A Geometric Approach to Solving Problems of Control Constraints: Theory and a DAE Framework

The paper deals with controlled mechanical systems in which the number of control inputs is equal to the number of desired system outputs, and is smaller than the number of degrees of freedom of the system. The determination of control input strategy that force the underactuated system to complete the partly specified motion is a challenging problem. In the present formulation, the outputs (performance goals), expressed in terms of system states, are treated as constraints on the system—called control or program constraints as distinct from contact constraints in the classical sense, and a mathematical resemblance of the inverse control problem to the constrained system dynamics is exploited. However, while the reactions of contact constraints act in the directions orthogonal to the respective constraint manifold, the available control reactions may have arbitrary directions with respect to the program constraint manifold, and in the extreme may be tangent. A specific methodology must then be developed to find the solution of such singular problems, related to a class of control tracking problems such as position control of elastic joint robots, control of cranes, and aircraft control in prescribed trajectory flight. The governing equations of the problem arise as a set of differential-algebraic equations (DAEs), and an effective method for solving the DAEs, based on backward Euler method, is proposed. The open-loop control formulation obtained this way is then extended by a closed-loop control law to provide stable tracking of the required reference trajectories in the presence of perturbations. Some examples of applications of the theory and results of numerical simulations are reported.     

Methods for constraint violation suppression in the numerical simulation of constrained multibody systems – A comparative study

Multibody systems are frequently modeled as constrained systems, and the arising governing equations incorporate the closing constraint equations at the acceleration level. One consequence of accumulation of integration truncation errors is the phenomenon of violation of the lower-order constraint equations by the numerical solutions to the governing equations. The constraint drift usually tends to increase in time and may spoil reliability of the simulation results. In this paper a comparative study of three methods for constraint violation suppression is presented: the popular Baumgarte’s constraint violation stabilization method, a projective scheme for constraint violation elimination, and a novel scheme patterned after that proposed recently by Braun and Goldfarb [D.J. Braun, M. Goldfarb, Eliminating constraint drift in the numerical simulation of constrained dynamical systems, Comput. Meth. Appl. Mech. Engrg., 198 (2009) 3151–3160]. The methods are confronted with respect to simplicity in applications, numerical effectiveness and influence on accuracy of the constraint-consistent motion.     

An effective solver for absolute variable formulation of multibody dynamics

This paper presents an effective and general method for converting the equations of motion of multibody systems expressed in terms of absolute variables and Lagrange multipliers into a convenient set of equations in a canonical form (constraint reaction-free and minimal-order equations). The method is applicable to open-loop and closed-loop multibody systems, and to systems subject to holonomic and/or nonholonomic constraints. Being aware of the system configuration space is a metric space, the Gram-Schmidt ortogonalization process is adopted to generate a genuine orthonormal basis of the tangent (null, free) subspace with respect to the constrained subspace. The minimal-order equations of motion expressed in terms of the corresponding tangent speeds have the virtue of being obtained directly in a resolved form, i.e. the related mass matrix is the identity matrix. It is also proved that, in the case of absolute variable formulation, the orthonormal basis is constant, which leads to additional simplifications in the motion equations and fits them perfectly for numerical formulation and integration. Other useful peculiarities of the orthonormal basis method are shown, too. A simple example is provided to illustrate the convertion steps.The research leading to this paper was supported in part by the State Committee for Scientific Research, Grant No. 3 0955 91 01     

A Geometric Unification of Constrained System Dynamics

A unified geometric formulation of the methods used for solving constrained system problems is given. Both holonomic and nonholonomic systems are treated in like manner, and the dynamic equations are expressible in either generalized velocities or quasi-velocities. Moreover, a wide range of unconstrained systems are uniformly regarded as generalized particles in the multi-dimensional metric spaces relating to their configuration. The derivation is grounded on the tensor calculus formalism and appropriate geometric interpretations are reported. In its useful matrix form, the formulation turns out short, elementary and general. This unified geometric approach to constrained system dynamics may deserve to become a generally accepted method inacademic and engineering applications.     

Modeling and computational issues in the inverse dynamics simulation of triple jump

The triple jump is a demanding field event consisting of an approach run, and then followed by a hop, a bound, and a jump. The three consecutive takeoffs are executed at high speed, during which a jumper must absorb extremely large impact forces. The purpose of this paper is to develop an effective formulation for the inverse dynamics simulation of all the jump phases separately. A planar model of the jumper is used, composed of 14 rigid segments connected by 13 hinge joints, and actuated by muscle forces in the lower limbs and resultant muscle torques in the upper body joints. The equations of motion of the model are obtained using a projective technique, allowing for effective assessment of the ground reactions as well as muscle forces and joint reaction forces in the lower limbs. Some numerical results of the inverse dynamics simulation of a triple jump are reported.     

