This paper adopts the helicopter simulator at laboratory in Yanshan University which is shown in Fig. 1. The helicopter flight simulator is mainly composed of four modules: a helicopter simulation cockpit system, a virtual scene system, a dynamic motion system and an audio system. The simulator's haptic interface is shown in Fig. 2. The haptic interface is driven by two servomotors, and two disc speed reducers are used for speed regulation. The motor adopts Kollmorge CKM04 AC permanent magnet synchronous servo motors, and its corresponding AKD series servo driver supports Modubus TCP communication. The parameters of the servo motors and the reducers are shown in Tables 1 and 2. The hardware system composition of the haptic interface is shown in Fig. 3.

First, the computer is connected to the switch, and the switch is connected with two motor drivers respectively through Enternet port. Then, C++ programmed code has been written to establish communication between servo motor and computer through Modbus TCP protocol. When the driver operates the haptic interface, the computer can obtain the current electricity value from the servo motor at each time. Then, the driving torque of the servo motor is obtained. At the same time, the photoelectric encoder of the servo motor can measure the stick's position and speed at the current time, which can be used for the dynamic calculation of the haptic interface.

First, coordinate system for the haptic interface is established for the haptic interface, as shown in Fig. 4. The intersection point of the two motor axes is defined as origin O. The two motor axes are set as x axis and y axis respectively. z axis is perpendicular to the xy plane.

The free-body diagram is shown in Fig. 5. The angle θ, θx, θy formed between the haptic interface and the z axis is the output angle of reducer 1 and reducer 2 respectively. the relationship of the three parameters can be expressed as follows: 1θ˙2=θ˙x2+θ˙y2.

When the haptic interface moves, the control handle, the haptic interface and the base of the haptic interface move at an angle θ, the motor 1, the reducer 1 and the rotating mechanism move at an angle θy.

In Fig. 5, m is the mass of the haptic interface system, lG is the distance between the center of mass of the haptic interface and the origin, and l is the distance between the top of the stick and the origin. τa1 and τa2 are the output torques by motor 1 and 2 through the disc reducer respectively; τf1 and τf2 are the friction torques on the x axis and y axis of the servo system respectively. The manual force of the pilot in the direction of x axis, y axis and z axis defined as Fx, Fy, Fz.

The Lagrange method is adopted to conduct dynamic analysis of the system. The Lagrange function can be formulated as 2ddt∂Ek∂q˙i-∂Ek∂qi+∂Ep∂qi=Fi(i=1,2). In Eq. (2), Ek is the kinetic energy of the haptic interface system; Ep is the potential energy of the haptic interface system; q˙1, q˙2 in the haptic interface system represent the output angles θx, θy of the two reducers respectively.

The total kinetic energy Ek for the haptic interface system can be expressed as: 3Ek=12J1θ˙y2+12J2θ˙2=12J2θ˙x2+12(J1+J2)θ˙y2. In Eq. (3), J1 is the sum of the moment of inertia of motor 1, reducer 1 and rotating mechanism; J2 is the sum of the moment of inertia of the control handle, haptic interface and haptic interface base.

The potential energy of the haptic interface Ep can be expressed as: 4Ep=mglGcos⁡θ. In combination with Fig. 5 and Eq. (1), the following equation can be obtained: 5cos⁡θ=1/1+tg2θx+tg2θy. According to the comprehensive Eqs. (2)–(5), can be obtained: 6J2θ¨x-mglGtgθxsec⁡2θx(1+tg2θx+tg2θy)3/2=τ1+τa1-τf1(J1+J2)θ¨y-mglGtgθysec⁡2θy(1+tg2θx+tg2θy)3/2=τ2+τa2-τf2. In Eq. (6), τ1 and τ2 are the torques exerted by the pilot on the x axis and y axis during the control process respectively.

