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Draftversion January12,2016 PreprinttypesetusingLATEXstyleemulateapjv.08/22/09 A HEXAPOD DESIGN FOR ALL-SKY SIDEREAL TRACKING Andra´s Pa´l1,2, La´szlo´ Mésza´ros1,2, Attila Jasko´1, Gyo¨rgy Mezo˝1, Gergely Csépańy3,2,1, Krisztiań Vida1, and Katalin Ola´h1 Draft version January 12, 2016 ABSTRACT In this paper we describe a hexapod-based telescope mount system intended to provide sidereal 6 ′′ tracking for the Fly’s Eye Camera project – an upcoming moderate, 21 /pixel resolution all-sky 1 survey. By exploiting such a kind of meter-sized telescope mount, we get a device which is both 0 capable of compensating for the apparent rotation of the celestial sphere and the same design can 2 be used independently from the actual geographical location. Our construction is the sole currently n operatinghexapodtelescopemountperformingdedicatedopticalimagingsurveywithasub-arcsecond a tracking precision. J Subject headings: Techniques: photometric – Instrumentation: miscellaneous 1 1 1. INTRODUCTION Telescope And Rapid Response System (Pan-STARRS, ] M One of the most challenging part of astronomical in- Kaiser et al. 2002) or the (currently under construction) Large Synoptic Survey Telescope (LSST, Ivezić et al. strumentation is to provide a mount on which the opti- I 2008) project are prominent examples of initiatives ex- . cal system of the telescope is installed. Basically, such h ploiting these features. In these projects it is possible a mount provides both a way to point the telescope to- p to map the visible sky within a cadence of a few days wards a certain celestial position and to compensate the - and detect variations from this timescale up to several o apparent rotation of the celestial sphere. In modern ob- years of the planned operations. These projects are also r servational astronomy the most commonly used equato- t noticeable in their largeétendue. s rial and alt-azimuth type mounts have two-stage serial Withtheincreasingtechnicalcapabilitiesitisnowpos- a kinematics. Equatorialmounts aremorefrequently used [ in autonomous or remotely operated sites while the az- sible to observe astronomicalobjects and phenomena on imuthmechanicsisthestandardsolutionfortheground- awide rangeoftime scalesfromsecondsto years. These 1 kinds of researchare only possible with surveys that not based telescopes with the largest apertures. v onlyrunfor years,but alsohavefrequenttemporalsam- 6 In order to accurately quantify the motion of such a pling. 9 telescope mechanics, pointing models are constructed. The goal of the Fly’s Eye project is to provide a con- 3 Such mathematical models characterize how the various ′′ tinuous, 21 /pixel resolution all-sky survey in multiple 2 mount axes should be actuated to attain the desired ce- opticalpassbandsbyimagingtheentirevisibleskyabove 0 lestial positions (what are usually defined in first equa- ◦ the altitude of30 with a cadenceofafew minutes. The . torial coordinate system: hour angle and declination for 1 scientific purpose of the Fly’s Eye device is to perform the current epoch). Pointing models exist for equatorial 0 time-domain observations by covering a time range of (Spillar et al.1993;Buie2003),andalt-azimuthmechan- 6 nearly six magnitudes with an imaging cadence of a few ics (see e.g. Granzer et al. 2012), as well as mathemati- 1 minutesuptoseveralyears,i.e. theplannedlengthofthe cal constructions can be generated for both types of se- : v rial kinematics (Pa´l et al. 2015). In addition, accurate operations. With this moderate resolution, individual i pointing models can also be derived if only the attitude stellar sources can be resolved with a sufficient signal- X to-noise ratio in the magnitude range of r = 7 15. (orientation) of mount axes are known with respect to − r The scientific goals of such a simultaneous all-sky sur- the vertical direction (Mésza´ros et al. 2014). However, a veyas Fly’s Eye areinduced by the continuous monitor- this type of pointing feedback can only be exploited in ing of the bright-end (r . 15) regime of astrophysical the case of equatorial mounts. variations and transients. Such signals can either be a Time-domain astronomy is a novel approach in astro- small amplitude variation of a brighter sources (where nomical research that recently became due to the ad- the signal-to-noise ratio is limited by the photon noise vances in instrumentation. These advances include the or by systematics) or a larger amplitude variation of a possibility for autonomous operation of large telescopes, fainter sources (where the signal-to-noiseratio is limited the employment of mosaic CCD detector arrays as well by the background/readoutnoise or by confusion). asthedevelopmentofhugecomputerdatabasesandpar- Regarding to the science domains, this magnitude allel data processing systems. The Panoramic Survey regimecoversa/brighterminorplanetsintheSolarSys- Electronicaddress: apal@szofi.net, lmeszaros@flyseye.net tem, including the homogeneous multicolor sampling of 1KonkolyObservatoryofMTAResearchCentreforAstronomy thebrightest 10,000objectsortheflybyofnear-Earth andEarthSciences, asteroids; b/∼stellar astrophysics, including multicolor KonkolyThegeMiklo´su´t15-17,BudapestH-1121,Hungary 2Department of Astronomy, Lorańd Eötvös University, photometry of pulsating and rotating variables, active PázmányP.stny. 1/A,BudapestH-1117,Hungary stars,eclipsingmultiplestellarsystems,transitingextra- 3ESO-Garching,Germany,D-85748,Karl-Schwarzschild-Str. 2 solarplanets;andc/detectionandobservationsofextra- 2 Pa´l et al. galactic phenomena such as brighter supernovae, optical counterparts of GRBs. In the Fly’s Eye design, the simultaneous surveying of the visible sky is performed by a mosaic array of 19 wide-field camera units where each of the cameras has ◦ a field-of-view of 26 in diameter. These cameras are assembled on a single, hexapod-based telescope mount. In addition, the total étendue of such a configuration is comparable to that of Pan-STARRS (i.e. 30deg2m2). ∼ Similar initiatives for all-sky surveying include the PASSproject(Deeg et al.2004). Thisprojecthasasim- ilar mosaic setup (i.e. it uses 15 cameras to cover the sky above h & 30◦), but the cameras are fixed to the ground. The Evryscope design (Law et al. 2015), like itsprototypeconcept(Law et al.2012)featuresasingle- axis setup responsible for sidereal tracking and employ 24 wide-field lens. Wide-field survey projects that have a comparable field-of-views to the Fly’s Eye were also successful in finding transiting extrasolar planet candi- dates(seee.g.Pepper et al.2007;Bieryla et al.2015,and the references therein). More specified details of the key scientific projects and design concepts can be found in Pa´l et al. (2013). These goals are in an overlapwith the goals of the Evryscope project (see also the detailed paper of Law et al. 2015). Hexapods are only seldom applied in astronomical instrumentation. Oneofthemostprominentusageofsuch a device is to provide an alignment and focusing mecha- Fig.1.