[AMR] Fix errors pointed out to me

2026-09-10 19:15:25 +02:00 · 2026-08-09 08:30:17 +02:00
parent cc5f111813
commit a8b3bd7b20
24 changed files with 50 additions and 47 deletions
@@ -38,7 +38,7 @@ $\begin{bmatrix}
 $\begin{smallmatrix}
        \psi = \arcsin\left( R_{21} \div \sqrt{1 - R_{31}^2} \right)\\
        \theta = \arcsin(-R_{31})\\
-        \varphi = \arcsin\left( R_{31} \div \sqrt{1 - R_{31}^2} \right)\\
+        \varphi = \arcsin\left( R_{32} \div \sqrt{1 - R_{31}^2} \right)\\
    \end{smallmatrix}$
 $[\vec{n}]^\times = \begin{bmatrix}
        0    & -n_3 & n_2  \\
@@ -51,7 +51,7 @@ For pitch axis $\theta = \pm 90\deg$, \bi{Gimbal Lock}, Jacobian \bi{singular}.
 \shortdefinition[Angle-Axis]
 $\vec{\alpha} = \alpha \vec{n}$ ($\vec{n}$ normal); To \bi{rotation matrix}:\\
 $\mat{R}(\alpha, \vec{n}) = \mat{I}_3 + \sin(\alpha)[\vec{n}]^\times + (1 - \cos(\alpha))([\vec{n}]^\times)^2$\\
-To \bi{quaternion}: $\vec{q} = [\vec{n}, \alpha]$
+To \bi{quaternion}: $\vec{q} = [\vec{n} \sin(\alpha \div 2), \cos(\alpha \div 2)]$


 \shortdefinition[Quaternions] $q = q_w + q_x i + q_y j + q_z k$ with\\
@@ -63,7 +63,7 @@ $\vec{q} = \begin{bmatrix}

 \bi{Mult} {\scriptsize
    $\vec{q} \otimes \vec{p} = \begin{bmatrix}
-            a(\vec{q}) \vec{v}(\vec{p}) + a(\vec{p}) + \vec{v}(\vec{q}) \times \vec{v}(\vec{p}) \\
+            a(\vec{q}) \vec{v}(\vec{p}) + a(\vec{p})\vec{v}(\vec{q}) + \vec{v}(\vec{q}) \times \vec{v}(\vec{p}) \\
            a(\vec{q}) a(\vec{p}) - \vec{v}(\vec{q})^\top \vec{v}(\vec{p})
        \end{bmatrix}$
 }
@@ -8,10 +8,10 @@ $ \begin{bmatrix}
        L_1 \sin(\theta_1) + L_2 \sin(\theta_1 + \theta_2)
    \end{bmatrix}$\\
 }
-Workspace $W$: $\theta_1, \theta_2 \in [-\pi, \pi]$.
+Configuration space: $\theta_1, \theta_2 \in [-\pi, \pi]$.
 \bi{Computation} Similar for $n$R sys (more angles, more lengths).
 Last dim is typically sum of angles (or equiv).
-\bi{For velocity kinematics}: See \ref{sec:manipulator-velocity-kinematics}. 
+\bi{For velocity kinematics}: See \ref{sec:manipulator-velocity-kinematics}.
 % Comp. Jacobian of this $nD$ position ($nD$ pos is vec). ${_W}\vec{t}_{WE} \mat{J}(\vec{\theta}) \cdot \vec{\dot{\theta}}$
 % \bi{Jacobian}: see \ref{sec:ms-lin}.

@@ -12,7 +12,7 @@ $\theta_1 = \varphi \pm \alpha$, $\theta_2 = \pm(\pi - \beta)$
 2. 2 sols from 2R for shoulder + elbow\\
 3. Solve for wrist joints (no infl. on pos)

-\bi{For Inv. velocity}: $\vec{\dot{\theta}} = J_S^{-1}(\vec{\theta}) {_W} \vec{t}_{WE}$
+\bi{For Inv. velocity}: $\vec{\dot{\theta}} = J_S^{-1}(\vec{\theta}) {_W} \dot{\vec{t}}_{WE}$

 % \shortdefinition[Denavit-Hartengerb (DH)] Serial chains
 % \[
@@ -11,7 +11,7 @@ With: $\pardiff{t}\vec{x}(t) = f_C(\vec{x}(t), \vec{u}(t))$ the model for the ro
 \hrule
 \vspace{0.5mm}

-\bi{Linearised} at $\vec{f}_C(\vec{\overline{x}}, \vec{\overline{y}}) = 0$, at \bi{equilibrium} {\scriptsize (if $\neq 0$, then add it)}:\\
+\bi{Linearised} at $\vec{f}_C(\vec{\overline{x}}, \vec{\overline{u}}) = 0$, at \bi{equilibrium} {\scriptsize (if $\neq 0$, then add it)}:\\
 $\delta \vec{\dot{x}}(t) = \mat{F}_C \delta \vec{x}(t) + \mat{G}_C \delta \vec{u}(t) + \mat{L}_C \vec{w}(t)$;
 $\delta \vec{z}(t) = \mat{H} \delta \vec{x}(t) + \vec{v}(t)$.

