Robot control
Differential flatness
Definition
Differential flatness is a property of some dynamical systems for which every state and input can be written from a chosen set of flat outputs and a finite number of their time derivatives. Planning those outputs can turn a difficult trajectory problem into a lower-dimensional one, but only when valid flat outputs and the required reconstruction are known.
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Plan in outputs that determine the full motion
For a differentially flat system, a flat-output trajectory and enough of its derivatives determine the robot's state and control input. The output is not merely a measured quantity. It must support an algebraic reconstruction of the complete dynamically feasible motion.
The MIT Underactuated Robotics notes give the formal reconstruction condition and work through planar and three-dimensional quadrotors. For the three-dimensional case, position and yaw can serve as flat outputs under the model used there.
Reduce trajectory generation to a smoother search
Instead of optimising every state and actuator trajectory independently, a planner can optimise smooth curves in the flat outputs and recover the other variables. Mellinger and Kumar used quadrotor flatness in minimum-snap trajectory generation, where snap is the derivative of jerk.
The approach is useful for aerial vehicles, wheeled systems, and some manipulators whose structure exposes suitable flat outputs. It differs from general trajectory optimisation: flatness is a system property that permits a special parameterisation, while a general optimiser can operate without it.
Flatness does not remove physical constraints
Many systems are not differentially flat, and finding flat outputs can require substantial algebra and modelling insight. The reconstruction may demand high-order derivatives, so a noisy or insufficiently smooth output trajectory can create unusable commands.
Actuator saturation, collision avoidance, aerodynamic effects, contacts, and state-estimation error still constrain the reconstructed motion. A planned flat trajectory also needs feedback for execution. The cited quadrotor work demonstrates a particular model, planner, controller, and experimental setting rather than a universal recipe for every flying robot.
Sources
Related terms
Trajectory optimization
Trajectory optimization finds a time-varying motion, and often control inputs, that minimizes an objective while satisfying specified constraints. Robot applications can include geometric, kinematic, and dynamic constraints.
Motion planning
Motion planning finds a robot movement from an initial state to a goal while satisfying constraints such as collision avoidance. A planner may produce a geometric path, a timed trajectory, or a sequence of controls.
Jerk
Jerk is the rate at which acceleration changes with time, or the third time derivative of position. Robotics uses jerk limits to constrain how abruptly a commanded motion changes acceleration.
Geometric control
Geometric control designs control laws using the intrinsic geometry of a system's configuration space rather than treating its state as unconstrained coordinates. In robotics, it is often used for rotation and rigid-body motion on manifolds such as SO(3) and SE(3).