Homework 3. Chapter 4. Vector bases and rotation matrices 3.1 Dynamics project rotation matrices Update your typed problem description and identifiers as necessary. Include schematics showing the bodies, reference frames, unit vectors, points, constraints, etc. Include calculations (e.g., submit your MotionGenesis file) for relevant rotation matrices. 3.2 Efficient calculation of the inverse of a rotation matrix The following rotation matrix R relates two right-handed, orthogonal, unitary bases. Calculate its inverse by-hand (no calculator) in less than 30 seconds. R = 0.3830-0.6634 0.6428 0.9237 0.2795-0.2620-0.0058 0.6941 0.7198 R -1 = 3.3 Calculating dot-products, cross-products, and angles between vectors The following a R b rotation table relates two sets of right-handed, orthogonal, unit vectors, namely,, and,,. Perform the calculations below to two (or more) significant digits. a R b b z b y 0.9623-0.0842 0.2588 0.1701 0.9284-0.3304-0.2125 0.3619 0.9077 (a) Efficiently determine the following dot-products. = = = = = = (b) Determine the angles between the following vectors. (, ) = (, ) = (, ) = (, ) = (c) Express the unit vector û in the direction of 3 +4 as shown below. û = + (d) Perform the following calculations involving v 1 =2 and v 2 = +. v 1 v 2 = ( v 1, v 2 )= v 1 v 2 = + = + (e) Express v = + in terms of,,. v = + + Copyright c 1992-2009 by Paul Mitiguy 325 Homework 3
3.4 Rotation tables for a landing gear system. The figures below show three versions of the same landing gear system that consists of a strut which has a simple rotation relative to a fuselage. In each figure, n x,, n z is a set of orthogonal unit vectors fixed in and,, is a set of orthogonal unit vectors fixed in. However, these unit vectors have a different orientation in each figure. Determine the a R n rotation table for each figure. ote, each figure has two missing vectors (e.g., n x and are missing from the first figure). Use the fact that each set of vectors is right-handed to add the missing vectors to each figure. Verify your by-hand results by using the MotionGenesis command Rotate. Show your work by submitting your MotionGenesis file SimpleRotationMatrices.all. 1 n z a R n n x n z (a) 1 0 0 0 cos(θ) sin(θ) 0 -sin(θ) cos(θ) θ n x a R n n x n z (b) θ n x a R n n x n z (c) θ 1 The MotionGenesis commands for (a) are: RigidFrame,; Variable q;.rotatex(,q). The MotionGenesis commands for (c) are: RigidFrame,; Variable q;.rotateegativez(,q). Copyright c 1992-2009 by Paul Mitiguy 326 Homework 3
3.5 Rotation matrices for a crane and wrecking ball. The figure to the right shows a crane whose cab supports a boom that swings a wrecking ball C. There are three sets of mutually perpendicular right-handed unit vectors, namely n x,, n z ;,, ; and c x, c y, c z. The point of this problem is to relate these sets of unit vectors. L θ θ C L C C o ote: To relate the,, and n x,, n z unit vectors, it is helpful to redraw these vectors in a geometrically suggestive way as shown below. x (a) Use the definitions of sine and cosine to express each of,, in terms of n x,, n z. θ = cos(θ )n x + sin(θ ) θ = = (b) Fill in the second and third rows of the b R n rotation table shown to the right by extracting the various coefficients of the n x,, n z unit vectors in the previous results. b R n n x n z cos(θ ) sin(θ ) 0 (c) Form b R n,therotation matrix relating,, to n x,, n z. Then form its transpose. = x n z n x n z = x (d) To relate the c x, c y, c z and n x,, n z unit vectors, redraw these vectors in a geometrically suggestive way and then use the definitions of sine and cosine to express each of c x, c y, c z in terms of n x,, n z. Use these expressions to form the c R n rotation table. c x = c y = c z = c R n n x n z c x c y c z (e) Use matrix multiplication to form the b R c rotation table, i.e., b R c = b R n n R c. Simplify the results with the following trigonometric identities. sin(θ + θ C ) = sin(θ )cos(θ C )+ sin(θ C )cos(θ ) cos(-θ C ) = cos(θ C ) cos(θ + θ C ) = cos(θ )cos(θ C ) sin(θ )sin(θ C ) sin(-θ C ) = -sin(θ C ) b R c c x c y c z Copyright c 1992-2009 by Paul Mitiguy 327 Homework 3
