To match the ever increasing standards of fresh products, and the need to reduce waste, we devise an alternative to the destructive and highly variable fruit ripeness estimation by a penetrometer. We propose a fully automatic method to assess the ripeness of mango which is non-destructive, allows the user to test multiple surface areas with a single touch and is capable of dissociating between ripe and non-ripe fruits. A custom-made gripper equipped with a capacitive tactile sensor array is used to palpate the fruit. The ripeness is estimated as mango stiffness extracted through a simplified spring model. We test the framework on a set of 25 mangoes of the Keitt variety, and compare the results to penetrometer measurements. We show it is possible to correctly classify 88% of the mango without removing the skin of the fruit. The method can be a valuable substitute for non-destructive fruit ripeness testing. To the authors knowledge, this is the first robotics ripeness estimation system based on capacitive tactile sensing technology.

Children with a diagnosis of Autism Spectrum Disorder (ASD) have serious difficulties in the development of their communicative and social skills. In recent years, robots have been tested in the therapy of autistic children as a promising tool for increasing their interest and motivation in the activities. In this paper we present the results of a pilot study with playful robot-child interaction developed for the therapy of diagnosed children aged between 3 and 5. The children were separated into an intervention and a control group. Their progress in development was measured before and after the intervention. Although the experience was unanimously considered as positive by parents and caregivers, we have found no significative differences between the intervention and control groups. Some observed trends demand more caution and additional studies for identifying not only the advantages but also the possible pitfalls of the use of robots in the therapy of autistic children.

The information used to determine the internal model of a robot system emerges from individual interactions with the environment. Knowledge about a specific internal model can be acquired by means of model learning techniques based on supervised machine learning tools, such as neural networks. One of the main challenges is to specify the starting point for the learning process. We propose a methodology for initial weight estimation based on the robot morphology. We use the generation of saccades in a robotic head as a case study to evaluate the performance of this approach. Our results suggest that this methodology, based on learning the relationship between the morphological parameters of the robot and those in its internal model, improves the quality of weight initialization, resulting in a considerable speed-up in the process of learning the internal model.

Saccades are fast eye movements that allow humans and robots to bring the visual target in the center of the visual field. Saccades are open loop with respect to the vision system, thus their execution require a precise knowledge of the internal model of the oculomotor system. In this work, we modeled the saccade control, taking inspiration from the recurrent loops between the cerebellum and the brainstem. In this model, the brainstem acts as a fixed-inverse model of the oculomotor system, while the cerebellum acts as an adaptive element that learns the internal model of the oculomotor system. The adaptive filter is implemented using a state-of-the-art neural network, called I-SSGPR. The proposed approach, namely recurrent architecture, was validated through experiments performed both in simulation and on an antropomorphic robotic head. Moreover, we compared the recurrent architecture with another model of the cerebellum, the feedback error learning. Achieved results show that the recurrent architecture outperforms the feedback error learning in terms of accuracy and insensitivity to the choice of the feedback controller.

The aim of this article is to investigate the effectiveness of the body language of a humanoid robot to induce emotions. It is based on the principles of positive psychology (PP), social learning, therapeutic robotics and mood induction procedures (MIPs). According to the Velten Method to induce a positive mood, the body language of a humanoid robot is used here as a modulated variable to test its efficacy inducing emotions. We have three hypotheses: (H1) Positive body language reinforces the positive attitude of the Velten positive statements; (H2) Body language which expresses the opposite attitude that the one expressed by Velten statements, that is a negative attitude, can vary negatively the mood induction results; (H3) The more positive the body language is, the higher the positive induction effect is; We have run experiments with 48 volunteers to test these hypotheses. Results show that the hypotheses (H1) and (H2) are correct, but it is not the case for (H3), which is not confirmed with an exaggerated expression of elated mood. Furthermore, our new combined MIP has a significant effect size to induce positive emotions.

Autonomous robots need to make sense of their surroundings, recognize objects, detect and, possibly, identify the people around them. Visual perception would be ideally suited for these tasks. Yet, visual perception remains one of the limiting factors of modern robotics: artificial vision systems tend to perform poorly in normal environments, where the scene and the illumination conditions are unpredictable. Evolution has faced similar problems, leading to surprisingly accurate visual capabilities, even in species with very small brains. We argue that the success of biological perception systems relies on three fundamental computational principles: (a) the continual coupling of perception and behavior; (b) the resulting emergence of multimodal cues; (c) and their efficient integration. Building on these computational principles, we describe a humanoid robot that emulates the dynamic strategy by which humans examine a visual scene. The proprioceptive and visual depth cues resulting from this strategy are integrated in statistically optimal manner into a unified representation. We show that this approach yields accurate and robust 3D representations of the observed scene.

