Potential Applications of Category Theory to Brake Design
Exploring the use of category theory in the design process of brakes, focusing on car brakes. The talk illustrates how category theory can serve as a modeling language for designing complex multi-technology brake systems. Requirements, architecture, sketching, idealization, and simulation are discussed in the context of brake design.
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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/hour What is a requirement for the design? (implicit) State space Subset of acceptable states (requirement) ( ? ? ? ??) (?) (?) (??) (? ? ??)
Simplified design process Requirements Architecture Sketch/geometry Modeling Manufacturing Qualification and Testing
Architecture Fwa Fw Fwb WHEEL WHEEL TAa Chassis A X L E A X L E Velocity Propulsion Wba Wfa Tp Brake System Ffa TAa WHEEL WHEEL Fwa Fwb Fhandle Port graph/Wiring Diagram - Nodes are typed interfaces
Sketch Involves L Geometric elements Parameters Can be represented as a Function R Parameters Geometry Sketch Sketch is now rendered as a CAD model (includes Materials, etc)
Idealization Simplification Negligible lateral load transfer Negligible longitudinal load transfer Negligible significant roll and pitch motion . Parameters Radius of the wheel rw Velocity (V)= 20 mph Power Pd= 5Hp Free body diagrams (forces) Fhuman L2 Pd, v L1 rw mg Rr Fdr Rf FBf Fdf Idealize Geometry Physics
Solve and Simulate Equations Force at Brake Rr = Rrf + Rrb = Cf mg (-v) Fbf = PowerD/V Cfmg (v) Pivot location L2/L1 = (Powerd - Cfmgv2)/ Fhuman L1 L2 Iteratively determine Fbr Dimensions of the Lever Sizing of the Lever dimensions for wood -(shape and measurements) Stress analysis (FEM) Simulated Behavior Desired Behavior Simulate Physics Geometry Sizing (update)
Manufacturing Process Plan Energy Energy Energy Wood Rubber Geometry Geometry Steel geometry Parts of the Brake Forging Sawing Forming Brake Lever Brakepad Pivot Pin Adhesive Waste Energy Waste Energy Brake- pad Pivot pin Brake lever Chassis Geometry Attachi ng Waste wood Attaching Brake lever assembled Process-plan Geometry Manufacturing Chassis+brake lever Brake system
Post-Manufacturing Qualifying Testing Geometry as measured Observed Behavior Manufacturing
Product design and manufacturing process R e q u I r e m e N t s Manufacturing Observed Behavior = Parameter Geometry Simulated Behavior Sketch Physical model Sizing (update)
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 (1) (2)
Changed geometry Change manufacturing Changed part geometry Change process plan Part 1 Geometry Part 2 Geometry Wood Wood Sawing Sawing Fastners Part 1 Part 2 Attach 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 R e q u i r E m e n t Manufacturing Observed Behavior Geometry Simulated Behavior Physical model
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 Body Propulsion Chassis Dashboard Seat Gear system Wheel Assembly Engine Fuel System Brake system Sensors Front Back Actuator Force Amplifier Brake pad AxleF Wheels AxleR Wheels Carry-weight Fwb Fwf Fwb Fwa Fw WHEEL WHEEL Weight -Car TAa WHEEL WHEEL E N G I N E Chassis G E A R S TAa A X L E A X L E Wba Chassis Wfa A X L E A X L E v Tp v Propulsion Wba Wfa ANCHOR PAD Tp FBw TAa Brake System Fpad TAa Ffa WHEEL WHEEL Fuel System AMPLIFIER WHEEL WHEEL Fwf Fw Fwa Fw b Fhandle Fhandle b
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 research Manufacturing Operad of String diagrams Requirement Morphism ?:?1, .?? ? 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 R e q u I r e m e n t s Manufacturing Parameter Values Behavior Geometry = Sketch Physical model Process of creating semantics for each subsystem/part and composing them to create 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 diagnosis Spencer Breiner, Olivier Marie-Rose, Blake S. Pollard, and Eswaran Subrahmanian, Operadic diagnosis in hierarchical systems, ACT Oxford Future 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.
