Dynamic modeling of a 2-dof parallel electrohydraulic-actuated homokinetic platform

Mechanism and Machine Theory 118 (2017) 1–13

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Research paper

Dynamic modeling of a 2-dof parallel electrohydraulic-actuated homokinetic platform William Schroeder Cardozo∗, Hans Ingo Weber PUC-Rio, R. Marquês de São Vicente, 225, RJ, Brasil

a r t i c l e

i n f o

Article history: Received 21 January 2017 Revised 17 July 2017 Accepted 24 July 2017

Keywords: Thrust vector control Electrohydraulic system Inertial measurement unit Kinematics Dynamics

a b s t r a c t Thrust vector control (TVC) is used to control the attitude of spacecraft. The most widespread method to connect the nozzle assembly and rocket consist of a gimbal joint in the case of liquid-propellant or a flexible bearing in the case of solid-propellant. Traditionally, two electrohydraulic linear actuators that are controlled by servovalves tilt the rocket nozzle. However, in this work, TVC is treated as a robotic system and the connection between the platform and the fixed base is achieved using a constant velocity joint. Instead of a servovalve, a novel proportional digital hydraulic valve is proposed. The position feedback is provided through a real-time attitude estimation based on the orientation matrix by using an inertial measurement unit (IMU). Platform kinematics modeling is performed. Dynamics modeling is developed using Newton-Euler formulation. Hydraulics modeling is presented with the description of the electrohydraulic system. Due to the linear behavior of the control valve, a simple pure proportional controller is considered. An experimental validation showed that the proposed system achieves good position tracking without instabilities. © 2017 Elsevier Ltd. All rights reserved.

1. Introduction In this paper, a two degrees of freedom (2-dof) parallel platform with two electrohydraulic actuators (EHAs) is developed for thrust vector control (T VC). T VC has been widely used for the attitude control of aircrafts [1]. In the TVC application, two hydraulic or two electromechanical actuators are usually used to deflect the rocket nozzle [2]. This deflection generates a moment that acts on the aircraft to control its attitude [3]. Aerospace applications require a high power density, high dynamic performance, robustness and overload capacity; hence, EHAs are a common choice [4]. An EHA system is composed of a valve to modulate the fluid flow rate based on an input signal and a hydraulic cylinder that converts the flow rate into linear displacements [5]. The TVC system must orient the nozzle on a desired trajectory for the vehicle and maintain the controllably during disturbances. Gust and shear wind are the main disturbances [6]. When using liquid propellant to steer a vehicle along its trajectory, a gimbaled thrust engine is the most common option [7]. In this work, a low-cost EHA platform is built. Usually, much of the cost of an electrohydraulic servo-system is due to the servovalves. For cost reduction, a novel Proportional Digital Hydraulic Valve (PDHV) is proposed to control the oil flow in the actuator chambers. A hydraulic servovalve requires an expensive super-clean fluid system [8], whereas the PDHV operates using an inexpensive fluid filter system. Another cost reduction is achieved using a constant velocity joint (CVJ)

∗

Corresponding author. E-mail addresses: [email protected] (W.S. Cardozo), [email protected] (H.I. Weber).

http://dx.doi.org/10.1016/j.mechmachtheory.2017.07.018 0094-114X/© 2017 Elsevier Ltd. All rights reserved.

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W.S. Cardozo, H.I. Weber / Mechanism and Machine Theory 118 (2017) 1–13

