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3 Chapter Kinematics As noted in the Introduction, the study of dynamics can be decomposed into the study of kinematics and kinetics. For the translational motion of a particle of mass m, this decomposition amounts to expressing Newton s second law, m = f r (3.1) a 2nd-order vector di erential equation, as the two 1st-order vector di erential equations r = p/m p = f (3.2) (3.3) Here r is the position vector of the particle relative to an inertial origin O, p = mr (or mv) is the linear momentum of the particle, and f is the sum of all the forces acting on the particle. Equation (3.2) is the kinematics di erential equation, describing how position changes for a given velocity; i.e., integration of Eq. (3.2) gives r(t). Equation (3.3) is the kinetics di erential equation, describing how velocity changes for a given force. It is also important to make clear that the dot , (), represents the rate of change of the vector as seen by a xed (inertial) observer (reference frame). In the case of rotational motion of a reference frame, the equivalent to Eq. (3.2) is not as simple to express. The purpose of this chapter is to develop the kinematic equations of motion for a rotating reference frame, as well as the conceptual tools for visualizing this motion. In Chapter 4 we develop the kinetic equations of motion for a rigid body. This chapter begins with the development of attitude representations, including reference frames, rotation matrices, and some of the variables that can be used to describe attitude motion. Then we develop the di erential equations that describe attitude motion for a given angular velocity. These equations are equivalent to Eq. (3.2), which describes translational motion for a given translational velocity. 3-1 Copyright C. D. Hall August 26, 2005 3-2 f X m $$$ $ x CHAPTER 3. KINEMATICS T r O E Figure 3.1: Dynamics of a particle 3.1 Attitude Representations In this section we discuss various representations of the attitude or orientation of a rigid body. We begin by discussing reference frames, vectors, and their representations in reference frames. The problem of representing vectors in di erent reference frames leads to the development of rotations, rotation matrices, and various ways of representing rotation matrices, including Euler angles, Euler parameters, and quaternions. 3.1.1 Reference Frames A reference frame, or coordinate system, is generally taken to be a set of three unit vectors that are mutually perpendicular. An equivalent de nition is that a reference frame is a triad of orthonormal vectors. Triad of course means three, and orthonormal means orthogonal and normal. The term orthogonal is nearly synonymous with the term perpendicular, but has a slightly more general meaning when dealing with other sorts of vectors (which we do not do here). The fact that the vectors are normalized means that they are unit vectors, or that their lengths are all unity (1) in the units of choice. We also usually use right-handed or dextral reference frames, which simply means that we order the three vectors in an agreed-upon fashion, as described below. The reason that reference frames are so important in attitude dynamics is that following the orientation of a reference frame is completely equivalent to following the orientation of a rigid body. Although no vehicle is perfectly rigid, the rigid body model is a good rst approximation for studying attitude dynamics. Similarly, no spacecraft (or planet) is a point mass, but the point mass model is a good rst approximation for studying orbital dynamics. We normally use a triad of unit vectors, denoted by the same letter, with subscripts 1,2,3. For example, an inertial frame would be denoted by 1 , 2 , 3 , an orbital frame i i i Copyright C. D. Hall August 26, 2005 3.1. ATTITUDE REPRESENTATIONS 3-3 by { 1 , o2 , o3 }, and a body- (or spacecraft-) xed frame by b1 , b2 , b3 . The hats o are used to denote that these are unit vectors. We also use the notation Fi , Fo , and Fb , to represent these and other reference frames. The orthonormal property of a reference frame s base vectors is de ned by the dot products of the vectors with each other. Speci cally, for a set of orthonormal base vectors, the dot products satisfy 1 1 = 1 1 2 = 0 1 3 = 0 i i i i i i 2 1 = 0 2 2 = 1 2 3 = 0 i i i i i i 3 1 = 0 3 2 = 0 3 3 = 1 i i i i i i which may be written more concisely as i j = i i 1 0 if if i=j i=j (3.5) (3.4) or even more concisely as i j = ij i i (3.6) where ij is the Kronecker delta, for which Eq. (3.5) may be taken as the de nition. We often nd it convenient to collect the unit vectors of a reference frame into a 3 1 column