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This paper presents a robot trajectory optimization formulation that builds upon numerical optimal control and Lie group methods. In particular, the inherent sparsity of direct collocation is carefully analyzed to dramatically reduce the number of floating-point operations to get first-order information of the problem. We describe how sparsity exploitation is employed with both numerical and analytical differentiation. Furthermore, the use of geometric algorithms based on Lie groups and their associated Lie algebras allow to analytically evaluate the state equations and their derivatives with efficient recursive algorithms. We demonstrate the scalability of the proposed formulation with three different articulated robots, such as a finger, a mobile manipulator and a humanoid composed of five, eight and more than twenty degrees of freedom, respectively. The performance of our implementation in C++ is also validated and compared against a state-of-the-art general purpose numerical optimal control solver.

This paper focuses on important implementation issues of numerical optimal control that are often overlooked. In particular, transcription methods should be carefully implemented for obtaining a discrete representation of the problem. For this purpose, we explain the algorithms to solve the minimum-effort problem by applying a direct collocation method based on B-Splines. In addition, we describe how to compute the gradient of the objective function as well as the Jacobian of the constraints without the use of finite differences and automatic differentiation. Geometric algorithms based on Lie groups and Lie algebra are examined to efficiently compute the analytical derivatives of the equations of motion of articulated robots. These ingredients allow the fast computation of dynamically feasible robot motions. We provide numerical comparisons with a biped robot to validate our recipe against classical direct collocation methods.

A visual memory (VM) is a topological map in which a set of key images organized in form of a graph represents an environment. In this paper, a navigation strategy for humanoid robots addressing the problems of localization, visual path planning and path following based on a VM is proposed. Assuming that the VM is given, the main contributions of the paper are: 1) A novel pure vision-based localization method. 2) The introduction of the estimated rotation between key images in the path planning stage to benefit paths with enough visual information and with less effort of robot rotation. 3) The integration of the complete navigation strategy and its experimental evaluation with a Nao robot in an unstructured environment. The humanoid robot is modeled as a holonomic system and the strategy might be used in different scenarios like corridors, uncluttered or cluttered environments.

In this paper, we propose an approach of following a visual path for humanoid navigation. The problem consists in computing appropriate robot velocities for the humanoid walking task from the visual data shared between the current robot view and a set of target images. Two types of visual controllers are evaluated: a position-based scheme and an image-based scheme. Both of them rely on the estimation of the homography model even for non-planar scenes. We assume that the sequence of target images is given and we focus on the controllers performance. Because classical visual path following controllers generate discontinuous robot velocities, we propose a generic controller (applicable for different types of visual feedback) to alleviate this issue, which is a main contribution of the paper. The stability of such controller is addressed theoretically and verified through experiments with a NAO humanoid robot.

This paper presents a strategy for estimating the geometry of an interest object from a monocular video sequence acquired by a walking humanoid robot. The problem is solved using a space carving algorithm, which relies on both the accurate extraction of the occluding boundaries of the object as well as the precise estimation of the camera pose for each video frame. For data acquisition, a monocular visual-based control has been developed that drives the trajectory of the robot around an object placed on a small table. Due to the stepping of the humanoid, the recorded sequence is contaminated with artefacts that affect the correct extraction of contours along the video frames. To overcome this issue, a method that assigns a fitness score for each frame is proposed, delivering a subset of camera poses and video frames that produce consistent 3D shape estimations of the objects used for experimental evaluation.

This paper proposes a general scheme based on terminal attractors to parametrize operational tasks with respect to time in continuous kinematic controllers. Commonly, a pseudo-inverse operator is applied to solve operational tasks. However, it is known that discontinuities in the control signals appear as a by-product of task transitions during motion execution. In particular, these transitions involve the insertion, removal and swapping of tasks. Recently, some methods have been reported that overcome such discontinuities in the control signals. In this work, we provide time parametrization capabilities to these continuous kinematic controllers in order to handle the time axis of operational tasks at will. Then, we show that the proposed scheme can be naturally extended for the case of prioritized kinematic tasks while preserving continuity under task transitions. Our scheme is validated in a real experiment with a NAO humanoid robot.

