To demonstrate the adaptability of the proceeding motion control algorithm in a six DOF motion simulator, as shown in Fig. 1.1, this study integrates the dynamics of several various vehicles, the collision detection algorithm, the VR programming techniques, the sound effects, the motion control system, the cabin operating system and the force feedback steering system, to perform a human-machine interactive motion in real time.
5.1 Dynamics of Vehicle
The dynamics of vehicle include operating mode, colliding mode and hopping mode. The operating mode must firstly calculate the engine’s torque output with respect to the engine’s rotation speed and the gear; then, the car’s size, weight, static sliding friction coefficient of wheels, dynamic sliding friction coefficient of wheels, drag force of wheels’ rolling friction with respect to the car speed, the wind drag force with respect to the car speed and the engine brake, all have to be considered; finally, suspension effects are also applied to the dynamics.
Colliding mode include the dynamics of car vs. wall colliding, car vs. car colliding and car vs.
other objects colliding. Hopping mode include the dynamics of starting hopping, hopping and grounding. Figs. 5.1 ~ 5.3 depict the results of these three dynamic modes.
5.2 Collision Detection
60
In order to make wheels a rtual ground, the wheels have to d
Virtual reality (VR) include the building of 3D models and textures of scene, data reading frame rate control, and special effects such as lighting, shadow, exp
5.5 Cabin Operating System and Force Feedback Steering System
system consists of the display system, lighting system, control button lways run on the surface of the vi
etect the ground and modify the positions of the wheels. Also to avoid the car crossing the virtual wall, another car or other objects, the car must detect if it collides the virtual wall, another car or other objects.
5.3 VR Programming and Sound Effects
of 3D model, drawing,
losion, smoke, spark, …, and etc. Sound effects consist of background music control, collision sound effect relative to the various collided material and the various collided force, the sound of wheels contacting ground depending on the ground material and the car speed, the sounds of other cars, …, and etc.
5.4 Motion Control System
By combining the proposed motion cuing algorithm with the master switching ECAM control yields the motion control system, as presented in chapter 3 and chapter 4.
The cabin operating
61
system and joystick system. The force feedback steering system is embedded in the cabin operating system, which includes the algorithm of wheels’ force feedback and the driving system of toque control.
62
Chapter 6 Conclusion
In chapter 2, the proposed disturbance estimator can effectively suppress the external disturbance and the high frequency measurement noise, trading off delay time and the robustness of the estimator. As a result, higher order polynomial fitting must be adapted for a cam profile with a farther travel distance. The cam profile tracking is formulated as optimization in real-time control. A deterministic and unique solution is derived for all possible cases of tracking control. The proposed method is effective for general motion tracking control and guarantees a global optimal solution for practical control.
In chapter 3, the proposed washout strategy is a general method and can be used in many simulators with different workspaces and driving systems. However, it is particular effective in a simulator with a small workspace. This approach is practical and efficient, especially for use in motion simulators used for entertainment, with restricted workspaces. Furthermore, this study establishes the performance index to conveniently quantify the efficiency of motion as the reference for realism. Repeated tests were performed online; they demonstrated that the proposed washout filter yields much more realistic motion cues than the classical technique for a motion simulator with a restricted workspace and an inexpensive driving system.
In chapter 4, the master switching electronic cam tracking control is combined with the motion planning of a six DOF motion simulator. The displacements of slaves of the electronic cam control system depend on the displacement of the master; the master switching method
63
selects the most heavily loaded ax al-time. The trajectory following spee
is to be the master in re
d yielded by the master switching method can be less than the speed yielded by the conventional (master fixed) method. Precision and robustness are the key concerns and the proposed method is sound. As aforementioned, by adjusting the proportional gain, a tradeoff exists between the robust stability and the velocity response of the control system. Using the well-known µ analysis of structured uncertainty, a most appropriate proportional gain may be chosen to satisfy the demand of control performance, provided robust stability is guaranteed. Furthermore, the poly-line curve-fitting method requires less computational time than
ll joint slide
the polynomial curve-fitting method, although the latter one may theoretically yield higher precision for a motion of low frequencies.
To perform a realistic human-machine interactive motion for entertainment demands, this dissertation finally roughly describes how to perform the system integration. That is to integrate the dynamics of several various vehicles, the collision detection algorithm, the VR programming techniques, the sound effects, the proceeding motion control system, the cabin operating system and the force feedback steering system.
This study uses the structural parametric method to model the dynamics of each ba
r; that may be just used to design the proportion gain, but is not direct and complete for designing a multivariable control system. Thus, the future research may focus on the dynamics and the design of multivariable controller for a six DOF motion simulator.
