Advances in Force and Moments Measurements by an Innovative Six-axis Load Cell

Abstract

In this paper the design, the accuracy assessment and the calibration procedure of an innovative six axis load cell are presented. The load cell is able to measure the three forces and three moments and has been designed at the Politecnico di Milano (Technical University of Milan) as a result of 10 years of research in this field. The sensing structural element of the six-axis load cell is a three spoke structure constrained to the frame of the load cell by means of special joints conceived to avoid friction. Strain gauges are conveniently located on highly stressed areas. Mathematical models, both analytical and numerical, are implemented and presented in order to describe the behaviour of the load cell, to optimize its parameters and to theoretically assess its accuracy. A special device has been designed and constructed in order to accurately and quickly calibrate the load cell. Particular attention has been devoted to the processing of the strain gauges signals. An electronic DSP board, located inside the load cell, has been realized able to compute the six components of the generalized force from the strain gauge signals and return six voltage signals proportional to the measured forces. Software selectable filters and gains are implemented on the DSP board.

Appendix

Let us consider:

• The vector S _j of the six reactions in the central node of the structure for each spoke (a total of 18 reactions).

  $$ {\mathbf{S}}_{j\left(j=1,2,3\right)}={\left[\begin{array}{cccccc}\hfill {S}_{xj}\hfill & \hfill {S}_{yj}\hfill & \hfill {S}_{zj}\hfill & \hfill {M}_{xj}\hfill & \hfill {M}_{yj}\hfill & \hfill {M}_{zj}\hfill \end{array}\right]}^T $$
  $$ \mathbf{S}={\left[\begin{array}{ccc}\hfill {\mathbf{S}}_1\hfill & \hfill {\mathbf{S}}_2\hfill & \hfill {\mathbf{S}}_3\hfill \end{array}\right]}^T $$

• The vector δ _j of the six components of displacement “δ” and rotation “φ” of each spoke at the central node of the structure.

  $$ {\boldsymbol{\updelta}}_{j\left(j=1,2,3\right)}={\left[\begin{array}{cccccc}\hfill {\delta}_{xj}\hfill & \hfill {\delta}_{yj}\hfill & \hfill {\delta}_{zj}\hfill & \hfill {\varphi}_{xj}\hfill & \hfill {\varphi}_{yj}\hfill & \hfill {\varphi}_{zj}\hfill \end{array}\right]}^T $$
  $$ \boldsymbol{\updelta} ={\left[\begin{array}{ccc}\hfill {\boldsymbol{\updelta}}_1\hfill & \hfill {\boldsymbol{\updelta}}_2\hfill & \hfill {\boldsymbol{\updelta}}_3\hfill \end{array}\right]}^T $$

• the rotation matrix R _t (α) (α = ± 2/3π) defined as follows

  $$ \begin{array}{l}{\boldsymbol{R}}_{\boldsymbol{t}}=\left(\alpha \right)=\left[\begin{array}{cc}\hfill \boldsymbol{R}\left(\alpha \right)\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill \boldsymbol{R}\left(\alpha \right)\hfill \end{array}\right]\hfill \\ {}\boldsymbol{R}\left(\alpha \right)=\left[\begin{array}{ccc}\hfill \cos \left(\alpha \right)\hfill & \hfill 0\hfill & \hfill \sin \left(\alpha \right)\hfill \\ {}\hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ {}\hfill - \sin \left(\alpha \right)\hfill & \hfill 0\hfill & \hfill \cos \left(\alpha \right)\hfill \end{array}\right]\hfill \end{array} $$

  (19)

In order to compute the 18 components of the vector S, a 18 equation system has been written in the following form

$$ \left\{\begin{array}{l}\mathbf{F}=\mathbf{I}\cdot {\mathbf{S}}_1+{\mathbf{R}}_{\mathbf{t}}\left(\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.\pi \right)\cdot {\mathbf{S}}_2+{\mathbf{R}}_{\mathbf{t}}\left(-\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.\pi \right)\cdot {\mathbf{S}}_3\hfill \\ {}{\boldsymbol{\updelta}}_1={\mathbf{R}}_{\mathbf{t}}\left(\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.\pi \right)\cdot {\boldsymbol{\updelta}}_2\hfill \\ {}{\boldsymbol{\updelta}}_1={\mathbf{R}}_{\mathbf{t}}\left(-\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.\pi \right)\cdot {\boldsymbol{\updelta}}_3\hfill \end{array}\right. $$

(20)

