Robotic Dynamics

Lecturer : Marco Hutter

Kinematics

position

Cartesian coordinate $\boldsymbol \chi_{Pc}=\begin{bmatrix}x\\y\\z\end{bmatrix}, \bf r=\begin{bmatrix}x\\y\\z\end{bmatrix}$
Cylindrical coordinate $\boldsymbol \chi_{Pz}=\begin{bmatrix}\rho\\\theta\\z\end{bmatrix},_{\mathcal A}\bf r=\begin{bmatrix}\rho \cos \theta \\\rho \sin\theta\\z\end{bmatrix}$
Spherical coordinate $\boldsymbol \chi_{Ps}=\begin{bmatrix}r\\\theta\\\phi\end{bmatrix},_{\mathcal A}\bf r=\begin{bmatrix}r\cos\theta\sin\theta\\ r\sin\theta\sin\phi\\ r\cos\phi\end{bmatrix}$

linear velocity

\begin{aligned} \dot{r} & = E_{P} (χ_{P}) {\dot{χ}}_{P} \\ {\dot{χ}}_{P} & = E_{P}^{- 1} (χ_{P}) \dot{r} \end{aligned}

Cartesian coordinate $\textbf E_{Pc} =\textbf E_{Pc}^{-1}= \mathbb I$
Cylindrical coordinate $\textbf E_{Pz}=\begin{bmatrix}\cos\theta&-\rho\sin\theta&0\\\sin\theta&\rho\cos\theta &0\\0&0&1\end{bmatrix}$ $\textbf E_{Pz}^{-1} = \begin{bmatrix}\cos\theta&\sin\theta&0\\-\sin\theta/\rho&\cos\theta/\rho &0\\0&0&1\end{bmatrix}$
Spherical Coordinate $\textbf E_{Ps}=\begin{bmatrix}\cos\theta\sin\phi & -r\sin\phi\sin\theta & r\cos\phi\cos\theta\\\sin\phi\sin\theta & r\cos\theta\sin\phi & r\cos\phi \sin\theta\\ \cos\phi & 0 & -r\sin\phi\end{bmatrix}$ $\textbf E_{Ps}^{-1}=\begin{bmatrix}\cos\theta\sin\phi & \sin\phi\sin\theta&\cos\phi\\-\sin\theta/(r\sin\phi)&\cos\theta/(r\sin\phi)&0\\(\cos\phi\cos\theta)/r&(\cos\phi\sin\theta)/r& -\sin\phi /r\end{bmatrix}$

rotation

Passive Rotation $_{\mathcal A}\textbf u = \textbf C_{\mathcal{AB}}\cdot_{\mathcal B}\textbf u$ Active Rotation $_{\mathcal A}\textbf v=\textbf R\cdot _{\mathcal A}\textbf u$
Elementary Rotation $\textbf C_x(\varphi)=\begin{pmatrix}1&0&0\\0&\cos\varphi&-\sin\varphi\\0&\sin\varphi&\cos\varphi\end{pmatrix}$ $\textbf C_y(\varphi) = \begin{pmatrix}\cos\varphi & 0 &\sin\varphi \\0&1&0\\-\sin\varphi&0&\cos\varphi\end{pmatrix}$ $\textbf C_z(\varphi)=\begin{pmatrix}\cos\varphi&-\sin\varphi&0\\\sin\varphi&\cos\varphi&0\\0&0&1\end{pmatrix}$
Euler Angle
- ZYZ Euler Angle $\textbf C = \textbf C_{z_1}\textbf C_{y}\textbf C_{z_2}$ $\boldsymbol \chi_{R,eulerZYZ}=\begin{pmatrix}\atan(c_{23}/c_{13})\\ \atan(\sqrt{c_{13}^2+c_{23}^2}/c_{33})\\\atan(c_{32}/(-c_{31}))\end{pmatrix}$
- ZXZ Euler Angle $\textbf C = \textbf C_{z_1}\textbf C_{x}\textbf C_{z_2}$ $\boldsymbol \chi_{R,eulerZXZ}=\begin{pmatrix}\atan(c_{13}/(-c_{23}))\\ \atan(\sqrt{c_{13}^2+c_{23}^2}/c_{33})\\\atan(c_{31}/c_{32})\end{pmatrix}$
- ZYX Euler Angle $\textbf C = \textbf C_{z}\textbf C_{y}\textbf C_{x}$ $\boldsymbol \chi_{R,eulerZYX}=\begin{pmatrix}\atan(c_{21}/c_{11})\\ \atan(-c_{31}/\sqrt{c_{32}^2+c_{33}^2})\\\atan(c_{32}/c_{33})\end{pmatrix}$
- XYZ Euler Angle $\textbf C = \textbf C_{x}\textbf C_{y}\textbf C_{z}$ $\boldsymbol \chi_{R,eulerXYZ}=\begin{pmatrix}\atan(-c_{23}/c_{33})\\ \atan(c_{13}/\sqrt{c_{11}^2+c_{12}^2})\\\atan(-c_{12}/c_{11})\end{pmatrix}$
Angle Axis $\boldsymbol \chi_{R,AngleAxis}=\begin{pmatrix}\theta\\\textbf n\end{pmatrix},\Vert \textbf n\Vert=1$
- Euler VectorRotation Vector $\varphi = \theta\cdot \textbf n\in \mathbb R^3$
- $\textbf C_{\mathcal {AB}} = \begin{bmatrix}n_x^2(1-c_\theta)+c_\theta&n_xn_y(1-c_\theta)-n_zs_\theta&n_xn_z(1-c_\theta)+n_ys_\theta\\n_xn_y(1-c_\theta)+n_zs_\theta& n_y^2(1-c_\theta)+c_\theta & n_yn_z(1-c_\theta)-n_xs_\theta\\ n_xn_z(1-c_\theta)-n_ys_\theta&n_yn_z(1-c_\theta)+n_xs_\theta&n_z^2(1-c_\theta)+c_\theta\end{bmatrix}$
- $\theta=\cos^{-1}\left(\frac{\text{Tr}(\textbf C)-1}{2}\right)$ $\textbf n=\frac{1}{2\sin\theta}\begin{pmatrix}c_{32} - c_{23}\\c_{13}-c_{31}\\c_{21}-c_{12}\end{pmatrix}$
Unit Quaternions $\boldsymbol \chi_{R,quat}=\boldsymbol \xi=\begin{pmatrix}\xi_0\\\check {\boldsymbol \xi}\end{pmatrix}, \sum_{i=0}^3\xi_i^2=1$
- $\xi_0 = \cos\left(\frac{\Vert \varphi\Vert}{2}\right)=\cos\left(\frac{\theta}{2}\right)$ $\check{\boldsymbol \xi}=\sin\left(\frac{\Vert\varphi\Vert}{2}\right)\frac{\varphi}{\Vert\varphi\Vert} = \sin\left(\frac{\theta}{2}\right)\textbf n$
- $\textbf C_{\mathcal {AD}} = \mathbb I + 2\xi_0\left[\check{\boldsymbol \xi}\right]_\cross + 2\left[\check{\boldsymbol\xi}\right]^2_{\cross} = (2\xi_0^2-1)\mathbb I+2\xi_0\left[\check{\boldsymbol \xi}\right]_\cross + 2\check {\boldsymbol \xi}\check{\boldsymbol \xi}^\top$ $\left[\check {\boldsymbol \xi}\right]_\cross = \begin{bmatrix}0&-\xi_3&\xi_2\\\xi_3&0&-\xi_1\\-\xi_2&\xi_1&0\end{bmatrix}$
- $\boldsymbol \xi = \frac{1}{2}\begin{pmatrix}\sqrt{\text{Tr}(\textbf C)+1}\\\text{sgn}(c_{32}-c_{23})\sqrt{c_{11}-c_{22}-c_{33}+1}\\\text{sgn}(c_{13}-c_{31})\sqrt{c_{22}-c_{33}-c_{11}+1}\\\text{sgn}(c_{21}-c_{12})\sqrt{c_{33}-c_{11}-c_{22}+1}\end{pmatrix}$
- Inverse $\boldsymbol \xi^{-1} = \boldsymbol \xi^\top=\begin{pmatrix}\xi_0\\-\check{\boldsymbol \xi}\end{pmatrix}$
- Multiplication $\boldsymbol \xi_{\mathcal{AC}} = \underbrace{\begin{bmatrix}\xi_0 & -\check{\boldsymbol \xi}^\top\\ \check{\boldsymbol \xi}&\xi_0\mathbb I+\left[\check {\boldsymbol \xi}\right]_\cross\end{bmatrix}}_{\textbf M_l(\boldsymbol \xi_{\mathcal AB})}\boldsymbol \xi_{\mathcal {BC}} = \underbrace{\begin{bmatrix}\xi_0 & -\check{\boldsymbol \xi}^\top\\\check{\boldsymbol \xi}&\xi_0\mathbb I-\left[\check {\boldsymbol \xi}\right]_\cross\end{bmatrix}}_{\textbf M_r(\boldsymbol \xi _{\mathcal BC})}\boldsymbol \xi_{\mathcal {AB}}$
- Vector Rotation $\begin{pmatrix}0\\_{\mathcal A}\textbf r\end{pmatrix} = \textbf M_l(\boldsymbol \xi_{\mathcal {AB}})\textbf M_r(\xi_{\mathcal {AB}}^\top)\begin{pmatrix}0\\_{\mathcal B }\textbf r\end{pmatrix}$
- Degree of Freedom : 3

