KinDER: A Physical Reasoning Benchmark

• The paper introduces KinDER, a benchmark designed to systematically evaluate embodied physical reasoning through 25 procedurally generated 2D and 3D environments.
• KinDER decouples perception and language influences by offering a Gymnasium-compatible API with parameterized skills, high-level demonstrations, and robust evaluation protocols.
• The benchmark reveals that symbolic planning (BP) achieves the highest success rate (0.57) while RL and imitation methods struggle with sparse rewards and complex dynamic tasks.

KinDER (Kinematic and Dynamic Embodied Reasoning) is a benchmark designed to systematically evaluate and advance physical reasoning in robotic learning and planning. Unlike pure visual or text-based benchmarks, KinDER directly targets the embodied setting, providing a suite of environments, tools, and evaluation protocols that specifically isolate the reasoning challenges intrinsic to kinematic and dynamic interactions between robots and their environment. The benchmark encompasses procedurally generated 2D and 3D environments, a Gymnasium-compatible API with parameterized skill libraries and demonstrations, and an evaluation suite covering diverse paradigms such as task-and-motion planning, reinforcement learning, imitation learning, and foundation-model-based approaches. KinDER's explicit focus is the quantification and comparison of physical reasoning abilities, with tasks and metrics disentangled from perception, language grounding, and application-specific confounds (Huang et al., 28 Apr 2026).

1. Design Principles and Benchmark Scope

KinDER is structured around three interlocking components: KinDERGarden, KinDERGym, and KinDERBench. The design objective of KinDERGarden is to provide 25 procedurally generated environments that collectively span kinematic and dynamic reasoning challenges in both 2D and 3D. Environments are grouped by type: Kinematic2D (6 tasks), Dynamic2D (4), Kinematic3D (5), and Dynamic3D (10). Each environment is sampled from an infinite procedural distribution $p_\theta$, with the parameter $\theta$ encapsulating randomizations such as object count, obstacle placement, and tool dimensions.

The five core physical reasoning challenges isolated by KinDER are:

1. Basic spatial relations: e.g., placing objects in spatially specified relationships.
2. Nonprehensile multi-object manipulation: e.g., pushing or sweeping several objects simultaneously.
3. Tool use: e.g., employing a stick or hook to manipulate distant actuators.
4. Combinatorial geometric constraints: e.g., packing or retrieving items in densely cluttered spaces.
5. Dynamic constraints: e.g., balancing, tossing, or dynamically maneuvering objects.

The explicit design intent is to decouple reasoning from perception and language, ensuring that performance metrics reflect underlying physical reasoning competencies rather than confounding factors. Environments such as "ConstrainedCupboard3D" randomly vary shelf slot arrangements, compelling agents to adapt geometric reasoning to each trial. The state space $\mathcal{S}$ utilizes object-centric representations mapping objects $o_i$ to feature vectors $x_i$, capturing pose (in SE(2)/SE(3)), velocity, shape/size, and joint values. Actions $\mathcal{A}$ span real-valued joint or end-effector controls.

2. Task Formulation, Constraints, and Reward Structures

KinDER encodes a range of kinematic, dynamic, and contact constraints to mimic realistic robotic scenarios. Typical kinematic reachability is defined for an $n$-DOF arm by the condition $$\|p_{\mathrm{ee}}(q) - p_{\mathrm{obj}}\| \le r_{\max}$$, where $p_{\mathrm{ee}}(q)$ is end-effector position as a function of joint configuration $q$, and $\theta$0 locates the target object.

Dynamic 3D environments implement rigid-body mechanics using MuJoCo:

$\theta$1

where $\theta$2 is the mass/inertia matrix, $\theta$3 captures Coriolis and centrifugal effects, $\theta$4 is gravity, and $\theta$5 represents the commanded torques.

2D environments modeled in Pymunk use complementarity constraints to enforce nonpenetration:

$\theta$6

Task success criteria $\theta$7 are task-specific, ranging from positional/orientational thresholds to set-based goals for multi-object tasks. The sparse reward function is:

$\theta$8

Cumulative reward for successful episodes is $\theta$9, incentivizing minimal-step solutions.

3. Parameterized Skills, Demonstrations, and API

KinDERGym provides a unified Python API compatible with OpenAI Gymnasium, supporting object-centric states, parameterized skills, teleoperation, and demonstration playback. High-level skills are encoded as options $\mathcal{S}$0, where:

• $\mathcal{S}$1 is the initiation set (preconditions),
• $\mathcal{S}$2 is a high-level policy parameterized over continuous variables $\mathcal{S}$3 (e.g., SE(3) grasp poses, motion waypoints),
• $\mathcal{S}$4 is a termination predicate.