The first equation refers to the force equilibrium at the central node (roots of the three spokes), while the other 2 equations refer to the displacement consistency at the central node between the spokes 1–2 and 1–3 respectively. The equation system can be written in a compact form:

$$ \left[\begin{array}{ccc}\hfill \mathbf{I}\hfill & \hfill {\mathbf{R}}_{\mathbf{t}}\left(\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.\pi \right)\hfill & \hfill {\mathbf{R}}_{\mathbf{t}}\left(-\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.\pi \right)\hfill \\ {}\hfill \boldsymbol{\Delta} \hfill & \hfill -{\mathbf{R}}_{\mathbf{t}}\left(\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.\pi \right)\cdot \boldsymbol{\Delta} \hfill & \hfill \mathbf{0}\hfill \\ {}\hfill \boldsymbol{\Delta} \hfill & \hfill \mathbf{0}\hfill & \hfill -{\mathbf{R}}_{\mathbf{t}}\left(-\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.\pi \right)\cdot \boldsymbol{\Delta} \hfill \end{array}\right]\cdot \mathbf{S}={\mathbf{A}}_{\mathbf{A}}\cdot \mathbf{S}=\left[\begin{array}{c}\hfill \mathbf{F}\hfill \\ {}\hfill \mathbf{0}\hfill \\ {}\hfill \mathbf{0}\hfill \end{array}\right] $$

(21)

Where Δ is a 6 × 6 matrix that links the generalized displacements vector δ _j to the reactions vector S _j of spoke j:

$$ \begin{array}{cc}\hfill {\boldsymbol{\updelta}}_j=\left({\boldsymbol{\Delta}}_{\mathbf{R}}+{\boldsymbol{\Delta}}_{\mathbf{D}}\right)\cdot {\mathbf{S}}_j=\boldsymbol{\Delta} \cdot {\mathbf{S}}_j\hfill & \hfill j=1,2,3\hfill \end{array} $$

(22)

The matrix Δ derives from the sum of two different contributions: Δ _R accounts for the elastic deformation of the sliding spherical joint at the spoke tip, and Δ _D represents the elastic deformation of the spoke itself (Fig. 24).

Fig. 24

Displacement δ_xj and rotation φ_yj due to a force S_xj on spoke j ((a) rigid displacement and rotation associated to the elasticity of the sliding spherical joints, (b) effect caused by the spoke elastic deformation)

Referring to the case of the deformation in the x direction due to a reaction force Sx in the same direction shown in Fig. 24, the contribution of the elastic deformation of the spoke and of the rigid motion due to the compliance of the joints of the generalized displacement δ_xj and of the generalized rotation φ_yj can be computed as follows. The total generalized rotation φ_yj reads

$$ {\varphi}_{Yj}\left({S}_{Xj}\right)={\varphi}_{Yj}\left({k}_{bt}\right)+{\varphi}_{Yj}\left({J}_Y\right) $$

(23)

where the first term of equation (23) expresses a rigid rotation of the spoke

$$ {\varphi}_{Yj}\left({K}_{bt}\right)=\frac{S_{Xj}\cdot L}{k_{bt}} $$

(24)

and the second term can be calculated by applying the virtual work principle

$$ {\varphi}_{Yj}\left({J}_y\right)={\displaystyle \underset{0}{\overset{l}{\int }}\frac{\left(R+\xi \right)\cdot {S}_{xj}}{E\cdot {J}_y\left(\xi \right)} d\xi} $$

(25)

The generalized displacement δ_xj reads

$$ {\delta}_{Xj}\left({S}_{Xi}\right)={\delta}_{Xj}\left({k}_{rx}\right)+{\delta}_{Xj}\left({k}_{bt}\right)+{\delta}_{Xj}\left({J}_Y\right)+{\varphi}_{Yj}\left({J}_Y\right)\cdot R $$

(26)

where the first two terms of equation (26) account for the rigid motion associated to the joints elasticity

$$ {\delta}_{Xj}\left({k}_{rx}\right)+{\delta}_{Xj}\left({k}_{bt}\right)=\frac{S_{Xj}}{K_{rX}}+{\varphi}_{Yj}\left({K}_{bt}\right)\cdot L $$

(27)

being φ _Yj (K _bt ) computed in equation (24). The third and fourth terms account for the elastic deformation of the spoke. In particular the third term can be obtained by applying the virtual work principle

$$ {\delta}_{Xj}\left({J}_Y\right)={\displaystyle \underset{0}{\overset{l}{\int }}\xi \cdot \frac{\left(R+\xi \right)\cdot {S}_{xj}}{E\cdot {J}_y\left(\xi \right)} d\xi +{\displaystyle \underset{0}{\overset{l}{\int }}\chi}\cdot \frac{S_{xj}}{A\left(\xi \right)} d\xi} $$

(28)

where χ is the shear factor and A(ξ) is the cross-section area. For a rectangular section the shear factor χ reads [60]

$$ \chi =\frac{12+11\cdot \nu }{10\cdot \left(1+\nu \right)} $$

(29)

The last term of equation (28) accounts for the shear contribution in the elastic deformation of the spoke [61]. The fourth term of equation (26) is given by considering the displacement of the rigid disk due to the rotation of the deformed spoke computed in equation (25).

