Vehicle position keeping system

• US 6,963,795 B2
• Filed: 07/16/2002
• Issued: 11/08/2005
• Est. Priority Date: 07/16/2002
• Status: Expired due to Fees

• Current Assignee: Honeywell International Inc.
• Original Assignee: Honeywell International Inc.
• Inventors: Haissig, Christine M., Euteneuer, Eric, Samanant, Paul P., Enns, Dale
• Primary Examiner(s): Keith, Jack
• Assistant Examiner(s): Mancho, Ronnie
• Application Number:
  US10/198,491
• Publication Number:
  US 20050165516A1
• Time in Patent Office: 1,211 Days
• Field of Search: 701 4- 9, 701/11, 701/221, 701/301, 701/213, 342/29, 342/30, 342/36, 342/455, 342/465, 342/179, 342/181, 342/107, 342/126, 342/104, 340/972
• US Class Current: 701/7
• CPC Class Codes: G05D 1/104 involving a plurality of aircrafts, e.g. formation flying traffic control systems for aircraft G08G5/00
• CPC Class Codes: G08G 5/0008 with other aircraft
• CPC Class Codes: G08G 5/0052 for cruising combined instruments indicating more than one navigational value G01C23/00

Abstract

A position keeping system, useable to maintain a position of a vehicle in a vehicle formation, to keep all formation members flying at a specified velocity and altitude, and to perform commanded maneuvers, determines guidance corrections based on a track state referenced to a leader or a virtual leader of the formation and on a formation position error indicating the error in the position of the vehicle in the formation. The guidance corrections are required so as to maintain the vehicles in the formation.

Claims

1. A method of keeping position of a vehicle in a vehicle formation comprising:

acquiring a track velocity input;

acquiring a formation error input;

determining a velocity output for the vehicle based on the track velocity input and the formation position error, wherein the velocity output indicates a velocity required to maintain the vehicle in the vehicle formation,

wherein the determining of a velocity output compromises performing a dynamic inversion based on the track velocity input and the formation position error input in order to determine the velocity output.

2. The method of claim 1 further compromising:

converting the velocity output to an acceleration output;

acquiring a commanded acceleration input; and

determining guidance corrections based on the acceleration output and the commanded acceleration input.

3. The method of claim 2 wherein the guidance corrections comprise an acceleration required to make formation corrections and to implement commanded maneuvers.

4. The method of claim 3 further comprising displaying the acceleration.

5. The method of claim 2 wherein the guidance corrections comprise a velocity to make formation corrections and to implement commanded maneuvers.

6. The methods of claim 5 further comprising displaying the velocity.

7. The method of claim 2 further comprising displaying the guidance corrections to an operator of the vehicle to enable the operator to make corrections to a position of the vehicle in the vehicle formation.

8. The method of claim 2 further comprising supplying the guidance corrections to an automatic pilot.

9. The method of claim 2 further comprising supplying the guidance corrections to a flight management system.

10. The method of claim 2 wherein the track velocity input comprises ground speed, track angle, and flight path angle, and wherein the commanded acceleration input comprises a commanded speed change, a commanded track angle rate, and a commanded flight path angle rate.

11. The method of claim 10 wherein the acquiring of a track velocity input comprises converting the ground speed, the track angle, and the flight path angle into directional track state components based on a reference frame, wherein the acquiring of a formation position error input comprises acquiring directional formation position error components based on the reference frame, and wherein the determining of a velocity output comprises:

adding the directional track states components to the directional formation position error components to produce directional output components; and

,

dynamically inverting the directional output components so as to produce the velocity output including a desired track angle, a desired ground speed, and a desired flight path angle.

12. The method of claim 2 wherein the tack velocity comprises track air speed, track heading angle, and track flight path angle, and wherein the commanded acceleration input comprises a commanded air speed change, a commanded heading angle rate, and a commanded flight path angle rate.

13. The method of claim 2 wherein the acquiring of a track velocity input comprises converting the track velocity input into a directional track velocity input, wherein the acquiring of a formation position error input comprises acquiring a directional formation position error input, and wherein the determining of a velocity output comprises:

adding the directional track velocity input to the directional formation position error input to produce a directional output; and

,

dynamically inverting the directional output so as to produce the velocity output.

14. The method of claim 13 wherein the guidance corrections comprise accelerations required to make formation corrections.

