Online Update of Safety Assurances Using Confidence-Based Predictions Kensuke Nakamura1and Somil Bansal2 Abstract Robots such as autonomous vehicles and assistive

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Online Update of Safety Assurances Using Conﬁdence-Based Predictions

Kensuke Nakamura1and Somil Bansal2

Abstract— Robots such as autonomous vehicles and assistive

manipulators are increasingly operating in dynamic environ-

ments and close physical proximity to people. In such scenarios,

the robot can leverage a human motion predictor to predict

their future states and plan safe and efﬁcient trajectories.

However, no model is ever perfect – when the observed

human behavior deviates from the model predictions, the robot

might plan unsafe maneuvers. Recent works have explored

maintaining a conﬁdence parameter in the human model to

overcome this challenge, wherein the predicted human actions

are tempered online based on the likelihood of the observed

human action under the prediction model. This has opened

up a new research challenge, i.e., how to compute the future

human states online as the conﬁdence parameter changes? In this

work, we propose a Hamilton-Jacobi (HJ) reachability-based

approach to overcome this challenge. Treating the conﬁdence

parameter as a virtual state in the system, we compute

a parameter-conditioned forward reachable tube (FRT) that

provides the future human states as a function of the conﬁdence

parameter. Online, as the conﬁdence parameter changes, we

can simply query the corresponding FRT, and use it to update

the robot plan. Computing parameter-conditioned FRT corre-

sponds to an (ofﬂine) high-dimensional reachability problem,

which we solve by leveraging recent advances in data-driven

reachability analysis. Overall, our framework enables online

maintenance and updates of safety assurances in human-robot

interaction scenarios, even when the human prediction model

is incorrect. We demonstrate our approach in several safety-

critical autonomous driving scenarios, involving a state-of-the-

art deep learning-based prediction model.

I. INTRODUCTION

When a robot operates in close proximity to humans, it

often employs a model of the human’s behavior to predict her

future actions, and subsequently, her future states. Given the

possible future states of the human, the robot may leverage

online motion planning methods to plan around these moving

obstacles, and generate real-time dynamically feasible and

safe trajectories. However, when the predictions deviate from

the true human behavior, the robot might conﬁdently plan

unsafe maneuvers and violate critical safety constraints.

To overcome this challenge, recent works have proposed

estimating a conﬁdence parameter in the human model based

on how well the predictions align with the observed human

behavior [1]. The conﬁdence parameter is then used to

dilate the possible future actions of the human, essentially

predicting every possible human action when the conﬁdence

is low. Even though promising, this approach is hard to scale

for many human-robot applications because an update in the

1Author is with the MAE Department at Princeton University;

k.nakamura@princeton.edu.2Author is with the ECE department at

USC; somilban@usc.edu. Project website: https://kensukenk.

github.io/OnlineConfidenceUpdate/

This research is supported in part by the NVIDIA Academic Hardware Grant

Program and the USC SURE Program.

Fig. 1: Top left: The FRT, without using conﬁdence esti-

mation (purple), fails to alert the ego (red) vehicle that a

collision is possible. Bottom left: The ego vehicle learns

the human’s true intent (i.e., making a U-turn) too late and

crashes into the human (blue) car. Top right: The FRT of the

human vehicle with conﬁdence estimation (cyan). Bottom

right: The use of conﬁdence estimation allows the ego

vehicle to detect a potential collision 2 seconds earlier than

without conﬁdence estimation, and safely stop before the

stop line. This allows the human to complete the maneuver

without any safety violations.

conﬁdence parameter requires recomputing the future human

states online, which can be computationally demanding even

for common, nonlinear dynamical systems.

In this work, we cast the computation of future human

states as a Hamilton-Jacobi (HJ) reachability problem [2]–[4]

and leverage recent advances in high-dimensional reachabil-

ity analysis to quickly update human state predictions online.

Speciﬁcally, in reachability analysis, the future human states

can be obtained by computing the forward reachable tube

(FRT) of the human – the set of all states that the human

can reach starting from its current state under predicted

control actions. Our key idea to update the FRT online is

to compute a parameter-conditioned FRT [5] of the human,

wherein a family of the FRT is computed and a member

of this family can be obtained by specifying the conﬁdence

parameter value. Thus, the FRT can be updated online with

a simple query of the parameterized family, corresponding

to the current conﬁdence estimate.

The parameterized FRT can be obtained by adding the

conﬁdence parameter (and other environment factors that

might change online) as “virtual states” to the human dy-

namics and computing the FRT for the augmented system

using the standard reachability tools. This, however, results

in a high-dimensional reachability problem, especially for

high-capacity predictive models that leverage semantic envi-

ronment factors for accurate predictions, which now become

additional parameters in the model that are only known on-

arXiv:2210.01199v4 [cs.RO] 5 Jun 2023

line. To overcome this challenge, we leverage DeepReach [6]

– a reachability toolbox that builds upon recent advances in

neural partial differential equation (PDE) solvers to compute

high-dimensional reachable sets. DeepReach along with a

parameterized FRT allows us to ensure safe human-robot

interaction despite erroneous predictions.