All the displacements and rotations resulting at each spoke root can be computed by applying the same approach, and the matrix Δ reads

$$ \boldsymbol{\Delta} =\left[\begin{array}{cccccc}\hfill \frac{L^2}{k_{bt}}+\frac{1}{k_{rX}}+\frac{1}{E}{I}_1+\frac{1}{G}{I}_{11}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{L}{k_{bt}}-\frac{1}{E}{I}_2\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill \frac{L^2}{k_b}+\frac{1}{k_{rY}}+\frac{1}{E}{I}_3+\frac{1}{G}{I}_{12}\hfill & \hfill 0\hfill & \hfill \frac{L}{k_b}+\frac{1}{E}{I}_4\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill \frac{1}{k_a}+\frac{1}{E}{I}_5\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill \frac{L}{k_b}+\frac{1}{E}{I}_6\hfill & \hfill 0\hfill & \hfill \frac{1}{k_b}+\frac{1}{E}{I}_7\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill -\frac{L}{k_{bt}}-\frac{1}{E}{I}_8\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{1}{k_{bt}}+\frac{1}{E}{I}_9\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{1}{k_t}+\frac{1}{G}{I}_{10}\hfill \end{array}\right] $$

Where the coefficients I_1–10 are summarized in Table 7

Table 7 Coefficients I_1–10 used in the matrix Δ

while the coefficients I₁₁ and I₁₂ have the following expression [61]

$$ {I}_{11}={I}_{12}={\displaystyle \underset{0}{\overset{l}{\int }}\chi \cdot \frac{1}{A\left(\xi \right)} d\xi} $$

(30)

The coefficients reported in equation (30) quantify the effect of the shear force on the spoke elastic deformation.

Matrix Δ is numerically computed and it provides a linear relationship between the generalized displacements vector δ _j and the reactions vector S _j as stated in equation (22).

Numerical results show that the shear forces provide an average variation of 5 % on the computed stiffness values.

From equation (21), vector S can be computed. The linear relationship between S and the bridges outputs can be derived as follows.

Vector S can be divided into three vectors S _j (j = 1,2,3), each one containing the 3 reaction forces and the 3 reaction moments applied at the tip of each spoke, and indicating with j the number of the spoke and with i the location of each of the 8 strain gauges on each spoke (i = 1a,1a′,1b,1b′,2a,2a′,2b,2b′, see Fig. 4), the stress components at each strain gauge location read

$$ {\boldsymbol{\upsigma}}_{ij}=\left[\begin{array}{c}\hfill {\sigma}_{xx}\hfill \\ {}\hfill {\sigma}_{yy}\hfill \\ {}\hfill {\sigma}_{zz}\hfill \\ {}\hfill {\tau}_{xy}\hfill \\ {}\hfill {\tau}_{xz}\hfill \\ {}\hfill {\tau}_{yz}\hfill \end{array}\right]={\boldsymbol{\Sigma}}_i\cdot {\mathbf{S}}_j $$

(31)

where Σ _i is a 6 by 6 matrix that expresses the relationship between the reactions at the spoke tip and the stress components at each strain gauge location. Matrix Σ _i is different for each of the 8 strain gauges on each spoke, but, being defined in a local reference frame, the 8 matrices Σ _i are the same for all of the spokes.