matrix of vectors, and we denote this object by = 2 i i i 3 1 i (3.7) This matrix is a rather special object, as its components are unit vectors instead of scalars. Hughes1 introduced the term vectrix to describe this vector matrix. Using this notation, Eq. (3.4) can be written as i i T i which is the 3 3 identity matrix. The superscript T on transposes the matrix from a column matrix (3 1) to a row matrix (1 3). The right-handed or dextral property of a reference frame s base vectors is de ned by the cross products of the vectors with each other. Speci cally, for a right-handed set of orthonormal base vectors, the cross products satisfy 1 1 = 0 1 2 = 3 1 3 = 2 i i i i i i i i 2 1 = 3 2 2 = 0 2 3 = 1 i i i i i i i i 3 1 = 2 3 2 = 1 3 3 = 0 i i i i i i i i (3.9) 1 0 0 = 0 1 0 =1 0 0 1 (3.8) Copyright C. D. Hall August 26, 2005 3-4 This set of rules may be written more concisely as i j = ijk k i i i where ijk is the permutation symbol, de ned as ijk CHAPTER 3. KINEMATICS (3.10) Equation (3.9) can also be written as i i T 1 for i, j, k an even permutation of 1,2,3 = 1 for i, j, k an odd permutation of 1,2,3 0 otherwise (i.e., if any repetitions occur) = 3 i i 2 0 (3.11) 3 i 0 1 i 2 i 1 i 0 The superscript is used to denote a skew-symmetric 3 3 matrix associated with a 3 1 column matrix. Speci cally, if a is a 3 1 matrix of scalars ai , then a1 0 a3 a2 a2 a = a3 0 a1 a= a3 a2 a1 0 = i (3.12) (3.13) Note that a satis es the skew-symmetry property (a )T = a . 3.1.2 Vectors A vector is an abstract mathematical object with two properties: direction and length (or magnitude). Vector quantities that are important in this course include, for example, angular momentum, h, angular velocity, , and the direction to the sun, . s Vectors are denoted by a bold letter, with an arrow (hat if a unit vector), and are usually lower case. Vectors can be expressed in any reference frame. For example, a vector, v, may be written in the inertial frame as v = v1 1 + v2 2 + v3 3 i i i (3.14) The scalars, v1 , v2 , and v3 , are the components of v expressed in Fi . These components are the dot products of the vector v with the three base vectors of Fi . Speci cally, Since the vectors are unit vectors, these components may also be written as i v1 = v cos 1 , v2 = v cos 2 , v3 = v cos 3 (3.16) v1 = v 1 , v2 = v 2 , i i v3 = v 3 i (3.15) where v = v is the magnitude or length of v, and j is the angle between v and j i for j = 1, 2, 3. These cosines are also called the direction cosines of v with respect to Copyright C. D. Hall August 26, 2005 3.1. ATTITUDE REPRESENTATIONS 3-5 3 i T v 1 i O E 2 i Figure 3.2: Components of a vector Fi . We frequently collect the components of a vector v into a column matrix v, with three rows and one column: v1 v = v2 v3 (3.17) A bold letter without an overarrow (or hat) denotes such a matrix. Sometimes it is necessary to denote the appropriate reference frame, in which case we use vi , vo , vb , etc. A handy way to write a vector in terms of its components and the base vectors is to write it as the product of two matrices, one the component matrix, and the other a matrix containing the base unit vectors. For example, v = [v1 v2 v3 ] 1 i 2 i 3 i = vi T i (3.18) Recall that the subscript i denotes that the components are with respect to Fi . Using this notation, we can write v in terms of di erent frames as v = vi T = vo T { } = vb T b i o (3.19) and so forth. There are two types of reference frame problems we encounter in this course. The rst involves determining the components of a vector in one frame (say Fi ) when the components in another frame (say Fb ) are known, and the relative orientation of the two frames is known. The second involves determining the components of a vector in Copyright C. D. Hall August 26, 2005 3-6 CHAPTER 3. KINEMATICS a frame that has been reoriented or rotated. Both problems involve rotations, which are the subject of the next section. 3.1.3 Rotations Suppose we have a vector v, and we know its components in Fb , denoted vb , and we want to determine its components in Fi , denoted vi . Since v = vi T = vb T b i (3.20) we seek a way to express in terms of b , say i =R b i where R is a 3 3 transformation matrix. Then we can write v = vi T = vi T R b = vb T b i (3.22) Comparing the last two terms in this equation, we see that vi T R = vb T Transposing both sides , we get RT vi = vb (3.24) (3.23) (3.21) Thus, to compute vi , we just need to determine R and solve the linear system of equations de ned by Eq. (3.24). If we write the components of R as Rij , where i denotes the row and j denotes the column, then Eq. (3.21) may be expanded to 1 = R11 b1 + R12 b2 + R13 b3 i 2 = R21 b1 + R22 b2 + R23 b3 