Planning collision-free trajectories in time-varying environments a two-level hierarchy Kamal Kant and Steven W. Zucker* Computer Vision and Robotics Laboratory McGill Research Centre for Intelligent Machines McGill University, Montr6al. Qu6bec, Canada We propose a two-level hierarchy for plan- ning collision-free trajectories in time varying environments. Global geometric algorithms for trajectory planning are used in conjunction with a local avoidance strategy. Simulations have been developed for a mobile robot in the plane among sta- tionary and moving obstacles. Essentially, the robot has a global geometric planner that provides a coarse global trajectory (the path and velocity along it), which may be locally modified by the low-level local avoidance module if local sensors detect any obstacles in the vicinity of the robot. This hierarchy makes effective use of the complementary aspects of the global tra- jectory planning approaches and the local obstacle avoidance approaches. Key words: Robotics - Trajectory plan- ning - Path planning - Obstacle avoidance - Velocity planning * Senior Fellow, Canadian Institute for Advanced Research 304 1 Introduction How does one get from "here" to "there"? This is an everyday version of the problem we shall be considering in this paper, especially in the form that arises in robotics. The ability to plan and exe- cute collision-free motions is fundamental for both manipulator arms and mobile robots. Starting from planning a collision-free motion to the actual control of limbs (or links) to achieve that motion involves descriptions at many different levels - geometry, dynamics, and control - and different constraints arise at each of these levels. These levels are broadly grouped under two levels: a higher "Planning level" and a lower (closer to actuators) "control level". The higher level concerns itself with specifying a desired trajectory given the task description, and is often called motion planning. One formulation of the motion planning problem is: determine the motion (path) of a given object from an initial con- figuration to a goal configuration, such that it avoids collisions with other stationary or moving obstacles in its environment. Traditionally re- searchers have concentrated on the case of static obstacles (for mobile robots, see Nilsson 1969; Thompson 1977; Moravac 1980; Giralt et al. 1979; Giralt 1984); for manipulator arms, see Udupa 1977; Lozano-P6rez 1979; Schwartz and Sharir 1982; Brooks 1983). Recently, however, approaches have been suggested to plan robot motion among moving obstacles, called trajectory planning (Kant and Zucker 1984, 1986; Reif and Sharir 1985; Erd- mann and Lozano-P6rez 1986; Kant 1987; Aronov 1987). Researchers have drawn their tools from ge- ometry, topology, and graph theory. We shall in- formally group these approaches under one name: geometric approaches to motion planning. These approaches are generally associated with "spatial reasoning". They assume a complete knowledge of the environment, build an explicit description of the constraints on robot motion due to the obstacles, and then search for a complete path. In time-vary- ing environments, these constraints are functions of time, and are represented as regions in space- time (see Sect. 3). This space-time is then searched for a trajectory, i.e. the path as a function of time. The lower control level, on the other hand, con- cerns itself with predetermined trajectories: it is re- sponsible for moving the robot along the trajectory to achieve various requirements such as accuracy, smoothness, etc. The issues here are kinematics, dynamics and control. However, there exist ap- proaches that have incorporated local obstacle avoidance at this lower control level (akin to reflex- The VisualComputer(1988) 2:304-313 9 Springer-Verlag 1988