Safety and Efficiency in Autonomous Vehicles through Online Learning

Zachary Sunberg

Postdoctoral Scholar

University of California

Autonomy is becoming possible and prevalent

How do we deploy with Confidence?

Waymo Image By Dllu - Own work, CC BY-SA 4.0, https://commons.wikimedia.org/w/index.php?curid=64517567

Shoettle and Sivak, "A Preliminary Analysis of Real-World Crashes Involving Self-Driving Vehicles" UMTRI-2015-34

image/svg+xml

Introduction

Lane Changing with Internal States

Solving Continuous POMDPs

POMDPs.jl

Future

image/svg+xml

Introduction

Lane Changing with Internal States

Solving Continuous POMDPs

POMDPs.jl

Future

Two Objectives for Autonomy

EFFICIENCY

SAFETY

Minimize resource use

(especially time)

Minimize the risk of harm to oneself and others

Hard Safety: Guaranteeing that there will be no harm

Soft Safety: Reducing the risk of harm

Two extremes:

"The two greatest risks in life are risking too much and risking too little" - Utah Avalanche Center Podcast

Objective Function

$$R(s_t, a_t) = R_\text{E}(s_t, a_t) + \lambda R_\text{S}(s_t, a_t)$$

Safety

Weight

Efficiency

\[\text{maximize} \quad E \left[ \sum_{t=0}^\infty \gamma^t R(s_t, a_t) \right]\]

Objective Function

Safety

Better Performance

Model \(M_2\), Algorithm \(A_2\)

Model \(M_1\), Algorithm \(A_1\)

Efficiency

$$R(s_t, a_t) = R_\text{E}(s_t, a_t) + \lambda R_\text{S}(s_t, a_t)$$

Safety

Weight

Efficiency

Uncertainty Expression

OUTCOME

MODEL

STATE

Markov Model

  • \(\mathcal{S}\) - State space
  • \(T:\mathcal{S}\times\mathcal{S} \to \mathbb{R}\) - Transition probability distributions

Markov Decision Process (MDP)

  • \(\mathcal{S}\) - State space
  • \(T:\mathcal{S}\times \mathcal{A} \times\mathcal{S} \to \mathbb{R}\) - Transition probability distribution
  • \(\mathcal{A}\) - Action space
  • \(R:\mathcal{S}\times \mathcal{A} \times\mathcal{S} \to \mathbb{R}\) - Reward

Partially Observable Markov Decision Process (POMDP)

  • \(\mathcal{S}\) - State space
  • \(T:\mathcal{S}\times \mathcal{A} \times\mathcal{S} \to \mathbb{R}\) - Transition probability distribution
  • \(\mathcal{A}\) - Action space
  • \(R:\mathcal{S}\times \mathcal{A} \times\mathcal{S} \to \mathbb{R}\) - Reward
  • \(\mathcal{O}\) - Observation space
  • \(Z:\mathcal{S} \times \mathcal{A}\times \mathcal{S} \times \mathcal{O} \to \mathbb{R}\) - Observation probability distribution
image/svg+xml

Introduction

Lane Changing with Internal States

Solving Continuous POMDPs

POMDPs.jl

Future

Sadigh, Dorsa, et al. "Information gathering actions over human internal state." Intelligent Robots and Systems (IROS), 2016 IEEE/RSJ International Conference on. IEEE, 2016.

Schmerling, Edward, et al. "Multimodal Probabilistic Model-Based Planning for Human-Robot Interaction." arXiv preprint arXiv:1710.09483 (2017).

Sadigh, Dorsa, et al. "Planning for Autonomous Cars that Leverage Effects on Human Actions." Robotics: Science and Systems. 2016.

Tweet by Nitin Gupta

29 April 2018

https://twitter.com/nitguptaa/status/990683818825736192

POMDP Formulation

\(s=\left(x, y, \dot{x}, \left\{(x_c,y_c,\dot{x}_c,l_c,\theta_c)\right\}_{c=1}^{n}\right)\)

\(o=\left\{(x_c,y_c,\dot{x}_c,l_c)\right\}_{c=1}^{n}\)

\(a = (\ddot{x}, \dot{y})\), \(\ddot{x} \in \{0, \pm 1 \text{ m/s}^2\}\), \(\dot{y} \in \{0, \pm 0.67 \text{ m/s}\}\)

