Quantum Closed Loop Inverse Kinematics Through Hilbert-Space Encoding of Robot Kinematics

Structured spatial data require quantum encodings that preserve geometric relations, expose measurable observables, and remain implementable on finite-depth hardware. This work introduces a quantum representation and circuit architecture for rigid-body transformations and specializes it to Denavit--Hartenberg kinematics of serial open-chain manipulators. Each translational contribution is factorized into a classical metric magnitude and a signed unit direction encoded by a single-qubit Bloch vector, while parameterized rotations reproduce the ordered propagation of frame directions. A selector register prepares probabilities proportional to the contribution magnitudes, and the reduced state of a designated readout qubit encodes their normalized weighted sum. The retained classical scale then reconstructs the metric end-effector position. Two additional readout qubits encode terminal-frame axes, providing a compact and geometrically interpretable pose interface. At the ideal expectation-value level, measured Pauli observables reproduce the corresponding classical kinematic quantities. Alternative circuit architectures realize the same representation with different tradeoffs in qubit count, circuit depth, controlled operations, and measurement requirements. Validation on a serial manipulator yields numerically negligible position and orientation reconstruction errors under ideal simulation. Finite-shot simulations, noisy executions, transpilation analysis, and a hardware demonstration further characterize statistical error, noise sensitivity, and implementation overhead without asserting computational advantage.

Robotic hand kinematic design involves combinatorial selection among multiple finger structures whose performance depends on both individual kinematic characteristics and inter-finger interactions. This study presents a potential-dexterity-based quadratic unconstrained binary optimization (QUBO) formulation for combinatorial optimization of robotic hand kinematic structures. Potential dexterity-related metrics were evaluated through kinematic analysis and encoded into a 27-variable QUBO problem incorporating individual finger performance, thumb-to-finger workspace-overlap interactions, one-hot selection constraints, and pairwise structural compatibility. Exhaustive evaluation of the design space established a reference optimum and enabled direct validation of the formulated objective. Discretization sensitivity analysis showed that the selected design remained unchanged across the tested joint-space sampling intervals and at a finer workspace voxel resolution, while sensitivity was observed under coarse voxelization. Simulated annealing consistently recovered the reference optimum at higher sampling counts, and direct execution on D-Wave quantum annealing hardware also recovered the reference optimum under all tested sampling conditions. These results demonstrate that the proposed formulation provides a common QUBO representation for constrained robotic hand kinematic structure selection and enables evaluation using exhaustive, classical annealing, and quantum annealing approaches.

The inverse kinematics of generic 3R robots has been investigated through multiple approaches, mainly algebraic methods involving the solution of certain equation sets. Previous geometric interpretations of the solution, characterized as the intersection of a pair of conics have been confined to the joint-space domain. In this article, we study the Inverse Kinematic Model (IKM) of 3R robots, using the advantages of Conformal Geometric Algebra (CGA) to provide further insights on its kinematic properties. Our approach directly yields a univariate polynomial in terms of theta_2 without the need to eliminate theta_1 and theta_3 by reframing the problem as the intersection of two circles, which are fundamental elements within this algebraic framework.