Flavien Paccot

Nicolas Andreff

Philippe Martinet

Revisiting the Major Dynamic Control Strategies of Parallel Robots

In this paper, we propose a comparison of the three major dynamic control strategies for a parallel robot, namely the linear control, the nonlinear feedforward control and the Computed Torque Control. These control schemes are first transposed in the Cartesian space. This control space is shown to be perfectly relevant for a parallel robot. Then, the control equations are analysed to exhibit the ordinary differential equation of the controlled error. This allows for obtaining a characteristic of the disturbances on this error. Finally, simulation and experiment are provided to verify the theoretical approach.

I. INTRODUCTION

A parallel robot is a complex mechanism with an awkward behaviour [START_REF] Merlet | Parallel Robots[END_REF]. Therefore, controlling such a system is not straightforward. Furthermore, required accuracies and speeds make this task all the more difficult. Concerning the two major applications of the parallel robots, the PAR4 manipulator reaches 170m.s2 in pick-and-place operations [START_REF] Nabat | PAR4: very high speed parallel robot for pick-and-place[END_REF] and 100µm tolerances are generally required in machining. To reach these requirements, performant mechanical design, modeling and identification should be achieved [START_REF] Merlet | Still a long way to go on the road for parallel mechanisms[END_REF]. Nevertheless, this is only useful when an adapted dynamic control method is used. The presented approach focuses on the three major ones.

The simplest and most widely used way to control a system is a linear PID controller without any a priori knowledge on the system. That is realised for most industrial parallel robot [START_REF] Merlet | Parallel Robots[END_REF]. Nevertheless, this is not enough sufficient with regards to the dynamic behaviour of a parallel robot. Actually, a linear PID controller is efficient with a linear system, which is achieved very locally for the dynamics of a parallel robot. Consequently a simple control scheme only composed of a linear PID controller, without any manipulator structure considerations, can not ensure good tracking performances at high speed [START_REF] Ouyang | Non linear PD control for trajectory tracking with consideration of the design for control methodology[END_REF], [START_REF] Honegger | Application of a nonlinear adaptative controller to a 6 dof parallel manipulator[END_REF]. The literature abounds of methods improving this simple control scheme either with a path generation taking into account the dynamics or with an adapted gain tuning [START_REF] Abdellatif | Adapted time-optimal trajectory planning for parallel manipulators with full dynamic modelling[END_REF], [START_REF] Zhiyong | A new method for tuning PID parameters of a 3-DOF reconfigurable parallel kinematic machine[END_REF]. However, tracking performances are low since the still linear control is associated with a nonlinear system. Therefore, a nonlinear control including a model of the dynamic behaviour can be employed.

A first nonlinear method adapts the controller gains with regards to the end-effector pose and the mechanism dynamic behaviour [START_REF] Ouyang | Non linear PD control for trajectory tracking with consideration of the design for control methodology[END_REF]. Nevertheless, a simpler solution is to include an Inverse Dynamic Model in the control scheme, allowing for computing the torques required for the desired motion. This model can be inserted either in a feedforward loop [START_REF] Honegger | Application of a nonlinear adaptative controller to a 6 dof parallel manipulator[END_REF] or between the controller and the process, in the case of the well-known Computed Torque Control [START_REF] Khalil | Modeling, identification and control of robots[END_REF].

To sum up, there are three major dynamic control strategies (linear control, nonlinear feedforward control and Computed Torque Control) on which more elaborated control can be built: gain tuning, performant path generation, adaptive control, robust control. . .

In this paper, we propose a comparison between the three major dynamic control strategies for parallel robots. This approach is based on a theoretical analysis of the control equations, where the dynamics of the controlled error are exhibited and discussed. The characteristics of the disturbance on these errors are thus obtained. The comparison is performed both in simulation and in experiments. Furthermore, we propose an improvement of the control schemes by employing the Cartesian space control. We show that this control space is more relevant for parallel robots than the traditionally used joint space.

This article is organised as follows: Section II is devoted to control, Section III presents the test-bed, Section IV concerns the simulations and experiments results.

II. MAJOR DYNAMIC CONTROL STRATEGIES REVISITED

This section focuses on three major dynamic control strategies met in the literature: simple linear control, nonlinear feedforward and Computed Torque Control. First of all, it is explained why a Cartesian space control is perfectly relevant for a parallel robot. Then, each control schemes are detailled and adapted in this control space.

A. Cartesian space control

Since the knowledge on parallel robots flows from the serial ones, control schemes used for the latter are generally reused without any adaptation. Therefore, in most cases, parallel robots are controlled in the joint space [START_REF] Honegger | Application of a nonlinear adaptative controller to a 6 dof parallel manipulator[END_REF], [START_REF] Ouyang | Non linear PD control for trajectory tracking with consideration of the design for control methodology[END_REF], [START_REF] Zhiyong | A new method for tuning PID parameters of a 3-DOF reconfigurable parallel kinematic machine[END_REF]. Nevertheless, next words show that this is not a relevant control space for such robots.

Firstly, parallel and serial robots present a duality [START_REF] Waldron | Series-parallel Dualities in Actively Coordinated Mechanisms[END_REF]. Whereas the state of serial robot is completely defined by its joint positions (algebraic Forward Kinematic Model), the behaviour of a parallel robot is only ruled by the end-effector pose, except in some rare cases where the Inverse Kinematic Model has no closed-form expression [START_REF] Dallej | 3d pose visual servoing relieves parallel robot control from joint sensing[END_REF]. Consequently, a model-based control for a parallel robot, done in the joint space, requires forward transformations from joint to Cartesian space. In the parallel robots case, these transformations have not a closed-form expression contrary to the serial one [START_REF] Merlet | Parallel Robots[END_REF]. Thus, the numerical estimations limit the control performances in term of stability, accuracy and speed.