Improved DAE formulation for inverse dynamics simulation of cranes

Cranes are underactuated systems with less control inputs than degrees of freedom. Dynamics and control of such systems is a challenging task, and the existence of solution to the inverse dynamics simulation problem in which an r-degree-of-freedom system with m actuators, m<r, is subject to m specified motion task (servo-constraints) is conditioned upon the system is differentially flat (all the system states and control inputs can be algebraically expressed in terms of the outputs and their time derivatives up to a certain order). The outputs are often designed as specified in time load coordinates to model a rest-to-rest maneuver along a trajectory in the working space, from the initial load position to its desired destination. The flatness-based methodology results then in the required control inputs determined in terms of the fourth time derivatives of the imposed outputs, and the derivations are featured by substantial complexity. The DAE formulation motivated in this contribution offers a more convenient approach to the prediction of dynamics and control of cranes executing prescribed load motions, and only the second time derivatives of the specified outputs are involved. While most of the inverse simulation formulations, both flatness-based and DAE ones, are performed using independent state variables, the use of dependent coordinates and velocities may lead to substantial modeling simplifications and gains in computational efficiency. An improved DAE formulation of this type is presented in this paper.     

Multibody modeling of human body for the inverse dynamics analysis of sagittal plane movements

A multibody methodology for systematic construction of a two-dimensional biomechanical model of a human body is presented, aimed at effective determination of the muscle forces and joint reaction forces in the lower extremities during sagittal plane movements such as vertical jump, standing long jump or jumping down from a height. While the hip, knee and ankle joints are modeled as enforced directly by the muscle forces applied to the foot, shank, thigh and pelvis at the muscle attachment points, the actuation of the other joints is simplified to the torques representing the respective muscle action. The developed formulation is applicable to both the flying and support phases, enhanced by an effective scheme for the determination of reaction forces exclusively in the lower extremity joints. The determination of reactions from the ground is also provided. The problem of muscle force redundancy in the lower extremities is solved by applying the pseudoinverse method, with post-processing procedures used to assure the muscle being tensile. Results of the inverse dynamics analysis of vertical jump are reported.     

Augmented Lagrangian Formulation: Geometrical Interpretation and Application to Systems with Singularities and Redundancy

A geometric interpretation of the augmented Lagrangian formulation ofBayo et al. (Comput. Methods Appl. Mech. Engrg. 71, 1988, 183–195), appliedto equations of motion in relative and Cartesian coordinates, ispresented. Instead of imposing constraints on a system in thetraditional sense, large artificial masses resisting in the constraineddirections are added, and the system motion is enforced to evolveprimarily in the directions with smaller masses (in the unconstraineddirections). Then, the residual motion in the constrained directions isremoved by applying the constraint reactions to the system, estimatedeffectively in few iterations. The formulation is comparatively simpleand leads to computationally efficient numerical codes. Usefulapplications of the formulation to the dynamic analysis of constrainedmultibody systems with possible singular configurations, massless linksand redundant constraints are shown. The theoretical background isfollowed by some remarks on the modeling precautions and assistedcomputational peculiarities of the method. The results of numericalsimulation of motion of a parallel five-bar and a parallel four-barlinkage are reported.     

Influence of selected modeling and computational issues on muscle force estimates

Knowledge of muscle forces and joint reaction forces during human movement can provide insight into the underlying control and tissue loading. Since direct measurement of the internal loads is generally not feasible, non-invasive methods based on musculoskeletal modeling and computer simulations have been extensively developed. By applying observed motion data to the musculoskeletal models, inverse dynamic analysis allow to determine the resultant joint torques, transformed then into estimates of individual muscle forces by means of different optimization procedures. Assessment of the joint reaction forces and other internal loads is further possible. Comparison of the muscle force estimates obtained for different modeling assumptions and parameters in the model can be valuable for the improvement of validity of the model-based estimations. The present study is another contribution to this field. Using a sagittal plane model of an upper limb with a weight carried in hand, and applying the data of recorded flexion and extension movement of the upper limb, the resultant muscular forces are predicted using different modeling assumptions and simulation tools. This study relates to different coordinates (joint and natural coordinates) used to built the mathematical model, muscle path modeling, muscle decomposition (change in number of the modeled muscles), and different optimization methods used to share the joint torques into individual muscles.