裝配圖機器人基礎(chǔ)教程,裝配,機器人,基礎(chǔ)教程 Chapter 3 Geometric description of the robot mechanism The geometric description of the robot mechanism is based on the usage of translational and rotational homogenous transformation matrices. A coordinate frame is attached to the robot base and to each segment of the mechanism, as shown in Figure 3.1. Then, the corresponding transformation matrices between the consecutive frames are determined. A vector expressed in one of the frames can be transformed into another frame by successive multiplication of intermediate transformation matrices. Vector a in Figure 3.1 is expressed relative to the coordinate frame x 3 , y 3 , z 3 , while vector b is given in the frame x 0 , y 0 , z 0 belonging to the robot base. A mathe- matical relation between the two vectors is obtained by the following homogenous transformation bracketleftbigg b 1 bracketrightbigg = 0 H 1 1 H 2 2 H 3 bracketleftbigg a 1 bracketrightbigg . (3.1) 3.1 Vector parameters of a kinematic pair Vector parameters will be used for the geometric description of a robot mechanism. For simplicity we shall limit our consideration to the mechanisms with either par- allel or perpendicular consecutive joint axes. Such mechanisms are by far the most frequent in industrial robotics. In Figure 3.2, a kinematic pair is shown consisting of two consecutive segments of a robot mechanism, segment i?1 and segment i. The two segments are connected by the joint i including both translation and rotation. The relative pose of the joint is determined by the segment vector b i?1 and unit joint vector e i , as shown in the Figure 3.2. The segment i can be translated with respect to the segment i?1 along the vector e i for the distance d i and can be rotated around e i for the angle ? i .The coordinate frame x i , y i , z i is attached to the segment i, while the frame x i?1 , y i?1 , z i?1 belongs to the segment i?1. T. Bajd et al., Robotics, Intelligent Systems, Control and Automation: Science 23 and Engineering 43, DOI 10.1007/978-90-481-3776-3_3, c?Springer Science+Business Media B.V. 2010 24 3 Geometric description of the robot mechanism z 0 y 0 b x 0 z 1 z 2 z 3 z 4 y 1 y 2 y 3 y 4 x 1 x 2 x 3 x 4 a 0 H 1 1 H 2 2 H 3 3 H 4 Fig. 3.1 Robot mechanism with coordinate frames attached to its segments e i z i–1 y i–1 x i–1 e i+1 z i y i J i e i–1 d i b i b i–1 x i i–1 i Fig. 3.2 Vector parameters of a kinematic pair The coordinate frame x i , y i , z i is placed into the axis of the joint i in such a way that it is parallel to the previous frame x i?1 , y i?1 , z i?1 when the kinematic pair is in its initial pose (both joint variables are zero ? i = 0andd i = 0). The geometric relations and the relative displacement of two neighboring segments of a robot mechanism are determined by the following parameters: e i – unit vector describing either the axis of rotation or direction of translation in the joint i and is expressed as one of the axes of the x i , y i , z i frame. Its components are the following e i = ? ? 1 0 0 ? ? or ? ? 0 1 0 ? ? or ? ? 0 0 1 ? ? ; b i?1 – segment vector describing the segment i?1 expressed in the x i?1 , y i?1 , z i?1 frame. Its components are the following 3.1 Vector parameters of a kinematic pair 25 b i?1 = ? ? b i?1,x b i?1,y b i?1,z ? ? ; ? i – rotational variable representing the angle measured around the e i axis in the plane which is perpendicular to e i (the angle is zero when the kinematic pair is in the initial position) d i – translational variable representing the distance measured along the direction of e i (the distance equals zero when the kinematic pair is in the initial position) If the joint is only rotational (Figure 3.3), the joint variable is represented by the angle ? i , while d i = 0. When the robot mechanism is in its initial pose, the joint angle equals zero ? i = 0 and the coordinate frames x i , y i , z i and x i?1 , y i?1 , z i?1 are parallel. If the joint is only translational (Figure 3.3), the joint variable is d i , while ? i = 0. When the joint is in its initial position, then d i = 0. In this case the coordinate frames x i , y i , z i and x i?1 , y i?1 , z i?1 are parallel irrespective of the value of the translational variable d i . By changing the value of the rotational joint variable ? i , the coordinate frame x i , y i , z i is rotated together with the segment i with respect to the