4.1 Simulation blocks
Fig. 4-1 The block diagram of a simulation system
The complete block diagram of simulation system is shown in Fig. 4-1. First, the forward acceleration and angular rate of vehicle are the inputs of car dynamic model and linear acceleration and angular rate of center of gravity of vehicle along X, Y and Z axes with respect to global coordinate are given. However, the specific force that driver experiences, the input of washout filter, happens when car is moving. It is a non-inertial coordinate. We need to transform the global coordinate into non-inertial coordinate with a point in car as reference point. The driver seat is selected. After transformation, the specific force is passed into washout filter and motion trajectory of Stewart platform is given. Then, the corresponding lengths of six actuators and the force of each actuator are derived by inverse kinematics and inverse dynamics.
Next, we transform the length of the actuator into the corresponding current signal.
The force given by inverse dynamics is the input of electro-hydraulic system. The hydraulic system is controlled by PID controller to reach the desired position. Finally, we use forward kinematics to transform the length of six actuators into Cartesian position and orientation of the Stewart Platform.
4.2 The electro-hydraulic system [19]
X
) ( X Fl
Fig. 4-2 Schematic diagram of electro-hydraulic control system
As shown in Fig. 4-2, the pump converts its energy of rotation into a flow. The flow is usable to the output device, hydraulic actuator. The relief valve sets a maximum pressure value in the system. When the pressure value exceeds the maximum value, oil is dumped to the tank to relieve the pressure. The servo valve controls or changes the flow into the hydraulic actuator. The input of the servo valve is current, i . From orifice law the load flow rate QI of the servo valve is given by
l s
I Ki P i P
Q = −sgn() (4.1)
where the input current i is limited by maximum input current imax imax
i ≤ . (4.2)
Pl is the load pressure across the cylinder. Let A be the piston ram area, Ct be the total leakage coefficient, Vt be the total volume of the valve and the cylinder chamber, β be the bulk modulus of the oil and X& be the velocity of the piston.
According to the continuity equation of the servo valve and cylinder chamber, we can obtain the formula as
l t l t
I V P
P C X A
Q )
(4 + β +
= & . (4.3)
And if we neglect the Coulomb friction between the piston and its sleeve, we can obtain the equation of motion of the piston as
F X B X M
APl = &&+ & + (4.4)
where B is the viscous damping coefficient, F is the external load disturbance which can be obtained from inverse dynamics.
4.3 Simulation results Case 1：
Forward acceleration is 5 m/s2 and it lasts for 4 seconds. Then forward acceleration decreases to 0 in one second. As shown in Fig. 4-3, yaw rate is zero. Fig.
4-3 and Fig. 4-4 show the response of linear acceleration and angular rate of center of gravity of vehicle along each axis with respect to global coordinate.
0 1 2 3 4 5 6 7 8 9 10
-3 -2 -1 0 1 2 3 4 5
Time (sec)
Linear acceleration (m/sec2)
X axis Y axis Z axis
Fig.4-3 Linear acceleration along x, y, and z axis
0 1 2 3 4 5 6 7 8 9 10
-0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08
Time (sec)
Angular rate (deg/sec)
X axis Y axis Z axis
Fig.4-4 Angular rate along x, y, and z axis
Once we make sure of the driving situation, we get the position and orientation of Stewart platform through coordinate transformation and washout filter. Then, we can get the corresponding length of each actuator by inverse kinematics. The simulation results are shown in Fig.4-7 and Fig. 4-8. They also include the real response of each actuator. Next, we can get the force of each actuator by inverse dynamics. The simulation results are shown in Fig. 4-5 and Fig. 4-6.
