Keywords: Cogging torque, Finite element analysis, PM machines, Valve positioning

In this paper, it is aimed to use the cogging torque of PM machines for providing the load torque in the valve positioning applications. The policy is to use the machine cogging torque to overcome the load torque at the steady-state condition (fixed valve position). Therefore, there is no excitation, and no armature Joule losses at the steady-state condition and consequently lower thermal stress for the machine. To have a more compact actuator, it is tried to increase the machine cogging torque. Two well-known types of PM machines, i.e. Surface-mounted PM (SPM) and Flux-Switching PM (FSPM) machines are studied to be applied in this application. Some techniques such as applying auxiliary slots and PM segmentation (for the SPM machine), and two-part rotor topology (for both SPM and FSPM machines) are proposed to increase the machine cogging torque. The impact of each technique on the developed cogging torque is studied by means of Finite Element Analysis (FEA). Finally, the SPM and FSPM machines are compared together in the terms of the torque density. The validity of the proposed concept of providing the load torque by the cogging torque is investigated by experimental tests.

I. INTRODUCTION

Usually, servomotors are applied in the positioning applications. In these applications, it is required to lock the rotor and keep it at a fixed position. In such a situation, the locking torque at the zero speed is provided by exciting the armature which results in losses and rising in temperature [START_REF] Yu | Development of a Radial-Flux Slotted Limited-Angle Torque Motor with Asymmetrical Teeth for Torque Performance Improvement[END_REF]. For the sake of reducing the energy consumption, the servo-motor could be powered off and the rotor is kept fix through different locking systems such as solenoid electromagnetic braking systems [START_REF] Kumar | Design of a new Electro-magnetic brake for actuator locking mechanism in Aerospace vehicle[END_REF] or self-locking gears [START_REF] Deng | Current-Sensor-Based Contact Stiffness Detection For Prosthetic Hands[END_REF][START_REF] Chu | Design and Control of a Multifunction Myoelectric Hand with New Adaptive Grasping and Selflocking Mechanisms[END_REF]. Equipping the servomotors with the self-locking gears or solenoid electromagnetic brake is a good and efficient solution with precise positioning. However, it increases the size and complexity of the system.

In this paper, it is tried to remove the self-locking systems in the positioning applications by using the cogging torque of the PM servo-motors as the self-locking torque. Although, cogging torque results in torque ripple and noise in non-zero speed applications, it is not an important issue in the positioning applications which require limited changes in the position and zero steady-state speed working conditions.

In the case of using the machine cogging torque as the steadystate locking torque, some considerations must be taken into account. At first, the machine cogging torque must be high enough to provide the locking torque and provide the load torque at the steady-state condition. Secondly, to change the rotor position, the electromagnetic torque provided by armature current must overcome both of the load and cogging torque. Thirdly, the servo-motor could be stabilized only on some discrete positions, which depends on the cogging torque period. Therefore, to increase the precision of the motor positioning, the period of the machine cogging torque must be decreased. Finally, in the case of changes of the load-torque in a steadystate locked position, there are some changes in the rotor position. The changes in the position by the load variation could be decreased by using a steeper cogging torque waveform. Therefore, in the case of adopting some tolerances in the rotor position, the machine cogging torque may be used to provide the load torque. In such a condition it is not necessary to energize the machine by the external power supply at the steady-state condition; therefore, there is no Joule loss and consequently lower thermal stress on the machine. In addition, in the case of losing the power supply, it is guaranteed that the rotor position is locked. Therefore, such a solution seems good in the terms of torque density, machine efficiency, reliability, and reducing the cost of the supplying inverter.

In most applications, it is tried to reduce the cogging torque as much as possible. However, the used techniques for reducing the machine cogging torque could be applied for increasing it too. The most effective parameter on the machine cogging torque is the combination of the machine slots and poles [START_REF] Jalali | Expansion of the Feasible Slot/Pole Combinations in the Fractional Slot PM Machines by Applying Three-Slot Pitch Coils[END_REF]. However, other design parameters, such as the slot opening and the geometry [START_REF] Ping | Distribution Characteristic and Combined Optimization of Maximum Cogging Torque of Surface-Mounted Permanent-Magnet Machines[END_REF] and the shape [START_REF] Boroujeni | Influence of magnet shaping on cogging torque of surface-mounted PM machines[END_REF] of the PMs have significant influences on the machine cogging torque. The machine cogging torque is reduced by means of other well-known techniques such as pole-shifting [START_REF] Bianchi | Design techniques for reducing the cogging torque in surface-mounted PM motors[END_REF][START_REF] Chaithongsuk | On the Use of Pulse Width Modulation Method for the Elimination of Flux Density Harmonics in the Air-Gap of Surface PM Motors IEEE Transactions on Magnetics[END_REF], PM-segmentation [START_REF] Lateb | Effect of Magnet Segmentation on the Cogging Torque in Surface-Mounted Permanent-Magnet Motors[END_REF][START_REF] Ashabani | Cogging force mitigation of tubular permanent magnet machines with magnet dividing[END_REF][START_REF] Zamani | Optimum arrangement of PMs in surface-mounted PM machines: cogging torque and flux density harmonics[END_REF], optimum PM arrangement [START_REF] Xia | Cogging torque modeling and analyzing for surface-mounted permanent magnet machines with auxiliary slots[END_REF], and applying auxiliary slots [START_REF] Ebadi | Analytical Energy-Based Approaches for Cogging Torque Calculation in Surface-Mounted PM Motors[END_REF].

Although the effect of design parameters on the machine Using and enhancing the cogging torque of PM machines in valve positioning applications Samad Taghipour Boroujeni cogging torque is studied deeply in the literature, they are focused on minimization of the cogging torque. In addition, selecting the number of the PM segments or the number of the auxiliary slots is not discussed in the terms of the Cogging Torque Periodicity (CTP). In this paper, two well-known types of PM machines, i.e. Surface Mounted PM (SPM) machine and Flux-Switching PM (FSPM) machines are studied as candidates for the valve positioning application. These machines are extensively researched to obtain a low-value of cogging torque and high electromagnetic torque. However, in this paper, due to the specification of the considered application, high values of cogging and electromagnetic torque is required simultaneously. For the sake of increasing cogging torque two novel works are presented in this paper. The first novelty is increasing the machine CTP in the SPM machines with the fixed slot/pole combination. The second one is using two-part machines to achieve simultaneously high cogging and high electromagnetic torque components in the FSPM machines. It is worth mentioning that the main idea of the work is to evaluate the effectiveness of some techniques on increasing the machine cogging torque, and detailed optimization of the machine geometrical parameters for obtaining the maximum possible cogging torque is out of the scope of the work. The considered techniques include the PM-segmentation, applying auxiliary slots and using two-part rotor machines.

Hereafter, in section 2, a description of the valve positioning application, the concept of using cogging torque in these applications, and the physics of cogging torque is given. In section 3, some techniques for increasing the cogging in the SPM and FSPM machines are presented. The efficacy of the techniques is evaluated by means of Finite Element Analysis (FEA). Finally, the SPM and FSPM machines are compared together in the term of the torque density. The validity of the proposed concept of providing the load torque by the cogging torque is investigated by experimental tests.

II. AIM AND SCOPE

A. Presentation of the application

In the valve positioning applications, it is desired to move the position of the valve to control the flow of oil, air, fuel or etc. Therefore, the position of the valve could be at some discrete angles instead of the exact continuous values. In these applications, the exact positioning may not be really important and some amount of error signal in the position could be tolerable. In addition, in these applications, a gear-box could be used to increase the number of discrete position steps and consequently the precision of the positioning.

In this paper, it is planned to use the machine cogging torque to overcome the load torque at the steady-state condition, i.e., when the valve is fixed (zero rotor speed). However, due to the load variation, for example, the changes in the pressure of the fluid inside the valve, the positioning process is not very exact. In Fig. 1 (a) typical waveforms of the machine cogging torque and the load torque are shown, where, τ c is the period of the cogging torque waveform. Definitely, to implement the proposed idea, the peak value of the cogging torque must be higher than the maximum of the load torque. When the machine is not energized, the rotor will be located only at points B or G as the stable working points in Fig. 1 (a). Therefore, the maximum tolerance of the positioning process is one-half of the cogging torque period (τ c /2). Actually, τ c is determined by the combination of the slot and poles of the machine.

In order to move the rotor position about one step forward, some current must be injected in the stator winding. Therefore, positive electromagnetic torque is generated (T Is-1 ), which shifts up the torque curve, and settles the rotor at point C. Injecting a higher stator current results in a higher electromagnetic torque (T Is-max ). In this case, the torque curve touches the load torque at point D. Increasing the provided electromagnetic torque a bit more, the rotor runs away from point D toward the points E and then F. After the point F, even with zero stator current (zero electromagnetic torque) the rotor rotates and hunts point G. To move the rotor position forward for another step, the same algorithm must be followed. To move the rotor back from point G for one step, the stator current must provide a negative electromagnetic torque (T Is-3 ) to pull back the rotor position to point H. Definitely, higher negative electromagnetic torque (T Is- 4 ) is required to reach point I (see Fig. 1 (b)). Finally, by increasing a bit more negative electromagnetic torque, the rotor rotates back toward point F and stabilized in point B as illustrated in Fig. 1 (b). Actually, after point F, no stator current is required to rotate back the rotor toward point B. It is worth mentioning that, this simple presented strategy for changing the rotor position is pessimistic because the dynamics of the rotor inertia is not considered. Therefore, changes in the rotor position are possible with the electromagnetic torque lower than T Is-max or T Is-4 for the forward and backward rotation, respectively.