By considering an elastic, isotropic material, the relationship between the stress and strain components at each strain gauge location is

$$ {\boldsymbol{\upvarepsilon}}_{ij}={\left(\begin{array}{c}\hfill {\varepsilon}_{xx}\hfill \\ {}\hfill {\varepsilon}_{yy}\hfill \\ {}\hfill {\varepsilon}_{zz}\hfill \\ {}\hfill {\gamma}_{\raisebox{1ex}{$ xy$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\hfill \\ {}\hfill {\gamma}_{\raisebox{1ex}{$ xz$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\hfill \\ {}\hfill {\gamma}_{\raisebox{1ex}{$ yz$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\hfill \end{array}\right)}_{ij}=\left[\begin{array}{cccccc}\hfill \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$E$}\right.\hfill & \hfill -\raisebox{1ex}{$\nu $}\!\left/ \!\raisebox{-1ex}{$E$}\right.\hfill & \hfill -\raisebox{1ex}{$\nu $}\!\left/ \!\raisebox{-1ex}{$E$}\right.\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill -\raisebox{1ex}{$\nu $}\!\left/ \!\raisebox{-1ex}{$E$}\right.\hfill & \hfill \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$E$}\right.\hfill & \hfill -\raisebox{1ex}{$\nu $}\!\left/ \!\raisebox{-1ex}{$E$}\right.\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill -\raisebox{1ex}{$\nu $}\!\left/ \!\raisebox{-1ex}{$E$}\right.\hfill & \hfill -\raisebox{1ex}{$\nu $}\!\left/ \!\raisebox{-1ex}{$E$}\right.\hfill & \hfill \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$E$}\right.\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$G$}\right.\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$G$}\right.\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$G$}\right.\hfill \end{array}\right]\cdot \left(\begin{array}{c}\hfill {\sigma}_{xx}\hfill \\ {}\hfill {\sigma}_{yy}\hfill \\ {}\hfill {\sigma}_{zz}\hfill \\ {}\hfill {\tau}_{xy}\hfill \\ {}\hfill {\tau}_{xz}\hfill \\ {}\hfill {\tau}_{yz}\hfill \end{array}\right)=\mathbf{H}\cdot {\boldsymbol{\Sigma}}_i\cdot {\mathbf{S}}_j $$

(32)

being H the matrix of the stress–strain relationship (Hooke law), j refers to the spoke number and i to the location of the strain gauge.

Finally, to get the deformation read by the strain gauge, the taper angles of the spoke have to be considered, i.e. the strain gauge orientation is different from the orientation of the local reference system of the spoke (see Fig. 25). The strain measured by the strain gauge can be expressed by considering the rotation of the strain tensor

Fig. 25

Strain gauge orientations with respect to the spoke local reference system. β1 is the taper angle of the spoke in the plane yz, β2 refers to the plane xz

$$ {\varepsilon}_{sg, ij}={\left\lfloor \begin{array}{cccccc}\hfill { \cos}^2\left({\widehat{n}}_x\right)\hfill & \hfill { \cos}^2\left({\widehat{n}}_y\right)\hfill & \hfill { \cos}^2\left({\widehat{n}}_z\right)\hfill & \hfill \cos \left({\widehat{n}}_x\right) \cos \left({\widehat{n}}_y\right)\hfill & \hfill \cos \left({\widehat{n}}_x\right) \cos \left({\widehat{n}}_z\right)\hfill & \hfill \cos \left({\widehat{n}}_y\right) \cos \left({\widehat{n}}_z\right)\hfill \end{array}\right\rfloor}_i\cdot {\boldsymbol{\upvarepsilon}}_{ij} $$

where n_x,n_y,n_z are the components of the direction of the strain gauge. Being q _i the vector containing the functions of n_x,n_y,n_z, the relationship can be expressed in a more compact form:

$$ {\varepsilon}_{sg, ij}={\mathbf{q}}_i\cdot {\boldsymbol{\upvarepsilon}}_{ij} $$

(33)

By replacing equation (32) in equation (33), the relationships between the vectors S _j and the corresponding strain gauge measures can be obtained:

$$ {\varepsilon}_{sg, ij}={\mathbf{q}}_i\cdot \mathbf{H}\cdot {\boldsymbol{\Sigma}}_i\cdot {\mathbf{S}}_j={\mathbf{D}}_{ij}\cdot {\mathbf{S}}_j $$

(34)

where i refers to the strain gauge and j to the spoke. A total of 24 equations of the type of equation (34) can be written for the 24 strain gauges. By considering a full Wheatstone bridge connection, the equivalent strains measured by the six bridges due to the bending of the spokes read

