Magnetometer-Free Joint Angle Estimation

As explained in Section 1, handling arbitrary sensor-to-segment mounting is a major challenge in gait analysis with inertial sensors. Manual measurements, as well as calibration poses and movements, are commonly suggested solutions. Furthermore, we pointed out that the use of magnetometers is typically limited by the assumption of a homogeneous magnetic field. In this section, we describe a set of methods for IMU-based joint angle estimation that allow us to face these two challenges in a new way. We will combine elements of the methods reviewed above, but unlike most previous attempts, we will:

avoid sensor-to-segment mounting assumptions;

require no manual measurements of any distances, etc.;

not rely on the accuracy with which the subject performs predefined postures or movements;

and avoid the use of magnetometers.

Instead of employing any of these commonly used assumptions and restrictions, we make use of the fact that the knee joint behaves approximately like a mechanical hinge joint. The kinematic constraints that result from this fact are exploited to obtain the position vector and the direction vector of the knee flexion/extension axis in the local coordinates of both sensors. As outlined above, this information is crucial to precise joint angle calculation. We will use it to fill the gap between the sensor coordinate systems and the joint-related coordinate systems in which the angles are denned. Subsequently, this will allow us to calculate flexion/extension joint angles on joints with a major axis of motion, for example the knee and the ankle during walking. All of the methods that we will introduce use only angular rates and accelerations, while the use of magnetometer readings is completely avoided.
Before we describe the respective algorithms, let us define the available measurement signals. Assume that two inertial sensors, one attached to the upper leg and the other attached to the lower leg, measure the accelerations, a₁(t),a₂(t)ϵ ℝ³, and angular rates, g₁(t),g₂(t)ϵ ℝ³, at some sample period, Δt. Additionally, we calculate the time derivatives ġ₁(t),ġ₂(t)ϵ ℝ³ of the angular rates via the third order approximation:
${\dot{g}}_{1/2}(t)\approx \frac{g_{1/2}(t-2\mathrm{\Delta }t)-8g_{1/2}(t-\mathrm{\Delta }t)+8g_{1/2}(t+\mathrm{\Delta }t)-g_{1/2}(t+2\mathrm{\Delta }t)}{12\mathrm{\Delta }t}$