1. The Use of Differential Geometry to Control and Guidance

2. Problem

3. Introduction
There are three phases to the engagement and interception of a target.
Launch phase -usually uncontrolled-the missile motor is initiated, and the missile boosted up to its operating velocity.
Mid-course Guidance Phase, for missiles that are not locked onto the target at launch. This is either straight line flight toward a point where it is anticipated that the missile sensor will lock onto the target, or a navigation phase where the missile is guided into an area that allows it to lock onto the target with its own sensor.
The terminal guidance phase where the missile is guided onto the target using its local sensor measurements.

4. Three Phases of Guidance
There are three Phases of missile guidance:
1. BOOST Phase:
- Get the missile up to SPEED!
- Missile leaves the launcher until Booster burned out
- Places the missile at a high altitude
- see the target
- receive target information.
2. Midcourse Guidance Phase:
- Missile closes the target
- Generally the longest phase
- General corrections to course and altitude
3. Terminal Phase
- Last phase; missile guidance is critical
- Final corrections, close the target, detonate warhead.
Boost Mid-Course Terminal

5. Introduction (cont.)
The guidance systems over the recent past years has been to rely more on precision and maneuverability with a small warhead (than on large missiles with a large warhead)
The sensors which they used in the missile and within the guided weapon system have become more efficient and sophisticated
From the other hand the targets have also become more stealthy and smart

6. Introduction (cont.)
Hence there is need to produce better guidance algorithms that can intercept smaller and more maneuverable targets that are difficult to detect and track

7. Geometry
The differential geometry approach offers a suite of possible trajectories that the missile can adopt to intercept the target
The well-known Proportional navigation (PN) is based on a straight line constant velocity target and defines a straight line constant velocity trajectory to intercept the target.
With the differential geometric approach a maneuvering target can be intercepted with either a straight line or a constant acceleration trajectory. A non-maneuvering target can likewise be intercepted using either a non-maneuvering trajectory or a trajectory with a constant lateral acceleration.

8. About Missiles Unguided Guided No course corrections
Elevation, ballistics, gravity, etc
Guided
Control system
Change direction
We will talk about Aerodynamic Guided Missiles, but these principles apply to other types of weapons, for example torpedoes!

9. Main Categories of Guidance-Control
1. Control Guidance
Command
Beam rider
2. Homing Guidance
Active
Semi-Active
Passive
3. Self Contained – “Smart Weapon”
Three categories of missile guidance control.
1. Controlled guidance
- the shooter (WCO, TAO, etc) actively controls the weapon, for a time.
- examples: Mk 48 wire guided, TOW
2. Homing guidance
- Distinguishing feature of the target to zero in on
- Active
- Harpoon, Phoenix, AMRAAM, Mk 46 and 48 (after wire breaks)
- Semi-active
- NATO Sea Sparrow, SM-2, SM-1P
- Passive
- Mk – 46 and 48 (magnetic influence, acoustic), HARM, SLAM, Sidewinder
3. Self-contained guidance
- Most sophisticated
- All detection, error generation and guidance corrections determined by the missile
- examples:
- TLAM, Trident

10. Beam Rider Control Guidance
Older method of
guidance control.
Tracking
Beam
Narrow
Guidance
Beam
Beam Rider Control is older method
a. Guidance is performed by keeping the missile inside a radar or laser beam cone.
b. The beam comes from a friendly source.
c. If beam remains on target, then the missile will follow the beam to the target.
d. Point and shoot type of control. NOT FIRE AND FORGET!
Disadvantage:
- Missile is always LAGGING behind the target!
- Must keep target illuminated the WHOLE TIME!

11. Homing Guidance Control
Target
Characteristic
Radar
Radar
MOST ACCURATE! TARGET GIVES THE DATA!
1. Homing Guidance Systems react to a target feature. Could be radio EM, Light, Heat, Acoustic
2. For consistency we will talk EM (RADAR)
Three types of homing guidance controls:
1. Active: Missile uses own radar to transmit and receive information about target to make changes in flight profile.
2. Semi Active: The target is illuminated by a friendly platform. Missile receives the reflection and uses that information to alter flight profile.
3. Passive: Target is the source of energy.
There are hybrid systems which combine command guidance with some active homing. Guidance systems are combined and use other systems for backup.
1. Active
2. Semi-Active
3. Passive

