Experimental Results

After having performed computer simulation, we also conducted real experiments to realistically demonstrate the feasibility of the proposed output feedback control scheme based on our proposed real-time forward kinematics solution using nonlinear observers.

The experiments are also categorized into two groups. The first part is to demonstrate the

0 0.2 0.4 0.6 0.8 1

-0.1 -0.05 0 0.05 0.1 0.15

Time (sec)

Translation along x axis (m)

(d) x-axis with uncertainty

Actual Desired

with uncertainty

observer performance, whereas the second part is to validate our proposed output feedback control. In the first experimental group, we employ tilt sensors and off-line Newton-Raphson algorithm as means to extract posture state of the moving platform.

Note that the tilt sensors are used to measure the pitch and roll angles, whereas the Newton-Raphson iterative algorithm can provide a rather precise forward kinematics solution after measuring the leg lengths by using LVDT (linear variable displacement transducers).

Observer Performance

The motion in this experiment is to execute a pitch angle rotation following a z-axis translation. The platform starts moving along z-axis for 10 cm and then experiences a 5 degree (0.0873 rad) oscillation in pitch angle with frequency 2 rad/sec. Figure 3.8(a) shows the z-axis trajectory of the estimated state and that calculated by the precise Newton-Raphson iteration algorithm and Figure 3.8(b) shows the difference between these two trajectories. It is shown that larger estimation error appears in the stage of upward heaving of the platform, but relatively less during the period of regulation. After post study, it is found that the rough 0.15 cm estimation error is due to the parametric error of platform height. Figures 3.9(a) and 3.9(b) show the estimation results vs. the results calculated by the precise Newton-Raphson iteration algorithm, whereas Figures. 3.9(c) and 3.9(d) illustrate the estimation result vs. the result measured by the tilt sensor. The estimation error

from comparison with the tilt sensor measurement, exhibiting phenomenon of oscillation about the origin, is mainly caused by the sensor noise. The estimation error from comparison with Newton-Raphson method exhibits a larger amount when the platform starts rotation. This is because the Newton’s iteration forward kinematics is not to measure the platform motion directly; instead, it is through calculation from measurements of the leg lengths. Thus, some little parametric errors could cause such phenomenon, and the larger range the platform moves about, the larger errors the estimation will experience.

However, the estimation error due to comparison with tilt sensor will not proportionally vary with the motion range, which suggests that the estimation should coincide with the real motion of the platform well.

(a) (b)

Figure 3.8 Experimental results of observer performance in z-axis translation: (a) estimation and Newton-Rhapson algorithm, (b) estimation error

0 5 10 15

(a) (b)

(c) (d)

Figure 3.9 Experimental results of observer performance in rotation about y-axis: (a) Estimation vs. Newton-Rhapson algorithm, (b) Estimation error due to Newton’s method, (c) Estimation vs. tilt sensor measurement, (d) Estimation error due to tilt sensor

Output Feedback Control

The final experiment we want to conduct is to show the performance of the output feedback control incorporating the observer based real-time forward kinematics solution.

The motion is a mixture of z-axis and x-axis translations. The platform first moves along z-axis for 5 cm from the initial position and is regulated at 0.6 m. Following the z-axis motion is a sinusoidal motion in x-axis with amplitude 5 cm and frequency 2 rad/sec. The experimental results are shown in Figure 3.10, where the dashed curves represent the command trajectories whereas the solid curves signify the estimated trajectories. Because

0 5 10 15

the feedback signals are from the incorporated nonlinear observer, apparently some time lag between the estimated trajectories and the desired ones have been revealed. In fact, this experiment has adopted a concept of boundary-layer modification for the sliding mode control so that significant vibrations (due to chattering) are successfully eliminated by appropriate choice of layer width. Up to now, this experiment successfully validates the feasibility of the proposed output feedback control built upon the observer based real-time forward kinematics solver.