As explained, to change the valve angular position, the machine is energized for a short time to provide the electromagnetic torque and settle the valve in the new angular position. Since the machine is excited with an impulse current waveform only in short-time intervals of the rotor transition, a high current density could be adopted inside the coils. This will result in a very compact machine. Applying six-step current waveforms is a good suggestion for providing the rotor transitions. It is worth mentioning that the cogging torque waveform may not be sinusoidal and includes some harmonics. But the shape of cogging torque waveform is not an important matter, because, the main point is that having a stable point cross-section between the load and cogging torque curve. The other thing is that the torque ripple component due to the timeand space-harmonics of the armature flux density is not an important issue in our discussion, because they affect only the electromagnetic torque and have no influence when the machine is unexcited in the steady-state condition (zero armature current, zero speed, and a fixed rotor position).

Since the load torque is provided by the machine cogging torque at the steady-state condition, providing enough cogging torque is a key issue. Furthermore, the precision of the positioning could be improved by reducing the cogging torque period or applying a step-down gearbox.

It is worth mentioning that the policy of using PM magnetic interaction as the steady-state electromagnetic torque is the basic working principle in the magnetic gears [START_REF] Jing | An Asymmetric Pole Coaxial Magnetic Gear with Unequal Halbach Arrays and Spoke Structure[END_REF][START_REF] Jing | Design, Analysis and Realization of a Hybrid-Excited Magnetic Gear during Overload[END_REF][START_REF] Rahideh | Two-dimensional analytical investigation of the parameters and the effects of magnetisation patterns on the performance of coaxial magnetic gears[END_REF].

B. Physics of the cogging torque

The machine cogging torque is due to the interaction of the PMs edges and the stator teeth in the SPM machine or the interaction of the stator and rotor tooth in the FSPM machine, which are mentioned as magnetic interaction, hereafter. It is a consequence of the basic Electromagnetics law that the generated torque on the rotor desires to reduce the stored magnetic energy as small as possible. In terms of co-energy [START_REF] St Boroujeni | Analytical modeling of no-load eccentric slotted surface-mounted PM machines: Cogging torque and radial force[END_REF], the machine cogging torque is given in [START_REF] Yu | Development of a Radial-Flux Slotted Limited-Angle Torque Motor with Asymmetrical Teeth for Torque Performance Improvement[END_REF], where θ r is the rotor angular position and W c is the stored magnetic co-energy inside the whole machine volume.

()

c cog r r W T    =  (1)
The cogging torque could be obtained by the Maxwell stress integration [START_REF] Žarko | Analytical calculation of magnetic field distribution in the slotted air gap of a surface permanent-magnet motor using complex relative air-gap permeance[END_REF] as given in [START_REF] Kumar | Design of a new Electro-magnetic brake for actuator locking mechanism in Aerospace vehicle[END_REF], where, B r and B ϕ are the radial and circumferential components of the air gap flux density due to the PMs, r is the radius of the considered Maxwell cylinder in the air gap and L stk is the machine stack length.

 =       d B B r L T r r r stk r cog ) ( ) ( ) ( 0 2 (2)
In addition, the cogging torque could be obtained by integrating the forces on the lateral sides of the machine slots as expressed in (3) [START_REF] Chen | Winding Configurations and Optimal Stator and Rotor Pole Combination of Flux-Switching PM Brushless AC Machines[END_REF], where R s is the stator bore radius, Q s is the number of the stator slots, θ ri and θ li are the angles of the right and left lateral sides of the i th stator slot.

   =  →         - = s li j li j s ri j ri j s Q i re e R re e R r stk cog ds B ds B L T 1 2 2 0 lim 2     
(3) However, using (1), ( 2) and ( 3) is not a good idea to predict the overall quality of the cogging torque waveform, especially in a machine with high numbers of poles and teeth. Using the general expression of cogging torque in ( 4) is more useful for evaluating the amplitude of the cogging torque, ( )

) cog s k s r k T GCD x Q T LCM x Q k   = =  1 ( , ) sin ( ,
where, T k is the amplitude of k th harmonic of the cogging torque in one CTP of the machine, GCD is the greatest common divisor operator, LCM is the least common multiplier operator, Q s is the number of the stator teeth. For the SPM and the FSPM machines, x equals to the number of the magnet blocks and the number of the rotor teeth, respectively. The value of GCD(x,Q s ) determines the CTP. Physically, the CTP shows the number of magnetic interactions in the machine with exactly identical cogging torque waveforms. It means a machine with higher CTP has the potential of producing a greater cogging torque. As well as the CTP, the value of the resultant cogging torque in one CTP of the machine is an important factor too. It depends on the level of the coherency of the magnetic interaction waveforms in one CTP of the machine. A higher level of coherency results in greater cogging torque. Actually, the level of coherency between the magnetic interactions in one CTP of the machine depends on geometrical parameters such as the slot-opening, the teeth arcs, and PM arcs. However, the CTP has a more significant impact on the machine cogging torque. Therefore, the cogging torque of the machine could be increased in two separate steps. At first, the CTP must be increased as much as possible by using an appropriate combination of the number of slots and poles. In the second step, coherency between the magnetic interactions in one CTP of the machine must be maximized by optimizing the geometrical parameters such as slot opening and PM or tooth arcs. It should be noted that the attempt of the presented work is focused on the first step, and increasing the level of coherency between the magnetic interactions in one CTP of the machine is not the scope of the paper. The mentioned steps are studied extensively and applied in literature to minimize the cogging torque of the PM machines. The first novelty of the presented work is increasing the CTP of the SPM machines with the fixed slot/pole combination.

Actually the pole/slot combination in the SPM machine and the stator-/rotor-teeth number in the FSPM defines the machine periodicity. In comparison with the integral-slot SPM machines, in the existing fractional-slot SPM machines, the machine periodicity, its CTP and, consequently the cogging torque is small [START_REF] Jalali | Expansion of the Feasible Slot/Pole Combinations in the Fractional Slot PM Machines by Applying Three-Slot Pitch Coils[END_REF]. Similarly, the proposed FSPM machines in the literature have a fractional ratio between the stator and the rotor teeth [START_REF] Chen | Winding Configurations and Optimal Stator and Rotor Pole Combination of Flux-Switching PM Brushless AC Machines[END_REF]. Therefore, the extant fractional-slot machines may not be good solutions for the valve positioning application. However, the other merits of fractional-slot machines, such as using concentrated coils with simple manufacturing and high value of slot filling factor, still encourage us to use fractionalslot machines for the valve positioning application. Since in the positioning applications a high amount of cogging torque is required, in the following, some techniques are proposed to increase the machine cogging torque in the fractional-slot SPM and FSPM machines.

III. INCREASING THE COGGING TOQUE

To study the impact of applying different techniques on increasing the cogging torque, two types of PM machines, i.e., SPM and FSPM machines are considered. As mentioned, the aim of the work is not to increase the level of the coherency between the magnetic interactions in one periodicity of the cogging torque. Actually, it is tried to increase the CTP of the SPM machine with predefined slot/pole combination. Therefore, the stator geometry is kept fix. The fixed data of these machines stators are given in Table 1. The geometry of these machines is shown in Fig. 2. In addition, it is desired to achieve the cogging torque equal to 3 N.m and its period less or equal to 15 o .

Hereafter, in the influence of the number of the PM segments in each pole and the number of auxiliary slots on the cogging torque of the SPM machine is investigated. Since it not possible to apply the mentioned methods in FSPM machines, the effect of applying another technique, i.e. using a two-part rotor is studied on the cogging torque of both SPM and FSPM machines. In order to have an insight into the efficacy of each method on increasing the machine cogging torque, at first the existing cogging torque waveforms of the SPM and FSPM machines with variant design parameters are obtained.

The cogging torque of the SPM machine with 4-, 8-and 10-poles and different ratios of the PM arc to the pole pitch (α PM ) are illustrated in Fig. 3 (a)-(c). The CTP of the 4-, 8-and 10-pole machines are 4, 4, and 2, respectively. Since in one CTP of the 4, 8, and 10-pole SPM machine, there are 1, 2, and 5 number of the magnet blocks with 2, 4, and 10 number of the PM edges, there will be 2, 4, and 10 number of magnetic interaction beetwen the PM edges and slots., respectively. Although the number of magnetic interactions in one CTP of the machine is higher in the 10-pole machine in comparison with the 4-and 8pole machines, they are not coherent waveforms. Therefore, it is expected that the amplitude of the resultant cogging torque in 4-and 8-pole machines is much higher than the 10-pole machine. On the other hand, the 4-pole machine is not interested, because it is not only an integral-slot machine but also its cogging torque period is higher than the supposed maximum angle tolerance (15°). Therefore, the 4-and 10-pole machines are not good candidates for the considered valve positioning application. Although, the electromagnetic torque density and the winding factor of the 12-slot 8-pole machine is lower than the 12-slot 10-pole machine [START_REF] Wang | Synthesis of High Performance Fractional-Slot Permanent-Magnet Machines With Coil-Pitch of Two Slot-Pitches[END_REF], due to its higher cogging torque this machine is considered for the valve positioning application, hereafter. From Fig. 3 (b), it can be found that for providing 3 N.m locking (load) torque about 300mm stack length of the 8-pole machine is required.