i 3 = R31 b1 + R32 b2 + R33 b3 i (3.25) (3.26) (3.27) Comparing these expressions with Eqs. (3.14 3.16), it is evident that R11 = 1 b1 , i R12 = 1 b2 , and in general, Rij = i bj . Using direction cosines, we can write i i R11 = cos 11 , R12 = cos 12 , and in general, Rij = cos ij , where ij is the angle between i and bj . Thus R is a matrix of direction cosines, and is frequently referred to i as the DCM (direction cosine matrix). As with Eq. (3.8), where we have T = 1, i i we can also write R as the dot product of with b T , i.e., i R= b i T (3.28) Recall that to transpose a product of matrices, you reverse the order and transpose each matrix. Thus, (ABT C)T = CT BAT . Copyright C. D. Hall August 26, 2005 3.1. ATTITUDE REPRESENTATIONS 3-7 If we know the relative orientation of the two frames, then we can compute the matrix R, and solve Eq. (3.24) to get vi . As it turns out, it is quite simple to solve this linear system, because the inverse of a direction cosine matrix is simply its transpose. That is, R 1 = RT To discover this fact, note that it is simple to show that i b = RT (3.30) (3.29) using the same R as in Eq. (3.21). Comparing this result with Eq. (3.21), it is clear that Eq. (3.29) is true. A matrix with this property is said to be orthonormal, because its rows (and columns) are orthogonal to each other and they all represent unit vectors. This property applied to Eq. (3.24) leads to vi = Rvb (3.31) Thus R is the transformation matrix that takes vectors expressed in Fb and transforms or rotates them into Fi , and RT is the transformation that takes vectors expressed in Fi and transforms them into Fb . We use the notation Rbi to represent the rotation matrix from Fi to Fb , and Rib to represent the rotation matrix from Fb to Fi . Thus vb = Rbi vi and vi = Rib vb (3.32) The intent of the ordering of b and i in the superscripts is to place the appropriate letter closest to the components of the vector in that frame. The ordering of the superscripts is also related to the rows and columns of R. The rst superscript corresponds to the reference frame whose base vector components are in the rows of R, and the second superscript corresponds to the frame whose base vector components are in the columns of R. Similarly the superscripts correspond directly to the dot product notation of Eq. (3.28); i.e., Rib = b T , and Rbi = b T . i i Looking again at Eqs. (3.25 3.27), it is clear that the rows of R are the components of the corresponding i , expressed in Fb , whereas the columns of R are the components i of the corresponding bj , expressed in Fi . To help remember this relationship, we write the rotation matrix Rib as follows: i1b T Rib = i2b T = i3b T b1i b2i b3i (3.33) Although the discussion here has centered on frames b and , the development i is the same for any two reference frames. Copyright C. D. Hall August 26, 2005 3-8 CHAPTER 3. KINEMATICS 3.1.4 Euler Angles Computing the nine direction cosines of the DCM is one way to construct a rotation matrix, but there are many others. One of the easiest to visualize is the Euler angle approach. Euler reasoned that any rotation from one frame to another can be visualized as a sequence of three simple rotations about base vectors. Let us consider the rotation from Fi to Fb through a sequence of three angles 1 , 2 , and 3 . We begin with a simple rotation about the 3 axis, through the angle 1 . We i }. Using the rules developed above denote the resulting reference frame as Fi , or {i for constructing Ri i , it is easy to show that the correct rotation matrix is Ri i so that vi = R3 ( 1 )vi (3.35) cos 1 sin 1 0 = R3 ( 1 ) = sin 1 cos 1 0 0 0 1 (3.34) The subscript 3 in R3 ( 1 ) denotes that this rotation matrix is a 3 rotation about the 3 axis. Note that we could have performed the rst rotation about 1 (a 1 i 2 (a 2 rotation). Thus there are three possibilities for the rst simple rotation) or i rotation in an Euler angle sequence. For the second simple rotation, we cannot choose , since this choice would amount to simply adding to 1 . Thus there are only two i3 choices for the second simple rotation. We choose 2 as the second rotation axis, rotate through an angle 2 , and call the i resulting frame Fi , or { }. In this case, the rotation matrix is i Ri so that vi = R2 ( 2 )vi = R2 ( 2 )R3 ( 1 )vi (3.37) i cos 2 0 sin 2 1 0 = R2 ( 2 ) = 0 sin 2 0 cos 2 (3.36) Now Ri i = R2 ( 2 )R3 ( 1 ) is the rotation matrix transforming vectors from Fi to Fi . Leonhard Euler (1707 1783) was a Swiss mathematician and physicist who was associated with the Berlin Academy during the reign of Frederick the Great and with the St Petersburg Academy during the reign of Catherine II. In addition to