[PPT] - Scalable Parametric Approximations of the Maximal Controlled PowerPoint Presentation

SLIDE 1

Scalable Parametric Approximations

f the Maximal Controlled Invariant Set

and Two Health Related Application Domains Ian M. Mitchell

Department of Computer Science University of British Columbia

SLIDE 2

Formal Methods & Reachability

Formal Methods seek to bring the rigour of mathematical

proof to the specification, development and verification of hardware and software systems

Safety is a commonly desired form of specification

– The system will not do something bad

Reachability is a common approach to analyzing safety

Dec 2014 Ian M. Mitchell (UBC Computer Science) 2

target (unsafe) backward reach set (uncontrollably unsafe) safe (under appropriate control)

SLIDE 3

Dec 2014 Ian M. Mitchell (UBC Computer Science) 3

Evolving Continuous Reachable Sets

Modified Hamilton-Jacobi partial differential equation

final set growing set Reachable set for the game of two identical vehicles

SLIDE 4

The Toolbox of Level Set Methods

Collection of Matlab routines to approximate viscosity

solutions of time-dependent Hamilton-Jacobi equations

– First publicly released implementation of state-of-the-art level set algorithms – Only HJ package that works in arbitrary dimension – Vectorized code achieves reasonable speed – Direct access to Matlab debugging and visualization – Download package include 25+ examples and 120+ page user guide – Can compute reachable sets for nonlinear systems with free parameters / inputs in best case and/or worst case fashion

Hamilton-Jacobi framework and level set algorithms have

limitations as a tool for formal verification

– Cost of nonparametric, grid-based representation of sets scales exponentially with state space dimension

Dec 2014 Ian M. Mitchell (UBC Computer Science) 4

SLIDE 5

Closed Loop Control of Anesthesia

Push to automate delivery of anesthesia in order to

reduce cost and number of adverse events

– Driven considerable improvement in sensing & modelling

Current systems are open-loop

– Closed loop would be better able to handle patient variability and surgical stimuli

Key element for FDA / Health Canada: confidence that

closed-loop system will maintain a suitable depth of hypnosis

Dec 2014 Ian M. Mitchell (UBC Computer Science) 5

SLIDE 6

Smart Wheelchair for Older Adults

Cognitively (and mobility) impaired older

adults in long term care (LTC) facilities

– Heterogenous population – Constrained but navigable environment

Assistance with multiple objectives

– Short term: Collision avoidance – Medium term: Wayfinding

Shared(?) control

– Autonomous navigation (with supervisory control) can cause confusion or agitation in this population – What to call it? Collaborative control? Human-in-the-loop?

Low cost sensors
User trials with target population
Reproducible research

Dec 2014 Ian M. Mitchell (UBC Computer Science) 6

SLIDE 7

Viability & Anesthesia Slides

Dec 2014 Ian M. Mitchell (UBC Computer Science) 7

SLIDE 8

Scalable Techniques for Viability Kernels in LTI Systems with Application to Automated Control of Anesthesia Ian M. Mitchell

Department of Computer Science University of British Columbia

December 2014

Ian M. Mitchell — 1

SLIDE 9

Background: Dynamic System and Constraints

     ˙ x = f(x, u), x(0) = x0, t ∈ [0, τ] =: T u(t) ∈ U (input constraint) K ⊆ X (target set/state constraint)

What is reachability analysis?

◮ [Tomlin, et al. 03; Aubin, et al. 11; Kurzhanski and Varaiya 00; Lygeros 04;

Blanchini and Miani 08; ...]

◮ Maximal vs. minimal reachability [Mitchell 07] Ian M. Mitchell — 2

SLIDE 10

Background: Reachability Constructs

Maximal reach tube

Reach♯

T(K, U) := {x0 ∈ X | ∃u(·), ∃t, x(t) ∈ K}

K

Ian M. Mitchell — 3

SLIDE 11

Background: Reachability Constructs

Maximal reach set

Reach♯

t(K, U) := {x0 ∈ X | ∃u(·), x(t) ∈ K}

K

Ian M. Mitchell — 4

SLIDE 12

Background: Maximal Reachability

Maximal reach tube vs. set [Lygeros 04; Mitchell 07]

Reach♯

T(K, U) =

t∈T Reach♯

t(K, U)

