Robot control
Hybrid zero dynamics
Definition
Hybrid zero dynamics is the reduced closed-loop dynamics that remains when a legged robot's controlled outputs are held at zero and that zero set is preserved through impacts. It provides a way to design and analyse periodic gaits in systems with continuous motion and discrete contact events.
Also known as: HZD, Hybrid zero dynamics control
Updated
Zero dynamics across contact events
In nonlinear control, zero dynamics describes the internal motion left after feedback drives chosen outputs and their required derivatives to zero. Walking is hybrid because a continuous swing phase is interrupted by discrete events such as foot impact. Hybrid zero dynamics adds the requirement that the zero-dynamics surface maps back into itself across those events.
The Grizzle and colleagues chapter develops this construction for underactuated bipedal robots. It uses virtual constraints to synchronise joints to an internal gait phase and obtain a lower-dimensional model without approximating away the system's underactuated coordinates.
Designing a periodic gait
A designer chooses outputs, a phase variable, and desired output trajectories. Optimisation can then search for a periodic orbit that satisfies the full robot dynamics, actuator limits, contact conditions, and impact consistency. Feedback drives the robot towards the corresponding zero-dynamics surface, while a return map can be used to study the orbit's step-to-step stability.
HZD can also sit inside a layered controller. Dai and colleagues generated full-order HZD walking trajectories offline for the point-foot ADAM humanoid, regulated step length online with a reduced pendulum model, and reported simulation and hardware experiments. Those results establish the demonstrated controller and robot combination, not a guarantee for every biped.
Model assumptions remain important
Hybrid invariance is a property of a chosen model, impact map, outputs, and controller. Unexpected compliance, foot slip, actuator saturation, state-estimation error, or a contact sequence different from the model can move the physical robot away from the intended manifold.
The method can provide a rigorous stability analysis for a nominal gait, but it does not by itself plan around obstacles or select safe footholds. Those functions may come from a separate planner, estimator, or higher-level controller.
Sources
Related terms
Underactuation
Underactuation means that a system’s available control inputs cannot independently command acceleration in every degree of freedom of its model. It often occurs when a mechanism has fewer independent actuators than degrees of freedom.
Bipedal locomotion
Bipedal locomotion is movement using two legs, with body motion coordinated through changing contacts between the feet and the environment. It includes walking and running.
Virtual constraint
A virtual constraint is a relation among a robot's configuration variables that is imposed by feedback instead of by a physical linkage. In legged locomotion, virtual constraints often coordinate joint trajectories around a state-based gait phase.
Control Lyapunov function
A control Lyapunov function is a scalar function of a controlled system's state for which an admissible control input can make the function decrease toward a target. It provides a way to design or constrain feedback controllers with a stated stability objective.