with an Inertial Measurement Unit (IMU) instead of a gimballed assembly with two angular transducers. The CVJ choice also reduces the moment on the centerline of the rocket [9]. Lazic and Ristanovic [10] proposed a test bench for a TVC system based on a fixed base robotic system. They have presented an analysis of the kinematics and a control strategy for a gimballed 2-dof parallel platform with two EHAs. Two angular transducers provide the angular position of the gimbal, a real-time DAQ system calculates the actuators position for the control algorithm, and two hydraulic servovalves receive the analog signal from the controller. Their experiments show a good position tracking. Wekerle [6] presented the requirements of an actuation system for TVC systems and presented in [11] a 2-dof mockup of a rocket motor nozzle with two EHAs. The actuators performances are identified in a mass-spring test bench. Ghosh et al. [12] developed a 2-dof parallel hydraulic actuated system for a heave-and-pitch motion simulator. Two lowcost commercial solenoid proportional valves control the hydraulic actuators. These types of valves have a dead-band and non-linear behavior [13]. Ghosh et al. have considered different types of controllers, and the self-tuning fuzzy proportional– integral–derivative (PID) with bias control showed the best performance. The PID controller was found to have the worst response [12]. Yu et al. [14] developed a novel rotary spool direct drive servovalve with axisymmetric flow. Experiments and computational fluid dynamics (CFD) simulations show the reduction of forces on the valve spool. The angular position estimation of an IMU is a common topic of research, with the variations of the Kalman filter (KF) representing the most common option for sensor fusing between accelerometer and gyroscope data [15–17]. Accuracy improvements are achieved if the physical constraints are considered during the angular position estimation [18–19]. KF is able to model the noise of the IMU and obtain an optimal estimation of the attitude [20]. However, these stochastic approaches of the KF require a recursive calculation, making its real-time implementation unacceptable for low-cost hardware. A frequency-based approach, for example, a complementary filter (CF), requires a smaller computational cost [21]. In a CF, the accelerometer data passes through a low-pass filter and the gyroscope data passes through a high-pass filter. The KF and CF have a maximum angular error of approximately less than 1° for most of the applications [22–26]. Pennestrí et al. [27] conducted a kinematic analysis and a numerical simulation of a CVJ with geometric errors. Kimata et al. analyzed the CVJ dynamics by taking into account the friction between moving parts [28]; furthermore, the results are validated with simulations and experiments [29]. These works focus on the powertrain problem. In the present work, the platform developed is similar to that of Lazic and Ristanovic [10]; however, a CVJ with an IMU is used instead of a gimbal with two rotary encoders and a novel PDHV is used instead of commercial servovalves. These differences reduce the system cost by more than one order of magnitude. The proposed controller is a simple proportional (P) controlled controller; however, it has an improved performance compared with the controller of Gosh et al. [12] because of the linear behavior of the PDHV. Accelerometer and gyroscope sensor data fusion is performed using a CF considering the kinematic constrains of the CVJ. In addition, the CF is based on attitude matrices (usually called rotation matrix) instead of quaternions, Euler angles, or Cardan (Tait-Bryan) angles, which are the typical choices. In this paper, first a kinematic description of the platform is presented. Hence, in the dynamic modeling, a full analysis is performed using the Newton-Euler formulation to obtain the equations of motion. The system model consists of the dynamic model of the platform, a proposed model for the hydraulic system and the proportional controller model. The method to calculate the attitude of the platform from accelerometer and gyroscope data is depicted afterwards. Finally, the simulation and experimental results are shown, also validating the system operation, and then, some concluding remarks are presented at the end. Fig. 1 shows an overview of the platform. 2. Kinematics Fig. 2 shows the kinematic scheme of the parallel two degrees of freedom platform (2dof) with a load. One of the shafts of the constant velocity joint (CVJ) is fixed to the fixed base, and the other shaft is attached to a moving platform. Point B is the center of the fixed base, and O is the center of the CVJ. Universal joints connect the hydraulic actuators to the base, and ball joints connect it to the platform. Points C31 and C32 are the centers of the actuators ball joints. C21 and C22 are the centers of the actuator universal joint crossheads. P and L are the centers of mass of the moving platform and load, respectively. C is the center of the top surface of the moving platform. Two references frames are defined: F is fixed to the base and S is attached to the moving platform. The rotation between the frames are summarized by the following indication, θ ( p)

F

F ixed Base (x,y,z )

−−→

S

Moving Plat f orm x ,y ,z

(

)

where p is the Euler vector and θ is the angle of rotation around p, which superposes F and S. Angle θ is the angle that tilts the platform, i.e., the angle between OC and the vertical. The position of p changes in the fixed (x, y) plane and can be written as follows:



p = cos ϕ

sin ϕ

T

0 ,

where ϕ is the angle between vector p and the x-axis.

(1)

W.S. Cardozo, H.I. Weber / Mechanism and Machine Theory 118 (2017) 1–13

3

Fig. 1. Homokinetic platform overview.

Fig. 2. Kinematic scheme.

The attitude matrix F TS leads from S to F (also called the coordinate transformation matrix or rotation matrix), and allows two different interpretations: when using sequential rotations, a vector r can be written in coordinates of F as F r or in S as S r, and the transformation is: F r = F T S S r. In the direct rotation (Euler), a vector r in F is moved (rotating with θ around p) to a new position, as given in the fixed frame as F r. Before moving, F and S are overlapped and the vector can be written as S r. The attitude matrix multiplies S r to obtain F r after the rotation. Again, one has F r = F T S S r, but in a completely different manner. For the Euler vector itself, because we are rotating around it, we have F p = S p = p. The main property of this matrix is its orthogonality. Using geometric considerations, it is possible to write this matrix in terms of the Euler vector (unitary) and the rotation angle: F

T S = E + p˜ sin θ + (1 − cos θ ) p˜2 .

