Mohamed Sorour

Andrea Cherubini

Philippe Fraisse

Robin Passama

Motion Discontinuity-Robust Controller for Steerable Mobile Robots

Keywords: Steerable Wheeled Mobile Robots, Pseudoomni Mobile Robots, Nonholonomic Omnidirectional Wheeled Mobile Robots, Kinematic Control

Steerable wheeled mobile robots (SWMR) are able to perform arbitrary 3D planar trajectories, only after initializing the steer joint vector to the proper values. These robots employ fully steerable conventional wheels. Hence, they have higher load carrying capacity than their holonomic counterparts, and as such are preferable for industrial applications. Industrial setups nowadays are being prepared for the emerging field of human-robot collaboration/cooperation. Such field is highly dynamic, due to fast moving human workers, sharing the operation space. This imposes the need for human safe trajectory generators, that can lead to frequent halts in motion, replanning, and to sudden, discontinuous changes in the position of the robot's instantaneous center of rotation (ICR). Indeed, this requires steer joint reconfiguration to the newly computed trajectory. This issue is almost ignored in the literature, and motivates this work. The authors propose a new ICR-based kinematic controller, that is capable of handling discontinuity in commanded velocity, while respecting the maximum joint performance limit. This is done by formulating a quadratic optimization problem with linear constraints in the 2D ICR space. The controller is also robust against representation and kinematic singularities. It has been tested successfully on the Neobotix-MPO700 industrial mobile robot.

I. INTRODUCTION

Steerable wheeled mobile robots gain mobility by employing fully steerable wheels, having two active joints, one for steering, and another for driving. Despite having only one degree of mobility (DOM) (defined here as the instantaneously accessible degrees of freedom DOF), corresponding to the rotation about the instantaneous center of rotation (ICR), such robots can perform complex 3D planar trajectories. This can be done only after orienting the wheels to the proper angles, depending on the trajectory to be performed. Fully omnidirectional robots, on the other hand, employing either castor, Swedish or omni wheels, have 3DOM and, as such, can instantaneously track any desired trajectory. However, this type of wheels has limited load carrying capacity, and may cause vibration during robot motion. For these reasons, SWMR are the preferred choice for industrial setups.

Due to their wide use in the industry, many research efforts have been made to enhance their performance, against kinematic (ICR at the steering joint axis) and representation (from the mathematical model) singularities. The latter have been solved, in the case of SWMR with three or more wheels, by using the 3D Cartesian velocity in deriving the kinematic model [START_REF] Giordano | On the kinematic modeling and control of a mobile platform equipped with steering wheels and movable legs[END_REF]. The former was addressed as a constraint to robot motion in [START_REF] Thuilot | Modeling and feedback control of mobile robots equipped with several steering wheels[END_REF], or by using repulsive potential fields in [START_REF] Dietrich | Singularity avoidance for nonholonomic, omnidirectional wheeled mobile platforms with variable footprint[END_REF], [START_REF] Connette | Singularity avoidance for over-actuated, pseudo-omnidirectional, wheeled mobile robots[END_REF], [START_REF] Schwesinger | A novel approach for steering wheel synchronization with velocity/acceleration limits and mechanical constraints[END_REF]. However, reduction in robot maneuverability, by 1 Interactive Digital Human group IDH, Laboratory for Computer Science , Micro-electronics and Robotics LIRMM, University of Montpellier CNRS, 860 rue Saint Priest, 34090 Montpellier, France.

firstname.lastname@lirmm.fr 2 PSA Peugeot-Citroen, Velizy Villacoublay, France. employing these methods, is not acceptable, and attempts were made to deal with kinematic singularity by handling the maximum joint limits [START_REF] Oftadeh | A novel time optimal path following controller with bounded velocities for mobile robots with independently steerable wheels[END_REF] and by changing coordinate system [START_REF] Connette | Singularity-free state-space representation for non-holonomic, omnidirectional undercarriages by means of coordinate switching[END_REF]. Steering mechanical limits were also studied in [START_REF] Schwesinger | A novel approach for steering wheel synchronization with velocity/acceleration limits and mechanical constraints[END_REF] and [START_REF] Chamberland | Motion planning for an omnidirectional robot with steering constraints[END_REF]. Simulations in [START_REF] Connette | Singularity-free state-space representation for non-holonomic, omnidirectional undercarriages by means of coordinate switching[END_REF] do not indicate the behavior of the steer joints in terms of respecting the velocity limits at kinematic singularity, while in [START_REF] Oftadeh | A novel time optimal path following controller with bounded velocities for mobile robots with independently steerable wheels[END_REF], it is shown that the steer rate will saturate and keep operating at the maximum limit.

On the other hand, to the best of the authors knowledge, no thorough investigation has been conducted on the issue of reorienting the wheels, once discontinuity in the robot trajectory occurs. Usually, steer reconfiguration is performed in a manual fashion, depending on the test trajectory. This is found in the literature under various designations, e.g. "initial phase" in [START_REF] Giordano | On the kinematic modeling and control of a mobile platform equipped with steering wheels and movable legs[END_REF], and "open-loop starting procedure" in [START_REF] Thuilot | Modeling and feedback control of mobile robots equipped with several steering wheels[END_REF]. Although an ICR-based controller is the most suited to handle such cases, the work in [START_REF] Oftadeh | A novel time optimal path following controller with bounded velocities for mobile robots with independently steerable wheels[END_REF] and [START_REF] Oftadeh | Time optimal path following with bounded velocities and accelerations for mobile robots with independently steerable wheels[END_REF] is limited by the assumption of continuous and differentiable desired signal, whereas in [START_REF] Schwesinger | A novel approach for steering wheel synchronization with velocity/acceleration limits and mechanical constraints[END_REF] the singularities imposed by the ICR-based model are handled by reducing the robot maneuverability.

Such situation -discontinuity in robot motion -is more likely to happen nowadays, in the emerging field of humanrobot collaboration. Mobile robots working in the vicinity of fast moving human workers, will usually encounter discontinuity in the online computed trajectory. In case of static obstacles, the online planner can be less restricted in the form of the output trajectory, so that smooth behavior can always be expected. Instead, here, sudden appearance of an active worker can result in prohibiting motion, and re-routing to follow other trajectories. In such cases, the state of the art will fail to provide proper solutions.

In this work, we propose a kinematic control framework that is:

1) robust against trajectory discontinuity, 2) capable of handling kinematic singularity, 3) compliant with the maximum steer joint limits in terms of velocity and acceleration (or jerk, seamlessly). The framework consists of two decoupled kinematic controllers: a Cartesian-velocity based controller, and an ICRbased one. The former is used to control the drive rate "wheel speed", employing a Cartesian space kinematic model. The latter controls the steering rate, while respecting the maximum steer joint limits, by using optimization to locate the "next sample time" ICR coordinates. The developed 3D Cartesian space kinematic model is free from representational singularity, while the kinematic singularities are being handled in the 2D ICR space controller. The benefit of using separate kinematic controllers for the drive and steering rates, is does not require mapping the 2D ICR-coordinate space to the 3D Cartesian space, hence avoids associated singularities and inconveniences. Thanks to the formulated optimization problem, discontinuity in robot velocity trajectory is handled, while respecting the steer joint limits.

