minimumsnap

Minimum snap is a trajectory optimization objective used in robotics and motion planning to produce smooth paths by minimizing the integral of the squared fourth derivative of position with respect to time, i.e., the snap. The approach is particularly popular in aerial robotics, such as quadrotor flight, where smoother trajectories reduce aggressive control inputs and improve tracking performance.

In practice, trajectories are often represented as piecewise polynomials. For each segment, the position is modeled as a polynomial in local time, typically of degree seven, yielding eight coefficients per spatial dimension. The objective minimizes J = sum over segments ∫ ||p''''(t)||^2 dt, where p'''' is the fourth derivative of the position. When segment times are fixed, this objective becomes a convex quadratic form in the polynomial coefficients, subject to boundary conditions that fix position, velocity, acceleration, and sometimes jerk at segment endpoints. Continuity is enforced across segments (commonly C^4 continuity).

A landmark approach is the minimum-snap trajectory generation method for quadrotors, introduced in the early 2010s, which has influenced many subsequent algorithms and implementations. The standard solution method uses convex optimization or quadratic programming to solve for the coefficients that minimize snap while satisfying waypoint and boundary constraints.

Variants of the method may relax or adjust the objective, incorporate time allocation as variables, or handle dynamic environments with re-planning. Minimum snap remains a foundational concept in high-drive robotics for producing smooth, dynamically feasible trajectories while balancing computational tractability with trajectory quality.