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Review of the Geometry of Screw Axes

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Review of the Geometry of Screw Axes Review of the geometry of screw Axes Nasser M. Abbasi Dept. of Mechanical engineering, UCI. 2006 Compiled on May 19, 2020 at 4:18am Contents 1 Denitions, Important formulas, and terminology 2 1.1 Planar displacement . 2 1.2 Spatial displacement . 2 1.3 Homogeneous Transformation . 2 1.4 The rotation axis . 2 1.5 A Pole or a Fixed point . 2 1.6 Rotational displacement . 3 1.7 Rodrigues vector 푏 .................................. 4 1.8 Screw.......................................... 4 1.9 Plücker coordinates of a line . 4 1.10 screw defined in terms of the plücker coordinates of its axis . 5 1.11 Important relations between Cayley Matrix 퐵 .................. 5 2 Introduction to the rotation matrix in 3D 6 3 Screw displacement 10 3.1 Derivation for expression for finding reference point for screw axis . 13 4 The Screw Matrix 16 5 The spatial displacement of screws 17 6 The screw axis of a displacement 24 7 References 25 Report writing at UCI during work on my MSc in Mechanical Engineering. 2006 1 2 1 Denitions, Important formulas, and terminology 1.1 Planar displacement General motion of a body in 2D that includes rotation and translation. We use [퐴] for the rotation matrix, the vector 풅 for the translation and the matrix [푇] for displacement. Hence we write 푿 = [퐴]풙 + 풅 1.2 Spatial displacement General motion of a body in 2D that includes rotation and translation. Another name for this is rigid displacement. Distance between points in a body remain unchanged before and after spatial displacement. 1.3 Homogeneous Transformation Recall that the transformation that defines spatial displacement, which is given by 푿 = [퐴]풙 + 풅 is non-linear due to the presence of the translation term 풅. It is more convenient to be able to work with linear transformation, therefore we add a fourth component to the position vector which is always 1 and rewrite the transformation, now calling it as 푇 which maps 푋 to 푥 as 푋 = 푇 푥 Or in full component form 푇 ⎧ ⎫ ⎡ ⎤ ⎧ ⎫ ⎪ ⎪ ������������������������� ⎪ ⎪ ⎪푋1⎪ ⎢푎11 푎12 푎13 푑1⎥ ⎪푥1⎪ ⎪ ⎪ ⎢ ⎥ ⎪ ⎪ ⎨⎪푋 ⎬⎪ ⎢푎 푎 푎 푑 ⎥ ⎨⎪푥 ⎬⎪ 2 = ⎢ 21 22 23 2⎥ 2 ⎪ ⎪ ⎢ ⎥ ⎪ ⎪ ⎪푋3⎪ ⎢푎31 푎32 푎33 푑3⎥ ⎪푥3⎪ ⎪ ⎪ ⎣⎢ ⎦⎥ ⎪ ⎪ ⎩ 1 ⎭ 0 0 0 1 ⎩ 1 ⎭ Or in short form 푇 ⎧ ⎫ ⎧ ⎫ ⎪ ⎪ �����������⎡ ⎤ ⎪ ⎪ ⎨⎪푿⎬⎪ ⎢ 퐴 풅⎥ ⎨⎪풙⎬⎪ ⎪ ⎪ = ⎢ ⎥ ⎪ ⎪ ⎩⎪ 1 ⎭⎪ ⎣000 1⎦ ⎩⎪1⎭⎪ 1.4 The rotation axis The set of points that remain fixed during the rotation defined by [퐴]. We use Rodrigues vector 풃 to define the rotation axis. 1.5 A Pole or a Fixed point A pole is point that remains fixed during planar or spatial displacement. Planar displacement (2D) have a pole, but 3D spatial displacement do not have a pole in general, since the 3 requirement for a pole is to have an inverse for [퐼 − 퐴] which for 3D is not possible since 퐴 has