[Modern Robotics] 강좌 2: 로봇 기구학 #2

 

 
 
https://www.coursera.org/learn/modernrobotics-course2-ko

현대 로봇공학, 강좌 2: 로봇 기구학

 
 
Inverse kinematics(IK): the calculation of the joint configuration from the end effector frame configuration(postion+orientation).
●For n-revolute joint serial robots, $T_{sb}\in SE(3)⟶\theta \in {\mathbb{R}}^n$
●Widely used representations:

●Unlike the forward kinematics of serial robots, it can have ①a unique solution, ②multiple solutions, or ③no solution
●Some systems like the 6R PUMA or the Standford arm have analytic IK solutions. 
●Most systems don't have an analytical solution, or we can't find one. So, numerical methods are used.
└e.g. Newton-Raphson method, Method of optimization)
●Even in cases where an analytic solution exists, numerical methods are used to improve accuracy.
 
 
 
Newton-Raphson method: An iterative numerical method for finding a function's root using the only linear part of the Taylor series approximation.
●Main concept: Finding a guess error(${\theta }^{k+1}-{\theta }^k$) from the result error($x-g({\theta }^k)$). (My own explanation)
●Only one root is found at a single set of iterations, and the result depends on the initial root guess. (Root that is the closest to the initial guess)
●Three possibilities exist: ①Find the desired root, ②Find the undesired root, ③Fail to find root(Divergence)
 
●For a algebraic equation $g(\theta )=x_d$, initial root guess ${\theta }^0$      ($\theta ,x_d\in \mathbb{R}$):
Repeating ${\theta }^{k+1}-{\theta }^k={(\frac{\partial g}{\partial \theta }({\theta }^k))}^{-1}\{x-g({\theta }^k)\}$
 
●For a matrix equation $g(\theta )=x_d$, initial root guess ${\theta }^0$      ($\theta ,x_d\in {\mathbb{R}}^n$):
Repeating ${\theta }^{k+1}-{\theta }^k=J^{-1}({\theta }^k)\{x_d-g({\theta }^k)\}$
 
 
 

(Moore-Penrose) Pseudo-inverse $A^{†}$: A generalization of matrix inverse that exists for any matrix(even for non-square, singular matrix=noninvertible matrix).
 
For $A\in {\mathbb{R}}^{m\times n}$, $m>n$ (tall matrix), $A^{†}=A^T(A{A}^T{)}^{-1}$
For $A\in {\mathbb{R}}^{m\times n}$, $m<n$ (fat matrix), $A^{†}=(A{A}^T{)}^{-1}{A}^T$
 
●The matrix equation $Ax=b$ and the solution $x=A^{†}b$ results in cases.
①If solutions exist, $x=A^{†}b$ provides the solution with the smallest norm.
②If no solutions exist, $x=A^{†}b$ provides the value that minimizes the error.
 
 
 
IK using Newton-Raphson method: Find a root of the IK matrix equation using the Newton-Raphson method.
●Main point: Convert result error transformation $T_{bd}\in SE(3)$ into result error twist $\nu \in {\mathbb{R}}^6$ with the matrix log of rigid-body motion.
$[{\nu }_s]=log(T_{bd}({\theta }^k))=log(T_{sb}^{-1}({\theta }^k)T_{sd})$
●FK $T_{sb}(\theta )$ is needed.
 
Space frame: ${\theta }^{k+1}-{\theta }^k=J_s^{†}({\theta }^k){\nu }_s$
 
Body frame: ${\theta }^{k+1}-{\theta }^k=J_b^{†}({\theta }^k){\nu }_b$   (Generally used)
 

 
 
 
+)Modern Robotics book example 6.1 Matlab code.

 
 
 
 
 
 
Closed chain: Any kinematic chain that contains loops
●Delta robot, Stewart platform $\in$ Parallel mechanism $\in$ Closed chain
●Not all joints are actuated
●Joints satisfy loop-constraint-equations. This makes an additional Jacobian called Constraint Jacobian.
●As opposed to open chains, IK has a unique solution while FK can have multiple solutions.
 
 

========================================================================
정확한 정보 전달보단 공부 겸 기록에 초점을 둔 글입니다.
틀린 내용이 있을 수 있습니다.
틀린 내용이나 다른 문제가 있으면 댓글에 남겨주시면 감사하겠습니다. : )
========================================================================