , (10) where! is the sphere's angular v elo cit y and L b eing the half angle of the cone formed b y the angular momen tum (its sine therefore b eing the v ertical pro jection). Increasing angular momen tum b y increasing v elo cit y Eq. 8 and 10 ha v e strong similarities in their form. They b oth con tain a rst factor that dep ends solely on the mass or inertia, a second factor relating to the orbit amplitude and a last on v elo cit y . This last factor pro vides the rst lev er with whic h to adjust the angular momen tum's amplitude. Indeed, in the case of planar mec hanisms, Eq. 7 and 8 sho w that angular momen tum is prop ortional to b oth v elo cit y and angular v elo cit y . F or spherical mec hanisms, the conclusion is iden tical. Increasing angular momen tum b y increasing momen t of inertia A second w a y to adjust the angular momen tum's amplitude is to pla y with the rst parts of Eq. 8 and 10. These dep end on the motion amplitude, whic h relates to the system's momen t of inertia. Indeed, for planar mec hanisms, Eq. 8 sho ws that the angular momen tum L increases quadratically with the orbit amplitude r . Ho w ev er, as highligh ted in Eq. 9, mr2simpli es to the momen t of inertia I of the p oin t mass. The amplitude of angular momen tum can therefore b e a jdusted b y mo difying the system's momen t of inertia. In the case of spherical mec hanisms, the reasoning is more subtle. Increasing the amplitude of the circular orbit results in a smaller angle of the cone sw ept b y the angular momen tum. This smaller angle leads to a more imp ortan t v ertical comp onen t of the angular momen tum (describ ed b y sin(L) ). This can b e seen as an increase of the system's momen t of inertia around the z -axis. 2.3.2.2.2 Exp orting a torque b y c hanging the angular momen tum's direction The previous section discussed the case where the angular momen tum's amplitude w as v aried in order to exp ort a torque. W e will no w examine the second manner to v ary this v ector: b y c hanging its direction. This option to exp ort a torque has already b een implemen ted in traditional attitude con trol systems, more sp eci cally in con trol momen t gyroscop es (CMGs). These systems consist in a wheel rotating at a constan t rate (usually b y means of ball b earings) whose direction is c hanged in order to exp ort the gyroscopic torque to the spacecraft. This option of exp orting torque has not b een further explored for sev eral reasons. First, at rst glance this w ould simply consist in one of the previously describ ed systems (with constan t angular 26 Instan t Lab | EPFL 2 FLEXURE WHEELS F OR SP A CECRAFT A TTITUDE CONTR OL momen tum) on gim bals. As designing these gim bals with exures is relativ ely simple, this option do es not bring an y extra no v elt y . Second, if the gim bals are used, the actuation will b e mobile as they are on the end of the gim bals, whic h complicates the implemen tation. 2.3.3 Detailed designs of new mec hanisms 2.3.3.1 Planar Mec hanisms Planar mec hanisms design has b een exensiv ely studied, particularly in the led of exure b ecause edm As discussed in Sect. 2.3.2, these planar mec hanisms can follo w circular orbits b y b eing used as 2-DOF oscillators or b y activ e con trol. A dding a Circular Motion Flexure is another option, but this requires an additional in termediate link age to a v oid o v er-constraining the mec hanism. Without losing generalit y , these mec hanisms will b e b e presen ted without sp ecifying whic h metho d (oscillations, activ e con trol or CMF) is used as all can b e implemen ted. 2.3.3.1.1 In v erter pan tographs A crucial asp ect of the abstract design is that masses are at opp osite ends of iden tical tra jectories. This b eha viour can b e understo o d as the t w o masses b eing sub ject to an in v ersion along b oth the x - andy -axes. A mec hanism capable of pro viding suc h prop ert y is the in v erting pan tograph. This mec hanism is based on the w ell kno wn pan tograph, mainly used for cop ying b y dra wing or engra ving. The mec hanism