5-7 Minutes About Online Planning in POMDPs and POSGs

Assistant Professor Zachary Sunberg

University of Colorado Boulder

April 18, 2026

Autonomous Decision and Control Laboratory

cu-adcl.org

Algorithmic Contributions
- Scalable algorithms for partially observable Markov decision processes (POMDPs)
- Motion planning with safety guarantees
- Game theoretic algorithms
Theoretical Contributions
- Particle POMDP approximation bounds
Applications
- Space Domain Awareness
- Autonomous Driving
- Autonomous Aerial Scientific Missions
- Search and Rescue
- Space Exploration
- Ecology
Open Source Software
- POMDPs.jl Julia ecosystem

PI: Prof. Zachary Sunberg

PhD Students

What do people think of when someone says "POMDP"?

Intractable

\([x, y, z,\;\; \phi, \theta, \psi,\;\; u, v, w,\;\; p,q,r]\)

\(\mathcal{S} = \mathbb{R}^{12}\)

\(\mathcal{S} = \mathbb{R}^{12} \times \mathbb{R}^\infty\)

(Offline) Reinforcement Learning | Online Planning

Online POMDP Planning

Environment

\(Q(b, a)\)

Belief \(b\)

Belief Updater

Planner

State \(s\)

Observation \(o\)

Online Planning Complements RL

Composition: Components learned offline can be combined online
Adaptation: The model used in an online planner can update without complete re-optimization
Combination with Offline Optimization: Online planners can be used as improvement operators in offline optimization
Explainability (?): At least all of the data used to make a decision is right there

[Deglurkar, Lim, Sunberg, & Tomlin, 2023]

Why is POMDP planning difficult?

Curse of History
Curse of dimensionality
1. State space
2. Observation space
3. Action space

Tree size: \(O\left(\left(|A||O|\right)^D\right)\)

Particle Filter POMDP Approximation

PF Approximation Accuracy

\[|Q_{\mathbf{P}}^*(b,a) - Q_{\mathbf{M}_{\mathbf{P}}}^*(\bar{b},a)| \leq \epsilon \quad \text{w.p. } 1-\delta\]

For any \(\epsilon>0\) and \(\delta>0\), if \(C\) (number of particles) is high enough,

[Lim, Becker, Kochenderfer, Tomlin, & Sunberg, JAIR 2023]

No direct dependence on \(|\mathcal{S}|\) or \(|\mathcal{O}|\)!

Why are POMDPs difficult?

Curse of History
Curse of dimensionality
1. State space
2. Observation space
3. Action space

[Lim, et al., 2023, JAIR]

Example: POMDP Tracking in Crowds

State:

Vehicle physical state
Human physical state
Human intention
Target location

[Gupta, Aladum et al., IROS 2026]

Two Caveats

Number of particles is too big for practical safety guarantees.
Depends on the Renyi Divergence \(d_\infty(\mathcal{P} \Vert \mathcal{Q})\), which often grows in high-dimensional problems.

Possible Solution to 2:

Rao-Blackwellized POMDP Planning

\(p(s_t \mid h_t)\)

\(= p(s^a_t \mid s^p_t, h_t) \, p(s^p_t \mid h_t)\)

Analytical update (e.g. Kalman Filter)

Particle Filter

[Lee, Wray, Sunberg, Ahmed, ICRA 2026]

Marginal

Observation-Conditioned

Multiple agents: How to be unpredictable

What is the best evader policy?

What is the best defender policy?

Depends on the best attacker policy?

Where has this been solved before?

Image: Russel & Norvig, AI, a modern approach

P1: A

P1: K

P2: A

P2: K

Conditional Distribution Info-Set Trees

[Becker & Sunberg, AAMAS 2025 Short Paper]

Conditional Distribution Info-Set Trees

[Becker & Sunberg, AAMAS 2025 Short Paper]

Open Source Software!

