Clustered Regression Control of a Biped Robot Model

Controlling of a biped walking mechanism is a very challenging multivariable problem, the system being highly nonlinear, high-dimensional, and inherently unstable. In almost any realistic case the exact dynamic equations of a walking robot are too complicated to be utilized in the control solutions, or even impossible to write in closed form. Data-based modelling methods try to form a model of the system using only observation data collected from the system inputs and outputs. Traditionally, the data oriented methods are used to construct a global black-box model of the system explaining the whole sample data within one single function structure. Feedforward neural networks, as presented in (Haykin, 1999), for example, typically map the input to the output with very complicated and multilayered grid of neurons and the analysis of the whole net is hardly possible. Local learning methods (Atkeson et al., 1997), on the other hand, offer a more structured approach to the problem. The overall mapping is formed using several local models, which have a simple internal structure but are alone valid only in small regions of the input-output space. Typically, the local models used are linear, which ensures the scalability of the model structure: Simple systems can be modelled, as well as more complex ones, using the same structure, only the number of the local models varies. In robotics, local modelling has been used quite successfully to form inverse dynamics or kinematic mappings that have then been applied as a part of the actual controller (Vijayakumar et al., 2002). However, when trying to cover the whole high-dimensional input-output space, the number of local models increases rapidly. Additionally, external reference signals are needed for the controller to get the system function as desired. To evaluate the assumption of simple local models, a feedback structure based on linear local models, clustered regression, is used here to implement the gait of a biped walking robot model. The local models are based on principal component analysis (see Basilevsky, 1994) of the local data. Instead of mapping the complete inverse dynamics of the biped, only one gait trajectory is considered here. This means that the walking behaviour is stored in the model structure. Given the current state of the system, the model output estimate is directly used as the next control signal value and no additional control solutions or reference signals are needed. The walking cycle can become automated, so that no higher-level control is needed. This text summarizes and extends the presentation in (Haavisto & Hyotyniemi, 2005).