his many contributions on the motion of rigid bodies, he was a major contributor in the elds of geometry and calculus. Many of our familiar mathematical notations are due to Euler, including e for the natural logarithm base, f () for functions, i for 1, for , and for summations. One of my favorites is the special case of Euler s formula: ei + 1 = 0, which relates 5 fundamental numbers from mathematics. Copyright C. D. Hall August 26, 2005 3.1. ATTITUDE REPRESENTATIONS 3-9 For the third, and nal, rotation, we can use either a 1 rotation or a 3 rotation. We choose a 1 rotation through an angle 3 , and denote the resulting reference frame Fb , or {b}. The rotation matrix is Rbi so that vb = R1 ( 3 )vi = R1 ( 3 )R2 ( 2 )R3 ( 1 )vi (3.39) 1 0 0 = R1 ( 3 ) = 0 cos 3 sin 3 0 sin 3 cos 3 (3.38) Now the matrix transforming vectors from Fi to Fb is Rbi = R1 ( 3 )R2 ( 2 )R3 ( 1 ). For a given rotational motion of a reference frame, if we can keep track of the three Euler angles, then we can track the changing orientation of the frame. As a nal note on Euler angle sequences, recall that there were three axes to choose from for the rst rotation, two to choose from for the second rotation, and two to choose from for the third rotation. Thus there are twelve (3 2 2) possible sequences of Euler angles. These are commonly referred to by the axes that are used. For example, the sequence used above is called a 3-2-1 sequence, because we rst rotate about the 3 axis, then about the 2 axis, and nally about the 1 axis. It is also possible for the third rotation to be of the same type as the rst. Thus we could use a 3-2-3 sequence. This type of sequence (commonly called a symmetric Euler angle set) leads to di culties when 2 is small, and so is not widely used in vehicle dynamics applications. Example 3.1 Let us develop the rotation matrix relating the Earth-centered inertial (ECI) frame Fi and the orbital frame Fo . We consider the case of a circular orbit, with right ascension of the ascending node (or RAAN), , inclination, i, and argument of latitude, u. Recall that argument of latitude is the angle from the ascending node to the position of the satellite, and is especially useful for circular orbits, since argument of periapsis, , is not de ned for circular orbits. We denote the ECI frame (Fi ) by { and the orbital frame (Fo ) by { }. Interi}, o mediate frames are designated using primes, as in the Euler angle development above. We use a 3-1-3 sequence as follows: Begin with a 3 rotation about the inertial 3 i axis through the RAAN, . This rotation is followed by a 1 rotation about the i1 axis through the inclination, i. The last rotation is another 3 rotation about the 3 i axis through the argument of latitude, u. We denote the resulting reference frame by { }, since it is not quite the desired o orbital reference frame. Recall that the orbital reference frame for a circular orbit has its three vectors aligned as follows: { 1 } is in the direction of the orbital velocity o vector (the v direction), { 2 } is in the direction opposite to the orbit normal (the h o direction), and { 3 } is in the nadir direction (or the r direction). However, the o Copyright C. D. Hall August 26, 2005 3-10 CHAPTER 3. KINEMATICS frame resulting from the 3-1-3 rotation developed above has its unit vectors aligned in the r, v, and h directions, respectively. Now, it is possible to go back and choose angles so that the 3-1-3 rotation gives the desired orbital frame; however, it is instructive to see how to use two more rotations to get from the { } frame to the { } frame. Speci cally, if we perform another o o 3 rotation about { 3 } through 90 and a 1 rotation about { 1 } through 270 , we o o arrive at the desired orbital reference frame. These nal two rotations lead to an interesting rotation matrix: Roo 0 1 0 0 1 0 1 0 0 = R1 (270 )R3 (90 ) = 0 0 1 1 0 0 = 0 0 1 (3.40) 1 0 0 0 0 1 0 1 0 Careful study of this rotation matrix reveals that its e ect is to move the second row to the rst row, negate the third row and move it to the second row, and negate the rst row and move it to the third row. So, the rotation matrix that takes vectors from the inertial frame to the orbital frame is Roi = Roo R3 (u)R1 (i)R3 ( ) which, when expanded, gives su c cu ci s su s + cu ci c cu si si s si c ci Roi = cu c + su ci s cu s su ci c su si (3.41) (3.42) where we have used the letters c and s as abbreviations for cos and sin, respectively. Now, we also need to be able to extract Euler angles from a given rotation matrix. This exercise requires careful consideration of the elements of the rotation matrix and careful application of various inverse trigonometric functions. suppose Thus, we