§2-2 Inverse Kinematics
Figure 2.1 shows a type of hexglider parallel manipulator. The centers of the ball joints on the mobile plate are denoted as , , and . is a equilateral-triangle with each side of a length a. B 1 B B is a equilateral-triangle on the fixed lower plate with each side of a length b.
S1~S specify the positions of six sliders. Each of the six linkages, with a length Q1 Q2 Q3 Q1Q2Q3
2 3
6
l , is linked to the movable upper plate through a ball joint. The other ends of
them are connected with a slider travelling on the guideway mounted on the lower fixed plate by a universal joint. Figure 2.2 shows the definitions of geometric relationships existing at the movable plate and the fixed plate. From the results of the geometric relationship, the coordinates of B1, B , and B3 of the fixed plate with respect to the base frame are：
2
b XB
6 3
1 = ， XB b 6
3
2 = ， XB b 3
3
3 =−
1 2
YB =−b ，
2 2
YB =b ， YB3 =0
1 =0
ZB ， ZB2 =0 ， ZB3 =0
The coordinates of Q1, Q , Q of the movable plate with respect to the top
The homogeneous transformation from the top to the base frames is described by the transformation matrix：
The coordinates of the origin of the top frame with respect to the base frame are denoted by [P 、PX Y、P ]Z T. Let α, β , γ represent the rotation angles defined by rotating the top frame first about the X axis with α degrees, then about the Y axis with β degrees, and finally about the Z axis with γ degrees, respectively. The coordinates of Q , =1, 2, 3, in terms of the base frame can be calculated through
i i
⎥⎥
Consequently, the coordinates of Q , , and Q with respect to the base frame can be computed
1 Q2 3
⎥⎥
⎥⎥
Figure 2.3 illustrates a ball joint has two adjacent equal-length linkages connected to two sliders movable on one of the three guideways mounted on the stationary plate. is the ball joint. and are linear sliders. The
2 i 2i 1 2 2
i 1
i
2 (b t cos ) l t sin
Sˆ = − ϕ − − − ϕ , 3i=1~ (2.8)
§2-3 Kinematics of RSSR Mechanisms
The conventional forward kinematics for the six degree-of-freedom manipulator needs complicated mathematic operations. The forward kinematic problem involves systems of highly non-linear equations. Many of these studies was established on the basis of a simplified structure to reduce the nonlinearity of the equations, such that an analytical approach can be performed to complete the solution set.
The proposed forward kinematics used in this chapter needs at first to decomposite the hexglider parallel manipulator into three sets of RSSR structures. After deriving the kinematic relationships existing at the RSSR mechanism mounting on any guideway of the hexglider parallel manipulator, we can intergrate all of the three ones and solve the simultaneous equations to describe the behavior of the hexglider manipulator.
As shown in Figure 2.4, the positions of the ball joints U1 and U2 can be expressed as follows:
⎥⎥
⎥
⎦
⎤
⎢⎢
⎢
⎣
⎡
−
⋅
−
=
r y
Rot f
Trans
U 0
0 ) , ( ) 0 , , 0 (
1 θ (2.9)
⎥⎥
The distance between U1 and U2 yields
)
Eq (2.11) can be expressed as follows
) computeded by Newton-Raphson method.
§2-4 Newton-Raphson Method
Consider a system of equations . Using the Taylor series
⎪⎩
X and nelecting terms of order two and higher, we obtain
(2.13) As described above, a system of equations is given by
⎪⎩ The system of equations in this study is given by
⎪⎩
Thus , the system of equations becomes
⎪⎪ To apply Cramer’s rule, we can obtain the solutions of the above linear system of equations ∆θi, ∆δi, ∆φi as follows
θ The recursive formulus yields
⎪⎩ recursive operations.
§2-5 Forward Kinematics
Applying Eq (2.11) to the forward kinematic analysis of the hexglider manipulator, a system of equation with respect to the attitude angles θ, δ , and φ can be obtained
As shown in Figure 2.5, the coordinates of ball joint U1, U2, U3 on the upper plate can be computeded with the solved attitude angles θ , δ , and φ
⎥⎥
Substituting the coordinates of ball joints U1, U2, U3 into Eq.(2.26), the coordinates of the center of the end-effector px, py, and pz can be obtained as follows
⎥⎥
The Euler rotation matrix Eq.(2.27) denotes a set of unit base vecters of the coordinate system describing the pose of the end-effector
⎥ ⎥
As a result, the positions of ball joints U1, U2, and U3 with respect to the coordinates of the center of the end-effector px, py, and pz, the radius of the tangent circle of the equilateral-triangle UR, and the unit pose vectors of the end-effector , , can be derived as Eq.(2.28), Eq.(2.29), andQ1Q2Q3
u u u
Eq.(2.30). computed from Eq.(2.31), Eq.(2.32), and Eq.(2.33).
