# Task-Distribution-Aware Counterweight Synthesis and Constrained Co-Design for Serial Manipulators

arXiv: https://arxiv.org/abs/2609.15082 · HTML: https://mlsanity.com/paper/2609.15082/

## Key facts

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- First posted on arXiv 14 Sep 2026.

## Abstract (arXiv)

Passive counterweights are simple gravity compensators, but a counterweight selected from a single pose is not generally optimal for the configurations and tasks a manipulator actually executes. This paper develops a task-distribution-aware synthesis framework in which the operating distribution ρ(q) enters the design explicitly. For a counterweight moment p=mc rc with gravity torque -gpφ(q), the weighted mean-square residual gravity torque has the closed-form minimizer p^*=Eρ[τgφ]/(gEρ[φ2]). If payload gravity torque is affine in payload mass, the optimum is also affine: p^*(mp,ρ)=p0^*(ρ)+mpKp(ρ). For fixed static moment, added counterweight inertia is Ic=prc while mass is mc=p/rc, so mass-radius selection is underdetermined unless physical constraints are specified. A recovered three-link manipulator is used as a case study. At rc=0.20 m, zero-payload equivalent optima are 0.672 kg for uniform joint-space operation, 0.683 kg for approximately uniform task-space operation, 0.713 kg for a representative pick-and-place family, and 0.952 kg for a high-gravity-biased distribution, a change of more than 40% caused solely by the operating distribution. Nondominated fronts show that preferred mass-radius pairs depend on declared engineering bounds. A rated-torque-referenced all-joint screen increases zero-payload feasible task-space coverage from 78.1% without compensation to 93.7% for the uniform-distribution design. A lumped point-mass trajectory study gives a provisional crossover from no counterweight at very aggressive motion to stronger compensation as motion slows. These actuator and dynamic results are engineering consequence studies rather than physical validation.