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Probabilistically Safe Robot Planning with Confidence-Based Human Predictions

Jaime F. Fisac, Andrea Bajcsy, Sylvia L. Herbert, David Fridovich-Keil, Steven Wang, Claire J. Tomlin, Anca D. Dragan

arXiv:1806.00109v1cs.ROcs.LG

TL;DR

Limited sensing and human data constrain online refinement of human models, motivating real-time estimates of model confidence that adapt the robot’s predictions. The paper combines these predictions with safe planning, reporting safer and more efficient trajectories in environments with unmodeled obstacles or human goals.

  • Problem

    Limited sensing and human data constrain how much a human model can be refined online without overfitting.

    Source: §I · p. 1

  • Method

    The robot infers β as confidence in its human-motion model and combines confidence-aware predictions with a probabilistic extension of FaSTrack safety planning.

    Source: §IV · p. 4, §V · p. 5

  • Results

    When human motion is well explained by the model, the three methods perform similarly; with unmodeled obstacles or human goals, β inference yields safer and more efficient trajectories.

    Source: §VII · p. 8

  • Takeaways & Limitations

    Combining worst-case tracking guarantees with probabilistic human predictions yields trajectories that are collision-free at run time with high probability.

    Source: §VII · p. 8

    The setting considers one robot and one human; more complex human motion, closed-loop interaction, and multiple humans remain future work.

    Source: §VII · p. 8

Abstract

from arXiv · show

In order to safely operate around humans, robots can employ predictive models of human motion. Unfortunately, these models cannot capture the full complexity of human behavior and necessarily introduce simplifying assumptions. As a result, predictions may degrade whenever the observed human behavior departs from the assumed structure, which can have negative implications for safety. In this paper, we observe that how "rational" human actions appear under a particular model can be viewed as an indicator of that model's ability to describe the human's current motion. By reasoning about this model confidence in a real-time Bayesian framework, we show that the robot can very quickly modulate its predictions to become more uncertain when the model performs poorly. Building on recent work in provably-safe trajectory planning, we leverage these confidence-aware human motion predictions to generate assured autonomous robot motion. Our new analysis combines worst-case tracking error guarantees for the physical robot with probabilistic time-varying human predictions, yielding a quantitative, probabilistic safety certificate. We demonstrate our approach with a quadcopter navigating around a human.

I. INTRODUCTION

Human-motion models can miss behavior that departs from their assumptions, putting safety at risk. The paper responds with real-time confidence estimation and a safe planner that incorporates uncertain human predictions.

I. INTRODUCTION

  • When human motion diverges from the robot’s model, predictions can become inaccurate and safety may be compromised.Examples include a person heading toward an unknown goal or avoiding an obstacle the robot does not know about.Source: §I · p. 1
  • The robot estimates confidence in its human model in real time and adapts its motion plan, using the Boltzmann model’s rationality coefficient as an indicator of predictive performance.The approach couples confidence-aware predictions with a provably safe planner to make plans conservative when appropriate but efficient when possible.Source: §I · p. 1

A. Human Modeling and Prediction

  • Rational-agent models infer an unknown objective from observed human actions, then predict future movement on the assumption that the person continues to pursue that objective.This paper introduces an online technique for estimating confidence in such a learned model.Source: A. Human Modeling and Prediction · p. 2

B. Safe Robot Motion Planning

  • Path-based planners can be efficient, but may ignore robot dynamics and disturbances, so a robot can deviate from its planned path and collide.FaSTrack instead provides a guaranteed tracking-error margin and an error-feedback controller under bounded external disturbance [11] [8].Source: B. Safe Robot Motion Planning · p. 2

III. PROBLEM STATEMENT AND APPROACH

The problem is to plan a robot’s motion to a preset goal while maintaining safe distance from a human. The section sets out the joint-state collision constraint, illustrates it with a quadcopter, and explains why the robot’s physical dynamics must also be considered.

