Derivation of EOM in polar coordinates Julio C. Gonzalez-Saenz School of System Engineering, University of Reading, Unit...

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Derivation of EOM in polar coordinates Julio C. Gonzalez-Saenz School of System Engineering, University of Reading, United Kingdom, RG30 2PN [email protected] January 15, 2015

1

Introduction

In this article the two-dimension equations of motion (EOM) of a spacecraft (S/C) in a rotating polar coordinate system centred at the Moon will be deduced. In many books [1, 2, 3, 4] and articles in the literature, the following equations are defined as the EOM in 2D: r˙ = vr v✓ ✓˙ = r Tmax k µm v ✓ 2 v˙r = sin( ) + !2r m r2 r Tmax k vr v✓ v˙✓ = cos( ) 2vr ! m r Tmax k m ˙ = Isp g0

(1) (2) 2!v✓

(3) (4) (5)

where r is the radio vector from the center of the rotating system to the S/C ✓ is angle between the S/C and the r axis vr is the velocity of the S/C in the radial direction v✓ is the velocity of the S/C in the ✓ direction v˙r is the acceleration of the S/C in the radial direction v˙r is the acceleration of the S/C in the ✓ direction Tmax is the maximum thrust from the engines k is the variable to control the start/stop sequence of the engines µm is the Moon’s gravitational constant (4902.78 km3 /s2 ) ! is the angular speed of the rotating system equivalent to the Moon’s own rotation speed (2.66 ⇥10 6 ) is the angle of the thrust vector with the local horizon Isp is the specific impulse of S/C engine g0 is the Earth gravity acceleration at see level (0.000981km/s2 ) The aim of this paper is presenting the formal derivation of the equations of motion (EOM) in Polar Coordinate since in the literature these equations are given but not clearly deduced. It will be shown that the equations of motion (1) - (5) are correctly defined for a reference system which is rotating with respect to an inertial reference system located at the center of the Moon. 1

Figure 1: Inertial Coordinate and Moon Centred Coordinate systems Figure 1 shows the reference system in which the equations of motion will be deduced. Notice that there is only the force due to the Thrust, T, of the engines. The unit vector in the r direction will be denoted by er and the unit vector in the ✓ direction will be denoted by e✓ .

2

Deduction of the Equations of Motion

Before defining the equations of motion, the concept of derivative of a vector in a rotating frame of reference with respect to an inertial reference system will be introduced. The inertial system will be the Moon-Centred frame, F, and the Moon-Centred, Moon-Fixed frame will be the rotating system B, the derivative of a vector in one frame of reference with respect to another frame of reference is given by the following formula: F

A=

B

A+!⇥A

(6)

As the origin of the two frames, the inertial and the rotating frame coincide in the same point, the relative velocity of the origin of the rotating frame with respect to the inertial frame is zero. Therefore, the velocity of the particle with respect to the inertial frame is given by: vP/F = vP/B + ! ⇥ r

(7)

where ! is pointing outward in the z-direction, ortogonal to r and ✓-axis. In figure 1 the spacecraft (S/C) is located at a distance r from the center of the inertial coordinate system (Moon) and and angle ✓ is measured clockwise in the ✓-axis. The particle is moving with a velocity v, (r) and with an acceleration a (r). The direction of the thrust is the angle and is measured counterclockwise from the ✓-Axis. vr is parallel to the r-Axis and v✓ is perpendicular to vr . The vector position of the spacecraft is given by: r = rer

(8)

The velocity of the spacecraft in the inertial system is the derivative of equation (9) with respect to t and is given by: r˙ = re ˙ r + re˙r + (! ⇥ rer ) ˙ ✓ ) + r(! ⇥ er ) = re ˙ r + r(✓e ˙ ✓ + (! ⇥ r) = re ˙ r + r✓e 2

(9) (10) (11)

