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

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 $\rho(q)$ enters the design explicitly. For a counterweight moment $p=m_c r_c$ with gravity torque $-gp\phi(q)$, the weighted mean-square residual gravity torque has the closed-form minimizer $p^*=E_\rho[\tau_g\phi]/(gE_\rho[\phi^2])$. If payload gravity torque is affine in payload mass, the optimum is also affine: $p^*(m_p,\rho)=p_0^*(\rho)+m_pK_p(\rho)$. For fixed static moment, added counterweight inertia is $I_c=pr_c$ while mass is $m_c=p/r_c$, so mass-radius selection is underdetermined unless physical constraints are specified. A recovered three-link manipulator is used as a case study. At $r_c=0.20$ 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.