15. The method of claim 1 wherein the acquiring of a track velocity input comprises converting the track velocity input into a directional track velocity input, wherein the acquiring of a formation position error input comprises acquiring a directional formation position error input, and wherein the determining of a velocity output comprises:

adding the direction track velocity input to the directional formation position error to produce a directional output; and

,

dynamically inverting the directional output so as to produce the velocity output.

16. The method of claim 15 further comprising displaying information required to make formation corrections.

17. The method of claim 1 wherein the track velocity input is based at least partially on information derived from a leader of the vehicle formation.

18. A method of keeping a position of a vehicle in a vehicle formation comprising:

receiving leader information over a communication link;

acquiring relative position information from a relative position measurement system;

selecting at least one of the leader information and the relative position information so as to provide selected information; and

,

determining guidance corrections based on the selected information, wherein the guidance corrections indicate corrections that are required to place the vehicle in the vehicle formation, wherein the selected information includes a track velocity and a formation position error related to the vehicle, wherein the determining of guidance corrections includes determining a velocity and the formation position error, wherein the velocity output comprises a velocity required to maintain the vehicle in the vehicle formation, wherein the selected information includes a commanded acceleration input, and wherein the determining of the guidance corrections includes;

converting the velocity output to an acceleration output; and

,

determining the guidance corrections based on the acceleration output and the commanded acceleration input,

wherein the track velocity comprises ground speed, track angle, and flight path angle, and wherein the commanded acceleration input comprises a commanded speed rate, a commanded track angle rate, and a commanded flight path angle rate.

19. The method of claim 18 wherein the relative position measurement system comprises a traffic collision avoidance system.

20. The method of claim 18 wherein the leader information is received from a formation leader.

21. The method of claim 18 wherein the leader information is received from another vehicle.

22. The method of claim 18 wherein the leader information is received from a ground station.

23. The method of claim 18 further comprising receiving maneuver command information via the communication link, wherein the selecting of at least one of the leader information and the relative position comprises selecting at least on of the leader information, command information so as to provide the selected information.

24. The method of claim 23 wherein the maneuver command information is received from a formation leader.

25. The method of claim 23 where in the maneuver command information is received from another vehicle.

26. The method of claim 23 wherein the maneuver command information is received from a ground station.

27. The method of claim 18 wherein the guidance corrections comprise a command speed guidance V_c and a commanded flight path angle guidance V_c required to make formation corrections.

28. The method of claim 27 further comprising displaying the commanded speed guidance V_c and the commanded flight path angle guidance V_c.

29. The method of claim 18 wherein the guidance corrections comprise a commanded bank angle μ

_c required to make formation corrections.

30. The method of claim 18 wherein the guidance corrections comprise a thrust guidance TC required to make formation corrections.

31. The method of claim 18 wherein the guidance corrections comprise an angle of attack guidance α

_c required to make formation corrections.

32. The method of claim 18 further comprising displaying the guidance corrections to an operator of the vehicle to enable the operator to make corrections to a position of the vehicle in the vehicle formation.

33. The method of claim 18 wherein the formation position error comprises directional formation position error rate components based on a reference frame, and wherein the determining of a velocity output comprises:

converting the ground speed, the track angle, and the flight path angle into directional track position rate components based on the reference frame;

adding the directional track position components to the directional formation position error or rate components to produce directional output rate components; and

,

dynamically inverting the directional output rate components so as to produce a desired track angle, a desired ground speed, and a desired flight path angle required to maintain the vehicle formation.

34. The method of claim 33 wherein the determining the guidance corrections based on the acceleration output and the commanded acceleration input comprises:

converting the desired track angle, the desired ground speed, and the desired flight path angle to a desired track angle rate, and desired speed rate, and a desired flight path angle rate; and

,

adding the desired track angle rate, the desired speed rate, and the desired flight path angle rate to the commanded track angle rate, the commanded speed rate, and the commanded flight path angle rate, respectively, in order to produce the guidance corrections.

35. The method of claim 34 further comprising displaying the guidance corrections to an operator of the vehicle.

36. A method of keeping position of a vehicle in a vehicle formation comprising:

acquiring a rate of the vehicle;

converting the vehicle rate to a directional vehicle rate;

acquiring a direction formation position error rate dependent upon an error in the position of the vehicle formation;

determining a desired vehicle rate based on the directional vehicle rate and the directional formation position error rate, wherein the desired vehicle rate indicates a rate required to maintain the vehicle in the vehicle formation;

converting the desired vehicle rate to a vehicle acceleration input;

acquiring a commanded acceleration; and

,

determining guidance corrections for the vehicle based on the vehicle acceleration input and the commanded acceleration, wherein the guidance corrections indicate guidance corrections required to maintain the vehicle in the vehicle formation and to respond to the commanded acceleration.