— A simple sketch depicting the main components of nism of secondary mirrors of large telescopes (see e.g. the hexapod. The upper panel shows a perspective view while thelowerpanelshowsthesimilargeometryasviewedfromabove. Geijo et al. 2006). The instrument named “Hexapod Thelower/largerdashedcirclerepresentsthebase(thatisusually Telescope” (Chini 2000) also exploits a similar mecha- fixed to the ground which alsodefines the reference framefor the nism for telescope pointing. This instrument is capable hexapod kinematics), the upper/smaller dashed circle represents ofperformopticalspectroscopy. Probablythemostspec- thepayloadplatform. Thisplatformiscontrolledbytheactuation ofthesixlegs,shownassolidlines. Thelegsareconnected tothe tacular instrument driven by a hexapod is the AMIBA base platform with joints, shown here as filled (on the base) and mount (Koch et al. 2009), a standalone radio telescope empty(onthepayloadplatform)circles. TheD3 symmetryofthe system. Radiointerferometerarraysalsoapplyhexapods hexapod can clearly be seen on the lower panel, showing the the (Huang, Raffin & Chen 2011). two planar hexagonal configuration of the universal joints. These hexagons arealsomarkedwithathindashedline. Inthis paper we describe ourdesignofa small, meter- sized hexapod mount dedicated to the all-sky Fly’s Eye and usually realized as a kind of joint (for instance, ball survey instrument. In Sec. 2 we detail how the mount jointoruniversaljoint). Fig.1displays asketchofthese itself is built and controlled. Sec. 3 describes the ba- basic components of the hexapods. sic equations needed for the computations regarding to In the following, we describe the kinematic properties hexapod motion control. In Sec. 4 we describe the of generic hexapods in more details. This description is computations and algorithms that provide the sidereal followedbythepresentationofourhexapoddesignedfor trackingitself. These computationsinclude the methods the Fly’s Eye project. neededforautomaticalignmentcalibration. InSec.5we summarize our results. 2.1. Hexapods at a glance 2. THEHEXAPODMOUNT One of the most relevant characteristics of a hexa- Hexapods, also known as Steward-platforms, employ pod is the domain of the 6 spatial motions in which the parallel kinematics to provide a complete, 6 degrees of payload platform is able to move (w.r.t. the hexapod freedom (DoF) spatial motion. As its name implies, the base and/or the reference frame). At the first glance, motionofahexapodis controlledviasix legsbyaltering the sizes of these domains (i.e. the total displacement their lengths. We note here that in the literature about and rotation of the platform) are defined by the size of hexapods, these legs are also called actuators, struts or the base/payload platforms, the locations of the control jacks. points on the base/payloadplatforms and the minimum These six legs connect the so-called base of the hexa- and maximum lengths of the legs. The difference be- podwith the payload platform. By extending orretract- tween the maximum and minimum lengths of the legs ing the legs, the payload platform will move w.r.t. the is referred also as travel length. More precisely, domain base. Forinstance,simultaneouslyincreasingthelengths sizes depend on the driving mechanism employed in the ofallofthelegswouldmovethepayloadupwardswhileit legs as well as the type of joints. would not rotate or offset horizontally. Throughout this As a rule of thumb, we can say that the size of the paper, the points which attain the connection between displacementdomainforthepayloadplatformisroughly the legs and the base/platform are called control points equal to the travel length while the size of the rotation All-sky Sidereal Tracking With Hexapods 3 Fig. 2.— Left panel: a computer-aided design (CAD) view of the hexapod skeleton, showing the base and payload platforms and the six legs. Each leg consists of a high precision worm gear driven ball screw linear actuator (that is driven by a stepper motor) and two universal joints. Right panel: the fully assembled hexapod in the lab, just prior delivery for first light tests. The hexapod can be seen withmounted electronics whileadditional control hardwareand single-boardcomputer (SBC) isfixed tothe base platform. Thepayload is a single imaging camera, filter wheel and lens. For simplicity, housekeeping and lens focusing are managed via spare Serial Peripheral Interface(SPI)and/orInter-Integrated Circuit(I2C)connectors onthebaseboard. domain in radians is roughly equal to the ratio of the the base, the 6 legs to these joints, the 6 upper joints to travel length to the characteristic size of the platforms: the legs and the platform to the 6 upper joints, we perform 6 4 assembly operations in total, which reduces ±Splatform∆ρ≈∆L (1) the effec×tive DoFs to 144 6 4 6=0. It means that − × × Here,S denotestheplatformsizewhile∆ρstands thehexapodcanindeedbeassembledinanunambiguous platform forthe rotationdomainsize and∆L is the travellength. manner. In addition, enabling the ball screw threads to Although any kind of hexapods can be constructed rotate in their bearing housing (i.e. to be driven by a with the topology shown in Fig. 1, practical designs al- motor or any external mechanism) allows us to control most always bear various symmetries. Namely, all of the hexapod. the legs are equivalent, by considering their mechani- One should note here that hexapods employing ball caldesignandminimum/maximumstrokeswhile the ar- screw driven actuatorsand