Kinodynamic Rollouts and Action Shielding for Intention-Aware Planning

 

William Pope

M.S. Thesis Defense

14 Nov 2022

Intention-Aware Planning

  • Environments with other agents pose a challenge for path planning
  • Agents act with intention, but their objectives are not known to others
  • By observing past behavior, we can reason about likely future behavior

Pedestrian Problem

  • Vehicle needs to navigate around pedestrians/obstacles to reach a goal
  • Assumptions:
    • Receives measurements of human positions
    • Dynamics are deterministic, knows its own state
    • Has emergency brake
    • Not-at-fault for collisions if stopped

POMDPs

  • Partially-observable Markov decision process (POMDP)
    • Sequential decision-making framework
    • State contains hidden elements
    • Belief – probability over possible true states, informed by observations
  • For the pedestrian problem:

 

s = \begin{bmatrix} \text{vehicle state} \\ \text{human positions} \\ \text{human intentions} \end{bmatrix}
o = \begin{bmatrix} \text{vehicle state} \\ \text{human positions} \end{bmatrix}
b = \begin{bmatrix} \text{intention probabilities} \end{bmatrix}
a = \begin{bmatrix} \text{vehicle controls} \end{bmatrix}

Online POMDP Planning

  • State/belief space is infinitely large, intractable to solve for global policy
  • Online planning:
    • Search local area only with forward tree search
    • Execute best action, take observation, update belief, repeat search

 

  • POMDP solver is time-constrained by planning frequency

Preceding Work

  • Baseline: POMDP speed control [Bai, H. (2015)]
    • Tree searches 1D space along pre-planned path
    • Rollout: adjust speed along path to avoid humans
  • Improved: POMDP speed/steering control [Gupta, H. (2022)]
    • Tree searches full state space
    • Rollout: adjust speed to avoid humans and find the goal
      • ​Uses multi-query motion planning to guide rollout
Holonomic Nonholonomic
Speed Bai, H. Bai, H.
Speed/Steering Gupta, H.

Planning under Differential Constraints

  • Vehicle trajectories must obey equations of motion for dynamical system

 

  • In nonholonomic systems, state evolution is dependent on past states

Holonomic planning

Nonholonomic planning

\dot{x} = f(x, u, t)

Dynamics Model

  • Bicycle model with velocity (\(v \geq 0\))
  • Inputs: steering, acceleration (discrete set)
x = \begin{bmatrix} x \\ y \\ \theta \\ v \end{bmatrix}
u = \begin{bmatrix} \phi \\ a \end{bmatrix}
\dot{x} = f(x,u) = \begin{bmatrix} \dot{x} \\ \dot{y} \\ \dot{\theta} \\ \dot{v} \end{bmatrix} = \begin{bmatrix} v \cos{\theta} \\ v \sin{\theta} \\ v \frac{1}{l}\tan{\phi} \\ a \end{bmatrix}

steering limited based on velocity

Contributions and Overview

Contributions

  • Extended POMDP planner to nonholonomic vehicles with new rollout policy
  • Improved planner safety around pedestrians with online reachability analysis
  • Demonstrated planner on new autonomous robot

 

Overview

  • Chapter I: Hamilton-Jacobi-Bellman Rollouts
  • Chapter II: Sparse Tree Shielding
  • Chapter III: Implementation/Experiments

Chapter I:

Hamilton-Jacobi-Bellman Rollouts

Tree Search Value Estimates

  • In online POMDP solvers, need to estimate value of each new node
    • Collects information beyond reach of tree
  • Value estimates:
    • Influence macro scale behavior of planner
    • Need to be representative of true POMDP execution/reward

DESPOT Rollout Policies

  • In DESPOT, estimate upper/lower bound on value for each node
    • UB:  simulate path to goal (no humans)
    • LB:  simulate path to goal (with humans, brake to avoid)
  • Requirements:
    • Reach goal near-optimally from any state
    • Avoid pedestrians along path (lower bound)
    • Generate path near instantly
    • Follow vehicle dynamics

Goal: create UB/LB rollout policies that meet requirements

Possible Policies

  • Random rollout
    • Will not find sparse reward in navigation problem
  • Graph-based methods
    • ​Difficult to satisfy differential constraints when connecting nodes
  • Forward tree search
    • Single-query, too slow for online rollouts​
  • Potential field methods
    • Subject to local minima issues, not guaranteed to reach goal

 

  • Need global optimal control policy
    • ​Hamilton-Jacobi-Bellman equation

Hamilton-Jacobi-Bellman Equation

V(x(t),t) = \max_{u}\{\int_{t}^{t_{f}} R(x(\tau),u(\tau)) d\tau\}
V(x(t),t) = \max_{u}\{\int_{t}^{t+dt} R(x(\tau),u(\tau)) d\tau + V(x(t+dt),t+dt)\}
  • Uses dynamic programming to solve Hamilton-Jacobi PDE for optimal control
  • Value: optimal reward-to-go from given state/time
  • Reformulated with dynamic programming
  • With optimal value function, can quickly get optimal control at any state
u^{*}(x(t),t) = \argmax_{u}(Q(x(t),u,t))
0 = \frac{\partial V}{\partial t} + \max_{u}\{\frac{\partial V}{\partial x}\cdot f(x,u) + R(x,u) \}

