From 550baa1e4929aa98e69d4da97205809943cce213 Mon Sep 17 00:00:00 2001 From: Atsushi Sakai Date: Sun, 23 Jan 2022 20:07:34 +0900 Subject: [PATCH] fix duplicated math lables (#633) --- .../move_to_a_pose_control.rst | 6 ++-- .../quintic_polynomials_planner.rst | 34 +++++++++---------- 2 files changed, 20 insertions(+), 20 deletions(-) diff --git a/docs/modules/control/move_to_a_pose_control/move_to_a_pose_control.rst b/docs/modules/control/move_to_a_pose_control/move_to_a_pose_control.rst index 8bb40f22..77ec682a 100644 --- a/docs/modules/control/move_to_a_pose_control/move_to_a_pose_control.rst +++ b/docs/modules/control/move_to_a_pose_control/move_to_a_pose_control.rst @@ -68,7 +68,7 @@ The distance :math:`\rho` is used to determine the robot speed. The idea is to s .. math:: v = K_P{_\rho} \times \rho\qquad - :label: eq1 + :label: move_to_a_pose_eq1 Note that for your applications, you need to tune the speed gain, :math:`K_P{_\rho}` to a proper value. @@ -92,9 +92,9 @@ The final angular speed command is given by .. math:: \omega = K_P{_\alpha} \alpha - K_P{_\beta} \beta\qquad - :label: eq2 + :label: move_to_a_pose_eq2 -The linear and angular speeds (Equations :eq:`eq1` and :eq:`eq2`) are the output of the algorithm. +The linear and angular speeds (Equations :eq:`move_to_a_pose_eq1` and :eq:`move_to_a_pose_eq2`) are the output of the algorithm. Move to a Pose Robot (Class) ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ diff --git a/docs/modules/path_planning/quintic_polynomials_planner/quintic_polynomials_planner.rst b/docs/modules/path_planning/quintic_polynomials_planner/quintic_polynomials_planner.rst index 2bced070..feec345b 100644 --- a/docs/modules/path_planning/quintic_polynomials_planner/quintic_polynomials_planner.rst +++ b/docs/modules/path_planning/quintic_polynomials_planner/quintic_polynomials_planner.rst @@ -18,7 +18,7 @@ We assume a one-dimensional robot motion :math:`x(t)` at time :math:`t` is formulated as a quintic polynomials based on time as follows: .. math:: x(t) = a_0+a_1t+a_2t^2+a_3t^3+a_4t^4+a_5t^5 - :label: eq1 + :label: quintic_eq1 :math:`a_0, a_1. a_2, a_3, a_4, a_5` are parameters of the quintic polynomial. @@ -31,44 +31,44 @@ End position, velocity, and acceleration are :math:`x_e, v_e, a_e` respectively. So, when time is 0. .. math:: x(0) = a_0 = x_s - :label: eq2 + :label: quintic_eq2 -Then, differentiating the equation :eq:`eq1` with t, +Then, differentiating the equation :eq:`quintic_eq1` with t, .. math:: x'(t) = a_1+2a_2t+3a_3t^2+4a_4t^3+5a_5t^4 - :label: eq3 + :label: quintic_eq3 So, when time is 0, .. math:: x'(0) = a_1 = v_s - :label: eq4 + :label: quintic_eq4 -Then, differentiating the equation :eq:`eq3` with t again, +Then, differentiating the equation :eq:`quintic_eq3` with t again, .. math:: x''(t) = 2a_2+6a_3t+12a_4t^2 - :label: eq5 + :label: quintic_eq5 So, when time is 0, .. math:: x''(0) = 2a_2 = a_s - :label: eq6 + :label: quintic_eq6 -so, we can calculate :math:`a_0, a_1, a_2` with eq. :eq:`eq2`, :eq:`eq4`, :eq:`eq6` and boundary conditions. +so, we can calculate :math:`a_0, a_1, a_2` with eq. :eq:`quintic_eq2`, :eq:`quintic_eq4`, :eq:`quintic_eq6` and boundary conditions. -:math:`a_3, a_4, a_5` are still unknown in eq :eq:`eq1`. +:math:`a_3, a_4, a_5` are still unknown in eq :eq:`quintic_eq1`. -We assume that the end time for a maneuver is :math:`T`, we can get these equations from eq :eq:`eq1`, :eq:`eq3`, :eq:`eq5`: +We assume that the end time for a maneuver is :math:`T`, we can get these equations from eq :eq:`quintic_eq1`, :eq:`quintic_eq3`, :eq:`quintic_eq5`: .. math:: x(T)=a_0+a_1T+a_2T^2+a_3T^3+a_4T^4+a_5T^5=x_e - :label: eq7 + :label: quintic_eq7 .. math:: x'(T)=a_1+2a_2T+3a_3T^2+4a_4T^3+5a_5T^4=v_e - :label: eq8 + :label: quintic_eq8 .. math:: x''(T)=2a_2+6a_3T+12a_4T^2+20a_5T^3=a_e - :label: eq9 + :label: quintic_eq9 -From eq :eq:`eq7`, :eq:`eq8`, :eq:`eq9`, we can calculate :math:`a_3, a_4, a_5` to solve the linear equations: :math:`Ax=b` +From eq :eq:`quintic_eq7`, :eq:`quintic_eq8`, :eq:`quintic_eq9`, we can calculate :math:`a_3, a_4, a_5` to solve the linear equations: :math:`Ax=b` .. math:: \begin{bmatrix} T^3 & T^4 & T^5 \\ 3T^2 & 4T^3 & 5T^4 \\ 6T & 12T^2 & 20T^3 \end{bmatrix}\begin{bmatrix} a_3\\ a_4\\ a_5\end{bmatrix}=\begin{bmatrix} x_e-x_s-v_sT-0.5a_sT^2\\ v_e-v_s-a_sT\\ a_e-a_s\end{bmatrix} @@ -80,10 +80,10 @@ Quintic polynomials for two dimensional robot motion (x-y) If you use two quintic polynomials along x axis and y axis, you can plan for two dimensional robot motion in x-y plane. .. math:: x(t) = a_0+a_1t+a_2t^2+a_3t^3+a_4t^4+a_5t^5 - :label: eq10 + :label: quintic_eq10 .. math:: y(t) = b_0+b_1t+b_2t^2+b_3t^3+b_4t^4+b_5t^5 - :label: eq11 + :label: quintic_eq11 It is assumed that terminal states (start and end) are known as boundary conditions.