A new approach to the incorporation of servo constraints in multibody dynamics

A new index reduction approach is developed to solve the servo constraint problems [2] in the inverse dynamics simulation of underactuated mechanical systems. The servo constraint problem of underactuated systems is governed by differential algebraic equations (DAEs) with high index. The underlying equations of motion contain both holonomic constraints and servo constraints in which desired outputs (specified in time) are described in terms of state variables. The realization of servo constraints with the use of control forces can range from orthogonal to tangential [3]. Since the (differentiation) index of the DAEs is often higher than three for underactuated systems, in which the number of degrees of freedom is greater than the control outputs/inputs, we propose a new index reduction method [1] which makes possible the stable numerical integration of the DAEs. We apply the proposed method to differentially flat systems, such as cranes [1,4,5], and non-flat underactuated systems. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical integration of differential-algebraic equations with mixed holonomic and control constraints

The present work aims at the incorporation of control (or servo) constraints into finite–dimensional mechanical systems subject to holonomic constraints. In particular, we focus on underactuated systems, defined as systems in which the number of degrees of freedom exceeds the number of inputs. The corresponding equations of motion can be written in the form of differential–algebraic equations (DAEs) with a mixed set of holonomic and control constraints. Apart from closed–loop multibody systems, the present formulation accommodates the so–called rotationless formulation of multibody dynamics. To this end, we apply a specific projection method to the DAEs in terms of redundant coordinates. A similar projection approach has been previously developed in the framework of generalized coordinates by Blajer & Kołodziejczyk [1]. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Inverse dynamics of underactuated multibody systems

The present work deals with controlled mechanical systems subject to holonomic constraints. In particular, we focus on underactuated systems, defined as systems in which the number of degrees of freedom exceeds the number of inputs. The governing equations of motion can be written in the form of differential-algebraic equations (DAEs) with a mixed set of holonomic and control constraints. The rotationless formulation of multibody dynamics will be considered [1]. To this end, we apply a specific projection method to the DAEs in terms of redundant coordinates. A similar projection approach has been previously developed in the framework of generalized coordinates by Blajer & Kołodziejczyk [2]. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Simulating underactuated multibody dynamics using servo constraints and variational integrators

In underactuated dynamical systems, the number of control inputs n_u is smaller than the number of degrees of freedom n_q. Real world examples include e. g. flexible robot arms or cranes. In these two exmples the goal is to prescribe the trajectory of an end effector and find the necessary control variables. One approach to model these problems is to introduce servo constraints in the equations of motion that enforce a given trajectory for some part of the system [1]. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Dynamics and control of underactuated mechanical systems: analysis and simple experimental verification

Underactuated mechanical systems are systems with fewer control inputs than the degrees of freedom, m < n, the relevant technical examples being e.g. cranes, aircrafts and flexible manipulators. The determination of an input control strategy that forces an underactuated system to complete a set of m specified motion tasks (servo-constraints) is a demanding problem. The solution is conditioned to differential flatness of the problem, denoted that all 2n state variables and m control inputs can algebraically be expressed, at least theoretically, in terms of the desired m outputs and their time derivatives up to a certain order. A more practical formulation, motivated hereafter, is to pose the problem as a set of differential-algebraic equations, and then obtain the solution numerically. The theoretical considerations are illustrated by a simple two-degree-of-freedom underactuated system composed of two rotating discs connected by a flexible rod (torsional spring), in which the pre-specified motion of the first disc is actuated by the torque applied to the second disc, n = 2 and m = 1. The determined control strategy is then verified experimentally on a laboratory stand representing the two-disc system. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Optimal Control of Differential–Algebraic Equations of Higher Index, Part 1: First-Order Approximations

This paper deals with optimal control problems described by higher index DAEs. We introduce a class of these problems which can be transformed to index one control problems. For this class of higher index DAEs, we derive first-order approximations and adjoint equations for the functionals defining the problem. These adjoint equations are then used to state, in the accompanying paper, the necessary optimality conditions in the form of a weak maximum principle. The constructive way used to prove these optimality conditions leads to globally convergent algorithms for control problems with state constraints and defined by higher index DAEs.     

On the determination op forces op constraint reaction : PMM vol. 39, n 6, 1975, pp. 1129–1134

The equations of motion of mechanical systems with multipliers are reduced to the form enabling the separation of these equations into two groups, the first group describing the motions of the system, and the second group defining the multipliers. Each multiplier is determined independently of the remaining multipliers, and this makes it easy to assess the dynamic effect of each constraint on the system. On the basis of this approach, we study the following problems: determination of the constraint reactions [1], study of the motion of controlled systems with prescribed constraints [2, 3] and utilization of the method of nonholonomic mechanical systems in the case when the first integrals exist [4].     

基于虚拟完整约束的欠驱动起重机控制方法

欠驱动系统的控制是非线性控制的一个重要领域,欠驱动系统指系统控制输入个数小于自由度个数的非线性系统.目前,欠驱动非线性系统动力学和控制研究的主要方法包括线性二次型最优控制方法和部分反馈线性化方法等,如何使系统持续的稳定在平衡位置一直是研究的难点.虚拟约束方法是指通过选择一个周期循环变化的变量作为自变量来设计系统的周期运动.该文以典型的欠驱动模型起重机为例,采用虚拟约束方法,使系统能够在平衡位置稳定或周期振荡运动.首先,通过建立虚拟约束,减少系统自由度变量;然后,通过部分反馈线性化理论推导出系统的状态方程;最后,通过线性二次调节器设计反馈控制器.仿真结果表明,重物在反馈控制下可以在竖直位置的附近达到稳定状态,反映了虚拟约束方法对欠驱动系统的有效性.     

