POLITO – LIM Solar cells orientation v.2.doc STEPS WP 1.I Sun Vector Computation and Solar Cells Orientation for Path Planning POLITO ‐ LIM Technical Report B. Bona, L. Carlone, M. Kaouk Ng CSPP – LIM – Politecnico di Torino Version: 2.0 11/02/2010 WPR POLITO ‐ LIM/2010/02.01 1/10
POLITO – LIM Solar cells orientation v.2.doc 1 Introduction This report describes a simplified model used to compute the relation between the rover pose on the Mars surface (i.e., its , x y position and 3D orientation) and the Sun position over the horizon. Sun position is responsible of solar illumination of the photovoltaic cells onboard the rover and consequently affects their recharge efficiency. Solar illumination will be simply modelled as a parameter ranging from 0 (null illumination, i.e., null cell recharge) to 1 (maximum illumination, i.e., maximum cell recharge). This parameter will be used as a weighting factor affecting the path planning strategy; the planning algorithms will take into consideration the Sun global illumination received by the rover along a candidate trajectory, therefore trying to maximize the total recharge of photovoltaic cells. Some simplifying assumptions will be introduced, as detailed in the following Sections. 2 Rover kinematic coordinates � In Figure 1 a rover is represented by a rigid body with an attached reference frame; the rover pose ( , p R ) � p = ( , , ) x y z is the position and R is the orientation; the pose varies in time according to the rover where motion, specified by the wheel commands. Two right ‐ hand reference frames are considered: a fixed frame � � � � � � R ( , i j k , ) R ( i , j , k ) and a rover centred frame . 0 0 0 0 m m m m R is usually located at the intersection of the prime meridian and the equator. The prime On Mars 0 meridian of Mars is defined by the location of the crater Airy ‐ 0 (De Vaucoulers et al., 1973). Although it might be tempting to refer to “standard” time on the Mars prime meridian as “Airy Mean Time” in analogy to Earth's “Greenwich Mean Time” (GMT), the latter term has been supplanted by Coordinated Universal Time (UTC) in international timekeeping services. Therefore we refer to the Mean Solar Time on the Mars prime meridian as Coordinated Mars Time, or MTC, by analogy to the terrestrial UTC. X axis is positive in the East direction (positive longitudes), Y axis is positive in the North direction (positive latitudes) Figure 1. Rover coordinates and reference frames R is located at the barycentre point of the rover, X points toward the positive motion, Z is orthogonal m to the main base frame and Y completes the frame. R may be expressed in R by different m 0 representations; the most widely used is that based on Roll ‐ Pitch ‐ Yaw (RPY) angles ( , θ θ θ , ) that x y z originates the following transform/rotation matrix R ; see B. Bona (2002) and B. Siciliano and O. Khatib (2008). WPR POLITO ‐ LIM/2010/02.01 2/10
POLITO – LIM Solar cells orientation v.2.doc ⎛ ⎞ r r r ⎟ ⎜ ⎟ ⎜ 11 12 13 ⎟ ⎜ ⎟ ⎜ = R r r r ⎟ (1) ⎜ ⎟ ⎜ 21 22 23 ⎟ ⎜ ⎟ ⎜ r r r ⎟ ⎝ ⎠ 31 32 33 with RPY ‐ A = θ θ r cos cos 11 y z = θ θ θ − θ θ r sin sin cos cos sin 12 x y z x z = θ θ θ + θ θ r cos sin cos sin sin 13 x y z x z = θ θ r cos sin 21 y z = θ θ θ + θ θ r sin sin sin cos cos (2) 22 x y z x z = θ θ θ − θ θ r cos sin sin sin cos 23 x y z x z = − θ r sin 31 y = θ θ r sin cos 32 x y = θ θ r cos cos 33 x y Not all the textbooks and manuals accept this relation; another “definition” of RPY gives origin to an alternative matrix form RPY ‐ B r = cos θ cos θ 11 y z r = − cos θ sin θ 12 x z r = sin θ 13 y r = sin θ sin θ cos θ + cos θ sin θ 21 x y z x z r = − sin θ sin θ sin θ + cos θ cos θ (3) 22 x y z x z r = − sin θ cos θ 23 x y r = − cos θ sin θ cos θ + sin θ sin θ 31 x y z x z r = cos θ sin θ sin θ + sin θ cos θ 32 x y z x z r = cos θ cos θ 33 x y We will use always representation RPY ‐ A in (2), but the reader is advised to check the adopted representation. 3 Solar illumination model ≤ σ ≤ In order to compute the solar illumination parameter 0 1 it is necessary to consider two vectors (see Figure 2): � to the photovoltaic cells array; a) the normal unit vector n � (also named sun vector ) connecting the local rover position to the local Sun b) the unit vector s position in the sky. WPR POLITO ‐ LIM/2010/02.01 3/10