Active eye movements can be exploited to learn the oculomotor command that allows a humanoid head to gaze at a visual target. In the field of adaptive robotics this transformation is commonly encoded by means of neural networks that are trained using ballistic eye movements. In this work, we encode the visuo-oculomotor transformation using radial basis function networks and we derive the equations to use such a transformation in visual servo control tasks. We study how the choice of the input (monocular vs binocular) influences the execution of ballistic and visually-guided eye movements. Analytic analysis and experimental results on a simulated model of the robot show that the proposed framework can be employed to execute both kind of movements.

The ability of any robot to robustly detect and recognize people is a fundamental issue to allow safe Human-Robot Interactions. This ability allows the system to deal with situations that are not determined a priori, while safely performing its tasks. Some research has been carried out on this topic, although most of the proposed approaches restrict the systems autonomy, flexibility and/or performance time. In this paper, we present a visual application for human detection and recognition without limiting the systems capabilities.

Do individual differences in dispositional behavioral tendencies, such as immersive tendency and need to belong, play a significant role in human-robot interaction? To answer this question, the present study conducted a 2 x 2 between-subjects experiment to examine the effects of immersive tendency (high vs. low) and need to belong (high vs. low) on individuals' perceptions of a social robot. Preliminary data analyses revealed that participants with a higher level of immersive tendency and need to belong showed greater attachment and trust towards the robot, and were more satisfied with their relationship with the robot than participants with a lower level of immersive tendency and need to belong. In addition, participants with a higher level of immersive tendency experienced greater feelings of social presence. Implications of notable findings are discussed.

A 2 x 2 between-subjects experiment was conducted to examine the effects of the type of artificial agent (robot vs. computer) and the role of the agent (ally vs. enemy) on people's perceptions and evaluations of the agent when playing a video game. Participants perceived that playing the game with a robot was more enjoyable and easier than playing with a computer. Regardless of the agent type, participants reported that playing the game was more enjoyable when the agent played as an ally rather than as their opponent. Implications of notable findings are discussed.

The way to autonomous robotic systems leads to deal with acting and interacting with real-life scenarios. Focusing on human-robot interaction, one of the first issues to be solved concerns safety since systems are becoming larger and more sophisticated and the damage caused can be considerable. So, robotic systems must exhibit levels of adaptability high enough to achieve their goals by guaranteeing the safety of all their surrounding elements, especially when they are human beings. With that aim, we propose a visual application that provides the system with information about all-embracing activity and, consequently, with the ability of safe interacting with human beings.

The aim of this paper is to improve the skills of robotic systems in their interaction with nearby objects. The basic idea is to enhance visual estimation of objects in the world through the merging of different visual estimators of the same stimuli. A neuroscience-inspired model of stereoptic and perspective orientation estimators, merged according to different criteria, is implemented on a robotic setup and tested in different conditions. Experimental results suggest that the integration of multiple monocular and binocular cues can make robot sensory systems more reliable and versatile. The same results, compared with simulations and data from human studies, show that the model is able to reproduce some well-recognized neuropsychological effects.

This study explores the effect of the robot's appearance vs. behavior (voice and gestures) on the way it is perceived as a machine-like instead of a human-like robot. A between-subjects experiment with four conditions was conducted. Results suggest that both the robot's behavior and appearance are important but, if they are contradictory, the robot's behavior is more powerful than the robot's appearance in the perception of the robot as more machine-like or human-like.

We have proposed a new approach for solving a fundamental issue in HRI, that is, how to properly detect and identify people in everyday environments since some conditions might make it a difficult task. For that, fisheye cameras are used since they provide panoramic vision and one or two of them allow to cover the whole workspace. A modified background maintenance approach was developed for fast, robust motion detection; while person identification for interaction is dealt with Viola-Jones classifier, although, instead of searching in the whole image, its input is only composed of the detected moving elements. Moreover, in order to avoid restricting the system autonomy by requiring the person has to face the system any time, once a person is identified as a target for interaction, they are tracked by using another designed method. We have also carried out an implementation of the proposed approach and a comparative experiment to assess its feasibility.