In addition, the machine cogging torque in the FSPM machine with the rotor teeth number of 10 and 14 and different ratios of the Rotor Teeth Arc (RTA) are illustrated in Fig. 4 (a) and (b). As seen in Fig. 4 to provide 3 N.m locking (load) torque about 750mm/430mm stack length of the 10-/14-rotor teeth machine is required.

The CTP of the FSPM machines with 10 and 14 is 2. On the other hand, the phases of the magnetic interaction of the rotor and stator teeth waveforms in one CTP of the machine are not coherent. Therefore, as expected and discussed previously, the cogging torque of these FSPM machines is too small, and this machine seems not to be a good candidate for valve positioning applications. Since the FSPM is supposed to be a symmetrical three-phase machine, other rotor geometries with higher CTPs (rotors with 9, 12, 15, and 18 number of the rotor teeth) could not be applied. So at the first look FSPM machine is not a good candidate for valve positioning applications. Hereafter, it is tried to increase the cogging torque of the SPM and FSPM machine.

A. With Applying auxiliary slots in the SPM machine

In this section, one auxiliary slot is considered in each stator tooth. In the [START_REF] Ebadi | Analytical Energy-Based Approaches for Cogging Torque Calculation in Surface-Mounted PM Motors[END_REF][START_REF] St Boroujeni | Analytical modeling of no-load eccentric slotted surface-mounted PM machines: Cogging torque and radial force[END_REF], applying auxiliary slots is presented as a technique for reducing the machine cogging torque, however, the possibility and condition of increasing the cogging torque is not investigated. The shape of the stator of the SPM machine with auxiliary slots is shown in Fig. 5 (a). The cogging torque of the 4-, 8-and 10-pole SPM machines is obtained by FEA and shown in Fig. 5 (b)-(d), respectively. It should be noted that the auxiliary slots are provided in all stator teeth uniformly with the width of the real slot openings and the height of 1mm. The GCD of the all stator slots and the rotor poles, and the maximum obtained cogging torque for the 4-, 8-and 10-pole SPM machines are given in Table 2, where, d+1 is the number of the slots per stator slot pitch.

As seen in Table 2, by applying the auxiliary slots, the GCD(2p,(d+1)Q s ) in the 4-pole and 10-pole machines does not change while in the 8-pole machine it increases two times. In other words, the number of coherent interactions between the PMs and stator teeth increased. Therefore, it is expected that against the 4-and 10-pole machines, the machine cogging torque is increased by applying auxiliary slots in the 8-pole machine. From the cogging torque waveforms in Fig. 5, it is seen that applying auxiliary slots does not change the period of the cogging torque in 8-pole machine because the LCM of the slots and poles does not change, while, in the 4-pole SPM machine the period of the cogging torque is halved because the LCM value is doubled by applying auxiliary slots. Finally, it could be stated that applying the auxiliary slots can increase the machine cogging torque if the value of GDC (2p, d+1) is increased (or the number of all teeth is a multiplier of the rotor poles). As seen in Fig. 5 to provide 3 N.m locking (load) torque about 170mm stack length of the 8-pole SPM machine with auxiliary slots is required.

B. Segmentation of the PMs in the SPM machine

PM segmentation is a well-known technique to mitigate the cogging torque in the SPM machines [START_REF] Bianchi | Design techniques for reducing the cogging torque in surface-mounted PM motors[END_REF][START_REF] Chaithongsuk | On the Use of Pulse Width Modulation Method for the Elimination of Flux Density Harmonics in the Air-Gap of Surface PM Motors IEEE Transactions on Magnetics[END_REF][START_REF] Lateb | Effect of Magnet Segmentation on the Cogging Torque in Surface-Mounted Permanent-Magnet Motors[END_REF][START_REF] Ashabani | Cogging force mitigation of tubular permanent magnet machines with magnet dividing[END_REF][START_REF] Zamani | Optimum arrangement of PMs in surface-mounted PM machines: cogging torque and flux density harmonics[END_REF]. However, here, it is used to increase the cogging torque. For this aim, the uniform segmentation, i.e., the similar equidistance magnet blocks, is adopted. The geometries with s=2, 3 and 4 are considered hereafter, where s is the number of PM segments per pole. The shape of the 8-pole SPM machine with 3 segments per pole is shown in Fig. 6 (a). The cogging torque of the segmented PM machines with 4-, 8-and 10-pole SPM machines are given in Fig. 6 (b)-(d), respectively. The GCD of the stator slots and the rotor PM segments and the maximum of the cogging torque waveform for the 4-, 8-and 10-pole SPM machines are given in Table . 3.

As seen in Table . 3, against the other segmented machines, in the 4-and 8-pole SPM machines with 3 segments PMs per pole, the number of PM pieces is an integer number of the stator slots and the GCD(Qs,2ps) is increased significantly (the maximum possible value i.e., 12), therefore, in these machines, the cogging torque increases very much. In this condition, the number of coherent interactions between the PMs and stator teeth is increased, considerably. From the cogging torque waveforms in Fig. 6, it is seen that the PM segmentation does not change the period of the cogging torque. Comparing the cogging torque waveforms in Fig. 6 by the ones in Fig. 5, it is found that the PM segmentation technique is more effective than applying auxiliary slots on increasing the machine cogging torque. As seen in Fig. 5 to provide 3 N.m locking (load) torque about 110mm stack length of the 8-pole SPM machine with 3 magnet segments per pole is required.

C. Using two-part machines

In the previous sections, the cogging and electromagnetic torque components are generated simultaneously in the whole axial length of the machine Although considering this idea and applying PM segmentation is an efficient method in increasing the SPM machine cogging torque, it cannot be applied to the FSPM machines effectively. In this part of the study, it is tried to separate the rotor of the machines in two axial parts and provide the cogging and electromagnetic torques separately in each axial part of the machine. The two parts include the same stator geometry (for the sake of simplicity in manufacturing), while they contain different rotor geometries which are magnetically isolated by applying an air distance between them to avoid 3D effects. The configuration of the two-part machine is illustrated in Fig. 7. In the proposed two-part machine SPM or FSPM machines could be applied. The first part of the rotor provides a big amount of cogging torque which is required in the steady-state condition. This part of the rotor is named CT rotor hereafter. The second part of the rotor mentioned as ET rotor produces the electromagnetic torque. ET rotor is responsible for rotating the rotor during transitions. Therefore, the number of rotor poles/teeth in the SPM/FSPM machine can be selected to maximize the CTP of the machine and consequently, increase the machine cogging torque with no concern regarding the balanced three-phase condition or the electromagnetic torque. Therefore, 24-pole SPM or 24-rotor tooth geometries could be applied for the CT rotor. In addition, the number of the rotor poles/teeth in the ET rotor of the SPM/FSPM machine is selected as 10/14, the normal SPM/FSPM machines, to provide balance three-phase condition with a high-value electromagnetic torque and a small amount of cogging torque. Hereafter, both SPM and FSPM machines are considered for the design of two-part machines.

1) Providing a high cogging torque

A rotor with a high-level of CTP should be selected for the CT rotor section. Since the CT rotor is not supposed to contribute on producing the electromagnetic torque as a 3-phase and symmetrical machine, there are more variant alternatives for the rotor topology. In other words, a very huge cogging torque could be produced by selecting an appropriate number of PMs and the rotor teeth in the SPM and FSPM machines, respectively. Actually, the number of the rotor poles (in the SPM machine) and the rotor teeth (in the FSPM machine) must be an integer number of the stator slots to reach a high CTP value. In this study, a 12-pole rotor is not selected because the period of its cogging torque is 30°, which is higher than the minimum acceptable period for the cogging torque waveform. Therefore, a 24-pole rotor with α PM =0.8 is selected and the cogging torque waveforms with and without applying auxiliary slots in the stator are obtained and shown in Fig. 8 (a). In this case, the CTP of the machine is 24 and 12 for the machines with and without auxiliary slots, respectively. By comparing the cogging torque waveforms of the 24-pole rotor machine (without using auxiliary slots) in Fig. 8 (a) and the 8-pole SPM machine with 3 segments per pole in Fig. 6 (b) (as the machine with highest cogging torque), it is inferred that the using twopart rotors does not increase significantly the cogging torque in the SPM machines. Alternatively, as shown in Fig. 7, for the CT rotor of the FSPM machine the number of the rotor teeth may be selected as 12 and 24, respectively. The cogging torque waveforms of the FSPM machines with 12 and 24 rotor teeth are shown in Fig. 8 (b). By comparing the cogging torque waveforms in Fig. 8 (b) with the waveforms in Fig. 4, it is found that using the two-part rotor is an effective technique that results in a very high cogging torque in the FSPM machines even more than SPM machines. As seen in Fig. 8 stack lengths are required in the CT rotor of the by SPM with and without auxiliary slots and FSPM machines, respectively. For providing 3N.m locking torque, using a two-part SPM machine is not a good idea because it needs 60mm axial length only for providing the cogging torque. However, using the 3segment PM rotor is a very nice idea in the SPM machines (110mm axial length for providing both cogging and electromagnetic torque components). Against the SPM, in the FSPM using a two-part rotor is logical (20mm for CT rotor).

