Movement curvature planning through force field internal models

Human motion studies have focused primarily on modeling straight point-to-point reaching movements. However, many goal-directed reaching movements, such as movements directed towards oneself, are not straight but rather follow highly curved trajectories. These movements are particularly interesting to study since they are essential in our everyday life, appear early in development and are routinely used to assess movement deficits following brain lesions. We argue that curved and straight-line reaching movements are generated by a unique neural controller and that the observed curvature of the movement is the result of an active control strategy that follows the geometry of one’s body, for instance to avoid trajectories that would hit the body or yield postures close to the joint limits. We present a mathematical model that accounts for such an active control strategy and show that the model reproduces with high accuracy the kinematic features of human data during unconstrained reaching movements directed toward the head. The model consists of a nonlinear dynamical system with a single stable attractor at the target. Embodiment-related task constraints are expressed as a force field that acts on the dynamical system. Finally, we discuss the biological plausibility and neural correlates of the model’s parameters and suggest that embodiment should be considered as a main cause for movement trajectory curvature.     

Motor control of voluntary arm movements

The motor control of pointing and reaching-to-grasp movements was investigated using two different approaches (kinematic and modelling) in order to establish whether the type of control varies according to modifications of arm kinematics. Kinematic analysis of arm movements was performed on subjects' hand trajectories directed to large and small stimuli located at two different distances. The subjects were required either to grasp and to point to each stimulus. The kinematics of the subsequent movement, during which subject's hand came back to the starting position, were also studied. For both movements, kinematic analysis was performed on hand linear trajectories as well as on joint angular trajectories of shoulder and elbow. The second approach consisted in the parametric identification of the black box (ARMAX) model of the controller driving the arm movement. Such controller is hypothesized to work for the correct execution of the motor act. The order of the controller ARMAX model was analyzed with respect to the different experimental conditions (distal task, stimulus size and distance). Results from kinematic analysis showed that target distance and size influenced kinematic parameters both of angular and linear displacements. Nevertheless, the structure of the motor program was found to remain constant with distane and distal task, while it varied with precision requirements due to stimulus size. The estimated model order of the controller confirmed the invariance of the control law with regard to movement amplitude, whereas it was sensitive to target size.     

眼－头运动协同中头部运动的双重模式控制

本文通过目标运动引起的眼－头运动协同的实验，测量和分析了头部运动的动态特性来探讨其头部运动的控制机制。研究结果揭示了眼－头协同的注视运动中头部运动的双重模式控制机制：在小幅度运动范围是线性比例控制，在大幅度运动范围是使用最大作用力的Ｂａｎｇ－Ｂａｎｇ开关控制。     

Sensitivity analysis and optimization for a head movement model

A sixth order nonlinear model for horizontal head rotations in humans is analyzed using an extended parameter sensitivity analysis and a global optimization algorithm. The sensitivity analysis is used in both the direct sense, as a model fitting tool, and in the indirect sense, as a guide to experimental design. Resolution is defined in terms of the sensitivity table, and is used to interpret the sensitivity results. Using sensitivity analyses, the head and eye movement systems are compared and contrasted. Controller signal parameters are the most influential. Their variations and effects on head movement trajectories and accelerations are investigated, and the conclusions are compared with clinical neurological findings. The global optimization algorithm, in addition to automating the fitting of various types of data, is combined with time optimality theory to give theoretical time-optimal inputs to the model.On leave from Department of Neurology, University of Hamburg, FRG; supported by Deutsche Forschungsgemeinschaft Bonn, FRG     

Analysis of an optimal control model of multi-joint arm movements

 In this paper, we propose a model of biological motor control for generation of goal-directed multi-joint arm movements, and study the formation of muscle control inputs and invariant kinematic features of movements. The model has a hierarchical structure that can determine the control inputs for a set of redundant muscles without any inverse computation. Calculation of motor commands is divided into two stages, each of which performs a transformation of motor commands from one coordinate system to another. At the first level, a central controller in the brain accepts instructions from higher centers, which represent the motor goal in the Cartesian space. The controller computes joint equilibrium trajectories and excitation signals according to a minimum effort criterion. At the second level, a neural network in the spinal cord translates the excitation signals and equilibrium trajectories into control commands to three pairs of antagonist muscles which are redundant for a two-joint arm. No inverse computation is required in the determination of individual muscle commands. The minimum effort controller can produce arm movements whose dynamic and kinematic features are similar to those of voluntary arm movements. For fast movements, the hand approaches a target position along a near-straight path with a smooth bell-shaped velocity. The equilibrium trajectories in X and Y show an ‘N’ shape, but the end-point equilibrium path zigzags around the hand path. Joint movements are not always smooth. Joint reversal is found in movements in some directions. The excitation signals have a triphasic (or biphasic) pulse pattern, which leads to stereotyped triphasic (or biphasic) bursts in muscle control inputs, and a dynamically modulated joint stiffness. There is a fixed sequence of muscle activation from proximal muscles to distal muscles. The order is preserved in all movements. For slow movements, it is shown that a constant joint stiffness is necessary to produce a smooth movement with a bell-shaped velocity. Scaled movements can be reproduced by varying the constraints on the maximal level of excitation signals according to the speed of movement. When the inertial parameters of the arm are altered, movement trajectories can be kept invariant by adjusting the pulse height values, showing the ability to adapt to load changes. These results agree with a wide range of experimental observations on human voluntary movements. Received: 4 December 1995 / Accepted in revised form: 17 September 1996     

