Lane Changing

- Compatibility

Requirements Engineering

The process of defining, documenting, and maintaining requirements
6 steps:
- Requirements inception （需求獲取）
- Requirements analysis（需求分析）
- Requirements specification（需求規格）
- Requirements validation（需求驗證）
- Requirements management（需求管理）

Requirements Engineering in our case

Original requirements: collision-free
- Issue: incompatible between systems
Requirements inception
- compatibility
Requirements analysis, specification, validation
- (research objective)
- guarantee compatibility under some conditions ?

Speed-up: 15% / 20%~30%

ca1

加速

等速

減速

\(v = v_{\min}\)

\(v = v_{\max}\)

\(x' - x = d_{\min}\)

\(\neg blue \land \neg red\)

\(\neg blue\)

\(x' - x = d_{\min}\)

變換車道（\(|x - x'| > d_{lc}\)）

ca1

加速

等速

減速

\(v = v_{\min}\)

\(v = v_{\max}\)

\(x < x' \land x' - x < d_{\min}\)

\(\neg blue \land \neg red\)

\(\neg blue\)

\(x < x' \land x' - x < d_{\min}\)

變換車道（\(|x - x'| > d_{lc}\)）

線性加速

等加速

等減速

等速max

等速min

Modeling Hybrid System

\(q_1\in Q\)

\(q_2\in Q\)

continuous state: \(X = \mathbb{R}^n, f\)

\(\dfrac{dx}{dt} = f_{q_1}(x)\)

constraints

\(x\in D(q_1)\)

\(x\in D(q_2)\)

\(x\in G(q_1, q_2)\)

\(e \in E\)

\(\dfrac{dx}{dt} = f_{q_2}(x)\)

\(x \gets R(e, x)\)

discrete state/graph: \(Q, E\)

constraints: \(D_{omain}, G_{uard}\)

def PeriodicSimulation(
    Q, #                  discrete state space
    X, # = R^n            continuous state space
    f, # = Q × X → R^n    vector field
    Init, # = Q × X       set of initial states
    D, # Q → P(X)         domain
    E, # in Q × Q         set of edges
    G, # E → P(X)         guard condition
    R, # E × X → P(X)     reset map
):
    dt = 0.000001
    q, x = pick(Init)
    while True:
        # edge case
        if x not in D(q):
            raise Exception("OUT_OF_DOMAIN")
        # change state (discrete transition)
        next = []
        for q2 in Q:
            if ((q, q2) in E) and (x in G(q, q2)) and (x in D(q2)):
                next.append(q2)
        # same state (continuous evolution)
        x2 = x + f(q, x) * dt
        if x2 in D(q):
            next.append(STAY)
        # test blocked
        if is_empty(next):
            raise Exception("BLOCKED")
        # choose nondeterministically
        q2 = pick(next)
        if q2 == STAY:
            x = x2
        else:
            x = pick(R((q, q2), x))
            q = q2

Issues

Blocking / Non-blocking
- hard to verify
Non-deterministic
- hard to verify
Composability
- all variables are states
Zeno execution?
- intentional deviation?

Simulation - Water Tank System

\(q_1\)

\(q_2\)

\(q_1\)

current

STAY

change state

\(dt\)

solution: binary search for the boundary

(discrete transition)

(continuous evolution)

import numpy as np
import random

def arr(*x): return np.array(x)
def pick(x): return random.choice(x)
def is_empty(x): return not bool(x)
def water(x, b = 1, l = 30, mx = 15):
    n = int(x / mx * l)
    s = '.' * n + ' ' * (l - n)
    b = int(b / mx * l)
    return s[:b] + '|' + s[b:]
DT = 0.001
eps = 1e-30
converge_threshold = 100

