For discrete-time nonlinear systems with uncertainty, this paper presents an interval analysis approach to design controller and compute the estimate of the closed-loop robust domain of attraction (RDOA). The dynamics of the system is modelled using difference inclusions. A robust negative-definite and invariant set (RNIS) in the state-control space is proposed. An RNIS is defined by the combination of a robust negative-definite set (RNS) and a robust controlled invariant set (RCIS), which leads to sufficient conditions for Lyapunov stability of the system. The estimate of RDOA can be obtained by projecting an RNIS along the state space. However, the RNIS is hard to obtain by its definition. Drawing inspiration from the RCIS-computation approach, we define a mapping that utilizes the predecessor operator in the state-control space to compute a set limit. Then, the RNIS can be obtained by finding the limit set for an RNS. The computations of RNS and the limit set are based on interval analysis. An algorithm to estimate the RNIS is introduced with rigorous convergence analysis. Finally, we formulate an optimization problem that is solvable, and enlarges the RNIS and the estimate of RDOA. The method is validated on examples of nonlinear systems subject to actuator saturation.

Local minima are a well-known drawback of image-based visual servoing systems. Up to now, there were no formal guarantees on their number, or even their existence, according to the considered configuration. In this work, a formal approach is presented for the exhaustive computation of all minima and unstable equilibria for a class of six well-known image-based visual servoing controllers. This approach relies on a new polynomial formulation of the equilibrium condition that avoids using the camera pose. By using modern computational algebraic geometry methods and an ad hoc symmetry breaking strategy, the formal resolution of this new equilibrium condition is rendered computationally feasible. The proposed methodology is applied to compute the equilibria of several classical visual servoing tasks, with planar and nonplanar configurations of four and five points. The effects of local minima and saddle points on the dynamics of the system are finally illustrated through intensive simulation results, as well as the effects of image noise and uncertainties on depths.

Research on continuum parallel robots has been essentially devoted to the computation of their geometricostatic models and of their performance in terms of workspace size, accuracy, compliance, force transmission, and manipulability. Their singularity analysis has been limited to the identification of a limited number of singular configurations, without any deep investigation of the physical phenomena occurring in these singularities. In this article, we define the singularity conditions for continuum parallel robots. We provide a straightforward interpretation of the phenomena occurring in singularities. Especially, we prove that some singularities appear when the robot potential energy has a local isovalue. Because of this property, we show that these singularities separate the stable configurations from the unstable ones in the workspace. Moreover, on such singularities, the robot can freely move along a given direction without any constraint under the action of small perturbations. We illustrate the singularity phenomena and their effects by simulations performed with two different continuum parallel robots.

This paper introduces an intrinsically safe parallel manipulator dedicated to fast pick-and-place operations, called R-Min. It has been designed to reduce the risk of injury during a collision with a human operator, while maintaining high speed and acceleration. The proposed architecture is based on a modification of the well-known planar five-bar mechanism, where additional passive joints are introduced to the distal links in order to create a planar seven-bar mechanism with two degrees of underactuation, so that it can passively reconfigure in case of collision. A supplementary passive leg, in which a tension spring is mounted, is added between the base and the end-effector in order to constrain the additional degrees of freedom. A prototype of this new collaborative parallel robot is designed and its equilibrium configurations under several types of loadings are analyzed. Its dynamics is also studied. We analyze the impact force occurring during a collision between our prototype and the head of an operator and compare these results with those that would have been obtained with a rigid five-bar mechanism. Simulation results of impact during a standard pick-and-place trajectory of duration 0.3 s show that a regular five-bar mechanism would injure a human, while our robot would avoid the trauma.

Finding the current pose of the end-effector of a parallel robot is a problem, since its forward geometric model generally has several solutions. Current methods to address this problem operate mainly under the assumption that the robot never changes its assembly mode nor gets close to Type 2 singularities. Nonetheless, recent works proved that a parallel robot can change its assembly mode, thanks to dedicated trajectory generation and control. Such a feature allows increasing the operational workspace of such manipulators. Hence, correctly tracking the end-effector pose while crossing Type 2 singularities is mandatory for a practical usage of this workspace enhancement method. However, on Type 2 singularities, several solutions of the forward geometric model merge, making current tracking methods ineffective. To fill this gap, we propose a two-step pose tracking methodology. First, a differential inclusion based on kinematics and dynamics is solved. Second, joint measurements are used to tighten resulting enclosures. The effectiveness of this method is discussed, thanks to experimental data gathered on a planar parallel robot.

Increasing the size of operationnal workspace is one of the main problems parallel robots are faced with. Among all the proposed solutions to that, crossing Type 2 singularities using dedicated trajectory generation and multi-model controller has a great potential. Yet, this approach is not sufficient for the robot to operate autonomously, as assembly mode detection during the motion currently requires additional redundant information. To tackle this problem, we propose an algorithm based on Interval Analysis (IA) that is able to track the end-effector of the robot even under assembly mode change. IA-based solvers for the forward kinematic problem of parallel robots are well known, but they cannot be used under assembly mode change. Compared to those classical approaches, the major modification introduced is the tracking of end-effector velocity in addition to its pose. Using this new information of velocity, the algorithm is capable to monitor the assembly mode change of the robot happening when the singularities are crossed. The behavior and the reliability of this algorithm are analyzed experimentally on a five-bar planar parallel mechanism.

Numerical methods based on interval arithmetic are efficient means to reliably solve nonlinear systems of equations. Algorithm bc3revise is an interval method that tightens variables' domains by enforcing a property called box consistency. It has been successfully used on difficult problems whose solving eluded traditional numerical methods.We present a new algorithm to enforce box consistency that is simpler than bc3revise, faster, and easily data parallelizable. A parallel implementation with Intel SSE2 SIMD instructions shows that an increase in performance of up to an order of magnitude and more is achievable.