The flux density distribution in the first part of the SPM and FSPM machines for two rotor positions are given in Fig. 9 (a) and (b), respectively. It is understood from Fig. 9 that in the first part of machines there is no flux linkage and back-EMF inside the stator coils due to the PMs and, consequently, this part of the machine has no influence on the electromagnetic torque production.

2) Providing a high electromagnetic torque The ET rotor is responsible for providing the required electromagnetic torque in the transient conditions. Therefore, the well-known classical fractional-slot SPM or FSPM with very low cogging torque and high electromagnetic torque density could be applied in the EM rotor. As shown in Fig. 8, a 10-pole rotor for the SPM or a 14-tooth rotor in the FSPM machines could be applied. Hereafter The stators of the SPM and FSPM are excited and the behavior of the provided torque in the CT and ET rotors of these machines are investigated separately. Considering the same current density inside the stator slots of the SPM and FSPM machines, the electromagnetic torque waveforms for DC and synchronized six-step armature currents in the second part of the machine are obtained by means of FEA and shown in Fig. 10 Finally, from the easiness of manufacturing point of view, it should be noted that using PM segmentation (with magnet blocks with similar dimensions) and applying two-part machines are applicable and straightforward techniques and could be implemented without applying changes in the stator laminations or coil windings. Regarding using auxiliary slot techniques in the SPM machines, although, it could be applied in the existing stator laminations, it requires some mechanical modifications efforts.

IV. CONCEPT VERIFICATION

To evaluate the applicability of using cogging torque as the load torque experimental tests are applied. The test rig is shown in Fig. 12a. In the test rig, an interior PM machine is used to provide the cogging torque, a synchronous reluctance machine is used to implement the load torque and the DSPACE MicroBox controller is used to control the PWM inverters. The measured cogging torque is illustrated versus the rotor position in Fig. 12b. In the test, the synchronous reluctance machine is controlled to provide a constant torque of 0.3 N.m. By exciting the phases of the PM machine with DC voltages (±5V), the rotor position is changed as shown in Fig. 12c. The sequence of the applied DC voltages is presented in Fig. 12c. The waveform of the measured torque is given in Fig. 12d. As seen in Fig. 12c after changing the rotor position, the stator terminals of the PM machine is short-circuited and the machine cogging torque provides the load torque.

V. CONCLUSION

In this paper, it is proposed to use the machine cogging torque to provide the load torque at the steady-state condition (i.e., zero speed and the fixed rotor potion) in the valve positioning system. Therefore, it is tried to increase the machine cogging torque as much as possible. Two types of PM machines, i.e, SPM, and FSPM machines with a fixed stator geometry is considered. The first novelty of the work is focused on increasing the machine CTP in an SPM machine with a fixed number of slot/pole combination. The influence of PM segmentation and using auxiliary slot techniques on increasing the CTP of the SPM machines are investigated. It is shown that PM segmentation is a more effective approach. It is shown that to increase the cogging torque in the PM-segmented SPM machines, the number of the rotor poles and the PM segments must be selected such that GCD(Q s ,2ps) is high as possible. In addition, it is found that in the case of applying auxiliary slots to reach a high value of the cogging torque, the number of the rotor poles and the auxiliary slots must be selected such that GCD((d+1)Q s ,2p) is high as much as possible. Finally, since increasing the machine CTP is not applicable in the FSPM machine, applying two-part machines is proposed and used to increase the cogging torque of FSPM machines. It is found that using a two-part machine in the FSPM results in a very high cogging torque. Using FEA, it is shown that the CT rotor in the first part of the machine produces only the cogging torque even in the case that stator armature is excited. During the rotor transitions, the stator current must be injected inside the armature coils to provided electromagnetic torque in the second part of the machine with the ET rotor. Finally, the concept of providing the load torque by the cogging torque is verified by experiments.

Fig. 1 .

 1 The load, cogging torque and electromagnetic torque of the PM machine proposed for the valve positioning application (a) forward and (b) backward rotation algorithms.

Fig. 2 .Fig. 3 .

 23 Fig. 2. Geometry of the 12-slot (a) 10-pole SPM and (b) 14-rotor tooth FSPM machines

Fig. 4 .Fig. 5 .

 45 Fig. 4. Cogging torque waveforms of FSPM machine with (a) 10, (b) 14 rotor teeth.

Fig. 6 .

 6 The SPM machine with 8-pole and 3-segment PMs rotor (a) and the cogging torque waveforms of (b) 4-pole, (c) 8-pole and (d) 10-pole SPM machine with segmented PMs.

  (a) and (b) to provide 3 N.m locking (load) torque about 100, 60, and 20mm

Fig. 7 .Fig. 8 .

 78 Fig. 7. The layout of the two-part SPM and FSPM machines.

  (a) and (b), respectively. To investigate the contribution of the first rotor part in the torque production in the transients, the stator of the first part of the SPM and FSPM machines are excited by the DC and sixstep synchronized currents and the obtained torque waveforms are shown in Fig.11 (a) and (b), respectively. Comparing the static, synchronized torque waveforms with the cogging torque of the first parts of the machines in Fig.11 (b), shows that no electromagnetic torque is generated by the interaction of the armature currents and PMs in the first part of the machines. The same result (having no PM flux linkage and consequently no electromagnetic torque component in the first part of the machine) is obtained from Fig.9.

9 .

 9 Flux density distribution for the rotor position of 0°(a), 7.5° (b) for the 4-pole SPM, and 0°(c) and 7.5° (d) for the FSPM machines.

Fig. 10 .

 10 Fig. 10. The total torque in the second part of the 10-pole SPM and 14-rotor teeth FSPM machines with (a) DC and (b) six-step synchronized currents.

Fig. 11 .

 11 Fig. 11. The total torque in the first part of the (a) 24-pole SPM and (b) 24rotor teeth FSPM machines with DC and six-step synchronized currents.

  1,2 , Noureddine Takorabet 2 , Smail Mezani 2 , and Thierry Lubin 2 , Peyman

	Haghgooie 2
	1) Department of Engineering, Shahrekord University, Shahrekord, Iran
	2) Université de Lorraine -GREEN, Nancy, France
	The corresponding author is S. Taghipour Boroujeni (s.taghipour@sku.ac.ir)

TABLE II THE

 II CHANGES IN CTP AND THE MAXIMUM COGGING TORQUE OF THE SPM

		MACHINES WITH AUXILIARY SLOT
	No. of poles	GCD((d+1)Q s ,2p)	T cog-max (N.m)
		d=0	d=1	d=0	d=1
	P=4	4	4	0.10	0.10
	P=8	4	8	0.10	0.17
	P=10	2	2	0.008	0.004


S L Happy
email: s-l.happy@inria.fr

Antitza Dantcheva
email: antitza.dantcheva@inria.fr

Francois Bremond
email: francois.bremond@inria.fr

Semi-supervised Emotion Recognition using Inconsistently Annotated Data

Expression recognition remains challenging, predominantly due to (a) lack of sufficient data, (b) subtle emotion intensity, (c) subjective and inconsistent annotation, as well as due to (d) in-the-wild data containing variations in pose, intensity, and occlusion. To address such challenges in a unified framework, we propose a self-training based semi-supervised convolutional neural network (CNN) framework, which directly addresses the problem of (a) limited data by leveraging information from unannotated samples. Our method uses 'successive label smoothing' to adapt to the subtle expressions and improve the model performance for (b) low-intensity expression samples. Further, we address (c) inconsistent annotations by assigning sample weights during loss computation, thereby ignoring the effect of incorrect ground-truth. We observe significant performance improvement in in-the-wild datasets by leveraging the information from the in-the-lab datasets, related to challenge (d). Associated to that, experiments on four publicly available datasets demonstrate large performance gains in cross-database performance, as well as show that the proposed method achieves to learn different expression intensities, even when trained with categorical samples.