This paper proposes a novel approach for the simultaneous localization and reconstruction of a 3D object. This reconstruction is done with a small humanoid platform by the only means of monocular vision.We combine (1) a stochastic control approach to decide how to move the robot such that the object can be both localized with respect to the robot and correctly reconstructed, and (2) a space carving approach for the 3D reconstruction of the object that merges the acquired monocular views into a coherent model. We present a set of very promising results obtained with a NAO platform.

Classical prioritized inverse kinematics (PIK) allows to generate admissible trajectories for redundant robotic systems according to a prioritized finite set of kinematic tasks to simultaneously handle multiple posture constraints. In this paper, we propose a novel regulation scheme that permits to define terminal attractors (TA) at each priority level for satisfying an additional constraint which is the final time. TA were first introduced in the context of movement neuroscience for studying the targeting problem in reaching movements. Based on the underlying properties of TA, we extend PIK methods for achieving desired time convergence of each task while guaranteeing smooth transitions between active and inactive tasks during execution. The proposed scheme is illustrated in simulation with branched robotic manipulators and verified with a real redundant robotic arm.

In a previous work [1], the authors have shown how the Principal Components Analysis (PCA) of a set of human walking paths provides sufficient information to derive a linear human-like path generator based on examples. The present work aims to provide an analysis of human walking paths from the perspective of multilinear algebra, using the n-mode Singular Value Decomposition (SVD). This decomposition subsumes the conventional PCA and extends its capabilities to study the interaction between the different vector spaces attached to the data. The training set consists of sixty routes performed three times by five different subjects for a total of 900 examples. The outcome of this decomposition is a novel multilinear model capable of revealing the global shape variations between the different vector spaces, particularly, exhibiting the planning strategies that each subject seemed to use to approach to the different targets. In addition, we sketch how the multilinear model may be used to synthesize new routes for the hypothetical case of a subject trying to reach a moving target.

We propose a scheme to deal simultaneously with local motion planning and dynamic control of redundant cooperative robots subject to holonomic, posture and loop-closure constraints. In contrast to previous contributions, an iterative method, that glue together the problem of kinematic motion planning with dynamic control, generates the sequence of feasible collision-free motions by combining in a single strategy the task-priority redundancy formalism with a joint model-free smooth force-motion control scheme based on a well-posed projection, which allows motions of the system only in the tangent subspace of the contact constrained manifold. Overall the new algorithm is able to compute fast and accurately local solutions involving severely constrained robotic motions. We have successfully verified in simulations the effectiveness of the proposed iterative method in the context of cooperative manipulation tasks.

We propose a simple and efficient online strategy based on inertial sensors to follow human arm movements in the task space by robotic platforms. The strategy is composed by the following ingredients: 2 inertial sensors attached to the human arm and forearm, the differential kinematics (DK) as well as the dynamic model of the corresponding robot and a motion control scheme. We use inertial sensors to obtain the orientation measurements of the human bodies in real-time. These body parameters together with the DK module allow us to map human hand trajectories into the joint space of the robot, where a standard PID control is implemented. Overall our strategy is able to transfer motion skills from human to a given robot by means of an online inertial sensor-guided motion control. Moreover, our method can be used for robot teleoperation applications in a master-slave configuration, with a human interface based on inertial sensors. We have successfully validated in simulations the effectiveness of the proposed strategy in a 2 degrees of freedom (dof) planar robot.

This paper presents a motion planner for a vigilant humanoid robot. In this context of surveillance, the robot task is to keep a distinctive point in the environment in sight during all of its motion. The method we propose consists of three main ingredients: (1) A motion planner for an appropriate simplified model of the walking robot, adapted to the particular needs of humanoid robots, that outputs an admissible path with local optimality properties. This path is guaranteed to satisfy the visibility constraints resulting both from the landmark and from the angular limits of the mechanical system; (2) A generic walking pattern generator that produces stable walking motions; (3) A generalized inverse-kinematics module to satisfy the remaining collisions and posture constraints, in particular the gaze direction. The effectiveness of this method is shown with several examples on the humanoid robot plataform HRP-2.