universaljoints suffer from a rangementofthe6controlpointsalsoshowsatriangular sideeffectcalledscrewrotationerror (seee.g.Koch et al. (D ) symmetry. This arrangementshowingthe D sym- 2009, Appendix B). This is implied by the the fact that 3 3 metry is in fact a planar hexagon with two alternating thesoleinternalDoFoftheactuatorscrewthreadiscou- side lengths while all of the internal angles are 120 de- pled with a rotation of the payload platform. This rota- grees. Hence, this geometry can be quantified by two tioneffectivelychangesthelengthoftheleg(i.e. thedis- numbers: the shorter and longer side lengths. Depend- tancebetweenthecontrolpoints)whilethescrewthread ing on the application, the home position of a particular is not driven at all. Since the length of the leg is usu- hexapod is defined either at the state when all of the allymeasuredindirectlybyencodersplacedonthescrew legs are retracted or when the legs are in middle posi- thread (see Koch et al. 2009, or this paper, Sec. 2.4), tion (i.e. halfway between fully extended and the fully this feedback yields a need for correction. While this retracted position). Considering these types of symme- side effectcaneither be treatedas anerror,knowing the tries, hexapods can be characterized by 1+2 2 = 5 orientation of the universal joints w.r.t. the platforms geometricparametersatits homeposition: the l×engthof helps us to properly quantify it. the legs, and the shorter and longer side lengths of the 2.2. Payload control point hexagons (both on the base and payload) More precise qualitative characterization is needed TheFly’sEyedesigncontains19camera+filterwheel when considering hexapods with a particular combina- + lens units mounted on a support frame, hence the ex- tion of a leg mechanism and a joint type. In the follow- pected weight of the payload is approximately 60kgs. ingwedetailtheimplicationsofusingaballscrewdriven The CCD detectors have a pixel size of 9µm 9µm, × electromechanicalactuator combined with two universal the imaging resolution is 4k 4k and we employ lenses × joints on both ends of it. First of all, one has to ensure with a focal length of f = 85mm and an f-number of that this type of hexapod can unambiguously be assem- f/1.2. Thisopticalanddetectorsetupyieldsaneffective ′′ bled. In the following, we give a short description why resolution of 21 /pixel. this combination yields a working hexapod. Any solid The characteristic size of the payload (i.e. the struc- mechanical part has 6 DoFs w.r.t. a reference frame. turethatsupportsthese19cameraunits)is inthe range A universal joint has two internal DoFs, hence 8 DoFs of a meter. From these parameters, one can estimate in total. Similarly to a nut on a screw, a ball screw the principal moments of inertia as well as the forces driven actuator (with a fixed thread) has a single in- acting on each actuator on average. The resolution and ternal DoF, i.e. it has 7 in total. Therefore, consider- repeatability of the actuators are implied by the desired ing a disassembled hexapod with a fixed base, we have sidereal tracking (and/or pointing) precision and accu- 2 6 8+6 7+6=144 DoFs for all of its parts. As- racy. Our requirement for the precision of the sidereal × × × ′′ sembling two arbitrary parts reduce the total DoFs by tracking is defined as 0.1 pixels, i.e. 2 . This is equiv- ′′ 6. Therefore, if we mount the 6 lower universaljoints to alent to 10microradians (note that 1 π/648000 radi- ≡ 4 Pa´l et al. Leg #1 (SW) Leg #2 (W) Leg #3 (NW) Leg #4 (NE) Leg #5 (E) Leg #6 (SE) Controller/UI Limit switches Stepper motor Rotary encoder Limit switches Stepper motor Rotary encoder Limit switches Stepper motor Rotary encoder Limit switches Stepper motor Rotary encoder Limit switches Stepper motor Rotary encoder Limit switches Stepper motor Rotary encoder SBC (TCP/IP to RS232) Real−time Real−time Real−time Real−time Real−time Real−time control logic control logic control logic control logic control logic control logic (AVR MCU) (AVR MCU) (AVR MCU) (AVR MCU) (AVR MCU) (AVR MCU) Rld+eahSvtoe2aul3g ss2reha ktimfoete eRfrop Sarimn4ng8da5 tter RS485transceiver USB debug/calib Optional I2C Power regulators(12V DC, local) RS485transceiver USB debug/calib Optional I2C Power regulators(12V DC, local) RS485transceiver USB debug/calib Optional I2C Power regulators(12V DC, local) RS485transceiver USB debug/calib Optional I2C Power regulators(12V DC, local) RS485transceiver USB debug/calib Optional I2C Power regulators(12V DC, local) RS485transceiver USB debug/calib Optional I2C Power regulators(12V DC, local) Power supply Fig. 3.— Block diagram of the hexapod mount system as it is implemented in the Fly’s Eye project. Each hexapod leg has its own embeddedmicrocontrollerunit(MCU),allowingustoconnectthelegsdirectlyviahigherlevelprotocols,suchasRS485. Thissetupenables afullystateless operationofthehexapod. ans). Therefore the characteristic size of the instrument TABLE 1 (see also equation 1) implies a precision of 10µm on Motion domains ofthe Fly’sEye hexapodin the various ≈ each leg by expecting this sidereal tracking precision of rotationanddisplacement directions. The direction 0.1 pixels. domains are definedaroundthe homeposition (where the length of the legsare 510mm). The motion domain limits Sinceourgoalistoperformsiderealtrackingduringthe forthe rotationsanddisplacements are roundedtothe exposures and re-adjust the hexapod platform between nearesttenth of a degree or millimeter,respectively. subsequent images, the rotational domain is roughly equivalent (in radians) to the exposure time multiplied Direction Notation −limit +limit unit by Earth’s sidereal angular rotation frequency. Hence, Roll P −9.5 +9.5 deg actuator travel length can be relatively small compared Pitch Π −9.1 +9.3 deg tothe characteristicsizeofthe hexapod. Agreatadvan- Yaw Ω −9.2 +9.2 deg tage of the small travel length during operations is that Left-right X −66 +72 mm all actuators are pushed, yielding no backlash in the ac- Forward-backward Y −75 +75 mm tuators. Up-down Z −78 +70 mm 2.3. Geometry As we detailed in Sec. 2.1, the geometry of a hexapod usingawormgearmechanismwithaspeedratioof25:1. bearing similar actuators and a D3 symmetry in both The pitch of the ball screw thread is 4mm, hence one platforms