End-point constraints in aiming movements: effects of approach angle and speed

 The present study focuses on two trajectory-formation models of point-to-point aiming movements, viz., the minimum-jerk and the minimum torque-change model. To date, few studies on minimum-jerk and minimum torque-change trajectories have incorporated self- or externally imposed end-point constraints, such as the direction and velocity with which a target area is approached. To investigate which model accounts best for the effects on movement trajectories of such – in many circumstances – realistic end-point constraints, we adjusted both the minimum-jerk and the minimum torque-change model so that they could generate trajectories of which the final part has a specific direction and speed. The adjusted models yield realistic trajectories with a high curvature near movement completion. Comparison of simulated and measured movement trajectories show that pointing movements that are constrained with respect to final movement direction and speed can be described in terms of minimization of joint-torque changes. Received: 7 July 1999 / Accepted in revised form: 8 January 2001     

Allometric trajectories of body and head morphology in three sympatric Arctic charr (Salvelinus alpinus (L.)) morphs

A study of body and head development in three sympatric reproductively isolated Arctic charr (Salvelinus alpinus (L.)) morphs from a subarctic lake (Skogsfjordvatn, northern Norway) revealed allometric trajectories that resulted in morphological differences. The three morphs were ecologically assigned to a littoral omnivore, a profundal benthivore and a profundal piscivore, and this was confirmed by genetic analyses (microsatellites). Principal component analysis was used to identify the variables responsible for most of the morphological variation of the body and head shape. The littoral omnivore and the profundal piscivore morph had convergent allometric trajectories for the most important head shape variables, developing bigger mouths and relatively smaller eyes with increasing head size. The two profundal morphs shared common trajectories for the variables explaining most of the body and head shape variation, namely head size relative to body size, placement of the dorsal and pelvic fins, eye size and mouth size. In contrast, the littoral omnivore and the profundal benthivore morphs were not on common allometric trajectories for any of the examined variables. The findings suggest that different selective pressures could have been working on traits related to their trophic niche such as habitat and diet utilization of the three morphs, with the two profundal morphs experiencing almost identical environmental conditions.     

Formation and control of optimal trajectory in human multijoint arm movement

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.     

Effect of Colony, Patch Distance, And Trajectory Sense on Movement Complexity in Foraging Ants

Animals display foraging trajectories when exploiting food patches. These displacements have been studied, particularly in organisms that forage from a central place. The complexity of a movement path may be analyzed by means of the fractal dimension, an index that estimates the tortuosity of a trajectory. In field experiments we studied the effect of trajectory sense, distance to the resource patch and colony on the movement complexity in a common ant of central Chile. We found that these factors and their interactions significantly affected the complexity of movement paths. We discuss whether mortality risk determines more linear trajectories when the resource patch is distant from the nest, and whether the information acquired from the environment by workers determines less complex return trips.     

Optimal Tracking Controller Design for a Small Scale Helicopter

A model helicopter is more difficult to control than its full scale counterpart. This is due to its greater sensitivity to control inputs and disturbances as well as higher bandwidth of dynamics. This work is focused on designing practical tracking controller for a small scale helicopter following predefined trajectories. A tracking controller based on optimal control theory is synthesized as a part of the development of an autonomous helicopter. Some issues with regards to control constraints are addressed.The weighting between state tracking performance and control power expenditure is analyzed. Overall performance of the control design is evaluated based on its time domain histories of trajectories as well as control inputs.     

Change in Auditory Image Movement Trajectories under Conditions of Direct Nonsimultaneous Masking

The authors studied fused auditory image (FAI) movement trajectories under conditions of direct nonsimultaneous masking. This movement was created by a gradual change in a dichotically presented series of clicks with interaural differences in stimulation from 0 to ±700 s or from ±700 to 0 s. Binaurally presented transmissions of wide-band noise served as maskers. The location and length of the trajectories were evaluated without a masker and with five values of the time lag between the signal beginning and masker end. When the test signal duration was 200 ms, the length of the trajectories was 33–44° without a masker. In the first test group, this trajectory lay close to the median line of the head without a masker (irrespective of the movement direction) and moved away from it under masking conditions. When the FAI moved from the median line towards the right or left ear, the initial part of the trajectory was masked; when the movement direction was opposite, the final part was masked. In the second group, the trajectories were located near the ears when the FAI moved from either ear and shifted towards the median line as a result of masking. When the movement direction was opposite, they were close to the median line and shifted towards the ear under masking conditions. When the FAI moved along all trajectories, their initial parts were masked.     

