Basics of Biomechanics #3: Calculating Signals

Welcome back to Basics of Biomechanics, a series of blog posts covering foundational topics in the field using practical, data-driven examples. In this post we will look at some details about calculating biomechanical signals from our measurement data. 

As we saw in the previous blog post, some biomechanical analysis can be performed directly with measured signals. For example, in the case of our ongoing countermovement jump analysis we can immediately extract force parameters (e.g., peak landing force) and timing parameters (e.g., flight time) using the measured time and vertical force signals (see below).

However, many parameters in biomechanics are extracted from signals that first need to be calculated. This means that it is important to understand the mathematical process of how these calculated signals are created. As an example, if we had collected 3D motion capture data in our jump test we would have access to a large number of signals produced by calculations in a kinematic body model (see the the skeletal avatar and joint angle signals below). In this case, the motion capture equipment is using inertial sensors to measure acceleration signals, angular velocity signals and magnetic field signals. These measured signals go through a detailed processing pipeline to eventually produce calculated signals of interest such as joint angles.

Some of the most complex calculations in biomechanics are found in musculoskeletal models. These take in multiple signals as inputs and include aspects of advanced physics, calculus and matrix algebra. Other calculations in biomechanics involve much more simple processing of individual signals such as scalar operations, element-wise operations, time-derivatives and integrals etc.

These kinds of basic calculations can be illustrated using our jump analysis example. With only time and force signal measurements we can generate a number of useful calculated signals using a very simple computational model based on vertical motion of the centre of mass (COM). The relationship between vertical force and vertical motion is governed by Newton’s 2^nd law:

Force = mass x acceleration

Based on Newton’s 3rd law, when the person is standing still on the treadmill before the jump the gravitational force acting on their centre of mass is counteracted by an equal and opposite vertical ground reaction force. There is thus no vertical acceleration of the COM because the vertical forces cancel out i.e. the nett force is zero. However, this obviously changes when the COM starts moving during the jump as illustrated by the diagram below. As with all computational models, it is important to the keep the underlying assumptions in mind. In this case, for example, we treat the human body as a point-mass (we ignore non-rigidity and energy dissipation within the body). We also assume that the person has not other contact with their environment, and that they follow a particular protocol.

Challenge #1: can you figure out where each acceleration phase of the jump (static, down, up) is on the graph of the force signal in the first video above? Clue: imagine the horizontal BW line on the graph

It is important to understand that the force plate (FP) measures the ground reaction force and not the nett force. So to track the changes in vertical motion of the COM (through acceleration) we need to track the the nett vertical force. If we know the jumper’s body weight (F_BW), we can calculate the signal for the nett force acting on the jumper’s COM throughout the jump using a simple scalar subtraction operation:

      F_NETT (t) = F_FP (t) – F_BW

we can assume that our force plate measurement equates to body weight (BW) when the jumper is in a stationary standing position on the force plate just before the Initiate event. We can then calculate the COM acceleration from F_NETT and body mass (m_BW) using a simple scalar division operation (note that F_BW is also used to calculate m_BW):

         a_COM (t)=  F_NETT (t) /  m_BW

Scalar operations such as offset and scaling are the most basic kind of calculation you can perform to calculate new signals, but they often provide very useful insights. One insight that is visible in the graph below (although quite intuitive) is that during the flight phase a_COM is a constant value of -9.81 m/s². You may also notice that the shape of the acceleration curve is identical to the force curve, which is always the case for offset and scaling operations.

Challenge #2: the relationship between force and motion as illustrated in the figure above is a fundamental concept in biomechanics. What are some of the reasons that this calculation might be risky here? Hint: can you think of anything that could go wrong with the estimation of body weight? What could happen to our acceleration curve if body weight were estimated 10 N too high or too low? How would that affect our calculations downstream?

Since we have measured a time signal as well as a force signal, the COM velocity signal (v_COM) can now be calculated by time-integrating a_COM from just before the Initiate event. Likewise, the COM position/displacement signal (d_COM) is calculated from v_COM. Integration is a recursive element-wise calculation involving two signals (force and time), but conceptually it is just the area under the curve.

Challenge #3: Notice that peak velocity occurs before take-off. Do you find this surprising? How would you explain it from a biomechanical stand-point? Hint: inspect the acceleration graph around Take-off for the mechanical reason and the position graph around Take-off for the physiological reason.

Another signal which is sometimes used in jump analysis for assessing explosive capacities in athletes is the power signal (magenta curve above). At each signal sample, power is calculated as the product of force and velocity:

Power (t) = F_FP (t) v_COM (t)

This is an example of using two calculated signals to generate a third signal using an element-wise operation. The figure above is also an example of a somewhat uncommon case where signals are only calculated for specific portions of the timeline using events – in this case, from Initiate to Landing. The reason for this is that the force plate does not measure velocity or position directly, so we have to assume an initial value for these signals when we start the time-integration calculation. As shown in the model diagram above, we assume that the velocity and displacement at Initiate are zero. This means that d_COM is not the height above the floor; rather a zero value represents the height during still standing just before Initiate. It also means that if the jumper is already moving at our Initiate point (the event is incorrect), v_COM will be incorrectly calculated. This will also significantly affect d_COM and and the power signal, as well as all the discrete outcomes related to these two signals. So if your signals depend upon your event detection, it is important to know this.

Summary

This post is a basic illustration of how useful a computational model can be for generating additional signals of interest in biomechanics. However, most model calculations make certain approximations of reality and it is important to be familiar with the equations and assumptions that are involved so that you can perform your interpretations “with your eyes open”. For the technically oriented biomechanist, it is important to develop competencies in numerical analysis techniques so that you can innovate and perform quality assurance on your data. However, even if the mathematical detail is not as relevant to you, there is great benefit in understanding the definitions and conceptual background to the calculations in your biomechanics pipelines.

Please feel free to leave a comment if you are interested in sharing your experiences, asking questions or giving constructive feedback on how this post presents the topic. We can also discuss the answers to the challenges!