Minimum Torque Change Criterion Research Articles

Optimality of a kip performance on the high bar: An example of skilled goal-directed whole-body movement

Characteristics of simple goal-directed tasks, such as hand reaching movements, have been well-studied from the perspective of optimization principles. However, it is still unclear what characteristics or control mechanisms of such movements are shared with more general movements. This paper focuses on a gymnastic maneuver on the high bar, referred to as the kip movement, a difficult goal-directed whole-body movement under a nonholonomic constraint. The kip movement of an expert gymnast was simply represented by a three-link planar model, and attempted to be reproduced under three optimization criteria: the minimum angle jerk criterion; the minimum torque change criterion that has been reported to explain the double-joint reaching movements of the hand; and the minimum effort criterion. Numerical analysis shows that (i) none of the criteria that assume only an initial point and a final point could reproduce a measured movement of the gymnast; however, (ii) the minimum torque change criterion that assumes the initial and final points, and an appropriate via-point could almost reproduce the measured movement, which is the best prediction among the three criteria with a via-point. The results may suggest that the hand reaching movements and the specific sections of the kip movement share a common characteristic roughly explained by the minimum torque change criterion. Moreover, by comparison of the solutions under the minimum torque change criterion with different via-points, a possible criterion for explaining the via-point allocation in a hierarchical manner is speculated to reduce the effort cost of the whole movement.

鉄棒運動の生成規範―トルク変化とエフォートに着目した二つのアプローチ

While the kip movements of expert gymnasts have been explained under the minimum torque-change criterion with assumption of a via-point, the meaning of the via-point has been unclear. On the other hand, expert gymnasts say that their movements are associated with effortless skill to achieve a task, in comparison with a movement without the via-point. We consider an integral of square of joint-torques as the effort cost, in addition to the commanded torque-change cost, and propose two approaches to reproduce the kip movement as follows: 1) a combined criterion that minimizes weighted sum of the torque-change and effort costs without assuming via-points, and 2) a hierarchical criterion that allocates via-points such that the effort cost of whole movement is minimized. The results of numerical analysis suggest that both the approaches can improve predictions of the kip movement compared with the results under the minimum torque-change model alone.

非ホロノミック拘束下における到達運動としての鉄棒でのけ上がり運動の解析

The criteria that explain the hand reaching movements of the human, based on the framework of the optimization problem, have a possibility of being applied to the other complex movements. In this paper, we attempt to explain a basic gymnastic maneuver on the horizontal bar, referred to as the kip motion, as a difficult goal-directed movement under a nonholonomic constraint. First, we show the results of the motion measurement of the kip motion performed by expert gymnasts, where the motions of each subject are very similar among the trials. Second, we construct a model of the gymnast performing the kip motion as a rigid 3-link with an unactuated joint at the hand and actuated joints at the shoulder and the hip, where the angle profiles of the actuated joints are approximated by fifth spline functions. Moreover, it is confirmed that the model's motion is approximately consistent with the measured motion. Third, by using the above model, the kip motion connecting the initial, via-, and final posture is generated with the minimum torque change criterion, which is known to explain the hand reaching movements. The results of the numerical analysis show that various kip motions are generated depending on imposed via-postures, and a very similar motion to the measured data is generated when an appropriate via-posture is imposed for each subject. The results suggest that the motion generation of the kip movements and the hand reaching movements could be understood by using a common criterion.

Optimization model predictions for postural coordination modes

This paper examines the ability of the dynamic optimization model to predict changes between in-phase and anti-phase postural modes of coordination and to evaluate influence of two particular environmental and intentional constraints on postural strategy. The task studied was based on an experimental paradigm that consisted in tracking a target motion with the head. An original optimal procedure was developed for cyclic problems to calculate hip and ankle angular trajectories during postural sway with a minimum torque change criterion. Optimization results give a good description of the sudden bifurcation phase between in-phase and anti-phase postural coordination modes in visual target tracking. Transition frequency and predicted effects of environmental and intentional constraints are also in line with experimental observations described in existing literature. In particular, these investigations pointed out that postural planning process can be related to the minimization of a dynamic cost criterion with an equilibrium constraint. In conclusion, the optimization technique is well suited for the prediction of postural modes of coordination and seems to offer many opportunities for better comprehension of neuromuscular movement control.

