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Joint stiffness concept clarification

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  • Joint stiffness concept clarification

    Dear all,

    I wish to clarify the use of stiffness in biomechanics, more specifically “joint stiffness” (and maybe extending to “leg stiffness”). I have read chapters from “Stiffness and Stiffness-like Measures. Latash and Zatsiorsky (2016) Biomechanics and Motor control”, and tried to find some answers in physics textbooks. This seems basic enough, but unfortunately, I do not have a satisfactory answer.

    Qn1: Is “stiffness” based on hook’s law a scalar or vector (

    Reason: Traditionally, joint stiffness in gait has been derived using various formulations of the form:

    ∆momentX/∆CardanAngleX (1)

    Where X = one axis, usually flexion-extension axis. However, moment is a vector yet cardan angle is not a vector. Should one be using the vector version of angles (ie angle of rotation X unit vector).

    Qn2: What if a coiled spring is deformed in three dimensions (or for that matter 6 directions), how does one calculate deformation force, deformation, and hence “stiffness”? By extension, how do we calculate a joint’s stiffness in 3D (or 6D)? Is it a right thing to do base on physical law? Is it meaningful? In my mind 3D/6D stiffness about a joint would be essential especially in prosthesis development?

    Solutions: If the above is right and meaningful, do we simply perform individual calculations of (1) along each axis to get kX, kY, kZ (assuming 3D only)? Does it make sense to get a kTOTAL using the hypotenuse of the components, to get a single value of 3D stiffness? This is like treating stiffness as a vector.

    Alternatively, one can take the hypotenuse of the moment (X,Y,Z) to get a single moment about a joint, take the hypotenuse of joint rotation (vector) and use equation (1), to get a single value of 3D stiffness? .

    Any help in clarification or pointing me to relevant articles would be of great help.


  • #2
    Re: Joint stiffness concept clarification

    Hey Bernard,

    To answer question one. “Stiffness” of the spring component of the joint could be modeled either as a scalar or vector. It would really depend on the phenomenon you are trying to model. As you stated modeling joint stiffness for the development of a prosthesis could possibly be crucial in 3 dimensions, however if you were performing a quiet standing assessment of A/P sway then modeling the joint (musculature/ligaments etc..) with so many degrees of freedom (in this case 6) would not be appropriate.

    Continuing on to Cardan angles, I think you might be forgetting about the other two rotations. You can view the Cardan angles as a 3X1 vector. α, β, γ (roll, pitch and yaw) in 3 dimensional space. The moment arm itself is actually a 3X1 vector as well since force can act in the x, y, and z axes. In the traditional method you described it is just easier to model joint stiffness by decreasing the complexity (dimensions) of the model. In such a case the moment arm is described by a 2X1 vector (X and Y) and the spring stiffness by a 1X1 vector or ‘scalar’. A simple spring can be modeled in 1 dimension (forward, backward) since it only stretches and relaxes, a rotation of the spring in the (X,Y) plane does not affect how it behaves.

    On to question 2. If a spring was to be deformed in 3 dimensions you would in essence end up with a differentiable equation i.e 3 equations with 3 unknowns (this is an oversimplification). Spring stiffness in each dimension would be dependent on the springs deformation in the other two dimensions. This only takes into account the proportionality constants. To be more precise you would also need to add stress and strain of the spring to the model.
    As you alluded to, it does not make sense to have a k-total, or for that matter a single value of 3D “stiffness” since the springs “stiffness” in one dimension is dependent on its “stiffness” in the other dimensions. If you were to hold 2 dimensions constant you could then give a stiffness value, but you would be right back to modeling the spring in a single dimension.
    Suffice it to say in most (but certainly not all) biomechanics/kinesiology work this is too complex a model to answer the researcher’s question.

    Hope I helped


    • #3
      Re: Joint stiffness concept clarification

      Hi Bernard,

      There was a lengthy thread/discussion on ankle joint stiffness a bit ago that might be helpful:



      • #4
        Re: Joint stiffness concept clarification

        Hi Bernard,

        If you are considering the 3 axes of the joint as well as both rotations and translations, you are talking about the 6-by-6 stiffness matrix.

        The stiffness coefficients in the diagonal define the relationship between forces/moments and displacements/angles about each axis. Interestingly, the stiffness coefficients out of the diagonal define coupling effects between the degrees of freedom. The stiffness matrix has been extensively in vitro studied for the inter-vertebral joints and the knee.

        In this stiffness matrix, it is important to express the forces/moments and displacements/angles about the same joint axes. It can be quite complex in the “joint coordinate system” (JCS) because the axes are not orthogonal. Some ideas on how to deal with that can be found in:
        O'Reilly OM, Metzger MF, Buckley JM, Moody DA, Lotz JC. On the stiffness matrix of the intervertebral joint: application to total disk replacement. J Biomech Eng. 2009;131(8):081007. doi: 10.1115/1.3148195.
        Fujie H, Livesay GA, Fujita M, Woo SL. Forces and moments in six-DOF at the human knee joint: mathematical description for control. J Biomech. 1996;29(12):1577-85.
        It means that the joint moments must be expressed in the JCS (another debate).
        However, this is required only if you want to interpret the stiffness coefficients as flexion, abduction, internal rotation. Otherwise, the stiffness matrix can be defined considering other axes (of the proximal joint) and other angle definition.