0 1 2 3 4 5 6 7 8 9 10
0 500 1000 1500 2000 2500 3000
Time (sec)
Force (N)
Actuator 1 Actuator 2 Actuator 3
Actuator 1 Actuator 2
Actuator 3
Fig.4-5 The force of actuator1, 2 and 3
0 1 2 3 4 5 6 7 8 9 10
0 500 1000 1500 2000 2500
Time (sec)
Force (N)
Actuator 4 Actuator 5 Actuator 6
Actuator 4
Actuator 5
Actuator 6
Fig.4-6 The force of actuator 4, 5 and 6
0 1 2 3 4 5 6 7 8 9 10 0.56
0.58 0.6 0.62 0.64 0.66 0.68 0.7 0.72 0.74 0.76
Time (sec)
Actuator Length (m)
planned real response
Actuator 1
Actuator 2
Actuator 3
Fig.4-7 The length of actuator 1, 2, and 3
0 1 2 3 4 5 6 7 8 9 10
0.54 0.56 0.58 0.6 0.62 0.64 0.66 0.68 0.7 0.72
Time (sec)
Actuator Length (m)
planned real response
Actuator 4 Actuator 5
Actuator 6
Fig.4-8 The length of actuator 4, 5, and 6
From Fig.4-7 and Fig.4-8, we can obtain the length errors of six actuators which are shown in Fig.4-9 to Fig.4-10. The simulation results show that the length error of each actuator is below 2 millimeter.
0 1 2 3 4 5 6 7 8 9 10
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8x 10-3
Time (sec)
Actuator Length Error (m)
Actuator 1 Actuator 2 Actuator 3
Fig.4-9 The length error of actuator 1, 2 and 3
0 1 2 3 4 5 6 7 8 9 10
0 0.2 0.4 0.6 0.8 1 1.2 1.4x 10-3
Time (sec)
Actuator Length Error (m)
Actuator 4 Actuator 5 Actuator 6
Fig.4-10 The length error of actuator 4, 5 and 6
Finally, Fig. 4-11 through Fig. 4-16 show the results when we use forward kinematics to transform the real lengths of six actuators into corresponding position and orientation of Stewart platform. In Fig. 4-11, we see that the platform moves forward about 0.11 meter and is pulled back to the initial position. The purpose of this trajectory is to give the pilot the feeling of moving forward and keep the platform in maximum workspace. When linear acceleration is decreasing, the platform moves backward to do the same thing.
0 1 2 3 4 5 6 7 8 9 10
-0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15
Time (sec)
Position along x axis(m)
real trajectory planned trajectory
Fig.4-11 Motion trajectory of Stewart platform along x axis
0 1 2 3 4 5 6 7 8 9 10
-1 -0.5 0 0.5 1 1.5x 10-3
Time (sec)
Position along y axis(m)
real trajectory planned trajectory
Fig.4-12 Motion trajectory of Stewart platform along y axis
0 1 2 3 4 5 6 7 8 9 10
0.52 0.525 0.53 0.535 0.54 0.545 0.55 0.555
Time (sec)
Position along z axis(m)
real trajectory planned trajectory
Fig.4-13 Motion trajectory of Stewart platform along z axis
0 1 2 3 4 5 6 7 8 9 10
-0.15 -0.1 -0.05 0 0.05 0.1
Time (sec)
Angle along x axis(degree)
real trajectory planned trajectory
Fig.4-14 Rotation angle of Stewart platform along x axis
In Fig. 4-15, we can see that the angle along y axis ( pitch ) is about -11 degrees and lasts for 3 seconds. In this trajectory, the car keeps doing the motion of linear acceleration. In order to let the pilot feels realistic, the platform rotates and uses gravity to give the pilot the corresponding specific force. When the motion of linear acceleration disappears, the platform rotates to its initial posture.
0 1 2 3 4 5 6 7 8 9 10
-12 -10 -8 -6 -4 -2 0 2
Time (sec)
Angle along y axis(degree)
real trajectory planned trajectory
Fig.4-15 Rotation angle of Stewart platform along y axis
0 1 2 3 4 5 6 7 8 9 10
-0.04 -0.02 0 0.02 0.04 0.06 0.08 0.1
Time (sec)
Angle along z axis(degree)
real trajectory planned trajectory
Fig.4-16 Rotation angle of Stewart platform along z axis
Fig. 4-17 and Fig. 4-18 show the errors of position and orientation of motion trajectory. From Fig. 4-17, the position errors along x, y, and z axes are below 1.5 millimeter. And the rotation angle errors along three axes are below 0.15 degree.