In this work, we aim to exhibit the geometrical shape primitives of human walking trajectories using a statistical model constructed through Principal Component Analysis. This analysis provides sufficient information to derive a linear human-like path generator based on examples. The examples are provided by a motion capture database of human walking trajectories. The proposed model captures the shape of trajectories in terms of path length and deformation. We have successfully applied our model to compute a good approximation of the reachable space of human walking. This can be done with a negligible computational cost since it is based on a linear combination of basis human paths.

This paper addresses the problem of understanding the shape of the locomotor trajectories for a human being walking in an empty space to reach a goal defined both in position and in direction. Among all the possible trajectories reaching a given goal what are the fundamental reasons to chose one trajectory instead of another (see Fig. (1))? The underlying idea to attack this question has been to relate this problem to an optimal control problem: the trajectory is chosen according to some optimization principle. This is our basic starting assumption. The subject being viewed as a controlled system, the question becomes what criteria is optimized? Is it the time to perform the trajectory? the length of the path? the minimum jerk along the path?... In this study we show that the human locomotor trajectories are well approximated by the geodesics of a differential system minimizing the L2 norm of the control. Such geodesics are made of arcs of clothoids. The clothoid or Cornu spiral is a curve, whose curvature grows with the distance from the origin. The study is based on an experimental protocol involving 7 subjects. They had to walk within a motion capture room from a fixed starting point, and to cross over distant porches from which both position in the room and orientation were changing over trials.

This paper deals with mechanical part assembly planning. The goal is to automatically compute a collision-free path for both the part to be assembled and the mannequin manipulating it. Two approaches are proposed according to the difficulty of the problem. Both are based on a general probabilistic diffusion algorithm working in the configuration space of the considered system. The first approach consists in first planning a path for the part alone and then in checking the feasibility of the solution by adding the mannequin. The second one considers the part grasped and the mannequin as a single system. While the first approach performs quickly the second one is able to solve more constrained and difficult cases. Both solutions are based on the same path planning library allowing the user to easily evaluate the proposed solutions. Experimental results based on feedback experiences in automotive industry are presented

This paper presents a motion planner to automatically compute animations for virtual (human, humanoid or robot) mannequins cooperating to move bulky objects in cluttered environments. The main challenge is to deal with 3D collision avoidance while preserving the believability of the agents behaviors. To accomplish the coordinated task, a geometric and kinematic decoupling of the system is proposed. This decomposition enables us to plan a collision-free path for a reduced system, then to animate locomotion and grasping behaviors in parallel, and finally to clean up the animation from residual collisions. These three steps are automatically applied making use of different techniques such as probabilistic path planning, locomotion controllers, inverse kinematics and path planning for closed-kinematic mechanisms.

This work presents a differential system which accurately describes the geometry of human locomotor trajectories of humans walking on the ground level, in absence of obstacles. Our approach emphasizes the close relationship between the shape of the locomotor paths in goal-directed movements and the simplified kinematic model of a wheeled mobile robot. This kind of system has been extensively studied in robotics community. From a kinematic perspective, the characteristic of this wheeled robot is the nonholonomic constraint of the wheels on the floor, which forces the vehicle to move tangentially to its main axis. Humans do not walk sideways. This obvious observation indicates that some constraints (mechanical, anatomical...) act on human bodies restricting the way humans generate locomotor trajectories. To model this, we propose a differential system that respects nonholonomic constraints. We validate this model by comparing simulated trajectories with actual (recorded) trajectories produced during goal-oriented locomotion in humans. Subjects had to start from a pre-defined position and direction to cross over a distant porch (position and orientation of the porch were the two manipulated factors). Such comparative analysis is undertaken by making use of numerical methods to compute the control inputs from actual trajectories. Three body frames have been considered: head, pelvis and trunk. It appears that the trunk can be considered as a kind of a steering wheel that steers the human body with a delay of around 0.2 second. This model has been validated on a database of 1,560 trajectories recorded from seven subjects. It opens a promising route to better understand the human locomotion via differential geometry tools successfully experienced in mobile robotics

This paper presents a motion planner for a digital mannequin carrying objects in cluttered environments. The goal is to account for 3-dimensional obstacle avoidance, eye-convincing locomotion as well as manipulation constraints within an integrated motion planning approach. This approach combines probabilistic path planning methods, a motion capture based walking controller and inverse kinematics techniques.