Parametric methods to approximate

◮ e.g. [Frehse, et al. 11; Girard and Le Guernic 08; Girard, et al. 06; Han and

Krogh 06; Kurzhanski and Varaiya 00; Kurzhanskiy and Varaiya 06]

◮ Scalable and computationally efficient (polynomial) Ian M. Mitchell — 5

SLIDE 13

Background: Minimal Reachability

Minimal reach tube

Reach♭

T(K, U) := {x0 ∈ X | ∀u(·), ∃t, x(t) ∈ K}

K

Ian M. Mitchell — 6

SLIDE 14

Background: Viability

Viability kernel (finite horizon)

V iabT(K, U) := {x0 ∈ X | ∃u(·), ∀t, x(t) ∈ K}

K

Infinite horizon viab kernel ≡ maximal controlled-invariant subset

Ian M. Mitchell — 7

SLIDE 15

Introduction

Viability kernel vs. minimal reach tube [Cardaliaguet, et al. 99]

(V iabT(K, U))c = Reach♭

T(Kc, U)

The only constructs to prove existence of safety control laws [Mitchell

07; Lygeros 04]

◮ Applications: [Lygeros, et al. 98; Tomlin, et al. 03; Bayen, et al. 07; Gillula,

et al. 10; Oishi, et al. 03; Aswani, et al. 11; Borrelli, et al. 10; Panagou, et al. 09; Del Vecchio, et al. 09; Ghaemi and Del Vecchio 11; ...]

Non-parametric methods to approximate

◮ [Mitchell, et al. 05; Cardaliaguet, et al. 99; Gao, et al. 06; Saint-Pierre 94] ◮ Computationally intensive (exponential) since grid-based Ian M. Mitchell — 8

SLIDE 16

Set-theoretic Methods

Efficient techniques (parametric) to compute maximal reach sets

Reach♯

t(K, U) := {x0 ∈ X | ∃u(·), x(t) ∈ K}

K

Approximate V iabT(K, U) via a nested sequence of sets reachable in

small sub-time intervals of T

Ian M. Mitchell — 10

SLIDE 17

Set-theoretic Methods: Continuous-Time

Start with an under-approximation K↓(P) of K

(P: interval partition of time; M: uniform bound on f) K↓(P) := {x ∈ K | dist(x, Kc) ≥ MP}

Recursively compute K0(P) from:

K|P|(P) = K↓(P), Kk−1(P) = K↓(P) ∩ Reach♯

tk−tk−1(Kk(P), U)

for k ∈ {1, . . . , |P|}.

Ian M. Mitchell — 11

SLIDE 18

Set-theoretic Methods: Continuous-Time

Start with an under-approximation K↓(P) of K

(P: interval partition of time; M: uniform bound on f) K↓(P) := {x ∈ K | dist(x, Kc) ≥ MP}

Recursively compute K0(P) from:

K|P|(P) = K↓(P), Kk−1(P) = K↓(P) ∩ Reach♯

tk−tk−1(Kk(P), U)

for k ∈ {1, . . . , |P|}.

Ian M. Mitchell — 11

SLIDE 19

Set-theoretic Methods: Continuous-Time

Start with an under-approximation K↓(P) of K

(P: interval partition of time; M: uniform bound on f) K↓(P) := {x ∈ K | dist(x, Kc) ≥ MP}

Recursively compute K0(P) from:

K|P|(P) = K↓(P), Kk−1(P) = K↓(P) ∩ Reach♯

tk−tk−1(Kk(P), U)

for k ∈ {1, . . . , |P|}.

Ian M. Mitchell — 11

SLIDE 20

Set-theoretic Methods: Continuous-Time

Start with an under-approximation K↓(P) of K

(P: interval partition of time; M: uniform bound on f) K↓(P) := {x ∈ K | dist(x, Kc) ≥ MP}

Recursively compute K0(P) from:

K|P|(P) = K↓(P), Kk−1(P) = K↓(P) ∩ Reach♯

tk−tk−1(Kk(P), U)

for k ∈ {1, . . . , |P|}.

