Motion Kinematics – Lecture Series 3 ME 4135 – Fall 2011 R. Lindeke

Outline Of Motion Kinematics Rigid Body Motions ◦ Includes rotation as well as translations The Full blow Homogenous Transformation Matrix ◦ Coupling origin movement with reorientation ◦ Physical Definition ◦ Making Use of its power ◦ Building its Inverse Compound HTM’s is Rigid Motion Screw Coordinates

Rigid Motion The Body Frame (B) has been coincidently displaced by a vector d and reoriented about Z G, X G and z i axes

Accounting for this overall change:

An Example: Find the global position of a body point: [.5, 1.25, 3] T if the Body frame has been subjected to the following ‘operations’. A Rotation about Z G of 30 ˚ followed by Rotation X G of 45 ˚ and a translation of [7,4,-10] T Like this one with some extras

(As found in MathCad:)

Trying Another – A Rotational /Translational Device Initially (a) B and G are coincident – in (b) the Device has been rotated and then the upper arm has been extended, and note that B has been translated and rotated in this second image

Accounting for these – Where is P i in G space for both cases CASE 1: P 1 defined wrt the origin Case 1.5: After Rotation (45 ˚) about Z G

And Finally: After an Elongation of 600 in the x B direction: Where G d x,B 1.5 is the motion of the elongation axis of the “Upper Arm” resolved to the Ground Space

Wouldn’t it be Nice if … Combining Rotational and Translational Effects into a Grand Transformation could be done This is the role of the Homogenous Transformation Matrix It includes a “Rotational Submatrix” a “Origin Translational Vector” a “Perspective Vector” and finally a “Spatial Scaling Factor”

Lets see how it can be used in the two jointed robot Example

Dropping into MathCad: And Note: To use the original positional vector we needed to append a scaling factor to it as seen here Thus the position of P2 in the Ground space is this vector: [1378.9,1378.9,900] just as we found earlier

What’s Next Equipped with the ideas of the HTM and individual effects “easily” separated we should be able to address multi-linked machines – like robots But, before we dive in let’s examine some other Motion Kinematic tools before we! ◦ Axis Angle Rotation and Translation ◦ Inverse Transformations ◦ Screw Motions – see the text, they are a general extension of Axis Angle Rot/trans motion

Turning about a body axis – Developing the Rodriguez Transformation sub-matrix We’ll consider rotation about and translation along a Vector u

Developing an HTM 1. Develop the unit vector in the direction of u 2. Develop the Rodriguez Rotation Matrix

Building Rodrigues Matix (MathCad)

Continuing with HTM 3. The Translational Vector: 4. The Transform:

The HTM in Use: (MathCad)

What of the Inverse of the HTM? It is somewhat like the Inverse of the orientation matrix The Rotational sub- matrix is just the transpose (since we are reversing the point of view when doing an inverse) The positional vector changes to:

Leading to: Note these are DOT Products of 2 vectors – or scalars!

Summary The Homogeneous Transformation Matrix is a general purpose operator that accounts for operations (rotations and translations) taking place between Ground and Remote Frames of reference As such, they allow us to relate geometries between these spaces and actually perform the operations themselves (mathematically) Finally, they can be studied to understand the relationships (orientation and position) of two like geometried – SO3 – coordinate frames

Summary Their Inverses are simply constructed since they represent the geometry of the Ground in the geometry defined in the Remote Frames space Thus they are powerful tools to study the effects of motion in simple situations, complex single spaced twisting /translating motion as well as multi- variable motion as is seen in robotics