2
F = ( OC R − OA
+ OC − OA
ℜ ⋅ CB
− L
(25)
f
i Rf )
( Rf
i Rf )
2
|
|
|
|
|
i m
R
i
i
F = FC
(26)
7
1
184
Parallel Manipulators, Towards New Applications
F = FC
(27)
8
2
The system comprises 6 polynomials of degree 4 and 2 quadratics. The highest degree
monomials are either xi4; xi3 xj or xi2 xj2. One more variable is added over the former
quaternion model. The variable choice is not intuitive.
4.2 Position based equations
We shall examine four formulations derived from the position based equations. Every
variable has the same units and their range is equivalent.
4.2.1 AFP1 - three point model with platform dimensional constraints
The 3 platform distinct points are usually selected as the three joint centers B 1, B 2 and B 3, fig.
5. The 6 variables are set as: OB i|Rf = [ xi, yi, zi] for i = 1 …3.
Using the relations 6, the constraint equations Li 2 = ║ A B
i
i | Rf║2, i = 1, …, 6 can be expressed
with respect to the variables x i, y i, z i, i = 1, 2, 3. Together with equations 30, they define an algebraic system with 9 equations in 9 unknowns { x 1, y 1, z 1, x 2, y 2, z 2, x 3, y 3, z 3}. The resulting kinematics chain system becomes:
F =
(28)
i
( x − OA
i
ix )2 + ( y − OA
i
iy )2 + ( x − OA
i
ix )2
2
− L , i = 1...3
i
2
2
F = B | jR − OA
− L j = …
(29)
b
|
j R
j
f
j ,
4
6
1
The mobile platform geometry yields the following three distance equations:
2
F = B B
− x − x
+ y − y
+ z − z
= B B
7
2
(
|
1 R
2
)2
1
( 2
)2
1
( 2
)
2
2
1
2
|
1
f
Rm
2
F = B B
− x − x
+ y − y
+ z − z
= B B
(30)
8
3
(
|
1 R
3
)2
1
( 3
)2
1
( 3
)
2
2
1
3
|
1
f
Rm
2
F = B B
− x − x
+ y − y
+ z − z
= B B
9
3
(
2| R
3
)2
2
( 3
)2
2
( 3
)
2
2
2
3
2|
f
Rm
Together with equations 30, they produce an algebraic system with 9 equations with 9
unknowns { x 1, y 1, z 1, x 2, y 2, z 2, x 3, y 3, z 3}. In all instances, it can be easily proven that this 6-6
FKP formulation yields 9 quadratic polynomials.
The system variable choice is relatively intuitive. Each equation polynomial is always
quadratic. However, the b 1 reference frame and the platform points B i in the b 1 frame require
computations, which usually result into coefficient size explosion. The variable number is
not minimal.
4.2.2 AFP2 - the three point model with platform constraints
The former system can be slightly modified by replacing the last mobile platform constraint
with a platform normal vector one. Hence, lets take the two mobile platform vectors
B B and B B , then the last constraint is calculated from these two vector multiplication:
1
2
1
3
Certified Solving and Synthesis on Modeling of the Kinematics. Problems of Gough-Type
Parallel Manipulators with an Exact Algebraic Method
185
2
2
F = B B
−
−
+
−
+
−
=
7
2
( x
x
|
1
2
)2
1
( y
y
2
)2
1
( z
z
2
1 )2
B B
R
2
|
1 R
f
m
(31)
2
2
F = B B
−
−
+
−
+
−
=
8
3
( x
x
|
1
3
)2
1
( y
y
3
)2
1
( z
z
3
1 )2
B B
R
3
f
|
1 Rm
F =
−
∗
−
+
−
∗
−
+
−
∗
−
−
∧
9
( x
x
3
1 )
( x
x
2
1 )
( y
y
3
1 )
( y
y
2
1 )
( z
z
3
1 )
( z
z
2
1 )
B B
B B
3
2| R
3
m
|
1 Rm
The result is still an algebraic system with nine equations in the former nine unknowns
{ x 1, y 1, z 1, x 2, y 2, z 2, x 3, y 3, z 3}. The 6-6 FKP formulation using this three point model is constituted by nine quadratic polynomials.
4.2.3 AFP3 - the three point model with constraints and function recombination
By rewriting the IKP as functions, the algebraic system comprises three equations and three
functions in terms of the nine variables: x 1, y 1, z 1, x 2, y 2, z 2, x 3, y 3, z 3, equation (29).
