ARO4090

For a cubic satellite with Ix=Iy=Iz=12.6 kg.m^2, calculate the amount of maximum gravity gradient torque.

0

d/dt Fi^T|Fi

0

How many sets of equilibrium points exist for a rigid body under no torque?

4

Which one is correct about a directional cosine matrix C12?

C12=C21-1 , C12=C21T

Which expression is a representation of Eulerian set 2-3-1?

C=Cx(φ)Cz(ψ)Cy(θ)

Select all that are an example of momentum exchange devices for attitude control.

CMG, Reaction Wheel, VSCMG

Which method is a passive control system to stabilize the attitude of a spacecraft? Dual spin stabilization or Stability Augmentation

Dual spin stabilization

A constant speed control moment gyro can control two axes of spacecraft attitude control.

False

Euler's Equations of Motion are represented in an inertial frame.

False

For a given angular momentum, the energy is maximum when rotating around the largest moment of inertia axis.

False

For the stability of the spacecraft in yaw motion, the stability in pitch motion is necessary.

False

Gravity gradient torque decreases faster than magnetic torque as we get away from the planet.

False

Gravity gradient torque does not depend on the shape of the spacecraft.

False

If η is zero is in the quartenion set, we cannot identify the ε vector associate with the rotation matrix.

False

In general, Euler's EOM is linear but coupled ODE.

False

In linearized EOM for attitude dynamics, including 3 axis RWA, and Gravity Gradient , all three equations for Eulerian angles are uncoupled.

False

In the motion of an axisymmetric spacecraft under no torque, the transverse angular velocity vector stays constant in magnitude and direction in the body frame.

False

Parallel axis theorem is only defined for principal axes frame.

False

Quaternion set (ε,η ) and (ε,-η ) represent the same rotation matrix.

False

Quaternions have singularity and cannot be uniquely identified for some rotations.

False

Rotation of the frames does not change the matrix of inertia.

False

The angular velocity vector is always equal to the rate of change of Eulerian angles.

False

The axis of symmetry is unique for a directional cosine matrix, but the angle of rotation depends on the vector that rotation is applied to.

False

The directional cosine matrix Cij is created by two frames Fi and Fj as Cij=FiFj

False

The free motion of a spacecraft in a torque-free environment can be asymptotically stable when perturbed by a small disturbance.

False

The matrix of inertia is only defined in the body frame.

False

The motion around the smallest moment of the inertia axis is the most stable motion for the spacecraft.

False

The nutation angle is defined between the angular momentum vector and the angular velocity vector.

False

The oblate object precesses in the same direction as the spin of the body.

False

The rotation rate about the axis of symmetry for an axisymmetric spacecraft under to external torque changes with time.

False

The trace of [a]x is always 1

False

The transformation of a matrix of inertia through rotation of the frames is called Parallel Axis Theorem.

False

We prefer using Eulerian angles for numerical calculations.

False

When a momentum bias spacecraft ( a spacecraft with one RW) is stable, the reaction wheel does not reach saturation.

False

What is the benefit of using quartenions instead of Eulerian angles?

Faster numerical calculations, Removing the singularity

Which of the following frames can be used to derive the EOM?

Fb

Which axis of motion in the spacecraft is inherently unstable under small perturbations?

Intermediate axis of Inertia

What is the shape of the polholde when the spacecraft is at one of its equilibrium?

It depends on the equilibrium point.

Which statement(s) is/are are correct about a single reaction wheel in the spacecraft?

It provides stability in 2 axes and control in 1 axis.

Which one of the following can be used to control the attitude motion of a spacecraft?

Magnetic control, gravity gradient, solar radiation pressure, aerodynamic torque

Which set of equations for gravity gradient motion are coupled?

Roll and Yaw

Which sensor(s) can be used in a rate of feedback system?

Star Tracker, IMU, Gyro

For a linear rotation transformation matrix C, a valid eigenvalue set should have:

The eigenvalues of a rotation matrix should include only one real number, 1, and only roots of unity which can be represented as e^±ai, where a is an integer. Also, C should have an odd number of eigenvalues

A derivative is a coordinate dependent operator.

True

A principal axes frames can be defined for any rigid body.

True

Any directional cosine matrix is a rotation matrix.

True

Eulerian angles can be calculated directly from the rotation matrix.

True

For the stability of the spacecraft in yaw motion, in roll motion is necessary.

True

Gravity gradient torque decreases faster than gravity force as we get away from the planet.

True

Gravity gradient torque depends on the orbit of the spacecraft around the planet.

True

Nutation angle is stable for an oblate object when there is a leak of energy due to non ideal rigid body deformation.

True

Spacecraft can be stabilized in all three axes by two reactions wheels.

True

State feedback is one of the common types of feedback control systems for attitude dynamics.

True

State feedback is one of the most common types of feedback control systems for attitude dynamics.

True

The attitude of the spacecraft changes the orbit of the spacecraft.

True

The determinant of any rotation matrix is 1

True

The directional cosine matrix Cij is created by two frames Fi and Fj as Cij=FjFi.

True

The matrix of inertia depends on the coordinate frame.

True

The precession rate for axisymmetric spacecraft under no torque stays constant.

True

The principal moments of inertia are the eigenvalues of any matrix of inertia defined at the center of mass.

True

The prolate object precesses in the same direction as the spin of the body.

True

There are 7 equilibrium points for a generally rigid body in a torque free environment.

True

When a moment bias spacecraft (a spacecraft with one RW) is in a torque-free environment, the reaction wheel does not reach saturation.

True

Which set of equations for gravity gradient motion are coupled?

roll and yaw

If u is an arbitrary vector, what is the result of trace(uuT)?

uTu , (|u|)^2