With the rapid development of robotics swarm technology, there are more tasks that require the swarm to pass through complicated environments safely and efficiently. Virtual tube technology is a novel way to achieve this goal. Virtual tubes are free spaces connecting two places that provide safety boundaries and direction of motion for swarm robotics. How to determine the design quality of a virtual tube is a fundamental problem. For such a purpose, this paper presents a degree of flowability (DOF) for two-dimensional virtual tubes according to a minimum energy principle. After that, methods to calculate DOF are proposed with a feasibility analysis. Simulations of swarm robotics in different kinds of two-dimensional virtual tubes are performed to demonstrate the effectiveness of the proposed method of calculating DOF.

Robotic swarm systems are now becoming increasingly attractive for many challenging applications. The main task for any robot is to reach the destination while keeping a safe separation from other robots and obstacles. In many scenarios, robots need to move within a narrow corridor, through a window or a doorframe. In order to guide all robots to move in a cluttered environment, a curve virtual tube with no obstacle inside is carefully designed in this paper. There is no obstacle inside the tube, namely the area inside the tube can be seen as a safety zone. Then, a distributed swarm controller is proposed with three elaborate control terms: a line approaching term, a robot avoidance term and a tube keeping term. Formal analysis and proofs are made to show that the curve virtual tube passing problem can be solved in a finite time. For the convenience in practical use, a modified controller with an approximate control performance is put forward. Finally, the effectiveness of the proposed method is validated by numerical simulations and real experiments. To show the advantages of the proposed method, the comparison between our method and the control barrier function method is also presented in terms of calculation speed.

Principal curves have been defined as "self consistent" smooth curves which pass through the "middle" of a d-dimensional probability distribution or data cloud. Recently, we [1] have offered a new approach by defining principal curves as continuous curves of a given length which minimize the expected squared distance between the curve and points of the space randomly chosen according to a given distribution. The new definition made it possible to carry out a theoretical analysis of learning principal curves from training data. In this paper we propose a practical construction based on the new definition. Simulation results demonstrate that the new algorithm compares favorably with previous methods both in terms of performance and computational complexity.

Principal curves have been defined as "self consistent" smooth curves which pass through the "middle" of a d-dimensional probability distribution or data cloud. Recently, we [1] have offered a new approach by defining principal curves as continuous curves of a given length which minimize the expected squared distance between the curve and points of the space randomly chosen according to a given distribution. The new definition made it possible to carry out a theoretical analysis of learning principal curves from training data. In this paper we propose a practical construction based on the new definition. Simulation results demonstrate that the new algorithm compares favorably with previous methods both in terms of performance and computational complexity.

Principal curves have been defined as "self consistent" smooth curves which pass through the "middle" of a d-dimensional probability distribution ordata cloud. Recently, we [1] have offered a new approach by defining principal curves as continuous curves of a given length which minimize the expected squared distance between the curve and points of the space randomly chosen according to a given distribution. The new definition made it possible to carry out a theoretical analysis of learning principal curves from training data. In this paper we propose a practical construction based on the new definition. Simulation results demonstrate that the new algorithm compares favorably with previous methods both in terms of performance and computational complexity.