Multi-Goal Path Optimization for Robotic Systems with Redundancy based on the Traveling Salesman Problem with Neighborhoods

Finding an optimal path for a redundant robotic system to visit a sequence of several goal locations is a complex optimization problem and poses two main technical challenges. Because of the redundancy in the system, the robot can assume an infinite number of goal configurations to reach each goal location. Therefore, not only an optimal sequence of the goals has to be defined, but also, for each goal, an optimal configuration has to be chosen among infinite possibilities. Second, the actual cost for the system to move from one configuration to the next depends on many factors, such as obstacle avoidance or energy consumption, and can be calculated only through the employment of specific path planning techniques.

We first address the optimization problem of finding an optimal sequence of optimal configurations, while assuming the cost function to be analytically defined. This problem is modeled as a Traveling Salesman Problem with Neighborhoods (TSPN), which extends the well-known TSP to more general cases where each vertex (goal configuration) is allowed to move in a given region (neighborhood). In the literature, heuristic solution approaches are available for TSPN instances with only circular or spherical neighborhoods. For more general neighborhood topologies, but limited to the Euclidean norm as edge weighting function, approximation algorithms have also been proposed. We present three novel approaches: (1) a global Mixed Integer Non Linear Programming (MINLP) optimizer and (2) a convex MINLP optimizer are modified to solve to optimality TSPN instances with up to 20 convex neighborhoods, and (3) a hybrid random-key Genetic Algorithm (GA) is developed to address more general problems with a larger number of possibly non-convex neighborhoods and with different types of edge weighting functions. Benchmark tests show that the GA is able to find the same optimal tour calculated by the MINLP solvers while drastically reducing the computational cost, and it always improves the best known solutions for available test problems with up to 1,000 neighborhoods.

Second, we integrate the GA with a probabilistic path planning technique to apply the proposed procedure to two practical applications. We minimize the time currently required by an industrial vision inspection system to complete a multi-goal cycle, where the neighborhoods are defined using piecewise cubic splines in a seven-dimensional configuration space. Afterwards, we optimize the flight path and the energy consumption of a quadrotor Unmanned Aerial Vehicle (UAV) on an urban survey mission. The specifications of the camera installed on the UAV are used here to define the neighborhoods as three-dimensional polyhedra.