In this article, we propose a new algorithm for solving the forward dynamics of multibody systems consisting of rigid bodies connected in arbitrary topologies by localized joints and/or soft links, possibly actuated or not. The simulation is based on the implicit time integration of the Lagrangian model of these systems, where the soft links are modeled by Cosserat rods parameterized by assumed strain modes. This choice imposes a predictor–corrector structure on the approach, and requires computing both the residual vector and the Jacobian of the residual vector of the dynamics constrained by the time integrator. These additional calculations are handled here with a new Newton–Euler recursive inverse dynamics algorithm and its linearized tangent version. The approach is illustrated with numerical examples from the Cosserat rod literature and from recent robotic applications.

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.

The decentralization of formations using onboard sensing is important for multirobot systems, improving the robustness and independence of fleet operations. Bearing measurements (obtainable from embedded cameras) are an attractive choice for use in decentralized formation control, however, this requires that the formation framework be bearing rigid. Rigidity may be checked numerically for a given formation framework, however, it remains difficult to determine the geometric conditions under which otherwise rigid formations become flexible. This article models the sensor and robot constraints in bearing formations of quadrotors as a kinematic mechanism with analogous properties to find geometric conditions for the degeneration of bearing rigidity (singularities) and the resulting uncontrollable motions. A classification of singularities based on graph substructures is developed, and it is shown that arbitrarily large formations may be designed for which all singularities lie within a known set of geometric conditions. An application on how to use the knowledge of all singularity cases in a formation for singularity-free control maintenance is provided.

Equilibrium stability assessment is a primary issue in continuum robots (CRs). The possible stable-to-unstable transitions that CRs may admit complicate the use of CRs in tasks where safety and human–robot interactions are mandatory. In this context, metrics measuring the distance from instability are essential but rarely developed. Existing metrics are frequently based on the evaluation of matrices involving mixed units, thus resulting in unit-dependent metrics. Moreover, the physical meaning of existing metric is hard to interpretate. This article proposes to use the magnitude of a force that brings instability to the CR equilibrium as a measure of the distance to instability. The major advantages of this metric are the intrinsic physical meaning, the practical interpretation of the results, and the well-defined unit of the measurements. The proposed metric (named directional critical load index) is based on the linearization of the eigenvalues of the reduced Hessian matrix of the total potential energy, which can be achieved regardless of the employed discretization technique. Three different case studies illustrate and demonstrate the main results of this article.

There is a growing interest on the study of continuum parallel robots (CPRs) due to their higher stiffness and better dynamics capacities than serial continuum robots (SCRs). Several works have focused on the computation of their geometrico- and kinemato-static models that can be sorted into two main categories. Models based on the continuous Cosserat equations are very accurate but assessing elastic stability with them is tricky, and discretized models allow easily checking the elastic stability, but they require a large number of elastic variables to be accurate. In this article, we extend an approach based on assumed strain modes developed for the dynamics of SCRs to the statics of CPRs. This method is able to predict the robot configuration with an excellent accuracy with a very limited number of elastic variables, contrary to other discretization methods. The method is also more than 100 times faster than finite differences for a better prediction accuracy. Finally, it is possible to assess the robot elastic stability by only checking the Hessian of the potential energy as for any discretization method, thus making the analysis of this property simpler than for the continuous Cosserat model. All results are validated through simulations on two case studies.

Sagging cable-driven parallel robots (CDPRs) are often modeled by using the Irvine's model. We will show that their configurations may be unstable, and moreover, that assessing the stability of the robot with the Irvine's model cannot be done by checking the spectrum of a stiffness matrix associated with the platform motions. In this article, we show that the static configurations of the sagging CDPRs are local extrema of the functional describing the robot potential energy. For assessing the stability, it is then necessary to check two conditions: The Legendre–Clebsch and the Jacobi conditions, both well known in optimal control theory. We will also 1) prove that there is a link between some singularities of the CDPRs and the limits of stability and 2) show that singularities of the platform wrench system are not singularities of the geometric model of the sagging CDPRs, contrary to what happens in rigid-link parallel robotics. The stability prediction results are validated in simulation by cross-validating them by using a lumped model, for which the stability can be assessed by analyzing the spectrum of a reduced Hessian matrix of the potential energy.

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.