angular velocity

_{A} ω_{AB} = E_{R} (χ_{R}) \cdot {\dot{χ}}_{R}

Transform $\left[_{\mathcal B}\omega_{\mathcal {AB}}\right]_\cross = \textbf C_{\mathcal {BA}}\cdot \left[_{\mathcal A}\omega_{\mathcal {AB}}\right]_\cross\cdot \textbf C_{\mathcal {AB}}$
Euler Angle $\textbf E_{R,euler \alpha\beta\gamma} = \begin{bmatrix}_{\mathcal A}\textbf e_A^{\mathcal \alpha}& _{\mathcal A}\textbf e_\beta^{\mathcal A'}& _{\mathcal A}\textbf e_\gamma^{\mathcal A''}\end{bmatrix}$
- $\textbf E_{R,eulerZYZ} = \begin{bmatrix}0&-\sin z_1&\cos z_1\sin y\\0 & \cos z_1 & \sin z_1\sin y\\ 1 & 0 & 0\end{bmatrix}$ $\textbf E^{-1}_{R,eulerZYZ} = \begin{bmatrix}-\cos y\cos z_1/\sin y & -\cos y \sin z_1/\sin y & 1\\ -\sin z_1 & \cos z_1 & 0\\ \cos z_1/\sin y & \sin z_1/ \sin y & 0\end{bmatrix}$
- $\textbf E_{R,euler ZXZ}=\begin{bmatrix}0&\cos z_1&\sin z_1 \sin x\\ 0 & \sin z_1 & -\cos z_1 \sin x\\ 1 & 0 & \cos x\end{bmatrix}$ $\textbf E^{-1}_{R,eulerZXZ} = \begin{bmatrix}-\cos x\sin z_1/\sin x & \cos x\cos z_1/\sin x & 1\\ \cos z_1 & \sin z_1 & 0 \\ \sin z_1 /\sin x & -\cos z_1/ \sin x & 0\end{bmatrix}$
- $\textbf E_{R,eulerZYX} = \begin{bmatrix}0 & -\sin z & \cos y \cos z \\ 0 & \cos z & \cos y \sin z\\ 1 & 0 & -\sin y\end{bmatrix}$ $\textbf E^{-1}_{R,eulerZYX} = \begin{bmatrix}\cos z \sin y /\cos y & \sin y \sin z /\cos y & 1\\ -\sin z & \cos z & 0 \\ \cos z / \cos y & \sin z /\cos y & 0\end{bmatrix}$
- $\textbf E_{R,eulerXYZ}=\begin{bmatrix}1 & 0 & \sin y \\ 0 & \cos x & -\cos y \sin x\\ 0 & \sin x & \cos x \cos y \end{bmatrix}$ $\textbf E^{-1}_{R,euler XYZ}=\begin{bmatrix}1&\sin x\sin y/\cos y & -\cos x\sin y /\cos y\\ 0 & \cos x & \sin x\\ 0 & -\sin x /\cos y & \cos x /\cos y\end{bmatrix}$
Angular Axis $\textbf E_{R,angleaxis}=\begin{bmatrix}\textbf n& \sin\theta\mathbb I+(1-\cos\theta)\left[\textbf n\right]_\cross\end{bmatrix}$ $\textbf E_{R,angleaxis}^{-1}=\begin{bmatrix}\textbf n^\top\\ -\frac{1}{2}\frac{\sin\theta}{1-\cos\theta}\left[\textbf n\right]^2_\cross-\frac{1}{2}\left[\boldsymbol n \right]_\cross\end{bmatrix}$
Rotation Vector $\textbf E_{R,rotationvector}=\begin{bmatrix}\mathbb I+\left[\varphi\right]_\cross\left(\frac{1-\cos\Vert\varphi\Vert}{\Vert\varphi\Vert^2}\right)+\left[\varphi\right]_\cross^2\left(\frac{\Vert \varphi \Vert -\sin\Vert \varphi\Vert}{\Vert \varphi\Vert^3}\right)\end{bmatrix}$ $\textbf E_{R,rotationvector}^{-1}=\left[\mathbb I -\frac{1}{2}\left[\varphi\right]_\cross + \left[\varphi\right]^2_\cross\frac{1}{\Vert\varphi\Vert^2}\left(1-\frac{\Vert\varphi\Vert}{2}\frac{\sin\Vert\varphi\Vert}{1-\cos\Vert\varphi\Vert}\right)\right]$
Unit Quaternions $\textbf E_{R,quat}=2\textbf H(\boldsymbol \xi)$ $\textbf E_{R,quat}^{-1}=\frac{1}{2}\textbf H(\boldsymbol \xi)^\top$ $\textbf H(\boldsymbol \xi) = \begin{bmatrix}-\check{\boldsymbol \xi}&\left[\check{\boldsymbol \xi}\right]_\cross + \xi_0 \mathbb I\end{bmatrix}\in \mathbb R^{3\times 4}$