(a) (b)

Figure 3.10 Experimental results of output feedback sliding mode control in translation along: (a) z-axis, and (b) x-axis

Chapter 4

Control of a Hydraulic Actuated Stewart Platform

4.1 Electro Hydraulic Actuator

The platform is suspended by six actuators which are hydraulic power elements. The piston of the actuator can extend by injection of hydraulic fluid. Varied lengths of the actuators create different postures of the moving platform. Each actuator is composed of two parts: cylinder and servovalve. Figure 4.1 shows the schematic diagram of an electrohydraulic servovalve, which exhibits inner control loop, in which the relation between the valve opening xv and the current (or voltage) input u is often modeled as a first-order system in two-stage electrohydraulic servovalves in some literatures. In this work, the servovalve dynamics are neglected since the parallel manipulator works in low frequency range and the servovalve opening xv is proportional to the control input.

Therefore, in the sequel, we will use the relation, xv=kau where ka is the servo amplifier gain and u is the current signal.

Figure 4.1 Schematic diagram of an electrohydraulic servovalve

Figure 4.2 Schematic diagram of hydraulic servo actuator

Figure 4.2 shows the schematic diagram of a hydraulic system composed of a four-way servo valve and a cylinder. With the supply pressure Ps and the spool displacement xv, the servovalve can create the chamber pressure P1, P2, and the related flow rates Q1, Q2 to push the piston of the cylinder. The system exhibits nonlinear dynamics.

Considering the leakage and the compressibility, the dynamics can be expressed as follows [28]: volumes. If the initial position of the piston is at the bottom of the cylinder, then V10=0 and V20=A2l0 where l0 is the piston stroke. The displacement of the piston is defined as xl=l-l0.

The fluid flow rates in the chambers of cylinder are related to the spool valve displacement and the pressure, and such relation can be expressed as

1

where Cd is the discharge coefficient, w is the servovalve area gradient, and ρ is the fluid density. When the valve opening xv is in positive direction, the supply pressure Ps charges pressure to P1 and another chamber’s pressure P2 leaks to the return pressure Pr. When xv is in the opposite direction, Ps charges pressure to P2 and then P1 leaks to Pr. This relation can be seen in Figure 4.2. The direction of displacement xv determines which side of the cylinder is to be charged or leaked. For a general four-way valve, the flows are equal, that is Q₁=Q₂ =Q x P P_L( , ,_{v 1 2})

.

Substituting this condition in (4.3) and let Pr=0 (the return

The force generated by the rod of cylinder depends on the pressures and is given as

1 1 2 2.

where

In this control scheme, input u of overall system is the actuator’s input current, hence the platform input is the force generated by the actuators. Thus, the mechanical dynamics (2.4) becomes

( ) ( , ) ( ) ^T( ) .

M q q C q q q G qɺɺ+ ɺ ɺ+ =J qτ (4.9) The input of the platform τ is composed of six forces FL1~FL6 generated by actuators and is

given as

[

1 6

]

T

L L

F F

τ = ⋯

.

Combining all the 6 actuators, the dynamics can be written

as this work since the parallel manipulator works in low frequency range, and hence we adopt a simplified model x_vⁱ =k u_{a i}ⁱ . Under this circumstance, (4.10) becomes

the input ut, we first note that matrix gA(xL) is always invertible, so that the left-hand-side (LHS) of (4.12) can be made equal to any form we like. More of this will be explained in the next section.

4.2 Controller Design

4.2.1 Backstepping Controller

Conventional joint-space control (as shown in Figure 4.3) of the Stewart platform is to perform the leg length tracking by first converting the desired posture command to the desired leg length commands via inverse kinematics. Here, the observer provides information of the actual posture so that we can control the platform in the task-space directly. In this paper, we propose a control scheme to achieve posture tracking while taking into account both the platform dynamics and the hydraulic actuators, as shown in Figure 4.4.