37. The method of claim 36 wherein the acquiring of a commanded acceleration comprises acquiring the commanded acceleration from the leader in the vehicle formation.

38. The method of claim 36 where in the acquiring of a commanded acceleration comprises acquiring the commanded acceleration from another vehicle.

39. The method of claim 36 wherein the acquiring of a commanded acceleration comprises acquiring the commanded acceleration from a ground station.

40. The method of claim 36 wherein the acquiring of a commanded acceleration comprises acquiring the commanded acceleration from a memory on the vehicle.

41. The method of claim 36 wherein the guidance corrections comprise velocity guidance.

42. The method of claim 36 where in the guidance corrections comprise acceleration guidance.

43. The method of claim 36 further comprising displaying the guidance corrections to an operator of the vehicle.

44. The method of claim 36 wherein the vehicle rate comprises a vehicle ground speed v_track, a vehicle track and angle x_track, and a vehicle flight path angle y_track, wherein the commanded acceleration comprises a commanded acceleration v_dottrack, x_dottrack, and y_dottrack, wherein the desired vehicle rate comprises a desired vehicle rate v^c, x^c, and y^c, wherein the converting of the vehicle rate to a directional vehicle rate comprises converting the vehicle ground speed, vehicle track angle, and vehicle flight path angle to a North position rate ξ

_dot^t, and East position rate η

_dot^t, and a vertical position rate h_dot^t, wherein the acquiring of a directional formation position error rate comprises acquiring a North formation position error rate K_ε

e_ε

, and east formation position error rate K_η

e_η

, and a vertical formation position error Rate K_he_h, wherein the determining of a desired vehicle rate comprises (i) adding the North position rate ξ

_dot^t and the North formation position error rate K_ε

e_ε

to produce a North error rate ξ

_dot^d, (ii) adding the East position rate η

_dot^t and the East error rate η

_dot^d, (iii) adding the vertical position rate ξ

_dot^t and the vertical formation position error rate K_he_h to produce a vertical error rate h_dot^d, and (iv) performing a dynamic inversion of the North error rate ξ

_dot^d, the East error rate η

_dot^d, and the vertical error rate h_dot^d in order to determining the desired vehicle rate v^c, x^c, and y^c, wherein the converting of the desired vehicle rate v^c, x^c, and y^c to a vehicle acceleration input comprises converting the desired vehicle rate to a vehicle acceleration input V_dot, X_dot, Y_dot, and wherein the determining of the guidance corrections comprises adding the vehicle acceleration input V_dot^t, X_dot^t, Y_dot^t, to the commanded acceleration V_dottrack, X_dottrack, Y_dottrack, to produce a commanded acceleration V_dot^d, X_dot^d, Y_dot^d, and determining the guidance corrections based on the commanded acceleration V_dot^d, X_dot^d, Y_dot^d.

45. The method of claim 44 further comprising displaying the guidance corrections to an operator of the vehicle.

46. The method of claim 44 wherein the vehicle rate is converted to the North position rate ξ

_dot^t, the East position rate η

_dot^t, and the vertical position rate h_dot^t based on the following equations;

ξ

_dot^t=v_trackcos(y_track)cos(x_track)

 

η

_dot^t=v_trackcos(y_track)sin(x_track)

h_dot^t=v_tracksin(y_track).

47. The method of claim 44 wherein the desired vehicle rate v^c, x^c, and y^c is determined based on the following equations:

$v^c=\mathrm{sign}<br><br>\sqrt{{\left({\mathrm{\xi <br><br>}}_{\mathrm{dot}}^d\right)}^2+{\left({\mathrm{\eta <br><br>}}_{\mathrm{dot}}^d\right)}^2+{\left(h_{\mathrm{dot}}^d\right)}^2}$ 

