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Equation 1 · Comparing the Main Approaches to Robotics and Embodied AI

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Q(s,a)←Q(s,a)+α[r+γmax⁡a′Q(s′,a′)−Q(s,a)]Q(s,a) \leftarrow Q(s,a) + \alpha \left[ r + \gamma \max_{a'} Q(s',a') - Q(s,a) \right]

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QQ

Symbol Q

Q is a part of this expression. Its role is fixed by the surrounding article and by the operations shown in the formula.

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ss

Symbol s

s is a part of this expression. Its role is fixed by the surrounding article and by the operations shown in the formula.

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aa

Symbol a

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α\alpha

Symbol α

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rr

Symbol r

r is a part of this expression. Its role is fixed by the surrounding article and by the operations shown in the formula.

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γ\gamma

Symbol gamma

gamma is a part of this expression. Its role is fixed by the surrounding article and by the operations shown in the formula.

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addition

addition

Add the term after the plus sign to the term or group before it.

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subtraction

subtraction

Subtract the following term or group from the preceding one. A leading minus marks a negative quantity.

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subscript

subscript

The lower label selects a particular version, component, or indexed member of the quantity. For example, x₀ and xₜ can be values at different positions.

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What the article says around this equation

A model-free update never references a transition model. A canonical form, temporal-difference learning, updates a value estimate purely from sampled transitions: Q(s,a)←Q(s,a)+α[r+γmax⁡a′Q(s′,a′)−Q(s,a)]Q(s,a) \leftarrow Q(s,a) + \alpha \left[ r + \gamma \max_{a'} Q(s',a') - Q(s,a) \right]. Nothing in this update requires knowing or estimating p(s' ∣\mid s, a) ; it only requires having experienced (s, a, r, s') . A model-based approach instead fits an explicit dynamics model p^θ(s′∣s,a)\hat{p}_\theta(s' \mid s, a) — a “world model” — and plans against it directly:

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