Étienne Picard
email: etienne.picard@irt-jules-verne.fr

Stéphane Caro
email: stephane.caro@ls2n.fr

Fabien Claveau
email: fabien.claveau@imt-atlantique.fr

Franck Plestan
email: franck.plestan@ec-nantes.fr

Pulleys and Force Sensors Influence on Payload Estimation of Cable-Driven Parallel Robots

The subject of this paper is about the use of a suspended Cable-Driven Parallel Robot (CDPR) for pickand-place operations of heavy and heterogeneous objects. The knowledge of the payload mass and its center of mass in realtime is an asset for robust control of the device, which is required to ensure a good stability, especially when the objects have different shapes, sizes and masses. Accordingly, this paper aims at experimentally evaluating the effects of (i) the pulleys modeling and (ii) the use of force sensors for the payload estimation. It turns out that the consideration of the pulleys into the geometric model of the robot improves the mass and center of mass estimations of the payload. A comparison is made between the estimation of cable tensions from force sensors and from motor currents. Finally, a torque controller with a feedforward term for real-time mass compensation is proposed and implemented on a CDPR prototype.

I. INTRODUCTION

Cable-Driven Parallel Robots (CDPRs) form a particular class of parallel robots whose moving platform is connected to a fixed base frame by cables. The cables are coiled on motorized winches. Passive pulleys may guide the cables from the winches to the cable exit points. Accordingly, the motion of the platform is controlled by modifying the cable lengths. CDPRs have several advantages such as a relatively low mass of moving parts and a potential large workspace. As a consequence, they can be used in several applications such as heavy payload handling, painting and large structures sandblasting [START_REF] Gagliardini | Discrete reconfiguration planning for Cable-Driven Parallel Robots[END_REF], cargo handling [START_REF] Holland | Cable array robot for material handling[END_REF], warehouse applications [START_REF] Hassan | Analysis of a Large-Workspace Cable-Actuated Manipulator for Warehousing Applications[END_REF], large-scale assembly and handling operations [START_REF] Pott | Large-scale assembly of solar power plants with parallel cable robots[END_REF], and fast pick-and-place operations [START_REF] Kawamura | High-speed manipulation by using parallel wire-driven robots[END_REF]. Other possible applications include the broadcasting of sporting events, haptic devices [START_REF] Fortin-Coté | An admittance control scheme for haptic interfaces based on cable-driven parallel mechanisms[END_REF], support structures for giant telescopes [START_REF] Yao | Dimensional optimization design of the four-cable-driven parallel manipulator in fast[END_REF], and search and rescue deployable platforms [START_REF] Merlet | A portable, modular parallel wire crane for rescue operations[END_REF]. Some recent works have dealt with the design and reconfiguration planning of reconfigurable CDPRs that can be used in cluttered industrial environments [START_REF] Gagliardini | Discrete reconfiguration planning for Cable-Driven Parallel Robots[END_REF].

In pick and place applications, the knowledge of the payload mass and center of gravity is useful to control the platform position accurately and ensure its stability, especially for heavy loads. A relationship between the tension of the cables and the payload characteristics was proposed in [START_REF] Kraus | Load identification and compensation for a Cable-Driven parallel robot[END_REF] for a position based control method able to compensate for cable elasticity, and tested on up to 40 kg payloads. Elastogeometrical model of CDPRs [START_REF] Baklouti | Dynamic and Oscillatory Motions of Cable-Driven Parallel Robots Based on a Nonlinear Cable Tension Model[END_REF] can be used to calculate the cable length changes under load but their computation cost can be high for real-time implementation in a controller. However, in order to get a good estimation of the payload we had better use a detailed robot model of the robot and the most reliable measurements for cable tensions. Previous works such as [START_REF] Gagliardini | Discrete reconfigurations of cable-driven parallel robots[END_REF] and [START_REF] Pott | Influence of Pulley Kinematics on Cable-Driven Parallel Robots[END_REF] proposed rigid CDPR models including pulleys with two degrees of freedom (DoF) and analyzed the effect of the pulleys onto the workspace size.

This paper deals with the analysis of the effect of the pulley geometric parameters onto the payload estimation as well as the comparison between dynamometers (force sensors) and motor currents to assess the cable tensions. Moreover, a control scheme including a feedforward term for payload mass estimation and compensation in real-time is proposed and implemented on the IRT Jules Verne CAROCA CDPR prototype.

The paper is organized as follows. Section II describes the moving-platform, the available measurements and the test trajectory used in the experiments. Section III expresses the CDPR models with and without the pulleys geometry as well as the payload mass and center of mass estimation method. Section IV presents a torque control scheme suitable for a unknown payload. Section V deals with the influence of the pulleys on the mass and center of mass estimations and a comparison between the results obtained from tension measured by the dynamometers versus tension estimations from the motor torques. To this end, experimental measures along the test trajectory are analyzed. Finally, conclusions are drawn and future work is presented in Sec. VI.

II. PROTOTYPE AND EXPERIMENTAL SETUP

This section presents the CDPR prototype for the experimental validations. This prototype is named CAROCA [START_REF] Gagliardini | Discrete reconfiguration planning for Cable-Driven Parallel Robots[END_REF] and shown in Fig. 1.

A. Hardware

CAROCA is a reconfigurable CDPR developed at IRT Jules Verne for industrial applications experimentation.

This prototype has pulleys of radius 150 mm with two-DoF that can be displaced in a discrete manner on its frame. Thanks to its reconfigurability, the robot can be assembled This work is part of the ROCKET project, which aims at displacing metal parts of highly variable shapes and mass, up to 700 kg, thanks to a robust control scheme. The MP size is 1.5 m×1.5 m×1 m and its mass equals to 366 kg (Fig. 1). Five magnets are embedded under the moving platform to pick metal parts.