The acceleration of the spacecraft is the time derivative of equation (12) and it is given by the following equation: ¨r = r¨er + r˙ e˙r + r˙ ✓e ˙ + ! ⇥ (r˙ + (! ⇥ r)) ˙ ✓ + r✓e ¨ ✓ + r✓˙e˙✓ + (!˙ ⇥ r) + (! ⇥ r) ˙ + (! ⇥ r) ˙ + (! ⇥ (! ⇥ r)) ˙ ✓ + r✓e ¨ ✓ + r✓˙e˙✓ + (!˙ ⇥ r) + (! ⇥ r) = r¨er + r˙ e˙r + r˙ ✓e ˙ + (! ⇥ (! ⇥ r)) ˙ ✓ + r˙ ✓e ˙ ✓ + r✓e ¨ ✓ r(✓) ˙ er + (!˙ ⇥ r) + 2(! ⇥ r) = r¨er + r˙ ✓e ˙ + (! ⇥ (! ⇥ r)). ˙ ✓ + (!˙ ⇥ r) + 2(! ⇥ r) = (¨ r r✓˙2 )er + (r✓¨ + 2r˙ ✓)e 2

(12) (13) (14) (15)

Now, considering the last 3 terms of equation (15). Because the Moon is moving with a constant angular speed !ez , the angular acceleration, !˙ = 0

(16)

It is already known that: (ez ⇥ er ) = e✓

(17)

(ez ⇥ ez ) = 0

(19)

(ez ⇥ e✓ ) =

er

(18)

˙ is equal to: The term 2(! ⇥ r) ˙ = 2(!ez ⇥ r) ˙ 2(! ⇥ r)

(20)

˙ ✓ )) = 2!(ez ⇥ (re ˙ r + r✓e ˙ z ⇥ e✓ )) = 2!(re ˙ z ⇥ er + r✓(e ˙ er )) = 2!(re ˙ ✓ + r✓( ˙ r = 2! re ˙ ✓ 2!r✓e

(21) (22) (23) (24)

The last term, (! ⇥ (! ⇥ r)) can be simplified: (! ⇥ (! ⇥ r)) = !ez ⇥ (!ez ⇥ rer ) = r! 2 ez ⇥ (ez ⇥ er ) 2

= r! ez ⇥ e✓ )

(25) (26) (27)

2

(28)

(29)

= (¨ r

˙ ✓ + 2! re ˙ r r! 2 er r✓˙2 )er + (r✓¨ + 2r˙ ✓)e ˙ ✓ 2!r✓e ˙ r + (r✓¨ + 2r˙ ✓˙ + 2! r)e r✓˙2 r! 2 2!r✓)e ˙ ✓.

= (¨ r

r✓˙2

(31)

=

r! er

Then, substituting (16), (24) and (28) in (15), ¨r = (¨ r

˙ r + (r✓¨ + r˙ ✓˙ + r˙ ✓˙ + 2! r)e 2!r✓)e ˙ ✓.

r! 2

(30)

˙ + r˙ ✓˙ which is equal to d(r✓)/dt ˙ ˙ Moreover, Now, r✓¨+ r˙ ✓˙ + r˙ ✓˙ can be grouped as (r✓¨+ r˙ ✓) + r˙ ✓. ˙ d(r✓)/dt is equivalent to a✓ , so equation (31) can be written equivalently: ¨r = (ar

r✓˙2

r! 2

˙ r + (a✓ + r˙ ✓˙ + ! r)e 2!r✓)e ˙ ✓.