STAY = -1
def PeriodicSimulation(
    Q, #                  discrete state space
    X, # = R^n            continuous state space
    f, # = Q × X → R^n    vector field
    Init, # = Q × X       set of initial states
    D, # Q → P(X)         domain
    E, # in Q × Q         set of edges
    G, # E → P(X)         guard condition
    R, # E × X → P(X)     reset map
):
    q, x = pick(Init)
    t = 0
    convergence = (None, 0)
    while True:
        dt = DT
        while dt > eps:
            if not D(q, x):
                raise Exception("OUT_OF_DOMAIN")

            next = []
            for q2 in Q:
                if ((q, q2) in E) and (G(q, q2, x)) and (D(q2, x)):
                    next.append(q2)

            x2 = x + f(q, x) * dt
            if D(q, x2):
                next.append(STAY)

            if is_empty(next):
                dt /= 2
                continue

            q2 = pick(next)
            if q2 == STAY:
                x = x2
                t += dt
            else:
                x = R((q, q2), x)
                q = q2

            if convergence[0] == x.tolist():
                convergence = (convergence[0], convergence[1] + 1)
            else:
                convergence = (x.tolist(), 1)
            # print(convergence)
            if convergence[1] > converge_threshold:
                raise Exception("CONVERGED")
            break

        if not (dt > eps):
            raise Exception("BLOCKED")

        print(f'{q = },\t{water(x[0])},\t{water(x[1])},\t{t = },\tln(dt) = {int(np.log(dt))}')
        # print(f'{q = },\t{x = },\t{t = },\t{dt = }')

w = 0.5
v0 = 0.2
v1 = 0.4
r0 = 1
r1 = 1

np.set_printoptions(precision=3)
PeriodicSimulation(
    Q = [0, 1],
    X = None,
    f = lambda q, x: [arr(w-v0, -v1), arr(-v0, w-v1)][q],
    Init = [(0, arr(7, 7))],
    D = lambda q, x: [x[1] >= r1, x[0] >= r0][q],
    E = [(0, 1), (1, 0)],
    G = lambda q1, q2, x: { (0, 1): x[1] <= r1, (1, 0): x[0] <= r0 }[(q1, q2)],
    R = lambda qs, x: x,
)

Ca1

\(f_A(x, v) = (v, a)\)

\(Q = \{\ A = \text{加速},\ C = \text{等速},\ D = \text{減速}\ \}\)

\(f_C(x, v) = (v, 0)\)

\(f_D(x, v) = (v, -a)\)

\(R(\cdot, x) = x\)

\(D(A) = \{V \leq v_{\max}\}\)

\(G(\cdot, \cdot) = ?\)

\(D(D) = \{V \geq v_{\min}\}\)

ca0

\(L_{ane\ change}\)

\(A_{ccelerate}\)

\(C_{onstant\ speed}\)

\(D_{eccelerate}\)

high

low

\(D_L\)

\(D_A\)

\(D_D\)

\(D_C\)

priority

Flow

\((x_1', v_1') = (0, 0)\)

\((x_1', v_1') = (v_1, a_{\max})\)

\((x_1', v_1') = (v_1, a_{\min})\)

\((x_1', v_1') = (v_1, 0)\)

\(D_L \coloneqq |x_1 - x_2| \geq d_{\min}\)

\(D_A \coloneqq \begin{cases}v_2 < v_{\max} - v_\text{mgn}, &\text{ if }x_1 > x_2\\ v_2 < v_{\min} + v_\text{mgn}, &\text{ else}\end{cases}\)

\(D_C \coloneqq (v_1 = v_{\min}) \lor (v_1 = v_{\max})\)

\(D_D \coloneqq \begin{cases}v_2 > v_{\max} - v_\text{mgn}, &\text{ if }x_1 > x_2\\ v_2 > v_{\min} + v_\text{mgn}, &\text{ else}\end{cases} \coloneqq \neg D_A\)

ca0

high

low

priority

\(D_L \coloneqq |x_1 - x_2| \geq d_{\min} \coloneqq \begin{cases}x_1 - x_2 \geq d_{\min}, &\text{if }x_1 > x_2\\ x_2 - x_1 \geq d_{\min}, &\text{else}\end{cases}\)

\(D_A \coloneqq \begin{cases}(v_1 < v_{\max}) \land (v_2 < v_{\max} - v_\text{mgn}), &\text{if }x_1 > x_2\\ (v_1 < v_{\max}) \land (v_2 < v_{\min} + v_\text{mgn}), &\text{else}\end{cases}\)

\(D_C \coloneqq (v_{\min} \leq v_1) \land (v_1 \leq v_{\max})\)