In the following, section II presents the Cartesian space kinematic model. The velocity discontinuity robust controller is detailed in section III. Experiments are depicted in section IV. Conclusions are finally given in section V.

II. KINEMATIC MODEL

The kinematic model presented in this section (detailed in previous work by the authors [START_REF] Sorour | Kinematic modeling and singularity treatment of steerable wheeled mobile robots with joint acceleration limits[END_REF]) is inspired by the pioneer work of Muir [START_REF] Muir | Kinematic modeling of wheeled mobile robots[END_REF], Campion [START_REF] Campion | Structural properties and classification of kinematic and dynamic models of wheeled mobile robots[END_REF], Betourne [START_REF] Betourne | Kinematic modelling of a class of omnidirectional mobile robots[END_REF] and Low et al. [START_REF] Low | Kinematic modeling, mobility analysis and design of wheeled mobile robots[END_REF]. The schematic of a SWMR is shown in Fig. 1 for a 4 wheeled robot. However, the model is generic for SWMR with number of wheels N ≥ 3.

A. Cartesian Space 3D Model Formulation

Let F I = (o I , x I , y I , z I ) be the inertial frame, F b = (o b , x b , y b , z b ) the mobile base frame, with origin o b located at the base geometric center, F hi = (o hi , x hi , y hi , z hi ) the i th hip frame (i = 1, . . . , N ), attached to the fixed part of the steering joint, and related to the base frame by a fixed transformation matrix, and F si = (o si , x si , y si , z si ) the steering frame, attached to the movable part. The hip and steering frames share the same origin, with relative orientation β i (the steering angle). Frame F wi = (o wi , x wi , y wi , z wi ) is attached to (but not rotating with) the i th wheel, assigned such that x wi points along the heading of the wheel, which rotates about y wi by the driving angle φ i . All the frames have the z axis pointing upwards. Let the mobile base pose w.r.t. the inertial frame define the 3D task space coordinates ξ = x y θ T . A left superscript is added to indicate the frame in which the pose is expressed, for instance I ξ and b ξ denote the robot pose, expressed in the inertial and base frames respectively. In the sequel, the left superscript is omitted in case of vectors expressed in the base frame to lighten the notation, unless otherwise specified.

Considering the i th wheel velocity v ci = v ti v ni 0 T at the ground contact point o ci (expressed in the wheel frame F wi ), with v ti and v ni respectively the i th tangential and normal velocities, it can be shown that [START_REF] Sorour | Kinematic modeling and singularity treatment of steerable wheeled mobile robots with joint acceleration limits[END_REF]:

v ti = f (β i ) b ξ + d βi -r w φi , (1)
v ni = g(β i ) b ξ, (2)
f (β i ) = c(β i ) s(β i ) d -h yi c(β i ) + h xi s(β i ) , g(β i ) = -s(β i ) c(β i ) h xi c(β i ) + h yi s(β i ) ,
where h xi = ±b and h yi = ±a denote the position of the i th hip frame origin o hi in the base frame, c( * ) and s( * ) short hand cos( * ) and sin( * ) respectively. The parameters d and r w are the steerable wheel offset and radius respectively. Setting v ti = 0 and v ni = 0, (1) and (2) represent the rolling with no slipping and the no lateral skidding kinematic constraints, respectively. The no skidding constraint imposes restrictions on the robot motion (the wheel cannot move sideways), and forces the existence of a unique ICR point, around which the base frame and all wheels must rotate. From such constraint, we construct the kinematic constraint matrix G(β) as: Equation ( 2) is differentiated w.r.t time, rearranged to evaluate the required steer joint rate when v ni = 0 as:

G(β) N ×3 = g(β 1 ) T ... g(β N ) T T . (3)
βi = -g(β i ) ξ dg(βi) d(βi) ξ . (4)
Then, the wheel drive rate is obtained from (1), when v ti = 0:

φi = 1 r w f (β i ) ξ + d r w βi . (5)
Equations ( 4) and ( 5) represent the SWMR inverse actuation kinematic model (IAKM).

B. Odometry Model

In order to compute the task space velocity response from the joint space velocity measurements, we need the odometry model, or the forward actuation kinematic model (FAKM):

ξ = F + (d) ( β)(r w φ -d β), (6)
F ( β) N ×3 = f ( β1 ) T ... f ( βN ) T T ,
where the hat symbol ˆindicates a measurement value and F + (d) ( * ) denotes the damped pseudo-inverse of F ( * ), evaluated using [START_REF] Nakamura | Inverse kinematic solutions with singularity robustness for robot manipulator control[END_REF] and [START_REF] Wampler | Manipulator inverse kinematic solutions based on vector formulations and damped least-squares methods[END_REF]:

F + (d) ( * ) = (F T ( * )F ( * ) + δ 2 I 3×3 ) -1 F T ( * ),
with δ ∈ R the damping factor. We use the damped pseudo-inverse to overcome algorithmic singularities, occurring when the mobile base moves with null angular velocity (all steer angles identical), as in such case, F ( β) loses rank.

III. DISCONTINUITY ROBUST ICR-POINT CONTROLLER

In this section, we present our control framework for injecting the steer joint initialization in the trajectory performed by SWMR, and for handling run-time trajectory discontinuities, which "depending on the application" can be often necessary, especially in human robot interaction tasks.

The whole framework is depicted in Fig. 2. The desired 3D Cartesian space robot motion ( ξ * , and ξ * ) is generated by a high level perception controller (out of the scope of this work). This is then mapped to the 2D ICR space, and the output desired ICR motion (ICR * , and İCR * ) is fed to the ICR velocity controller, along with the current ICR coordinates ICR curr . The output reference signal ICR ref is then used by an optimization algorithm to decide the "next sample time" ICR coordinates ICR next that will minimize the quadratic cost error:

ICR ref -ICR next 2 2
while respecting the joint performance limits formulated as linear constraints. We use the ICR ref rather than ICR * in the cost function in order to obtain smooth behavior since it is error dependant. The corresponding steer joint reference signal β ref is then evaluated (while fixing the numeric issues involved) and differentiated, to obtain the βref that is sent to the robot low level controller. At the same time, a decoupled robot twist controller is implemented, the initial output of which ξref(init) is projected onto the null space of the "next sample time" robot configuration (represented by the kinematic constraint matrix G(β ref )), to obtain the feasible control signal ξref . The reference wheel rate φref is then obtained using the IAKM in [START_REF] Schwesinger | A novel approach for steering wheel synchronization with velocity/acceleration limits and mechanical constraints[END_REF]. The colored blocks in Fig.

(2) are detailed in each of the following subsections.

A. ICR Velocity Controller

In Fig. 2, the desired base frame motion is the output of the high level perception block. As shown in Fig. 3, the corresponding desired ICR coordinates, expressed in base frame, ICR * = X * Y * T , are usually evaluated using:

X * = -ẏ * / θ * , Y * = ẋ * / θ * .
However, this formula is singular in pure translation motion.