has t w o degrees-of-freedom and consists in an end eector whose motion is restricted to the xy -plane and an in termediate end eector whose motion is a p erfect scaled-do wn v ersion of the end eector's as sho wn in Fig. 26. The scale can b e adjusted, but only the ratio 1/2 is of in terest in this case. The transformation from this standard pan tograph to the in v erting pan tograph is ac hiev ed b y using the in termediate end eector as ground and the initial end eector and ground b ecome t w o coupled end eectors with in v erted motion, as sho wn in Fig. 27. Masses can then b e placed on the end eectors to genereate the angular momen tum. intermediate end-eﬀectorend-eﬀector Figure 26: Standard pan tograph mec hanism in its neutral p osition (left) and dis- placed con guration (righ t). It can b e seen that the t w o end eectors ha v e the same motion, but with a ratio of 1/2 27 Instan t Lab | EPFL 2 FLEXURE WHEELS F OR SP A CECRAFT A TTITUDE CONTR OL end-eﬀectors Figure 27: In v erter pan tograph mec hanism in its neutral p osition (left) and dis- placed con guration (righ t). It can b e seen that the t w o end eector ha v e p erfectly in v erted mo v emen t. If p oin t masses are placed on the end eectors, this mec hanism is not a Flexure Wheel as the orbits of the t w o masses do not coincide. Indeed, this is the same con guration as detailed in Sect. 4.2 and leads to an angular momen tum that is not constan t, but oscillates around a xed v alue (see Sect. 4.2 for complete calculation). T o solv e this issue, t w o solutions ha v e b een found: a rst in whic h solids are used instead of p oin t masses and a second in whic h in v erted pan tographs are com bined. Croissan ts, don uts and cup cak es The fact that the orbits of the t w o masses do not coincide can b e solv ed b y no longer using p oin t masses, but solids with particular shap es whic h ha v e their cen tres of mass placed outside of themselv es. This enables the t w o cen tres of mass on circular motion to b e arbitrarily placed while a v oiding collision, since there is no matter at the lo cation of the cen tre of mass. Suc h a design is sho wn in Fig. 28 in whic h t w o 'croissan t' shap ed solids pro vide the coinciding circular orbits. These croissan ts ha v e additional w eigh ts (the blac k circles) to bring the cen tre of mass to the desired p osition. Another design is sho wn in Fig. 29 in whic h a 'don ut' and -follo wing the pastry theme- 'cup cak e' shap ed solids are used. Both of these are circular and are arranged concen trically when the pan tograph is in its neutral p osition, making their cen tres of mass coincide. croissant-shaped solid centre of mass of each croissant coincideaddi�onal weight to posi�on the centre of mass Figure 28: In v erter pan tograph mec hanism (dotted line) in whic h the p oin t masses at the end eectors ha v e b een replaced b y croissan t-shap ed solids whose cen tres of mass are coinciden t. 28 Instan t Lab | EPFL 2 FLEXURE WHEELS F OR SP A CECRAFT A TTITUDE CONTR OL Figure 29: In v erter pan tograph with don ut and cup cak e shap ed masses. The don ut (ligh t grey) is xed to the pan tograph's left end eector while the cup cak e (dark grey) is xed to the righ t end eector. These t w o solids b eing concen tric, their cen tres of mass coincide. Using solids do es ho w ev er bring up another issue. Up to no w, the masses placed and the in v erting pan tograph's end eector w ere assumed to b e punctual. F or this reason, the orien tation of these masses w as irrelev an t as p oin t masses ha v e no size and therefore no inertia when rotating around themselv es. When considering solids xed to the pan tograph's end eectors, their orien tations will c hange throughout the motion, whic h leads to additional angular momen tum and p ossible collisions b et w een the masses (if not dimensioned prop erly). T o solv e this issue, a second in v erting pan tograph needs to b e added as w ell as additional branc hes of arms to main tain the masses' orien tation in a giv en direction. This branc h consists in arms that run parallel to those of the pan tograph whic h blo c ks the rotation of the solids, as sho wn in Fig. 30 and 31. Figure 30: Double in v erting pan tograph moun ted in parallel. The dashed part is the ground and the t w o masses are in gra y . 