X^c′

=tan<sup>−

1</sup>(η

_dot^d/ε

_dot^d) $\begin{array}{c}\mathrm{sign}=<br><br>1<br><br>\text{\hspace{1em}<br><br>}<br><br>\mathrm{if}<br><br>\text{\hspace{1em}<br><br>}<br><br><br><br>x^{c^{\mathrm{\prime <br><br>}}}<br><br>\le <br><br>\mathrm{\pi <br><br>}/2<br><br>\text{\hspace{1em}<br><br>}\\ =<br><br>-1<br><br>\text{\hspace{1em}<br><br>}<br><br>\mathrm{if}<br><br>\text{\hspace{1em}<br><br>}<br><br><br><br>x^{c^{\mathrm{\prime <br><br>}}}<br><br>><br><br>\mathrm{\pi <br><br>}/2\\ X^{c^{\mathrm{\prime <br><br>}}}=<br><br>{\tan }^{-1}<br><br>\left({\mathrm{\eta <br><br>}}_{\mathrm{dot}}^d/{\mathrm{\epsilon <br><br>}}_{\mathrm{dot}}^d\right)\\ X^c=<br><br>X^c<br><br>{\mathrm{\eta <br><br>}}_{\mathrm{dot}}^t<br><br>\text{\hspace{1em}<br><br>}<br><br>\mathrm{if}<br><br>\text{\hspace{1em}<br><br>}<br><br><br><br>x^{c^{\mathrm{\prime <br><br>}}}<br><br>\le <br><br>\mathrm{\pi <br><br>}/2\\ =<br><br>\mathrm{\pi <br><br>}-X^{c^{\mathrm{\prime <br><br>}}}<br><br>\text{\hspace{1em}<br><br>}<br><br>\mathrm{if}<br><br>\text{\hspace{1em}<br><br>}<br><br><br><br>x^{c^{\mathrm{\prime <br><br>}}}<br><br>><br><br>\mathrm{\pi <br><br>}/2\\ Y^c=<br><br>{\sin }^{-1}<br><br>\left(h_{\mathrm{dot}}^d/v^c\right).\end{array}$ 

Y^c=sin<sup>−

1</sup>(h_dot^d/v^c).

48. The method of claim 44 wherein the guidance corrections comprise v_c, y_c, μ

_c, α

_c, and T_c, wherein the guidance corrections v_c, y_c, μ

_c, α

_c, and T_c are determined based on the following equations;

v_c=v+┐

v_dot^d

y_c=y+┐

y_dot^d

μ

_c=tan<sup>−

1</sup>(v cos(y)x_dot^d)/(vy_dot^d+g cos(y))

tan α

_c=((mvy_dot^d+mg cos(y/cos(μ

_c)−

L/(mv_dot^d+D+mg sin(y))

T_c=(mv_dot^d+D+mg sin(y)/cos(α

_c)

wherein v and y represent a current state of the vehicle, wherein L and D are modeled according to the following equations;

L=qSC_L

D=qSC_D

C_L=C_Lα

₀+C_Lα

α

C_D=C_Dmin+KC_L²

wherein q is dynamic pressure, and wherein K, C_Dmin, c_Lα

₀, C_Lα

, and S are constants.

49. The method of claim 48 further comprising displaying v_c, y_c, μ

_c, α

_c, T_c, V_dot^d, X_dot^d, and Y_dot^d to an operator of the vehicle.

50. The method of claim 44 wherein the vehicle rate comprises a vehicle ground speed V_track, and a vehicle flight path angle V_track wherein the vehicle rate is converted to the North position rate ξ

_dot^t, the East position rate η

_dot^t, and the vertical position rate h_dot^t based on the following equations;

ξ

_dot^t=V_track cos(Y_track)cos(X_track)

η

_dot=V_track cos(Y_track)sin(X_track)

h_dot^t=V_track sin(Y_track)

wherein the desired vehicle rate v_c, x_c, y_c and is determined by the following equations;

$v^c=\mathrm{sign}<br><br>\sqrt{{\left({\mathrm{\xi <br><br>}}_{\mathrm{dot}}^d\right)}^2+{\left({\mathrm{\eta <br><br>}}_{\mathrm{dot}}^d\right)}^2+{\left(h_{\mathrm{dot}}^d\right)}^2}$ 