64
Appendix A
The physical parameters of the motor may be dynamically varied, so the effect of parameter uncertainty must also be discussed. The well-known analysis of the modeling uncertainty is the µ analysis [39]. A more direct method is to analyze the sensitivities (S , KGc
Gc
S andJ S ) of the transfer function BGc G to the motor’s uncertain parameters, K , J and B , c respectively, here w
B
Figures A.1 (a) ~ (c) show the magnitudes of the three sensitivities in relation to the input frequency, where the parameters of the master motor are all set as in Section 3.5. According to Eqs.(A.2) and (A.3), the magnitudes of the sensitivities, S and GKc S , are both small for GJc low-frequency motion. Figures A.1 (a) and (b) reveal that the magnitudes of the sensitivities,
Gc
hermore, according to Fig.A.1(c), the magnitude of the sensitivity S is less than BGc 0.00086 over the entire frequency domain. From Eqs.(A.1) ~ (A.3) and the foregoing discussion, the low time constant (ℑ) of the disturbance estimator suppresses the sensitivities,
Gc
S , K S and JGc S . BGc
65
frequency (ω ) at various time constants (ℑ)
(a)
Appendix B
The following analysis of the structured singular value follows reference [39].
Suppose that the uncertainty block is given by
, where Then, the closed-loop system is well-posed and internally stable iff sup ∆( 1( ))≤1
∈ µ ω
Hence, by the theorem of Packard and Doyle [43], at each frequency ω ,
⎟⎟
where the function σ(⋅) expresses the maximum singular value of . Since the minimization is convex in (see, Doyle [44, 45]), the optimal can be found by a search; however, two approximations [39] to can be obtained easily by approximating the right-hand side of Eq. (B.4):
B.1 First approximation
(⋅)
dω
log d ω
d ω
67
)
with minimized given
ω ⎠
B.2 Alternative approximation can be obtained by using the Frobenius norm:
.
or, alternatively,
⎟
⎟⎟
⎟
⎠
⎞
⎜⎜
⎜
⎝
⎛
⎥⎥
⎥
⎦
⎤
⎢⎢
⎢
⎣
⎡
∆ ≤ ~1 ( ) ( )
)
~ ( ) ( ))
( (
22 21
12 11
1 ω ω
ω σ ω
ω µ
ω
ω
j G j
d G
j G d j
G j
G (B.10)
These approximated µ are now determined.
69
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75
Fig. 1.1 Flow chart of the system tive motion
Fig. 1.2 The motion control system for a six DOF motion simulator integration for the human-machine interac Vehicle Dynamics Virtual Reality Collision Detection
Sound Effects Motion Control
System
Cabin Operating system (force feedback steering, joystick, button)
Vehicle Dynamics
Motion Cuing Control (motion trajectory generation)
Trajectory Tracking Control (ECAM controller)
6-axis control inputs 6-axis reference inputs
6-axis driving system
6-axis position outputs linear accelerations angular velocities
76
Fig. 2.1(a) Block diagram of the proposed ECAM system for mathematical representation
Slave #1 Driver &
...
Slave #2 Driver &
Slave #n Driver &
Disturbance
PC-based Programming *HA8506 multi-axis motion control card is the product of Hurco Co.
77
Fig. 2.1(b) Block diagram of the proposed disturbance estimator for one practical embodiment
ia
) /(
1 Js+B K
Kˆ
) 1 /(
1 ℑs+
ℑ / Jˆ K
R s Lf f
ˆ ˆ ˆ +
τL ω
τˆL
+
+ + +
-+
Bˆ +
- +
- Vref
f
fs R
L + DA 1
AD AD
ℑ ˆJ/
K a
Micro-Processor
78
Fig. 2.2 The estimation of external disturbance
Fig. 2.3 The external disturbance estimator nd external disturbance eliminated control a K 1/(Js+B)
Fig. 2.4 Temporal relations between the two proposed procedures s
rad / ω
T 2T kT
…
(k-N-1)T kT
…
(k-N-2)T 0
(k-N)T (k-1)T
Time t
Time t (k-1)T
s rad / ω
(I) k≤ N +1, ω0~ωk−1 are the recorded data and ωˆ is the unknown (estimated) k
(II) k> N +1, ωk−N−2 ~ωk−1 are the recorded data and ωˆ is the unknown (estimated) k
1
ω2
−1
ωk
ω
ωˆ k
ω0
−1
−N
ωk ωk−N
−1
ωk
ωˆ k
−2
−N
ωk
80
5 ,
pk+1
Fig. 2.5 Cubic B-spline curve rk+1 ,j(u), j = 1 to 4, and its control points, pk+1 ,0 ~pk+1 ,6 )
ˆ ( pk+1 ,6 = fL xk+2
2 ,
pk+1
3 ,
pk+1
4 ,
pk+1
4th
1st
3rd 2nd
0
1
the 4th curve segment rk+1 ,4(u) the start of the 4th curve segment, u = 0
the end of the 4th curve segment, u = 1
1 ,
pk+1 0 ,
pk+1
Note: Modulate the spline curve by adjusting the control pointpk+1 ,5.
