Wojciech Blajer

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.

Walking without impacts as a motion/force control problem

The paper deals with the synthesis of control for impactless bipedal walking. In order of avoid impulses, both the specified motion of the biped and its ground reactions are controlled, yielding a combined motion and force control problem. A method for modeling and solving such problems is proposed. and then illustrated by the example of an impactless planar walk of a seven-link bipedal robot. Some numerical results of the motion simulation are reported.

On the Determination of Joint Reactions in Multibody Mechanisms

In this paper some existing codes for the determination of joint reactions in multibody mechanisms are first reviewed. The codes relate to the DAE (differential-algebraic equation) dynamics formulations in absolute coordinates and in relative joint coordinates, and to the ODE (ordinary differential equation) formulations obtained by applying the coordinate partitioning method to these both coordinate types. On this background a novel efficient approach to the determination of joint reactions is presented, naturally associated with the reduced-dimension formulations of mechanism dynamics. By introducing open-constraint coordinates to specify the prohibited relative motions in the joints, pseudoinverse matrices to the constraint Jacobian matrices are derived in an automatic way. The involvement of the pseudo-inverses leads to schemes in which the joint reactions are obtained directly in resolved forms-no matrix inversion is needed as it is required in the classical codes. This makes the developed schemes especially well suited for both symbolic manipulators and computer implementations. Illustrative examples are provided.

An Orthogonal Complement Matrix Formulation for Constrained Multibody Systems

A method is proposed for the automatic generation of an orthogonal complement matrix to the constraint matrix for the dynamic analysis of constrained multibody systems. The clue for this method lies in the determination of local constraint matrices and their orthogonal complements relative to the local reference frames of particular constrained points. These matrices are then transformed into the systems configuration space in order to form the final constraint matrix and its orthogonal complement. The avoidance of singularities in the formulation is discussed. The method is specially suited for the dynamic analysis of multibody systems with many constraints and/or closed-loops.

Dynamic analysis of constrained multibody systems using inverse kinematics

Abstract This paper draws attention to the advantages that may arise in the dynamic analysis of constrained multibody systems by applying special algorithms of inverse kinematics developed in the field of robotics. The algorithms result in explicit (recursive) relations for the arbitrary chosen dependent coordinates as functions of the remaining (independent) ones. Then analogous velocity and acceleration relations are available. Using these explicit closing condition forms, minimal-dimension governing equations of a constrained system can be formulated conveniently. The avoidance of singularities in the analysis is discussed. An illustrative example is included.

A DAE Formulation for the Dynamic Analysis and Control Design of Cranes Executing Prescribed Motions of Payloads

The dynamic behavior and control of cranes executing prescribed motions of payloads are strongly affected by the underactuated nature of the robotic systems, in which the number of control inputs/outputs is smaller than the number of degrees-of-freedom. The outputs are specified in time load coordinates, which, expressed in terms of the system states, lead to servo-constraints on the system. The problem can then viewed from the perspective of constrained motion. It is noticed however that servo-constraints differ from passive constraints in several aspects. Mainly, they are enforced by means of control forces which may have any directions with respect to the servo-constraint manifold, and in the extreme (some of them) may be tangent. A specific methodology must be developed to solve the’ singular’ inverse dynamics problem. In this contribution, a theoretical background for the modeling of the partly specified/actuated motion is given. The initial governing equations, arising as index five differential-algebraic equations, are transformed to a more tractable index three form by projecting the dynamic equations into the orthogonal and tangent subspaces with respect to the servo-constraint manifold in the crane velocity space. A simple numerical code for solving the resultant differential-algebraic equations, based on backward Euler method, is then proposed. The feedforward control law obtained this way is enhanced by a closed-loop control strategy with feedback of the actual errors in load position to provide stable tracking of the required reference load trajectory in presence of perturbations. A rotary crane executing a load prescribed motion serves as an illustration. Some results of numerical experiments/simulations are reported.

An improved inverse dynamics formulation for estimation of external and internal loads during human sagittal plane movements.

Planar musculoskeletal models are common in the inverse dynamics analysis of human movements such as walking, running and jumping. The continued interest in such models is justified by their simplicity and computational efficiency. Related to a human planar model, a unified formulation for both the flying and support phases of the sagittal plane movements is developed. The actuation involves muscle forces in the lower limbs and the resultant muscle torques in the other body joints. The dynamic equations, introduced in absolute coordinates of the segments, are converted into useful compact forms using the projective technique. The solution to a determinate inverse dynamics problem allows for the explicit determination of the external reactions (presumed to vanish during the flying phases) and the resultant muscle torques in all the model joints. The indeterminate inverse dynamics problem is then focused on the assessment of muscle forces and joint reaction forces selectively in the supporting lower limb. Numerical results of the inverse dynamics simulation of sample sagittal plane movements are reported to illustrate the validity and effectiveness of the improved formulation.

Diversity of Servo-Constraint Problems for Underactuated Mechanical Systems: A Case Study Illustration

Underactuated mechanical systems are systems with fewer control inputs than degrees of freedom. Determination of an input control strategy that forces an underactuated system to complete specified in time outputs (servo-constraints), whose number is equal to the number of inputs, can be a challenging task. Diversity of the servo-constraint problems is discussed here using a simple spring-mass system mounted on a carriage (two degrees of freedom, one control input, and one specified in time output). A formulation of underactuated system dynamics which includes the output coordinates is motivated, with the governing equations arising either as ODEs (ordinary differential equations) or DAEs (differential-algebraic equations). Solutions to the servo-constraint problem are then discussed with reference to so-called non-flat systems (with internal dynamics) and differentially flat systems (no internal dynamics). Some computational issues related to the ODE and DAE formulations are finally discussed, and relevant simulation results for the sample case study are reported.

Dependent versus Independent Variable Formulation for the Dynamics and Control of Cranes

Cranes are underactuated mechanical systems with fewer control inputs than the degrees of freedom. Their usual performance goal is to execute a desired load trajectory, which is specified by as many outputs as the control inputs. A solution to the inverse simulation problem, in which the control of the underactuated system required to execute the partly specified motion is determined, is a challenging task. The inverse simulation study is usually formulated in independent variables. In this paper a dependent variable formulation is reported, advantageous in many aspects. The resultant governing equations appear as simple index-three differential-algebraic equations, and an effective numerical code for their solution is discussed.

Assessment of external and internal loads in the triple jump via inverse dynamics simulation.

The triple jump is a demanding athletics event that, after an approach run, consists of three consecutive phases: the hop, the bound, and the jump. During the involved three take-off actions a jumper is exposed to increased risk of injury due to the high impact forces from the ground and powerful muscle/tendon efforts, which are further reflected in the internal loads of the lower limb joints. While external ground reactions can possibly be measured using force platforms, in vivo measurements of the internal loads are practically not feasible. The purpose of the paper is to present the development of an effective formulation for the inverse dynamics simulation of the triple jump, based on the jumper dynamical model and non-invasive kinematic recordings of the movement. The developed simulation model serves for the analysis of all the triple jump phases, irrespective of whether the jumper is in flight or in contact with the ground with one of his feet, and is focused on effective assessment of the external reactions on the supporting leg as well as the muscle forces and joint reaction forces in the leg. Some numerical results of inverse dynamics simulation of the triple jump are reported.