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Problem 13. Frame {B}\{B\} is initially coincident with frame {A}\{A\}. Frame $\{...
Aug 6, 2024
Solution by Steps
step 1
Identify the rotation vector R^\hat{R} and the angle of rotation θ\theta. Here, R^=[0.744−0.620.248]\hat{R} = \left[\begin{array}{c} 0.744 \\ -0.62 \\ 0.248 \end{array}\right] and θ=60∘\theta = 60^\circ
step 2
Convert the angle θ\theta from degrees to radians: θ=60∘=π3\theta = 60^\circ = \frac{\pi}{3} radians
step 3
Use the Rodrigues' rotation formula to find the rotation matrix RR. The formula is R=I+sin⁡(θ)K+(1−cos⁡(θ))K2R = I + \sin(\theta)K + (1 - \cos(\theta))K^2, where KK is the skew-symmetric matrix of R^\hat{R}
step 4
Construct the skew-symmetric matrix KK from R^\hat{R}: K=[0amp;−0.248amp;0.620.248amp;0amp;−0.744−0.62amp;0.744amp;0]K = \left[\begin{array}{ccc} 0 & -0.248 & 0.62 \\ 0.248 & 0 & -0.744 \\ -0.62 & 0.744 & 0 \end{array}\right]
step 5
Calculate sin⁡(θ)\sin(\theta) and cos⁡(θ)\cos(\theta): sin⁡(π3)=32\sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2} and cos⁡(π3)=12\cos\left(\frac{\pi}{3}\right) = \frac{1}{2}
step 6
Substitute sin⁡(θ)\sin(\theta), cos⁡(θ)\cos(\theta), and KK into the Rodrigues' rotation formula to find RR
step 7
For part (1), since the position of point PP is [000]\left[\begin{array}{c} 0 \\ 0 \\ 0 \end{array}\right], the transformation matrix BAT{ }_{B}^{A} T is simply the rotation matrix RR
step 8
For part (2), the position of point PP is [200]\left[\begin{array}{c} 2 \\ 0 \\ 0 \end{array}\right]. The transformation matrix BAT{ }_{B}^{A} T will include both the rotation matrix RR and the translation vector
step 9
Construct the transformation matrix BAT{ }_{B}^{A} T for part (2) as: BAT=[Ramp;t0amp;1]{ }_{B}^{A} T = \left[\begin{array}{cc} R & \mathbf{t} \\ 0 & 1 \end{array}\right], where t\mathbf{t} is the translation vector
step 10
Calculate the translation vector t\mathbf{t} by applying the rotation to the position of point PP: t=R[200]−[200]\mathbf{t} = R \left[\begin{array}{c} 2 \\ 0 \\ 0 \end{array}\right] - \left[\begin{array}{c} 2 \\ 0 \\ 0 \end{array}\right]
Answer
The transformation matrix for part (1) is the rotation matrix RR.
The transformation matrix for part (2) is BAT=[Ramp;t0amp;1]{ }_{B}^{A} T = \left[\begin{array}{cc} R & \mathbf{t} \\ 0 & 1 \end{array}\right].
Key Concept
Rodrigues' rotation formula
Explanation
Rodrigues' rotation formula is used to compute the rotation matrix given an axis and an angle of rotation.
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