Alternatively, here we propose to use:

X * = R ∞ tanh -ẏ * θ * + sign( θ * )δ 1 /R ∞ , Y * = R ∞ tanh ẋ * θ * + sign( θ * )δ 1 /R ∞ , (7)
sign( θ * ) = +1, ∀ θ * ≥ 0 -1, ∀ θ * < 0 ,
with R ∞ a large positive scalar representing the radius of curvature when the ICR point is very far. R ∞ should be chosen carefully, to avoid numerical instability. On the other hand, δ 1 is an infinitesimally small positive scalar value. Equation ( 7) is both singularity free, and provides bounded ICR * components thanks to the tanh() function. Similarly, the desired ICR velocity, İCR * can be computed using:

Ẋ * = Vmax tanh -ÿ * θ * + ẏ * θ * ( θ * ) 2 + δ 1 / Vmax , Ẏ * = Vmax tanh ẍ * θ * -ẋ * θ * ( θ * ) 2 + δ 1 / Vmax , (8)
where Vmax limits İCR * . On the other hand, the current ICR coordinates ICR curr = X curr Y curr T are computed using the current "measured" steer joint configuration, β. We assume that this configuration corresponds to a unique valid ICR point, i.e., that the low level robot controller properly tracks the reference steer rate βref , and as such the measured steer joint values correspond to a "unique" ICR curr . We use the pair of steer joints with the biggest angular difference as depicted in Fig. 3 for numeric robustness, whose indexes i, j ∈ {1, . . . , N } are given by:

[i, j] =        max i,j βi -βj , ∀ βi -βj ≤ π/2 max i,j βi -βj -π , ∀ βi -βj > π/2
, then,

Y curr =    R ∞ c( βi ), ∀ βi < β th hxi-hxj +hyit( βi)-hyj t( βj ) t( βi)-t( βj )
, otherwise ,

X curr = R ∞ s( βi ), ∀ βi < β th h xi -(Y curr -h yi )t( βi ), otherwise , (9)
where β th is a small threshold value, and t( * ) short-hands the tan( * ) function. The result of ( 9) is saturated to ±R ∞ to obtain X curr and Y curr . Finally, the ICR-Point Velocity Controller is formulated as:

İCR ref = İCR * + λICR err , ICR err = ICR * -ICR curr , ( 10
)
where λ is a positive scalar proportional gain, and ICR err is the error between the desired and current ICR coordinates.

B. Kinematic Singularity Treatment

Kinematic singularity occurs whenever the ICR point approaches any of the steering axes (where Y ref ≈ h yi , and X ref ≈ h xi ), since evaluating the singular steer angle using:

β i = arctan 2 Y ref -h yi , X ref -h xi -π/2,
will result in undefined value. In such case, the steering rate grows unbounded. This is handled in the literature either by constraining the robot velocity space [START_REF] Thuilot | Modeling and feedback control of mobile robots equipped with several steering wheels[END_REF], [START_REF] Dietrich | Singularity avoidance for nonholonomic, omnidirectional wheeled mobile platforms with variable footprint[END_REF], [START_REF] Connette | Singularity avoidance for over-actuated, pseudo-omnidirectional, wheeled mobile robots[END_REF], [START_REF] Schwesinger | A novel approach for steering wheel synchronization with velocity/acceleration limits and mechanical constraints[END_REF], so that the ICR never passes by any steering axis, or by saturating the steer rate at singularity [START_REF] Oftadeh | A novel time optimal path following controller with bounded velocities for mobile robots with independently steerable wheels[END_REF]. In previous work, we developed a method to dampen such effect: the steer rate slows down, and is zeroed, as the ICR approaches and then reaches the steer axis [START_REF] Sorour | Kinematic modeling and singularity treatment of steerable wheeled mobile robots with joint acceleration limits[END_REF]. However, it cannot be applied here, as we do steer joint control in the 2D ICR space.

Instead, here, we construct a singularity zone (circle with fixed radius R zone ) around each steering axis (the 2 nd steer joint zone in Fig. 4). If the ICR enters the circle, ICR ref is set on the opposite side, along the straight line normal to the singular wheel. Thanks to this approach, the motion of the singular joint is minimal. The ICR ref is modified as:

X ref = X curr + 2R zone cos(β s + π/2), Y ref = Y curr + 2R zone sin(β s + π/2), (11)
with R zone , the zone radius, and β s , the singular steer angle.

C. Optimization to obtain a feasible ICR

The reference control signal ICR ref obtained from ( 10) is error driven. Whenever trajectory discontinuity occurs, or when sending the initial motion commands where the error is maximum, excessive joint velocity/acceleration is required. This corresponds to an ICR reference point that is far from the current one. In this section, a quadratic programming optimization is formulated to determine the best feasible "next sample time" ICR coordinates ICR next among a feasible set. Such set is constructed by formulating the maximum and minimum "next sample time" steer angles as linear constraints depending on the current steer joint state and on its maximum performance (here, velocity and acceleration) limits. Then, ICR next will replace ICR ref in computing the steer joint commands.

Fig. 5 shows the feasible change, in one sample time period t s , in each of the steer joints (in positive and negative directions) based on: 1. the ICR curr , and 2. the maximum steer joint velocity and acceleration. Indeed, the feasible set of ICR next is within the intersection of the extreme feasible changes of all steer joints. The feasible region for ICR next shown in Fig. 5.a corresponds to a stationary ICR curr , which can move arbitrarily in any direction within the indicated feasible set. Instead, Fig. 5.b depicts the case of moving ICR curr (i.e., İCR curr = 0). In such case, the maximum steering deceleration constraints (shown in Fig. 5.b as β i(t+ts)min+ and β i(t+ts)min-for minimum change in steer angle in positive and negative directions, respectively) will apply, to further restrict the feasible set, dividing it into four regions based on the current steer rate direction. Figure 5.c shows a particular case of Fig. 5.b where β2 > 0 and β3 > 0. Also, the case of discontinuous change in the desired motion trajectory is depicted, where a new reference ICR point ICR ref appeared while the ICR curr is following an old one ICR ref (old) . Thanks to this approach, discontinuity in velocity command can be handled seamlessly with the same formulation, and no steer joint reconfiguration is needed.

