Adaptive Robotic System for 3D Profile Grinding/Polishing 71 rotation,are independent variables of the index function.As the airfoil profiles are odd shaped,there is no analytical expression available for the index function.As a matter of fact,the value of the index function can only be obtained through some numerical methods.Therefore,any existing minimisation method which uses analytical expression or any derivatives of the index function will not be applicable to our problem. To solve airfoil optimal profile fitting problem,we developed an intuitive direct search method.It turns out that the new method is an improved version of Hooke-Jeeves pattern search method [12],with a faster convergence rate.The basic idea is that we take a point in the search space as a base point and explore around it to find a right direction and a right step to move the base point towards the minimum point.In order to obtain a fast convergence rate,the direction is adjusted and the step is increased repeatedly before each move of the base point.The following is the outline of the multi-dimensional minimisation algorithm.An illustrative flow chart of the algorithm is shown in Figure 9. Step 1 Set initial base point and initial search span.The search span may be different for each coordinate direction in the search space. Step 2 Test the index function values of the surrounding points of base point with search span.Find the best point(i.e.with the lowest index)among the surrounding points.In this step,all the coordinate directions in the search space have to be exhausted. Step 3 Compare base point with best point.If base point is better (i.e. with a smaller index)than best point (which means the minimum point is within the search span),jump to Step 9.Otherwise, continue. Step 4 Take best point as direction point. Step 5 Along the direction from base point to direction point,compute temporary base point by extrapolation with a ratio,say 2.5.That means temporary base point will be away from base point 2.5 times of the distance between base point and direction point. Step 6 Test the index function values of the surrounding points of temporary base point with search span.Find best point among the surrounding points.Again,all the coordinate directions in the search space have to be considered in this step. Step 7 Compare direction point with best point.If best point is better than direction point (which means that the direction of last extrapolationAdaptive Robotic System for 3D Profile Grinding/Polishing 71 rotation, are independent variables of the index function. As the airfoil profiles are odd shaped, there is no analytical expression available for the index function. As a matter of fact, the value of the index function can only be obtained through some numerical methods. Therefore, any existing minimisation method which uses analytical expression or any derivatives of the index function will not be applicable to our problem. To solve airfoil optimal profile fitting problem, we developed an intuitive direct search method. It turns out that the new method is an improved version of Hooke-Jeeves pattern search method [12], with a faster convergence rate. The basic idea is that we take a point in the search space as a base point and explore around it to find a right direction and a right step to move the base point towards the minimum point. In order to obtain a fast convergence rate, the direction is adjusted and the step is increased repeatedly before each move of the base point. The following is the outline of the multi-dimensional minimisation algorithm. An illustrative flow chart of the algorithm is shown in Figure 9. Step 1 Set initial base point and initial search span. The search span may be different for each coordinate direction in the search space. Step 2 Test the index function values of the surrounding points of base point with search span. Find the best point (i.e. with the lowest index) among the surrounding points. In this step, all the coordinate directions in the search space have to be exhausted. Step 3 Compare base point with best point. If base point is better (i.e. with a smaller index) than best point (which means the minimum point is within the search span), jump to Step 9. Otherwise, continue. Step 4 Take best point as direction point. Step 5 Along the direction from base point to direction point, compute temporary base point by extrapolation with a ratio, say 2.5. That means temporary base point will be away from base point 2.5 times of the distance between base point and direction point. Step 6 Test the index function values of the surrounding points of temporary base point with search span. Find best point among the surrounding points. Again, all the coordinate directions in the search space have to be considered in this step. Step 7 Compare direction point with best point. If best point is better than direction point (which means that the direction of last extrapolation