To summarize, the key contributions of this work are: (1)

incorporating conﬁdence estimation in high-capacity human

prediction models, e.g., models based on deep neural net-

works. The proposed framework allows us to exploit the

predictive power of these models to plan efﬁcient robot

trajectories, yet ensure safety when the predictions cannot be

trusted; (2) developing a Hamilton-Jacobi reachability frame-

work to update the model conﬁdence and the corresponding

safety assurances online for safe human-robot interaction.

II. RELATED WORK

Human Modeling and Prediction. It is a common view-

point that humans are rational agents, that is, that humans

act with intent. A common model used in human-robot inter-

action domains is the Boltzmann model, which captures the

notion that humans are exponentially more likely to choose

actions that maximize some reward function [1], [7], [8].

However, reward functions that incorrectly specify human

intent can lead to overly conﬁdent incorrect predictions.

Furthermore, the reward functions used to model the human’s

goals often fail to capture semantic information that impacts

human decision-making. Speciﬁcally, in contexts such as

autonomous driving, semantic information like stop signs or

crosswalks shape how humans make decisions. One way to

leverage semantic information and make predictions in the

continuous action space is to use a neural network-based

human model. These models have enabled inference and

planning around human arm motion [9], [10], navigation

[11], [12], and autonomous driving [13]–[15] (see [16] for

a survey). However, data-driven approaches are in general

subject to incorrect predictions in scenarios not captured in

the training data. In this work, our goal is to ensure safe

human-robot interaction despite erroneous predictions.

Safe Motion Planning. The notion of safety in the context of

human-robot interaction is well studied [17], [18]. Works in

[7], [19] use backward reachability to ﬁnd the set of unsafe

states and utilize them within a model predictive control

framework to plan efﬁcient trajectories. Although some of

these works [7] track human model conﬁdence, the safety-

enforcing backward reachable set is typically computed for

a ﬁxed set of parameters. [1], [20]–[22] add ﬂexibility by

precomputing a small discrete bank of reachable sets that

reﬂect different potential beliefs of the human model. The

system switches between these reachable sets based on which

one ﬁts the robots estimate best at runtime. However in

practice, the parameters that affect the model predictions (and

subsequently the unsafe set) are not known a priori and must

be observed/estimated online, such as semantic information

in the environment and the model conﬁdence parameter.

Any precomputed bank of reachable sets will suffer from

being overly conservative in such scenarios. In this work,

we propose a method to update such reachable sets online

in an effective fashion.

III. PROBLEM SETUP

We consider a robot operating in a human occupied

space. We assume that the robot has full knowledge of the

environment, and the robot and human states.

A. Agent Dynamics

We model each agent as a dynamical system, where we

denote the robot and human states as xR∈Rnand xH∈

Rmrespectively. Their individual dynamics and controls are

as follows: ˙

xi=f(xi,ui)i∈[R,H]

We also let ξ(τ;xi,ui(·), t)denote the agent state at time τ

starting at the state xiat time tand applying control ui(·)

over the time horizon [t, τ].

The robot is assumed to have some objective or task, such

as reaching a goal state, that it needs to plan and execute

a trajectory for. While the robot performs its task, it is

imperative for it to never incur any safety violations. We

denote by Cthe set of states the robot should avoid to ensure

safety, e.g., because they imply physical collisions with the

human. In this work, we will compute Cvia evaluating a

forward reachable tube of the human.

Running example: We introduce a running example for

illustration throughout the paper. We consider a scenario

where an autonomous car is interacting with a human-driven

vehicle at a trafﬁc intersection. We model both agents in this

scenario as extended unicycles where ˙x = [ ˙x, ˙y, ˙

θ, ˙v]⊺=

[vcos θ, v sin θ, u1, u2]⊺. The vehicle controls are given by

steering rate and acceleration. The unicycle model is widely

used in the literature for modeling autonomous vehicles [14],

[15]. Given a collision radius of Rcol = 1.5m, we deﬁne C

as the positions of the autonomous vehicle that are within a

distance of Rcol of the human vehicle.

B. Human Prediction Model

In human-robot interaction scenarios, the robot typically

maintains a model of human behavior in order to aid in the

prediction of their future states. In this work, we are par-

ticularly interested in the settings where the human motion

predictors might be high-capacity models that use semantic

information about the environment as an input (e.g., the

roadgraph and trafﬁc light state in the context of autonomous

driving), along with the human states (and possibly their

history) to generate continuous distributions over human

controls. We assume that at each time step t, the robot has

a prediction for each time step over the prediction horizon

[t, t +T]in terms of multivariate Gaussian distribution over

human control actions:

ut:t+T

H∼ N (µt:t+T,Σt:t+T)(1)

Here, µt:t+Tand Σt:t+Tare the vectors and matrices of

appropriate dimensions that represent the mean and covari-

ance for the human control actions from time tto t+T.

Such prediction representations are common in the literature,

especially when the model is data-driven (e.g., [14] and [15]).

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OnlineUpdateofSafetyAssurancesUsingConfidence-BasedPredictionsKensukeNakamura1andSomilBansal2Abstract—Robotssuchasautonomousvehiclesandassistivemanipulatorsareincreasinglyoperatingindynamicenviron-mentsandclosephysicalproximitytopeople.Insuchscenarios,therobotcanleverageahumanmotionpredictortopredic...

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Online Update of Safety Assurances Using Confidence-Based Predictions Kensuke Nakamura1and Somil Bansal2 Abstract Robots such as autonomous vehicles and assistive

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