All cases can be addressed by formulating a quadratic optimization problem, subject to linear constraints, to minimize the error between the ICR reference signal ICR ref and the best feasible ICR at the next sample time

ICR next = X next Y next
T , being also the decision variable:

minimize ICRnext ICR ref -ICR next 2 2
subject to (-1)

q i(min) A i(min) (ICR next -h i ) ≥ 0, (-1) q i(max) +1 A i(max) (ICR next -h i ) ≤ 0,
where * 2 2 is the squared Euclidean norm, A i(max/min) = cot(β i(max/min) ) 1 , h i = h xi h yi T , and q i(max/min) is a parameter indicating the β i(max/min) quadrature: The ICR ref is obtained from ( 10) by numeric integration, or from [START_REF] Muir | Kinematic modeling of wheeled mobile robots[END_REF] in case of kinematic singularity. Using ICR ref instead of ICR * provides smooth damped behavior, since its computation depends on ICR err (refer again to [START_REF] Sorour | Kinematic modeling and singularity treatment of steerable wheeled mobile robots with joint acceleration limits[END_REF]).

q i(max/min) = 0, if β i(max/min) ∈ 1 st ∨ 4 th quadrant, 1, if β i(max/min) ∈ 2 nd ∨ 3 rd quadrant.
The constraints are the straight lines of slopes β i(max) and β i(min) :

βi(max)(ref) = βi + βmax * t s , βi(max)(ref) < βmax βmax , otherwise , βi(min)(ref) = βi -βmax * t s , βi(min)(ref) > -βmax -βmax , otherwise , β i(max/min) = β i + β(max/min)(ref) * t s . ( 12
)
Finally, using the optimal ICR next , the next best steer angles are computed:

β i(next) = arctan 2 Y next -h yi , X next -h xi - π 2 . (13)

D. Fixing Numeric Issues

Two numeric issues can be observed, when employing the previously described controller. The first originates from the arctan 2() function in [START_REF] Betourne | Kinematic modelling of a class of omnidirectional mobile robots[END_REF] whenever β i(next) moves between 4 th and 1 st , or between 2 nd and 3 rd quadrants. This results in 2π jumps in the computed angle. The second occurs when the ICR passes by kinematic singularity: moving on a straight line passing by a steer joint axis, and then moving from one side of that axis to the other, will require a π jump in β i(next) . This problem has been avoided in our previous work [START_REF] Sorour | Kinematic modeling and singularity treatment of steerable wheeled mobile robots with joint acceleration limits[END_REF], by employing the steer joint rate. However, a different method is necessary here, since we compute the joint angle rather than its rate in [START_REF] Betourne | Kinematic modelling of a class of omnidirectional mobile robots[END_REF]. To handle these two issues, we use ∆β i = β i(next) -β i to detect and fix the jumps:

βi(ref) =      (∆β i + lπ)/t s , ∆β i < -lπ + 2 βmax t s (∆β i -lπ)/t s , ∆β i > lπ -2 βmax t s ∆β i /t s , otherwise, (14)
where l ∈ {1, 2} depending on the case being handled, and βi(ref) is the joint-space steering reference/command signal.

E. The Driving Rate

Computing the wheel drive rate φi(ref) from ( 5), requires the reference velocity vector ξref equivalent to ICR next . However, this is not feasible, since one information is missing when mapping from the 2D space of ICR next to the 3D space of ξref . To avoid such inconvenience, we propose a separate robot Space Velocity Controller, to compute the driving rate. First, a simple proportional controller is used to set the initial value of the reference signal ξref(init) :

ξref(init) = K p ( ξ * -ξ), (15)
with K p a positive scalar gain, and ξ * the desired robot velocity output by the task controller. The corresponding ξref(init) (obtained by numeric integration of ( 15)) may be incompatible with the "next sample time" robot configuration obtained by substituting β ref (obtained by integrating ( 14)) in the kinematic constraint matrix [START_REF] Dietrich | Singularity avoidance for nonholonomic, omnidirectional wheeled mobile platforms with variable footprint[END_REF], where

G ref = G(β ref ).
Hence, ξref is obtained by projecting ξref(init) onto the null-space of G ref :

ξref = (I -G + ref (d) G ref ) ξref(init) . (16)
Substituting by ξref in (5), we finally obtain φi(ref) .

IV. EXPERIMENTS In this section, the benchmark test trajectory used to evaluate the proposed discontinuity robust controller is detailed and validated experimentally on the industrial mobile robot Neobotix MPO700 (shown in Fig. 6). All parameters used in the experiments are provided in Table I. A video of the experiment can be found on the IDH YouTube channel1

A. Benchmark trajectory

The benchmark trajectory proposed in [START_REF] Sorour | Kinematic modeling and singularity treatment of steerable wheeled mobile robots with joint acceleration limits[END_REF], depicted in Fig. 7, will be employed to test the robustness of the proposed controller against both representation and kinematic singularities. It consists of 5 distinct velocity trajectories, with connections not tailored. Hence, the performance against commanded velocity discontinuity can also be verified. In the following, we detail each of the 5 velocity trajectories (the desired acceleration is the derivative of the provided velocity). 1) Parabolic ICR position profile: This test excites steer joint motion in the vicinity of kinematic singularity, to check if it will respect the joint limits. A parabolic ICR-point motion with vertex at one of the steering axes is used, and implemented smoothly using a constant velocity profile:

   ẋ * = 0.5 * (2 ẏ + h x2 ) 2 + h y2 ẏ * = T c4b ( ẏi = 0, ẏf = -h x2 , t i = 2, t f = 6, t) θ * = 0.5. (17)
Here, T c4b is a 4 th order trajectory, taking initial velocity ẏi , final velocity ẏf , initial time t i , final time t f , and current time t as arguments:

T c4b =        a0 + a1δt1 + a2δt 2 1 + a3δt 3 1 + a4δt 4 1 , ti ≤ t < t1 a5, t1 ≤ t < t2 a0 + a1δt3 + a2δt 2 3 + a3δt 3 3 + a4δt 4 3 , t2 ≤ t < t f .
Coefficients a 0 ... a 5 , are computed by setting the initial and final acceleration/jerk (boundary conditions) to zero, δt 1 = t -t i , δt 2 = t -t 1 , and δt 3 = t -t 2 , while t 1 = t i + 0.1 * (t f -t i ), and t 2 = t f -0.1 * (t f -t i ). Before employing the velocity profile in (17), the following velocity command is applied, to guarantee that the parabolic ICR profile starts at the correct initial condition:

ξ * = 0.5 * h 2 x2 + h y2 0 1 T , 0 ≤ t < 2.
2) ICR point at kinematic singularity: This test reveals the behavior of the steer joint exactly at kinematic singularity. Will it respect the maximum steer joint performance limits? Will it keep operating at the maximum limits or move at "low" velocity? The latter is preferable, in terms of energy consumption and hardware safety. We used: 3) Pure linear motion: Tests the robustness against representational singularity in ICR-based models and controllers, that usually employ θ in the denominator, by sending "zero angular velocity" and arbitrary linear velocity v ar : This tests the performance of Cartesian space kinematic models, developed using only two steerable wheels, since an ICR motion on the line connecting them will result in undefined motion for the other steering joints. This test is realized by the following command sequence:

ξ * = 0.5 * h y1 -h x1 1 T , 6 ≤ t < 8.
ξ * = v ar1 v ar2 0 T , 8 ≤ t < 10.
ξ * = 0.5 * h y1 -h x1 1 T , 10 ≤ t < 12, ξ * = 0.5 * h y2 -h x2 1 T , 12 ≤ t < 14, ξ * = 0.5 * h y3 -h x3 1 T , 14 ≤ t < 16, ξ * = 0.5 * h y4 -h x4 1 T , 16 ≤ t < 18, ξ * = 0.5 * h y2 -h x2 1 T , 18 ≤ t < 20, ξ * = 0.5 * h y1 -h x1 1 T , 20 ≤ t < 22, ξ * = 0.5 * h y4 -h x4 1 T , 22 ≤ t < 24, ξ * = 0.5 * h y3 -h x3 1 T , 24 ≤ t < 26, ξ * = 0.5 * h y1 -h x1 1 T , 26 ≤ t < 28.
5) Zero Velocity: Tests the robustness against representation singularity in Cartesian velocity based models employing ẋ, ẏ, and θ in the denominator of (4). We use: ξ * = 0 0 0 T , 28 ≤ t < 30.

