Step sequence planning for a biped robot by means of a cylindrical shape model and a high-resolution 2.5D map
Research highlights
► Efficient evaluation of step feasibility is achieved by a cylindrical robot model. ► Fast step sequence planning using high-resolution 2.5D occupancy grids is achieved. ► The applied numerical navigation function supports stepping over obstacles. ► Navigation function allows local planning without getting stuck in local minima. ► Stepping over obstacles and stair climbing are treated in a unified manner.
Introduction
Humanoid robots are a particularly interesting group of mobile robots because of their resemblance to humans and their potential ability to operate in environments created for humans. In comparison to wheeled robots, humanoid robots can move more freely in indoor and outdoor environments. They can climb stairs, step over or onto obstacles. However, motion planning for a biped walking robot is a rather demanding task because of the complex kinematics of such machines and the many degrees of freedom involved.
One approach to this problem is to decompose the motion planning problem into two subproblems. The first subproblem involves generation of feasible stepping actions for handling particular environmental situations, while the second considers planning of step sequences from a starting configuration to a goal configuration by concatenation of appropriate steps. Feasible steps can be generated on-line [1] or acquired from a walking primitive database generated off-line [2], [3]. The methods for the generation of the steps are not discussed herein. It is assumed that the control system of the robot is able to generate an appropriate step for certain arrangements of walking surfaces and obstacles on the walking trail. The focus of this work is the problem of generating a sequence of steps that enables a biped robot a collision-free walk from a start to a goal position in a static indoor environment, such as the one indicated in Fig. 1. In addition to being collision-free, the planned step sequence must be such that the supporting foot of the robot always rests upon a flat horizontal surface. Furthermore, the SSP must be capable of using the locomotion abilities of a walking robot to step over obstacles and change the walking level, i.e. by climbing stairs, as shown in Fig. 1.
Given a discrete set of walking primitives, SSP can be performed by building a search tree using a dynamic programming technique, where each node in this search tree corresponds to a robot configuration defined by its foot placements, as proposed in [4], [5]. A node is expanded by adding a set of nodes to the search tree, where each added node represents the robot configuration which can be reached from the expanded configuration by a single feasible step. Step sequences are generated using two graph search methods, -search and best-first search. -search is guided by a heuristic which assigns to each configuration of the robot considered in the search procedure the estimated remaining cost of the path from the current configuration to the goal region. If an admissible heuristic is used to estimate the remaining cost to the goal, -search provides an optimal path. Instead of an Euclidean distance, which is commonly used in -search for estimation of the remaining cost to the goal, an alternative heuristic is applied in [6]. Path planning backwards from the goal to the currently considered configuration with a standard mobile robot planner is performed as a precomputation step, while the length of the obtained path is used as the estimated remaining cost to the goal.
The heuristic based on a standard mobile robot planner used in [6] for estimation of the remaining cost to the goal takes into account more information about the environment than an Euclidean distance metric. It is demonstrated in [6] that it can significantly speed up the planning process, but only insofar as the optimal path for the considered walking robot can be followed by a wheeled robot. Because of the abilities of a walking robot to step over small obstacles, the cost of the path which can take the robot to the goal position can in some cases be much lower than the cost estimated by planning a path for a wheeled robot. Let us, for example, consider the case where a thin and very long obstacle is located between the current and the goal state of the robot. Although the walking robot can reach the goal with one or two steps by stepping over the obstacle, a standard mobile robot planner would plan a path around the obstacle. The length of this rather long path would be taken as the estimated remaining cost to the goal, thus resulting in an overestimate. Furthermore, in a case where the only way a given goal state can be reached from a current state of the robot is to climb stairs, there is no path for a wheeled robot which can be used for estimating the remaining cost to the goal. In general, the more an environment requires the biped’s capabilities to step over or onto obstacles, the less informed the proposed heuristic will be.
In this paper, an alternative method for estimating the remaining cost to the goal is proposed. Instead of using a standard mobile robot planner, we propose application of path planning based on a numerical navigation function computed using the free space representation for walking robots, as earlier presented in [7], [8]. This method is expected to provide much better estimation of the remaining cost to the goal, since it takes into account the capabilities of a walking robot to step over or onto obstacles. Furthermore, the same free space representation can be used to achieve a very efficient checking of feasibility of steps, i.e. to verify whether a particular step can be executed at a given position in the robot’s environment.