This workshop will address the impact of robot morphology on HRI from the perspective of Interaction Science, which encompasses theory and design of human interaction with technology. Anthropomorphic designs, which are common, have to be balanced with the "uncanny valley effect," since different morphologies suggest different affordances to users, triggering a variety of cognitive heuristics and thereby shaping their interactions with robots. We expect progress towards more human-acceptable interactions with robots by understanding the cognitive, behavioral, organizational, and contextual factors of morphology in HRI, as well as new meta-theories and design guidelines. We emphasize a highly multi-disciplinary approach, by involving participants from social sciences, engineering, and design.

This paper presents an approach for integrating vision, tactile and force sensors in a robotic manipulation framework. Having an initial estimation of the object pose in the environment, a position-based visual servoing loop controls the hand for task execution, based on the input received from a model-based articular pose estimator following the Virtual Visual Servoing approach. The visual control is combined with another control signal obtained from tactile feedback, through a set of selection matrices that can be modified at runtime in order to select the best modality for a given cartesian degree of freedom. The result of the preliminary integration is modified by an impedance force controller, in charge of performing the task motion along the task direction, at the same time that forces are regulated on the rest of directions. The design of the controller allows to perform the task even if a sensor is not available or provides inaccurate data. Several experiments are performed, first by considering only force feedback, and then adding vision and, finally, tactile information. Errors in the estimation of the object initial position are manually introduced in the experiments, and results show how the vision-tactile-force combination is able to deal with them, performing much better than the vision-force and force-alone approaches.

This paper describes how the use of panoramic cameras can dramatically simplify safety issues for a robot arm moving in close proximity to human beings, since they can simultaneously observe a 360° field of view. We present in this context an approach to visual servoing in which both the manipulator as well as any other moving object are tracked. Reliability and robustness are enhanced by adaptative background modelling and global illumination change detection.

Integration of multiple visual cues provides natural systems with superior abilities in dealing with nearby objects. This research is aimed at verifying if robotic systems could also benefit from the merging of different visual cues of the same stimulus. A computational model of stereoscopic and perspective orientation estimators, merged according to different criteria, is implemented on a robotic setup and tested in different conditions. Experimental results suggest that the principle of cue integration can make robot sensory systems more reliable and robust. The same results compared with data from human studies show that the model is able to reproduce some well-known neuropsychological effects.

In robotics, the manipulation ofa prioriunknown objects involves several steps and problems that must be carefully considered and solved by proper planning and control algorithms. For example, once suitable contact points have been computed, the control system should be able to track them in the approach phase, i.e., while the relative position/orientation of the object and the gripper of the robotic system change due to the approaching movement of the robot toward the object. This correspondence paper proposes a practical method for the tracking of grasp points in image space that is based on transferring previously computed grasp points from an initial image to subsequent ones and on the analysis of the new grasp configuration. Three different options are proposed for this transference. Experimental results show the interesting practical performance of the general procedure.

In this paper we present a robot that is able to perform daily manipulation tasks in a home environment, such as opening doors and drawers. Taking as input a simplified object model and the task to perform, the robot automatically finds a grasp suitable for the task and performs it. For this, we identify a set of hand preshapes and classify them according to the grasp wrench space they generate. Given a task, the robot selects the most suitable hand preshape and automatically plans a set of actions in order to reach the object and to perform the task, taking continuously into account the task forces. The concept of hand preshape is extended for the inclusion of a task frame, which is a concept from task planning, thus filling the gap between the grasp and the task.

In this paper, we present a novel approach for sensor-guided robotic execution of everyday tasks, which is amenable to be integrated in current mobile manipulators and humanoid robots. We consider a robot which is observing simultaneously his hand and the object to manipulate, by using an external camera (i.e. robot head). Task-oriented grasping algorithms are used in order to plan a suitable grasp on the object according to the task to perform. A new vision/force coupling approach [1] is used in order to, first, guide the robot hand towards the grasp position and, second, perform the task taking into account external forces. Experimental results on a real robot are presented which validate our approach.

This work deals with the problem of dependable physical interaction through manipulation in partially-known everyday human environments. We present a modular software architecture that allows the definition and compliant execution of manipulation tasks under the Task Frame Formalism. Several software modules implemented within this architecture, enable higher levels of adaptability and robustness, as well as the incremental incorporation of more complex skills in a modular fashion. The whole system is validated making a real robot arm with a three-finger hand perform two different manipulation tasks: opening a doorhandle and taking a book out of a bookshelf.

