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Path-constrained robot trajectory planning in the vicinity of ordinary singularities is studied in this paper. First, the desired end-effector path is parameterized with an untimed variable q0, by which the robot kinematics can be formulated as a nonlinear equation. The solution to this equation forms a spatial curve in the augmented joint space. Then the path-tracking problem can be fulfilled by computing the curve solution numerically. This produces a pseudo trajectory with respect to an artificial independent parameter s, which represents the arc-length of the solution curve. The trajectory with respect to time can be obtained naturally by arbitrarily assigning s(t). Numerical simulations are carried out to verify the algorithm.