The presence of Type 2 singularities in parallel robots severely affects their performances, mainly because the platform motion control is partially lost. It also leads to a size reduction of the operational workspace. Moreover, the dynamic model of the parallel mechanism degenerates and locally, the robot becomes underactuated in the singularity. It has been proven that it is possible to cross Type 2 singularities by respecting a dynamic criterion. Nevertheless, the controllers designed up to now require a pre-planned optimized trajectory including this criterion, and as a result, this strategy can only be used by qualified users. In order to avoid this drawback and to cross these types of singularities even if the trajectory is not pre-planned, this paper proposes a controller based on virtual constraints. Furthermore, the controller is integrated in a multi-control architecture in order to switch between a classical computed torque control far from the singularity and the virtual-constraint-based control law near to the singularity locus. Experimental results on a five-bar mechanism validated the automatic Type 2 singularity crossing.

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.

The determination of the singularity cases in visual servoing is a tricky problem, which is unsolved for most of the image-based approaches. In order to avoid singularities, redundant measurements may be used. However, they lead to the presence of local minima. Moreover, they do not always ensure that singularities can be avoided. Here, we show that a concept named the “hidden robot,” which was formerly used for understanding the singularities of a vision-based controller dedicated to parallel robots, can be used for interpreting the singularities in the visual servoing of image points. These singularity cases were already found in the case in which three points are observed, but we show that the hidden robot concept considerably simplifies the analysis by using geometric interpretations of the mapping degeneracy and tools provided by the mechanical engineering community. Moreover, to the best of our knowledge, for the first time, we provide the singularity conditions when more than three points are observed. We also discuss how these tools could be extended in order to find the singularity cases of other visual servoing techniques (e.g., when lines are observed).

Previous works on parallel robots have shown that their visual servoing using the observation of their leg directions was possible. There were, however, found two main results for which no answer was given. These results were: 1) the observed robot that is composed of n legs could be controlled in most cases using the observation of only m leg directions (m <; n), and 2) in some cases, the robot did not converge to the desired end-effector pose, even if the observed leg directions did (i.e., there was not a global diffeomorphism between the observation space and the robot space). Recently, it was shown that the visual servoing of the leg directions of the Gough-Stewart platform and the Adept Quattro was equivalent to controlling other virtual robots that have assembly modes and singular configurations different from those of the real ones. These hidden robot models are tangible visualizations of the mapping between the observation space and the real robots Cartesian space. Thanks to this concept, all the aforementioned points pertaining to the studied robots were answered. In this paper, the concept of the hidden robot model is generalized for any type of parallel robots controlled using visual servos based on the observation of elements other than the end-effector, such as the robot legs into motion. It is shown that the concept of the hidden robot model is a powerful tool that gives useful insights about the visual servoing of robots and that it helps define the necessary features to observe in order to ensure the controllability of the robot in its whole workspace. All theoretical concepts are validated through simulations with an Adams mockup linked to Simulink.

In order to increase the workspace size of parallel robots (largely reduced by the presence of singularities) several solutions have been proposed. One promising solution consists in the definition of optimal trajectories that ensure the non degeneracy of the dynamic model in the singularity and therefore are able to cross the Type 2 singularities. Those works are based on the computation of the optimal trajectories and assume that the robot can perfectly track the desired trajectory. Nevertheless, this assumption cannot be verified in reality due to modelling errors which largely impact the control law used to follow the desired trajectory. Therefore, if the optimal trajectory is not perfectly tracked, the dynamic model can degenerate near the Type 2 singularities and the robot might stay blocked. In order to solve that problem, this paper proposes a multimodel approach that allows parallel robots to cross the Type 2 singularities without any torque discontinuity. The main idea is to shift near singularities from the full robot dynamic model to another simplified one that can never degenerate. The proposed control law is then coupled with an optimal trajectory planning methodology that makes the singularity crossing more robust to modelling errors. The proposed approach is validated experimentally on a prototype of Five-bar planar parallel mechanism.

As the end-effector pose is an external property of a parallel robot, it is natural to use exteroceptive sensors to measure it in order to suppress inaccuracies coming from modelling errors. Cameras offer this possibility. So, it is possible to obtain higher accuracy than in the case of classic control schemes (based on geometrical model). In some cases, it is impossible to directly observe the end-effector, but the leg directions can instead be used. In this case, however, unusual results were recorded, namely: (i) the possibility of controlling the robot by observing a number of legs less than the total number of legs, and that (ii) in some cases, the robot does not converge to the desired end-effector pose, even if the observed leg directions did. These results can be explained through the use of the hidden robot concept, which is a tangible visualisation of the mapping between the observed leg direction space (internal property) and Cartesian space (external property). This hidden robot has different assembly modes and singular configurations from the real robot, and it is a powerful tool to simplify the analysis of the aforementioned mapping. In this paper, the concept of hidden robot model is generalised for any type of parallel robot controlled through visual servoing based on observation of the leg directions. Validation has been accomplished through experiments on a Quattro robot with 4 dof.

