Potential applications of category theory to design: Example of brakes - PowerPoint Presentation

Potential applications of category theory to design: Example of brakes Eswaran Subrahmanian (CMU/NIST) AMS Sectional Meeting: Session on Applied Category theory November 9-10, 2019

Design of brakes The purpose of the talk to is illustrate the process of design of brakes using CT as a potential modeling language Brakes are used in a large number of contexts. We will concentrate on brakes for car(t)s Brakes have evolved over time from simple devices to complex multi-technology devices

Requirements for a Brake What is a requirement? External environment for brakes requirement: The weight of the vehicle Material of the wheels (Mat - Coefficient of friction) Driving surface material (coefficient for Friction)Specified speed of the Vehicle for stopping conditions:Stopping distance at a given velocity: 20 feet at 20 miles/hourWhat is a requirement for the design?(implicit) State spaceSubset of acceptable states (requirement)          

Simplified design process Requirements Architecture Sketch/geometry Modeling ManufacturingQualification and Testing

Architecture WHEEL WHEEL WHEEL WHEEL Propulsion A X LE A XLEChassis Brake System F handle T p W fa W ba T Aa T Aa F fa F wa F wa F wb F w F wb Velocity Port graph/Wiring Diagram - Nodes are typed interfaces

S ketch Involves Geometric elements Parameters Can be represented as a Function Parameters Geometry Sketch Sketch is now rendered as a CAD model (includes Materials, etc ) L R

Idealization Simplification Negligible lateral load transfer Negligible longitudinal load transfer Negligible significant roll and pitch motion……….ParametersRadius of the wheel rw Velocity (V)= 20 mphPower Pd= 5HpFree body diagrams (forces) mg P d , v r w F dfRr Fdr Rf F Bf F human L2 L1 Geometry Physics Idealize

Solve and Simulate Equations Force at Brake R r = Rrf + Rrb = Cf mg (-v)Fbf = PowerD/V – C fmg (v) Pivot locationL2/L1 = (Powerd - Cfmgv2)/ FhumanIteratively determineFbrDimensions of the LeverSizing of the Lever dimensions for wood -(shape and measurements)Stress analysis (FEM) L1 L2 Physics Simulated Behavior Simulate Geometry Sizing (update)   Desired Behavior

Manufacturing Process Plan Wood Sawing Brake lever Rubber Forming Geometry Geometry Steel ForginggeometryBrake-padAttaching Brake leverassembledPivot pinAttachingChassis Chassis+brake lever Brake system Geometry Manufacturing Geometry Adhesive Process-plan Waste wood Energy Waste Energy Parts of the Brake Brake Lever Brakepad Pivot Pin Energy Energy Waste Energy

Post-Manufacturing Manufacturing Qualifying Geometry as measured Observed Behavior Testing

Product design and manufacturing process Parameter Sketch Geometry Physical model Manufacturing Simulated Behavior Testing Simulate idealize Process plan Qualifying RequIreme NtsObservedBehavior   Sizing (update) =

Design Feedback Loops

Physics based models do not meet requirements Physics based models don’t meet requirements Change the specification (1) Change Parameter Value (example: Move Pivot Point) Change the geometry (two parts) Change parameters and Mapping Physical models (2)Optimization Similar to above for shape (2) (1)

Changed geometry  Change manufacturing Changed part geometry Change process plan Wood Sawing Part 2 Geometry Wood Sawing Part 1 Geometry Part 2 Attach Part 1 Fastners Brake Lever Assembled

Physics model and testing diverge Physics model and testing diverge Tune parameters (change in the Simulation) More complex model (change “Idealize”) Modify Requirements Geometry Physical model SimulatedBehaviorSimulateidealize Manufacturing Testing Process planObservedBehavior   R equirEment  

Product Evolution(change in requirement and architecture)

Evolution of Brake systems: change in amplification and control

Modularity and Composition Port Graphs form the arrows of Operad Operad is a category with tree shaped composition Subsystem/part decomposition defines System Architecture by composing sub-system architecture Vehicle Chassis BodySeatWheelAssemblyAxleFWheels FrontBack AxleRPropulsionEngineGearsystem Wheels Brake system Actuator Force Amplifier Brake pad Dashboard Sensors WHEEL WHEEL WHEEL WHEEL A X LE A X LE Chassis F handle T p W fa W ba T Aa T Aa F Bw F wb F wf F wf F wb EN G I N E G EARS Weight -Car PAD AMPLIFIER F pad ANCHOR v Carry-weight WHEEL WHEEL WHEEL WHEEL Propulsion A X LE A X LE Chassis Brake System F handle T p W fa W ba T Aa T Aa F fa F wa F wa F wb F w F wb v Fuel System Fuel System

Everything is an operad in the product development Process Parameters: Aggregate across branches Specification for parts gives the specifications for the whole Geometry Geometry of the vehicle is the union of the geometry of all parts and subsystems Physics models??? Open area for researchManufacturing Operad of String diagramsRequirementMorphism is a Wiring diagram operad then Req( Req( Req(Y) For each morphism in the Architecture we need to go through the whole process, to align the models with the real world test observation Comp

Architecture is syntax and the rest are semantics Parameter Values Sketch Geometry Physical model Manufacturing Behavior Testing Simulate idealize Process plan Qualifying RequIrements Process of creating semantics for each subsystem/part and composing them tocreate an architecture with the desired semantics =

References and Future work String diagrams for process planning Spencer Briener , Albert Jones and Eswaran Subrahmanian, Categorical models for process planning, Computers and Industry, November, 2019. Operads for architecture description and diagnosisSpencer Breiner, Olivier Marie-Rose, Blake S. Pollard, and Eswaran Subrahmanian, Operadic diagnosis in hierarchical systems, ACT OxfordFuture work Relation of Lenses to design representation update problem

Thank you sub@cmu.edu This work is done with Spencer Breiner and Blake Pollard at NIST.