Problem Set 7#
Problem 7.1#
In a reference frame \((\bf i, j, k)\), a vector \(\mathbf{v}\) has the form:
Express this vector in a reference frame obtained by a rotation by a positive angle \(\phi\) about the \(k\) axis. How would this vector look if \(\mathbf{v}\) had a component in the \(\bf k\) direction?
Consider now the effect of two rotations: a rotation of \(\phi\) about the \(\bf i\) axis followed by a rotation of \(\psi\) about the \(\bf k\) axis. Write down the rotation matrix for the net rotation. Draw a diagram showing this sequence of rotations.
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Problem 7.2#
A spacecraft is rotated about an axis \(\mathbf{e} = (0.6, 0.0, 0.8)\). The angle of rotation is \(\phi = 60^\circ\). Compute the quaternion that describes this maneuver.
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Problem 7.3#
A spacecraft executes a maneuver of \(\phi_1 = 30^\circ\) about an axis \(\mathbf{e}_1 = (0.5, 0.5, 0.707)\). After it has turned through \(20^\circ\), one of the magnetorquers fires by an error. This causes a new rotation of the spacecraft of \(\phi_2 = 14^\circ\) about an axis \(\mathbf{e}_2 = (0.9, 0.11, 0.422)\). Using the quaternion product, find the Euler axis and rotation angle for the net rotation.
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Problem 7.4#
A spacecraft frame has its \(z\) axis pointing away from the Earth and its \(y\) axis pointing along its orbit. The spacecraft executes a rotation around \(z\) (yaw) of \(\phi = 60^\circ\) followed by a rotation around \(y\) (roll) of \(\theta = 20^\circ\). Determine the attitude matrix \(A\) of the spacecraft after these two rotations. To return to its original orientation in space, the spacecraft performs a yaw of \(-20^\circ\) followed by a roll of \(-60^\circ\). Will this return the spacecraft as desired?
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