README.md

layout	title
page	Advanced Robotics

• Math/Physics
• FK/IK
• Dynamics
• Controls
• Motion Planning

Rotation

Q: What's the properties of rotation matrix?

A:

• A rotation matrix will always be a square and orthogonal matrix.
  • Column vectors are mutually orthogonal (orthonormal frame)
  • Column vectors have unit norm
• the transpose will be equal to the inverse of the matrix $R^T=R^{-1}$.
• The determinant of a rotation matrix will always be equal to 1. $det(R)=1$.
• Right-handed coordinate system
  • for clockwise rotation, a negative angle is used.
• Multiplication of rotation matrices will result in a rotation matrix.

Q: how to determine a matrix is a rotation matrix. given $ R^A_B = \begin{bmatrix} \cos \theta & 0& \sin \theta \ \sin \phi \sin\theta & \cos\phi & -\sin\phi\cos\theta\ -\cos\phi\sin\theta & \sin\phi & \cos\phi\cos\theta \ \end{bmatrix} $

A: calculate the $det(R^A_B)=1$ and prove $R^{-1} = R^T \Leftrightarrow RR^T=I$

Jacobian/FK/IK

Q: List one advantage and one disadvantage for using the Jacobian transpose ($J^T$) method, versus of the Jacobian Moore-Penrose pseudoinverse ($J^+$) when computing inverse kinematics solutions.

A:

	pros	cons	principles
$J^{-1}$	most precise	not exist when J drops rank
$J^T$	Simple computation (numerically robust) No matrix inversions	Needs many iterations until convergence in certain configurations Unpredictable joint configurations Non conservative	Project difference vector Dx on those dimensions q which can reduce it the most
$J^+=J^T(JJ^T)^{-1}$	Computationally fast (second order method) pseudoinverse gives the best possible solution in the sense of least squares the matrix ($I − J^+J$) performs a projection onto the nullspace of J, which can be exploited to achieve secondary goals	The pseudoinverse tends to have stability problems in the neighborhoods of singularities worked very poorly whenever the target positions were out of reach Matrix inversion necessary (numerical problems) Unpredictable joint configurations Non conservative	Shortest path in q-space
Damped least squares	pros	cons	principles

Q: Determine the relationship between angular velocities and time derivatives of the Euler angles.

A: $\omega_e = T(\phi_e)\dot{\phi_e}$

• For ZYZ: $$ T = \begin{bmatrix} 0& -s_\phi & c_\phi s_\theta \ 0& c_\phi & s_\phi s_\theta \ 1& 0& c_\theta\ \end{bmatrix} $$

$det(T) = -s^2_\phi s_\theta - c^2_\phi s_\theta=-s_\theta=0$. There exist angular velocities which cannot be expressed by means of $\dot \phi_e$ Reach Representation sigularity when $\theta=0,\pi$

• For ZYX (yaw pitch roll): $$ T=\begin{bmatrix} 0& -s_\phi & c_\phi c_\theta \ 0& c_\phi & s_\phi c_\theta \ 1& 0& -s_\theta\ \end{bmatrix} $$

$det(T) = -s^2_\phi c_\theta - c^2_\phi c_\theta=-c_\theta=0$ Reach Representation sigularity when $\theta=\pm\pi/2$

Dynamics

Q: How to derive Lagrange equation for robot?

A:

1. Compute position $p_i$, linear velocity $v_i=\frac{d p_i}{dt}$ and acceleration $a_i=\frac{d v_i}{dt}$, angular velocity $\omega_i$, acceleration $\alpha_i=\frac{d \omega_i}{dt}$ vector of CoM each joint.
2. Write / assume moment of Inertia $Ii$, mass $m_i$ of each link; Make assumptions that coordinate frame is aligned with principle axes so that Off diagonals (prodcts of inertia) are all ZEROs. Diagonal terms (moments of inertia) are inertia about each axis. (SEE Dynamics Q1.)
3. Calculate kinetic $\frac{1}{2}m_iv_i^Tv_i \pink+ \frac{1}{2}w_i^TI_iw_i$ AND potential energy $m_igh_i$ for each link.
4. Derive the Lagrange equation $L = KE-PE$.
5. Derive equation of motion with dynamics $\frac{d}{dt} (\frac{\partial{L}}{\partial{\dot{\theta}}}) -\frac{\partial{L}}{\partial{\theta}} = F_{ext} = \sum\tau$ and get $M,B,C,G$. (SEE Inverse Dynamics)

