1 Fundamentals of Robotics Linking perception to action 2. Motion of Rigid Bodies 南台科技大學電機工程系謝銘原

2 Chapter 2. Motion of Rigid Bodies 2.2 Cartesian Coordinate Systems 2.2 Cartesian Coordinate Systems 2.3 Projective Coordinate Systems 2.3 Projective Coordinate Systems 2.4 Translational Motions 2.4 Translational Motions 2.5 Rotational Motions 2.5 Rotational Motions 2.6 Composite Motions 2.6 Composite Motions Homogeneous Transformation Differential Homogeneous Transformation Successive Elementary Transformation Successive Elementary Rotations Euler Angles Equivalent Axis and Angle of Rotation

3 2.2 Cartesian Coordinate Systems Two references for describing the motions of a rigid body – Two references for describing the motions of a rigid body – The time reference (for velocity and acceleration, t ) The spatial reference (for position and orientation, X, Y, Z ) Cartesian Coordinate System – O 0 – X 0 Y 0 Z 0 Cartesian Coordinate System – O 0 – X 0 Y 0 Z 0

4 2.3 Projective Coordinate Systems Geometric operations Geometric operations Translation, Rotation, Scaling, Projection Typical projection – Typical projection – Display of 3D objects onto a 2D screen Visual perception (onto an image plane) Projective coordinates ( k – homogeneous coordinate ) Projective coordinates ( k – homogeneous coordinate )

5 2.4 Translational Motions Linear Displacement Linear Displacement The relative distance (displacement) Between the origin of frame 0 and the origin of frame 1

6 Translational transformation The Cartesian coordinates The Cartesian coordinates The equivalent projective coordinates The equivalent projective coordinates Homogeneous motion transformation matrix

Linear Velocity and Acceleration The displacement vector The displacement vector The corresponding linear velocity and acceleration The corresponding linear velocity and acceleration

8 2.5 Rotational Motion Rotational motion – Rotational motion – The rotation of a rigid body about any straight line (a rotation axis) Force, or torque Circular displacement Circular displacement

9 Rotation matrix Rotation matrix, R – inverse Rotation matrix, R – inverse Initial configuration Initial configuration After rotation After rotation

10 Example 2.4 Rotational transformation Rotational transformation from frame1 to frame0 Inverse (from frame0 to frame1)

11 The homogeneous motion transformation matrix Rotational motion Rotational motion Translational motion Translational motion

Circular Velocity and Acceleration Circular velocity Circular velocity caused by a rigid body’s angular velocity Circular acceleration Circular acceleration Vector is parallel to the rotation axis and its norm is equal to atat anan tangential acceleration vector centrifugal acceleration vector

Composite Motions Any complex motion can be treated as the combination of translational and rotational motions. Any complex motion can be treated as the combination of translational and rotational motions.

Homogeneous Transformation Homogeneous Transformation Homogeneous Transformation with

Differential Homogeneous Transformation Homogeneous Transformation Homogeneous Transformation By differentiating, By differentiating,

16 In a matrix form Homogeneous Transformation Homogeneous Transformation Differential Homogeneous Transformation Differential Homogeneous Transformation

Successive Elementary Translations Three successive translations Three successive translations

Successive Elementary Rotations Three successive rotations of frame 1 to frame 0 Three successive rotations of frame 1 to frame 0

19 The equivalent projective coordinates In robotics, the above equation describes the forward kinematics of a spherical joint having three degrees of freedom.

20 Useful expression Imagine now that  x,  y, and  z can undergo instantaneous variations with respect to time. Imagine now that  x,  y, and  z can undergo instantaneous variations with respect to time. The property of skew-symmetric matrix The property of skew-symmetric matrix We can derive the following equalities: We can derive the following equalities:

Euler Angles The three successive rotations The three successive rotations 1st elementary rotation can choose X, Y, or Z axes as its rotation axis. 2nd, 3rd, both have two axes to choose from. Total, 3*2*2 = 12 possible combinations These sets are commonly called Euler Angles.

22 Example 2.5 ZYZ Euler Angles 1st – about Z axis, rotation angle  1st – about Z axis, rotation angle  2nd – about Y axis, rotation angle  2nd – about Y axis, rotation angle  3rd – about Z axis, rotation angle  3rd – about Z axis, rotation angle   are called ZYZ Euler Angles  are called ZYZ Euler Angles

Equivalent Axis and Angle of rotation Euler angles Euler angles The set of minimum angles which fully determine a frame’s rotation matrix with respect to another frame. Each set has three angles which make three successive elementary rotations. Thus, the orientation of a frame, with respect to another frame, depends on three independent motion parameters even through the rotation matrix is a 3*3 matrix with 9 elements inside.

24 In robotics It is necessary to interpolate the orientation of a frame from its initial orientation to an actual orientation so that a physical rigid body (e.g. the end-effector) can smoothly execute the rotational motion in real space. It is necessary to interpolate the orientation of a frame from its initial orientation to an actual orientation so that a physical rigid body (e.g. the end-effector) can smoothly execute the rotational motion in real space.

25 Equivalent axis of rotation Now, imagine that there is am intermediate frame i which has the Z axis that coincides with the equivalent rotation axis r. Now, imagine that there is am intermediate frame i which has the Z axis that coincides with the equivalent rotation axis r.

26 Rotation If r is the equivalent axis of rotation for the rotational motion between frame 1 and frame 0: If r is the equivalent axis of rotation for the rotational motion between frame 1 and frame 0: The orientation of frame i with respect to frame 0, The orientation of frame i with respect to frame 0,

27 The solutions for r and  To derive, To derive, If  = 0, R z = I 3*3 If  = 0, R z = I 3*3 given