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Supplementary Note on Rotation Matrix 1 Representation Of ECE5463 Spring 2018 Supplementary Note on Rotation Matrix Wei Zhang Department of Electrical and Computer Engineering Ohio State University This note summarizes and further clarifies the three major uses of a rotation matrix discussed in Lecture note 3: 1. to represent an orientation; 2. to change the reference frame in which a vector or a frame is represented; 3. to rotate a vector or a frame; Consider three frames fag, fbg, and fcg with unit axes f^xa; ^ya; ^zag, f^xb; ^yb; ^zbg, and f^xc; ^yc; ^zcg, respectively. 1 Representation of Orientation: Let R be a rotation matrix: 2 3 r11 r12 r13 R = r1 r2 r3 = 4 r21 r22 r23 5 r31 r32 r33 where r1; r2; r3 denote the three column vectors and rij denotes its (i; j) entry. We call R a repre- sentation of orientation of fbg relative to fag if the columns of R are the coordinates of unit axes f^xb; ^yb; ^zbg in fag, i.e., 8 h i ^x = r ^x + r ^y + r ^z = r > b 11 a 21 a 31 a ^xa ^ya ^za 1 > <> h i ^yb = r12^xa + r22^ya + r32^za = ^xa ^ya ^za r2 (1) > > h i :>^zb = r13^xa + r23^ya + r33^za = ^xa ^ya ^za r3 If R is defined as above, then each of its columns describes how the corresponding fbg-frame unit axis is oriented relative to frame fag. Therefore, R can be used to represent the orientation of one frame relative to another. Note that in matrix form, equation (1) can be equivalently written as ^xb ^yb ^zb = ^xa ^ya ^za R (2) 1 2 Change Reference Frame Let Rab be a rotation matrix that represents the orientation of fbg relative to fag. Let pa and pb be the coordinates of the same physical point p in frames fag and fbg, respectively. Since they represent the same physical point, we must have that ^xa ^ya ^za pa = ^xb ^yb ^zb pb = ^xa ^ya ^za Rabpb where the second equality is due to equation (2). Thus, we can conclude that pa = Rabpb Therefore, Rab not only represents the orientation of fbg relative to fag, but also can be used to change the coordinates of the same point p from reference frame fbg to frame fag Following a similar argument, we can see that Rab can be used to change the reference frame in which another frame is represented. To see this, let Rbc represent the orientation of fcg relative to fbg. Using equation (2) twice, we can get ^xc ^yc ^zc = ^xb ^yb ^zb Rbc = ^xa ^ya ^za RabRbc (3) According to the definition of a rotation matrix, equation (3) implies that RabRbc represents the orientation of frame fcg relative to frame fag. By our subscript convention, we can write this as Rac = RabRbc In the above equation, Rbc is still interpreted as the orientation of frame fcg relative to fbg. The matrix Rab here is viewed as changing reference frame for the representation of the orientation of fcg. In other words, multiplying Rab changes the orientation of fcg from \relative to fbg" to \relative to fag". 3 Rotation Operator A rotation matrix can also be viewed as an operator that rotates a frame or a point. Let R be the matrix satisfies equation (1). Then R represents the orientation of fbg relative to fag. Further assume that the frame fbg is obtained by rotating fag about an axis! ^ by an angle θ. Rotating a frame: R can be equivalently viewed as an operator that rotates fag to obtain fbg. It can be seen from (2) that such an operation is accomplished by postmultiplying the axes f^xa; ^ya; ^zag by R. Here, the rotation axis! ^ is expressed in fag (i.e. the coordinates of the! ^ vector is specified with respect to fag). Given the above set up, what does RR mean? Using (2), we have ^xa ^ya ^za RR = ^xb ^yb ^zb R Therefore, RR can be viewed as rotating fag about! ^ by θ and then rotating fbg about! ^ by θ. It is important to know that the rotation axes for the two rotations have the same coordinates but correspond to different physical vectors in general. For the first operation,! ^ is expressed in fag, while for the second rotation operation,! ^ is expressed in fbg. 2 In general, for an arbitrary frame fcg with orientation represented by Rc relative to some fixed frame, the multiplication RcR means rotating fcg about the axis! ^ by an angle θ, where! ^ is expressed in frame fcg. In other words, RcR is the orientation of the new frame obtained by the rotation operation. Rotating a vector: Now consider the matrix vector multiplication Rp, where R is defined in the same way as above. Previously, we have viewed R as a change of reference frames for the representations of the same physical vector in different frames. Now we want to view this as rotating p about! ^ by an angle θ. The rotated vector is given by p0 = Rp. Note that only one frame is involved in this interpretation, namely, p and p0 are the coordinates of two different points (before and after rotation) in the same frame. The rotation axis! ^ must also be expressed in the same frame. In order for the above interpretation to work, the vectors p, p0 and! ^ have to be expressed in the same frame. If there are more than 1 frame involved, we need to do a change of coordinate system. To see this, let pc and pd be the coordinates of a physical point p in frames fcg and fdg, respectively. Define 0 00 pc = Rpc; pd = Rpd According to our previous interpretation, Rpc means rotating p about! ^ (expressed in fcg) by θ. 0 0 0 Let the rotated vector be p with coordinates pc and pd in frames fcg and fdg, respectively. Similarly, 00 Rpd means rotating p about! ^ (expressed in fdg) by θ. Let the resulting rotated vector be p with 00 00 coordinates pc and pd in frames fcg and fdg, respectively. Note that the same R matrix corresponds to different rotation operations (i.e. p0 6= p00) depending on the frames we are working with. 0 Now let's focus on the first rotation operation that rotates p to p . Let Rcd be the orientation of 0 −1 0 fdg relative to fcg. Then we know pd = Rcd pc. Therefore, we have the following 0 −1 0 −1 −1 pd = Rab pc = Rcd Rpc = (Rcd RRcd)pd (4) Recall that p0 is obtained by rotating p about! ^ (expressed in fcg) by θ. This operation can be represented by R if we are working with frame fcg (i.e. the same frame in which! ^ is expressed). −1 Equation (4) indicates that this operation can be equivalently represented by (Rcd RRcd) if we work with frame fdg. 3.
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