(f) Verify your by-hand result by submitting the file CraneRotationMatrices.all which results from running the following MotionGenesis file. % File: CraneRotationMatrices.al %----------------------------------- RigidFrame,, C Variable q, qc.rotatez(, q ) % rotates about the "z-axis" relative to by an angle q C.RotateZ(, qc ) % C rotates about the "z-axis" relative to by an angle qc C =.GetRotationMatrix(C) % Rotation matrix relating and C Save CraneRotationMatrices.all Quit (g) Calculating rotation matrices with MotionGenesis is easier/harder than doing it by hand. 3.6 Vertical displacement of a bifilar pendulum (useful for calculating moment of inertia). The following figure shows a rigid human bone suspended by two inextensible cables 1 and 2, each of which is attached to a flat horizontal ceiling. Cable 1 attaches to the ceiling at point 1 of and to the bone at point 1 of. Cable 2 attaches to the ceiling at point 2 of and to the bone at point 2 of. Point o of is centered between 1 and 2. Point o of is centered between 1 and 2. Point cm ( s center of mass) and point o always lie directly below o. Initially, i lies directly below i (i=1, 2), respectively. is rotated by an angle θ about the vertical line through o and o. Relate y (the distance between o and o ) to L, h, andθ (defined in the following table). = 0 Complete the following table by calculating a numerical value for y (3 significant digits). Description Symbol Value Distance from 1 to 2 L 1m Distance from i to i (i=1, 2) h 1m s rotation angle in θ 135 Distance between o and o y m 1 o 2 Optional : Sketch y versus θ for 0 θ<180. 1 2 1 y (meters) 0.8 0.6 0.4 1 o 2 0.2 0 0 20 40 60 80 100 120 140 160 180 theta (degrees) cm Calculate a numerical value for ẏ (3 significant digits) when θ =0.5 rad sec. ẏ = m sec Copyright c 1992-2009 by Paul Mitiguy 328 Homework 3
3.7 Clinical determination of pelvis orientation (described in Section 4.10) (a) One way to mathematically describe the orientation of a rigid body (pelvis) in a reference frame (gait laboratory) is to first align,, with,,, respectively, and then subject to right-handed successive-rotations characterized by θ t, θ o,andθ r. For purposes of discussion, these successive-rotations are called TOR. Form b R a, the rotation matrix relating,, to,,. 2 b R a cos θ r cos θ t +sinθ o sin θ r sin θ t sin θ r cos θ o sin θ o sin θ r cos θ t sin θ t cos θ r sin θ o sin θ t cos θ r sin θ r cos θ t cos θ o cos θ r sin θ r sin θ t +sinθ o cos θ r cos θ t sin θ t cos θ o -sin θ o cos θ o cos θ t (b) nother way to mathematically describe the orientation of in is to align,, with,,, respectively, and then subject to right-handed successive-rotations characterized by θ r, θ o,andθ t. For purposes of discussion, these successive-rotations are called ROT. Form b R a, the rotation matrix relating,,,to,,. b R a cos θ r cos θ t sin θ o sin θ r sin θ t sin θ r cos θ t +sinθ o sin θ t cos θ r -sin θ t cos θ o -sin θ r cos θ o cos θ o cos θ r sin θ o sin θ t cos θ r +sinθ o sin θ r cos θ t sin θ r sin θ t sin θ o cos θ r cos θ t cos θ o cos θ t (c) Clinically, thepelvis elevation angle φ is defined as the angle of above the horizontal plane perpendicular to.expressφ in terms of θ r, θ o,andθ t,firstbyusingtor and then ROT. 3 TOR successive-rotations: φ = 90 acos (sin θ r sin θ t + sinθ o cos θ r cos θ t ) ROT successive-rotations: φ = 90 acos (sin θ o ) = θ o when -90 θ o 90 (d) Clinically, thepelvis progression angle ψ is defined as the angle of behind the vertical plane perpendicular to.expressψ in terms of θ r, θ o,andθ t,firstwithtor and then ROT. 4 TOR successive-rotations: ψ = 90 acos (sin θ r cos θ t sin θ o sin θ t cos θ r ) ROT successive-rotations: ψ = 90 acos (sin θ r cos θ o ) (e) Clinically, thepelvis lean angle γ is defined as the angle of below the horizontal plane perpendicular to.expressγ in terms of θ r, θ o,andθ t,firstbyusingtor and then ROT. TOR successive-rotations: γ = 90 acos (sin θ t cos θ r sin θ o sin θ r cos θ t ) ROT successive-rotations: γ = 90 acos (sin θ t cos θ o ) (f) The orientation of in is uniquely determined by the ROT angles θ r, θ o,andθ t. True/False The orientation of in is uniquely determined by φ, ψ, andγ. True/False The ROT angles θ r, θ o,andθ t are uniquely determined by the orientation of in. True/False φ, ψ, andγ are uniquely determined by the orientation of in. True/False 2 Consider using the MotionGenesis command.rotate(, odyyxz, θ t, θ o, θ r) and pass iour MotionGenesis file. 3 The results are simplified by noting acos [sin θ o] = acos [cos(90 θ o)] = 90 θ o. 4 The results are simplified by noting acos(- x) = 180 acos(x). Copyright c 1992-2009 by Paul Mitiguy 329 Homework 3