0 1 2 3 4 5 6 7 8 9 10
-1.5 -1 -0.5 0 0.5 1 1.5x 10-3
Time (sec)
position error (m)
x axis y axis z axis
Fig.4-17 Position error of motion trajectory
0 1 2 3 4 5 6 7 8 9 10
-0.1 -0.05 0 0.05 0.1 0.15
Time (sec)
Angle error (degree)
x axis y axis z axis
Fig.4-18 Rotation angle error of motion trajectory
Case 2：
The second simulation case is that a car moves forward and turns left at the same time. Fig. 4-19 and Fig. 4-20 show the linear acceleration and angular rate of center of gravity of vehicle along each axis with respect to global coordinate. Forward acceleration is 5 m/s2 and it lasts for 4 seconds. Then forward acceleration decreases to 0 in one second. As shown in Fig. 4-20 yaw rate is 0.15 and it lasts for 4 seconds.
Then yaw rate decreases to 0 in one second.
0 1 2 3 4 5 6 7 8 9 10
-3 -2 -1 0 1 2 3 4 5
Time (sec)
Linear acceleration (m/sec2)
X axis Y axis Z axis
Fig.4-19 Linear acceleration along x, y, and z axis
0 1 2 3 4 5 6 7 8 9 10
-0.1 -0.05 0 0.05 0.1 0.15
Time (sec)
Angular rate (deg/sec)
X axis Y axis Z axis
Fig.4-20 Angular rate along x, y, and z axis
Once we get the linear acceleration and angular rate of center of gravity of vehicle, we can obtain the corresponding motion trajectory of Stewart platform through coordinate transformation and washout filter. Then, the length of each actuator is given by inverse kinematics. The simulation results shown in Fig.4-23 and Fig.4-24 are the corresponding lengths of six actuators. Besides, the force of each actuator are shown in Fig. 4-21 and Fig. 4-22.
0 1 2 3 4 5 6 7 8 9 10
0 500 1000 1500 2000 2500
Time (sec)
Force (N)
Actuator 1 Actuator 2 Actuator 3
Actuator 1 Actuator 2
Actuator 3
Fig.4-21 The force of actuator 1, 2, and 3
0 1 2 3 4 5 6 7 8 9 10
0 500 1000 1500 2000 2500
Time (sec)
Force (N)
Actuator 4 Actuator 5 Actuator 6 Actuator 4
Actuator 5
Actuator 6
Fig.4-22 The force of actuator 4, 5, and 6
0 1 2 3 4 5 6 7 8 9 10 0.56
0.58 0.6 0.62 0.64 0.66 0.68 0.7 0.72 0.74 0.76
Time (sec)
Actuator Length (m)
planned real response
Actuator 1
Actuator 2
Actuator 3
Fig.4-23 The length of actuator 1, 2, and 3
0 1 2 3 4 5 6 7 8 9 10
0.54 0.56 0.58 0.6 0.62 0.64 0.66 0.68 0.7 0.72
Time (sec)
Actuator Length (m)
planned real response
Actuator 4 Actuator 5
Actuator 6
Fig.4-24 The length of actuator 4, 5, and 6
From Fig. 4-23 and Fig. 4-24, we can obtain the length errors of six actuators which are shown in Fig.4-25 and Fig.4- 26. The simulation results show that the length error of each actuator is below 1.6 millimeter.
0 1 2 3 4 5 6 7 8 9 10
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6x 10-3
Time (sec)
Actuator Length Error (m)
Actuator 1 Actuator 2 Actuator 3
Fig.4-25 The length error of actuator 1, 2, and 3
0 1 2 3 4 5 6 7 8 9 10
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6x 10-3
Time (sec)
Actuator Length Error (m)
Actuator 4 Actuator 5 Actuator 6
Fig.4-26 The length error of actuator 4, 5, and 6
Finally, Fig. 4-27 through Fig. 4-32 show the results that we use forward kinematics to transform the actual lengths of six actuators into corresponding position
and orientation of Stewart platform. From Fig. 4-27, we see that the platform moves forward about 0.11 meter and is pulled back to the initial position. The purpose of this trajectory is to give the pilot the feeling of moving forward and keep the platform in maximum workspace. However, the car turns left at the same time. From Fig. 4-32, we can see that the platform rotates along z axis until yaw rate is zero. The simulation results are pretty conform the driving situation.