B. Results and Discussion

Applying the benchmark test trajectory shown in Fig. 8a and Fig. 8b to the proposed controller, we obtain the steer joint velocity and acceleration commands in Fig. 8c and Fig. 8d, respectively. As shown, these commands respect the maximum joint performance limits. The corresponding Cartesian space velocity is also shown in Fig. 8a and Fig. 8b, where a deviation from the trajectory is autonomously performed to accommodate for the "planned discontinuity". When implemented on the Neobotix-MPO700, the steer joint velocity response is obtained as in Fig. 9a, showing some overshoot due to imperfect embedded controller. However, 8e and Fig. 9b, respectively. In Fig. 8c, we can see that during the parabolic ICR motion, while passing the singular configuration, the second steer joint velocity grows, while respecting the maximum limit. In the second test where the robot must rotate about the first steer joint axis (kinematic singularity), we can observe that: 1. The singular joint does not rotate at the maximum limit, which is favorable as noted earlier, 2. Low amplitude vibration exists in the steering joints other than the singular one (see also test 4). This is due to the singularity treatment algorithm, which keeps pushing the ICR ref from one side of the singular zone circle to the other. This vibration can be further lowered in amplitude, by decreasing the controller sample time t s , beyond the response time of the mechanical system (the control loop can run at 2 milliseconds sample time on a CORE i7 processor). However this was not achievable without rewriting the codes of the embedded low level controller, a task that we plan among future work. The frequency of such vibration can also be attenuated by taking into account the maximum steering jerk in the QP formulation. It is worth noting that this kinematic controller assumes a sufficiently good low level joint space dynamic controller implemented on the robot in hand. Otherwise, low acceleration values are assumed with maximum performance parameters like Vmax , βmax , and βmax computed at the worst case loading scenario based on the application in hand.

V. CONCLUSIONS

A motion-discontinuity robust controller has been developed and successfully tested on an industrial mobile robot. A discontinuous benchmark test trajectory that excites representation and kinematic singularities has been performed by the proposed controller with success. Maximum steer joint performance limits are taken into account and shown to be well respected throughout the experiments. In future work, the drive joint maximum performance limits will be added to the proposed framework.

Fig. 4 :

 4 Fig. 4: ICR point approaching the singularity zone (in red) of the 2 nd steer joint. ICR motion direction is indicated by the top arrow.

Fig. 5 :

 5 Fig. 5: Feasible set of solutions for the ICR coordinates, respecting the maximum steer joint performance with: (a) stationary ICR, (b) moving ICR (general case), and (c) moving ICR (particular case) with β2 > 0 and β3 > 0, ts is the sample time.

Fig. 6 :

 6 Fig. 6: Photograph of the Neobotix MPO700 industrial SWMR.

Fig. 7 :

 7 Fig. 7: Schematic model of the 5 benchmark test trajectories.

4 )

 4 Straight line motion of the ICR between steering axes: Consists in moving the ICR along a straight line connecting any two steering axes (i.e., six straight lines for the 4 SWMR).

  Linear velocity: desired and output by the proposed controller.

  Angular velocity: desired and output by the proposed controller.

  Output steering velocity command of the proposed controller.

  Output steering acceleration command of the proposed controller.

  Output drive speed command of the proposed controller.

Fig. 8 :

 8 Fig. 8: Results of the proposed controller along the "discontinuous" benchmark trajectory.

  Output steering velocity response of the proposed controller.

  Output drive speed response of the proposed controller.

Fig. 9 :

 9 Fig.9: Joint space response results of the proposed controller, applied to the MPO700 industrial mobile robot.

  Relation between the 3D Cartesian velocity space and the 2D ICR coordinate space. The steer joint pair with the biggest angular difference δβij = |βi -βj| is used to compute ICRcurr.

	𝒚 𝑏			𝜃	𝐼𝐶𝑅
	𝑌 𝐼𝐶𝑅	𝛿𝛽 13	𝛿𝛽 34 𝛿𝛽 23		𝛿𝛽 24
					𝛿𝛽 12
				𝛿𝛽 14
	𝑦
					𝒙 𝑏
	𝑜 𝑏	𝑥			𝑋 𝐼𝐶𝑅
			𝒗 =	𝑥 𝑦	=	𝑌 𝐼𝐶𝑅 𝜃 -𝑋 𝐼𝐶𝑅 𝜃
	Fig. 3:

TABLE I :

 I Robot and Controller parameters used in the experiments.

						③ ICR at infinity	𝒚 𝑏	① Parabolic ICR passing by
						𝐼𝐶𝑅	the steer axis
						④ Straight ICR
						profile between
						steering axes	② ICR at the
						⑤ Zero base frame velocity	𝑜 𝑏	steer axis 𝒙 𝑏
	a	0.19m	b	0.24m	d	0.045m
	rw	0.09m	δ	0.001	R∞	10m
	δ 1	1 -9	Vmax	10m/s	β th	0.005rad.
	λ	3.7	Rzone	0.015m	βmax	2rad./s
	βmax	25rad./s 2	ts	25ms	Kp	2

https://youtu.be/P9aIIwz6qLE

ACKNOWLEDGMENTS

This work is supported by the French region Occitanie (project CoBot@LR) and by the PSA Robotics Department.


S Armstrong

A D Anjos

J T M Baines

C P Bee

M Biglietti

J A Bogaerts

V Boisvert

M Bosman

B Caron

P Casado

A Dos Anjos

G Cataldi

D Cavalli

M Cervetto

G Comune

P Conde Muino

A De Santo

M Diaz Gomez

M Dosil

N Ellis

D Emeliyanov

B Epp

S Falciano

A Farilla

S George

V Ghete

S Gonzalez

M Grothe

S Kabana

A Khomich

G Kilvington

N Konstantinidis

A Kootz

A Lowe

L Luminari

T Maeno

J Masik

A Di Mattia

C Meessen

A G Mello

G Merino

R Moore

P Morettini

A Negri

N Nikitin

A Nisati

C Padilla

N Panikashvili

F Parodi

V Perez Reale

J L Pinfold

P Pinto

Z Qian

S Resconi

S Rosati

C Sanchez

C Santamarina

D A Scannicchio

C Schiavi

E Segura

J M De Seixas

S Sivoklokov

R Soluk

E Stefanidis

S Sushkov

M Sutton

S Tapprogge

E Thomas

F Touchard

B Venda Pinto Aa

V Vercesi

P Werner

S Wheeler

F J Wickens

W Wiedenmann

M Wielers Cc

G Zobernig

Implementation and performance of the high level trigger electron and photon selection for the ATLAS experiment at the LHC

établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Implementation and Performance of the High Level

Trigger Electron and Photon Selection for the ATLAS Experiment at the LHC Carlo Schiavi on behalf of the ATLAS High Level Trigger Group * [START_REF]ATLAS HLT group[END_REF] Abstract-The ATLAS experiment at the Large Hadron Collider (LHC) will face the challenge of efficiently selecting interesting candidate events in pp collisions at 14 TeV center of mass energy, while rejecting the enormous number of background events, stemming from an interaction rate of up to 10 9 Hz. The First Level trigger will reduce this rate to around O(100 kHz). Subsequently, the High Level Trigger (HLT), which is comprised of the Second Level trigger and the Event Filter, will need to further reduce this rate by a factor of O( 103 ). The HLT selection is software based and will be implemented on commercial CPUs, using a common framework built on the standard ATLAS object oriented software architecture. In this paper an overview of the current implementation of the selection for electrons and photons in the HLT is given. The performance of this implementation has been evaluated using Monte Carlo simulations in terms of the efficiency for the signal channels, rate expected for the selection, data preparation times, and algorithm execution times. Besides the efficiency and rate estimates, some physics examples will be discussed, showing that the triggers are well adapted for the physics programme envisaged at LHC. The electron and photon trigger software is also being exercised at the ATLAS 2004 Combined Test Beam, where components from all ATLAS subdetectors are taking data together along the H8 SPS extraction line; from these tests a validation of the selection architecture chosen in a real on-line environment is expected.

Index Terms-ATLAS, Trigger, Electron, Photon, Higgs

I. INTRODUCTION

T HE ATLAS (A Toroidal LHC ApparatuS) experiment will start taking data in 2007 at the Large Hadron Collider (LHC), a pp collider, which is currently under construction at the European Organization for Nuclear Research (CERN). At the LHC, protons will collide at a centre of mass energy of 14 TeV, with a bunch spacing of 25 ns and a design luminosity of 10 34 cm -2 s -1 . ATLAS [START_REF] Zobernig | ATLAS Detector and Physics Performance Technical Design Report[END_REF] is a multipurpose experiment, designed to cover various aspects of high energy physics phenomenology: discovering new physical phenomena, like Higgs bosons or supersymmetrical particles, predicted by theories compatible with the current experimental observations; performing precision Standard Model (SM) studies, like measurements of the t quark and W boson masses; detecting possible unexpected signals from unpredicted physics scenarios. To achieve this goal, moving from the inside out, the ATLAS detector is equipped as follows: tracking and particle identification detectors (Pixel, SCT silicon strips and TRT straw tubes), forming the so called Inner Detector (ID) [START_REF]ATLAS Inner Detector Community[END_REF]; Liquid Argon (LAr) electromagnetic and Tile and LAr hadronic calorimeters [START_REF]Calorimeter Performance Technical Design Report[END_REF]; outer muon system [START_REF][END_REF], designed both for tracking (Monitored Drift Tubes and Cathode Strip Chambers) and trigger purposes (Resistive Plate and Thin Gap Chambers). At design luminosity, ∼ 23 pp interaction will occur at every bunch crossing, in addition to any interesting physics events; together with the bunch spacing of 25 ns, this poses stringent design demands on both the detectors and the Trigger and Data Acquisition (TDAQ) system [START_REF]ATLAS High-Level Trigger, Data Acquisition and Controls Technical Design Report[END_REF]. From the hardware side, this means, as an example, that every subdetector must be equipped with built-in pipeline memories,

ATL-DAQ-CONF-2005-014

to temporarily store events while the first level trigger decision is taken; furthermore the entire detector, which is 22 m tall and 44 m long, must be synchronized to better than 25 ns, in order to ensure a correct event building. On the other side, the high number of detector channels, O(10 8 ) in total, leads to a mean event size of ∼ 1.5 MB; this means that the TDAQ has to face a very challenging networking task, while at the same time has to limit the final event storage rate to a maximum value of ∼ 200 Hz. This last requirement must be fulfilled through the event rejection performed by the trigger system.

II. THE ATLAS TRIGGER SYSTEM

In the beginning of this section, the general ATLAS trigger scheme is described; subsequently more detail is given to the High Level Trigger (HLT) strategy and implementation.

A. General trigger scheme

In the ATLAS experiment, reduction of the 1 GHz interaction rate down to the 200 Hz maximum event data storage rate will be provided through three different trigger selection layers, as shown in figure 1. The hardware-based First Level trigger (LVL1) performs a preliminary rejection using only reduced granularity data coming from the calorimeters and the muon detectors; it operates within a 2 µs latency, producing an average output rate of 75 kHz, upper limited to 100 kHz. Further event selection is then performed by software tools running on dedicated commercial processor farms and is divided in two layers, the Second Level trigger (LVL2) and the Event Filter (EF), collectively referenced as HLT. The choice of having two different software selection levels, though increasing the complexity of the system, has the effect of significantly reducing the computing power needs and of improving the flexibility of the event selection scheme. Reconstruction at LVL2, seeded by information collected at LVL1, can exploit full granularity information from all AT-LAS subdetectors; it is performed processing in parallel data contained inside one or more geometrical regions identified at LVL1, the so called Regions of Interest (RoI), each corresponding on average to the 2% of the total event. Event selection is designed in order to provide an output rate below 2 kHz, and the LVL2 decision must be taken with a mean processing time of 10 ms; obviously the limited execution time greatly contraints the LVL2 reconstruction algorithms, which have to be optimized for timing performance and whose complexity is strongly constrained. The EF has much looser time constraints of the order of 1 s. It can in its turn be seeded by the LVL2 results, or the whole event can be analysed, as an example for the missing transverse energy triggers. More complex reconstruction strategies, including bremsstrahlung and converted photon recovery, can be adopted at this stage.

B. The HLT event selection

The main two concepts characterizing event selection at the HLT in the ATLAS experiment are the RoI-guided reconstruction and the step processing steering. The Region of Interest (RoI) mechanism is used, mainly at LVL2, to significantly reduce the amount of data to be analyzed, and thus processing time, while retaining the full rejection power of a software selection based on full granularity data from all the detectors. All the information gathered at LVL1, like the geometrical position in η and φ of the tagged objects and the thresholds they passed, are combined into a single RoI record which is then sent to the LVL2 framework. As described in more detail in the next subsections, only data produced by detector modules inside the LVL1 RoI will be asked to the Data Flow software (DF), which takes care of all data movement needs from data acquisition to the final storage, and passed to the LVL2; this way a lot of network bandwidth and processing time can be saved. Another feature peculiar to the ATLAS HLT event selection strategy is the early rejection, achieved through event processing in alternate steps of event reconstruction and hypothesis testing on the reconstructed quantities [START_REF] Comune | The Algorithm Steering and Trigger Decision mechanism of the ATLAS High Level Trigger, ATLAS Internal Note[END_REF]; this means that an event can be rejected after any of these steps, if the reconstructed features do not fulfil required criteria or signatures. This method proves particularly powerful at LVL2, where the different RoIs are processed in parallel; as a matter of fact if a minimum number of RoIs is required to pass the selection cuts in order to accept the event, as soon as too many RoIs are prematurely rejected and the minimum required number cannot be met, all the parallel reconstruction tasks taking care of each RoI are stopped and the whole event is rejected.