@@ -19,23 +19,17 @@ $\mat{F}_C$ system (often a Jac. / Taylor, vars are $x_i$. If no $x$ in $\mat{F}
 $\mat{G}$ input gain (often Jac / Taylor, vars are $u_i$, if no $u_i$, then lin.), $\vec{w}$ process noise, $\mat{H}$ is measurement, $\vec{v}$ measurement noise,
 both zero-mean \bi{Gauss. White Noise Proc.}. $\vec{u}_k$ inputs at time $k$.

-% TODO: Taylor approximation
-\vspace{0.5mm}
-\hrule
-\vspace{0.5mm}
-
-
 \bi{Discretized}
 $\vec{x}_k = \vec{f}(\vec{x}_{k - 1}, \vec{u}_k, \vec{w}_k)$;
 $\vec{z}_k = \vec{h}(\vec{x}_k) + \vec{v}_k$

 \bi{Linearised}:
 $\delta \vec{x}_k = \mat{F} \delta \vec{x}_{k - 1} + \mat{G}_k \delta \vec{u}_k + \mat{L}_k \vec{w}_k$;
-$\delta \vec{z}_k = \mat{H}_k \delta \vec{x}_k$
+$\delta \vec{z}_k = \mat{H}_k \delta \vec{x}_k + \vec{v}(t)$

 For discretized, iterative integral from $t_{k - 1}$ to $t_k$

-Linearization happens typically with one of the below:
+Numerical integration typically happens with one of the below:

 \shortdefinition[Euler-Forward] $\vec{x}_k = \vec{x}_{k - 1} + \Delta t \vec{f}_C(\vec{x}_{k - 1}, \vec{u}_{k - 1}, t_{k - 1})$

@@ -13,7 +13,7 @@
    \item For left pertubing\\
          $\mat{\dot{R}}_{IB} = [{_I} \omega_{IB}]^\times \mat{R}_{IB}$
    \item For right pertubing
-          $\mat{\dot{R}}_{IB} = \mat{R}_{IB} [{_I} \omega_{IB}]^\times$
+          $\mat{\dot{R}}_{IB} = \mat{R}_{IB} [{_B} \omega_{IB}]^\times$
    \item Constant angular velocity ($\exp{[\Delta \vec{\alpha}]^\times} = \delta \mat{R}(\Delta \vec{\alpha}) \quad \vec{\alpha} = {_I}\vec{\omega}_{IB} \delta t$)\\
          $\mat{R}_{IB}(t + \Delta t) = \exp{[\Delta \vec{\alpha}]^\times} \mat{R}_{IB}(t)$
 \end{itemize}
@@ -41,7 +41,7 @@
              }$
 \end{itemize}

-\bi{IMU} (Outputs {\color{blue} ${_S}\vec{\tilde{a}}$} (accel.), {\color{red} ${_S}\vec{\tilde{\omega}}$} (rot. accel.))\\
+\bi{IMU} (Outputs {\color{blue} ${_S}\vec{\tilde{a}}$} (accel.), {\color{red} ${_S}\vec{\tilde{\omega}}$} (rot. vel.))\\
 ${_W}\vec{\dot{t}}_S = {_W} \vec{v}$;
 $\quad \displaystyle \vec{\dot{q}}_{WS} = \frac{1}{2} \vec{q}_{WS} \otimes
    {\scriptsize
@@ -2,7 +2,7 @@
 \label{sec:rigid-body-dynamics}
 \shortdefinition[Newton II] For fin. body w/ mass $m$ and inertia mat. $I$, with force $\vec{F}$ and torque $\vec{T}$ on \bi{Centre of Mass} (CoM), in body frame:
 \begin{align*}
-    {_B}\vec{F} & = m({_B} \vec{\dot{v}}_{CoM}) + m_B \vec{\omega} \times {_B}\vec{v}_{CoM}           \\
+    {_B}\vec{F} & = m({_B} \vec{\dot{v}}_{CoM}) + m ({_B}\vec{\omega} \times {_B}\vec{v}_{CoM})       \\
    {_B}\vec{T} & = \mat{I}({_B} \vec{\dot{\omega}}) + {_B} \vec{\omega} \times \mat{I}_B\vec{\omega}
 \end{align*}

@@ -25,7 +25,7 @@ if over-actuated, with $J^+ = J^\top(J J^\top)^{-1}$ (Moore Penrose-Inverse)

 \newpage
 \subsection{Legged robots}
-{\scriptsize Legged robots don't need to maintain stability!}
+{\scriptsize Legged robots don't need to maintain static stability!}

 \shortdefinition[Static Walking] 3 or more legs on ground, stable if frozen, safe, slow, inefficient.