DATA TRANSFORMS
Non-Linear Transforms
With the acquisition of EMG signals from various sources (online datasets and custom-built EMG sensor datasets) that have been thoroughly pre-processed (see the Wavelet Filtering section in the Mathematical Backgrounds page for more information), we must convert the EMG signal into an useful form that our intended visualization system (Simulink and Simscape) can easily interpret. In particular, we must transform the EMG signal (which is measured in voltage for both the online datasets and custom-built EMG sensor datasets) into a torque signal. Inputting the torque signal to a joint (for example, the elbow) allows us to leverage the capabilities of Simscape as a multibody physics simulation by abstracting away the complex kinematic and dynamic relationships of different muscle groups and physiological structures into a black-box system in which a torque signal input will result in a corresponding motion in space. Luckily, this signal transformation can conveniently be handled by applying a transform to the EMG signal.
A transform converts a signal in one domain to another signal in another domain, in the sense that the physical phenomenon described by the signal is not changing, but merely that the perspective through which we observe the signal is shifting. As described above, our project requires that we observe the EMG signals as a torque signal rather than a voltage signal. Having identified the need for a transform, our team had to decide whether we were going to search for a pre-existing transform for converting EMG voltage signals into torque signals, or develop our own model. We decided to pursue the former option because the scope of our project does not include extracting transforms from complex biomechanical models of the human body.
Fortunately, the prior art for EMG signal conversion is extensive, with the earliest papers on this topic written in 1952. Our review of the prior art quickly revealed that non-linear transforms
A research paper written by Khalil Ullah, and Jung-Hoon Kim [14] proposed a mathematical model estimating the torque from a given emg signal. With 5 subjects taking part in the experiment, they measured the participants at various effort levels and recorded their outputs. With this data at hand, they noted that the correlation between the torque and emg tended to be nonlinear as well as exponential. They proposed several models to chart this data and compared the outputs of these models to the actual output. The model that produced the highest level of accuracy was:
𝞃 = (u^a) * exp(b - c * u)
Where “u” represents the emg signal and “a,b,c” are variables determined in the table below. With this expression from the paper, we can approximate the emg data we have collected into some torque force.
It is important to note that the model proposed by Ullah et. al. is not the only EMG signal-to-torque transform in the prior art, and that other non-linear models do exist. Furthermore, these models are heavily dependent on the setups that individual research groups use to collect the data for developing their models. However, it appears that the exponential trend is a common theme across different research papers. For example, the plot of expected versus measured outputs of the non-linear model produced by Ullah et. al. is shown below, where the exponential exhibits a concave curvature [14]:
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Bhadane et. al. also reported a non-linear, exponential that has a similar curvature to Ullah et. al., as shown below (note that the axes for the plot generated by Bhadane et. al. are flipped with respect to the plot generated by Ullah et. al.) [5]:
However, the data collection setups used by the two research groups are different, as it can be seen in the pictures below. After observing that the EMG signal-to-torque transform exhibits similar behavior across different setups, our team determined that using the non-linear transform model provided by Ullah et. al. could be applicable to our project as well.
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Linear Transforms
After the non-linear model converting EMG signal to torque was established, we inputted the resulting torque signal into a simplified human arm as seen in Figure 1 comprising a single revolute joint at the elbow (as opposed to the ball-and-socket joint providing three rotations). As such, the Shoulder Joint and Bicep serve as the ground frame of the Elbow Joint. Similarly, the Hand is rigidly attached to the Forearm.
Figure 1. CAD model of the human arm simplified to a single revolute joint at the elbow.
The following linear transform relating torque to the angular displacement of the arm about a single revolute axis is an example of what is used by Simscape to process the model because we made the simplifying approximation. 𝜏(t) is torque, I(t) is moment of inertia, B(t) is damping coefficient of the elbow joint, K(t) is stiffness of the elbow joint, and θ(t) is angular displacement. [22]
By inputting a torque at the elbow joint, one can solve the above set of differential equations to get the angular position, velocity, and acceleration of the elbow joint, which is the underlying working principle (perhaps a simplified version) of the Simulink model that Simscape is using to produce the visual rendering.
To fully define the Simulink model, we had to select several parameters of the arm geometry along with those embedded in the linear transform described in Equation 1. Most of the arm parameters were directly extracted from the anthropometric survey conducted by Gordon et. al. [23]. The density of the arm, as well as the local geometries of the arm, were approximated to fit the anthropometric measurements. More specifically, the density of the arm was approximated to be the density of ABS plastic, which is 1052 kg/m^3 (provided by the CAD software). We selected ABS as an approximation because it was one of the materials whose density was the closest to that of water, which is ~1000 kg/m^3, and the human body is mostly made of water.
The moment of inertia parameter of the linear transform was then subsequently calculated by the CAD software based on the arm geometry and density. While Equation 1 represents the moment of inertia as a function varying in time, we considered the moment of inertia of the arm to be constant because of the underlying assumption that the arm is rigidly modeled in the CAD model. We also had to make a similar decision for the stiffness and damping coefficient parameters of the linear transform due to the elbow joint’s non-constant stiffness and damping characteristics (see Figure 2) [22]. We selected the upper bound of the stiffness and damping coefficient values to ensure that the dynamics of the arm are not too fast (which can introduce unexpected values)
Figure 2. Non-constant damping and stiffness parameters of the elbow joint [22].
The following list shows our finalized parameter selections:
Elbow-to-Wrist Length: 11.30” [23]
Shoulder-to-Elbow Length: 14.32” [23]
Body part density: 1052 kg/m^3
Stiffness (K): 10 Nm/rad [22]
Damping Coefficient (B): 1 Nm/rad/s [22]
By using these parameters in the model, we were able to produce a realistic rendering of the arm moving in response to the Gaussian-filtered custom EMG data that we collected with the EMG sensor (see Figure 3).
Figure 3. The output of the non-linear transform with a Gaussian smoothing filter for the custom EMG dataset. This signal was inputted to the Simulink model as an input torque to actuate the elbow joint.