Ian M. Mitchell — 11

SLIDE 21

Set-theoretic Methods: Continuous-Time

Start with an under-approximation K↓(P) of K

(P: interval partition of time; M: uniform bound on f) K↓(P) := {x ∈ K | dist(x, Kc) ≥ MP}

Recursively compute K0(P) from:

K|P|(P) = K↓(P), Kk−1(P) = K↓(P) ∩ Reach♯

tk−tk−1(Kk(P), U)

for k ∈ {1, . . . , |P|}.

Ian M. Mitchell — 11

SLIDE 22

Set-theoretic Methods: Continuous-Time

Start with an under-approximation K↓(P) of K

(P: interval partition of time; M: uniform bound on f) K↓(P) := {x ∈ K | dist(x, Kc) ≥ MP}

Recursively compute K0(P) from:

K|P|(P) = K↓(P), Kk−1(P) = K↓(P) ∩ Reach♯

tk−tk−1(Kk(P), U)

for k ∈ {1, . . . , |P|}.

Ian M. Mitchell — 11

SLIDE 23

Set-theoretic Methods: Continuous-Time

Start with an under-approximation K↓(P) of K

(P: interval partition of time; M: uniform bound on f) K↓(P) := {x ∈ K | dist(x, Kc) ≥ MP}

Recursively compute K0(P) from:

K|P|(P) = K↓(P), Kk−1(P) = K↓(P) ∩ Reach♯

tk−tk−1(Kk(P), U)

for k ∈ {1, . . . , |P|}.

Ian M. Mitchell — 11

SLIDE 24

Set-theoretic Methods: Continuous-Time

Guaranteed under-approximation: K0(P) ⊆ V iabT(K, U)
Arbitrarily precise by choosing a sufficiently fine partition of time:

V iabT(

K, U) ⊆
P∈P(T)

K0(P) ⊆ V iabT(K, U)

Ian M. Mitchell — 12

SLIDE 25

Set-theoretic Methods: Discrete-Time

Particular form of the continuous-time case
Recursively compute K0 from:

Kn = K, Kk−1 = K ∩ Reach♯

1(Kk, U)

for k ∈ {1, . . . , n}

Compute exactly: K0 = V iabT∩Z+(K, U)
Closely related to discrete algorithms in e.g. [Saint-Pierre 94; Cardaliaguet,

et al. 99; Blanchini and Miani 08]

Ian M. Mitchell — 13

SLIDE 26

Set-theoretic Methods: Computational Algorithms

For LTI systems L(x) = Ax + Bu there are several parametric options

Polytopes: exact, but the algorithms do not scale
Ellipsoids

◮ Only deals with ellipsoids (fixed complexity) ◮ Efficient, scalable, guaranteed under-approximation ◮ Implementation (Ellipsoidal Toolbox [Kurzhanskiy and Varaiya 06]) ◮ Generalizable to discriminating kernels ◮ Safety-preserving control synthesis ◮ Key set operations are only approximate

Support vectors

◮ User choice of complexity ◮ Efficient, scalable, guaranteed under-approximation ◮ Exact representation of key set operations ◮ Available implementations (eg: CVX [Grant and Boyd 08], SpaceEx

[Frehse et al 2011])

◮ Extension to discriminating kernels and control synthesis? Ian M. Mitchell — 14

SLIDE 27

Set-theoretic Methods: Ellipsoids

Ellipsoidal techniques (under-)approximating the maximal reach set:

K Reach♯

t(K, U)

ℓ(τ) ℓ(τ − t) [Kurzhanski and Varaiya 00; Kurzhanski and Valyi 96]

Ian M. Mitchell — 15

SLIDE 28

Piecewise Ellipsoidal Algorithm (CT)

Given P ∈ P(T) form an under-approximation K↓(P) of K
For a fixed terminal direction ℓτ ∈ M do the recursion

K∗[ℓτ]

k−1 = maxvol(K|P|(P) ∩ Reach♯[ℓτ] tk−tk−1(K∗[ℓτ] k

(P), U) for k ∈ {1, . . . , |P|} with K∗[ℓτ]

|P| (P) = K|P|(P) = K↓(P).

