The inversion of differential kinematics dates back to Whitney 59 under the name of resolved motion rate control. The adoption of the pseudoinverse of the Jacobian is due to Klein and Huang. 30 More on the properties of the pseudoinverse can be found in Boullion and Odell. 4 The use of null- space joint velocities for redundancy resolution was proposed in Liégeois, 33 and further refined in Maciejewski and Klein 36 and Yoshikawa 60 concerning the choice of objective functions. The reader is referred to Nakamura 39 for a complete treatment of redundant robots. The adoption of the damped least-squares inverse was independently presented in Nakamura and Hanafusa 40 and Wampler. 58 More about kinematic control in the neighborhood of kinematic singu- larities can be found in Chiaverini. 9 The technique for estimating the smallest singular value of the Jacobian is due to Maciejewski and Klein, 37 and its modification to include the second smallest singular value was achieved by Chiaverini. 10 The use of the damped least-squares inverse for redundant robots was presented in Egeland et al. 21 The user-defined accuracy strategy was proposed in Chiaverini et al. 12 and further refined in Chiaverini et al. 13 A review of the damped least-squares inverse kinematics with experiments on an industrial robot was recently presented. 16 Closed-loop inverse kinematics algorithms are discussed in Sciavicco and Siciliano. 51 The orig- inal Jacobian transpose inverse kinematics algorithm was proposed in Sciavicco and Siciliano; 49 the choice of suitable gains for achieving robustness to singularities was discussed in Chiacchio and Siciliano. 7 Singular value decomposition of the Jacobian transpose is due to Chiaverini et al. 14 Combining the Jacobian transpose solution with the pseudoinverse solution was proposed in Chiac- chio and Siciliano. 8 References on the augmented task space approach are Egeland, 20 Samson et al., 48 Sciavicco and Siciliano, 50 and Seraji. 52 The occurrence of artificial singularities was pointed out in Baillieul, 2 and their properties were studied in Chiacchio et al. 6 The task priority strategy was originally proposed in Nakamura et al. 41 and has recently been refined in Chiaverini 11 concerning robustness to artificial singularities. The use of the Jacobian transpose for the constraint task was presented in Chiaverini et al. 15 and Siciliano. 53 The expression of the end-effector orientation error based on an angle/axis description of orientation is due to Luh et al. 35 and its properties were studied in Lin. 34 The use of a quaternion-based orientation error is due to Yuan. 61 More about the possible definitions of the orientation error can be found in Caccavale et al. 5 References 1. Angeles, J., Spatial Kinematic Chains: Analysis, Synthesis, Optimization, Springer-Verlag, Berlin, 1982. 2. Baillieul, J., Kinematic programming alternatives for redundant manipulators, in Proc. 1985 IEEE Int. Conf. Robotics and Automation, St. Louis, MO, 1985, 722. 3. Bottema, O. and Roth, B., Theoretical Kinematics, North Holland, Amsterdam, 1979. 4. Boullion, T. L. and Odell, P. L., Generalized Inverse Matrices, Wiley, New York, 1971. 5. Caccavale, F., Natale, C., Siciliano, B., and Villani, L., Resolved-acceleration control of robot manipulators: A critical review with experiments, Robotica, 16, 565, 1998. 6. Chiacchio, P., Chiaverini, S., Sciavicco, L., and Siciliano, B., Closed-loop inverse kinematics schemes for constrained redundant manipulators with task space augmentation and task priority strategy, Int. J. Robotics Res., 10, 410, 1991. 7. Chiacchio, P., and Siciliano, B., Achieving singularity robustness: An inverse kinematic solution algorithm for robot control, in Robot Control: Theory and Applications, IEE Control Engineering Series 36, Warwick, K. and Pugh, A., Eds., Peter Peregrinus, Herts, U.K., 149, 1988. 8. Chiacchio, P. and Siciliano, B., A closed-loop Jacobian transpose scheme for solving the inverse kinematics of nonredundant and redundant robot wrists, J. Robotic Systems, 6, 601, 1989. 9. Chiaverini, S., Inverse differential kinematics of robotic manipulators at singular and near-singular configurations, in Prepr. 