1 )
Chapter 3
Modeling of Motion Errors
§3.1 Introduction
The SP-120 is a six degree of freedom parallel manipulator which is presently the one used for modeling the false circular cotour of motion errors resulting from the individual error sources of the manipulater. With the results of simplified forward kinematics, the position deviations generated by geometric error sources, especially assembly errors, are derived mathmatically in this chapter. Simutaneously, the parameter errors relating to the geometric error sources are described as well.
§3.2 Error Descriptions
All of the errors are coupled in any parallel manipulator. Therefore, the total possible errors can be summed and expressed in terms of the Taylor expansion as follows:
Total error
) ( ...
) ( )
(
) ..., ,
,
( ,
x E c x
E c x E c
c c
c x E
m m 2
2 1
1
m 2
1
+ +
+
=
=
(3.1)
Geometric resources of motional deviations, including translate and angular ones, resulting from the manufacturing or assembly of guideways and linkages
are classified and modeled with the analytical results of the kinematics in the thesis.
As shown in Fig 3.1, the geometric resources of motional deviations under investigation can be classified and described as follows.
(1) Linkage length error
The length errors of the six linkages due to the the manufacturing tolerances may affect the pose accuracy of the manipulator.
(2) Slider positioning error
The positioning errors of the six sliders S1 to S6 are usually generated from the controller, and they might exist at the initialized command data.
(3) Assembly errors for the guideways
Three guideways are assembled on the stationary plate. Three ball joints are mounted on the mobile plate as well. The improper positions of fixing guideways or ball joints may lead to assembly errors. The assembly errors can be categorized into the following items:
1. Guideway translation error ﹔In Figure 3.2 (a), the guideway is translated a positional error along the X , Y , or Z axis.
2. Guideway rotation error (I) ﹔In Figure 3.2 (b), the guideway is rotated an angular error by the center of the guide way on XY or YZ plane.
3. Guideway rotation error (II) ﹔In Figure 3.2 (c), the guideway is rotated an angular error about the center of the tri-angle, consist of three guideways and fixed on the base plate, on XY plane.
The angular and positional deviations of the end-effector in the machine coordinate system are defined. There are thirty three deviations in a hexglider manipulator. In the following, We begin to model the above deviations with the analytical results of the kinematics.
§3.3 Error Modeling
(1) Translation errors of the guide way
a) Translation error along X-axis [Type I]
In Figure 3.3, a translation error λXI along the X-axis of the ight guideway is added. The position of the ball joint U2 is taken from Eq.(2.10), and the true position of the ball joint U1 becomes
1 U ) 0 , 0 , ( Trans '
1
U = λXI ⋅ (3.2) After simplifing the equation, the distance between U1’ and U2 yields
) rSin 2 qSin g
3 (
Cos qrCos 2 Sin qrSin fqSin
3 grSin
3 fg r q g f b
XI
2 2 2 2 2
θ δ
λ
δ θ δ
θ δ
θ
−
− +
− +
−
−
− + + +
=
(3.3) Eq (3.3) can be expressed as
) 2qrCosCos δ
Sin
Figure 3.4 shows a translation error λXII along the X-axis of the left guideway is added .We can get the position of the ball joint U1 from Eq.(2.9) and derive the true position of the ball joint U2
2 The distance between U1 and U2’ yields
)
2qrCos θqrC
Sin
b) Translation error along Y-axis [Type I]
In Figure 3.5, a translation error λYI along the Y-axis of the right guideway is added. The position of the ball joint U2 remains the same and is taken from Eq.(2.10). The true position of the ball joint U1 becomes
1
Simplify the distance b between U1’ and U2 and obtain
Figure 3.6 shows a translation error λ along the Y-axis of the left YII guideway is added .We can get the position of the ball joint U1 form Eq.(2.9) and derive the true position of the ball joint U2
⎥⎥
Simplify the distance b between U1 and U2’ and obtain
) Eq (3.12) can be expressed as
)
c) Translation error along Z-axis
[Type I]
In Figure 3.7, a translation error λ along the Z-axis of the ZI right guideway is added. The position of the ball joint U2 remains the same and is taken from Eq.(2.10). The true position of the ball joint U1 becomes
1 Eq (3.15) can be expressed as
)
Figure 3.8 shows a translation error λ along the Z-axis of the left ZII guideway is added .We can get the position of the ball joint U1 form eq.(2.9) and derive the true position of the ball joint U2
