【正文】 ws to the wall by pulling the endeffectors (EE),described below, down toward the floor and checking the reaction forces. The two remaining motorized DOFs whose axes are parallel to the wall’s plane are designed for determining the distance of the robot from the wall (Motor 3) and the angular constraint for the EE(Motor 4). This design of the leg provides the robot with good gait first two motors in each leg drive the robot’s the attachment of the hooks and upon determination of the distance from the wall (by Motors 3 and 4) of every leg, the robot’s movement is made by the first two motors in each leg. This movement is similar to the movement of rockclimbers who use their fingers to grasp cracks in a rock face and activate their shoulder and elbow muscles to advance. The structure of the robot allows it to move in any desired direction (360176。 angle in relation to a wall’s plane. Such rotation prevents the gripping device’s body from colliding with the wall. Since all the legs are fixed to the wall, the orientation of the legs must change as the robot moves its central body. The hooks attach to the wall and a change in orientation will apply torque on the gripping device about the axis perpendicular to the wall. This torque can cause the leg to disengage from the wall. In order to prevent this, a passive DOF was added to the gripping device’s axis. Thus, the gripping device is attached to the leg by two miniature bearings, creating 18 a 1 DOF axis. A small balancing weight was added to the gripping device in order to keep it horizontal as it approached the wall seeking to attach itself. . Kinematics The first step in designing the robot’s motion was to analyze its kinematics. Thus, a systematically analytical method was needed for acquiring the robot’s orientation data based on the position feedbacks obtained from the servo motors . Direct kinematics The use of direct kinematics makes it possible to pinpoint the position of the leg EEs as a function of the leg joint angles. Based on the joint angles, the EE positions can be calculated in relation to the global frame. In order to analyze the kinematics, a set of frames is attached to the system (Fig. 3). The robot moves relative to Frame W, the global frame. Frame 0, positioned on the robot’s central body, keeps its parallelism to frame W. Frame B is fixed to the robot’s central body. Frame L, fixed on the first motor of every leg, keeps its parallelism to frame B. Frames i (i = 1, 2, 3, 4) are frames placed on motor i’s axis and rotates with it. It is assumed that the robot moves in a plane parallel to the wall. 19 Since all legs are similar, although in a mirror view, the position of the EE is first located in relation to the position of the first motor (frame L). It is then transformed into the central body frame the leg is fully stretched sideways, all the angles are set to zero. Let frame 4 be the EE frame. The vector rL which expresses the position of the EE position at frame L is: where is a homogeneous transformation matrix from frame i to frame j, r4 is the position of the EE related to frame 4. Hence, the EE position with respect to the frame L: where Li is the length of the ith link, θi is the angle between link i ? 1 and link i. As there are four legs mirrored at each side, then for every leg,rL is mapped to frame B and is expressed by the vector rB: is the homogeneous transformation matrix from frame L to frame B where rotations by ?By around x axis and by ?Bx around y axis are made. Each leg’s constants, ?Bx and ?By, are given by the position of the leg around the central body and can be either 0176。. We use frame B to represent the position of the leg EEs in relation to the current central body position of the robot. However,because the legs are similar, all the legs movements will be controlled in frame L by the same global function . Inverse kinematics To position the EE at a desired location, we use the inverse kinematics (IK) which defines the legs’ matching angles. This means that a certain configuration would give the desired position of the leg EEs. The IK is used for a single leg relative to CLIBO’s central body. The IK is used to reach a desired EEs position in relative to the central body according to the legs’ matching orientation of the EE remains constant due to the balancing weight mentioned previously. The calculation of the IK is made with the assumption that the central body’s orientation remains vertical at all time. This assumption is accurate due to orientation angle correction of the central body which will be made within the motion algorithm, as will be 20 detailed later. Moreover, the distance of the central body from the wall is constrained to the defined value Z. By construction of the leg, the two lateral joints are responsible for the distance from the wall and the approach angle of the the two joints which are closer to the central base are responsible for the location of the contact point in the X–Y these constraints and assumptions, there are four different solutions for the desired angles, two for θ1,θ2 and two for θ3, , as we search for the same configuration solution for all legs, frame L of every leg is fixed as a mirror view to its the position of the EE as (X, Y)T , the inverse kinematics,puted using the leg geometrics, is calculated in frame L. Let variable E be the projection of the distance from frame 2’s origin to the EE on the global frame’s x–y plane (Fig. 3). From the law of cosines, θ2 is: From the law of sines, θ1 will be: when the leg is attached to the wall, the distance from the wall Z remains constant. Therefore, the sum of θ3 and θ4 which defines the distance Z, remains constant and is given by κ. From (2): From (6), we can extract θ3: Therefore, θ4 will be: This method