1992 IEEE Int. Conf. Robotics Automation — Tutorial on Redundancy: Performance Indices, Singularities Avoidance, and Algorithmic Implementations, Nice, 1992. 10. Chiaverini, S., Estimate of the two smallest singular values of the Jacobian matrix: Application to damped least-squares inverse kinematics, J. Robotic Systems, 10, 991, 1993. 8596Ch19Frame Page 484 Tuesday, November 6, 2001 9:56 PM © 2002 by CRC Press LLC 11. Chiaverini, S., Singularity-robust task-priority redundancy resolution for real-time kinematic con- trol of robot manipulators, IEEE Trans. Robotics Automation, 13, 398, 1997. 12. Chiaverini, S., Egeland, O., and Kanestrøm, R. K., Achieving user-defined accuracy with damped least-squares inverse kinematics, in Proc. 5th Int. Conf. Advanced Robotics, Pisa, I, 672, 1991. 13. Chiaverini, S., Egeland, O., Sagli, J. R., and Siciliano, B., User-defined accuracy in the augmented task space approach for redundant manipulators, Lab. Robotics Automation, 4, 59, 1992. 14. Chiaverini, S., Sciavicco, L., and Siciliano, B., Control of robotic systems through singularities, in Advanced Robot Control, Lecture Notes in Control and Information Science 162, Canudas de Wit, C., Ed., Springer-Verlag, Berlin, 285, 1991. 15. Chiaverini, S., Siciliano, B., and Egeland, O., Redundancy resolution for the human-arm-like manipulator, Robotics Autonomous Systems, 8, 239, 1991. 16. Chiaverini, S., Siciliano, B., and Egeland, O., Review of the damped least-squares inverse kine- matics with experiments on an industrial robot manipulator, IEEE Trans. Control Systems Tech- nology, 2, 123, 1994. 17. Craig, J. J., Introduction to Robotics: Mechanics and Control, 2nd ed., Addison-Wesley, Reading, MA, 1989. 18. Denavit, J. and Hartenberg, R. S., A kinematic notation for lower-pair mechanisms based on matrices, ASME J. Appl. Mech., 22, 215, 1955. 19. Dombre, E. and Khalil, W., Modélisation et Commande des Robots, Hermès, Paris, 1988. 20. Egeland, O., Task-space tracking with redundant manipulators, IEEE J. Robotics Automation, 3, 471, 1987. 21. Egeland, O., Sagli, J. R., Spangelo, I., and Chiaverini, S., A damped least-squares solution to redundancy resolution, in Proc. 1991 IEEE Int. Conf. Robotics Automation, Sacramento, CA, 945, 1991. 22. Featherstone, R., Position and velocity transformations between robot end-effector coordinates and joint angles, Int. J. Robotics Res., 2(2), 35, 1983. 23. Goldenberg, A. A., Benhabib, B., and Fenton, R. G., A complete generalized solution to the inverse kinematics of robots, IEEE J. Robotics Automation, 1, 14, 1985. 24. Hollerbach, J. M., Wrist-partitioned inverse kinematic accelerations and manipulator dynamics, Int J. Robotics Res., 2(4), 61, 1983. 25. Hunt, K. H., Kinematic Geometry of Mechanisms, Clarendon, Oxford, U.K., 1978. 26. Khalil, W., A system for generating the symbolic models of robots, in Postpr. 4th IFAC Symp. Robot Control, Capri, 416, 1994. 27. Khalil, W. and Bennis, F., Automatic generation of the inverse geometric model of robots, Robotics and Autonomous Systems, 7, 1, 1991. 28. Khalil, W. and Kleinfinger, J. F., A new geometric notation for open and closed-loop robots, in Proc. 1986 IEEE Int. Conf. Robotics Automation, San Francisco, CA, 1174, 1986. 29. Khatib, O., A unified approach for motion and force control of robot manipulators: The operational space formulation, IEEE J. Robotics Automation, 3, 43, 1987. 30. Klein, C. A. and Huang, C. H., Review of pseudoinverse control for use with kinematically redundant manipulators, IEEE Trans. Systems, Man, Cybernetics, 13, 245, 1983. 31. Klema, V. C. and Laub, A. J., The singular value decomposition: Its computation and some applications, IEEE Trans. Automatic Control, 25, 164, 1980. 32. Lee, H. Y. and Liang, C. G., Displacement analysis of the general 7-link 7R mechanism, Mecha- nism Machine Theory, 23, 219, 1988. 33. Liégeois, A., Automatic supervisory control of the configuration and behavior of multibody mechanisms, IEEE Trans. Systems, Man, Cybernetics, 7, 868, 1977. 34. Lin, S. K., Singularity of a nonlinear feedback control scheme for robots, IEEE Trans. Systems, Man, Cybernetics, 19, 134, 1989. 