Kinematics of robotsM. C. Martín Montes Rivera

Kinematics

Kinematics applied to robots it is the science that studies movement of a robot without

considering forces

There are 2 kinds of kinematical problems in robots, first is known as forward kinematics and

second inverse kinematics.

Forward kinematics it is to get the position and orientation of end effector with joint

coordinates.

Inverse Kinematics allows to get the required joint coordinates to reach a desired

position and orientation

Forward Kinematics

To find a position or orientation of an end effector robot, requires of a

homogeneous matrix, which have this information.

With this homogeneous matrix it is possible to get equations depending of

joint coordiantes.

Example with a few DOF

Robots with more DOF

Methodology using transformation matrixes which are simpler to understand

and program in a computer.

In general industrial robots have n degrees of freedom, n joints and n links,

which means that every link-joint it’s a DOF.

Transformation matrix that changes two consecutive coordinate system it is

represented by (last one to actual).

Examples

Exercise

Obtain the equations that represents position of end effector robot

Denavit-Hartenberg Algorithm

Denavit-Hartenberg (D-H) AlgorithmFollowing directions according to its joints, revolute axis

point out of the rotation and prismatic in the same

direction.

X axis must be selected pointing to end effector

Example

Exercise D-H and Kinematic equations

Exercise D-H and kinematic equations

Exercise D-H and kinematic equations

Exercise D-H and kinematic equations

Exercise D-H and kinematic equations

Exercise D-H and kinematic

equations

Orientation Angles

Orientation

Considering that a vector with magnitude k and components given by

If there is a rotation around angle theta in OZ axis, it is required to perform

the bellow operations in order to make concurrency of k with OZL, and then

get the rotation angle.

And with the next equivalences

After perform operations and considering that

Homogeneous matrix obtained is:

Then all components are equaled to

Orientation

Considering every angle between 0º and 180º and equations 3.74 and 3.75 can

be obtained

With alpha for x angle, beta

for y angle and theta for zeta

angle.

References

Fundamentos de Robótica: Antonio Barrientos, Luis Felipe Peñín, Carlos Balaguer, Rafael Aracil

Robot Dynamics and Control: Mark W. Spong, Seth Hutchinson, and M. Vidyasagar