Trajectory

Trajectory planning starts with start and end configurations and generates a smooth path through intermediate points. The system must specify position, velocity, and acceleration at each point in time. Common approaches use polynomial functions (quintic polynomials provide smooth acceleration profiles) or splines that pass through waypoints. Constraints include maximum velocity, acceleration, and jerk (rate of change of acceleration) to ensure motors can execute the trajectory and movements are smooth. Time-optimal trajectories minimize duration while respecting constraints. Cartesian trajectories specify end-effector path in task space, while joint-space trajectories plan each joint independently. The resulting trajectory is discretized into small time steps and sent as position commands to joint controllers.

Trajectory generation enables smooth, natural-looking arm movements in humanoid robots during manipulation tasks. Walking gaits use leg trajectories planned for each step. Pick-and-place operations require precise trajectories avoiding obstacles. Tool use applications plan trajectories for operating screwdrivers, hammers, or brushes effectively. Whole-body movements coordinate trajectories across all joints simultaneously. Dynamic movements like jumping or running use carefully optimized trajectories to maintain balance.