机器人手臂的运动解析

Behind every robotic movement An industrial robot is a general-purpose manipulator that consists of several rigid bodies, called links, con- nected in series by revolute or prismatic joints (Figure 1). One end of the chain is attached to a supporting base, while the other end is free and equipped with a tool to manipulate objects or perform assembly tasks. The mo- tion of the joints results in relative motion of the links. Mechanically, a robot is composed of an arm (or primary frame) and a wrist subassembly plus a tool and is designed to reach a workpiece located within its work volume. The work volume is a sphere of influence for a robot whose arm can deliver the wrist subassembly unit to any point within the sphere. The arm subassembly typically consists of three degree-of-freedom movements, which together place or position the wrist unit at the workpiece. The wrist subassembly unit usually consists ofthree rotary motions, often called pitch, yaw, and roll, and their combination orients the tool according to the configuration of the ob- ject to ease pickup. Hence for a six-joint robot, the arm subassembly is the positioning mechanism, while the wrist subassembly is the orientation mechanism. Many commercially available industrial robots are widely used in simple material handling, spot/arc weld- ing, and parts assembly, including the Unimate 2000B and PUMA 260/550/560 series robots by Unimation Inc.; the T3 by Cincinnati Milacron; the Versatran by Prab; the Asea robot; and the Sigma by Olivetti of Italy. These robots, which exhibit their characteristics in mo- tion and geometry, fall into one of four basic motion- defining categories (Figure 1): * Cartesian coordinate (three linear axes), * cylindrical coordinate (two linear and one rotary axes), * spherical or polar coordinate (one linear and two rotary axes), and is a series of complex geometric evaluations and equations that describe motion dynamics. Conformance to system goals a must also be considered. Robot Arm Kinematics, Dynamics, and Control C. S. George Lee, University of Michigan, Ann Arbor * revolute or articulated coordinate (three rotary axes). Most automated manufacturing tasks are done by special-purpose machines that are designed to perform prespecified functions in a manufacturing process. The inflexibility of these machines makes the computer-con- trolled manipulators more attractive and cost-effective in various manufacturing and assembly tasks. Today's in- dustrial robots, though controlled by mini-/micro- computer, are basically simple positional machines. They execute a given task by playing back prerecorded or preprogrammed sequences of motions that have been previously guided or taught by a user with a hand-held control/teach box. Moreover, because the robots are equipped with few or no external sensors (both contact and noncontact), they cannot obtain vital information about their working environment. More research needs to be directed towards improving the overall performance of the manipulator systems, and one way is through the study of robot arm kinematics, dynamics, and control. Robot arm kinematics deals with the geometry ofrobot arm motion with respect to a fixed-reference coordinate system as a function of time without regard to the forces/moments that cause the motion. Thus it deals with the spatial configuration of the robot as a function of time, in particular the relations between the joint-variable space and the position and orientation of a robot arm. The kinematics problem usually consists oftwo subprob- lems-the direct and inverse kinematics problems. The direct kinematics problem is to find the position and orientation of the end effector of a manipulator with respect to a reference coordinate system, given the joint- angle vector , = (D1, t92, t93, t94, tq5, 96)t of the robot arm. The inverse kinematics problem (or arm solution) is to calculate the joint-angle vector e given the position and 001 8-9162/82/1200-406200.75 ( 1982 IEEE62 COMPUTER orientation of the end effector with respect to the refer- ence coordinate system. Since the independent variables in a robot arm are the joint angles, and a task is generally stated in terms of the base or world coordinate system, the inverse kinematics solution is used more frequently in computer applications. The direct kinematics results in a 4 x 4 homogeneous transformation matrix that relates the spatial configuration between neighboring links. These homogeneous transformation matrices are useful in deriving the dynamic equations of robot arm motion. Robot arm dynamics deals with the mathematical for- mulations of the equations of robot arm motion. The dynamic equations of manipulator motion are a set of equations describing the dynamic behavior of the manip- ulator. Such equations of motion are useful for computer simulation of robot arm motion, the design of suitable control equations for a robot arm, and the evaluation of the kinematic design and structure of a robot arm. The purpose of robot arm control is to maintain the dynamic response of a computer-based manipulator in accordance with some prespecified system performance and goals. In general, the control problem consists of ob- taining suitable dynamic models ofthe physical robot arm for designing the controller and specifying corresponding control laws or strategies to achieve the desired system response and performance. This article details the com- puted torque technique in the joint-variable space and an adaptive control strategy. Robot arm kinematics Vector and matrix algebra* are used to develop a syste- matic and generalized approach to describing and repre- senting the location of robot arm links with respect to a fixed reference frame. Since the robot arm links can rotate and/or translate with respect to a reference coor- dinate frame, a body-attached coordinate frame is estab- lished at the joint