# Quasi-Passive Resistive Exosuit for Space Activities: Proof of Concept
## 2 Materials and Methods
### Proof of Concept
#### 2.2.3 Soft Sensors Technology
The magnetometers, embedded into the IMU, are very sensitive to electromagnetic noise, and provide low reliability in any scenarios where electromagnetic materials are located [93]. For this reason, the knee angle measurements rely only on accelerometer and gyroscope outputs [94]. We assume that the knee is a perfect revolute joint, whose rotation axis is perpendicular to the sagittal plane (shown in Figure 5), however, this assumption that the knee can be modelled as a perfect revolute joint introduces an important constraint for the algorithm. Base on this assumption, the movement of the linked bodies is a rotation  
Figure 4: Sensing kneepad integrated into the XoSoft exosuit prototype. (**A**) Schematics of the sensorized kneepad with three embedded capacitive strain sensors. (**B**) Kneepad with sensors covered by textile patches and with coaxial cables embedded in an elastic sleeve. (**C**) Raw soft sensor outputs during normal walking tests. around the joint axis. Consequently, the coordinates of the joint axis, expressed in their local frames, are different for each IMU although they are constant in time [95]. The knee angle is subsequently computed by assuming that the rotation axis of the knee joint is invariant with respect to the local frame of the IMUs. This implies that, if \(k_{1}\) represents a unitary vector in the direction of the knee rotation axis in the local frame \(1\), and \(k_{2}\) represents the same unitary vector in the local frame \(2\), the knee joint angular velocity at time \(k\), \(\theta_{neck}\), can be computed as follows  
\[\theta_{knee_{k}}=\omega_{1k}^{T}K_{1}-\omega_{2k}^{T}K_{2} \tag{4}\]  
where \(\omega_{1k}\) and \(\omega_{2k}\) are the angular velocities of bodies \(1\) and \(2\) measured by the corresponding IMUs at time k. Then, the knee angle \(\theta_{knee}\) at time \(k\) can be obtained from the linear approximation  
\[\theta_{knee_{k}}=\theta_{knee_{k-1}}+\theta_{knee_{k}}\Delta h \tag{5}\]  
Equation (4) requires knowledge of the local coordinates of the \(\mathbf{k}_{1}\) and \(\mathbf{k}_{2}\) vectors. Then, the spheroidal coordinates of these vectors (\(\mathbf{k}_{1}\), \(\mathbf{k}_{2}\)) are computed using [PERSON]'s method which minimizes the error vector [95]. Vectors \(\mathbf{k}_{1}\) and \(\mathbf{k}_{2}\) (with unitary modules), expressed using spheroidal coordinates, can be iteratively computed using [PERSON]'s method to minimize the error vector. This method used to estimate the relative angle between two IMUs is known Least Square method (LSQ). A significant problem with integration, however, is that inaccuracies inherent in the measurements quickly accumulate and rapidly degrade the accuracy [96]. To improve the knee angle tracking, the algorithm incorporates: sensor random drift models, prior knowledge of the physical constraints and human natural range of motion, and zero-velocity updates. This gives better joint angle estimates and mitigates the effect of sensor drift on the estimated heading angles during long periods of movement. Due to the difficulties, for the LSQ method, to mitigate with a running calibration phase the drift, biases and noise, an extended Kalman filter (EKF) based method is required. Although the development of the EKF is beyond the scope of this work, the main equations are reported for completeness. The nonlinear state-space discrete form representation of the process and measurement models are expressed by  
\[ \begin{split}\mathbf{x}_{k+1}&=\mathbf{f}(\mathbf{ x}_{k},\mathbf{u}_{k},k)+\mathbf{w}_{k}\\ \mathbf{z}_{k}&=\mathbf{h}(\mathbf{x}_{k},k)+ \mathbf{v}_{k}\end{split}  \tag{6}\]  
where \(\mathbf{x}\), \(\mathbf{u}\) and \(\mathbf{z}\) are the state, control and measurement vectors and \(\mathbf{w}\sim N(0,\mathbf{Q})\) and \(\mathbf{v}\sim N(0,\mathbf{R})\) represent uncorrelated Gaussian processes with zero mean and covariance matrices, \(\mathbf{Q}\) and \(\mathbf{R}\).