0 1 2 3 4 5 6 7 8 9 10
-0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15
Time (sec)
Position along x axis(m)
real trajectory planned trajectory
Fig.4-27 Motion trajectory of Stewart platform along x axis
0 1 2 3 4 5 6 7 8 9 10
-0.015 -0.01 -0.005 0 0.005 0.01 0.015
Time (sec)
Position along y axis(m)
real trajectory planned trajectory
Fig.4-28 Motion trajectory of Stewart platform along y axis
0 1 2 3 4 5 6 7 8 9 10 0.52
0.525 0.53 0.535 0.54 0.545 0.55 0.555
Time (sec)
Position along z axis(m)
real trajectory planned trajectory
Fig.4-29 Motion trajectory of Stewart platform along z axis
0 1 2 3 4 5 6 7 8 9 10
-0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15
Time (sec)
Angle along x axis(degree)
real trajectory planned trajectory
Fig.4-30 Rotation angle of Stewart platform along x axis
0 1 2 3 4 5 6 7 8 9 10 -12
-10 -8 -6 -4 -2 0 2
Time (sec)
Angle along y axis(degree)
real trajectory planned trajectory
Fig.4-31 Rotation angle of Stewart platform along y axis
0 1 2 3 4 5 6 7 8 9 10
-0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
Time (sec)
Angle along z axis(degree)
real trajectory planned trajectory
Fig.4-32 Rotation angle of Stewart platform along z axis
Fig. 4-33 and Fig. 4-34 show the errors of position and orientation of motion trajectory. From Fig. 4-33, the position errors along x, y, and z axes are below 1.5 millimeter. And the orientation errors along each axis are below 0.15 degree.
0 1 2 3 4 5 6 7 8 9 10
-2 -1.5 -1 -0.5 0 0.5 1 1.5x 10-3
Time (sec)
position error (m)
along x axis along y axis along z axis
Fig.4-33 Position error of motion trajectory
0 1 2 3 4 5 6 7 8 9 10
-0.1 -0.05 0 0.05 0.1 0.15
Time (sec)
Angle error (degree)
along x axis along y axis along z axis
Fig.4-34 Rotation angle error of motion trajectory
Chapter 5 Conclusions
In this thesis, we have analyzed a simulator based on Stewart platform. First, a car dynamic model was obtained and it was made sure of what kind of driving situation to be simulated. Then, we could obtain linear acceleration and angular rate of the center of gravity of the vehicle along x, y, and z axes. Using dynamics in non-inertial coordinates, force of a reference point in car was given. It was passed into washout filter and trajectory of Stewart platform was obtained. The force that each actuator has to be taken, external disturbance, is known by inverse dynamics of Stewart platform once we get the motion trajectory of platform. Because high external disturbance causes oscillations of each actuator, we designed a PID controller to decrease the effect of high external disturbance. We have known that I controller has good performance on decreasing oscillation in high frequency and steady state error.
And D controller has performance on decreasing transient state error but it is poor at dealing with noise in high frequency. So, we have to choose a I controller with high gain and a D controller with low gain. The simulation results also showed that the oscillation can be almost ignored.
Herein only a simple dynamic model of vehicle has been used. In the future, a more complex model including the character of pedal or other factor such as air resistance or rough road can be considered. Combination with the technology of virtual reality, let people steeped in the powerful effect of sound and image. Moreover, the dynamic models of vehicle can be changed to other transportation suck as airplane, boat, or truck. In this way the simulation will be more realistic and simulation situation can be more diversified.
In this thesis, we designed a PID controller to control Stewart platform and got pretty good performance. However, the gain of PID controller may be not suitable one day because the oldness of mechanism. Furthermore the hydraulic system is a highly non-linear system. Many nonlinear control methods such as adaptive control, fuzzy control, or robust control can be applied on it.
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