# Simulation of Manipulator Dynamics and Adjusting to Functional Movements

• Miomir Vukobratović
• Veljko Potkonjak
Part of the Communications and Control Engineering Series book series (CCE, volume 1)

## Abstract

The methods for computer-aided formation and solution of the mathematical models of active mechanisms (described in Chapter II and in [1–15]) have served as a basis for the development of the algorithm for the simulation of manipulator dynamics. We shall first indicate the main ideas and problems, and then describe the general algorithm for the simulation of dynamics [16, 17]. Later, we shall adjust this algorithm to some classes of practical tasks, i.e., to functional movements, particularly because of the efficiency of handling the algorithm. At the same time we shall analyse the need for a certain number of degrees of freedom of the manipulator as well as their use in the various categories task [18]. Something more should be said about the number of manipulator d.o.f. and about the number of d.o.f. necessary for performing a particular manipulation task. At first, we note the difference between the number of d.o.f. of the whole manipulator (considered as a dynamical system), and the number of d.o.f. of the gripper (considered as the last rigid body in the chain). These two numbers need not be equal. The number of manipulator d.o.f. depends on the number of joints and the number of d.o.f. in each joint. For instance, if a manipulator represents an open chain without branching and consists of n segments and n one-d.o.f. joints, then it has n degrees of freedom.

