Rotation Matrices 2

WCR 2017-04-21

Outline Markers and Measured Orientation Vectors Rotation-of-Points Matrix Rotation-of-Coordinate-System Matrix Summary Helical Axis and Angles Simulating Movement with Helical Angles Euler Angles Simulating Movement with Euler Angles Shoulder Joint Hip Joint

Markers and Measured Orientation Vectors

Marker locations are determined in the global (lab) reference system (GRS), using motion capture techniques. These marker locations are used to estimate orthogonal unit vectors iprox,jprox,kprox for the proximal segment and idistal,jdistal,kdistal for the distal segment, at each time point. Since the markers are in the GRS, the unit vectors are also expressed in the GRS. We assume vectors are expressed as column vectors. We also assume that i,j,k of the proximal and distal segments are defined so that they are aligned in the neutral position. Another way of saying this is that the joint angles (flexion/extension, abduction/adduction, internal/external rotation) are assumed to be zero when i,j,k point in the same directions for both segments. (The ankle joint may be an exception to this general rule.)

Our goal is to use our knowledge of the unit vector orientations at each time to determine the angles of flex/ext, abd/add, and IR/ER, of the distal segment relative to the proximal, at each time.

At each time, we arrange the orthogonal unit vectors of the proximal and distal segments into two 3x3 matrices. Each column is a unit vector expressed in the GRS:

𝑅𝑅𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 = (𝒊𝒊𝒑𝒑𝒑𝒑𝒑𝒑𝒑𝒑 𝒋𝒋𝒑𝒑𝒑𝒑𝒑𝒑𝒑𝒑 𝒌𝒌𝒑𝒑𝒑𝒑𝒑𝒑𝒑𝒑) = �𝑖𝑖𝐺𝐺𝐺𝐺𝐺𝐺𝑝𝑝,𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 𝑗𝑗𝐺𝐺𝐺𝐺𝐺𝐺𝑝𝑝,𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 𝑘𝑘𝐺𝐺𝐺𝐺𝐺𝐺𝑝𝑝,𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑖𝑖𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺,𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 𝑗𝑗𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺,𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 𝑘𝑘𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺,𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑖𝑖𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺,𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 𝑗𝑗𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺,𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 𝑘𝑘𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺,𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝

� (1)

𝑅𝑅𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 = (𝒊𝒊𝒅𝒅𝒊𝒊𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝒅 𝒋𝒋𝒅𝒅𝒊𝒊𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝒅 𝒌𝒌𝒅𝒅𝒊𝒊𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝒅) = �𝑖𝑖𝐺𝐺𝐺𝐺𝐺𝐺𝑝𝑝,𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 𝑗𝑗𝐺𝐺𝐺𝐺𝐺𝐺𝑝𝑝,𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 𝑘𝑘𝐺𝐺𝐺𝐺𝐺𝐺𝑝𝑝,𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑖𝑖𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺,𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 𝑗𝑗𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺,𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 𝑘𝑘𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺,𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑖𝑖𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺,𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 𝑗𝑗𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺,𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 𝑘𝑘𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺,𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑

� (2)

Rotation-of-Points Matrix

The rotation matrix that will move the vectors in the proximal segment into alignment with the distal segment is Rp2d :

𝑅𝑅𝑝𝑝2𝑑𝑑𝑅𝑅𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 = 𝑅𝑅𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 (3)

𝑅𝑅𝑝𝑝2𝑑𝑑𝑅𝑅𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑅𝑅𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝−1 = 𝑅𝑅𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑅𝑅𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝−1 (4)

𝑅𝑅𝑝𝑝2𝑑𝑑𝐼𝐼 = 𝑅𝑅𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑅𝑅𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝−1 (5)

𝑅𝑅𝑝𝑝2𝑑𝑑 = 𝑅𝑅𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑅𝑅𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑇𝑇 (6)

In going from eq. 5 to eq. 6, we used the fact that R-1=RT when R is a rotation matrix. Note that the rotation matrix Rp2d rotates the vectors, or (equivalently) the points on the segment. Rp2d is not a rotation of the coordinate system. The rotation matrices in Winter, 4th ed., chapter 7, are rotations-of-coordinate-systems, which leave the points themselves unmoved. Therefore we calculate Rp2d according to the equation 6. Saying it again, slightly differently, because it is important to understand: Rp2d is the matrix which, if we multiply it times the unit vectors in the proximal segment (as they are at one instant), will give us a set of unit vectors that are identical to the distal segment unit vectors, as they are at that instant. At most joints (the ankle being an exception), we assume the distal and proximal segment unit vectors are aligned in the neutral position. In that case, Rp2d tells us how the distal segment is rotated relative to its neutral orientation.

Rotation-of-Coordinate-System Matrix

An alternate approach to quantifying the relative orientation of segments is to consider rotation of the coordinate reference system, instead of rotation of the distal segment relative to the proximal. Mathematically, there is no particular reason to prefer rotation-of-points or rotation-of-coordinate-systems: either approach will successfully account for rotations in three dimensions. The rotation-of-points approach is, perhaps, easier to visualize, since one rotates the segment itself. However, the rotation-of-coordinate-system approach has a practical advantage for biomechanics: when we do successive rotations of the coordinate system, the second and third rotations will be relative to the rotated (distal segment) axes, because the rotation of coordinates “takes the axes along”. Rotations-of-points do not “take the axes along”, so successive rotations are always about the original axes, and not about the moving distal segment axes.

Imagine rotating the coordinate system along with the distal segment. When we multiply the rotation-of-coordinate-system matrix by a vector, we get the expression of that vector in the rotated reference system. Therefore, the rotation-of-coordinate-system matrix representing the combined movement of the distal segment relative to the proximal will express the unit vectors

of the proximal segment (represented by the identity matrix) in the distal segment coordinate system. Put another way, the columns of the rotation-of-coordinate-system matrix express what iprox, jprox, and kprox “look like” to an observer in the distal segment coordinate system. We know, from experimental observation of markers, Rprox and Rdistal, whose columns are the unit vectors of the proximal and distal segments, expressed in the GRS. The following equations show how we can use that information to express iprox, jprox, kprox in terms of idistal, jdistal, kdistal.

𝒊𝒊𝒑𝒑𝒑𝒑𝒑𝒑𝒑𝒑 = 𝒊𝒊𝒅𝒅𝒊𝒊𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝑞𝑞11 + 𝒋𝒋𝒅𝒅𝒊𝒊𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝑞𝑞21 + 𝒌𝒌𝒅𝒅𝒊𝒊𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝑞𝑞31 (7)

𝒋𝒋𝒑𝒑𝒑𝒑𝒑𝒑𝒑𝒑 = 𝒊𝒊𝒅𝒅𝒊𝒊𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝑞𝑞12 + 𝒋𝒋𝒅𝒅𝒊𝒊𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝑞𝑞22 + 𝒌𝒌𝒅𝒅𝒊𝒊𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝑞𝑞32 (8)

𝒌𝒌𝒑𝒑𝒑𝒑𝒑𝒑𝒑𝒑 = 𝒊𝒊𝒅𝒅𝒊𝒊𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝑞𝑞13 + 𝒋𝒋𝒅𝒅𝒊𝒊𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝑞𝑞23 + 𝒌𝒌𝒅𝒅𝒊𝒊𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝑞𝑞33 (9)

The above three equations can be written as a matrix equation:

(𝒊𝒊𝒑𝒑𝒑𝒑𝒑𝒑𝒑𝒑 𝒋𝒋𝒑𝒑𝒑𝒑𝒑𝒑𝒑𝒑 𝒌𝒌𝒑𝒑𝒑𝒑𝒑𝒑𝒑𝒑) = (𝒊𝒊𝒅𝒅𝒊𝒊𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝒅 𝒋𝒋𝒅𝒅𝒊𝒊𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝒅 𝒌𝒌𝒅𝒅𝒊𝒊𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝒅)�𝑞𝑞11 𝑞𝑞12 𝑞𝑞13𝑞𝑞21 𝑞𝑞22 𝑞𝑞23𝑞𝑞31 𝑞𝑞32 𝑞𝑞33

� (10)

where iprox, etc., are column vectors. Matrix Q≡(qij) is the rotation-of-coordinate-system matrix. The columns of Q are the proximal coordinate system unit vectors, expressed in terms of the distal coordinate system. We apply equations 1 and 2 to equation 10:

𝑅𝑅𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 = 𝑅𝑅𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑄𝑄 (11)

𝑅𝑅𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑−1𝑅𝑅𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 = 𝑅𝑅𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑−1𝑅𝑅𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑄𝑄 (12)

𝑄𝑄 = 𝑅𝑅𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑇𝑇𝑅𝑅𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 (13)

Compare equation 6 to equation 13. Equation 6 is the rotation-of-points matrix that moves the proximal segment unit vectors into alignment with the distal coordinate system vectors. Q in equation 13 is the rotation-of-coordinate-system matrix, that expresses the proximal segment unit vectors in terms of the distal segment coordinate system.

