## 8 Conclusions  
Theo Jansen linkage is an appealing mechanism to implement a bio-inspired motion for a legged robot. The oval orbit generated by the Theo Jansen linkage, possessing a transversal axis longer than a lateral axis, achieves energy efficient walking as comparing to the circular orbit generated by the four-bar linkage. However, the patterns of orbits that are produced by the TJL may vary according to its ensemble dimension and can be casted into four groups, such as bell curves, ovals, sharp-pointed ovals, and lemniscates. Some orbits can be used as the foot trajectories for legged robots but some cannot. Overall speaking, the ovals or bell orbits are legitimate and the sharp-pointed ovals are partly legitimate, nevertheless the lemniscates are illegitimate. All of academic works relevant to TJLs mainly focus on the analysis of forward problems, like kinematics, dynamics, and controls. None have ever addressed the issue of design that has to rely on the knowledge of inverse problems. To design a Theo Jansen linkage, which can generate a foot trajectory to achieve an energy efficient walking, relies on successfully sizing its eight links. It is equivalent to determine its six dimensional parameters according to the specification of orbits. Due to the complexity of TJLs, it is impossible to derive any inverse function analytically. As resorting to the numerical scheme, the heterogeneous patterns of orbits hinder the Levenberg-Marquardt method from prevailing because it is lacking capability to tell the patterns of orbits. In order to discriminate the orbits, the machine learning technique, called SVM, is employed. The SVM gives a geometric perspective that is finding the equation for a hyperplane to best separate the different classes in the feature space, solves the convex optimization problem analytically, and hence always returns the same optimal hyperplane. Obtaining a sufficient amount of training data that adequately reflects the characteristics of the field data is always the bottleneck of supervised learning. The breakthrough that we made is to apply the image processing to label a vast amount of training data without bias. The proposed method is divided into two steps. First, a preliminary model is built with a small portion of data to conduct a pilot study on discovering the knowledge of data. Second, an advanced  
Figure 16: Orbits generated by Theo Jansen Linkage (TJLs) randomly selected from the database of dimensions. model is built to deal with the large scale data. Consequently, two models of SVM are trained and the two databases concerned with properties and dimensions of all eligible designs can be established. The models of SVM are used to quickly identify the eligibility of a design data and the databases provide abundant of feasible data to which a designer can refer as designing the TJLs. The inverse problem concerned with design is fulfilled by searching the orbits in compliance with the specification of obliqueness and slenderness from the database of properties and using their correspondent identity numbers to list all the candidates of TJLs from the database of dimensions. With the aid of this proposed method, the TJLs have been successfully designed and implemented on a legged robot. There are two topics of interest for future research. First, SVM is one of discriminant technique among learning machines. In contrast to ANN (Artificial Neural Network), hyperplanes are highly dependent on the initialization and termination criteria. SVM distinguishes itself from ANN, in that it always returns the same optimal hyperplane with a large margin. Due to its successful application, we shall advance to other ML techniques, like K-Means clustering, K-Nearest Neighbors, Naive Bayes, and Random Forest, etc., and evaluate their merits and demerits. Second, the structure of a TJL can be improved by replacing two isosceles right triangles with two right triangles.