Symbolic computation of robot models for geometric parameters identification with singularity analysis

The computation of the geometrical parameters identification model of a robot arm leads to an intrinsic extended Jacobian matrix. This matrix is often singular and its manual computation is tedious and may lead to error. A computer assisted procedure is developed to compute this matrix symbolically and to check its singularity. The program GPIM is developed in Pascal and works on PC's for this purpose. Jacobian singularity is checked using its symbolic expressions with the aid of a proposed table for robot arm parameters to obtain a minimum set of identified parameters. This program can be used for simple chain robot arms having revolute (R) and/or prismatic (P) joints including the case of consecutive near parallel axes. TH8 robot arm of type RPPRRR having two pairs of consecutive near parallel axes is studied to show the program efficiency and how singularity is eliminated.Nomenclature a_i Denavit-Hartenberg parameter. - C_i,C_i andC₁₂ cos _i, cos _i and cos (₁ + ₂). - J₀, J Basic Jacobian and Jacobian matrices. - J_a, J_r, J, J and J Sub-Jacobian matrices. - J_0a, J_0r, J₀, J₀ and J₀ Intrinsic sub-Jacobian matrices. - m Work space dimensions. - n Number of moving links of the robot arm. - p Number of pairs of consecutive near parallel axes. - P_i,i+1 and P_li [3×1] position vectors. - Q_pi, Q_ai, Q_ri, Q_i, Q_i and Q_i [4×4] differential operator matrices. - r_i Denavit-Hartenberg parameter. - R_l, R_i, R_i+1 and R_n+1 Base, links i–1, i and n coordinate frames. - R_i,i+1 and R_1i [3×3] orientation matrices. - S_i, S_i and S₁₂ sin _i, sin _i and sin ₁ + ₂. - T_i,i+1 and T_1i [4×4] homogeneous transformation matrices. - X [m×1] robot arm operational coordinates. - 1st columns of the orientation matrices, and associated tensors. - 2nd columns of the orientation matrices, and associated tensors. - 3rd columns of the orientation matrices, and associated tensors. - _i Denavit-Hartenberg parameter. - _i Twist angle parameter. - X Changes in robot arm operational coordinates. - , ls and rr Changes in robot arm geometrical parameters, and their least square and ridge regression estimated changes. - _i Denavit-Hartenberg parameter. - , ^c and ⁿ Robot geometrical parameters and their correct and nominal values. - pi, ai, ri, i, i and i [4×4] differential operator matrices. - and ^–1 [3×3] conversion matrix and its inverse.