X^c′

=tan<sup>−

1</sup>(η

_dot^d/ξ

_dot^d) $\begin{array}{c}\mathrm{sign}=<br><br>1<br><br>\text{\hspace{1em}<br><br>}<br><br>\mathrm{if}<br><br>\text{\hspace{1em}<br><br>}<br><br><br><br>x^{c^{\mathrm{\prime <br><br>}}}<br><br>\le <br><br>\mathrm{\pi <br><br>}/2<br><br>\text{\hspace{1em}<br><br>}\\ =<br><br>-1<br><br>\text{\hspace{1em}<br><br>}<br><br>\mathrm{if}<br><br>\text{\hspace{1em}<br><br>}<br><br><br><br>x^{c^{\mathrm{\prime <br><br>}}}<br><br>><br><br>\mathrm{\pi <br><br>}/2\\ X^{c^{\mathrm{\prime <br><br>}}}=<br><br>{\tan }^{-1}<br><br>\left({\mathrm{\eta <br><br>}}_{\mathrm{dot}}^d/{\mathrm{\xi <br><br>}}_{\mathrm{dot}}^d\right)\\ X^c=<br><br>X^{c^{\mathrm{\prime <br><br>}}}<br><br>\text{\hspace{1em}<br><br>}<br><br>\mathrm{if}<br><br>\text{\hspace{1em}<br><br>}<br><br><br><br>x^{c^{\mathrm{\prime <br><br>}}}<br><br>\le <br><br>\mathrm{\pi <br><br>}/2\\ =<br><br>\mathrm{\pi <br><br>}-X^{c^{\mathrm{\prime <br><br>}}}<br><br>\text{\hspace{1em}<br><br>}<br><br>\mathrm{if}<br><br>\text{\hspace{1em}<br><br>}<br><br><br><br>x^{c^{\mathrm{\prime <br><br>}}}<br><br>><br><br>\mathrm{\pi <br><br>}/2\end{array}$ 

Y^c=sin<sup>−

1</sup>(h_dot^d/v^c)

wherein the guidance corrections comprise v_c, y_c, μ

_c, α

_c, and T_c, wherein the guidance corrections v_c, y_c, μ

_c, α

_c, and T_c, are determined based on the following equations;

v_c=v+┐

v_dot^d

y_c=y+┐

y_dot^d

μ

_c=tan<sup>−

1</sup>(v cos(y)x_dot^d)/(vy_dot^d+g cos(y))

tan α

_c=((mvy_dot^d+mg cos(y))/cos(μ

_c)−

L/(mv_dot^d+D+mg sin(y))

T_c=(mv_dot^d+D+m+g sin(y)/cos(α

_c)

wherein v and y represent a current state of the vehicle, wherein L and D are modeled according to the following equations;

L=qSC_L

D=qSC_D

C_L=C_Lα

₀+C_Lα

α

C_D=C_Dmin+KC_L²

wherein q is dynamic pressure, and wherein K, C_Dmin, C_Lα

₀, C_Lα

, and S are constants.

51. The method of claim 50 further comprising displaying v_c, y_c, μ

_c, α

_c, and T_c, v_dot^d, x_dot^d, Y_dot^d to an operator of the vehicle.

52. The method of claim 36 wherein the determining of a desired vehicle rate comprises performing a dynamic inversion based on the directional vehicle rate and the directional formation position error rate so as to produce the desired vehicle rate.

53. The method of claim 36 wherein the converting of the desired vehicle rate to a vehicle acceleration input comprises:

subtracting a current vehicle state from the desired vehicle rate to produce a subtraction result; and

,

converting the subtraction result to the vehicle acceleration input.

54. The method of claim 53 wherein the determining of a desired vehicle rate comprises performing a dynamic inversion based on the directional vehicle rate and the directional formation position error rate so as to produce the desired vehicle rate.

55. A method of keeping position of a vehicle in a vehicle formation comprising:

receiving a track velocity input comprising v_track, x_track, and ytrack;

receiving a commanded maneuver comprising v_dottrack, x_dottrack, y_dottrack; and

,

determining guidance corrections based on the track velocity and the commanded maneuver during position keeping only, during a simultaneous altitude change maneuver, during a simultaneous speed change maneuver, during a simultaneous turn maneuver, during a sequential altitude change maneuver, during a sequential speed change maneuver, during a sequential turn maneuver, during a simultaneous altitude and speed change maneuver, during a simultaneous altitude change and turn maneuver, during a simultaneous turn and speed change maneuver, during a simultaneous turn and altitude and speed change maneuver, during any and all pending maneuvers, during sequential altitude and speed change maneuvers, during a sequential altitude change and turn maneuver, during a sequential altitude change and turn maneuver, during a sequential turn and speed change maneuver, and during a sequential turn and altitude and speed change maneuver.

56. The method of claim 55 wherein at least some of the sequential maneuvers are performed based on a time delay, and wherein the time delay is equal to the time required for the follower to cover a desired distance between the position of the follower when a leader begins the maneuver and the position of the leader where the leader begins the maneuver.