Secondly, a parallel robot can have several end-effector poses for one joint configuration [START_REF] Husty | An algorithm for solving the direct kinematic of the Gough-Stewart platforms[END_REF]. Therefore, a disturbance on the end-effector trajectory can shift its pose without changing the joint configuration, especially in the neighborhood of singularities or naturally in the cuspidal robots case [START_REF] Chablat | Working modes and aspects in fully parallel manipulators[END_REF]. By servoing the joint positions, the return to the end-effector pose reference is not achieved while the joint reference tracking is still ensured. On the opposite, a Cartesian space control brings back the end-effector pose to its reference (see Figure 1).

Thirdly, considering the complex structure of a parallel robot, the joint motion has to be compatible with the endeffector one. Whatever a suitable joint trajectory can be planned with the Inverse Kinematic Model, the joint tracking errors are not necessarily compatible with the end-effector motion in the joint space control case. Consequently, internal torques can appear, like in the two-arm robots case (see Figure 2 and [START_REF] Dauchez | Hybrid control of a twoarm robot handling firmly a single rigid object[END_REF]). By servoing the end-effector pose, the active joint torques are filtered throughout the Forward Instantaneous Kinematics matrix. Consequently a Cartesian space control cancels the internal torques leading to better accuracy and structure damage preventing [START_REF] Dauchez | Hybrid control of a twoarm robot handling firmly a single rigid object[END_REF].

Last but not least, the Cartesian space is the user's task space where trajectories are planned in Computer Aided Manufacturing. Consequently, to our mind, a parallel robot should be controlled in the Cartesian space instead of the joint space. Nevertheless, this assumes to have a fast measure of the end-effector pose, in a dynamic control context.

Current classical measurement means, such as a laser tracker, are quite expensive and integration in a control scheme is not straightforward [START_REF] Vincze | A laser tracking system to measure position and orientation of robot end-effectors under motion[END_REF]. On the opposite, com-puter vision seems to be perfectly relevant [START_REF] Gangloff | Asservissements visuels rapides d'un robot manipulateur à six degrés de libertés, Suivi de cible et de profilé[END_REF], [START_REF] Aider | Simultaneous object pose and velocity computation using a single view from a rolling shutter camera[END_REF]. However, speed and accuracy should be improved to meet required performances for a parallel robot. While waiting for a reliable, fast, accurate and flexible measurement mean, the forward kinematic model can be used to give an estimation of the end-effector pose. In special cases [START_REF] Krut | I4: a new parallel mechanism for scara motions[END_REF], [START_REF] Gogu | Fully-isotropic T3R1-type parallel manipulators[END_REF], parallel robots have a closed-form forward kinematic model avoiding the numerical estimation issues listed above. In the other cases, adapted computation methods can be employed [START_REF] Merlet | Solving the froward kinematic of a Gough-type parallel manipulator with interval analysis[END_REF]. Furthermore, by adding extra sensors, the metrological redundancy decreases computation time and the number of solutions [START_REF] Marquet | Enhancing parallel robot with redundant sensors[END_REF], [START_REF] Baron | The direct kinematics of parallel manipulators under joint-sensor redundancy[END_REF]. Nevertheless, a model can not compete with a measure. Indeed, it can be biased with structure deformations and assembly errors, which have a great influence with regards to the numerous joints [START_REF] Wang | On the accuracy of a Stewart platformpart I: The effect of manufacturing tolerances[END_REF]. A kinematic identification can reduce this issue [START_REF] Daney | Self calibration of Gough platform using leg mobility constraints[END_REF], [START_REF] Merlet | Parallel Robots[END_REF].

In the further paragraphs, a measure of the end-effector pose is assumed to be available. Therefore, all presented control schemes are Cartesian space controls.

B. Dynamic modeling

To analyse the presented control schemes, the inverse dynamic modeling of a parallel robot is used. The Lagrangian formulation is generally employed to set up dynamic control laws. In the serial case, it is expressed as: Γ(q, q, q) = A(q)q + H(q, q) + G(q) + Γ f ( q) + Γ ext [START_REF] Merlet | Parallel Robots[END_REF] where q are the active joint variables, Γ are the torques applied in the active joints, A is the inertia matrix, H is a vector containing Coriolis and centrifugal forces, G is a vector containing the gravity forces, Γ f is the friction influence and Γ ext are the external forces applied on the robot.

This expression is generally directly reused for parallel robot. However, the dynamic behaviour of a parallel robots depends on the end-effector pose and time derivatives [START_REF] Khalil | General solution for the dynamic modeling of parallel robots[END_REF], [START_REF] Dasgupta | A general strategy based on the Newton-Euler approach for the dynamic formulation of parallel manipulators[END_REF]. It is hence more relevant to express the model above as [START_REF] Dasgupta | A general strategy based on the Newton-Euler approach for the dynamic formulation of parallel manipulators[END_REF]:

Γ(X, Ẋ, Ẍ) = A(X) Ẍ + H(X, Ẋ) + G(X) + Γ f ( Ẋ) + Γ ext (2)
where X is the end-effector pose.

The Cartesian space formulation allows for computation saving [START_REF] Dasgupta | A general strategy based on the Newton-Euler approach for the dynamic formulation of parallel manipulators[END_REF] leading to faster, more accurate and more stable control. In the further paragraphs, this formulation is used.