(32)

Using the equations (1) and (2) and replacing them in equation (32): ¨r = (ar

v✓ 2 r

r! 2

2!v✓ )er + (a✓ +

3

vr v✓ + 2!vr )e✓ . r

(33)

Using Newton’s 2nd Law, (Fr er + F✓ e✓

m

P

F = ma, v✓ 2 r

µm er )/m = (ar r2

r! 2

2!v✓ )er + (a✓ +

vr v✓ + 2!vr )e✓ . r

(34)

simplifying (34), µm v✓ 2 = (a r! 2 2!v✓ ) r r2 r vr v✓ F✓ /m = (a✓ + + 2!vr ). r Substituting v˙r = ar in (35) and v˙✓ = a✓ in (36) Fr /m

Fr /m

µm v✓ 2 = ( v ˙ r! 2 2!v✓ ) r r2 r vr v✓ F✓ /m = (v˙✓ + + 2!vr ) r

(35) (36)

(37) (38)

Resolving, µm v✓ 2 + + r! 2 + 2!v✓ r2 r vr v✓ v˙✓ = F✓ /m 2!vr r Equation (1) and (2) can be deduced directly from equation (11): v˙r = Fr /m

˙ ✓ + (! ⇥ r) vr er + v✓ e✓ = re ˙ r + r✓e

(39) (40)

(41)

vr = r˙

(42)

v✓ = r✓˙

(43)

The component perpendicular to vr and v✓ is zero; therefore: r˙ = vr v✓ ✓˙ = r

3

(44) (45)

Conclusion

It has been demonstrated that system of equation (1) - (5) represent the equations of motion of a particle in a reference system moving with respect to an inertial coordinate system centred in the Moon. Equation (5) is the rocket equation deduced by Tsiolkovsky towards the end of the 19th century which the deduction can be found in any rocket book, for example [5].

References [1] A. E. Bryson, Dynamic Optimization. Adyson Wesley Longman, Inc., 1999. [2] A. E. Bryson and Y.-C. Ho, Applied Optimal Control. Taylor & Francis, 1975. [3] B.-G. Park and M.-J. Thak, “Three-dimensional trajectory optimization of soft lunar landing from the parking orbit with considerations of the landing site,” International Journal of Control, Autimation and Systems, vol. 9, no. 6, pp. 1164–1172, 2011. [4] B.-G. Park, J.-S. Ahn, and M.-J. Thak, “Two-dimensional trajectory optimization for soft lunar landing considering a landing site,” International Journal of Aerospace & Space Sci., vol. 12, no. 3, pp. 288–295, 2011. [5] H. D. Curtis, Orbital Mechanis for Engineering Students. Elsevier, 2005.

4

1

Introduction

In this article the two-dimension equations of motion (EOM) of a spacecraft (S/C) in a rotating polar coordinate system centred at the Moon will be deduced. In many books [1, 2, 3, 4] and articles in the literature, the following equations are defined as the EOM in 2D: r˙ = vr v✓ ✓˙ = r Tmax k µm v ✓ 2 v˙r = sin( ) + !2r m r2 r Tmax k vr v✓ v˙✓ = cos( ) 2vr ! m r Tmax k m ˙ = Isp g0

(1) (2) 2!v✓

(3) (4) (5)

where r is the radio vector from the center of the rotating system to the S/C ✓ is angle between the S/C and the r axis vr is the velocity of the S/C in the radial direction v✓ is the velocity of the S/C in the ✓ direction v˙r is the acceleration of the S/C in the radial direction v˙r is the acceleration of the S/C in the ✓ direction Tmax is the maximum thrust from the engines k is the variable to control the start/stop sequence of the engines µm is the Moon’s gravitational constant (4902.78 km3 /s2 ) ! is the angular speed of the rotating system equivalent to the Moon’s own rotation speed (2.66 ⇥10 6 ) is the angle of the thrust vector with the local horizon Isp is the specific impulse of S/C engine g0 is the Earth gravity acceleration at see level (0.000981km/s2 ) The aim of this paper is presenting the formal derivation of the equations of motion (EOM) in Polar Coordinate since in the literature these equations are given but not clearly deduced. It will be shown that the equations of motion (1) - (5) are correctly defined for a reference system which is rotating with respect to an inertial reference system located at the center of the Moon. 1

Figure 1: Inertial Coordinate and Moon Centred Coordinate systems Figure 1 shows the reference system in which the equations of motion will be deduced. Notice that there is only the force due to the Thrust, T, of the engines. The unit vector in the r direction will be denoted by er and the unit vector in the ✓ direction will be denoted by e✓ .