This paper deals with the problem of dependable physical interaction through manipulation in partially-known everyday human environments. We present a modular software architecture that allows the definition and compliant execution of manipulation tasks under the task frame formalism. We show the details of several software modules implemented within this architecture, that enable higher levels of adaptability and robustness, as well as the incremental incorporation of more complex skills in a modular fashion. The whole system is validated making a real robot arm with a three-finger hand perform a complex manipulation task: taking a book out of a bookshelf. Results show how the presented framework is suitable for easily defining and performing a great variety of manipulation tasks

This paper describes design and implementation aspects of an autonomous robot that is intended to perform service tasks in a library. We first address problems concerning the arrangement of its components into a compact mobile manipulator suitable for our needs. Then, software issues are also discussed, related to the techniques that are necessary in order to navigate and to reliably manipulate books in a library. In this context, two methods are presented, concerning visual tracking and force/vision control. The first one is aimed at locating a book in a bookshelf by means of vision. For this, a tracking algorithm is implemented, which takes into account the spatio-temporal constraints of the problem. Book tags are located in the image and processed using open source optical character recognition (OCR) software, until the desired book is found. The second of the methods addresses the problem of extracting the book from the shelf. A hybrid force/vision control algorithm has been implemented, which uses vision for guiding the gripper to the correct book, and force for detecting contacts.

This paper presents and analyzes 12 quality measures that characterize robotic grips according to their stability and reliability. The measures are designed to assess three-finger grips of two-dimensional parts performed in a real environment, taking into account both theoretical aspects and unavoidable uncertainties of a grasping action. They build on the existing literature and on physical and mechanical considerations. The measures constitute a feature space that pattern recognition methods can use in order to classify robotic grips according to their quality. Six of the measures depend on the actual finger configuration of the gripper, and they have shown to be critical for better characterization. The kinematics of the Barrett Hand have been used. As a validation step, the measures are merged in two global quality values (with different practical applicability) that can be used to rank feasible candidate grips.

Learning techniques in robotic grasping applications have usually been concerned with the way a hand approaches to an object, or with improving the motor control of manipulation actions. We present an active learning approach devised to face the problem of visually-guided grasp selection. We want to choose the best hand configuration for grasping a particular object using only visual information. Experimental data from real grasping actions is used, and the experience gathering process is driven by an on-line estimation of the reliability assessment capabilities of the system. The goal is to improve the selection skills of the grasping system, minimizing at the same time the cost and duration of the learning process.

The positioning of a gripper with respect to an object is one of the basic steps involved in the manipulation of that object. In particular, in a grasping task the fingers of the gripper should be placed at a set of selected grasp points belonging to the object. In this paper, a two-finger gripper and a stereo pair of cameras mounted in an eye-in-hand configuration are considered for this task. A visual servoing control system is proposed for the guidance of the positioning movement in which the grasp points are tracked in image space and used as input features for the control law. This tracking is based on the use of an invariant description of the grasp with respect to the object. The control law is restricted to control the degrees of freedom that produce movements that respect the invariance required by the grasp tracking.

This paper presents and analyses ten criteria that assess the quality of a set of three-finger grips suitable for dextrous manipulation on real 2D parts. The set of candidate hand configurations is the result of a previous process of grasp generation from the object image. The proposed criteria include six that depend on the actual finger configuration of the gripper. The kinematics of the Barrett hand has been used. The criteria are merged to give a global quality value that can be used to select the best grip to execute. Experimental results include tests on stability and the effect of parameter variation.

This paper deals with visually guided grasping of unmodeled objects for robots which exhibit an adaptive behavior based on their previous experiences. Nine features are proposed to characterize three-finger grasps. They are computed from the object image and the kinematics of the hand. Real experiments on a humanoid robot with a Barrett hand are carried out to provide experimental data. This data is employed by a classification strategy, based on the k-nearest neighbour estimation rule, to predict the reliability of a grasp configuration in terms of five different performance classes. Prediction results suggest the methodology is adequate.

