PaperlayerView exact revision on arXiv

Source-linked AI summary

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

Andrea Morghen, Pierluigi Arpenti, Roberto Schiattarella, Giovanni Acampora, Bruno Siciliano

arXiv:2610.33715v1cs.RO

TL;DR

Robotics needs quantum representations that preserve the geometry and configuration dependence of robot motion, rather than recasting only task objectives as generic quantum optimization. This paper constructs a quantum representation of rigid-body and serial-manipulator kinematics and uses its pose and Jacobian information in QLIK. It establishes analytical equivalence between the reconstructed geometric Jacobian and its classical counterpart under ideal readout.

  • Problem

    Robotics quantum methods need representations that preserve kinematic geometry and configuration dependence, while current NISQ devices face qubit, decoherence, gate-error, sampling, and compilation constraints.

    Source: §I · p. 1

  • Method

    The paper represents rigid-body transformations and their ordered composition along configuration-dependent serial chains, reconstructs the geometric Jacobian from quantum-readout derivatives, and uses pose and Jacobian information in QLIK.

    Source: §I · p. 2

  • Results

    The reconstructed manipulator geometric Jacobian is analytically equivalent to its conventional counterpart under ideal expectation-value readout.

    Source: §I · p. 2

  • Takeaways & Limitations

    The framework provides a geometrically interpretable interface between rigid-body kinematics and quantum computation without claiming computational quantum advantage.

    Source: §I · p. 2

    The stated analytical equivalence between the reconstructed Jacobian and the classical Jacobian holds under ideal expectation-value readout.

    Source: §I · p. 2

Abstract

from arXiv · show

The integration of quantum computing into robotics requires mathematical representations that preserve the geometric structure and configuration dependence of robot motion. This paper introduces a structure-preserving quantum representation of rigid-body transformations and serial-manipulator kinematics, establishing a correspondence between classical kinematic models and parameterized quantum circuits. Building on the Bloch-sphere representation, spatial directions are encoded into single-qubit states, while quantum rotations reproduce the ordered composition of reference-frame transformations. Translational contributions are combined through quantum-state aggregation, with their metric scales retained classically, enabling the reconstruction of end-effector position and orientation from Pauli expectation values. The formulation preserves the original joint variables and accommodates different kinematic conventions and joint types. Beyond forward kinematics, we establish that the manipulator geometric Jacobian can be reconstructed from quantum-observable derivatives using the parameter-shift rule, with analytical equivalence to its classical counterpart under ideal readout. To demonstrate the applicability of this representation to robot-level algorithms, we introduce Quantum Closed Loop Inverse Kinematics, a hybrid closed-loop method that exploits quantum-derived pose and Jacobian information. The proposed framework provides a geometrically interpretable foundation for developing and investigating quantum and hybrid computational methods for robotic kinematics and motion.

I. INTRODUCTION

The paper develops a geometrically interpretable quantum representation of configuration-dependent manipulator kinematics, including pose and differential information, without claiming computational quantum advantage. It also introduces QLIK to use quantum-derived pose and Jacobian information in closed-loop inverse kinematics.

I. INTRODUCTION

  • The paper identifies a gap in unified quantum representations that connect rigid-body transformations, their ordered composition along serial chains, and differential information for local task-space motion.Source: §I · pp. 1–2
  • The representation composes rigid-body transformations along configuration-dependent serial chains while keeping quantum readouts associated with robot pose quantities.It builds on a Bloch-sphere correspondence between Pauli expectation values, spatial rotations, and three-dimensional vectors.Source: §I · p. 2
  • The manipulator geometric Jacobian is reconstructed from parameter-shift derivatives of quantum pose readouts, with analytical equivalence under ideal expectation-value readout.Source: §I · p. 2
  • Quantum Loop Inverse Kinematics (QLIK) uses the proposed quantum representation to evaluate pose and Jacobian information in a hybrid closed-loop inverse-kinematics scheme.Source: §I · p. 2
  • The described experimental campaign covers multiple manipulators in ideal simulation and quantum-hardware tests, plus QLIK inverse-kinematics experiments on a physical manipulator.Source: §I · p. 2
  • The framework is methodological: it preserves configuration-dependent pose and differential information without claiming computational quantum advantage over classical robotics.The motivation includes current NISQ constraints, including limited qubit counts, decoherence, gate errors, finite sampling, and compilation overhead.Source: §I · pp. 1–2

II. RELATED WORK

The review distinguishes quantum methods that solve robotic tasks from methods that represent manipulator kinematics themselves. It positions the paper's contribution in the latter category.

II. RELATED WORK

  • The review separates solver-level quantum optimization from quantum-state representations of manipulator kinematics.Source: §II · p. 2
  • Encoding candidate joint configurations for optimization does not itself represent the kinematic map from those configurations to task-space quantities.Source: §II · p. 2
  • The paper positions its rigid-body and serial-kinematic representation against both solver-level optimization and direct kinematic representations.Source: §II · p. 2

A. Quantum Optimisation for Robotic Manipulators

Prior quantum approaches to robotic manipulation primarily use quantum computation for search, optimization, or learned function approximation rather than for an analytic, continuous kinematic representation.

