Principled Domain Extension for Analytic IK
Abstract
Motion planning algorithms often inherently model configuration space (C-space) as a Euclidean space. But
when a robot must satisfy end-effector constraints during its motion (e.g. carrying an object with two
hands), the set of valid configurations collapses to a measure-zero submanifold in the ambient configuration
space. Recent work has examined the use of analytic inverse kinematics (analytic IK), a closed form mapping
from end-effector space to joint angles, to construct charts with a large domain. By planning in
that parameterized space, the constraint is eliminated, and the path adheres to the manifold by
construction. The mapping is made well-defined by including self-motion parameters as input to resolve the
many-to-one problem of IK.
Recent works have applied analytic IK to trajectory optimization under
end-effector constraints. Decision variables describe a trajectory in parameterized space, costs and
constraints are imposed in C-space, and gradient-based optimization is enabled by differentiating through
the IK function. A key challenge is the limited domain of an IK function: nonlinear trajectory optimizers do
not guarantee workspace reachability at each iteration of the optimization. Bespoke IK solutions
made for particular manipulators can be made to return approximate solutions outside of the reachable
workspace, but popular automatic IK solvers return "no solution" for non-reachable targets. This is
catastrophic for general nonlinear optimizers.
We extend the domain of an analytic IK function by
leveraging least-squares solutions, yielding a closed-form Jacobian for end-effector poses outside the
reachable workspace. This preserves the gradient flow outside of the reachable workspace, allowing the
optimizer to restore feasibility. Inspired by the numerical IK literature, we present efficient
approximations to this Jacobian. Empirically, we demonstrate that the overall framework begins to close the
gap between bespoke and automatically generated analytic IK solutions.