57. The method of claim 56 wherein v_track=v_com, x_track=x_com, y_track=0, v_dottrack=0, x_dottrack=0, and y_dottrack=0 during position keeping only, wherein v_track=v_com, x_track=x_com, y_track=sin⁻

1 (h_dotcom/v_track), v_dottrack=0, x_dottrack=0, and y_dottrack=0 during a simultaneous altitude change maneuver, wherein v_track=v_L(t), x_track=x_com, y_track=0, v_dottrack=v_dotcom, x_dottrack=0, and y_dottrack=0 during a simultaneous speed change maneuver, wherein v_track=v_L(t)±

Δ

v_turn(t), x_track=x_L, y_track=0, v_dottrack=0, x_dottrack=x_dotcom, and y_dottrack=0 during a simultaneous turn maneuver, where v_track=v_com, x_track=x_com, y_track=sin<sup>−

1</sup> (h_dotcom/v_track), v_dottrack=0, x_dottrack=0, and y_dottrack=0 during a sequential speed change maneuver, wherein v_track=v_L(t), x_track=x_com, y_track=0, v_dottrack=v_dotcom, x_dottrack=0, and y_dottrack=0 during a sequential altitude change maneuver, wherein v_track=v_L(t−

t_delay), x_track=x_com, y_track=0, v_dottrack=v_dotcom, x_dottrack=0, and y_dottrack=0 during a sequential speed change maneuver, wherein v_track=v_L(t−

t_delay)±

Δ

v_turn(t−

t_delay), x_track=x_L(t−

t_delay), y_track=0, v_dottrack=0, x_dottrack=x_dotcom, and y_dottrack=0 during a sequential turn maneuver, wherein v_track=v_L(t), x_track=x_com, y_track=sin<sup>−

1</sup> (h_dotcom/v_track), v_dottrack=v_dotcom, x_dottrack=0, and y_dottrack=0 during a simultaneous altitude and speed change maneuver, wherein v_track=v_L(t)±

Δ

V_turn(t), x_track=x_L(t), y_track=sin<sup>−

1</sup> (h_dotcom/v_track), v_dottrack=0, x_dottrack=x_dotcom, and y_dottrack=0 during a simultaneous altitude change and turn maneuver, wherein v_track=v_L(t)±

Δ

v_turn(t), x_track=x_L(t), y_track=0 during a simultaneous turn and speed change maneuver, wherein v_track=v_L(t)±

Δ

v_turn(t), x_track=x_L(t), y_track=sin<sup>−

1</sup> (h_dotcom/v_track), v_dottrack=v_dotcom, x_dottrack=x_dotcom, and y_dottrack=0 during a simultaneous turn and altitude and speed change maneuver, wherein v_track=(v_L^PK(t_startment)−

x_F^PK(t))/(t_startment−

t), x_track=x_com, and y_track=0, v_dottrack=0, x_dottrack=0, y_dottrack=0, during any and all pending maneuvers, wherein v_track=v_com(t−

t_delay), x_track=x_com, y_track=sin⁻

(h_dotcom/v_track), v_dottrack=v_dotcom, x_dottrack=0, and y_dottrack=0 during a sequential altitude and speed change maneuver, wherein v_track=v_L(t−

t_delay), x_track=x_L(t−

t_delay), y_track=sin<sup>−

1</sup> (h_dotcom/v_track), v_dottrack=0, x_dottrack=x_dotcom, and y_dottrack=0 during a sequential altitude change and turn maneuver, wherein v_track=v_L(t−

t_delay)±

Δ

v_turn(t), x_track=x_L(t−

t_delay), y_track=0, v_dottrack=v_dotcom, x_dottrack=x_dotcom, and y_dottrack=0 during a sequential turn and speed change maneuver, and wherein v_track=v_L(t−

t_delay)±

Δ

v_turn(t−

t_delay), x_track=x_L(t−

t_delay), y_track=sin<sup>−

1</sup> (h_dotcom/v_track), v_dottrack=v_dotcom, x_dottrack=x_dotcom, and y_dottrack=0 during a sequential turn and altitude and speed change maneuver, wherein the com subscript designates a leader value, wherein the L subscript designates a leader value, wherein the F subscript designates a follower value, wherein h_dotcom designates a command altitude rate, wherein dotcom subscript designates a commenced rate value, wherein t_startment designates the time at which the leader starts the maneuver, wherein t_delay designates the time required for a follower to cover a distance between the position of the follower when a leader begins the maneuver and the position of the leader where the leader begins the maneuver, wherein x_L^PK designates a position of a leader in a position keeping reference frame, and wherein x_F^PK designates a position of a follower in the position keeping reference frame.