It is worth mentioning that the dynamics of the hydraulic actuators, as revealed in Section 4.1, are quite complex. Traditional control methods of hydraulic Stewart platform intend not to include the actuator dynamics, but it may not attain the level of high performance control. Therefore, the proposed backstepping controller to be described shortly will try to resolve this and to provide a more complete control set-up.

u ^{τ y}

( )

_d

h q q

_d

+

−

Figure 4.3 Conventional joint-space control scheme

u τ

τ

d

y

q ˆ

q

d

+

−

e

τ ^ɶ

Figure 4.4 Observer-based backstepping Control Scheme

In this section, a backstepping control law is proposed to control the hydraulic-actuated parallel manipulator. The posture states of the platform are non-measurable so that an observer is incorporated in this control scheme. In other words, instead of the real posture q(t), the estimated posture ˆ( )q t is used in the control law. The

notations introduced to simplify the derivation are defined as ˆ the actuator force τ . The convergence of e means estimation ˆq approaches to the desired posture qd, but it does not guarantee that the real state q converges to qd. This issue will be discussed in the stability analysis of the overall system.

The backstepping control law includes the platform controller and actuator controller, where the former is a function of the estimated posture since the real one is unavailable.

The platform controller is given as

ˆ ˆ ˆ ˆ ˆ

( ) ( ) ( , ) ( )

T

d J q M q qr C q q qr G q K e K sp d

τ = ^{− } ɺɺ + ɺ ɺ + − − ^ (4.14)

whereas the actuator controller is designed as

1 ˆ

( , ) ( ) ( , ) ^T( )

A L L A L Lt V t d

f x xɺ +g x Q x P =τɺ −J q s−K_ττɶ (4.15) where K_p, K_d and K_τ are positive definite diagonal matrices. It is worthy to note that the

right-hand-side (RHS) of (4.15) will be available from (4.14) and the observer design in Section 3.1, which implies that such actuator controller is a feasible one. The estimated state qˆ is obtained from the nonlinear observer (3.6) designed for the platform dynamics

(3.5). The nonlinear observer is expressed as

ˆ ( )ˆ ( )ˆ 1( ) [ˆ ( )]ˆ xɺ= f x +g xτ+Q⁻ x K y h x−

(4.16) where τ is taken as the input of the observer in this control scheme. The design of the observer is shown in Section 3.1.

In the first platform controller (4.14), we design the desired force τ_d to make the platform tracks the desired posture, and then backstep to design the actuator controller (4.15) with τ_d being treated as the command of the actuators.

Since the functions f_A(x xɺ and _L, _L) g_A(x_L) are known with the measured actuator lengths xL, the input signal ut, proportional to the valve opening xV, can be calculated from (4.5) and (4.15). It is obvious that the sign of the flow rate QL of the single hydraulic

actuator is the same as that of xv, and hence the real input u can be explicitly expressed as

1/ 2

which is the current signal given to the valve. Up to now, we can clearly see that the input u can be explicitly specified for the corresponding actuator controller as shown in (4.15).

Note that the input u mentioned in Chapter 2 & 3 is the actuator’s force which inputs to the

mechanical dynamics (2.4) of the platform. Here, the system includes the actuator’s dynamics such that the input u of overall system is current signal which inputs to the servovalve.

4.2.2 Stability Analysis

In the stability analysis, we will show that the proposed controller employing the estimated state from the aforementioned nonlinear observer, which solves the forward kinematics problem in real time, will render the entire Stewart platform system stable, and in turn the tracking error will converge exponentially.

Theorem 4.1 Consider the dynamics model of the electrohydraulic Stewart platform

consisting of platform dynamics as described in (3.1)~(3.5) and actuator dynamics as shown by (4.12) with the nonlinear observer as described by (4.16). The backstepping control law as shown in (4.14) and (4.15) is applied to such complex cascaded system such that the platform posture will track the desired trajectory of the posture state, namely,

[ ^T, ^{T T}]

d d d

x = q qɺ . Then, if the resulting actuator output τ is well within its limit range, then the resulting closed-loop system will ensure asymptotical convergence of the state tracking errors e eɺ, while maintaining the overall system’s exponentially stability. ▓

Proof. Consider the observer described by (4.30), and then the posture error dynamics

expressed in terms of s will be

[ ]

The error dynamics of the actuator can also be expressed as

( )ˆ JT q s K_τ

τɺɶ= − − τɶ (4.20)

which arises from (4.10) and (4.15).