35. Luh, J. Y. S., Walker, M. W., and Paul, R. P. C., Resolved-acceleration control of mechanical manipulators, IEEE Trans. Automatic Control, 25, 468, 1980. 36. Maciejewski, A. A. and Klein, C. A., Obstacle avoidance for kinematically redundant manipulators in dynamically varying environments, Int. J. Robotics Res., 4(3), 109, 1985. 8596Ch19Frame Page 485 Tuesday, November 6, 2001 9:56 PM © 2002 by CRC Press LLC 37. Maciejewski, A. A. and Klein, C. A., Numerical filtering for the operation of robotic manipulators through kinematically singular configurations, J. Robotic Systems, 5, 527, 1988. 38. McCarthy, J. M., An Introduction to Theoretical Kinematics, MIT Press, Cambridge, MA, 1990. 39. Nakamura, Y., Advanced Robotics: Redundancy and Optimization, Addison-Wesley, Reading, MA, 1991. 40. Nakamura, Y. and Hanafusa, H., Inverse kinematic solutions with singularity robustness for robot manipulator control, ASME J. Dynamic Systems, Measurement, Control, 108, 163, 1986. 41. Nakamura, Y., Hanafusa, H., and Yoshikawa, T., Task-priority based redundancy control of robot manipulators, Int. J. Robotics Res., 6(2), 3, 1987. 42. Orin, D. E. and Schrader, W. W., Efficient computation of the Jacobian for robot manipulators, Int. J. Robotics Res., 3(4), 66, 1984. 43. Paul, R. P., Robot Manipulators: Mathematics, Programming, and Control, MIT Press, Cambridge, MA, 1981. 44. Paul, R. P. and Zhang, H., Computationally efficient kinematics for manipulators with spherical wrists based on the homogeneous transformation representation, Int. J. Robotics Res., 5(2), 32, 1986. 45. Pieper, D. L., The Kinematics of Manipulators under Computer Control, memo. AIM 72, Stanford Artificial Intelligence Laboratory, 1968. 46. Raghavan, M. and Roth, B., Inverse kinematics of the general 6R manipulator and related linkages, ASME J. Mechanical Design, 115, 502, 1990. 47. Renaud, M., Calcul de la matrice jacobienne necessaire à la commande coordonnee d’un manip- ulateur, Mechanism and Machine Theory, 15, 81, 1980. 48. Samson, C., Le Borgne, M., and Espiau, B., Robot Control: The Task Function Approach, Oxford Engineering Science Series 22, Clarendon, Oxford, U.K., 1991. 49. Sciavicco, L. and Siciliano, B., Coordinate transformation: A solution algorithm for one class of robots, IEEE Trans. Systems, Man, Cybernetics, 16, 550, 1986. 50. Sciavicco, L. and Siciliano, B., A solution algorithm to the inverse kinematic problem for redundant manipulators, IEEE J. Robotics Automation, 4, 403, 1988. 51. Sciavicco, L. and Siciliano, B., Modelling and Control of Robot Manipulators, 2nd ed., Springer, London, 2000. 52. Seraji, H., Configuration control of redundant manipulators: Theory and implementation, IEEE Trans. Robotics Automation, 5, 472, 1989. 53. Siciliano, B., Solving manipulator redundancy with the augmented task space method using the constraint Jacobian transpose, in Prepr. 1992 IEEE Int. Conf. Robotics and Automation — Tutorial on Redundancy: Performance Indices, Singularities Avoidance, and Algorithmic Implementations, Nice, 1992. 54. Spong, M. W. and Vidyasagar, M., Robot Dynamics and Control, Wiley, New York, 1989. 55. Tsai, L. W. and Morgan, A. P., Solving the kinematics of the most general six- and five-degree- of-freedom manipulators by continuation methods, ASME J. Mechanisms, Transmission, Automa- tion Design, 107, 189, 1985. 56. Vukobratovi´c, M., Introduction to Robotics, Springer, Berlin, 1989. 57. Vukobratovi´c, M. and Kir´canski, M., Kinematics and Trajectory Synthesis of Manipulation Robots, Scientific Fundamentals of Robotics 3, Springer, Berlin, 1986. 58. Wampler, C. W., Manipulator inverse kinematic solutions based on vector formulations and damped least-squares methods, IEEE Trans. Systems, Man, Cybernetics, 16, 93, 1986. 59. Whitney, D. E., Resolved motion rate control of manipulators and human prostheses, IEEE Trans. Man–Machine Systems, 10, 47, 1969. 60. Yoshikawa, T., Manipulability of robotic mechanisms, Int. J. Robotics Res., 4(2), 3, 1985. 61. Yuan, J. S C., Closed-loop manipulator control using quaternion feedback, IEEE J. Robotics Automation, 4, 434, 1988. 