Especially, the torque output τa by the motor through the disc reducer is related to the reduction ratio i of the reducer and the output torque Ta of the motor, as: 7τa=iTa, Ta can be deduced from Eq. (7): 8Ta=1.5Pn[ψrIq+(Ld-Lq)IdIq]. In Eq. (8), Pn is polar logarithm; ψr is the flux chain of rotor magnetic pole; Ld, Lq is the armature inductance of axis d and axis q; Id, Iq is the current of d axis and q axis.

Since the armature inductance of d axis and q axis of the motor used same value, namely: 9Ld=Lq. From Eqs. (7), (8) and (9) can be obtained as follows: 10τa=1.5iPnψrIq. At this time, the torque and the q axis current have a linear relationship, which can be expressed as: 11τa=iKtIq. Combined with Eqs. (6) and (11), and using mglG replaced by the gravity moment TG, the following equation can be obtained: 12τ1=J2θ¨x-TGtgθxsec⁡2θx(1+tg2θx+tg2θy)3/2-i1KtIq1+τf1τ2=(J1+J2)θ¨y-TGtgθysec⁡2θy(1+tg2θx+tg2θy)3/2-i2KtIq2+τf2. Thus, the manual force of the pilot can be expressed as: 13F=FxFyFz=1l0cos⁡θycos⁡θx0-sin⁡θx-sin⁡θyτ1τ2=1l0cos⁡θycos⁡θx0-sin⁡θx-sin⁡θy⋅J2θ¨x-TGtgθxsec⁡2θx(1+tg2θx+tg2θy)3/2-i1KtIq1+τf1(J1+J2)θ¨y-TGtgθysec⁡2θy(1+tg2θx+tg2θy)3/2-i2KtIq2+τf2.

In the haptic interface structure, there are complex frictions within the servo motor and between the reducer. The friction torque generated by these frictions has a great influence on the accurate modelling of the haptic interface. Therefore, the friction torque at each moment must be accurately obtained to ensure the accuracy of the calculated force and torque at each moment.

The classic Coulomb friction model friction indicated that friction force is related to the positive pressure acting on the object. However, due to the lubrication in the servo system, the influence of viscous friction must be considered.

Stribeck proposed to model the servo friction as a combination of Coulomb friction and viscous friction. He has given a quantitative formulation for friction with respect to velocity (Balogh and Krstic, 2004). Similarly, the Stribeck formula can be converted into the formula of friction torque regarding motor speed as proposed in (Iwasaki et al., 1999): 14τf=[τc+(τs-τc)e-(ωωs)2]sgn(ω)+Bωω, In Eq. (14), ω is the motor speed; τc is Coulomb friction torque; τs is the maximum static friction torque; ωs is Stribeck speed; Bω is the coefficient of viscous friction.

The Stribeck curve is related to the speed of the servo system. In the low-speed area, it is affected by Coulomb friction and viscous friction, and the friction torque decreases. This area is called boundary lubrication friction area. With the increase of speed, the influence of viscous friction is much higher than Coulomb friction. At this time, the friction torque is approximately proportional to the rotational speed. This operation condition is defined as liquid lubrication condition.

There are two problems in applying the Stribeck friction model to the control of haptic interface force control: on the one hand, it is difficult to describe the change of haptic interface friction torque when the velocity is zero; on the other hand, the change of velocity near the zero point will cause a sudden change of friction torque, which will lead to a jitter in the reversing process. Therefore, approximate treatment of the low speed phase of the Stribeck friction model is required. This paper proposes to use a sigmoid function to deal with Stribeck friction model (Ciliza and Tomizuka, 2007).

The Stribeck friction torque model improved by sigmoid function can be expressed as follows: 15τf=1-e-γω1+e-γωτc+(τs-τc)e-(ωωs)2sgn(ω)+Bωω. According to the requirements of the model and accuracy, the value of γ is defined as 500 in the proposed model.