I. INTRODUCTION

Facial expression recognition aims at inferring emotions based on visual cues from face images. In spite of recent advancements in this field, following set of challenges remain open. (a) Lack of sufficient annotated data is a major limitation in emotion recognition. At the same time, we have that manual annotation is subjective (e.g., intra and interindividual variation), as well as time-consuming. (b) Often large affective datasets are created by crowdsourcing [START_REF] Li | Reliable crowdsourcing and deep localitypreserving learning for expression recognition in the wild[END_REF], which results in noisy and inconsistent annotation due to inexperienced annotators. Utilizing wrongly labeled samples often negatively affects the model performance. Contrarily, (c) the in-the-lab datasets are created under constraint environments with proper annotation by experts. However, such datasets predominantly contain only peak expression images posed by actors, which is far from realistic. Consequently, a model trained with such data often fails when applied to real-world applications mainly due to low sample size and lack of expression intensity variation in the training data. Further, unlike 'object classification', multiple emotions may coexist in real-life data, raising false alarms in blended or mixed emotions [START_REF] Zhou | Emotion distribution recognition from facial expressions[END_REF], [START_REF] Dantcheva | Expression recognition for severely demented patients in music reminiscence-therapy[END_REF]. Given that emotion classes are not mutually exclusive but collectively exhaustive, it is important for the model to learn accurate emotion intensities for all emotion-classes. Finally, (d) in contrast to in-thelab data, real-world data (in-the-wild) contains images with different illumination, facial pose, occlusion, stemming from various sensor-types, and other undesired factors, such as low resolution and blurred images.

Over the past years, convolutional neural networks (CNNs) have achieved outstanding performances in many domains of application, including facial expression recognition [START_REF] Jung | Joint fine-tuning in deep neural networks for facial expression recognition[END_REF], [START_REF] Zhao | Peak-piloted deep network for facial expression recognition[END_REF], [START_REF] Ding | Facenet2expnet: Regularizing a deep face recognition net for expression recognition[END_REF], [START_REF] Meng | Identity-aware convolutional neural network for facial expression recognition[END_REF], due to their powerful capacity for modeling complex input patterns. Transfer learning addresses the limited data problem in emotion recognition [START_REF] Ding | Facenet2expnet: Regularizing a deep face recognition net for expression recognition[END_REF], [START_REF] Ng | Deep learning for emotion recognition on small datasets using transfer learning[END_REF] by using a pretrained network trained with large face data in the context of face recognition. Some works exploit the ample unlabeled face data along with the labeled data in a weakly or semisupervised approach. CNN models based on semi-supervised learning (SSL) [START_REF] Happy | A weakly supervised learning technique for classifying facial expressions[END_REF], [START_REF] Zhang | Enhanced semi-supervised learning for multimodal emotion recognition[END_REF] have been explored, demonstrating the effectiveness of emotion classification frameworks by leveraging information from unlabeled data. In self-training [START_REF] Rosenberg | Semi-supervised self-training of object detection models[END_REF], an initial model selects the high confidence samples from unlabeled data, which is further utilized to update the model. In this work, we address the limited data problem using both transfer learning and self-training, thereby maximally exploiting the information in unlabeled data.

Training with one-hot vectors forces CNNs to predict one of the classes with high confidence. The presence of noisy or inadequate data overfits the model with one-hot encoding. Label smoothing [START_REF] Szegedy | Rethinking the inception architecture for computer vision[END_REF] renders models less confident about their associated predictions and acts as a natural regularizer. We here use successive label smoothing, in order to allow the model to automatically adapt to different emotion intensities. In other words, we use label smoothing on the model predictions for the labeled data and utilize them for further parameter update. Thus, our model output is consistent with the emotion intensity, by adapting to subtle and mixed emotion intensities.

Towards adapting the model to noisy and inconsistent annotated samples, the literature suggests eliminating hard samples [START_REF] Wang | Data dropout: Optimizing training data for convolutional neural networks[END_REF], [START_REF] Wang | Instance-based deep transfer learning[END_REF] or assigning weights to samples [START_REF] Jiang | Mentornet: Learning data-driven curriculum for very deep neural networks on corrupted labels[END_REF] for further parameter tuning. In our SSL approach, we assume that the labeled data contains noisy annotations, whereas the selected unlabeled data used in training process is clean. Thus, we assign weight to each labeled sample based on its training loss, which indicates the sample importance. Our model allows the sample weight values to grow or shrink dynamically based on model parameters at every iteration. By decreasing the weight of hard examples, the model becomes robust to inconsistent and noisy labeled data during model training.

We firstly train an initial model to achieve sufficient accuracy in predicting emotion classes. This model is further employed for the purpose of classifying unlabeled data and including a fraction of high confidence samples in the training process in the subsequent iteration. This allows us to leverage information from ample unlabeled data towards addressing the low sample size problem. The combination of successive label smoothing and the use of predicted class probability of unlabeled data as the true distribution, enables the model to automatically adapt to different emotion intensities. Thus, prediction scores from our model resemble the emotion class probabilities, allowing the recognition of low intensity and blended emotions. Further, we assign sample weights based on the training loss and the model is updated considering the importance of each sample. Thus, noisy and inconsistent labeled data is ignored in the training process, due to their low sample importance. The combined effect of SSL, successive label smoothing, and sample importance assignment learns a robust model, which inherently tackles the issues raised by the in-the-wild data. Contribution In summary, our main contributions are the following.

• We propose a self-training based SSL approach to improve emotion recognition performance by leveraging information from the unlabeled data. As opposed to using the predicted class of unlabeled samples as groundtruth [START_REF] Lee | Pseudo-label: The simple and efficient semi-supervised learning method for deep neural networks[END_REF], we use the predicted class distribution as the true distribution in our model. • We use successive label smoothing to adapt to expression images with different intensities. We demonstrate that our model adapts to expression intensity from the data, annotated with discrete expression categories. • We incorporate insights from sample weighting [START_REF] Jiang | Mentornet: Learning data-driven curriculum for very deep neural networks on corrupted labels[END_REF], [START_REF] Wang | Instance-based deep transfer learning[END_REF], in order to make our model robust against noisy and inconsistent annotation. Contrarily to former approaches [START_REF] Jiang | Mentornet: Learning data-driven curriculum for very deep neural networks on corrupted labels[END_REF], [START_REF] Wang | Instance-based deep transfer learning[END_REF], our model initially assigns equal weight to each sample, which can further increase or decrease based on the training and validation loss. • We report performances, outperforming state-of-the-art results on four datasets. We show that the performance of in-the-wild datasets improve largely from using the in-the-lab datasets as unlabeled data. The proposed method also achieves a significant improvement in cross-database experiments.

II. RELATED WORK

A. Deep Learning-based Facial Expression Recognition

Given the powerful ability to model complex class representation, CNNs have outperformed and replaced handcrafted feature extraction methods in many applications, including facial expression recognition [START_REF] Zhao | Peak-piloted deep network for facial expression recognition[END_REF], [START_REF] Liu | Facial expression recognition via a boosted deep belief network[END_REF], [START_REF] Lopes | Facial expression recognition with convolutional neural networks: coping with few data and the training sample order[END_REF], [START_REF] Zeng | Facial expression recognition with inconsistently annotated datasets[END_REF]. Most CNN methods attempted to resolve emotion recognition challenges inspired by methods, proposed in early literature. For example, CNN based models have been proposed for improving emotion recognition by encoding temporal appearance and geometry features [START_REF] Jung | Joint fine-tuning in deep neural networks for facial expression recognition[END_REF], by jointly learning feature representation-selection-classification [START_REF] Liu | Facial expression recognition via a boosted deep belief network[END_REF], by embedding facial representation of both peak -nonpeak intensity samples [START_REF] Zhao | Peak-piloted deep network for facial expression recognition[END_REF], by jointly learning expression and identity related features [START_REF] Meng | Identity-aware convolutional neural network for facial expression recognition[END_REF], etc.

To address the limited data-size problem, Oquab et al. [START_REF] Oquab | Learning and transferring mid-level image representations using convolutional neural networks[END_REF] proposed the transfer of mid-level image representation for related source and target domains. The network parameters learned on large-scale data generalizes the input space and transferring such parameters could significantly improve the performance of a task with a limited amount of training data. To improve emotion recognition performance with limited data, Ng et al. [START_REF] Ng | Deep learning for emotion recognition on small datasets using transfer learning[END_REF] used the network weights trained on ImageNet. Similarly, FaceNet2ExpNet [START_REF] Ding | Facenet2expnet: Regularizing a deep face recognition net for expression recognition[END_REF] finetuned FaceNet in order to capture high level expression semantics. In our framework, VGG-Face model [START_REF] Parkhi | Deep face recognition[END_REF] is used to achieve knowledge transfer. VGG-Face uses a VGG-16 architecture and was trained with 2.6 million face images for face recognition applications and is a popular network choice for face related applications [START_REF] Ding | Facenet2expnet: Regularizing a deep face recognition net for expression recognition[END_REF].

B. Semi-Supervised Learning (SSL)

SSL methods use both labeled and unlabeled data to overcome the limited data problem. Coates et al. [START_REF] Coates | Learning feature representations with kmeans[END_REF] provided an intuition that data distribution from unlabeled data can be leveraged towards improving model performance. The performance of stacked auto-encoders was improved by initially training with unlabeled data followed by fine-tuning the model using labeled data [START_REF] Rasmus | Semi-supervised learning with ladder networks[END_REF]. Moreover, the use of adversarial networks, which learns from abundant unlabeled data [START_REF] Radford | Unsupervised representation learning with deep convolutional generative adversarial networks[END_REF], has been very well accepted.