are given a speci c rotation matrix with nine speci c numbers. We can extract the three angles associated with Roi as developed above as follows: i = cos 1 ( R23 ) u = tan 1 ( R33 /R13 ) = tan 1 ( R21 /R22 ) (3.43) (3.44) (3.45) 3.1.5 Euler s Theorem, Euler Parameters, and Quaternions The Euler angle sequence approach to describing the relative orientation of two frames is reasonably easy to develop and to visualize, but it is not the most useful approach for many vehicle dynamics problems. Another of Euler s contributions is the theorem that tells us that only one rotation is necessary to reorient one frame to another. This theorem is known as Euler s Theorem and is formally stated as Copyright C. D. Hall August 26, 2005 3.1. ATTITUDE REPRESENTATIONS Euler s Theorem. The most general motion of a rigid body with a xed point is a rotation about a xed axis. 3-11 Thus, instead of using three simple rotations (and three angles) to keep track of rotational motion, we only need to use a single rotation (and a single angle) about the xed axis mentioned in the theorem. At rst glance, it might appear that we are getting something for nothing, since we are going from three angles to one; however, we also have to know the axis of rotation. This axis, denoted a, is called the Euler axis, or the eigenaxis, and the angle, denoted , is called the Euler angle, or the Euler principal angle. For a rotation from Fi to Fb , about axis a through angle , it is possible to express bi the rotation matrix R , in terms of a and , just as we expressed Rbi in terms of the Euler angles in the previous section. Note that since the rotation is about a, the Euler axis vector has the same components in Fi and Fb ; that is, Rbi a = a (3.46) and the subscript notation (ai or ab ) is not needed. We leave it as an exercise to show that Rbi = cos 1 + (1 cos )aaT sin a (3.47) where a is the column matrix of the components of a in either Fi or Fb . Equation (3.46) provides the justi cation for the term eigenaxis for the Euler axis, since this equation de nes a as the eigenvector of Rbi associated with the eigenvalue 1. A corollary to Euler s Theorem is that every rotation matrix has one eigenvalue that is unity. Given an Euler axis, a, and Euler angle, , we can easily compute the rotation bi matrix, R . We also need to be able to compute the component matrix, a, and the angle , for a given rotation matrix, R. One can show that = cos 1 a = 1 (trace R 1) 2 (3.48) (3.49) 1 RT R 2 sin So, we can write a rotation matrix in terms of Euler angles, or in terms of the Euler axis/angle set. There are several other approaches, or parameterizations of the attitude, and we introduce one of the most important of these: Euler parameters, or quaternions. We de ne four new variables in terms of a and . q = a sin q4 2 = cos 2 (3.50) (3.51) Copyright C. D. Hall August 26, 2005 3-12 CHAPTER 3. KINEMATICS The 3 1 matrix q forms the Euler axis component of the quaternion, also called the vector component. The scalar q4 is called the scalar component. Collectively, these four variables are known as a quaternion, or as the Euler parameters. We use the notation q to denote the 4 1 matrix containing all four variables; that is, q = qT q4 T (3.52) A given a and correspond to a particular relative orientation of two reference frames. Thus a given q also corresponds to a particular orientation. It is relatively easy to show that the rotation matrix can be written as 2 R = q4 qT q 1 + 2qqT 2q4 q (3.53) We now have three basic ways to parameterize a rotation matrix: Euler angles, Euler axis/angle, and Euler parameters. Surprisingly there are many other parameterizations, some of which are not named after Euler. However, these three su ce for the topics in this course. To summarize, a rotation matrix can be written as R = Ri ( 3 )Rj ( 2 )Rk ( 1 ) R = cos 1 + (1 cos )aaT sin a R = 2 q4 qT q 1 + 2qqT 2q4 q We also need to express q in terms of the elements of R: 1 q4 = 1 + trace R 2 R23 R32 1 q = R31 R13 4q4 R12 R21 (3.54) (3.55) (3.56) (3.57) (3.58) The subscripts i, j, k in the Euler angle formulation indicate that any of the twelve Euler angle sequences may be used. That is, using set notation, k {1, 2, 3}, j {1, 2, 3}\k, and i {1, 2, 3}\j. Before leaving this topic, we need to establish the following rule: Rotations do not add like vectors. The Euler axis/angle description of attitude suggests the possibility of representing a rotation by the vector quantity . Then, if we had two sequential rotations, say a 1 a1 and 2 a2 , then we might represent the net rotation by the vector sum of these two: 1 a1 + 2 a2 . This operation is not valid, as the following example illustrates. Suppose that 1 