(2)

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Here, a special notation is used for an antisymmetric matrix (∼) of order 3. Because the vectors are (3 × 1) and the (3 × 3) antisymmetric matrix has 3 different elements, we can always associate a matrix p˜ to a vector p:

 

p=

px py pz



⇒ p˜ =

0 pz −py

−pz 0 px



py −px . 0

(3)

The angular velocity of a body is a result of the variation in time of the attitude matrix and can be defined as the antisymmetric matrix [30]: F F

ω˜ S = F T˙ S S T F .

(4)

Calculating the result for (2), the angular velocity of the moving platform is obtained; using body fixed coordinates (S), it is given by S F

ωS = θ˙ p + sin θ p˙ − (1 − cos θ ) p˜ p˙ ,

(5)

where the dots represent the local (S) derivative of the Euler vector. The position of the center of mass of the platform and the load it is carrying, using S coordinates, is given by:



S O rP

= 0

S O rL

= 0



0

lp

0

lL

T

T

,

(6)

.

(7)

In the fixed frame F, it results in F O rP

= F T S SO rP ,

(8)

F O rL

= F T S SO rL .

(9)

Eq. (10) describes the velocity of the center of mass of the moving platform, and Eq. (11) describes the velocity of the load, S

vP = SF ω˜ S SO rP ,

(10)

S

vL = SF ω˜ S SO rL

(11)

The lengths of the hydraulic actuators 1 (HA1) and 2 (HA1) are given by Eqs. (12) and (13), respectively:









la1 = CF21 rC31 ,

(12)

la2 = CF22 rC32 .

(13)

The longitudinal displacement speed of each EHA is given by the projection of the ball joint center velocity on a unit vector parallel to the rod using the dot product: F

C l˙a1 = F vCT31  21

rC

31 , F rC31  C21

F

C l˙a2 = F vCT32  22

rC

32 , F rC32  C22

(14)

(15)

where the velocity of the C31 and C32 points are given by

Fr O C31

F

vC31 = FF ω˜ S FO rC31 ,

(16)

F

vC32 = FF ω˜ S FO rC32 .

(17)

and FO rC32 are the position vectors of the ball joint centers C31 and C32 , respectively. The speed of the actuators is used in the numerical simulations to compute the friction force on the EHA. The linear and angular velocities of the platform are used according to their dynamics to calculate the linear and angular momentum of the moving platform. Before presenting the dynamic model, the hydraulic control system is discussed.

W.S. Cardozo, H.I. Weber / Mechanism and Machine Theory 118 (2017) 1–13

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Fig. 3. Hydraulic scheme.

3. Hydraulic control system Each platform actuator is controlled by one PDHV, as shown in Fig. 3. By changing the valve rotor angle θ V , the pressure drop and flow rate through the valve change, which change the pressure in the pipeline and in the actuator chambers, thereby generating a force that moves the piston. In Fig. 3, pS and QS are the supply pressure and flow rate, respectively. The arrows in the pipeline indicate the positive flow sense. p1 , p2 , p3 and p4 are the pressures on the valve connections. Q1 , Q2 , Q3 and Q4 are the flow rates. pA and pB are the pressures in chambers A and B, respectively; Ae and Ac are the areas of the embolus and the crow, respectively; Fa is the force acting on the rod; and l˙a is the piston speed. For a simpler kinematic analysis of the platform, la is the distance between the centers of the mounting joints of the actuators. From the measured platform position, the actual actuators position la is calculated. The controller compares the actual positions with the desired ones and changes the valve rotor angle θ V proportionally to the error. 4. Dynamics Because of the symmetries of the moving platform and the load, their inertia matrices about its centers of mass are given by

 S

0 I1P 0

0 0 , I3P

(18)

I1L 0 0

0 I1L 0

0 0 . I3L

(19)

IP =

 S

IL =



I1P 0 0



The vector of angular momentum of the moving platform and the load using the S frame about its center of gravity is introduced by S

HP = S IP SF ωS ,

(20)

S

HL = S IL SF ωS .

(21)

The linear momentum vectors of the platform and the load are given by Eqs. (22) and (23) using the S frame: S

G P = m P S vP ,

(22)

S

G L = m L S vL ,

(23)

where mP and mL are the platform and the load masses, respectively.