\(D_D \coloneqq \begin{cases}(v_1 > v_{\min}) \land (v_2 > v_{\max} - v_\text{mgn}), &\text{ if }x_1 > x_2\\ (v_1 > v_{\min}) \land (v_2 > v_{\min} + v_\text{mgn}), &\text{ else}\end{cases}\)

\(D_L\)

\(D_A\)

\(D_D\)

\(D_C\)

\((x_1', v_1') = (0, 0)\)

\((x_1', v_1') = (v_1, a_{\max})\)

\((x_1', v_1') = (v_1, a_{\min})\)

\((x_1', v_1') = (v_1, 0)\)

\(L_{ane\ change}\)

\(A_{ccelerate}\)

\(C_{onstant\ speed}\)

\(D_{eccelerate}\)

\(v_{\min}=5,\;v_{\max}=25,\;v_{\text{mgn}}=3,\;d_{\min}=5,\;a_{\max}=3,\;a_{\min}=-2\)

Re-implementation

Model

\(q_1\in Q\)

\(q_2\in Q\)

continuous state: \(X = \mathbb{R}^n, f\)

\(\dfrac{dx}{dt} = f_{q_1}(x)\)

constraints

\(x\in D(q_1)\)

\(x\in D(q_2)\)

\(x\in G(q_1, q_2)\)

\(e \in E\)

\(\dfrac{dx}{dt} = f_{q_2}(x)\)

\(x \gets R(e, x)\)

discrete state/graph: \(Q, E\)

constraints: \(D_{omain}, G_{uard}\)

Original Method

Bottleneck

Input: state \(q, x_0\); next state \(q'\)

Output: minimum \(\tau\) for transition \((q, x(\tau)) \rightarrow q'\)

\(\iff \min \tau: \begin{cases}x(0) &= x_0\\ x'(t) &= f_q(x(t))\\ x(\tau) &\in G_{q\rightarrow q'} \cap D_{q'} \end{cases}\)

\(\begin{cases}x(0) &= x_0\\ x'(t) &= f_q(x(t))\\ x(\tau) &\in G_{q\rightarrow q'} \cap D_{q'} \end{cases}\)

from sympy import symbols, Eq, Function, dsolve

X_0 = (10, 5)
A = 2
def f(x, v):
    return (v, A)

t = symbols("t")
X = symbols("X:2", cls=Function)
X = [x(t) for x in X]

equations = [
    Eq(x.diff(t), fx)
for x, fx in zip(X, f(*X))]

sol = dsolve(equations, X, ics={
    x.subs(t, 1e-30): x_0
for (x, x_0) in zip(X, X_0) })
print(sol)

min(sympy.solve(...).as_set().boundry) ?

\(\begin{cases}x(0) &= x_0\\ x'(t) &= f_q(x(t))\\ x(\tau) &\in G_{q\rightarrow q'} \cap D_{q'} \end{cases}\)

Cond = Cond & Cond
     = Cond | Cond
     = !Cond
     = f(X) >= 0
     (f is continuous)

Constraints on \(G\) and \(D\)

min(sympy.solve(...).as_set().boundry) ?

\(X \in G \iff \)Cond is true

Cond = Cond & Cond
     = Cond | Cond
     = !Cond
     = f(X) >= 0
     (f is continuous)


Decelerate = 車速 > v_min
           = !(v_min - v1 >= 0)

Accelerate = 車速 < v_max & (行駛於前 | 前車距離 > d_min)
           = !(v1 - v_max >= 0) & (x1 - x2 >= 0 | !(d_min - (x2 - x1) >= 0))

\(X \in G_{q\rightarrow q'} \cap D_{q'} \iff (f(X)\geq0 \lor g(X)\geq0)\land (\neg (h(X)\geq0))\)

\(\land\)

\(\lor\)

\(\neg\)

\(h(X) \geq 0\)

\(g(X) \geq 0\)

\(f(X) \geq 0\)

\(f\geq0\)

\(g\geq0\)

\(h\geq0\)

\(t\)

\(f\geq0\)

\(g\geq0\)

\(h\geq0\)

\(t\)

\(X \in G_{q\rightarrow q'} \cap D_{q'} \iff (f(X)\geq0 \lor g(X)\geq0)\land (\neg (h(X)\geq0))\)

