The free end of the link chain consisting of the manipulator end effector。 According to the application of robot, the end effector can be a starting point, the torch, electromagnet, or other device。 We usually by mechanical hand position description framework description tool, which is connected to the end effector, relative to the base, the base of the mobile manipulator。 In the study of mechanical operation of a very basic problem is the kinematics。 This is to compute the position of mechanical static geometric problems in hand terminal positioning。 Specifically, given a set of joint angles, the forward kinematics problem is to compute the position and orientation relative to the base of the tool holder。 Sometimes, we think this is a change from the joint space is described as a manipulator position that Cartesian space description。 "This problem will be discussed in the 3 chapter。 In the 4 chapter, we will consider the inverse kinematics problem。 The problems are as follows: the end effector position and direction of the manipulator, computing all possible joint angle, can be used to achieve the position and direction of a given。 (see Figure 1。7。) This is a practical problem of manipulator is fundamental。 This is quite a complex geometry problem, the conventional solution in tens of thousands of humans and other biological systems time every day。 In a case like a robot simulation system, we need to create computer control algorithm can make the calculation。 In some ways, the solution to this problem is the most important element in the operating system。

This is quite a complex geometry problem, the conventional solution in tens of thousands of humans and other biological systems time every day。 In a case like a robot simulation system, we need to create computer control algorithm can make the calculation。 In some ways, the solution to this problem is the most important element in the operating system。

We can use this problem as a mapping on 3D Descartes "position" space "position" in the robot joint space。 This need will occur when the 3D spatial objects outside the specified coordinates。 Lack of this kind of algorithm some early robot, they just transfer (sometimes by hand) required for the position, and then be recorded as a common set of values (i。e。, as a position in joint space for later playback)。 Obviously, if the playback position and motion pattern recording and joint of the purely robot in Cartesian space, no algorithm for the joint space is necessary。 However, the industrial robot is rare, the lack of basic inverse kinematics algorithm。 The inverse kinematics problem is not a simple forward kinematics of A。 The equation of motion is nonlinear, their solution is not always easy (or even possible in a closed form)。 At the same time, the existing problems of solutions and multiple solutions occur。 The study of these problems provides an appreciation of what the human mind nervous system is achieved when we, there seems to be no conscious thought, object movement and our arms and hands operation。 Manipulator is a solution of the presence or absence of a given definition of work area。 A solution for the lack of means of mechanical hands can not reach the desired position and orientation, because it is in the manipulator working area。

In addition to static positioning problem, we can analyze the robot motion。 Usually, the analysis in the actuator velocity, it is convenient to define a matrix called the Jacobi matrix of the manipulator。 The speed of Jacobi matrix specified in Descartes from the velocity mapping space and joint space。 (see Figure 1。8。) This mapping configuration of the manipulator changes the natural changes。 At some point, called a singularity, this mapping is not to make the transformation。 This phenomenon are important to the understanding of the mechanical hand designers and users。

Figures：

FIGURE 1。5: Coordinate systems or "frames" are attached to the manipulator and to