HJB PDE

Discrete Time HJB

  • For discrete time systems, HJB is equivalent to value iteration
V(s) = \max_{a}\{R(s,a) + V(s')\}
  • To solve over continuous state space, need to discretize into grid of states \((s)\)
  • Next state \((s')\) usually not on grid, so value needs to be interpolated
  • Generally requires discrete action set to optimize

Solving for Value Function

  • Reward function (same as POMDP)
    • if goal: \(R(s) = +1000.0\)
    • if obstacle: \(R(s) = -1e6\)
    • all actions: \(R(a) = -1.0\)

 

  • Semi-Lagrangian PDE method: compute one-step lookahead at every node until values converge
  • Produces \(n\)-dimensional value array
  • Can be discontinuous
  • Expensive due to dimensionality

Planning with Value Function

  • For any state in continuous space, take action that maximizes interpolated Q-value

Faster Planning

  • Rollout has sizable impact on POMDP computation cost
  • Can speed up policy with offline information
    • Exact: interpolates new Q-values for exact continuous state
    • Approx: uses stored Q-values from nearest neighbor in state grid

Reactive-HJB Policy

  • In order to simulate rollout path with humans, apply reactive acceleration based on distance to nearest human in scenario

 

  • Rollout still needs to reach goal, so reactive controller is coupled with HJB
    • RC chooses acceleration
    • HJB chooses steering
    • If RC accel is unsafe, rollout reverts to pure HJB policy

HJB as Pedestrian Problem Rollout

  • Upper bound:
    • Use exact HJB policy to simulate optimal human-free path 
    • Runs once per node
  • Lower bound:
    • Use approximate reactive policy to simulate path avoiding humans
    • Runs once per scenario per node

Rollout Policy Comparison

  • Lower bound policies reach goal, regardless of reactive acceleration input
Policy Runtime [μs]
HJB 18.064
Reactive-HJB 6.083
Approx Reactive-HJB 1.915
|A| = 21, |A_{RC}| = 7

POMDP w/ HJB Rollouts

Performance by Rollout Choice

  • No significant difference observed for faster lower bound policy
  • Any improvement in tree size may be negated by looser bound
Lower Bound Policy Average Time-to-Goal [s] Average Node Rate [1/s]
Reactive-HJB 17.9 759.4
Approx Reactive-HJB 16.5 820.0

N = 10

Chapter II:

Sparse Tree Shielding

Pedestrian Collision Avoidance

  • Future human collisions are predicted/penalized in POMDP tree​
  • However,
    • DESPOT explores finite number of observation branches
    • Collision penalty may be washed out by nearby safe nodes
    • Vehicle assumed to have instant emergency brake

Goal: add safety mechanism to handle low probability collisions and finite braking ability

Ye, N., et al

Reach-Avoid Games

  • Players:
    • Vehicle (P1) wants to reach goal
    • Humans and obstacles (P2) "want" to cause collisions

 

  • At a given state:
    • Vehicle wins if there exists a safe path to goal in finite time
    • Obstacles win if all of vehicle's future paths can be blocked

 

  • To ensure vehicle safety, must only visit states with known safe future paths

[Chen, M.]

Human Reachable Set

  • Vehicle path is not safe if it intersects human's reachable set in space/time
  • Approximating human FRS:
    • From observation, propagate human toward all possible goals
    • Define FRS as convex hull of states at each time step

Vehicle Safe Set

  • Vehicle has two known invariant safe sets:
    • In goal
    • In free space with zero velocity (not at fault)

 

  • For any other state to be safe, must have a path to these sets that does not intersect obstacles or human FRS
    • Only need to prove existence of one such path

 

  • More simply: vehicle must be able to safely stop from any visited state

Online Action Shielding

  • DESPOT produces Q-values for all actions at \(t_{k+1}\)
  • Shield filters out actions with human FRS intersections prior to \(t_{stop}\)
  • Agent executes best action from remaining safe action set

POMDP w/ Shield

Performance

Humans in Environment Average Time-to-Goal [s] Interventions / Actions Collisions / Trials
10 (off) 15.4 n/a 0.1
10 (on) 23.0 0.207 0
20 (off) 17.2 n/a 0.2
20 (on) ​31.3 0.280 0
  • Shield is able to prevent collisions, but at significant time cost

N = 10

Chapter III.a:

Experimental Robot

Marmot

  • Built car-like robot to test planners in real-time/real-world environment
    • Runs POMDP planner on-board
    • Collects state data from motion capture system

Hardware

  • Based on F1Tenth open-source design
  • Modified for ADCL uses
    • Larger computer, dual batteries, more sensor mounts

Hardware stack

Intel NUC i5

Software

  • Using ROS1
  • POMDP implemented in Julia (RobotOS.jl)
  • Planner and belief updater in separate nodes
  • Minimal integration effort for new planners

Online decision loop

\(f_{loop} = 2 \text{ Hz}\)

POMDP-ROS Demonstration

Chapter III.b:

BellmanPDEs.jl

BellmanPDEs.jl

  • Created Julia package for HJB solver/planner
  • Intended for standalone planning or POMDP rollouts

github.com/william-pope/BellmanPDEs.jl

Conclusion

  • HJB rollout enables use of planner in broader applications
  • Action shielding provides safety while preserving planner behavior

 

Future work

  • Implement more efficient HJB solvers for larger environments
  • Add disturbances and low level controller to vehicle dynamics 
  • Include more social context in humans/planner

Questions

William Pope

william.pope@colorado.edu

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