C. Linear control

In most cases, industrial parallel robots are controlled with a simple linear control [START_REF] Merlet | Parallel Robots[END_REF], [START_REF] Honegger | Application of a nonlinear adaptative controller to a 6 dof parallel manipulator[END_REF]. This control scheme is well-known allowing for an easy maintenance. In addition, it can be used for a parallel robot as well as for a serial one. We propose an adaptation of this control scheme in the Cartesian space (see Figure 3). Nevertheless, such a control scheme does not allow for good dynamic accuracy [START_REF] Denkena | Advanced position and force control concepts for the linear direct driven hexapod PaLiDA[END_REF], [START_REF] Honegger | Application of a nonlinear adaptative controller to a 6 dof parallel manipulator[END_REF]. In fact, a desired position in the workspace can be reached with accuracy, with the prerequisite that a good kinematic identification is performed. However, the tracking

+ - + + Path Linear Feedforward PID uP ID u f f Robot X d M e Γ X
Fig. 3. Linear control in the Cartesian space with PID controller and feedforward performances deteriorate quickly when the trajectory speed increases. Indeed, by using a simple linear control, the hypothesis of a linear dynamic behaviour is made:

Γ(X, Ẋ, Ẍ) = M Ẍ + F v Ẋ + F s + ΓP ID ( 3
)
where M is the simplified inertia matrix, F v is the viscous friction matrix, these two matrices are supposed to be constant and diagonal, F s is the dry friction vector. The term ΓP ID is the resulting error between Equation 2 and Equation 3. It can be expressed as:

ΓP ID = (A(X) -M ) Ẍ + H(X, Ẋ) + G(X) +Γ f ( Ẋ) -F v Ẋ -F s + Γ ext (4)
A linear feedforward term and a friction compensation can be used. The torque control signal is then expressed as:

Γ(X d , Ẋd , Ẍd ) = M Ẍd + F v Ẋ + F s + M u P ID (5)
where X d is desired trajectory, u pid is the signal generated by the linear controller:

u pid = K d ė + K p e + K i edt (6)
where e = X d -X is the error signal. Since dry friction depends on the speed sign, let us remark that this feedforward is not completely linear. Considering Equations 3 and 5, the error signal has the following behaviour:

ë + K d ė + K p e + K i edt = M + ΓP ID (7)
The gain tuning aims at obtaining a system with a triple negative pole, (p + ω) -3 , with ω chosen smaller than the mechanical resonance frequency of the robot [START_REF] Khalil | Modeling, identification and control of robots[END_REF]:

K p = 3ω 2 K d = 3ω K i = ω 3 (8)
where K p is the proportional gain matrix, K d the derivative gain matrix and K i the integral gain matrix. Consequantly, under the hypothesis of negligible disturbances ΓP ID , the error signal presents an exponential decay determined by the gain tuning.

With the linear behaviour and small disturbance hypotheses, a good dynamic accuracy can be expected. Nevertheless, the dynamics of a parallel robot are strongly nonlinear especially because of a dynamic coupling between legs [START_REF] Merlet | Parallel Robots[END_REF], [START_REF] Khalil | General solution for the dynamic modeling of parallel robots[END_REF], [START_REF] Dasgupta | A general strategy based on the Newton-Euler approach for the dynamic formulation of parallel manipulators[END_REF]. Therefore, the first hypothesis is achieved only

+ - + + Path Γ(X d , Ẋd , Ẍd ) PID uP ID Robot X d e Γ X A(X)
Fig. 4. Nonlinear feedforward in the Cartesian space very locally and at low speed. In addition, the disturbances ΓP ID are mainly generated by the dynamic coupling (see Equation 4) where the first two terms concern the dynamic coupling between legs). In this case, the second hypothesis is also only verified at low speed. In conclusion, the hypotheses are too restrictive, grounding a linear control approach unable to meet high speed and high accuracy requirements. To improve the dynamic accuracy, nonlinear control taking into account the dynamic behaviour of the robot needs hence to be used.

D. Nonlinear feedforward control

In order to obtain good tracking accuracy at high speed, the control scheme should include a model of the robot dynamics. The aim is a linearisation of the dynamic behaviour of the robot by a compensation of the latter. Therefore, a simple linear controller on the linearized dynamics can thus ensure optimal performances. In the parallel robot case, the compensation of the robot dynamics in the feedback loop is difficult. Indeed, it depends on the end-effector pose, which is generally estimated with heavy computation. Consequently, Honegger proposed a joint space nonlinear feedforward control, compensating the robot dynamics with regards to the only reference Cartesian trajectory [START_REF] Honegger | Application of a nonlinear adaptative controller to a 6 dof parallel manipulator[END_REF].

Here, an adaptation of this control scheme in the Cartesian space is proposed (see Figure 4) and an analysis of the control equations. Using the Lagrangian formulation, the estimated dynamic behaviour of the robot can expressed as:

Γ(X, Ẋ, Ẍ) = A(X) Ẍ + H(X, Ẋ) + G(X) + Γ f ( Ẋ) + Γ (9)
where A is the estimated inertia matrix, H a vector containing the estimated Coriolis and centrifugal forces, G the estimed gravity forces and Γ f the estimated friction. The term Γ is the resulting error between Equation 2 and Equation 9. It can be expressed as:

Γ = (A(X) -A(X)) Ẍ + H(X, Ẋ) -H(X, Ẋ) +G(X) -G(X) + Γ f ( Ẋ) -Γ f ( Ẋ) + Γ ext (10
) With the hypothesis of small errors e = Xd -X and time derivative ė, the zero order approximation around X d of Equation 9 leads to:

Γ(X, Ẋ, Ẍ) = A(X) Ẍ + H(X, Ẋ) + G(X) + Γ f ( Ẋ) + Γ = A(X d -e) Ẍd -A(X)ë + H(X d -e, Ẋd -ė) + G(X d -e) + Γ f ( Ẋd -ė) + Γ = A(X d ) Ẍd -A(X)ë + H(X d , Ẋd ) G(X d ) + Γ f ( Ẋd ) + Γ + o(e, ė) (11)
where o(e, ė) are the higher order terms. Notice that the second term can be written as A(X d )ë for a joint space control where X is not accessible. By analysing Equation 11, the Inverse Dynamic Model in Equation 9 applied to the reference X d can be noticed. Therefore, by using these terms in a feedforward term, a suitable torque control signal is:

Γ(X, Ẋ, Ẍ) = A(X d ) Ẍd + H(X d , Ẋd ) G(X d ) + Γ f ( Ẋd ) + A(X)u P ID (12
) where u P ID is still the control signal defined in Equation 6.