can fully be characterized by 5 independent revolution in the driven axis is equivalent to 0.16mm geometric parameters at its home position. Due to our travel length. In order to precisely control the motion, intended usage (i.e. performing sidereal tracking), we weemployedsteppermotorswith200steps. Onefullstep declared the hexapod home position when all of the ac- is therefore corresponds to 0.8µm of stroke or retract. tuators are in their middle position. The motors are driven by a high precision controller Theconstructionforourhexapod(bothasacomputer- featuring sine-cosine microstepping down to 1/16 step- aided design model as well as its fully built version) is pings as well as a digital motor current control circuit. shown in the panels of Fig. 2. These images show that A direct feedback from motor motion is provided by a our choice for the hexapod contains a similar base and full-turn Hall-sensor based encoder with a 12-bit resolu- payload platform. The adjacent sides of the hexagons tion (i.e. one full revolution is divided into 4096 ticks). defined by the control points on both platforms have Thisresolutioniscomparabletothe3200microstepsper lengths of 97.41mm and 742.59mm by design. The revolution provided by the motor controller. The actua- lengths of the legs were set to 510.00mm at the home tors are also protected by redundant limit switches. position. Our choice for linear actuators was a model The motor controller as well as the Hall-encoder is with a net (safe) travel length of 100mm. Therefore, driven by an onboard microcontroller unit (MCU). The ∼ the control point distances ℓi (1 i 6) can be var- firmware of this MCU provides for each of the legs an ≤ ≤ ied in the domain of 460mm . ℓi . 560mm. The full embedded control and therefore the legs are capable of rotation and displacement motion domains of our hexa- performing motion in a fully autonomous manner. The pod design as implied by the geometry described here firmwareallowsmotorpositioningwithconstantspeedas are summarized in Table 1. wellas ramping with constant accelerationand constant jerk. In other words, strokes can be controlled up to a 2.4. Hardware and electronics cubic function of the time. The duration of motion se- AswementionedinSec.2.1,ourhexapodemployslegs quences can be commanded in the units of 1/256swhile withtwouniversaljointsonbothendswhilethelegitself the coefficients of the cubic motion are commanded in is formed by a high precision ball screw driven linear units of motor microsteps. actuator. The threadof the actuatoris drivenindirectly The timing resolution of the motion control is 10 − All-sky Sidereal Tracking With Hexapods 5 15µs. The actual resolution depends on the load of the MCU, i.e. in practice, it depends on the polynomial order of the currently running motion sequence. Dur- Coldstart ing sidereal tracking, the typical stroke/retract speeds are in the range of . 0.03mm/s, which is equivalent to . 0.2turn/sec in the driven shaft (see above for the actual values for thread pitch and gear ratios). Since one turn in the driven shaft is divided into 3200 motor mi- Firmware yes Internet crosteps, sidereal tracking imply a . 600microstep/sec upgrade stepping frequency. Hence, the jitter in the motor speed is less than 1% by considering microsteps (and less than 0.1% if we consider full motor steps). In addition, the no TCP/IP correspondingmotorcontrolfrequencyof60 100kHzis yes − alsocomparabletothepulsewidthmodulationfrequency of the motor driver circuit (which is set to 35kHz). Core no ∼ Inadditiontotheaforementionedfeatures,theembed- boot MDSNACD dedelectronicsalsosupportamicroelectromechanicalac- celerometer system (MEMS accelerometer) on each leg using asimilar dataacquisitionschemeas itis described yes in Mésza´roset al. (2014). Since the tilt of the legs vary RS485 as the hexapod moves, these accelerometers provide an independent means of retrieving the state of the instru- Wat hdog ment. Due to its complexity, the mathematics of this timeout hexapod pointing model based on such accelerometers RS485( om- are going to be described in a separate paper. manding) no 2.5. Control subsystem MultiplexedI/O SPI(Rotary The microcontrollers are connected to an RS485 bus (SPI,I2C,RS485,ADC) en oder) system in a similar fashion used in our former projects (Mésza´ros et al. 2014). The RS485 bus system allows Steppermo- I2C(MEMS the user to upload the coefficients of the elementary cu- tion ontrol a elero, bicfunctionsindependentlyforthe6legswhiletheactual thermo: on demand) motion can be started simultaneously with a broadcast Commandpro essing RS485 command. The MCUs are capable of storing a (ifre v'dfromRS485) queue with 8 such cubic motion sequences in total while new sequences can be uploaded during motion (if the queue has not been fully loaded). This implementation no yes Reboot allowsasmooth,continuousandfully traceablehexapod motion control. The MCU firmware also includes fea- turesregardingthesynchronizationofthemotion(which Fig.4.— A simpleflowchart showing the main structure of the is needed because of the different primary internal clock MCUfirmwareaswellasthecommunicationchannelsbetweenthe frequencies). The topology of the electronics are dis- peripherals(includingtheconnectionwiththeRS485bussystem). playedintheblockdiagramsofFig.3. InFig.4weshow Oneofthemostnotablefeatureofthestructureisthelackofinter- the main loops running on the MCU firmware as well as ruptrequests: allof the peripherals areaccessed inamultiplexed form, keeping the duration of a single processing cycle relatively the main communication channels between peripherals, constant. the RS485bus system andthe higher level controlchan- nels(whichis,inpractice,aTCP/IP-basedprotocol,im- a hexapod is capable of performing arbitrary rotations plemented in a separate hardware). It should be noted (withinitsallowedrotationdomain),itisanidealinstru- that in the MCU core, some of the peripheral communi- ment to use it for arbitrary geographical locations and cationsrunconstantly(pollingand/ordrivingtheRS485 without the need to be perfectly alignedto the principal bus, reading the magnetic encoder, motor step control, directions (i.e. unlike a conventional equatorial mount etc.) while other peripherals are driven only on demand whichneeds to be alignedproperlyto the celestialpole). (such as reading the thermometer values, accessing the Inordertoquantifymotionsofahexapodsystem,first, MEMS accelerometer). However, the external protocol let us investigate the “small motion” limit of this con- principle is always master-slave, independently whether struction (with the geometry described above). Let us the addressed part is running in a loop or only on de- quantify the attitude of the platform by the orthogonal mand. Inotherwords,thisimplementationyieldsafully transformation matrix O, then let us introduce the roll, autonomous operation but on the other hand, the hexa- pitch and yaw angles P, Π and Ω as pod itself (i.e. the six legs, one by one) must be polled in order to figure out the current status. 