and 2 are both 90 , then the vector sum of the two supposed rotation vectors would be /2( 1 + a2 ). Since vector addition is commutative, a the resulting rotation vector does not depend on the order of performing the two rotations. However, it is easy to see that the actual rotation resulting from the two individual rotations does depend on the order of the rotations. Thus the rotation vector description of attitude motion is not valid. Copyright C. D. Hall August 26, 2005 3.2. ATTITUDE KINEMATICS 3-13 3.2 Attitude Kinematics In the previous sections, we developed several di erent ways to describe the attitude, or orientation, of one reference frame with respect to another, in terms of attitude variables. The comparison and contrast of rotational and translational motion is summarized in Table 3.1. The purpose of this section is to develop the kinematics Table 3.1: Comparison of Rotational and Translational Motion Variables Kinematics D.E.s Translational Motion Rotational Motion (x, y, z) ( 1 , 2 , 3 ) (a, ) (q, q4 ) r = p/m ? ? ? di erential equations (D.E.s) to ll in the ? in Table 3.1. To complete the table, we rst need to develop the concept of angular velocity. 3.2.1 Angular Velocity The easiest way to think about angular velocity is to rst consider the simple rotations developed in Section 3.1.4. The rst example developed in that section was for a 32-1 Euler angle sequence. Thus we are interested in the rotation of one frame, Fi , with respect to another frame, Fi , where the rotation is about the 3 axis (either 3 i or 3 ). Then, the angular velocity of Fi with respect to Fi is i i i i i = 1 3 = 1 3 (3.59) Note the ordering of the superscripts in this expression. Also, note that this vector quantity has the same components in either frame; that is, i i i = i i i This simple expression results because it is a simple rotation. For the 2 rotation from Fi to Fi , the angular velocity vector is i i 0 = 0 1 (3.60) i i = 2 2 = 2 2 (3.61) which has components i i i i = i i 0 = 2 0 (3.62) Copyright C. D. Hall August 26, 2005 3-14 CHAPTER 3. KINEMATICS Finally, for the 1 rotation from Fi to Fb , the angular velocity vector is i bi = 3 b1 = 3 1 with components bi = bi b i 3 = 0 0 (3.63) (3.64) Thus, the angular velocities for simple rotations are also simple angular velocities. Now, angular velocity vectors add in the following way: the angular velocity of Fb with respect to Fi is equal to the sum of the angular velocity of Fb with respect to Fi , the angular velocity of Fi with respect to Fi , and the angular velocity of Fi with respect to Fi . Mathematically, bi = bi + i i + i i (3.65) However, this expression involves vectors, which are mathematically abstract objects. In order to do computations involving angular velocities, we must choose a reference frame, and express all these vectors in that reference frame and add them together. Notice that in Eqs. (3.60,3.62, and 3.64), the components of these vectors are given in di erent reference frames. To add them, we must transform them all to the same frame. In most attitude dynamics applications, we use the body frame, so for this example, we develop the expression for bi in Fb , denoting it bi . b The rst vector on the right hand side of Eq. (3.65) is already expressed in Fb in Eq. (3.64), so no further transformation is required. The second vector in Eq. (3.65) i i is given in Fi and Fi in Eq. (3.62). Thus, in order to transform i i (or i i ) into Fb , we need to premultiply the column matrix by either Rbi or Rbi . Both matrices give the exact same result, which is again due to the fact that we are working with simple rotations. Since Rbi is simpler [Rbi = R1 ( 3 ), whereas Rbi = R1 ( 3 )R2 ( 2 )], we use i i = Rbi i i , or b i i b i 0 0 1 0 0 = 0 cos 3 sin 3 2 = cos 3 2 0 sin 3 cos 3 0 sin 3 2 (3.66) Similarly, i i must be premultiplied by Rbi = R1 ( 3 )R2 ( 2 ) hence i i i b sin 2 1 0 cos 2 0 sin 2 1 0 0 1 0 = 0 cos 3 sin 3 0 0 = cos 2 sin 3 1 (3.67) sin 2 0 cos 2 0 sin 3 cos 3 1 cos 2 cos 3 1 Copyright C. D. Hall August 26, 2005 3.2. ATTITUDE KINEMATICS 3-15 Now we have all the angular velocity vectors of Eq. (3.65) expressed in Fb and can add them together: i bi = bi + b i + i i b b b 3 0 sin 2 1 = 0 + cos 3 2 + cos 2 sin 3 1 0 sin 3 2 cos 2 cos 3 1 3 sin 2 1 = cos 3 2 + cos 2 sin 3 1 sin 3 2 + cos 2 cos 3 1 (3.68) (3.69) (3.70) 1 sin 2 0 1 = cos 2 sin 3 cos 3 0 2 cos 2 cos 3 sin 3 0 3 (3.71) The last version of this equation is customarily abbreviated as bi = S( ) b (3.72) where = [ 1 2 3 ]T , and S( ) obviously depends on which Euler angle sequence is used. For a given Euler angle sequence, it is relatively straightforward to develop the appropriate S( ). Often it is clear