§5-1 Error Estimation
Let x be the slider positions, x∈ℜ6, all p of the kinematic parameters of the parallel manipulator can be contained in the vectorβ ∈ℜp, and the reading from the ball bar, y∈ . Since the ball bar is a passive joint, the end effector pose
ℜT
ρ will be determined by the kinematic parameters and the slider positions, )
, ( β
ρ = forkin x . Once the pose is determined, the length of the ball bar is fixed through the inverse kinematics by y=invkin(ρ,β). These relationships can be combined to give
y =invkin(forkin(x,β),β)= f(x,β) (5.1) The function f is a numerical function that combines the inverse and forward kinematics of the parallel manipulator.The relationship given by the above equation holds for the actual values y, x, andβ . In reality, these values are corrupted by unknown errors. Therefore, Eq. (5.1) is more accurately stated as
y + ~y = f(x+ ~x,β + β~) (5.2) where the notation stands for nominal values in the case of β , slider positions of x , the ball bar reading of y , and the tilda notation represents small
deviations. Over a series of i =1...n poses, Eq. (5.2) becomes
~)
~ ,
~ = ( + β + β
+ i i i
i y f x x
y (5.3) The goal of the diagnosis is the way to find a combination of deviations (errors) that satifies Eq. (5.3). The diagnosis should not just find any combination of deviations, but it should find the “best” combination. By formulating the diagnosis as a least square minimization, deviations will be estimated. In this chapter, it is the major work we will concentrate on.
§5-2 Least Square Method
Consider a system of linear equations : Ax = b
Where A ∈Rmxn, b∈Rm, m n, and rank A = n. Note that the number of unknows, n, is no longer than the number of equations, m. If b does not belong to the range of A; that is , if b
≥
∉R(A), then this system of equations is said to be insistent or overdetermined. In such case, there is no solution to the above set of equations. Our goal is to find the vector (or vectors) x miniming Ax−b 2
Let x*be a vector that minimizes Ax−b ; that is, for all x∈Rm, b
Ax− 2 ≥ Ax*−b 2
We refer to the vector as a least-squares solution to Ax = b. In the case where Ax = b has a solution, then the solution is a least-squares solution.
Otherwise, a least-squares solution minimizes the norm of the difference x*
between the left- and right-hand sides of thhe equation Ax = b. The unique vector x* that minimizes Ax−b 2 is given by the solution to the equation ATAx=ATbA (5.4) that is,
b A A A
x* =( T )−1 T (5.5) In view of considering a fitting function with a form of the linear combination of several known functions , the fitting function can be written as follow :
yˆ
fi yˆ
yˆ=f(x,c1,c2,...,cm)=c1f1(x)+c2f2(x)+...+cmfm(x) (5.6) where the dependent varible is linear with respect to coefficients , ,…, . Let x is a set of independent varibles , , ,…, , and is a set of corresponding dependent varibles , , ,…, . Meanwhile, y is the set of datum , , ,…, obtained from the experiment. Consequently, the residuals
can be experssed as :
yˆ c1 c2 cm
x1 x2 x3 xn yˆ
yˆ1 yˆ2 yˆ3 yˆn
y1 y2 y3 yn
ri
⎪⎪
⎪⎪
⎩
⎪⎪
⎪⎪
⎨
⎧
=
−
=
−
=
−
∑
∑
∑
=
=
=
n n n m
1 k
k
2 1 2 m
1
k k
1 1 1 m
1 k
k
r y ) x ( f c
...
...
...
r y ) x ( f c
r y ) x ( f c
(5.7)
For the sake of minimizing the sum of the squares of the residuals, Eq (5.7) should be satisified.
0
Equation (5.8) can be formulated in a matrix form of linear algebratic equations in terms of m undetermined coefficient as follows:
[ ] [ ] [ ]
§5-3 Simulation and Diagnosis
The computer program based on the model we have developed in previous sections simulates a variety of contouring error conditions. These simulation results presented in polar plots are shown in Fig.3.16~Fig.3.20. A phenomona observed in the simulation results reveals that the error contour rotates as the guide way rotates in any error condition we have discussed before. The experimental data based on DBB test are shown in Fig.4.8. as well.