29 Instan t Lab | EPFL 2 FLEXURE WHEELS F OR SP A CECRAFT A TTITUDE CONTR OL Figure 31: T op view of a exure implemen tation of the previous double in v erting pan tograph. The dashed part is the ground and the t w o end eectors are in gra y . Com bining in v erting pan tographs As explained in Sect. 4.2, if t w o masses are at opp osite ends of circular orbits that do not coincide, the total angular momen tum is not constan t, but oscillates around a xed v alue. T o eliminate this oscillation, a second pair of masses is added so that the alternating comp onen ts of their angular momen ta cancel out while their a v erages are added. These t w o pairs can b e arbitrarily placed with resp ect to one another. Ho w ev er, they can b e adv an tageously arranged side b y side. When lo oking from left to righ t, the rst t w o masses are in v erted, the second and third follo w iden tical orbits and the third and fourth masses are again in v erted. These motions can again b e implemen ted with t w o in v erting pan tographs moun ted in series, as sho wn in Fig. 32. xy Figure 32: Flexure implemen tation of t w o in v erting pan tographs moun ted in series. The masses on circular orbits are the four dark circles. The dashed parts are the ground 30 Instan t Lab | EPFL 2 FLEXURE WHEELS F OR SP A CECRAFT A TTITUDE CONTR OL 2.3.3.1.2 Com bining masses on circular and linear motions This t yp e of design is based on that of Fig. 17 in the sense that a mass on a circular motion has its cen tre of gra vit y comp ensated b y a set of masses on linear motion. In this design, the mass on circular motion crosses the path of those in translation, whic h w ould in practice result in collisions. T o b ypass this issue, the mass on circular motion is connected to a set of in v erting lev ers, eac h holding a coun ter-mass (Fig. 33). A coun ter-mass mo v es in opp osition to the mass on circular motion only when the latter mo v es in the direction of the corresp onding in v erting lev er. By ha ving these in v erting lev ers displa y ed symmetrically around the mass on circular orbit, the cen tre of gra vit y remain xed and the cen tre of the circular orbit. Suc h an arrangemen t with 4 in v erting lev ers and coun ter-masses is sho wn in Fig. 34. Figure 33: In v erting lev er with a coun ter-mass. Figure 34: Other con guration equiv alen t to a reaction wheel. Left: mec hanism in its neutral p osition. Righ t: mec hanism on a circular orbit. 2.3.3.2 Spherical Mec hanisms The abstract design presen ted in Sect. 2.3.1.2 require t w o asp ects whic h will b e dealt with separately: a exure mec hanism pro viding the kinematics of a CV-join t and com bining suc h mec hanisms to obtain the constan t angular momen tum. Creating a single spherical mec hanism CV-join ts ha v e b een extensiv ely studied in general mec hanical design and a n um b er of designs ha v e already b een disco v ered, suc h as the Rzeppa join t (Fig. 35). Ho w ev er, their implemen tation with exures has only b een recen tly ac hiev ed in the IsoSpring 31 Instan t Lab | EPFL 2 FLEXURE WHEELS F OR SP A CECRAFT A TTITUDE CONTR OL pro ject (see [15]). Again follo wing the analogy of a planet, this design consists in a rst set of three equatorial b eams whic h xes the xy -p osition of the sphere as w ell as blo cs its rotation around the z -axis (Fig. 36a). The last degree-of-freedom to remo v e is translation in the z -direction and is tak en care of b y a p olar b eam placed either b eneath or ab o v e the sphere (Fig. 36b). Figure 35: A Rzeppa CV-join t [15] y x (a) Equatorial b eams d Lp� z x (b) P olar b eam, in a displaced state Figure 36: Flexure CV-join t of the Spherical IsoSpring. The fabrication of this design is ho w ev er relativ ely complex. Indeed, ev en though the three equatorial b eams are easy to man ufacture or assem ble (as they lie in the same plane), the p olar b eam adds complexit y . It w ould b e highly desirable to ha v e a design in whic h all the exures lie in the same plane. Luc kily , suc h a design exists and uses the equiv alence b et w een a b eam and a corner blade (Fig. 37). In this design, the p olar b eam is replaced b y its equiv alen t corner blade, whic h then lies in the same plane as the equatorial b eams. This coplanarit y of equatorial b eams and p olar corner blade means that the exure can b e man ufactured in a single part (monolithic) as sho wn in Fig. 38. 