0 Ω Π − 3. HEXAPODMOTION O=exp Ω 0 −P (2) Π P 0 The hexapod is used primarily for sidereal tracking −  during exposures in an all-sky survey instrument. Since while the offset vector between the base and platform 6 Pa´l et al. centersis ∆=(X,Y,Z). Note thatin its home position, 64 12 correspondingto the geometry describedin Sec. 2.3, the roll, pitch and yaw angles are P = Π = Ω = 0, the 48 10 horizontal displacement offsets are X = 0 and Y = 0 while the distance between the centers of the platform 32 8 reference points is Z =348.35mm. During routine operations with a hexapod, includ- ingthenormalastronomicalimageacquisitionsequences 16 6 with the all-sky survey instrument, we have to compute the actuator lengths ℓk (k = 1,...,6) as the function of 0 4 theattitudeandoffsetparametervector(P,Π,Ω,X,Y,Z 0 16 32 48 64 4 6 8 10 andviceversa. The formercomputationcanbe deduced Fig.5.—Leftpanel: asmallstampshowingaregionof64×64 with simple vector arithmetics. Using the notations in- pixels, i.e. approximately 23′ ×23′ area of the sky. Image has been acquiredduringan exposure of 130 seconds, usinganf/1.8, troduced earlier, the actuator lengths, we got f = 85mm lens and the hexapod for sidereal tracking. Right panel: the point-spread function of the stellar profiles shown on ℓk = ∆+O jPk jBk , (3) the stamp of the left panel. The PSF is clearly symmetric at its · − core,the wingsareduetothe aberrationofthelens. Note thatif (cid:13) (cid:13) (cid:13) (cid:13) the sidereal tracking by the hexapod would completely be turned where denotes the Euclid norm (length) of the vec- off, the lengths of the stellar trails during this 130 seconds long tors. Hke·rke the vectors jP and jB are constants and de- exposurewouldbe∼61pixelslong,i.e. comparabletothesizeof k k thestampitself. finesthecontrolpointoffsetsfortheactuatorkw.r.t. the payloadplatformandhexapodbase,respectively. Dueto matrix transposition. This computation yields the presence of the aforementioned screw rotation error, the true actuator length ℓk will slightly differ from the +0.295 0 0 0 +0.279 0 commanded actuator length. 0 +0.295 0 −0.279 0 0   Boththe qualitativeandthe quantitativebehaviourof LTL 0 0 +0.565 0 0 0 hexapod motion can well be analyzed via the derivative ≈ 0 −0.279 0 +1.600 0 0   +0.279 0 0 0 +1.600 0  matrix  0 0 0 0 0 +2.799   ∂(ℓ ,ℓ ,ℓ ,ℓ ,ℓ ,ℓ ) (6) L= 1 2 3 4 5 6 . (4) ∂(P,Π,Ω,X,Y,Z) which clearly shows that the determinants of LTL and hence that of L are non-zero. This matrix tells us how the lengths of the hexapod legs The inverse problem related to Eq. 3 is the computa- alter as one varies the payloadattitude and/or displace- tion of the payload displacement offset and the attitude ment and vice versa, i.e. how the attitude and displace- w.r.t. the hexapod base once the actuator lengths are ment changes when some of the legs are actuated. The known. This problem can also be effectively computed hexapoddo notsuffer fromany parametricsingularityif inaniterativemannerbyinvolvingtheNewton–Raphson the determinant of L differs from zero at any arbitrary algorithm and the derivatives given by L. The effec- but allowed values for ℓk and/or the (P,Π,Ω,X,Y,Z) tiveness of this algorithm is due to the fact that LTL parameter vector. (and hence L) varies only slightly in the full domain For the geometry used in the Fly’s Eye design (see of 460mm . ℓi . 560mm. Namely, if we denote the above, in Sec. 2.3), the matrix L at the hexapod’s home derivative matrix of Eq. 4 at the home position (where position is going to be ℓi = 510mm) by L0, then the root mean square of the elements in the matrix LTL LTL is around 0.07, i.e. +0.033 +0.287 +0.254 −0.254 −0.287 −0.033 less by a factor of 4 40−th0an0the elements of the −0.312 +0.127 +0.185 +0.185 +0.127 −0.312 matrix in Eq. 6. ∼ −   L −0.307 +0.307 −0.307 +0.307 −0.307 +0.307 (5) ≈+0.365 −0.730 +0.365 +0.365 −0.730 +0.365 4. SIDEREALTRACKING −0.633 0 +0.633 −0.633 0 +0.633 +0.683 +0.683 +0.683 +0.683 +0.683 +0.683 Sincehexapodsarecapableofperformingarbitraryro-   tations,thistypeoftelescopemountcanbeusedwithout where, for simplicity, the roll, pitch and yaw angles P, anymodificationsatanygeographicallocationaswellas ΠandΩaremeasuredinmilliradiansinsteadofradians. the installationproceduredoes notrequireprecise align- This matrix implies some of the expected characteristics ment. Forinstance,expectingthatthedeviceisinstalled of the qualitative motion of the hexapod. For instance, precisely at the poles, the apparent rotation of the ce- the lastrowis constant,meaning thataverticaloffsetof lestial sphere is compensated by a purely yaw rotation, +1mm in the payload platform position needs a stroke i.e. when P = Π = 0 and dΩ/dt = 2π/P (where sidereal of 0.683mm in all of the legs. However, one should P is the sidereal rotation period of Earth). Sim- sidereal ≈ note thatatransformationwhichessentiallyswapthe X ilarly, considering a device installed at the equator, the and Y directions is not a member of the D symmetry celestialsphere wouldperform a purely rollrotation,i.e. 3 group. Hence, the lines in the above matrix correspond- dP/dt=2π/P while the two other angles are zero sidereal ingtotheX andY directions(orsimilarly,rollandpitch throughout the motion. If the hexapod is installed at rotations)donotshowany(implied)correlationbetween temperate latitudes, then the motion would be a com- the respective elements. bination of roll and yaw rotations. If the hexapod is In order to see that the determinant of L differs