2

Deduction of the Equations of Motion

Before defining the equations of motion, the concept of derivative of a vector in a rotating frame of reference with respect to an inertial reference system will be introduced. The inertial system will be the Moon-Centred frame, F, and the Moon-Centred, Moon-Fixed frame will be the rotating system B, the derivative of a vector in one frame of reference with respect to another frame of reference is given by the following formula: F

A=

B

A+!⇥A

(6)

As the origin of the two frames, the inertial and the rotating frame coincide in the same point, the relative velocity of the origin of the rotating frame with respect to the inertial frame is zero. Therefore, the velocity of the particle with respect to the inertial frame is given by: vP/F = vP/B + ! ⇥ r

(7)

where ! is pointing outward in the z-direction, ortogonal to r and ✓-axis. In figure 1 the spacecraft (S/C) is located at a distance r from the center of the inertial coordinate system (Moon) and and angle ✓ is measured clockwise in the ✓-axis. The particle is moving with a velocity v, (r) and with an acceleration a (r). The direction of the thrust is the angle and is measured counterclockwise from the ✓-Axis. vr is parallel to the r-Axis and v✓ is perpendicular to vr . The vector position of the spacecraft is given by: r = rer

(8)

The velocity of the spacecraft in the inertial system is the derivative of equation (9) with respect to t and is given by: r˙ = re ˙ r + re˙r + (! ⇥ rer ) ˙ ✓ ) + r(! ⇥ er ) = re ˙ r + r(✓e ˙ ✓ + (! ⇥ r) = re ˙ r + r✓e 2

(9) (10) (11)

The acceleration of the spacecraft is the time derivative of equation (12) and it is given by the following equation: ¨r = r¨er + r˙ e˙r + r˙ ✓e ˙ + ! ⇥ (r˙ + (! ⇥ r)) ˙ ✓ + r✓e ¨ ✓ + r✓˙e˙✓ + (!˙ ⇥ r) + (! ⇥ r) ˙ + (! ⇥ r) ˙ + (! ⇥ (! ⇥ r)) ˙ ✓ + r✓e ¨ ✓ + r✓˙e˙✓ + (!˙ ⇥ r) + (! ⇥ r) = r¨er + r˙ e˙r + r˙ ✓e ˙ + (! ⇥ (! ⇥ r)) ˙ ✓ + r˙ ✓e ˙ ✓ + r✓e ¨ ✓ r(✓) ˙ er + (!˙ ⇥ r) + 2(! ⇥ r) = r¨er + r˙ ✓e ˙ + (! ⇥ (! ⇥ r)). ˙ ✓ + (!˙ ⇥ r) + 2(! ⇥ r) = (¨ r r✓˙2 )er + (r✓¨ + 2r˙ ✓)e 2

(12) (13) (14) (15)

Now, considering the last 3 terms of equation (15). Because the Moon is moving with a constant angular speed !ez , the angular acceleration, !˙ = 0

(16)

It is already known that: (ez ⇥ er ) = e✓

(17)

(ez ⇥ ez ) = 0

(19)

(ez ⇥ e✓ ) =

er

(18)

˙ is equal to: The term 2(! ⇥ r) ˙ = 2(!ez ⇥ r) ˙ 2(! ⇥ r)

(20)

˙ ✓ )) = 2!(ez ⇥ (re ˙ r + r✓e ˙ z ⇥ e✓ )) = 2!(re ˙ z ⇥ er + r✓(e ˙ er )) = 2!(re ˙ ✓ + r✓( ˙ r = 2! re ˙ ✓ 2!r✓e

(21) (22) (23) (24)

The last term, (! ⇥ (! ⇥ r)) can be simplified: (! ⇥ (! ⇥ r)) = !ez ⇥ (!ez ⇥ rer ) = r! 2 ez ⇥ (ez ⇥ er ) 2

= r! ez ⇥ e✓ )

(25) (26) (27)

2

(28)

(29)

= (¨ r

˙ ✓ + 2! re ˙ r r! 2 er r✓˙2 )er + (r✓¨ + 2r˙ ✓)e ˙ ✓ 2!r✓e ˙ r + (r✓¨ + 2r˙ ✓˙ + 2! r)e r✓˙2 r! 2 2!r✓)e ˙ ✓.