12. About homing guidance
The expression homing guidance is used to describe a missile system that can sense the target by some means, and then guide itself to the target by sending commands to its own control
Homing is useful in tactical missiles where considerations such as autonomous (or fire-and-forget) operation usually require sensing of target motion to be done from the interceptor missile itself

13. About homing guidance (cont.)
Homing guidance is a term used to describe a guidance process that can determine the position (or certain position parameters) of the target (e.g., an aircraft, ship, or tank) and can formulate its own commands to guide itself to the target
A homing system is a specialized form of guidance, which entails selecting, identifying, and following a target through some distinguishing characteristic of the target. Such identifying characteristics as heat or sound from a factory, light from a city, or reflections of radar waves from a ship or aircraft are used as the source of intelligence to direct the missile to the target

14. Self-Contained Guidance Systems
Consists of two parts:
Preset
Updating

15. Guided Flight Paths Preset Variable Constant Programmed Pursuit
Constant Bearing
Proportional Navigation
Preset
Constant:
Fixed before launch.
Once launched, profile can not be changed
Programmed
Programmed phases (torpedo search)
Variable
Pursuit: ‘Chases’ the target
Constant Bearing: Predict target bearing, fire at intercept point
Proportional Navigation: Rate of change information
Line-of-Sight (being phased out): Beam rider!

16. Pursuit Path Points the target at all times.
Pursuit Guided flight path
1. The weapon will remain pointed at the target at all times.
2. The weapon constantly heads along the line of sight from the weapon to the target.
Disadvantage:
Weapon needs lots of gas!!

17. Constant Bearing Path X Initial Target Path Intercept Point New Path
Constant Bearing Guided Flight Path
Weapons anticipates where the target will be. As a result:
1. Weapon is aimed ahead of the target and will intercept the target at a point in space further down the track.
2. Weapon will alter course to maintain this constant bearing
Constant Bearing
Path

18. Proportional Navigation
Initial
Range
Proportional Navigation.
1. A receiver measures the rate of change of the line of sight to the target, new steering commands are generated.
The weapon chooses a course in which the rate of change of the weapons heading is directly proportion to the rate of rotation of the line-of-sight from the weapon to the target.
Very similar to Constant Bearing

19. Tools from Differential Geometry
Frenet-Serret Frame
We could define in a curve the triple t, n, b (vectors)
Where t is the tangent vector, n is the normal vector and b is the bi-normal vector.
The velocity vector v = v(t) over time t is expressed as a multiple v = v(t) of the unit tangent vector t = t(t), i.e., v(t) = v(t)t(t).

21. Tools from Differential Geometry (cont.)
The normal vector n is obtained by differentiating t and is therefore orthogonal to t.
We have k(t) = κ(t)n(t), where n is the unit normal vector and κ is the curvature.
The curvature measures the local deviation of the curve from the rectilinear progression along the tangent line. Indeed, its inverse is the radius of best fitting circle in the plane spanned by the tangent and normal vectors

22. Tools from Differential Geometry (cont.)
This coordinate frame (which is local) is completed by the bi-normal vector τ(t)=τ(t)b(t), where b=txn is the unit bi-normal vector.
The torsion τ measures locally how much the curve deviates from the plane spanned by the tangent and normal vectors
It is simpler (to calculations) to use the arc length instead of time t

23. Frenet Serret equations

24. Geometry of guidance

25. The target is flying in a straight line at constant velocity
The missile is flying at velocity , also in a straight line.
Both trajectories are assumed to intercept at the impact point at point I.
Impact triangle: The target, the missile and the impact point form a triangle
Sight line: the line from missile head to the target

26. Let’s consider T as the time in which the target has traveled in a straight line and at constant velocity from its initial position to the impact point. The length of this trajectory will be:

27. In order for the missile to arrive at the impact point at the same time as the target,
it must travel a distance in the same time T,
The ratio of the trajectory lengths is:

28. This equation shows that in order to impact on a target flying at constant velocity in a straight line, the missile must maneuver until the trajectory lengths of the impact triangle are in the same ratio as the target and missile velocities.
As the target velocity and heading are estimated, and targets can maneuver, there must be an active control system to acquire and maintain this impact geometry.