B. Measurements

The robot is equipped with Tractel TM dynamometers located between the cables and the anchor points of the platform to give a direct measurement of the efforts applied by the cables onto the platform, as shown in Fig. 1. These sensors output a current of magnitude between 4 and 20 mA proportional to the tension in the corresponding cable, up to 25000 N. They have been calibrated for the typical working range of the robot comprised between 0 and 5000 N while using a reference force sensor.

Hardware such as motors and control board are standard industrial components commercialized by B&R Automation TM . Angular positions and velocities are available for each motor and the delivered torques are assessed from the motor currents. The robot programming is done in Automation Studio 4.1 TM , with the code running in a 10 ms real-time cyclic. The control schemes of the robot are developed and simulations are performed with MATLAB R and Simulink R .

In the absence of external sensors and because of the complexity of the direct geometric model in CDPR, no information on the actual pose of the platform is available. However, in order to calculate position errors, it is possible to estimate the position and orientation of the MP by solving the direct geometric model through a Least Squares Method, as presented in [START_REF] Schenk | Application of a differentiator-based adaptive super-twisting controller for a redundant cable-driven parallel robot[END_REF], for example using the lsqnonlin function in MATLAB.

C. Test trajectory

The trajectory presented in Fig. 2 and Fig. 3 simulates a typical pick-and-place operation corresponding to the targeted application, and was used as a basis in the results presented in Sec. V. The platform takes 30 seconds to move from point A to point E, then 30 seconds from E back to A. The trajectory is generated using s-curves, which ensure continuous velocity and acceleration trajectory profiles [START_REF] Germain | Optimal design of the irsbot-2 based on an optimized test trajectory[END_REF]. This test is first performed on the CDPR with the empty MP. In order to evaluate the robustness of the task toward variable payloads, the platform is then loaded by an additional mass of either 122 kg or 249 kg, denoted as M 1 and M 2 , respectively. Note that the mass variation is significant, namely +33% and +68% with respect to the MP mass.

III. MODELING

The geometric, kinematic and dynamic models of CAROCA are written thereafter.

A. Model without pulleys

Fig. 4 depicts the main geometric parameters of a CDPR and its i th loop-closure equation, i ∈ {1, . . . , m}, m being the number of cables attached to the MP. F b is the robot base frame, and F p is the MP frame.

Cable exit points are denoted as A i , while cable anchor points are denoted as B i . Vector b a i points from point O to point A i and is expressed in frame F b . Vector p b i point from point P to point B i and is expressed in frame F p . Vector b p is the position vector of point P, the MP geometric center, expressed in F b .

In a simplified model, the pulleys are treated as points and the points A i are considered fixed in the frame F b . Their exact position is usually set to the intersection between the vertical rotation axis of the pulley and the horizontal plane containing the axis of the pulley sheave, as depicted in Fig. 5 Vector l i represents the i th cable vector and points from B i to A i :

b l i = l i b u i = b a i -b p -b R p p b i = b a i -b p -b b i (1)
with b R p the rotation matrix from frame F b to frame F p . l i is the length of the i th cable and u i is the unit vector of the i th cable vector, defined as

l i = l i 2 u i = l i l i 2 (2)
where . 2 denotes the Euclidean norm of a vector. The static equilibrium of the platform is given by

W I t + w e + w g = 0 (3)
W I being named as the wrench matrix of the robot [START_REF] Gagliardini | Determination of a dynamic feasible workspace for cable-driven parallel robots[END_REF] and expressed as

W I = u 1 . . . u i . . . u m b b 1 × u 1 . . . b b i × u i . . . b b m × u m (4)
t is the cable tension vector, w g is the wrench applied to the platform due to gravity and w e is an external wrench expressed in the fixed reference frame F b .

The kinematic Jacobian matrix J of the CDPR is obtained from W I as:

J = -W T I (5)

B. Model including pulleys

To improve the accuracy of the pulley geometry, we introduce A i as the moving exit point of the cable onto the pulley sheave. The parameterization for a two-DoF pulley is given in Fig. 5. The i th cable is tangent to its pulley at point A i . l ci is the distance from point A i to point B i . l pi is the length of the i th cable part embracing the shape of the i th pulley sheave. Point A i is the intersection point between the vertical rotation axis of the i th pulley and the horizontal plane containing the axis of its sheave.

Details about the calculations of the position of A i and angles θ i , α i , β i and γ i can be found in [START_REF] Gagliardini | Discrete reconfigurations of cable-driven parallel robots[END_REF] and [START_REF] Pott | Influence of Pulley Kinematics on Cable-Driven Parallel Robots[END_REF]. We now write:

b o i = b a i + r p b x p b x p = b R pulley p x p (6)
where b R pulley is the rotation matrix between F b and a new frame F pulley attached to the pulley and r p is the inner radius of the sheave groove.

By including the pulleys geometry, we obtain a new wrench matrix W II :

W II = e 1 . . . e i . . . e m b b 1 × e 1 . . . b b i × e i . . . b b m × e m ( 7
)
where e i is the new unit vector of the i th cable vector, defined as:

l ci = l ci 2 e i = l ci l ci 2 (8) with the cable vector b l ci from A i to B i b l ci = b a i -r p b z p + b e i × b y p -b b i (9)

C. Dynamic model

From [START_REF] Gagliardini | Determination of a dynamic feasible workspace for cable-driven parallel robots[END_REF], the dynamic model of a CDPR can be written as Wt -I p v -Cv + w e + w g = 0 [START_REF] Baklouti | Dynamic and Oscillatory Motions of Cable-Driven Parallel Robots Based on a Nonlinear Cable Tension Model[END_REF] with I p the spatial inertia of the platform and C the matrix of the centrifugal and Coriolis wrenches. In what remains, the wrench matrix is either W I or W II depending on the geometric model used in the control scheme.