32 Instan t Lab | EPFL 2 FLEXURE WHEELS F OR SP A CECRAFT A TTITUDE CONTR OL Figure 37: Equiv alence b et w een a b eam and a corner blade. Left: exible b eam of circular cross section with its degrees-of-freedom (t w o translations and three rotations). Righ t: corner blade with its degrees-of-freedom, whic h are the same as the b eam. Equatorial beams Polar corner blade Ground Mobile part Figure 38: Monolithic exure CV-join t consisting of three equatorial b eams and one p olar corner-blade. Concerning the oscillating mass, spherical masses are to b e used. Suc h masses ha v e the prop ert y that their inertia are iden tical along eac h of their principal axes. Note: this is equiv alen t to their inertia matrix b eing diagonal. Suc h a spherical mass can b e obtained in a v ariet y of manners. The trivial approac h is to actually use a sphere (either hollo w or lled), but other shap es are equiv alen t from the p ersp ectiv e of their momen ts of inertia. F or example, a system of t w o discs can ac hiev e this prop ert y if they are separated b y the righ t amoun t as sho wn in Fig. 39. Other p ossibilities include ha ving p oin t masses placed at the ap exes of regular p olyhedrons suc h as a tetrahedron. 33 Instan t Lab | EPFL 2 FLEXURE WHEELS F OR SP A CECRAFT A TTITUDE CONTR OL = = Figure 39: Examples of spherical masses. Left to righ t: a sphere, t w o discs, a tetrahedron. Com bining and coupling spherical mec hanisms As in the planar mec hanisms, all three meth- o ds of use (CMF, 2-DOF oscillator, activ e con trol) are implemen table, but spherical mec hanisms are particularly suited for the Circular motion Flexure. Indeed, referring to the CMF with rigid segmen ts and lo calized exures of Fig. 23, it can b e seen that this mec hanism already con tains t w o CV-join ts in an ti-phase and on circular motion. T o con v ert this CMF in to a Flexure Wheel, one only needs to add a spherical mass to eac h of these CV-join ts as sho wn in Fig. 40. The en tire Flexure Wheel therefore consists in the t w o CV-join ts, b oth with their resp ectiv e spherical mass, link ed together b y a spherical join t. Sphericalmasses Figure 40: Left: initial Circular Motion Flexure of Fig. 23. Righ t: resulting Flexure Wheel after adding the t w o spherical masses Another notew orth y design, whic h this time can also b e used as a 2-DOF oscillator or activ e con trol, is to in v ert the motion of t w o spherical mec hanisms with corner blades (or their equiv alen t ro ds). The in v ersion is done in t w o steps and requires t w o additional in termediate spherical mec hanisms, but whose spherical masses are negligible. The rst consists in linking a spherical mass to one of these in termediate spherical mec hanisms b y means of t w o corner blades, as sho wn in Fig.41a. This arrangemen t in v erts the motion of these t w o mec hanisms only in one direction ( x -axis for example) and they therefore b eha v e as if they w ere the re ection of one another in a mirror, as sho wn in Fig.41b. The second step consists 34 Instan t Lab | EPFL 2 FLEXURE WHEELS F OR SP A CECRAFT A TTITUDE CONTR OL in rep eating this mirroring eect in order to in v ert the motion along the other direction ( y -axis). The nal result is sho wn in Fig. 42 whic h consists in four spherical mec hanisms, where t w o neigh b ouring spherical mec hanisms mirror eac h other while t w o diagonally opp osed spherical mec hanisms are en tirely in v