3D inverse dynamics in non-orthonormal segment coordinate system

Abstract

The net joint forces and moments may be computed by several 3D inverse dynamic methods. To do so, an orthonormal segment coordinate system (SCS) is generally mandatory. However, the segment axes ought to be selected following anatomical, functional, and inertial requirements that are hardly compatible with orthogonal axes. An alternative method based on generalized coordinates allows computing inverse dynamics using directly a set of basic points and unitary vectors. A segment definition is put forward in order to follow all of the anatomical, functional, and inertial requirements and the inverse dynamics is performed in a non orthonormal segment coordinate system (NSCS). The NSCS seems a convenient definition in biomechanics as far as anatomical, functional and inertial axes are concerned, but providing that the 3D joint forces and moments are still computable. The inverse dynamic method in NSCS is applied to the gait of a knee valgus subject and compared to a classical inverse dynamic method. The inverse dynamic method in NSCS shows comparable results but implies further clinical interpretations.

Appendix

The interpolation matrix allows computing the position of a point in the ICS, knowing its coordinates in the NSCS ${{\left({P_{i}, {\mathbf{u}}_{i}, {\left({{\mathbf{r}}_{{P_{i}}} - {\mathbf{r}}_{{D_{i}}}} \right)},{\mathbf{w}}_{i}} \right)}}$ and the generalized coordinates of the segment. For example, the position vector of the centre of mass ${\bf r}_{C_{i}}$ in the ICS can be expressed:

$$ {\mathbf{r}}_{{C_{i}}} - {\mathbf{r}}_{{P_{i}}} = {n_{i}^{C_i}}_{1} {\mathbf{u}}_{i} + {n_{i}^{C_i}}_{2} {\left({{\mathbf{r}}_{{P_{i}}} - {\mathbf{r}}_{{D_{i}}}} \right)} + {n_{i}^{C_i}}_{3} {\mathbf{w}}_{i}. $$

(15)

This expression (Eq. 15) is written in matrix form:

$$ {\mathbf{r}}_{C_i } = \underbrace{\left[ {\begin{array}{*{20}c} {{n_i^{C_i}}_1 {\mathbf{E}}_{3 \times 3} } & {\left({1 + {n_i^{C_i}}_2} \right){\mathbf{E}}_{3 \times 3} } & { - {n_i}^{C_{i}}_2 {\mathbf{E}}_{3 \times 3}} & {n_i}^{C_{i}}_3{\mathbf{E}}_{3 \times 3} \\ \end{array}} \right]}_{{\left[{{\mathbf{N}}_i^{C_{i}}}\right]}} \cdot{\left[{\begin{array}{*{20}c} {{\mathbf{u}}_i } \\ {{\mathbf{r}}_{P_i }} \\ {{\mathbf{r}}_{D_{i}}}\\{{\mathbf{w}}_i } \\ \end{array} } \right]}. $$

(16)

Four points, P _i , D _i , U _i and W _i , have remarkable coordinates in the NSCS (for i = 1, 2 and 3):

$$ {\mathbf{n}}^{{P_{i}}}_{i} = {\left[{\begin{array}{*{20}c} {0} \\ {0} \\ {0} \\ \end{array}} \right]},\quad {\mathbf{n}}^{{D_{i}}}_{i} = {\left[{\begin{array}{*{20}c} {0} \\ {{- 1}} \\ {0} \\ \end{array}} \right]},\quad {\mathbf{n}}^{{U_{i}}}_{i} = {\left[{\begin{array}{*{20}c} {1} \\ {0} \\ {0} \\ \end{array}} \right]}\, \hbox{and}\, {\mathbf{n}}^{{W_{i}}}_{i} = {\left[{\begin{array}{*{20}c} {0} \\ {0} \\ {1} \\ \end{array}} \right]}. $$

The corresponding remarkable interpolation matrices are:

$$ {\left[{{\mathbf{N}}^{{P_{i}}}_{i}} \right]}^{\rm T} = {\left[{\begin{array}{*{20}c} {{{\mathbf{0}}_{{3 \times 3}}}} \\ {{{\mathbf{E}}_{{3 \times 3}}}} \\ {{{\mathbf{0}}_{{3 \times 3}}}} \\ {{{\mathbf{0}}_{{3 \times 3}}}} \\ \end{array}} \right]},\quad {\left[{{\mathbf{N}}^{{D_{i}}}_{i}} \right]}^{\rm T} = {\left[{\begin{array}{*{20}c} {{{\mathbf{0}}_{{3 \times 3}}}} \\ {{{\mathbf{0}}_{{3 \times 3}}}} \\ {{{\mathbf{E}}_{{3 \times 3}}}} \\ {{{\mathbf{0}}_{{3 \times 3}}}} \\ \end{array}} \right]},\quad {\left[{{\mathbf{N}}^{{U_{i}}}_{i}} \right]}^{\rm T} = {\left[{\begin{array}{*{20}c} {{{\mathbf{E}}_{{3 \times 3}}}} \\ {{{\mathbf{0}}_{{3 \times 3}}}} \\ {{{\mathbf{0}}_{{3 \times 3}}}} \\ {{{\mathbf{0}}_{{3 \times 3}}}} \\ \end{array}} \right]}\, \hbox{and}\, {\left[{{\mathbf{N}}^{{W_{i}}}_{i}} \right]}^{\rm T} = {\left[{\begin{array}{*{20}c} {{{\mathbf{0}}_{{3 \times 3}}}} \\ {{{\mathbf{0}}_{{3 \times 3}}}} \\ {{{\mathbf{0}}_{{3 \times 3}}}} \\ {{{\mathbf{E}}_{{3 \times 3}}}} \\ \end{array}} \right]}. $$

These interpolation matrices can be used to compute the generalized forces applied at these points. Concerning moments, the generalized moments cannot be computed with interpolation matrices [25]. Moments have first to be replaced by couple of opposite forces.