A. Quantum Optimisation for Robotic Manipulators

  • Prior solver-level approaches include Grover-based control search, QUBO inverse kinematics on annealing hardware, and parameterized-circuit surrogates paired with Grover search.The cited examples are a six-jointed manipulator, planar inverse kinematics, and discretized configurations and design parameters, respectively.Source: A. Quantum Optimisation for Robotic… · p. 2
  • Quantum-inspired neural-network and metaheuristic inverse-kinematics methods are related but do not require quantum-circuit execution.Source: A. Quantum Optimisation for Robotic… · p. 2
  • These methods do not derive an analytic, continuous, joint-parameterized quantum kinematic representation that yields both pose and local differential information.Source: A. Quantum Optimisation for Robotic… · p. 2

B. Quantum Representations of Manipulator Kinematics

Prior work represents selected robotic geometric quantities with quantum states and observables, but the reviewed approaches do not provide the paper's unified construction of serial kinematics and Jacobian recovery.

B. Quantum Representations of Manipulator Kinematics

  • Zioui et al. connected robotic-arm and rigid-body orientation to single-qubit rotations, demonstrating the approach on SCARA and articulated manipulators.The approach remains primarily orientational and does not compose complete rigid-body transformations along a general configuration-dependent chain.Source: B. Quantum Representations of… · pp. 2–3
  • Fazilat et al. used quaternion–Pauli correspondences for ABB IRB 140 forward kinematics, but performed translational composition classically and did not derive the manipulator Jacobian from quantum readouts.Source: B. Quantum Representations of… · p. 3
  • Otani et al. reconstructed end-effector position from encoded link directions, but their specific-geometry formulation lacked general rigid-body composition and reported residual position errors on the order of one metre, even for the entangled formulation.The reviewed work also did not establish configuration-wise equivalence with classical forward kinematics.Source: B. Quantum Representations of… · p. 3
  • The paper identifies the unresolved combination as complete rigid-body transformations, ordered composition along configuration-dependent chains, and geometric-Jacobian recovery from quantum-readout derivatives.Source: B. Quantum Representations of… · p. 3

III. METHODOLOGICAL BACKGROUND

This section establishes the classical serial-manipulator notation and quantum concepts used in the formulation, including frame-axis propagation and differential readout.

  • The formulation starts with notation for serial-manipulator pose and differential kinematics.Source: §III · p. 3
  • It emphasizes propagation of frame axes along an ordered kinematic chain.Source: §III · p. 3
  • The quantum background covers Bloch-vector rotations, reduced-state linearity for weighted spatial aggregation, and the parameter-shift rule for differential readout.Source: §III · p. 3

A. Serial-Manipulator Kinematics

Serial-manipulator kinematics maps joint configurations to terminal poses by composing frame transformations in chain order; the geometric Jacobian describes the terminal differential motion used in inverse kinematics.

A. Serial-Manipulator Kinematics

  • The terminal forward-kinematics map composes configuration-dependent transforms along the ordered frame sequence.The chain contains revolute or prismatic joint coordinates; multiplication order follows the frame sequence.Source: A. Serial-Manipulator Kinematics · p. 3
  • The terminal pose comprises orientation and position, with its orientation represented by three terminal-frame axes expressed in the base frame.Source: A. Serial-Manipulator Kinematics · p. 3
  • The geometric Jacobian relates joint rates to terminal translational and angular velocities, and its translational block is the derivative of terminal position with respect to joint configuration.Source: A. Serial-Manipulator Kinematics · p. 4
  • Inverse kinematics seeks a configuration satisfying a target terminal pose, while differential methods iteratively update configuration through the geometric Jacobian.Source: A. Serial-Manipulator Kinematics · p. 4

B. Single-Qubit Spatial Encoding and Rotations

The section uses the Bloch-vector representation to encode spatial directions in single-qubit states and to make spatial rotations correspond to quantum-state rotations.

B. Single-Qubit Spatial Encoding and Rotations

C. Reduced-State Aggregation and Linearity of Pauli Readout

Tracing out a selector subsystem yields a readout state whose Bloch vector is the probability-weighted average of the branch Bloch vectors.

C. Reduced-State Aggregation and Linearity of Pauli Readout

  • A selector and readout subsystem are prepared across branches, each associated with a single-qubit state and a probability.Source: C. Reduced-State Aggregation and… · p. 4
  • The reduced readout state gives a probability-weighted average of the branch Bloch vectors, generally producing a mixed state even when every branch state is pure.This follows from reduced-state and expectation-value linearity, not coherent vector addition within a single qubit.Source: C. Reduced-State Aggregation and… · p. 4

D. Parameter-Shift Differentiation

This section extends the parameter-shift rule from gate-angle derivatives to derivatives with respect to external circuit parameters. It sums contributions from dependent gate occurrences; derivatives are exact with exact expectation values, while finite-shot estimates introduce statistical uncertainty.

D. Parameter-Shift Differentiation

  • The parameter-shift rule gives gate-angle derivatives from expectation values with that gate occurrence shifted by ±π/2.Other circuit parameters remain unchanged during each shift.Source: D. Parameter-Shift Differentiation · p. 4
  • For derivatives with respect to an external parameter, independently shift each dependent gate occurrence and sum the chain-rule contributions.For affine gate-angle dependence, the chain-rule factor is the affine coefficient.Source: D. Parameter-Shift Differentiation · pp. 4–5
  • The resulting derivative is exact under exact expectation-value evaluation, while finite-shot shifted estimates introduce statistical uncertainty.For vector-valued readouts, apply the rule independently to each measured component.Source: D. Parameter-Shift Differentiation · p. 5
Loading 2610.33715v1…