Summary

We have demonstrated two ways to represent the relative orientation of the distal and proximal segments: by the rotation-of-points (Rp2d) or by the rotation-of-coordinate-systems (Q). We need to estimate the proximal and distal segment unit vectors in the GRF (Rprox and Rdistal) to estimate Rp2d or Q. The next step is to use Rp2d or Q to estimate the joint angles,

Helical Axis and Angles

In the helical approach, we will do a single rotation about an axis which may not be aligned with any of the principle axes in either segment. It is always true that for any non-zero rotation of a

segment, there is a single axis n=(nx;ny;nz) and an associated angle φ such that a rotation by angle φ about n will move the proximal i,j,k into the distal i,j,k. (If the distal segment is in the neutral position relative to the proximal, then the angle φ will be zero and the axis will be undefined.) If n is a unit vector (which we can assume without loss of generality), then the components of n along the X, Y, and Z axes tell us how to “allocate” φ to flexion/extension, abduction/adduction, and internal/external rotation. Now we show how to determine n=(nx;ny;nz) from Rp2d. (Remember that we have already determined Rp2d from marker locations.)

The rotation-of-points matrix arising corresponding to rotation by φ about n=(nx;ny;nz) is

𝑅𝑅ℎ𝑒𝑒𝑑𝑑𝑑𝑑𝑒𝑒𝑑𝑑𝑑𝑑 =

�𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 − 𝑢𝑢𝑝𝑝2(1 − 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐) 𝑢𝑢𝑝𝑝𝑢𝑢𝐺𝐺(1 − 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐) − 𝑢𝑢𝐺𝐺𝑐𝑐𝑖𝑖𝑠𝑠𝑐𝑐 𝑢𝑢𝑝𝑝𝑢𝑢𝐺𝐺(1 − 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐) + 𝑢𝑢𝐺𝐺𝑐𝑐𝑖𝑖𝑠𝑠𝑐𝑐

𝑢𝑢𝑝𝑝𝑢𝑢𝐺𝐺(1 − 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐) + 𝑢𝑢𝐺𝐺𝑐𝑐𝑖𝑖𝑠𝑠𝑐𝑐 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 − 𝑢𝑢𝐺𝐺2(1 − 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐) 𝑢𝑢𝐺𝐺𝑢𝑢𝐺𝐺(1 − 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐) − 𝑢𝑢𝑝𝑝𝑐𝑐𝑖𝑖𝑠𝑠𝑐𝑐𝑢𝑢𝑝𝑝𝑢𝑢𝐺𝐺(1 − 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐) − 𝑢𝑢𝐺𝐺𝑐𝑐𝑖𝑖𝑠𝑠𝑐𝑐 𝑢𝑢𝐺𝐺𝑢𝑢𝐺𝐺(1 − 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐) + 𝑢𝑢𝑝𝑝𝑐𝑐𝑖𝑖𝑠𝑠𝑐𝑐 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 − 𝑢𝑢𝐺𝐺2(1 − 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐)

� (14)

which can also be written as

𝑅𝑅ℎ𝑒𝑒𝑑𝑑𝑑𝑑𝑒𝑒𝑑𝑑𝑑𝑑 = �𝑒𝑒02 + 𝑒𝑒12 − 𝑒𝑒22 − 𝑒𝑒32 2(𝑒𝑒1𝑒𝑒2 + 𝑒𝑒0𝑒𝑒3) 2(𝑒𝑒1𝑒𝑒3 − 𝑒𝑒0𝑒𝑒2)

2(𝑒𝑒1𝑒𝑒2 − 𝑒𝑒0𝑒𝑒3) 𝑒𝑒02 − 𝑒𝑒12 + 𝑒𝑒22 − 𝑒𝑒32 2(𝑒𝑒2𝑒𝑒3 + 𝑒𝑒0𝑒𝑒1)2(𝑒𝑒1𝑒𝑒3 + 𝑒𝑒0𝑒𝑒2) 2(𝑒𝑒2𝑒𝑒3 − 𝑒𝑒0𝑒𝑒1) 𝑒𝑒02 − 𝑒𝑒12 − 𝑒𝑒22 + 𝑒𝑒32

� (15)

(http://mathworld.wolfram.com/EulerParameters.html, retreived 2013-03-22) where

e0=cos(φ/2) (16a)

e1=nxsin(φ/2), e2=nysin(φ/2), e3=nzsin(φ/2) (16b)

The helical angle approach finds the one single axis n=(nx;ny;nz), about which we can rotate the segment to move it from its neutral to its observed position, and it finds the angle of rotation, φ. We wish to find the unit vector n and the rotation angle φ which will make the theoretical rotation-of-points matrix Rhelical equal to the measured rotation-of-points matrix Rp2d:

𝑅𝑅ℎ𝑒𝑒𝑑𝑑𝑑𝑑𝑒𝑒𝑑𝑑𝑑𝑑 = 𝑅𝑅𝑝𝑝2𝑑𝑑 (17)

when

𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 = 𝑝𝑝11+𝑝𝑝22+𝑝𝑝33−12

(18)

𝑐𝑐𝑝𝑝𝑝𝑝𝑒𝑒𝑑𝑑𝑑𝑑𝑝𝑝 = 𝑎𝑎𝑐𝑐𝑐𝑐𝑐𝑐 �𝑝𝑝11+𝑝𝑝22+𝑝𝑝33−12

� (19)

It can be shown by algebra (http://www.kwon3d.com/theory/jkinem/helical.html, retrieved 2013-03-24) that each column of the following matrix is a scaled version of the rotation axis, n.

𝑁𝑁 = ({𝑠𝑠1} {𝑠𝑠2} {𝑠𝑠3}) = 12�𝑅𝑅ℎ𝑒𝑒𝑑𝑑𝑑𝑑𝑒𝑒𝑑𝑑𝑑𝑑 + 𝑅𝑅ℎ𝑒𝑒𝑑𝑑𝑑𝑑𝑒𝑒𝑑𝑑𝑑𝑑𝑇𝑇� + 𝐼𝐼(1 − 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐) (20)

where I is the identity matrix. Combining equations 17, 18, and 20, we get

𝑁𝑁 = ({𝑠𝑠1} {𝑠𝑠2} {𝑠𝑠3}) = 12�𝑅𝑅𝑝𝑝2𝑑𝑑 + 𝑅𝑅𝑝𝑝2𝑑𝑑𝑇𝑇� + 𝐼𝐼 �1 − 𝑝𝑝11+𝑝𝑝22+𝑝𝑝33−1

2� (21)

𝑁𝑁 = ({𝑠𝑠1} {𝑠𝑠2} {𝑠𝑠3}) = 12�𝑅𝑅𝑝𝑝2𝑑𝑑 + 𝑅𝑅𝑝𝑝2𝑑𝑑𝑇𝑇 + 𝐼𝐼�1 − (𝑟𝑟11 + 𝑟𝑟22 + 𝑟𝑟33)�� (22)

To minimize round-off errors, estimate n by choosing the column from (22) above which has the largest magnitude. Normalize that column vector to have unit length:

𝒏𝒏 = 𝑛𝑛𝑚𝑚𝑚𝑚𝑚𝑚‖𝑛𝑛𝑚𝑚𝑚𝑚𝑚𝑚‖

(23)

where nmax = column of N with largest magnitude = maxnorm(n1, n2, n3).