Considering Equations 11 and 12, the error signal has the following behaviour:

ë + K d ė + K p e + K i edt = A(X) + Γ + o(e, ė) (13)
where A + is the pseudo inverse of A.

The gain tuning is the same as Equation 8. With the hypothesis of negligible disturbances Γ, the error has a exponential decay determined by the gain tuning.

In this case, there is a double integrator between the controller output u P ID and the controller input e:

u P ID = ë = Ẍ -Ẍd (14)
Therefore, the control scheme allows for a tangent linearisation of the robot dynamics. Thus, a linear PID controller allows for optimal dynamic performances. Nevertheless, this is only achieved under the hypothesis of negligible disturbances Γ and small error signal. The disturbances Γ are modeling errors mainly due to assembly errors and unmodeled frictions. These errors can be reduced with an adapted modeling method [START_REF] Khalil | General solution for the dynamic modeling of parallel robots[END_REF] and a good dynamic identification [START_REF] Gautier | Extended Kalman filtering and weighted least squares dynamic identification of robot[END_REF], [START_REF] Olsen | A new method for estimating parameters of a dynamic robot model[END_REF]. Let us remark that in a joint space control case, these disturbances are increased by the use of a numerical forward transformation expressing the joint errors in the Cartesian space. The most restricting hypothesis concerns the error signal. Indeed, high gains are needed to have small errors. However, the gain tuning is limited by the resonance frequency, rather low for industrial parallel robots. High gains are thus prohibited. To decrease errors with higher efficiency, robust techniques can be employed such as adaptive control [START_REF] Honegger | Application of a nonlinear adaptative controller to a 6 dof parallel manipulator[END_REF]. In addition, the feedforward term can include higher order approximations of Equation 9.

Nevertheless, these solutions require more complex models with more computation. Instead of using a nonlinear feedforward, the dynamics compensation can be also ensured in the feedback loop as detailed below.

E. Computed Torque Control

The so-called Computed Torque Control is widespread for serial robots [START_REF] Luh | Resolved acceleration control of mechanical manipulators[END_REF], [START_REF] Khalil | Modeling, identification and control of robots[END_REF]. It allows for excellent tracking performances at high speed. Nevertheless, the performances are not as good as expected in the parallel case [START_REF] Vivas | Predictive functionnal control of a parallel robot[END_REF], [START_REF] Lee | Control design for a Stewart platform using small workspace characteristics[END_REF]. Indeed, this control is generally done in the joint space (see Figure 5 where IKM is the algebraic Inverse Kinematic Model, F KM is a solution to the forward kinematic problem, D is a solution to the forward instantaneous kinematic problem). The use of forward numerical transformations results in a lack of accuracy, speed and stability. To improve these issues, we propose a Cartesian space Computed Torque Control (see Figure 6), without any numerical estimation. We show that this control is perfectly relevant for a parallel robot and allows for better performances than the nonlinear feedforward control scheme.

+ + + - + + Path PID IKM(X d ) Γ(X, Ẋ, Ẍ) FKM(q) D(X) D(X) D(X, Ẋ) d 2 dt d dt Robot X d X Ẋ e Γ q
By reporting result of Equation 14 directly in Equation 9, a suitable torque control law is expressed as:

Γ(X, Ẋ, Ẍ) = A(X) Ẍd + uP ID + H(X, Ẋ)+G(X)+ Γ f ( Ẋ) (15)
Considering Equations 9 and 15, the error signal has the following behaviour:

ë + K d ė + K p e + K i edt = A(X) + Γ ( 16
)
where Γ is expressed in Equation [START_REF] Dallej | 3d pose visual servoing relieves parallel robot control from joint sensing[END_REF]. The gain tuning is the same as Equation 8. With the hypothesis of negligible disturbances Γ, the error has a exponential decay determined by the gain tuning.

The interesting aspects of the nonlinear feedforward control, such as an error with an exponential decay and a linearisation of the dynamics, are still present. Let us remark that the linearisation is now exact instead of tangent. Nevertheless, the restrictive assumption of small errors is removed between Equation 13and Equation 16. The remaining assumption concerns only the negligible modeling errors. With the prerequisite of a good model associated with a good dynamic identification, the Computed Torque Control can ensure the best performances. Of course, this control scheme can be improved in terms of robustness with regards to modeling errors with predictive or H methods [START_REF] Vivas | Predictive functionnal control of a parallel robot[END_REF], [START_REF] Lee | Control design for a Stewart platform using small workspace characteristics[END_REF].

In addition, such methods should be easier to set up now than in the former control schemes, since there are fewer perturbations.

F. Summary

In conclusion of this theoretical approach, a discussion of each control scheme use with regards to the application can be set up. Firstly, when the application requires either low speed with high accuracy or vice versa in a small part of the workspace, a linear control is enough. In addition, this a simple control with an easy set-up. Nevertheless, one should be careful with the path generation and the gain tuning (which can be done locally for better performances). Secondly, when speed and accuracy requirements are not met with linear control, nonlinear control should be used. The nonlinear feedforward control allows for an off-line dynamics compensation. In addition, it can be used in the joint space when the measure of the end-effector pose is not available. Nevertheless, high gains or robust techniques are required to make this control scheme useful. However, high gains are rarely conceivable, especially in high speed machining where robots are quite heavy. Furthermore, the robust techniques are not always relevant in an industrial context since they require great knowledge in automatic control. On the opposite, a Computed Torque Control does not require robust techniques to ensure good tracking with the prerequisite of a performant dynamic model and a good identification. Nevertheless, it requires more computation resources. Fortunately, actual computers allows for sufficient ones.