R(s, a, s') = \text{in\_goal}(s') - \lambda \left(\text{any\_hard\_brakes}(s, s') + \text{any\_too\_slow}(s')\right)
R(s,a,s)=in_goal(s)λ(any_hard_brakes(s,s)+any_too_slow(s))R(s, a, s') = \text{in\_goal}(s') - \lambda \left(\text{any\_hard\_brakes}(s, s') + \text{any\_too\_slow}(s')\right)

Ego physical state

Physical states of other cars

Internal states of other cars

Physical states of other cars

  • Actions filtered so they can never cause crashes
  • Braking action always available

Efficiency

Safety

Human Behavior Model: IDM and MOBIL

\ddot{x}_\text{IDM} = a \left[ 1 - \left( \frac{\dot{x}}{\dot{x}_0} \right)^{\delta} - \left(\frac{g^*(\dot{x}, \Delta \dot{x})}{g}\right)^2 \right]
x¨IDM=a[1(x˙x˙0)δ(g(x˙,Δx˙)g)2] \ddot{x}_\text{IDM} = a \left[ 1 - \left( \frac{\dot{x}}{\dot{x}_0} \right)^{\delta} - \left(\frac{g^*(\dot{x}, \Delta \dot{x})}{g}\right)^2 \right]
g^*(\dot{x}, \Delta \dot{x}) = g_0 + T \dot{x} + \frac{\dot{x}\Delta \dot{x}}{2 \sqrt{a b}}
g(x˙,Δx˙)=g0+Tx˙+x˙Δx˙2abg^*(\dot{x}, \Delta \dot{x}) = g_0 + T \dot{x} + \frac{\dot{x}\Delta \dot{x}}{2 \sqrt{a b}}

M. Treiber, et al., “Congested traffic states in empirical observations and microscopic simulations,” Physical Review E, vol. 62, no. 2 (2000).

A. Kesting, et al., “General lane-changing model MOBIL for car-following models,” Transportation Research Record, vol. 1999 (2007).

A. Kesting, et al., "Agents for Traffic Simulation." Multi-Agent Systems: Simulation and Applications. CRC Press (2009).

All drivers normal

Outcome only

Omniscient

Mean MPC

QMDP

POMCPOW

Simulation results

Assume normal

Outcome only

Omniscient

Mean MPC

QMDP

POMCPOW

Correlation Trend

Robustness

Next Steps: Testing against Learned Models

image/svg+xml

Introduction

Lane Changing with Internal States

Solving Continuous POMDPs

POMDPs.jl

Future

Information State

History: all previous actions and observations

\[h_t = (b_0, a_0, o_1, a_1, o_2, ..., a_{t-1}, o_t)\]

Belief: probability distribution over \(\mathcal{S}\) encoding everything learned about the state from the history

\[b_t(s) = P(s_t=s \mid h_t)\]

A POMDP is an MDP on the belief space

Belief Example: Laser Tag

Solving MDPs and POMDPs - The Value Function

$$\mathop{\text{maximize}} V_\pi(s) = E\left[\sum_{t=0}^{\infty} \gamma^t R(s_t, \pi(s_t)) \bigm| s_0 = s \right]$$

$$V^*(s) = \max \left\{R(s, a) + \gamma E\Big[V^*\left(s_{t+1}\right) \mid s_t=s, a_t=a\Big]\right\}$$

\pi: \mathcal{S} \to \mathcal{A}
π:SA\pi: \mathcal{S} \to \mathcal{A}

Involves all future time

Involves only \(t\) and \(t+1\)

\(a \in \mathcal{A}\)

C. H. Papadimitriou and J. N. Tsitsiklis, “The complexity of Markov decision processes,” Mathematics of Operations Research, vol. 12, no. 3, pp. 441–450, 1987

But POMDPs are still PSPACE-Complete

Online Decision Process Tree Approaches

State Node

Action Node

(Estimate value function here)

Monte Carlo Tree Search

Image by Dicksonlaw583 (CC 4.0)

POMCP

  • Uses simulations of histories instead of full belief updates

 

  • Each belief is implicitly represented by a collection of unweighted particles

 

Silver, David, and Joel Veness. "Monte-Carlo planning in large POMDPs." Advances in neural information processing systems. 2010.

Ross, Stéphane, et al. "Online planning algorithms for POMDPs." Journal of Artificial Intelligence Research 32 (2008): 663-704.