an eigenvalue of 1. The following diagram is an example of a pole under planar displacement. Z Pole. This point has the same coordinates in both frames of references Y For spatial displacement (3D), one condition, when satisfied will result in a fixed point. This condition occurs when the translation vector 풅 is perpendicular to the rotation vector 풃. The pole in this case is given by 풃 × (풅 − 풃 × 풅) 풄 = 2풃 ⋅ 풃 Not only is this point 풄 fixed, by any point on the line 풄+푡푺 is also fixed. Where 푡 is a parameter and 푺 is a unit vector. Hence we can a fixed line under such a spatial displacement, and this line is called the rotation axis. 1.6 Rotational displacement This is a special case of spatial displacement when the translation vector 풅 is perpendicular to the rotation axis 풃. To make the dierence more clear, the following diagram is an illustration of spatial displacement which is not rotational displacement, and one which is. Rotation F Axis F Rotational Spatial displacement displacement 4 1.7 Rodrigues vector 푏 Defines the rotation axis under [퐴]. Using 푺 as the unit vector along 풃, then 풃 = 푘푺 where 푘 휃 is the length of vector 풃 given by tan � � where 휃 is the rotation angle around the rotation 2 axis. Hence we can write 휃 풃 = tan � � 푺 2 We see that the 휃 ‖풃‖ = tan � � 2 휋 Hence at 휃 = ±1800 , ‖풃‖ = tan �± � which goes to infinity. A plot of the function tan (훼) is 2 휋 below showing the discontinuities at ± 2 1.8 Screw Screw is defined as a pair of vectors (푾 , 푽)푇 such that 푾 ⋅ 푽 ≠ 0 and |푾 | ≠ 1. The pitch of 푾 ⋅푽 the screw 푝 is defined as 휔 푾 ⋅푾 1.9 Plücker coordinates of a line Given a line defined by a parametric equation 퐿 (푘) = 푪 +푘푺, where 퐶 is a reference⎧ point⎫ and ⎪ ⎪ ⎨⎪ 푺 ⎬⎪ 푆 is a unit vector along the line, the Plücker coordinates of this line is given by ⎪ ⎪ 푪 × 푺 ⎩⎪푪 × 푺⎭⎪ represents the moment of the line around the origin of the reference frame. The following diagram helps to illustrates this. 5 L C S line k F j s i C S Line L has Plücker coordinates C S 1.10 screw dened in terms of the plücker coordinates of its axis We said above that a screw is defined as a pair of vectors (푾 , 푽)푇. Given the plücker coordinates of the screw line (푾 , 푪 × 푾)푇, then we can write the pair of vectors that defines the screw associated with this screw line as follows ⎧ ⎫ ⎪ ⎪ ⎨⎪ 휔푺 ⎬⎪ ⎪ ⎪ ⎩휔푪 × 푺 + 휔푝휔푺⎭ Where 푝휔 is the screw pitch and 휔 = |푾| 1.11 Important relations between Cayley Matrix 퐵 These are some important formulas put here for quick reference. 