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Tham khảo tài liệu 'introductory robotics - selig part 5', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 48 Introductory Robotics 7T. The points in the plane determined by these values are given by - COSỚ2 1 gives x2 y2 1 Z2 2 cos 82 1 gives x2 y2 1 I2 2 These are the equations of two concentric circles on one the arm is at full stretch while to reach the other the arm must double back on itself see fig. 5.2. Beyond the outer circle and inside the smaller one the solutions for SÍI1Ớ2 become complex and it is clear that we cannot reach such points with a real arm. The annular region is the projection of the robot s work space onto the plane see section 3.3. It is the space that the robot can reach and work in. The work space of any robot is always bounded by curves or surfaces on which the number of postures is different from the body of the work space. Such points are called singular points however singular points may also occur in the interior of the work space. A better characterization of singular points is points where the robot loses one or more degrees-of-freedom. In the case of the planar manipulator it is easy to see that on the boundary of the work space the arm has no freedom to move in a radial direction. So far we have said nothing about the design parameters II and 2. In fact the relative sizes of the links do not affect the number of postures except in the very special case that II l . In this case there are still generally two postures for every point in the work space but the inner boundary has now shrunk to a point the origin. If we try to place the tip of the manipulator at X 0 y 0 then certainly we must have cos 82 1 and sin Ớ2 0 but our method for finding Ớ1 breaks down. It is quite clear though that there is no restriction on 81 so instead of one or two postures this point in the work space has a whole circle of postures. This kind of singularity with a continuous family of postures is particularly difficult to deal with when it comes to controlling the robot. Unfortunately all six axis robots that have been designed or built have such .
