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Table 5 Rules of the RB classification method to determine the different categories of compensation: Trunk Forward (TF), \(Y=0\); Trunk Rotation (TR), \(Y=1\); Shoulder Elevation (SE), \(Y=2\); Other (O), \(Y=3\). For normal movements \(Y=4\)

From: A low-cost virtual coach for 2D video-based compensation assessment of upper extremity rehabilitation exercises

Scenario Rules
Trunk forward (TF)/Trunk backward (O)
 S1 \({\hat{Y}} = {\left\{ \begin{array}{ll} 0 &{} \text {if } \, \Delta H^t > th_{TF} \\ 3 &{} \text {if } \, \Delta H^t < -th_{TF} \\ 4 &{} otherwise \end{array}\right. }\)  
 S2 and S3 \({\hat Y = \left\{ {\begin{array}{*{20}{ll}} 0& {\text{if }}\, {a^t}\left( {p_8^1,p_1^1,p_1^t} \right) > t{h_{TF}}\\ & \quad \wedge \Delta {x^t}\left( {p_{2/5}^t,p_1^t} \right) > 0 \\ \\ 3& {\text{if }}\, {a^t}\left( {p_8^1,p_1^1,p_1^t} \right) > t{h_{TF}}\\ & \quad \wedge \Delta {x^t}\left( {p_{2/5}^t,p_1^t} \right) < 0 \\ \\ 4& {otherwise} \end{array}} \right. }\)  
Trunk rotation (TR) and Shoulder elevation (SE)
 S1 \(\hat Y = \left\{ {\begin{array}{*{20}{ll}} 2& {\text{if}}\, (1){\text{ }}{a^t}(p_{2/5}^1,p_1^1,p_{2/5}^t) > t{h_{SE}}\\ & \quad \wedge (2){\text{ }}{a^t}(p_{5/2}^1,p_1^1,p_{5/2}^t) < t{h_{SE}} \\ 1&{\text{if}}\, (1) > t{h_{SE}} \\ & \quad \wedge (2) > t{h_{SE}} \wedge (1) - (2) \approx 0 \\ \\ {1 \wedge 2} &{\text{if}}\, (1) > t{h_{SE}} \\ & \quad \wedge (2) > t{h_{SE}} \wedge (1) - (2) \gg 0 \\ \\ 4&{{\text{otherwise}}} \end{array}} \right.\)  
Trunk rotation (TR)
 S2 \({\hat{Y}} = {\left\{ \begin{array}{ll} 1 &{} \text {if } \Delta x^t(p^t_{2/5},p^t_{1}) > th_{TR} \\ 4 &{} otherwise \end{array}\right. }\)  
 S3 \({\hat{Y}} = {\left\{ \begin{array}{ll} 1 &{} \text {if } |\Delta d^t(p^t_{2},p^t_{5})| > th_{TR} \\ 4 &{} otherwise \end{array}\right. }\)
Shoulder elevation (SE)
 S2 and S3 \({\hat{Y}} = {\left\{ \begin{array}{ll} 2 &{} \text {if } \Delta y^t(p^t_{2/5},p^t_{1}) > th_{SE} \\ 4 &{} otherwise \end{array}\right. }\)
Trunk tilt (O)
 S1 \({\hat{Y}} = {\left\{ \begin{array}{ll} 3 &{} \text {if } a^t(p^1_{8},p^1_{1},p^t_{1}) > th_{O} \\ 4 &{} otherwise \end{array}\right. }\)
 S2 and S3 \({\hat{Y}} = {\left\{ \begin{array}{ll} 3 &{} \text {if } |\Delta H^t | > th_{TI} \\ 4 &{} otherwise \end{array}\right. }\)