Given that the center of mass of the platform, denoted as G, does not coincide with the origin of frame F p , the wrench w g due to the gravity acceleration g = [0, 0, -g] is defined as

w g = mI 3 M Ŝp g (11)
with m the mass of the platform, g = 9.81 m.s -2 , I 3 the 3 × 3 identity matrix, MS p = b R p [Mx p , My p , Mz p ] T the first momentum of the moving platform defined with respect to frame F b . The vector S p = [x p , y p , z p ] T defines the position of G in F p . M Ŝp is the skew-symmetric matrix associated to MS p . Note that the other terms of (10) are detailed in [START_REF] Gagliardini | Determination of a dynamic feasible workspace for cable-driven parallel robots[END_REF].

D. Payload mass (m) and center of mass (x g ,y g ) estimation

We describe here a method to obtain a CDPR payload mass and center of mass, knowing the cable tensions.

The following assumptions are made: 1) the orientation of the moving platform is constant 2) the linear velocity of the moving-platform is low 3) the only wrench applied to the platform is due to the carried payload, namely, the metal plate 4) the cables are straight and inelastic These hypothesis are acceptable in our pick-and-place application of heavy parts, where it is desirable to keep the platform leveled and the dynamics of the trajectories low. It should be noted that a similar approach is presented in [START_REF] Kraus | Load identification and compensation for a Cable-Driven parallel robot[END_REF], but with cables in polymer and lighter payloads.

Equation [START_REF] Baklouti | Dynamic and Oscillatory Motions of Cable-Driven Parallel Robots Based on a Nonlinear Cable Tension Model[END_REF] can then be rewritten as:

Wt + w g = 0 ( 12
)
From ( 11) and ( 12), w g can be expressed as:

w g =         0 0 -mg -mgy G mgx G 0         = -Wt = -W         t 1 . . . t i . . . t 8         (13)
Knowing the tensions t in each cable and the wrench matrix W, it is then possible to calculate the payload mass m and the Cartesian coordinates x G and y G expressed in F p of the center of mass of the set composed of the MP and the carried payload.

Cable tensions can be either measured using force sensors or estimated from the motor currents as discussed in Sec. V. The wrench matrix W depends from the platform position. As explained in Sec. II-B it is only possible to estimate the MP pose by solving the direct geometric model with Least Squares Method. However, the computation cost and the uncertainties can be detrimental for a real-time mass estimation method. As a consequence, in this paper the errors between real and desired platform pose are supposed to be small enough along the trajectory in order to use the desired positions in the calculation of W.

IV. TORQUE CONTROLLER WITH REAL-TIME MASS

COMPENSATION

From the evaluation of the payload mass, a new controller with a compensation term updated with the real-time estimation is presented thereafter.

The control scheme implemented, depicted in Fig. 6, is based on a proportional-derivative strategy (PD) that is well established in robotics including CDPRs [START_REF] Kawamura | High-speed manipulation by using parallel wire-driven robots[END_REF], with X d the 6dimensional vector containing the desired Cartesian position and orientation of MP, Ẋd the desired MP twist, q d the 8dimensional vector of desired angular positions of the motors, τ m the 8-dimensional vector of desired motor torques. K p and K d are the proportional and derivative gains of the controller, respectively, and have been tuned identically for all motors such that the robot achieves acceptable accuracy and stability with the MP alone.

While a PD controller is the basis of robot control, a feedforward term is commonly included in CDPR control strategies [START_REF] Lamaury | Dual-space adaptive control of redundantly actuated cable-driven parallel robots[END_REF], [START_REF] Vafaei | Integrated controller for an over-constrained cable driven parallel manipulator: KNTU CDRPM[END_REF] to predict the dynamics of the platform and improve the robot accuracy. Based on the dynamic model, it is possible to add a term to the controller that will anticipate the MP dynamics and compensate for the payload mass along the trajectory. The final controller is denoted as (PDFF), as is includes the feedforward term calculated in real-time.

From [START_REF] Pott | Influence of Pulley Kinematics on Cable-Driven Parallel Robots[END_REF], this feedforward term is defined as

τ f f = mrW + [ Ẍd + g] R (14)
with τ f f the feedforward torque, m the total mass of the MP and its payload, R the gearbox ratio, r the drum radius. W + denotes the Moore-Penrose pseudo-inverse of W. The MP Cartesian and angular acceleration Ẍd has been considered to anticipate the robot dynamics. The inverse geometric model (IGM) and inverse kinematic model (IKM) in the experiments used Model I. Model II will be used in the future experimental tests and should lead to better accuracy. A friction compensation was implemented in each actuation chain to compensate the losses in the transmission systems:

τ f = F c sign( qd ) + F v qd , (15)
with τ f the generated friction compensation torque and qd the desired motor rate vector. F c and F v are respectively the vectors containing the static and viscous friction coefficients separately identified for each motor.

V. SENSITIVITY ANALYSIS BASED ON EXPERIMENTAL

RESULTS

This section analyzes the effect of the pulleys geometric parameters onto the payload estimation as well as the comparison between force sensors and motor currents for cable tensions and payload estimation. Moreover, the performance of the controller described in Sec. IV are presented thereafter.