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Eq. (24) shows the angular momentum of the system composed of the moving platform and embarked load about the center of the main crosshead, which is a fixed point, using the S frame [30]. S

HO = S HP + SO r˜P S GP + S HL + SO r˜L S GL

(24)

Using Euler’s Law, Eq. (25) gives the torque S

SM

O

acting on the moving platform.

˜ S S HO MO = H˙ O + SF ω S

(25)

The torque acting on the moving platform is generated by the force vector of the actuators, by the gravity force and by the universal joint constrains, as shown in Eq. (26). S

MO = S T F



mP FO r˜P + mL FO r˜L

F



g + FO r˜C31 F Fa1 + FO r˜C32 F Fa2 + S MJ

where S TF is the rotation matrix that leads from the F frame to the S frame, F g = [ 0 S M is the torque due to the CVJ constraint, given by Eq. (27). J

(26) 0

−g ]T is the gravity vector and

 

S

MJ =

0 0 MJ

(27)

Using Eqs. (28) and (29), the angular accelerations are calculated as

θ¨ = ϕ¨ =

τ1 cos ϕ + τ2 sin ϕ + c3 sin θ ϕ˙ 2 c1

,

τ2 cos ϕ − τ1 sin ϕ − c2 ϕ˙ θ˙ c1 sin θ

(28)

(29)

where:

  τ1    τ1 = S T F mP FO r˜P + mL FO r˜L F g + FO r˜C31 F Fa1 + FO r˜C32 F Fa2 , τ1 2

2

(30)

c1 = I1P + m p l p + I1L + mL lL ,

(31)

c2 = 2c1 cos θ + (1 − cos θ )(I3P + I3L ),

(32)

c3 = c1 cos θ + (1 − cos θ )(I3P + I3L ).

(33)

Neglecting the actuators inertia, the forces of the HA1 and HA2 are given by F

F

Fa1 =

r

O C31  Fa1 , F rC31 

(34)

O

F

F

Fa2 =

r

O C32  Fa2 , F rC32 

(35)

O

where

Fr O C31

|FO rC31 |

and

Fr O C32

|FO rC32 |

are unit vectors in the HA1 and HA2 directions, respectively. Fa1 and Fa2 are the forces magnitudes

of the HA1 and HA2 [8], respectively, given by

Fa1 = pA1 Ae − pB1 Ac − Ff ,

(36)

Fa2 = pA2 Ae − pB2 Ac − Ff

(37)

where Ff is the friction force. In the analyzed conditions, the friction force is well represented as a function of the actuator speed using a Stribeck curve because the slip velocity on the actuator is significant and a relatively small amount of static/kinetic friction appears in the transition [31]. A more precise model, for example, the LuGre model, shows the time dependence of the friction and could be used in a more general condition or for real-time friction compensation [32]. In the present work, the friction force is obtained experimentally at constant actuator speed without a load; the fitted function used (continuous line) is depicted in Fig. 4 with the experimental data (vertical bars). The vertical bars have this shape because the experiment is performed for a discrete number of actuator speeds and the variation is due to noise and inaccuracy of the used model. Fig. 5 shows the valve opening over time for this experiment.

W.S. Cardozo, H.I. Weber / Mechanism and Machine Theory 118 (2017) 1–13

7

60

40

20

f

F (N)

0

-20

-40

-60

-80 -0.2

-0.15

-0.1

-0.05 0 0.05 Actuator speed (m/s)

0.1

0.15

0.2

Fig. 4. Actuator friction force.

Fig. 5. Valve opening in no-load experiment.

The pressure drop trough a PDHV is calculated using a variable-area orifice model [13],

Qi j = αDi j Ai j

2 

ρ







pi − p j sign pi − p j ,

(38)

where Qij is the flow rate between connections i and j; these indices vary from 1 to 4 according to the valve connection depicted in Fig. 3. α Dij is the discharge coefficient. Aij is the orifice area, which is a function of the valve rotor angle θ V . In this work, the pipeline pressure loss due to friction is negligible and is not considered. However, the pump is considered to be capable of maintaining a constant supply pressure during the experiments. This hypothesis was validated.

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W.S. Cardozo, H.I. Weber / Mechanism and Machine Theory 118 (2017) 1–13

Fig. 6. Chamber A flow in no-load experiment.

The continuity equation applied on the hydraulic scheme, considering an incompressible flow, gives the oil flow in the actuators chambers:

Qa = Q12 − Q23 ,

(39)

Qb = Q14 − Q43 .

