Joint Angle Estimation Using Accelerometers and Magnetometers

Magnetometers

Tracking a human body's joint angle empowers the clinical studies for post-operative analysis and prediction of an unhealthy as well as healthy subjects’ possibility of injury. To this end, joint angles of a human jumping and landing are estimated in a daily environment where professional medical facilities are not available. Joint angle estimation along with the inertial data of the human body movement can form a strong tool for activity detection. In this section, an attempt is made to address the problem by finding the joint angle changes during an epileptic seizure.

Current approach to estimate the joint angles rely on the complimentary features of both accelerometers/gyroscopes and compute the best estimate. Indeed, other portable sensors, such as magnetometers, may be included in the integration in order to improve the overall quality. However, the provided estimates may be corrupted by different types of errors such as the induced error of angular rate signal integration. To overcome these problems, a method of angle estimation without integration of the angular rate signal is proposed. This method has been originally proposed by Kun et al [112] and is composed of inertial sensor difference based and virtual sensor difference based algorithms. Virtual sensors are sensors that are not physical and are imagined in order to relate the acceleration of the arm to the acceleration of the forearm at their intersection point which is the elbow joint in the present case. Hence, the difference of the two accelerometer terms of the virtual sensors is thought to be solely due to the rotation of the coordinate systems. Considering this fact, the rotation matrix can be defined for relating the two accelerometer terms and this can eventually lead to the joint angle estimation. As shown in Figure 4-1, three accelerometers are attached to the forearm, out of which two are in the same plane in accordance with the configuration proposed in chapter 3. The equivalent acceleration measured by the accelerometer including the gravitational component is given by

, ‎4-1

Figure ‎4-1: The elbow virtual sensor and three forearm physical sensors

where is the equivalent acceleration at point where the th sensor is located, g is the gravitational acceleration, is the position vector associated with point relative to the global coordinate system (O-XYZ) and is the rotational radii about the origin of the forearm at point O which corresponds to the elbow location in the forearm. Based on the proposed configuration, following equations are derived for the acceleration of each sensor:

, ‎4-2

, ‎4-3

. ‎4-4 In Eq. ‎4-2 to Eq. ‎4-4, represents the virtual sensor acceleration while and

denote the accelerations of the two sensors placed in the same plane of the forearm.

Using equations ‎4-2 to ‎4-4, the vector of acceleration at the rotation joint can be obtained as follows: ‎ 4-5

To analyze the elbow joint rotation angles, an algorithm based on the difference between double virtual sensors implanted in the elbow joint, called virtual-sensor difference based, is employed. The forearm and the arm segments are assumed as rigid segments connected with a 3-DOF joint. Three physical sensors in locations determined by the optimization technique in chapter 3 and a virtual sensor in ‘green’ are fixed on the forearm as shown in Figure 4-1. In the same manner, three physical sensors and a virtual sensor are attached on the arm. The corresponding axes of the two physical sensors in the same plane and the virtual sensor in each segment are in the same direction. Hence, the accelerations of the two virtual sensors can be calculated from the accelerations measured by the pair of physical sensors associated with each group, respectively, using the physical-sensor-difference based algorithm as explained earlier in this section.

It is clear from the fundamentals of dynamics, when a multi segment rigid body is moving in space each point on that body has a unique acceleration; hence, the two virtual sensors in the elbow joint must have equal accelerations in the same coordinate frame. As shown in Figure 4-2, two virtual sensors are placed on the elbow joint at the same position and plane but with different orientation; one in the longitudinal direction of the arm and another in the longitudinal direction of the forearm. On the same position and plane at the elbow joint, the two virtual sensors attached in different orientations measure two groups of accelerations. The difference between the acceleration vectors represents the angular change associated with the joint connecting the two segments, which can

illustrate the rotation angles of the elbow joint. The relationship between the two accelerations measured by the two virtual sensors can then be formulated as

, ‎4-6

where R is the rotation matrix between the two virtual sensors, which also represents the rotation matrix between the forearm and arm segments.

Figure ‎4-2: Analysis of the elbow joint angle using the double virtual sensors considered to be on the elbow

It may be noted that for calculating the rotation angles, at least two vectors relating the two planes are required. For this purpose, two magnetometers in conjunction with the accelerometers are used to measure the magnetic field data attached on the forearm and arm, with the corresponding axes in the same directions as those of the accelerometers. Following the same procedure for calculating the rotational angles from accelerations using the virtual accelerometers in the elbow joint, two virtual magnetometers attached with different orientations in the elbow joint must physically have a unique magnetic field data. The orientation difference between the vectors of magnetic field data represents the elbow joint angular change and hence can be employed in order to

calculate the joint angles. The relationship between the measured magnetic field data is then governed by

, 4-7

Hence, based on physical-sensor-difference-based algorithm and virtual-sensor- difference-based algorithm, the rotational angles of the elbow joint can be calculated from 4-5 to 4-7, and then the rotation matrix R can be obtained. Once the rotation matrix R is obtained the angles can be computed from the following expression

, 4-8

In the same manner, two physical sensors can be placed on the chest so that the angles of the shoulder can be decided.