Motion Kinematics – Lecture Series 3

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 ZG, XG and zi axes

Accounting for this overall change:

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 ZG of 30˚ followed by Rotation XG of 45˚ and a translation of [7,4,-10]T An Example: 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 Pi in G space for both cases • CASE 1: P1 defined wrt the origin • Case 1.5: After Rotation (45˚) about ZG

And Finally: After an Elongation of 600 in the xB direction: Where Gdx,B1.5 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 • Develop the unit vector in the direction of u • Develop the Rodriguez Rotation Matrix

Building RodriguesMatix (MathCad)

Continuing with HTM • The Translational Vector: • 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

Motion Kinematics – Lecture Series 3