2
The distance between U1 and U2’ yields
) Eq (3.18) can be expressed as
)
a) Rotatation error on XY plane [Type I]
In Figure 3.9, a rotatation error εXYI on the XY-plane is added to the right guideway. The position of the ball joint U2 remains the same and is taken from Eq.(2.10). The true position of the ball joint U1 becomes
⎥⎥ Eq (3.21) can be expressed as
]
Figure 3.10 shows a rotatation error εXYII on the XY-plane is added to
the guideway B2B3. We can get the position of the ball joint U1 form Eq.(2.9) and derive the true position of the ball joint U2
⎥⎥ Simplify the distance b between U1 and U2’ and obtain
]
Eq (3.24) can be expressed as
]
b) Rotatation error on YZ plane [Type I]
In Figure 3.11, a rotatation error ε on the YZ-plane is added to the YZI right guideway. The position of the ball joint U2 remains the same and is taken from Eq.(2.10). The true position of the ball joint U1 becomes
⎥⎥
Simplify the distance b between U1’ and U2 and obtain
] Eq (3.27) can be expressed as
]
Figure 3.12 shows a rotatation error εYZII on the YZ-plane is added to the left guideway. We can get the position of the ball joint U1 form Eq.(2.9) and derive the true position of the ball joint U2
⎥⎥
Simplify the distance b between U1 and U2’ and obtain
Eq (3.30) can be expressed as
] The deviations due to rotating the guide way on the XZ plane will not be derived because the joint of the RSSR mechanism is free.
(c) Rotatation error about the center of the base on XY plane [Type I]
Figure 3.13 shows a rotatation error ε about the center of the base OI on the XY-plane is added to the right guideway. We can get the position of the ball joint U2 form Eq.(2.9) and derive the true position of the ball joint U1
Simplify the distance between U1’ and U2 and obtain
Eq (3.33) can be expressed as
]
Figure 3.14 shows a rotatation error ε about the center of the base OII on the XY-plane is added to the left guideway. We can get the position of the ball joint U1 form Eq.(2.9) and derive the true position of the ball joint U2
⎥⎥
Simplify the distance between U1 and U2’ and obtain
] Eq (3.36) can be expressed as
]
[ θ δ θ δ
ε
δ θ δ
θ δ
θ
Sin qrSin 3 qSin ) f BR 3 2 ( rSin ) g BR 3 2 ( ) g f ( BR 2 fg 3
Cos qrCos 2 Sin qrSin fqSin
3 grSin 3 fg b r q g f F
OII
2 2 2 2 2 12
−
−
⋅ +
−
⋅ + +
⋅
⋅
−
−
− +
−
−
−
− + + +
=
(3.37) Table 3.1 summarizes the kinematic equations relating to a variety of geometric errors for the RSSR mechanism. Systems of equations, retrived and correlated from the kinematic equations listed in Table 3.1, in terms of the attitude angles θ , δ , and φ for any error condition discussed above can be obtained as shown in Table 3.2. Due to the geometric error, the new coordinates of the ball joints U1’, U2’, U3’ on the upper mobile plate can be computed with the solved attitude angles θ , δ , φ of a corresponding system of equations listed in Table 3.2. Figure 3.15 concludes a procedure of modeling the deviations along the radial direction of the circular contour under the effects of the geometric errors.
§3.4 Error diagnosis
The simulation plots of the radial deviations varing along the circular contour under the interaction of the error sources discussed in the previous section are presented as Figure 3.16 ~ Figure 3.20. These data are used as the reference data for identifying the deviations. In practice, a combination of these individual errors may lead to the incorrect pose of the end effector. For improving the position and orientation accuracy of the end effector, diagnosing the parameter errors relating to the faulty manufacture and assembly can be
beneficial to either revising the incorrect parts or adjusting the improper assembly of the mechanism, or compensating errors in the controller.
At first, the correlated parameter errors (deviations) in mathematic models are defined and the radial circular motion errors due to various kinds of individual error sources are derived respectively. Then the measured data, the composite error, based on the DBB test is obtained experimentally and applied to the estimation of error parameters with the least square method, which will be metioned in chapter 5. Referring to these error parameters, deviations of the end-effector of the six degree-of-freedom manipulator may be compensated in the controller repeatly so that the position and orientation precision of it can be promoted effectively.