29. The range of impact triangles (possible)

30. This shows the locus of possible impact triangles, where the missile position lies on a circle of radius (the impact circle), and the missile velocity vector lies along the radius of the impact circle
The ratio of velocities is important

31. Example
The impact circle for velocity ratio γ=2. The missile is moving left to right and ranges of 1 to10 km, in 1 km steps, are chosen as the impact point

32. the only condition that is required for a no-maneuvering intercept is that the ratio of the trajectories is the same as the ratio of the missile and target velocities
intercept geometry:

33. Using the sine rule, we have
As the target angle to the sight line (θt) varies, the intercept point I will change.
The locus of the intercept point can be determined by using trigonometry on the two triangles of the previous figure.

34. On the triangle (M I N), made up of the missile position M, the intercept point I, and the intercept of the
normal from the intercept point onto the sight line N we have
On the triangle (T I N), replacing the missile position with the target position. Hence we have

35. Then we have Or
Completing the square:

36. This equation represents a circle with radius
and center
This circle represents the locus of the intercept points and can be used to assess the guidance algorithm

37. Geometry-Kinematics
There is a sensor which determines the relative motion and position of the target and missile. This sensor is located in the nose of the missile for homing guidance.
That’s why the sight line between the target and the missile is an important measure of the relative geometry

39. Defining derivatives with respect to time and with respect to arc length by the relations

40. Differentiating:
Both the missile and target velocity is constant, and by noting that the sightline range r vector can be expresses in terms of sightline coordinates defined by basis vectors ts and ns, where ts is the basis vector along the sightline and ns is the basis vector normal to the sightline, as shown in the previous figure, the equation can be written in the form
(*)

41. This equation represents the components of the target velocity relative to the missile.
Components of the relative velocity along and normal to the sight line are given by projection onto the basis vectors ts and ns

42. Missile to target relative acceleration is given by
differentiating (*)
By using Frenet Serret equations, make a rotation of Frenet Serret frame around the bi-normal vector b (in order to have a right-handed triplet (t,n,b)) we conclude:

43. Components along and normal to the sightline can be determined by projection onto the basis vectors ts and ns. For a missile producing a lateral acceleration fm

44. The acceleration components along and normal to the sightline can be determined as

45. Geometry-guidance
The geometry of the intercept configuration is determined by the requirement to match the missile path length sm to the target path length st
If we assume that velocities of target and missile are constants, then these are related:

46. Geometry-guidance (cont.)
Let’s consider a constant manoeuvre target and a zero curvature intercepting missile
The trajectories are as the following figure

47. Geometry-guidance (cont.)
The intercept point I determined by considering the target manoeuvre arc and the straight line missile trajectory geometry.
So using the intercept triangle TIM, we could define the relative positions of the missile M, the intercept point I and the target T.

48. Geometry-guidance (cont.)
We can determine now the intercept triangle by the missile tangent vector tm and the target arc chord defined by the vector tLt

49. Geometry-guidance (cont.)
From the figure the intercept condition can be represented by a vector in the form
The arc length is given by

50. Geometry-guidance (cont.)
and the chord length Lt is given by
and

51. Geometry-guidance (cont.)
Since we supposed that the target curvature approaches zero, we have
So the arc chord Lt will approach the arc length st
The arc chord vector tLt has a length Lt which is a function of the arc angle θtα.

52. Geometry-guidance (cont.)
As the engagement progresses, the arc angle θtα will tend to zero, with both the missile M and the target T points moving to the stationary intercept point I.
For the manoeuvring target case we can write now
(1)

53. Geometry-guidance (cont.)
This equation represents the matching condition for the engagement that can be solved by applying the cosine rule to the intercept triangle TIM
From this we can conclude that

54. Geometry-guidance (cont.)
The arc length st and the arc angle θtα are related by

55. Geometry-guidance (cont.)
By substituting the previous expression into eqaution (1) we have the required missile direction
Flying the missile along this direction defined by this tangent vector will then guarantee interception of the target.

56. References
B. A. White, R. Zbokowski, and A.Tsourdos. Direct intercept guidance using differential geometry concepts. IEEE Transactions on Aerospace and Electronic Systems, 43(3), July 2007.
Ryoo, C. K., Cho, H. J.,&Tahk, M. J. (2006). Time-to-go weighted optimal guidance with impact angle constraints. IEEE Transactions on Control Systems Technology,14(3), 483–492
Shin, H. S., Tahk, M. J., A. Tsourdos, A., & White, B. A. (2010). Earliest intercept geometry guidance to improve mid-course guidance in area air-defence. International Journal of Aeronautical and Space Science, 11(2), 118–125
White, B. A.&Tsourdos, A. (2011). Modern missile guidance design: Anoverview. In IFAC Automatic Control in Aerospace