According to Eq. (13), the total force applied to the haptic interface is directly related to the magnitude of the gravity torque TG. In order to obtain an accurate model, it is necessary to identify the gravity torque TG. This is achieved with an experiment conducted on the motor 1 when its movement is on xz plane. Set the haptic interface to rotate at a constant speed, and then θ¨x is zero. The haptic interface was not moved in the y direction, so θy is zero. The driver has no additional force to the haptic interface, accordingly also τ2 is zero. Since the velocity is constant, τf1 is constant. In addition, sin⁡θx is relatively small in practical application, accordingly θx can be used instead. The experimental process is shown in Fig. 6 and Eq. (6) can be converted into: 16i1KtIq1+TGθx-τf1=0.

At this time, θx maintain a liner relationship with Iq1 have a linear relationship. Discretizing Eq. (16) one can obtain: 17θx(k)=τf1TG-i1KtIq1(k)TG. A following experiment consists in setting the motor speed at 40 rpm (Rotation per minute), make the haptic interface rotate at a constant speed, take the values of θx and Iq1 in multiple groups. A curve fitting of the results in such experiments is shown in Fig. 7. The identified gravity torque is 11.14 N m. The toolbox of Curve Fitting Tool in Matlab is used to get the result.

When we choose Stribeck friction model, it contains four unknowns: τc, τs, ωs and Bω, these four parameters need to be identified. Similarly, the motor 1 is set as the speed mode and a specific motor speed is given to make the haptic interface rotate at a constant speed. At the same time the driver has no additional force to the haptic interface, in this case, θy, θ¨xand τ1 are equal to zero. Therefore, according to Eq. (6), the following equation can be obtained: 18i1KtIq1+TGsin⁡θx-τf1=0. According to Sect. 2.1, the current value, rotation angle and rotation speed of the motor can be read at each moment. It is can be concluded from Eq. (18), the friction torque τf1 at the corresponding rotational speed θ˙x can be obtained. The value of friction torque τf1 can be obtained by the different output angle θ˙x.

Genetic algorithm is an algorithm designed according to Darwinian evolution theory: through continuous natural selection, survival of the fittest, so as to obtain the most adaptable results (Liu, 2006). The process of parameter identification mainly includes generation of initial population, selection of race parameters, setting of fitness function, selection, crossover, variation and termination.

The search scope of parameters τc, τs, ωs and Bω can obtained by tracing method. The iteration interval is [4–5], [4–5], [0–0.1] and [20–40] respectively. The population size was set as 100, and the crossover probability and mutation probability were set as 0.4 and 0.1 respectively. The selection of GA parameters has been carried out by considering the peculiarities of the specific identification problem as well as several preliminary identification tests. In particular, after preliminary identification tests we selected 0.4 as crossover parameter and 0.1 as variation parameters. The selected crossover parameter gives a preference to parental individuals on filial individuals. The selected variation value is motivated by a limited influence of variation on this specific parameter identification. Numerical simulations have been carried out also with other values of the crossover parameter (e.g. 0.4 and 0.8). Results led to similar outcomes of the identification procedure.

The multi-group friction torques obtained through Eq. (18) can be expressed as: 19τ^f1=τ^c+(τ^s-τ^c)e-(θ˙xω^s)2sgn(θ˙x)+B^ωθ˙x. In Eq. (8), τ^c, τ^s, ω^s and B^ω are the four parameters to be identified respectively, and τ^f1 is the theoretical friction torque calculated by the motor speed. Take the target function: 20J=∑i=1n(τf1-τ^f1)2. The actual identification task of genetic algorithm is to solve the minimum value of objective function.

The minimum target value is defined as 10-6 here, and the maximum evolutionary algebra is 10 000 generations. In each generation, the population is sorted in ascending fitness order, and the optimal solution of this generation is recorded as the elite. Parental individuals were selected in the fitness sequence for crossover, and the offspring were mutated. Finally, when the fitness of a certain generation of elite meets the requirements (J≤10-6) or the evolutionary algebra is more than 10 000 generations, the whole selection process of genetic algorithm is terminated.