Before proceeding further, check that the values for n and φprelim generate a rotation matrix equal to Rp2d, by plugging n and φprelim into equation 14 and computing Rhelix. If Rhelix does not match Rp2d to good precision, set φ=-φprelim. This is necessary because the estimation of φ by equation 19 may result in a sign error for φ. You might think that n would then point the opposite way, in which case we would still get correct answers for angles, but this is not always the case. We do not want small round-off differences between matrices to make us erroneously conclude that the sign of ϕprelim needs changing. Therefore we use equations 14 to compute two rotation-of-points matrices: Rhelix(n,+ϕprelim) and Rhelixn,-ϕprelim). We choose the sign of ϕ that gives a matrix that is “closer to” the matrix Rp2d , the rotation of points matrix computed with equation 6. We define “closer to” by computing

𝑑𝑑+ = 𝑠𝑠𝑐𝑐𝑟𝑟𝑛𝑛�Rℎ𝑒𝑒𝑑𝑑𝑑𝑑𝑝𝑝�𝑠𝑠, +𝑐𝑐𝑝𝑝𝑝𝑝𝑒𝑒𝑑𝑑𝑑𝑑𝑝𝑝� − R𝑝𝑝2𝑑𝑑� (23a)

and

𝑑𝑑− = 𝑠𝑠𝑐𝑐𝑟𝑟𝑛𝑛�Rℎ𝑒𝑒𝑑𝑑𝑑𝑑𝑝𝑝�𝑠𝑠,−𝑐𝑐𝑝𝑝𝑝𝑝𝑒𝑒𝑑𝑑𝑑𝑑𝑝𝑝� − R𝑝𝑝2𝑑𝑑� (23b)

where norm denotes the 2-norm, or equivalently, the Euclidean norm, of a matrix. If d- is smaller than d+, we choose ϕ = -ϕprelim ; otherwise, we choose ϕ = +ϕprelim.

The projections of φn onto the proximal segment axes are the respective amounts of flex/ext, abd/add, and IR/ER. If +X points right (flex/ext axis), +Y points forward (add/abd axis), and +Z points up (IR/ER axis), then

Flexion/extension: 𝜃𝜃𝑋𝑋 = 𝑐𝑐nx (24)

Adduction/abduction: 𝜃𝜃𝑌𝑌 = 𝑐𝑐ny (25)

Internal/external rotation: 𝜃𝜃𝑍𝑍 = 𝑐𝑐nz (26)

The algorithm just described, for finding the helical axis and angles, fails when the proximal and distal segments are exactly aligned. It is not surprising that it fails, because when the segments are aligned, the rotation angle should be zero, and in that case there is no uniquely defined axis of rotation. Mathematically, the problem occurs in equation 23: n=nmax/||nmax||. When the proximal and distal segments are aligned, Rp2d is the identity matrix, and when Rp2d=I, the matrix N in equations 20-22 is the zero matrix, and so the vector nmax has length 0, and equation 23 has an undefined (divide by zero) result. This problem is handled by checking for Rp2d=I. If Rp2d=I, the helical angles are set equal to zero, and the axis can be arbitrarily set to n=(1;0;0), or can be estimated by interpolating from nearby time points at which the axis is well defined.

Simulating Movement With Helical Angles

Simulation of movement using helical angles is fairly straightforward. Create a 3xN matrix P, whose columns Pj, j=1..N, are the N points in an object. Choose the axis of rotation n and the desired rotation angle φ. Use equations 15, 16, and 14 to create the rotation-of-points matrix R. Rotation about the origin moves the points P to Protated:

𝑃𝑃𝑝𝑝𝑝𝑝𝑑𝑑𝑑𝑑𝑑𝑑𝑒𝑒𝑑𝑑 = 𝑅𝑅𝑃𝑃 (27)

where P and Protated are both expressed in the same coordinate system, such as the global reference system. Rotation about a point C is achieved as follows, assuming P, C, and Protated are all expressed in the same coordinate system:

𝑃𝑃𝑝𝑝𝑝𝑝𝑑𝑑𝑑𝑑𝑑𝑑𝑒𝑒𝑑𝑑 = 𝑅𝑅�𝑃𝑃 − (𝐶𝐶 …𝐶𝐶)� + (𝐶𝐶…𝐶𝐶) (28)

where (C…C) denotes a 3xN matrix composed of N replicates of the column vector C.

Euler Angles

One can decompose a rotation in three dimensions into three successive rotations about axes that are anatomically defined. This is in contrast to the helical approach, in which a single rotation is done about an axis that is not anatomically defined. When using Euler angles, we must define axes and choose the order of rotations. The order of rotation matters. The order dependence can be understood mathematically as a result of the fact the matrix multiplication is not commutative: if A and B are matrices, it is generally true that AB≠BA. The order dependence can be understood physically as meaning that, for example, 45° flexion followed by 45° abduction (about the rotated ab/adduction axis) will result in a different segment orientation than 45° abduction followed by 45° flexion (about the rotated flexion/extension axis). The angles of rotation are called Euler angles.

We choose to use matrices that represent rotating the coordinate system while leaving the points in place: the rotation-of-coordinate-system matrices. We choose this even though it may be easier to visualize the action of rotation-of-points, which moves the segment itself. To recapitulate the reason, which was also described above: if we do successive rotations of points, each successive rotation is about the original coordinate axes, since points are always expressed in the original coordinate system. This means that the combined rotation matrix RpointsXYZ = RpointsZRpointsYRpointsX, for example, represents rotation about proximal X, then proximal Y, then proximal Z. That’s not what we want. We want to rotate about X (which is the same for proximal and distal segments initially), then about Y’= distal segment Y, then about Z”=twice-rotated distal segment Z. For this reason, we must use rotation-of-coordinate-system matrices.

There are two kinds of rotation sequences that can be used to account for any rotation in three dimensions: Euler sequences and Cardan sequences. Euler sequences use a shared axis in the aligned proximal and distal segments for the first rotation, then an axis in the distal segment, which has been rotated by the first rotation, for the second rotation, and finally, the same distal segment axis that was used for the first rotation is used again for the final rotation – but due to the first two rotations, this axis now points in a different direction than it did initially. An example of such a sequence is X-Y’-X’, or simply X-Y-X. This sequence indicates a rotation about X, followed by a rotation about the rotated Y axis (called Y’ to indicate that it moved during the first rotation), and then about the rotated X axis (denoted X’). There are six possible Euler sequences: X-Y-X, X-Z-X, Y-X-Y, Y-Z-Y, Z-X-Z, Z-Y-Z. Euler sequence rotations can not be accomplished with rotation-of-points. If one attempts to do so, one finds that the first and final rotation axes are the same, and thus there are really only two independent axes of rotation. A rotation sequence with two pre-defined axes is not sufficiently general to account for most rotations in three dimensions.

The second type of rotation sequence is called a Cardan sequence. A Cardan sequence involves rotation about three different axes. The first one is shared between proximal and distal segment, since they are presumed to be aligned initially. The second and third rotations are about the other two orthogonal axes, which are fixed in the distal segment. Because they are fixed in the distal segment, they move with each rotation. An example of a Cardan sequence is X-Y’-Z”, or simply X-Y-Z, denoting an initial rotation about the X axis, then about the rotated Y-axis in the distal segment (denoted Y’), and finally about the twice-rotated Z-axis (Z”) of the distal segment. There are six possible Cardan sequences: X-Y-Z, X-Z-Y, Y-X-Z, Y-Z-X, Z-X-Y, Z-Y-X.

