Fig. 3.

Schematic of the inverse kinematics process for whole-arm movements. Inputs include angles $\left[a,e,r\right]$ (representing azimuth, elevation, and roll) of each sensor ( $\mathrm{E}$– $\mathrm{H}$) and the rotation matrices between each sensor and its BCS ( $\mathrm{A}$– $\mathrm{D}$) established during calibration. The output consists of the three joint angles ( $\left[\alpha ,\beta ,\gamma \right]$) for each of the shoulder ( $s$), elbow-forearm ( $e$), and wrist ( $w$) joints. The inverse kinematics process includes the four steps described above, each represented by a column of boxes: (1) $\mathit{aer}\to R_{\text{SCS}}$ converts sensor angles into rotation matrices describing the orientation of each SCS with respect to the universal frame, (2) $R_{\text{SCS}}\to R_{\text{BCS}}$ multiplies each SCS rotation matrix by its calibration matrix, yielding the rotation matrices describing the orientation of each BCS related to the universal frame, (3) $R_{\text{BCS}}\to R_{\text{JCS}}$ multiplies the rotation matrices of adjacent BCS to obtain JCS rotation matrices, and (4) $R_{\text{JCS}}\to \alpha \beta \gamma$ extracts joint angles from each JCS rotation matrix. The leading superscript and subscript of rotation matrices indicate the original and final CS; for example, ${}_B{}^{A}R$ is the rotation matrix that describes $B$ relative to $A$ (see Appendix A for more detail).