An efficient multipoint 5-axis tool positioning method for tensor product surfaces

Abstract

In this work, a robust tool-drop algorithm coupled with the bisection method is used to speed up the computations in the drop and tilt method (DTM), a multipoint tool positioning technique for 5-axis machining with a toroidal end mill. In the DTM, the tool is dropped on the surface along a predetermined tool-axis direction until it touches the surface tangentially. The position of the circular tool insert (pseudo-insert circle) through the first point of contact is then used to compute the tool-tilt angle that causes the tool to touch the surface tangentially at a second point. A previous implementation of DTM used Newton’s method to solve the system of equations for tilting the tool, but took many iterations and multiple initial seed solutions to converge to a valid solution for tool-tilt orientation. Using the bisection method coupled with the generalized tool-drop algorithm instead of a direct use of Newton’s method to determine the tool-tilt angle improves performance without compromising the accuracy. The method was implemented in C+ +, and a comparative study with an earlier method for several bi-quadratic and bi-cubic Bézier surfaces demonstrates the reliability of the developed method.

Appendix: Newton’s method

Newton’s method iteratively computes the value of an error function $F(X, \hat {t},T)$ by numerically estimating the values of X = [u,v,h] to obtain the solution for the point of contact between the dropping toroidal tool and the surface S(u,v). The error function solved using Newton’s method for tool drop for is

$$ F(X, \hat{t},T)=S(u,v)+R_{i} \hat{n}(u,v) + R_{o} \hat{j}(u,v, \hat{t},T) - T - h\,\hat{t}. $$

(1)

For the set of three non-linear simultaneous rational equations given by $F(X, \hat {t}, T)=[F.x, F.y, F.z]$ the forward difference approximation of the derivative is

$$\frac{\partial F(X, \hat{t}, T)}{\partial X} = \frac{F(X_{i + 1}, \hat{t}, T) - F(X_{i}, \hat{t}, T)}{X_{i + 1} - X_{i}}.$$

Knowing a value of X _i , the value of X_i+ 1 for which $F(X_{i + 1}, \hat {t}, T)= 0$ is

$$X_{i + 1}=X_{i}-[J]^{-1}F(X_{i}, \hat{t}, T)$$

where J is the Jacobian matrix:

$$J= \left[ \begin{array}{ccc} \frac{\partial F.x}{\partial u} & \frac{\partial F.x}{\partial v} & \frac{\partial F.x}{\partial h}\\ \\ \frac{\partial F.y}{\partial u} & \frac{\partial F.y}{\partial v} & \frac{\partial F.y}{\partial h} \\ \\ \frac{\partial F.z}{\partial u} & \frac{\partial F.z}{\partial v} & \frac{\partial F.z}{\partial h} \end{array} \right]. $$

The elements of the Jacobian J are the partial derivatives of the x,y, and z components of $F(X, \hat {t}, T)$ with respect to u,v, and h; their expansions are given in Appendix A.2.

The pseudocode for Newton’s method is given in Fig. 13.

Fig. 13

Pseudocode for Newton’s method

1.1 A.1 Upper bound number of iterations in Newton’s algorithm

Newton’s method is sensitive to the initial seed values of surface parameters (u,v) and for a given initial seed solution X _i = [u _i ,v _i ,h _i ], it may not yield required solution for the point of contact within the bounds of the Bézier surface given by u _{m i n} ≤ u ≤ u _{m a x} and v _{m i n} ≤ v ≤ v _{m a x} . Thus, an upper bound on the number of iterations used in Newton’s method, num _N , is required. If Newton’s method fails to converge to a solution within num _N iterations then the search for the solution is stopped and Newton’s algorithm is re-initiated with a new initial seed solution as given in line 4.5.b of the pseudocode for tool-tilt in Fig. 6.