With either kind of sequence, the rotation can be represented by multiplying three rotation matrices.

The matrices that represent rotations of the coordinate system about an axis (rather than rotations of the points themselves) are as follows:

𝑅𝑅𝑋𝑋(𝜃𝜃𝑝𝑝) = �1 0 00 𝑐𝑐𝑐𝑐𝑐𝑐𝜃𝜃𝑝𝑝 𝑐𝑐𝑖𝑖𝑠𝑠𝜃𝜃𝑝𝑝0 −𝑐𝑐𝑖𝑖𝑠𝑠𝜃𝜃𝑝𝑝 𝑐𝑐𝑐𝑐𝑐𝑐𝜃𝜃𝑝𝑝

� (29)

𝑅𝑅𝑌𝑌�𝜃𝜃𝐺𝐺� = �𝑐𝑐𝑐𝑐𝑐𝑐𝜃𝜃𝐺𝐺 0 −𝑐𝑐𝑖𝑖𝑠𝑠𝜃𝜃𝐺𝐺

0 1 0𝑐𝑐𝑖𝑖𝑠𝑠𝜃𝜃𝐺𝐺 0 𝑐𝑐𝑐𝑐𝑐𝑐𝜃𝜃𝐺𝐺

� (30)

𝑅𝑅𝑍𝑍(𝜃𝜃𝐺𝐺) = �𝑐𝑐𝑐𝑐𝑐𝑐𝜃𝜃𝐺𝐺 𝑐𝑐𝑖𝑖𝑠𝑠𝜃𝜃𝐺𝐺 0−𝑐𝑐𝑖𝑖𝑠𝑠𝜃𝜃𝐺𝐺 𝑐𝑐𝑐𝑐𝑐𝑐𝜃𝜃𝐺𝐺 0

0 0 1� (31)

The rotation matrices above operate on points represented as column vectors, and therefore the rotation matrix is to the left of the column vector, or the matrix of points. When several rotations are done, the matrix for the first rotation is right-most and the last rotation is left-most. This allows the first (right-most) matrix to operate first on the point or points. For example,

expression of point 𝑃𝑃0 = �𝑝𝑝𝑝𝑝𝑝𝑝𝐺𝐺𝑝𝑝𝐺𝐺� in a coordinate system that is rotated about X, then Y’, then Z’’ is

represented with matrices as follows:

𝑃𝑃𝑓𝑓𝑑𝑑𝑛𝑛𝑑𝑑𝑑𝑑 = 𝑅𝑅𝑍𝑍(𝜃𝜃𝐺𝐺)𝑅𝑅𝑌𝑌�𝜃𝜃𝐺𝐺�𝑅𝑅𝑋𝑋(𝜃𝜃𝑝𝑝)𝑃𝑃0 = 𝑅𝑅𝑍𝑍(𝜃𝜃𝐺𝐺)𝑅𝑅𝑌𝑌�𝜃𝜃𝐺𝐺�𝑃𝑃1 = 𝑅𝑅𝑍𝑍(𝜃𝜃𝐺𝐺)𝑃𝑃2 (32)

𝑃𝑃𝑓𝑓𝑑𝑑𝑛𝑛𝑑𝑑𝑑𝑑 = 𝑅𝑅𝑑𝑑𝑝𝑝𝑑𝑑𝑑𝑑𝑑𝑑𝑃𝑃0 (33)

where Rtotal is the rotation-of-coordinate-system matrix, and P0 and Pfinal are the same vector, expressed in the initial (proximal) and final (distal) coordinate systems respectively. Rtotal depends on the rotation sequence chosen, as shown in the following table.

Cardan Sequences Rtotal = X then Y’ then Z” 𝑅𝑅𝑋𝑋𝑌𝑌𝑍𝑍 = 𝑅𝑅𝑍𝑍(𝜃𝜃𝐺𝐺)𝑅𝑅𝑌𝑌�𝜃𝜃𝐺𝐺�𝑅𝑅𝑋𝑋(𝜃𝜃𝑝𝑝) Y then X’ then Z” 𝑅𝑅𝑌𝑌𝑋𝑋𝑍𝑍 = 𝑅𝑅𝑍𝑍(𝜃𝜃𝐺𝐺)𝑅𝑅𝑋𝑋(𝜃𝜃𝑝𝑝)𝑅𝑅𝑌𝑌�𝜃𝜃𝐺𝐺� X then Z’ then Y” 𝑅𝑅𝑋𝑋𝑍𝑍𝑌𝑌 = 𝑅𝑅𝑌𝑌�𝜃𝜃𝐺𝐺�𝑅𝑅𝑍𝑍(𝜃𝜃𝐺𝐺)𝑅𝑅𝑋𝑋(𝜃𝜃𝑝𝑝) Z then X’ then Y” 𝑅𝑅𝑍𝑍𝑋𝑋𝑌𝑌 = 𝑅𝑅𝑌𝑌�𝜃𝜃𝐺𝐺�𝑅𝑅𝑋𝑋(𝜃𝜃𝑝𝑝)𝑅𝑅𝑍𝑍(𝜃𝜃𝐺𝐺) Y then Z’ then X” 𝑅𝑅𝑌𝑌𝑍𝑍𝑋𝑋 = 𝑅𝑅𝑋𝑋(𝜃𝜃𝑝𝑝)𝑅𝑅𝑍𝑍(𝜃𝜃𝐺𝐺)𝑅𝑅𝑌𝑌�𝜃𝜃𝐺𝐺� Z then Y’ then X” 𝑅𝑅𝑍𝑍𝑌𝑌𝑋𝑋 = 𝑅𝑅𝑋𝑋(𝜃𝜃𝑝𝑝)𝑅𝑅𝑌𝑌�𝜃𝜃𝐺𝐺�𝑅𝑅𝑍𝑍(𝜃𝜃𝐺𝐺) Euler Sequences

X then Y’ then X” 𝑅𝑅𝑋𝑋𝑌𝑌𝑋𝑋 = 𝑅𝑅𝑋𝑋(𝜃𝜃𝑝𝑝2)𝑅𝑅𝑌𝑌�𝜃𝜃𝐺𝐺�𝑅𝑅𝑋𝑋(𝜃𝜃𝑝𝑝1) Y then X’ then Y” 𝑅𝑅𝑌𝑌𝑋𝑋𝑌𝑌 = 𝑅𝑅𝑌𝑌�𝜃𝜃𝐺𝐺2�𝑅𝑅𝑋𝑋(𝜃𝜃𝑝𝑝)𝑅𝑅𝑌𝑌�𝜃𝜃𝐺𝐺1� X then Z’ then X” 𝑅𝑅𝑋𝑋𝑍𝑍𝑋𝑋 = 𝑅𝑅𝑋𝑋(𝜃𝜃𝑝𝑝2)𝑅𝑅𝑍𝑍(𝜃𝜃𝐺𝐺)𝑅𝑅𝑋𝑋(𝜃𝜃𝑝𝑝1) Z then X’ then Z” 𝑅𝑅𝑍𝑍𝑋𝑋𝑍𝑍 = 𝑅𝑅𝑍𝑍(𝜃𝜃𝐺𝐺2)𝑅𝑅𝑋𝑋(𝜃𝜃𝑝𝑝)𝑅𝑅𝑍𝑍(𝜃𝜃𝐺𝐺1) Y then Z’ then Y” 𝑅𝑅𝑌𝑌𝑍𝑍𝑌𝑌 = 𝑅𝑅𝑌𝑌�𝜃𝜃𝐺𝐺2�𝑅𝑅𝑍𝑍(𝜃𝜃𝐺𝐺)𝑅𝑅𝑌𝑌�𝜃𝜃𝐺𝐺1�

Z then Y’ then Z” 𝑅𝑅𝑍𝑍𝑌𝑌𝑍𝑍 = 𝑅𝑅𝑍𝑍(𝜃𝜃𝐺𝐺2)𝑅𝑅𝑌𝑌�𝜃𝜃𝐺𝐺�𝑅𝑅𝑍𝑍(𝜃𝜃𝐺𝐺1) Table 1. Rotation-of-coordinate-system matrices for all Cardan and Euler sequences.

