STEPS WP 1.I. Sun Vector Computation and Solar Cells Orientation for Path Planning. POLITO LIM Technical Report

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1 POITO IM Sola cell oientation v.2.doc STEPS WP 1.I Sun Vecto Coputation and Sola Cell Oientation fo Path Planning POITO IM Technical Repot B. Bona,. Calone, M. Kaouk Ng CSPP IM Politecnico di Toino Veion: 2. 11/2/21 WPR POITO IM/21/2.1 1/1

2 POITO IM Sola cell oientation v.2.doc 1 Intoduction Thi epot decibe a iplified odel ued to copute the elation between the ove poe on the Ma uface (i.e., it x, poition and 3D oientation) and the Sun poition ove the hoizon. Sun poition i eponible of ola illuination of the photovoltaic cell onboad the ove and conequentl affect thei echage efficienc. Sola illuination will be ipl odelled a a paaete anging fo (null illuination, i.e., null cell echage) to 1 (axiu illuination, i.e., axiu cell echage). Thi paaete will be ued a a weighting facto affecting the path planning tateg; the planning algoith will take into conideation the Sun global illuination eceived b the ove along a candidate tajecto, theefoe ting to axiize the total echage of photovoltaic cell. Soe iplifing auption will be intoduced, a detailed in the following Section. 2 Rove kineatic coodinate In Figue 1 a ove i epeented b a igid bod with an attached efeence fae; the ove poe ( pr, ) whee p = (,, xz) i the poition and R i the oientation; the poe vaie in tie accoding to the ove otion, pecified b the wheel coand. Two ight hand efeence fae ae conideed: a fixed fae R ( i, j, k ) and a ove cented fae R ( i, j, k ). On Ma R i uuall located at the inteection of the pie eidian and the equato. The pie eidian of Ma i defined b the location of the cate Ai (De Vaucoule et al., 1973). Although it ight be tepting to efe to tandad tie on the Ma pie eidian a Ai Mean Tie in analog to Eath' Geenwich Mean Tie (GMT), the latte te ha been upplanted b Coodinated Univeal Tie (UTC) in intenational tiekeeping evice. Theefoe we efe to the Mean Sola Tie on the Ma pie eidian a Coodinated Ma Tie, o MTC, b analog to the teetial UTC. X axi i poitive in the Eat diection (poitive longitude), Y axi i poitive in the Noth diection (poitive latitude) Figue 1. Rove coodinate and efeence fae R i located at the bacente point of the ove, X point towad the poitive otion, Z i othogonal to the ain bae fae and Y coplete the fae. R a be expeed in R b diffeent epeentation; the ot widel ued i that baed on Roll Pitch Yaw (RPY) angle ( θ, θ, θ ) that x z oiginate the following tanfo/otation atix R ; ee B. Bona (22) and B. Siciliano and O. Khatib (28). WPR POITO IM/21/2.1 2/1

3 POITO IM Sola cell oientation v.2.doc R = (1) with RPY A = coθ co θ = in θ in θ co θ co θ in θ = co θ in θ co θ + in θ in θ = co θ in θ = in θ in θ in θ + co θ co θ = co θ in θ in θ in θ co θ = in θ = in θ co θ = co θ co θ z z x x (2) Not all the textbook and anual accept thi elation; anothe definition of RPY give oigin to an altenative atix fo RPY B = coθ co θ z = co θ in θ x z = in θ = in θ in θ co θ + co θ in θ = in θ in θ in θ + co θ co θ = in θ co θ x = co θ in θ co θ + in θ in θ = co θ in θ in θ + in θ co θ = co θ co θ x We will ue alwa epeentation RPY A in (2), but the eade i advied to check the adopted epeentation. 3 Sola illuination odel In ode to copute the ola illuination paaete σ 1 it i necea to conide two vecto (ee Figue 2): a) the noal unit vecto n to the photovoltaic cell aa; b) the unit vecto (alo naed un vecto) connecting the local ove poition to the local Sun poition in the k. (3) WPR POITO IM/21/2.1 3/1