preceding segment i? 1 and the corresponding x i?1 , y i?1 , z i?1 frame. By changing the translational variable d i , the displacement is translational, where only the distance between the two neighboring frames is changing. e i z i–1 y i–1 e i+1 z i y i x i d i e i+1 e i–1 z i–1 e i y i–1 z i y i x i J i b i–1 b i b i–1 x i–1 e i–1 x i–1 b i i–1 i–1 i i Fig. 3.3 Vector parameters of a kinematic pair 26 3 Geometric description of the robot mechanism The transformation between the coordinate frames x i?1 , y i?1 , z i?1 and x i , y i , z i is determined by the homogenous transformation matrix taking one of the three possible forms regarding the direction of the joint vector e i . When the unit vector e i is parallel to the x i axis, there is i?1 H i = ? ? ? ? 10 0 d i + b i?1,x 0cos? i ?sin? i b i?1,y 0sin? i cos? i b i?1,z 00 0 1 ? ? ? ? , (3.2) when e i is parallel to the y i axis, we have the following transformation matrix i?1 H i = ? ? ? ? cos? i 0sin? i b i?1,x 010d i + b i?1,y ?sin? i 0cos? i b i?1,z 000 1 ? ? ? ? (3.3) and when e i is parallel to the z i axis, the matrix has the following form i?1 H i = ? ? ? ? cos? i ?sin? i 0 b i?1,x sin? i cos? i 0 b i?1,y 001d i + b i?1,z 01 ? ? ? ? . (3.4) In the initial pose the coordinate frames x i?1 , y i?1 , z i?1 and x i , y i , z i are parallel (? i = 0andd i = 0) and displaced only for the vector b i?1 i?1 H i = ? ? ? ? 100b i?1,x 010b i?1,y 001b i?1,z 000 1 ? ? ? ? . (3.5) 3.2 Vector parameters of the mechanism The vector parameters of a robot mechanism are determined in the following four steps: Step 1 – The robot mechanism is placed into the desired initial (reference) pose. The joint axes must be parallel to one of the axes of the reference coordinate frame x 0 , y 0 , z 0 attached to the robot base. In the reference pose all values of joint variables equal zero, ? i = 0andd i = 0, i = 1,2,...,n Step 2 – The centers of the joints i = 1,2,...,n are selected. The center of joint i can be anywhere along the corresponding joint axis. A local coordinate frame x i , y i , z i is placed into the joint center in such a way that its axes are parallel to the axes of the reference frame x 0 , y 0 , z 0 . The local coordinate frame x i , y i , z i is displaced together with the segment i 3.2 Vector parameters of the mechanism 27 Step 3 – The unit joint vector e i is allocated to each joint axis i = 1,2,...,n.Itis directed along one of the axes of the coordinate frame x i , y i , z i . In the direction of this vector the translational variable d i is measured, while the rotational variable ? i is assessed around the joint vector e i Step 4 – The segment vectors b i?1 are drawn between the origins of the x i , y i , z i frames, i = 1,2,...,n. The segment vector b n connects the origin of the x n , y n , z n frame with the robot end-point Sometimes an additional coordinate frame is positioned in the reference point of a gripper and denoted as x n+1 , y n+1 , z n+1 . There exists no degree of freedom between the frames x n , y n , z n and x n+1 , y n+1 , z n+1 , as both frames are attached to the same segment. The transformation between them is therefore constant. The approach to geometric modeling of robot mechanisms will be illustrated by an example of a robot mechanism with four degrees of freedom shown in Figure 3.4. The selected initial pose of the mechanism together with the marked positions of the joint centers is presented in Figure 3.5. The corresponding vector parameters and joint variables are gathered in Table 3.1. The rotational variables ? 1 , ? 2 and ? 4 are measured in the planes perpendicular to the joint axes e 1 , e 2 and e 4 , while the translational variable d i is measured along h 0 h 1 l 1 l 2 h 3 d 3 l 3 l 4 0 1 1 2 2 3 3 4 4 J 4 J 2 J 1 Fig. 3.4 Robot mechanism with four degrees of freedom 28 3 Geometric description of the robot mechanism y 0 x 0 z 0 x 4 x 3 x 2 x 1 y 4 y 3 y 2 y 1 z 4 z 3 z 2 z 1 l 2 y 0 x 0 z 0 e 2 b 4 e 3 b 2 e 4 e 1 b 1 b 3 b 0 Fig. 3.5 Positioning of the coordinate frames for the robot mechanism with four degrees of freedom the axis e 3 . Their values are zero when the robot mechanism is in its initial pose. In Figure 3.6 the robot manipulator is shown in a pose where all four variables are positive and nonzero. The variable ? 1 represents the angle between the initial and 3.2 Vector parameters of the mechanism 29 Table 3.1 Vector parameters and joint variables for the robot mechanism in Figure 3.5 i 1234 ? i ? 