III. PROBLEM STATEMENT AND APPROACH

  • The setting has one robot moving toward a preset goal alongside one human who expects the robot to avoid her; the quadcopter example illustrates the approach.The theory is presented for a general single-human, single-robot setting.Source: §III · p. 2

A. Motion Model

  • The human and robot are represented by states and controls, and the keep-out set contains joint states that imply collision and must be avoided.The robot must reason about the human’s future motion when constructing its plan.Source: A. Motion Model · p. 2
  • In the example, a quadcopter navigates a room toward a goal while a walking human moves in the plane; the keep-out region also excludes states outside the room bounds.The example uses a 3D position state for the quadcopter and assumes the human walks at about 1 m/s or remains still.Source: A. Motion Model · p. 2

B. Robot Dynamics

  • A simplified planning model may not capture the physical robot’s dynamics, making its planned trajectory difficult to track or infeasible for the real robot.The example’s planner generates nominal position trajectories from a higher-fidelity quadcopter model and projects away velocity; the flight model is for the near-hover regime.Source: B. Robot Dynamics · pp. 2–3

C. Predictive Human Model

The human model predicts actions from a learned or designer-specified utility, while treating β as an indicator of how well that model explains observed motion. The planning problem combines probabilistic human-motion predictions with robot tracking uncertainty to find efficient trajectories below an acceptable collision risk.

C. Predictive Human Model

  • β controls how strongly the modeled utility aligns expected human actions, and is interpreted here as model confidence rather than the person’s actual rationality.The reward parameters θ may be learned from demonstrations or encoded by designers; the human policy assigns higher likelihood to actions with higher expected utility.Source: C. Predictive Human Model · p. 3

D. Probabilistic Safe Motion Planning Problem

  • The human is assumed not to react to the robot, which plans to reach a preset goal while maintaining a safe distance from one human.The paper presents the theory for a general single-human, single-robot setting and illustrates it with a quadcopter navigating around a walking human.Source: C. Predictive Human Model · p. 3, D. Probabilistic Safe Motion Planning… · p. 3
  • The safety target is an acceptable probability of collision, not absolute safety; the demonstrated planner uses a 1% collision-probability threshold.The acceptable threshold is left to system designers or operators because probabilistic predictions can assign nonzero likelihood to many future human states.Source: D. Probabilistic Safe Motion Planning… · p. 3
  • The safety analysis combines worst-case robot-state uncertainty with probabilistic uncertainty in human motion to assess collision risk over time.The planning problem accounts for robot and human states, both uncertain as they evolve.Source: D. Probabilistic Safe Motion Planning… · pp. 3–4
  • The proposed approach first infers model confidence and predicts human motion probabilistically, then extends a safe planner to produce a quantitative probabilistic safety certificate.The robot seeks efficient plans that keep collision probability below the selected threshold.Source: D. Probabilistic Safe Motion Planning… · pp. 3–4

IV. CONFIDENCE-AWARE HUMAN MOTION PREDICTION

Human-motion predictions can degrade when behavior exceeds a model’s predictive power, and delayed recognition can harm safety. The paper treats β as model confidence and updates a Bayesian belief about it from observed human actions to adapt predictions in real time.

IV. CONFIDENCE-AWARE HUMAN MOTION PREDICTION

  • Predictions can perform poorly when human behavior outstrips the model, so the robot needs to detect degradation quickly.The model’s fit may vary across situations and people, motivating corresponding changes in robot conservativeness.Source: §IV · p. 4
  • β indicates the predictive capability of the human model, allowing the robot to adapt predictions and its motion plan to current model reliability.The paper calls these confidence-aware, introspective predictions.Source: §IV · p. 4

A. Real-time Inference of Model Confidence

  • At each time step, the robot uses a newly observed human action as evidence to update its Bayesian belief over β.In practice, it derives the action by inverting the human motion model; the update is designed to run quickly using about 10 β values on a log scale.Source: A. Real-time Inference of Model… · p. 4
  • Figure 2 contrasts predictions at low and high confidence with the Bayesian-confidence predictions for direct goal movement and spill avoidance.The top row depicts movement toward a goal; the bottom row depicts a course change to avoid a spill.Source: Figure 2 · p. 4

B. Human motion prediction

The robot propagates its belief over model confidence to predict human occupancy probabilistically over future time steps. The paper also describes ways to combine confidence inference with online parameter updates, including joint Bayesian inference and a bootstrapped alternative.