If P0 and Pfinal are direction vectors, as is the case when we are estimating joint angles from segment unit vectors, then it does not matter if the two systems share an origin or not, since direction vectors are independent of origin. If P0 and Pfinal are position vectors, then the origin matters, and equation 33 will be correct if and only if the proximal and distal systems share an origin.

We will now complete the explanation of how to use Rtotal and the matrix Q, which we obtain from motion capture data, to estimate Euler angles. We use the rotation-of-coordinate-system matrices (equations 29-31) for RX, RY, and RZ to compute the theoretical Rtotal for our preferred rotation sequence. Then we assign values to the angles that will make the theoretical matrix Rtotal equal to the experimentally measured matrix Q.

Euler example 1

If we choose to rotate about X, then Y’, then Z”, then we see from Table 1 that

𝑅𝑅𝑑𝑑𝑝𝑝𝑑𝑑𝑑𝑑𝑑𝑑 = 𝑅𝑅𝑋𝑋𝑌𝑌𝑍𝑍 = 𝑅𝑅𝑍𝑍(𝜃𝜃𝐺𝐺)𝑅𝑅𝑌𝑌�𝜃𝜃𝐺𝐺�𝑅𝑅𝑋𝑋(𝜃𝜃𝑝𝑝) (34)

which after plugging in from equations 29-31 becomes

𝑅𝑅𝑋𝑋𝑌𝑌𝑍𝑍 = �𝑐𝑐𝑐𝑐𝑐𝑐𝜃𝜃𝑍𝑍 𝑐𝑐𝑖𝑖𝑠𝑠𝜃𝜃𝑍𝑍 0−𝑐𝑐𝑖𝑖𝑠𝑠𝜃𝜃𝑍𝑍 𝑐𝑐𝑐𝑐𝑐𝑐𝜃𝜃𝑍𝑍 0

0 0 1��

𝑐𝑐𝑐𝑐𝑐𝑐𝜃𝜃𝑌𝑌 0 −𝑐𝑐𝑖𝑖𝑠𝑠𝜃𝜃𝑌𝑌0 1 0

𝑐𝑐𝑖𝑖𝑠𝑠𝜃𝜃𝑌𝑌 0 𝑐𝑐𝑐𝑐𝑐𝑐𝜃𝜃𝑌𝑌��

1 0 00 𝑐𝑐𝑐𝑐𝑐𝑐𝜃𝜃𝑋𝑋 𝑐𝑐𝑖𝑖𝑠𝑠𝜃𝜃𝑋𝑋0 −𝑐𝑐𝑖𝑖𝑠𝑠𝜃𝜃𝑋𝑋 𝑐𝑐𝑐𝑐𝑐𝑐𝜃𝜃𝑋𝑋

� (35)

𝑅𝑅𝑋𝑋𝑌𝑌𝑍𝑍 = �𝑐𝑐𝑐𝑐𝑐𝑐𝜃𝜃𝑍𝑍 𝑐𝑐𝑖𝑖𝑠𝑠𝜃𝜃𝑍𝑍 0−𝑐𝑐𝑖𝑖𝑠𝑠𝜃𝜃𝑍𝑍 𝑐𝑐𝑐𝑐𝑐𝑐𝜃𝜃𝑍𝑍 0

0 0 1��

𝑐𝑐𝑐𝑐𝑐𝑐𝜃𝜃𝑌𝑌 𝑐𝑐𝑖𝑖𝑠𝑠𝜃𝜃𝑋𝑋𝑐𝑐𝑖𝑖𝑠𝑠𝜃𝜃𝑌𝑌 −𝑐𝑐𝑐𝑐𝑐𝑐𝜃𝜃𝑋𝑋𝑐𝑐𝑖𝑖𝑠𝑠𝜃𝜃𝑌𝑌0 𝑐𝑐𝑐𝑐𝑐𝑐𝜃𝜃𝑋𝑋 𝑐𝑐𝑖𝑖𝑠𝑠𝜃𝜃𝑋𝑋

𝑐𝑐𝑖𝑖𝑠𝑠𝜃𝜃𝑌𝑌 −𝑐𝑐𝑖𝑖𝑠𝑠𝜃𝜃𝑋𝑋𝑐𝑐𝑐𝑐𝑐𝑐𝜃𝜃𝑌𝑌 𝑐𝑐𝑐𝑐𝑐𝑐𝜃𝜃𝑋𝑋𝑐𝑐𝑐𝑐𝑐𝑐𝜃𝜃𝑌𝑌� (36)

𝑅𝑅𝑋𝑋𝑌𝑌𝑍𝑍 = �𝑐𝑐𝑐𝑐𝑐𝑐𝜃𝜃𝑌𝑌𝑐𝑐𝑐𝑐𝑐𝑐𝜃𝜃𝑍𝑍 𝑐𝑐𝑖𝑖𝑠𝑠𝜃𝜃𝑋𝑋𝑐𝑐𝑖𝑖𝑠𝑠𝜃𝜃𝑌𝑌𝑐𝑐𝑐𝑐𝑐𝑐𝜃𝜃𝑍𝑍 + 𝑐𝑐𝑐𝑐𝑐𝑐𝜃𝜃𝑋𝑋𝑐𝑐𝑖𝑖𝑠𝑠𝜃𝜃𝑍𝑍 −𝑐𝑐𝑐𝑐𝑐𝑐𝜃𝜃𝑋𝑋𝑐𝑐𝑖𝑖𝑠𝑠𝜃𝜃𝑌𝑌𝑐𝑐𝑐𝑐𝑐𝑐𝜃𝜃𝑍𝑍 + 𝑐𝑐𝑖𝑖𝑠𝑠𝜃𝜃𝑋𝑋𝑐𝑐𝑖𝑖𝑠𝑠𝜃𝜃𝑍𝑍−𝑐𝑐𝑐𝑐𝑐𝑐𝜃𝜃𝑌𝑌𝑐𝑐𝑖𝑖𝑠𝑠𝜃𝜃𝑍𝑍 −𝑐𝑐𝑖𝑖𝑠𝑠𝜃𝜃𝑋𝑋𝑐𝑐𝑖𝑖𝑠𝑠𝜃𝜃𝑌𝑌𝑐𝑐𝑖𝑖𝑠𝑠𝜃𝜃𝑍𝑍 + 𝑐𝑐𝑐𝑐𝑐𝑐𝜃𝜃𝑋𝑋𝑐𝑐𝑐𝑐𝑐𝑐𝜃𝜃𝑍𝑍 𝑐𝑐𝑐𝑐𝑐𝑐𝜃𝜃𝑋𝑋𝑐𝑐𝑖𝑖𝑠𝑠𝜃𝜃𝑌𝑌𝑐𝑐𝑖𝑖𝑠𝑠𝜃𝜃𝑍𝑍 + 𝑐𝑐𝑖𝑖𝑠𝑠𝜃𝜃𝑋𝑋𝑐𝑐𝑐𝑐𝑐𝑐𝜃𝜃𝑍𝑍

𝑐𝑐𝑖𝑖𝑠𝑠𝜃𝜃𝑌𝑌 −𝑐𝑐𝑖𝑖𝑠𝑠𝜃𝜃𝑋𝑋𝑐𝑐𝑐𝑐𝑐𝑐𝜃𝜃𝑌𝑌 𝑐𝑐𝑐𝑐𝑐𝑐𝜃𝜃𝑋𝑋𝑐𝑐𝑐𝑐𝑐𝑐𝜃𝜃𝑌𝑌�(37)

Choose the angles θX,θY,θZ to make RXYZ equal the measured Q, from equation 13:

𝑄𝑄 = 𝑅𝑅𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑−1𝑅𝑅𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 = �𝑞𝑞11 𝑞𝑞12 𝑞𝑞13𝑞𝑞21 𝑞𝑞22 𝑞𝑞23𝑞𝑞31 𝑞𝑞32 𝑞𝑞33

� (38)

RXYZ equals Q when

𝜃𝜃𝑍𝑍 = 𝑎𝑎𝑎𝑎𝑎𝑎𝑠𝑠2(−𝑞𝑞21, 𝑞𝑞11) (39)

𝜃𝜃𝑋𝑋 = 𝑎𝑎𝑎𝑎𝑎𝑎𝑠𝑠2(−𝑞𝑞32, 𝑞𝑞33) (40)

𝜃𝜃𝑌𝑌 = 𝑎𝑎𝑎𝑎𝑎𝑎𝑠𝑠2(𝑞𝑞31, +�𝑞𝑞112 + 𝑞𝑞212 ) (41)

where atan2(y,x) = four quadrant arctangent function. With the above definitions, θX and θZ will be in the range [-π,+π], and θY will be in the range [-π/2,+π/2].