[Krusniak et al. AAMAS 2026]

Decisions.jl

Arbitrary Dynamic Decision Networks

POMDPs.jl

using POMDPs, QuickPOMDPs, POMDPTools, QMDP

m = QuickPOMDP(
    states = ["left", "right"],
    actions = ["left", "right", "listen"],
    observations = ["left", "right"],
    initialstate = Uniform(["left", "right"]),
    discount = 0.95,

    transition = function (s, a)
        if a == "listen"
            return Deterministic(s)
        else # a door is opened
            return Uniform(["left", "right"]) # reset
        end
    end,

    observation = function (s, a, sp)
        if a == "listen"
            if sp == "left"
                return SparseCat(["left", "right"], [0.85, 0.15])
            else
                return SparseCat(["right", "left"], [0.85, 0.15])
            end
        else
            return Uniform(["left", "right"])
        end
    end,

    reward = function (s, a)
        if a == "listen"
            return -1.0
        elseif s == a # the tiger was found
            return -100.0
        else # the tiger was escaped
            return 10.0
        end
    end
)

solver = QMDPSolver()
policy = solve(solver, m)

Thank You!

www.cu-adcl.org

Funding orgs: (all opinions are my own)

Part V: Open Source Research Software

Good Examples

Open AI Gym interface
OMPL
ROS

Challenges for POMDP Software

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
POMDPs are computationally difficult.

Explicit

Black Box

("Generative" in POMDP lit.)

\(s,a\)

\(s', o, r\)

Previous C++ framework: APPL

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

Open Source Research Software

Performant
Flexible and Composable
Free and Open
Easy for a wide range of people to use (for homework)
Easy for a wide range of people to understand

C++

Python, C++

Python, Matlab

Python, C++

2013

We love [Matlab, Lisp, Python, Ruby, Perl, Mathematica, and C]; they are wonderful and powerful. For the work we do — scientific computing, machine learning, data mining, large-scale linear algebra, distributed and parallel computing — each one is perfect for some aspects of the work and terrible for others. Each one is a trade-off.

We are greedy: we want more.

2012

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

Mountain Car

partially_observable_mountaincar = QuickPOMDP(
    actions = [-1., 0., 1.],
    obstype = Float64,
    discount = 0.95,
    initialstate = ImplicitDistribution(rng -> (-0.2*rand(rng), 0.0)),
    isterminal = s -> s[1] > 0.5,

    gen = function (s, a, rng)        
        x, v = s
        vp = clamp(v + a*0.001 + cos(3*x)*-0.0025, -0.07, 0.07)
        xp = x + vp
        if xp > 0.5
            r = 100.0
        else
            r = -1.0
        end
        return (sp=(xp, vp), r=r)
    end,

    observation = (a, sp) -> Normal(sp[1], 0.15)
)

using POMDPs
using QuickPOMDPs
using POMDPPolicies
using Compose
import Cairo
using POMDPGifs
import POMDPModelTools: Deterministic

mountaincar = QuickMDP(
    function (s, a, rng)        
        x, v = s
        vp = clamp(v + a*0.001 + cos(3*x)*-0.0025, -0.07, 0.07)
        xp = x + vp
        if xp > 0.5
            r = 100.0
        else
            r = -1.0
        end
        return (sp=(xp, vp), r=r)
    end,
    actions = [-1., 0., 1.],
    initialstate = Deterministic((-0.5, 0.0)),
    discount = 0.95,
    isterminal = s -> s[1] > 0.5,

    render = function (step)
        cx = step.s[1]
        cy = 0.45*sin(3*cx)+0.5
        car = (context(), circle(cx, cy+0.035, 0.035), fill("blue"))
        track = (context(), line([(x, 0.45*sin(3*x)+0.5) for x in -1.2:0.01:0.6]), stroke("black"))
        goal = (context(), star(0.5, 1.0, -0.035, 5), fill("gold"), stroke("black"))
        bg = (context(), rectangle(), fill("white"))
        ctx = context(0.7, 0.05, 0.6, 0.9, mirror=Mirror(0, 0, 0.5))
        return compose(context(), (ctx, car, track, goal), bg)
    end
)

energize = FunctionPolicy(s->s[2] < 0.0 ? -1.0 : 1.0)
makegif(mountaincar, energize; filename="out.gif", fps=20)