As mentioned last year we have started to work on the direct kinematics (DK) of cable-driven parallel robot (CDPR) having sagging cables (i.e. being elastic and having their own mass). The direct kinematics consists in determining the pose(s) of the platform for given cable length $L_0$. Sagging cable may be modeled by the Irvine textbook planar model where the upper attachment point $A$ of the cable is supposed to be grounded and be the origin of a planar frame with horizontal and vertical axis. The model provides the coordinates of the lowest attachment point $B$ of the cable as functions of the cable length $L_0$ at rest and of the horizontal and vertical components $F_x,F_z$ of the force applied at $B$:

where $E$ is the Young modulus of the cable material (which characterizes its elasticity), $\mu$ its linear density and $A_0$ the area of a cross-section of the cable that are all supposed to be known. Note that if $x_b,z_b$ and $L_0$ are known the 2 unknowns $F_x,F_z$ of these equations may be found using neural networks 17.

We have proposed last year a preliminary version of a DK solver using AI and we have finalized this year a very efficient DK solver 16.

The pose of a CDPR is defined by a vector $𝐗$ which has 6 components: three coordinates of the center of mass of the platform and 3 angles which describes the platform orientation. For a CDPR with $n$ cables we have $2n+6$ unknowns (the $2n$$F_x,F_z$ and the components of $𝐗$). As the $x_b,z_b$ of each cable can be expressed as functions of $𝐗$ we have $2n$ Irvine equations. In a static case we must ensure the mechanical equilibrium of the platform, which provide 6 equations in $𝐗$ and in the $F_x,F_z$. Hence solving the DK consists in solving a square system $ℱ$ of $2n+6$ equations, that has usually several solutions whose number cannot be determined in advance. For solving the DK we assume that we have been able to determine solutions of $ℱ$ (not necessarily all of them) for a few set of $n{L}_0$. Hence we have a set of $(L_0^j,S_1^j,S_2^j,...S_m^j)$ where $S_k^j$ are solutions of $ℱ$ for the set of lengths $L_0^j$. These data can be represented graphically: an horizontal line, called level, represents a set of lengths $L_0^j$ and points, called nodes, on this line represents the $S_k^j$. A node is there represented by a pair ($L_0^j,S_k^j$). Consider two nodes at different levels ($L_0^j,S_k^j$), ($L_0^u,S_l^u$). We define a linear path in the $L_0$ space as $L_0=L_0^j+\lambda (L_0^u-L_0^j)$ and we use a continuation scheme with $S_k^j$ as starting point and $\lambda$ as continuation parameter for determining a solution of $ℱ$ for $L_0=L_0^u$. Starting from $\lambda =0,S=S_k^j$ we incrementally increase $\lambda$ and use Newton with $S$ as guess to determine a solution $S$ of $ℱ$. This scheme stops if either $\lambda =1$ or if Newton encounters a singularity. Note that if $\lambda =1$, then $S$ may be a solution that was unknown for $L_0^u$.

If two nodes ($L_0^j,S_k^j$), ($L_0^u,S_l^u$) are connected with this scheme we establish an edge between them, allowing to establish a directed graph called the solution graph. In this graph we may identify the non redundant circuits i.e. a list of successively connected nodes that contains at most one node at a given level. In robotics terms these circuits define a path in the $L_0$ space and an associated singularity-free path for the platform that allow to connect the pose of the first node of the circuit to the pose of the last one and all poses of the nodes of the circuit belong to the same variety called an aspect. Poses belonging to the same aspect have the property that any two poses in the same aspect can be joined by following a singularity-free path in the $L_0$ space. The non redundant circuits will be used to obtain learning sets for the neural networks that we will create: when creating the edges with the continuation scheme we may store sets of samples ($L_0,S$) using storage rules such that

If not enough samples have been obtained for a given circuit we may add samples just by selecting a stored sample ($L_0^u,S^u$), choosing an arbitrary unit vector $𝐯$ in the $L_0$ space and applying a continuation scheme $L_0=L_0^u+\lambda 𝐯$ starting at $\lambda =0$, solution $S^u$ while storing samples along this path by using the same storage rules.

Regarding the neural networks we use multi-layer perceptron (MLP) with a specific training procedure. It is partly based on a decrease of the MSE loss but when a new MLP is obtained we run a verification procedure to obtain the success rate of the MLP. For calculating this rate the verification procedure runs the MLP on each sample of the training set and use the solution prediction of the MLP as guess for a Newton scheme. The success rate is then calculated as the percentage of samples such that the verification procedure leads to the expected solution of $ℱ$ (e.g if the success rate is 100, then the verification procedure obtains the expected solution for all samples in the training set). Clearly if during the training we find a MLP that has a 100% success rate, then it is not necessary to continue the training and furthermore given the size of our problem the training time is low (a few minutes). Consequently as we have no a-priori knowledge on the number of layers, number of neurons in the layers and activation function that may lead to a MLP that has a 100% success rate we may perform a systematic approach by testing all combinations of MLP that have from 2 to 10 layers, between 10 to 200 neurons with a step size of 20 and activation function among a set of 3 classical one (ReLU, LeakyReLU, CELU). We also use an adaptive learning rate usually starting at ${10}^{-3}$ and decreasing by 2 the learning rate if after 15 optimizer step the loss has not decreased and stopping the learning if the learning rate is lower than ${10}^{-6}$. For a given training set we stop the systematic MLP exploration as soon as a MLP with 100% success rate is obtained. If a 100% success rate is not reached we retain the MLP having the highest success rate and we consider in sequence the sample(s) whose solution has not been found and use them as starting point for creating a new learning set using random unit vectors in the $L_0$ space and a continuation process to obtain new samples. This set is used to train a new MLP that will allow to find the missed solution and possibly other missed solution(s). New MLPs are then added until all solutions of the training set are found. After having processed all the training sets we obtain a set of MLPs such that the the MLP prediction used as a guess for Newton (a procedure called local solver) will provide the solution for all the samples of the training sets.