Quantitative examinations for human arm trajectory planning in three‐dimensional space

AbstractThe following two characteristics of point‐to‐point human arm movement on a plane have been well demonstrated: (1) The path is roughly a straight line but slightly curved. (2) The velocity profile is bell shaped with a single peak. Several models have been proposed to explain these features. Four criteria for trajectory planning based on optimal principles have been proposed: the minimum hand jerk criterion, the minimum angle jerk criterion, the minimum torque change criterion, and the minimum commanded torque change criterion. It has been shown that trajectories generated by the minimum commanded torque change model correspond well with measured trajectories in a horizontal and sagittal work space. However, it was very difficult to establish a robust method for obtaining an optimal trajectory based on the minimum commanded torque change criterion, because of the need to solve a nonlinear optimization problem. Recently, Wada and colleagues proposed a new method to stably calculate optimal trajectories based on the minimum commanded torque change criterion. The method can obtain trajectories satisfying the Euler–Poisson equations with sufficiently high accuracy. We show that optimal trajectories based on the minimum commanded torque change criterion in three‐dimensional space can be calculated by applying the method. Finally, we show that the measured trajectory is closest to the minimum commanded torque change trajectory by statistical examination of many point‐to‐point trajectories in three‐dimensional space. © 2003 Wiley Periodicals, Inc. Syst Comp Jpn, 34(7): 43–54, 2003; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/scj.10323

Quantitative examinations for multi joint arm trajectory planning—using a robust calculation algorithm of the minimum commanded torque change trajectory

In previous research, criteria based on optimal theories were examined to explain trajectory features in time and space in multi joint arm movement. Four criteria have been proposed. They were the minimum hand jerk criterion (by which a trajectory is planned in an extrinsic-kinematic space), the minimum angle jerk criterion (which is planned in an intrinsic-kinematic space), the minimum torque change criterion (where control objects are joint links; it is planned in an intrinsic-dynamic-mechanical space), and the minimum commanded torque change criterion (which is planned in an intrinsic space considering the arm and muscle dynamics). Which of these is proper as a criterion for trajectory planning in the central nervous system has been investigated by comparing predicted trajectories based on these criteria with previously measured trajectories. Optimal trajectories based on the two former criteria can be calculated analytically. In contrast, optimal trajectories based on the minimum commanded torque change criterion are difficult to be calculated, even with numerical methods. In some cases, they can be computed by a Newton-like method or a steepest descent method combined with a penalty method. However, for a realistic physical parameter range, the former becomes unstable quite often and the latter is unreliable about the optimality of the obtained solution. In this paper, we propose a new method to stably calculate optimal trajectories based on the minimum commanded torque change criterion. The method can obtain trajectories satisfying Euler–Poisson equations with a sufficiently high accuracy. In the method, a joint angle trajectory, which satisfies the boundary conditions strictly, is expressed by using orthogonal polynomials. The coefficients of the orthogonal polynomials are estimated by using a linear iterative calculation so as to satisfy the Euler–Poisson equations with a sufficiently high accuracy. In numerical experiments, we show that the optimal solution can be computed in a wide work space and can also be obtained in a short time compared with the previous methods. Finally, we perform supplementary examinations of the experiments by Nakano, Imamizu, Osu, Uno, Gomi, Yoshioka et al. (1999). Estimation of dynamic joint torques and trajectory formation from surface electromyography signals using a neural network model. Biological Cybernetics, 73, 291–300. Their experiments showed that the measured trajectory is the closest to the minimum commanded torque change trajectory by statistical examination of many point-to-point trajectories over a wide range in a horizontal and sagittal work space. We recalculated the minimum commanded torque change trajectory using the proposed method, and performed the same examinations as previous investigations. As a result, it could be reconfirmed that the measured trajectory is closest to the minimum commanded torque change trajectory previously reported.

A theory for cursive handwriting based on the minimization principle.

