Calibration Woes – Spatial Alignment of Systems

When Space Slips

A few years ago, I read a blog post from the International Women in Biomechanics group, by Katie Bradley. I can’t seem to find it on their new website since they moved hosts, but the original post remains in my bookmarks and in my memory- accessible here for now: link.

In it, Katie writes about the importance of a very basic step in motion capture: setting your origin relative to force plates. This is often one of the first things done in a motion capture lab after turning on the system, and something I discussed briefly in my first post about calibration.

In most motion capture labs, you will find squares or rectangles built into the ground that are essentially quite fancy scales. They can measure an incredible range of forces output in six components: force in each axis (X, Y, Z) as well as the rotational moment around those axes ( $M_x, M_y, M_z$ ). They are useful not only as scales, but also because the measured forces along the axes allow you to calculate and draw conclusions on what produced the force. We can get metrics such as A center of pressure can further be calculated from this data, and all of this force data gives us the external boundary conditions required for calculating inverse dynamics- without it the equations become statically indeterminate or become estimates. Combined with the kinematics from the motion capture system, we have a full method of calculating inverse dynamics.

The core of the problem is that we have two systems that must cooperate together but are deaf and blind to one another. We have force plates that only care where things are within the confines of the plates edges, and we have a motion capture system that is designed to track things but cannot see the force plates. To be able to use them together, we need to establish the same zero point for both systems, as well as directions for that zero point. By aligning them, we have a motion capture system that can track the things we care about and a force plate system to measure ground reaction forces when interacted with, and the ability to do calculations using both.

But how exactly can we align them? With a magic wand, of course!

An older Vicon 5 marker passive wand – note the axes are not to current Vicon convention

Okay, a little less than magic. This figure is an older variant Vicon passive wand, but the principles are that the wand has a right angle which can be aligned to something, and which also defines the X, Y, and Z axes.

The wand can be perceived by the motion capture system, on account of its markers or LEDs. The wand can be squared to the edge of a force plate, so that we can use a physical reference point relative to the plates that the motion capture system can see. By placing the wand on the corner of a plate and setting the origin in the software, we move the motion capture system’s zero point to the corner of the force plate in physical space. From here, we intuitively know where the force plate needs to appear in the software, and we can figure out where any successive plates are by taking physical measurements and translating those into the virtual positions of the force plates in our motion capture software.

Force plates visualized in the Vicon Nexus 2 software, but in incorrect locations. The wand is highlighted in blue.

For example, in this image, we can see that we have a few force plates in the middle of the floor grid. The center of the grid represents the origin point, with the axes shown in the bottom left corner; they are also slightly visible in the middle of the grid.

In this case, we have three force plates, but they do not have their positions set correctly, and so they are all overlapping. Physically, the wand, highlighted in blue, has been placed in the corner of force plate 2, in between force plates 1 and 2, so we know where the force plates should be relative to where the origin is set. The software doesn’t know where the force plates are supposed to be, so we need to instruct it.

Force plates visualized in the Vicon Nexus 2 software, in the correct locations. The wand is highlighted in blue.

First, we will work with force plate 2, since we used it as a physical reference point by squaring the wand to the corner of the force plate. In this case, the position of the force plate is easy to derive- it just needs to be moved by half of its width and length so that the corner of the plate matches the corner of the wand.

Force plate 1 is similarly easy, just needing to be moved by half its width and length. However, we also need to account for the gap between the force plates, so that is considered when inputting its position in the X direction (red axis).

Force plate 3 requires a bit more arithmetic, needing to be moved a full 1.5 plate lengths away from the origin, plus the gap between plate 2 and 3.

Again, these positions are all in reference to where the motion capture system’s origin is defined, and where the force plate origin is defined. The force plate coordinate system is defined at the center top of the plate surface, so we move the position accordingly. Once that is done, we’ve defined the relationship between the two systems. If any of these positions or orientations are incorrect, we lose alignment between the systems and the calculations done assuming they are aligned become inaccurate.

Now that we’ve defined the background, let’s return to the subject at hand: spatial misalignment.

Let’s say that we’ve come in for the morning, ready to collect some data. We calibrate the system as usual, set the origin in the same way we’ve always done, but today after we put the wand down, the wand shifts just slightly out of alignment with the corner of the force plate- only one degree.

Misalignment between physical and virtual coordinate systems, illustrated in PowerPoint.

As you can see, the further away from the origin, the larger the difference is between the actual, physical force plate location and the force plate position in our L frame defined motion capture system. How could this impact our calculations down the line, when we have someone walking over the force plates? Could it influence our data and cause us to draw incorrect conclusions?

Time for math. Let’s assume the force plates are in a perfectly straight line and are of standard size, each 1L in length and 1W in width. The furthest corner would be at coordinates (3L, 1W) from the origin- not accounting for the gap between the plates. What is the difference between C (the actual, physical furthest force plate corner from the origin) and C’ (the corner location in the motion capture system)? How much worse can it get if you have N force platforms?