81
Fig. 2.6 Flow chart of the optimal solution process
solution bounds of Eq. (19b) solution bounds of Eq. (19c) solution bounds of Eq. (19d)
: solution set of the inequality in Eq. (19b)—velocity cons
: solution set of the inequality in Eq. (19c) —acceleration constraint C: solution set of the inequality in Eq. (19d) —jerk constraint
* : The idea
: The optimal position command,
* : Optimal position, , coincided with the ideal position command, Note: The above categories are in the extreme case that
Fig. 2.7 The location of the optimal position command, , for all different cases (a)
84 15
17 19 21 25 27 29
1 201 401 601 8 1001 1201 1401 160
Fig. 2.8(a) Simulated angular velocity of the master 31
33
23
01 1 1801
30.91.131 1.2 31.3 31.4 31.5 31.6 31.7
Fig. 2.8(b) Si
Time (
3 3
1 201 401 601 801 1001 1201 1401 1601 1801
mulated angular velocity of the master using disturbance estimator feedback control (zoom in)
× 1 ms) angular velocity of the maste
s rad /
r
Result using the disturbance estimator feedback control
Result without a disturbance estimator Reference
velocity
angular velocity of the master
s rad /
Reference
velocity Result using the disturbance estimator feedback control
Time (× 1 ms)
-6
1 201 401 601 801 1001 1201 1401 1601 1801
-2
Fig. 2.9 The errors between the fed torque (
181
Fig. 2.9 The errors between the fed torque ( 0
Fig. 2 10(a) The tracking error of the master’s position for the zero-order interpolation method
Fig. 2.10(b) The tracking error of the master for the third-order polynomial tracking method.
.
-15 -10 -5 0 5 10 15
1 8 15 22 29 36 43 50 57 64 71 78 85 92 99 106 113 120 127 134 141 148 155 162 169 176 183 190
Sampling Time (10 ms ) Master position error (encoder's counts)
-2 -1 0 1 2
1 8 15 22 29 36 43 50 57 64 71 78 85 92 3 34 141 148 155 62 169 176 183 190
Sampling Time (10 msec )
Master position error (encoder's counts)
99 106 11 120 127 1 1
86
-2 -1 0 1 2
1 8 15 22 29 36 43 50 57 64 71 78 85 92 99 106 113 120 127 134 141 148 155 162 169 176 183 190
Sampling Time (10 msec )
Master position error (encoder's counts)
Fig. 2.10 (c) The tracking error of the master for the fourth-order polynomial tracking method
-1 0 1
1 8 15 22 29 36 43 50 57 64 71 78 85 92 99 106 113 120 127 134 141 148 155 162 169 176 183 190
Sampling Time (10 msec )
Master position error (encoder's counts)
Fig. 2.10 (d) The tracking error of the master for the fifth-order polynomial tracking method
87
0 0.5 1 1.5 2 2.5 x 104 0
0.5 1 1.5
2x 105
0 50 100 150 200
0 0.5 1 1.5
2x 105
0 0.5 1 1.5 2 2.5
x 104
0 50 100 150 200
Fig. 2.11 The piecewise tracking trajectory of the electronic cam motion (Input)
(CAM) (Output)
(a)
Master position (counts) Master position (counts)
Time (10 ms)
Time (
×10
ms) Slave position (counts)
Slave position (counts)
(b) (c)
88
Fig. 2.12 (a) Cam profile error with the zero-order(conventional) tracking method (The maximum travel distance: 200000 encoder’s counts)
-20 -15
-500 400 -300 -200 -100 0 100 200 300 400 500
1 8 15 22 29 36 43 50 57 64 71 78 85 92 99 106 113 120 127 134 141 148 155 162 169 176 183 190
Sampling Time (10 msec )
Slave position error (encoder's counts)
-1 10 15 20
0 -5 0 5