The rest of the paper is organized as follows. A survey of the recently published motion planning approaches for biped walking robots is given in Section 2. In Section 3, a novel robot model based on the bounding approximation proposed in [7] is presented. This robot model allows efficient determination of the feasibility of performing a particular action at a given location in a 2.5D environment map. Efficiency is achieved by transforming the 2.5D map into a suitable free space representation presented in [7] and searching for a low-cost path in this free space. Properties of this free space representation relevant for SSP are discussed in Section 4 and a novel SSP algorithm which uses these properties is presented in Section 5. The efficiency of the proposed approach is evaluated by a series of simulations performed for eight typical walking scenarios. Details of the simulations and results are presented in Section 6, where the performance of the proposed algorithm is compared to some of the recently reported SSP methods.
Section snippets
Related work
Various approaches to the SSP problem have been reported recently. Some of them are reviewed in this section in order to make the reader aware of basic ideas applied in this field, some of which are used as guidelines in the design of the method proposed in this paper.
An approach to the problem of computing global navigation strategies for biped humanoids which involves an iterated discrete search over a set of valid foot placements [4], [5] has already been discussed in the introduction. In
Biped walking robot model
In this section, a novel mathematical model of biped walking robot suitable for footstep planning is presented. Trajectory planning and execution is not considered in this paper. It is assumed that the robot is capable of performing stable motion whose properties are specified in this section. The robot standing in a particular pose in a 3D scenario is modeled by a configuration vector defining its footstep positions in the double support phase relative to a world coordinate system . A step
Environment model
Modeling a walking robot by hulls described in Section 3 allows efficient collision-checking using a specially designed environment map, as described in the following. The proposed environment representation can be created from a 2.5D map . In the following, the 2.5D map is represented by a function which assigns to each point of the -plane of the world coordinate system a value representing the height of the lowest free point, i.e. a point not occupied by an object, at
Step sequence planning
Let us define a feasible sequence as a sequence of feasible configurations , where , for . The SSP problem can be formulated as a search for a feasible sequence , with , …, and where is the initial configuration and . The SSP task is successfully completed if a feasible sequence is found which ends in any pose defined by the set . This set can be regarded as a region
Simulations and evaluation
A SSP program based on the Algorithm 1 is developed and evaluated by a series of simulations. The program consists of three main parts. The first part creates the free space representation from a given 2.5D map using the algorithm presented in [21]. The second part computes the NNF by wave propagation, as explained in Appendix E. Finally, the third part performs SSP using the free space representation created by the first part and the values of NNF computed by the second part. The total
Discussion and conclusions
In this work, a novel SSP method for biped walking robots is presented. The method is based on the free space representation proposed in [7], which is created from a 2.5D map of the robot’s environment. This free space representation relies on a bounding approximation of the robot’s shape by a set of cylindrical solids that enables efficient detection of obstacle free regions as well as paths over obstacles for which feasible step combinations exist. Both simplifications result in a significant
Acknowledgements
The authors would like to thank the anonymous reviewers of this paper for their invaluable remarks and suggestions.
Robert Cupec graduated at the Faculty of Electrical Engineering, University of Zagreb, Croatia in 1995, where he received the M.Sc. degree in 1999. He received Dr.–Ing. degree from the Institute of Automatic Control Engineering, Technische Universität München, Munich, Germany, in 2005. Currently, he is employed as an assistant professor at the Faculty of Electrical Engineering, Josip Juraj Strossmayer University of Osijek, Croatia. His research interests include mobile robotics and robot vision.
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Robert Cupec graduated at the Faculty of Electrical Engineering, University of Zagreb, Croatia in 1995, where he received the M.Sc. degree in 1999. He received Dr.–Ing. degree from the Institute of Automatic Control Engineering, Technische Universität München, Munich, Germany, in 2005. Currently, he is employed as an assistant professor at the Faculty of Electrical Engineering, Josip Juraj Strossmayer University of Osijek, Croatia. His research interests include mobile robotics and robot vision.
Ivan Aleksi graduated at Faculty of Electrical Engineering, Josip Juraj Strossmayer University of Osijek, Croatia, in 2006. Currently, he is a Teaching and Research Assistant at the Faculty of Electrical Engineering, Osijek. His research interests include FPGA-based systems and mobile robotics. He is a member of IEEE.
Günther Schmidt is a Professor emeritus at the Institute of Automatic Control Engineering of the Technische Universität München, Germany. His main research interests are control theory and mobile robotics, telepresence and neuroprosthetic sytems. Dr. Schmidt is a recipient of various scientific awards.