Bézier Curve Trajectory

A smooth polynomial curve shaped by a handful of control points, often used to describe robot trajectories.

The Bézier curve was popularized for car-body design in the 1960s by Renault engineer Pierre Bézier, while Citroën's Paul de Casteljau had independently found the underlying computation even earlier. An n-th order curve is defined by n+1 control points P₀…Pₙ: B(s) = Σᵢ C(n,i)(1−s)ⁿ⁻ⁱ sⁱ Pᵢ, with the parameter s running from 0 to 1 and C(n,i) the binomial coefficient. Three properties make it well suited to trajectories: the curve passes exactly through the first and last control points; the tangent direction at the start and end is set by the adjacent control points, which makes it easy to join with the segments before and after; and the whole curve stays within the convex hull formed by the control points, so keeping the control points clear of obstacles keeps the curve clear too. The intermediate control points don't lie on the curve — they only pull its shape. In robotics it's commonly used for swing-foot trajectories and end-effector path smoothing; hybrid zero dynamics methods also use Bézier polynomials to describe reference gait trajectories. Long paths are usually built from several low-order segments rather than one high-order curve.

ExampleMIT's open-source Cheetah-Software uses a cubic Bézier curve to generate the swing-leg trajectory: the horizontal direction transitions smoothly from the start point to the landing point, while the vertical direction rises to the step height in the first half and descends back to the ground in the second, with the curve's derivative giving foot velocity and acceleration directly.

Also called

Bezier Curve, Bézier Polynomial

Related

Swing Foot Trajectory Planning · Cubic Spline Interpolation · Quintic Polynomial Interpolation · Minimum-Jerk Trajectory · Waypoint · Hybrid Zero Dynamics

Sources

Wikipedia: Bézier curve
MIT Cheetah-Software: FootSwingTrajectory.cpp（Currently uses Bezier curves like Cheetah 3 does）