Section: New Results

Robotics

Analysis of Cable-driven parallel robots

Participants : Alain Coulbois, Artem Melnyk, Jean-Pierre Merlet [correspondant] , Yves Papegay.

We have continued the analysis of suspended CDPRs for control and design purposes. This analysis is heavily dependent on the behavior of the cable. Three main models can be used: ideal (no deformation of the cable due to the tension, the cable shape is a straight line between the attachments points), elastic (cable length changes according to the tension to which it is submitted, straight line cable shape) and sagging (cable shape is not a line as the cable is submitted to its own mass). The different models leads to very different analysis with a complexity increasing from ideal to sagging. All cables exhibit sagging but the sagging effect is often neglected if the CDPR is relatively small while it definitively cannot be neglected for large CDPRs. The most used sagging model is the Irvine model  [19]. This is a non algebraic planar model with the upper attachment point of the cable is supposed to be grounded: it provides the coordinates of the lowest attachment point $B$ of the cable if the cable length $L_0$ at rest and the force applied at this point are known. It takes into account both the elasticity and deformation of the cable due to its own mass. A drawback of this model is that we will be more interested in a closed-form of the $L_0$ for a given pose of $B$ (for the inverse kinematics of CDPR) and in alternate form of the model that will provide constraint on the force components (for the direct kinematics). We have proposed new original formulations of the Irvine model in [15] (best paper award of the Eucomes conference) and have shown that their use drastically improve the solving time for both the inverse and direct kinematics (i.e finding all possible solutions for both problems) that are required for CDPRs control. Still the solving time of the direct kinematics is too large for the real-time direct kinematics and in that case only the current pose of the platform is of interest. For that purpose it is of interest to add sensors on the robot beside the measurement of cable lengths in order to improve the solving time by using additional constraints and possibly ending up with a single solution. But these measurements are uncertain although we may assume that the measurement errors are bounded. It is necessary to determine these error bounds for a practical use of these measurement and we have conducted an experimental investigation of various additional measurements [12]: a mechanical system for measuring the angle of the cable plane with respect to a reference axis, cable angulation with accelerometers glued on the cable, a “poor man lidar” on the platform for optically determining several cables angulation, accelerometers on the platform and cable tensions with strain gauges while the pose of the platform was estimated accurately by using a metrology arm and laser range-meters. This investigation has shown that:

• the friction in the mechanical system leads to large errors for the cable plane angle (up to 30 degrees). For later measure we have bypassed this system

• even for small and medium-sized CDPRs the sagging effect cannot be neglected for estimating cable angulation

• accelerometers on the cable and the lidar system have a good accuracy (between 1 and 5 degrees)

• cable tension measurement is very approximate even with high accuracy strain gauges and cannot be used for control purposes.

We have also continued to investigate calculation of planar cross-sections of the workspace for CDPR with sagging cables, i.e. when 4 of the 6 platform pose parameters are fixed leaving only 2 free parameters. Brand new algorithms have been developed, based on a continuation approach [12],[13]. The main idea is that almost everywhere the workspace border is a one-dimensional variety so that if one of the free parameters is fixed, then a pose on the border should satisfy a square equation system constituted of the kinematic equations and the constraints equations (e.g. that a cable length is equal to a given maximum limit). Pose on the border are obtained by choosing an arbitrary pose that has an inverse kinematic solution that satisfy the constraints in the workspace and then moves incrementally along one of the free axis using a certified Newton scheme for finding the inverse kinematics solution until the constraint equations are almost satisfied in which case the certified Newton scheme is used to determine exactly (i.e with an arbitrary accuracy) a pose that lies on the border. Then a continuation scheme is used to find new poses on the border until we reach a pose at which a new set of constraints is satisfied i.e. a starting point for a new border arc. The border is then composed of several polygonal arcs that approximate the real border. The scheme is devised so that we completely master the difference between the real workspace area and the region defined by the polygonal approximation of the border. If necessary we may reduce this difference by adding new vertices on the border polygon. An important point is that the constraints define border arcs but also singularity curves (i.e. pose at which the direct kinematics equations are singular) and a specific continuation scheme has been developed to determine those arcs. Indeed the cancellation of the determinant of the jacobian of the direct kinematic equations is part of the equations that are satisfied on this type of border arc but this determinant cannot be obtained in closed-form. Consequently we have devised a certified Newton scheme that just require to evaluate the determinant and its derivatives at a given pose. A consequence of the existences of such arcs is that the workspace may have several aspects i.e. workspace region that can be reached only for a given inverse kinematics solution and is unreachable for the other one(s).

Cable-Driven Parallel Robots for large scale additive manufacturing

Participants : Jean-Pierre Merlet, Yves Papegay [correspondant] .

Easy to deploy and to reconfigure, dynamically efficient in large workspaces even with payloads, cable-driven parallel robots are very attractive for solving displacement and positioning problems in architectural building at large scale seems to be a good alternative to crane and industrial manipulators in the area of additive manufacturing. We have co-founded in 2015 years ago the XtreeE (www.xtreee.eu) start-up company that is currently one of the leading international actors in large-scale 3D concrete printing.

We have been contacted this year by artists interested in mimicking the 3D additive manufacturing process on a large scale with glass micro-beads for a live art performance to be held in 2019 (www.lestanneries.fr/exposition/monuments-larmes-prince). We have been working on the design of the robotics system, namely a cables parallel robots with autonomous refilling capabilities.

Robotized ultrasound probe

Participant : Jean-Pierre Merlet.

In collaboration with the EPIONE project we have started investigation the development of a portable robotized cardiac ultrasound probe that may be used while performing an effort test. A first step, somewhat surprising was the necessity to instrument an existing probe in order to determine what are the forces that the doctor exert on the probe during an investigation and the maximal angulation of the probe (apparently this data has not been measured beforehand). We add an accelerometer (for measuring the angle) and a force sensor in a 3D-printed covering of the probe and recorded the data during several experiments. We were then planing to develop a small, portable 3 d.o.f. rotational parallel robot whose range of motion was within the maximum angles that has been determined experimentally and was able to sustain the force exerted by the doctor. Unfortunately there was not a general consensus between the doctors and the company manufacturing the probe on the number of d.o.f. that was requested for the robot (which clearly have a drastic influence on the mechanical design and on the dimensional synthesis of the robot) so that the project is on stand-by.

Parallel robot performances and uncertainties

Participants : Jean-Pierre Merlet, Hiparco Lins Vieira [correspondant] .

The purpose of this study, which is the PhD subject of H. Lins Vieira, is to develop interval analysis-based algorithm for determining if some performance requirements for parallel robots (e.g. on workspace, accuracy, load lifting ability) can be guaranteed in spite of the unavoidable manufacturing and control uncertainties of the system.