from not precisely aligned, corrections with small respective zero,letus computethe matrixLTL. Here,()T denotes angular speeds are needed in all of the three principal · All-sky Sidereal Tracking With Hexapods 7 directions. angular speeds are computed as As a foreword, we note here that the algorithm de- S∆x tailed in this section works if the conditions of the pre- ω = , (8) 1 sented linear treatment meets with the accuracy of the ∆T initial alignment of the hexapod. In practice, our expe- S∆y ω = , (9) rience was that the employment of standard tools (e.g. 2 ∆T classic bubble leveler, digital compass based on MEMS ∆ϕ magnetometers embedded in a smartphone, etc.) yields ω3= , (10) ∆T a precision comparable to or less than one degree. This accuracy is sufficient to apply our linear algorithm and where the (linear part of the) differential plate transfor- ′ ′ yield the sub-arcsecond tracking precision. However, we mation (x,y) (x,y ) between the pairs is described → should also note that equations (2) and (3) work on the as full hexapod motion domain (see also Table 1). There- ′ x ∆x cos∆ϕ sin∆ϕ x xc fore, even if the misalignment is in the range of several = + − − (11) degrees, the similar tracking precision can be achieved (cid:18)y′(cid:19) (cid:18)∆y(cid:19) (cid:18)sin∆ϕ cos∆ϕ (cid:19)·(cid:18)y yc(cid:19) − by an algorithmwhich composesthe matrix exponentof S istheplatescale(intheunits ofarcsecondsperpixel), Eq. (2) with the linear terms described in the following. and xc and yc are the field centroid coordinates (usu- In the following subsections, we present a simple algo- ally the half of the image size dimensions). If tracking rithmhowahexapodcanbecalibratedinordertoobtain is nearly fine, then these values of ∆x, ∆y (field cen- adjustment parametersneeded for proper siderealtrack- ter offsets) and ∆ϕ (field rotation) are usually small – ing. This description is followed by the presentation of some pixels or some tens of pixels, or equivalently some a series of tests validating our algorithm and, in gen- arcminutes–,sothelineardifferentialastrometrictrans- eral, the usability of a hexapod-based mount for all-sky formation is still reasonable. tracking. In the next step, we perturb Eq. (7) with adequately large offsets A , A and A as 1 2 3 4.1. Self-calibration One of the most straightforwardways to calibrate the dℓ(kA) =n (cosϕ +A )∂ℓk+ sidereal 0 1 hexapod motion parameters is to involve the sky itself dt (cid:20) ∂P as a reference frame. First, let us assume that the hexa- ∂ℓk ∂ℓk pod, located at the geographical latitude ϕ0, is roughly +(sinϕ0+A2)∂Ω +A3∂Π(cid:21), (12) orientedandalignedto the cardinaldirections as wellas the base is nearly horizontal. In this case, the speed of In a similar fashionas it was described above,the corre- the leg k at a linear approximation is need to be sponding ωm(A) drift and field rotationspeeds canalso be retrieved by exactly the same procedure. It can be con- dℓk =n cosϕ ∂ℓk +sinϕ ∂ℓk , (7) sidered that the apparent drift and field rotation vanish sidereal 0 0 dt (cid:18) ∂P ∂Ω(cid:19) if the parameters A , A and A are set to satisfy the 1 2 3 equation qwuheenrceynosfidEeraearlth=an2dπ/tPhesidpeareratli,althdeersividaetriveeasl ∂anℓkg/u∂laPr afrned- ωm+ 3 Ak∂ωm(A) =0 (13) ∂ℓk/∂Ω can be taken from Eqs. (4) and (5). kX=1 ∂Ak In our construction, the hexapod can directly be com- mandedtomovethelegsaccordingtothespeedsdefined for all m = 1, 2 and 3. Using real measurements on the by Eq. (7), see Sec. 2.5 above. Since the real placement stellar field seen by a camera mounted on the hexapod, of the hexapod is not perfect, the tracking which em- the partial derivative ∂ωm(A)/∂Ak can be measured nu- ploys the speeds defined by this equation would also not mericallybyinvolvingtheformerlyobtainedseriesoftest be perfect. If one obtains a series of images, the drift as images. Namely, this derivative can be approximated as wellas the field rotationcaneasily be measuredby finding the plate solutions for these images. We note here ∂ωm(A) ωm(Ak) ωm − . (14) thatthehexapodalsoactsasafieldrotatordevice,hence ∂Ak ≈ Ak plate solution should also include information about the field rotation. Here, these “series” should mean at least Hereωm(Ak) correspondstoadriftandfieldrotationspeed two images, however, the longer the series are, the more derived from an image series when the hexapod was accuratethe determinationofthe plate driftandthe ap- moved according to Eq. (12) and Ak is non-zero while parent field rotation. Both the plate drift as well as the the other two Ax (x = k) values are zero. All in all, field rotation has a unit of radians per second, i.e. the four image series shou6ld be taken in total, correspond- actual apparent drift and/or field rotation between the ing to the speeds ωm, ωm(A1), ωm(A2) and ωm(A3). We note subsequent images first must be converted to radians as here that the derivatives of Eq. (14) can even better be well as divided by the image cadence. If the series con- approximated using symmetric first derivatives instead tains more than two images, these parameters are de- of this one-sided numerical derivative. Namely, this ap- rived by simple linear least squares regression. Let us proximation is denote these three angularspeeds by ω , ω andω . For 1 2 3 instance,ifallofthesethreeparametersareobtainedus- ∂ωm(A) ωm(+Ak) ωm(−Ak) ing a pair of images with a time separationof ∆T,these − . (15) ∂Ak ≈ 2Ak 8 Pa´l et al. In order to obtain the derivatives in this way, six image would be close to unity and the vector invariant of its sequences are needed to be taken instead of four. logarithm(i.e. theinverseofEq.2)tellsustherespective One of the most important property of the above pro- roll,pitchandyawoffsetsthatneededtobecommanded cedure is that it does not need any kind of assumption to the hexapod. In practice, this vector invariantcan be from the underlying hexapod. Namely, the procedure computed as follows. Let us define also compensates for the screw rotation error and even we do not have to precisely know the hexapod geometry x=O32 O23, (18) − as well. In addition, we need to use at least three hexa- y=O O , (19) 13 31 − pod legs during the calibration procedure, i.e. the num- z=O O , (20) 21 12 ber of involved hexapod legs