C. The HLT data access scheme

The goal of minimizing the processing time and network transfers during HLT operation is also pursued through the data access scheme adopted, shown in figure 3. Every time an HLT algorithm is seeded by a RoI, the correponding geometrical information is passed to the Region Selector service in order to be transformed into a list of detector elements contained inside the RoI, as depicted in figure 2; addressing of detector elements, corresponding to subdetector Fig. 3. HLT algorithm data access scheme. modules or read-out channels, is univocally performed using the so called detector element identifiers. To optimize the timing performance, the mapping of the geometrical (η, φ) region into the unique identifiers for the readout units and their corresponding detector elements is implemented using fast look-up tables. The list of identifiers received from the Region Selector is then passed to the Transient Event Store and then to the Data Access tools which request data to the DF software only for the Readout Buffers (RoB) containing the required detector elements. During this stage, raw data also undergoes the conversion from binary read-out channel information to higher level objects containing more physical and algorithm-oriented data, such as calorimeter cell information or ID space points. Decoded data, ordered by detector element, is then passed back to the Transient Event Store [START_REF] Calafiura | The Store-Gate: a Data Model for the Atlas Software Architecture, ATLAS Internal Note[END_REF], a service also used by the offline reconstruction, and eventually to the algorithm that issued the request; transient data is also temporarily cached to avoid another access to the DF and another decoding process in case it will be reasked, as an example by another HLT algorithm.

III. ELECTRON AND PHOTON SELECTION

In this section the physical importance of an effective electron and photon selection is discussed along with their signatures, with particular emphasis on the ATLAS HLT implementation.

A. Physical relevance of electron and photon signatures

Many interesting physical phenomena at LHC, such as H → 4e, lead to a final state containing isolated electrons and/or photons, which provide very clean signatures; they are therefore very important in the reconstruction of possible discovery channels with a high statistical significance. Moreover, electron selection will prove very valuable also for detector calibration and alignment and during the commissioning phase; as an example, a common channel like Z → ee is a multipurpose tool with many key applications. Electron and photon reconstruction mainly exploits data coming from the EM LAr calorimeter and the ID tracking systems; this way not only the kinematical properties of these particles are obtained, taking into account bremsstrahlung effects and photon conversions, but also significant particle identification and rejection of fake e/γ candidates can be achieved, which proves essential for online event selection, as described in detail in the next subsection.

B. Implementation at the HLT

The procedure for the electron and photon selection at the HLT is guided by the Region of Interest mechanism. In particular, LVL2 reconstruction uses information on the transverse energy and the direction of the electromagnetic clusters selected by the LVL1 trigger, so that typically around 2% of the whole event needs to be further analysed. First the energy and position measurements obtained at LVL1 are refined; the leakage into the hadronic calorimeter is evaluated and variables related to the shower shape in the electromagnetic calorimeter are used to perform preliminary particle identification. If a candidate is found to be consistent with an electron, track reconstruction is performed in the ID; cluster to track association is done using (η, φ) matching criteria, achieving further rejection against fake candidates; in case the matching was successful, the E T /p T ratio between the transverse energy measured in the EM calorimeter and the transverse momentum of the corresponding ID track is evaluated for particle identification.

In the case of photon candidates, reconstructed EM clusters undergo tighter shower shape cuts. If the objects under analysis fulfil the required signatures, the event and its LVL2 result are passed to the EF, where information on the complete event is available, along with more precise calibrations and alignment constants. Even if selection at the EF follows the same scheme described for LVL2 operation, the looser timing constraints enable to employ more sophisticated reconstruction algorithms, taking care of bremsstrahlung recovery for electrons and conversion reconstruction for photons.

IV. RESULTS

In this section the performance of the HLT e/γ selection on both single trigger menus and their application to interesting physical channels is reported. Then preliminary results on timing are shown and the perspectives for testing the electron and photon selection on real data are described.

A. Single trigger menu studies

The performance of the e/γ trigger menus has been evaluated on Monte Carlo simulated samples for which the detector response has been simulated in detail; results are given in terms of the efficiency for the real electron and photon signals and of the expected output rates, directly related to the rejection power for fake candidates. Table I shows the efficiency and expected rate for the trigger menu selecting single isolated electrons with a transverse energy (E T ) exceeding 25 GeV (e25i) at initial luminosity (L = 2 × 10 33 cm -2 s -1 ); errors, as also in the following, only take into account the statistical uncertainty contribution, while it should be noted that the uncertainties in the QCD dijet cross-sections at the LHC are of the order of 2-3. Results have been evaluated on a simulated sample containing single electrons with p T = 25 GeV over the full tracking rapidity range |η| < 2.5. The efficiencies and rates are evaluated, after each HLT selection step, with respect to a LVL1 output efficiency of 95% and a LVL1 EM cluster rate of 12 kHz, which are preliminary values. Results show that a final electron selection efficiency of 80% with respect to LVL1 can be achieved with a final rate around 35 Hz. The final selected sample mainly contains real electrons (44% from electrons from b and c quark decays, 21% from converted photons, 19% from W → eν decays, 1% from Z → ee decays). Only 25% of the rate is coming from fake clusters.

B. Application to physical channels

The aim of any trigger system is to provide the necessary event rate reduction, while preserving as much as possible the selection efficiency for interesting physical channels; a constant crosscheck on fully simulated events has hence to be performed, to ensure this goal is correctly met. In particular, for the HLT e/γ selection, studies have been performed on important physics channels such as H → 4e and H → γγ (whose combination is crucial for Higgs discovery in the low mass region 100 GeV < m H < 150 GeV), Z → ee and W → eν.

Table II shows the selection efficiencies for two samples of fully simulated Monte Carlo H → 4e and H → γγ events. In this case, the efficiency is defined as the ratio of the final number of events selected by each trigger menu with respect to an initial sample, preselected according to geometrical and kinematical criteria, as used by the offline analyses [START_REF] Zobernig | ATLAS Detector and Physics Performance Technical Design Report[END_REF], [START_REF] Perez-Reale | Triggering Standard Model Higgs processes in the ATLAS experiment[END_REF]. In particular, both electrons and photons must lay in the |η| < 2.45 region, but outside the barrel-endcap transition region 1.37 < |η| < 1.52. The final efficiencies obtained for each trigger configuration demonstrate that the trigger menus for electrons and photons are well adapted for the Higgs search programme envisaged at LHC.