Euler example 2

If we reverse the order of the first and second rotations, we will rotate about Y, then X’, then Z”. Using the same approach as before, we now obtain

𝑅𝑅𝑑𝑑𝑝𝑝𝑑𝑑𝑑𝑑𝑑𝑑 = 𝑅𝑅𝑌𝑌𝑋𝑋𝑍𝑍 = 𝑅𝑅𝑍𝑍(𝜃𝜃𝐺𝐺)𝑅𝑅𝑋𝑋(𝜃𝜃𝑝𝑝)𝑅𝑅𝑌𝑌�𝜃𝜃𝐺𝐺� (42)

which after plugging in from equations 29-31 becomes

𝑅𝑅𝑌𝑌𝑋𝑋𝑍𝑍 = �𝑐𝑐𝑐𝑐𝑐𝑐𝜃𝜃𝑍𝑍 𝑐𝑐𝑖𝑖𝑠𝑠𝜃𝜃𝑍𝑍 0−𝑐𝑐𝑖𝑖𝑠𝑠𝜃𝜃𝑍𝑍 𝑐𝑐𝑐𝑐𝑐𝑐𝜃𝜃𝑍𝑍 0

0 0 1��

1 0 00 𝑐𝑐𝑐𝑐𝑐𝑐𝜃𝜃𝑋𝑋 𝑐𝑐𝑖𝑖𝑠𝑠𝜃𝜃𝑋𝑋0 −𝑐𝑐𝑖𝑖𝑠𝑠𝜃𝜃𝑋𝑋 𝑐𝑐𝑐𝑐𝑐𝑐𝜃𝜃𝑋𝑋

��𝑐𝑐𝑐𝑐𝑐𝑐𝜃𝜃𝑌𝑌 0 −𝑐𝑐𝑖𝑖𝑠𝑠𝜃𝜃𝑌𝑌

0 1 0𝑐𝑐𝑖𝑖𝑠𝑠𝜃𝜃𝑌𝑌 0 𝑐𝑐𝑐𝑐𝑐𝑐𝜃𝜃𝑌𝑌

� (43)

𝑅𝑅𝑌𝑌𝑋𝑋𝑍𝑍 = �𝑐𝑐𝑐𝑐𝑐𝑐𝜃𝜃𝑍𝑍 𝑐𝑐𝑖𝑖𝑠𝑠𝜃𝜃𝑍𝑍 0−𝑐𝑐𝑖𝑖𝑠𝑠𝜃𝜃𝑍𝑍 𝑐𝑐𝑐𝑐𝑐𝑐𝜃𝜃𝑍𝑍 0

0 0 1��

𝑐𝑐𝑐𝑐𝑐𝑐𝜃𝜃𝑌𝑌 0 −𝑐𝑐𝑖𝑖𝑠𝑠𝜃𝜃𝑌𝑌𝑐𝑐𝑖𝑖𝑠𝑠𝜃𝜃𝑋𝑋𝑐𝑐𝑖𝑖𝑠𝑠𝜃𝜃𝑌𝑌 𝑐𝑐𝑐𝑐𝑐𝑐𝜃𝜃𝑋𝑋 𝑐𝑐𝑖𝑖𝑠𝑠𝜃𝜃𝑋𝑋𝑐𝑐𝑐𝑐𝑐𝑐𝜃𝜃𝑌𝑌𝑐𝑐𝑐𝑐𝑐𝑐𝜃𝜃𝑋𝑋𝑐𝑐𝑖𝑖𝑠𝑠𝜃𝜃𝑌𝑌 −𝑐𝑐𝑖𝑖𝑠𝑠𝜃𝜃𝑋𝑋 𝑐𝑐𝑐𝑐𝑐𝑐𝜃𝜃𝑋𝑋𝑐𝑐𝑐𝑐𝑐𝑐𝜃𝜃𝑌𝑌

� (44)

𝑅𝑅𝑌𝑌𝑋𝑋𝑍𝑍 = �𝑐𝑐𝑐𝑐𝑐𝑐𝜃𝜃𝑌𝑌𝑐𝑐𝑐𝑐𝑐𝑐𝜃𝜃𝑍𝑍+𝑐𝑐𝑖𝑖𝑠𝑠𝜃𝜃𝑋𝑋𝑐𝑐𝑖𝑖𝑠𝑠𝜃𝜃𝑌𝑌𝑐𝑐𝑖𝑖𝑠𝑠𝜃𝜃𝑍𝑍 𝑐𝑐𝑐𝑐𝑐𝑐𝜃𝜃𝑋𝑋𝑐𝑐𝑖𝑖𝑠𝑠𝜃𝜃𝑍𝑍 −𝑐𝑐𝑖𝑖𝑠𝑠𝜃𝜃𝑌𝑌𝑐𝑐𝑐𝑐𝑐𝑐𝜃𝜃𝑍𝑍 + 𝑐𝑐𝑖𝑖𝑠𝑠𝜃𝜃𝑋𝑋𝑐𝑐𝑐𝑐𝑐𝑐𝜃𝜃𝑌𝑌𝑐𝑐𝑖𝑖𝑠𝑠𝜃𝜃𝑍𝑍−𝑐𝑐𝑐𝑐𝑐𝑐𝜃𝜃𝑌𝑌𝑐𝑐𝑖𝑖𝑠𝑠𝜃𝜃𝑍𝑍 + 𝑐𝑐𝑖𝑖𝑠𝑠𝜃𝜃𝑋𝑋𝑐𝑐𝑖𝑖𝑠𝑠𝜃𝜃𝑌𝑌𝑐𝑐𝑐𝑐𝑐𝑐𝜃𝜃𝑍𝑍 𝑐𝑐𝑐𝑐𝑐𝑐𝜃𝜃𝑋𝑋𝑐𝑐𝑐𝑐𝑐𝑐𝜃𝜃𝑍𝑍 𝑐𝑐𝑖𝑖𝑠𝑠𝜃𝜃𝑌𝑌𝑐𝑐𝑖𝑖𝑠𝑠𝜃𝜃𝑍𝑍 + 𝑐𝑐𝑖𝑖𝑠𝑠𝜃𝜃𝑋𝑋𝑐𝑐𝑐𝑐𝑐𝑐𝜃𝜃𝑌𝑌𝑐𝑐𝑐𝑐𝑐𝑐𝜃𝜃𝑍𝑍

𝑐𝑐𝑐𝑐𝑐𝑐𝜃𝜃𝑋𝑋𝑐𝑐𝑖𝑖𝑠𝑠𝜃𝜃𝑌𝑌 −𝑐𝑐𝑖𝑖𝑠𝑠𝜃𝜃𝑋𝑋 𝑐𝑐𝑐𝑐𝑐𝑐𝜃𝜃𝑋𝑋𝑐𝑐𝑐𝑐𝑐𝑐𝜃𝜃𝑌𝑌�(45)

