Minimum Commanded Torque Change Research Articles

Investigation of isochrony phenomenon based on the computational theory of human arm trajectory planning

The isochrony principle is a well-known phenomenon whereby the speed of human arm movement is regulated to increase as its trajectory distance increases. However, the relationship between the trajectory planning and the isochrony phenomenon has never been sufficiently explained. One computational study derived the algorithm for estimating the optimal movement segmentation and its duration based on the framework of the minimum commanded torque change criterion. By extending this finding, we can consider the hypothesis that the human arm trajectory is generated based on the minimum commanded torque change criterion to ensure that the duration average of the commanded torque changes (DCTCs) are equivalent between certain movement segmentations, rather than to satisfy the isochrony phenomenon. To test this hypothesis, we measured the behavioral performance of hand movement tasks in which subjects write eight-shaped and double-elliptical-shaped trajectories including two similar shaped arcs of different sizes (hereafter called large and small loops). Our results indicate that the human arm movement is planned in such a manner that the DCTCs for the large and small loops are equivalent during writing of the double-elliptical-shaped trajectories regardless of the arc size. A similar tendency was also observed for the data during the eight-shaped movements, although the ratio of the DCTCs slightly changed depending on the arc size conditions. Thus, our study provides experimental evidence that the isochrony phenomenon is ensured through the computational process of trajectory planning.

Trajectory formation based on the minimum commanded torque change model using the EulerPoisson equation

A minimum commanded torque change criterion based on the optimization principle is proposed as a model that accounts for human voluntary motion. It is shown that the trajectory of human arm motion ...

Trajectory formation based on the minimum commanded torque change model using the Euler-Poisson equation

A minimum commanded torque change criterion based on the optimization principle is proposed as a model that accounts for human voluntary motion. It is shown that the trajectory of human arm motion can be well reproduced by the model. In the point-to-point movement, the calculation of the torque based on the minimum commanded torque change criterion requires a highly nonlinear calculation, and it is difficult to determine the optimal trajectory. As solution methods, a Newton-like method and a steepest descent method have been proposed. However, an optimal solution cannot be obtained by these methods, for several reasons. This paper proposes a method in which the trajectory of the joint angle is analytically represented by a system of orthogonal polynomials, and the coefficients of the orthogonal polynomials are estimated by a linear iterative calculation so that the parameters satisfy the Euler–Poisson equation, as a necessary condition for the optimal solution. As a result of numerical experiments, it is shown that a solution satisfying the Euler–Poisson equation with high numerical accuracy is obtained in a short time, regardless of the parameters such as those of the boundary conditions. © 2005 Wiley Periodicals, Inc. Syst Comp Jpn, 36(2): 92–103, 2005; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/scj.20014

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

A computational trajectory formation model for the human sagittal plane movement at various motion durations

We have confirmed from human arm movement measurement on the sagittal plane that the highest position in the vertical direction where the hand passes through changes according to the motion duration. In this paper, we propose an estimation algorithm of the position where the hand passes through according to the variation in the motion duration. We add the algorithm into the hardware of the trajectory formation model, i.e., the forward inverse relaxation model, which is the trajectory formation model based on the minimum commanded torque change criterion. Finally, we show that the model can reproduce the same trajectories as the human trajectories at various motion durations on the sagittal plane.

An examination of a relation between hand height position and motion duration on the human sagittal plane movement based on the minimization principle

We confirmed from human arm movement measurement on the sagittal plane that the highest position in the vertical direction where the hand passes through changes according to the motion duration. The minimum commanded torque change criterion is one of the optimal criteria for a trajectory formation in a reaching movement. We examined whether the criterion can explain the highest position change on the sagittal plane movement for various motion durations. This shows the possibility that the position where the hand passes through, which changes according to the motion duration, can be estimated by using the differential coefficient of the criterion with respect to the hand height with the amount of change in the hand height of measurement trajectory.

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.

Quantitative examinations of internal representations for arm trajectory planning: minimum commanded torque change model.

Quantitative examinations of internal representations for arm trajectory planning: minimum commanded torque change model. A number of invariant features of multijoint planar reaching movements have been observed in measured hand trajectories. These features include roughly straight hand paths and bell-shaped speed profiles where the trajectory curvatures between transverse and radial movements have been found to be different. For quantitative and statistical investigations, we obtained a large amount of trajectory data within a wide range of the workspace in the horizontal and sagittal planes (400 trajectories for each subject). A pair of movements within the horizontal and sagittal planes was set to be equivalent in the elbow and shoulder flexion/extension. The trajectory curvatures of the corresponding pair in these planes were almost the same. Moreover, these curvatures can be accurately reproduced with a linear regression from the summation of rotations in the elbow and shoulder joints. This means that trajectory curvatures systematically depend on the movement location and direction represented in the intrinsic body coordinates. We then examined the following four candidates as planning spaces and the four corresponding computational models for trajectory planning. The candidates were as follows: the minimum hand jerk model in an extrinsic-kinematic space, the minimum angle jerk model in an intrinsic-kinematic space, the minimum torque change model in an intrinsic-dynamic-mechanical space, and the minimum commanded torque change model in an intrinsic-dynamic-neural space. The minimum commanded torque change model, which is proposed here as a computable version of the minimum motor command change model, reproduced actual trajectories best for curvature, position, velocity, acceleration, and torque. The model's prediction that the longer the duration of the movement the larger the trajectory curvature was also confirmed. Movements passing through via-points in the horizontal plane were also measured, and they converged to those predicted by the minimum commanded torque change model with training. Our results indicated that the brain may plan, and learn to plan, the optimal trajectory in the intrinsic coordinates considering arm and muscle dynamics and using representations for motor commands controlling muscle tensions.