Generates an ellipsoidal set K∗[ℓτ]

(P) such that

ℓτ∈M

K∗[ℓτ] (P) := K∗

0(P) ⊆ V iabT(K, U)

Ian M. Mitchell — 16

SLIDE 29

Applications: Flight Envelope Protection (CT, 4D)

Longitudinal aircraft dynamics [Source: Bryson 94]

A =     −0.003 0.039 −0.322 −0.065 −0.319 7.740 0.020 −0.101 −0.429 1     B =

0.010

−0.180 −1.160 T subject to u(t) ∈ U := [−13.3◦, 13.3◦], x(t) ∈ K := E

2.18
,

1075.84

67.24 42.7716 76.0384

,

∀t ∈ [0, 2].

We choose |P| = 400, |M| = 8

Ian M. Mitchell — 17

SLIDE 30

Applications: Flight Envelope Protection (CT, 4D)

Level-Set: 5.5 h Piecewise Ellipsoidal: 10 m

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[Produced via ET and LS Toolbox]

Ian M. Mitchell — 18

SLIDE 31

Applications: Safety in Anesthesia Automation (DT, 7D)

Discrete-time Laguerre models (6D); patient’s response to rocuronium
Safety constraint: therapeutic bounds on output (pseudo-occupancy

level), i.e. desired clinical effect

Input constraint: actuator bounds (hard bounds on drug infusion rate)
Compute viability kernel for a 30 min surgery (patient #80)

A =        0.9960 0.0080 0.9960 −0.0080 0.0080 0.9960 0.0079 −0.0080 0.0080 0.9960 −0.0079 0.0079 −0.0080 0.0080 0.9960 0.0079 −0.0079 0.0079 −0.0080 0.0080 0.9960        B = 0.0894 −0.0890 0.0886 −0.0883 0.0879 −0.0876T C = 18.5000 8.2300 3.5300 4.3400 3.7000 3.0700

Ian M. Mitchell — 19

SLIDE 32

Applications: Safety in Anesthesia Automation (DT, 7D)

Reformulate by projecting the output bounds onto the state space

while making the control action regulatory.

Dynamics are augmented and transformed to a coordinate system of

dimension seven

Compute for |M| = 30 directions (15 resulted in non-empty ellipsoids)
Approximation in such high dimensions using non-parametric methods

is not currently possible

Ian M. Mitchell — 20

SLIDE 33

Applications: Safety in Anesthesia Automation (DT, 7D)

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Level Set: ? Piecewise Ellipsoidal: 15 m

[Produced via ET]

Ian M. Mitchell — 21

SLIDE 34

Set-theoretic Methods: Support Vectors

Support functions provide polytopic

verapproximation in specified

directions Corresponding support vectors provide polytopic underapproximation in specified directions Switch to discrete time algorithm for simplicity

Ian M. Mitchell — 22

SLIDE 35

Support Functions and Vectors

Key operations are exact: Reach♯

1(K, U) = A−1(K ⊕ U)

For matrix A and support function σK : Rd → R of convex set K,

such that σK(ℓ) = maxx∈K x · ℓ σAK(ℓ) = σK(AT ℓ) σK⊕U(ℓ) = σK(ℓ) + σU(ℓ) σK∩ ˆ

K(ℓ) = inf w∈Rd

σK(ℓ − w) + σ ˆ

K(w)

The (set of) support vectors νK of convex set K in direction ℓ is

νK(ℓ) = arg maxx∈K x · ℓ νAK(ℓ) = AνK(AT ℓ) νK⊕U(ℓ) = νK(ℓ) ⊕ νU(ℓ) νK∩ ˆ

K(ℓ) = νK(ℓ − ˆ

w) + ν ˆ

K( ˆ

w) where ˆ w = arg inf

w∈Rd

σK(ℓ − w) + σ ˆ

K(w)

Ian M. Mitchell — 23

SLIDE 36

Scaling the Support Vector Algorithm

Support function representation of Kn+1 = A−1(Kn ⊕ U) ∩ K0 σKn+1(ℓ) = inf

w∈Rd

σK0(ℓ − w) + σU(A−T w) + σKn(A−T w)
Convex function can be computed efficiently
Recursive function requires too many convex optimizations
Instead, unroll the recursion

σKn(ℓ) = inf w∈Rnd ξ(ℓ, w) w =

w1

w2 · · · wn T ξ(ℓ, w) = σK0(ℓ − wn) +

n−1

k=1

σK0(A−T wk+1 − wk) + σK0(A−T w1) +

n

k=1

σU(A−T wk)