(40)

Eqs. (41) and (42) show the relationships between the flow rate in the chambers and the speed of the actuators:

Qa = Ae l˙a ,

(41)

Qb = Ac l˙a .

(42)

During the experiment that identifies the friction force, the discharge coefficient from Eq. (38) is also estimated using Eqs. (38)–(42). Fig. 6 shows the oil flow in the actuator chamber A calculated using the speed measure in Eq. (41) (black line) and the fitted value for a set of discharge coefficients (grey line). Fig. 7 shows the oil flow in the actuator chamber B calculated using the speed measure in Eq. (42) (black line) and the fitted value the set of discharge coefficients (gray line). Hence, it is concluded that the chosen set of discharge coefficients and Eq. (38) can be used for oil flow modeling in the hydraulic system. The experimental results presented in this section are for HA1, and they are very similar for HA2. 5. Angular position measurement To compute the orientation of the platform, a 6-dof IMU is attached to the moving platform. The accelerometer data consist of the linear acceleration of the sensor measured in the embarked frame (S a),

 

S

S

Xa Ya Za

S

s= a− g=

(43)

where Xa , Ya and Za are the accelerometer data from the IMU. The gravity in the embarked frame S is given by



S

S

g= T

FF



− sin ϕ sin θ g = g cos ϕ sin θ . cos θ

(44)

W.S. Cardozo, H.I. Weber / Mechanism and Machine Theory 118 (2017) 1–13

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Fig. 7. Chamber B flow in no-load experiment.

Considering S a = 0 in Eq. (43), we have

  Xa Ya Za





− sin ϕ sin θ = g cos ϕ sin θ . cos θ

(45)

Hence, if θ = 0,ϕ from the accelerometer data is given by

ϕa = a tan 2(−Xa , Ya ).

(46)

For θ , two options are available:

θa = a tan 2(Ya /cos ϕ , Za ),

(47)

θa = a tan 2(−Xa /sin ϕ , Za ).

(48)

To be distant from the sine and cosine singularities, Eq. (47) is used only when kπ − π /4 ≤ ϕa ≤ kπ + π /4, where k is an integer. With ϕ a and θ a , the attitude matrix by the accelerometer data is presented as



F

TaS

=

1 − (1 − cos θa )sin ϕa (1 − cos θa ) sin ϕa cos ϕa − sin ϕa sin θa 2

(1 − cos θa ) sin ϕa cos ϕa 1 − (1 − cos θa )cos2 ϕa cos ϕa sin θa



sin ϕa sin θa − cos ϕa sin θa . cos θa

(49)

The attitude matrix is integrated using Eq. (4), F

˜ S. T˙ S = F T S SF ω

(50)

The IMU gyroscope gives SF ωS . Eq. (51) shows how the attitude matrix from the gyroscope is integrated, F

˜ S t, TgS = F T S + F T S SF ω

(51)

where the sample period is t = 0.02s. The sensor fusion is preformed using a complementary filter, F

S TCF = wF TgS + (1 − w )F TaS ,

where the gyroscope weight is w = 0.98.

(52)

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W.S. Cardozo, H.I. Weber / Mechanism and Machine Theory 118 (2017) 1–13 Table 1 Parameters. Parameter

Value

lP mP I1P I3P lL mL I1L I3L F g S O rC31 F O rC21 S O rC32 F O rC22

260.0 · 10−3 m 4.52 kg 75.7 · 10−3 kg m2 16.9 · 10−3 kg m2 421.1 · 10−3 m 17.32 kg 53.6 · 10−3 kg m2 45.8 · 10−3 kg m2 [0 0 −9.81]T m/s2 [−65 0 207.9]T · 10−3 m [−135 0 −30]T · 10−3 m [0 −65 207.9]T · 10−3 m [0 −135 −30]T · 10−3 m

Fig. 8. Simulated and experimental platform orientation.