III. PRESENTATION OF THE ISOGLIDE-4 T3R1

To validate the above discussion, the proposed modeling and control scheme are applied to the Isoglide-4 T3R1. This parallel robot is a fully-isotropic one with decoupled motion (see Figure 7 and [START_REF] Gogu | Fully-isotropic T3R1-type parallel manipulators[END_REF]). It is a four degrees of freedom robot with three translations and one rotation. It is designed for High Speed Machining. Hence, stiffness requirements impose an important weight: 31kg per leg and 14kg for the end-effector. A desired maximum acceleration of 20m.s -2 associated with heavy weight creates an important dynamic coupling between the legs. This test-bed is well suited to the validation of the approach, since its weight prevents us from neglecting the dynamics. Moreover, its closed-form kinematic models remove from the control problem the troubles associated to kinematic nonlinear couplings [START_REF] Rizk | Precision study of a decoupled four degrees of freedom parallel robot including manufacturing and assembling errors[END_REF]. They also compensate for the current technological lack of reliable and accurate high-speed sensor of the end-effector pose.

IV. RESULTS

A. Simulation

A comparison between linear, nonlinear feedforward and Computed Torque controls is proposed in simulation. Each control is tested on a square trajectory in the XY plan. A fifth degree polynomial interpolation is used with several maximal accelerations, ranging from 1m.s -2 to 20m.s -2 . The dynamic behaviour of the robot is simulated with the forward dynamic model (FDM). Realistic noises are included such as a 50µm error on the geometrical parameters (considering the required manufacturing tolerances for this robot), a 10% error on dynamic parameters (as generally achieved in dynamic identification) and a 1µm accuracy on joint sensors (see Figure 8). For a fair comparison, gains are tuned with ω = 5Hz for each control scheme (see Equation 8).

The results confirm the assertions above. Figure 9 shows a comparison between the three control schemes at 3m.s -2 , which is a typical high-speed machining process acceleration. As expected, the Computed Torque Control ensures the best performances. The linear control can not compensate for the dynamic coupling between leg, even at such a low speed. Concerning the nonlinear feedforward control, the tracking performances are worse than expected. Indeed, the hypothesis of small errors is not verified here. To improve the latter performances, higher gains can be set up (ω = 15Hz). However, the stability bounds are reached making this tuning unsuitable. Therefore, a robust controller should be used.

According to Figure 9(b), it can be noticed that no overshoot appears with the Computed Torque Control, contrary to the other controls. This is interesting in terms of path generation. In fact, corners are generally made smooth, with speed decreasing and circular connection, to avoid these As the Isoglide-4 T3R1 is designed for High-Speed Machining operations, a straightness measure on axis X,Y,Z and an orientation error measure are judicious. These measures are summarized in Table I for the three control schemes along the same trajectory. It can be noticed that the Computed Torque Control allows for reaching required tolerances in the XYZ plan (100µm) with machining typical accelerations (above 5m.s -2 ) contrary to the other control schemes.

B. Experiments

We propose now an experimental verification of the simulations realised above. Nevertheless, we aim at obtaining a control without robust techniques, which is industrially more interesting. According to the poor performances of the nonlinear feedforward control without robust controller, the latter is thus not implemented. All the more as it requires as much implementation effort as the Computed Torque Control. Therefore, only Computed Torque and linear control are compared. The end-effector trajectory is measured with a 512 × 512 camera running at 250Hz. This exteroceptive measure allows for focusing on the real motion of the endeffector contrary to an estimation with a forward kinematic model which is biased by structure defects and deformations. In order to prove the camera accuracy, a comparison between the latter and a laser interferometer is proposed (see Figure 10). The Figure 11 gives the measures from the interferometer and the camera on the same deviation. The camera is close to the interferometer with a 26µm average error between the two sensors. It validates the further results.

Figure 12 shows a comparison between linear and Com- The trajectory is executed segment by segment. According to Figure 12, it can be noticed that great improvements are realized. Indeed, due to heavy inertia, the dynamic coupling between legs is not negligible even at 3m.s -2 . Using Computed Torque Control improves tracking since straightness errors are approximately divided by 7 for X-axis displacements and 10 for Y-axis displacements (see Table II where segment 1 is the left edge of square, segment 2 the right edge, segment 3 the bottom edge and segment 4 the top edge).

The simulation and experiments show similar behaviour. Contrary to the simulation, the experimental results take into account the structure defects and deformations. The performances are thus lower in experiments than in simulation.

V. CONCLUSION AND PERSPECTIVES

Parallel robots have a different behaviour from serial ones. Therefore, the control schemes have to be adapted and not directly reused. In this paper, we revisited the three major dynamic control strategies of a parallel robot. First of all, we showed that the Cartesian space control is perfectly suitable for parallel robots. Next, each control scheme was transposed in this control space. Then, the three control schemes are compared through a theoretical analysis of control equations. Simulation and experiments on a particular test-bed were provided to prove our assertions. As a conclusion, a Cartesian space Computed Torque Control offers the best performances even at machining speed, although generally considered as quasi-static and not dynamic. From a theoretical point of view, this result can be generalised to every parallel robot with a closed-form Inverse Kinematic Model. Nevertheless, it assumes to have a competitive measure of the end-effector pose.

Fig. 1 .Fig. 2 .

 12 Fig. 1. Comparison between Cartesian and joint space control

Fig. 5 .Fig. 6 .

 56 Fig. 5. Computed Torque Control in the joint space

Fig. 7 .Fig. 8 .

 78 Fig. 7. Global view of the Isoglide-4 T3R1

Fig. 9 .

 9 Fig. 9. Comparison of the three control schemes on a 100mm square at 3m.s -2

Fig. 10 .

 10 Fig. 10. Straightness measure with an high speed camera and a laser interferometer

Fig. 11 .