C. Timing performance

As said before, timing performance is one of the most crucial issues related to HLT selection and reconstruction algorithms, especially at LVL2, and has thus to be continuously optimized to meet the design target. As an example, here are reported the results obtained from a study on the execution times per RoI for LVL2 calorimeter cluster reconstruction; these also include the corresponding data preparation, which means the raw data accesses and the preprocessing needed by the HLT algorithms; all the estimations refer to execution on a LVL2 dedicated test-bed in a farm of 2.2 GHz processors and are performed operating inside ∆η × ∆φ = 0.2 × 0.2 RoIs. Figures 4 and5 show the main contributions to the total latency for a sample of di-jet events at initial luminosity. On figure 4 the total latency is reported and the dotted curve, corresponding to the integral of the distribution, shows that 95% of the events are processed within 5 ms. The four main contributions to the total latency are then shown, again as integrals of distributions, on figure 5. In order of decreasing importance we find data preparation and conversion, framework overheads, network access time, and algorithmic processing; results range between the 500µs pure algorithm execution and the 1.8 ms within which data access is completed for 95% of the events. Timing measurements for LVL2 tracking using the Pixel and SCT detectors have also been performed; the results hereafter reported have been obtained on initial luminosity single electron data sets and again refer to the reconstruction within a ∆η × ∆φ = 0.2 × 0.2 RoI. This time the measurement was made using a machine equipped with a 2.8 GHz processor. The results obtained for a LVL2 Silicon tracking algorithm are shown in figure 6, and refer to the algorithmic part only, excluding data access and preparation; they show that the mean execution time is around 560µs, and track reconstruction is completed within 1 ms in 95% of the events. Continuous effort is put in improving all these timings, even if they are already well within the 10 ms target mean LVL2 decision time, in order to be sure that the same will hold once all the selection components will be exercised as a full chain; these results are even more encouraging if we imagine to scale them with a factor taking into account the improved computing power that will be available at startup, granted by the usage of 8 GHz processor farms.

D. Tests with real data

An important step towards the real operation during the experiment commissioning phase is the validation of the ATLAS TDAQ infrastructure in real or reconstructed online envionments. Many activities on specific test-beds have already been performed and are still ongoing, and a good chance of testing the whole data taking and trigger chains on real online data is provided by the ATLAS Combined Test Beam (CTB) activity carried on this year at the H8 SPS extraction line at CERN. Components from all ATLAS subdetectors (among which the ID and the EM LAr calorimeter, crucial for electron and photon selection) are aligned along the beam line and are currently taking data together; the geometry and the distance between subdetectors are designed to emulate a slice of the real ATLAS barrel detector. Figure 7 shows, an event taken at the CTB and, in particular, the hits produced by particles in the ID subdetectors. From the computing point of view, data acquisition, event selection and reconstruction are based on the official ATLAS software, that will evolve in the final software for the experiment. Among the goals of the ATLAS HLT community was to have an entire e/γ selection slice working at the Combined Test Beam. Online operation tests were performed both for the algorithms used in this slice (like the LVL2 track reconstruction and EF calorimeter clustering ones), for the e/γ hypothesis testing Although running the algorithms directly at the CTB provides interesting feedback on the TDAQ infrastructure, accurate HLT algorithms characterization will have to be done offline, where precise detector alignment and calibration will be available, enabling to correctly tune the algorithms. In particular it will be possible to study the results from runs with different particle composition: electrons, pions, muons and photons. Furthermore, comparing HLT reconstruction with the offline one, many studies on algorithm and selection performance will be achievable: efficiency studies for both the detectors and the reconstruction; particle identification studies, using data from single detectors, in particular the TRT and the EM calorimeter, or from combined E T /p T evaluation; track to cluster (η, φ) matching; effects of bremsstrahlung on tracking and tests of the possible EF correction; studies on photon conversions.

V. CONCLUSION

The LHC environment poses challenging design issues to both the detectors and the TDAQ infrastructure.

As far as the trigger system is concerned, the ATLAS collaboration has adopted a three level strategy, in order to reduce the initial interaction rate of 10 9 Hz to a final event storage rate of about 200 Hz. The choice of having two software selection layers, altough leading to a more complex design, significantly reduces the needs for processing power and network bandwith and, at the same time, increases the flexibility of the selection strategies. The goal of reducing computing power and networking needs is also evident in the HLT choice of using the RoI concept and the corresponding data access scheme, where only a few percentage of event data needs to be transfered and processed. An effective selection of electrons and photons at the HLT is crucial, since they are involved as final signatures in many new physics phenomena expected at the LHC energies. Using Monte Carlo data sets for which the detector response was simulated in detail, the performance of the electron and photon trigger menus has been evaluated in terms of their selection efficiency and rate, and of timing performance both for the data preparation and the algorithmic processing. The preliminary results obtained look promising and in agreement with physics and technical design requirements. In order to crosscheck the correct tuning of the e/γ trigger configuration, studies have been performed on the selection of important physics channels such as H → 4e or H → γγ. The results show that the current trigger menus for electrons and photons are correctly set-up to ensure the envisaged ATLAS physics programme.

Fig. 1 .

 1 Fig. 1. Block diagram of the ATLAS Trigger/DAQ system.

Fig. 2 .

 2 Fig. 2. The Region Selector tool: each geometrical (η, φ) region is converted into a set of detector elements to be analyzed by LVL2 algorithms.

Fig. 4 .Fig. 5 .

 45 Fig. 4.EM calorimeter clustering: total execution time for LVL2 RoI processing of di-jet events at inital luminosity; results obtained on a 2.2 GHz processor.

Fig. 6 .

 6 Fig.6. Inner Detector tracking: total algorithmic execution time for LVL2 single electron reconstruction at inital luminosity; tracking is performed using data from the Pixel and SCT detectors within a ∆η × ∆φ = 0.2 × 0.2 RoI; results obtained on a 2.8 GHz processor.

Fig. 7 .

 7 Fig. 7. Example of tracks reconstructed at the CTB from the hits produced by the corresponding particle in the ID subdetectors: from the left to the right are visible the three pixel layer, the four SCT layers and the TRT.

  algorithm and for infrastructural components (e.g. testing the propagation of LVL2 resuts to the EF); an online monitoring histogram produced by the EF calorimeter cluster reconstruction algorithm is shown in figure8.

Fig. 8 .

 8 Fig. 8. EF calorimeter clustering: monitoring histogram for the transverse energy measurement performed through cluster reconstruction in the EM LAr calorimeter; distributions are compared for a photon and a pion run.

  TABLE II TRIGGER EFFICIENCIES FOR THE H(130 GeV) → 4e AND H(120 GeV) → γγ CHANNELS AT INITIAL AND DESIGN LUMINOSITY.

	Trigger Item	Luminosity H → 4e (%)	H → γγ (%)
	e25i	initial	96.5 ± 0.2
	2e15i	initial	95.8 ± 0.2
	e25 or 2e15i	initial	96.7 ± 0.2
	e30i	design	96.0 ± 0.4
	2e20i	design	94.5 ± 0.4
	e30 or 2e20i	design	95.5 ± 0.3
	2γ20i	initial		74.0 ± 0.6
	γ60i or 2γ20i	initial		83.0 ± 0.5