Power to the planes?

I’ve just noticed that my blog is still displaying its Christmas card which feels a bit poor as we move into February. Thought I’d reflect on an issue that came up just before Christmas when we were looking at some joint power data from a cohort of amputees. My colleague first presented the data in “components” in the different planes. I commented that I regarded this as wrong and asked her to plot out the total joint power. She came back to me a little later to say that she couldn’t see how to compute this directly within Visual3D.  This caused us to look at their wicki which appeared to confirm what she had found with an explanation that this is how joint power was presented in Move3D, the software out of which Visual3D evolved, and that this “has become common in the biomechanics community”. My initial reaction was that “common in the biomechanics community” does not necessarily correlate with “correct”. To do Visual3D justice, the wicki also points out that you can calculate total power from its “components” but not vice versa which at least makes sense, even if you question how appropriate it is to refer to these as “components” . (Rather bizarrely feedback from C-motion after publication of this post makes it clear that a JOINT POWER SCALAR function has been available in the software for quite a long time as well as the original calculation of “components” and it was the wicki that was/is misleading! The original wording of this paragraph has been modified in acknowledgement of this).

Knee power

The confusion is quite widespread. When Vicon first produced Polygon it only allowed a graphing of total power and then one day I noticed an option to plot the different “components”. I dropped them an e-mail pointing out the mistake and was told that it wasn’t a mistake but a feature that a number of their customers had requested. It was clear that consumer demand was a more important driver of product development than the rigour of the biomechanics!

So what is the issue? As defined in physics, power is what we call a scalar, it cannot be related to any particular direction or plane. Think of it as a bit like your age, another scalar, it doesn’t really make any sense to talk about having age in a particular direction or plane does it. Well, to the classically trained physicist (me!) then talking about sagittal plane power doesn’t make any more sense than talking about sagittal plane age!

Or is it that simple? The quantities in physics that are related to direction are called vectors (position, speed, acceleration are common examples in biomechanics). Vectors are generally represented as a set of three number which are the components in a particular direction. Thus speed (v) is written (vx, vy, vz) with vx representing the component of speed in the x-direction. Joint power is the product of two such vectors, moment (mx, my, mz) and angular velocity (ωx, ωy, ωz) and under the laws of vector multiplication this gives the equation:

P = m.ω = mx ωx+my ωy.+mz ωy

and, although the physicist doesn’t think it has any significance, it is clear that the total power does appear to be made of three separate terms that involve quantities measured along different directions. It is these three terms that have come to be known as the “components” of power. (Notice that throughout this article I’ve put putted inverted commas around “component” when I’ve used it differently to the conventional definition in physics).

So if it is very clear what the terms mean, does it matter if we just choose to use it even if the physicists don’t think we should? I think the answer to this is “yes, it does matter” (I would though, I’m trained as a physicist). To me the whole point of biomechanics is that it allows us to understand the way the body works using rules and relationships that have been developed in the context of wider physics and engineering and which we know are true in all practical circumstances. If we start using terms which are not part of that understanding, no matter how convenient, then we lose that guarantee that they relate to each other in any particular way. It may seem sensible when you set out, but sooner or later it will lead you into trouble.

Power, in this context for example, is the amount of energy generated in a given time. The “components” of power (e.g. mx ωx ) can be negative as well as positive so if, for example, the x “component” is positive and the y and z “components” are negative, then the amount of energy generated in a given time in the x plane (if this is how it is regarded) is greater than the total energy generated in all the planes. This just doesn’t make sense. Are we saying that power is being generated at a joint in one plane at the same time as it is being absorbed in the other planes?! I hope even the non-physicists who read this can appreciate the problem.

The problem with calculating and using “components” of joint powers is that we don’t know under what other circumstances they lead us to nonsensical conclusions. Stick to the rules of physics and we know our conclusions will always be valid (as long as we’ve applied them properly of course!)

One defence of “sagittal plane joint power” which I have a little sympathy with is that, because the components of both angular velocity and moment tend to be considerably greater in the sagittal plane than others, the “sagittal plane joint power” is generally quite a good approximation to the total joint power. Given that in the modern world all these numbers just pop out of the computer anyway though its not at all clear how this is useful. If you want to know the total joint power why not calculate the total joint power? You also need to be careful that if you justify “sagittal plane power” as a good approximation to total joint power, then all you can really say about the transverse and coronal “plane powers” is that they represent the error in this approximation. Attributing physical significance to poorly defined error terms in a calculation is always going to end in tears.

In passing it may be worth commenting that kinetic energy can also be defined as a product of two vectors,

KE = ½mv.v = vx vx+vy vy.+vz vy

but I’ve never heard anyone talking of kinetic energy having components in different directions!



  1. Dear Richard,
    I would first like to say that this is a very nice post that I enjoyed reading and that I was also a bit curious to find out your thoughts regarding this emerging and probably too appealing methodology.
    I see your point and admit that sometimes the reasoning behind does not justify the use of the separate power. Power is a scalar and should remain as is. Nevertheless, I believe it can have interesting applications. I take the example of my work (hip OA/THA) and concerning the hip joint. As noticed by Eng and Winter (Kinetic analysis of the lower limbs during walking: what information can be gained from a three-dimensional model? Journal of Biomechanics 1995), about 20% of the total power is coming from the frontal plane. I do agree that it might be difficult to approach the separate power as saying that during stance phase of gait, power at the hip is negative. But looking at separate power, power is negative within the sagittal plane while being positive in the frontal plane. Clinically I don’t think it is awkward to mention that the hip flexors, in the plane of forward progression, absorbs energy while the hip abductors generate energy to stabilize the pelvis. Therefore I believe that such methodology should be used with caution and not be reported as a total amount of power within a plane in comparison to total power but can still be appropriate when regarded within specific gait phases to explain global power pattern changes. This is in the case of hip OA or THA a nice approach to give insights into the abductor problem.