\(f\geq0\)

\(g\geq0\)

\(h\geq0\)

\(t\)

\(X \in G_{q\rightarrow q'} \cap D_{q'} \iff (f(X)\geq0 \lor g(X)\geq0)\land (\neg (h(X)\geq0))\)

\(f\geq0\)

\(g\geq0\)

\(h\geq0\)

\(t\)

\(X \in G_{q\rightarrow q'} \cap D_{q'} \iff (f(X)\geq0 \lor g(X)\geq0)\land (\neg (h(X)\geq0))\)

Assumptions

Solving for minimum positive root is easy
- \((f(X_t)\geq0 \lor g(X_t)\geq0)\land (\neg (h(X_t)\geq0))\)
- vs \(f(X_t) = 0\)
Cond is general enough

Cond = Cond & Cond
     = Cond | Cond
     = !Cond
     = f(X) >= 0
     (f is continuous)

========REQUIREMENT===================
Root(F=>T)

========BASE CASES====================
Condition: f(x) > 0
T = min(nsolve(sols + f(x), [t] + X, [0] + X0))
F=>T = f(X0) > 0 ? inf : T
T=>F = f(X0) > 0 ? T : inf

========RECURSIVE CASES===============
Boolean: !A
F=>T = A(T=>F)
T=>F = A(F=>T)

Boolean: A&B
F=>T = (F&F,F&T,T&F)=>(T&T)
    ?= max(A(F=>T), B(F=>T))
T=>F = (T&T)=>(F&F,F&T,T&F)
     = min(A(T=>F), B(T=>F))

Boolean: A|B
F=>T = (F|F)=>(T|T,F|T,T|F)
     = min(A(F=>T), B(F=>T))
T=>F = (T|T,F|T,T|F)=>(F|F)
    ?= max(A(T=>F), B(T=>F))

Tree DP optimization?

\(\land\)

\(\lor\)

\(\neg\)

\(h(X) \geq 0\)

\(g(X) \geq 0\)

\(f(X) \geq 0\)

a ? b : c
= (a & b) | ((!a) & c)

Graph optimization

\(\lor\)

\(\land\)

\(a\)

\(b\)

\(a\)

\(b\)

\(c\)

\(\neg\)

a ? b : c
= (a & b) | ((!a) & c)

Graph optimization

\(\lor\)

\(\land\)

\(a\)

\(b\)

\(a\)

\(b\)

\(c\)

\(\neg\)

Graph optimization

\(f_1(X) \geq 0\)

\(f_2(X) \geq 0\)

\(f_3(X) \geq 0\)

\(f_4(X) \geq 0\)

\(f_5(X) \geq 0\)

\(Cond_1\)

\(Cond_2\)

\(Cond_3\)

\(Cond_4\)

push the timeline synchronously

Original:
- \(f(X) \geq 0 \implies\) T
- \(f(X) < 0 \implies\) F
Improved:
- \(|f(X)| \leq \epsilon \implies\) X
- \(f(X) > \epsilon \implies\) T
- \(f(X) < \epsilon \implies\) F

Simulation improvement

Accelerates to \(25\) m/s (max speed)
\(v = 25.0000000001\) m/s
\(D_C = \neg \text{too\_fast} \land \neg \text{too\_slow}\)
Can't maintain constant speed

round-off error

modes: A, B, C

w/o error

w/ error

？

Solutions

Enforce boundary:
- Whenever blocked, move back to boundary
Model check:
- Verify if behavior is defined on or across boundaries

？

https://link.springer.com/chapter/10.1007/978-3-540-30206-3_10

Robustness and Implementability of Timed Automata

Simplification

Domain \(D = \top\)
Guard \(G = \bigwedge (a_i \leq x_i \leq b_i)\) (linear & independent)
Flow \(f(X) = (1, 1, \dots, 1)\) (clock)

