Edwin Francis Cárdenas¹, Luis Miguel Mendez², Jorge Sofrony Esmeral³

¹ Electromechanical Engineer, M.Sc., Universidad Nacional de Colombia, Colombia. ecardenasc@unal.edu.co

² Mechanical Engineer, M.Sc., Assistant Professor, Universidad Nacional de Colombia, Colombia. lmmendezm@unal.ecu.co

³ Electrical Engineer, Ph.D., Assistant Professor, Universidad Nacional de Colombia, Colombia. jsofronye@unal.edu.co

ABSTRACT

Reliable path-planning and generation of collision-free trajectories has become an area of active research over the past decade where the field robotics has probably been the most active area. This paper' s main objective is to analyse the advantages and disadvantages of two of the most popular techniques used in collision-free trajectory generation in n-dimensional spaces. The importance of analysing such techniques within a generalised framework is evident as path-planning is used in a variety of fields such as designing medical drugs, computer animation and artificial intelligence and, of course, robotics. The review provided in this paper starts by drawing a historical map of path-planning and the techniques used in its early stages. The main concepts involved in artificial potential fields and probabilistic roadmaps will be addressed as these are the most influential methods and have been widely used in specialised literature.

Keywords: robotics, path planning, obstacle avoidance.

Received: October 2th 2009

Accepted: November 4th 2010

Introduction

The widespread use of robots in a variety of industrial processes during the 1980s and 1990sbrought the need for generating trajectories to the foreground which were free from collisions between a robot and the environment it interacted with (Goldman, 1990). Although this task was initially performed by different specialists, autonomy has become an important aspect of robotics in more recent years and hence the need to develop more efficient algorithms. The task of collision-free trajectory planning (CFTP) comes naturally to humans and a variety of living organisms, but an abstraction of generating such a trajectory from start to goal position is not trivial and may become highly complex to implement.

The first attempt to plan the path of a robot was presented by Lozano-Pérez (Lozano, 1981). After the seminal work by Jean-Claude Latombe (Latombe, 1991), many researchers have addressed the problem of CFTP, but very few publications have closed the gap between newly-developed methods and the theoretical aspects involved.

Recent technological developments have increased processing and computational capability, allowing novel path-planning techniques to be implemented as part of state-of-the-art applications such as robot-aided surgery (Sim, 2001; Goo, 2005), 3D animation (Sim, 2001;Goo, 2005), virtual actors in movies and video games (Funge, 2004), robot-assisted museum tours and hospital navigation (Maio, 1995), modelling chemicals and their molecular structure (Kirillova, 2008) and deep space exploration using mobile robots such as rovers (Bresina, 2005; Carsten, 2009).

Although several geometric methods can be used for solving basic path-planning problems, Latombe has highlighted three methods (Latombe, 1991): artificial potential fields (APF), roadmaps and cell decomposition. researchers have developed a technique termed probabilistic roadmaps during the last decade (Choset, 2005). It should be mentioned that geometric methods may be limited for robots having a large number of degrees of freedom or cinematic restriction sin highly complex environments due to long computational times and convergence issues.

Trajectory generation algorithms require the definition of two spaces: W denotes the work space and Q denotes the configuration space. W is the environment or geometric Cartesian space where a robot interacts with the “real” world; Q is defined as the set of all possible robot configurations defined by a cinematic relationship (Spong, 2006) and can be seen as an n-dimensional space, where n is the number of robot joints.

It should be stressed that configuration space Q allows a robot to be represented as a point in an n-dimensional space; t is thus possible to generate a continuous trajectory from an initial condition (i.e. starting configuration) to a final state (i.e. goal configuration)

Consider the case of a 2 degree-of-freedom (DOF) planar robot, where Q ⊂R² is defined as the space of all possible values which each joint (in this case q₁ and q₂) may take. In the particular case of a 2 DOF robot, Q is a two-dimensional space (Figure 1); for robots with 6 DOF,Q ⊂R² and the search have to be done in aspace having at least 6thorderdimensionality.

A robot is said to have n-DOF if its configuration has minimal representation with n parameters (i.e. a robot’s DOF are equal to the number of joints or Q dimensionality) (Spong, 2006).