transformation $\textbf T_{\mathcal {AB}} = \begin{bmatrix}\textbf C_{\mathcal {AB}} & _{\mathcal A}\textbf r_{\mathcal{AB}}\\\textbf 0 & 1\end{bmatrix}$

Inverse $\textbf T_{\mathcal {AB}}^\top = \begin{bmatrix}\textbf C^\top_{\mathcal{AB}} &\overbrace{-\textbf C_{\mathcal {AB}}^\top \textbf r_{\mathcal {AB}}}^{_{\mathcal B}\textbf r_{\mathcal {BA}}}\\ \textbf 0 & 1\end{bmatrix}$
Multiplication $\textbf T_{\mathcal {AC}}=\textbf T_{\mathcal {AB}} \textbf T_{\mathcal {BC}}$

transformation acceleration

Acceleration $\ddot {\textbf r}_{P} = \ddot{\textbf r}_{B} + \dot \omega_B\cross \textbf r_{BP } + \omega_B\times(\omega_B\cross \textbf r_{BP})$
Moving System $\mathcal B$ $_{\mathcal B}\dot{\textbf r}_{\mathcal {BP}} = _{\mathcal B}\dot{\textbf r_{\mathcal{AP}}} + _{\mathcal B}\omega_{\mathcal {AB}}\times _{\mathcal B}\textbf r_{\mathcal {AP}}$

task-space coordinate

End-Effector/Operational Space Coordinate $\boldsymbol \chi_e=\begin{pmatrix} \boldsymbol \chi_{e_P}\\ \boldsymbol \chi_{e_R}\end{pmatrix}$

forward Kinematics

T_{IE} (q) = T_{I 0} \cdot \prod_{k = 1}^{n_{j}} T_{k - 1, k} (q_{k}) \cdot T_{n_{j} E}

differential Kinematics $\dot{\boldsymbol \chi}_e = \textbf J_{eA}(\textbf q)\dot {\textbf q}$ $\ddot{\boldsymbol\chi}_e = \textbf J_{eA}(\textbf q)\ddot {\textbf q} + \dot{\textbf J}_{eA}(\textbf q)\dot{\textbf q}$

Analytical Jacobian $\textbf J_{eA}=\begin{bmatrix}\textbf J_{eA_P}\\\textbf J_{eA_R}\end{bmatrix}=\begin{bmatrix}\frac{\partial \boldsymbol \chi_{e_P}}{\partial \textbf q}\\ \frac{\partial \boldsymbol \chi _{e_R}}{\partial \textbf q}\end{bmatrix}\in \mathbb R^{m_e\times n_j}$
- $\dot{\boldsymbol \chi}_e = \textbf J_{eA}(\textbf q)\dot {\textbf q}$
Geometric Jacobian $\textbf J_{e0}\in \mathbb R^{6\times n_j}$
- $\textbf w_e = \textbf J_{e0}(\textbf q) \boldsymbol \omega$
- $\textbf J_{c0} = \textbf {J}_{b0} +\textbf J_{bc0}$
- $\textbf J_{e0}(\textbf q)=\textbf E_e(\boldsymbol\chi)\textbf J_{eA}(\textbf q)$ $\textbf E_e(\boldsymbol\chi)=\begin{bmatrix}\textbf E_P&\\&\textbf E_R\end{bmatrix}\in \mathbb R^{6\times m_e}$
Revolute Joint $_{\mathcal I}\textbf J_{e0}=\begin{bmatrix}_{\mathcal I}\textbf n_1\cross _{\mathcal I}\textbf r_{1(n+1)}&\cdots & _{\mathcal I}\textbf n_n\cross _{\mathcal I}\textbf r_{n(n+1)}\\_{\mathcal I}\textbf n_1&\cdots & _{\mathcal I}\textbf n_n\end{bmatrix}$
Prismatic Joint $_{\mathcal I}\textbf J_{e0}=\begin{bmatrix}_{\mathcal I}\textbf n_1&\cdots & _{\mathcal I}\textbf n_n\\ \textbf 0 &\cdots &\textbf 0\end{bmatrix}$
Analytical Jaocbian and Geometric Jacobian :

Inverse Differential Kinematics $\dot {\textbf q} = \textbf J_{e0}^+\textbf w_e^*$ $\textbf w_e^*$ is (desired) end-effector velocity

$m\ge n$ $A^+ = (A^\top A)^{-1}A^\top=\underset{A}{\text{argmin}}\Vert Ax-b\Vert_2^2$
$m\le n$ $A^+ = A^\top (AA^\top)^{-1}$
$A^+_d = A^\top (AA^\top + \lambda ^2 \mathbb I)^{-1}=\underset{A}{\text{argmin}}\Vert Ax-b\Vert_2^2+\lambda^2 \Vert x\Vert_2^2$
$\text{rank}(\textbf J_{e0})<6$
- solution : 1. damped, 2. redundancy

Multi-task Inverse Differential Kinematics $task_i =\{\textbf J_i, \textbf w_i^*\}$