Error Formulation

drift on clocks \(\epsilon\)
- \((S_1, X_1) \overset{t \text{ secs}}{\longrightarrow} (S_2, X_2)\) \(\Rightarrow\) \((S_1, X_1) \overset{t(1-\epsilon) \sim t(1+\epsilon)}{\longrightarrow} (S_2, X_2)\)
imprecision on guards \(\Delta\)
- \(a_i \leq x_i \leq b_i\) \(\Rightarrow\) \(a_i - \Delta \leq x_i \leq b_i + \Delta\)

mode A

mode B

mode C

mode D

偏差率 (\(\frac{Err}{\Delta t}\))
\(f_A(X) \cdot \nabla G_{A \rightarrow B}\)

Lane Changing - Compatibility

short summary

Phase 1

Work on Existing Method

Phase 2

Redesign Model

Phase 3

Identify Difficulties

Work on Existing Method

Achievement

Speed up MATLAB code

Identified Issues

No formal definition of lane-changing systems
Imprecise implementation
Hard to maintain the original code

Redesign Model

Achievement

Model hybrid systems based on this paper
Add restrictions for Guard / Domain conditions
Re-implement our paper's algorithm
Complete discrete/continuous simulation
Build visualization tools

Cond = Cond & Cond
     = Cond | Cond
     = !Cond
     = f(X) >= 0
     (f is continuous)

Identify Difficulties

Limited power of equation solvers
Unavoidable floating point errors
Errors are expected in real world interactions

Possible Solution 1

Set-based verification

Flow*
Plenty of existing work
No convenient tools yet
Explosive states?
May still work in most cases (trivial cases)

Possible Solution 2

Confidence score

Provide confidence score (robustness) about the simulated trace

May assist in downstream task decisions
Can't guarantee safety

https://ieeexplore.ieee.org/document/9867937

Possible Solution 3

Restrictions on lane-changing systems

How to determine restrictions?
Linear boundaries?
The models in some papers seem oversimplified

https://link.springer.com/chapter/10.1007/978-3-540-30206-3_10

Another way to measure robustness?

(against real world errors)

Robustness + Falsification

Idea

Error => multiple possible traces

=> sample different traces

Assumptions

\(\tau = \Delta t\) between two steps
\(\tau\) follows uniform distribution between 0 and \(\epsilon\) (e.g. 0.001)
deterministic hybrid system

init

\(\epsilon\)

init

minimum

robustness

\(\epsilon\)

init

minimum

robustness

sample break points

\(\epsilon\)

minimum

robustness

control

mode

control

mode

\(0.4\epsilon\)

\(p = 0.6\)

\(p = 0.4\)

Add-ons

must sample between every control mode change

1 step

0.2 step

1 step

Robustness Improvement

TODO:

some small bugs
binary search for boundary (after sampling)
use dijkstra in robustness improvement
optimize for control mode oscillation

Ca0

Ca1

Agg

Ca0

Ca1

Agg

https://docs.google.com/presentation/d/1qnIwSruxLhmJxVC2EQ7cFsvaHSBqF9y9XfIJwkXZkBk/preview

sample

\(\tau\)

\(t\)

\(\tau\)

p=0.5

p=0.25

\(\tau\)

\(t\)

low \(R\)

high \(R\)

sample

p=0.25

p=0.75

\(y\)

\(\tau\)

lane changing - compatibility

By thomaswang2003

lane changing - compatibility

Lane Changing

- Compatibility

Requirements Engineering

Requirements Engineering

Requirements Engineering in our case

Modeling Hybrid System

Issues

Simulation - Water Tank System

Ca1

Re-implementation

Model

Model

Original Method

Bottleneck

min(sympy.solve(...).as_set().boundry) ?

Constraints on \(G\) and \(D\)

min(sympy.solve(...).as_set().boundry) ?

Assumptions

Tree DP optimization?

Graph optimization

Graph optimization

Graph optimization

Simulation improvement

round-off error

round-off error

Solutions

Robustness and Implementability of Timed Automata

Simplification

Error Formulation

Lane Changing - Compatibility

short summary

Phase 1

Work on Existing Method

Phase 2

Redesign Model

Phase 3

Identify Difficulties

Work on Existing Method

Work on Existing Method

Redesign Model

Redesign Model

Identify Difficulties

Possible Solution 1

Possible Solution 2

Possible Solution 3

Another way to measure robustness?

(against real world errors)

Robustness + Falsification

Idea

Assumptions

Add-ons

Robustness Improvement

lane changing - compatibility

More from thomaswang2003