Several problems of practical interest in robotics can be modelled as the convolution of functions on the Euclidean motion group. These include the evaluation of reachable positions and orientations at the distal end of a robot manipulator arm. A natural inverse problem arises when one wishes to design rather than to model manipulators. Namely, by considering a serial-chain robot arm as a concatenation of segments, we examine how statistics of known segments can be used to select, or design, the remainder of the structure so as to attain the desired statistical properties of the whole structure. This is then a deconvolution density estimation problem for the Euclidean motion group. We prove several results about the convergence of these deconvolution estimators to the true underlying density under certain smoothness assumptions. A practical implementation to the design of planar robot arms is demonstrated.