 11 Fig. 11. Comparison between laser interferometer and camera

Fig. 12 .

 12 Fig. 12. Comparison between single-axis and CTC controller measured with an high speed camera on a 100mm XY square

TABLE I MEASURE

 I OF THE AVERAGE STRAIGHTNESS ALONG AXIS X, Y AND Z AND ORIENTATION ERROR ALONG AXIS θ FOR THE THREE CONTROL SCHEMES ON A 100mm SQUARE FOR SEVERAL ACCELERATIONS

TABLE II Measured

 II

straightness in µm on square segment with an high speed camera

LASMEA -UMR 6602 CNRS, Blaise Pascal University, Clermont-Ferrand, France

LAMI, IFMA, Aubiere, France

ISRC, Sungkyunkwan University, Suwon, South Korea


David Hurther

Ulrich Lemmin

Koen Branckaert

A field study of transport and mixing in a river using an acoustic Doppler velocity profiler

de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION

The physical-biological-chemical-geological interplay in rivers is strongly affected by transport and mixing processes. Effective resource management and river engineering require a good understanding of these processes. Unfortunately, at present, this is often still based on 1-D section averaged mean parameterization. This situation, however, is no longer satisfactory in order to assure water of high quality in view of conflicting interests:

• the impact of pollutants by ever more toxic substances • the need for re-aeration due to the oxygen demand caused by waste discharge • the variation of bed friction velocity and the renaturalization of river systems • the contaminant run-off from diffusive sources on the watershed • the determination of realistic bed load transport rates for the control of river bed dynamics. Only a limited number of field studies is reported in the literature and results often deviate from simple 1-D or 2-D assumptions widely used today. In particular, the dynamics of turbulence and coherent structures and their distribution across a river section need to be addressed in order to understand why these deviations occur. Coherent structures and secondary currents are transport processes. They act faster than small scale turbulent mixing and produce potentially harmful local maxima and minima in the distribution of pollutants and oxygen. They will influence the dynamics within the water column, the interaction with the bed and the exchange across the air-water interface. Therefore in order to progress in the understanding of river dynamics, field measurements must incorporate the whole spectrum of scales from large scale mean motion to small scale turbulence.

Here we report on a field study in a medium-sized, lowland river with the aim of investigating parameters which control transport and mixing in rivers. They will be interpreted in the framework of existing knowledge on open-channel hydrodynamics in order to determine to what extent rivers, where the Reynolds number is typically higher and where the idealized conditions of laboratory studies seldom exist, may be different from laboratory openchannels.

A field study of transport and mixing in a river using an Acoustic Doppler Velocity Profiler

Koen Blanckaert

LRH, Ecole Polytechnique Fédérale Lausanne, Lausanne, Switzerland ABSTRACT: A 3-D Acoustic Doppler Velocity Profiler (ADVP) is used for measuring closely spaced profiles across the river Poisine (on a straight river reach with mean water depth ≈ 0.5 m, width ≈ 12 m, flat gravel bed with estimated D 50 ≈ 6 cm). The mean streamwise velocity profiles show logarithmic profiles all across the half section, decreasing towards the bank. Most of the three turbulence intensities profiles between the center (B/y=0.5) and B/y=0.24 agree relatively well with laboratory profiles. Closer to the bank (B/y<0.25) strong deviations in the profile form indicate "side-wall" effects. A cumulant discard method is applied and conditional statistics and conditional sampling are used to compare the experimental relative covariance contributions from the four quadrants in all three planes. In the center, all quadrant distributions correspond to those observed in laboratory open-channel flow. Near the bank, significant differences between the results for different depths occur in all planes. Thus, the flow near the bank cannot be considered two-dimensional.

2 BACKGROUND [START_REF] Nikora | Turbulence characteristics of New Zealand gravel-bed rivers[END_REF] demonstrated the presence of a logarithmic longitudinal mean velocity profile and the inertial subrange in the turbulence domain in gravel bed rivers. In an artificial irrigation channel, [START_REF] Nikora | Flow turbulence over fixed and weakly mobile gravel beds[END_REF] took measurements in the center. They observed that the streamwise velocity is usually well represented by a logarithmic profile. They found coherent structures, dominated by ejections and sweeps, over the whole water depth for different discharge conditions. [START_REF] Ferguson | Markov analysis of velocity fluctuations in gravel-bed rivers[END_REF] and [START_REF] Roy | Scales of turbulent coherent flow structures in a gravel-bed river[END_REF] describe flow over rough beds in rivers which have similar characteristics as those which we investigate. They verify the existence of coherent structures in rivers. [START_REF] Smart | Turbulent velocity profiles and boundary shear in grave bed rivers[END_REF] points out that -different from laboratory channel studies -in gravel bed rivers the depth, the surface slope and the bed roughness vary across the channel. In such cases, local values of bed shear stress are important because the sum of transport rates across the section may be considerably different from the transport rate calculated with section averaged parameters. [START_REF] Powell | Hydraulic interpretation of cross-stream variations in bed-load transport[END_REF] have shown that in coarse grained rivers on the average, transport rates in the center were about twice those recorded at the margins. These local differences in bedload transport are a direct indication of the local differences in bed-shear stress. Measurements of coherent structures in open channels and rivers have not been able to resolve the very near surface features of these structures because of the difficulty of tracking fine scales so close to a mobile interface. [START_REF] Pan | A numerical study of free surface turbulence in channel flow[END_REF] have carried out a direct numerical simulation for this region. They found that surface boils are linked to coherent structures in the water column below. The impinging coherent structures and other vortices attach themselves to the free surface for a limited time creating "patches," below which lie greatly elevated vorticities. [START_REF] Moog | Air-water gas transfer in uniform channel flow[END_REF] have studied the effect of surface boils on air-water gas exchange and have found that burst motions can transport turbulent energy to the free surface, creating active zones for gas transfer in which embedded small scale eddies may renew the surface many times, enhancing the gas transfer. Surface boils therefore provide for river reaeration and affect water quality management concepts in the context of river recovery from biochemical oxygen demand created by waste discharge.

