## 2 [PERSON] Linkage and Foot Trajectories  
The TJL can be built either using eight links or twelve links. It depends on the way of fabricating two triangluar links of the TJL. If each triangular link is machined into a whole piece, then the TLJ is an eight-bar linkage. If each of them is assembled by three straight bars, the TLJ will be a twelve-bar linkage. Regardless of eight or twelve, the degrees of freedom of the TJLs turn out to be the same, i.e., one. The TJL in this paper is the eight-bar linkage, as shown in Figure 1a, where the TJL is normalized with respect to its link-1, so that link-1, 2, 3, 6, 7 are straight bars with lengths 1, a, b, d, d,respectively; link-4, 5 are isosceles right triangles with sides of length \(c\); link-8 is a rectangular fixture with dimensions \(\rm{S_{x}}\) and \(\rm{S_{y}}\). In Figure 1b, a robot is mounted with TJLs to achieve the legged motion. The kinematics of the TJL can be derived according to the Denavit-Hartenberg [16; 17] convention along with some trigonometric constraints. Hence, the foot trajectory at the point E is a function of the input variable \(\theta\) defined, as follows. \[x=-c\sin\theta_{1}+d\cos(\theta_{1}+\theta_{2}) \tag{1}\]  
\[y=-[c\cos\theta_{1}+d\sin(\theta_{1}+\theta_{2})] \tag{2}\]  
where  
\[\theta_{1}=\beta_{1}+\beta_{3}+\beta_{4}-90^{\lx@math@degree} \tag{3}\]  
\[\theta_{2}=270^{\lx@math@degree}-(\beta_{4}+\beta_{6}) \tag{4}\]  
\[\beta_{1}=\tan^{-1}(\frac{S_{y}}{S_{x}}) \tag{5}\]  
\[\beta_{2}=\beta_{1}+\theta \tag{6}\]  
\[\beta_{3}=\tan^{-1}(\frac{\sin\beta_{2}}{S_{0}-\cos\beta_{2}}) \tag{7}\]  
\[\beta_{4}=\cos^{-1}(\frac{S_{1}^{2}+c^{2}-a^{2}}{2S_{1}c}) \tag{8}\]  
\[\beta_{6}=\cos^{-1}(\frac{S_{1}^{2}+d^{2}-b^{2}}{2S_{1}d}) \tag{9}\]  
\[S_{0}=\sqrt{S_{x}^{2}+S_{y}^{2}} \tag{10}\]  
\[y=-[c\cos\theta_{1}+d\sin(\theta_{1}+\theta_{2})] \tag{2}\]0  
With the aid of kinematic Equations (1)-(11), the orbit of a TJL can be determined provided that the six parameters, i.e., a, b, c, d, \(\rm{S_{x}}\), \(\rm{S_{y}}\), are given. This is so called the forward problem. However, designing a TJL to generate a legitimate orbit is an inverse problem, i.e., given the specification of an orbit to determine six parameters, a, b, c, d, \(\rm{S_{x}}\), \(\rm{S_{y}}\). It is hopeless to derive any analytical function to deal with this inverse problem. Hence, the numerical schemes have to be applied. \[y=-[c\cos\theta_{1}+d\sin(\theta_{1}+\theta_{2})] \tag{2}\]1 [PERSON] Linkages (**a**) Dimensions (**b**) Implemented on a robot. The LM (Levenberg-Marquardt) method [9; 10] is a standard numerical scheme in solving the inverse problems. Supposedly, the six unknown parameters can be obtained by minimizing an objective function, which is the square of residual between the desired and actual orbits. However, let us take a glimpse of Figure 2, where the orbits are generated by randomly resizing TJLs. The patterns of orbits can be roughly casted into four patterns, i.e., bell curves, ovals, sharp-pointed ovals, and lemniscates, as shown in \[y=-[c\cos\theta_{1}+d\sin(\theta_{1}+\theta_{2})] \tag{2}\]2  
Some orbits, named as legitimate class, can be used as the foot trajectories for legged robots, but some, named as illegitimate class, cannot. Overall speaking, the ovals or bell orbits are legitimate and the sharp-pointed ovals are partly legitimate, nevertheless the lemniscates are illegitimate. Unfortunately, the LM method is unable to tell the patterns of orbits. Hence, its algorithm does not guarantee that its solution always stays within the legitimate class. In other words, the LM method is a numerical scheme good for the case of homogeneity, i.e., all orbits belonging to the legitimate class. As encountered with the case of heterogeneity, i.e., legitimate orbits mingled with illegitimate ones, it becomes so incompetent. Since neither the analytical method nor the standard numerical scheme can prevail, we resort to leverage the ML (Machine Learning) in the attempt of drawing out the knowledge or model from those heterogeneous data, i.e., orbits.