된사람 되기

 The following example is the puma 560 manipulator whose joint 5 value equal to zero.

 In mathematics, we say the matrix has reduced the rank.

We can consider the velocity ellipse of the robot.

For a set of joint angles which lie on a circle in the plane of all possible joint angle velocities,

we mapped that into an ellipse in the space of all possible end-effector velocities.

We can invert the Jacobian to get the joint angles according to the spatial velocity. 
(If the Jacobian is not singular)

For example in Puma560, the singular configuration occurs when the joint angle q5 equals to zero.

If the joint 4 and joint 6 rotate by counter direction, their movements compensate each other.

Under this circumstance, the end-effector pose of the Puma560 will not change!

A singular Jacobian matrix indicates that the end-effector motion is unable to move in a particular direction..

The jacobian is updated every time step.

We know  the inverse of the rotation matrix is the transpose of it.

Now we can get the vector omega since we know about the structure of skew symmetric matrix.

Combine the translation part and rotation part.

... for all joints.

Now this is a Jacobian matrix.

Each column means the effect on the spatial velocity of the robot due to each joint velocity.

* ref

[1] https://moocs.qut.edu.au/courses/814/learn

DENAVIT-HARTENBERG PARAMETERISATION

* ref

[1] https://moocs.qut.edu.au/courses/814/learn

* ref

[1] https://moocs.qut.edu.au/courses/814/learn

갈릴레오가 서로 다른 질량의 물체를 떨어뜨리는 실험을 했다는 것은 유명하다.

선조의 이러한 실험은 아인슈타인이 Equivalence principle을 정리하는 데 큰 도움이 되었을 것이다.

재미있는 것은 지구에 살고 있는 우리가 가만히 서있는 것과 1g (중력가속도) 만큼의 가속도로 발사되는 로켓에 타고 있는 것은 같은 가속도를 갖고 있다는 것이다. 

또한 가만히 떨어뜨린 물체가 갖는 가속도와 가만히 서있는 우리가 갖는 가속도 역시 같다.

우리 역시 중력에 의해 지구의 중심부로 떨어지고 있지만, 지면에 의해 떨어지지 않게 받쳐지고 있다고 보면 되겠다. (선반 위에 있는 물건처럼..)

그래서 우리의 발에서는 우리의 무게를 지탱하는 힘(반력)이 느껴지는 것이다.

중력가속도가 없다면 우리의 발이 그런 무게를 느낄 필요가 없어진다. 

(우주에서 무중력상태에 떠다니는 것들을 생각해보라!)

이러한 직관은 수식적으로도 증명이 되어있으니 궁금한 사람들은 Wikipedia를 찾아보면 되겠다. 

* ref

[1] https://moocs.qut.edu.au/courses/814/learn

[2] http://en.wikipedia.org/wiki/Equivalence_principle

Common technique to generate trajectory in robotics

but it has some performance problems..

more efficient technique to generate trajectory is the trapezoidal profile..

however it has a problem that robot can be unstable because of its discrete acc/deceleration.

(generally, this has lower bandwidth relatively.. it can be ignored)

* Ref :

[1] https://moocs.qut.edu.au/courses/814/learn

linear interpolation 방식으로 position은 간단하게 interpolation할 수 있었으나

orientation의 경우, rotation matrix를 interpolation하면 아래와 같은 문제가 발생한다. 

따라서 Euler angle이나 RPY를 interpolation해야 하는데, 이 경우에도 문제는 있다..

바로 아래와 같은 direction problem이다.

그렇다면 우리는 효율적인 orientation interpolation 방법에 대해 궁금해질 차례이다.

아래를 보면 그 해답을 알 수 있다.

* Ref :

[1] https://moocs.qut.edu.au/courses/814/learn

KINEMATIC control of a robot manipulator consists of solving the control problem in two stages, i.e., the desired task trajectory is transformed via the inverse kinematics into corresponding joint trajectories, which then constitute the reference inputs to some joint space control scheme [1]. This approach differs from operational space control [2] in the sense that manipulator kinematics is handled outside the control loop, thus allowing the problem of kinematic singularities and/or redundancy to be solved separately from the motion control problem. Moreover, most built-in controllers of industrial robots are based on joint servos and, thus, they can be easily embedded into a kinematic control strategy.

* Ref

[1] Fabrizio Caccavale, Student Member, IEEE, Stefano Chiaverini, and Bruno Siciliano, Senior Member, IEEE, Second-Order Kinematic Control of Robot Manipulators with Jacobian Damped LeastSquares Inverse: Theory and Experiments, IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 2, NO. 3, SEPTEMBER 1997.