Optimization in target movement simulations

Abstract

This paper discusses a methodology for fixed-time simulations of optimal movements of a mechanical system, between specified initial and target configurations, without any a priori knowledge on the trajectory between those. It is primarily aimed at human movement simulations with muscular controls. The basic formulation considers both displacements and forces as unknowns during the movement, connects them, and utilizes a finite element time discretization for solving the whole fixed-time interval simultaneously. Through a consistent interpolation of all kinetic and kinematic variables, the formulation becomes general, needing only minimal input for description of a particular problem, but also eliminating errors inherent in many forms of time-integration. The same consistency allows systematic formulations of a large class of optimization cost functions, primarily focussing on the mechanical behavior of the system rather than on the matching of previously measured movements. It thereby allows the use of robust and efficient general optimization algorithms. Kinetic and kinematic constraints can restrict the movement. As an example of the general setting, a simplified human movement is studied, with different choices of controls (joint moments or muscular tensions), and with different optimization criteria. The example shows that the simulation results are strongly dependent on these choices, in particular that smoothness of movement demands forces considerably higher than the strictly minimum ones. A larger example shows that more complex constraints can be handled within the setting proposed, but also the effects from the fixed-time assumption.

Introduction

When analyzing and simulating human movement, two basic approaches can be used [7]. The inverse dynamics situation analyzes a recorded movement, and deduces the forces needed to create this movement from the anatomical data. A forward dynamics approach postulates the forces acting on the body and calculates the resulting movement. In both cases, forces—the term including joint-affecting moments—obviously vary during a movement sequence. In real movement, this variation is governed by neural control, where it can be assumed that a basic force variation is related to motion planning, but a correction mechanism takes care of involuntary deviations in real-time.

The movement plan can be assumed to be related to some optimality condition. A common choice for characterization of human movement is to measure the optimality by some cost function for the forces needed to create the movement [4], [6]. Another interesting possibility is to emphasize the physiological demand for smooth motion patterns. Variations of the minimum-jerk model formulated by Flash and Hogan [12], have been used both in the diagnosis of pathology [23], and as a tool for evaluating the kinematics of both human and robotic or prosthetic movement [25], [28], [30]. Hogan and Flash [18] discuss whether any criterion is valid for all kinds of human movements and in all situations; in particular they note that kinematic smoothness can be the sole objective, disregarding the forces needed to create movement.

A forward simulation method, where the time variations of both displacements and forces are a priori unknown, has been discussed and evaluated by Eriksson [8], [9], [10]. The mechanical formulation governing the movement can be of any complexity, and non-linear structural formulations, together with arbitrary time variations of the loads can be introduced [14]. This paper extends the approach in order to find an ideal movement between two specified configurations. The formulation therefore considers a priori unknown controls acting on the system. These controls can be stated in different ways, for instance as joint resultants, or as individual muscle forces or tensions, in the human movement case. The algorithm allows mechanically redundant control force systems [11]. Under certain conditions, the combination of introduced controls and resulting movement can be chosen to be—in some sense—optimal in the fulfilment of the target movement. The basic hypothesis of the present work is that the dynamic formulation can be used together with a general optimization algorithm. Although this optimization can be questioned as the answer to the sensorimotor control of human movement [33], it can give important information on the envelope of possible movements, the main shortcoming being the assumption that controls can be infinitely quickly regulated, but also the lack of feed-back mechanisms, essential for real human movement [34].

The method chosen has some similarities with work by Kaplan and Heegaard [21], [22], but uses fewer state variables and other collocation expressions. The consistent interpolation of kinetic and kinematic variables gives high accuracy in the time-integration [10]: higher than when the dynamical problem is seen as two first-order evolution equations, loosely coupled. Through a consistent local interpolation of both control and state variables, clear and efficient algorithmic forms are obtained. It also makes easier the usage of general optimization routines. Compared to the references given, where the method is only verified for problems where the kinematic history is defined in advance, and convergence thereby easily obtained, the present form is aimed to be globally convergent from basic facts only. Although in very different settings, the present work shares some objectives with, e.g., [26], [29]. The latter reference includes the muscle activation as variables, and allows a free time interval for a movement, but uses a forward integration in time of several variables, and a numerical differentiation of all relevant quantities, which is computationally extremely demanding for larger problems, and is sensitive to good initial approximations to the solution. The work done by Johansson and Magnusson [20], which primarily considers the optimization in following a pre-planned motion trajectory, can also be used for the situations considered in the present work, at least under certain restrictions. A survey of other similar applications and methods is given by Betts [2].

The paper gives the background for the basic formulation, and the inherent temporal interpolation. Although the basic form can be used in several contexts, only optimally controlled problems for a human musculo-skeletal models are considered in this paper. The resulting generality of the formulation is emphasized, with only minimal problem-specific description needed. The generality is verified through a comparison of different criteria and measures for optimality, utilizing robust and efficient standard softwares for the numerical optimization. Kinetic and kinematic restrictions for the controls and displacement coordinates are introduced, for more physiologically relevant solutions.