**3. Kinematics**

The kinematics of the TJL (Figure 5) can be derived according to the Denavit–Hartenberg [26] convention, along with certain trigonometric constraints. Hence, the orbit of the foot tip is a function of the input variable θ, also referred to as the crank angle, and defined as follows:

$$\mathbf{x}(\theta) = A\_1 \cos \theta\_1 + L\_4 \cos(\theta\_1 + \theta\_2) + S\_2 \cos(\theta\_1 + \theta\_2 + \theta\_3) \tag{1}$$

$$\mathbf{y}(\boldsymbol{\theta}) = -[A\_1 \sin \theta\_1 + L\_4 \sin(\theta\_1 + \theta\_2) + S\_2 \sin(\theta\_1 + \theta\_2 + \theta\_3)] \tag{2}$$

where

$$\text{So} = \sqrt{S\_x^2 + S\_y^2} \tag{3}$$

$$\beta\_1 = \tan^{-1} \left( \frac{\text{S}\_{\text{Y}}}{\text{S}\_{\text{x}}} \right) \tag{4}$$

$$
\beta\_2 = \beta\_1 + \theta \tag{5}
$$

$$\mathbf{S}\_1 = \sqrt{\mathbf{S}\_0^2 + \mathbf{L}\_1^2 - 2\mathbf{L}\_1\mathbf{S}\_0\cos\beta\_2} \tag{6}$$

$$\beta\_3 = \tan^{-1} \left( \frac{\mathcal{L}\_1 \sin \beta\_2}{\mathcal{S}\_0 - \mathcal{L}\_1 \cos \beta\_2} \right) \tag{7}$$

$$\beta\_4 = \cos^{-1} \left( \frac{\mathbf{S}\_1^2 + \mathbf{A}\_2^2 - \mathbf{L}\_2^2}{2 \mathbf{S}\_1 \mathbf{A}\_2} \right) \tag{8}$$

$$\beta\_5 = \cos^{-1}(\frac{A\_1^2 + A\_2^2 - A\_3^2}{2A\_1A\_2})\tag{9}$$

$$
\theta\_1 = \beta\_1 + \beta\_3 + \beta\_4 + \beta\_5 - 180^\circ \tag{10}
$$

$$\beta\_6 = \cos^{-1} \left( \frac{\mathbf{S}\_1^2 + \mathbf{L}\_4^2 - \mathbf{L}\_3^2}{2\mathbf{S}\_1\mathbf{L}\_4} \right) \tag{11}$$

$$
\theta\_2 = \beta\_7 = 360^\circ - (\beta\_4 + \beta\_5 + \beta\_6) \tag{12}
$$

$$\beta \rho = \cos^{-1}(\frac{B\_1^2 + B\_3^2 - B\_2^2}{2B\_1 B\_3}) \tag{13}$$

$$\beta\_{10} = \tan^{-1}(\frac{(B\_3 + E)\sin\theta\_9}{B\_1 - (B\_3 + E)\cos\theta\_9})\tag{14}$$

$$
\theta\_3 = 180^\circ - (\beta\_7 + \beta\_{10}) \tag{15}
$$

$$S\_2 = \sqrt{\left(B\_3 + E\right)^2 + B\_1^2 - 2B\_1 \left(B\_3 + E\right) \cos \theta\_{\theta}} \,. \tag{16}$$

**Figure 5.** Theo Jansen Linkage (**a**) dimensions and (**b**) geometrical variables.

The orbits patterns produced by TJLs might end up as bell curves, ovals, sharp-pointed ovals, or lemniscates, depending on the assembly of their variant sizes. Not all generated orbits are suitable to serve as foot trajectories for the legged robots. Hence, dimensioning the TJL appropriately is nontrivial, since the orbits patterns generated by a TJL may vary according to its assembled dimensions, and can be cast into four groups, including bell curves, ovals, sharp-pointed ovals, and lemniscates. In general, the ovals or bell orbits are legitimate, while the sharp-pointed ovals are partly legitimate, and the lemniscates are illegitimate. After analyzing and balancing requirements, the design data of *L4*, *A*1, *A*2, *A*3, *B*1, *B*2, *B*3, *E*, *Sx*, and *Sy* were set to be 140, 135, 140, 190, 135, 200, 150, 55, 135, and 30, respectively.

In Table 1, four cases of links with different lengths: *L*1, *L*2, and *L*3, are illustrated along with the respective data of foot trajectories, demonstrating how they are influenced by the variant links. Although all four cases include legitimate orbits, there are some other concerns to be taken into account in order to achieve a satisfactory design. The maximum step size is defined as the distance between the leftmost point and the rightmost point. Similarly, the maximum step height is defined as the distance between the topmost point and the bottommost point. Although both of these are different from the actual step size and height, they are able to show the effectiveness of the following actions. Regarding the actual step size and height, a slightly more complicated calculation is required, which will be presented in the next section.

Indicatively, in the trajectory of Case 1 (Figure 6a), where the inclination is too deep, the corrective measure, taken by Case 2, is to reduce the lengths of *L*2 and *L*3. Nonetheless, as mentioned earlier, the stability of the hexapod is ensured by the intersection of two support polygons. Given the fact that, the larger the stride, the smaller the intersected polygon will be, the hexapod will eventually be destabilized if the stride is too large. In order to ensure stability, the action taken by Case 3 is to shorten the step size, by reducing the length of *L*1 (Figure 6b). After these efforts, the hexapod is ready to walk. However, its step height is relatively small, so the further action, taken by Case 4, is to increase the step height (Figure 6c), by the fourth set of data in Table 1.


**Table 1.** Four sets of design data for *L*1, *L*2, and *L*3.

**Figure 6.** Actions: (**a**) inclination decreased, (**b**) step size reduced, and (**c**) step height increased.