First, we define the coordinates of the corner. With N force plates, the coordinates are:

C = (nL,W, 0)

where n is the number of platforms in the row, W is the platform width, and L is the platform length along the x axis.

Next, we’ll write the rotation matrix. A rotation about the vertical (Z) axis by angle θ is:

R(\theta)= \begin{bmatrix} \cos\theta & \quad -\sin\theta & \quad 0\\ \sin\theta & \cos\theta & 0\\ 0 & 0 & 1 \end{bmatrix}

Theta is the angle our L frame is off by.

To get C’, we multiply R(θ) by our corner coordinate, C. If there was translation, we would add that too, but not in this example.

C’ = R(\theta) * C

\begin{bmatrix} C’_x \\ C’_y \\C’_z \end{bmatrix} = \begin{bmatrix} nL\cos\theta -W*\sin\theta \\ nL\sin\theta + W*\cos\theta \\ 0 \end{bmatrix}

To get the error, we simply subtract the new apparent position from the true position.

\Delta X = C’_x – C_x = nL(\cos(\theta )- 1) – W * \sin \theta

\Delta Y = C’_y – C_y = nL\sin(\theta ) + W ( \cos( \theta) – 1)

Since this is a small angle, we can use the small-angle approximation, where $\cos \theta \approx 1$ and $\sin \theta \approx \theta$ (θ in radians), leaving us with:

\Delta X \approx -W * \theta

\Delta Y \approx nL*\theta

Assuming standard force plate dimensions of W = 0.4m and L = 0.6m, and n = 3, with our 1 degree error:

\Delta X≈-0.4*0.01745 \approx -0.00698m

\Delta Y \approx (3 * 0.6) * 0.01745 \approx 0.0314 m

All of this means that if the L frame is rotated 1 degree, the far corner appears roughly 7mm offset along the X axis and about 31mm along the Y axis. This doesn’t seem like a huge difference, but looking closely at $\Delta Y$ , we can see that the difference scales with the number of plates we have. The error grows with distance from the origin.

How does it affect our math downstream, in a joint moment?

Let’s try to reduce the math to be as simple as possible, and assume we have a vertical point force of 100 N applied to the physical center of the third force plate, at point P. Point P’ is where the motion capture system thinks the center of the force plate is.

We first find the center of the third plate, which would be:

P_x = L(n-0.5) \qquad P_y = W/2

Now we find the change in P using the same formulas:

P’_x = L(n-0.5)\cos\theta -0.5W*\sin\theta \qquad P’_y = L(n-0.5)\sin\theta + 0.5W\cos\theta

\Delta X = P’_x – P_x =L(n-0.5)(\cos(\theta)-1) – 0.5W*\sin(\theta)

\Delta Y = P’_y – P_y = L(n-0.5)\sin(\theta ) + 0.5W ( \cos( \theta) – 1)

Plugging in our width, length, and number of plates, and ditching the small angle approximation:

ΔX = 0.6(3-0.5)(\cos(1\degree) – 1) – 0.5*0.4*\sin(1\degree) \qquad \approx -0.0037m

\Delta Y = 0.6(3-0.5) \sin(1\degree)+ 0.5 * 0.4 (\cos(1\degree) – 1) \qquad \approx 0.0262m

With the 100 N of point force applied directly to the physical center of the plate, the force plate measures the correct center of pressure (COP) in its own local coordinate system, with zero moment about the x and y axes of the force plate coordinate system. This is represented by point P.

The motion capture system sees the foot apply 100N of force to the physical center of the plate, and the markers show at point P in the motion capture system. However, in merging the two systems, we transform that measurement into the global motion capture coordinate system using the wrong L frame orientation. The force data remains accurate, but the relationship between the force plate and the motion capture coordinate system is not. As a result, the center of pressure is drawn at P’ instead of its true location P, which introduces an error into the moment arm used for inverse dynamics.

Considering only the larger Y-direction position error and the vertical force of 100 N, the resulting moment about the x axis for the foot segment centered at P is approximately 2.62 Nm. Note that this is not the total joint moment- this is an erroneous moment introduced into the inverse dynamics calculation.

\tau = r \cross F = 0.0262m \cross 100N \approx 2.62 Nm

This is an oversimplification of the problem- but the key idea is that the placement of the L-frame affects calculations downstream by changing the COP location. A small rotation of the calibration L frame can have a noticeable effect on your joint moments.

In most labs, the force plates aren’t perfectly level. They may not be perfectly in line, either. There will also be gaps between the force plates that need to be accounted for, and human error in placing the L Frame. If you have big arrays of force plates that aren’t necessarily next to each other, the error in placement can be particularly large, which is where devices like HasMotion’s CalTester come in.

Labs that have instrumented treadmills- treadmills with force plates built into them- have further considerations to make, as usually there is no edge to square to, the treadmill itself is able to incline, and the force plates aren’t directly beneath the surface of the belt, leading to more specialized methods of setting origins and updating platform positions.

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