To compare the computational efficiency of the bisection DTM with the direct numerical DTM version [19] the upper bound for num _N is kept at 50. The comparison of the number of iterations used by Newton’s method in the vertical tool drop (to determine the first point of contact) and the tilted-axis drop in the numerical DTM and the bisection-based DTM (to compute the second point of contact) is given in Tables 6 and 7. For computing first point of contact, most valid solutions of Newton’s algorithm converged within 29 iterations while in case of computing the second point of contact (using both bisection DTM as well as direct numerical DTM) the maximum number of iterations were as high as 50.

1.2 A.2 Jacobian matrix for Newton’s method

The elements of Jacobian J matrix are the derivatives of x,y, and z components of $F(X, \hat {t}, T)$ with respect to variables u,v, and h. These derivative terms may have complex expressions depending on the order of the given part surface and need to be determined with significant accuracy. The mathematical expressions of various Jacobian terms are given as:

$$ \frac{\partial F}{\partial u}=\frac{\partial S(u,v)}{\partial u}+R_{i}\frac{\partial \hat{n}(u,v)}{\partial u}+R_{o}\frac{\partial \hat{j}(u,v)}{\partial u} $$

(2)

$$ \frac{\partial F}{\partial v}=\frac{\partial S(u,v)}{\partial v}+R_{i}\frac{\partial \hat{n}(u,v)}{\partial v}+R_{o}\frac{\partial \hat{j}(u,v)}{\partial v} $$

(3)

$$ \frac{\partial F}{\partial h}=-\hat{t}. $$

(4)

where

$$\frac{\partial S(u,v)}{\partial u} = \sum\limits_{i = 0}^{n} \sum\limits_{j = 0}^{n} P_{i,j}\frac{\partial{{B_{i}^{n}}(u)}}{\partial{u}}{B_{j}^{n}}(v)$$

$$\frac{\partial S(u,v)}{\partial v} = \sum_{i = 0}^{n} \sum_{j = 0}^{n} P_{i,j}{B_{i}^{n}}(u)\frac{\partial{{B_{j}^{n}}(v)}}{\partial{v}}.$$

$$\frac{\partial\hat{n}(u,v)}{\partial{u}}= \frac{\partial}{\partial{u}}\left[{\frac{\frac{\partial S}{\partial u}\times \frac{\partial S}{\partial v}}{\Arrowvert\frac{\partial S}{\partial u}\times \frac{\partial S}{\partial v}\Arrowvert}}\right]$$

$$\frac{\partial\hat{n}(u,v)}{\partial{v}}= \frac{\partial}{\partial{v}}\left[{\frac{\frac{\partial S}{\partial u}\times \frac{\partial S}{\partial v}}{\Arrowvert\frac{\partial S}{\partial u}\times \frac{\partial S}{\partial v}\Arrowvert}}\right]$$

$$\frac{\partial\hat{j}(u,v)}{\partial{u}}= \frac{\frac{\partial j(u,v)}{\partial u}}{\Arrowvert j(u,v)\Arrowvert} -\frac{j(u,v)}{\{\Arrowvert j(u,v)\Arrowvert\}^{3}}\cdot\left\{ j(u,v)\cdot\frac{\partial j(u,v)}{\partial u}\right\}$$

$$\frac{\partial\hat{j}(u,v)}{\partial{v}}= \frac{\frac{\partial j(u,v)}{\partial v}}{\Arrowvert j(u,v)\Arrowvert} -\frac{j(u,v)}{\{\Arrowvert j(u,v)\Arrowvert\}^{3}}\cdot\left\{ j(u,v)\cdot\frac{\partial j(u,v)}{\partial v}\right\}$$

where

$$\frac{\partial j(u,v)}{\partial{u}}=\left\{(\frac{\partial S(u,v)}{\partial u}\cdot\hat{t})\cdot\hat{t} - \frac{\partial S(u,v)}{\partial u}\right\}$$

$$\frac{\partial j(u,v)}{\partial{v}}= \left\{(\frac{\partial S(u,v)}{\partial v}\cdot\hat{t})\cdot\hat{t} - \frac{\partial S(u,v)}{\partial v}\right\}$$

and $j(u,v)=[T-S(u,v)-\{(T-S(u,v))\cdot \hat {t}\}\cdot \hat {t}]$.