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机械毕业设计全套
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JX03-224@粗纱机升降平衡机构设计,机械毕业设计全套
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Coordinate Transformation and. Resolved Motion Rate Control A robot is made up of links connected by joints. In kinematic analysis, it is considered to be a chain of links and joints. At one end of the chain is the supporting base;at the other end is the end effector or hand. Controlling a robot requires that the end effector or hand be moved to a specific point in space to carry out a task. In performing the task, the robots end effector must move through a particular designated path. This section discusses a simplified mathematical method that is useful in describing the relationship of the end effector to the robot base coordinates. Both the position in space nd the orientation or direction of the robots end effector must be described and controlled. Determining the position and orientation of any joint in a robot relative to its base coordinate set requires the transformation of coordinates through all other joints between the base reference and the joint whose coordinates are being determined. If the robot has six joints, or degrees of freedom, it is necessary to set up six coordinate transfers, one for each joint. Each transformation relates the coordinates of one joint to the coordinates of the previous joint in the chain of links and joints. 1. Joint-to-World Transformations World coordinates are defined as the base reference coordinates of the robot. These coordinates are taken through the base joint of the robot or at a known distance from it. Base coordinates, by convention, are defined asx0,y0, and z0 in the 0 coordinate frame. Joint coordinates are defined as the set of coordinates centered on a particular jint. In a sliding or prismatic joint, one coordinate of the coordinate set is along the direction of motion. In a rotary or revolute joint, one coordinate is parallel to the axis of the joint. Figure 6.23 shows the relationship between the successive joints in a robot. Joint I -1 could be taken as the base reference joint because it could be any joint. The next joint toward the end effector (along the chain of links) has the number 1 assigned; each successive joint has an assigned number identification that is one greater than the . preceding joint. Links between joints are assigned in a similar way. These assignments are shown in Figure 6.23. In this particular case, a Cartesian (x, y, z) system is used. If we choose to set I equal to 1, the first link is link 1, and the first joint has the coordinates of x0 , y0, z0and can be considered fixed in the reference base. nts 2. Coordinate Frame Parameters Each coordinate set is also called a frame. Every frame is determined by four parameters that describe how it relates to a previous frame. This mathematical approach was first developed by Denavit and Hartenberg in 1955. It provides the necessary parameters in homogeneous matrices to perform the transformations between coordinate systems in a remarkably simple way (Denavit and Hartenberg 10 , as described in Section 6.6.2. There are two distances and two angles in each set of four parameters. These parameters are called theta sub-i (Os) s sub-i (s1) , sub-i (a1) , and alpha sub-i (a1) , as shown in Figure 6.23. Frames are oriented and determined by two rules: 1. The z1 axis lies along the axis of motion of the /th joint. 2. The X, axis is normal (perpendicular) to the , axis and is pointing away from it. In a cartesian coordinate system,the x axis is always normal(perpendicular) to the z axis. Thus, rule 2 above means that the x1 axis is normal to both the z, axis and the Z axis and goes from the zi_1 axis to the Z1 axis. Trace out this relationship in figure 6.23 to make sure that this point is clear. Note that the coordinate system is attached to the corresponding link even though its z axis direction is determined by the direction of the joint. The ith frame moves with the ith link; the nth frame (assigned to the hand or end effector) moves with the hand. There are n + 1 fra mes on an n joint arm. The four parameters for each ith joint are defined as follows (check each of these in Figure 6.23): 1. The parameter 0, is the angle from the axis to the x1 axis measured around the , axis going in the positive direction specified by the right-hand rule. The right-hand rule specifies that moving from the x axis to the y axis around z in the clockwise direction (the direction of advance of a right-handed screw) will be positive. Moving from y to z clockwise around x is ntsalso positive, as is the movement from z to x clockwise around y. Going in the counterclockwise direction gives a negative value for the angle. 