Artificial potential fields

The philosophy behind APF was first described by (Khatib, 1985) and used as an “on-line” strategy for collision avoidance in mobile robots having proximity sensors. The main idea behind APF is to conceive the robot as an electrically-charged particle influenced by a charged field generated by the goal position and the obstacles (i.e. the particle is attracted by the goal configuration and repelled by obstacles through potential fields U_att and U_rep respectively). The resulting potential field is the sum of these two components and can be generated by

[1]

Particle motion can be stated as an optimisation problem having a global minimum at q_goal and a set of initial conditions q_start. The most common approach is to use gradient descent methods to solve this problem.

Attractive potential field

Several criteria may be used to select an appropriate attractive potential field. Nonetheless, the most commonly used potential is a mixture of conic and parabolic convex hulls. It should be noticed that potential grows according to the distance between the particle (the robot) and q_goal. An expression is given by

[2]

The Euclidian (norm) distance between actual configuration q and goal configuration q_goal is given by d(q, q_goal) = llq_goal - qll the user-defined parameter ζ is used to scale the effect of the attractive potential. The resulting attractive field can be determined in one of two ways: initially, the field varies linearly with distance when d(q, q_goal) is above a certain threshold but once d(q, q_goal) ≤ d*_goal, the attractive field becomes quadratic, ensuring that the field is continuously differentiable around q_goal.

Repulsive potential field

The repulsive field' s main purpose is tokeepa particle (i.e. the robot as a point in Q) away from any obstacle. The potential field’s magnitude is inversely proportional to the distance between the obstacle and the particle; the field is thus defined in terms of the shortest distance d_i(q) to an obstacle given by QO_i. In order to avoid any collisions with the i-^th obstacle, the potential field must asymptotically repel the particle in the vicinity of the obstacle. Bearing this in mind, the potential can be defined as

[3]

Q*_i∈R is a threshold that allows a robot to ignore any faraway obstacles. Scalar η is a weighted repulsive field. These constants are generally chosen by trial and error. Figure 2 depicts a circular obstacle’s repulsive field. Figure 3 shows d_i(q,) measured from co, which is the nearest point to obstacle QO_i and whose gradient is given by

[4]

A drawback to such an approach is that the calculation of the shortest distance in W is not always trivial; an alternative approach (which has been explored in several texts) is thus to define a set of points belonging to the robot (i.e. control points). A potential field U_j is then generated for each point (r_j) within the control set. After determining the effect of the field on each point, total field effect in W can be mapped as forces acting on a robot in the Q space. The combined effect of the field acting on the robot is calculated as the sum of all the forces in the q space, where the robot is represented as a particle. It should be noticed that summing the forces in the W space and then transforming this vector to the Q space is not equal to summing the transformed forces in the Q space. The latter is the correct way.

Combined field effect

Considering a robot to be one or more rigid bodies (in the case of robotic manipulators), several control points r_j can be determined, as expressed above. The effect of having more than one obstacle must also be considered. As a result, the total artificial potential field is given by

[5]

The term J_j(q) is the Jacobian matrix for each control point r_j. The Jacobian matrix is widely used in robotics and allows forces and speeds to be mapped in the work-space and transferred to the configuration space (Craig, 1989). The gradient of the attractive and repulsive potential fields of the j-_th component are calculated for each control point by means of the expressions give in (2) and (3). This results in the following expressions

[6]

[7]

It is reasonable to have a number of control points equal to or greater than the number of DOF. In the case of a rotational-joint manipulator, it is common to place this control point on the joints’ coordinate frames.

The subsequent configuration must be calculated once the effect of the potential field on a robot has been determined for a given step; this can be achieved by using a gradient descent algorithm which has been widely used for solving optimisation problems. The direction of the resulting forces produced by the field is calculated using “down-hill” velocities taking the particle to a minimum state.

The general idea of gradient descent methods is simple and intuitive; a start is made from an initial configuration and then a “small” step is taken in the gradient’s negative direction; this results in a new configuration. The gradient is calculated in this new configuration and a step is taken in this gradient’s negative direction. This process is repeated until the gradient reaches zero. Figure 4 gives a basic gradient descent algorithm which has been effectively implemented in APF path-planning applications.

The variable q(i) is the vector of joint values in the i-^th iteration; the solution to the path-planning problem consists of a sequence of configurations {q(0), q(1), q(2),…, q(i)}. The scalar value α(i) must be sufficiently small so that a robot does not “jump” over obstacles. As the gradient may never reach zero in practice, a small tolerance value ∈ must be introduced; the size of this scalar depends on the application at hand.

