RB-1007 Trajectory Generation is a robotics knowledge-base entry cataloguing the algorithms, mathematical representations, and computational methods used to synthesise smooth, dynamically feasible motion trajectories for robotic systems, specifying how a robot must move through configuration space or task space from an initial to a goal state while satisfying kinematic, dynamic, and environmental constraints. It covers polynomial spline interpolation, time-optimal trajectory planning, jerk-limited profiles, and learned trajectory generation approaches, forming the bridge between high-level task planning and low-level joint control execution.
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- Trajectory generation has been a core topic in robotics since the early industrial manipulator era of the 1970s, when the need to move robot arms smoothly between pick-and-place positions without exciting mechanical vibrations motivated the first polynomial interpolation methods. Paul (1981) formalised cubic and quintic polynomial joint-space trajectories that guarantee continuity of position, velocity, and acceleration across via-points. The time-optimal trajectory problem — finding the fastest feasible trajectory subject to actuator torque and velocity limits — was later formalised by Bobrow, Dubowsky, and Gibson using the Phase Plane method, remaining a foundational result.
- A trajectory generation system takes a set of via-points or a planned path from a motion planner (such as RRT or PRM variants) and produces a time-parameterised curve q(t) specifying joint angles, velocities, accelerations, and optionally jerk at every instant. Cubic splines provide C2 continuity (continuous up to acceleration), while quintic splines additionally constrain jerk. Trapezoidal velocity profiles are commonly used for simpler point-to-point moves, while S-curve profiles limit jerk to reduce mechanical stress. Online trajectory generators adapt in real time to unexpected perturbations or replanning events using reflexxes-style libraries that guarantee kinematic feasibility within one control cycle.
- For humanoid robots and legged systems, trajectory generation extends to full-body motion synthesis, where the challenge is maintaining dynamic balance (Zero Moment Point or Divergent Component of Motion criteria) while executing limb trajectories. Learning-based approaches — motion imitation from human capture data, diffusion models generating trajectory distributions, and reinforcement-learning policies that output joint trajectories directly — are rapidly supplementing classical methods, particularly for dexterous manipulation and locomotion in unstructured environments.
- In 2024-2025 the field is being transformed by large-scale robot learning, with foundation models trained on diverse trajectory datasets enabling generalisation across robot morphologies and task types without manual trajectory programming. Diffusion-based trajectory planners such as Diffuser and variants are achieving competitive performance on manipulation benchmarks. Real-time neural trajectory generation on embedded processors is becoming feasible through model distillation, enabling on-device replanning at kilohertz control rates without cloud round-trips.