Chapter 1 Introduction Introduction
2.8 Numerical Simulation and Experimental Validation
2.8.1 Validation for SMP I
The kinematic and dynamic models are utilized to validate the design and performance of motion control. Numerical simulations are computed, and the results are compared to experiments with a motion platform in an open-loop (OL) control for two cases. First, orientation control with two positions of the cockpit is compared: symmetric case (L1 = L2 = L3) and asymmetric case (L1 ≠ L2 ≠ L3). Second, the position along each X-, Y-, and Z-axis is controlled, respectively. In addition, the coupled rotational motion due to the translational motion is investigated.
2.8.1.1 Numerical Simulation
The motion control based on the kinematic and dynamic analysis is numerically simulated based on an OL shown in Fig. 2.11. Viscous friction is included to decelerate motion as the platform is slowed down. The gravitational torque can be assumed as small and negligible by assuming the center of mass and rotation are coincident. In addition, the platform is designed as the human sits inside as close to the center so as to minimize changing the center of mass and moment of inertia. The moment of inertia of the cockpit is estimated as a spherical shell without a human on board for the numerical simulation.
Fig. 2.11 shows the OL control system for the numerical simulation where X represents the position vector of the cockpit. The desired torque is computed from the inverse dynamics with
, , , , ,
d d d d d d
q q q X X X in (2.20). In the simulation, the dynamics based on Euler angle causes mathematical singularity at θ = ±90 degree. However, the motion platform for the experimental demonstration is made in actual scale size, and human can be on board. For safety, the cockpit is opened on the top as shown in Fig. 5 and cannot be rotated more than θ = ±90 deg (the range of operation becomes −90° ≤ ϕ, θ ≤ 90° and −180° ≤ ψ ≤ 180°). The mathematical singularity of Euler angle rotation in the dynamic model is not necessary to be considered. However, the singularity can be avoided by using quaternions instead of Euler angle rotation for the completed spherical cockpit.
Similar to the rotational motion, the translational motion can be numerically simulated. In particular, the kinematic coupling between orientation and position is pre-computed for accurate rotation.
Furthermore, the OL simulation can be utilized to calculate the required motor inputs from inverse kinematics calculated by (2.7) and (2.13). In the numerical simulation, the system operated by Td is halted at desired orientation from motor break Cν with viscous friction. The parameters of the motion platform are detailed in Tables 2.3 and 2.4.
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Figure 2.11. Open-loop control system for numerical simulation.
Table 2.3. Parameters of Simulation and Experiment.
Position of linear stage Symmetric case Asymmetric case
L1 0.6 m 0.6 m
L2 0.6 m 0.8 m
L3 0.6 m 0.5 m
Moment of inertia of cockpit: Ia = 41.3 kg•m2; It = 41.3 kg•m2 Desired angular velocity: ωd = 0.15 rad/s
Desired angular acceleration (deceleration): αd = 0.15 rad/s2 Viscous frictional coefficient and motor break: Cν = 6.291 Nm•s Given external torque: ||Text||= 6.35 Nm
Table 2.4. List of Specifications of Motion Platform.
hs height of small sphere 0.35 m
M mass of cockpit 80 kg
rw radius of roller 0.0375 m
rs radius of small sphere 0.11 m
RC radius of cockpit 0.88 m
Lmin Minimum length of L 0.285 m
Lmax Maximum length of L 0.745 m
Lhome Length of home position 0.535 m
38 2.8.1.2 Experimental Setup
Fig. 2.12 shows the motion platform capable of supporting human on board. The cockpit sphere is made from composite material, the surface of which is rough, so that torque for orientation control is delivered to minimize the slipping motion between the spheres. Moreover, although both kinematics and dynamics are based on a rigid body, indicating the point contact between the small spheres and the cockpit, the actual contact becomes surface and causes slipping. Various materials of the small spheres have been investigated, such as epoxy, rubber, sandpaper, etc. Finally, a urethane ball as the small sphere has been chosen to minimize slipping motion. Each urethane ball with a hardness of 90~95 can support heavy loads of at least 300kg. The servomotors to rotate each small sphere are installed with a 60:1 gear ratio to reduce speed and increase torque; MSME042G1A and PGX62-H-60 are selected as the servomotor and gearbox shown in Fig. 2.12(b).
