J-PARSE is an algorithm for smooth first-order inverse kinematic control of a serial manipulator near kinematic singularities. The commanded end-effector velocity is interpreted component-wise, according to the available mobility in each dimension of the task space. First, a substitute "Safety" Jacobian matrix is created, keeping the aspect ratio of the manipulability ellipsoid above a threshold value. The desired motion is then projected onto non-singular and singular directions, and the latter projection scaled down by a factor informed by the threshold value. A right-inverse of the non-singular Safety Jacobian is applied to the modified command. In the absence of joint limits and collisions, this ensures safe transition into and out of low-rank configurations, guaranteeing asymptotic stability for reaching target poses within the workspace, and stability for those outside. Velocity control with J-PARSE is benchmarked against approaches from the literature, and shows high accuracy in reaching and leaving singular target poses. By expanding the available workspace of manipulators, the algorithm finds applications in teleoperation, servoing, and learning. Videos and code are available at https://jparse-manip.github.io/.

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