## 1 Introduction  
The particle filter [1] is an important probabilistic method for state tracking and estimation [2; 3], being widely utilized in robotics. For instance, particle filters are widely applied in mobile robots' self-localization [4; 5] and visual tracking [6; 7]. Three major factors greatly affect the performance of particle filters; namely, measurement model, state space, and motion model. The likelihood function in a measurement model is used for calculating the weights of particles and contributes to accurate state estimation. The particle filter distributes particles in the state space and performs estimation in the state space. The state space affects the number of particles that determines the computational cost of estimation. In addition, the motion model is applied in the state space for state tracking and enables the particle state update to the next state. Figure 1 shows a real-time human motion imitation system using a particle filter with data acquired from a Microsoft Kinect sensor. The joint angles of the humanoid robot are estimated in real time to mirror the human motions. Note that a humanoid robot usually comprises dozens of joints, such as the NAO humanoid robot shown in Figure 2, which has 25 actuated joints (degrees of freedom--DOFs). Conventionally, a particle filter would be applied to the 25-dimensional (25D) state space, including all the joints, to estimate the corresponding angles in the humanoid robot for  
imitation. This high-dimensional state space demands increasing the number of particles to ensure adequate accuracy. Consequently, the computational cost becomes relatively high, as state estimation with a particle filter is very complex for high-dimensional state spaces. Usually, a particle filter applied to a high-dimensional state space requires many particles to perform accurate estimation, and the computational cost increases exponentially. The computational cost becomes a serious problem for humanoid robots given their usual structure with many DOFs. Hence, reducing the state space efficiently helps to constrain the computational cost of the particle filter. Such reduction can be achieved by replacing the high-dimensional state space with a low-dimensional and more efficient state space via various methods [8]. The goal of our work is to construct a real-time humanoid robot imitation system with low computational cost using a particle filter and a Gaussian process dynamical model (GPDM). One of the contributions of the work lies in combining a particle filter and GPDM. The high-dimensional state space is replaced by a GPDM learned, low-dimensional, latent state space to reduce the computational cost of state estimation using a particle filter. A GPDM learned model defines the motion model of the particle filter. The architecture of the proposed system is illustrated in Figure 3. Motion data of a humanoid robot are collected as sequences of joint angles. A low-dimensional latent space and a motion model are constructed based on the motion data. Then, a human demonstrator performs a motion in front of a motion capture system. Our system explores an appropriate set of joint angles for the humanoid robot to imitate the demonstrated motion in real time. Principal component analysis (PCA) [9] is an efficient and simple method for dimensionality reduction of data. Although it enables learning in low-dimensional latent space, it cannot provide a motion model for particle state tracking and may not be suitable for the nonlinear behavior of humanoid robot joints. Alternatively, kernel PCA [10] allows efficiently learning nonlinear data. Still, it also lacks a motion model for state tracking. Hence, a dynamical motion model should be constructed to realize particle state tracking in the latent state space. Gaussian process dynamical models (GDPMs) [11] are efficient for motion learning and suitable for dynamical motion modeling in the latent state space. [PERSON] et al. [12] and [PERSON] et al.