4 POITO IM Sola cell oientation v.2.doc Then σ i coputed a Figue 2. Relation between cell aa noal vecto and un vecto T T if > σ = n n (4) T if n Both vecto ut be expeed in the ae coodinate efeence fae R, that in ou cae can be eithe R o R. The ola adiation affecting the cell echage a depend on an othe facto, a the k obfucation due to dut o othe upended paticle, but thee effect ae not conideed hee. 3.1 Vecto coputation Accoding to onboad un eno availabilit, thee ae fou poibilitie fo coputing the elevant vecto. Thee ae indicated in Table 1. Refeence fae R R Vecto n the coputation i odeatel if onboad un eno ae coplex; one ut know the available, the coputation of intantaneou poe ( pr, ) of the in R i uele; if un eno ove in the inetial efeence ae unavailable, the coputation fae. Uuall thi i povided b i odeatel difficult, eling on the onboad odoetic eno diect coputation of Sun path on the k (ee below) the coputation i ea, ince the if onboad un eno ae cell aa ae onboad the ove and n available, the coputation of i a pioi known in R i taightfowad; if un eno ae unavailable it i odeatel difficult, eling on diect coputation of Sun path on the k (ee below). Tanfoation of fo R to R i necea. Table 1. Coputation of noal unit vecto and un vecto WPR POITO IM/21/2.1 4/1

5 POITO IM Sola cell oientation v.2.doc et u aue that the obot poe i available in R a poition p and atix R given b the ove oll pitch aw (RPY) angle, a pecified in (2); the atix R allow to tanfo vecto in R, a Unit vecto noal to cell aa A outlined above, two poibilitie aie: v v = Rv = R v R to vecto in (5) T a) Noal unit vecto i expeed in the ove efeence fae R. In thi cae the unit vecto i eail coputed, ince the ola panel ae uuall poitioned on the x plane a in Figue 1; theefoe n = k, whee k i the z axi unit vecto of R. Othewie n k, but all the ae it i eail available, and can be conideed fixed wt the ove efeence fae (tilting panel ae not taken into account). b) Noal unit vecto i in the inetial efeence fae R. Thi cae will not be conideed hee Sun vecto Having choen to epeent n in R a R, we ut theefoe copute in R a and then tanfo it in T = R 4 Sun path algoith and un vecto coputation The poition of the Sun in the k i given b two angle: the aziuth angle φ () t and the elevation angle θ () t, a hown in Figue 1. Aziuth o ola aziuth angle φ () t i the angle between the line OP, i.e., the pojection on the gound plane π of the line OS fo the obeve to the Sun, and the line fo the obeve to the geogaphical South o Noth, accoding to the convention ued. A poitive aziuth angle geneall indicate that the Sun i Eat of South o Noth, and a negative aziuth angle geneall indicate the Sun i Wet of South o Noth. Sola elevation angle θ () t i the angle between the line OS connecting the obeve with the Sun and the hoizontal line OP. A elation that copute both angle i deived taking into account the vaiou atonoical paaete affecting the Sun aziuth and height. Thee paaete depend on eveal facto; the pincipal one ae: WPR POITO IM/21/2.1 5/1 1. The eaon of the ea. Seaon ae annual change in tepeatue on a planet caued b a cobination of two facto: axial tilt (i.e., ecliptic obliquit ε ) and vaiable ditance fo the Sun. On Eath, axial tilt deteine neal all of the annual vaiation, becaue Eath' obit i neal cicula. But Ma ha a high obital eccenticit; indeed the ditance fo the Sun to Ma vaie fo 1.64 AU to 1.36 AU ove a atian ea. Thi lage vaiation, cobined with an axial obliquit lightl geate than Eath' (ee Table 2), give ie to eaonal change fa geate than thoe on Eath.

