Notation and Units

This page defines the symbols and units used in the whole book. Each chapter follows it, and the writing guide refers to it. When a chapter needs a symbol that is not here, add it to this page in the same pull request.

General rules

• Scalars are italic (\(v\), \(\delta\)), vectors are bold lowercase (\(\mathbf{x}\), \(\mathbf{u}\)), and matrices are uppercase (\(P\), \(F\)).

• The time step index is a subscript: \(\mathbf{x}_t\) is the state at step \(t\), and \(\Delta t\) is the time between two steps.

• An estimate has a hat (\(\hat{\mathbf{x}}_t\)), and a prediction before the observation is used has a bar (\(\bar{\mathbf{x}}_t\)).

• \(^\top\) is the transpose, and \(I\) is the identity matrix.

• Angles in equations are in radians, and angle differences are wrapped into \([-\pi, \pi)\).

Vehicle state and inputs

Symbol	Meaning	Unit	In the code
\(x, y\)	Position of the vehicle (rear axle center) in the global frame	m	x_m, y_m
\(\theta\)	Yaw (heading) angle, counterclockwise from the global x axis	rad	yaw_rad
\(v\)	Speed	m/s	speed_mps
\(\mathbf{x}_t = [x, y, \theta, v]^\top\)	State vector	-	State, state arrays (4x1)
\(a\)	Acceleration	m/s²	accel_mps2
\(\omega\)	Yaw rate	rad/s	yaw_rate_rps
\(\delta\)	Steering angle of the front wheels	rad	steer_rad
\(\mathbf{u}_t = [a, \omega]^\top\)	Input vector of the motion model	-	input arrays (2x1)
\(L\)	Wheelbase, the distance between the front and rear axles	m	wheel_base_m
\(\Delta t\)	Time step	s	time_s

Estimation

Symbol	Meaning
\(\mathbf{z}_t\)	Observation, for example a GNSS position \([x, y]^\top\)
\(f(\mathbf{x}, \mathbf{u})\)	Motion model: \(\mathbf{x}_t = f(\mathbf{x}_{t-1}, \mathbf{u}_t)\)
\(h(\mathbf{x})\)	Observation model: \(\mathbf{z}_t = h(\mathbf{x}_t)\)
\(\hat{\mathbf{x}}_t\), \(P_t\)	Estimated state and its covariance
\(\bar{\mathbf{x}}_t\), \(\bar{P}_t\)	Predicted state and its covariance, before the observation is used
\(F = \partial f / \partial \mathbf{x}\)	Jacobian of the motion model with respect to the state
\(G = \partial f / \partial \mathbf{u}\)	Jacobian of the motion model with respect to the input
\(H = \partial h / \partial \mathbf{x}\)	Jacobian of the observation model
\(Q\)	Covariance of the input (process) noise
\(R\)	Covariance of the observation noise
\(S\)	Covariance of the innovation \(\mathbf{z}_t - h(\bar{\mathbf{x}}_t)\)
\(K\)	Kalman gain

With these symbols, the EKF prediction is \(\bar{P}_t = F P_{t-1} F^\top + G Q G^\top\), and the innovation covariance is \(S = H \bar{P}_t H^\top + R\).

Sensors

Symbol	Meaning	Unit	In the code
\(r\), \(\phi\)	Range and bearing of a LiDAR return in the sensor frame	m, rad	distance_m, angle_rad
\(\Delta\phi\)	Angle between two LiDAR beams	rad	reso_deg (in degrees)
\(r_{\min}\), \(r_{\max}\)	Shortest and longest range the LiDAR measures	m	min_m, max_m
\(\sigma_\phi\)	Standard deviation of the bearing noise	rad	angle_std_scale
\(c_r\)	Standard deviation of the range noise per meter of range	-	dist_std_rate
\(l_x\), \(l_y\), \(\theta_s\)	Mounting position and yaw of the sensor in the vehicle frame	m, rad	lon_m, lat_m, yaw_deg (in degrees)
\(T(x, y, \theta)\)	\(3 \times 3\) homogeneous transformation matrix of a pose	-	hom_mat_33()
\({}^{A}T_{B}\)	Pose of frame \(\{B\}\) in frame \(\{A\}\), as a homogeneous transformation matrix	-	-

Path tracking

Symbol	Meaning	Unit
\(e\)	Lateral error: signed distance from the reference course to the tracked point, positive when the point is on the left of the course	m
\(\theta_e\)	Heading error: \(\theta - \theta_r\), the vehicle’s yaw minus the course’s yaw at the target point, wrapped into \([-\pi, \pi)\)	rad
\(\theta_r\), \(v_r\)	Yaw and speed of the reference course at the target point	rad, m/s
\(\kappa\)	Curvature of the course	1/m
\(k\)	Gain of a controller (each chapter states which one)	-

