inverse kinematics

This work addresses the global, exact computation of one-dimensional self-motion manifolds (SMMs) for redundant manipulators. To overcome challenges—including SMM multi-connectivity, non-convexity, and the absence of natural redundancy in non-redundant subsystems—we propose a numerical continuation method based on ordinary differential equations (ODEs). We design an explicit fixed-step ODE integrator that jointly incorporates initial-condition search and redundancy-induction mechanisms, enabling automatic tracing of all isolated SMM branches without post-hoc inverse-kinematics optimization. Our method is the first to support generalized configurations including prismatic joints. Extensive validation on canonical robotic platforms demonstrates high accuracy, strong robustness, and computational efficiency. The results expose fundamental limitations of conventional SMM algorithms regarding topological completeness and task adaptability, establishing a verifiable, globally consistent framework for representing the solution space—thereby advancing redundancy resolution, obstacle-avoidance planning, and human–robot collaboration.

This work proposes a novel framework for inverse kinematics (IK) optimization that addresses the high failure rates commonly caused by the nonlinear relationship between joint variables and end-effector poses, as well as non-convex constraints such as obstacle avoidance. By introducing analytical IK solutions as a change of variables within the optimization process, the method uniquely combines the precision of analytical approaches with the flexibility of numerical optimization, substantially simplifying the problem structure. Evaluated across three mainstream optimizers, the approach demonstrates significantly higher success rates than conventional optimization techniques and baseline methods in complex tasks—including obstacle avoidance, grasp selection, and humanoid robot stability—thereby achieving an effective unification of analytical and optimization-based IK strategies.

Simulating large-scale articulated rigid-body systems remains challenging for conventional rigid-body solvers due to geometric nonlinearities and numerical stiffness. This work proposes a co-rotational framework based on Affine Body Dynamics (ABD) that decouples geometric nonlinearities through a linear kinematic mapping and projects high-dimensional body coordinates onto a dual space spanned by the minimal joint degrees of freedom. By combining implicit integration with KKT system solves, the method enforces exact constraint satisfaction and ensures physically accurate motion propagation. It supports diverse topologies—including chains, trees, closed loops, and irregular networks—and leverages pre-factorization of constant-coefficient matrices to achieve significant computational efficiency. The approach enables interactive simulation of systems comprising hundreds of thousands of rigid bodies on a single CPU core, maintaining high stability and accuracy even with large time steps.

Analytical inverse kinematics (IK) solutions for robotic manipulators have long suffered from reliance on manual derivation, numerical ill-conditioning, and inefficiency of symbolic computation. Method: This paper proposes a fully automated analytical IK generation framework. It re-models the kinematic chain based on geometric relationships among joint axes (e.g., intersecting or parallel), systematically classifies subproblems for structural decomposition, and integrates geometric modeling, kinematic classification, and subproblem mapping to build a high-performance C++ core engine with a Python interface. Contribution/Results: The method enables one-click analytical IK generation, achieving derivation speeds orders of magnitude faster than conventional symbolic tools (e.g., Maple or Mathematica). Online IK evaluation incurs sub-millisecond latency (<1 ms) and demonstrates superior accuracy and computational efficiency compared to state-of-the-art baselines such as IKFast.

To address the real-time computational demands of optimal robot control—particularly differential flatness-based control—for high-order kinematics and inverse dynamics, this paper proposes a unified algorithm grounded in spatial screw theory and Lie group/Lie algebra formalism. The method achieves linear-time complexity (O(n)) for both forward and inverse kinematics up to the fourth order, as well as second-order inverse dynamics, via recursive forward/backward propagation and rigid-body screw modeling, fully supporting vectorized parameter inputs. The algorithm is compact, analytically exact, and real-time capable. Experimental validation on a Franka Panda 7-DOF manipulator demonstrates significant speedup over conventional approaches, enabling millisecond-level response required for high-order closed-loop control. The core contribution is the first unified O(n) framework integrating fourth-order kinematics and second-order inverse dynamics, establishing a foundational advancement for real-time differential flatness control of serial manipulators.

本文提出一种新方法，通过逆函数定理计算解析IK参数化的梯度，以解决机器人在运动学约束下的轨迹规划问题。

This study addresses the limited accuracy of analytical methods and the initial-value sensitivity of numerical approaches in the inverse kinematics of offset-redundant manipulators by proposing a two-stage hybrid solving strategy. First, an approximate model generates candidate solutions. Subsequently, split conformal prediction is introduced to establish an upper bound on calibration difficulty, which, combined with a lightweight learned predictor, efficiently selects the optimal seed. The Levenberg–Marquardt algorithm then refines this solution on the full model. Achieving computation times below 40 microseconds and a 100% target-reaching success rate, the proposed method demonstrates superior real-time performance and reliability, as validated through extensive simulations and humanoid robot experiments.

本文结合代数和几何方法，分析了通用3R机器人的运动学特性，并提出了正交机器人具有四个逆运动学解的充分必要条件。

本文扩展了刚体动力学一阶导数算法，以适应闭链系统，并通过约束嵌入方法解决了现有算法在处理复杂关节类型时的局限性。

This study addresses the challenges of low sampling efficiency and computational bottlenecks inherent in projection-based methods for constrained robot motion planning. To overcome these limitations, this work proposes reparameterizing the planning space via analytical inverse kinematics to construct a vectorized motion planner, thereby exploiting parallel computing opportunities to transcend existing performance ceilings. The proposed approach achieves microsecond-level planning for high-dimensional systems, delivering a tenfold speedup over state-of-the-art methods and fundamentally restructuring robotic manipulation pipelines.