The moment M at proximal point P _i can be replaced by three forces about the three axes ${{\left({{\mathbf{u}}_{i}, {\mathbf{r}}_{{P_{i}}} - {\mathbf{r}}_{{D_{i}}}, {\mathbf{w}}_{i}} \right)}}$ and applied at the three points U _i , W _i and D _i :

$$ {\mathbf{M}} = {\mathbf{w}}_{i} \times {\left({f_{1} {\mathbf{u}}_{i}} \right)} + {\mathbf{u}}_{i} \times {\left({f_{2} {\left({{\mathbf{r}}_{{P_{i}}} - {\mathbf{r}}_{{D_{i}}}} \right)}} \right)} + {\left({- {\left({{\mathbf{r}}_{{P_{i}}} - {\mathbf{r}}_{{D_{i}}}} \right)}} \right)} \times {\left({f_{3} {\mathbf{w}}_{i}} \right)}. $$

(17)

This expression (Eq. 17) is written in matrix form:

$$ {\mathbf{M}} = {\underbrace {{\left[{\begin{array}{*{20}c} {{{\mathbf{w}}_{i} \times {\mathbf{u}}_{i}}}& {{{\mathbf{u}}_{i} \times {\left({{\mathbf{r}}_{{P_{i}}} - {\mathbf{r}}_{{D_{i}}}} \right)}}}& {{- {\left({{\mathbf{r}}_{{P_{i}}} - {\mathbf{r}}_{{D_{i}}}} \right)} \times {\mathbf{w}}_{i}}} \\ \end{array}} \right]}}_{{{\left[{{\mathbf{B}}^{*}_{i}} \right]}}}}{\left[{\begin{array}{*{20}c} {{f_{1}}} \\ {{f_{2}}} \\ {{f_{3}}} \\ \end{array}} \right]}. $$

(18)

Three opposite forces are applied at endpoint P _i in order to maintain equilibrium. Thus, six concentrated forces are required to represent the moment and the generalized moment is:

$$ \begin{aligned} {\left[{{\mathbf{N}}^{*}_{i}} \right]}^{\rm T} {\mathbf{M}} &= {\left[{{\mathbf{N}}^{{W_{i}}}_{i}} \right]}^{\rm T} {\left( {f_{1} {\mathbf{u}}_{i}} \right)} + {\left[{{\mathbf{N}}^{{U_{i}}}_{i}} \right]}^{\rm T} {\left({f_{2} {\left({{\mathbf{r}}_{{P_{i}}} - {\mathbf{r}}_{{D_{i}}}} \right)}} \right)} + {\left[{{\mathbf{N}}^{{D_{i}}}_{i}} \right]}^{\rm T} {\left({f_{3} {\mathbf{w}}_{i}} \right)} \\ &\quad + {\left[{{\mathbf{N}}^{{P_{i}}}_{i}} \right]}^{\rm T} {\left({- f_{1} {\mathbf{u}}_{i} - f_{2} {\left( {{\mathbf{r}}_{{P_{i}}} - {\mathbf{r}}_{{D_{i}}}} \right)} - f_{3} {\mathbf{w}}_{i}} \right)} \\ \end{aligned}.$$

(19)

Therefore, the pseudo-interpolation matrix is (Eq. 12):

$$ {\left[{{\mathbf{N}}^{*}_{i}} \right]}^{\rm T} = {\left[{\begin{array}{*{20}c} {{{\mathbf{0}}_{{3 \times 1}}}}& {{{\left( {{\mathbf{r}}_{{P_{i}}} - {\mathbf{r}}_{{D_{i}}}} \right)}}}& {{{\mathbf{0}}_{{3 \times 1}}}} \\ {{- {\mathbf{u}}_{i}}}& {{- {\left( {{\mathbf{r}}_{{P_{i}}} - {\mathbf{r}}_{{D_{i}}}} \right)}}}& {{- {\mathbf{w}}_{i}}} \\ {{{\mathbf{0}}_{{3 \times 1}}}}& {{{\mathbf{0}}_{{3 \times 1}}}}& {{{\mathbf{w}}_{i}}} \\ {{{\mathbf{u}}_{i}}}& {{{\mathbf{0}}_{{3 \times 1}}}}& {{{\mathbf{0}}_{{3 \times 1}}}} \\ \end{array}} \right]}{\left[{{\mathbf{B}}^{*}_{i}} \right]}^{{- 1}}. $$