8596Ch19Frame Page 486 Tuesday, November 6, 2001 9:56 PM © 2002 by CRC Press LLC 20 Robot Dynamics 20.1 Fundamentals of Robot Dynamic Modeling Basic Ideas • Robot Geometry • Equations of Dynamics 20.2 Recursive Formulation of Robot Dynamics Velocities and Accelerations of Robot Links • Elimination of Reactions — Minimization of Dynamic Model Form • Calculation of Direct and Inverse Dynamics 20.3 Complete Model of Robot Dynamics Dynamic Model of a DC-Driven Robot • Generalized Form of the Dynamic Model 20.4 Some Applications of Computer-Aided Dynamics Dynamics and Robot Design • Dynamics in On-Line Control 20.5 Extension of Dynamic Modeling — Some Additional Dynamic Effects Robot Dynamics — Problems and Research • Dynamics of Robot in Constrained Motion • Robot in Contact with Dynamic Environment • Effects of Elastic Transmissions Appendix: Calculation of Transformation Matrices We start our discussion on robot dynamics from the standpoint that successful design and control of any system require appropriate knowledge of its behavior. This is certain, but we should discuss what is meant by “appropriate knowledge.” Let us consider a robot as an example of a technical system. Appropriate knowledge of its behavior may, but need not, include the mathematical model of its dynamics. In the earlier phases of robotics development, design was not based on the exact calculation of robot dynamics but followed experience from machine design. Control did not take into account many dynamic effects. Large approximations were made to reduce the problem to the well-known theory of automatic control. The undeveloped robot theory could not support a more exact approach. For a long time, the practice of robotics (design, manufacture, and implementation) grew independently of the theory that was too academic. However, this did not stop manufacturers from producing many successful robots. Presently, the need for complex, precise, and fast robots requires a close connection between theory and practice. Regarding the application of robot dynamics, the main breakthrough was made with the development of computer-aided methods for dynamic modeling. 1-3 Such methods allowed fast and user-friendly calculations of all relevant dynamic effects. In this way dynamic modeling and simulation became the essential tools in robot design. The other possibility for application of robot dynamics is the synthesis of the so-called dynamic control. In this subsection we first discuss the principles of dynamic modeling, the approach to the description of dynamics, and the derivation of the mathematical model. Then, special attention is Miomir Vukobratovi´c Mihajlo Pupin Institute Viljko Potkonjak University of Belgrade 8596Ch20Frame Page 487 Tuesday, November 6, 2001 9:54 PM © 2002 by CRC Press LLC paid to computer-based methods. To complete the information on robot dynamics, the mechanism model should be supplemented with the driving system model. After that, we briefly describe the application of the dynamic model. One of the promising directions is the development of CAD systems for robots. The other is dynamic control. Some extension of robot dynamics is made and discuss different effects that were not included in the initial model, trying to locate those of main importance (contact problems, elastic deformations, friction, impact, etc.). 20.1 Fundamentals of Robot Dynamic Modeling 20.1.1 Basic Ideas From the notion dynamic modeling we understand the system of differential equations that describes robotic dynamic behavior. We expect the reader to possess the knowledge necessary to understand the derivation of the model. However, we will try to give enough information at an adequate level of presentation to allow readers to follow the text easily. Here we consider a manipulation robot as an open and simple kinematic chain (as shown in Figure 20.1) consisting of n rigid bodies (robot links) interconnected by means of n one-degree- of-freedom (one-DOF) joints. A joint allows one relative rotation (revolute joint) or one relative translation (linear joint). Because the complete chain has n DOFs, its dynamics can be described by means of n differential equations of motion. They are second-order equations. This set is called the dynamic model . Several approaches have been used to describe system dynamics: laws of linear momentum and angular momentum. 