B. Human motion prediction

  • The robot combines its belief over β with the human action model and dynamics to recursively predict a distribution of human occupancy at future times.The update applies the modeled action likelihood at each future state.Source: B. Human motion prediction · p. 4
  • The running example shows confidence-aware predictions for direct movement toward a known goal and for a detour around a coffee spill.These cases contrast predictable behavior with a departure from the modeled path.Source: B. Human motion prediction · p. 4
  • When β may change as behavior evolves, a hidden Markov model smooths its belief between measurement updates, downweighting older observations.The time update gradually equalizes the belief over β.Source: B. Human motion prediction · p. 4

C. Integrating Model Confidence into Online Model Updates

  • Confidence inference remains necessary during online model updates; the ideal approach jointly maintains a Bayesian belief over β and utility parameters θ.This can be practical when parameters take values from a finite discrete set, such as distinct driver modes.Source: C. Integrating Model Confidence into… · p. 5
  • When the quadcopter’s known-destination model encounters an unknown destination or other unmodeled behavior, it becomes more cautious as behavior departs from its predictions.Figure 3 illustrates inferred confidence alongside destination reasoning; the robot is confident when motion is explained by the model and uncertain when it is not.Source: C. Integrating Model Confidence into… · p. 5, Figure 3 · p. 5
  • If full Bayesian parameter inference is impractical, the robot can estimate parameters separately and update a bootstrapped belief over β; declining confidence can indicate when re-estimation may help.The running example estimates walking speed using a running average of recent motion-capture measurements.Source: C. Integrating Model Confidence into… · p. 5

V. SAFE PROBABILISTIC PLANNING AND TRACKING

The safety analysis extends FaSTrack’s deterministic-obstacle certificates to probabilistic certificates for uncertain dynamic obstacles such as humans.

V. SAFE PROBABILISTIC PLANNING AND TRACKING

  • The extension combines FaSTrack’s robust safety certificates with probabilistic human predictions to certify plans around uncertain dynamic obstacles.The analysis accounts for the physical robot’s trajectory deviations as well as uncertainty in human motion.Source: §V · p. 5

A. Background: Fast Planning, Safe Tracking

FaSTrack provides a worst-case tracking-error bound that lets a planner verify safety around a simplified robot model. The planner can use this bound without explicitly modeling the robot’s high-fidelity dynamics.

A. Background: Fast Planning, Safe Tracking

B. Robust Tracking, Probabilistic Safety

Safety checking must account for both worst-case robot tracking error and uncertain human motion. The planner therefore evaluates probabilistic collision risk using an expanded collision region and a real-time approximation over future times.

B. Robust Tracking, Probabilistic Safety

  • Probabilistic human predictions require collision checks that return collision probability, so the planner can distinguish sufficiently safe trajectories from unsafe ones.Source: B. Robust Tracking, Probabilistic Safety · p. 6
  • FaSTrack’s tracking-error bound expands the robot’s possible states, and HE(xR) collects human states that could collide with any state in that bound.The collision region can be represented as a rectangle around the quadcopter, where any human position could imply collision.Source: B. Robust Tracking, Probabilistic Safety · p. 6
  • The trajectory-level collision probability is approximated from marginal risks over time because exact evaluation requires the joint distribution over human trajectories.The method uses positive correlation from motion continuity to seek a tight lower bound, then takes the tightest bound across conditioning orders.Source: B. Robust Tracking, Probabilistic Safety · p. 6
  • The approximation uses the highest marginal collision probability along the trajectory, which can be optimistic; continual replanning as predictions update may mitigate underestimated risk.The construction also provides a collision-probability result for a robot with worst-case tracking error E.Source: B. Robust Tracking, Probabilistic Safety · p. 6

C. Safe Online Planning under Uncertain Human Predictions

The online planner checks predicted collision probabilities at future trajectory points and rejects unsafe candidates. It continually replans from updated human predictions while tracking within FaSTrack’s error bound.