FIELD STUDY

We have carried out several field studies in small turbulent rivers under fluvial conditions on the lower Swiss plateau. The results from the river Poisine (on a straight river reach with mean water depth ≈ 0.6 m, width ≈ 12 m, gravel bed with estimated D 50 ≈ 6 cm) which are presented here are typical for our findings in other rivers. The study reach has been "corrected" and resembles closely a wide laboratory channel. The shorelines are straight and the bottom is flat with very subdued bed topography. The roughness height is small compared to the water depth and with a minimum of scour holes. In such sections, form drag is commonly assumed to be minimal [START_REF] Prestegaard | Variables influencing water surface slopes in gravel-bed streams at bankful stage[END_REF]. In this study, we have used the Acoustic Doppler Velocity Profiler (ADVP) which we have developed and which is capable of taking instantaneous velocity profiles of all three velocity components with a resolution of turbulence scales. The parameters we have investigated are the cross section distributions of: a) the mean values of the three velocity components, b) the velocity variances, c) the shear stresses and d) the dynamics of coherent structures and their relation to bottom shear stress. Measurements were carried out at a large number of verticals across the cross-section with the ADVP system suspended from a portable bridge placed above the water surface. The distance of the ADVP from a shore reference point is marked and repeated measurements with the same setting were made. At each point, the ADVP is first optimized for the particular conditions and 3-D instantaneous profile measurements are then carried out for about 5 min. at each point. The instrument can detect the river bottom and the instantaneous water depth is therefore known at all times. Here, we briefly resume the characteristics of the focalized 3-D acoustic Doppler velocity profiler. Details are documented in the literature (Roland & Lemmin, 1998;[START_REF] Hurther | A constant beam width transducer for 3-D acoustic Doppler profile measurements in open channels[END_REF]. The instrument works at an acoustic frequency of 1 MHz. Profiles are obtained by subdividing the water depth into a number of consecutive equal-sized range bins. The bin location is determined by the time of flight with respect to the acoustic transducer. The instrument is capable of resolving the 3-D velocity vector at up to 256 points in each profile simultaneously with a time resolution of normally 15 Hz, but at least 5 Hz. The actual resolution depends to some extent on the river flow conditions. However, the resolution we can obtain is always satisfactory for the study of turbulence under the conditions specified by [START_REF] Nezu | Turbulence in open-channel flows[END_REF]eq. 3.1). Coherent structures and secondary currents occur at a much lower fre-quency and are therefore well resolved. The instantaneous profiling capability of the ADVP is an important feature in river studies. A great number of vertical profiles across the river section are needed for sufficient resolution. With single point instruments this can never be realized within a reasonable time (one day). Measurement campaigns which need longer time periods are logistically impractical and risk significant variations in river discharge.

DATA ANALYSIS

The data analysis addresses the objectives indicated above. The basis for the analysis is a set of computer programs which we have developed over the years for the treatment of ADVP data and in particular for the measurements in rivers. They allow us to test for all profile laws in mean, turbulence and shear stress distributions related to the logarithmic velocity profile.

For the analysis of coherent structures we use the technique which Nakagava and [START_REF] Nakagawa | Prediction of the contribution to the Reynolds stress from bursting events in openchannel flow[END_REF] developed. The equations for the relative covariance term u'w' are conditionally sampled over the four quadrants of the (u'w') plane. Events in quadrant one to four are identified as outward interaction, ejection, inward interaction and sweep respectively. A third-order Gram-Charlier distribution was found sufficient to quantify the quadrant covariance fraction and time fraction over most of the water column under laboratory conditions [START_REF] Hurther | Shear stress statistics and wall similarity analysis in turbulent boundary layers using a high resolution 3-D ADVP[END_REF]. Experimental results of fractional contribution to the corresponding shear from each quadrant will be compared to results from a cumulant discard Gramm-Charlier distribution applied to the u'w' term. This analysis is designed to determine whether and in what way rivers may deviate from laboratory open-channels and the importance of coherent structures.

RESULTS

Mean profiles

Above 0.2 h, the mean streamwise velocity profiles (Fig. 1) show a structure which is not far from a logarithmic profile all across the half section. In this layer the mean velocity decreases continuously from the center towards the bank. Below 0.2 h there are local deviations which are sometimes important, particularly in the near bank zone. From our observations of the bed roughness, they most likely result from form roughness elements. In the near bank zone, however, the deviation to almost no flow in this lower layer is more systematic and seems to indicate that the flow in this zone is not two-dimensional.

The mean profiles of the three relative turbulence intensities are compared to the profiles seen in the center of a uniform, laboratory open-channel flow (Fig. 2). In most of the profiles between the center (B/y=0.5) and B/y=0.24 the intensities agree relatively well with those from the laboratory study, particularly for the transversal component. The streamwise component is slightly larger, while the vertical component is always larger. Stronger deviations are seen below 0.2h where often no clear profile form can be established. Nevertheless, one can assume that in this part of the cross-section the Mean profiles of the Reynolds stresses and the turbulence intensities are shown in figure 3. A behaviour similar to that of the turbulence intensities is seen in that above 0.2h and for y/B>0.25 the values obtained from the river follow the curves from the laboratory but the intensities are often slightly higher, particularly for the turbulent kinetic energy. The strongest deviations are once more observed below 0.2h. In the near bank zone we find again greater deviations over the whole water column. In this zone, the shear stress term u'v' becomes quite large and positive above 0.2h.

