As a quick primer, we seek to learn the gait state, the set of variables that describes how a person walks, as it evolves during locomotion. The gait state as defined in our prior work is composed of the gait phase, a mechanical variable taking values between 0 and 1 that tells you how far along you are in a stride; phase rate, the time derivative of phase that describes the speed of progression through stride; stride length, the distance traveled during stride; and incline, the inclination of the ground. Together, these gait state variables are capable of parametrizing a wide variety of walking gaits (though by no means are you strictly limited to these, you can include more!). The gait state vector x(t) is then:

with phase, phase rate, stride length, and incline, left to right respectively.

At the heart of any Kalman Filter lies a measurement model that relates the (hidden) state vector, in our case the gait state, to some directly measurable quantities. In our past work, we used a variety of lower-limb kinematic variables. In this experiment, I tried out the following measurements: foot angle, shank angle, thigh angle, and pelvis angle, along with their respective time derivatives. These represent a plausible set of variables that could be estimated using the sensors on a lower-limb robot like an exoskeleton. To obtain the relationship between the gait state and these kinematics, I used the data from an open-source walking data-set, which contained different walking conditions on an instrumented treadmill over various speeds and inclines. This data-set also contained the ‘ground-truth’ gait state variables corresponding to each condition’s kinematics, and thus could be used to regress a continuous gait model that mapped gait state to kinematics, which was precisely what I needed. In short, I used constrained least-squares to fit a series of Bezier polynomial functions in 4D space to obtain this gait model shown below:

Additionally, having the dataset like this let me do something else pretty cool for the Kalman Filters. Ordinarily, Kalman Filters have a set of fixed gains in their process noise matrices and measurement noise matrices, which capture uncertainty in the dynamics process and measurement sensors respectively. The measurement noise matrix is typically informed by sensor spec sheets and lets the filter know how much trust it should place in each measurement. The process noise matrix then acts as your tunable parameters to control how fast the Filter updates, and thus, your response times and stability. However, what if the sensors had different levels of trustworthiness depending on the state vector? In that case, the measurement noise matrix would be heteroscedastic and could change its values dynamically (depending on how you look at it this could count as gain scheduling). For instance, consider the fact that, during normal locomotion, your foot’s global angle in the sagittal plane must be roughly equal to the incline of the ground slope during early stance. Consequently, you’d expect the foot angle signal to be highly informative of incline during that period of phase.

With this idea in mind, I also developed a heteroscedastic noise model within the Kalman Filters, which took the covariances of all the measurements as a function of phase and constructed a series of noise matrices from them. As the gait phase evolved, so too did the elements of the noise matrix (right).

As expected, the foot angle covariance (blue) noticeable decreases during early stance between 0.2 and 0.4 phase. This informs the Filter that the foot signal should be trusted more during early stance when the foot is contacting the ground. Similarly, the covariances between the foot angle and the other kinematics also drops during this region. This heteroscedastic model produced some really good estimates of ramp during testing, and let us have faster response times without sacrificing accuracy.