C. Safe Online Planning under Uncertain Human Predictions

  • At each future time, the planner computes marginal collision probability and rejects a candidate when that probability exceeds threshold Pth.For search-tree planners, rejecting a node also rejects trajectories extending from it.Source: C. Safe Online Planning under… · p. 6
  • The planner continually regenerates its plan as human predictions update, while a feedback controller keeps the robot within tracking-error bound E.Planning and tracking continue until the robot reaches its goal.Source: C. Safe Online Planning under… · p. 6
  • With inferred model confidence, the robot avoids collisions and replans for greater separation when the human’s motion departs from the model; fixed confidence is either overly conservative or overly aggressive.Fig. 4 contrasts large detours at low β, dangerous proximity at high β, and a slight deviation with inferred confidence.Source: C. Safe Online Planning under… · p. 6, Figure 4 · p. 7

VI. DEMONSTRATION WITH REAL HUMAN TRAJECTORIES

The quadcopter was evaluated on recorded human trajectories across complete and mismatched models. Adaptive β inference maintained safety and improved efficiency in the unmodeled-obstacle and unmodeled-goal scenarios, while producing safe and efficient plans in the complete-model case.

VI. Demonstration with Real Human Trajectories

  • The demonstration used 48 walking trajectories from 16 people across three environments, plus a Crazyflie 2.0 hardware demonstration.Trajectories were measured on the ground plane at roughly 235 Hz; the authors note that a more realistic setting would require alternative state estimation such as lidar.Source: §VI · p. 7
  • The evaluation compared adaptive β inference with fixed β values of 0.05 and 10, using minimum human–quadcopter distance for safety and travel time for efficiency.Simulation replayed recorded human trajectories with near-hover quadcopter dynamics and a FaSTrack controller applied at 100 Hz.Source: §VI · p. 7
  • The reported demonstration used motion-capture state estimates; a more realistic sensing setup would require alternatives such as lidar.Source: §VI · p. 7

Complete Model

  • With a complete model and a known human goal, adaptive β produced safe and efficient plans, as did the high-confidence method.Direct goal-directed motion led the adaptive robot to infer high model confidence; the low-confidence method also performed comparably on both metrics.Source: §VI · p. 7

Unmodeled Obstacle

  • In the unmodeled coffee-spill scenario, β inference almost always maintained a safer distance and reached the goal faster than either fixed-confidence baseline.The paper reports increased minimum distance and faster goal arrival; Fig. 5 summarizes safety and efficiency for the complete and unmodeled-obstacle environments.Source: §VI · pp. 7–8, Figure 5 · p. 8

Unmodeled Goal

  • In the unmodeled-goal scenario, inferred β achieved safety similar to both fixed-β methods and was several seconds faster.The human first follows known goals, then returns to the start in a way that appears irrational to the robot.Source: §VI · p. 8, Figure 6 · p. 8

VII. DISCUSSION & CONCLUSION

The paper treats the human model’s rationality coefficient as an indicator of predictive confidence and combines confidence-aware predictions with probabilistic safety analysis. It reports safer and more efficient trajectories in environments with unmodeled obstacles or human goals, while identifying more complex human motion, closed-loop interaction, and multiple humans as future work.

VII. DISCUSSION & CONCLUSION

  • The method interprets the human model’s rationality coefficient as predictive confidence and combines probabilistic human predictions with worst-case analysis in a high-probability runtime safety certificate.The certificate builds on FaSTrack’s theoretical analysis.Source: §VII · p. 8
  • In environments with unmodeled obstacles or human goals, inferring β yields safer and more efficient trajectories than fixed-β approaches using the same probabilistically safe planner.Fig. 7 presents efficiency results for the unmodeled goal scenario.Source: §VII · p. 8, Figure 7 · p. 8
  • Future work should investigate more complex human motion, closed-loop interaction models, and navigation around multiple humans.Source: §VII · p. 8
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