Motion Planning: Wild Frontiers

The basic problem of computing a collision-free path for a robot among known obstacles is well-understood and reasonably well-solved; however, deficiencies in the problem formulation itself and the demand of engineering challenges in the design of autonomous systems raise important questions and topics for future research. The shortcomings of basic path planning become clearly visible when considering how the computed path is typically used in a robotic system. It has been known for decades that effective autonomous systems must iteratively sense new data and act accordingly; recall the decades-old Sense Plan Act (SPA) paradigm. Figure 1 shows how a computed collisionfree path τ : [0, 1] → Cfree is usually brought into alignment with this view by producing a feedback control law. Step 1 produces τ using a path planning algorithm. Step 2 then smooths τ to produce σ : [0, 1] → Cfree, a path that the robot can actually follow. For example, if the path is piecewise linear, then a car-like mobile robot would not be able to turn sharp corners. Step 3 reparameterizes σ to make a trajectory q̃ : [0, tf ] → Cfree that nominally satisfies the robot dynamics (for example, acceleration bounds). In Step 4, a state-feedback control law is designed that tracks q̃ as closely as possible during execution. This results in a policy or plan, π : X → U . The domain X is a state space (or phase space) and U is an action space (or input space). These sets appear in the definition of the control system that models the robot: ẋ = f(x, u) in which x ∈ X and u ∈ U . One clear problem in this general framework is that a later step might not succeed due to an unfortunate, fixed choice in an earlier step. Even if it does succeed, the produced solution may be horribly inefficient. This motivates planning under differential constraints, which essentially performs Steps 1 and 2, or Steps 1, 2, and 3 in one shot; see Section II. The eventual need for feedback in Step 4 motivates the direct computation of a feedback plan, covered in Section III. Another issue with the framework in Figure 1, which is perhaps more subtle, is that this fixed decomposition of the overall problem of getting a robot to navigate has artificially inflated the information requirements. The framework requires that powerful sensors, combined with strong