Answer Happy

5:48 PM Mon May 9 7 89% 5. (25%) Consider planning a trajectory using a piecewise cubic spline. The trajectory must satisfy the boundary conditions: an initial position x(0) = xo, the initial velocity and acceleration are zero, and final position x(TX) = xp, the final velocity and acceleration are also zero. a) What is the minimum number of cubic polynomials should be used in this case? Proof why? What assumptions did you have to make in this situation? b) If the maximum acceleration and jerk (third order time derivative of position with respect to time) are limited, how these limits will affect the trajectory to be planned? c) Which type of trajectory will yield the minimum time? Why? You can use simple schematics to show your idea and cite the formula provided in the lecture notes and textbook as you wish. You do not need to draw all position, velocity, and acceleration, but draw only the important one among them to illustrate your points.