*This is the M.Sc. thesis of Mr. Olli Haavisto, graduated in 2004*

Walking robots offer challenging problems in the field of system modeling and control. The mechanical structures of the walkers are usually complex, being composed of many joint-connected parts which impact also with the surrounding environment. Therefore the mathematical models easily appear to be highly nonlinear and high-dimensional preventing the effective use of traditional modeling and control methods.

Now the interesting question is, could the difficulties be avoided by considering only the input and output data of the system, that is, with the help of data-based techniques. If the relation between the control values given and the resulting state of the robot could somehow been captured, the actual internal structure of the system could be forgotten.

This master's thesis work handled the modeling and control of a simple biped walking robot. There were two main goals:

- Development and simulation of the exact mathematical model of the biped.
- Data-based modeling and control of the simulated robot walking.

First, a simple biped robot model was determined to serve as a starting point for data-based modeling and control. Also a Matlab/Simulink simulation tool was developed to provide easy data collection and model simulation. After that, a data-based model structure named *clustered regression* was applied to model and control the gait of the simulated robot.

The biped was modeled to operate in two-dimensional space, and it has a torso and two identical legs with knees. The figure below shows the structure of the biped and the constants and coordinates used to describe the configuration of the system. Also the input forces (*F*) and moments (*M*) are shown.

The walking surface is modeled by the external forces (collected in vector *F*), which are calculated using PD controllers when the leg is touching the ground. The actual control of the biped is done with the moments (in vector *M*) applied to the joints of the robot.

With these assumptions the system has seven degrees of freedom and is *holonomic*, which means that there does not exist any additional "hard" constraints. Now the dynamic equations of the biped can be derived using Lagrangian techniques. If the seven coordinate values are collected to the column vector *q*, the equations can be presented in the form

Here *A* is the 7 x 7 inertia matrix and *b* a 7 x 1 vector consisting of the right hand sides of the equations. The complete formulas are available in Matlab form and in pdf.

To enable the data collection from the biped walking movement, a PD control scheme for the system was developed. The controller contains four independent SISO PD controllers, which follow continuously updated reference signals. The animation (click on the figure to animate) shows the resulting gait, which is by no means optimal as the parameters of the controllers were tuned by hand. Now input and output data of the system could be collected, the input being the four dimensional moment vector values and the output the 16 dimensional biped state vector.

The idea of the data-based approach was to model the cyclic trajectory that the walking biped is following in the state and output space. More precisely, the goal was to form a regression mapping from the current system state (output of the biped system) to the next control vector value (input of the system) during the gait. As the walking trajectory was symmetrical with respect to left and right leg, it sufficed to model only half of the cycle, that is, one step.

Using the data collected from the PD controlled gait a clustered regression structure was taught. The data was first divided into 20 consecutive *clusters*, each corresponding to an *operating point* of the system. The following figure shows the clustered teaching data points projected to three state variables. The circles show the operating point locations and are drawn in the same color as the data clusters belonging to them. Also the system states in the operating point centers are shown. Following the trajectory, one can see how the rearmost leg of the robot is gradually swinging forward as the robot takes one step. Switching the left and right leg in the final state, the initial state is reached again.

The state-to-control-mapping was modeled separately in every cluster using linear *principal component regression* models. The output estimate of the whole regression model was calculated as a weighted combination of these local models, the nearest ones having the biggest weights.

The accuracy of the model was tested by applying it directly to control the biped: The control estimate corresponding to the measured system state was used as the next control input for the biped. It appeared that this kind of clustered regression controller was able to keep the biped walking. Additionally, the resulting gait was quite similar to the PD controlled one, as shown by these animations:

PD controlled gait Learned gait

The main difference between the two gaits is that the clustered regression controlled biped walks a bit slower.

To conclude the work done in this thesis, it can be stated that both of the goals mentioned above were at some level reached. The simulation of the mathematical biped model succeeded and a working simulator environment was developed. It was also shown that the clustered regression modeling was accurate enough to reproduce the PD controlled gait.

Unfortunately, the clustered regression controlled biped was detected to be quite sensitive to disturbances, which was, however, the case with the PD controlled one, too. The unsuccessful attempts to optimize the learned control by updating the regression model also suggested that some modifications to the control scheme are probably required in the future.