7.2 HOMOGENEOUS MATRICES

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This method still uses geometry to determine the position of the robot, but it is put into an ordered method using matrices.

Consider the planar robot below,

The basic approach to this method is,

1. On the base, each joint, and the tool of the robot, attach a reference frame (most often x-y-z). Note that the last point is labels `T' for tool. This will be a convention that I will generally follow.

2. Determine a transformation matrix to map between each frame. It is important to do this by assuming the joints are in their 0 joint positions. Put the joint positions in as variables.

3. Multiply the frames to get a complete transformation matrix.

The position and orientation can be read directly from the homogenous transformation matrix as indicated above.

To reverse the transform, we only need to invert the transform matrix - this is a direct result of the loop equation.

7.2.1 Denavit-Hartenberg Transformation (D-H)

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Designed as more specialized transforms for robots (based on homogenous transforms)

Zi-1 axis along motion of ith joint

Xi axis normal to Zi-1 axis, and points away from it.

Basic transform is,

1. rotate about Zi-1 by thetai (joint angle)

2. translate along Zi-1 by di (link offset)

3. translate along Xi by ai (link length)

4. rotate about Xi by alphai (link twist)

We can see how the D-H representation is applied using the two link manipulator from before

7.2.2 Orientation

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The Euler angles are a very common way to represent orientation in 3-space.

The main problem in representing orientation is that the angles of rotation must be applied one at a time, and by changing the sequence we will change the final orientation. In other words the three angles will not give a unique solution unless applied in the same sequence every time.

By fixing a set of angles by convention we can then use the three angles by themselves to define an orientation.

The convention described here is the Euler angles.

The sequence of orientation is,

Therefore to reorient a point in space we can apply the following matrix, to the position vectors, or axes vectors, (there will be more on these matrices shortly)

We can find these angles given a set of axis before and after.

7.2.3 Inverse Kinematics

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Basically we can find the joint angles for the robot based on the position of the end effector.

This is not a simple problem, and there are few reliable methods. This is partly caused by the non-unique nature of the problem. At best there are typically multiple, if not infinite numbers of equivalent solutions. The 2 dof robot seen before has two possible solutions.

We can do simple inverse kinematics with trigonometry.

If we have more complicated problems, we may try to solve the problem by examining the transform matrix,

7.2.4 The Jacobian

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A matrix of partial derivatives that relate the velocity of the joints, to the velocity of the tool.

The inverse Jacobian is used for motion control

Find the Jacobian and inverse Jacobian for the 2 dof robot.