Coherent structures

Similar to our laboratory studies, a cumulant discard method is applied to describe the statistical properties of the covariance term ′ ′ u w along the mean flow relative to their time means. Conditional statistics and conditional sampling are used to compare the experimental relative covariance contributions from the four quadrants in the streamwise plane. Results for the vertical at B/h= 0.5 in the center of the river are compared to those at B/h=0.05 close to the bank in figure 4. At each vertical, results for the same three relative depth at mid-depth are presented. The change from the center to the bank is striking. In the center, the quadrant distribution corresponds to the one observed in the laboratory open-channel flow [START_REF] Hurther | Shear stress statistics and wall similarity analysis in turbulent boundary layers using a high resolution 3-D ADVP[END_REF] with quadrants two and four clearly dominating along the mean flow ′ ′ u w . Thus, ejections (quadrant two) and sweeps (quadrant four) are the dominant structures. There is only a slight difference between the curves from the different depths. However, the sweep curves fall off faster than in the laboratory channel with respect to the threshold level H. Increasing the level of H allows to select progressively stronger fractional covariance events. Contributions are already nearly negligible for values of H=10. This level is reached in the laboratory study for H=20. The opposite trend is seen for the ejections. In the laboratory study values had fallen to less than 0.05 for H=20 while they remain at about 0.1 or above in the river. As was observed in the laboratory, there is no preferential quadrant orientation in the cross flow direction ′ ′ v w (not shown here). As in the laboratory study, values taper off to below 0.1 only for very high values of H around 90. This means that coherent structures are aligned in the streamwise direction. Near the bank, sweeps (quadrant four) show an intensity and a distribution which is very similar to that in the center of the river. This means that downward oriented structures behave similarly all across the river. Significant differences are seen for the ejections (quadrant two) where the distribution is strongly modified from that observed in the center. In quadrant two and also mirrored but at smaller amplitude in quadrant three, curves do not taper off with increasing selection criterion H. Instead, they remain constant over a wide range of H. Thus, there is no balance between the ejections and the sweeps in the near bank zone. Furthermore, the behaviour at the three selected depths is quite different. The distribution between the quadrants remains balanced for the ′ ′ v w plane (not shown). Thus, as was already seen for the mean profiles of turbulence and Reynolds stress above, it is obvious that the flow near the bank cannot be considered two-dimensional. This is further supported by the decrease in strength for the terms in the u v ©© plane. They approach the same level as those in the ′ ′ u w plane indicating that this shear term becomes important in the near bank area.

SUMMARY AND CONCLUSIONS

The overall objective of this study was to provide a first insight into transport and mixing dynamics in medium scale rivers, where at present little is known due to the difficulty in measuring the relevant parameters with sufficient resolution and precision. In order to attain this objective we have carried out field measurements in rivers using our Acoustic Doppler Velocity Profiler. By measuring instantaneous three-dimensional profiles at a large number of verticals across river sections, we have investigated the mean field and the fluctuating field of all three velocity components. Our still limited set of river field studies either indicates or suggests that compared to open channel laboratory studies:

• vertical mean value distributions of streamwise velocity are similar in both cases. • the vertical distribution of the turbulence parameters changes strongly across a section. • the distribution of parameters such as bed friction velocity may vary across a section. • coherent structures are present in rivers and may affect mixing and transport processes. At the same time these measurements indicate that different from laboratory flow:

• the mean profile structure across a river section does not support the concept of 2-D flow. • the dynamics of coherent structures change across the cross-section.

There are still questions which need to be addressed in order to understand the dynamics of river flow:

• Can the deviations be explained by a change in bottom roughness alone? • What is the effect of secondary currents; are they locally or globally important enough to rule out the application of 2-D concepts?

From the high scatter of all data below 0.2h it is obvious that it will be rather difficult to obtain reliable bottom shear velocity information by classical methods of profile extension. This is not surprising because most laboratory studies for which this concept was developed were carried out with rather uniform roughness throughout the channel. The bottom roughness in rivers varies over a rather wide range. We have recently used our ADVP measurements in laboratory open channel flow to show the universality of the concept of wall similarity [START_REF] Hurther | Shear stress statistics and wall similarity analysis in turbulent boundary layers using a high resolution 3-D ADVP[END_REF]. We will also test it for the river condition because it may provide a new way to determine the effective bottom roughness from the TKE balance in midwater depth. Classical concepts of transport and mixing reach their limits, when as is often the case today, the fine-scale dynamics are required. This is particularly true in the bottom boundary layer of the river where pollutants and waste material come in contact with sediment. They risk affecting the oxygen budget in the near bottom layer. This is the zone in which biological activities are predominant and which may suffer severe damage as a consequence.

The dominant time and length scales of the turbulent flow and the residence time of the pollutant in that layer need to be determined. This problem also relates to sediment resuspension and sediment transport. We have recently shown that using an extension of the ADVP instrument, the particle flux can be determined by measuring particle concentration at the same time. These measurements can give new insight into particle transport [START_REF] Hurther | Discussion of "Equilibrium near bed concentration of suspended sediment" by Cao[END_REF]. It is expected that the information which will be obtained from detailed measurements in rivers will contribute to the improvement of numerical predictive models of pollutant mixing, which in turn will improve effective water resource management. We will continue these measurements and their analysis and extend them into other rivers in order to give more statistical significance to the first results we have provided here.

ACKNOWLEDGEMENT

This work was funded by the Swiss National Science Foundation under Grant 20-39494.93. We are grateful for the support.

Figure 1 .

 1 Figure 1. Cross-section distribution of the mean streamwise velocity profiles in the river Poisine.

Figure 2 .

 2 Figure 2. Cross-section distribution of the relative turbulence intensity profiles in the river Poisine.

Figure 3 .

 3 Figure 3. Cross-section distribution of the Reynolds stress and turbulent energy profiles in the river Poisine.

Figure 4 .

 4 Figure 4. Quadrant distribution of shear stresses at the center (left) and the bank (right; y/B = 0.05) of the river Poisine. The horizontal axis is the selection criterion H.