Questions: Trajectory Planning and Motion Planning

Question 1

A robot must move from configuration A to configuration B. Joint-space trajectory planning connects A and B with a cubic polynomial: θ(t) = θ₀ + (3/t_f²)·(t_f - t)·(θ_f - θ₀)·t - (2/t_f³)·(t_f - t)·t²·(θ_f - θ₀). At the start (t=0) and end (t=t_f), the velocity is zero. Why is this constraint important?

• A. To ensure the robot doesn't move too fast, which would damage it
• B. To ensure the trajectory starts and ends with zero velocity (smooth starts and stops), preventing jerk and mechanical stress
• C. To ensure the trajectory passes through the intermediate configuration θ_mid at t = t_f/2
• D. To ensure the robot doesn't hit obstacles at the beginning and end of the motion

Answer: (B)

This is why cubic polynomials are popular: they have continuous acceleration. Piecewise-linear trajectories (connecting configurations with straight lines in joint space) have discontinuous acceleration at the connection points, requiring infinite torque. Splines with higher-order continuity (quintic, B-splines) smooth acceleration changes further.

Question 2

A path-planning algorithm finds a collision-free path from start to goal, but doesn't account for dynamics. In the context of real robot execution, what additional check is needed?

• T. True
• F. False

Answer: True

A collision-free path in configuration space guarantees the path exists, but not that the robot can follow it given its speed and acceleration limits. When the trajectory is parameterized with time and accelerations, the trajectory itself (not just the path) might collide with obstacles if the robot moves too fast through tight spaces. More subtly, accelerations can shift the robot's center of mass or configuration slightly, potentially causing self-collision. Therefore, the full trajectory in space-time should be checked for collision, especially for fast or highly dynamic motions.