Constraint preserving integrators for general nonlinear higher index DAEs

Summary. In the last few years there has been considerable research on numerical methods for differential algebraic equations (DAEs) where is identically singular. The index provides one measure of the singularity of a DAE. Most of the numerical analysis literature on DAEs to date has dealt with DAEs with indices no larger than three. Even in this case, the systems were often assumed to have a special structure. Recently a numerical method was proposed that could, in principle, be used to integrate general unstructured higher index solvable DAEs. However, that method did not preserve constraints. This paper will discuss a modification of that approach which can be used to design constraint preserving integrators for general nonlinear higher index DAEs. Received August 25, 1993 / Revised version received April 7, 1994     

Model-based control strategies for systems with constraints of the program type

The paper presents a model-based tracking control strategy for constrained mechanical systems. Constraints we consider can be material and non-material ones referred to as program constraints. The program constraint equations represent tasks put upon system motions and they can be differential equations of orders higher than one or two, and be non-integrable. The tracking control strategy relies upon two dynamic models: a reference model, which is a dynamic model of a system with arbitrary order differential constraints and a dynamic control model. The reference model serves as a motion planner, which generates inputs to the dynamic control model. It is based upon a generalized program motion equations (GPME) method. The method enables to combine material and program constraints and merge them both into the motion equations. Lagrange’s equations with multipliers are the peculiar case of the GPME, since they can be applied to systems with constraints of first orders. Our tracking strategy referred to as a model reference program motion tracking control strategy enables tracking of any program motion predefined by the program constraints. It extends the “trajectory tracking” to the “program motion tracking”. We also demonstrate that our tracking strategy can be extended to a hybrid program motion/force tracking.     

Geometric Properties of Runge-Kutta Discretizations for Nonautonomous Index 2 Differential Algebraic Systems

We analyze Runge-Kutta discretizations applied to nonautonomous index 2 differential algebraic equations (DAEs) in semi-explicit form. It is shown that for half-explicit and projected Runge-Kutta methods there is an attractive invariant manifold for the discrete system which is close to the invariant manifold of the DAE. The proof combines reduction techniques to autonomou index 2 differential algebraic equations with some invariant manifold results of Schropp [9]. The results support the favourable behavior of these Runge-Kutta methods applied to index 2 DAEs for t = 0.     

Numerical integration of rigid body dynamics in terms of quaternions

Unit–quaternions (or Euler parameter) are known to be well–suited for the singularity–free parametrization of finite rotations. Despite of this advantage, unit quaternions were rarely used to formulate the equations of motion (exceptions are the works by Nikravesh [1] and Haug [2]). This might be related to the fact, that the unit–quaternions are redundant, which requires the use of algebraic constraints in the equations of motion. Nowadays robust energy consistent integrators are available for the numerical solution of these differential–algebraic equations (DAEs). In the present work a mechanical integrator for the quaternions will be derived. This will be done by a size–reduction from the director formulation of the equations of motion, which also has the form of DAEs. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Trajectory sensitivity analysis of hybrid systems with sliding motion

This paper concerns hybrid control systems exhibiting the sliding motion. It is assumed that the system’s motion on the switching surface is described by index-2 differential–algebraic equations (DAEs), which guarantee the accurate tracking of the sliding motion surface. For those systems the sensitivity analysis is performed with the help of solutions to system’s linearized equations. The paper states conditions under which the solutions to the linearized equations for original DAEs and the solutions to linearized equations for underlying ordinary differential equations (ODEs) exhibit similar properties. Due to the presence of sliding motion, we restrict the class of admissible control functions to piecewise differentiable functions. The presented sensitivity analysis might be useful in deriving the weak maximum principle for optimal control problems with hybrid systems exhibiting sliding motion and in establishing the global convergence of algorithms for solving those problems.     

Systematic Modelling Using Lagrangian DAEs

A treatment for formulating equations of motion for discrete engineering systems using a differential-algebraic form of Lagrange's equation is presented. The distinguishing characteristics of this approach are the retention of constraints in the mathematical model and the consequent use of dependent coordinates. A derivation of Lagrange's equation based on the first law of thermodynamics is featured. Nontraditional constraint classifications for Lagrangian differential-algebraic equations (DAEs) are defined. Model formulation is systematic and lays a foundation for developing DAE-based tools and algorithms for applications in dynamic systems and control.     

A consistent direct transcription method for optimal control problems in multibody dynamics

The present work deals with optimal control problems governed by differential-algebraic equations (DAEs). In particular, the control effort, which is necessary for moving a multibody system from one configuration to another, will be minimized. The orientation of the rigid bodies will be described using directors, which facilitates the integration of the equations of motion with an energy-momentum consistent time-stepping scheme [1]. This type of structure-preserving integrators offer outstanding numerical stability and robustness properties in comparison to the often applied generalized coordinates formulation. In the context of optimal control, other kinds of consistent integrators have been applied previously in [2] and [3]. We will test the different formulations with two numerical examples, a 3-link manipulator and a satellite. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)