Examples

Figures 5 and 6 show how APFs are applied to path-planning. The rectangular mobile robot shown in Figure 5(a) is in a twodimensional space Q and a three-dimensional space W defined by the (x, y) position of centroid and orientation angle θ (i.e. the robot has a q(x, y, θ) configuration at any given instant). Control points r_j are defined in W for each of the rectangle’s vertices. Using the gradient descent method, the shortest distances between j control points and obstacles are calculated in each iteration and the attractive force of the goal configuration on each control point is used for calculating the total force acting on the particle.

Figure 5(b) shows various positions of the path generated by an APF algorithm using the gradient descent method. It can be observed that the repulsive and attractive fields must be calculated for each control point r_j; hence, if there are a large number ofobstacles, and the number of control points is also large, computational cost increases and so does the calculation time. APF path planning methods are thus well-suited for on-line applications in mobile robotics where the minimum distance to the obstacles is fedback by a distance sensor array. One of the advantages of APF techniques is that they can be easily implemented in highly dimensional spaces. For example, an aircraft can be treated in a six dimensional space corresponding to its six DOF.

Two particular patterns were observed during simulation. It was observed that some vibrations were present when the robot tried to avoid an obstacle near a local minimum. This vibration resulted from large step size α and is common in gradient descent methods. Secondly, the robot did not reach goal configuration as the gradient descent method never reaches U(q(i))= 0. Postprocessing algorithms can be used to avoid such pattern as they smooth-out such vibrations and allow a robot to reach its goal position exactly.

The methodology applied for articulated robot is the same as mentioned previously where the main difference is control point location. Control points are usually located at the end effectors, the joints and some extra control points on the body of each joint. The latter is recommended to avoid any unwanted joint contact with any of the obstacles due to irregular geometries. Figure 6 shows the results of applying an APF path-planning algorithm to a planar three-joint robot.

Simulating redundant-joint robots has shown that even though local minima may exist for a particular control point, the attractive/ repulsive fields ' combined effect on the remaining points lets the system escape such “trapped” configuration. Escaping such trapped configurations requires a large number of iterations which can be smoothed-out later on by post-processing algorithms.

Probabilistic roadmaps

Since their first appearance, probabilistic roadmaps (PRM) have been shown to be well-suited to the problem of finding collisionfree trajectories in large dimension spaces and highly-cluttered environments (Kavraki, 1996).

The main idea behind PRM is simple. A set of node points is randomly distributed within collision-free space Q_free. Connections between these nodes (links) are then established by using local planners where only the links lying within the collision-free space are used. The set of all nodes and links is called a roadmap. The initial configuration is added through a link connecting q_start to the nearest node. Once a roadmap has been determined, a path can be established from q_start to q_goal by using algorithms such as A* or Dijkstra. The shortest path between start and goal configurations can thus be found (see Figure 7)). A link is a straight line joining two nodes and represents a series of successive robot configurations in work space W. A valid link must guarantee that no configuration composing it collides with any of the obstacles. This may be seen as a disadvantage.

The computational cost of PRM algorithms relies heavily on the number of times the collision detection algorithm has to be invoked. Many algorithms thus aim at reducing the number of links in the roadmap. This has given rise to algorithms exploring their environment through expansive trees, for example expansive-space trees (EST) and rapidly-exploring random trees (RRT). For a detailed description of EST algorithms, see the work of David Hsu (Hsu, 2000). Although RRT algorithms are widely used, no work has explained recent developments in detail.

Rapidly-exploring random trees

Rapidly-exploring random trees (RRT) was introduced by LaValle (LaValle, 1999; LaValle, 2001) and soon proved an efficient way of exploring configuration space and thus generating roadmaps joining q_start and q_goal. The basic RRT algorithm constructs a tree Tby using nodes and links that gradually growing in a random fashion; they stop once start and goal configurations become joined. The result of this procedure is shown in Figure 8. In the RRT algorithm, a random, collision-free configuration q_rand is created within a neighbourhood of the tree. Once the configuration has been created, the tree tries to grow towards this configuration by using an EXTEND function, as shown in Figure 9

The nature of RRT allows for rapid expansion in regions where no obstacles are present. This is the case as obstacle-free regions present a greater chance of feasible point q_rand being found. This can be observed in Figure 10 depicting the expansion of a tree.