58. The method of claim 55 wherein the determining of guidance corrections comprises determining the guidance corrections additionally during a drop maneuver alone and with any of the maneuvers except a turn maneuver.

59. A method of deterring a formation error vector e in a position of a vehicle with respect to a leader L in a vehicle formation comprising:

determining a position vector Δ

x based upon a position x_v of the vehicle and a position x_L of the leader;

comparing the position vector Δ

x to a desired relative position vector Δ

x_ref, wherein the desired relative position vector Δ

x_ref, designates a desired relative position between the vehicle and the leader; and

,

determining the formation error vector e based on a result or the comparison.

60. The method of claim 59 further comprising displaying the formation error vector e to the vehicle.

61. The method of claim 59 wherein certain sequential maneuvers of the vehicle are delayed by a time delay t_delay, and wherein the time delay t_delay is equal to the time required for the vehicle to cover a distance between the location of the vehicle when a leader begins the certain sequential maneuvers and the location of the leader where the leader begins the sequential maneuvers.

62. The method of claim 61 further comprising displaying the formation error vector e to the vehicle.

63. The method of claim 59 wherein the formation error vector e, the position vector Δ

x, the position x_v of the vehicle, the position x_L of the leader, and the desired relative position vector Δ

x_ref are based on a position keeping reference frame PK such that e=e^PK, Δ

x=Δ

x^PK, x_v=x_v, x_L=x_L, and Δ

x_ref=Δ

x_ref^PK.

64. The method of claim 63 wherein e^PK=Δ

x^PK−

Δ

x_ref^PK, and Δ

x^PK=x_L^PK(t)−

x_v^PK(t) for position keeping.

65. The method of claim 64 further comprising displaying the formation error vector e^PK to the vehicle.

66. The method of claim 63 wherein e^PK=Δ

x^PK−

Δ

x_ref^PK, and Δ

x^PK=x_L^PK(t)−

x_v^PK(t) for simultaneous maneuvers.

67. The method of claim 63 wherein e^PK=Δ

x^PK−

Δ

x_ref^PK, and Δ

x^PK=x_L^PK(t−

t_delay)−

x_v^PK(t) for sequential maneuvers, and wherein t_delay is equal to the time required for the vehicle to cover a distance between the location of the vehicle when a leader begins the sequential maneuvers and the location of the leader where the leader begins the sequential maneuvers.

68. The method of claim 63 e^PK−

Δ

x^PK−

Δ

x_ref^PK, and Δ

x^PK=x_L^PK(t_startman)−

x_v^PK(t_startman) for pending maneuvers, and wherein (t_startman) is the time at which the leader starts a maneuver.

69. The method of claim 59 wherein the determining of a position of vector Δ

x based upon a position x_v of the vehicle and a position x_L of the leader comprises determining the position vector Δ

x based upon a position Δ

_v of the vehicle, a position x_L of the leader, and a commanded altitude.

70. A method of executing a sequential maneuver by a follower F following a leader L in a vehicle formation comprising:

once the leader begins execution of the sequential maneuver, waiting to execute the sequential maneuver;

during the waiting, guiding the follower to a location occupied by the leader at the time that the leader began the sequential maneuver; and

,

initiating execution of the sequential maneuver when the follower arrives at the location.

71. The method of claim 70 wherein the guiding of the follower comprises:

determining a formation error vector e relating to a position of the follower in relation to the leader in the vehicle formation; and

,

setting a longitudinal component of the formation vector to 0 during the waiting.

72. The method of claim 70 wherein the guiding of the follower comprises:

determining a delay time t_delay required for the follower to reach the location; and

,

controlling speed so that, upon expiration of the delay time t_delay, the follower reaches the location.

73. The method of claim 72 wherein the determining of a delay time t_delay comprises determining the delay time t_delay according to the following equation:

t_delay=Δ

x_nom(t_startman)/c_com(t_startman)

where t_startman is the time at which the leader starts the sequential maneuver, x_nom is the nominal desired longitudinal separation between the follower and the leader at the time that the leader starts the sequential maneuver, and v_com is a commanded ground speed.

74. The method of claim 72 wherein the guiding of the follower comprises:

determining a formation error vector e relating to a position of the follower in relation to the leader in the vehicle formation; and

,

setting a longitudinal component of the formation vector to 0 during the waiting.