POLITO – LIM Solar cells orientation v.2.doc Figure 2. Relation between cells array normal vector and sun vector Then σ is computed as � � � � ⎧ ⎪ if n s > 0 n s T T ⎪ ⎪ σ = ⎨ � � (4) ⎪ if ≤ 0 n s T 0 ⎪ ⎪ ⎩ Both vectors must be expressed in the same coordinate reference frame R , that in our case can be either R or R . m 0 The solar radiation affecting the cells recharge may depend on many others factors, as the sky obfuscation due to dust or other suspended particles, but these effects are not considered here. 3.1 Vector computation According to onboard sun sensors availability, there are four possibilities for computing the relevant vectors. These are indicated in Table 1. Vectors � � n s Reference frames the computation is moderately if onboard sun sensors are � available, the computation of s complex; one must know the instantaneous pose ( , p R of the ) R is useless; if sun sensors in 0 R rover in the inertial reference are unavailable, the computation 0 frame. Usually this is provided by is moderately difficult, relying on the onboard odometric sensors direct computation of Sun path on the sky (see below) the computation is easy, since the if onboard sun sensors are � available, the computation of s cell arrays are onboard the rover � is a priori known and n R is straightforward; if sun in m sensors are unavailable it is R moderately difficult, relying on m direct computation of Sun path on the sky (see below). � from Transformation of s R 0 R is necessary. to m Table 1. Computation of normal unit vector and sun vector WPR POLITO ‐ LIM/2010/02.01 4/10
POLITO – LIM Solar cells orientation v.2.doc � and matrix R given by the rover roll ‐ R as position p Let us assume that the robot pose is available in 0 pitch ‐ yaw (RPY) angles, as specified in (2); the matrix R allows to transform vectors in R to vectors in m R , as 0 � � = v Rv 0 m � � (5) = v R v T m 0 3.1.1 Unit vector normal to cells array As outlined above, two possibilities arise: R . a) Normal unit vector is expressed in the rover reference frame m In this case the unit vector is easily computed, since the solar panels are usually positioned on the xy � � � � � n = k , where k is the z axis unit vector of n ≠ k R . Otherwise plane as in Figure 1; therefore , m but all the same it is easily available, and can be considered fixed wrt the rover reference frame (tilting panels are not taken into account). R . This case will not be considered here. b) Normal unit vector is in the inertial reference frame 0 3.1.2 Sun vector � in � in Having chosen to represent n R , we must therefore compute s R as and then transform it in m 0 0 R as m � � = s R s T 0 4 Sun path algorithm and sun vector computation φ s t ( ) The position of the Sun in the sky is given by two angles: the azimuth angle and the elevation angle θ s t ( ) , as shown in Figure 1. ��� � φ s t ( ) is the angle between the line OP Azimuth or solar azimuth angle , i.e., the projection on the ground ��� plane π of the line OS from the observer to the Sun, and the line from the observer to the geographical South or North, according to the conventions used. A positive azimuth angle generally indicates that the Sun is East of South or North, and a negative azimuth angle generally indicates the Sun is West of South or North. ��� θ s t ( ) is the angle between the line OS Solar elevation angle connecting the observer with the Sun and the ��� � horizontal line OP . A relation that computes both angles is derived taking into account the various astronomical parameters affecting the Sun azimuth and height. These parameters depend on several factors; the principal ones are: 1. The season of the year. Seasons are annual changes in temperature on a planet caused by a ε ) and variable distance from the Sun. combination of two factors: axial tilt (i.e., ecliptic obliquity 0 On Earth, axial tilt determines nearly all of the annual variation, because Earth's orbit is nearly circular. But Mars has a high orbital eccentricity; indeed the distance from the Sun to Mars varies from 1.64 AU to 1.36 AU over a martian year. This large variation, combined with an axial obliquity slightly greater than Earth's (see Table 2), gives rise to seasonal changes far greater than those on Earth. WPR POLITO ‐ LIM/2010/02.01 5/10
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