erted. (a) neutral p osition (b) displaced Figure 41 (a) neutral p osition (b) displaced Figure 42 2.4 Demonstrator protot yp es and results 2.4.1 Obtaining the circular motion T o k eep the protot yp es as simple as p ossible for these rst exp erimen ts, asp ects suc h as using a circular motion exure and frictionless actuation ha v e b een left aside. The actuation and circular motion w ere implemen ted with a standard rotating motor. T o pro vide the circular orbit of xed amplitude, the mass on circular orbit has a pin whic h lo dges in to a lev er via a hole eccen tric to the motor shaft. As the motor turns, the pin is held in this hole and therefore follo ws circular orbits of xed amplitude. It can b e noted that this standard rotating motor con tributes to the total angular momen tum, since its shaft is on angular rotation (i.e. the motor b y itself can b e seen as a standard reaction wheel). 35 Instan t Lab | EPFL 2 FLEXURE WHEELS F OR SP A CECRAFT A TTITUDE CONTR OL In order to remo v e this undesired con tribution, a second motor and crank are added. They are not connected to an ything and turn at the same sp eed as the rst motor, but in opp osite direction. Their only role is to coun ter the angular momen tum generated b y the rst driving motor so that the total angular momen tum is solely pro duced b y the Flexure Wheel. Note: this metho d of suppressing the motor's angular momen tum w as only implemen ted in the rst protot yp e. Pin Crank MotoraxisDC motor Figure 43: Pin and hole mec hanism that restricts to circular motion. Note: in this construction a screw w as used as the pin 2.4.2 Protot yp e 1: t w o serial in v erting pan tographs This protot yp e is based on the design presen ted in Sect. 2.3.3.1.1 consisting in t w o serially arranged in v erting pan tographs. Its exure structure is made of t w o 5mm sheets of POM that ha v e b een laser cut and screw ed together to form the rigid segmen ts as w ell as crossed leaf-spring piv ots. The masses consist in four brass cylinders of appro x. 35[mm] in diameter and 15[mm] high. Note: these masses are naturally not punctual, but this asp ect can b e neglected as it is only a demonstrator protot yp e. A stepp er motor forms the actuation b y means of the lev er and pin mec hanism presen ted in the previous section. The diameter of the orbits has b een set to appro x. 12[mm] b y the hole in whic h the masses' pin lo dges and the mec hanism runs at 240[rpm]. A second motor has b een added in order to suppress the additional un w an ted angular momen tum brough t b y the rst motor, as sho wn in Fig. 45b. 36 Instan t Lab | EPFL 2 FLEXURE WHEELS F OR SP A CECRAFT A TTITUDE CONTR OL (a) Isometric view (b) T op view Figure 44: Ov erview of the rst protot yp e (a) The driving crank with the pin lo dged in a hole (b) The extra motor and crank whic h suppress the angular momen tum of the rst motor. They are not link ed to the Flexure Wheel. Figure 45: Pin and lev er actuation of the rst protot yp e. 2.4.3 Protot yp e 2: one mass on circular orbit and t w o masses to stabilise the cen tre of mass This design is similar to that of Fig. 34 with the dierence that there are only t w o masses to k eep the cen tre of mass xed. This design can also b e seen as that of Fig. 17, but where the t w o masses in translation do not oscillate around the cen tre of the circular orbit, but ha v e an oset. The consequence is that the cen tre of mass is not lo cated at the cen tre of the circular orbit, but has an oset. This leads to the fault of this mec hanism, whic h is an alternating angular momen tum, though on a v erage p ositiv e. This mec hanism is therefore not rigorously a Flexure Wheel, but is still capable of putting a spacecraft in rotation, though not at a constan t rate (i.e. if this mec hanism is suddenly put in to motion, the spacecraft's orien tation will b e the sup erp osition of rotation at a constan t rate and shaking in rotation). 