120ο
60ο
The simulation of the diagnosis using the data generating from the synthesis of theoretical error contours is illustrated as follows. A set of values of parameter errors may be given at first to generate the corresponding contouring errors and these contouring errors can be summed to generate a synthetic contouring error. Then we estimate the values of these parameter errors of the synthetic contouring error inversely by means of the least square technique. In comparison with the given values and the identified ones of these parameter errors, we can judge that the estimation (diagnosis) can be judged to be valid or not.
Li
∆
The matrix A is prepared using the reference data corresponding to each deviation as shown in Figure 3.16 ~ Figure 3.20. So far as the diagnosis is concerned, the error model and the measurement strategy should be properly designed to yield a nonsigular matrix ATA such that the identified parameters are completely observable. As the solutions of altitude angles θ , δ , φ are obtained through the Newton recursive formula and the circular contouring error
is computed with Li
∆ θ, δ , and φ, the contouring error term will be expressed with numerical forms instead of polynomial forms. Without exact solutions, it is not easy to judge what variables or functions are related to each contouring error and whether there exist the relationship of linear independent among all of them. The error estimation begins with the sorting and merging of parameter errors into independent parameters. In the following
Li
∆
Li
∆
practical simulations for diagnosis, we set the guideway B3B1 as the reference one without errors and obtain results as follows:
(1)In case of taking five geometric errors, including three linkage length errors and two guideway translation errors to execute the error estimation, It can be seen that the identified values of the five motion errors are corresponding to the given ones (shown as Table 5.1).
(2)In case of taking six geometric errors, including three linkage errors and three guideway translation errors to execute the error estimation, we can find that a part of the identified values of the six motion errors are not corresponding to the given values (shown as Table 5.2).
(3)In case of taking four geometric errors, including four linkage length errors, we can find that a part of the identified values of the four motion errors are not corresponding to the given values (shown as Table 5.3).
(4)In case of taking six geometric errors, including three linkage length errors, two guideway translation errors, and a guideway rotating error to execute the error estimation, It can be seen that the identified values of the six motion errors are corresponding to the given ones (shown as Table 5.4).
(5)In case of taking seven geometric errors, including three linkage errors, two guideway translation errors, and two guideway rotation errors to execute the error estimation, we can find that the identified values of the seven motion errors are corresponding to the given values (shown as Table 5.5).
(6)In case of taking eight (or nine) motion errors, including three linkage length errors, two guideway translation errors, and three (or four) guideway rotation errors to execute the error estimation, we can find that a part of the identified values of the nine motion errors are not corresponding to the given values (shown as Table 5.6 ~ Table 5.7).
As a result, we can identify seven geometric errors, including three linkage length errors, two guideway translation errors, and two guideway rotation errors.
The simulation result of the error estimation with data based on DBB test for the manipulator SP-120 is shown as Table 5.8.
Chapter 6 Conclusion
In the study, the error modeling based on the numerical method has been performed to analyze the parameter errors corresponding to geometric errors for the hexglider manipulator. We use the Newton-Raphson method to reduce the complexity of the nonlinear equations properly by means of transferring them into linear equations. Then, the least-square technique has been applied to evaluating the coefficients with regard to the linear combination of the modeling errors in order to diagnose the geometric errors resulting in motional deviations.
As for the concrete implement and verification of the error modeling and diagnosis, a computer-aided analysis program has been developed for estimating parameter errors based on the circular contour error measurement using the kinematical double ball bar. The developed software has been applied to a practical case of the hexglider manipulator.
In spite that we can identify geometric errors and make an attempt to approximate the testing curve of a true hexglider manipulator based on the double ball bar test, however, there are still a few error sources, except for geometric errors, existing at a hexglider manipulator and left to be identified for improving the accuracy of a hexglider manipulator. Therefore, it is worthwhile
to make everlasting efforts on exploring the issue in the future.
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