The Linear Wrench

A multi-fingered gripper, a legged robot, or a team of cooperating manipulators all share the same mathematical problem: more actuators than degrees of freedom. The system is redundantly actuated — there are infinitely many ways to distribute forces among the actuators that achieve the same net effect. Choosing among them requires solving an optimization or inverting a matrix at every timestep.

A closed-form solution for wrench distribution scales linearly with the number of applied wrenches, avoiding matrix inversion entirely. The solution exploits the algebraic structure of the redundancy — the null space of the force mapping — to decompose the problem into independent components that can be summed.

The result also identifies significant errors in current state-of-the-art approaches. Existing methods produce force distributions that appear correct for single-wrench cases but accumulate systematic errors when multiple wrenches act simultaneously, because they fail to account for interactions in the null space.

The through-claim: the standard approach — solve the optimization at each timestep — hides the problem’s structure behind numerical machinery. The closed-form solution reveals that wrench distribution is not fundamentally an optimization problem but a linear algebra problem in disguise. The linear scaling with wrench count follows directly from the decomposition — each wrench contributes independently to the solution. The errors in existing methods come from treating a linear structure as if it were nonlinear, adding computational cost and systematic bias simultaneously.