1-4 Lagrange’s equations, 5,6 and Gauss’ principle. 7,8 All approaches lead to the same dynamic model but the model formation procedure is different. Here, we use the laws of linear momentum and angular momentum. This approach is often called Newton–Euler equations. In the authors’ opinion it is the most appropriate for the majority of readers. Let us introduce one position coordinate for each joint, angle in the revolute joint, and longitudinal displacement in the linear joint. This set of coordinates uniquely describes the position of the chain. We usually call this set the internal coordinates (or joint coordinates, or generalized coordinates). If the coordinate for joint S j is marked by q j , then the complete position vector is (20.1) 20.1.2 Robot Geometry At this point we have to decide the mathematical presentation of robot geometry and kinematics. Up to now, two ways have been defined. One is based on the Rodrigues’ formulae of finite rotation and the other uses the Denavit–Hartenberg parameters. The latter method is more widely accepted FIGURE 20.1 Robot as a simple and open chain. 1 2 n S n 1 S 2 S qqq q n T = [] 12 L 8596Ch20Frame Page 488 Tuesday, November 6, 2001 9:54 PM © 2002 by CRC Press LLC because it allows simpler expression of transformation matrices and probably faster calculation of robot kinematics. However, if the intention is to discuss dynamics, it should be stressed that the first method is more appropriate. It is more general and follows the rigid body motion approach used in all standard textbooks on mechanics. For these reasons, we utilize the method based on Rodrigues’ formulae. Figure 20.2 shows one link of the robot chain, the j -th one. Joint S j is shown as revolute and S j+ 1 as linear. To define the link geometry, it is necessary to describe the position and the orientation of the joints with respect to the mass center (MC). The motion direction in each joint is defined by means of an axis, that is, by a unit vector. It can describe rotation or translation, depending on the type of joint. Thus corresponds to joint S j and to . The relative position of MC with respect to the joints is defined by means of vectors and as shown in the Figure 20.2. MC is marked by C j . During robot motion, positions of all links, and accordingly, geometry vectors expressed in the immobile external frame, change. However, if geometry vectors are considered relative to the corresponding link, they become constant and represent the property of the link itself. To express these constant values, we introduce a Cartesian system fixed to the link and with the origin in the MC (link-fixed frame). The axes are x j , y j , and z j . The system may be oriented in an arbitrary way but is most suitable if its axes coincide with the so-called principal axes of inertia. Consider now vector . It can be expressed by means of three constant projections onto the axes of the frame fixed to the link j: , and . For this triple we introduce the notation (20.2) The tilde “~” above the letter indicates that the vector is expressed in the link-fixed frame. Notation (without tilde) denotes three projections onto the axes of an external immobile frame. If the same is applied to vectors and, two constant triples are obtained (20.3) (20.4) Vector is constant if expressed in the frame fixed to link j and a suitable notation is needed for these projections. Notation indicates that the vector is considered relative to link j + 1 (analogously to relation (20.2)). Hence, a new notation is introduced to indicate the projections onto link j : (20.5) FIGURE 20.2 Geometry of a link. S j S j+1 e r j , j j j r j , j+1 e j+1 C z x y j j j j r e j r e j+1 S j+1 r r jj, r r jj, +1 r e j ee j x j y jj , e j z j ˜ ,, r eeee jj x j y j z jjj = () r e j r r jj, r r jj, , +1 ˜ ,, ,,,, r rrrr jj jj x jj y jj z jjj = () ˜ ,, ,,,, r rrrr jj jj