for each link. The direct kinematics problem is then reduced to finding a transformation matrix that relates the body-attached coordinate frame to the reference coordinate frame. A 3 x 3 rotation matrix 'Vectors are represented in lowercase bold letters, and matrices are in up- percase bold. Figure 1. Various robot arm categories. December 1982 ,-ff CARTESIAN OR x-y-z SPHERICAL CYLINDRICAL REVOLUTE 63 can be used to describe the rotational operations of the body-attached frame with respect to the reference frame. The homogeneous coordinates are then used to represent position vectors in a 3-D space, and the rotation matrices are expanded to 4 x 4 homogeneous transformation ma- trices to include the translational operations of the body- attached coordinate frames. This matrix representation of a rigid mechanical link to describe the spatial geometry of a robot arm was first used by Denavit and Hertenberg. I The advantage of using the Denavit-Hartenberg represen- tation of linkages is its algorithmic universality in deriving the kinematic equation of a robot arm. Rotation matrices. A 3 x 3 rotation matrix can be de- fined as a transformation matrix that operates on a posi- tion vector in a 3-D Euclidean space and maps its coor- dinates expressed in a rotated coordinate system OUVW (body-attached frame) to a reference coordinate system OXYZ. Figure 2 shows two righthand rectangular coor- dinate systems, namely the OXYZ coordinate system with OX, OY, and OZ as its coordinate axes, and the OUVW coordinate system with OU, OV, and OW as its coor- dinate axes. The origins of both coordinate systems coin- cide at point 0. The OXYZ coordinate system is fixed in the 3-D space and is considered the reference frame. The OUVW coordinate frame is rotating with respect to the reference frame OXYZ. Physically we can consider the OUVWcoordinate system as a body-attached coordinate frame. That is, it is permanently and conveniently at- tached to the rigid body (e.g., an aircraft or a link of a robot arm) and moves together with it. Let (ixs jy, kz) and (iu, iv, kw) be the unit vectors along the coordinate axes of the OXYZ and OUVWsystems, respectively. A point p in the space can be represented by its coordinates with respect to both coordinate systems. For ease of discus- sion, assume that p is at rest and fixed with respect to the OUVW coordinate frame. Then point p can be repre- sented by its coordinates with respect to the OUVWand OXYZ coordinate systems, respectively, as puvw =- (Pup p, pw) t and Pxyz = (PxPY PZ)f (1) pxyz and puvw represent the same point p in the space with reference to different coordinate systems. We want a 3 x 3 transformation matrix A that will transform the coordinates of pusw to the coordinates expressed with respect to the OXYZ coordinate system, after the OUVW coordinate system has been rotated. That is, PXYZ= A puvw (2) Note that the point PUVW has been rotated together with the OUVWcoordinate system. Recalling the definition of the component of a vector, we have Puvw = Puil + Pvjv + pwkw (3) andp, py, andpz represent the components of p along the OX, OY, and OZ axes, respectively, or the projections of p onto the respective axes. Thus using the definition of a scalar product and equation 3, we have Px= ix. P = ixiupu +ix jvPv+ ix kwpw py=jy-p=jy-iupu+jy-jvpv+jy-kwpw PZ=kz-p=kziupu+kzjpv+kzkwpw or expressed in matrix form [Px] xiIxju ix kw Pu Py = Jy'iu Jyiju jy*kw Pv Pz kziu kzjv kz.kw Pw and (4) (5) 'x, u 'x jv ix kw A= jyiu jy iv jy kw (6) Lkz iu kz jv kz kw Similarly, we can obtain the coordinates of puvw from the coordinates of pxyz Pvw =B Pxyz Pu Fu lx uAJy i ukkz Px Pv = Jv*ix JvuJy Jv* kz py (7) LPw kw*ix kw jy kw*kzj LPZJ Since dot products are commutative, we can see that from equations 6 and 7, B=A-= At (8) and BA=AtA=A -A=13 (9) The transformation is called an orthogonal transforma- tion, and since the vectors in the dot products are all unit vectors, it is also called an orthonormal transformation. The prime interest of developing the above transforma- tion matrix is to find the rotation matrices that represent rotations of the OUVWcoordinate system at each of the three principal axes in the reference coordinate system OXYZ. If the OUVWcoordinate system is rotated at an a angle about the OXaxis to arrive at a new location in the space, then point puUw having coordinates (pu. pv Pw)I with respect to the OUVW system will have different coor- dinates (Px py, pz) t with respect to the reference system OXYZ. The necessary transformation matrix Rxa, is called the rotation matrix about the OXaxis with a angle. Rx a can be derived from the above transformation matrix concept, that is: Figure 2. Coordinate systems for a rigid body. PxyZ = Rx, * Puuw COM PUTER z P y (10) 64 and Ix.iu ix-jv ix kw [ 0 0 1 Rx,a jy iu jy jv jy-kw = 0 cosa -sinla (11) kz*iu kzj*v kz kwj L0 sina cosaj (Note that ix= iu. ) Similarly the 3 x 3 rotation matrices for rotation about the O Yaxis with 'p angle and about the OZ axis with tQ angle are, respectively (Figure 3), cos'p 0 sinmp cos - sint O Ry"< 0 1 0 R;Rz = sind cost 0 (12) sin'p 0 cosj 0 0 1 Rx, Ry, and Rz,,o are called the basic rotation matrices. Other finite rotation matrices can be found from these matrices. Example. Find the resultant rotation matrix that represents a rotation of 'p angle about the OY axis fol- lowed by a rotation of i5 angle about the O Waxis follow- ed by a rotation of a angle about the OU axis. Solution R=Ry,s0Rw,6,Ru,a C(p 0 S(1 co -SWO 1 0 0 = 0 10IO SW C9 O 0 Cc, -Sa 0SpO Cfj E 0] L0 Sa cac CPC SpSa-C'pSOCa CsOSdSa+SpCa1 = fsoc0ca - C jSo -SPCtg S9Osdca + C0 (17) 0 0 OOs s The physical Cartesian coordinates of the vector are: =x =y z 5PX=S , Py=Ys Pz= , W=5-= 1 (18) Therefore the fourth diagonal element in the homo- geneous transformation matrix has the effect of globally reducing the coordinates if s> 1 and enlarging the coor- dinates if 0