Preconditions and effects are specified using PDDL-style operators, grounded via learned relational classifiers (e.g., On(obj, table)), and samplers generate $\mathcal{S}$5, such as IK solutions.

Demonstration data is amassed through:

• 2D mouse/keyboard or PS5 controller teleoperation,
• 3D iPhone web app or Meta Quest VR headset interfaces,
• planner-generated rollouts using bilevel planning.

Over 100 demonstrations are available for the “core” 10 environments, each storing time-indexed object-centric states and executed actions.

4. Baseline Algorithms and Evaluation Protocol

KinDERBench evaluates 13 baselines, spanning symbolic, learning-based, and hybrid approaches. The methods include:

Paradigm	Example Methods	Average SR (8 tasks)
Task & Motion Planning	Bilevel Planning (BP)	0.57
LLM/VLM Planning	LLMPlan, VLMPlan (zero-shot); LLMCon/VLMCon (in-context)	≈0.34/≈0.43
Model-based Control	MPC, Model-Based RL (MBRL)	0.32/0.08
Generative Planning	Generative Skill Chaining (GSC)	0.26
Reinforcement Learning	PPO, SAC	0.13/0.02
Imitation Learning	Diffusion Policy (DP), DP with Env. States (DPES)	0.24/0.25
Vision-Language-Action	Fine-tuned VLA ($\mathcal{S}$6)	0.32

BP achieves the highest overall success rate, but its performance declines on tasks with high object count and complex geometric reasoning requirements. LLM/VLM methods perform better with in-context examples. RL methods, notably PPO and SAC, struggle with sparse rewards and long-horizon requirements, while imitation and diffusion methods demonstrate generalization in contact-rich scenarios. Inference times range from milliseconds (policy-gradient methods) to minutes (symbolic planning and MPC) (Huang et al., 28 Apr 2026).

5. Bridging Simulation and Reality: Real-to-Sim-to-Real Validation

To assess the fidelity of KinDER’s simulations, real-to-sim-to-real (RS2R) studies were conducted using the Shelf3D task executed by a TidyBot++ mobile manipulator. Object states were initialized via an overhead RGBD camera and open-vocabulary detector. Simulated environments were configured to match the perceived object-centric state. Planning was performed using BP to generate abstract symbolic plans and corresponding motion trajectories. The robot executed trajectories using an onboard controller, achieving successful shelf-packing in 7 out of 10 real-world trials, suggesting strong correspondence between simulated and physical execution when perception noise is moderate.

6. Insights, Limitations, and Future Directions

Current results indicate that high-engineering-cost symbolic planners (BP) are most reliable, but performance degrades as scene complexity increases. LLM-/VLM-based planners benefit significantly from in-context prompt augmentation, but zero-shot planning remains unreliable. RL methods are handicapped by sparse rewards and exploration challenges; dense reward shaping aids, but is not generalizable across tasks. MPC is effective when supplied with accurate dynamics, whereas model-based RL struggles to fit environment transitions.

Imitation learning and fine-tuned VLAs generalize better on manipulation tasks involving contact and tool use, especially when informed by large-scale pretrained priors. However, nonprehensile dynamics and complex clutter—especially breakaway tool-use under occlusion—remain unsolved.

This suggests future research will leverage KinDER to synthesize parameterized skills, integrate planning and learning for scalable task and motion planning, adopt perception-noise extensions, and broaden coverage to multi-robot and partially observable domains. KinDER’s fully open-source release provides a platform for systematic community-driven advancement in embodied physical reasoning (Huang et al., 28 Apr 2026).

7. Relationship to Related Benchmarks

KinDER is distinguished from vision-centric physics reasoning benchmarks such as SeePhys by its embodied, interactional formulation. Where SeePhys systematically explores vision-essential versus vision-optional diagrammatic physics problems and quantifies the coupling of diagram understanding to scientific reasoning (Xiang et al., 25 May 2025), KinDER’s environments directly embed the physical constraints faced by real robots executing in a simulated world. Both benchmarks structure their instances to stratify by reasoning challenge and difficulty tier, and both advocate robust composite evaluation metrics and multimodal task decomposition. A plausible implication is that design features from SeePhys—such as VE/VOpt categorization, stratified difficulty, and explicit CoT evaluation—could be systematically adopted to extend KinDER’s diagnostic power.

KinDER enables direct, application-agnostic measurement of physical reasoning ability, making it a foundational tool for research focused on the intersection of embodiment, kinematics/dynamics, and artificial intelligence (Huang et al., 28 Apr 2026).