Dynamic decoupling of robot manipulators. Springer
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In this chapter, a review of the main methods permitting to achieve the dynamic decoupling of robot manipulators is presented. The design approaches based on the variation of mechanical parameters are disclosed via three sub-groups: decoupling of dynamic equations via mass redistribution; decoupling of dynamic equations via actuator relocation and decoupling of dynamic equations via addition of auxiliary links. The last approach is illustrated via two examples. In the first solution, the optimal design is achieved via gears used as counterweights. It is allows a considerable reduction of the total masses of links of the decoupled manipulator. In the second solution, the dynamic decoupling of robot manipulators is achieved by using an epicyclic gear train. Special attention is paid to the dynamic decoupling of robot manipulators through the use of the double integrator. The second-order linear and time-invariant dynamical system, called double integrator, is one of the most fundamental systems in control applications. It can be considered as single-degree-of-freedom translational and rotational motion. The present review considers in detail the aim of this solution, as well as the advantages of the joint application development inclosing mechanical and control solutions. 1.1 Dynamics and Control of Manipulators It is known that the manipulator dynamics are highly coupled and nonlinear. The complicated dynamics results from varying inertia, interactions between the different joints, and nonlinear forces such as Coriolis and centrifugal forces. The aim of dynamic decoupling of manipulators is to ensure the conditions which allow one to obtain decoupled and linear dynamic equations. It simplifies the optimal control V. Arakelian (&) J. Xu J. P. Le Baron Department of Mechanical and Control Systems Engineering, INSA, 20 avenue des Buttes de Coësmes, CS 70839, 35708 Rennes Cedex, France e-mail: vigen.arakelyan@insa-rennes.fr V. Arakelian J. P. Le Baron LS2 N/ECN, 1 rue de la Noë, BP 92101, 44321 Nantes Cedex 03, France © Springer International Publishing AG, part of Springer Nature 2018 V. Arakelian (ed.), Dynamic Decoupling of Robot Manipulators, Mechanisms and Machine Science 56, https://doi.org/10.1007/978-3-319-74363-9_1 1 and accumulation of energy in manipulators. There are two ways to create the dynamically decoupled manipulators: via optimal mechanical design or control. Let us firstly consider the dynamics and control of serial manipulators in order to show the dynamic decoupling by using the control of the double integrator. An effective way to deal with the problem of high complex coupled dynamics is the decoupling. In fact, this concept was first proposed by Morgan in 1964 when he tried to introduce the design method of the typical control theory into the MIMO (Multi-Input and Multi-Output) linear system [1]. Morgan searches the necessary condition in order that the closed-loop transfer function matrix is a full rank diagonal rational matrix. As known, the form of the dynamic equation of the serial manipulator can be written as: s 1⁄4 MðhÞ€hþCðh; _ hÞ _ hþ gðhÞ ð1:1Þ where s is a n 1 torque vector applied to the joints of the manipulator; h; _ h and €h are n 1 vectors representing the angular positions, velocities and accelerations, respectively; MðhÞ is n n inertia matrix; Cðh; _ hÞ is n 1 vector of Coriolis and centrifugal effects; gðhÞ is the torque vector due to gravity. The first term on the right hand side represents the inertia torque. It can be divided into two parts: the main diagonal elements of the matrix and the off-diagonal elements of the matrix. The first part is generated by the acceleration of the corresponding joint. The second part is the interactive inertia torque caused by the accelerations of the other joints. This interactive inertia torque is linearly proportional to acceleration. The second term represents the nonlinear velocity torques resulting from Coriolis and centrifugal effects. Generally, the dependence of the inertia matrix on the arm configuration produces these nonlinear velocity torques. For an arbitrary arm configuration, the inertia matrix MðhÞ is reduced to a diagonal matrix if the off-diagonal elements of the matrix MðhÞ are all zero. Then it is defined as decoupled inertia matrix. Hence, the control system can be treated as a set of SISO (Single-Input and Single-Output) subsystems. When the second term Cðh; _ hÞ _ h disappears, the inertia matrix is constant for all arbitrary arm configurations (the matrix MðhÞ 1⁄4 M is independent of joint displacements). In this case, the inertia matrix is referred to as configuration invariant inertia matrix. The significance of this form is that the linear control methods which are much simpler and easier to implement can be adopted. However, the most desirable form for the manipulator dynamics is the one with decoupled and configuration-invariant inertia matrix where the effects of gravity are compensated by mechanical engineering, that is