x jj y jj z jjj ++++ = () 1111 r e j+1 ˜ r e j+1 r eeee j j x j y j z jjj ~ ,, + +++ = () 1 111 8596Ch20Frame Page 489 Tuesday, November 6, 2001 9:54 PM © 2002 by CRC Press LLC Generally, for any vector having index j , the tilde “~” above the letter ( ) indicates the projections onto frame j , while the tilde under the letter ( ) indicates the projections onto the preceding frame, j – 1. Notation without the tilde ( ) indicates projections onto the external immobile frame. Now, it should be stated that four vectors, define the geometry of link j . To define the geometry of the complete chain, one has to prescribe these four vectors for all links. It is still necessary to distinguish between the revolute and linear joints. For this purpose we introduce the indicator s j for each joint: (20.6) Now, it is possible to define the joint coordinates more precisely. We consider the revolute joints first. If S j is a revolute joint, then coordinate q j represents the angle of rotation measured from the extended position. The exact definition is shown in Figure 20.3. The angle lies in a plane perpen- dicular to axis . The negative projection of defines the extended position ( q j = 0) and the angle is measured to the projection of . Figure 20.3a shows the extended position and Figure 20.3b the rotated position. Suppose now that joint S j is linear. Coordinate q j defines the length of translation along and its precise definition requires previous introduction of the zero position. This zero-point can be adopted anywhere on the axis of translation. It is marked by in Figure 20.4. Once adopted, this point determines the vector . Coordinate q j is defined as the displacement with the proper sign with respect to (see Figure 20.4). It is necessary to introduce an additional vector. It follows from Figure 20.4. that (20.7) or, more generally, (20.8) For a linear joint ( s j = 1) relation (20.8) becomes (20.7) and for revolute joints ( s j = 0) it reduces to . Thus, expression (20.8), i.e., vector , may replace for any type of joint. FIGURE 20.3 Definition of a coordinate in a revolute joint. (a) (b) (extended joint) C C j j , j j j -1 , j e j j -1 j -1 q j = 0 C C j j , j j -1 , j e j j -1 q j r r r r (a) (b) (extended joint) C C j j , j j j -1 , j e j j -1 j -1 q j = 0 C C j j , j j -1 , j e j j -1 q j r r r r (a) (b) (extended joint) C C j j , j j j -1 , j e j j -1 j -1 q j = 0 C C j j , j j -1 , j e j j -1 q j r r r r r a j ˜ r a j r a j ~ r a j ˜ ,, ˜ , ˜ , ~ ,, r r rr ee r r j j jj jj + + 1 1 s S S j j j =      0, if is a revolute joint 1, if is a linear joint. r e j r r jj−1, r r jj, r e j ′ S j r r jj−1, ′′′ SS jj r e j ′ = ′ SC r jj jj, . ′ = ′′′ + ′′ =+rSSSCqer jj j j j j j j jj,, rr ′ =+ rr r rrsqe jj jj j j j,, . ′ = rr rr jj jj,, ′ r r jj, r r j j , 8596Ch20Frame Page 490 Tuesday, November 6, 2001 9:54 PM © 2002 by CRC Press LLC After introducing different frames (link-fixed and external immobile) the question arises about the possibility of transforming a vector from one frame to another. Let us consider a vector . For the transition between the frames, transformation matrices are applied: From the j -th link-fixed frame to the external one dim A j = 3 × 3 (20.9) and in the opposite direction (20.10) From the j -th link-fixed frame to the ( j – 1)-th one (20.11) and in the opposite direction (20.12) It could be noted that the matrices are orthogonal and thus the inverse equals the transpose. A detailed explanation of how to calculate the transformation matrices is given in this chapter’s appendix. 