Light-Dark Problem

\begin{aligned} & \mathcal{S} = \mathbb{Z} \quad \quad \quad ~~ \mathcal{O} = \mathbb{R} \\ & s' = s+a \quad \quad o \sim \mathcal{N}(s, s-10) \\ & \mathcal{A} = \{-10, -1, 0, 1, 10\} \\ & R(s, a) = \begin{cases} 100 & \text{ if } a = 0, s = 0 \\ -100 & \text{ if } a = 0, s \neq 0 \\ -1 & \text{ otherwise} \end{cases} & \\ \end{aligned}
S=Z  O=Rs=s+aoN(s,s10)A={10,1,0,1,10}R(s,a)={100 if a=0,s=0100 if a=0,s01 otherwise\begin{aligned} & \mathcal{S} = \mathbb{Z} \quad \quad \quad ~~ \mathcal{O} = \mathbb{R} \\ & s' = s+a \quad \quad o \sim \mathcal{N}(s, s-10) \\ & \mathcal{A} = \{-10, -1, 0, 1, 10\} \\ & R(s, a) = \begin{cases} 100 & \text{ if } a = 0, s = 0 \\ -100 & \text{ if } a = 0, s \neq 0 \\ -1 & \text{ otherwise} \end{cases} & \\ \end{aligned}

State

Timestep

Accurate Observations

Goal: \(a=0\) at \(s=0\)

Optimal Policy

Localize

\(a=0\)

QMDP

\[Q_{MDP}(b, a) = \sum_{s \in \mathcal{S}} Q_{MDP}(s,a) b(s) \geq Q^*(b,a)\]

Equivalent to assuming full observability on the next step

Will not take costly exploratory actions

$$Q_\pi(s,a) = R(s, a) + \gamma^t E\left[V_\pi (s')\right]$$

Belief Trajectories

QMDP Predicted Trajectories

 

[  ] An infinite number of child nodes must be visited

[  ] Each node must be visited an infinite number of times

Solving continuous POMDPs - POMCP fails

[1] Adrien Coutoux, Jean-Baptiste Hoock, Nataliya Sokolovska, Olivier Teytaud, Nicolas Bonnard. Continuous Upper Confidence Trees. LION’11: Proceedings of the 5th International Conference on Learning and Intelligent OptimizatioN, Jan 2011, Italy. pp.TBA. <hal-00542673v2>

POMCP

Limit number of children to

\[k N^\alpha\]

Necessary Conditions for Consistency [1]

 

POMCP

POMCP-DPW

POMCP-DPW converges to QMDP

Proof Outline:

  1. Observation space is continuous → observations unique w.p. 1.


  2. (1) → One state particle in each belief, so each belief is merely an alias for that state


  3. (2) → POMCP-DPW = MCTS-DPW applied to fully observable MDP + root belief state


  4. Solving this MDP is equivalent to finding the QMDP solution → POMCP-DPW converges to QMDP



  5.  

Sunberg, Z. N. and Kochenderfer, M. J. "Online Algorithms for POMDPs with Continuous State, Action, and Observation Spaces", ICAPS (2018)

POMCP-DPW

 

[  ] An infinite number of child nodes must be visited

[  ] Each node must be visited an infinite number of times

[  ] An infinite number of particles must be added to each belief node

Necessary Conditions for Consistency

 

Use \(Z\) to insert weighted particles

POMCP

POMCP-DPW

POMCPOW

Ye, Nan, et al. "DESPOT: Online POMDP planning with regularization." Journal of Artificial Intelligence Research 58 (2017): 231-266.

Next Step: Planning on Weighted Scenarios

image/svg+xml

Introduction

Lane Changing with Internal States

Solving Continuous POMDPs

POMDPs.jl

Future

POMDPs.jl - An interface for defining and solving MDPs and POMDPs in Julia

Challenges for POMDP Software

  1. POMDPs are computationally difficult.
  2. There is a huge variety of
    • Problems
      • Continuous/Discrete
      • Fully/Partially Observable
      • Generative/Explicit
      • Simple/Complex
    • Solvers
      • Online/Offline
      • Alpha Vector/Graph/Tree
      • Exact/Approximate
    • Domain-specific heuristics