In this work, the base is considered fixed. However, in a real TVC system, the attitude of the base also requires estimation. This analysis is beyond the scope of the present work. 6. Simulation and experiments The experimental data are compared with the numerical simulation and desired position. The trajectory is defined using the articulation angle θ and ϕ angle. In the first case, θ = 10◦ and the angle ϕ changes with constant angular velocity ϕ˙ = π rad/s from 0 to 2π . Table 1 shows the parameters of the platform and the load. Fig. 8 shows θ and ϕ from the experimental (Act) data compared with the numerical simulation (Sim) and the desired position (Des). The deviation between the simulated and experimental result for θ is 0.9° at maximum and 0.01° in average. The ϕ deviation is 4.9° at maximum and 1.1° in average. The error between the actual and the desired position is 0.7° for θ and 27.2° for ϕ in the average. In this case, Fig. 9 shows the actuators length, where the plots limits are the actuator limits. The average error between the simulated and the experimental actuators length results is 0.3 mm, and this error reaches 2.3 mm at maximum. The deviation from the desired position is 1.3 mm in the average and 12 mm at maximum. The top view of the actual table center trace is shown in Fig. 10 with the platform limits (Bound). The difference between the actual (or the simulated) position and the desired position is due the extremely simple control algorithm. This proportional control with higher gain leads to stability problems, mainly chattering.

W.S. Cardozo, H.I. Weber / Mechanism and Machine Theory 118 (2017) 1–13

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Fig. 9. Simulated and experimental platform actuator length.

Fig. 10. Top view of the platform table center trace.

Another case is analyzed to evaluate the accuracy of the system to achieve a desired position. In this case, the desired orientation of the platform alternates each second between (θ = 15◦ , ϕ = 0◦ ) and (θ = 15◦ , ϕ = 90◦ ), thus ϕ is a square wave with 45° of amplitude and 1 Hz frequency and θ is constant. Fig. 11 shows the numerical, simulated (Sim) and the experimental (Act) result for θ and ϕ with the desired (Des) as reference. This case reveals that after a settling time, the deviation between the actual and the desired position of the actuator is less than 0.9 mm in maximum, and 0.5 mm in the average. In this case, where a long time is given to reach the desired position, the velocity is not an issue.

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Fig. 11. Actuators length when alternating between two points.

7. Conclusion In this work, a 2-dof EHA platform was developed for a low-cost TVC system. A novel control valve was validated to control the actuator position. A real-time attitude estimation approach using an IMU and based on the orientation matrix was tested. In addition, the use of a CVJ as support for the moving platform was validated. In this paper, the kinematic and dynamic models of the platform were presented. The models of the electrohydraulic system and controller were also shown. These models were used to perform numerical simulations. With the aim of performing experiments, a real-time attitude estimation system was developed. The results showed that the presented model provides good insight into the behavior of the experimental system. The experimental validation showed that the platform with the novel control valve and the proposed attitude estimation follow the desired trajectories without instabilities during the tests. In the first case, the table center trajectory is circular. The results revealed that the table center draws the desired circle; however, there is a significant phase angle between the desired and the actual (or the simulated) result. This phase error is due the fully proportional controller algorithm, where the gain is limited by stability problems, mainly chattering. The actuators speed in this case reaches about 50% of the value that is achieved with a fully open valve. Therefore, the speed in this case is significant. In addition, to analyze the position accuracy of the system in a situation where the speed is not important, another case is considered. In the second case, it was demanded to the platform to alternate between two desired positions and it was given a long time for the system reach the desired position. The results for this case show a better accuracy of the system after a settling time, when compared with the circular trajectory. Therefore, the enhanced position accuracy in low speeds seems more interesting for upper stage TVCs in vacuum, where high precision and not velocity is the challenge. Acknowledgements The present work was carried out with the support of CNPq, National Council of Scientific and Technological Development - Brazil. References [1] C.B.F. Ensworth, Thrust vector control for nuclear thermal rockets, 49th Joint Propulsion Conference and Exhibit, 2013. [2] Y. Li, H. Lu, S. Tian, Z. Jiao, J.T. Chen, Posture control of electromechanical-actuator-based thrust vector system for aircraft engine, IEEE Trans. Indust. Electron. 59 (9) (2012) 3561–3571. [3] T. Wekerle, E.G. Barbosa, C.M. Batagini, L.E.V.L. Costa, L.G. Trabasso, Brazilian thrust vector control system development: status and trends; propulsion and energy forum, 52nd AIAA/SAE/ASEE Joint Propulsion Conference, Salt Lake City, UT, 2016. [4] Y. Li, Mathematical modeling and characteristics of the pilot valve applied to a jet-pipe/deflector-jet servovalve, Sens. Actuators A 245 (2016) 150–159. [5] A.G. Brito, Characterization of electrohydraulic actuators: technical and experimental aspect, Experimental Techniques, Society for Experimental Mechanics, 2015. [6] T. Wekerle, A. Brito, L.G. Trabasso, Requirements for an aerospace actuation system derived from the control design point of view, 23rd ABCM International Congress of Mechanical Engineering, Rio de Janeiro, RJ, Brazil, December 6–11, 2015.

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