Equal Priority $\dot {\textbf q} =\underbrace{\begin{bmatrix}\textbf J_1\\\vdots\\\textbf J_{n_t}\end{bmatrix}^+}_{\bar{\textbf J}^+ }\underbrace{\begin{pmatrix}\textbf w_1^*\\\vdots\\ \textbf w^*_{n_t}\end{pmatrix}}_{\bar{\textbf w}} = \underset{\dot q}{\text{argmin}}\Vert \bar{\textbf J}\dot{\textbf q}-\bar{ \textbf w}\Vert_2$
- $\bar{\textbf J}^{+W} = (\bar{\textbf J}^\top \textbf W\bar{\textbf J})^{-1}\bar{\textbf J}^\top \textbf W$ $\textbf W=\text{diag}(w_1,\cdots,w_m)$
Prioritization $\dot {\textbf q} = \sum_{i=1}^{m}\bar{\textbf N}_{i-1}\dot {\textbf q}_i$ $\dot {\textbf q}_i = (\textbf J_i\bar{\textbf N}_{i-1})^+\left(\textbf w_i^* - \textbf J_i\sum_{k=1}^{i-1}\bar{\textbf N}_{k-1}\dot {\textbf q}_k\right)$ $\bar{\textbf N}_i=\mathbb I - \textbf J_i^+\textbf J_i$ is the null space
- $m=2$ $\dot {\textbf q}=\textbf J_1^+\textbf w_1^* + \textbf N_1(\textbf J_2\textbf N_1)^+(\textbf w_2^* - \textbf J_2\textbf J_1^+\textbf w_1^*)$

Inverse Kinematics

Numerical Solution :
1. $\textbf q\gets \textbf q^0$
2. $\Vert \boldsymbol \chi _e^* - \boldsymbol \chi _e(\textbf q)\Vert> \epsilon$ do
  1. $\Delta \boldsymbol \chi_e\gets \boldsymbol \chi_e^*-\boldsymbol \chi_e(\textbf q)$ $_{\mathcal I} \Delta \varphi = \text{rotVec}(\textbf C_{\mathcal{IB}}\textbf C_{\mathcal {IA}}^\top)$
  2. $\textbf q\gets \textbf q + \alpha \textbf J_{eA}^+(\textbf q)\Delta \boldsymbol \chi_e$
Trajectory Control
- $\dot q^* = \textbf J^+_{e0_P}(\textbf q^t)\cdot (\dot{\textbf r}_e^*(t) + k_{p_P}\Delta \textbf r_e^t)$
- $\dot {\textbf q}^* = \textbf J_{e0_R}^+(\textbf q^t)\cdot (\omega_e^*(t)+k_{p_R}\Delta \varphi)$

floating base kinematics $n_n=n_b+n_j$ $n_b$ $n_j$ actuated joint coordinate

Generalized Velocity $\textbf u = \begin{pmatrix}_{\mathcal I}\textbf v_{ \mathcal B}\\_{\mathcal B}\boldsymbol \omega_{\mathcal {IB}}\\\dot \varphi_1\\\vdots \\ \dot\varphi_{n_j}\end{pmatrix} \in \mathbb R^{6+n_j}$ $\mathcal B$ $\mathcal I$ is the inertial frame
Forward Kinematics $_{\mathcal I}\textbf r_{\mathcal{IQ}}(\textbf q) = _{\mathcal I}\textbf r_{\mathcal {IB}}(\textbf q) + \textbf C_{\mathcal{IB}}(\textbf q)\cdot_{\mathcal B}\textbf r_{\mathcal{BQ}}(\textbf q)$ $\textbf C_{\mathcal{IB}}$ $\mathcal B$
Differential Kinematics $_{\mathcal I}\textbf J_Q(\textbf q) = \begin{bmatrix}_{\mathcal I}\textbf J_P \\ _{\mathcal I}\textbf J_R\end{bmatrix}= \begin{bmatrix}\mathbb I & -\textbf C_{\mathcal {IB}}\cdot\left[_{\mathcal B}\textbf r_{BQ}\right]_\cross & \textbf C_{\mathcal{IB}}\cdot _{\mathcal B}\textbf J_{P_{q_j}}(\textbf q_j)\\\textbf 0 & \textbf C_{\mathcal {IB}}& \textbf C_{\mathcal{IB}}\cdot _{\mathcal B}\textbf J_{R_{q_j}}(\textbf q_j)\end{bmatrix}$
Contact and Constraint $\textbf r_c = \text{const}\quad\dot{\textbf r}=\ddot{\textbf r}_c=0$ $\Leftrightarrow$ $_{\mathcal I}\textbf J_{C_i}\textbf u=\textbf 0\quad_{\mathcal I}\textbf J_{C_i}\dot {\textbf u}+_{\mathcal I}\dot{\textbf J}_{C_i}\textbf u = \textbf 0$ $\textbf J_c\in \mathbb R^{3n_c\times n_n}$ $n_c$ is the number of contacts