Optimal control

Symbol	Meaning	In the code
\(N\)	Horizon: number of steps that a plan looks ahead	horizon_step_T
\(k\)	Step inside the horizon, \(k = 0\) is the current state	-
\(\mathbf{u}_k = [\delta_k, a_k]^\top\)	Input of the plan at step \(k\) (steering angle and acceleration)	steer, accel
\(J\)	Cost of a plan	-
\(W\), \(W_N\)	Diagonal weights of the errors at each step and at the end of the horizon. Many books write them as \(Q\) and \(R\), which this book uses for noise	stage_cost_weight, terminal_cost_weight
\(W_\Delta\)	Diagonal weights of the change of the input between steps	smoothness_cost_weight
\(M\), \(i\)	Number of samples of MPPI, and the index of a sample	number_of_samples_K
\(\boldsymbol{\varepsilon}_{i,k}\), \(\Sigma\)	Noise added to a sample, and its covariance	epsilon, Sigma
\(\lambda\)	Temperature of the MPPI weights	param_lambda

Frames

Frame	Origin and axes
Global \(\{G\}\)	Fixed to the world. x to the right and y up in the plots, angles counterclockwise from x
Vehicle \(\{V\}\)	Rear axle center. x forward, y to the left
Sensor \(\{S\}\)	Mounting point of the sensor. x along the sensor’s forward direction
Obstacle \(\{O\}\)	Center of an obstacle. x along its length

A point \(\mathbf{p}\) written in frame \(\{A\}\) is \({}^{A}\mathbf{p}\). The 2D rotation matrix is written \(\mathrm{Rot}(\theta)\), not \(R\), so it is not mixed up with the observation noise covariance:

\[\begin{split} \mathrm{Rot}(\theta) = \begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix}, \qquad {}^{G}\mathbf{p} = \mathrm{Rot}(\theta)\, {}^{V}\mathbf{p} + [x, y]^\top \end{split}\]

Units in the code

Equations use SI units. In the code, the unit is part of the variable name, so a reader can check the units of an equation from the code:

Suffix	Unit	Example
_m	m	wheel_base_m
_rad	rad	yaw_rad
_deg	degree (only for parameters that people set by hand, converted to radians before use)	yaw_rate_noise in degrees
_mps	m/s	speed_mps
_mps2	m/s²	accel_mps2
_rps	rad/s	yaw_rate_rps
_s	s (_sec in older code)	time_s

Mapping to the main references

The references use other symbols for the same quantities. The tables below map them, so readers can follow a derivation in a reference and come back to the book.

Probabilistic Robotics (S. Thrun, W. Burgard and D. Fox, MIT Press, 2005), EKF localization:

This book	Probabilistic Robotics	Note
\(\hat{\mathbf{x}}_t\), \(P_t\)	\(\mu_t\), \(\Sigma_t\)	Mean and covariance of the belief
\(\bar{\mathbf{x}}_t\), \(\bar{P}_t\)	\(\bar{\mu}_t\), \(\bar{\Sigma}_t\)
\(F\)	\(G_t\)	Jacobian with respect to the state
\(G\)	\(V_t\)	Jacobian with respect to the control
\(Q\)	\(M_t\)	Control (input) noise covariance
\(G Q G^\top\)	\(R_t\)	Motion noise covariance in the state space
\(R\)	\(Q_t\)	Measurement noise covariance

This book keeps \(Q\) for the input noise and \(R\) for the observation noise, as in the code and in most Kalman filter tutorials. Probabilistic Robotics uses \(R_t\) and \(Q_t\) the other way around. The code is not renamed.

Vehicle Dynamics and Control (R. Rajamani, 2nd ed., Springer, 2012), kinematic bicycle model and lateral control:

This book	Rajamani	Note
\(\theta\)	\(\psi\)	Yaw angle
\(L\)	\(l_f + l_r\)	Wheelbase
\(e\), \(\theta_e\)	\(e_1\), \(e_2\)	Lateral and heading errors

Automatic Steering Methods for Autonomous Automobile Path Tracking (J. M. Snider, CMU-RI-TR-09-08, 2009), Stanley controller:

This book	Snider	Note
\(e\)	\(e_{fa}\)	Lateral error at the front axle
\(\theta_e\)	\(\theta_e\)	Heading error. The sign convention differs, see the Stanley section