Ian M. Mitchell — 24

SLIDE 37

Support Vector Results

Three compartment LTI model of Propofol metabolism
Third order Pad´

e approximation of input delay yields six dimensional state space

State constraint K given by a tight ellipsoid inscribing the box

[1, 6] × [0, 10]2 × [−100, +100]3

Eighteen purposefully chosen direction vectors

Ian M. Mitchell — 25

SLIDE 38

Discrete Time Double Integrator

Consider standard continuous time double integrator discretized with uniform timestep ρ and input constraint u ∈ U = [−u0, +u0] x1(t + 1) x2(t + 1)

=

1 ρ 1 x1(t) x2(t)

+

1

2ρ2

ρ

u(t)
In subsequent simulations we choose ρ = 0.1 time units, u0 = 0.3 and

a time horizon of 40 steps

Ian M. Mitchell — 26

SLIDE 39

Ellipsoidal

K = {x | x2 ≤ 0.5} (light grey)
Ellipsoidal approximation (dark blue) with uniformly sampled

directions

True viability kernel (black)

−0.4 −0.2 0.2 0.4 0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0.1 0.2 0.3 0.4 0.5

|L| = 5, execution time 105s.

−0.4 −0.2 0.2 0.4 0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0.1 0.2 0.3 0.4 0.5

|L| = 20, execution time 280s.

Ian M. Mitchell — 27

SLIDE 40

Support Vector (and Function)

K = {x | x2 ≤ 0.5} (not shown)
Support vector (dark blue) and corresponding support function (light

grey) with uniformly sampled directions

True viability kernel (black)

−0.4 −0.2 0.2 0.4 0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0.1 0.2 0.3 0.4 0.5 x1 x2

|L| = 5, execution time 28s.

−0.4 −0.2 0.2 0.4 0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0.1 0.2 0.3 0.4 0.5 x1 x2

|L| = 20, execution time 56s.

Ian M. Mitchell — 28

SLIDE 41

Scaling with Dimension

Compare execution time over ten steps for a discrete time model of a chain of integrators

Exact polytopic method
Ellipsoidal algorithm in a single

direction

Support vector algorithm in 2n

standard basis vectors (positive and negative directions)

5 10 15 20 25 30 35 40 20 40 60 80 100 120 140 160 180 200 state dimension run time (s) polytope ellipsoid support vector

Ian M. Mitchell — 29

SLIDE 42

So What About the Smart Wheelchair?

Desired properties seemed so similar to my earlier aircraft

control verification problems, but they were not

Dec 2014 Ian M. Mitchell (UBC Computer Science) 8

Aircraft Wheelchair Simple physics-based models No models for proprietary embedded control system Simple safety problems (regulator defined safety zone) Unclear risk metric (how close is too close?) Open environments (airports) with accurate sensors (ADS-B, GPS) Complex dynamic environments (pedestrians) and noisy sensors (RGBD cameras) Autopilot behavior can be analyzed

ffline

Shared control requires online risk assessment Highly trained pilots familiar with complex visual interfaces Cognitively impaired older adults and limited interface options

SLIDE 43

Understanding the Target Population

Wizard of oz study allows testing without a fully

implemented system

– Teleoperator pretends to be the intelligent wheelchair – Collect qualitative and quantitative data to obtain user feedback and inform continuing design work

Test with target population

– 10 participants from 3 LTC facilities in Vancouver – 5+ driving sessions to test multiple repetitions of 3 control policies on 5 typical driving tasks – Record lots of sensor data to get a robot’s eye view (and release it so that other researchers can use it)

Preliminary findings

– Participants almost always want some control – Desired assistance depends on participant, task, environment and probably time – Feedback to participant about type, degree, cause and timing of intervention is challenging

Dec 2014 Ian M. Mitchell (UBC Computer Science) 9

The Wizard [Baum, 1900]

SLIDE 44

What Worked?

Nonlinear but not scalable: Hamilton-Jacobi

nonparametric representation

Linear but scalable: ellipsoidal and support vector

parametric representations

– Demonstrated on seven dimensional anesthesia models

Extensions for ellipsoidal representations

– Discriminating kernels: worst-case robustness to uncertainty – Control synthesis: sampled data control signals which maintain safety

Dec 2014 Ian M. Mitchell (UBC Computer Science) 10

SLIDE 45

What Needs Work?