2. The parameter S, is the distance from the origin of the ( I - 1)th frame to the intersection of the axis with the x1 axis. This distance is measured along the z,_1 axis. 3. The parameter a1 is the shortest distance from the Z, axis to the z, axis. Note that this distance is measured along a line which makes an angle O with the X axis. The two slashes (1/) through these two lines in figure 6.23 indicate that they are parallel. If the center of the coordinates for the z. axis lies on the z1_ axis, which is often true, the value of a1 is zero. In the Stanford arm, for example, illustrated in Figure 6.24 all the a1 values are zero. 4. The parameter a is the offset angle from the z,. axis to the zi axis about the x, axis, again using the right-hand rule. Three back slashes () on each line are used to indicate that the reference line for cr s parallel to the z axis. In the simple robot configurations that we will consider, the distances between axes may become zero. In most cases also, the angles will be either zero or multiples of 90 degrees. Any point in the ith coordinate frame ( x , y, , z1 ) can be transformed to the ( i-1)th system by using the homogeneous matrix defined by the parameters previously determined in the matrix equation where A(,_1) iS the 4 X 4 homogeneous matrix derived from the values of the four joint parameters. Multiplying the coordinate vector of the ( I -1)th frame by the A(l_ 1) matrix generates the coordinate vector of the ith frame. The value of the generalized A matrix is given by Paul 321 as nts Transformation of hand coordinates to world coordinates can be done by successively multiplying together the individual homogeneous matrices, the A matrices, representing the transformations between coordinate frame. These A matrices are obtained by substituting in the preceding matrix the values found for the four parameters in Table 6.5. The T matrix is the total transformation matrix and is defined by where the subscripts for the A matrices indicate the initial and final coordinate systems, respectively. Using the T matrix, we can convert the hand coordinate vector to the world coordinate vector: This matrix was first applied to the Stanford Arm, designed by V. Sheinman in 1969. The Stanford Arm is shown in Figure 6.24. Nodesin the schematic diagram represent the six joints of the arm. Five of the joints are rotary (hinge joints) with angles of rotation O 02 04 05 , and 06 The remaining joint, at position 3, has a prismatic (sliding) joint with a length, s3 , that is variable. Note that three of the rotary joints coincide, so that S4 and S5 are zero. These three joints act as a ball-and-socket joint to support the hand. There are four ntslinks in the arm: the vertical column of length S1 , the horizontal cylinder with length S2 , the sliding boom with variable length S3 , and the wristto-hand length S6. Cartesian coordinates are used to describe the locations of each joint. These coordinates are defined as described in the four parameter rules illustrated in Figure 6.23. Note that all the a, angles are fixed at either 0, -90, or +90 degrees, so that their sines and cosines are constants of values 0, 1, or 1. By examining Figure 6.24 carefully and using the four parameter rules of Section 6.6.2, Part 2, we can tabulate the values of each of the four parameters at each joint as listed in Table 6.5. Table 6.5 Parameter values for Stanford arm By inserting the parameter values from Table 6.5 into the A(1_ 1)1 matrix of Equation (61) for each joint we obtain the D-H matrices for the Stanford Arm. It has become customary to substitute C, for cos and S1 for sin O in these matrices in order to reduce the space required and improve the readability of the matrix equations. Those substitutions have been made in the following matrices for the Stanford Arm. nts Multiplying these matrices together in the reverse sequence gives us the T matrix described by Equation (62). We start by multiplying A45 by A56 to obtain A46 , multiply A34 by A46, , and continue this process to finally obtain the T matrix. The first result is： Repeating this process, we obtain the T06 matrix, which relates the position of the end of the arm to the base as nts In the preceding equations, the values for s1 and s6 have been set to zero, so the effective base is at the center of the cylinder and the end of the arm is taken as the end of the wrist. This arrangement simplifies and clarifies the equations. The reverse problem, determining the joint angles required to position the arm to a particular location and orientation in space, is complex and will not be covered in detail here. In the next section, some useful control methods are described. More detail is available in the references cited there. Controlling an industrial robot arm along a desired trajectory is Control done by applying feedback control to the servo motors that drive the Methods individual links so that a desired path will be followed despite external .Resolved Motion Rate Control Resolved motion rate control (RMRC) was developed to provide a way to coordinate the simultaneous motion of several joints at once. Given a desired path to be followed, RMRC is used to control the individual joints in order to achieve the desired path. This method was developed by Whitney and his associates at Draper Laboratory 441. Several alternative previous methods are reviewed in this reference. RMRC allows commands to be applied in a wide variety of coordinate systems, and the axes being controlled can be placed wherever desired, even outside the manipulator. Several varieties of a command may b
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