### Keywords

Torque Sine Lution Compressibility Dinates

## Preview

### References

1. [1]
Stepanenko Yu., “Method of the Analysis of Spatial Articulated Mechanisms” (in Russian), Mekhanika mashin, Vol. 23, Moscow, 1970.Google Scholar
2. [2]
Juridic D., Vukobratovid M., “Mathematical Modelling of a Biped Walking System”, ASME Publ. 72-WA/BHF-13.Google Scholar
3. [3]
Vukobratovid M., Stepanenko Yu., “Mathematical Models of General Anthropomorphic Systems”, Math. Biosciences, Vol. 17, pp. 191–242, 1973.
4. [4]
Stepanenko Yu., Vukobratovid M.,Dynamics of Articulated Open-Chain Active Mechanisms, Math. Biosciences, Vol. 28, No 1/2, 1976.Google Scholar
5. [5]
Vukobratovid M., “Computer Method for Mathematical Modelling of Active Kinematic Chains via Generalized Coordinates”, Journal of IFToMM Mechanisms and Machine Theory, Vol. 13, No 1, 1978.Google Scholar
6. [6]
Vukobratovid M., Legged Locomotion Robots and Anthropomorphic Mechanisms, Monograph, Mihailo Pupin Institute, 1975, P.O.B. 15, Beograd, Yugoslavia.Google Scholar
7. [7]
Orin D., Vukobratovid M., R.B. Mc Ghee., G.Hartoch, G.Hartoch, “Kinematic and Kinetic Analysis of Open-Chain Linkages Utilizing Newton-Euler Methods”, Math. Biosciences, Vol. 42, pp. 107–130, 1978.Google Scholar
8. [8]
Luh J.Y.S., Walker M.W., Paul R.P.C., “On Line Computational Scheme for Mechanical Manipulators”, Trans. of ASME, Journal of Dynamic Systems, Measurement and Control, June 1980, Vol. 102/09.Google Scholar
9. [9]
Vukobratovid M., Potkonjak V., “Contribution to the Forming of Computer Methods for Automatic Modelling of Active Spatial Mechanisms Motion”, PART1: “Method of Basic Theoremes of Mechanics”, Journal of Mechanisms and Machine Theory, Vol. 14, No 3, 1979.Google Scholar
10. [10]
Vukobratovid M., Potkonjak V., “Contribution to the Computer Methods for Generation of Active Mechanisms via Basic Theoremes of Mechanics”, Teknitcheskaya kibernetika ANUSSR, No 2, 1979.Google Scholar
11. [11]
Vukobratovid M., “Computer Method for Mathematical Modelling of Active Kinematic Chains via Euler’s Angles”, Journal of IFT0MM Mechanisms and Machine Theory, Vol. 13, No 1, 1978.Google Scholar
12. [12]
Vukobratovid M., Potkonjak V., “Contribution to Automatic Forming of Active Chain Models via Lagrangian Form”, Journal of Applied Mechanics, No 1, 1979.Google Scholar
13. [13]
Hollerbach J.M., “A Recursive Formulation of Lagrangian Manipulator Dynamics”, Proc. JACC, June, 1980, San Francisco.Google Scholar
14. [14]
Potkonjak V., Vukobratovid M., “Two Methods for Computer Forming of Dynamic Equations of Active Mechanisms”, Journal of Mechanism and Machine Theory, Vol. 14, No 3, 1979.Google Scholar
15. [15]
Paul R.C., Modeling, Trajectory Calculation and Servoing of a Computer Controlled Arm, A.1 Memo 177, Sept. 1972, Stanford Artificial Intelligence Laboratory, Stanford University, Sept. 1972Google Scholar
16. [16]
Vukobratovid M., Potkonjak V., Hristid D., “Dynamic Method for the Evaluation and Choice of Industrial Manipulators”, Proc 9th International Symposium on Industrial Robots, Washington, 1979.Google Scholar
17. [17]
Vukobratovid M., Potkonjak V., “Contribution to Computer-Aided Design of Industrial Manipulators Using Their Dynamic Properties”, Journal of IFToMM, Mechanisms and Machine Theory, Vol. 16, No 2, 1982.Google Scholar
18. [18]
Vukobratovi6 M., Potkonjak V., “Transformaiton Blocks in the Dynamic Simulation of Typical Manipulator Configurations”, IFToMM Journal of Mechanism and Machine Theory, No 3, 1982.Google Scholar
19. [19]
Lurie A.I., Analytical Mechanics (in Russian), Gostehizdat, Moscow, 1961.Google Scholar
20. [20]
Vukobratovid M., Stokid D., “Engineering Approach to Dynamic Control of Industrial Manipulators”, PART 1: Synthesis of Nominal Regimes“, (in Russian), Journal of Mashinovedenia ANUSSR, No 3, 1981.Google Scholar
21. [21]
Vukobratovid M., Hristie D., Stokid D., “Synthesis and Design of Anthropomorphic Manipulator for Industrial Application, Journal of Industrial Robot, Vol. 5, No 2, 1978.Google Scholar
22. [22]
Vukobratovid M., Stokid D., “Contribution to the Decoupled Control of Large-Scale Mechanical Systems”, IFAC Automatica, Jan. 1980Google Scholar
23. [23]
Vukobratovid M., Stokid D., “One Engineering Concept of Dynamical Control of Manipulators”, Trans. of ASME, Journal of Dynamic Systems, Measurement and Control, Special issue, June 1981.Google Scholar
24. [24]
Vukobratovid M., Borovac B., Stokid D., “Influence Analysis of the Actuator Mathematical Model Complexity on the Manipulator Control Synthesis”, IFToMM Journal of Mechanisms and Machine Theory, (in the press).Google Scholar
25. [25]
Athans M., Falb P., Optimal Control, Mc Graw-Hill Book Company, 1966Google Scholar
26. [26]
Bryson A., Yu-Chi Ho, Applied Optimal Control, New York, 1975.Google Scholar
27. [27]
Kahn M.E., Roth B., “The Near-Minimum-Time Control of Open-Loop Articulated Kinematic Chains”, Jour. of Dynamic Systems, Measurement and Control, Trans. of the ASME, Sept. 1971.Google Scholar
28. [28]
Cvetkovid V., Vukobratovie M., “Contribution to Controlling Non-Redundant Manipulators”, Proc. of III CISM-IFToMM Symp., Udine, 1978.Google Scholar
29. [29]
Kirćanski M., “Contribution to Synthesis of Nominal Trajectories of Manipulators via Dynamic Programming”, International Conf. on Systems Engineering, Coventry, Sept. 1980.Google Scholar
30. [30]
Bellman R., Dynamic Programming, Princeton University Press, Princeton N.J., 1957.Google Scholar

## Authors and Affiliations

• Miomir Vukobratović
• 1
• Veljko Potkonjak
• 2