20.1.3 Equations of Dynamics We start by considering dynamics from one of the links, the j -th one. For this purpose we fictively interrupt the chain in joints S j and . The disconnection in joint S j is shown in Figure 20.5a. The FIGURE 20.4 Definition of coordinate in a linear joint. rr ' CC S '' r q e S ' C rr j j , j j , j j j j j j -1 j j -1 , j r a j r r aAa jjj = ˜ , ˜ r rr aAaAa jjjj T j == −1 r r aAa j jjj ~ , ˜ = −1 ˜ . , ~ , ~ , ~ r rrr aAaAaAa jjj j jj j jj T j === −− − −11 1 1 S j+1 8596Ch20Frame Page 491 Tuesday, November 6, 2001 9:54 PM © 2002 by CRC Press LLC mutual influence of the two links is expressed in terms of a force and a couple. denotes the force and denotes the moment of the couple. They act in the forward direction (from link j – 1 to link j ) while and act in the backward direction (from j to j – 1). Figure 20.5b shows the link j together with all forces and couples acting upon it. We use the law of linear momentum (Newton’s law) to describe the motion of MC: (20.13) where m j is the link mass, is MC acceleration, and is the acceleration due to gravity (9.81 m/s 2 ). The law expresses the equilibrium of the inertial and real forces. Now, we discuss the rotation of the link about its MC. It can be described by the law of angular momentum (Euler’s equations): (20.14) where and are the angular acceleration and angular velocity. The tilde “~” indicates that the vectors are expressed in the frame fixed to the link j . is the tensor of inertia calculated for the axes of the link-fixed frame. In a general case, the tensor has the form (20.15) However, if the frame axes are placed to coincide with the principal inertial axes, then the tensor takes the diagonal form (20.16) where is the inertial moment with respect to axis x j and analogously holds for and . FIGURE 20.5 Extraction of one link from the chain. M S S F - M S - F M S F - M S - F S r C - F S F S r M S - M S g m (a) (b) j , j+1 j j j+1 j+1 j j , j j -1 j j j j j j r F S j r M S j − r F S j − r M S j mw mg F F jj j S S jj r r rr =+− +1 r w j r g ˜ ˜˜ ˜ ˜ ˜ ˜ ˜ ˜ ~ ,, ~ JJMMrFrF jj j j j S S jj S jj S j j j j rr r r r r r r r εω ω+× () =− − ′ ×+ × + + + 1 1 1 ˜ r ε j ˜ r ω j ˜ J j ˜ J JJJ JJJ JJJ j xx xy xz yx yy yz zx zy zz jjj jjj jjj =             ˜ J J J J j x y z j j j =             J x j J y j J z j 8596Ch20Frame Page 492 Tuesday, November 6, 2001 9:54 PM © 2002 by CRC Press LLC It should be mentioned that Equation (20.13) is written in the external frame, while (20.14) is written in the link-fixed frame. Equation (20.13) could be simply rewritten and expressed in the link-fixed frame, while (20.14) cannot change the frame so easily. It would have to be multiplied by transformation matrix A j . We now discuss force and couple transmitted through the revolute joint S j (Figure 20.6). First, we consider the “pure” reactions, force and couple, that follow from the mechanical connec- tion of the two links. Because the joint permits one rotation (about ), the reaction force (from link j – 1 to link j ) may be of arbitrary direction, while the reaction couple is perpendicular to axis (that is, ). In the revolute joint a driving torque acting about axis exists. In the vector form, the drive is . Thus, the total force and couple transmitted through the joint are (20.17) Consider now a linear joint S j (Figure 20.7). The total force transmitted through the joint has two components, the reaction force , and the driving force . The reaction is perpendicular to the axis (that is, ). The total couple consists of reaction only and can be of arbitrary direction. Thus, it holds that (20.18) Equations (20.17) and (20.18) can be written in a unique way that fits both the revolute and the linear joints: (20.19) FIGURE 20.6 The total force and the total couple in a revolute joint. j j -1 e j M S F S R F M R τ j j j j j = e j () r F S j () r M S j r F S j r M S j r e j r F R j r M R j r e j r r Me Rj j ⊥τ j r e j τ jj e r rr FF SR jj = rr r MM e SRjj jj =+τ. r F S j r F R j τ jj e r r e j r r Fe Rj j ⊥ r M S j rr r FF e SRjj jj =+τ rr MM SR jj = rr r FFse SRjjj jj =+τ rr r MM se SR jjj jj =+−().1 τ 8596Ch20Frame Page 493 Tuesday, November 6, 2001 9:54 PM © 2002 by CRC Press LLC [...]