4.7.1.3.3. Acceleration
Marker in { } is optional. If the user does not define optional markers, those markers use Ground.InertiaMarker. The marker name is defined as follows:
(m1{,m2}{,m3}{,m4}) : (bodyname.markername{,bodyname.markername}{,bodyname.markername}{,bodyname.markername}).
m1 becomes action marker.
If m2 is defined, m2 becomes base marker.
If m3 is defined, m3 becomes reference marker.
If m4 is defined, m4 becomes the reference marker which is considering a change of Orientation.
Relative acceleration between action and base marker(\({{\mathbf{\ddot{d}}}^{'}}\)) is computed like below formula.
\({{\mathbf{\ddot{d}}}^{'}}=\mathbf{A}_{m3}^{T}\left[ {{\mathbf{\omega }}_{m4}}\times {{\mathbf{\omega }}_{m4}}\times \left( {{\mathbf{r}}_{m1}}-{{\mathbf{r}}_{m2}} \right) \right]-\mathbf{A}_{m3}^{T}\left[ {{{\mathbf{\dot{\omega }}}}_{m4}}\times \left( {{\mathbf{r}}_{m1}}-{{\mathbf{r}}_{m2}} \right) \right]-2\mathbf{A}_{m3}^{T}\left[ {{\mathbf{\omega }}_{m4}}\times \left( {{{\mathbf{\dot{r}}}}_{m1}}-{{{\mathbf{\dot{r}}}}_{m2}} \right) \right]+\mathbf{A}_{m3}^{T}\left( {{{\mathbf{\ddot{r}}}}_{m1}}-{{{\mathbf{\ddot{r}}}}_{m2}} \right)\)
\(\mathbf{r}\): Position vector of marker with respect to ground Inertia.
\(\mathbf{A}\): Orientation matrix of marker with respect to ground inertia.
\(\mathbf{\dot{r}}\): Velocity vector of marker with respect to ground inertia.
\(\mathbf{\omega }\): Angular velocity vector of marker with respect to ground inertia.
\(\mathbf{\ddot{r}}\): Acceleration vector of marker with respect to ground inertia.
\(\mathbf{\dot{\omega }}\): Angular acceleration vector of marker with respect to ground inertia.
Generally, m3 is the same as m4. Then, above formula means acceleration vector of action marker relative to base marker in the view of reference marker.
Relative angular acceleration between action and base marker(\({{\mathbf{\dot{\omega }}}^{'}}\)) is computed like below formula.
\({{\mathbf{\dot{\omega }}}^{'}}=\mathbf{A}_{m3}^{T}\left( -{{\mathbf{\omega }}_{m4}}\times \left( {{\mathbf{\omega }}_{m1}}-{{\mathbf{\omega }}_{m2}} \right)+{{{\mathbf{\dot{\omega }}}}_{m1}}-{{{\mathbf{\dot{\omega }}}}_{m2}} \right)\)
Likewise, m3 is the same as m4 generally. Then, above formula means angular acceleration vector of action marker relative to base marker in the view of reference marker.
4.7.1.3.3.1. ACCM
The ACCM function returns the absolute value for the acceleration of one marker or the relative acceleration between two markers.
Format
ACCM(Marker1{, Marker2}{, Marker3})
Arguments definition
Marker1 |
The name or argument number of a marker to be calculated |
Marker2 |
|
Marker3 |
|
Formulation
\(\text{ACCM}={{\left( \left[ ^{\text{(mk3)}}{{{\vec{a}}}_{\text{mk1}}}{{-}^{\text{(mk3)}}}{{{\vec{a}}}_{\text{mk2}}} \right]\cdot \left[ ^{\text{(mk3)}}{{{\vec{a}}}_{\text{mk1}}}{{-}^{\text{(mk3)}}}{{{\vec{a}}}_{\text{mk2}}} \right]\, \right)}^{1/2}}\)
\(^{\text{(mk3)}}{{\vec{a}}_{\text{mk1}}}\): Acceleration vector of Marker1 relative to the angular velocity and the angular acceleration of Marker3
\(^{\text{(mk3)}}{{\vec{a}}_{\text{mk2}}}\): Acceleration vector of Marker2 relative to the angular velocity and the angular acceleration of Marker3
Example
4.7.1.3.3.2. ACCX
The ACCX function returns the x-axis acceleration of one marker or the x-axis relative acceleration between two markers.
