Nonholonomic Motion Planning of Mobile Robots

Introduction

Wheeled mobile robots have attracted a lot of interest recently due to useful practical tasks such as robot fork trucks handling palettes in factories or obstacle avoidance. Based on the location of the load, a trajectory must be generated which ends precisely in front of, and aligned with the fork holes. The robot velocity must be zero at terminal point. It is often practically desirable that the mobile robot should avoid an obstacle while staying on the road. In such a case the robot should approach the obstacle as close as possible, which may lead to the necessity of determining the collision-free paths of minimum lengths. The aforementioned tasks belong to a class of motion planning problems. Motion planning is concerned with obtaining open loop controls which steer a platform from an initial configuration (state) to a final one, without violating the nonholonomic constraints. In addition, control constraints resulting from the physical abilities of the actuators driving the wheels should also be taken into account during the motion. Moreover, if there exist obstacles in the work space (state inequality constraints), the controls should steer the robot in such a way as to avoid collisions with obstacles. In such a context, two basic approaches to solving this problem may be distinguished. One is based on global methods of nonholonomic motion planning, which may be divided into discretized and continuous. The discretized technique is a sequential search of a graph whose edges are generated based on a discretized control space. Graph search methods proposed in [1, 2, 3] generate the globally optimal motion. The drawback of using graph-search techniques for trajectory generation is the resolution lost due to discretization of state and/or control space. Global continuous methods may be categorized into two classes. The first class uses an optimal control theory to generate robot trajectories. The Pontryagin maximum principle has been used in [4] to determine the time optimal trajectory in the unobstructed work space. The resulting controls are discontinuous and bang-bang. Using the calculus of variations and parameterization of controls, works [5, 6] convert the continuous optimal control formulation into an equivalent nonlinear programming problem. The second class of nonholonomic motion planning techniques offered in works [7, 8] is based on the use of Newton’s method. Its adaptation to collision avoidance is contained in [9]. However, it requires a time consuming iterative computational procedure. Moreover, only local optimization of a performance index is carried out when searching for the robot trajectory. A near realtime optimal control trajectory generator is presented in [10] which solves eleven first-order differential equations subject to the state constraints. Application of averaging method to motion planning is presented in [11]. The author of [11] proposed the use of truncated Fourier series to express the steering input and found the solution based on a shooting method. A suitable transformation of nonholonomic constraints and trajectory parameterization has been proposed in [12] to avoid obstacles. Nevertheless, this method does not take into account control constraints. In the context of mobile manipulators with nonholonomic platform, motion planning algorithms at the control feedback level have been proposed in works [13, 14]. The second approach to the nonholonomic motion planning is based on local methods. Among them, a method based on a Lie algebra of system evaluated at a given point is the most representative. It has been developed in work [15]. However, the computation of the GCBHD formula [16] seems to be time consuming. In order to adequately react to the environment, based on sensed information gathered during the robot movement, a nonholonomic motion plan must be generated in real time. Such amotion (desired trajectory) is simultaneously provided to online controllers (see e.g. [17, 18]), which may considerably increase precision control. It is obvious that almost all the aforementioned algorithms are not suitable to realtime implementations due to their computational complexity.