# Semi-Supervised Manifold Alignment Using Parallel Deep Autoencoders
## 5 Experiments and Results
### Experiments on 2-, 3- and 4-Manifold Alignment
#### 5.2.1 Double Pendulum Datasets
For each case, two datasets, \(X\) and \(Y\), were generated that represented the motion of two similar double pendulums that differed only in different limb lengths and limb length ratios:  
1. Pendulum X: \(\left(u_{2}/u_{1}\right)=0.75/1.25=0.60\)
2. Pendulum Y: \(\left(u_{2}/u_{1}\right)=1.25/1.56=0.80\)  
Figure 4: (a) shows the 2D-2D version of the double pendulum where both limbs are rotating in a two-dimensional plane. \(x\) and \(y\) are the local coordinate axes of limb \(u_{1}\). \(x^{\prime}\) and \(y^{\prime}\) are the local axes of limb \(u_{2}\). \(\left(e_{x},e_{y}\right)\) are the end-effector coordinates. (b) shows the 2D-3D version of the double pendulum where limb \(u_{1}\) is in a two-dimensional plane and limb \(u_{2}\) is rotating on a sphere in three-dimensional space. \(x\) and \(y\) are the local coordinate axes of limb \(u_{1}\). \(x^{\prime},y^{\prime}\) and \(z^{\prime}\) are the local coordinate axes of limb \(u_{2}\), and \(\left(e_{x},e_{y},e_{z}\right)\) are the end-effector coordinates. (c) shows the 3D-3D version of the double pendulum where both limbs are rotating on spheres in three-dimensional space. \(x\), \(y\) and \(z\) are the local axes of limb \(u_{1}\). \(x^{\prime}\), \(y^{\prime}\) and \(z^{\prime}\) are the local axes of limb \(u_{2}\), and \(\left(e_{x},e_{y},e_{z}\right)\) are the end-effector coordinates. The data were generated in simulation for the 2D-2D, 2D-3D and 3D-3D cases as follows (Figure 4):
1. 2D-2D motion: The pendulum has two Degrees-Of-Freedom (DOF), that is, both limbs \(u_{1}\) and \(u_{2}\) rotate in the two-dimensional (\(x\)-\(y\))-plane, each of them describing a circle. In Figure 4a, \(\theta_{1}\) and \(\theta_{2}\) are the rotation angles of limbs \(u_{1}\) and \(u_{2}\) at joints \(J_{1}\) and \(J_{2}\), respectively. Accordingly, the manifold representing the dynamics of the 2D-2D case is the cross-product of two circles, \(S^{1}\times S^{1}\), which is homeomorphic to the two-dimensional torus, that is a 2-manifold. 2. 2D-3D motion: The pendulum has three DOFs, where limb \(u_{2}\) can rotate on a two-dimensional sphere \(S^{2}\) in three-dimensional space, while \(u_{1}\) is restricted to rotate on a circle \(S^{1}\) in a two-dimensional plane. That is, the manifold representing the dynamics of the 2D-3D case is homeomorphic to \(S^{1}\times S^{2}\), which is a 3-manifold. As the pendulum moves in 3D space, the end-effector has the 3D coordinates \(\left(e_{x},e_{y},e_{z}\right)\). In Figure 4b, \(\theta_{y}\) and \(\theta_{z^{\prime}}\) are the angles of \(u_{2}\) with axes \(y^{\prime}\) and \(z^{\prime}\), respectively, and describe the motion on the sphere \(S^{2}\). \(\theta_{1}\) is the angle between the \(x\)-axis and \(u_{1}\) and describes the two-dimensional rotation of the sphere's centre in the (\(x\)-\(y\))-plane. 3. 3D-3D motion: In this case, the pendulum has four DOFs, where both limbs can rotate on two-dimensional spheres in 3D space. In Figure 4c, \(\theta_{y}\) and \(\theta_{z}\) are the angles of \(u_{1}\) with the \(y\) and \(z\) axes, respectively, and \(\theta_{y^{\prime}}\) and \(\theta_{z^{\prime}}\) are the angles of \(u_{2}\) with the \(y^{\prime}\) and \(z^{\prime}\) axes, respectively. Accordingly, we expected that the manifolds representing the dynamics of the 3D-3D case were homeomorphic to \(S^{2}\times S^{2}\), which is a 4-manifold. For each case, two datasets, \that is, we restricted the experiments to the case \(u_{2}<u_{1}\). The feature vectors for each sample point (or instance) were calculated from the kinematics at the joints and the coordinates of the end-effector. The end-effector coordinates were calculated using forward kinematics.