37 Instan t Lab | EPFL 2 FLEXURE WHEELS F OR SP A CECRAFT A TTITUDE CONTR OL Concerning its fabrication, this protot yp e uses aluminium pro les as rigid segmen ts whic h are then link ed together b y short spring steel blades. The actuation has the same pin and hole arrangemen t as the rst protot yp e, but uses a DC motor. (a) Isometric view (b) T op view Figure 46: Ov erview of the second protot yp e (a) Links to the mass on circular orbit (b) Crossed leaf spring piv ots Figure 47: Construction details of the second protot yp e 2.4.4 Exp erimen tal setup The main goal of the setup w as to v alidate whether the exure protot yp es w ere able to put themselv es in to rotation without an y external in uence. The second goal w as to ev aluate whether the system w as statically balanced (i.e. no shaking in translation). The usual test b enc h for suc h exp erimen ts consists in ha ving the system oat on a at surface via an air cushion as it almost en tirely remo v es friction while still allo wing rotation and translation in the plane. F or simplicit y and a v ailabilit y reasons, the setup consisted in susp ending the protot yp es on the end of a string. Indeed, suc h a setup still allo ws for rotation as w ell as translation to some exten t. 38 Instan t Lab | EPFL 2 FLEXURE WHEELS F OR SP A CECRAFT A TTITUDE CONTR OL Figure 48: Exp erimen tal setup consisting in the t w o demonstrator protot yp es sus- p ended b y strings. The fact of susp ending on a string do es ho w ev er bring more complexit y to the exp erimen t than with an air b earing. Indeed, as the protot yp e rotates, torsion in the string app ears and creates a restoring torque, whic h aects the protot yp e's rotation. It has b een c hosen that the actuation suddenly puts the exure mec hanism in to motion at a constan t rate. This results in an instan taneous increase of the angular momen tum, whic h means that the exp orted torque that protot yp e receiv es is a single impulse. The exp ected exp erimen tal pro cedure is as follo ws:  The en tire system is at rest. This means that neither the exure mec hanism is in motion and neither the protot yp e is rotating on the end of the string.  The actuation is turned on, whic h sets the exure mec hanism in to motion.  The protot yp e starts to rotate around the string.  As the protot yp e rotates, the string's restoring torque increases  A t some p oin t the protot yp e slo ws do wn, stops and starts to rotate the other w a y . 2.4.5 Results The results of the qualitativ e exp erimen tal setup consist in the follo wing gures and in a video recording of the exp erimen t. The video recording giv es a b etter understanding of the exp erimen t and p erformances and it is a v ailable at: h ttps://doi.org/10.5281/zeno do.5148075 39 Instan t Lab | EPFL 3 CONCLUSION AND FUTURE W ORK The follo wing gures illustrate the main results of the video. Indeed, it can b e seen that the protot yp es ha v e turned clo c kwise after ha ving turned on the motors. (a) Before turning the motors on (b) After ha ving turned the motors on, still in mo- tion Figure 49: T op view of protot yp e 1 b efore and after ha ving turned on the motors (a) Before turning the motor on (b) After ha ving turned the motor on, still in mo- tion Figure 50: T op view of protot yp e 1 b efore and after ha ving turned on the motor 3 Conclusion and future w ork This pap er started with an o v erview of ball b earing reaction wheels and their issues. The fundamen tal prop erties of reaction wheels w ere then highligh ted using systems mec hanics. This step enabled other t yp es of motion to b e considered as a replacemen t. These in v en ted alternativ e t yp es of motion consisted in a p oin t mass on circular motion and spherical mec hanisms on circular orbits around their cen tre of 40 Instan t Lab | EPFL 3 CONCLUSION AND FUTURE W ORK mass. T o fully matc h the b eha viour of a reaction wheel, sev eral masses or spherical mec hanisms are com bined. It w as then sho wn ho w these t yp es of motion could b e implemen ted with exures as w ell as ho w circular motion could b e reac hed. Last, the pro of-of-concept protot yp es w ere presen ted along with their results. One