## 6 Joint Angle Prediction Using Gaussian Process Dynamical Motion Models
Figure 13: Prediction of joint angles for a humanoid robot using similarity measurement. Humanoid Robot Joint Angle Estimation Using a Gaussian Process Dynamical Motion Model and Particle Filter  
The proposed particle filter was then applied to the learned 3D state space shown in Figure 7(a). In the conventional particle filter detailed in Section 3, each particle is defined as \(\mathbf{p_{t}^{q}}=[\theta_{t}^{1},\theta_{t}^{2},\cdots,\theta_{t}^{M}]\), with \(M\) being number of the humanoid robot joint angles. Using the GPDM, the particles are distributed over the 3D state space, and each particle is given by \(\mathbf{p_{t}^{q}}=[x_{t}^{1},x_{t}^{2},x_{t}^{3}]\). Table 1 shows the differences between conventional particle filter method and our proposed system. The high-dimensional state space is replaced by a 3D state space. The computational cost is greatly reduced when replacing the high-dimensional state space by the 3D GPDM learned state space. Figure 15 shows the particle filter estimation system using the learned GPDM, where particles are distributed in the 3D state space. The current state, \(\mathbf{p_{t}}\), can be obtained from the previous state, \(\mathbf{p_{t-1}}\), using learned GPDM dynamical mapping \(f_{x}\) in the 3D state space. Once the humanoid robot states are estimated in the 3D state space, the joint angles are reconstructed using GPDM decoder \(f_{\theta}\). Figure 16 shows a sequence of real-time joint angle estimation using the GPDM and particle filter. This real-time estimation outperforms that using only the particle filter, illustrated in Figure 6. The experiment results of real-time joint angle estimation using only particle filter in Figure 6 shows that it delayed about 2 s when the Nao robot imitated human motions. This is because the computational cost is relatively high when a particle filter performs estimation in high-dimensional state space. It cannot be avoided that the system takes much time to estimate humanoid robot joint angles in high-dimensional state space and delay exists. By combining a particle filter and a GPDM, we reduced the high-dimensional joint angle space into the GPDM learned 3D state space, and the computational cost was greatly reduced. The GPDM motion model was also adopted to make accurate the estimation \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Method** & **State Space** & **Number of Particles** & **Motion Model** & **Measurement Model** \\ \hline PF & \(M\) & \(Q\) & Gaussian & Cosine similarity \\ \hline PF + GPDM & 3 & \(Q\) & GPDM & Cosine similarity \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparison of conventional particle filter (PF) and our GPDM based particle filter. Figure 14: Motion simulation for predicted joint angles using similarity measurement. of humanoid robot joint angles. The imitation experiments show that the humanoid robot moves almost simultaneously with the human, and the delay is reduced greatly by applying the estimation in the 3D state space. These results verify that the learned GPDM enable efficient humanoid robot state estimation in the 3D state space with low computational cost for real-time imitation of human motion. Figure 16: Joint angle estimation for robot imitation using GPDMs and particle filter. Figure 15: Particle filter estimation with GPDM.