This paper presents work in progress towards a complete system working to assist users in a library. With this aim, the system must be capable to looking for a specific book in a shelf, asked by any user, and whether it is found, deliver it as soon as possible to the user. To get its objectives the system integrates automatic object recognition, visually guided grasping, and force feedback, among other advanced capabilities. Implementation details about the main modules developed presently are shown. Finally, after success in preliminary results obtained in our campus, we are encouraged to fallow working in this way to obtain the complete prototype.

A very interesting robotics area is the high-level tasks' specification, and the way a user-robot interface can facilitate programming complex actions to be executed on real environments. This paper shows a novel contribution in the user-robot interaction domain, and particularly in the web robotics field. The user, who is in the control loop, is able to specify the robot actions over a simulated scenario (running off-line), assisted with 3D virtual and augmented reality. Then, once the task is defined, it can be easily executed over the real scenario (running online). To get its main goal (i.e. task specification) working in real life scenarios, the system incorporates automatic object recognition and visually-guided grasping among others capabilities.

This paper presents an implemented vision-based strategy for computing three-finger stable grasps on unknown planar objects. Its only input is an image of a real unknown curved object instead of synthetic polygonal models. Grasp regions are identified for achieving tolerance around contact points so that errors in finger positioning as well as a finite finger size are considered. It can find expanding and squeezing grasps involving internal and external regions, including holes. Theoretical constraints for force-closure are met while assuring robustness and real-time performance under real-world conditions.

Research in grasping and manipulation has mainly focused on the theoretical framework of the grasp analysis and synthesis, however there are few works in the practical implementation of these approaches. We propose several methods and criteria for constraining visually-selected planar grasps to the particular kinematics of a three finger gripper, the Barrett Hand. The method relies on the features of an object provided by a vision system, and uses several criteria for classifying the possible grasps according to the configurations of the gripper that fit them. It has been implemented on the UMass Torso and the results are discussed.

In this paper, a strategy that combines the tracking of an object and the selected grasp on it is presented. This strategy is intended to provide enough information to a control system for executing the grasp. The tracking of the object uses the ICondensation and AARM prediction algorithms to predict the current object position and shape in the image space. Using a set of quality criteria, stable grasps are searched for on the object contour and the best one is selected. The selected grasp is tracked together with the object contour in the sequence of images.

A key issue in robotics is the development of the ability to grasp unknown objects. This ability requires a grasp determination mechanism that, based on the analysis of the description of the object, determines how it can be stably grasped. In this paper, a grasp determination method is presented that computes a set of grasps that comply with the force-closure condition. Its input is a set of contours, extracted from the vision data, describing the shape of the object. The internal holes of the object are taken into account, so the algorithm can find grasps on them. The algorithm also finds expansion and squeezing grasps, which are executed by opening and closing the gripper fingers.

The paper presents a general object representation for efficient collision detection. Spatial representation is a crucial factor when motion planning is applied to real environments with complex objects, since the efficiency of the collision detection algorithms depends on the spatial representation used for the agents and obstacles in the scene. Our spatial model consists of a hierarchy of representations that approximates the object at different levels of detail. The input to the system is any object that could be described using non-homogeneous generalized cylinders. An algorithm that automatically converts between polyhedral approximations and generalized cylinders extends the applicability of the approach. Compared with related approaches in the literature, our system can deal with concave and curved objects, it is spatially balanced, complete, stable and converges to a zero-error model. Experimental results show that better approximations are obtained (in terms of quality), and computation times are even two orders of magnitude less. The approach has been applied to solve collisions in a scene where human beings and robots interact. This is a fundamental requirement towards human-safe service robotics.

This paper first reviews the literature about the very important but neglected issue of safety in service-robots applications. Most of the few existing approaches are limited because they lack flexibility. We argue that human-tailored safety schemes are greatly needed, and propose two strategies in the line we think human-safe robots should be conceived. A fast method to derive the robot's incremental movements given a workspace force applied to the robot constitutes the basis of the robot behaviors that underlie the proposed strategies. Our simulations show that our strategies are safe, human-friendly and exhibit real-time performance. Finally, we discuss the many ways in which our work can be extended.

Mobile manipulators are mobile robot bases with at least one mounted robot arm which function in an integrated manner. A single arm mobile manipulator guided by speech commands is presented. The purpose of the mobile manipulator is to reach concrete locations in its environment and to pick up objects. The user can interact verbally with the system to help it localize and identify new objects, as well as incorporate them to its database of learned objects. The overall system is capable of recognizing speech commands, interpret them and execute the corresponding action. Human-robot interaction via speech has also been tested using different examples. Our approach supports the fact that the key to successful service robots is an adequate human-robot interface.