A. Pulley effect onto payload estimation

The study of the effect of the pulleys geometry on the payload estimation is realized along the test trajectory described in Sec. II.C using the cable tensions measured by the dynamometers. Since we want to establish a method that is compatible with real-time computation, the wrench matrix of the platform is computed using the desired poses of the platform. If the processor can solve the direct geometric model fast enough, the mass estimation from the estimated position will be further improved.

Figure 7 presents the estimated mass of the payload along the trajectory for the three different payloads with both the basic model, Model I, and the model including the geometry of the pulleys, Model II. It is noteworthy that the payload mass estimation is better with the model including the geometry of the pulleys than with the model which does not consider the pulleys. The changes in the estimation can be explained by the new wrench matrix and computed cable angles from the MP to the exit points on the pulleys. This leads to higher and more stable values for the mass estimations along the trajectory. As shown in Tab. I and Figs. 8 to 9, the average mass estimation error drops from 4.5% to around 1% when considering the pulleys and the standard deviation is reduced by more than 20%. The variations in the estimation from t = 0 s to t = 5 s, t = 25 s to t = 35 s and t = 55 s to t = 60 s are due to the acceleration of the MP in the upward and downward phases and to vibrations which caused cable tensions fluctuations. The position of the center of mass along the x-axis and the y-axis of the MP frame F p have been calculated with Models I and II and are illustrated in Fig. 10 for the heaviest payload, i.e., with mass M 2 . From Fig. 10, we can notice that there is an offset of the center of mass of the set composed of the moving-platform MP and the metal plate M 2 , which is consistent with the real position of payload M 2 under the moving-platform as shown in Fig. 11.

According to the complexity of the pulleys model, its use in the mass estimation is therefore of interest and Model II is used in the remainder of the paper.

B. Tension estimations versus dynamometer measurements

Parallel robots are not all equipped with dynamometers, since the integration of additional sensors might be expensive. However, it may be essential to implement a payload compensation method by estimating the cable tensions. In this section, the influence of the cable tensions measurements on the payload estimation is studied. The experiments were made with the heaviest payload, i.e. MP+M 2 of 615kg, as illustrated in Fig. 11.

Motor torques, obtained from the motors currents, can be used to estimate the cable tensions, knowing the gearbox ratio and the drum radius. Figure 12 shows (i) the cable 1 tension measured by the dynamometer located between the There is an offset between the center of mass of M 2 and axis z p along axis x p cable 1 exit point and the MP; (ii) the cable 1 tension assessed from the raw motor 1 torque; (iii) the cable 1 tension estimated from the raw motor 1 torque and the friction model expressed in [START_REF] Gagliardini | Determination of a dynamic feasible workspace for cable-driven parallel robots[END_REF], to reduce the perturbation caused by friction on the measure. It first appears that the dynamometer offers a measure with much less noise that the estimations from the raw motor torque. However, we can observe that the estimation from the motor torque and the friction model is very close to the dynamometer measurement, especially from t = 5 s to t = 25 s and t = 35 s to t = 55 s when the velocity of the platform is almost constant. The discontinuities in cable 1 tension profile obtained from the motor 1 torque are due to changes of motor 1 rotation direction. These results confirm that the dynamometers offer a cleaner measure of tension, but also highlight the possibility that with the proper filter and a better identification of the friction in the motors, motor torques could give a usable reading of the cable tension.

The mass estimation along the trajectory shown in Fig. 13 shows again that the estimation is the most accurate with the dynamometer measurements. The impact of the noise in the cable tension estimation from motor torques is amplified in the mass estimation, as seen in Table II. While the mean of the estimated mass error is smaller than 6% with both the raw torques and the torques minus the friction, the maximum error reaches respectively 57% and 30% of the payload mass with those measures. On the other hand the dynamometers lead to a maximum error slightly above 4%. The largest errors occur along the vertical motion of the platform, in the acceleration phases and especially when the motors direction changes, for which the mass estimation error is suddenly inverted. In the central sections of the paths the mass estimation from the torques is closer to the dynamometers results. Finally, the center of mass is estimated for payload MP+M 2 and shown in Fig. 14. Here only the estimations from the dynamometers to the motors torques minus friction are compared since it consistently offered better results than the raw torques. The same conclusions can be made than with the mass estimation as a more stable estimation was obtained from the dynamometer measurements than with the torque, but with a fairly similar shape. A filter might lead to usable readings in the absence of force sensors.

C. Control method performance

Experiments have been realized on the CAROCA prototype along the test trajectory described in Sec. II-C with two controllers:

1) the PD controller presented in Fig. 6, with real-time update of the mass in the feedforward term (PDFF) 2) a PD controller without mass compensation (PD) For this comparison both controllers used the same gains K p and K d . Position errors were computed by solving the direct geometric model including the pulleys with the Least Square Method. The MP position errors along the test trajectory with the two controllers and the three payloads are shown in Fig. 15. As the impact of the payload mass is directed along the gravity, we focus on the platform position accuracy along the z-axis. The PD controller suffers from an immediate error at the beginning of the trajectory, as the initial output torque of the PD is null and the platform position drops under the load. Between the lightest and the heaviest payloads, the errors of the PD controller increased up to 20 mm. On the other hand, the PDFF controller has an absolute position error smaller than 5 mm, for every payload. Although there are still some variations in the position accuracy, this behavior is stable enough to pick and place the heaviest metal plate, shown in Fig. 11, with a good accuracy. As a conclusion, while the performance of the PD controller are affected by the mass increment, the real time mass compensation allows for a more precise control of the platform position.