Dynamics

Generalized Equation of Motion

M (q) \ddot{q} + b (q, \dot{q}) + g (q) = S^{⊤} τ + J_{c} (q)^{⊤} F_{c}

$\textbf q,\dot{\textbf q},\ddot{\textbf q}\in \mathbb R^{n_q}$ : generalized position, velocity, acceleration
$\textbf M(\textbf q)\in \mathbb R^{n_q\times n_q}$ $\textbf M=\sum_{i=1}^{n_b}(_{\mathcal A}\textbf J_{S_i}^\top m_{\mathcal A}\textbf J_{S_i}+_{\mathcal B}\textbf J_{R_i}^\top~ _{\mathcal B}\Theta_{S_i} ~_{\mathcal B}\textbf J_{R_i})$ generalized mass matrix
$\textbf b(\textbf q,\dot{\textbf q})\in \mathbb R^{n_q}$ $\textbf b=\sum_{i=1}^{n_b}\left(_{\mathcal A}\textbf J_{S_i}^\top m_{\mathcal A}\dot{\textbf J}_{S_i}\dot{\textbf q}+~_{\mathcal B}\textbf J_{R_i}^\top\left(_{\mathcal B}\Theta_{\mathcal B}\dot{\textbf J}_{R_i}\dot{\textbf q}+~_{\mathcal B}\boldsymbol\Omega_{S_i}\cross ~_{\mathcal B}\Theta_{S_i}~_{\mathcal B}\boldsymbol \Omega_{S_i}\right)\right)$ $\boldsymbol \Omega_{S_i} = \textbf J_{R_i}\dot{\textbf q}$
$\textbf g(\textbf q)\in \mathbb R^{n_q}$ $\textbf g=\sum_{i=1}^{n_b}-~_{\mathcal A}\textbf J_{S_i}^\top ~_{\mathcal A}\textbf F_{g,i}$ gravitational terms
- $\textbf g = \sum_i -\textbf J_i^\top m_i\begin{pmatrix}0\\0\\-9.81\end{pmatrix}$
$\tau\in \mathbb R^{n_\tau}$ $\tau = \sum_{k=1}^{n_A} \tau_{a,k}\tau_{ext}$ external generalized forces,
- $\tau_{a,k}=(\textbf J_{S_k} - \textbf J_{S_{k-1}})^\top\textbf F_{a,k}+(\textbf J_{R_k}-\textbf J_{R_{k-1}})^\top \textbf T_{a,k}$ $\textbf T$ is torque here
- $\tau_{ext}=\sum_{j=1}^{n_{f,ext}}\textbf J_{P,j}^\top \textbf F_j+\sum_{j=1}^{n_{m,ext}}\textbf J_{R,j}^\top\textbf T_{ext, j}$
$\textbf S\in \mathbb R^{n_\tau\times n_q}$ : selection matrix of actuated joints
$\textbf F_c\in \mathbb R^{n_c}$ : external Cartesian forces (e.g. from contacts)
- Soft Contact Model $\textbf F_c = k_p(\textbf r_c-\textbf r_{c0})+k_d\dot{\textbf r}_c$
- Contact Force from Constraint $\textbf r_c = \text{const}\quad\dot{\textbf r}=\ddot{\textbf r}_c=0$ $\textbf F_c = (\textbf J_c\textbf M^{-1}\textbf J_c^\top)^{-1}\left(\textbf J_c\textbf M^{-1}(\textbf S^{\top}\tau-\textbf b-\textbf g)+\dot{\textbf J}_c\textbf u\right)$
$\textbf J_c(\textbf q)\in \mathbb R^{n_c\times n_q}$ : Geometric Jacobian corresponding to external force
Kinect Energy $\mathcal T (\textbf q,\dot{\textbf q})=\frac{1}{2}\dot{\textbf q}^\top\underbrace{\left(\sum_{i=1}^{n_b}(\textbf J_{S_i}^\top m\textbf J_{S_i}+\textbf J_{R_i}^\top \Theta_{S_i}\textbf J_{R_i})\right)}_{\textbf M(\textbf q)}\dot {\textbf q}$
Potential Energy $\mathcal U_g = -\sum_{i=1}^{n_b} \textbf r_{S_i}^\top \textbf F_{g_i}$

Dynamics of Floating Base System

Contact Force
- $\textbf F_c = k_p (\textbf r_c - \textbf r_{c0}) + k_d \dot{\textbf r}_c$
- $\textbf r_c = \text{const}$ $\dot{\textbf r}_c =\textbf J_c\textbf u = \textbf 0$ $\ddot{\textbf r}_c = \textbf J_c \dot{\textbf u}+\dot{\textbf J}_c\textbf u = \textbf 0$
  $\textbf F_c = (\textbf J_c\textbf M^{-1}\textbf J_c^\top)^{-1}\left(\textbf J_c\textbf M^{-1}(\textbf S^\top \tau -\textbf b-\textbf g) + \dot{\textbf J}_c \textbf u\right)$
Constraint Consistent Dynamics :
- $\textbf N_c = \mathbb I - \textbf M^{-1}\textbf J_c^\top (\textbf J_c \textbf M^{-1}\textbf J_c^\top)^{-1} \textbf J_c$
- $\textbf N_c^\top (\textbf M\dot{\textbf u}+\textbf b+\textbf g) = \textbf N_c^\top \textbf S^\top \tau$
Impulse Transfer
- $\boldsymbol \Lambda_c = (\textbf J_c\textbf M^{-1}\textbf J_c^\top)^{-1}$
- $\Delta u =\textbf u^+-\textbf u^0= - \textbf M^{-1}\textbf J_c^\top (\textbf J_c\textbf M^{-1}\textbf J_c^\top)^{-1}\textbf J_c\textbf u^{-}$
- $\textbf u^+ = \textbf N_c \textbf u^{-1}$
- $E_{loss} = -\frac{1}{2}\Delta \textbf u^\top_c\boldsymbol \Lambda_c \Delta \textbf u$

Joint Space Dynamic Control

Gravity Compensation $\tau^* = k_p(\textbf q^* - \textbf q) + k_d(\dot{\textbf q}^*-\dot{\textbf q})+\textbf g(\textbf q)$
Inverse Dynamics Control $\ddot{\textbf q}^* = k_p(\textbf q^*-\textbf q)+k_d(\dot{\textbf q}^*-\dot{\textbf q})$ $\omega=\sqrt{k_p},D=\frac{k_d}{2\sqrt{k_p}}$

Task Space Dynamic Control $\dot {\textbf w}_e = \begin{pmatrix}\ddot{\textbf r}\\\dot{\boldsymbol \omega}\end{pmatrix}=\textbf J_e\ddot{\textbf q} + \dot{\textbf J}_e\dot{\textbf q}$

Multi-task Decomposition :
- $\ddot {\textbf q} =\begin{bmatrix}\textbf J_1\\\vdots\\\textbf J_{n_t}\end{bmatrix}^+\left(\begin{pmatrix}\dot{\textbf w_1}^*\\\vdots\\ \dot{\textbf w^*_{n_t}}\end{pmatrix} - \begin{bmatrix}\dot{\textbf J}_1\\\vdots\\\dot{\textbf J}_{n_t}\end{bmatrix}\dot{\textbf q}\right)$
- Prioritization $\ddot {\textbf q} = \sum_{i=1}^{n_t}\textbf N_{i}\ddot {\textbf q}_i$ $\ddot {\textbf q}_i = (\textbf J_i\textbf N_{i})^+\left(\dot{\textbf w_i}^* - \dot{\textbf J}_i\dot{\textbf q}-\textbf J_i\sum_{k=1}^{i-1}\textbf N_k\ddot {\textbf q}_k\right)$ $\textbf N_i=\mathbb I - \textbf J_i^+\textbf J_i$ is the null space
End-effector Dynamics $\Lambda _e \dot{\textbf w}_e + \boldsymbol \mu + \textbf p=\textbf F_e$
- $\Lambda_e = (\textbf J_e\textbf M^{-1}\textbf J_e^\top)^{-1}$ : end-effector inertia
- $\boldsymbol \mu = \Lambda_e\textbf J_e\textbf M^{-1}\textbf b - \Lambda_e \dot{\textbf J}_e\dot{\textbf q}$ : end-effector centrifugal/Coriolis
- $\textbf p = \Lambda_e \textbf J_e\textbf M^{-1}\textbf g$ : end-effector gravitational
- $\tau = \textbf J_e^\top \textbf F_e$
End-effector Motion Control $\tau^* = \textbf J^\top(\Lambda_e\dot{\textbf w}_e^*+\boldsymbol \mu + \textbf p)=\textbf J^\top \Lambda_e\dot{\textbf w}_e^*+\textbf b-\textbf J^\top \Lambda_e\dot{\textbf J}_e\dot{\textbf q}+\textbf g$
Operational Space Control $\tau^*=\textbf J^\top(\Lambda \textbf S_M\dot{\textbf w}_e + \textbf S_F\textbf F_c + \boldsymbol \mu +\textbf p)$
- $\Sigma_p =\begin{bmatrix}\sigma_{px}&0&0\\0&\sigma_{py}&0\\0&0&\sigma_{pz}\end{bmatrix}$ $\Sigma_r = \begin{bmatrix}\sigma_{rx}&0&0\\0&\sigma_{ry}&0\\0&0&\sigma_{rz}\end{bmatrix}$ $\sigma_i=1$ $0$
- $\textbf S_M=\begin{bmatrix}\textbf C^\top \Sigma_p\textbf C&0\\0&\textbf C^\top \Sigma_r \textbf C\end{bmatrix}$ $\textbf S_F=\begin{bmatrix}\textbf C^\top (\mathbb I -\Sigma_p)\textbf C&0 \\ 0& \textbf C^\top (\mathbb I-\Sigma_r)\textbf C\end{bmatrix}$