Self-training or incremental SSL method [START_REF] Rosenberg | Semi-supervised self-training of object detection models[END_REF], [START_REF] Lee | Pseudo-label: The simple and efficient semi-supervised learning method for deep neural networks[END_REF] is popular in deep learning classification tasks where the unlabeled data augment the limited annotated data. Self-training starts with optimizing model parameters with labeled data followed by classifying and selecting unlabeled data with high confidence scores for further training of the model. Co-training is a similar process, where multiple classifiers are trained to iteratively learn and generate additional training labels. Zhang et al. [START_REF] Zhang | Enhanced semi-supervised learning for multimodal emotion recognition[END_REF] proposed an enhanced multi-modal cotraining algorithm for semi-supervised emotion recognition.

With similar motivation, our work implements selftraining to leverage information from unlabeled data. However, instead of using the predicted class of unlabeled samples as ground-truth, we use the predicted class distribution as the true distribution in our model. This allows the model to partially adapt to the expression data of different intensities.

C. Methods with Noisy and Inconsistent Labels

Presence of noisy and inconsistent data can easily overfit a CNN model. Expression annotations are very subjective and datasets created from crowd-sourcing are often inconsistent and noisy. To avoid the risk of learning from the noisy data, a multi-task network is proposed in [START_REF] Veit | Learning from noisy large-scale datasets with minimal supervision[END_REF] to jointly learn the cleaning of the noisy annotation and classifying to the correct class. Li et al. [START_REF] Li | Learning from noisy labels with distillation[END_REF] proposed a unified distillation framework to learn from noisy labels by leveraging a knowledge graph.

Another way to address inconsistently labeled data is dataset resampling, i.e., minimizing the weighted loss during training using sample weights. Jiang et al. [START_REF] Jiang | Mentornet: Learning data-driven curriculum for very deep neural networks on corrupted labels[END_REF] proposed MentorNet to learn the weights of the examples using long short term memory based on the training loss. The effect of each training points on the CNN model's performance was exploited in [START_REF] Koh | Understanding black-box predictions via influence functions[END_REF] by analyzing the change of model parameters due to removal of a data point. Data dropout [START_REF] Wang | Data dropout: Optimizing training data for convolutional neural networks[END_REF] was proposed to optimize the training data by removing the unfavorable or bad training samples. The idea behind instance-based transfer learning [START_REF] Wang | Instance-based deep transfer learning[END_REF] was to use a pre-trained Our approach assigns weight to each labeled sample which reflects its importance during model update. In contrast to other methods in the literature, the sample weight in our approach is modified in each iteration based on the training and validation losses. Our model allows both downweighting the hard examples with high loss (usually noisy annotation) and up-weighting the easy examples with clean annotation. Thus, our model is robust to outliers and noisy data.

III. PROPOSED METHOD

The block diagram of the proposed work is shown in Figure 1. The idea behind our work is to train an initial model for expression classification, which is afterward used to classify the unlabeled data. A few samples from unlabeled data with high prediction scores are selected (with corresponding predictions as ground truth) and treated as train data for further model update. The predicted class probabilities for the train data are considered to update the ground truth vector of the train data. In addition, each training sample is assigned with a weight according to its importance in updating the model parameters. Thus, our framework has four main parts:

(1) initial model training: obtaining the model parameters for fairly accurate expression classification, (2) exploiting unlabeled data for model update: leveraging the unannotated data to improve the model performance by self-training, (3) ground-truth distribution correction for train data (label update): allowing the model to adapt to different emotion intensities while maintaining certain confidence in prediction, and (4) assigning sample importance: computing the sample weights and ignoring the inappropriate samples during model update. We proceed to discuss the details of these steps in this section.

A. Initial Model Training

The task of an expression recognition framework is to correctly predict the emotion class or the emotion class probabilities. Given the sample and label pair (x i , y i ), the classification network learns to accurately predict class probabilities p(k|x i , θ), where θ are the network parameters and k ∈ {1, 2, ..., K} represents K classes. For a soft-max layer, we have p(k|x i , θ) = exp(z k ) K j=1 exp(zj ) , where z j are the unnormalized log probabilities. In supervised learning, the ground-truth distribution q(k|x i ) is used to train the network parameters (θ) by minimizing the cross-entropy loss function

L = xi∈X l f i (θ) = - xi∈X l K k=1 log(p(k|x i , θ))q(k|x i ). (1)
Here f i (θ) denotes the loss incurred by x i and X l represents the set of training samples. Usually, one-hot encoding is used for classification models, which takes the form q(y i |x i ) = 1 and q(k|x i ) = 0 for all k = y i . For clarity, we drop the subscript and use x instead of x i to denote a sample.

The initial model (θ * ) is utilized to predict pseudo class probabilities pertaining to unlabeled data during the selftraining process. Thus, the initial model needs to be fairly accurate to avoid erroneous predictions, which in turn will result in a poorer model due to error accumulation. We trained the initial model using the labeled data for sufficient epochs depending upon the dataset until adequate performance is achieved.

Label Smoothing: Unlike object classification, expression categories are highly related, interconnected, as well as can occur simultaneously. For instance, multiple expression classes coexist during emotion transition or mixed emotions. As the expression datasets contain expressive images of various intensities, it would be best to train a classifier with normalized emotion intensity vectors as the ground truth. However, such datasets do not include the emotion intensity information. Training a model with one-hot vector usually results in overfitting the model. Therefore, the presence or absence of multiple objects can be formulated with one-hot vectors, whereas the expression recognition model should consider soft class assignment vectors.

Strong supervision with one-hot encoding over-fits the model on the training data and fails to generalize for the unseen data. Label smoothing [START_REF] Szegedy | Rethinking the inception architecture for computer vision[END_REF] provides a solution to this situation by replacing the one-hot encoding with a smooth distribution. Though it renders low confidence prediction, the model becomes more regularized and adaptable to unseen data. We implemented the label smoothing as,

q (k|x) = 1 -, k = y K-1 , k = y, (2)
where ∈ [0, 1] is the label smoothing hyperparameter. While setting = 0 refers to one-hot encoding, setting a large value might result in learning a poor performing model.

B. Exploiting Unlabeled Data for Model Update

The proposed semi-supervised method uses a fraction of the unlabeled data along with the labeled samples to update the network. We used a self-training procedure inspired by [START_REF] Lee | Pseudo-label: The simple and efficient semi-supervised learning method for deep neural networks[END_REF], where the class labels of the unlabeled data are estimated using the network predictions. Unlike in [START_REF] Lee | Pseudo-label: The simple and efficient semi-supervised learning method for deep neural networks[END_REF], we use the predicted probability distribution as the ground-truth distribution.

Suppose X u denotes the unlabeled data, q l stands for ground-truth distribution of labeled data, and p u denotes the network prediction probabilities for unlabeled data. Note that the initial model (θ * ) is trained with X l and q l . Afterward, we update the model parameters iteratively using a portion of data from both X l and X u simultaneously. The unlabeled samples are selected from the pool of unlabeled data predicted with high confidence scores ( X u ), as suggested in [START_REF] Rosenberg | Semi-supervised self-training of object detection models[END_REF]. Instead of using all the high confidence samples, we select a few samples randomly from X u in each iteration to avoid over-fitting the model.

Maintaining a proper balance between the number of labeled and unlabeled data is crucial to avoid error accumulation in self-training process. In our implementation, we randomly replace a fraction of X l (typically 1 -10% of number of labeled samples in X l ) with the unlabeled data from X u in each epoch. After the t-th update, we obtain the prediction scores using θ t for both X l and X u , denoted by p t l and p t u , respectively. We obtain the unlabeled data with high prediction scores using a threshold value (τ ), given by

X t u = {x|x ∈ X u and max k p t u (k|x) > τ}; X t u ⊂ X u .
(3) From the pool of samples in X t u , we select a few randomly and replace with the random samples from X l . Thus, the train data for (t +1)-th iteration becomes X t+1 = {X t+1 l , X t+1 u }, where X t+1 l ⊂ X l and X t+1 u ⊂ X t u are selected randomly. In our implementation, X t+1 u consists of equal amount of data from each class to avoid the trivial solution of predicting the dominant class for unlabeled data in successive iterations. We empirically found that τ in the range (0.7, 0.95) is suitable for different datasets, as it promises dominant class structure while adopting to moderate expression intensities.

C. Label update

The ground-truth distribution of the labeled data is cautiously modified to make the model adapt to expression samples of different intensities. It is carried out by updating the class distribution based on the model prediction. We apply label smoothing on the model predictions for the labeled data and utilize them for further parameter update, which we call successive label smoothing. In contrast, the predicted class probabilities for the unlabeled data are directly used for model training.