Analysis of control strategies for VIVA OpenHBM with active reflexive neck muscles

Modeling muscle activity in the neck muscles of a finite element (FE) human body model can be based on two biological reflex systems. One approach is to approximate the Vestibulocollic reflex (VCR) function, which maintains the head orientation relative to a fixed reference in space. The second system tries to maintain the head posture relative to the torso, similar to the Cervicocolic reflex (CCR). Strategies to combine these two neck muscle controller approaches in a single head-neck FE model were tested, optimized, and compared to rear-impact volunteer data. The first approach, Combined-Control, assumed that both controllers simultaneously controlled all neck muscle activations. In the second approach, Distributed-Control, one controller was used to regulate activation of the superficial muscles while a different controller acted on deep neck muscles. The results showed that any muscle controller that combined the two approaches was less effective than only using one of VCR- or CCR-based systems on its own. A passive model had the best objective rating for cervical spine kinematics, but the addition of a single active controller provided the best response for both head and cervical spine kinematics. The present study demonstrates the difficulty in completely capturing representative head and cervical spine responses to rear-impact loading and identified a controller capturing the VCR reflex as the best candidate to investigate whiplash injury mechanisms through FE modeling.

     

A biologically inspired hierarchical goal directed navigation model

We propose an extended version of our previous goal directed navigation model based on forward planning of trajectories in a network of head direction cells, persistent spiking cells, grid cells, and place cells. In our original work the animat incrementally creates a place cell map by random exploration of a novel environment. After the exploration phase, the animat decides on its next movement direction towards a goal by probing linear look-ahead trajectories in several candidate directions while stationary and picking the one activating place cells representing the goal location. In this work we present several improvements over our previous model. We improve the range of linear look-ahead probes significantly by imposing a hierarchical structure on the place cell map consistent with the experimental findings of differences in the firing field size and spacing of grid cells recorded at different positions along the dorsal to ventral axis of entorhinal cortex. The new model represents the environment at different scales by populations of simulated hippocampal place cells with different firing field sizes. Among other advantages this model allows simultaneous constant duration linear look-ahead probes at different scales while significantly extending each probe range. The extension of the linear look-ahead probe range while keeping its duration constant also limits the degrading effects of noise accumulation in the network. We show the extended model’s performance using an animat in a large open field environment.     

Behavioral states help translate dispersal movements into spatial distribution patterns of floaters

Within the field of spatial ecology, it is important to study animal movements in order to better understand population dynamics. Dispersal is a nonlinear process through which different behavioral mechanisms could affect movement patterns. One of the most common approaches to analyzing the trajectories of organisms within patches is to use random-walk models to describe movement features. These models express individual movements within a specific area in terms of random-walk parameters in an effort to relate movement patterns to the distributions of organisms in space. However, only using the movement trajectories of individuals to predict the spatial spread of animal populations may not fit the complex distribution of individuals across heterogeneous environments. When we empirically tested the results from a random-walk model (a residence index) used to predict the spatial equilibrium distribution of individuals, we found that the index severely underestimated the spatial spread of dispersing individuals. We believe this is because random-walk models only account for the effects of environmental conditions on individual movements, completely overlooking the crucial influence of behavior changes over time. In the future, both aspects should be accounted for when predicting general rules of (meta)population abundance, distribution, and dynamics from patterns of animal movements.     

Toward the Quantification of a Conceptual Framework for Movement Ecology Using Circular Statistical Modeling

To analyze an animal’s movement trajectory, a basic model is required that satisfies the following conditions: the model must have an ecological basis and the parameters used in the model must have ecological interpretations, a broad range of movement patterns can be explained by that model, and equations and probability distributions in the model should be mathematically tractable. Random walk models used in previous studies do not necessarily satisfy these requirements, partly because movement trajectories are often more oriented or tortuous than expected from the models. By improving the modeling for turning angles, this study aims to propose a basic movement model. On the basis of the recently developed circular auto-regressive model, we introduced a new movement model and extended its applicability to capture the asymmetric effects of external factors such as wind. The model was applied to GPS trajectories of a seabird (Calonectris leucomelas) to demonstrate its applicability to various movement patterns and to explain how the model parameters are ecologically interpreted under a general conceptual framework for movement ecology. Although it is based on a simple extension of a generalized linear model to circular variables, the proposed model enables us to evaluate the effects of external factors on movement separately from the animal’s internal state. For example, maximum likelihood estimates and model selection suggested that in one homing flight section, the seabird intended to fly toward the island, but misjudged its navigation and was driven off-course by strong winds, while in the subsequent flight section, the seabird reset the focal direction, navigated the flight under strong wind conditions, and succeeded in approaching the island.     

