Robot control
Reachability analysis
Definition
Reachability analysis computes sets of states that a dynamical system can reach, or from which it can reach a target, under specified controls, time limits, and disturbances. Robots use it to reason about feasibility, collision avoidance, and conditional safety guarantees.
Also known as: Reachable-set analysis, Reachable set analysis
Updated
Sets of possible future or past states
A forward reachable set contains states that can result from a starting set over a chosen time horizon. A backward reachable set contains states from which the system can reach a target. The definition also specifies who chooses the controls and how disturbances or another agent act. Changing those quantifiers can turn an avoidable collision set into an inevitable one.
Hamilton-Jacobi reachability represents a target or unsafe set with a value function and propagates it through the system dynamics. The overview by Bansal and colleagues defines a reach-avoid set as states from which a target can be reached while time-varying state constraints remain satisfied.
Feasibility and safety applications
A robot can use a backward reachable set to identify states from which braking cannot prevent contact, or states from which a goal remains attainable despite bounded disturbance. A controller derived from the value function can then act near the computed boundary. Reachability can also supply a terminal set or safety layer for model predictive control.
Reachability analysis is not the same as a geometric workspace. A workspace records end-effector poses permitted by geometry and joint limits. A dynamical reachable set includes time, velocity, actuation, and disturbance assumptions. It also differs from ordinary motion planning: a planner seeks one feasible motion, while reachability characterizes a set of states under stated choices and uncertainties.
Guarantees depend on tractable models
The cited overview explains that grid-based Hamilton-Jacobi calculations scale exponentially with the number of state variables. A full humanoid state is therefore far beyond a direct dense-grid calculation. Decomposition, reduced models, local approximations, learned approximations, or pairwise analyses can reduce cost, but they change what has been computed or certified.
Any guarantee is conditional on the model, bounds, target set, horizon, numerical resolution, and available controls. An obstacle absent from the state, an underestimated disturbance, or actuator saturation outside the model can invalidate the conclusion. Reachability should state those assumptions rather than label the whole robot safe.
Sources
Related terms
Control barrier function
A control barrier function is a mathematical function used to express a safe set for a dynamical system and constrain control inputs so the system remains inside that set. It is commonly used as a safety filter around a nominal robot controller.
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.
Model predictive control
Model predictive control repeatedly optimizes future actions using a system model, applies the next part of the solution, and replans from updated state information. It can account for objectives and constraints over a finite prediction horizon.
Workspace
A robot workspace is the set of positions or poses its end-effector can reach under specified geometric and joint constraints. Its meaning depends on whether orientation is included and which base and tool configuration are assumed.