This paper presents a state identification approach which eliminates ambiguities caused by sensing and uncertainty. In manipulation tasks, the identification of contact states based on force and position sensing is affected by ambiguities. The approach uses recurrent neural networks to learn an internal representation of the finite state automata defined by the sensor patterns. The technique is demonstrated using a simulated learning task. State identification is combined with other techniques in a sensor-based learning architecture for robotic manipulation. Results are presented for simulated tasks. The system is able to manage ambiguous states which previously were impossible to learn.

Practical robot motion planning for multiple moving objects in complex scenarios requires very fast collision detection. The efficiency of the algorithms is critically dependent on the representation that is used to model the objects. We present a hierarchical spatial representation based on spheres that is very adequate for speeding up collision detection. The objects can be non-convex and curved and they are not partitioned into convex ones. We present results that show some of its main features: spatial balance, stability and convergence. Comparative results in terms of quality and time are provided.

Practical robot motion planning for multiple moving objects in large complex scenarios requires very fast collision detection. We present an efficient algorithm for collision detection between multiple moving objects intended to be used for robot motion planning. It is based on our spherical representation for non-convex and curved object that are not partitioned into convex ones. We present results that show the stability of the approach with respect to the number of polygons used to model the scene. Numerical evidence of the efficiency of the algorithm is also shown by several experiments, in which we obtain average intersection detection times of around one millisecond using 8 or 11 processors for two closely moving robots involving 1,900 polygons.

We present an integrated vision-guided grasping system for service robots. Our system integrates computer vision to capture the shape of the objects, online grasp determination based on that shape, and image-based control for grasp execution. Novel techniques are presented to solve the problems concerned under the imposed resource constraints, namely, for information reduction and segmentation in image processing, strategies for grasp determination as well as vision-guided control for grasp execution. A preshaping heuristic strategy is combined with symmetry and local shape analysis. Also, the surface of contact between fingers and object is considered and a novel technique for sampling the visuomotor Jacobian is introduced. Experimental validation results are provided showing how the robot arm can efficiently and stably grasp unknown everyday objects.

Robots are extremely difficult to program in uncertain environments. Sensors are required to get feedback and detect the actual world state. In order to adapt to new situations, robots must be able to learn from examples or from their own experience. Learning can be accelerated if the available a priori task knowledge is used. The proposed sensor-based architecture combines learning with programmed modules. The correspondence between qualitative states and actions is learnt. Programming is used to decrease the complexity of the learning task. Numerical and qualitative processing are integrated in a suitable scheme for a wide range of robot tasks. Examples for a real insertion task are provided, where force sensing and the relative position are sufficient to successfully learn the task. The solution is thus valid for any other target location.

Since collision detection is becoming a bottleneck for real-world applications involving increasingly complex geometries, more efficient ways of detecting collisions are called for. Hierarchies of detail based on spheres seem a powerful approach to overcome this problem. A representation for non-convex objects with curved surfaces is presented, which is independent of the number of features used for the polyhedral model of the object. A new algorithm for collision detection between many moving objects is described. Experiments with two different robots show the efficiency of the method.

A novel approach based on the use of force sensors for motion in contact with uncertainty in task planning is presented. A neural network monitors the force signals measured by a sensor mounted in the robot wrist. This network is able to learn without need of a teacher the different contact states of the system. The method is intended to work properly in complex real-world situations, for which a geometric analytical model may not be feasible, or too difficult. In this paper the authors study the two-dimensional peg-in-hole problem and a real example of a complex insertion task in a flexible manufacturing system.

By combining the simplicity of the sphere and the power of the motion of hierarchy of detail, a novel model is proposed with applications to collision avoidance and detection in 3D. The model is based on a double spherical representation for solid bodies. First, each element making up the robot and the obstacles is approximated by a set of exterior spheres which are automatically defined. Second, another set composed of interior spheres is generated. These representations define a hierarchy, since they can be redefined as many times as necessary; starting with two spheres per element, the approximation may be improved until it contains hundreds of spheres. Moreover, they converge to a zero-error representation. The proposed models leads to a simple treatment for the problem of collision detection, and it is further applied to collision-free path planning for manipulators in 3D.>