This paper presents a symbolic and recursive calculation of the dynamic model of flexible parallel robots. In order to reduce the computational time, it is necessary to minimize the number of operators in the symbolic expression of the model. Some algorithms have been proposed for the rigid case, for parallel robots with lumped springs or for serial robots with distributed flexibilities, but to the best of our knowledge, nothing has been developed for parallel robots with distributed flexibilities. This paper aims at filling this gap. In order to minimize the number of operations, the Newton- Euler principle is used and combined with the principle of virtual powers. The Jacobian matrices defining the kinematic constraints are computed using recursive calculations that decrease the number of operators. The proposed algorithm is used to compute the elastodynamic model of a planar parallel robot. The obtained results, compared with those obtained with commercial softwares, show the validity of the proposed algorithm.

This paper presents new insights about the sensor-based control of Gough-Stewart (GS) platforms. Previous works have shown that it was possible to control the GS platform by observing its legs directions instead of using the encoders values or the measurement of the platform pose. It was demonstrated that observing only three legs directions was enough for the control but no physical explanations were given. Moreover, sometimes, the GS platform was not converging to the desired pose and the reasons of these divergences were not disclosed. This paper aims at answering to this two opened problems. It is shown that observing three leg directions involves controlling the displacement of a hidden robot whose models differs from those of the usual GS platform. This robot has assembly modes and singular configurations different from those of the GS platform. This involves that the legs to observe should be chosen carefully in order to avoid inaccuracy problems. In this sense, the accuracy analysis of the new robot is performed to show the importance of the leg selection. All these results are validated on a GS platform simulator created using ADAMS/Controls and interfaced with Matlab/Simulink.

Off-line dynamic identification requires the use of a model linear in relation to the robot dynamic parameters and the use of linear least squares technique to calculate the parameters. Most of time, the used model is the Inverse Dynamic Identification Model (IDIM). However, the computation of its symbolic expressions is extremely tedious. In order to simplify the procedure, the use of the Power Identification Model (PIM), which is dramatically simpler to obtain and that contains exactly the same dynamic parameters as the IDIM, was previously proposed. However, even if the identification of the PIM parameters for a 2 degrees-of-freedom (DOF) planar serial robot was successful, its fails to work for 6 DOF industrial robots. This paper discloses the reasons of this failure and presents a methodology for the identification of the robot dynamic parameters using the PIM. The method is experimentally validated on an industrial 6 DOF Stäubli TX-40 robot.

Off-line robot dynamic identification methods are generally based on the use of the Inverse Dynamic Identification Model (IDIM), which calculates the joint forces/torques (estimated as the product of the known control signal - the input reference of the motor current loop - by the joint drive gains) that are linear in relation to the dynamic parameters, and on the use of linear least squares technique to calculate the parameters (IDIM-LS technique). However, as actuation redundant parallel robot are overconstrained, their IDIM has infinity of solutions for the force/torque prediction, depending of the value of the desired overconstraint that is a priori unknown in the identification process. As a result, the IDIM cannot be used for the identification procedure. On the contrary the Power Identification Model (PIM) of any types of robot manipulator has a unique formulation and contains the same dynamic parameters as the IDIM. This paper proposes to use the PIM of actuation redundant robots for identification purpose. The identification of the inertial parameters of a planar parallel robot with actuation redundancy, the DualV, is then carried out using its PIM. Experimental results show the validity of the method.

Previous works on the Gough-Stewart (GS) platform have shown that its visual servoing using the observation of its leg directions was possible by observing only three of its six legs but that the convergence to the desired pose was not guarantied. This can be explained by considering that the visual servoing of the leg direction of the GS platform was equivalent to controlling another robot, the 3-UPS that has assembly modes and singular configurations different from those of the GS platform. Considering this hidden robot model allowed the simplification of the singularity analysis of the mapping between the leg direction space and the Cartesian space. In this paper, the work on the definition of the hidden robot models involved in the visual servoing using the observation of the robot leg directions is extended to another robot, the Adept Quattro. It will be shown that the hidden robot model is completely different from the model involved in the control of the GS platform. Therefore, the results obtained for the GS platform are not valuable for this robot. The hidden robot has assembly modes and singular configurations different from those of the Quattro. An accuracy analysis is performed to show the importance of the leg selection. All these results are validated on a Quattro simulator created using ADAMS/Controls and interfaced with Matlab/Simulink.

