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(2) Where Fn is the normal component, Rn is the normal component ratio, Ft is the tangential component, and Rt is the tangential force ratio. The various coefficients are experimentally determined by measuring the force for various depths of cut and feed-rate. It showed the presence of average size burrs has minimal effect on the cutting force and the burr is not considered in (2). The tangential cutting force is proportional to the material removal rate (MRR) [7]. When chamfering a 45-deg angle into a part, the MRR can be expressed as: () (1) cc b tool tng toolMRR A A V A R V =+ = + (3) tng bc RAA = (4) Where Ac and Ab denote the cross sectional area of the chamfer and the burr, Rtng and Rnorm are the surface area tangential ratio and the normal ratio, respectively, and Vtool is the velocity of the grinding tool along the workpiece edge. The surface area ratios Rtng and Rnorm can be used to determine the tangential and normal force variations[7]. For the worst burrs, (1) can be expressed as (1 ) n n norm F RF R =⋅⋅+ , (1 ) tt tng FRF R =⋅⋅+ (5) From (5), we can know that large variations are expected in the components of the cutting force due to variations in the burr area for a given feedrate and chamfer area. Therefore, constant force control is very limited and several repetitions of deburring procedures are needed. 2 Dynamic model of system 2.1 Dynamic model of the robot In general, a dynamic equation of an n-joint constrained manipulator can be expressed as () (, ) e Dqq hqq τ τ = ++
(6) where,nqR ∈ denotes the joint angle vector; ()nnDq R ×∈论文网 is the non-singular inertia matrix, and (, )nhqq R ∈ is a nonlinear function of the centrifugal, coriolis, gravity and friction force; nR τ ∈ represents the joint torque vector of input control; mnJR ×∈ is the Jacobian matrix connecting the velocities of robot end-effector and the velocities of robot generalized coordinates; me F R ∈ is the external force exerted from the environment to the end-effector in contact movements. m is the number of contact force components. The relationship of the velocity and acceleration between joint space and Cartesian space can be expressed as () x Jqq = , () x Jqq Jq = + (7) From (6) and (7) the robot dynamic equation model in Cartesian space can be obtained as ** e D xh FF + =− (8) Where 11* TDJDJ−−= , 11** ThJhDJJx−−=− . Let () x q denote the position vector of the end effector of the manipulator. The geometrical constraint can be modeled by (()) 0 x q φ ≤ (9) To express fn and ft on the chamfer surface, we define two unit orthogonal vectors, n(x) and t(x), represent the unit vector normal to the constraint surface and tangential to constraint surface, respectively. These two vectors are given by , ) ( / ) ( ) ( x x x nTφ φ ∇ ∇ = x x x t / ) ( = (10) The external force, Fe, given in (8) is the contact force due to the constraint, and can be written as () () entF nx f tx f = + (11) 2.2 Impedance control Impedance control is a general approach to robot motion and force control that attempts to make a manipulator behave as a mass-spring-dashpot system whose parameters can be specified arbitrarily. The desired impedance property of the end-effector can be given as de M XBXKXF F + +=− +++