1 ? 2 0 ? 4 d i 00d 3 0 i 1234 0100 e i 0010 1001 i 123 4 5 000 00 b i?1 0 l 1 l 2 l 3 l 4 h 0 h 1 0 ?h 3 0 d 3 J 1 J 2 J 4 x 0 y 0 z 0 x 1 y 1 z 1 x 2 y 2 z 2 x 3 y 3 z 3 x 4 y 4 z 4 l 2 Fig. 3.6 Determining the rotational and translational variables for the robot mechanism with four degrees of freedom 30 3 Geometric description of the robot mechanism momentary y 1 axis, the variable ? 2 the angle between the initial and momentary z 2 axis, variable d 3 is the distance between the initial and actual position of the x 3 axis, while ? 4 represents the angle between the initial and momentary x 4 axis. The selected vector parameters of the robot mechanism are inserted into the ho- mogenous transformation matrices (3.2)–(3.4) 0 H 1 = ? ? ? ? c1 ?s10 0 s1 c100 001h 0 0001 ? ? ? ? , 1 H 2 = ? ? ? ? 10 0 0 0 c2 ?s2 l 1 0 s2 c2 h 1 00 0 1 ? ? ? ? , 2 H 3 = ? ? ? ? 100 0 010d 3 + l 2 001 0 000 1 ? ? ? ? , 3 H 4 = ? ? ? ? c4 ?s40 0 s4 c40 l 3 001?h 3 0001 ? ? ? ? . An additional homogenous matrix describes the position of the gripper reference point where the coordinate frame x 5 , y 5 , z 5 can be allocated 4 H 5 = ? ? ? ? 1000 010l 4 0010 0001 ? ? ? ? . This last matrix is constant as the frames x 4 , y 4 , z 4 and x 5 , y 5 , z 5 are parallel and displaced for the distance l 4 . Usually this additional frame is not even attached to the robot mechanism, as the position and orientation of the gripper can be described in the x 4 , y 4 , z 4 frame. When determining the initial (home) pose of the robot mechanism we must take care that the joint axes are parallel to one of the axes of the reference coordinate frame. The initial pose should be selected in such a way that it is simple and easy to examine, that it corresponds well to the anticipated robot tasks and that it mini- mizes the number of required mathematical operations included in the transforma- tion matrices. As another example we shall consider the SCARA robot manipulator whose geometric model was developed already in the previous chapter and is shown in Figure 2.10. The robot mechanism should be first positioned into the initial pose 3.2 Vector parameters of the mechanism 31 y 0 x 0 z 0 e 1 e 2 b 1 b 0 e 3 b 2 Fig. 3.7 The SCARA robot manipulator in the initial pose in such a way that the joint axes are parallel to one of the axes of the reference frame x 0 ,y 0 ,z 0 . In this way the two neighboring segments are either parallel or perpendic- ular. The translational joint must be in its initial position (d3 = 0).TheSCARA robot in the selected initial pose is shown in Figure 3.7. The joint coordinate frames x i ,y i ,z i are all parallel to the reference frame. Therefore, we shall draw only the reference frame and have the dots indicate the joint centers. In the centers of both rotational joints, unit vectors e 1 and e 2 are placed along the joint axes. The rotation around the e 1 vector is described by the variable ? 1 , while ? 2 represents the angle about the e 2 vector. Vector e 3 is placed along the translational axis of the third joint. Its translation variable is described by d 3 . The first joint is connected to the robot base by the vector b 0 . Vector b 1 con- nects the first and the second joint and vector b 2 the second and the third joint. The variables and vectors are gathered in the three tables (Table 3.2). In our case all e i vectors are parallel to the z 0 axis, the homogenous transfor- mation matrices are therefore written according equation (3.4). Similar matrices are obtained for both rotational joints. 32 3 Geometric description of the robot mechanism Table 3.2 Vector parameters and joint variables for the SCARA robot manipulator i 123 ? i ? 1 ? 2 0 d i 00d 3 i 123 000 e i 000 11-1 i 123 000 b i?1 0 l 2 l 3 l 1 00 0 H 1 = ? ? ? ? c1 ?s100 s1 c100 001l 1 0001 ? ? ? ? . 1 H 2 = ? ? ? ? c2 ?s200 s2 c20l 2 0010 0001 ? ? ? ? . For the translational joint, ? 3 = 0 must be inserted into equation (3.4), giving 2 H 3 = ? ? ? ? 100 0 010 l 3 001?d 3 000 1 ? ? ? ? . With postmultiplication of all three matrices the geometric model of the SCARA robot is obtained 0 H 3 = 0 H 1 1 H 2 2 H 3 = ? ? ? ? c12 ?s12 0 ?l 3 s12?l 2 s1 s12 c12 0 l 3 c12 + l 2 c1 001l 1 ?d 3 000 1 ? ? ? ? . We obtained the same result as in previous chapter, however in a much simpler and more clear way.