We propose a trajectory planning and control theory which provides explanations at the computation, algorithm, representation, and hardware levels for continuous movement such as connected cursive handwriting. The hardware is based on our previously proposed forward-inverse-relaxation neural network. Computationally, the optimization principle is the minimum torque-change criterion. At the representation level, hard constraints satisfied by a trajectory are represented as a set of via-points extracted from handwritten characters. Accordingly, we propose a via-point estimation algorithm that estimates via-points by repeating trajectory formation of a character and via-point extraction from the character. It is shown experimentally that for movements with a single via-point target, the via-point estimation algorithm can assign a point near the actual via-point target. Good quantitative agreement is found between human movement data and the trajectories generated by the proposed model.

The Cascade Neural Network Model and a Speed-Accuracy Trade-Off of Arm Movement.

We propose a hybrid neural network model of aimed arm movements that consists of a feedforward controller and a postural controller. The cascade neural network of Kawato, Maeda, Uno, and Suzuki (1990) was employed as a computational implementation of the feedforward controller. This network computes feedforward motor commands based on a minimum torque-change criterion. If the weighting parameter of the smoothness criterion is fixed and the number of relaxation iterations is rather small, the cascade model cannot calculate the exact torque, and the hand does not reach the desired target by using the feedforward control alone. Thus, one observes an error between the final position and the desired target location. By using a fixed weighting parameter value and a limited iteration number to simulate target-directed arm movements, we found that the cascade model generated a planning time-accuracy trade-off, and a quasi-power-law type of speed-accuracy trade-off. The model provides a candidate neural mechanism to explain the stochastic variability of the time course of the feedforward motor command. Our approach also accounts for several invariant features of multijoint arm trajectories, such as roughly straight hand paths and bell-shaped speed profiles.

A neural network model for arm trajectory formation using forward and inverse dynamics models

The minimum torque-change model predicts and reproduces human multi-joint movement data quite well. However, there are three criticisms of the current neural network models for trajectory formation based on the minimum torque-change criteria: (1) their spatial representation of time, (2) back propagation is essential, and (3) they require too many iterations. Accordingly, we propose a new neural network model for trajectory formation based on the minimum torque-change criterion. Our neural network model basically uses a forward dynamics model, an inverse dynamics model, and a trajectory formation mechanism, which generates an approximate minimum torque-change trajectory. It does not require spatial representation of time or back propagation. Furthermore, there are less iterations required to obtain an approximate optimal solution. Finally, our neural network model can be broadly applied to the engineering field because it is a new methodfor solving optimization problems with boundary conditions.

Trajectory formation of arm movement by cascade neural network model based on minimum torque-change criterion

We proposed that the trajectory followed by human subject arms tended to minimize the time integral of the square of the rate of change of torque (Uno et al. 1987). This minimum torque-change model predicted and reproduced human multi-joint movement data quite well (Uno et al. 1989). Here, we propose a neural network model for trajectory formation based on the minimum torque-change criterion. Basic ideas of information representation and algorithm are (i) spatial representation of time, (ii) learning of forward dynamics and kinetics model and (iii) relaxation computation based on the acquired model. The model can resolve ill-posed inverse kinematics and inverse dynamics problems for redundant controlled object as well as ill-posed trajectory formation problems. By computer simulation, we show that the model can produce a multi-joint arm trajectory while avoiding obstacles or passing through viapoints.

Formation and control of optimal trajectory in human multijoint arm movement. Minimum torque-change model.

In this paper, we study trajectory planning and control in voluntary, human arm movements. When a hand is moved to a target, the central nervous system must select one specific trajectory among an infinite number of possible trajectories that lead to the target position. First, we discuss what criterion is adopted for trajectory determination. Several researchers measured the hand trajectories of skilled movements and found common invariant features. For example, when moving the hand between a pair of targets, subjects tended to generate roughly straight hand paths with bell-shaped speed profiles. On the basis of these observations and dynamic optimization theory, we propose a mathematical model which accounts for formation of hand trajectories. This model is formulated by defining an objective function, a measure of performance for any possible movement: square of the rate of change of torque integrated over the entire movement. That is, the objective function CT is defined as follows: (formula; see text) We overcome this difficult by developing an iterative scheme, with which the optimal trajectory and the associated motor command are simultaneously computed. To evaluate our model, human hand trajectories were experimentally measured under various behavioral situations. These results supported the idea that the human hand trajectory is planned and controlled in accordance with the minimum torque-change criterion.