Similarly, the same servomotor without a gearbox is used for linear motion. The motor is connected to a ball screw of the linear stage to change rotational motion to linear motion. In addition, for reference as homing and safety, proximity sensors to limit the range of motion are installed on the linear guide.
All six motors are connected to each servo driver with encoder feedback and controlled accurately;
MBDHT2510 is used for the servo driver. The NI cRIO-9082, a real-time embedded programmable controller, is used as a main controller to control all motor drivers simultaneously. It features 533MHz real-time processor and contains 256MB of DDR2 RAM for fast data processing. The chassis with FPGA modules executing control algorithms and NI-9512 modules for digital control on the chassis are attached on the cRIO. The six modules are used to interface stepper servo drivers and control angular position, velocity, acceleration, and deceleration of each servomotor. It also has incremental encoder inputs for position feedback and a full set of motion, including inputs from the home switch and forward and reverse limit switches. The control hardware is summarized in Fig. 2.13. The OL control for the rotational and translational motion is shown in Fig. 2.11. The experimental parameters are detailed in Table 2.3.
39 (a)
(b) (c)
Figure 2.12. Actual model of spherical motion platform I. (a) SMP I. (b) Three small spheres. (c) Linear stage with guide.
40 Figure 2.13. Control hardware system.
41 2.8.1.3 Orientation Control
Experiments for the orientation control are demonstrated as in the numerical simulation. Two configurations of the cockpit sphere with respect to three small spheres are investigated. Each case is composed of three independent rotational motions. In symmetric and asymmetric cases, Li (i=1,2,3) and other parameters are detailed in Tables 2.3 and 2.4. The six motors are controlled by using LabVIEW softmotion. MPU6050 inertial measurement unit (IMU) is used to measure the orientation of the cockpit using a wireless network, XBee module, at 3Hz sampling time. In each case, ϕ, θ, and ψ are controlled independently in an OL based on kinematics and dynamics. Each command is a 30 deg step input for the orientation control. The desired angular position, velocity, acceleration, and deceleration for each motor are solved from the inverse kinematics. The velocity of the cockpit is set to be 8.6 deg/s. The acceleration and deceleration are each set to be 8.6 deg/s2 to minimize slipping. The initial position of Li is set to Lhome as in Table 2.4.
Fig. 2.14 shows the comparison of the cockpit orientation between the simulation and the experiment. In the symmetric case, as shown in Figs. 2.14(a), (c), and (e), the experimental results show rotational motion can be independently controlled along each axis. The maximum error is less than 6 deg, and the error is mainly caused due to slip among spheres, inaccuracy of the measurement sensor, and imperfect cockpit sphere. Similarly, in the asymmetric case, the control is also accurate, as shown in Figs. 2.14(b), (d), and (f). The maximum error is less than 7 deg, slightly larger than the symmetric case. The errors for each comparison are detailed in Table 2.5. The errors could be compensated by a feedback control for the orientation and position of the cockpit.
Table 2.5. Maximum Error for Rotational Motion (degree).
Symmetric case Asymmetric case Motion coupling
Roll 5.72 3.25 2.47
Pitch 2.54 0.06 1.97
Yaw 2.61 6.49 0.29
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(a) (b)
(c) (d)
(e) (f)
Figure 2.14. Comparison of rotational motion. (S indicates symmetry and AS asymmetry). (a) ϕ = 30° (S). (b) ϕ = 30° (AS). (c) θ = 30° (S). (d) θ = 30° (AS). (e) ψ = 30° (S). (f) ψ = 30° (AS).
0 1 2 3 4 5 6
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10 20 30
sim sim sim
exp exp exp
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degree degree
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43 2.8.1.4 Position Control
The translational motion for position control is demonstrated in an OL similar to the orientation control. The kinematic analysis of translational velocity in (2.7)‒(2.18) shows that the translational motion is coupled with the rotational motion. Although the coupled motion in operation is undesired and relatively difficult to control, the design becomes compact and lightweight, offering a fast response in motion control.
The rotation coupled with the translation motion is validated by comparing both experiments and numerical simulations in three cases: the cockpit is moving along the X-, Y-, and Z-axes independently.
Each command is a 100mm step input for the position control. The desired velocity of the cockpit is set to be 20mm/s. The desired acceleration and deceleration are set to be 50mm/s2, respectively. The input
∆Li can be obtained from the inverse kinematics. The linear stage moves 50mm according to 100000 input pulse (= 10rev) of the motor.
The experiment results of the translational motion are shown in Fig. 2.15(a), (c), and (e). The coupled rotation angle due to each translational motion is estimated from the forward kinematics in (2.15) and validated by the experiments. The cockpit is rotated to 9.68 deg according to translation 100mm along the X- and Y-axes on the XY plane. Unlike the translational motion on the XY plane, the rotation and translation are fully decoupled along the Z-axis, as shown in Fig. 2.15(f). The maximum error is less than 3 deg. The errors for each motion are detailed in Table IV. The comparison results show the design, analysis, and control performance of the motion platform, although the errors in motion control occurred due to mechanical imperfections and uncertainties such as slipping and frictional force etc. Accuracy in motion control can be improved by a feedback controller.
2.8.1.5 Active Driving Control to Compensate for Coupled Rotation
The rotation due to the translational motion is validated by experimental results with and without the ADC. The experiment is implemented with three arbitrary cases to guarantee motion feasibility.
Each input command (mm) for movement of linear stage is applied as [100 ‒50 ‒50]T, [0 100 ‒100]T, and [100 100 100]T for Figs. 2.16(a), (c), and (e), respectively. The desired velocity (mm/s) is set to be [12.5 6.25 6.25]T, [0 12.5 12.5]T, and [12.5 12.5 12.5]T. The desired acceleration and deceleration for each motor are set to be 200mm/s2 for fast response. The linear stage moves 5mm according to 1 revolution (= 10000 input pulse) of the motor.
The results of the experiment with the ADC are shown in Fig. 2.16(b), (d), and (f) according to the translation of the cockpit as shown in Figs. 2.16(a), (c), and (e), respectively. The coupled rotation angle according to translation is estimated from (2.62), and compensated from forward kinematics. The maximum error with the control compensation is smaller than 0.47°. The comparison results show the
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coupled motion along ϕ- and θ-axes compensated from ADC in real-time, although small errors occurred due to mechanical imperfections, slipping, etc. Similarly, the compensation can be applied to the SMP II by utilizing (2.62)‒(2.64) with Na=4, and finally, the coupled motion can be eliminated.
(a) (b)
(c) (d)
(e) (f)
Figure 2.15. Comparison of translation and rotation according to motion coupling. (a) X = 100mm.
(b) ϕ = 9.68°. (c) Y = 100mm. (d) θ = 9.68°. (e) Z = 100mm. (f) ψ = 0°.
0 1 2 3 4 5 6 7
Time (s) 0
20 40 60 80
100 X
Y Z
p (mm)p (mm) degree
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(a) (b)
(c) (d)
(e) (f)
Figure 2.16. Comparisons of translation and rotation according to motion coupling for motion platform I. (C indicates compensation and N no compensation). (a) X = 95.7 mm and Z = ‒2.8 mm. (b) Rotation due to translation. (c) X = 6.4mm, Y = 116.2 mm. (d) Rotation due to translation. (e) Z = 73.5 mm. (f) Rotation due to translation.
0 2 4 6 8 10
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X Y Z
0 2 4 6 8 10
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p (mm)
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-6 -5 -4 -3 -2 -1 0
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