Kinematics of Motion, Part II: Angular Movement

Settle in class, today's lesson will build upon the physics we learned in a previous post.  Last time we discussed the merits of multibody dynamics and kinetic links, and how they play a role in human movement.  While that's all well and good we need to get into the basics of how we can analyze our movements.  When we take a step, what are the velocities/acceleration/forces of our foot/shank/thigh?  What are the angular velocities and accelerations and moments of inertia of the moving parts?  Why do we even care?  Well if you want to get strong and fast, or teach others to get strong and fast, you'd better at least know how we move.

These guys might be able to help with this article

That Was Fast

All movements no matter how small or big have an associated velocity and acceleration.  The quick and dirty of it is velocity is the rate of movement - how far you travel in a certain amount of time, given in the units of distance per time (km/h, m/s, miles/h, etc) - where average velocity = d/t.  Acceleration is defined as the rate of velocity - how fast you are moving in a certain time, given in the units of velocity per time (m/s^2, etc) - average acceleration = v/t.

As an example let's say you walk 10 m and it takes you 5 seconds to do so.  Your velocity for the entire 10 m trip (average velocity) is 2 m/s (10m / 5s).  Likewise, let's say you're driving in a car now; currently you're travelling 10 m/s, but you want to go faster so you speed up to 30m/s and it takes you 5 seconds to do so.  Your acceleration during that period is 4 m/s^2 (30-10/5).

This is all well and good but moving in a straight line, like sprinting down a 100 m path or driving down the street; but what about when moving in a circle or along an arc?  This is where angular velocity and angular acceleration come into play.

The general principles still stay the same; the angular velocity is a rate of distance but now the distance in no longer a straight line, but an arc.  We define angular velocity as 'w', where w = 2πf, and f is the frequency of revolution (revolutions per second).  That's all well and good if we know the rotational frequency, but sometimes we don't.  Instead we may be presented with the tangential velocity (v) - a straight line velocity (like what we discussed in the first paragraph) that comes off the arc we are traveling in.  Given the tangential velocity we can still calculate the angular velocity, as long as we have the arc's radius (r); w = v/r.


Angular acceleration works in a very similar matter.  Where straight line acceleration was the rate of velocity, angular acceleration is the rate of angular velocity.  We define angular acceleration α = w/t, or α = a/r; where a is the straight line acceleration. 


Complications


So if that wasn't complicated enough, I'm going to F it up one more notch.  Picture this: you're being drive in a bus traveling at 100 km/h.  You decide to stretch your legs and go for a walk, so you walk forward inside the bus at 2 km/h.  Relative to an observer sitting inside the bus, he sees you moving at 2 km/h.  However, an observer outside of the bus will see you move at 102 km/h.  This is because to the observer on the bus, his world (the bus) is stationary and you are the only object moving.  To the observer outside, his world is the outside environment, and the bus is moving with you inside it, thus your net velocity will be the buses' plus yours.


The same scenario occurs with angular velocity and acceleration too.  Unfortunately, we can't just add/subtract the values from one another and call it a day like with the straight line values, so we have to do some fancy math to figure things out.  This effect is known as the Coriolis Effect; the deflection of an object when moving along a rotating frame of reference.  


Before I lose you, that pretty much means what I said in the sentence before; if you're standing on a merry-go-round rotating clockwise from the top but you walk in a counter clockwise fashion, if the merry-go-round's angular velocity is greater than yours, to an observer watching you from the ground you'll still be moving clockwise.  However, to an observer standing with you on the merry-go-round you're moving counter clockwise?  Why is this?  Coriolis Effect!

Coriolis Effect:  NOT used to determine which way the water drains

I won't get into the calculations and formulas for involving the Coriolis Effect into dynamic force, but when analyzing movements we can now determine the overall velocity output of a multi-body segment provided they are moving independently of each other.  Even in practice engineers/biomechanists don't calculate these effects by hand but rather plug them into a computer; physically calculating this is usually reserved for those with time on their hands, or 2nd year engineering students.  If you fall under one of those two categories you can check it out here.

Pictured: A horribly disproportionate leg

Application

Alright, so why do you need to know this stuff?  Simple - all our movements are governed by laws of Kinematics.  Throwing a ball, sprinting, jumping, etc can all be expressed using the concepts I've outlined above.  

In this house we obey the laws of Thermodynamics!

As outlined in a previous post the human body can be represented by pins (joints) and segments (limbs). Each time a limb moves it is rotating about a pin joint; even with an overall linear motion (ie. a punch) each limb is rotating about its joint and thus has rotational kinematic properties - a direction of rotation, inertial centre of mass, angular velocity and angular acceleration (assuming a dynamic situation).  Therefore, we can state that all human movement can be expressed as angular velocities and moments of inertia of various limb segments.


When you take a step, what is occurring in your lower body?  Your thigh, shank (shin) and foot all rotate and move independently.  More specifically your hip is moving in flexion, so your thigh is rotating in a direction; your knee is moving in extension, so your shank rotates as well; your foot is dorsiflexed in anticipation of the heel strike, so your foot rotates.  When we walk no one walks with all three body sections moving in complete unison unless you wear a pair of super tight jeans and walk like Frankenstein.


All of the limb rotations and movements are not in sync with each other; the hip flexes initially, then the knee extends some short time after, then finally before the foot strikes the ground the foot dorsiflexes to allow heel strike.  Hence they will have their own relative angular velocities and accelerations, relative to an observer sitting on their surfaces.  


Just like with the bus example, to an observer standing back and watching you walk, there will be an absolutely angular velocity and acceleration of your foot.  That's where the Coriolis Effect calculations will come in; people who may use this include soccer analysts, examining a striker's foot prior to ball contact or a running coach looking at foot placement and movement.


For sport specific application, the actual velocities of each limb segment play a large role.  For the sprinter high hip flexion power is needed to drive off fast.  Therefore, the thigh will be moving with an extremely high angular velocity and acceleration, absolutely and relative to the shank.  Taking this one step further, what muscles are required to create high hip flexion forces?  Knowing this we can suggest exercises for the sprinter to perform.  We can apply this practice to all portions of sprinting gait to optimize the sprinter's performance.  The same goes for analyzing a soccer kick, baseball pitcher's throw or any other movement where limb segments are moving independent of one another.


Conclusion

Thus ends your physics lesson for the day.  The take home point is that in real life movement limb segments will move independently of each other.  No one walks like Kramer in tight jeans, or swings their arms like Raquel Welch.  So when we take this into consideration we need to apply principles of Coriolis Effect into our analysis when determining output velocites/accelerations.  This becomes useful for those analyzing movements such as walking, running, sprinting, throwing a ball, kicking a ball, etc. In later physics related posts I'll get into subjects such as static vs. dynamic force scenarios, power output and in depth sport specific applications.



References:

http://hyperphysics.phy-astr.gsu.edu/hbase/rotq.html

Hibbeler RC. Engineering Mechanics: Dynamics. Pearson Prentice Hall, 2010

Kadaba MP, Ramakrishnan HK, Wootten ME. Measurement of lower extremity kinematics during walking, J Orth Res 1990;8(3):383-392

Cavagna GA, Komarek L, Mazzoleni S. The mechanics of sprint running. J Phys 1971;217:709-721

Young M. Maximal Velocity Sprint Mechanics, United States Military Academy, Human Performance Consulting