theoretical p oin t m ust b e noted: the presen ted mec hanisms ha v e only b een studied in a xed frame (therefore that is not sub ject to rotations) and the p ossibilit y of them disturbing an already rotating spacecraft has not b een studied. This is the case for the rst protot yp e. Indeed, when all the masses are aligned (as in Fig. 32) the momen t of inertia along the x -axis is at its minim um. Ho w ev er, during motion, this momen t of inertia will c hange. It can then b e understo o d that if the spacecraft is already rotating around the x -axis, the spacecraft will see its momen t of inertia c hange, whic h disturbs its rotation. F uture w ork therefore in v olv es further study of these balancing asp ects. F uture w ork also includes the fabrication of a Circular Motion Flexure and its in tegration in to a pro of-of-concept protot yp e for the spherical mec hanisms, similar to the t w o that w ere presen ted. The actuation metho ds will also ha v e to b e studied. Last, it w ould b e v ery in teresting to study the scaling la ws at pla y in these Flexure Wheels. In- deed, their monolithic fabrication allo ws for miniaturisation. This could b e a p oten tial adv an tage o v er magnetic b earings for whic h miniaturisation pro v es to b e c hallenging [13]. 41 Instan t Lab | EPFL 4 APPENDICES 4 App endices 4.1 App endix: Angular momen tum of a mass on uniform circular motion W e consider a particles of mass m on a circular orbit of radius r within thexy -plane. The mass follo ws this orbit with a v elo cit y v , whic h results in an angular v elo cit y of !=v=r . A Omv,� R a L =mrv+mav cos( ) = constAvrt OL =mrv =const��� �� ������ r Figure 51: P article of mass m on a uniform circular motion of radius R . When ev aluated from the orbit's cen tre O , the angular momen tum LO is simply giv en b y LO= mrv =m!r2. Ho w ev er, when ev aluating from an arbitrary p oin t A at a distance a from the cen tre, the general de nition of angular momen tum is required: LA=mrv , (11) where r is the particle's p osition and v its v elo cit y . As r andv are b oth within the xy -plane, their cross-pro duct is exclusiv ely along the z -axis. W e will therefore only consider the z -comp onen t of the angular momen tum LA;z . W e therefore ha v e: LA;z=mrvsin(=2 + ) (12) =mrvcos(  ) (13) =mrv(cos( ) cos( ) + sin( ) sin( )) (14) Moreo v er, since v=!r and:8 >< >:rcos( ) =a+Rcos( ) rsin( ) =Rsin( ) =!t, (15) w e ha v e: LA;z=m!R (acos( ) +Rcos2( ) +Rsin2( )) (16) (17) whic h leads to: LA;z=m!R (acos(!t) +R) (18) Conclusion: The angular momen tum of a mass with uniform circular motion is not constan t when ev aluated at an arbitry p oin t, but oscillates around a xed v alue. 42 Instan t Lab | EPFL 4 APPENDICES 4.2 App endix: Angular momen tum of t w o masses on uniform circular mo- tion W e no w consider t w o particles A andB of equal mass m eac h on circular orbits of same amplitude within thexy -plane. These orbits are cen tred on the x -axis at equal distance from the origin, as sho wn in Fig. 52. The distance b et w een the orbit cen tres is 2a . Omv,� R ar m v,�Ra rXY Figure 52: System consisting in t w o masses at opp osite ends of iden tical circular orbits. Moreo v er, the masses are at opp osite ends of their resp ectiv e orbits (i.e. when particle A is at the top-left end of its orbit, particle B is at the b ottom-righ t.) The fact that their displacemen ts are opp osed means that that the cen tre of mass of the whole system do es not mo v e. As the cen tre of mass is immobile, the total angular momen tum do es not dep end on the p osition of the frame from whic h it is ev aluated (see Eq. 1). W e c hose to ev aluate the angular momen tum at the origin (halfw a y b et w een the t w o orbits). T o calculate the total angular momen tum, w e use the previous result of Eq. 18 and apply it to b oth particles A andB . Thez -comp onen t LA;z of massA 's angular momen tum is b y de nition: LA;z=m!R (acos(!t) +R) . (19) F or particle B ,LB;z of massB is then easily deducted b y replacing a b ya and b y+ (since