STEPS WP 1.I Sun Vector Computation and Solar Cells Orientation for - - PDF document

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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

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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

• where

( , , ) x y z = p

• is the position and R is the orientation; the pose varies in time according to the rover

motion, specified by the wheel commands. Two right‐hand reference frames are considered: a fixed frame

( , , ) i j k R

and a rover centred frame

( , , )

m m m m

i j k

• R

. On Mars

R is usually located at the intersection of the prime meridian and the equator. The prime

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

m

R is located at the barycentre point of the rover, X points toward the positive motion, Z is orthogonal

to the main base frame and Y completes the frame.

m

R may be expressed in R by different

representations; the most widely used is that based on Roll‐Pitch‐Yaw (RPY) angles ( ,

, )

x y z

θ θ θ

that

• riginates the following transform/rotation matrix R ; see B. Bona (2002) and B. Siciliano and O. Khatib

(2008).

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11 12 13 21 22 23 31 32 33

r r r r r r r r r ⎛ ⎞ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ = ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎟ ⎜ ⎝ ⎠ R

(1) with RPY ‐ A

11 12 13 21 22 23 31 32 33

cos cos sin sin cos cos sin cos sin cos sin sin cos sin sin sin sin cos cos cos sin sin sin cos sin sin cos cos cos

y z x y z x z x y z x z y z x y z x z x y z x z y x y x y

r r r r r r r r r θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ = = − = + = = + = − = − = =

(2) Not all the textbooks and manuals accept this relation; another “definition” of RPY gives origin to an alternative matrix form RPY ‐ B

11 12 13 21 22 23 31 32 33

cos cos cos sin sin sin sin cos cos sin sin sin sin cos cos sin cos cos sin cos sin sin cos sin sin sin cos cos cos

y z x z y x y z x z x y z x z x y x y z x z x y z x z x y

r r r r r r r r r θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ = = − = = + = − + = − = − + = + =

(3) We will use always representation RPY‐A in (2), but the reader is advised to check the adopted representation.

3 Solar illumination model

In order to compute the solar illumination parameter 0

1 σ ≤ ≤

it is necessary to consider two vectors (see Figure 2): a) the normal unit vector n

to the photovoltaic cells array;

b) the unit vector s

(also named sun vector) connecting the local rover position to the local Sun

position in the sky.

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Figure 2. Relation between cells array normal vector and sun vector Then σ is computed as

if if σ ⎧ ⎪ > ⎪ ⎪ = ⎨ ⎪ ≤ ⎪ ⎪ ⎩ n s n s n s

T T T

(4) Both vectors must be expressed in the same coordinate reference frame R , that in our case can be either

m

R or R .

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.

Reference frames Vectors

n

• s
• R

the computation is moderately complex; one must know the instantaneous pose ( ,

) p R of the

rover in the inertial reference

• frame. Usually this is provided by

the onboard odometric sensors if onboard sun sensors are available, the computation of s

• in

R is useless; if sun sensors

are unavailable, the computation is moderately difficult, relying on direct computation of Sun path

• n the sky (see below)

m

R

the computation is easy, since the cell arrays are onboard the rover and n

is a priori known

if onboard sun sensors are available, the computation of s

• in

m

R is straightforward; if sun

sensors are unavailable it is moderately difficult, relying on direct computation of Sun path

• n the sky (see below).

Transformation of s

from R

to

m

R is necessary.

Table 1. Computation of normal unit vector and sun vector

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Let us assume that the robot pose is available in

R as position p and matrix R given by the rover roll‐

pitch‐yaw (RPY) angles, as specified in (2); the matrix R allows to transform vectors in

m

R to vectors in R , as

m m

= = v Rv v R v

• T

(5)

3.1.1 Unit vector normal to cells array

As outlined above, two possibilities arise: a) Normal unit vector is expressed in the rover reference frame

m

R .

In this case the unit vector is easily computed, since the solar panels are usually positioned on the xy plane as in Figure 1; therefore

= n k

• , where k
• is the z axis unit vector of

m

R . Otherwise ≠ n k

• ,

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). b) Normal unit vector is in the inertial reference frame

R . This case will not be considered here. 3.1.2 Sun vector

Having chosen to represent n

in

m

R , we must therefore compute s in R as and then transform it in

m

R as = s R s

• T

4 Sun path algorithm and sun vector computation

The position of the Sun in the sky is given by two angles: the azimuth angle

( )

s t

φ

and the elevation angle

( )

s t

θ

, as shown in Figure 1. Azimuth or solar azimuth angle

( )

s t

φ

is the angle between the line OP

• , 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. Solar elevation angle

( )

s t

θ

is the angle between the line OS

• 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

combination of two factors: axial tilt (i.e., ecliptic obliquity

ε ) and variable distance from the Sun.

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.

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• 2. The rover position, measured by latitude and longitude: they are related to the position p

, but

noting that latitude and longitude are given as angles (usually in degrees), while the position p

is

given as linear quantities in meters. During the rover motion latitudes and longitudes vary very little: Mars has a mean circumference of 21,343 km, therefore a degree of longitude corresponds to a little more than 59 km, and 1 km corresponds to approximately 1.01’ of longitude.