3.8 2D orientation of a rigid body. β Right-handed sets of orthogonal unit vectors,, and,, are fixed in reference frames and, respectively. Initially, and are oriented so a i = b j (i, j=x, y, z). Two vectors of interest are a vector α fixed in and a vector β fixed in, which can always be expressed as α = 4 +1 α β = 4 +1 Find the R rotation table knowing that β is initially aligned with α and then is subjected to a right-handed rotation in about a vector in the direction of α β, so that after the rotation β is β = 1 +4 a R b 0.4706-0.8824 0 0 1 3.9 3D orientation of a rigid body. β Right-handed sets of orthogonal unit vectors,, and,, are fixed in reference frames and, respectively. Initially, and are oriented so a i = b j (i, j=x, y, z). Two vectors of interest are a vector α fixed in and a vector β fixed in, which can always be expressed as α α = 4 +1 +3 β = 4 + 1 +3 Find the R rotation table knowing that β is initially aligned with α and then is subjected to a right-handed rotation in about a vector in the direction of α β, so that after the rotation β is β = 1 +4 +3 a R b 0.7263-0.5045-0.4669 0.6494 0.7263 0.2254 0.2254-0.4669 0.8551 3.10 Optional : Rotation and scaling of a vector. Referring to the previous problem, suppose β is both rotated and scaled ( β is not fixed in ). Knowing that β is initially aligned with α and then is subjected to a right-handed rotation in about a vector in the direction of α β, so that after the rotation β is β = 1 +4 +1 find the angle θ between the initial and final orientation of β and the scaling factor between the initial and final magnitude of β. θ = 59.44 scalingfactor = 0.832 Copyright c 1992-2009 by Paul Mitiguy 330 Homework 3
3.11 Spaceship docking maneuver and orientation. The figure below shows a space shuttle which is in the process of docking with the space station. To properly dock, the shuttle must reorient itself from the orientation shown in to the orientation shown in. To describe this reorientation maneuver, it is helpful to fix a right-handed set of mutually perpendicular unit vectors,, in andanothersetfixedin so that a i = b i (i=x, y, z) prior to the reorientation. fter the maneuver, = - and the angle between and is 45, i.e., the b R a rotation table relating,, to,, is ote: The matrix b R a below is the transpose of the matrix a R b in equation (4.10). b R a az 0 0-1 b 2 y 2 b 2 z 2 2 2 0-2 2 0 (a) One way to perform this maneuver is with the ody-xyz angles q x, q y,andq z, i.e., subject to successive right-handed rotations of q x, q y, q z,findq x, q y,andq z. 5 Either q x =90 q y =45 q z = -90 or q x = -90 q y = 135 q z =90 (b) nother way to change orientation is with ody-yx angles, i.e., subject to successive righthanded rotations of r y and r x. Using your physical intuition, find r y and r x. r y =90 r x =45 (c) nother way to characterize the orientation of in is with the Euler parameters ɛ 0, ɛ 1, ɛ 2, ɛ 3. Find the values of the Euler parameters associated with this maneuver. ɛ 0 =0.6533 ɛ 1 =0.2706 ɛ 2 =0.6533 ɛ 3 = -0.2706 (d) Yet another way to change orientation is to subject to a simple right-handed rotation characterized by θλ, whereθ is an angle and λ is a unit vector. Find θ and λ. θ =98.42 = 0.3574 b 1 +0.863 b 2 + -0.3574 b 3 λ = 0.3574 a 1 +0.863 a 2 + -0.3574 a 3 (e) Optional : The normalized eigenvector of the rotation matrix b R a corresponding to an eigenvalue of 1 is parallel to the vector λ. True/False. Hint: See Sections 4.14.3 and 4.2. 5 Helpful information for Homework 3.11 and the transpose of b R a is in Sections 4.9.2, 4.12.7, and in equation (4.13). Copyright c 1992-2009 by Paul Mitiguy 331 Homework 3