Format
ACCX(Marker1{, Marker2}{, Marker3}{, Marker4})
Arguments definition
Marker1 |
The name or argument number of a marker to be calculated |
Marker2 |
|
Marker3 |
|
Marker4 |
|
Formulation
\(\text{ACCX}=\left[ ^{\text{(mk4)}}{{{\vec{a}}}_{\text{mk1}}}{{-}^{\text{(mk4)}}}{{{\vec{a}}}_{\text{mk2}}} \right]\cdot {{\hat{x}}_{\text{mk3}}}\)
\(^{\text{(mk4)}}{{\vec{a}}_{\text{mk1}}}\): Acceleration vector of Marker1 relative to the angular velocity and angular acceleration of Marker4
\(^{\text{(mk4)}}{{\vec{a}}_{\text{mk2}}}\): Acceleration vector of Marker2 relative to the angular velocity and angular acceleration of Marker4
\({{\hat{x}}_{\text{mk3}}}\): x-direction unit vector of Marker3
Example
4.7.1.3.3.3. ACCY
The ACCY function returns the y-axis acceleration for one marker or the y-axis relative acceleration between two markers.
Format
ACCY(Marker1{, Marker2}{, Marker3}{, Marker4})
Arguments definition
Marker1 |
The name or argument number of a marker to be calculated |
Marker2 |
|
Marker3 |
|
Marker4 |
|
Formulation
\(\text{ACCY}=\left[ ^{\text{(mk4)}}{{{\vec{a}}}_{\text{mk1}}}{{-}^{\text{(mk4)}}}{{{\vec{a}}}_{\text{mk2}}} \right]\cdot {{\hat{y}}_{\text{mk3}}}\)
\(^{\text{(mk4)}}{{\vec{a}}_{\text{mk1}}}\): Acceleration vector of Marker1 relative to the angular velocity and angular acceleration of Marker4
\(^{\text{(mk4)}}{{\vec{a}}_{\text{mk2}}}\): Acceleration vector of Marker2 relative to the angular velocity and angular acceleration of Marker4
\({{\hat{y}}_{\text{mk3}}}\): y-direction unit vector of Marker3
Example
4.7.1.3.3.4. ACCZ
The ACCZ function returns the z-axis acceleration of one marker or the z-axis relative acceleration between two markers.
Format
ACCZ(Marker1{, Marker2}{, Marker3}{, Marker4})
Arguments definition
Marker1 |
The name or argument number of a marker to be calculated |
Marker2 |
|
Marker3 |
|
Marker4 |
|
Formulation
\(\text{ACCZ}=\left[ ^{\text{(mk4)}}{{{\vec{a}}}_{\text{mk1}}}{{-}^{\text{(mk4)}}}{{{\vec{a}}}_{\text{mk2}}} \right]\cdot {{\hat{z}}_{\text{mk3}}}\)
\(^{\text{(mk4)}}{{\vec{a}}_{\text{mk1}}}\): Acceleration vector of Marker1 relative to the angular velocity and angular acceleration of Marker4
\(^{\text{(mk4)}}{{\vec{a}}_{\text{mk2}}}\): Acceleration vector of Marker2 relative to the angular velocity and angular acceleration of Marker4
\({{\hat{z}}_{\text{mk3}}}\): z-direction unit vector of Marker3
Example
4.7.1.3.3.5. WDTM
The WDTM function returns the absolute value for the angular acceleration of one marker or the relative angular acceleration between two markers.
