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We introduce Visual Inverse Kinematics (VIK), which finds kinematically feasible joint configurations that satisfy vision-based constraints, bridging the gap between inverse kinematics (IK) and visual servoing (VS). Unlike IK, no explicit end-effector pose is given, and unlike VS, exact image measurements may not be available. In this work, we develop a formulation of the VIK problem with a field of view (FoV) constraint, enforcing the visibility of an object from a camera on the robot. Our proposed solution introduces a virtual kinematic chain that connects the physical robot and the object, transforming the FoV constraint into a joint angle kinematic constraint. Along the way, we introduce multiple vision-based cost functions to fulfill different objectives. We solve this formulation of the VIK problem using a method that involves a semidefinite program (SDP) constraint followed by a rank minimization algorithm. The performance of this method for solving the VIK problem is validated through simulations. 

Inverse kinematics (IK) is an important problem in robot control and motion planning; however, the nonlinearity of the map from joint angles to robot configurations makes the problem nonconvex. In this paper, we propose an inverse kinematics solver that works in the space of rotation matrices of the link reference frames rather than joint angles. To overcome the nonlinearity of the manifold of rotation matrices $$\mathbf{SO}(3)$$, we propose a semidefinite programming (SDP) relaxation of the kinematic constraints followed by a fixed-trace rank minimization via maximization of a convex function. Along the way, we show that the feasible set of an IK problem is exactly the intersection of a convex set and fixed-trace rank-1 matrices. Thanks to the use of matrices with fixed trace, our algorithm to obtain rank-1 solutions has guaranteed local convergence. Unlike some traditional solvers, our method does not require an initial guess, and can be applied to robots with closed kinematic chains without ad-hoc modifications such as splitting the kinematic chain. Compared to other work that performs SDP relaxation for IK problems, our formulation is simpler, and uses variables with smaller sizes. We validate our approach via simulations on a closed kinematic chain constituted by two robotic arms holding a box, comparing against a standard IK method.