Choose the angles θX,θY,θZ to make RYXZ equal the measured Q, from equation 13:

𝑄𝑄 = 𝑅𝑅𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑−1𝑅𝑅𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 = �𝑞𝑞11 𝑞𝑞12 𝑞𝑞13𝑞𝑞21 𝑞𝑞22 𝑞𝑞23𝑞𝑞31 𝑞𝑞32 𝑞𝑞33

� (46, same as 38)

RYXZ equals Q when

𝜃𝜃𝑍𝑍 = 𝑎𝑎𝑎𝑎𝑎𝑎𝑠𝑠2(𝑞𝑞12, 𝑞𝑞22) (47)

𝜃𝜃𝑌𝑌 = 𝑎𝑎𝑎𝑎𝑎𝑎𝑠𝑠2(𝑞𝑞31, 𝑞𝑞33) (48)

𝜃𝜃𝑋𝑋 = 𝑎𝑎𝑎𝑎𝑎𝑎𝑠𝑠2(−𝑞𝑞32, +�𝑞𝑞312 + 𝑞𝑞332 ) (49)

where atan2(y,x) = four quadrant arctangent function. With the above definitions, θY and θZ will be in the range [-π,+π], and θX will be in the range [-π/2,+π/2].

Simulating Movement With Euler Angles

The simulation of movement using successive rotations about axes that move with the segment is not as simple as simulation with helical angles. We cannot use rotation-of-points matrices, because they rotate about axes that remain fixed in the proximal (not-rotating) segment. If Rpoints is the rotation-of-points matrix, and Rtotal is the rotation-of-coordinate-system matrix from Table 1, then it follows from equation 32b that

𝑃𝑃𝑓𝑓𝑑𝑑𝑛𝑛𝑑𝑑𝑑𝑑 = 𝑅𝑅𝑑𝑑𝑝𝑝𝑑𝑑𝑑𝑑𝑑𝑑�𝑅𝑅𝑝𝑝𝑝𝑝𝑑𝑑𝑛𝑛𝑑𝑑𝑑𝑑𝑃𝑃0� = 𝑃𝑃0, (50)

because if we first rotate the point P0 with Rpoints, then rotate the coordinate system by an equal amount with Rtotal, the final point Pfinal, which is expressed in the rotated coordinate system, will have to equal the original point P0, which is expressed in the unrotated system. Equation 32b is true if and only if

𝑅𝑅𝑝𝑝𝑝𝑝𝑑𝑑𝑛𝑛𝑑𝑑𝑑𝑑 = 𝑅𝑅𝑑𝑑𝑝𝑝𝑑𝑑𝑑𝑑𝑑𝑑−1 = 𝑅𝑅𝑑𝑑𝑝𝑝𝑑𝑑𝑑𝑑𝑑𝑑𝑇𝑇, (51)

where we note that for rotation matrices, the inverse equals the transpose. If the rotation is done about a point C other than the origin, and C is expressed in the proximal coordinate system, then the rotated point Protated, expressed in the proximal coordinate system, is given by

𝑃𝑃𝑝𝑝𝑝𝑝𝑑𝑑𝑑𝑑𝑑𝑑𝑒𝑒𝑑𝑑 = 𝑅𝑅𝑑𝑑𝑝𝑝𝑑𝑑𝑑𝑑𝑑𝑑𝑇𝑇(𝑃𝑃0 − 𝐶𝐶) + 𝐶𝐶. (52)

We can rotate N points in a segment at once by making a 3xN matrix P, whose columns Pj are the N points. Then

𝑃𝑃𝑝𝑝𝑝𝑝𝑑𝑑𝑑𝑑𝑑𝑑𝑒𝑒𝑑𝑑 = 𝑅𝑅𝑑𝑑𝑝𝑝𝑑𝑑𝑑𝑑𝑑𝑑𝑇𝑇�𝑃𝑃 − (𝐶𝐶 …𝐶𝐶)� + (𝐶𝐶…𝐶𝐶) (53)

where P and Protated are now 3xN matrices whose columns are points expressed in the proximal coordinate system, and (C…C) is a 3xN matrix formed by N replicates of the column vector for point C. To simulate rotation by an Euler sequence, specify the desired angles and rotation sequence. Then evaluate Rtotal using Table 1 and equations 29-31. Then use equation 53 to compute the coordinates of the rotated points in the proximal coordinate system.

Shoulder Joint

Shoulder joint recommendations of the International Shoulder Group (ISG) were published by Wu et al., J Biomech 38 (2005) 981–992. The recommendations include description of a “thoracohumeral joint”. A more anatomically correct description involves the thoracoclavicular, claviculoscapular, and scapulohumeral joints. It is difficult to track the 3D motions of clavicle and scapula with skin markers, due to movement artifacts. For this reason, the thoracohumeral joint is often used to describe shoulder motion. The markers and axis definitions for the thorax and humerus are shown below. GH is the glenohumeral joint center, determined by regression or spherical fitting.

Figure 2 of Wu et al., 2005, below, shows the thorax coordinate system.

Yt: The line connecting the midpoint between PX and T8 and the midpoint between IJ and C7, pointing upward. Zt: The line perpendicular to the plane formed by IJ, C7, and the midpoint between PX and T8, pointing to the right. Xt: The common line perpendicular to the Zt-and Yt-axis, pointing forwards.

Figure 5 of Wu et al., 2005, below, shows the humerus coordinate system, recommendation 1.

Yh1: The line connecting GH and the midpoint of EL and EM, pointing to GH. Xh1: The line perpendicular to the plane formed byEL, EM, and GH, pointing forward. Zh1: The common line perpendicular to theYh1- and Zh1-axis, pointing to the right.

Wu et al., 2005, also give a second humerus coordinate recommendation, in which the long axis of the flexed forearm is used to define the Xh axis, rather than using the EL and EM markers. The disadvantage of recommendation 1 is that it uses EL and EM to define Zh. Since EL and EM are close together, small marker placement variation or skin movement can make large angle differences. The disadvantage of the recommendation 2 is that it requires tracking the forearm, and it doesn’t work if the elbow is extended.

Figure 7 of Wu et al., 2005, below, shows the ISG recommendation for decomposing thoracohumeral motion into three rotations: Yh-Xh-Yh.

The first rotation is about Yh. This is referred to in Figure 7 as “Plane of elevation” because this rotates the plane in which the humerus will be elevated in the middle rotation. The middle rotation is about Xh, and is elevation. Note that Xh points anteriorly in the neutral or initial position. Therefore, if the first rotation is zero, the rotation about Xh is abduction/adduction. (It is called negative elevation because elevating the right humerus is a negative rotation about Xh.) The final rotation is about Yh again: a final axial (internal/external) rotation. Wu et al., 2005, chose the Yh-Xh-Yh sequence because it is well known that if one allows both flex/ext (i.e rotation about Zh) and abd/add (rotation about Xh), the angles obtained are different depending on the sequence chosen. The solution of Wu et al. is to do only one “elevation”, and so avoid the issue of how much is abd/add and how much is flex/ext. A sagittal plane motion of the humerus, which is traditionally and clinically regarded as flexion/extension, is described by the ISG recommendation as 90° internal rotation, then elevation, then 90° external rotation.

The rotation matrix representation of the ISG recommended sequence is

𝑅𝑅𝐼𝐼𝐺𝐺𝐺𝐺 = 𝑅𝑅𝑌𝑌(𝜃𝜃2)𝑅𝑅𝑋𝑋(𝑐𝑐)𝑅𝑅𝑌𝑌(𝜃𝜃1) (54)

where θ1 is the initial rotation about the humerus long axis (“rotation of the plane of elevation”), φ is the elevation, and θ2 is the final rotation about the humerus long axis (internal/external rotation). Note that this is an Euler sequence, since the first and final rotation axes are the same, in the distal segment.