Off-line robot dynamic identification methods are based on the use of the Inverse Dynamic Identification Model (IDIM), which calculates the joint forces/torques that are linear in relation to the dynamic parameters, and on the use of linear least squares technique to calculate the parameters (IDIM-LS technique). The joint forces/torques are calculated as the product of the known control signal (the current reference) by the joint drive gains. Then it is essential to get accurate values of joint drive gains to get accurate identification of inertial parameters. In the previous works, it was proposed to identify each gain separately. This does not allow taking into account the dynamic coupling between the robot axes. In this paper the global joint drive gains parameters of all joints are calculated simultaneously. The method is based on the total least squares solution of an over-determined linear system obtained with the inverse dynamic model calculated with available current reference and position sampled data while the robot is tracking one reference trajectory without load on the robot and one trajectory with a known payload fixed on the robot. The method is experimentally validated on an industrial Stäubli TX-40 robot.

Off-line robot dynamic identification methods are mostly based on the use of the Inverse Dynamic Identification Model (IDIM), which calculates the joint force/torque that is linear in relation to the dynamic parameters, and on the use of linear least squares technique to calculate the parameters (IDIM-LS technique). The joint forces/torques are calculated as the product of the known control signal (the current reference) by the joint drive gains. Then it is essential to get accurate values of joint drive gains to get accurate identification of inertial parameters. In this paper it is proposed a new method for the identification of the total joint drive gains in one step. A new inverse dynamic model calculates the current reference signal of each joint j that is linear in relation to the dynamic parameters of the robot, to the inertial parameters of a known mass fixed to the end-effector, and to the inverse of the joint j drive gain. This model is calculated with current reference and position sampled data while the robot is tracking one reference trajectory without load on the robot and one trajectory with the known mass fixed on the robot. Each joint j drive gain is calculated independently by the weighted LS solution of an over-determined linear systems obtained with the equations of the joint j. The method is experimentally validated on an industrial Sta¿ubli RX-90 robot.

In this paper, a new methodology for the optimal design of parallel kinematic machine tools is proposed. This approach is based on the concept of the maximal inscribed parallelepiped and uses technology-oriented constraints that are motivated by particular applications. This methodology is applied on two translational parallel robots with three degrees-of-freedom (DOF): the Y-STAR and the UraneSX. An analysis of the size of their workspace as a function of the design constraints is made. It is shown that, for identical workspaces with similar properties, the size of the legs of the UraneSX are greater than for the Y-STAR, thus leading to larger deformations. However, the footprint surface needed in order to install the Y-STAR is about two times bigger than for the UraneSX. Therefore, it may be interested to use the UraneSX in order to save some place on ground in manufacturing centres.

It is known that a parallel manipulator with a singular configuration can gain one or more degrees of freedom and become uncontrollable. That is it might not reproduce a stable motion under a prescribed trajectory. However, it is proved experimentally that there is possible passing through the singular zones. This was simulated and shown on numerical examples and illustrated on several parallel structures. In this paper, we determine the optimal dynamic conditions generating a stable motion inside the singular zones. The obtained results show that the general condition for passing through a singularity can be defined as follows: the end-effector of the parallel manipulator can pass through the singular positions without perturbation of motion if the wrench applied on the end-effector by the legs, and external efforts of the manipulator are orthogonal to the twist along the direction of the uncontrollable motion. This condition is obtained from the inverse dynamics and analytically demonstrated by the study of the Lagrangian of a general parallel manipulator. Numerical simulations are carried out using the software ADAMS and validated by experimental tests.

This paper presents the singularity analysis of four 3-DOF symmetric zero-torsion parallel mechanisms. These mechanisms are composed of three identical legs ending with a spherical joint that is constrained to move in one of three equally spaced planes intersecting at one line. The computation of the singularity loci is based on the degeneracy of the system of screws applied on the platform by the legs. The whole study is based on the use of a special orientation representation, previously introduced under the name of Tilt-and-Torsion angles. This representation is briefly introduced. Then the interdependence between the Cartesian coordinates of the general class of parallel mechanisms is derived. Finally, the singularity loci are derived and the size of the workspace taking into account all singular configurations is shown.