For example for a CDPR having 8 cables (and consequently the DK has 22 equations) we have used 11 initial sets of $L_0$ with between 8 and 24 DK solutions to establish 1133 MLPs that cover all solutions of the training sets. However some of these MLPs are redundant (they provide solutions that will also be discovered by other MLPs) and it is interesting to reduce the number of necessary MLPs while still getting all solutions. For that purpose let $𝒰$ be the set of all samples over all training sets. We run in sequence all MLPs on $𝒰$ and determine the MLP that provides the largest number of solutions ${𝒱}_1$ over $𝒰$ and store it in a list. We repeat this procedure for the remaining MLPs over the set of samples in $𝒰-{𝒱}_1$ until this list of samples is empty. In our example the final number of MLPs is 138 and we are able to establish a DK solver by using the local solver for each MLP.

We may then assess the quality of this DK solver on verification sets that provide all DK solutions for a given set of $L_0$. These sets are established as the training sets by selecting random unit vectors in the $L_0$ space but at the opposite of the training set the continuation scheme follows all solutions from the starting point, which implies that the number of DK solutions for each sample is the same than the number of solution of the starting point. We also ensure that the $L_0$ of the samples in the verification set are all different from the one used in the training sets. The result are very good with about 99.9% of the samples for which all solutions are found. But two cases may occur:

After this update all solutions are found for all samples of the verification set. We may then repeat the procedure with a new verification set. However we cannot guarantee that the current DK solver will find all solutions in all case. For example we have found examples where we have assessed 10 verification sets with 100% success rate but for the 11-th one solution has been missed, imposing the creation of a new MLP.

The computation time for this DK solver may be decomposed into the time for establishing the necessary MLPs and the exploitation time for solving one instance of the DK. In a sequential process about 100 hours are necessary to establish the solver while solving one instance of the DK takes about 1 second (to be compared to the 20+ hours that are required when using interval analysis or continuation). However these times may be improved as there are numerous steps for establishing the MLPs that may benefit from a parallel implementation while the local solvers of DK solver may be run in parallel. This year we have investigated the use of the IA processor JETSON Nano but the result was somewhat disappointing because of the limited floating point power of the GPU. Anyhow this AI solver is intended to be used when numerous DK instances have to be solved and here as soon as at least 5 DK instances have to be solved the AI solver is faster than the alternate methods.

Finally as mentioned in section 7.2 the principle of the DK solver is generic and may be used for any parametric square equations system.

The kinematics equations governing the behavior of CDPR with sagging cables have been presented presented in section 8.1.1. They are used for control purpose as they establish the relationship between the actuated variables (the cable lengths $L_0$) and the pose of the CDPR platform under the assumptions that the $L_0$ can be measured accurately. Special winches with cable guide have been designed for that purpose but have the drawback of restricting the CDPR workspace and of not being completely reliable as the guide is working only under the assumption that there is a sufficient tension in the cable, a condition that cannot always be enforced. At the opposite simple multi-layered winch offers a compact solution while providing a very large workspace but with the drawback that the cable length deduced from the drum rotation is more approximate. We have proposed to have redundant sensing i.e. to complement the cable length measurements with additional sensors:

These two sensing methods have been implemented on one of our prototype and have shown to be very effective. Here we investigate if this sensing redundancy may also allow to assess the wear of the cables which will decrease the $E$ of the cables with the objective of improving the accuracy of the CDPR by using more accurate $E$ and of preventive maintenance e.g. changing cables before their breakdown.

When controlling the cable lengths to reach a given pose ${𝐗}_{𝐢}$ we assume that the $E$ of all cables are identical and equal to $E_i$. If this assumption is wrong the CDPR will reach a pose ${𝐗}_{𝐫}\ne {𝐗}_{𝐢}$ and the lidars will allow to calculate the distance $Q_d$ between ${𝐗}_{𝐫},{𝐗}_{𝐢}$ while the angular sensors will measure a change $Q_a$ compared to the inclination of the cable at ${𝐗}_{𝐢}$. These measurements have an error margin and therefore can be exploited if the changes are larger than given thresholds $\Delta Q_d,\Delta Q_a$.