Let V1 be defined as

also be easily computed by

The property of the drift-observability stated in Section 3.1 leads to the fact that the map ( ) another function V2 which includes the force error of the actuators is defined here as

2 To account for the observation error, we define another function V3 as

1 3

V = ΣzɶT ⁻ zɶ (4.26)

where the positive definite symmetric matrix Σ satisfies an H^∞ Riccati-like inequality as described in [23]. Given that the drift-observability is fulfilled and the input τ is bounded (Remark 4.1), then the time derivative of V3 is proven [23] to be

1

3 2 ^T 2 3

Vɺ ≤ − ξzɶ Σ⁻ zɶ= − ξ⋅V (4.27)

where ξ>0 is a positive constant. Let V=V₁+V₂+V₃, and then the time derivative of the integrated Lyapunov function V is given by the summation of (4.23), (4.25) and (4.27) as

ˆ ˆ

tend to zero exponentially, we can conclude that our tracking goal will be achieved while maintaining system’s stability. This completes the proof. ▓

Remark 4.1 Boundedness of the input is another convergence condition of the proposed

observer. The input to the platform is the force output of the actuators. After applying the

relation as shown in (4.4), the output of each actuator in (4.6) becomes

1 1 2( 1) ( 1 2) 1 2 . simulation are given in Tables 3.1 and 4.1, and the reference trajectory is chosen as in Table 4.2. The simulation results illustrated in Figure 4.5 show the performance of the proposed control scheme, whose control objective is to ensure convergence of the estimation and tracking errors. Such simulation results reveal the effectiveness of the observer as well as

the controller.

Table 4.1 Simulation data of Hydraulic Actuators Hydraulic Actuator Parameters

A₁ Cylinder area 8.0425×10^-4 m² A₂ Cylinder area (rod) 3.9408×10^-4 m²

β_e Bulk modulus 1.379 Gpa

l_o Nominal leg length 0.6079 m

Cd discharge coefficient 0.61

w servovalve area gradient 0.02 m

ρ Fluid density 800 Kg/m³

Ps Supply presure 6.5 Mpa

Table 4.2 Reference trajectory of simulation

Sway 0.05sin(7t) m

Surge 0.05cos(7t) m

Heave 0.85m

Pitch 0.262sin(5t) rad

Roll 0.262cos(5t) rad

Yaw 0 rad

Specifically, the simulation is carried out with the initial state

[ ]

(0) 0 0 0.55 0 0 0^T

q = which is the original position when the linear actuators are fully contracted and with an integration step of 1.0 ms. The observer gain K used here is to assign the same set of eigenvalues λ= − × 15 3 3²^T for 6 blocks (such as that described in (3.28)) and the controller parameters

3 3 3 that the moving platform has 0.1 m error in the x and z axis translation at t=0. The tracking error converges to zero in z-axis as shown in Figure 4.5, where the solid lines and dash lines show the desired and the actual trajectories, respectively. In γ-axis, the steady state error is about ±0.01 rad. Due to larger amplitude motions with respect to x-, y-, α-, and β-axes respectively, larger tracking errors can be observed in these axes. In the steady state, there are tracking errors about ±0.004 and ±0.005m in x- and y-axes, respectively, and ± 0.03 rad in α- and β-axes. The simulation involves the model with hydraulic actuators and the nonlinear observer so that the amount of computation is relatively heavy. Figure 4.6 shows the estimation errors in x- and z-axes in this simulation. Both estimation error in these two axes are set at 0.1 m in initial conditions, and then converge to zero in about 0.1sec, as shown in Figure 4.6.

Figure 4.5 Simulation of position tracking

Figure 4.6 Simulation of estimation errors

4.4 Experimental Results

The proposed observer and controller are experimentally implemented on the 6-DOF hydraulic manipulator as shown in Figure 4.7. The actuators connected to the base and the moving platforms of the manipulator are six single-rod hydraulic cylinders. Each hydraulic actuator has the supply pressure 6.52×10⁶ Pa and can cause the maximum force 5244 N.

