## Reflexive stability of the skiing robotResearch activity duration: 2010 -Research area: Automation and Intelligent Control of Robots Keywords: Robotics, Activity leader: Tadej Petrič |

Andrej Gams

Leon Žlajpah

The proposed method for maintaining the stability is based on a modified prioritized kinematic control, which allows a smooth and continuous transition between priorities. For the skiing robot, the primary task is the stability, the secondary task is the direction and the third task is the pose of the robot. The proposed algorithm, allows arbitrary joint movements of the skiing robot without any regard for the consequential movement of the stability index, which was determined through the zero-moment-point (ZMP), as long as the selected stability criteria is in a predefined range. On the other hand, it constrains the movement when this criterion approaches a critical condition, i.e. when the stability index approaches the critical values. The transition between the two tasks is smooth and reversible, encapsulated in a single modified prioritised task control equation.

To design the skiing robot controller it was necessary to determine which aspects of skiing are important. By considering only the carving skis, which exploit their shape and do not require skidding in order to make turns, we modelled the lower extremities of a skier with two artificial legs and the upper body. This mechanism can change the relative inclination angle between the skis and the skiing surface. Since the radius of the carving skis is defined by their geometry, denoted by

where L is the ski length and h is the geometry coefficient as shown in Fig.1, it is evident that we can control the turning radius by controlling the angle between the skis and the skiing surface.

The skiing robot has three main objectives with the following priorities: first, to ensure stability; second, to control the direction of the skiing; and third to maintain the desired pose. The simplified block scheme of the skiing control system is presented in Fig.2.

Since the robot is relatively stable in the sagital plain due to the length of the skis, we focus only on lateral stability. The skiing robot is in the most stable position when the forces on both skis are equally distributed. We denote this position with MSP (Most Stable Point) as shown in Fig.3. On the other side, LSP (Last Stable Point) represents the point when all forces act only on one ski. If such a situation occurs, one of the skis looses contact with the ground and the pose becomes unstable.

where m_i is the mass of the i-th link, v is the velocity in the direction of the skiing, R_i is the radius of the turn for each body segment and g is the gravity vector. The relationship between the ZMP and the joint-space velocity is given by

The Jacobian is a function of the actual joint angles and it maps the velocity in the joint space to the velocity of ZMP. By denoting the stability index as function of a normalized ZMP to the power of an odd number

the stability controller can be defined as

where K_p is the proportional gain and z_{n,d} is the desired value of z_n. The commanded joint-space velocity for the skiing robot is defined as

where N_z' is defined as

The formulation of the null space matrix N_z' allows an arbitrary joint movement. Consequently, the ZMP can be moved inside the support polygon. By changing the value of n we can control how close to the boundary of the support polygon the ZMP is allowed to move before triggering the stability control.