# Coordinate Basis for Rigid Bodies

Contents

### Introduction

A **coordinate basis** is a set of x, y and z orthogonal axes which correspond to the axes of an object. It is useful to think of a coordinate basis as representing the orientation of an object in the room. Coordinate bases can be used to compute the relative angles between two rigid bodies, or between one body and the room.

Coordinate bases are constructed to have x, y and z axes which point in directions which have a useful interpretation in our measurement application. For biomechanics, the goal is usually to define coordinate bases so that they correspond to anatomical directions (e.g. the long axis of a limb, the medio-lateral direction, the anterior-posterior direction). Within the ODIN software suite, this process allows users to define useful measures such as the Euler angles between two limbs once the relevant coordinate bases are defined.

A coordinate basis can also be combined with a 3D reference point (marker, centroid, or virtual marker) to create a **coordinate system**, which is useful when we wish to compute the location of a point whose coordinates might be known relative to another point on the moving object.

The ODIN software suite allows the user to define one or more coordinate bases within any analysis protocol. To define a coordinate basis, we need to know the locations of any two points on the line of one of the **primary axes** (X, Y, or Z), and any two points on a **U-vector**, which can be any vector in one of the orthogonal planes containing the principal axis we have chosen.

Once the four points are defined, ODIN uses the Gram-Schmidt process to take the finite, linearly independent set of vectors (S) between the points and produce an orthogonal set (S’) that spans the same dimensional subspace. The vectors of S’ then point along the xyz orthogonal axes that define the coordinate basis.

### ODIN’s Coordinate Basis Panel

The Coordinate Basis panel within ODIN guides the user through the process to define a coordinate basis using three or four 3D reference points. Each 3D reference point can be a real or measured marker location, or a calculated point (virtual markers, centroids).

The “Coordinate Basis Configuration” dialogue box is displayed Figure 1 and is accessible through the path: “Calculations / New Coordinate Basis…”

**Figure 1: Coordinate System Panel**

### Configuration of Coordinate System Using Three Reference Points

In this example, we have an object as shown in the following diagram, where the arrows labelled X, Y, Z represent some meaningful axis directions, and the points labelled p, q, r are markers we have placed on the object (these may also be virtual markers or centroids).

We can see that point’s **p** and **q** both lie on our required Y axis, and we can also see that the line from **p** to **r** lies in the YZ plane. Therefore, we have sufficient information to define a coordinate basis using Y as our primary axis, and the YZ plane for the U-Vector.

First, select YZ plane in the “Coordinate Basis Configuration” dialogue box. Then select the primary axis start as **p** and end as **q**. Finally, define the U-vector in the YZ plane by selecting U-vector start as **p** and U-vector end as **r**.

### Configuration of Coordinate Basis Using 4 Reference Points

There is no particular reason that the start of the U-vector needs to be the same point as the start of the primary axis vector, as long as the line of the U-vector lies in the required plane, or a plane parallel to it, and is itself not parallel to the primary axis. For example, in the following diagram we have a situation where we have point’s **p** and **q** on the primary axis as before, but the required line in the YZ plane is from point’s **r** to **s**.