Kinematics Physical reality is already a graph. Software just learns how to read it. From Physical Machine to Computational Object A Kinematic Architecture for Arbitrary Mechanical Systems 1. Problem Statement The core problem is how to extend a general-purpose software architecture to arbitrary mechanical devices without reducing any of their inherent properties: ● Speed ● Reliability ● Efficiency ● Dexterity ● Repairability ● Safety The software must not interfere with or replace existing control systems. Instead, it must operate as a parallel cognitive layer that adds value immediately and scales with additional integration. 2. Immediate Value from Minimal Input Ideally, the system should begin providing value from the lowest possible level of information — even a single image of the machine. From that alone, the software should be able to: ● Identify rigid bodies ● Identify joints ● Infer kinematic structure ● Construct a symbolic model ● Run kinematic reasoning Further integration (CAD, telemetry, manuals) should refine accuracy, but not change the fundamental architecture. 3. Example System: Hydraulic Excavator The example shown is a John Deere tracked hydraulic excavator with a breaker attachment. Without any manufacturer data, we can already extract: ● Rigid Bodies ● Base (tracks + upper house) ● Boom ● Stick ● Tool carrier ● Breaker body ● Breaker chisel Joints / Degrees of Freedom J1: House rotation (revolute) J2: Boom pitch (revolute J3: Stick pitch (revolute) J4: Tool pitch (revolute) J5: Breaker orientation (revolute) J6: Breaker internal hammer (prismatic) J7: Track translation (prismatic) This yields a 7-DOF articulated mechanical system. 4. Kinematic Structure as a Graph Once identified, the machine becomes a constraint graph: Nodes: rigid bodies Edges: joints Root: grounded base Leaves: end effectors This representation is: ● Tree-structured ● Loop-free ● Deterministic ● Fully solvable At this point the physical machine has already become a pure computational object. 5. Geometric Constraints From geometry alone we can infer: ● Link length ratios ● Joint limits ● Collision boundaries ● Reach envelope ● Singularities Even rough estimates produce: Correct workspace shape Correct dependency structure Correct failure modes This is sufficient for: ● Forward kinematics ● Inverse kinematics ● Reachability analysis ● Collision checking No sensors required. 6. Dynamic / Physical Constraints From machine class we can infer approximate physical bounds: ● Total mass ● Tool mass ● Maximum torques ● Hydraulic pressure limits ● Energy per action ● Stability envelope These become numerical bounds inside the model: ● Max joint velocity ● Max acceleration ● Force limits ● Energy limits All expressed as constraints, not control logic. 7. Symbolic → Numeric Pipeline The system follows a fixed transformation pipeline: Image / CAD / Documentation ↓ Constraint Graph ↓ Symbolic Model ↓ Numeric Tensor Model ↓ Simulation / Planning / Reasoning This pipeline is independent of: ● Manufacturer ● Machine type ● Industry ● Control system 8. Elixir + Nx Mapping In Elixir, the entire machine reduces to a data structure: Kinematics becomes pure linear algebra: T = T1 @ T2 @ T3 @ T4 Where each transform is a tensor. No special robotics framework required. No ML required. No training required. 9. What CAD Adds (But Is Not Required) CAD improves: ● Precision ● Inertia models ● Mass distribution ● Cylinder geometry ● True joint limits But nothing changes architecturally. CAD refines parameters — it does not change the computational structure. 10. Why This Scales to All Machines Every mechanical system reduces to: 1. Rigid bodies 2. Joints 3. Constraints 4. Actuators 5. Objectives Which is mathematically: > A constraint satisfaction problem over manifolds. This is the same structure for: ● Excavators ● CNC machines ● Robots ● Aircraft ● Power grids ● Manufacturing lines --- 11. The Key Insight The moment a physical system is represented as: > Graph + Constraints + Solvers It becomes: ● Searchable ● Verifiable ● Simulatable ● Optimizable ● Explainable Autonomy is no longer a special problem. Safety is no longer a special problem. Planning is no longer a special problem. They all become queries over the same constraint graph. 12. Final Thesis This architecture does not replace machines. It does not override control systems. It does not impose new complexity.