75. The method of claim 74 wherein the determining of the a delay time t_delay comprises determining the delay time t_delay according to the following equation:

t_delay=Δ

x_nom(t_startman)/v_com(t_startman)

where t_startman is the time at which the leader starts the sequential maneuver, Δ

x_nom is the nominal desired longitudinal separation between the follower and the leader at the time that the leader starts the sequential maneuver, and v_com is a commanded ground speed.

76. A method of keeping position of a vehicle in a vehicle formation comprising:

determining guidance corrections to execute a combined maneuver of the vehicle in the vehicle formation, wherein the combined maneuver comprises a combination of at least two simple maneuvers, and wherein the combined maneuver is commanded by a leader of the vehicle formation; and

,

guiding the vehicle through the combined maneuver based on the guidance corrections,

wherein the determining of guidance corrections comprises determining guidance corrections based on a commanded ground speed, a commanded flight path angle, and a commanded track angle.

77. The method of claim 76 wherein the combined maneuver comprises combined speed and altitude changes.

78. The method of claim 76 wherein the combined maneuver comprises combined speed change and turn.

79. The method of claim 76 wherein the combined maneuver comprises a combined altitude change and turn.

80. The method of claim 76 wherein the combined maneuver comprises a combined speed and altitude change and turn.

81. A method of keeping a position of a vehicle in a vehicle formation comprising:

determining guidance corrections to execute a commanded maneuver of the vehicle in the vehicle formation and to maintain the position of the vehicle in the vehicle formation; and

,

displaying the guidance corrections on a primary flight display to an operator of the vehicle,

wherein the displayed guidance corrections comprise a course change countdown, and wherein the course change countdown indicates time remaining in a pending maneuver.

82. The method of claim 81 wherein the displayed guidance corrections comprise a speed target for the vehicle.

83. The method of claim 81 wherein the displayed guidance corrections comprise an altitude target for the vehicle.

84. The method of claim 81 wherein the displayed guidance corrections comprising a heading target for the vehicle.

85. The method of claim 81 wherein the displayed guidance corrections comprise a speed target and an altitude target for the vehicle.

86. The method of claim 81 wherein the displayed guidance corrections comprise a speed target and an altitude target for the vehicle.

87. The method of claim 81 wherein the displayed guidance corrections comprise a heading target and an altitude target for the vehicle.

88. The method of claim 81 wherein the displayed guidance corrections comprise a speed target, a heading target, and an altitude target for the vehicle.

89. The method of claim 81 wherein the displayed guidance corrections comprise a longitudinal track deviation of the vehicle.

90. The method of claim 81 wherein the displayed guidance corrections comprise a lateral deviation of the vehicle.

91. The method of claim 81 wherein the displayed guidance corrections comprise a vertical deviation of the vehicle.

92. The method of claim 81 wherein the displayed guidance corrections comprise a longitudinal deviation and a lateral deviation of the vehicle.

93. The method of claim 81 wherein the displayed guidance corrections comprise a longitudinal deviation and a vertical deviation of the vehicle.

94. The method of claim 81 wherein the displayed guidance corrections comprise a lateral deviation and a vertical deviation of the vehicle.

95. The method of claim 81 wherein the displayed guidance corrections comprise a longitudinal deviation, a lateral deviation, and a vertical deviation of the vehicle.

96. A method of keeping position of a vehicle in a vehicle formation comprising:

determining guidance corrections to execute a combined maneuver of the vehicle in the vehicle formation, wherein the combined maneuver comprises a combination of at least two maneuvers, and wherein the combined maneuver is commanded by a leader of the vehicle formation; and

guiding the vehicle through time combined maneuver based on the guidance corrections,

wherein the determining of guidance corrections comprises determining guidance corrections based on a commanded acceleration, a command flight path angle, and a commanded track angle rate.

97. A method of keeping a position of a vehicle in a vehicle formation comprising:

determining guidance corrections to execute a commanded maneuver of the vehicle in the vehicle formation and to maintain the position of the vehicle in the vehicle formation; and

displaying the guidance corrections on a primary flight display to an operator of the vehicle,

wherein the displayed guidance corrections comprise a course change countdown, and wherein the course change countdown indicates time remaining in a pending maneuver wherein the displayed guidance corrections comprise a course change countdown, and wherein the course change countdown indicates the time remaining until a simultaneous maneuver is performed.