Format
WDTM(Marker1{, Marker2}{, Marker3})
Arguments definition
Marker1 |
The name or argument number of a marker to be calculated |
Marker2 |
|
Marker3 |
|
Formulation
\(\text{WDTM=}{{\left( \left[ ^{\text{(mk3)}}{{{\dot{\vec{\omega }}}}_{\text{mk1}}}{{-}^{\text{(mk3)}}}{{{\dot{\vec{\omega }}}}_{\text{mk2}}} \right]\cdot \left[ ^{\text{(mk3)}}{{{\dot{\vec{\omega }}}}_{\text{mk1}}}{{-}^{\text{(mk3)}}}{{{\dot{\vec{\omega }}}}_{\text{mk2}}} \right]\, \right)}^{1/2}}\)
\(^{\text{(mk3)}}{{\dot{\vec{\omega }}}_{\text{mk1}}}\): Angular acceleration vector of Marker1 relative to the angular velocity and angular acceleration of Marker3
\(^{\text{(mk3)}}{{\dot{\vec{\omega }}}_{\text{mk2}}}\): Angular acceleration vector of Marker2 relative to the angular velocity and angular acceleration of Marker3
Example
4.7.1.3.3.6. WDTX
The WDTX function returns the x-axis angular acceleration of one marker or the x-axis relative angular acceleration between two markers.
Format
WDTX(Marker1{, Marker2}{, Marker3}{, Marker4})
Arguments definition
Marker1 |
The name or argument number of a marker to be calculated |
Marker2 |
|
Marker3 |
|
Marker4 |
|
Formulation
\(\text{WDTX}=\left[ ^{\text{(mk4)}}{{{\dot{\vec{\omega }}}}_{\text{mk1}}}{{-}^{\text{(mk4)}}}{{{\dot{\vec{\omega }}}}_{\text{mk2}}} \right]\cdot {{\hat{x}}_{\text{mk3}}}\)
\(^{\text{(mk4)}}{{\dot{\vec{\omega }}}_{\text{mk1}}}\): Angular acceleration vector of Marker1 relative to the angular velocity and angular acceleration of Marker4
\(^{\text{(mk4)}}{{\dot{\vec{\omega }}}_{\text{mk2}}}\): Angular acceleration vector of Marker2 relative to the angular velocity and angular acceleration of Marker4
\({{\hat{z}}_{\text{mk3}}}\): x-direction unit vector of Marker3
Example
4.7.1.3.3.7. WDTY
The WDTY function returns the y-axis angular acceleration of one marker or the y-axis relative angular acceleration between two markers.
Format
WDTY(Marker1{, Marker2}{, Marker3}{, Marker4})
Arguments definition
Marker1 |
The name or argument number of a marker to be calculated |
Marker2 |
|
Marker3 |
|
Marker4 |
|
Formulation
\(\text{WDTY}=\left[ ^{\text{(mk4)}}{{{\dot{\vec{\omega }}}}_{\text{mk1}}}{{-}^{\text{(mk4)}}}{{{\dot{\vec{\omega }}}}_{\text{mk2}}} \right]\cdot {{\hat{y}}_{\text{mk3}}}\)
\(^{\text{(mk4)}}{{\dot{\vec{\omega }}}_{\text{mk1}}}\): Angular acceleration vector of Marker1 relative to the angular velocity and angular acceleration of Marker4
\(^{\text{(mk4)}}{{\dot{\vec{\omega }}}_{\text{mk2}}}\): Angular acceleration vector of Marker2 relative to the angular velocity and angular acceleration of Marker4
\({{\hat{y}}_{\text{mk3}}}\): y-direction unit vector of Marker3
Example
4.7.1.3.3.8. WDTZ
The WDTZ function returns the z-axis angular acceleration of one marker or the z-axis relative angular acceleration between two markers.
Format
WDTZ(Marker1{, Marker2}{, Marker3}{, Marker4})
Arguments definition
Marker1 |
The name or argument number of a marker to be calculated |
Marker2 |
|
Marker3 |
|
Marker4 |
|
Formulation
\(\text{WDTZ}=\left[ ^{\text{(mk4)}}{{{\dot{\vec{\omega }}}}_{\text{mk1}}}{{-}^{\text{(mk4)}}}{{{\dot{\vec{\omega }}}}_{\text{mk2}}} \right]\cdot {{\hat{z}}_{\text{mk3}}}\)
\(^{\text{(mk4)}}{{\dot{\vec{\omega }}}_{\text{mk1}}}\): Angular acceleration vector of Marker1 relative to the angular velocity and angular acceleration of Marker4
\(^{\text{(mk4)}}{{\dot{\vec{\omega }}}_{\text{mk2}}}\): Angular acceleration vector of Marker2 relative to the angular velocity and angular acceleration of Marker4
\({{\hat{z}}_{\text{mk3}}}\): z-direction unit vector of Marker3
Example