$$ \begin{array}{l}{\varepsilon}_1=\left(\left({\mathbf{D}}_{1a1}+{\mathbf{D}}_{1a\prime 1}\right)-\left({\mathbf{D}}_{1b1}+{\mathbf{D}}_{1b\prime 1}\right)\right)\cdot {\mathbf{S}}_1={\overline{\mathbf{D}}}_{11}\cdot {\mathbf{S}}_1\hfill \\ {}{\varepsilon}_2=\left(\left({\mathbf{D}}_{2a1}+{\mathbf{D}}_{2a\prime 1}\right)-\left({\mathbf{D}}_{2b1}+{\mathbf{D}}_{2b\prime 1}\right)\right)\cdot {\mathbf{S}}_1={\overline{\mathbf{D}}}_{21}\cdot {\mathbf{S}}_1\hfill \\ {}{\varepsilon}_3=\left(\left({\mathbf{D}}_{3a2}+{\mathbf{D}}_{3a\prime 2}\right)-\left({\mathbf{D}}_{3b2}+{\mathbf{D}}_{3b\prime 2}\right)\right)\cdot {\mathbf{S}}_2={\overline{\mathbf{D}}}_{12}\cdot {\mathbf{S}}_2\hfill \\ {}{\varepsilon}_4=\left(\left({\mathbf{D}}_{4a2}+{\mathbf{D}}_{4a\prime 2}\right)-\left({\mathbf{D}}_{4b2}+{\mathbf{D}}_{4b\prime 2}\right)\right)\cdot {\mathbf{S}}_2={\overline{\mathbf{D}}}_{22}\cdot {\mathbf{S}}_2\hfill \\ {}{\varepsilon}_5=\left(\left({\mathbf{D}}_{5a3}+{\mathbf{D}}_{5a\prime 3}\right)-\left({\mathbf{D}}_{5b3}+{\mathbf{D}}_{5b\prime 3}\right)\right)\cdot {\mathbf{S}}_3={\overline{\mathbf{D}}}_{13}\cdot {\mathbf{S}}_3\hfill \\ {}{\varepsilon}_6=\left(\left({\mathbf{D}}_{6a3}+{\mathbf{D}}_{6a\prime 3}\right)-\left({\mathbf{D}}_{6b3}+{\mathbf{D}}_{6b\prime 3}\right)\right)\cdot {\mathbf{S}}_3={\overline{\mathbf{D}}}_{23}\cdot {\mathbf{S}}_3\hfill \end{array} $$

that can be rearranged in a more compact notation

$$ {\mathbf{E}}_{\mathbf{b}}=\left[\begin{array}{l}{\varepsilon}_1\hfill \\ {}{\varepsilon}_2\hfill \\ {}{\varepsilon}_3\hfill \\ {}{\varepsilon}_4\hfill \\ {}{\varepsilon}_5\hfill \\ {}{\varepsilon}_6\hfill \end{array}\right]={\left[\begin{array}{lll}{\overline{\mathbf{D}}}_{11}\hfill & \mathbf{0}\hfill & \mathbf{0}\hfill \\ {}{\overline{\mathbf{D}}}_{21}\hfill & \mathbf{0}\hfill & \mathbf{0}\hfill \\ {}\mathbf{0}\hfill & {\overline{\mathbf{D}}}_{12}\hfill & \mathbf{0}\hfill \\ {}\mathbf{0}\hfill & {\overline{\mathbf{D}}}_{22}\hfill & \mathbf{0}\hfill \\ {}\mathbf{0}\hfill & \mathbf{0}\hfill & {\overline{\mathbf{D}}}_{13}\hfill \\ {}\mathbf{0}\hfill & \mathbf{0}\hfill & {\overline{\mathbf{D}}}_{23}\hfill \end{array}\right]}_{6x18}\cdot {\left[\begin{array}{l}{\mathbf{S}}_1\hfill \\ {}{\mathbf{S}}_2\hfill \\ {}{\mathbf{S}}_3\hfill \end{array}\right]}_{18x1}=\overline{\mathbf{D}}\cdot \mathbf{S}=\overline{\mathbf{D}}\cdot \widehat{\mathbf{M}}\cdot \mathbf{F} $$

(35)

where $\widehat{\mathbf{M}}$ is obtained by extracting the first six columns of the inverse of matrix A _A of equation (21).

Finally, it is possible to compute the calibration matrix M _thp

$$ \boldsymbol{\Delta} \mathbf{V}=\frac{V}{4}k{\mathbf{E}}_{\mathbf{b}}=\left(\frac{V}{4}k\overline{\mathbf{D}}\cdot \widehat{\mathbf{M}}\right)\mathbf{F}={{\mathbf{M}}_{\mathbf{thp}}}^{-1}\cdot \mathbf{F} $$

(36)

By simply inverting the terms in equation (36), the calibration matrix M _thp can be derived

$$ \mathbf{F}={\mathbf{M}}_{\mathbf{thp}}\cdot \boldsymbol{\Delta} \mathbf{V} $$

(37)