Potential applications of category theory to design: Example of brakes

Eswaran Subrahmanian (CMU/NIST)AMS Sectional Meeting: Session on

Applied Category theoryNovember 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

WHEEL

WHEEL

WHEEL

WHEEL

Propulsion

AXLE

AXLE

Chassis

BrakeSystem

Fhandle

TpWfa Wba

TAa

TAa Ffa

Fwa

Fwa

FwbFw

Fwb

Velocity

• Port graph/Wiring Diagram - Nodes are typed interfaces

Sketch

Involves Geometric elements Parameters

Can be represented as a Function

Parameters GeometrySketch

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

• ……….

• Parameters• Radius of the wheel rw

• Velocity (V)= 20 mph• Power Pd= 5Hp

• Free body diagrams (forces)

mg

Pd, v

rw

Fdf RrFdrRf FBf

FhumanL2

L1

Geometry PhysicsIdealize

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

• Iteratively determine• Fbr

• Dimensions of the Lever

Sizing of the Lever dimensions for wood -(shape and measurements)

• Stress analysis (FEM)

L1

L2

PhysicsSimulatedBehavior

SimulateGeometry

Sizing (update)

 Desired Behavior

Manufacturing Process PlanWood

Sawing

Brake lever

Rubber

Forming

Geometry Geometry Steel

Forging

geometry

Brake-pad

Attaching

Brake leverassembled

Pivotpin

Attaching

Chassis

Chassis+brake leverBrake system

Geometry Manufacturing

Geometry

Adhesive

Process-plan

Wastewood

Energy

WasteEnergy

Parts of the Brake

• Brake Lever• Brakepad• Pivot Pin

Energy Energy

WasteEnergy

Post-Manufacturing

Manufacturing

QualifyingGeometryas measured

ObservedBehavior

Testing

Product design and manufacturing process

ParameterSketch

Geometry

Physical model

Manufacturing

SimulatedBehavior

Testing

Simulate

idealize

Proce

ss pl

an

Qualifying

RequIremeNts

ObservedBehavior 

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 2Geometry

Wood

Sawing

Part 1Geometry

Part 2

Attach

Part 1

Fastners

Brake LeverAssembled

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

SimulatedBehavior

Simulate

idealize

Manufacturing TestingProce

ss

plan ObservedBehavior

 

RequirEment

 

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

Body

SeatWheelAssembly

AxleF

Wheels

FrontBack

AxleR

Propulsion

EngineGearsystem

Wheels

Brake system

ActuatorForceAmplifier Brake

pad

Dashboard

Sensors

WHEEL

WHEEL

WHEEL

WHEEL

AXLE

AXLE

Chassis

Fhandle

TpWfa Wba

TAa

TAa FBw

Fwb

Fwf

Fwf

Fwb

ENGINE

GEARS

Weight -Car

PAD

AMPLIFIER

Fpad

ANCHOR v

Carry-weight

WHEEL

WHEEL

WHEEL

WHEEL

Propulsion

AXLE

AXLE

Chassis

BrakeSystem

Fhandle

TpWfa Wba

TAa

TAa Ffa

Fwa

Fwa

FwbFw

Fwb

v

FuelSystem

FuelSystem

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 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

ParameterValues

Sketch

Geometry

Physical model

Manufacturing

Behavior

Testing

Simulateidealize

Proce

ss 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 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.