Julia - Speed

Challenges for POMDP Software

  1. POMDPs are computationally difficult.
  2. There is a huge variety of
    • Problems
      • Continuous/Discrete
      • Fully/Partially Observable
      • Generative/Explicit
      • Simple/Complex
    • Solvers
      • Online/Offline
      • Alpha Vector/Graph/Tree
      • Exact/Approximate
    • Domain-specific heuristics

Explicit

Generative

\(s,a\)

\(s', o, r\)

Previous C++ framework: APPL

"At the moment, the three packages are independent. Maybe one day they will be merged in a single coherent framework."

image/svg+xml

Introduction

Lane Changing with Internal States

Solving Continuous POMDPs

POMDPs.jl

Future

Future Research

Deploying autonomous agents with Confidence

Practical Safety Gaurantees

Trusting Visual Sensors

Algorithms for Physical Problems

Testing on Physical Vehicles

TODO: MAKE ICON

TODO: MAKE ICON

Trusting Information from Visual Sensors

Environment

Belief State

Convolutional Neural Network

Control System

Architecture for Safety Assurance

Algorithms for the Physical World

Weaknesses of the algorithms I have researched:

1. Data-driven models on modern parallel hardware

2. Multidimensional continuous action spaces

image/svg+xml

CPU Image By Eric Gaba, Wikimedia Commons user Sting, CC BY-SA 4.0, https://commons.wikimedia.org/w/index.php?curid=68125990

Practical Safety Guarantees

  • Reachability-based guarantees are usually infeasible
  • Probabilistic guarantees involve low-probability distribution tails

Testing with Physical Vehicles

Texas A&M (HUSL):

TREX 600 RC Helicopter

Stanford (VAIL)

Colorado (RECUV)

Autorotation

"Expert System" Autorotation Controller

Example Controller: Flare

 

\[\alpha \equiv \frac{KE_{\text{available}} - KE_{\text{flare exit}}}{KE_{\text{flare entry}} - KE_{\text{flare exit}}}\]

\[TTLE = TTLE_{max} \times \alpha\]

\[\ddot{h}_{des} = -\frac{2}{TTI_F^2}h - \frac{2}{TTI_F}\dot{h}\]

Bad because:

  • No optimality justification
  • Controller must be adjusted by hand to accommodate any changes
  • Difficult to communicate about
  • Parameters not intuitive

Autorotation Simulation Results

References

image/svg+xml

Acknowledgements

The content of my research reflects my opinions and conclusions, and is not necessarily endorsed by my funding organizations.

TODO: HSL PICTURE

Thank You!

Convex Optimization

Boyd, Stephen, and Lieven Vandenberghe. Convex optimization. Cambridge university press, 2004.

Convexity <=> Exact Solution Tractable

all \(f\) convex

Energy Kites

  • Need safety guarantees
  • Every bit of efficiency is extra energy
  • Complex and partially observable dynamics
    • Aircraft dynamics
    • Tether state
    • Wind

Solving MDPs and POMDPs - Offline vs Online

ONLINE

OFFLINE

Value Iteration

Sequential Decision Trees

Solving MDPs and POMDPs - The Value Function

$$\mathop{\text{maximize}} V_\pi(s) = E\left[\sum_{t=0}^{\infty} \gamma^t R(s_t, a_t) \bigm| s_0 = s, a_t = \pi(s_t) \right]$$

$$V^*(s) = \max \left\{R(s, a) + \gamma E\Big[V^*\left(s_{t+1}\right) \mid s_t=s, a_t=a\Big]\right\}$$

\pi: \mathcal{S} \to \mathcal{A}
π:SA\pi: \mathcal{S} \to \mathcal{A}

Involves all future time

Involves only \(t\) and \(t+1\)

\(a \in \mathcal{A}\)

Handle continuous state space via \(V(s) \approx \tilde{V}(s; \theta) \)

\(45^\circ\)

\(-45^\circ\)

\(\phi=0^\circ\)

Sunberg, Zachary N., Mykel J. Kochenderfer, and Marco Pavone. "Optimized and trusted collision avoidance for unmanned aerial vehicles using approximate dynamic programming." Robotics and Automation (ICRA), 2016 IEEE International Conference on. IEEE, 2016.

Value Function

Policy

A better way: MPC Value Iteration

\(R\) quasi-convex (e.g. \(x_t \in \) safe landing)

Feasible sets convex

Job Talk

By Zachary Sunberg

Job Talk

  • 12
Loading comments...

More from Zachary Sunberg