VI. CONCLUSION

This paper dealt with the pulleys and force sensors influence on payload estimation of cable-driven parallel robots. First, the effect of the pulleys to the payload estimation was experimentally evaluated. Then, a comparison between the force sensors and the information coming from the motor currents to assess the cable tensions was made. It should be noted that the consideration of the pulleys into the geometric model of the robot substantially improves the mass and center of mass estimations of the payload at a low computational cost. Moreover, the cable tension values are much more accurate when they are measured with dynamometers instead of being assessed from the motor torques. A better friction model of the transmission systems and a filter are required to get a better estimation of the cable tensions from the motor currents. Besides, a torque controller with a feedforward term for real-time mass compensation was proposed and implemented on a CDPR prototype located at IRT Jules Verne. The experimental results confirm the interest in payload estimation for CDPR robust control. The consideration of cable elasticity and the determination of an improved friction model for the CDPR transmission systems are part of our future work.

Fig. 1 :

 1 Fig. 1: Top. The moving-platform (MP) equipped with five magnets to pick metal parts and eight dynamometers to measure cable tensions. Bottom. A motor, a winch and a pulley.

Fig. 2 :

 2 Fig. 2: Top and side views of the test trajectory.

Fig. 3 :

 3 Fig. 3: Test trajectory in blue and CDPR prototype configuration with pulleys in orange.

Fig. 4 :

 4 Fig. 4: CDPR geometric parameterization.

Fig. 5 :

 5 Fig. 5: Top and side views of a two-DoF pulley.

Fig. 6 :

 6 Fig. 6: Control scheme with feedforward for real time mass estimation and compensation (PDFF).

Fig. 7 :

 7 Fig. 7: Estimated mass (kg) with the two models defined in Sec. III.A and III.B and for the three payloads along the test trajectory; the cable tensions are measured with dynamometers.

Fig. 8 :Fig. 9 :

 89 Fig. 8: Mean of the estimated mass along the trajectory with the two models defined in Sec. III.A and III.B and the three payloads.

Fig. 10 :Fig. 11 :

 1011 Fig. 10: Center of mass position estimation in frame F p for the heaviest payload MP+M 2 (615 kg) along the test trajectory with Models I and II (Top). along x-axis (Bottom). along y-axis.

Fig. 12 :

 12 Fig.12:Cable tension 1 (N) with dynamometers measurements, raw motor torque and torque minus frictions, for the heaviest payload MP+M 2 (615 kg) along the test trajectory.

Fig. 13 :

 13 Fig.13: Estimated mass (kg) along the trajectory from dynamometers measures, raw motor torques and motors torques minus friction model, for the heaviest payload MP+M 2 (615 kg)

Fig. 14 :

 14 Fig.14: Center of mass position estimation in frame F p for the heaviest payload MP+M 2 (615 kg) along the test trajectory from dynamometer measurements and motor torques minus frictions (Top). along x-axis (Bottom). along y-axis.

Fig. 15 :

 15 Fig. 15: Platform position error along z-axis versus time obtained with the two controllers, for three masses. Left. MP (366 kg). Center. MP+M 1 (488 kg). Right. MP+M 2 (615 kg).

TABLE I :

 I Estimated mass along the test trajectory with Models I and II.

	Model	MP	MP+M 1	MP+M 2
	Mean mass (kg)
	Real mass	366	488	615
	Model I	349.8	468.0	587.1
	Model II	368.9	493.5	619.1
	Error (%)
	Model I	4.4	4.1	4.5
	Model II	0.8	1.1	0.7
	Standard deviation (kg)
	Model I	2.8	4.3	6.2
	Model II	2.2	3.0	4.3

TABLE II :

 II Estimated mass along the test trajectory with dynamometer measurements and motor torques.

	Measurements	MP	MP+M 1	MP+M 2
	Real mass (kg)	366	488	615
	Mean mass (kg)
	Raw torques	387.8	515.0	643.8
	Torques minus friction	387.7	466.3	643.7
	Dynamometers	368.8	493.4	619.1
	Mean error (%)
	Raw torques	5.97	5.54	4.7
	Torques minus friction	5.93	5.51	4.7
	Dynamometers	0.8	1.1	0.7
	Maximum error (%)
	Raw torques	56.58	48.1	44.5
	Torques minus friction	30.1	28.1	26.8
	Dynamometers	3.4	4.2	3.7
	Standard deviation (kg)
	Raw torques	66.5	71.9	78.5
	Torques minus friction	20.2	25.1	31.3
	Dynamometers	2.2	3.0	4.3

ACKNOWLEDGMENTS

The authors wish to associate the industrial and academic partners of this project, namely, STX France, Clemessy, B&R Automation, Clarté, IRT Jules Verne and CNRS.

*This work is supported by IRT Jules Verne (French Institute in Research and Technology in Advanced Manufacturing Technologies for Composite, Metallic and Hybrid Structures) in the framework of ROCKET project.


Jiuchun Gao
email: jiuchun.gao@ls2n.fr

Anatol Pashkevich
email: anatol.pashkevich@imt-atlantique.fr

Marco Cicellini
email: marco.cicellini@eleves.ec-nantes.fr

Stéphane Caro
email: stephane.caro@ls2n.fr

Optimization of Robot and Positioner Motions in Manufacturing of High-pressure Composite Vessels

Keywords: Redundant robotic system, Time-optimal motion, High-pressure composite vessel. 

