Robot Kinematics

Traditional robotic hand metrics focus on static properties such as workspace, manipulability, and grasp stability. However, these metrics do not directly measure dexterity under the standard definition in robotic manipulation: the ability to continuously change an object's pose within the hand while maintaining contact from an initial grasp. We introduce Kinematic Rolling Manipulation Ability (KaRMA), a kinematic-only metric for fine manipulation that quantifies reachable in-hand translation and reorientation of a spherical test object within a two-finger precision pinch through feasible rolling motions. KaRMA enforces joint limits, collision constraints, rolling contact, and antipodal force feasibility, then investigates reachable in-hand object poses via breadth-first search over translation and rotation primitives. KaRMA reports three scores: translational coverage (KaRMA-T), rotational coverage (KaRMA-R), and sensitivity to the initial grasp (KaRMA-S). We evaluate KaRMA on 16 widely used robotic hands and compare against static baselines, showing that KaRMA separates hands that rank identically under static proxies, reveals translation-rotation tradeoffs invisible to existing baselines, and is qualitatively consistent with selected published task benchmarks where Jacobian-based metrics can be misleading.

Robotic systems that interact with the physical world must reason about kinematic and dynamic constraints imposed by their own embodiment, their environment, and the task at hand. We introduce KinDER, a benchmark for Kinematic and Dynamic Embodied Reasoning that targets physical reasoning challenges arising in robot learning and planning. KinDER comprises 25 procedurally generated environments, a Gymnasium-compatible Python library with parameterized skills and demonstrations, and a standardized evaluation suite with 13 implemented baselines spanning task and motion planning, imitation learning, reinforcement learning, and foundation-model-based approaches. The environments are designed to isolate five core physical reasoning challenges: basic spatial relations, nonprehensile multi-object manipulation, tool use, combinatorial geometric constraints, and dynamic constraints, disentangled from perception, language understanding, and application-specific complexity. Empirical evaluation shows that existing methods struggle to solve many of the environments, indicating substantial gaps in current approaches to physical reasoning. We additionally include real-to-sim-to-real experiments on a mobile manipulator to assess the correspondence between simulation and real-world physical interaction. KinDER is fully open-sourced and intended to enable systematic comparison across diverse paradigms for advancing physical reasoning in robotics. Website and code: https://prpl-group.com/kinder-site/

In robot control, planning, and learning, there is a need for rigid-body dynamics libraries that are highly performant, easy to use, and compatible with CPUs and accelerators. While existing libraries often excel at either low-latency CPU execution or high-throughput GPU workloads, few provide a unified framework that targets multiple architectures without compromising performance or ease-of-use. To address this, we introduce frax, a JAX-based library for robot kinematics and dynamics, providing a high-performance, pure-Python interface across CPU, GPU, and TPU. Via a fully-vectorized approach to robot dynamics, frax enables efficient real-time control and parallelization, while supporting automatic differentiation for optimization-based methods. On CPU, frax achieves low-microsecond computation times suitable for kilohertz control rates, outperforming common libraries in Python and approaching optimized C++ implementations. On GPU, the same code scales to thousands of instances, reaching upwards of 100 million dynamics evaluations per second. We validate performance on a Franka Panda manipulator and a Unitree G1 humanoid, and release frax as an open-source library.

Real-time humanoid teleoperation requires inverse kinematics (IK) solvers that are both responsive and constraint-safe under kinematic redundancy and self-collision constraints. While differential IK enables efficient online retargeting, its locally linearized updates are inherently basin-dependent and often become trapped near joint limits, singularities, or active collision boundaries, leading to unsafe or stagnant behavior. We propose a GPU-parallelized, continuation-based differential IK that improves escape from such constraint-induced local minima while preserving real-time performance, promoting safety and stability. Multiple constrained IK quadratic programs are evaluated in parallel, together with a self-collision avoidance control barrier function (CBF), and a Lyapunov-based progression criterion selects updates that reduce the final global task-space error. The method is paired with a visual skeletal pose estimation pipeline that enables robust, real-time upper-body teleoperation on the THEMIS humanoid robot hardware in real-world tasks.

Underground mining robots are increasingly modeled for planning, operator training, and digital-twin workflows, where reliable actuator-level kinematics is needed to reduce hazardous in situ trials. Unlike typical open-chain industrial manipulators, representative mining machines are often linear-actuator-driven closed-chain mechanisms with planar four-bar linkages, making reusable kinematic modeling and real-time FK/IK solving challenging. We present \hl{MineRobot}, an actuator-centered framework for modeling and solving the kinematics of this representative mechanism class. MineRobot introduces the Mining Robot Description Format (MRDF), a domain-specific representation that parameterizes mining-robot kinematics with native semantics for actuators and loop closures. It then contracts planar four-bar substructures into generalized joints and extracts, for each actuator, an Independent Topologically Equivalent Path (ITEP) classified into four canonical types. Based on this decomposition, per-type solvers are composed into a sequential forward-kinematics (FK) pipeline, while inverse kinematics (IK) is formulated as a bound-constrained actuator-length optimization solved by a Gauss--Seidel-style update scheme. By converting coupled closed-chain kinematics into small topology-aware solves, MineRobot reduces robot-specific hand derivations and supports efficient repeated FK/IK computation without treating each query as a full coupled constraint-solving problem. Experiments on representative underground mining robots demonstrate real-time FK performance and robust IK convergence within the tested operating ranges, supporting the use of MineRobot as an actuator-centered kinematic layer for planning, training, and digital-twin workflows.