• 3. The actual time, i.e., the actual local hour measured wrt the day length (sol), that on Mars is slightly

larger that on Earth; 1 sol = 24h39.5min. Obliquity =

ε

Defined as the planet's axial tilt wrt the ecliptic plane. Earth Mars 23°26'21".4119 = 0.4090926 rad (

sin ε = 0.397776995)

25°19' = 0.441859 rad (

sin ε = 0.397776995)

Table 2. Obliquity of Earth and Mars In order to provide a fast but plausible algorithm for Sun vector computation, the Earth parameters were used instead of the Mars ones.

4.1.1 Local Sun elevation

For any given point on Earth's surface, the Sun elevation angle

( )

s t

θ

can be computed as:

sin cos cos cos sin sin

s L L

H θ δ Φ δ Φ = +

where

• H is the hour angle; 1 hour = 15o;

H =

at 12:00 local hour,

H <

morning hours,

H >

afternoon hours.

• δ is the Sun declination, computed as

( )

360 23.45 sin 284 365 N ⎛ ⎞ ⎟ ⎜ ⎟ + ⎜ ⎟ ⎜ ⎟ ⎟ ⎜ ⎝ ⎠

• .
• N is the day of the year; e.g., March 21st is N =80, June 21st is N =172.
• L

Φ is the local latitude.

The sun elevation angle for N =80,

L

Φ =45 o is given in Figure 3

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Figure 3. Sun elevation for N =80,

L

Φ =45 o 4.1.2 Local Sun azimuth

The second element of the Sun location is its azimuth

( )

s t

φ

. Azimuth angle is positive if toward East, while is negative if toward West. There are two different ways to compute this angle, as follows

sin sin sin cos cos cos

s L s s L

θ Φ δ φ θ Φ − =

(6)

• r

cos cos sin sin cos cos cos

L L s s

H δ Φ δ Φ φ θ − =

(7) The Sun azimuth for N =80,

L

Φ = 45 o is given in

Figure 4.

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Figure 4. Sun azimuth for N =80,

L

Φ = 45 o

4.2 Sun vector

Sun vector is a unit vector

1 = s

• computed in reference frame

R as

01 02 03

sin cos cos cos sin

s s s s s

s s s φ θ φ θ θ ⎛ ⎞ ⎛ ⎞ ⎟ ⎟ ⎜ ⎜ ⎟ ⎟ ⎜ ⎜ ⎟ ⎟ ⎜ ⎜ ⎟ ⎟ ⎜ ⎜ = = − ⎟ ⎟ ⎜ ⎜ ⎟ ⎟ ⎜ ⎜ ⎟ ⎟ ⎜ ⎜ ⎟ ⎟ ⎟ ⎟ ⎜ ⎜ ⎝ ⎠ ⎝ ⎠ s

• (8)

According to (4) one must compute n s

T

in

m

R ; assuming

( )

1 = n T

and applying relation (5), one

• btains

( )

01 13 23 33 02 13 01 23 02 33 03 03

s r r r s r s r s r s s σ ⎛ ⎞ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ = = = + + ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎟ ⎜ ⎝ ⎠ n R s

• T

T

(9) and, in particular, assuming RPY angles are those in RPY‐A (2)

( ) ( ) ( )

sin cos cos sin cos sin sin cos cos cos sin sin sin cos sin cos cos

s s x y z x z s s x y z x z s x y

σ φ θ θ θ θ θ θ φ θ θ θ θ θ θ θ θ θ = + − − +

(10) while assuming RPY angles are those in RPY‐B (3)

sin cos sin cos cos sin cos sin cos cos

s s y s s x y s x y

σ φ θ θ φ θ θ θ θ θ θ = + +

(11)

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As an example, we have computed

s and σ for N =80,

L

Φ =45 o; the results are reported in Figure 5

and Figure 5. Sun vector

s for N =80,

L

Φ = 45 o

Figure 6. Solar illumination parameter σ for N =80,

L

Φ =45 o

5 Final comments

The landing sites on Mars of the two NASA Rovers, Spirit and Opportunity, are reported in Table 3

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Rover Name Latitude Longitude Spirit 14.57°S 175.48°E Opportunity 1.95°S 354.47°E Table 3. NASA MER’s landing sites A realistic simulation of the computation of the various quantities described in this Report can be carried

• ut considering a rover position not too far from the equator ( approximately 0° Latitude) and around 0°
• Longitude. The simulation can start at mid‐morning and end a couple of hours later (10:00‐12:00 or 12:00‐

14:00). Figure 7. Spirit (left) and Opportunity (right) landing sites

6 References

Bona B. (2002) Modellistica dei Robot Industriali (2002), CELID, Torino. Siciliano B., Khatib O. (Eds.) (2008), Springer Handbook of Robotics, Springer. De Vaucouleurs G., Davies M.E., Sturms F.M. (1973), “Mariner 9 areographic coordinate system.” J.

• Geophys. Res. 78, 4395‐4404.

7 Acronyms and Abbreviations

AU = Astronomical Units MER = Mars Exploration Rover MTC = Coordinated Mars Time RPY = Roll‐Pitch‐Yaw UTC = Coordinated Universal Time wrt = with respect to