Viability algorithms

– Support vector representation: robustness, control synthesis, continuous time – Ellipsoidal representation: rigorous implementation – Hamilton-Jacobi representation: scalability…

Anesthesia problem

– Dealing with imperfect state estimates

Smart wheelchair problem

– Risk assessment metrics compatible with human intuition and behavior – Verification of systems with learned (and evolving) models – Verification (or design of verifiable) human-automation shared control

Reproducible research

– See HSCC repeatability evaluation

Dec 2014 Ian M. Mitchell (UBC Computer Science) 11

SLIDE 46

Reproducibility: A Personal Example

Study of safe flap settings during aircraft final approach to runway

– Publication: Bayen, Mitchell, Oishi & Tomlin, “Aircraft Autolander Safety Analysis Through Optimal Control-Based Reach Set Computation” in AIAA Journal of Guidance, Control & Dynamics, 30(1): 68–77 (2007).

October 2013 Ian M. Mitchell (UBC Computer Science) 12

g a V T D L mg

inertial frame wind frame body frame

z

[ cos , sin ] ( ) [ sin , cos ] sin ( ) ( ) m V mg d m V mg d V V z V t T D T L a a a g a g g g

 

                        

1 1

with mode switching without mode switching

SLIDE 47

Could You Send Me the Code?

Which directory was that in?

~/OldStanfordGagarin/Cyghome/Source/HS01/Landing/ ~/OldStanfordGagarin/Cyghome/Source/Projection/Working/ ~/OldStanfordGagarin/Cyghome/Source/JCP/ ~/OldStanfordGagarin/Cyghome/Papers/AIAA02/Source/ ~/OldStanfordGagarin/Winhome/VisualStudioProjects/LandingHighD/ ~/OldVonBraun/CygHome/Papers/AIAA03/Landing/Source ~/OldVonBraun/CygHome/Papers/AIAA03/Landing/Shriram

Which parameters did I use?

// 70% of 160e3 is 112e3 // assumes fixed thrust at minT (see Flow::hamiltonian() function) //const GradValue ModeMinT = 0e3; //const GradValue ModeMaxT = 160e3; //const GradValue ModeMinT = 32e3; //const GradValue ModeMaxT = 32e3;

October 2013 Ian M. Mitchell (UBC Computer Science) 13

SLIDE 48

Thanks to Many Collaborators

Anesthesia:

– Shahab Kaynama (UBC PhD, now Clearpath Robotics) – John Maidens (UBC MSc, now UC Berkeley) – Meeko Oishi (UBC, now U New Mexico) – Guy Dumont (UBC ECE)

Smart Wheelchairs:

– Pooja Viswanathan (U of T postdoc) – Alan Mackworth (UBC CS) – Alex Mihailidis (U of T) – William C. Miller (UBC OS & OT) – Jim Little (UBC CS) – Rosalie Wang (U of T) – Junaed Sattar (UBC postdoc, now Clarkson) – Paula Rushton (U Montreal) – Ben Mortenson (UBC OS & OT) – Bikram Adhikari (UBC MSc) – Pouria TalebiFard (UBC MSc) – Eric Rothfels (UBC BSc, now Waterloo) – The rest of the CanWheel Team – Advanced Mobility Products – Staff at G. F. Strong & long term care centers

Dec 2014 Ian M. Mitchell (UBC Computer Science) 14

SLIDE 49

Funding

NSERC Discovery, postgraduate and USRA grants
Anesthesia

– NSERC Collaborative Health Research Projects #CHRPJ- 350866 & #CPG127770

Smart Wheelchair

– CanWheel, the CIHR Emerging Team in Wheeled Mobility for Older Adults grant #AMG-100925 – Alzheimer Society Research Program – People & Planet Friendly Home (an ICICS & TELUS initiative) – CFI LOF / BC KDF grant #13113

Dec 2014 Ian M. Mitchell (UBC Computer Science) 15

SLIDE 50

Scalable Parametric Approximations

f the Maximal Controlled Invariant Set

and Two Health Related Application Domains

For more information contact

Ian M. Mitchell

Department of Computer Science University of British Columbia mitchell@cs.ubc.ca http://www.cs.ubc.ca/~mitchell http://www.canwheel.ca