Closed-Loop Systems
Closed-loops are inherent in many practical robotic systems. In this chapter, analyses of closed-loop systems are presented using the dynamic formulation given in Chaps. 5,6 and 7.
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Closed-Loop Systems
Closed-loops are inherent in many practical robotic systems. In this chapter, analyses of closed-loop systems are presented using the dynamic formulation given in Chaps. 5, 6, and 7.
8.1 Tree-Type Representation of Closed-Loop Systems Figure 8.1a shows schematic diagram of a general closed-loop system. The closedloop system has links and c joints, and there exists l D (c ) independent kinematic loops. One of the approaches to analyze a closed-loop system is to convert it into equivalent tree-type architecture by cutting suitable joints. A closed-loop system with l independent loops is required to cut at l joints in order to convert it into a tree-type system as shown in Fig. 8.1b. For complex systems, the joints to be cut may be identified using the concept of Cumulative DOF (CDOF) in graph theory (Deo 1974) as suggested by Chaudhary and Saha (2007). The concept will be illustrated in Sect. 8.4. The cut opened joints are then substituted with suitable constraint forces denoted with œ’s, which are also known as Lagrange multipliers. These multipliers are treated as external forces to the resulting open tree-type system. As a result, the problem is converted into solving a tree-type system with externally applied constraint forces. Therefore equations of motion of the closedloop system in terms of the externally loaded open-loop system are written as IqR C CqP D £ C £F C JT
(8.1)
In Eq. (8.1), J represents the l n constrained Jacobian matrix (Nikravesh 1988) for the closed-loop system, where l is the total number of constraints imposed by the joints of the l loops and n is total number of joint variables of the open tree-type system. It is defined in a way so that JqP D 0. Moreover, œ is the l-dimensional vector of Lagrange multipliers representing the constraint forces at the cut joints. It is worth noting here that Eq. (8.1) can further be written in terms of the independent coordinates, i.e., DOF of the closed-loop system at hand, by eliminating S.V. Shah et al., Dynamics of Tree-Type Robotic Systems, Intelligent Systems, Control and Automation: Science and Engineering 62, DOI 10.1007/978-94-007-5006-7 8, © Springer ScienceCBusiness Media Dordrecht 2013
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8 Closed-Loop Systems
a
b
Subsystem III: Serial type
Loop
λ2
λ3
Loop
λ1 Loop
Subsystem II: Serial type
Subsystem I: Tree type
Fig. 8.1 Tree-type representation of a closed-loop system. (a) Closed-loop system. (b) Tree-type representation of (a)
the Lagrange multipliers (Shabana 2001). This, however, will not allow one to use the recursive algorithms obtained in Chaps. 6 and 7 for solving the closed-loop system. Hence, the latter approach will not be followed in this chapter.
8.2 Dynamic Formulation Dynamic formulation of a closed-loop system, namely, inverse and forward dynamics formulation, is explained in the following sections.
8.2.1 Inverse Dynamics Once the equivalent tree-type system of a closed-loop system is obtained, one can solve for the Lagrange multipliers and the actuated forces or torque
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