Trajectory Interpolation

Filling in a continuous, timed trajectory between a few given key points, such as a start, an end, and waypoints.

Trajectory interpolation computes, between a set of given points (start, end, waypoints), the position each control cycle should be at, and where needed, the velocity and acceleration too. The simplest is linear interpolation, which causes velocity to jump abruptly at waypoints. Cubic polynomials keep velocity continuous but can still leave acceleration discontinuous; quintic (fifth-order) polynomials additionally keep acceleration continuous, giving smoother motion. B-splines don't necessarily pass exactly through every point, but the resulting trajectory is guaranteed to stay within the convex hull of those points, which helps respect joint limits. Orientation can't be linearly interpolated directly on Euler angles, so spherical linear interpolation (slerp) of quaternions is used instead. Another common use is ‘upsampling’: when a model or a teleoperation device produces points slowly but the low-level controller expects fast commands, interpolation fills the gap.

ExampleIn Diffusion Policy's push-T task on a UR5, the policy issues 10 end-effector position commands per second, and the controller linearly interpolates them up to the robot's required 125 Hz, while capping end-effector speed under 0.43 m/s and restricting position to a region at least 1 cm above the tabletop.

Also called

Interpolation, Motion Interpolation

Related

Waypoint · Cubic Spline Interpolation · Quintic Polynomial Interpolation · Spherical Linear Interpolation (SLERP) · Time Parameterization · Action Chunking

Sources

Lynch & Park, Modern Robotics (2017 preprint), 9.3 Polynomial Via Point Trajectories
Diffusion Policy: Visuomotor Policy Learning via Action Diffusion（附录 UR5 robot station） (Chinese)