Notation · Probabilistic Robotics via Rust

The book follows the notation of Thrun, Burgard and Fox so that readers can move between this text and the literature without translating. Where the field has moved on — chiefly estimation on manifolds — the notation is extended rather than replaced.

The estimation loop

Notation used in this chapter
Symbol	Meaning
$x_t$	State at time t — usually the robot pose, sometimes pose plus map
$u_t$	Control (or odometry reading) applied between t-1 and t
$z_t$	Measurement received at time t
$m$	The map
$x_{0:t}$	The trajectory: every state from 0 through t
$\bel(x_t) = p(x_t \mid z_{1:t}, u_{1:t})$	Belief: the posterior over the state given everything observed so far
$\belbar(x_t)$	Predicted belief, after the control but before the measurement
$\eta$	A normalizer, written generically wherever one is needed
$p(x_t \mid x_{t-1}, u_t)$	Motion model — the probability of landing in a state, given where you were and what you commanded
$p(z_t \mid x_t)$	Measurement model — the probability of a reading, given the state that produced it

Gaussians

Notation used in this chapter
Symbol	Meaning	Note
$\mu_t, \Sigma_t$	Moments form: mean and covariance
$\Omega_t = \Sigma_t^{-1}, \; \xi_t = \Sigma_t^{-1}\mu_t$	Canonical (information) form: information matrix and information vector	Cheap where moments form is expensive, and vice versa — Chapter 6
$R_t$	Motion noise covariance	R for motion, always
$Q_t$	Measurement noise covariance	Q for measurement, always
$G_t$	Jacobian of the motion model with respect to the state
$H_t$	Jacobian of the measurement model with respect to the state
$K_t$	Kalman gain
$S_t$	Innovation covariance

Nonparametric representations

Notation used in this chapter
Symbol	Meaning	Note
$\mathcal{X}_t = \{ x_t^{[i]}, w_t^{[i]} \}_{i=1}^{M}$	A weighted particle set of M hypotheses
$M_{\text{eff}}$	Effective sample size — how many particles are really carrying the belief
$\ell_{t,i}$	Log odds of occupancy for grid cell i	Log odds turns the Bayes update into addition — Chapter 13

Geometry and manifolds

Notation used in this chapter
Symbol	Meaning	Note
$T \in \SEtwo$	A planar pose, as an element of the special Euclidean group
$\exp(\tau), \; \log(T)$	Exponential and logarithm maps between the tangent space and the group
$x \bplus \tau, \; a \bminus b$	Retraction and its inverse: how you add a small increment to a pose, and subtract two poses	The operators that make Gaussian filters work on rotations — Chapters 3 and 7
$\Ad_T$	Adjoint: moves a tangent vector between the left and right conventions
$\alpha_1 \ldots \alpha_6$	Motion model noise parameters
$c_t$	Correspondence variable: which landmark produced which measurement
$f_t^i = (r, \phi, s)\T$	A detected feature: range, bearing, signature

Decision making

Notation used in this chapter
Symbol	Meaning
$\pi$	Policy: a mapping from state (or belief) to action
$r(x, u)$	Immediate reward for taking action u in state x
$\gamma$	Discount factor
$V(x)$	Value function
$b$	Belief state — the POMDP’s notion of what is known

Color

Color carries meaning in this book, consistently across prose, equations, figures, and code comments. A term tinted blue in an equation is the same quantity as the blue curve in the figure beside it.

Prior
Prediction
Measurement
Posterior
Truth