1 8 15 43 50 57 64 71 78 85 92 113 120 1 155 162 169 176 183 190
Sampling Time(10 msec )
Slave position error (encoder's counts)
Fig. 2.12(b) ethod
(The maximum travel distance: 200000 encoder’s counts)
22 29 36 99 106 27 134 141 148
Cam profile error with the third-order polynomial tracking m
89
-3 -2 -1 0 1 2 3
1 8 15 22 29 36 43 50 57 64 71 78 85 92 99 106 113 120 127 134 141 148 155 162 169 176 183 190
Sampling Time (10 msec )
Slave position error (encoder counts)
(The counts)
counts)
(The counts)
Fig. 2.12(c) Cam profile error with the fourth-order polynomial tracking method Fig. 2.12(c) Cam profile error with the fourth-order polynomial tracking method
maximum travel distance: 200000 encoder’s maximum travel distance: 200000 encoder’s
-2 -1 0
1 8 15 22 29 36 43 50 57 64 71 78 85 92 99 106 113 120 127 134 141 148 155 162 169 176 183 190
Sampling Time(10 msec )
Slave position error (encoder's counts)
Fig. 2.12(d) Cam profile error with the fifth-order polynomial tracking method (The maximum travel distance: 200000 encoder’s counts)
90
Fig. 2.13(a) The tracking result of the slave velocity, acceleration and jerk purely based on the Lagrange polynomial curve-fitting with no optimization
( 40 ) /× π rad s
-12000 -6000 0 6000 1
0 8 15 23 30 38 46 53 61 68 76 84 91 99 106 114 122 129 137 144 152 160 167 175 182 190
-2 -1 0 1 2 3
0 8 15 23 30 38 46 53 61 68 76 84 91 99 106 114 122 129 137 144 152 160 167 175 182 190
Actual Velocity Actual Accelera1tion Actual Jerk
-3
Sampling Time (10 msec)
-60 -40 -20 0 20 40 60
0 8 15 23 30 38 46 53 61 68 76 84 91 99 106 114 122 129 137 144 152 160 167 175 182 190
( 40 ) /× π rad s2
Sampling Time (10 msec)
Sampling Time (10 msec) 2000
( 40 ) /× π rad s3
91
-3 -2 -1 0 1 2 3
0 8 15 23 30 38 46 53 61 68 76 84 91 99 106 114 122 129 137 144 152 160 167 175 182 190
( 40 ) /× π rad s
-20 -15 -10 -5 0 5 10
0 8 15 23 30 38 46 53 61 68 76 84 91 99 106 114 122 129 137 144 152 160 167 175 182 190
-1000 -500 0 500
0 8 15 23 30 46 53 61 68 76 84 91 99 106 114 122 129 137 144 152 160 167 175 182 190
Fig. 2.13(b) The result of the slave velocity, acceleration and jerk applying the optimization Sampling Time (10 msec)
1000
38
( 40 ) /× π rad s2
) /rad s3
π Actual Velocity Actual Acceleration Actual Jerk
Sampling Time (10 msec)
Sampling Time (10 msec) ( 40×
92
Fig. 3.1 Classical linear washout filter (referred to Nahon and Reid, 1990) ωAA
aVR
ωHP
aHP
as
HP Filter HP Filter
LP Filter
Tilt Coordinate
Rate Limit
scale ωs
sω
scale sa
gI
TF
+ + aAA Rotation
Matrix (S to I)
ωVR
CLWF
+ +
93
94
Fig. 3.2 Block diagram of motion-cueing system, using the proposed control strategy
95
Transformation (S to J)
Eq. (1) ~ (3)
Will the cockpit be ou workspace?
t of the Perform washout
motion planning Eq. (14) ~ (31)
Perform the motion directly out using washout motion planning Tran mation (S to J) with
E sfor
q. (1) ~ (3)
No Yes
Washout Function
F structure (to be continued) Self-Tuning Process
ωref pω 1/s
φref
aref
s2
/
a 1 p
ω
a Pref
Fig. 3.3(a) AW
6-Axis Coordinates Output If the cockpit will be out
of the workspace ?
Re-scale the present angular velocity ω and linear acceleration a (Eq. (32))
Loop Count = 0 If the commands fed to
the driving system exceed the critical value?