The RRT algorithm starts exploring the neighbourhood of q_start as the root of T to find a collision-free trajectory. If this tree’s growth reaches goal configuration, or a maximum number of iterations is reached, the algorithm stops and a path is drawn from q_goal to q_start; this process is depicted in Figure 8. A continuous path can thus be established starting from the branches, extending through the tree trunk and finally reaching the root.

Modifications have been made to the basic RRT in recent years and particle results have been detailed in LaValle (LaValle, 2006). The algorithm termed RRT_ Connect, developed by Kuffner and LaValle (Kuffner, 2000), has been shown to be one of the most reliable variations. It is also worth mentioning that the RRT_Connect algorithm has been used in a vary of applications with encouraging results.

The main idea behind RRT_ Connect is to use two trees: one beginning from the start configuration, the other starting from goal configuration. Both trees grow towards each other simultaneously, allowing better convergence rates, as well as less computational cost due to the reduced number of collision checks.

Figure 11 shows the result of applying the RRT_ Connect algorithm to a search space in R². The environment contains three obstacles represented by irregular polygons. Notice the resemblance with the configuration space of the 2 DOF planar robot depicted in Figure 1). For paths generated using the RRT algorithm, and its variations, it is recommended to use postprocessing algorithms to reduce irregular topologies and sharp turns; the objective is to smooth the path without compromising calculation time due to overly complex post-processing algorithms. It has been noted that RRT algorithms are extremely fast in open spaces (i.e. an environment having few obstacles) (LaValle, 2001; Yershova, 2005) and it has been shown to be more efficient than other probabilistic methods (Brandt, 2006).

Conclusions and future work

This article has presented the fundamental theoretical concepts needed for implementing two of the most commonly-used collision-free path-planning strategies for robotic systems in an ndimensional search space. The first methodology, APF, allows a quick response to its environments and usually generates a smooth path requiring little post-processing. Its most noticeable weakness concerns regarding local minima and possible entrapment that may arise (Latombe, 1991) and the number of control points. If a reduced number of control points is used, the algorithm exhibits better convergence rates but the robot may collide with objects if we consider that it is a rigid body and not just a particle (Choset, 2005); on the other hand, if a large number of control points are used, then computational times may increase and make the algorithm extremely slow. It is worth mentioning that APF methods require trial and error tuning of several constant parameters which may influence the type of path generated, thus an ad-hoc method for their selection is necessary. Its use in the field of robotics has shown that the algorithm is highly dependent on the method used to calculate the system’s Jacobian matrix and the algorithm used to calculate the shortest distance between a control point and obstacles represented as polygons.

The PRM method involves basic RRT and the RRT_Connect variant. The main strength of such algorithms is that they are conceived so that they can easily be applied to multidimensional search spaces. It is not necessary to have full information about the work space (Choset, 2005). It has also been shown that convergence rate is very small if the number of obstacles is small and that the algorithm can escape trapped configurations (i.e. escape local minima). It must be mentioned that RRT algorithms produce extremely reliable solutions (LaValle, 2001) since the search space is explored methodically. Some drawbacks include constant collision-detection algorithm use where algorithm must be invoked several times per iteration before producing a collision-free path. In many practical cases, the generated path has highly irregular topology, making it difficult to follow. Consequently, post-processing algorithms must be used to smooth-out the trajectory and increased computational times must be expected. There is no guarantee that the generated path is optimal and a second post-processing stage must be implemented to generate acceptable curves using advanced geometrical applications (Angeles, 2007;Kwangjin, 2008a;Kwangjin, 2008b).

Extensive simulation-based comparison has shown that RRT and its variations are more reliable in large dimensionality spaces where a collision-free path is sought. A drawback is that even though a collision-free path is always found, the computational cost may be high. It is worth mentioning that AFP methods can be easily applied to mobile robots, including aerial vehicles, even in dynamic environments.

The following points must be born in mind when implementing any of the algorithms described in this paper. The shortest distance between the particle and different convex and non-convex geometries must be found for APF methods, something which is not always simple or intuitive in the field of computational geometry.

Collision-detection libraries must be used for RRT algorithms and the choice of library has a big impact on computational time. Once the path generated by RRT methods has been smoothened using post-processing algorithms, trajectories usually lie very close to the obstacle, a property that is not usually desirable.

Recent interest in PRM methods, more specifically RRT, has given rise to a variety of applications which include medical applications (Jijie, 2008), exploration in highly irregular terrain (Cortes, 2008b) and mobile robots' autonomous navigation systems (Kuwata, 2008).

References