(12) Where  Fd is the desired force and  ,,mm MBK R ×∈ are the desired inertia, viscosity and stiffness matrices of the end-effector, respectively.  X +  and  X   + are the adjustments of the impedance control,md F R ∈ , denotes the desired end-point force vector.  The control law for implementing the above impedance control concept is   ** F DU h =+                           (13) rD UX KEKpE =+ +                           (14) Where  KD and  Kp are the desired positive-definite diagonal matrices for damping and stiffness gains, E and  E    are the error between the actual position and the position command, respectively.  3  Intelligent force/position controller for robotic deburring The following constraints may be considered important during robotic deburring process. First of all, the desired workpiece contour of the finished product, and the maximum deflection allowed (tolerance) must be known which is crucial to setup the control parameters.  As has been shown above and the section one, only tangential or normal force control is not enough for the robotic deburring and the impedance control is limited for the stability problem.  Only impedance control is also not enough to get desired contour. 3.1 Intelligent force/position controller In this paper, a new force/position strategy is proposed (Fig.1). In industry manufacturing, burrs sizes of the workpiece are mostly uniform, and some bigger burrs and cavities are seldom encountered. The new strategy is based on the normal force control method, and when the bigger burrs and the cavity defects are encountered, the tangential force control is used to remove burrs entirely and avoid damaging workpiece. 3.2 The fuzzy impedance scheme In this paper, only straight edge deburring tasks are discussed, and the considered operation dimension is two. The impedance control is used to achieve the force control in two directions. For example, in the normal direction, the impedance model (9) can be rewritten as  dn en n n n f fmxbxkx −= + +      +++           
  (15) By the (12), mn , bn , kn is the weights determining the contributions of  , x x      ++ and  x +  to the overall force difference, respectively. It is obvious that they can be tuned for realization of the force/position control for deburring process by the fuzzy logic. The fuzzy controller has two inputs. First input is  position deviation  pd E XX =− , and the second is the velocity deviation  p d E XX =−     . Output of the fuzzy controller is the change of the stiffness coefficient. The fuzzy rules are described by   IF  pi E  is  Aj and  pi E    is  Cj then  ni k +  is  Lk  , Where i represents the variable of direction (tangential or normal direction), j is the number of input variable and k is the number of output variables. Fuzzy impedance controllerDeburring OperationImpedance Control−−Robotτ Position Controller−Fuzzy Adaptive Mechanism Fuzzy velocity controllerf f E E   ,p p E E   ,e FF ∆k ∆x ∆ x   ∆x x   ,−d Fd d x x   ,d xd x   Fig.1  Block scheme of the intelligent force/position control 3.3 The fuzzy velocity scheme To maintain the tangential and normal force in desired range and avoid causing instability of the system, the feedrate in tangential direction should be controlled correctly. The fuzzy impedance control only can adjust the velocity in small range and the fuzzy velocity controller is used to deal with the extreme situations, and we define three fuzzy sets small, medium and  big, which is the  vmin  ,  vnorm and  vmax  , respectively. Fuzzy controller has two inputs. First input of the fuzzy controller is force deviation () fde E FFk = − , and the force increment () ( 1) fe e EFkFk = −− + . Outputs of the fuzzy controller are the changed velocity. The fuzzy rules are described by IF  fi E  is  j P  and  fi F +  is  Qj then  vi is  Rk  , Where i represents the variable of direction (tangential or normal direction), j is the number of input variable and k is the number of output variables.  During most time of the process the reference feedrate is kept as constant and the robot motion is corrected in the normal direction with the help of fuzzy impedance control. The desired force is given based on the desired chamfer and the impedance parameters are modified by the fuzzy logic algorithm according position. The parameter  k is crucial to the stability of end effector for frequency reasons shown above. Its change region is limited and can not be used to remove the bigger burrs and cavity defects. The fuzzy velocity controller is used to adjust the feedrate according to the force deviation, and the feedrate will be increased when a cavity is encountered and decreased when a bigger burr is encountered. In this way, the bigger burrs can be removed completely and the workpiece can avoid to be damaged when a cavity is encountered.  