Inverse Dynamics for Floating-Base Systems

Hierarchical Least Square Optimization $\underset{\textbf x}{\text{min}}\Vert \textbf A_i\textbf x - \textbf b_i\Vert_2$ with priority
$\textbf x = \begin{bmatrix}\ddot {\textbf q}^\top &\boldsymbol\tau^\top & _{\mathcal I}\textbf F_E^\top\end{bmatrix}^\top$
1. $n_T$ : number of tasks
2. $\textbf x=\textbf 0$ : initial optimal solution
3. $\textbf N_1 = \mathbb I$ : initial null-space projector
4. $i=1\to n_T$ do
  1. $\textbf x_i\gets (\textbf A_i\textbf N_i)^+(\textbf b_i-\textbf A_i\textbf x)$
  2. $\textbf x\gets \textbf x + \textbf N_i\textbf x_i$
  3. $\textbf N_{i+1}=\mathcal N\left(\left[\textbf A_1^\top\cdots\textbf A_i^\top\right]^\top\right)$ $\mathcal N(\textbf A)=\mathbb I-\textbf A^+ \textbf A$ $\mathcal N(\textbf A)\textbf A = \textbf 0$

Legged Robot

Input $\textbf q,\dot{\textbf q}$

Optimization Target $\ddot {\textbf q}, \textbf F_c, \boldsymbol \tau$

Tasks :

$\begin{bmatrix}\textbf M(\textbf q)&-\textbf J_c&-\textbf S^\top\end{bmatrix}\begin{bmatrix}\ddot {\textbf q}\\\textbf F_c\\\boldsymbol \tau \end{bmatrix} = - \textbf b(\dot{\textbf q},\textbf q)-\textbf g(\textbf q)$
$\textbf w^*_e$ $\begin{bmatrix}\textbf J_e&\textbf 0 &\textbf 0\end{bmatrix}\begin{bmatrix}\ddot {\textbf q}\\\textbf F_c\\\boldsymbol \tau \end{bmatrix} = \dot{\textbf w}_e -\dot{\textbf J}_c \dot{\textbf q}$ $\dot {\textbf w}_e = k_p(\textbf r_e^*-\textbf r_e)-k_d(\textbf w_e^*-\textbf w_e)$
Torque minimize $\begin{bmatrix}\textbf 0&\textbf 0&\mathbb I\end{bmatrix}\begin{bmatrix}\ddot {\textbf q}\\\textbf F_c\\\boldsymbol \tau \end{bmatrix} = \textbf 0$
Torque limits $\begin{bmatrix}\textbf 0&\textbf 0&\mathbb I\end{bmatrix}\begin{bmatrix}\ddot {\textbf q}\\\textbf F_c\\\boldsymbol \tau \end{bmatrix} \le \textbf 1 \cdot\tau_{max}$ $\begin{bmatrix}\textbf 0&\textbf 0&-\mathbb I\end{bmatrix}\begin{bmatrix}\ddot {\textbf q}\\\textbf F_c\\\boldsymbol \tau \end{bmatrix} \le -\textbf 1 \cdot\tau_{max}$
Contact Force minimize $\begin{bmatrix}\textbf 0&\mathbb I & \textbf 0\end{bmatrix}\begin{bmatrix}\ddot {\textbf q}\\\textbf F_c\\\boldsymbol \tau \end{bmatrix} = \textbf 0$
Friction Cone $\begin{bmatrix}\textbf 0&\begin{matrix}\begin{bmatrix}0&-1\\1-\mu & -1-\mu\end{bmatrix}&\textbf 0\\ \textbf 0 & \begin{bmatrix}0&-1\\1-\mu & -1-\mu\end{bmatrix}\end{matrix} & \textbf 0\end{bmatrix}\begin{bmatrix}\ddot {\textbf q}\\\textbf F_c\\\boldsymbol \tau \end{bmatrix} \le \textbf 0$ $2D$ $x-z$ problem

Optimization

[HO]Hierarchical Least Square Optimization

Rotorcrafts

\begin{matrix} \underset{M}{\underset{⏟}{[\begin{matrix} m I & 0 \\ 0 & Θ \end{matrix}]}} \underset{\ddot{q}}{\underset{⏟}{[\begin{matrix} \dot{ν} \\ \dot{ω} \end{matrix}]}} + \underset{b}{\underset{⏟}{[\begin{matrix} ω \times m ν \\ ω \times Θ ω \end{matrix}]}} = \underset{τ}{\underset{⏟}{[\begin{matrix} F \\ \underset{torque}{\underset{⏟}{M}} \end{matrix}]}} \end{matrix}

\begin{matrix} _{B} F = \underset{_{B} F_{G}}{\underset{⏟}{C_{IB}^{⊤}_{I} [\begin{matrix} 0 \\ 0 \\ m g \end{matrix}]}} + \underset{_{B} F_{A e r o}}{\underset{⏟}{\sum_{i = 1}^{4}_{B} [\begin{matrix} 0 \\ 0 \\ - T_{i} \end{matrix}]}} \end{matrix}

\begin{matrix} _{B} M = \underset{_{B} M_{T}}{\underset{⏟}{_{B} [\begin{matrix} l (T_{4} - T_{2}) \\ l (T_{1} - T_{3}) \\ 0 \end{matrix}]}} + \underset{_{B} Q}{\underset{⏟}{_{B} [\begin{matrix} 0 \\ 0 \\ \sum_{i = 1}^{4} Q_{i} (- 1)^{i} \end{matrix}]}} \end{matrix}