Control

Q: Compare different control

	Equation	When
Feedforward	$M(q_d)\ddot{q}_d + C(q_d,\dot{q}_d)+G(q_d) = \tau$	Ideal case.
Feedback	$\tau = \underbrace{k_p(\theta_d-\theta)}{\text{spring}}\underbrace{-k_v\dot{\theta}}\text{damper}$	Inverse dynamics control. Minimise steady-state error.
PID	$\tau = \underbrace{k_p(\theta_d-\theta)}{\text{P:Proportional }}\underbrace{-k_v\dot{\theta}}\text{ D:Derivative}+\underbrace{\red{k_i\int(\theta_d-\theta)dt}}_\text{I:Integral}$	NOT safe for robot. Integral.
Gravity compensation	$\tau = \underbrace{k_p(\theta_d-\theta)}{\text{P:Proportional }}\underbrace{-k_v\dot{\theta}}\text{ D:Derivative}+\underbrace{rmg\theta}_\text{gravity compensation}$	Model based. Need to know G
Linearlisation	$\tau = M(q)\red{(\ddot q_d+K_P(q_d-q)+K_D(\dot q_d-\dot q))} + C(q,\dot{q})+G(q)$	tracking complete trajectory. Error guaranteed to 0
Task space control	$\tau = J^TF+(I-J^T\bar{J}^T)\tau_0$	Desired motion in task space ( end effector ). Null space projection for secondary task.
Impedance control	$M(q_d)\ddot{q}_d + C(q_d,\dot{q}d)+G(q_d) = \tau \red\pm J^T_CF{ext}$	External forces applied by(+)/to(-) robot. Can be at any joint.
Force control	$\tau = J^TF_d+ ~g(q)$
Hybrid Force-motion control	$\tau = J_b^TF_b$	seperate control of force and motion. Force in one direction and motion in another. e.g. open a door(door's moving). Task space controller with constraints. Projection matrix $P(\theta)$ and $1-P$ project motion and force to subspace.

Motion planning

Q: Comparisions between different methods?

A:

	pros	cons	others
Potential Fields	Smooth, obstacle-free trajectory	in highly-cluttered environments, Attraction and repulsion forces can balance; The robot can get stuck
Voronoi Graphs	solve stuck	Paths are not smooth, requires post-processing	Focus on the free space
Regular Cell Decomposition	The solution is obstacle-free; The result can be used for multiple starting positions;	The solution still requires post-processing(not smooth); Cell decomposition can prune valid solutions;	wavefront propagation algorithm
Sampling Based Roadmaps	A single roadmap can be used for multiple problems(Multiple-query methods); Single queries are very fast to compute;	This method could fail to find an existing solution; Dynamic environments require new roadmaps or costly “maintenance work”;
Rapidly-exploring Random Trees (RRTs)	Generally fast; Probabilistically complete (but convergence not guaranteed); Solutions respect differential constraints;	Non-optimal; Generated trajectories may have discontinuities; Difficult to obtain high end-pose accuracy; single-query method, in non-holonomic planning (in general, under differential constraints), connecting configurations and checking for collisions is very time-consuming;	configuration space is sampled randomly
Lattice Based Methods	Time-consuming computation off-line; Leverage decades of AI search algorithms; Deterministic results; Can generate optimal solutions;	Lattices can be extremely large; Discretized sampling for (q_I , q_G );