F =
(32)
i
( x − OA
i
ix )2 + ( y − OA
i
iy )2
2
− l , i = ...
1 3
i
2
2
C = B | kR − O A
− l i = …
(33)
b
|
k R
i
f
i ,
4
6
1
Hence, three constraints are derived from the following three functions, [Faugère and
Lazard 1995]. Two functions can be written using two characteristic platform vector norms
between the B 1, B 2 distinct points and the B 1, B 3 ones. The last function comes from these
vector multiplication.
2
2
F = B B
−
−
+
−
+
−
=
7
2
( x
x
|
1
2
)2
1
( y
y
2
)2
1
( z
z
2
1 )2
B B
R
2
f
|
1 Rm
(34)
2
2
F = B B
−
−
+
−
+
−
=
8
3
( x
x
|
1
3
)2
1
( y
y
3
)2
1
( z
z
3
1 )2
B B
R
3
f
|
1 Rm
F =
−
∗
−
+
−
∗
−
+
−
∗
−
−
∧
9
( x
x
3
1 )
( x
x
2
1 )
( y
y
3
1 )
( y
y
2
1 )
( z
z
3
1 )
( z
z
2
1 )
B B
B B
3
2| R
3
m
|
1 Rm
Furthermore, the three last equations ( F 7, F 8, F 9) are computed by the following function
sequential combinations:
F 7 = − C 7 + F 1 + F 2
F 8 = − C 8 + F 1 + F 3
(35)
F 9 = 2∗ C 9 + F 7 + F 8 − 2∗ F 1
The formulation is completed with other function combinations obtained by the following
algorithm leading to three middle equations ( F 4, F 5, F 6). Let d 7 = ║ B B
2
1 |Rmj║, d 8 =
║ B B
B B | Rm║ ^ ║ B B
3
1 | Rm║2 and d 9 = ║
3
2
3
1 |Rm║, then for i = 4, 5, 6, we compute:
C = C − a 2 ∗ C − b 2 ∗ C − c 2 ∗ C ∗ C − C 2 − a ∗ b ∗ 2 ∗ C
i
i
B
7
B
8
B
( 7 8 9 ) B B ( 9)
i
i
i
i
i
C = 2 ∗ C − a ∗ F − 2 ∗ F − b ∗ F + F − F
i
i
B
( 7
1 )
B
( 1 2 7)
(36)
i
i
F = C − 2 ∗ c 2 ∗ d ∗ F + F − F − 2 ∗ c 2 ∗ d ∗ F + F − F
i
i
B
7
( 1 2 8)
B
8
( 1 2 7)
i
i
− 2 ∗ c 2 ∗ d ∗ F − F + F − 2 ∗ F + 2 ∗ a + b −1 ∗ F − F ∗ a − F ∗ b B
9
( 7 9 8
1 )
( B Bi )
i
i
1
7
i
B
8
i
B
186
Parallel Manipulators, Towards New Applications
The result is an algebraic system with nine equations with the nine unknowns. The 6-6 FKP
formulation using this modified three point model includes six quadratic and three quartic
polynomials. The system includes polynomials of higher degree than for the former two
position based models. Computations cause to coefficient expansion.
4.2.4 AFP4 - the six point model
The six mobile platform B i joints can be used in defining 18 variables, [Rolland 2003]. Taking
the IKP equations (8), a position based variation is obtained:
2
l =
(37)
i
( x − OA
i
ix )2 + ( y − OA
i
iy )2 + ( x − OA
i
ix )2 , i = 1...6
The system is completed with 12 distance constraint equations selected among the distinct B i
passive platform joints. Here are some examples:
2
B B
=
i
( x − x
R
i
)2 + y − y + z − z = B B
i = …
|
1
1
( i
)2
1
( i 1)
2
2
,
1
6
i
|
1
f
Rm
2
2
B B
=
(38)
j
( x − x
j
)2 + y − y + z − z = B B
j = …
2
( j
)2
2
( j 2)2
,
3
6
|
2
j
R
|
2
f
Rm
2
B B
=
k
( x − x
R
k
)2 + y − y + z − z = B B
k = …
|
3
3
( k
)2
3
( k 3)
2
2
,
4
6
k
|
3
f
Rm
The formulation results in 18 polynomials in the 18 unknowns:
{ x 1,