wiki/ lv2cguidance

Guidance

Purpose: The LV2.3 rocket design is intended to be a testbed for development of our avionics, recovery, and control systems. We have successfully planned, designed, built, and tested systems in each of these areas. This project represents our next progression from simple control systems such as roll control to a considerably more complex six-degree-of-freedom control system. This step in our development process will likely result in some failures because it is more complex (by orders of magnitude) than anything we have done so far. From our failures and successes, we aim to actively guide our rocket.

There will be a final step in this process, which will be guiding our rocket in an unstable configuration (no fins) using only thrust vectoring. With that, we will push toward orbit.

Requirements:

Need:

Current Design

The current fin design is shown below. There are still details to be worked out in the fin construction, travel stops, servos, etc. Those design details are captured in the analysis sections that follow.

Current Tasks:

*General/Needs to be categorized

-- *Lower

-- *Gearbox

-- *linkages

-- *Controls

Analysis

Body Equations

Fin Equations

Body Forces at Supersonic Speeds

Body Forces at Subsonic Speeds

Forces produced by an aerodynamic fin are highly non-linear and depend on many factors. These dependencies are too complex to include in the overall control models, so fin equations are separated to abstract them from the guidance system. At higher altitudes, fin forces will be replaced by thruster forces, so separating them now is a logical choice.

Our design uses an 8 inch fin with a 60 degree sweep and a flat trailing edge. The trailing edge is perpendicular to the airframe centerline (0 notch ratio).

The fin produces a lift force of 62 N (14 lbs) per degree at Mach 1.2, 1855 N (417 lbs) at a 30 degree AOA.

Fin Forces at Supersonic Speeds

Fin Forces at Subsonic Speeds

The above equations require known atmospheric conditions, which can be approximated with a standard atmosphere as a function of altitude only.

Air Density at Altitude

Mass Moment of Inertia Calculations

Our project requires the full inertia tensor.

I_{T} =
\left [ \begin{matrix}
I_{xx}  &  I_{xy}  &  I_{xz}  \\
I_{yx}  &  I_{yy}  &  I_{yz}  \\
I_{zx}  &  I_{zy}  &  I_{zz}  \\
\end{matrix} \right ]

This is obtained from solid models used to design the rocket.

Needed:

Measured Mass Moment of Inertia

The mass moment of inertia is measured experimentally in (approximately) the principle axes. Inertia measurements in off axis orientations can be made to confirm the principle axis orientations.

(MEASUREMENTS)

(CALCS)

Then, the results are checked against the model.

(DATA)

Performance Requirements

The performance requirements describe how quickly the rocket and control system must track position and orientation commands. This can be expressed in terms of bandwidth. Bandwidth describes how quickly the rocket must track an orientation command, up to a pre-defined level of error in tracking.

Further, the performance requirements describe how accurately a particular position must be reached; possibly in meters or kilometers for our application.

The closed loop bandwidth of the rocket and controller depends on the size of the rocket and our estimate of disturbances that occur during flight. One answer to the bandwidth requirement has been "as high as possible". Since our rocket doesn't need to track a moving or accelerating object, "as high as possible" is a poor choice. A better choice is to determine what control bandwidth would result in reasonable forces on the rocket, considering the size and mass properties of that particular flight.

To do this, we compute the forces required to force the rocket to track a sine wave orientation command of a variety of frequencies. Next, we choose a command frequency (bandwidth) that will result in forces which are within the capabilities of our airframe. This is repeated in each of the principle axes of the inertia tensor.

Control Bandwidth Analysis

For our application, position accuracy of the rocket is determined by the goal of the flight. A test flight of a small sounding rocket has position accuracy tolerances only to evaluate the control. An orbital insertion flight, however, will require some analysis to determine positional accuracy requirements.

(ANALYSIS)

Estimate of Control Forces

Control forces must be evaluated to determine the following:

Fin Forces

Presently assuming a working force of 470 lbs (From analysis above)

Control Rod Forces

From the fin dimensions and aerodynamics, the fin center of pressure is 1.25 inch behind the middle of the root chord. With the fin pivot at the root chord, the 470 lbs max fin force at 30 degree AOA will produce 588 inch lbs of torque. Since the control rods are mounted at a distance of (CHECK THIS) 1.5 inches from the fin pivot, the normal working control rod force requirement is 392 lbs.

For the type of control rod we have selected, we did some analysis of the rods' strength and spring constant so that it could be considered in the dynamic analysis.

Control Rod Analysis

Control Fin Servo

Due to the control fin forces expected (above), servos with significant force/torque are required. For the longitudinal rocket dynamics, the servo speed could be very slow in comparison with the roll control system previously designed and flown. For the roll axis, however, we can expect even faster dynamics that previously seen due to the increased size of the fins. The fin size has increased from 4 60 degree fins at 2.5 inch chord, to 3 60 degree fins at an 8 inch chord. This increases the area available for roll by 7.68, but since the fin actuation travel is 5 degrees instead of 15 degrees, the actual increase is only 2.56.

What that means is, our roll axis response will require at least as much processing power as the original roll control, and our servos should remain as fast as they were on roll control. This is a servo capable of producing 588 inch lbs of torque at the fin, and actuating through 10 degrees of rotation in 0.06 seconds. A quick power analysis indicates that 193 watts are required to do this. Assume motor efficiency of 78%, that gives 250 watts.

Fin Actuation Requirements Summary
Parameter SI Imperial
Torque 66.43 [N m] 588 [inch lbs]
Angle of Rotation 0.1745 [radians] 10 [degrees]
Transit Time 0.060 [seconds] 60 [ms]
Power Required 250 [watts] 0.335 [HP]

We have chosen to create a custom designed servo using a DC brush motor, a custom motor control, and a custom gearbox.

Control Servo Design

CFT note

\left [ \begin{matrix}
\overset{\centerdot}{\phi}   \\
\overset{\centerdot}{\theta} \\
\overset{\centerdot}{\psi}
\end{matrix} \right ] =
\left [ \begin{matrix}
1  &  tan \theta sin \phi  &  tan \theta cos \phi  \\
0  &  cos \phi             &  -sin \phi            \\
0  &  sec \theta sin \phi  &  sec \theta cos \phi
\end{matrix} \right ]
\left [ \begin{matrix}
\omega_{x} \\
\omega_{y} \\
\omega_{z}
\end{matrix} \right ]