Loop counts. >
preset loop counts ? Washout Function
Yes Yes
No
No
Fig. 3.3(b) AWF structure (to continue) Yes
No
Self-Tuning Process
Redo AWF
96
Fig. 3.4 Prototype SP-120 G
x
y z
α γ β
97
Fig. 3.5 Kinematical skeleton of simulator platform SP-120 O
X Y
Z G
2
1 q
q =
4
3 q
q =
6
5 q
q =
2
1 S
S =
p1
p2
p 3
p4 p5
p 6
4
3 S
S = S5 =S6
x
y
z
L L
L L
L L
x5
y5
z 5
y 3 x 3 z 3
98
(a)
(b)
Fig. 3.6 Restoration process in the dead zone washout filter
.
t1 t2
t2
Acceleration curve
Acceleration Velocity curve
Velocity curve
period n restoratio total
period on decelerati
2
0
0
0 0
curve
t1
:
1 : t t
99
Fig. 3.7 Washout trajectory along the i-axis, where the subscript i represents the three mutually orthogonal axes (x-, y-, and z-axis)
t d tf
PMAX
t2, v0,i
a ) ( P t
aref
(t
) (t ai
i
i MAX ,
P
) vi
i ,
threshold
100
101
Fig. 3.8 Segmental data concerning linear accelerations along the x-axis
Fig. 3.9 Seg ental errors of linear accelerations along the x-axis, between the static scaled VR dynamic output ( ) and the simulator’s output ( ) using the (a) CLWF and (b) the proposed strategies
Time (× 0.05 sec)
: the simulator’s output using the proposed washout filter : the simulator’s output using the CLWF
m
: the scaled VR dynamic output
: the simulator’s output using the propo
-3 -2 -1 0 1 2 3
1 17 33 49 65 81 97 113 129 145 161 177 193 209 225 241 257 273 289 sed washout filter : the simulator’s output using the CLWF
s rad / ω
Fig. 3.10 Three segmental data concerning Euler’s angular velocities
Fig. 3.11 Segmental errors of Euler’s angular velocities between the static scaled VR dynamic output (
ω
s,x) and the simulator’s output (ωx) using (a) the CLWF and (b) the proposed strategiesTime (× 0.05 sec)
s rad /
-3-2 01 23
1 17 33 49 65 81 97 113 129 145 177 7 273 289
-1
161 193 209 225 241 25
s rad /
Time (× 0.05 sec) (b)
Time (× 0.05 sec) -3-2
-101
1 17 33 49 65 81 97 113 129 145 161 177 193 209 225 241 257 273 289
23
(a)
102
-400 -300 -200 -100 0 100 200 300 400
1 17 33 49 65 81 97 113 129 145 161 177 193 209 225 241 257 273 289
Fig. 3.12 Three segmental data concerning linear accelerations along the x-axis
ig. 3.13 Segmental errors of linear accelerations along the static scaled VR dynamic output ( ) and the simulator’s output ( ) using (a) the optimal pair
F the x-axis between
x
as, ax
of weighting parameters (1, 0) and (b) the pair of weighting parameters (1, 0.5) -400
-200 0 200 400
1 17 33 49 65 81 97 113 129 145 161 177 193 209 225 241 257 273 289
-400 -200 0 200 400
1 17 33 49 65 81 97 113 129 145 161 177 193 209 225 241 257 273 289 Time (× 0.05 sec)
: the scaled VR dynamic output
: the sim λ λ
: the sim
x
as,
ulator’s output using the optimal pair( a* =1, ω* =0) ulator’s output using the pair (λa =1, λω =0.5)
/ s2
cm a
/ s2
cm
/ s2
cm
(a)
(b) Time (× 0.05 sec) Time (× 0.05 sec)
103
Fig. 3.15 ler
Fig. 3.14 Three segmental data concerning Euler’s angular velocities
Segmental errors of Eu ’s angular velocities between the static scaled VR dynamic output (ω ) and the simulator’s output (s,x ωx ) using (a) the optimal pair of
eters (1, 0) and (b) the pair of weighting parameters (1, 0.5) weighting param
-3 -2 -1 0 1 2 3
1 17 33 49 65 81 97 113 129 145 161 177 193 209 225 241 257 273 289 s
rad /
ω : the scaled VR dynamic output ω s,x
: the simulator’s output using the optimal pair (λa* =1, λω* =0) ,
1 λω =0.5 ) : the simulator’s output using the pair (λa =
-3 -2 -1 0 1 2 3
1 17 33 49 65 81 97 113 129 145 161 177 193 209 225 241 257 273 289
-3 -2 -1 0 1 2 3
1 17 33 49 65 81 97 113 129 145 161 177 193 209 225 241 257 273 289 Time (× 0.05 sec)
s rad /
(a)
) ) Time (× 0.05 sec
Time (× 0.05 sec (b)
s rad /
104
Motion Planning cobian Matrix J
Calculate the heaviest loaded axis as the new master
Fig. 4.1 Master switching method for q axes ECAM control
Fig. 4.1 Master switching method for q axes ECAM control