Human motion provides rich priors for training general-purpose humanoid control policies, but raw demonstrations are often incompatible with a robot's kinematics and dynamics, limiting their direct use. We present a two-stage pipeline for generating natural and dynamically feasible motion references from task-space human data. First, we convert human motion into a unified robot description format (URDF)-based skeleton representation and calibrate it to the target humanoid's dimensions. By aligning the underlying skeleton structure rather than heuristically modifying task-space targets, this step significantly reduces inverse kinematics error and tuning effort. Second, we refine the retargeted trajectories through progressive kinodynamic trajectory optimization (TO), solved in three stages: kinematic TO, inverse dynamics, and full kinodynamic TO, each warm-started from the previous solution. The final result yields dynamically consistent state trajectories and joint torque profiles, providing high-quality references for learning-based controllers. Together, skeleton calibration and kinodynamic TO enable the generation of natural, physically consistent motion references across diverse humanoid platforms.

Vision-Language-Action (VLA) models build a token-domain robot control paradigm, yet suffer from low speed. Speculative Decoding (SD) is an optimization strategy that can boost inference speed. Two key issues emerge when integrating VLA and SD: first, SD relies on re-inference to address token errors, which is computationally expensive; second, to mitigate token errors, the acceptance threshold in SD requires careful adjustment. Existing works fail to address the above two issues effectively. Meanwhile, as the bridge between AI and the physical world, existing embodied intelligence has overlooked the application of robotic kinematics. To address these issues, we innovatively combine token-domain VLA models with kinematic-domain prediction for SD, proposing a kinematic-rectified SD framework named KERV. We employ a kinematics-based Kalman Filter to predict actions and compensate for SD errors, avoiding costly re-inference. Moreover, we design a kinematics-based adjustment strategy to dynamically rectify the acceptance threshold, addressing the difficulty of threshold determination. Experimental results across diverse tasks and environments demonstrate that KERV achieves 27%~37% acceleration with nearly no Success Rate loss.

Cross-robot policy learning -- training a single policy to perform well across multiple embodiments -- remains a central challenge in robot learning. Transformer-based policies, such as vision-language-action (VLA) models, are typically embodiment-agnostic and must infer kinematic structure purely from observations, which can reduce robustness across embodiments and even limit performance within a single embodiment. We propose an embodiment-aware transformer policy that injects morphology via three mechanisms: (1) kinematic tokens that factorize actions across joints and compress time through per-joint temporal chunking; (2) a topology-aware attention bias that encodes kinematic topology as an inductive bias in self-attention, encouraging message passing along kinematic edges; and (3) joint-attribute conditioning that augments topology with per-joint descriptors to capture semantics beyond connectivity. Across a range of embodiments, this structured integration consistently improves performance over a vanilla pi0.5 VLA baseline, indicating improved robustness both within an embodiment and across embodiments.

We present a theoretical and numerical analysis of the kinematics for the "Transpressor", a cuspidal 6R robot. It admits up to 16 inverse kinematics solutions which are described geometrically. For special target poses, we provide the solutions analytically and present a simple numerical solver for the general case. Moreover, an analytical estimate of the Jacobian determinant on a path between two solutions proves cuspidality for a class of robots similar to the transpressor.

This study proposes an intelligent multi-agent framework built on LLMs and VLMs and specifically tailored to robotics. The goal is to integrate the strengths of LLMs and VLMs with computational tools to automatically analyze and solve problems related to robotic manipulators. Our developed framework accepts both textual and visual inputs and can automatically perform forward and inverse kinematics, compute velocities and accelerations of key points, generate 3D simulations of the robot, and ultimately execute motion control within the simulated environment, all according to the user's query. To evaluate the framework, three benchmark tests were designed, each consisting of ten questions. In the first benchmark test, the framework was evaluated while connected to GPT-4o, DeepSeek-V3.2, and Claude-Sonnet-4.5, as well as their corresponding raw models. The objective was to extract the forward kinematics of robots directly from textual descriptions. The results showed that the framework integrated with GPT-4o achieved the highest accuracy, reaching 0.97 in computing the final solution, whereas the raw model alone attained an accuracy of only 0.30 for the same task. Similarly, for the other two models, the framework consistently outperformed the corresponding raw models in terms of accuracy. The second benchmark test was identical to the first, except that the input was provided in visual form. In this test, the GPT-4o LLM was used alongside the Gemini 2.5 Pro VLM. The results showed that the framework achieved an accuracy of 0.93 in obtaining the final answer, which is approximately 20% higher than that of the corresponding raw model. The third benchmark test encompassed a range of robotic tasks, including simulation, control, velocity and acceleration computation, as well as inverse kinematics and Jacobian calculation, for which the framework achieved an accuracy of 0.97.

Inverse Kinematics (IK) is a core problem in robotics, in which joint configurations are found to achieve a (collision-free) desired end-effector pose. Modern IK solvers face a fundamental trade-off: analytical methods are fast but lack generality, while numerical optimization-based methods are broadly applicable but prone to local minima and high computational costs. To overcome this challenge, we introduce HJCD-IK, a GPU-accelerated, sampling-based hybrid solver. By pairing a novel orientation-aware greedy coordinate descent initialization with Jacobian-based polishing and a parallel collision filter, our method achieves up to order-of-magnitude gains in speed and accuracy over state-of-the-art solvers, consistently finding collision-free solutions on the accuracy-latency Pareto frontier, while producing a diverse distribution of high-quality samples. We validate our solver on a physical Franka manipulator and release our code open-source.