The paper proposes a technique for optimal motions generation in redundant robotic tape winding systems for manufacturing of high-pressure composite vessels. It also presents two industrial case studies dealing with the motion generation for basic types of winding lamina, helical and circumferential. The developed technique transforms the original continuous problem into a discrete one where the desired time-optimal motions are presented as a shortest path on the task graph satisfying the problem-specific acceleration and velocity constraints imposed on the joint coordinates. It allows generating the optimal trajectories using enhanced dynamic programming algorithm that considers both of these constraints. Efficiency of the developed technique is confirmed by simulation studies.

INTRODUCTION

High-pressure vessels are containers designed to hold gases or liquids under a pressure substantially higher than the ambient one. Common pressure vessels are made of steel. However, since rolled and forged parts are welded together during fabrication, some mechanical properties could be adversely affected. The latter requires increasing of the thickness to resist the high pressure. For this reason, such allmetal construction is usually very heavy, weighing approximately 1.4 kg/L. To save the weight, there are some other manufacturing techniques based on composite materials (see Fig. 1). In particular, the construction of metal liner with full composite overwrap (generally aluminium with a carbon fiber composite) and all-composite structure comprising a polymer liner with carbon fiber composite allows achieving the weight parameter from 0.3 to 0.5 kg/L [START_REF] Legault | Pressure vessel tank types. Compositesworld[END_REF][START_REF] Boiler | Rules for Construction of Pressure Vessels[END_REF][START_REF] Hoa | Analysis for Design of Fiber Reinforced Plastic Vessels[END_REF]. For fabricating composite vessel, automated tape winding technique is usually used [START_REF] Frketic | Automated manufacturing and processing of fiberreinforced polymer (FRP) composites: An additive review of contemporary and modern techniques for advanced materials manufacturing[END_REF]. Relevant technological process can be implemented by using either specifically designed CNC machines or robotic systems. Existing CNC machines have no limitations on the component size, but they are expensive and require large work-floor areas. In contrast, the robotic systems are relatively cheap and flexible. However, they are usually kinematically redundant because of excessive number of actuated axes that are provided by a 6-dof robot, a 1-dof positioner and a 1-dof linear track. For this reason, optimal coordination of the manipulator motions with the positioner/track movements is an important issue in the robotic tape winding processes.

In literature, there are a number of works that deal with redundancy resolution. Relevant techniques are usually based on the pseudo inverse of the kinematic Jacobian [START_REF] Flacco | Discrete-time redundancy resolution at the velocity level with acceleration/torque optimization properties[END_REF]. However, they can be hardly applied to the considered problem because they do not allow generating optimal trajectories satisfying real-life industrial requirements [START_REF] Kazerounian | Redundancy resolution of serial manipulators based on robot dynamics[END_REF]. Alternatively, there are also several techniques based on conversion of the original continuous problem to a discrete one. The simplest one is able to generate time-optimal trajectories for point-to-point motions and was applied to the spot-welding [START_REF] Gueta | Compact design of a redundant manipulator system and application to multiple-goal tasks with temporal constraint[END_REF][START_REF] Gueta | Practical point-to-point multiple-goal task realization in a robot arm with a rotating table[END_REF] . A slightly different method was proposed in [START_REF] Pashkevich | Multiobjective optimization of robot motion for laser cutting applications[END_REF] for the laser cutting and arc-welding applications where the motion amplitude for the actuated axes was minimized but the tool speed was assumed to be constant.

For the considered process, where the tool speed variations are allowed in certain degree, a discrete optimization based methodology was proposed in our previous work [START_REF] Gao | Optimization of the robot and positioner motion in a redundant fiber placement workcell[END_REF], Gao et al., 2017a). It allows the user to convert the original problem to the combinatorial one taking into account particularities of the fiber placement technology and to generate time-optimal coordinated motions for the robot and positioner. However, the technique was applied to a planar benchmark example only, with a single redundant variable. In this work, an extension of the previous results is proposed allowing dealing with the optimal motion coordination for robotic systems for two basic types of winding paths (helical and circumferential, respectively) in the manufacturing of high-pressure composite vessels.

ROBOT-BASED MANUFACTURING OF HIGH-PRESSURE COMOPSITE VESSELS

A typical robotic platform for automated winding of the composite tape is presented in Fig. 2. Here, the vessel liner to be covered by the thermoplastic reinforcement is manipulated by the positioner, which is able to change its orientation in order to improve accessibility of certain zones by the technological tool. This tool is attached to the robot flange and ensures placement of the tape reinforcement in desired locations. The robot is installed on a translational linear track allowing adjusting its base location while processing relative large products. In robot-based high-pressure composite vessel fabrication, preparation of the manufacturing process includes a number of stages presented in Fig. 3, where the motion coordination of all robotic system components is one of the most difficult procedures. Within this process, the desired winding path is generated using a dedicated CAM system and it is presented in a discrete form (as an augmented line, usually). Further, the obtained set of task points is transformed into the task graph that describes all possible combinations of the robot, positioner and linear track coordinates. Then, the motion generator produces the optimal trajectories that correspond to the "shortest" path on the task graph. Finally, the obtained motions are converted into the program for the robot control system. More details concerning these procedures can be found in [START_REF] Gao | Optimal Motion Planning in Redundant Robotic Systems for Automated Composite Lay-up Process[END_REF]. Fig. 3. Manufacturing process preparation for high-pressure composite vessel.

KINEMATIC MODEL OF THE ROBOTIC TAPE WINDING SYSTEM

To describe the winding task, let us present it as a sequence of the frames

 () { , 1,2,... } i task F i
n , in such a way that the X-axis is directed along the path direction and Z-axis is normal to the workpiece surface pointing outside of it. Using these notations, the task locations can be described by a set of 4×4 homogenous transformation matrices and the considered tape winding task is formalized as follows:

   (1) ( ) ( )
; 1,2,...

w w i w n task task task in TTT (1)
where all vectors of positions and orientations are expressed with respect to the workpiece frame (see superscript "w"). To execute the given winding path, the technological tool must sequentially visit the frames (1) as fast as possible.

