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title: "Belief iLQR"
source: https://www.jemoka.com/posts/kbhilqr/
---

Motivation Imperfect sensors in robot control: partial observations Manipulators face tradeoff between sensing + acting curse of dimensionality and curse of history.
Belief-Space Planning Perhaps we should plan over all possible distributions of state space, making a belief-state MDP.
But: this is a nonlinear, stochastic dynamic. In fact: there maybe stochastic events that affects dynamics.
Big problem:
dim(belief) &gt;&gt; dim(state) dim(belief) &gt;&gt; dim(action) Belief iLQR &ldquo;determinize and replan&rdquo;: simplify the dynamics at each step, plan, take action, and replan
tracks belief via observations simplifies belief state dynamics based on linear MLE When the dynamics is linear, you can use Linear-Quadratic Regulator to solve. This results in a worse policy but will give you a policy.
Previous Work &ldquo;just solve most-likely state&rdquo;: doesn&rsquo;t take action to explore and understand the state. &ldquo;belief roadmap&rdquo;: not really planning in the belief space itself Approach Belief Update We use Baysian updates for the state probably updates:
\begin{equation} P(s_{t+1}) = \eta P(o_{t+1}|s_{t+1}) \int_{x} p(_{t+1}|x, a_{t}) P(s) \end{equation}
and then the actual beliefs are updated with Extended Kalman Filter.
Importantly, the Extended Kalman Filter usually requires us to take an expectation of each observation O over all O; instead, we assume that the future states are uniform linearly distributed.
Belief Update Cost Ideally, we want to lower covariance of the belief vectors in order to be more confident.
first term: reduce large trajectories (verify) second: stabilization Replanning Strategy while b not at goal: # replan at where we are at now (b, a, mean_b) = create_initial_plan(b); for depth d: a_t = solve_lqr_for_plan_at_time(b, a, mean_b) o = environment.step(a_t) b = extended_kalman(b, a, o) if mean(b) &gt; max_allowed_belief_uncertainty: break
