Trajectory control of flexible manipulators

Abstract

We present some feedback control techniques recently developed for the exact solution of trajectory tracking problems for manipulators with flexible elements. Two classes are considered: i) robots with rigid links but with elastic transmissions, in which flexibility is concentrated at the joints, and ii) robots with lightweight and/or long arms, where flexibility is distributed along the links. For robots with elastic joints, we introduce a generalized inversion algorithm for the synthesis of a dynamic feedback control law that gives input-output decoupling and full state linearization. For robots with flexible links, the end-effector trajectory tracking problem is solved based on the iterative computation of the link deformations associated with the desired output motion, combined with a state trajectory regulator. For both robot models, the control design is performed directly on the second-order dynamic equations. I n t r o d u c t i o nModeling robot manipulators as rigid mechanical systems is an idealization that becomes unrealistic when higher performance is requested. Tasks involving fast motion and/or hard contact with the environment are expected to induce deflections in the robot components, eventually exciting an oscillatory behavior. There are two sources of vibration in robot manipulators: joint flexibility, due to the elasticity of motion transmission elements such as harmonic drives, belts, or long shafts [26], and link flezibility, introduced by a long reach and slender/lightweight construction of the arm [6,17]. In order to be able to counteract the negative effects of flexibility, advanced robot control systems should be designed on the basis of a more complete dynamic model of the robot (see, e.g. [27] and [5])In robotic systems with flexible elements, output trajectories are typically defined beyond the structural flexibility, i.e. in terms of link motion for robots with elastic joints or at the level of manipulator tip for robots with link flexibility. We address in this chapter the stable and accurate reproduction of such trajectories using model-based state feedback control. Standard tools for solving trajectory tracking problems in nonlinear systems, such as feedback linearization, input-output decoupling, or inversion control (see, e.g. [19]), are not sufficient in these cases, so that the application of more advanced control techniques should be investigated.

Integrated mechanical and control design of underactuated multibody systems

“…Examples include rigid multibody systems with passive joints and planar flexible manipulators, where the shape functions of the flexible bodies are chosen according to clamped boundary conditions, see e.g. [5]. It should be noted that the calculations presented in this paper can also be applied to other cases of B a , B u .…”

“…Using these nonlinear control concepts the analysis of the mechanical design of underactuated multibody systems might show that they possess internal dynamics. Internal dynamics arise for example often in the case of flexible manipulators [5,12,18] or in multibody systems with passive joints [11,20]. In contrast manipulators with joint elasticity [5] and cranes [3] are often differentially flat, possessing no internal dynamics.…”

Servo-constraint realization for underactuated mechanical systems

“…The design of a feedback controller for the mixed orthogonal-tangential and pure tangential realizations of servo-constraints is in general more challenging. Higher-order controllers, discussed in, e.g., [29,36,37], might be more appropriate. For differentially flat servo-constraint problems, this order should be equal to the order r introduced in the flatness-based solution (21), and a feedback controller should be based on …”

“…Under usual modeling assumptions, the flatness order for the mentioned flat problems is r = 4, yieldingũ(t) = u(ỹ,ẏ,ÿ) for the p controls engaged in the orthogonal realization of p servo-constraints (rank(HM −1 B) = p), and u(t) = u(ỹ,ẏ,ÿ,ỹ (3) ,ỹ (4) ) for the other f − m controls engaged in the tangential realization of the remaining f − m servo-constraints. This is why, for the flat problems and mixed orthogonal-tangential realization of servo-constraints, the feedback loop for the orthogonal-realization control can be designed according to (8), while in the feedback loop for tangential-realization control higher-order derivatives of the output errors should be involved [29,30]. These issues will not be addressed in the present study, and the flat and non-flat problems will only be illustrated with the followed case studies.…”

“…For instance, differentially flat are servo-constraint problems for cranes executing a load prescribed motion [5,7,26,27] and for aircrafts in prescribed trajectory flight [28], both characterized by mixed orthogonal-tangential realization of servo-constraints. Differentially flat is also the trajectory tracking problem for flexible joint manipulators [16,29], with pure tangential realization of the servo-constraints. Under usual modeling assumptions, the flatness order for the mentioned flat problems is r = 4, yieldingũ(t) = u(ỹ,ẏ,ÿ) for the p controls engaged in the orthogonal realization of p servo-constraints (rank(HM −1 B) = p), and u(t) = u(ỹ,ẏ,ÿ,ỹ (3) ,ỹ (4) ) for the other f − m controls engaged in the tangential realization of the remaining f − m servo-constraints.…”