In any task location, the spatial configurations of the robot, positioner and linear track can be described by the joint coordinates R q , P q and L q . So, the task locations can be expressed using the direct kinematic functions of these components, which are further denoted as

() RR g q , () PP gq
and () LL gq . This allows us to write the kinematic equations describing the given winding task in the following form

        ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1,2,... ; World i i Tool Lbase L L R R task World i w i Pbase P P task g q g g q i n T q T TT (2)
where all notations are defined in Fig. 2. It is clear that the above equations cannot be solved for R q , P q and L q in unique way because the robotic system is kinematically redundant. On the other side, it gives some freedom for optimization of the coordinated motions of the robotic manipulator with the positioner and linear track movements.

GENERATION OF OPTIMAL MOTIONS FOR ROBOTIC MANIPULATOR AND POSITIONER

To present the problem of optimal motion generation in a formal way, let us define the function ()

)) ( ( )) ( ( )) () () R L P R i P i L i t q t q t World Tool Lbase L L i R R i task World w i Pbase P P i task R P L R P L i R P L R P L R P L i find t q t q t i n such that T st g q t g t g q t t t q q T q T TT q q q qq                                    max ,, (
( ), ( ), ( ) 0 0, ; 1, 2, ...

R P L R P L R P L i R P L Ri R i P i L i n t cond t C colls t q t q t where t t T i n q q q q Jq q (3)
where the main objective is to minimize the total motion time using full capacities of the redundant robotic system, which are limited by the maximum velocity/acceleration values for the actuated joints. Besides, the collision constraints () colls  as well as the distance to singularities () cond  are also taken into account. Because of specific constraints, the above presented continuous optimization problem cannot be solved in a straightforward way. For this reason, the considered problem is converted into a discrete form by sampling the redundant variables corresponding to the positioner and the linear track. Then, using ideas proposed in our previous work [START_REF] Gao | Optimization of the robot and positioner motion in a redundant fiber placement workcell[END_REF] and applying sequentially the direct kinematics of the positioner and linear track as well as the inverse kinematics of the robot, one can get the configuration states for the robotic system in joint space (see Fig. 4). This allows generating an extended task graph where all task locations are ordered in time. This graph contains all possible configurations of the considered robotic system for executing the given tape winding path, and the desired time-optimal solution of the relevant optimization problem is presented as the shortest path connecting the initial and the final layers.

The structure of this 3D graph is presented in Fig. 5 where the nodes are equal to the technological tool displacement time from the i th to the (i+1) th task point, which is restricted by the maximum velocities and accelerations of the robot, the positioner and the linear track. It should be also noted that some of the nodes are excluded from the graph because of violation of the collision or singularity constraints as well as the joint limits. These nodes are marked as "inadmissible" ones in Fig. 5, and they are not connected to any neighbour. So, the objective function to be minimized (robot motion time) can be presented as the sum of the edge weights ) min

       11 1 2 1 2 ( , , ) ( , , 1) 1 1 ( ,
i i i i k k i k k i task task n i T dist LL (4)
that are computed as

           11 1 2 1 2 11 1 2 1 2 ( , , ) ( , , 1) ( , , ) ( , , 1) max
( , ) max i i i i i i i i k k i k k i jj k k i k k i task task j qq dist q

LL

(5)

i.e. taking into account the maximum allowable velocities

 1 1 2 2 1 2 1 2 1 2 ( , , 1) ( , , 2) ( , , ) { } { } { } nn k k k k k k n task task task L L L
that contains the actuated coordinates of the robot, positioner and linear track. It worth mentioning that the above expression straightforwardly takes into account the velocity constraints, while the acceleration constraints are verified by means of the second order approximation applied to the corresponding functions

                ( 1) ( ) 1 max 11 2 ii i j i j j i i i i t q t q q t t t t (6)
where

     11 1 2 1 2 ( , , ) ( , , 1) () i i i i k k i k k i i j j j q q q
and  i t are computed as the travel time between the nodes 12 ( , , ) To find the desired optimal path, the conventional optimization techniques can be hardly applied because of extremely high computing time [START_REF] Gao | Optimization of the robot and positioner motion in a redundant fiber placement workcell[END_REF] Besides, these techniques are not able to take into account the acceleration constraints that are very essential here. For these reasons, a dedicated problem-oriented algorithm has been developed for this problem. This algorithm is based on the dynamic programming principle, aiming at sequentially finding the shortest paths for the problems of lower dimensions, i.e. from and the distances between the task locations with the indices i and i+1,

           1 2 1 2 1 2 1 2 , 12 ( , , 1) ( , ) , , 1 , , min
,

kk k k i k k , i k k i k k i task task d d dist LL (7)
This expression is applied recursively, starting from the second layer of the task graph ( 2 i  ) and finishing by the last one ( in  ). So, the desired optimal path can be obtained after selection of the minimum length

 12 , , 1 k k i d
corresponding to the last layer. It should be mentioned that this algorithm can be also simplified to a one redundant variable case by fixing the robot base on the linear track, when the workpiece is small.

INDUSTRIAL IMPLEMENTATION OF GENERATED OPTIMAL MOTIONS

To demonstrate advantages of the proposed technique, let us apply it to an industrial problem that deals with fabrication of a composite vessel with the maximum pressure of 500 bar. The considered high-pressure vessel is composed of a cylindrical part and two elliptical domes at both ends of the cylinder (see Fig. 1). The cylinder has the diameter of 168 mm and the length of 400 mm. The domes have the depth of 40 mm. The vessel is covered by several layers of the thermoplastic tape, which are winded by the robot following certain patterns and consolidated in situ.