1) Updating q t l : By performing successive label smoothing on labeled data, the model adapts to the expression samples of various intensities. However, incorrect predictions on X l can accumulate error and reduce model performance in subsequent iterations. It is important to maintain the prediction confidence of labeled data, while learning the necessary information from the unlabeled data. In order to achieve that, we scrutinize p t l after each iteration and force the model to rectify its prediction errors on X l . We maintain the confidence of the model by forcing the model to correctly predict the expression class with a prediction score greater than certain threshold (α), which is given by,

q t+1 l (k|x) = ⎧ ⎪ ⎨ ⎪ ⎩ p t l (k|x), if { max k p t l (k|x) > α and argmax k p t l (k|x) = y } g(p t l (k|x)), otherwise (4)
where

g(p t l (k|x)) = α, if k = y 1-α K-1 , if k = y.
(

) 5
Essentially, this means that we use the prediction probabilities as true distribution if the model predicts the correct expression class (y) with a probability above threshold α. The failure cases are assigned with ground-truth distribution with label smoothing (as shown in equation ( 5)). Thus, α determines the minimum probability that can be assigned to the correct class.

2) Updating q t u : The ground-truth distribution of unlabeled data is updated to its predicted probabilities, i.e., q t+1 u (k|x) = p t u (k|x). Thus, the model sees the unlabeled data with corresponding updated ground-truth distribution after each update.

The model is updated from θ t to θ t+1 in a supervised manner using {X t+1 l , X t+1 u } and the corresponding updated ground-truth probabilities {q t+1 l , q t+1 u }. This forces the model to generate closely similar probabilities every time it accepts a particular sample as input. Intuitively, the model will train itself to correctly classify the supervised data, while incorporating the variations from the unlabeled data into the model. Here the parameter α controls the prediction confidence of labeled data. Choosing the value of α ≈ 1 restricts the model to learn for definite dominant expressions. However, as discussed before, data of different expressionintensities might have a similar label and the value of α should be in the range 0.6 to 0.9, in order to allow the model to adapt to different intensities.

D. Assigning Sample Importance

The presence of wrongly annotated or inappropriate training data hinders the model from obtaining optimal parameters and negatively affects the model performance. Such samples are assigned with low sample importance or sample weight, thereby avoiding their effect on the model update. This is carried out by penalizing the loss incurred by the labeled training samples (X l ) by the corresponding sample weight (0 ≤ w i ≤ 1).

The loss function used at tth iteration is given by

L = xi∈X t l w t i f i (θ t ) + xi∈X t u f i (θ t ). (6)
Comparing with equation ( 1), here we minimize the weighted loss, where the sample weights are learned using the training loss. As we assume the selected unlabeled data is clean, the corresponding loss is not penalized. Further, we assume that all the labeled samples are equally important initially, i.e., w 0 i = 1, ∀x i ∈ X l . After training the initial model θ * , the sample weights are updated in every iteration t based on the loss values incurred by individual sample. Further, we use the overall loss obtained for the validation set, L t val = xi∈X val f i (θ t ), to decide the worthiness of the sample weights. Notice that the validation loss is computed without the sample weights.

We update w t i if the trend of average validation loss starts increasing, i.e., L t val > L t-1 val , where L t val = 1 T t t=t-T +1 L t val is the average validation loss over previous T iterations. We used T = 3 in all experiments. The weight update is carried out by

w t+1 i ← w t i -Δw t i ; where ( 7
)
Δw t i = c. tanh f i (θ t ) -f avg (θ t ) σ . ( 8
)
Here f avg (θ t ) is the average training loss at tth iteration, whereas 0 ≤ c ≤ 1 and σ > 0 are scalars that control the weight update in consecutive iterations. Note that Δw t i can take both positive and negative values, subsequently decreasing or increasing the sample weights. Δw t i takes positive values if the loss incurred due to a sample is higher than the average loss, and vice versa. We further clip the sample weights in the range [0, 1] to avoid unpleasant computations.

IV. EXPERIMENTS AND RESULTS

A. Implementation Details Datasets

Both in-the-lab (CK+ [START_REF] Lucey | The extended cohn-kanade dataset (ck+): A complete dataset for action unit and emotion-specified expression[END_REF], RaFD [START_REF] Langner | Presentation and validation of the radboud faces database[END_REF]) and in-the-wild (RAF [START_REF] Li | Reliable crowdsourcing and deep localitypreserving learning for expression recognition in the wild[END_REF], AffectNet [START_REF] Mollahosseini | Affectnet: A database for facial expression, valence, and arousal computing in the wild[END_REF]) datasets are used in our experiments. The in-the-wild datasets contain data collected from uncontrolled environments and thus cover real-world expressions with various facial poses, illuminations, emotion intensity, occlusion, and other factors. Whereas the in-the-lab datasets are mostly posed in a controlled environment and contain exaggerated expressions of frontal faces.

All these datasets (except CK+) contain annotation for seven expressions, namely: anger, disgust, fear, happy, neutral, sad, and surprise. CK+ [START_REF] Lucey | The extended cohn-kanade dataset (ck+): A complete dataset for action unit and emotion-specified expression[END_REF] contains 327 sequences annotated with 7 expression labels (six basic emotions and contempt). The contempt class is excluded in our experiments as we aim for cross-database evaluation and comparison purposes [START_REF] Liu | Facial expression recognition via a boosted deep belief network[END_REF]. Following the literature [START_REF] Liu | Facial expression recognition via a boosted deep belief network[END_REF], [START_REF] Ding | Facenet2expnet: Regularizing a deep face recognition net for expression recognition[END_REF], [START_REF] Meng | Identity-aware convolutional neural network for facial expression recognition[END_REF], we use the 10-fold cross-validation while using the first frame as neutral and the last three frames of each sequence of CK+ as the corresponding expression label. The frontal faces (1407 samples) are used from RaFD dataset for our experiments. We conduct five-fold cross-validation for RaFD evaluation.

AffectNet and RAF are collected from the internet by using certain emotional terms in various search engines. RAF [START_REF] Li | Reliable crowdsourcing and deep localitypreserving learning for expression recognition in the wild[END_REF] contains manually annotated images, out of which 12271 are listed in RAF-train and 3068 samples in RAF-test. Affect-Net [START_REF] Mollahosseini | Affectnet: A database for facial expression, valence, and arousal computing in the wild[END_REF] has around 400000 labeled data in AffectNet-train and 5000 samples in AffectNet-test. The agreement between two annotators in AffectNet is found to be 60%, which explains the complexity and subtlety of the expressions in this dataset. Following the experimental settings of [START_REF] Li | Patch-gated cnn for occlusionaware facial expression recognition[END_REF] and [START_REF] Kervadec | Cake: Compact and accurate k-dimensional representation of emotion[END_REF], we only use the seven classes for these datasets.

Preprocessing

Preprocessing steps involve face detection (using MTCNN [START_REF] Zhang | Joint face detection and alignment using multitask cascaded convolutional networks[END_REF]) and face alignment, in order to position both eyes at a fixed distance parallel to the horizontal axis. The training set is augmented using slight zooming, horizontal flipping, less than 10% vertical and horizontal shifting, as well as rotating the images randomly in the range of ±10 degrees. For both these in-the-wild datasets, we use the aligned images provided by the dataset developers. We normalize the training data to zero mean and unit variance in all experiments.

Network

We use the pre-trained VGG-Face model proposed by Parkhi et al. [START_REF] Parkhi | Deep face recognition[END_REF] initially introduced for face recognition. It consists of thirteen convolutional layers followed by two fully connected layers. The aligned faces are re-sized to 224 × 224 resolution and feed to the CNN models. We set the softmax layer to the number of expression classes (in our case seven: anger, disgust, fear, happy, neutral, sadness, and surprise). The initial model weights are obtained from the pretrained VGG-Face model, achieving knowledge transfer. We use Adam optimizer with the suggested weights β 1 = 0.9, β 2 = 0.999 and a learning rate of 1e-4 in our model. All models are trained using a batch size of 32. The experiments are carried out using NVIDIA 1080 GPU with CUDA to improve the speed.

Parameter Settings

We conduct several experiments by varying from 0.05 -0.2, and varying both τ and α in the range 0.6-0.9 to select the suitable values of corresponding parameters. Empirically we find = 0.05, τ = 0.9 and α = 0.7 to be adequate for all the experiments, irrespective of the database type and mode of evaluation. The weight update parameters are also set empirically to c = 0.3 and σ = 2. In all our experiments, we replace 2% of training data with high confidence unlabeled samples. We use 10% of the train data as the validation data in all experiments. TABLE III: Cross-dataset accuracy for seven expression classes. Note that we utilized the test dataset as the unlabeled data.

(a) without ST and SW X X X X X X X The average accuracy obtained for different datasets by the proposed method for self-training (ST) and/or sample weight assignment (SW) is presented in Table I. We report the accuracy for four different conditions based on the utilization of ST and SW.

As can be seen from Table I, the accuracy of the framework without ST and SW is the lowest for all datasets. Importantly, one can notice performance improvements by using either of ST or SW alone. This clearly demonstrates the effectiveness of ST and SW in emotion recognition. When ST and SW are applied together, we obtained the best performance in all datasets. This proves the effectiveness of combining the two approaches. As can be observed from Table I, we obtain close to perfect performances in both the in-the-lab datasets. However, the accuracy for the in-the-wild datasets is comparatively very poor. We obtain an accuracy of 57.37% only in AffectNet dataset.

