- Piecewise Polynomial¶
A piecewise function is function of the form
where . It is naturally possible to define piecewise functions on unbounded domains too, but in typical applications the domain is bounded.
A piecewise polynomial is a piecewise function where each of the is a polynomial, possibly vector-valued. With suitable number of pieces and polynomial degree, piecewise polynomials make a highly expressive function class that is still simple and numerically stable.
In Crazyswarm, we use degree-7 polynomials with 4-dimensional vector-valued output: (x, y, z) position and yaw angle. The quadrotor’s differential flatness property makes it possible to compute other states (attitude, acceleration, angular velocity) from these four values.
A collection of desired values for some or all of the quadrotor’s state that the feedback controller should try to achieve. For example, a setpoint may specify position, velocity, and acceleration – or just position.