# A Hexapod Robot with Non-Collocated Actuators
## 5 Modeling and Analysis  
A simplified model is proposed here to analyze the motion of this hexapod. As a result of tripod gaits, each set of tripod legs can be grouped together and regarded as a single leg, and hence the hexapod can be reduced to a biped model with absolute stability whose initial posture is shown in Figure 11, where G represents the center of mass of the robot; \(\mathrm{O}_{0}\), \(\{\mathrm{i}_{0},\mathrm{j}_{0},\mathrm{k}_{0}\}\) stand for the origin and base vectors of the inertial frame; L, H, D are length, height, and distance; \(\mathrm{P}_{1},\ \mathrm{P}_{2}\) denote the points at the feet corresponding to the bottom deck and upper deck, respectively. Figure 10: Sequence of motions to change course. Except for the indicial convention adapted to this particular configuration, a series of reference frames is launched according to the Denavit-Hartenberg convention as shown in Figure 12. The center of mass G is initially aligned with the origin \(O_{0}\) of inertial frame-0, and there is a bifurcation after frame-3 so that indexes 4b and 5b stand for the frames attached to G and \(P_{2}\). \(\theta_{1}\) and \(\theta_{2}\) stand for the outputs of the bottom deck and upper deck motors, respectively. \(\varnothing_{1}\), \(\varnothing_{2}\), and \(\varnothing_{3}\) are cross-angles between adjacent frames according to their geometrical configuration. Hence, the homogeneous transformation from frame-0 to frame-3 is defined in Equation (1):  
\[T_{3}^{0}=\text{Rot}(z,\varnothing_{3})Trans(L,0,0)Rot\left(x,90^{\circ}\right) Rot\left(z,90^{\circ}\right)Trans(H,0,0)Rot(z,\varnothing_{1})Trans(L,0,0). \tag{1}\]  
Likewise, the transformations \(T_{4a}^{3}\) and \(T_{5b}^{3}\) are defined in Equations (2) and (3), respectively:  
\[T_{4a}^{3}=Rot\left(z,\theta_{1}+90^{\circ}\right)Trans(D,0,0), \tag{2}\]  
\[T_{5b}^{3}=Rot(z,\varnothing_{2})Trans(L,0,0)Rot(z,\theta_{2}+90^{\circ})Trans( H,0,0). \tag{3}\]  
The compound transformations of \(T_{0}^{4a}\) and \(T_{5b}^{5b}\) can be obtained as follows:  
\[T_{4a}^{0}=T_{3}^{0}T_{4a}^{3}=\left[ \begin{array}{cccc}0&cos\varnothing_{3} &sin\varnothing_{3}&Lcos\varnothing_{3}(1-\cos\theta_{1})\\ 0&sin\varnothing_{3}&-cos\varnothing_{3}&Lsin\varnothing_{3}(1-\cos\theta_{1} )\\ -1&0&0&H-D+Lsin\theta_{1}\\ 0&0&0&1\end{array} \right], \tag{4}\]  
\[T_{5b}^{0}=T_{3}^{0}T_{5b}^{3}=\left[ \begin{array}{cccc}0&cos\varnothing_{3} &sin\varnothing_{3}&Lcos\varnothing_{3}(1-\cos\theta_{1}-\cos\theta_{2})\\ 0&sin\varnothing_{3}&-cos\varnothing_{3}&Lsin\varnothing_{3}(1-\cos\theta_{1} -\cos\theta_{2})\\ -1&0&0&L(sin\theta_{1}+sin\theta_{2})\\ 0&0&0&1\end{array} \right], \tag{5}\]  
Hence, the coordinates of G and \(P_{2}\) can be defined in Equations (6) and (7), respectively:  
\[\text{G}=\left[ \begin{array}{c}x_{G}\\ y_{G}\\ z_{G}\end{array} \right]=\left[ \begin{array}{c}Lcos\varnothing_{3}(1-\cos \theta_{1})\\ Lsin\varnothing_{3}(1-\cos\theta_{1})\\ H-D+Lsin\theta_{1}\end{array} \right], \tag{6}\]  
\[P_{2}=\left[ \begin{array}{c}x_{P2}\\ y_{P2}\\ z_{P2}\end{array} \right]=\left[ \begin{array}{c}Lcos\varnothing_{3}(1-\cos \theta_{1}-\cos\theta_{2})\\ Lsin\varnothing_{3}(1-\cos\theta_{1}-\cos\theta_{2})\\ L(sin\theta_{1}+sin\theta_{2})\end{array} \right]. \tag{7}\]  
Figure 11: Biped model. Observing the whole periodic motions (i.e., forward and backward as shown in Figures 13 and 14), the robot performs linear motion at the time events \(t_{-3},t_{-1},t_{1},t_{3}\), while it is either resting or managing a turn at the time events \(t_{-4},t_{-2},t_{0},t_{2},t_{4}\). The central pattern G described in Equation (6) in fact belongs to the time event \(t_{1}\). According to Figures 13 and 14, the complete description of the central pattern must be defined piecewise as follows:  
\[\text{G}=\left[ \begin{array}{c}x_{G}\\ y_{G}\\ z_{G}\end{array} \right]\] \[=\left\{