Inverse Dynamics and Trajectory Tracking of Underactuated Multibody Systems

Stefan Reichl

Research output: Types of ThesesDoctoral Thesis


Inverse dynamics problems arise in several mechanical systems. The aim is to calculate the inputs of a system in order that the outputs are identical to predefined or measured target signals. The motivation for inverse methods is related to practical applications in robotics, cranes or test rigs in the automotive and agricultural industry. A multibody system is called underactuated if the number of control inputs is less than the number of degrees of freedom. The control of underactuated systems is much more challenging compared to fully actuated systems. The thesis considers four mathematical methods regarding to inverse problems in underactuated multibody systems. The method of virtual iteration is based on a linearization of the nonlinear system and an inverse computation of the excitations in the frequency domain. The algorithm is suitable for large multibody systems and finite element models, which are nearly linear. The second method formulates the equations of motion as differential-algebraic equations and introduces so called control or servo constraints. This results in a system of high index, which can be solved by appropriate numerical algorithms. The inverse problem can also be formulated as an optimal control problem. The basis is a cost functional, which includes the system outputs and the targets. The goal is to minimize this performance measure. Here it is distinguished between indirect and direct methods. In indirect optimal control the necessary optimality conditions are derived and the resulting boundary value problem has to be solved. Direct methods discretize the system and reformulate the optimal control problem to static optimization problems. The fourth method under consideration is a flatness-based trajectory tracking control. In specific systems the state and input variables can be parameterized by the outputs and their time derivatives up to a certain order. Such systems are called differentially flat and the outputs are known as flat outputs. The considered methods are applied to academic and industrial examples. A nonlinear oscillator, an underactuated planar crane and an underactuated rotary crane are studied. Finite element models and hybrid multibody systems of a steel converter, a trailed cultivator and a plough are representative examples of industrial problems regarding to inverse dynamics. The different methods are compared with respect to their applicability and efficiency.
Original languageEnglish
Publication statusPublished - 2011


  • underactuated multibody system
  • inverse dynamics
  • virtual iteration
  • optimal control
  • control constraints
  • differentially flat system


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