With the LVDTs and pressure sensors, we can acquire the information of the length and force of each actuator. The results are separated to two parts: first one is to demonstrate the observer performance and the second part is to validate the control scheme. In the first part, we compare the observed trajectories with those obtained from the Newton-Raphson algorithm to validate the observer’s correctness. Note that the Newton-Raphson iterative algorithm can provide a rather precise forward kinematics solution by measuring the leg

4.4.1 Observer Performance

The observer is built to retrieve the platform posture from the lengths of the six

cylinders, and therefore its inputs include

experiment, the LVDTs provide joint lengths whereas the pressure sensors can provide

piston pressures that can be transformed to actuator forces with simple calculation (4.

The motion in the first experi

0.0873 rad and frequency 2 rad/s. Figure

line) compared with the trajectory calculated by the Newton

The difference from comparison with Newton

when the platform is changing direction of motion. The peak of the difference is about

±0.01 rad. Some small parametric errors in dynamics may increase the estimation error.

Figure 4.7 The experimental setup

Observer Performance

The observer is built to retrieve the platform posture from the lengths of the six

cylinders, and therefore its inputs include the joint lengths and the actuator forces. In the

experiment, the LVDTs provide joint lengths whereas the pressure sensors can provide

piston pressures that can be transformed to actuator forces with simple calculation (4.

The motion in the first experiment is to execute a pitch angle rotation

0.0873 rad and frequency 2 rad/s. Figure 4.8 shows the performance of estimation (dotted line) compared with the trajectory calculated by the Newton-Raphson iteration algorithm.

omparison with Newton-Raphson method exhibits a greater value when the platform is changing direction of motion. The peak of the difference is about

rad. Some small parametric errors in dynamics may increase the estimation error.

The observer is built to retrieve the platform posture from the lengths of the six

the joint lengths and the actuator forces. In the

experiment, the LVDTs provide joint lengths whereas the pressure sensors can provide

piston pressures that can be transformed to actuator forces with simple calculation (4.6).

rotation with amplitude shows the performance of estimation (dotted

Raphson iteration algorithm.

Raphson method exhibits a greater value

when the platform is changing direction of motion. The peak of the difference is about

rad. Some small parametric errors in dynamics may increase the estimation error. For

the larger range the platform moves, the larger error the estimation will experience. The comparison with with Newton-Raphson algorithm shows the precision of the estimation.

Figure 4.8 Experimental results of observer performance in rotation about y-axis

4.4.2 Controller Performance

The second part is to show the performance of the observer-based backstepping control which is as shown in Figure 4.4. The actuator forces obtained from pressure sensors are used as input to the observer and are also used in the actuator controller simultaneously. In

5 6 7 8 9 10

(b) Difference between estimation and Newton’s method

the experiments, feedback signals need to be filtered since the parameters of the hydraulic dynamics may enlarge the noises greatly and cause the control signal obstructed by the noises.

The motion in first experiment is to track a ramp signal with slope π for π seconds and then to regulate at 10 m along z-axis. Figure 4.9 shows the results of regulation in z-axis. In Figure 4.9(a), the estimated trajectory tracks and converges to the desired trajectory. The tracking error is shown in Figure 4.9(b) where the maximum error is about 0.007 m. To obtain good position tracking performance and to avoid larger vibration at the same time, we increase the position gains Kd, Kp and reduce the force gain K^τ. The comparison of the desired force and the actual force of one actuator is shown in Figure 4.9(c).

0 5 10 15

Figure 4.9 Experimental results of regulation

In the second case, the platform first moves along z-axis for 5 cm from the initial position and is regulated at 0.52 m. The following is a sinusoidal motion with 5 cm amplitude and 2 rad/s frequency. The experimental results are shown in Figure 4.10.

Because the feedback signals are generated from the incorporated nonlinear observer, apparently some time lag between the estimated trajectories and the desired ones have been revealed. In this experiment, we also decrease the force feedback gain to avoid large

Because the feedback signals are generated from the incorporated nonlinear observer, apparently some time lag between the estimated trajectories and the desired ones have been revealed. In this experiment, we also decrease the force feedback gain to avoid large

在文檔中 應用於虛擬實境之六自由度並聯式液壓平台之姿態 控制與沖淡濾波器設計 (頁 65-0)