$T_i$ $i$ $T_i = b\omega_{p,i}^2$ generate lift for keeping the rotorcraft in the air
$Q_i$ $i$ $Q_i=d\omega_{p,i}^2$ rotor drag
$_{\mathcal B}M_T$ $M_T = \begin{bmatrix}-\sin\theta\\\sin\phi\cos\theta\\\cos\phi \cos\theta\end{bmatrix}mg$
$_{\mathcal B}Q$ : Drag torques/moment
$\textbf C_{\mathcal {IB}}$ : transition from earth frame to body frame ,
$_{\mathcal B}\boldsymbol \omega$ $_{\mathcal B}\boldsymbol \omega =_{\mathcal B} \begin{bmatrix}p\\q\\r\end{bmatrix}=\begin{bmatrix}\dot \phi\\\dot \theta\\\dot \psi\end{bmatrix}$
$\omega_{p,i }$ $i$
$_{\mathcal B}\boldsymbol \nu$ $_{\mathcal B}\boldsymbol \nu = _{\mathcal B}\begin{bmatrix}u\\v\\w\end{bmatrix} = _{\mathcal B}\begin{bmatrix}\dot x\\\dot y \\\dot z\end{bmatrix}$
$l$ : Distance of the propeller from the central of gravity

Control of Quadrotor

\begin{array}{lr} Θ_{x x} \dot{p} = q \cdot r (Θ_{y y} - Θ_{z z}) + U_{2} & (1) \\ Θ_{y y} \dot{q} = r \cdot p (Θ_{z z} - Θ_{x x}) + U_{3} & (2) \\ Θ_{z z} \dot{r} = U_{4} & (3) \\ m \dot{u} = m (r \cdot v - q \cdot w) - \sin θ m g & (4) \\ m \dot{v} = m (p \cdot w - r \cdot u) + \sin ϕ \cos θ m g & (5) \\ m \dot{w} = m (q \cdot u - p \cdot v) + \cos ϕ \cos θ m g - U_{1} & (6) \end{array}

[\begin{matrix} U_{1} \\ U_{2} \\ U_{3} \\ U_{4} \end{matrix}] = \underset{A}{\underset{⏟}{[\begin{matrix} b & b & b & b \\ 0 & - l b & 0 & l b \\ l b & 0 & - l b & 0 \\ - d & d & - d & d \end{matrix}]}} [\begin{matrix} ω_{p, 1}^{2} \\ ω_{p, 2}^{2} \\ ω_{p, 3}^{2} \\ ω_{p, 4}^{2} \end{matrix}]

Equilibrium $_{\mathcal B}\boldsymbol \nu = \phi = \theta=\textbf 0$
- $_{\mathcal B} \dot{\boldsymbol \omega} = \boldsymbol \Theta^{-1}\begin{bmatrix}U_2&U_3&U_4\end{bmatrix}^\top$
- $\begin{array}{l} U_{1} = T^{*} \\ U_{2} = (ϕ^{*} - ϕ) k_{p, r o l l} - \dot{ϕ} k_{d, r o l l} \\ U_{3} = (θ^{*} - θ) k_{p, p i t c h} - \dot{θ} k_{d, p i t c h} \\ U_{4} = (ψ^{*} - ψ) k_{p, y a w} - \dot{ψ} k_{d, y a w} \end{array}$
Altitude Control $x=y=u=v=0$
- $\dot w = g- \frac{1}{m}\underbrace{\cos\phi\cos\theta ~U_1}_{T_z}$
- $T_z = -k_p(z^*-z)+k_d\dot z - mg$
Position Control
- $_{\mathcal B}{\dot {\boldsymbol \nu}} = \frac{1}{m}\begin{bmatrix}T_x&T_y&T_z\end{bmatrix}^\top + \begin{bmatrix}0&0&g\end{bmatrix}^\top$

Hexacopter

$\textbf C_{\mathcal{IB}}$ : rotation matrix from navigation to body frame
$\textbf p_i$ $i$ $\textbf p_i = [l_{xi},l_{yi},0]^\top$
$\textbf g$ $\textbf g = \textbf C_{\mathcal{IB}}^\top [0,0,g]^\top$
$\textbf T$ $\textbf T = \sum_{i=1}^6[0,0,b\omega_{p,i}^2]$
$\textbf M_Q$ $\textbf M_Q = \sum_{i=1}^6 [0,0,d\omega_{p,i}^2](-1)^{i+1}$
$\textbf M_T$ $\textbf M_T = \sum_{i=1}^6[0,0,b\omega_{p,i}^2]\times\textbf p_i$

\begin{matrix} A = (\begin{matrix} b & b & b & b & b & b \\ b l s_{30} & b l & b l s_{30} & - b l s_{30} & - b l & - b l s_{30} \\ - b l c_{30} & 0 & b l c_{30} & b l c_{30} & 0 & - b l c_{30} \\ d & - d & d & - d & d & - d \end{matrix}) \end{matrix}

MAV Control

$_{\mathcal B}\boldsymbol \omega_{ref} = \text{PID}(_{\mathcal B}\boldsymbol \nu_{ref},_{\mathcal B}\boldsymbol \nu)$
$U_1 = mg\\ \begin{bmatrix}U_2,U_3,U_4\end{bmatrix}^\top=\text{PID}(_{\mathcal B}\boldsymbol \omega_{ref},~_{\mathcal B}\boldsymbol \omega,~_{\mathcal B}\dot{\boldsymbol \omega})$
$\omega_p^2 = \textbf A^+ \begin{bmatrix}U_1&U_2&U_3&U_4\end{bmatrix}^\top$

Propeller Aerodynamics

$u_1 = u_2$ : no change of speed across rotor/propeller disc
$\rho A_0 V = \rho A_R(V+u_1) = \rho A_R(V+u_2)=\rho A_3 u_3$ : change in pressure
$u_3 = 2u_1$ : far wake slipstream velocity is twice the induced velocity
$F_{\text{Thrust}} = \rho A_R(V+u_1)u_1$
$P_{\text{Thrust}} = \frac{1}{2}\rho A_R(V+u_1)(2V + u_3) u_3$
Hover case $P = \frac{F^{3/2}_{\text{Thrust}}}{\sqrt{2\rho A_R}} = \frac{(mg)^{3/2}}{\sqrt{2\rho A_R}}$

