A robotic tracking arm consists of two segments, OA OA\,OA and ABABAB, each of length 16 cm. The segment OA OA\,OA is pivoted at the origin OOO. The configuration of the arm is defined by the angle θ \theta\,θ that OA OA\,OA makes with the positive xxx-axis.
The arm is geared such that the second segment AB AB\,AB always makes an angle of 2θ 2\theta\,2θ with the positive xxx-axis, where 0≤θ≤π2\displaystyle 0 \le \theta \le \frac{\pi}{2}0≤θ≤2π.
Show that the xxx-coordinate of the end effector B B\,B is given by
x=16(cosθ+cos2θ) x = 16(\cos \theta + \cos 2\theta) x=16(cosθ+cos2θ)Using a double angle identity, show that
x=16(2cos2θ+cosθ−1) x = 16(2\cos^2 \theta + \cos \theta - 1) x=16(2cos2θ+cosθ−1)The expression in part (b) can be written in the form x=32(cosθ+14)2−18\displaystyle x = 32\left(\cos \theta + \frac{1}{4}\right)^2 - 18x=32(cosθ+41)2−18. Determine the value of θ \theta\,θ for which x x\,x is a minimum, and calculate this minimum value.
Calculate the exact distance OB OB\,OB from the origin to the end effector when x x\,x is at its minimum value.