An alternative to the ISG recommendation is to do a Cardan sequence. We will now introduce an alternative set of axis definitions used by some investigators including James Richards. In this set of axis definitions, +X points to the subject’s right, +Y points forward, and +Z points upward. Using the Richards et al. axes, positive rotation about X is shoulder flexion, positive about Y is right humerus ADduction, and positive about Z is right humerus internal rotation.

If the final rotation is IR/ER, and if we use the axis definitions of Richards et al., then there are two Cardan sequences to choose from Xh-Yh-Zh or Yh-Xh-Zh. The Xh-Yh-Zh sequence correspond to flex/ext, then add/abd, then IR/ER (positive/negative in each case, for the right shoulder). Its matrix representation is

𝑅𝑅𝑋𝑋𝑌𝑌𝑍𝑍 = 𝑅𝑅𝑍𝑍(𝜃𝜃𝑍𝑍)𝑅𝑅𝑌𝑌(𝜃𝜃𝑌𝑌)𝑅𝑅𝑋𝑋(𝜃𝜃𝑋𝑋) (55)

The Yh-Xh-Zh sequence correspond to add/abd, then flex/ext, then IR/ER. Its matrix representation is

𝑅𝑅𝑌𝑌𝑋𝑋𝑍𝑍 = 𝑅𝑅𝑍𝑍(𝜃𝜃𝑍𝑍)𝑅𝑅𝑋𝑋(𝜃𝜃𝑋𝑋)𝑅𝑅𝑌𝑌(𝜃𝜃𝑌𝑌) (56)

where ±θX=flexion/extension, ±θY=adduction/abduction, and ±θZ=internal/external rotation. We showed above (equations 37-41) that RXYZ equals Q when

𝜃𝜃𝑍𝑍 = 𝑎𝑎𝑎𝑎𝑎𝑎𝑠𝑠2(−𝑞𝑞21, 𝑞𝑞11) (39, again)

𝜃𝜃𝑋𝑋 = 𝑎𝑎𝑎𝑎𝑎𝑎𝑠𝑠2(−𝑞𝑞32, 𝑞𝑞33) (40, again)

𝜃𝜃𝑌𝑌 = 𝑎𝑎𝑎𝑎𝑎𝑎𝑠𝑠2(𝑞𝑞31, +�𝑞𝑞112 + 𝑞𝑞212 ) (41, again)

where atan2(y,x) = four quadrant arctangent function.

Hip Joint

We can use a similar approach at the hip as at the shoulder. With the pelvis and femur in the neutral position, Xpelvis and Xfemur point to the subject’s right, Ypelvis and Yfemur point forward, and Zpelvis and Zfemur point upward. We can decompose hip motion with a sequence of flexion/extension followed by adduction/abduction followed by internal/external rotation. This means we rotate coordinates about Xpelvis=Xfemur, then about rotated Yfemur, then about twice-rotated Zfemur. The rotation matrix representation of this sequence is

𝑅𝑅𝑋𝑋𝑌𝑌𝑍𝑍 = 𝑅𝑅𝑍𝑍(𝜃𝜃𝑍𝑍)𝑅𝑅𝑌𝑌(𝜃𝜃𝑌𝑌)𝑅𝑅𝑋𝑋(𝜃𝜃𝑋𝑋) (55, again)

where ±θX=flexion/extension, ±θY=adduction/abduction, and ±θZ=internal/external rotation, for the right hip joint. For the left hip joint, ±θX=flexion/extension, ±θY=ABduction/ADduction, and ±θZ=EXternal/INternal rotation. Unit vectors for pelvis and thigh segments are identified from markers. The unit vectors, expressed as column vectors in the GRS, form the matrices

𝑅𝑅𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 = 𝑅𝑅𝑝𝑝𝑒𝑒𝑑𝑑𝑝𝑝𝑑𝑑𝑑𝑑 = (𝒊𝒊𝒑𝒑𝒑𝒑𝒅𝒅𝒑𝒑𝒊𝒊𝒅𝒅 𝒋𝒋𝒑𝒑𝒑𝒑𝒅𝒅𝒑𝒑𝒊𝒊𝒅𝒅 𝒌𝒌𝒑𝒑𝒑𝒑𝒅𝒅𝒑𝒑𝒊𝒊𝒅𝒅) (57)

𝑅𝑅𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 = 𝑅𝑅𝑓𝑓𝑒𝑒𝑝𝑝𝑓𝑓𝑝𝑝 = (𝒊𝒊𝒇𝒇𝒑𝒑𝒇𝒇𝒇𝒇𝒑𝒑 𝒋𝒋𝒇𝒇𝒑𝒑𝒇𝒇𝒇𝒇𝒑𝒑 𝒌𝒌𝒇𝒇𝒑𝒑𝒇𝒇𝒇𝒇𝒑𝒑). (58)

Q is defined as in equation 38:

𝑄𝑄 = 𝑅𝑅𝑓𝑓𝑒𝑒𝑝𝑝𝑓𝑓𝑝𝑝−1𝑅𝑅𝑝𝑝𝑒𝑒𝑑𝑑𝑝𝑝𝑑𝑑𝑑𝑑 = �

𝑞𝑞11 𝑞𝑞12 𝑞𝑞13𝑞𝑞21 𝑞𝑞22 𝑞𝑞23𝑞𝑞31 𝑞𝑞32 𝑞𝑞33

� (38, again)

and angles θX, θY, θZ (flex/ext, ad/ab, IR/ER respectively) are determined using equations 39-41.

Grood and Suntay and Knee Joint

Grood & Suntay (1983), J Biomech Engin 105: 136-144, proposed a “joint coordinate system” for the knee joint which has been widely cited. They proposed a standard rotation sequence and gave formulas for the rotation matrix in terms of “clinical angles”. They also gave equations for changing from tibial to femoral coordinate system when there is translation as well as rotation at the knee joint. They think about knee joint angles as arising from rotation about an axis fixed in the proximal segment, followed by rotation about a floating axis, followed by rotation about an axis fixed in the distal segment. The floating axis is perpendicular to the two body-fixed axes. They say that their approach gives a sequence-independent description of knee joint.

I do not think their proposed scheme was as new as they thought it was. I also think it is not sequence-independent, and their proposal does not “eliminate much of the confusion relative to nomenclature”, as they think. They use a specific sequence of rotations. Their equations for rotation matrix elements in their Appendix would be different if they had used a different sequence, which indicates that their claim of sequence-independence is not true. Their coordinate axes for thigh and shank are +X=right, +Y=anterior, and +Z=superior. They propose to rotate about X, then Y, then Z. This means flexion/extension, then AB/ADduction, then internal/external rotation. What they call the “floating axis” is really just the once-rotated Y axis, the axis of AB/ADduction. They call it “floating” because it is not aligned with any of the principle axes at the start or at the end. But that is standard for the axis of the middle rotation of a Cardan sequence. It is also worth noting that the “clinical angles” β and γ of Grood & Suntay 1983 are not defined positively by the RH rule in the right knee, as is usually the case.

Matlab files rot_coordsXYZ.m return points in rotated coordinate system rot_coordsXYZmtx.m return rotation-of-coordinate-system matrix rot_helixmtx.m return rotation-of-points matrix for any helical rotation rot_pointsXYZ.m return rotated points (rotations not in moving segment axes) shouldersim1.m simulation code using rot_pointsXYZ.m shouldersim2.m simulation code using rot_coordsXYZmtx.m shouldersimdata1.txt simulated data made with shouldersim1.m shouldersimdata2.txt simulated data made with shouldersim2.m euleranglesXYZ.m compute Euler angles for X-Y’-Z”, from Rprox, Rdist euleranglesYXZ.m compute Euler angles for Y-X’-Z”, from Rprox, Rdist helicalangles.m compute helical angles from Rprox, Rdist shoulder_angle_estimation.m estimate Euler and helical angles from shoulder marker data