... expression r r r ˙ ω j = ω j −1 + q j (1 − s j )e j © 2002 by CRC Press LLC (20.23) 8596Ch20Frame Page 496 Tuesday, November 6, 2001 9:54 PM showing the recursive character of angular velocity Making the derivative of (20.23) one obtains the expression for acceleration: r r r r r ˙˙ ˙ ε j = ε j −1 + (q j e j + q j (ω j × e j ))(1 − s j ) (20.24) r For the MC position vector, rc j, the recursive expression... new link is added to the chain (e.g., link j) Transformation matrix Aj is calculated to allow expressing the geometry vectors in the external frame Now, applying the recursive expressions (20.40) to (20.43), matrices Λ j, Γj, Ξ, j and ∆ j are found starting from Λ j–1, Γj–1, Ξ j–1, and ∆ j–1 © 2002 by CRC Press LLC 8596Ch20Frame Page 498 Tuesday, November 6, 2001 9:54 PM , Sk k rk,k Ck r k , k+1 S k+1... the reaction, thus giving the scalar expression for the driving torque r r τ j = MS j e j = r n ˜ ∑  A  J˜ ε k k k k= j r r r r r r ˜ ˜ ˜ + ω k × Jk ω k  + rk( j ) × mk (wk − g ) e j   ( ) (20.51) where expression (20.45) multiplied by Ak, and expressions (20.46) and (20.44) are substituted To summarize, in the case of a linear joint, the drive is expressed in form (20.50) and for a revolute... and the principal references For historical reasons, we discuss design issues first and then control 20.4.1 Dynamics and Robot Design Computer-aided kinematics enables the transformation of robot coordinates from internal (joints) to external (end-effector) and vice versa Computer procedures to calculate dynamics solve the © 2002 by CRC Press LLC 8596Ch20Frame Page 506 Tuesday, November 6, 2001 9:54... such a procedure represents a very useful tool in the robot design process A designer can quickly check a large number of different configurations He or she can vary parameters to see their influence on some dynamic characteristics This is of considerable help in fast and successful design The next step is made if the limits that we impose in the design (e.g., maximal elastic deformation) are given to the... described by mechanical Equation (20.62) and electrical Equation (20.63) If we neglect friction and inductivity, then these two equations could be combined to give the form (20.70) This model is expressed in terms of motor shaft angle θj Mj that appears in the model represents motor output torque The model holds for each motor, j = 1, …, n Dynamics of robot links are described by model (20.22) and expressed... 0       j  j M  M  M hn   Hn  τ n        (20.60) where © 2002 by CRC Press LLC 8596Ch20Frame Page 501 Tuesday, November 6, 2001 9:54 PM are of dimensions: τ (n × 1), H (n × n), h (n × 1) Thus, we have come to the previously stated form of the dynamic model, that is, Equation (20.22) Expressions (20.54) and (20.58) offer the possibility of recursive calculation of matrices H... be solved This result enables calculation of the mechanical stresses and elastic deformation (bending and torsion of links) For such calculation, some supplementary dynamic blocks (approximate or exact) would be needed to accomplish the model explained here These possibilities show that robot dynamics can be successfully applied in the design of robot mechanical structure (geometry, dimensions, cross... the accuracy of motion However, good controllability still makes DC motors very popular The dynamics of a DC motor that drives a robot joint Sj is described by the following relations expressing the mechanical and electrical equilibrium: ˙ θ J j ˙˙ j = CM j i j − Bj θ j − M j u j = Rj i j + L j di j dt (20.62) ˙ + CE j θ j (20.63) where θj is the angle of the motor shaft rotation, ij is the armature... research, however, saw redundancy as a possibility to improve robot dynamic performance.47,48 The biomechanical approach to the solution of redundancy of a humanoid robot arm was proposed in Potknojak et al.49 Special kind of redundancy appears in so-called systems with variable geometry.50,51 The mechanism is designed to have an augmented number of DOFs (more than the kinematics of the task requires) However, . 9:54 PM © 2002 by CRC Press LLC It should be mentioned that Equation (20 .13) is written in the external frame, while (20.14) is written in the link-fixed frame. Equation (20 .13) could be simply. multibody mechanisms, IEEE Trans. Systems, Man, Cybernetics, 7, 868, 1977. 34. Lin, S. K., Singularity of a nonlinear feedback control scheme for robots, IEEE Trans. Systems, Man, Cybernetics, 19, 134 , 1989. 35 PM © 2002 by CRC Press LLC 37. Maciejewski, A. A. and Klein, C. A., Numerical filtering for the operation of robotic manipulators through kinematically singular configurations, J. Robotic Systems,