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Register a new userThe Affine Geometric Heat Flow and Motion Planning for Dynamic Systems
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We present a new method for motion planning for control systems. The method aims to provide a natural computational framework in which a broad class of motion planning problems can be cast; including problems with holonomic and non-holonomic constraints, drift dynamics, obstacle constraints and constraints on the magnitudes of the applied controls. The method, which finds its inspiration in recent work on the so-called geometric heat flows and curve shortening flows, relies on a hereby introduced partial differential equation, which we call the affine geometric heat flow, which evolves an arbitrary differentiable path joining initial to final state in configuration space to a path that meets the constraints imposed on the problem. From this path, controls to be applied on the system can be extracted. We provide conditions guaranteeing that the controls extracted will drive the system arbitrarily close to the desired final state, while meeting the imposed constraints and illustrate the method on three canonical examples.
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We propose in this paper a motion planning method for legged robot locomotion based on Geometric Heat Flow framework. The motion planning task is challenging due to the hybrid nature of dynamics and contact constraints. We encode the hybrid dynamics and constraints into Riemannian inner product, and this inner product is defined so that short curves correspond to admissible motions for the system. We rely on the affine geometric heat flow to deform an arbitrary path connecting the desired initial and final states to this admissible motion. The method is able to automatically find the trajectory of robots center of mass, feet contact positions and forces on uneven terrain.
The problem of motion planning for affine control systems consists of designing control inputs that drive a system from a well-defined initial to final states in a desired amount of time. For control systems with drift, however, understanding which final states are reachable in a given time, or reciprocally the amount of time needed to reach a final state, is often the most difficult part of the problem. We address this issue in this paper and introduce a new method to solve motion planning problems for affine control systems, where the motion desired can have indefinite boundary conditions and the time required to perform the motion is free. The method extends on our earlier work on motion planning for systems without drift. A canonical example of parallel parking of a unicycle with constant linear velocity is provided in this paper to demonstrate our algorithm.
We propose a novel method for motion planning and illustrate its implementation on several canonical examples. The core novel idea underlying the method is to define a metric for which a path of minimal length is an admissible path, that is path that respects the various constraints imposed by the environment and the physics of the system on its dynamics. To be more precise, our method takes as input a control system with holonomic and non-holonomic constraints, an initial and final point in configuration space, a description of obstacles to avoid, and an initial trajectory for the system, called a sketch. This initial trajectory does not need to meet the constraints, except for the obstacle avoidance constraints. The constraints are then encoded in an inner product, which is used to deform (via a homotopy) the initial sketch into an admissible trajectory from which controls realizing the transfer can be obtained. We illustrate the method on various examples, including vehicle motion with obstacles and a two-link manipulator problem.
For homogeneous bilinear control systems, the control sets are characterized using a Lie algebra rank condition for the induced systems on projective space. This is based on a classical Diophantine approximation result. For affine control systems, the control sets around the equilibria for constant controls are characterized with particular attention to the question when the control sets are unbounded.
For a right-invariant and controllable driftless system on SU(n), we consider a time-periodic reference trajectory along which the linearized control system generates su(n): such trajectories always exist and constitute the basic ingredient of Corons Return Method. The open-loop controls that we propose, which rely on a left-invariant tracking error dynamics and on a fidelity-like Lyapunov function, are determined from a finite number of left-translations of the tracking error and they assure global asymptotic convergence towards the periodic reference trajectory. The role of these translations is to avoid being trapped in the critical region of this Lyapunov-like function. The convergence proof relies on a periodic version of LaSalles invariance principle and the control values are determined by numerical integration of the dynamics of the system. Simulations illustrate the obtained controls for $n=4$ and the generation of the C--NOT quantum gate.
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