ECE5463 Spring 2018 Supplementary Note on

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 ;

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 {a}, {b}, and {c} with unit axes {ˆxa, ˆya, ˆza}, {ˆxb, ˆyb, ˆzb}, and {ˆxc, ˆyc, ˆzc}, respectively.

1 Representation of Orientation:

Let R be a rotation matrix:   r11 r12 r13   R = r1 r2 r3 =  r21 r22 r23  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 {b} relative to {a} if the columns of R are the coordinates of unit axes {ˆxb, ˆyb, ˆzb} in {a}, i.e.,  h i ˆx = r ˆx + r ˆy + r ˆz = r  b 11 a 21 a 31 a ˆxa ˆya ˆza 1   h i ˆyb = r12ˆxa + r22ˆya + r32ˆza = ˆxa ˆya ˆza r2 (1)   h i ˆzb = r13ˆxa + r23ˆya + r33ˆza = ˆxa ˆya ˆza r3

If R is defined as above, then each of its columns describes how the corresponding {b}-frame unit axis is oriented relative to frame {a}. 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     ˆxb ˆyb ˆzb = ˆxa ˆya ˆza R (2)

1 2 Change Reference Frame

Let Rab be a rotation matrix that represents the orientation of {b} relative to {a}. Let pa and pb be the coordinates of the same physical point p in frames {a} and {b}, respectively. Since they represent the same physical point, we must have that       ˆxa ˆya ˆza pa = ˆxb ˆyb ˆzb pb = ˆxa ˆya ˆza 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 {b} relative to {a}, but also can be used to change the coordinates of the same point p from reference frame {b} to frame {a} 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 {c} relative to {b}. Using equation (2) twice, we can get       ˆxc ˆyc ˆzc = ˆxb ˆyb ˆzb Rbc = ˆxa ˆya ˆza RabRbc (3)

According to the definition of a rotation matrix, equation (3) implies that RabRbc represents the orientation of frame {c} relative to frame {a}. By our subscript convention, we can write this as

Rac = RabRbc

In the above equation, Rbc is still interpreted as the orientation of frame {c} relative to {b}. The matrix Rab here is viewed as changing reference frame for the representation of the orientation of {c}. In other words, multiplying Rab changes the orientation of {c} from “relative to {b}” to “relative to {a}”.

3 Rotation

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 {b} relative to {a}. Further assume that the frame {b} is obtained by rotating {a} about an axisω ˆ by an angle θ. Rotating a frame: R can be equivalently viewed as an operator that rotates {a} to obtain {b}. It can be seen from (2) that such an operation is accomplished by postmultiplying the axes {ˆxa, ˆya, ˆza} by R. Here, the rotation axisω ˆ is expressed in {a} (i.e. the coordinates of theω ˆ vector is specified with respect to {a}). Given the above up, what does RR mean? Using (2), we have     ˆxa ˆya ˆza RR = ˆxb ˆyb ˆzb R

Therefore, RR can be viewed as rotating {a} aboutω ˆ by θ and then rotating {b} aboutω ˆ by θ. It is important to know that the rotation axes for the two have the same coordinates but correspond to different physical vectors in general. For the first operation,ω ˆ is expressed in {a}, while for the second rotation operation,ω ˆ is expressed in {b}.

2 In general, for an arbitrary frame {c} with orientation represented by Rc relative to some fixed frame, the multiplication RcR means rotating {c} about the axisω ˆ by an angle θ, whereω ˆ is expressed in frame {c}. 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 aboutω ˆ 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 axisω ˆ must also be expressed in the same frame. In order for the above interpretation to work, the vectors p, p0 andω ˆ 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 {c} and {d}, respectively. Define 0 00 pc = Rpc, pd = Rpd

According to our previous interpretation, Rpc means rotating p aboutω ˆ (expressed in {c}) by θ. 0 0 0 Let the rotated vector be p with coordinates pc and pd in frames {c} and {d}, respectively. Similarly, 00 Rpd means rotating p aboutω ˆ (expressed in {d}) by θ. Let the resulting rotated vector be p with 00 00 coordinates pc and pd in frames {c} and {d}, 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 {d} relative to {c}. 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 aboutω ˆ (expressed in {c}) by θ. This operation can be represented by R if we are working with frame {c} (i.e. the same frame in whichω ˆ is expressed). −1 Equation (4) indicates that this operation can be equivalently represented by (Rcd RRcd) if we work with frame {d}.

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