Motion Planning With Gamma-Harmonic Potential Fields

📝 Abstract

This paper extends the capabilities of the harmonic potential field (HPF) approach to planning. The extension covers the situation where the workspace of a robot cannot be segmented into geometrical subregions where each region has an attribute of its own. The suggested approach uses a task-centered, probabilistic descriptor of the workspace as an input to the planner. This descriptor is processed, along with a goal point, to yield the navigation policy needed to steer the agent from any point in its workspace to the target. The approach is easily adaptable to planning in a cluttered environment containing a vector drift field. The extension of the HPF approach is based on the physical analogy with an electric current flowing in a nonhomogeneous conducting medium. The resulting potential field is known as the gamma-harmonic potential (GHPF). Proofs of the ability of the modified approach to avoid zero-probability (definite threat) regions and to converge to the goal are provided. The capabilities of the planer are demonstrated using simulation.

💡 Analysis

This paper extends the capabilities of the harmonic potential field (HPF) approach to planning. The extension covers the situation where the workspace of a robot cannot be segmented into geometrical subregions where each region has an attribute of its own. The suggested approach uses a task-centered, probabilistic descriptor of the workspace as an input to the planner. This descriptor is processed, along with a goal point, to yield the navigation policy needed to steer the agent from any point in its workspace to the target. The approach is easily adaptable to planning in a cluttered environment containing a vector drift field. The extension of the HPF approach is based on the physical analogy with an electric current flowing in a nonhomogeneous conducting medium. The resulting potential field is known as the gamma-harmonic potential (GHPF). Proofs of the ability of the modified approach to avoid zero-probability (definite threat) regions and to converge to the goal are provided. The capabilities of the planer are demonstrated using simulation.

📄 Content

Motion Planning with Gamma-Harmonic Potential Fields Ahmad A. Masoud Electrical Engineering Department, King Fahad University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia, e-mail: masoud@kfupm.edu.sa Abstract: This paper extends the capabilities of the harmonic potential field (HPF) approach to planning. The extension covers the situation where the workspace of a robot cannot be segmented into geometrical subregions where each region has an attribute of its own. The suggested approach uses a task-centered, probabilistic descriptor of the workspace as an input to the planner. This descriptor is processed, along with a goal point, to yield the navigation policy needed to steer the agent from any point in its workspace to the target. The approach is easily adaptable to planning in a cluttered environment containing a vector drift field. The extension of the HPF approach is based on the physical analogy with an electric current flowing in a nonhomogeneous conducting medium. The resulting potential field is known as the gamma-harmonic potential (GHPF). Proofs of the ability of the modified approach to avoid zero-probability (definite threat) regions and to converge to the goal are provided. The capabilities of the planner are demonstrated using simulation.

Keywords: probabilistic robotics, autonomous agents, motion planning, navigation, risk minimization, harmonic potential Nomenclature HPF: Harmonic potential field GHPF: Gamma-harmonic potential field BVP: Boundary value problem LFC: Liapunov function candidate P(x): a probabilistic field describing of the fitness of the point x to carry-out a task S: workspace of the agent So: definite threat (P(x)=0) subset of S Sa: the admissible region of the probabilistic workspace (Sa=S-So) O: forbidden regions in deterministic worspaces T: an infinitesimally small set in S u(x): navigation policy xs: starting point of motion xT: target point N: the empty set D(t): the time paramerized trajectory laid by the planner H(x): the hessian matrix L: the gradient operator L2: the laplacian operator L@: the divergence operator ‘: boundary of the deterministic workspace V(x): potential field F(x): conductivity J(x): electric current density Q(x): a vector field describing the drift force in the workspace =: Liapunov function : time derivative of = Ξ E: minimum invariant set I. Introduction: Designing an autonomous agent is a challenging multi-disciplinary task [1]. Special attention is being paid to the propulsion, data acquisition and communication systems used by an agent. However, the biggest challenge seems to be in designing a proper planning module. The function of this module is to unite these sub-systems into one goal-oriented unit. There is a long list of conditions a planner must satisfy. These conditions are necessary in order to generate a sequence of instructions which the actuators of motion may execute to successful completion of an assignment. However, the conditions on handling and representing mission data seem to be the most stringent [2]. There are two core requirements a representation should satisfy. They are 1- compatibility with the manner in which data is being processed and action is being generated, 2- updatability of the representation. Updatability requires that the validity of the existing portion of the representation not be conditioned on the future data that could be received. This allows for the incremental construction of a representation. Another important issue is to increase the diversity of environment-related, operator-supplied information which the planner is capable of processing in order to yield the guidance signal.

Most planners assume divisible environments that may be partitioned into subsets of homogeneous attributes. The most common scheme is to have a binary partition of admissible sets and forbidden ones. Binary partitions may be constructed using geometric structures like circles [3], occupancy maps, Voronoi partitions [4], grids and graphs [5], samples [9] and trees [6]. There are situations where the environment in which an agent is operating is not divisible. For example, a plane flying through turbulent atmosphere [7] experiences a degree of turbulence where clear space is diffused into turbulent space with no sharp boundaries separating the two. Also, in the case of mobile robots operating in rough terrains [8], the description should be based on the degree of difficulty in negotiating the terrain. It is highly unlikely that success can be attained by basing the actions on a binary geometric partition of the environment into admissible and forbidden regions. An alternative to the geometric approach is the use of a soft representation that consists of a field reflecting, at each point in the environment, the probability of achieving the task. Probabilistic representations are ideal for encoding the information in non-divisible environments. These representations may also be u

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