what angular velocity vector and reference frame we are working with, and we drop the sub and superscripts on . Thus if the s are known, then they can be integrated to determine the s, and then the components of can be determined. This integration is entirely analogous to knowing x, y, and z, and integrating these to determine the position x, y, and z. In this translational case however, one usually knows x, etc., from determining the T velocity; i.e., v = [x y z] . The velocity is determined from the kinetics equations of motion as in Eq. (3.3). Similarly, for rotational motion, the kinetics equations of motion are used to determine the angular velocity, which is in turn used to determine the s, not vice versa. In the next section, we develop relationships between the time derivatives of the attitude variables and the angular velocity. 3.2.2 Kinematics Equations We begin by solving Eq. (3.72) for , which requires inversion of S( ). It is straightforward to obtain: 0 sin 3 / cos 2 cos 3 / cos 2 1 1 = 0 cos 3 sin 3 2 = S 1 sin 3 sin 2 / cos 2 cos 3 sin 2 / cos 2 3 (3.73) Thus, if we know the s as functions of time, and have initial conditions for the three Euler angles, then we can integrate these three di erential equations to obtain the Copyright C. D. Hall August 26, 2005 3-16 CHAPTER 3. KINEMATICS s as functions of time. Careful examination of S 1 shows that some of the elements of this matrix become large when 2 approaches /2, and indeed become in nite when 2 = /2. This problem is usually called a kinematic singularity, and is one of the di culties associated with using Euler angles as attitude variables. Even though the angular velocity may be small, the Euler angle rates can become quite large. For a di erent Euler angle sequence, the kinematic singularity occurs at a di erent point. For example, with symmetric Euler angle sequences, the kinematic singularity always occurs when the middle angle ( 2 ) is 0 or . Because the singularity occurs at a di erent point for di erent sequences, one way to deal with the singularity is to switch Euler angle sequences whenever a singularity is approached. Hughes1 provides a table of S 1 for all 12 Euler angle sequences. Another di culty is that it is computationally expensive to compute the sines and cosines necessary to integrate Eq. (3.73). As we indicated in Section 3.1.5, other attitude variables may be used to represent the orientation of two reference frames, with the Euler axis/angle set, and quaternions, being the most common. We now provide the di erential equations relating (a, ) and q to . For the Euler axis/angle set of attitude variables, the di erential equations are = aT 1 a = a cot a a 2 2 (3.74) (3.75) The kinematic singularity in these equations is evidently at = 0 or 2 , both of which correspond to R = 1 which means the two reference frames are identical. Thus it is reasonably straightforward to deal with this singularity. There is, however, the one trig function that must be computed as varies. For Euler parameters or quaternions, the kinematic equations of motion are q= 1 2 q + q4 1 qT = Q( ) q (3.76) There are no kinematic singularities associated with q, and there are no trig functions to evaluate. For these reasons, the quaternion, q, is the attitude variable of choice for most aerospace vehicle attitude dynamics applications. 3.3 3.4 Visualization Summary This chapter provides the basic background for describing reference frames, their orientations with respect to each other, and the transformation of vectors from one frame to another. Copyright C. D. Hall August 26, 2005 3.5. SUMMARY OF NOTATION 3-17 3.5 Summary of Notation There are several subscripts and superscripts used in this and preceding chapters. This table summarizes the meanings of these symbols. Symbol v 1 , 2 , 3 i i i Fi Meaning vector, an abstract mathematical object with direction and length the three unit base vectors of a reference frame i i i the reference frame with base vectors 1 , 2 , 3 typically Fi denotes an inertial reference frame whereas Fb denotes a body- xed frame, and Fo denotes an orbital reference frame a column matrix whose 3 elements are the unit vectors a column matrix whose 3 elements are the components of the vector v expressed in Fi a column matrix whose 3 elements are the components of the vector v expressed in Fb rotation matrix that transforms vectors from Fi to Fb a column matrix whose 3 elements are the Euler angles 1 , 2 , 3 an angular velocity vector the angular velocity of Fb with respect to Fi the angular velocity of Fb with respect to Fi expressed typically used in Euler s equations the angular velocity of Fb with respect to Fi expressed not commonly used