Using in-the-lab datasets as unlabeled data

We conduct experiments to improve the performance of inthe-wild datasets by leveraging the information from in-thelab datasets. Therefore, we report the performance of RAF and AffectNet in two conditions: (1) the test split of the same dataset is used as the source for unlabeled data, and (2) one of the in-the-lab dataset is utilized as the unlabeled data source. For example, to evaluate the performance of AffectNet, we use AffectNet-train for training the model while evaluating with AffectNet-test. However, the unlabeled data used during model training can come from either (1) AffectNet-test or (2) CK+ or RaFD. Both cases are reported in Table II. Here all the results are obtained using both ST and SW.

From Table II, we observe performance improvement in both RAF and AffectNet by using the in-the-lab datasets as the source of unlabeled data. We obtain 84.94% and 62.25% accuracy on RAF and AffectNet respectively. The model performance is especially improved by 5% for AffectNet by using unlabeled samples from RaFD. The presence peak intensity samples in in-the-lab datasets might be the reason behind this performance boost. Our model selects the high confidence unlabeled samples with the corresponding predictions as true distribution. This assumption is mostly true for in-the-lab datasets as the chances of peak expressive images getting classified into correct classes is high with a fair performing model. Thus, the unlabeled samples selected in the process drives the model to accurately present the expression prototypes while learning the variations of the in-the-wild labeled data.

The classification results on some AffectNet samples are presented in Figure 2. One can observe that the predictions for different correctly classified samples (see Figure 2a) resemble the emotion intensities. The increase in prediction scores for happy and sad classes are demonstrated in the top row of Figure 2a. We believe that the ability of the model to adapt to the sample intensity is due to two factors: (1) successive label smoothing, and (2) using the predicted class probabilities as true distribution for unlabeled samples.

Figure 2b and 2c displays the samples wrongly classified by our model. However, the predicted classes in Figure 2b are more appropriate than the ground-truth annotation. Similarly, Figure 2c represents the images with blended emotions, where the model predicts the second dominant class. The presence of noisy annotation in test split (AffectNet-test and RAF-test) makes the evaluation of such datasets more difficult.

The effectiveness of the SW is demonstrated in Figure 3. Our model automatically assigns lower sample weights to the noisy samples as shown in the top row in Figure 3. The samples wrongly annotated with happy class are also assigned with low sample weight. This demonstrates the robust training of the model against inconsistently labeled data. In addition to the noisy data, the model assigns low weight to the difficult samples with lower emotion intensity.

C. Cross-Database Evaluation

Table III reports the performance of seven class classification for cross-dataset evaluation. We present the results in the ascending order of dataset size (CK+ <RaFD <RAF <AffectNet). One interesting trend that can be observed from Table III is that the models trained with large datasets perform well on small ones. This has also been observed in [START_REF] Liu | Facial expression recognition via a boosted deep belief network[END_REF] and [START_REF] Lopes | Facial expression recognition with convolutional neural networks: coping with few data and the training sample order[END_REF], where this trend is considered as the effect of fewer variations in training data due to low sample size. Furthermore, the occlusion, pose and illumination variations are absent in-the-lab datasets; thus, it is difficult to train the complex model architecture without causing significant overfitting.

The effect of ST and SW is also validated by the crossdatabase performance as shown in Table III. As can be observed, the performance of the model improves by using ST, which further improves by using both ST and SW. For instance, the AffectNet → RaFD model performance is improved from 84% to 94% when both ST and SW are implemented. In the same way, RaFD → CK+, RAF → RaFD, and RAF → AffectNet sees an performance gain of 7%, 20%, and 2% respectively among others. The assigned sample weights for happy and sad class in ascending order.)

D. Comparison with Other Methods

We compare our results with recent CNN-based methods and state-of-the-art results. All the results reported here are average classification accuracy for the seven expression classes. Our results outperform the state-of-the-art in both the in-the-lab datasets. As these datasets contain frontal faces with peak expressions, the performance of the framework is close to perfect. However, the accuracy drops drastically when in-the-wild datasets are considered. The proposed method outperforms the state-of-the-art (in Table VIb) at a margin of 4% for AffectNet dataset. However, the performance achieved in IPA2LT [START_REF] Zeng | Facial expression recognition with inconsistently annotated datasets[END_REF] is better than the proposed method for RAF dataset (see Table VIa). Note that samples from both AffectNet and RAF are combined during the training process in [START_REF] Zeng | Facial expression recognition with inconsistently annotated datasets[END_REF], which might provide the model an upper hand. Methods Accuracy BAE-BNN-3 [START_REF] Sun | An efficient unconstrained facial expression recognition algorithm based on stack binarized auto-encoders and binarized neural networks[END_REF] 96.93 TLCNN+FOS [START_REF] Zhou | Action unit selective feature maps in deep networks for facial expression recognition[END_REF] 97.75 Carcagni et al. [START_REF] Carcagnì | Facial expression recognition and histograms of oriented gradients: a comprehensive study[END_REF] 98.5 Proposed 99.71 (a) RAF dataset Methods Accuracy CAKE [START_REF] Kervadec | Cake: Compact and accurate k-dimensional representation of emotion[END_REF] 68.9 DLP-CNN [START_REF] Li | Reliable crowdsourcing and deep localitypreserving learning for expression recognition in the wild[END_REF] 74.2 Vielzeuf et al. [START_REF] Vielzeuf | An occam's razor view on learning audiovisual emotion recognition with small training sets[END_REF] 80 PG-CNN [START_REF] Li | Patch-gated cnn for occlusionaware facial expression recognition[END_REF] 83. 57.31 CAKE [START_REF] Kervadec | Cake: Compact and accurate k-dimensional representation of emotion[END_REF] 58.1 AlexNet [START_REF] Mollahosseini | Affectnet: A database for facial expression, valence, and arousal computing in the wild[END_REF] 58 Proposed 62.25

V. CONCLUSIONS In this paper, we proposed a unified semi-supervised learning model for expression recognition which uses (a) self-training, exploiting information from abundant unlabeled data, (b) successive label smoothing, allowing the model to adapt to the emotion intensities and perform well on low intensity data, as well as (c) sample weight assignment, in order to avoid learning from noisy and inconsistent data. We experimentally validated the performance improvement with self-training and sample weight assignment. Experiments conducted on four public datasets indicate (i) excellent model performance out-performing state-of-the-art for most datasets, (ii) ability to learn different expression intensities, even when trained with categorical samples, (iii) ability to learn from inconsistent and noisy data, (iv) significant performance improvement on in-the-wild datasets by leveraging the information from the in-the-lab datasets, and (v) large performance gains in cross-database performance.

Fig. 1 :

 1 Fig. 1: Workflow of the proposed expression recognition method.

  (a) Correctly recognized samples from AffectNet dataset. Prediction scores are provided below the images which closely resemble the emotion intensities. (b) Samples for which model predication is more appropriate than the ground-truth. (c) Mis-classified samples. The predicted class is the second dominant emotion for the blended expression samples.

Fig. 2 :

 2 Fig. 2: Classification examples from AffectNet dataset.

Fig. 3 :

 3 Fig. 3: Training samples from RAF data. The value below the images depicts the sample importance learned by our framework. (Top row: Noisy annotations for happy class. Low sample importance neglects the effect of noisy inconsistent samples on model learning. Middle and Bottom row:The assigned sample weights for happy and sad class in ascending order.)

TABLE I :

 I Comparison of average classification accuracy for different combinations of self-training (ST) and sample weight assignment (SW).Here the unlabeled samples come from the test-split of the same dataset. The best accuracy is reported in bold.

	Datasets	without ST and SW	with ST only	with SW only	with both ST and SW
	CK+	99.19	99.19	99.27	99.43
	RaFD	99.29	99.5	99.49	99.71
	RAF	83.44	83.6	83.83	84.61
	AffectNet	55.48	56.62	57.31	57.37

TABLE II :

 II Performance improvement on RAF and AffectNet datasets by using in-the-lab datasets as unlabeled data source.

	(using both ST and SW)
	(a) RAF dataset		(b) AffectNet dataset
	Unlabeled data source Accuracy	Unlabeled data source Accuracy
	RAF-test	84.61	AffectNet-test	57.37
	CK+	84.81	CK+	61.54
	RaFD	84.94	RaFD	62.25

TABLE IV :

 IV Comparison of classification accuracy on CK+.

	Methods	Validation settings	Accuracy
	LOMo [34]	7 class	95.1
	IACNN [7]	7 class	95.37
	BDBN[17]	6 class 1	96.7
	DTAGN [4]	7 class	97.25
	PPDN [5]	6 class	97.3
	Facenet2expnet [6] 6 class	98.6
	PPDN [5]	7 class	99.3
	Proposed	7 class	99.43

TABLE V :

 V Comparison of classification accuracy on RaFD.

TABLE VI :

 VI Performance comparison on RAF and AffectNet.

Eight-fold cross-validation is performed.

Trained with both AffectNet and RAF train set

This work was funded by the French Government (National Research Agency, ANR), under Grant ANR-17-CE39-0002.