Motion control of musculoskeletal systems with redundancy

Motion control of musculoskeletal systems for functional electrical stimulation (FES) is a challenging problem due to the inherent complexity of the systems. These include being highly nonlinear, strongly coupled, time-varying, time-delayed, and redundant. The redundancy in particular makes it difficult to find an inverse model of the system for control purposes. We have developed a control system for multiple input multiple output (MIMO) redundant musculoskeletal systems with little prior information. The proposed method separates the steady-state properties from the dynamic properties. The dynamic control uses a steady-state inverse model and is implemented with both a PID controller for disturbance rejection and an artificial neural network (ANN) feedforward controller for fast trajectory tracking. A mechanism to control the sum of the muscle excitation levels is also included. To test the performance of the proposed control system, a two degree of freedom ankle–subtalar joint model with eight muscles was used. The simulation results show that separation of steady-state and dynamic control allow small output tracking errors for different reference trajectories such as pseudo-step, sinusoidal and filtered random signals. The proposed control method also demonstrated robustness against system parameter and controller parameter variations. A possible application of this control algorithm is FES control using multiple contact cuff electrodes where mathematical modeling is not feasible and the redundancy makes the control of dynamic movement difficult.     

Expectation-Maximization Binary Clustering for Behavioural Annotation

The growing capacity to process and store animal tracks has spurred the development of new methods to segment animal trajectories into elementary units of movement. Key challenges for movement trajectory segmentation are to (i) minimize the need of supervision, (ii) reduce computational costs, (iii) minimize the need of prior assumptions (e.g. simple parametrizations), and (iv) capture biologically meaningful semantics, useful across a broad range of species. We introduce the Expectation-Maximization binary Clustering (EMbC), a general purpose, unsupervised approach to multivariate data clustering. The EMbC is a variant of the Expectation-Maximization Clustering (EMC), a clustering algorithm based on the maximum likelihood estimation of a Gaussian mixture model. This is an iterative algorithm with a closed form step solution and hence a reasonable computational cost. The method looks for a good compromise between statistical soundness and ease and generality of use (by minimizing prior assumptions and favouring the semantic interpretation of the final clustering). Here we focus on the suitability of the EMbC algorithm for behavioural annotation of movement data. We show and discuss the EMbC outputs in both simulated trajectories and empirical movement trajectories including different species and different tracking methodologies. We use synthetic trajectories to assess the performance of EMbC compared to classic EMC and Hidden Markov Models. Empirical trajectories allow us to explore the robustness of the EMbC to data loss and data inaccuracies, and assess the relationship between EMbC output and expert label assignments. Additionally, we suggest a smoothing procedure to account for temporal correlations among labels, and a proper visualization of the output for movement trajectories. Our algorithm is available as an R-package with a set of complementary functions to ease the analysis.     

Analysis of kinematic invariances of multijoint reaching movement

 There is a no unique relationship between the trajectory of the hand, represented in cartesian or extrinsic space, and its trajectory in joint angle or intrinsic space in the general condition of joint redundancy. The goal of this work is to analyze the relation between planning the trajectory of a multijoint movement in these two coordinate systems. We show that the cartesian trajectory can be planned based on the task parameters (target coordinates, etc.) prior to and independently of angular trajectories. Angular time profiles are calculated from the cartesian trajectory to serve as a basis for muscle control commands. A unified differential equation that allows planning trajectories in cartesian and angular spaces simultaneously is proposed. Due to joint redundancy, each cartesian trajectory corresponds to a family of angular trajectories which can account for the substantial variability of the latter. A set of strategies for multijoint motor control following from this model is considered; one of them coincides with the frog wiping reflex model and resolves the kinematic inverse problem without inversion. The model trajectories exhibit certain properties observed in human multijoint reaching movements such as movement equifinality, straight end-point paths, bell-shaped tangential velocity profiles, speed-sensitive and speed-insensitive movement strategies, peculiarities of the response to double-step targets, and variations of angular trajectory without variations of the limb end-point trajectory in cartesian space. In humans, those properties are almost independent of limb configuration, target location, movement duration, and load. In the model, these properties are invariant to an affine transform of cartesian space. This implies that these properties are not a special goal of the motor control system but emerge from movement kinematics that reflect limb geometry, dynamics, and elementary principles of motor control used in planning. All the results are given analytically and, in order to compare the model with experimental results, by computer simulations. Received: 6 April 1994/Accepted in revised form: 25 April 1995