can be decreased until the − corresponding linear combination of the L matrix com- r= x2+y2+z2,and (21) ponents have a rank of 3 (otherwise, Eq. 13 would also t=pTr(O)=O +O +O . (22) 11 22 33 be singular). This fact has an important consequence which can be highly relevant during the operation of an Thenthe components ofthe logarithmofO aregoingto autonomousand/orremoteobservatorysuchastheFly’s be either Eye device: the sidereal tracking can be attained even if P x arg(t 1,r) one, two or three of the hexapod legs are stuck (due to, Π= − y (23) for instance, mechanical or electronic failures). r Ω z Moreover, in the case of astronomical imaging, the     need for only 3 DoFs out of the 6 allows us to vary the or identically zero when x=y =z =0. displacementofthehexapodpayloadindependentlyfrom itsrotation. Thiscanbeusedtoprogramthehexapodat 4.3. Field tests different displacementoffsets in order to avoidthe wear- ing of the actuator’s ball screws at certain positions. In order to test both the constructed hexapod as well as the previously described algorithm, we attain a series 4.2. Absolute calibration of tests with a single camera mounted on the hexapod payload platform (see also the right panel of Fig. 2 for The above presentedalgorithmcanbe extended to at- the actual setup). tainanabsolutepositioningofthehexapodw.r.t. thece- During the determination of the numerical derivatives lestial reference frame. In order to accomplish this step, needed by Eq. 13, we mounted an f/1.8, f = 85mm one has to obtain the apparentcoordinatesof the stellar lens to a small-format camera having a pixel resolution field seen by the cameras. After deriving the astromet- of 4k 4k and a pixel size of 9µm 9µm. This is an ric solution of the field in J2000 system, the equatorial × × equivalentsetuptothefinalconstructionoftheFly’sEye coordinates(right ascensionand declination) are needed mosaicsystemwith the exceptionthatFly’s Eyecamera to be computedafter correctingforprecession,nutation, units are using an f/1.2 lens instead of f/1.8. During aberrationand refraction using the standard procedures thesetestruns,thelenswerefocusedusingthesparetest (Meeus 1998;Wallace2008). These coordinatesarethen portsandpins foundonthe hexapodcontrolelectronics. needed to be compared with the local sidereal time and Fourimageserieshavebeentakenaccordingtotheone- the geographical latitude. In a similar fashion, the field sided numerical derivative needed by the approximation rotationwithrespecttotheJ2000referenceframeshould in Eq. (14). All of the image series contain four individ- alsobeconvertedtothefieldrotationw.r.t. theapparent ualframesacquiredwith20secondsofexposuretime. In north direction. total,thisprocedurerequired4 4frameswhilethegross Itshouldbenotedthattheconversionbetweenthedif- × time was roughly 10 minutes in total. However, during ferencesinthecelestialcoordinatesorfieldrotationsand these experiments, the overall duty cycle was not ideal the respective roll, pitch and yaw offsets must be done (for instance, the re-positioning of the hexapod between with caution due to the singularity in the parameteriza- each series was not performed in parallel with the im- tion of the the celestial coordinate systems. Namely, let age readout and so). Based on these series of images we us define the matrix function A(α,δ,ρ) as were able to obtain the field drift and field rotation pa- cosα sinα0 rameters ωm and ωm(Ak). These parameters were derived − A(α,δ,ρ)=sinα cosα 0 (16) usingthe actualframe data(exposure durationsandob- × 0 0 1 servationtime instances as found in the respective FITS   header keywords) as well as the differential astrometric cosδ0 sinδ 1 0 0 − solutions obtained with the aid of the FITSH package × 0 1 0 ·0cosρ−sinρ, (Pa´l2012). Oncewederivedthe Ak coefficients,weused sinδ0 cosδ 0sinρ cosρ theselegspeeds(definedbyEq.12)toprogramthehexa-     pod motion and obtain images with significantly longer wheretheanglesα,δandρcorrespondtotherightascen- exposure times comparable to the proposed cadence of sion,declinationandfieldrotation. For simplicity, letus the Fly’s Eye device. One of the small stamps and the denote the field centroid coordinates (as obtained from corresponding point-spread function (PSF) is depicted the J2000 astrometric solution, see the details above) in Fig. 5. One can clearly see that the procedure works with the same symbols. Let us denote the local sidereal smoothlyanditisadequatefortheintendedgoalsofthe time (derived from UT1) by ϑ. Then, it can be shown Fly’s Eye all-sky survey. We repeated this experiment that the matrix severaltimesbydeliberatelystirringthehexapodaswell O=A(α,δ,ρ)TA(ϑ,ϕ ,0) (17) as we conducted similar tests on subsequent nights. 0 All-sky Sidereal Tracking With Hexapods 9 In order to better characterize the quality of the side- legs have an effect on the roll and pitch angles (see the real tracking, we performed an experiment by replacing first two rows in the matrix L in Eq. 5), such tests can the f/1.8, f =85mm lens by an f/8, f =800mm cata- be used to characterize the behaviour of each leg nearly dioptricoptics. Theusageofthelattertypeoflensyields independently – even if only 2 out of the 6 DoFs of the ′′ a resolution of 2.3 /pixel, i.e. roughly 9.4 times finer hexapod can be measured directly by such a tiltmeter than the intended Fly’s Eye plate scale. Once the Ak device. However, one should be aware that temperature coefficients were properly derived using the f = 85mm inhomogeneties in the device itself could play an even setup, the lens were replaced and a 3 minute long se- moresignificantroleinthepreciseresponseofthedevice ∼ quence were taken with exposure times of 30 seconds. to temperature changes. In order to accurately charac- Since this setup yields a smaller field-of-view, we obtain terizethisdependence,webuiltandmountthermometers only the field centroid drifts. Our tests show that even inallofthelegcontrolelectronics(seeFig.3,wherethese with this finer resolution, the root-mean-square (RMS) thermometersareconnectedtotheI2Cbus)aswellasin deviation of the frame sequences were always less than manypointsofthe hexapodpayloadplatform(including ′′ 0.3 pixels, i.e. 0.7 . the 19 cameraunits in the final design). We expect that Accordingto the data acquisitionstrategyofthe Fly’s the frequent re-calibration(see Sec. 4.2) in parallel with Eye device (Pa´l et al. 2013), frames are going to be syn- thermal data acquisition will provide sufficient informa- chronized to Greenwich sidereal time. Hence, frames in tionto performsub-arcsecondaccuracyduringroutinely every approx. 