the masses are at opp osite ends of their resp ectiv e orbit). The angular momen ta of eac h mass is therefore iden tical: LA;z=LB;z=m!R (acos(!t) +R) . (20) Thez -comp onen t Lz of the total angular momen tum is therefore the sum of LA;z andLA;z : )Lz= 2m!R (acos(!t) +R) (21) Conclusion: The angular momen tum of a t w o-mass system on circular orbit is not constan t, but oscillates around a xed v alue. Note: This oscillation of angular momen tum can b e remo v ed b y adding an second set of masses so that the angular momen tum oscillations will cancel eac h other out. 43 Instan t Lab | EPFL 4 APPENDICES 4.3 App endix: Angular v elo cit y of a CV-join t on circular orbits In this section, the angular v elo cit y of a spherical mec hanism on circular motion will b e studied. As these mec hanisms ha v e already b een presen ted in [15], the same notation describing the sphere's p osition will b e used. These consist in angles  and' whic h describ e resp ectiv ely ho w m uc h the sphere is tilted and in whic h direction. The approac h that will b e dev elop ed is gradual. Indeed, the starting p oin t is a CV-join t in the con guration of a shaft coupling (where b oth extremities of the join t rotate). The next step in v olv es placing this system in a rotating frame. As a v eri cation, the results shall b e compared to [15] whic h uses a more direct approac h. In this case, the kinematics of the mo ving mass are describ ed with Euler angles and the resulting angular v elo cit y is calculated. Starting from the shaft coupling The starting p oin t is a CV-join t in the con guration in whic h it is most commonly found: as a shaft coupling. Indeed, ha ving a rotating motor driv e another shaft is a t ypical situation in mec hanics. These shafts ma y not b e p erfectly aligned and ha v e an angle b et w een them. It is ho w ev er still desirable for the driv en output shaft to ha v e a constan t angular v elo cit y (for an input shaft with constan t angular v elo cit y)and this prop ert y is the de nition of a CV-join t. This situation is depicted in Fig. 53 in whic h the CV-join t is sc hematically sho wn b y a Cardan join t. In this gure, the input and output shafts form an angle  . Moreo v er, as it is a CV-join t and as the shaft axes are not aligned, b oth parts ha v e an angular v elo cities with the same amplitude !0 , but do not p oin t in the same direction. The input shaft has an angular v elo cit y v ertically up w ards along the z -axis !1=!0^z while the output shaft has its angular v elo cit y !2 forming an angle  with thez -axis. Note: Ev en though a Cardan join t is not a CV-join t, it is still used for visual purp oses as kno wn CV-join ts are more elab orate and di cult to visualise. yz x� �1�2 Figure 53: CV-join t in the con guration of a shaft coupling The next step consists in placing this rotating shaft coupling on a v ertically rotating frame in order to cancel out the rotation of the input shaft. Indeed, the spherical mec hanism on circular motion assumes the base (here the input shaft) to b e xed. The result of adding suc h a rotating frame is that the input shaft in no w xed and the output shaft no w follo ws the circular motion of a spherical mec hanism. Kinematically sp eaking this consists in adding an angular v elo cit y of amplitude !1 do wn w ards along thez -axis to b oth the input and output shafts. Fig. 54 sho ws the shaft coupling placed on the rotating frame and Fig. 55 isolates eac h shaft while highligh ting their resp ectiv e total angular v elo cities. 44 Instan t Lab | EPFL 4 APPENDICES � �1�2 ��1yz x Figure 54: CV-join t in the con guration of a shaft coupling, moun ted on a rotating frame to cancel the rotation !1 of the input shaft, therefore creating a spherical mec hanism on circular motion ���� �1�2 �yz x ��1��1 Figure 55: Output shaft (left) and input shaft (righ t) of Fig. 54 with their resp ec- tiv e o v erall angular v elo cities. �� �� ������ � OA BC Figure 56: Detailed view of the angular v elo cities of the output shaft of Fig. 55. 45 Instan t Lab | EPFL