= (¨ r

r✓˙2

(31)

=

r! er

Then, substituting (16), (24) and (28) in (15), ¨r = (¨ r

˙ r + (r✓¨ + r˙ ✓˙ + r˙ ✓˙ + 2! r)e 2!r✓)e ˙ ✓.

r! 2

(30)

˙ + r˙ ✓˙ which is equal to d(r✓)/dt ˙ ˙ Moreover, Now, r✓¨+ r˙ ✓˙ + r˙ ✓˙ can be grouped as (r✓¨+ r˙ ✓) + r˙ ✓. ˙ d(r✓)/dt is equivalent to a✓ , so equation (31) can be written equivalently: ¨r = (ar

r✓˙2

r! 2

˙ r + (a✓ + r˙ ✓˙ + ! r)e 2!r✓)e ˙ ✓.

(32)

Using the equations (1) and (2) and replacing them in equation (32): ¨r = (ar

v✓ 2 r

r! 2

2!v✓ )er + (a✓ +

3

vr v✓ + 2!vr )e✓ . r

(33)

Using Newton’s 2nd Law, (Fr er + F✓ e✓

m

P

F = ma, v✓ 2 r

µm er )/m = (ar r2

r! 2

2!v✓ )er + (a✓ +

vr v✓ + 2!vr )e✓ . r

(34)

simplifying (34), µm v✓ 2 = (a r! 2 2!v✓ ) r r2 r vr v✓ F✓ /m = (a✓ + + 2!vr ). r Substituting v˙r = ar in (35) and v˙✓ = a✓ in (36) Fr /m

Fr /m

µm v✓ 2 = ( v ˙ r! 2 2!v✓ ) r r2 r vr v✓ F✓ /m = (v˙✓ + + 2!vr ) r

(35) (36)

(37) (38)

Resolving, µm v✓ 2 + + r! 2 + 2!v✓ r2 r vr v✓ v˙✓ = F✓ /m 2!vr r Equation (1) and (2) can be deduced directly from equation (11): v˙r = Fr /m

˙ ✓ + (! ⇥ r) vr er + v✓ e✓ = re ˙ r + r✓e

(39) (40)

(41)

vr = r˙

(42)

v✓ = r✓˙

(43)

The component perpendicular to vr and v✓ is zero; therefore: r˙ = vr v✓ ✓˙ = r

3

(44) (45)

Conclusion

It has been demonstrated that system of equation (1) - (5) represent the equations of motion of a particle in a reference system moving with respect to an inertial coordinate system centred in the Moon. Equation (5) is the rocket equation deduced by Tsiolkovsky towards the end of the 19th century which the deduction can be found in any rocket book, for example [5].

References [1] A. E. Bryson, Dynamic Optimization. Adyson Wesley Longman, Inc., 1999. [2] A. E. Bryson and Y.-C. Ho, Applied Optimal Control. Taylor & Francis, 1975. [3] B.-G. Park and M.-J. Thak, “Three-dimensional trajectory optimization of soft lunar landing from the parking orbit with considerations of the landing site,” International Journal of Control, Autimation and Systems, vol. 9, no. 6, pp. 1164–1172, 2011. [4] B.-G. Park, J.-S. Ahn, and M.-J. Thak, “Two-dimensional trajectory optimization for soft lunar landing considering a landing site,” International Journal of Aerospace & Space Sci., vol. 12, no. 3, pp. 288–295, 2011. [5] H. D. Curtis, Orbital Mechanis for Engineering Students. Elsevier, 2005.

4

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