In this paper, we study quasi-static manipulation of a planar kinematic chain with a fixed base in which each joint is a linearly-elastic torsional spring. The shape of this chain when in static equilibrium can be represented as the solution to a discrete-time optimal control problem, with boundary conditions that vary with the position and orientation of the last link. We prove that the set of all solutions to this problem is a smooth manifold that can be parameterized by a single chart. For manipulation planning, we show several advantages of working in this chart instead of in the space of boundary conditions, particularly in the context of a sampling-based planning algorithm. Examples are provided in simulation.