The process of genetic algorithm is shown in Fig. 8, and its identification process is shown in Fig. 9. Figure 10 shows the comparison between the theoretical friction moment calculated by the identified parameters and the actual friction moment. The measurement result of the friction torque at the moment of θ˙x<0 is the same as the moment of θ˙x>0, but the symbol is opposite, so it was omitted. Here we only figure out the absolute value of the friction torque. The identified parameters at the end of program operation are shown in Table 1. The identification of friction model parameters in the y axis direction was the same as that in the x axis. The identification results are listed in Table 3. Several cases have been considered in the identification process for the friction torque model parameters to obtain a reliable mean value of all the parameters. Considering the characteristics of the proposed identification process we can estimate a confidence interval of 0.0001 % (the minimum target value is in Eq. 20), which can be considered suitable for the specific application, also in comparison with commonly available accuracy of commercial sensors.

According to Eq. (14), the Stribeck friction torque model can be simplified as: 21τf=τcsgn(ω)+Bθω. Firstly, the motor 1 is analyzed. When the pilot does not operate the haptic interface, Eq. (6) can be converted into: 22i1KtIq1+TGθx-τcsgn(θ˙x)-Bθθ˙x=J2θ¨x. Thus, the structure block diagram of Eq. (21) can be described and is shown as in Fig. 11.

After sorting out Fig. 11, the simplified structure block diagram of motor driven haptic interface is shown in Fig. 12.

According to Fig. 12, the continuous function can be expressed as: 23θs(s)i1KtIq1(s)-τc(s)=1J2s2+Bθs-TG. The zero order holder method is adopted for Z transformation in Eq. (22), and the following equation can be obtained: 24θx(z)i1KtIq1(z)-τc(z)=Z1-e-TssG(s)=1Bθ2+4J2TGz-1p2(z-ep2T)-z-1p1(z-ep1T)+1p1-1p2. In which, 25p1=-Bθ-Bθ2+4J2TG2J2p2=-Bθ+Bθ2+4J2TG2J2. By discretizing Eq. (24), the following equation can be obtained: 26θx(k)=(ep1T+ep2T)θx(k-1)-e(p1+p2)Tθx(k-2)+i1Ktp1p2Bθ2+4J2TG(Iq1(k-1)-d(k-1))⋅(p2-p1+p1ep2T-p2ep1T)+i1Ktp1p2Bθ2+4J2TG(Iq1(k-2)-d(k-2))⋅[p1ep1T-p2ep2T+(p2-p1)e(p1+p2)T]. In which, 27d(k)=τc+θ˙x>0τc-θ˙x<00θ˙x=0. In Eq. (27), τc+ represents positive Coulomb friction and τc- represents negative Coulomb friction.

Set the torque mode through the motor driver, set the specific current size, and drive the haptic interface to rotate. When the motor speed reaches a certain value, take multiple groups of current value Iq1 and rotation angle θx.

For the obtained data, the genetic algorithm in Sect. 3.2 can be used to identify the moment of inertia. The estimated J2 value among search range [0.1–1.1]. The population size was set as 100, and the crossover probability and mutation probability were set as 0.4 and 0.1 respectively. The selection of GA parameters has been carried out by considering the peculiarities of the specific identification problem as well as several preliminary identification tests. In particular, the selected crossover parameter has been defined to give a preference to parental individuals on filial individuals. The selected low variation value is motivated by a limited influence of variation on this specific parameter identification.

The multiple groups of rotation angles θ^x(k) calculated by Eq. (26) can be expressed as: 28J=∑k=1n(θx(k)-θ^x(k))2. The minimum target value is given as 10-4, and the maximum evolution algebra is 10 000 generations. The evolution of its optimal solution is shown in Fig. 13. The J2 of the recognized moment of inertia is 0.2887 kg m2.

Performming the same operation on the motor 2 as above, and the identified moment of inertia J1+J2 is 0.2989 kg m2. Therefore, J1 is 0.0102 kg m2.