4  Simulation results and discussion To demonstrate the performance of proposed algorithm, a simulation study has been carried out using a three links rotary robot manipulator whose parameters  are taken from the first three links of a Puma560 arm. The workpiece profile is defined to have an average burr, a step burr and a cavity as shown in Fig.2. Where xc is the position of the workpiece from CAD, the ideal edge prior to chamfering, xe is the actual position of the workpiece with the burrs,  xd is the desired position of chamfered edge, x is the actual position of the tool. This paper uses the model determined for deburring of low-carbon steel parts[8]. d x xe xc x Toolxy oFeedWorkpiece Fig.2  The profile of workpiece with burrs and a cavity The deburring feed-rate is set to 20mm/s for the desired chamfering (2mm). The force corresponding to the desired cutting depth is 6.35N by (1).  4.1 Conventional impedance control Results of the conventional impedance control for robotic deburring without using fuzzy logic are shown in Fig.3. Fd is the desired force in normal direction and is specified at constant magnitude which is 6.35N, Fe is the actual force in deburring process. The result shows the transition phase is characterized by relatively high overshoot and impaction of force. In addition, from the simulation results, it can be seen that the contour of chamfered surface is a replication of the original surface and a uniform chamfer depth of 2 mm is produced with the constant contact force.    0.1005 0.1 0.0995 0.099 0.0985 0.098 0.0975 0.097 Position / m Time / s 0  1 2 3 4 5 6  (a)  The Position Response  7 6 5 4 3 2 1 0 Force / N Time / s 0  1 2 3 4 5 68 9 10  (b)  The Force Response Fig.3 Results of conventional impedance control applied to deburring 4.2 Intelligent force/position control without fuzzy velocity controller  Results of the proposed intelligent force/position control without the fuzzy velocity controller are shown in Fig.4. Compared with Fig.3 (b), the overshoot of the force is reduced and touch stability is improved. From the Fig.4 (a), it can be seen that the small burr is removed mostly, but the bigger needs once more deburring operation to be removed completely. When the robot end-effector encounters the cavity, the workpiece is also damaged. 0.10050.1 0.09950.0990.09850.0980.09750.097Position / m Time / s 0 1 2 3 4 5 6 (a)  The Position Response 7 6 5 4 3 2 1 0 Force / N Time / s 0 1 2 3 4 5 68 9 10 (b)  The Force Response Fig.4  Results of the proposed controller  without fuzzy velocity controller applied to deburring 4.3 Intelligent force/position controller Results of the intelligent force/position controller are shown in Fig.5. Where the impedance parameters are modified when the burr or cavity is encountered and the fuzzy velocity controller is used to limit the cutting force in tolerance, the results show better quality surface is achieved. In addition, it is obvious that the transient impact is reduced. In this process, when an average burr is encountered, the fuzzy impedance controller is used to remove burrs by adjusting the impedance parameters, and when the chamfer depth is increased from 2 mm to 2.5 mm(big burr), the fuzzy velocity controller modifies the feedrate and the smooth contour is achieved. Similarly, when a cavity exists, if the force is maintained with a constant magnitude, the workpiece will be damaged as shown in Fig.3 (a). Thus, the results indicate that a more uniform surface is achieved with the intelligent force/position control strategy as compared with the case of conventional control methods.    0.1005 0.1 0.0995 0.099 0.0985 0.098 0.0975 0.097 Position / m Time / s 0  1 2 3 4 5 6  (a)  The Position Response  7 6 5 4 3 2 1 0 Force / N Time / s 0  1 2 3 4 5 68 9 10  (b)  The Force Response Fig.5  Results of the proposed intelligent  force / position control applied to deburring 5  Conclusions In this paper, an empirical force model taking into account the burrs effect and an intelligent force/position strategy for robotic deburring are developed. Only tangential or normal force control is not enough for the robotic deburring and the impedance control is limited for its stability problem. In the paper, we present an intelligent force/position controller in which the impedance parameters and the feedrate can be adjusted according to the burrs size to obtain better quality contour and improve control performance.  The simulation results show that the presented chamfering and deburring controller could generate an appropriate response to burr variations and other operating conditions instantaneously, yielding a desired chamfered edge. The proposed strategy can be applied for the other robotic manufacturing cells like grinding and polishing. Reference [1] Liu M H. Force-Controlled Fuzzy-Logic-Based Robotic deburring [J]. Control Engineering Practice (S0967-0661), 1995, 3(2): 189-201. [2] Su S F, Horng T J, Young K Y. Evolutionary-based Virtual Training in Extracting Fuzzy Knowledge for Deburring Tasks [J]. Journal of the Chinese Institute of Engineers, Transactions of the Chinese Institute of Engineers (S0257-9731), 2004, 27(2): 193-202. [3] Liu H Y, Wang L, Wang F. Fuzzy Force Control of Constrained Robot Manipulators based on Impedance Model in Unknown Environment [J]. Dongbei Daxue Xuebao/Journal of Northeastern University (S1005-3026), 2005, 26(8): 766-769. [4] Chen Y Y, Ji Z, Wang B D,  et al. High Precision Fuzzy Impedance Control of Free-form Surfaces Polishing Robotic Arm based on Position Control [C]// IEEE/ASME International Conference on Advanced Int 模糊逻辑的机械手智能力/位控制英文文献和中文翻译(3):http://www.youerw.com/fanyi/lunwen_52553.html