23 hours and 56 minutes should show ex- observations. actly the same field (neglecting the effect of precession All in all, we can conclude that our hexapod-based and nutation). Since the hexapod would make several sky tracking system is safely capable of providing sub- hundreds of individual movements (including tracking, arcsecond tracking precision. In addition, the hexa- re-positioning and homing) between two frames with a pod based on this construction is capable of perform- cadenceofexactlyonesiderealday,evenafew dayslong ingmeasurementsatthe precisionlevelneededby astro- such series will provide us information about the longer geodetics(seealsoHirt et al.2014,formoredetailsabout term repeatability of the hexapod configuration. Dur- such applications). ing our initial test runs, we performed this analysis as well. We found that the RMS deviation between these 5. SUMMARY correspondingframesis around0.10 0.12pixels for the This paper described how a hexapod can be used as − f =85mm lens. This deviation was obtained for a week a part of an all-sky survey instrument to perform side- longofconsecutiverun(i.e. thedataacquisitionwasnot real tracking. We show how can the astrometric infor- interrupted by bad weather, but the system was shut mation provided by the images be exploited in order to down during daytime). While the primary purposes of attain sub-arcsecond tracking precision. As a hexapod this hexapod mount and the classical telescope mounts provides a complete, 6 degrees-of-freedom motion con- ′′ are different, we can conclude that this RMS of 2 trol, the a survey instrument based on this type of par- ∼ is well comparable to large, meter-class telescope point- allel kinematics can be used without any modifications ing residuals (see e.g. Pa´l et al. 2015, and the references at arbitrary geographical location and without the need therein). Wenoteherethatduringthisone-weekrun,the of precise alignment. We also demonstrated the math- re-calibration procedure (described in Sec. 4.1) was not ematical background of the fault tolerant capabilities of performed at all, i.e. the same calibration constants A , a hexapod-based astronomical telescope mount. These 1 A and A were involved while commanding the hexa- capabilities are among the major advantages of using a 2 3 pod. Therefore, we can expect that more frequent at- hexapod in a fully autonomously and remotely operated titude re-calibration decrease this residual, even in the astronomicalinstrument since the sky tracking can con- order of fractions of arcseconds (see in the following). tinuously be performed even if one, two or three of the hexapod legs are stuck, most likely because of an elec- 4.4. Laboratory tests tronicormechanicalfailure. Ourmountdesignpresented Further laboratory tests were also performed using a hereformsthecorepartoftheFly’sEyeproject,anongo- veryhighresolutiontiltmeter (HRTM) manufacturedby ing moderate resolution all-sky time-domain variability Lippmann Geophysikalische Messgera¨te (L-GM). Such survey. sensors are used in geodetics, more specifically, in astro- The benefits of a hexapod with respect to a conven- geodetic measurements (see e.g. Hirt & Seeber 2008). tional telescope mount are the following. First, con- Thesesensorsprovideawhite noiseoutputforpitchand ventional mounts are needed to be extremely precisely rolltiltsintherangeof0.01 0.02′′/√Hzandcanevenbe alignedtothecardinaldirections–andifnot,eveninthe used4formetrology,i.e. qua−ntifyandvalidatedevicesca- framework of an isotropic pointing model (Spillar et al. 1993;Buie2003;Pa´l et al.2015),therequiredcorrections pableofalterattitude. We performedrepeatabilitytests haveacomplexitywhichiscomparabletothatofahexa- with our hexapod when such a HRTM was mounted on pod. Inaddition,eitheranimpreciselyalignedequatorial the payload platform. Based on these quick tests, we mount or an alt-az mount require a field rotation mech- conclude that the resolution in the commanded attitude of this hexapod is in the range of . 0.1′′ and the long- anism for compensating these errors. While in the case of a small field-of-view telescope, this third stage of se- termvariationsdue to the ambienttemperature changes ′′ ◦ rialkinematicsprovidingthefieldrotationcompensation are in the range of 1 / C. Due to the parallel nature ∼ is significantly smaller than the drivers of the two main of the hexapod and the fact that actuation on all of the axes, it is not true for an all-sky instrument. In other 4 beyond their intended purposes, such as levelling, seismology words,off-the-shelffieldrotatorscannotbeappliedatall orthemeasurementoflunisolartides for an all-sky device. Another benefit of using hexapods 10 Pa´l et al. instead of conventionalmounts is the fact that the char- canprovideacost-effectiveandelegantwayoflong-term acteristic alignment of the payload platform is always maintenance of all-sky instrumentation which also fea- horizontalinitshomeposition. Inthecaseofanequato- ture precise sidereal tracking. rialmount it is not true and eventhis direction depends on the actual geographical location of the installation. Therefore, the same design cannot be used at different We thank Hans Deeg and Zˇeljko Ivezić for the useful locations if the payload is a camera assembly perform- discussionsregardingtothedesignitselfandthepractical ing all-sky surveying. In the case of alt-az mounts, the scientific implications of the Fly’s Eye project. Our re- preferred direction of an all-sky camera (i.e. looking to- searchissupportedmainlybytheHungarianAcademyof wardsthezenith)coincideswiththegimballockposition Sciences via the grantLP2012-31. Additional support is of the telescope mount (even considering a field rotation also received via the OTKA grants K-109276, K-104607 mechanism),hencealt-azmountsareintrinsicallynotca- andK-113117. Inourproject,weinvolvednumerousfree pable of performing such a tracking. 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