the angular velocity of Fb with respect to Fi expressed there are applications for this form i vi vb Rbi of Fi bi bi b bi i bi a in Fb in Fi in Fa 3.6 References and Further Reading Most satellite attitude dynamics and control textbooks cover kinematics only as a part of the dynamics presentation. Pisacane and Moore2 is a notable exception, providing a detailed treatment of kinematics before covering dynamics and control. Shuster3 provided an excellent survey of the many attitude representation approaches, including many interesting historical comments. Wertz s handbook4 also covers some of this material. Kuipers5 provides extensive details about quaternions and their applications, but does not include much material on dynamics applications. Copyright C. D. Hall August 26, 2005 3-18 BIBLIOGRAPHY Bibliography [1] Peter C. Hughes. Spacecraft Attitude Dynamics. John Wiley & Sons, New York, 1986. [2] Vincent L. Pisacane and Robert C. Moore, editors. Fundamentals of Space Systems. Oxford University Press, Oxford, 1994. [3] Malcolm D. Shuster. A survey of attitude representations. Journal of the Astronautical Sciences, 41(4):439 517, 1993. [4] J. R. Wertz, editor. Spacecraft Attitude Determination and Control. D. Reidel, Dordrecht, Holland, 1978. [5] Jack B. Kuipers. Quaternions and Rotation Sequences. Princeton University Press, Princeton, 1999. 3.7 Exercises 1. Verify the validity of Eq. (3.8) by direct calculation. 2. Develop the rotation matrix for a 3-1-2 rotation from Fa to Fb . Your result should be a single matrix in terms of 1 , 2 , and 3 [similar to Eq. (3.42)]. 3. Using the relationship between the elements of the quaternion and the Euler angle and axis, verify that Eq. (3.53) and Eq. (3.47) are equivalent. 4. Select two unit vectors and convince yourself that the statement Rotations do not add like vectors described on p. 3-12 is true. 5. Develop S( ) for a 2-3-1 rotation from Fi to Fb so that bi = S( ) . Where b is S( ) singular? 6. A satellite at altitude 700 km is pointing at a target that is 7 away from nadir (i.e., = 7 ). What is the range D to the target, and what is the spacecraft elevation angle . 7. A satellite in a circular orbit with radius 7000 km is intended to target any point within its instantaneous access area (IAA) for which the elevation angle is greater than 5 . What must the range of operation be for the attitude control system? 8. The Tropical Rainfall Measuring Mission is in a circular orbit with altitude 350 km, and inclination i = 35 . If it must be able to target any point that is within its IAA and within the tropics, then what must the ACS range of Copyright C. D. Hall August 26, 2005 3.8. PROBLEMS 3-19 operations be, and at what minimum elevation angle must the sensor be able to operate? 3.8 Problems 0.45457972 0.43387382 0.77788868 ab 0.29351236 R = 0.34766601 0.89049359 0.82005221 0.13702069 0.55564350 Consider the following rotation matrix Rab that transforms vectors from Fb to Fa : 1. Use at least two di erent properties of rotation matrices to convince yourself that Rab is indeed a rotation matrix. Remark on any discrepancies you notice. 2. Determine the Euler axis a and Euler principal angle , directly from Rab . Verify your results using the formula for R(a, ). Verify that Ra = a. 3. Determine the components of the quaternion q, directly from Rab . Verify your results using the formula for R( ). Verify your results using the relationship q between q and (a, ). 4. Derive the formula for a 2-3-1 rotation. Determine the Euler angles for a 2-3-1 rotation, directly from Rab . Verify your results using the formula you derived. 5. Write a short paragraph discussing the relative merits of the three di erent representations of R. 6. This problem requires numerical integration of the kinematics equations of motion. You should have a look at the MatLab Appendix. Suppose that Fb and Fa are initially aligned, so that Rba (0) = 1. At t = 0, Fb begins to rotate with angular velocity b = e 4t [sin t sin 2t sin 3t]T with respect to Fa . Use a 1-2-3 Euler angle sequence and make a plot of the three Euler angles vs. time for t = 0 to 10 s. Use a su ciently small step size so that the resulting plot is smooth. Use the quaternion representation and make a plot of the four components of the quaternion vs. time time for t = 0 to 10 s. Use a su ciently small step size so that the resulting plot is smooth. Use the results of the two integrations to determine the rotation matrix at t = 10 s. Do this using the expression for R( ) and for R( ). Compare the q results. Copyright C. D. Hall August 26, 2005 3-20 BIBLIOGRAPHY Copyright C. D. Hall August 26, 2005

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