Transpose Jacobian-based controllers represent an attractive approach to robot control in Cartesian space. These controllers attempt to drive the robot end-effector posture to a specified desired position and orientation without solving either the inverse kinematics nor computing the robot inverse Jacobian. A wide class of transpose Jacobian-based regulators obtained from the energy shaping plus damping injection technique is analyzed in this paper. Our main theoretical contribution is the introduction of a novel analysis which does not invoke any assumption on Jacobian singularities to ensure local asymptotic stability for a family of nonredundant robots. The performance of four transpose Jacobian-based regulators is illustrated via experimental tests conducted on a direct-drive vertical arm.