6 POITO IM Sola cell oientation v.2.doc 2. The ove poition, eaued b latitude and longitude: the ae elated to the poition p, but noting that latitude and longitude ae given a angle (uuall in degee), while the poition p i given a linea quantitie in ete. Duing the ove otion latitude and longitude va ve little: Ma ha a ean cicufeence of 21,343 k, theefoe a degee of longitude coepond to a little oe than 59 k, and 1 k coepond to appoxiatel 1.1 of longitude. 3. The actual tie, i.e., the actual local hou eaued wt the da length (ol), that on Ma i lightl lage that on Eath; 1 ol = 24 h 39.5 in. Obliquit = ε Defined a the planet' axial tilt wt the ecliptic plane. Eath Ma 23 26'21".4119 = ad 25 19' = ad in ε = ) in ε = ) ( ( Table 2. Obliquit of Eath and Ma In ode to povide a fat but plauible algoith fo Sun vecto coputation, the Eath paaete wee ued intead of the Ma one ocal Sun elevation Fo an given point on Eath' uface, the Sun elevation angle θ () can be coputed a: t in θ = coh co δco Φ + in δin Φ whee H i the hou angle; 1 hou = 15 o ; H = at 12: local hou, H < oning hou, H > aftenoon hou. 36 δ i the Sun declination, coputed a in ( 284) 365 N +. N i the da of the ea; e.g., Mach 21 t i N =8, June 21 t i N =172. Φ i the local latitude. The un elevation angle fo N =8, Φ = 45 o i given in Figue 3 WPR POITO IM/21/2.1 6/1

7 POITO IM Sola cell oientation v.2.doc Figue 3. Sun elevation fo N =8, Φ = 45 o ocal Sun aziuth The econd eleent of the Sun location i it aziuth φ () t. Aziuth angle i poitive if towad Eat, while i negative if towad Wet. Thee ae two diffeent wa to copute thi angle, a follow in θ in Φ in δ coφ = (6) coθ co Φ o The Sun aziuth fo N =8, coφ coh co δin Φ in δco Φ = (7) co θ Φ = 45 o i given in Figue 4. WPR POITO IM/21/2.1 7/1

8 POITO IM Sola cell oientation v.2.doc 4.2 Sun vecto Figue 4. Sun aziuth fo N =8, Φ = 45 o Sun vecto i a unit vecto = 1 coputed in efeence fae R a T Accoding to (4) one ut copute n obtain inφ co θ 1 = = coφ co θ (8) 2 in θ 3 in R ; auing n T = ( 1) 1 T T σ = nr = ( ) = and appling elation (5), one (9) and, in paticula, auing RPY angle ae thoe in RPY A (2) ( ) ( ) ( ) σ = inφ co θ co θ in θ co θ + in θ in θ coφ co θ co θ in θ in θ in θ co θ + in θ co θ co θ x while auing RPY angle ae thoe in RPY B (3) σ = inφ coθ in θ + coφ co θ in θ co θ x + in θ co θ co θ x (1) (11) WPR POITO IM/21/2.1 8/1

9 POITO IM Sola cell oientation v.2.doc A an exaple, we have coputed and σ fo N =8, Φ = 45 o ; the eult ae epoted in Figue 5 and Figue 5. Sun vecto fo N =8, Φ = 45 o Figue 6. Sola illuination paaete σ fo N =8, Φ = 45 o 5 Final coent The landing ite on Ma of the two NASA Rove, Spiit and Oppotunit, ae epoted in Table 3 WPR POITO IM/21/2.1 9/1

10 POITO IM Sola cell oientation v.2.doc Rove Nae atitude ongitude Spiit S E Oppotunit 1.95 S E Table 3. NASA MER landing ite A ealitic iulation of the coputation of the vaiou quantitie decibed in thi Repot can be caied out conideing a ove poition not too fa fo the equato ( appoxiatel atitude) and aound ongitude. The iulation can tat at id oning and end a couple of hou late (1: 12: o 12: 14:). 6 Refeence Figue 7. Spiit (left) and Oppotunit (ight) landing ite Bona B. (22) Modellitica dei Robot Indutiali (22), CEID, Toino. Siciliano B., Khatib O. (Ed.) (28), Spinge Handbook of Robotic, Spinge. De Vaucouleu G., Davie M.E., Stu F.M. (1973), Maine 9 aeogaphic coodinate te. J. Geoph. Re. 78, Acon and Abbeviation AU = Atonoical Unit MER = Ma Exploation Rove MTC = Coodinated Ma Tie RPY = Roll Pitch Yaw UTC = Coodinated Univeal Tie wt = with epect to WPR POITO IM/21/2.1 1/1