First we have to perform a sensitivity analysis i.e to determine how much change in the $E$ is required to obtain $Q_d\ge \Delta Q_d$ and/or $Q_a\ge \Delta Q_a$. Indeed if measurable changes in the $Q$ are obtained only when the $E$ becomes very low our objectives cannot be reached. Clearly the changes in $Q_a,Q_d$ are dependent upon the pose and upon the way the $E$ of the cables are changing. We have therefore selected a set of representative cable lengths $L_0$ and solved the DK for these lengths which provides the different possible poses of the platform. We have then defined a variation law for the $E$ which uses a unit vector $𝐯$ in the $E$ space (with only negative of null components) such that the vector of the $E$ is equal to $𝐄={𝐄}_{𝐢}+\lambda 𝐯$. Consequently $Q_d$ and the $Q_a$ of the cables are functions of $\lambda$ and a continuation scheme may be used to follow the curves $Q_d\left(\lambda \right),Q_a\left(\lambda \right)$ for determining the first ${\lambda }_d$ such that $Q_d\left({\lambda }_d\right)\ge \Delta Q_d$ and ${\lambda }_a$ such $Q_a\left({\lambda }_a\right)\ge \Delta Q_a$. Clearly the ${\lambda }_a,{\lambda }_d$ are functions of the vector $𝐯$: we have decided to test the wear of a single cable (one component of $𝐯$ equal to -1, the others to 0) and random $𝐯$. We have also discarded cases where the ${\lambda }_d,{\lambda }_a$ correspond to less than 30% of the $E_i$. The analysis is not yet complete but we have found cases where we have measurable changes in $Q$ for moderate changes in the $E$ so that it appears that changes in the $E$ are observable.

After considering the sensitivity we will have to solve the inverse problem, i.e. determining the current value of all the $E$ of the cables. We will have to determine how many $Q$ are required to safely determine these values. Fortunately change in the $E$ is a slow process so that this estimation may be based on records of the $Q$ obtained during trajectories. Still the inverse problem implies solving a set of non-linear equations and our approach will be based on AI. Indeed we may obtain training set in simulation and we believe that, as for the DK, we will have to structure the data and to train several networks for obtaining reliable estimations.

A large number of industrial robots are used mostly only for simple pick-and-place operation, taking an object at a given position and moving it to another one. In most cases this task is performed by serial robot which are not energy efficient (beside the load the robot actuation has to move part of the robot own weight) while using a computer and various electronic boards that are largely under exploited which constitutes a waste of energy and resources. We intend to investigate the use for this task of cable-driven parallel robot, which are 25% more energy efficient, and to design an electro-mechanical mechanism for executing pick-and place trajectories while getting rid of the computer and of most electronic resources.

Human activity recognition (HAR) is a major topic in the team. We are focusing on monitoring mobility and displacements (we are not yet interested in recognizing the individual action of our subject) using a sensor-based approach, excluding vision which is intrusive and even prohibited in some places for legal reasons. For that purpose we have in the previous years developed a smart barrier combining redundant passive infrared motion detectors and infrared distance sensors. Smart barriers have been implemented in Ehpad Valrose, a new retirement house in which a specific infrastructure has been put in place to accommodate research works and in Institut Claude Pompidou, an Alzheimer day care hospital from 2019 to 2020. These two long term experiments have allowed us to determine that essential points in HAR are to determine what is possible to measure, the sensor types, how to retrieve and process sensor data, how effective are the quality of measurements on a long term basis and the level of monitoring that is acceptable for frail peoples and their helpers while providing significant and reliable data for the medical community in spite of the uncertainties both in the measurements and in the system modeling. These samples of questions will become central in our work.

We are interested in and currently investigating the use of another type of barriers, based on Lidar's. Measurements produced by lidars are richers and allow a better reliability on the number and the directions of simultaneous crossings.

Three years ago, through an international scientific cooperation with Israeli scientists located at the Ben-Gurion University of the Negev, and as explained in an activity report provided previously (2020), we developed a probabilistic model whose aim was to understand a strange - and up to now not understandable - reproductive behavior of the potter wasp insect Delta dimidiatipenne (Hymenoptera). Females of this insect lay their eggs in mud chambers provisioned with caterpillars they capture to feed their young. Experimental observations indicate that many of the caterpillars collected by this wasp are actually parasitized by a small gregarious parasitoid wasp and are providing a lower amount of food for the wasp progeny. As a result, all players in the interaction perish - the young potter wasp cannot fully exploit the caterpillars and presumably starve to death; and the small parasitoids complete their development, but cannot break out of the mud and remain trapped in the sealed pot. Following such observation, we developed a probabilistic model trying to understand under what environmental conditions such a striking phenomenon (i.e., bringing back to the nest parasitized caterpillars), can be maintained, despite the high cost to all players. After several problems to develop this modeling work (some of them were due to the current political situation in Israel), we were able to submit a manuscript to a good international journal (Behavioural Processes). By the end September, we received a decision (minor revision) and a revised version of the text was resubmitted by mid-November. We are now waiting on the editorial decision from the journal.

Several other activities were developed this (and the previous) years, all lead to publications in international journals: