Chapter 1 Introduction Introduction
2.2 Spherical Motion Platform I Based on Three Spherical Wheels Wheels
2.2.3 Kinematic Model
The cockpit sphere is rotating with the small spheres as well as translating along the linear guide to control six DOFs motion. The kinematics of the SMP I can be expressed by the angular velocities of six motors. An inertial frame with XIYIZI coordinates is defined at the intersection point of linear stages on the base, and a body frame with xbybzb coordinates is at the center of the cockpit, as shown in Fig.
2.2(a). The moving coordinate system xiyizi with xi-and yi-axes parallel and perpendicular to the ith linear guide, respectively, and zi-axis parallel to ZI-axis is attached to the center of the ith small sphere Pi.
2.2.3.1 Geometry
For motion analysis of the cockpit, positions of the small sphere and the cockpit are defined by geometric relation. The center Pi (i=1, 2, 3) and the position of cockpit center P0 can be expressed in (2.1) and (2.2) in the inertial frame.
2 2
cos ( 1) sin ( 1)
3 3
T iLi i Li i hs
P (2.1)
where Li is the distance from the center O to the center of the ith small sphere; hs is the height of the center of the small sphere from the base. In order to calculate P0, a principle of the circumcenter should be applied. Fig. 2.3 shows the circumcenter O(x0,y0) calculated from A(x1,y1), B(x2,y2), and C(x3,y3) by using (2.2) and (2.3).
Figure 2.3. Circumcenter between triangle and circle.
2 2 2
1 2 3 2 3 1 3 1 2 2 3 3 1 1 2
0
1 2 3 2 3 1 3 1 2
2
x y y x y y x y y y y y y y y
x x y y x y y x y y
(2.2)
16
2 2 2
1 2 3 2 3 1 3 1 2 2 3 3 1 1 2
0
1 2 3 2 3 1 3 1 2
2
y x x y x x y x x x x x x x x
y y x x y x x y x x
(2.3)
Note that three lines connecting Pi form a triangle and that C0 is a circumcenter of the triangle and the projected point of P0 into the triangle in Fig. 2.2(a). Thus, P0 can be computed with the given Li in (2.4). H is the height of the center of the cockpit from the center of the small sphere in (2.5).
0 0, 0, 0,
2 2
2 3 1 2 3 2 3 1 2 3 1 2 3
1 2 2 3 3 1 1 2 2 3 3 1
+ 3 + + 2 +
2 + + 6 + + +
T
X Y Z
T
s
P P P
L L L L L L L L L L L L L
h H
L L L L L L L L L L L L
P
(2.4)
s C
2
0,X 1,X
2 0,Y 1,Y
2H r R P P P P (2.5)
where rs and R are the radii of the small sphere and the cockpit, respectively, in Fig. 2.2(b) and (c). In addition, Li (i = 1, 2, 3) has geometric constraint limit in (2.6).
Lmin < L1, L2, and L3 < Lmax (2.6) where Lmin and Lmax are the minimum and the maximum distance allowed to move within the linear guide. The limits are pre-determined based on the design by keeping from the collision between the small spheres and contact between the cockpit and the base, respectively.
2.2.3.2 Rotational Motion
The rotation of the cockpit is driven by three motors through three small spheres to the cockpit sphere. It can be assumed that the power of the motors is transferred to the cockpit through the roller and the small sphere by friction enable it to roll without slipping since the cockpit sphere is heavy enough to transfer the power without losing contact between the cockpit and the small spheres. Then, angular velocities between the motor and the small sphere can be computed by the radius ratio between the roller and the small sphere. The angular velocity of the cockpit Ω in (2.7) is expressed as the sum of two components. One is due to rotation, and the other is translation. Furthermore, it is determined by angular velocities of the small sphere as well as the position of the cockpit with respect to three small spheres expressed in terms of the relative position of the cockpit Pi0 = P0 – Pi , from (2.1), (2.4)‒(2.5).
3
, 1
s
S T n i
i C
r
R
Ω Ω Ω ω (2.7)
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0
, , 0 , 2
0
( ) i
n i s i i s i
i
P
ω ω P ω
P (2.8)
ωs,i = ωm,i + ωf,i (2.9)
, w
m i i
s
r
r
ω ω (2.10)
cos 45 0 sin 45
Ti ai i
ω (2.11)
cos sin 0 2
1sin cos 0 and
0 0 1 3
i i
i i i i
b b i
b b b
Γ (2.12)
where ΩS and ΩT are angular velocities of the cockpit for rotation and translation driven by motors respectively. ωs,i and ωi are the angular velocities of the ith small sphere and motor respectively as shown in Fig. 2.4. ωi lies on the linear guide with 45°. ωn,i is the normal component of ωs,i with respect to Pi0. ωm,i and ωf,i are angular velocities of the small sphere and free rotation, respectively. ωf,i is due to rotation coupled from translational motion in (2.9). rw is the radius of the roller attached to the motor, and ai is the angular velocity of the motor. Γi is a coordinate transformation matrix from x1y1z1 to xiyizi
for each linear stage.
Figure 2.4. Schematic diagram of the rotational system.
18 2.2.3.3 Translational Motion
Translation of the cockpit driven by linear stages along linear guides can be expressed in terms of Li in (2.13)‒(2.14).
[ ]
X N L (2.13)
0, ( 1,2,3)
i i
L i
N P (2.14)
whereXrepresents velocity of the cockpit. N = [N1 N2 N3] represents translation of the cockpit according to variation of Li; L [ ]L L L 1 2 3 Trepresents velocity of the linear guide.
As the linear stage move, the small sphere also spins accordingly. Consequently, the linear motion of the stage generates rotation of the cockpit. The angular velocity of the cockpit due to the translation can be expressed in (2.15).
[ ]
T
Ω K L (2.15)
, i T i
Li
Ω
K (2.16)
, i[ ][0 1 0]T
f i i
s
L
r
ω Γ
(2.17)
where ΩT = ΩT,1 + ΩT,2 + ΩT,3 and ΩT,i represents angular velocity due to translation of the ith linear stage. K = [K1 K2 K3]T represents the rotation of the cockpit according to variation of L. The ith column vector of [K] as Ki can be computed by the normal component of ωf,i with respect to Pi0 in (2.7), (2.8), and (2.17). ωf,i is the angular velocity of the ith small sphere driven along the linear stage, parallel to the yi axis and perpendicular to the xi axis.
From (2.13)‒(2.17), the required translation of the linear stage can be computed from the translational motion of the cockpit as well as the coupled rotation due to the translation expressed in (2.18).
1 1
[ ] [ ] T
L N X K Ω (2.18)
where [N]‒1 always exists but [K]‒1 may have singularity due to geometric and motion constraints discussed in Dynamics. Table 2.2 shows the platform workspace limits computed from the forward kinematics using (2.7)‒(2.14) without energy loss caused by slip. The range of the coupled motion is also computed from (2.15)‒(2.17).
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Table 2.2. Maximum Workspace and Velocities of the SMP I.
Rotation ϕ θ ψ
Max. rotation angle (deg) ±180 ±90 ±180
Max. angular velocity (deg/s) ±32.3 ±37.4 ±57.6
Max. coupling effect (deg) ±45 ±57.6 0
Translation X Y Z
Max. linear displacement (m) ±0.37 ±0.43 0.30
Max. linear velocity (m/s) ±0.50 ±0.58 ±0.57