Mid-air motion planning of robot using heat flow method with state constraints

Geometric Heat Flow Method for Legged Locomotion Planning

“…For example, for constraint (16b) we have h = h c −f terr (p x , p y ) ≤ 0. We modify slightly the method in [6] to encode the activation/deactivation of the state constraints in different phases. To this end, denote by h j (x), j ∈ Z + ≤ k c , the j-th scalar constraint function with k c the number of such constraints.…”

“…Thus, whereas PDEs such the HJB scale exponentially in the dimension of the system, the AGHF scales polynomially. However in our implementation, choice of larger λ, α, β can result in longer solving time, since the pdepe function determines the integration step size ∆s automatically without option to customize [6]. For example, with large value of λ, the magnitude of dx(•,s) ds is generally large, as a result, the solver uses smaller ∆s, increasing the solving time.…”

“…In this paper, we show that the geometric method we proposed for motion planning extends naturally to handle hybrid dynamics. Precisely, we show how the Ansatz developed in [4,5,6], contending that motion planning problems can be encoded into Riemannian metrics, can be applied and using fast solvers for parabolic partial differential equations (PDE), we can obtain efficiently natural motions.…”

“…These methods were extended in [5], in which the Affine Geometric Heat Flow (AGHF) is introduced to handle systems with drift. Motions for robot systems are generated using AGHF in [19,6]. The present work is the first attempt at formulating the hybrid legged locomotion problem into AGHF framework and shows the potential to extend to more complicated systems and scenarios.…”