$T$ : Thrust force
- Aerodynamic force perpendicular to propeller plane
- $|T|=\frac{\rho}{2}A_PC_T(\omega_PR_P)^2$
$Q$ : Drag torque
- torque around rotor plane in the opposite direction of the propeller spinning direction
- $|Q| =\frac{\rho}{2}A_PC_Q(\omega_PR_P)^2 R_P$
- $C_T$ $C_Q$ depend on blade pitch angle, reynolds number ...
$H$ : hub force
- $V_H$
- $|H| = \frac{\rho}{2}A_PC_H(\omega_PR_P)^2$
$R$ : Rolling moment
- around flight direction
- $|R| = \frac{\rho}{2}A_PC_R(\omega_PR_P)^2R_P$
- $C_H$ $C_R$ $\mu = \frac{V}{\omega_PR_P}$

[BEMT]Blade Elemental and Momentum Theory : calculate forces for each element and sum them up

Fixed-Wing

Ailerons (rolling)
Elevator (pitching)
Rudder (yawing)

$\alpha$ $\atan(w/u)$
- stall $\begin{cases}\alpha\uparrow \to c_L\uparrow & \alpha <\text{stall} \\ \alpha^\uparrow \to c_L\downarrow & \alpha > \text{stall}\end{cases}$ stall is not safe
$\beta$ $\beta = \asin(v/V)$
$\theta$ : pitch angle
$\phi$ $x$ -axis
$\psi$ : yaw angle
$\gamma$ $\textbf v_a$
$\xi$ $\textbf v_a$
$\textbf v_w$ : wind velocity
$\textbf v$ : ground-based inertial velocity
$L$ $L=\frac{1}{2}\rho V^2 S{c_L}(\alpha)$ $S$ is the surface area
$D$ $D=\frac{1}{2}\rho V^2 S{c_D}$ $\textbf v_a$
- $\underset{\alpha}{\text{max}}\frac{c_L(\alpha)}{c_D(\alpha)}$
- $\underset{\alpha}{\text{max}}\frac{c_L(\alpha)^3}{c_D(\alpha)^2}$
$L_m$ $L_m = \frac{1}{2}\rho V^2 S{bc_l}$ $b$ is the wing span
$M_m$ $M_m = \frac{1}{2}\rho V^2 S{\bar cc_m}$ $c$ is the chord,
$N_m$ $N_m = \frac{1}{2}\rho V^2 S{bc_n}$

Steady Level Turning Flight

$_{\mathcal B}\dot{\textbf v}_a = ~_{\mathcal B}\dot{\omega}=0$
$\gamma = 0$
$\phi = \text{const}$ :

\begin{array}{l} L \cos ϕ = m g \\ L \sin ϕ = m \frac{V^{2}}{R} = m R \dot{ξ} \\ D = T \end{array} \to \begin{array}{l} \dot{ξ} = \frac{g \tan ϕ}{V} \\ L \propto \frac{1}{\cos ϕ} \\ V \propto \sqrt{\frac{1}{\cos ϕ}} \end{array}

$\mathcal L_1$ Guidance

\begin{array}{l} a_{s} = \frac{V^{2}}{R} = 2 \frac{V^{2} \sin η}{L_{1}} \\ \dot{ϕ} \approx \dot{ξ} = ϕ_{d} = atan (\frac{a_{s}}{g}) \end{array}

Total Energy Control System

$\dot E_{spec} = \frac{\dot E_{tot}}{mgV}=\frac{\dot V}{g}+\sin\gamma\approx \frac{\dot V}{g}+\gamma$
$\dot E_{dist} = \gamma - \frac{\dot V}{g}$

Modeling for Control (Linearized Plant)

Longitudinal Plant
$\Delta \delta_e, \Delta\delta_T$
$\Delta u, \Delta w, \Delta q,\Delta\theta$
- $\mathbb C$ $\omega=5\text{rad}/s$
  exchange between kinetic and potential energy: slow
- $\mathbb C$ $\omega = 0.6~\text{rad}/s$
  oscillation of angle of attack : fast
Lateral Plant
$\Delta \delta_a, \Delta \delta_r$
$\Delta v,\Delta p, \Delta r, \Delta\phi,\Delta\psi$
- $\mathbb R$ : unstable
- $\mathbb C$ $\omega = 5\text{rad}/s$
- $\mathbb R$

Statements

Kinematics

$\cross$ $T_{\mathcal {AB}}$ $\mathcal B$ $\mathcal A$ $v$
$\checkmark$ $C_{\mathcal {AB}}$ $\mathcal B$ $\mathcal A$ $v$
$\cross$ : A planar two-link robot arm has a unique solution to the inverse kinematic problem
$\cross$ : For a planar two-link robot arm, the differential inverse kinematic algorithm always converges to the same solution irrespective of initial configuration
$\cross$ : floating base in 3-dimensional space there exists a choice of generalized coordinates such that the analytical and geometric orientation Jacobian are equal

Dynamics

$\cross$ : Given a perfect model, a robotic arm controlled by a PD controller with gravity compensation can achieve zero tracking error for any desired trajectory.
$\cross$ $7\times 7$

Legged Robot

$\checkmark$ : For a bipedal system with two point feet on the ground, every torque command results in a unique acceleration
$\cross$ $\dot u^∗_{consistent}$ is achieved by a unique torque command

Rotor Craft

$\cross$ : A classic hexacopter (multi-rotor with 6 propellers) with all propellers spinning in the same plane is a fully actuated platform
$\checkmark$ : The yaw motion for a quadcopter is controlled by the drag moment of the propeller
$\cross$ : The attitude dynamics of a quadcopter can be stabilized by a proportional controller only
$\cross$ : The hub force on a rotor in froward flight results mostly due to an imbalance of the lift forces on the advancing and the retreating blade.
$\checkmark$ : BEMT can be used to model propeller characteristics, where momentum theory enables solving for induced velocities.
$\cross$ : A swashplate has generally three degree of freedom . One to control the cyclic pitch and two to control the collective pitch.
$\cross$ : A rotor in forward motion has a reverse flow region on the advancing blade
$\cross$ : In a front-rear rotor configuration, the yaw motion is steered by differential drag torques of the rotors.
$\checkmark$ : According to the momentum theory, the power consumption decrease to zero by increasing the disc area to infinity

Fixed Wing

$\cross$ : The magnitude of the GPS velocity can be used directly as an airspeed measurement
$\cross$ : The heading of a conventional aircraft is controlled primarily by the rudder
$\checkmark$ : Wind disturbances are typically modeled/mitigated within the guidance- level loops of a fixed-wing autopilot
$\checkmark$ : Minimum airspeed demand during a coordinated turn increases as the turning radius decreases, assuming constant angle of attach
$\cross$ : in a coordinate turn, the sideslip force causes the needed centripetal acceleration