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The minimum torque norm control scheme for redundant manipulators has long been known to exhibit instabilities. In this paper, an analysis is made of a class of acceleration-level redundancy resolution schemes that includes the minimum torque, minimum acceleration, and other well known schemes. It is proved that divergence of joint velocity norm to infinity in finite time is possible, and in fact does occur for self motions of mechanisms with one degree of redundancy under almost all such schemes in the class. The schemes that do not diverge in finite time are associated with conserved quantities and include minimum acceleration norm and dynamically consistent redundancy resolution schemes. Besides the divergence, all these schemes can exhibit local instabilities in the form of abrupt increases in joint velocities and torques. Simulations are presented that illustrate the types of divergence and instability. The details of the analysis should be useful to designers seeking to modify these resolution schemes. For example, it is proved that linear dissipation or space velocity cannot stabilize the algorithms.