푇 (풙1 + 풙2) [퐵] (풙1 + 풙2) = 0 See (4A) below for derivation of the above where ⎡ ⎤ ⎢ 0 −푏 푏 ⎥ ⎢ 푧 푦 ⎥ ⎢ ⎥ [퐵] = ⎢ 푏푧 0 −푏푥⎥ ⎣⎢ ⎦⎥ −푏푦 푏푥 0 and also 휃 [퐵] = tan [푆] 2 Where ⎡ ⎤ ⎢ 0 −푠 푠 ⎥ ⎢ 푧 푦 ⎥ ⎢ ⎥ [푆] = ⎢ 푠푧 0 −푠푥⎥ ⎣⎢ ⎦⎥ −푠푦 푠푥 0 6 From [퐵] we obtain a row vector and call it 푇 풃 = �푏푧, 푏푦, 푏푧� (This is rodrigues vector) From [푆] we obtain a row vector and call it 푇 풔 = �푠푥, 푠푦, 푠푧� (This is unit vector along the screw axis). We also have 휃 [퐵] = tan [푆] 2 And in terms of the vectors 풃, 풔 the above becomes 휃 풃 = tan 풔 2 2 Introduction to the rotation matrix in 3D We seek to derive an expression for the rotation matrix in 3D. Consider a point 푥1 in 3D being acted upon by a rotation matrix 퐴. Let the final coordinates of this point be 푥2. We consider the position vectors of these points, so we will designate these points by their position vectors 풙1 and 풙2 from now on. The following diagram illustrate this. z z Z’ Z’ x2 Y’ x1 Y’ y y Rotation A x x X’ X’ BEFORE AFTER rotation around x axis by angle |x1 | |x2 | Hence we have that 풙2 = 퐴풙1 Since |풙1| = |풙2| we can write 풙1 ⋅ 풙1 = 풙2 ⋅ 풙2 풙1 ⋅ 풙1 − 풙2 ⋅ 풙2 = 0 (풙1 − 풙2) ⋅ (풙1 + 풙2) = 0 (1) The geometric meaning of the above last equation is shown in the following diagram 7 2 x Z’ 1 x x 1 x 2 x2 x1 Y’ y x X’ To introduce the rotation matrix 퐴 into the equations, we can write 풙1 −풙2 as 풙1 −퐴풙1. Hence 풙1 − 풙2 = [퐼 − 퐴] 풙1 (2) Similarly, we obtain 풙1 + 풙2 = [퐼 + 퐴] 풙1 (3) From (3) we obtain −1 [퐴 + 퐼] (풙1 + 풙2) = 풙1 Substitute the above expression for 풙1 into (2) we obtain −1 풙1 − 풙2 = [퐴 − 퐼][퐴 + 퐼] (풙1 + 풙2) (3A) We now call the matrix [퐴 − 퐼] [퐴 + 퐼]−1 as 퐵 퐵 = [퐴 − 퐼] [퐴 + 퐼]−1 (3B) We can also write from above the following 퐴 = [퐼 − 퐵]−1[퐼 + 퐵] (3C) Now rewrite (3A) as 풙1 − 풙2 = [퐵] (풙1 + 풙2) (4) 8 What is the geometric meaning of [퐵]? we see that it is an operator that acts on vector 풙1 + 풙2 to produce the vector 풙1 − 풙2, however from the diagram above we see that the vector 풙1 − 풙2 is perpendicular to 풙1 + 풙2 and scaled down. Hence 푇 (풙1 + 풙2) [퐵] (풙1 + 풙2) = 0 (4A) This implied that 퐵 is skew-symmetric and have the form given by ⎡ ⎤ ⎢ 0 −푏 푏 ⎥ ⎢ 푧 푦 ⎥ ⎢ ⎥ 퐵 = ⎢ 푏푧 0 −푏푥⎥ (4A) ⎣⎢ ⎦⎥ −푏푦 푏푥 0 푇 Matrix 퐵 can be written as a column vector called 풃 = [푏푥, 푏푦, 푏푧] such that for any vector 푦 we have [퐵] 풚 = 풃 × 풚 The vector 풃 is called Rodrigues vector. Let the vector 풙1 + 풙2 = 풚 then we write [퐵] 풚 = 풃 × 풚 Where 풃 is some vector perpendicular to 풚 such that its cross product with 푦 results in [퐵] 풚 The vector 풃 is the vector that defines the rotation axis. The length of this vector is 푘 and a unit vector along 풃 is called 푺, Hence we can write 풃 = 푘푺 Now we solve for 퐵. We will use equation (3B) to find the form of 퐵 for dierent rotations. Consider for example the 3D rotation around the 푥 axis given by ⎡ ⎤ ⎢1 0 0 ⎥ ⎢ ⎥ ⎢ ⎥ 퐴 = ⎢0 cos 휃 − sin 휃⎥ ⎣⎢ ⎦⎥ 0 sin 휃 cos 휃 From (3B) we obtain 9 Z Z’ Y’ Y b x X’ Rodrigues vector b defines the rotation axis For example for 휃 = 450 the 퐵 matrix is ⎡ ⎤ ⎢0 0 0 ⎥ ⎢ ⎥ ⎢ ⎥ 퐵 = ⎢0 0 −0.414214⎥ ⎣⎢ ⎦⎥ 0 0.414214 0 Hence we see that 푏푧 = 0, 푏푦 = 0, 푏푥 = −0.41421 hence ⎡ ⎤ ⎢0.41421⎥ ⎢ ⎥ ⎢ ⎥ 풃 = ⎢ 0 ⎥ ⎣⎢ ⎦⎥ 0 Hence geometrically, Rodrigues vector is along the 푥 axis and in the positive direction as illustrated in the following diagram.
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