    1. Yes, I’m inclined to agree Christophe. I think it’s best to think of joint power as muscle power. I did this in my old software, Motion Toolbox:

      By multiplying the power by the sign of the velocity I was able to show concentric contraction in green and eccentric in red. I can’t see what is wrong with this, and the result would be clinically meaningless if you used total power.

      1. Not sure I follow the concentric-eccentric, Chris, surely a concentric contraction generates power and an eccentric one absorbs it. Is there any more to it than that.

        Also not so sure about being too confident in the relationship between joint power and muscle power. How do we cope with bi-articular muscles? I can feel another post coming on!

      2. I confess I also can’t recall where the sign multiplication comes in but I’m it needed some such jiggerypokery.

        I take your point too about 2-Joint muscles. I think this nicely illustrates the gait analysis dilemma. A lot of concmepts are relatively straightforward until you start to think about them more deeply (e.g. with 3D instead of 2D or in simulations). In the end it seems to me that we end up destroying all that we’ve learned and are back to square one again!

      3. Dear Richard, I do completely agree with you that this way of approaching power becomes problematic for bi-articular muscles but this one of the drawbacks of gait analysis. It’s that we usually about talk about net joint moment or power being then representative of muscle group action and not any individual muscles. I indeed see another post coming 🙂

  2. Richard,

    Visual3D computes Joint Power as a set of 3 terms and as a scalar; these are two different model based items. it is unfortunate that you did not ask someone at C-Motion before making such a gaffe.


    1. Scott,

      OK good to know this.

      It would be good if Visual 3D’s own documentation described its products then I wouldn’t have to!
      This is what the current ( accessed 12:15 on today) C-motion wiki entry on JOINT_POWER says:

      Joint Power is a scalar term computed at:

      Power = [Mx,My,Mz] . [wx ,wy,wz]
      Power = Mx.wx + My.wy + Mz.wz

      but it has become common in the biomechanics community to resolve Power into a segment coordinate system, but note that this does not result in Joint Power being a vector. Visual3D inherited many decisions made in the NIH MOVE3D software including this one. It is straightforward for the user to compute the magnitude of the power signal, so we decided to leave the calculation as 3 components. If we had elected to compute Joint Power as a scalar, we would make it very difficult for those users that want it resolved into components.

      I notice that there is now an extra entry on JOINT_POWER_SCALAR which only appears to have been created after this article was first posted.

  3. oh Richard…….

    To identify a bad link on our wiki (note that the linked page was JOINT_POWER not JOINT_POWER_SCALAR) is a poor excuse.

    If you had actually done your homework and looked at Visual3D or our text book chapter http://www.c-motion.com/textbook you will see that we understand this concept and have included two model based items for power for many years.


  4. This is interesting. I discovered gait analysis when I arrived in Melbourne and was taught by you so I have only seen power as a scalar value (which it is 🙂 ). I must admit I also never wondered about what the 3 terms of the scalar product would tell me. The following is my contribution to the debate.
    In the linear case (force and displacement rather than moment and rotation) power is the change of work with time and work W = F.l (both vectors, can’t find how to add arrows) where the scalar product expresses the contribution of F along l, that is W = |F|*|l|*cos(a) with a the angle between F and l. Work was formulated to represent the effect of a force on the change in kinematic energy and is equal to W = DEk = ½ m Dv2. Although kinematic energy is a scalar we could decompose the change of speed in 3 terms expressed along the axes of a coordinate system as: DEk = ½ m (Dvx2 + Dvy2 + Dvz2). If we calculate work in a one dimensional space (quite a boring space) Wx = Fx.lx =|Fx|*|lx|*[cos(ax) = 1] = Fx * lx and Wx = ½ m Dvx2 so the term of the scalar product along x is equal to the change of kinematic energy along x. We can check that this ‘work’ in the 3 dimensions space with W = Wx + Wy + Wz = Fx * lx + Fy * ly + Fz * lz = ½ m (Dvx2 + Dvy2 + Dvz2).
    The error would be to consider the 3 terms of the scalar product as 3 components of a vector but we don’t have to do that.
    This decomposition could help the interpretation if we had little torque generators in well-defined planes to motorise our joints. For example we could properly dimension each motor instead of using the same motor for each plane, we could also design smart wiring of the motors to absorb as much resisting energy as possible. We don’t have motor but muscles and the moments they generate are not in well-defined planes, so in the end I don’t think it is going to help much. What could be helping is to calculate the work/power (with the help of the moment arm) of individual muscle in some planes during walking. Unfortunately that requires much more than just decomposing a scalar product into 3 terms.

  5. Hi Richard,
    I am a biomedical engineering and I am finishing my univerisity in these months… as you can imagine I am not so expert in practise research, I need to discover, try and learn a lot! My thesis deals with the development of a multi-segment foot model for biomechanical analysis during different human motion. For the evaluation of joint power I had so many doubts, but your post gives me a final answer. I agree with you, talking about power components could be uncorrect; maybe it could be better if I talk about “contribution” from net moment components, what do you think?

    Now that I have discovered this blog I’ll follow you, for sure!


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