The winding patterns were generated using the dedicated software Composicad. For winding of the considered pressure vessel, there are two basic types of lamina: helical and circumferential. The helical winding is typically lower than 54°, whereas the circumferential one is usually done at an angle close to 90° [START_REF] Vasiliev | Composite Pressure Vessels: Design, Analysis, and Manufacturing[END_REF][START_REF] Shen | A filament-wound structure technology overview[END_REF]. After all of the helical and circumferential laminas are placed on the vessel surface, a connector lamina could be also added to join them. In this work, the developed motion generation technique was applied to the two basic types of lamina.

Fig. 6. Robotic platform for automated thermoplastic tape winding process.

Relevant robotic tape winding workcell (Fig. 6) is composed of 6-axis serial robot KUKA KR210 R3100, 1-axis translational linear track KUKA KL2000 and 1-axis rotational positioner AFPT 550. Here, there are two extra degrees of freedom provided by the positioner and linear track. However, for the considered manufacturing task that deals with rather small vessel, the robot can easily execute the tape winding process without changing its base location.

Besides, the minimum travel distance of the linear track is comparable with the vessel size. So, for this product, there is no reason to activate the linear track during the winding.

Task #1: A helical winding path

The first task deals with a helical winding lamina. A sample of the path locations ensuring a single circuit helical winding was created using Composicad. Relevant data contain the description of a discrete 3D-augmented curve. They include Cartesian positions and normal directions of the winding path with respect to the workpiece frame as shown in Fig. 7. To generate the time-optimal coordinated motions of robotic manipulator and positioner, the joint coordinate spaces of the positioner was discretized with the step of 1°. Then, the proposed time-optimal motion generation technique was applied for the case of a single redundant variable. Relevant computational results are presented in Fig. 8. As follows from them, the developed technique allows generating timeoptimal helical winding trajectory in manufacturing of the vessel with the minimum motion time 4.2 s, which is almost 3 times better than the result provided by Composicad.

Task #2: A circumferential winding path

The second task deals with a circumferential winding lamina. A sample of the path locations ensuring a single circuit circumferential winding was created in a similar way. The relevant discrete 3D-augmented curve is shown in Fig. 9. By sampling the joint coordinate spaces of the positioner with the step of 1°, the developed time-optimal motion generation technique for the case of a single redundant variable was applied again. Relevant computational results are presented in Fig. 10. As follows from them, the developed technique allows generating time-optimal circumferential winding trajectory in manufacturing of the composite vessel with the minimum motion time 4.5 s.

It should be mentioned that after discrete optimization, a dedicated smoothing technique might be also used to locally improve the velocity and acceleration profiles and ameliorate the actuator working conditions.

The above presented algorithm was tested using Matlab 2016b environment (running at Intel®i5 CPU@2.67GHz 2.67GHz). It took about one minute to generate the timeoptimal motions for each task.

CONCLUSIONS

This paper presents a technique dealing with the optimal motion coordination in redundant robotic tape winding systems for manufacturing of high-pressure composite vessels. In contrast to previous results, this work focuses on generation of the time-optimal coordinated motions for the two basic types of winding lamina, i.e. helical and circumferential. In the frame of the developed technique, the original continuous optimization problem is converted into a combinatorial one where the desired time-optimal motions are presented as a specific shortest path on the task graph.

Then, the desired time-optimal motions are obtained using the developed enhanced dynamic programming algorithm that takes into account the actuator capabilities (coordinate limits, velocities and accelerations) as well as the kinematic and geometric constraints allowing avoiding collisions and singular configurations of the manipulator. The proposed technique is illustrated by industrial case studies dealing with two winding tasks (helical and circumferential curves). In future, this technique will be generalized for a connector lamina that allows transitions between the two basic types of lamina considered in this work.

Fig. 1 .

 1 Fig. 1. Manufacturing of the high-pressure composite vessel with thermoplastic tape covering.

Fig. 2 .

 2 Fig. 2. Robotic workcell for manufacturing of high-pressure composite vessel

  the robot, positioner and linear track joint coordinates as functions of time [0, ] tT  . Additionally, let us introduce a sequence of time instants 12 { , ,... } n t t t corresponding to the cases when the technological tool visits the task locations (1). It allows presenting the considered optimal motion coordination problem as

Fig. 4 .

 4 Fig. 4. Transformation of the original continuous problem into discrete form.

  kk are related to the sampled coordinates of the positioner and linear track respectively.

Fig. 5 .

 5 Fig. 5. Task graph corresponding to the motion generation problem with two redundant variables. Using such presentation, the original continuous problem (1) is converted into a specific shortest-path problem on the graph, where all three successive nodes satisfy the acceleration constraints and the distances between two nodes 12 ( , , ) k k i task L

  us to present the acceleration constraints on the desired trajectory of the considered robotic system in the following form:

Fig. 7 .

 7 Fig. 7. Discrete representation of the helical winding path in the form of the augmented line.

Fig. 8 .

 8 Fig. 8. Time-optimal velocity profiles for Task #1.

Fig. 9 .

 9 Fig. 9. Discrete representation of the circumferential winding path in the form of the augmented line.

Fig. 10 .

 10 Fig. 10. Time-optimal velocity profiles for Task #2.

ACKONWLEDGEMENT

This work is supported by the China Scholarship Council (Grant N°210404490018). The author also acknowledged CETIM for the motivation of this research work.