A method for measuring endpoint stiffness during multi-joint arm movements

Current methods for measuring stiffness during human arm movements are either limited to one-joint motions, or lead to systematic errors. The technique presented here enables a simple, accurate and unbiased measurement of endpoint stiffness during multi-joint movements. Using a computer-controlled mechanical interface, the hand is displaced relative to a prediction of the undisturbed trajectory. Stiffness is then computed as the ratio of restoring force to displacement amplitude. Because of the accuracy of the prediction (< 1 cm error after 200 ms) and the quality of the implementation, the movement is not disrupted by the perturbation. This technique requires only 13 as many trials to identify stiffness as the method of Gomi and Kawato (1997, Biological Cybernetics 76, 163-171) and may, therefore, be used to investigate the evolution of stiffness during motor adaptation.     

Reconstruction of equilibrium trajectories during whole-body movements

The framework of the equilibrium-point hypothesis was used to reconstruct equilibrium trajectories (ETs) of the ankle, hip and body center of mass during quick voluntary hip flexions (`Japanese courtesy bow') by standing subjects. Different spring loads applied to the subject's back were used to introduce smooth perturbations that are necessary to reconstruct ETs based on a series of trials at the same task. Time patterns of muscle torques were calculated using inverse dynamics techniques. A second-order linear model was employed to calculate the instantaneous position of the spring-like joint or center of mass characteristic at different times during the movement. ETs of the joints and of the center of mass had significantly different shapes from the actual trajectories. Integral measures of electromyographic bursts of activity in postural muscles demonstrated a relation to muscle length corresponding to the equilibrium-point hypothesis. Received: 3 March 1997 / Accepted in revised form: 2 November 1998     

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.     

Three-dimensional kinematic analysis of influence of hand orientation and joint limits on the control of arm postures and movements

The problems related to kinematic redundancy in both task and joint space were investigated for arm prehension movements in this paper. After a detailed analysis of kinematic redundancy of the arm, it is shown that the redundancy problem is ill posed only for the control of hand orientation. An experiment was then designed to investigate the influence of hand orientation on the control of arm movements. Since movements must be made within the limits of the joints, the influence of these limits was also analyzed quantitatively. The results of the experiment confirm that the increase of movement time because of the change of object orientation is due to the lengthening of the deceleration phase disproportionately to the rest of the movement. The variation of hand path due to the change of object orientation was observed as being surprisingly small for some subjects as opposed to the large range of object orientation, implying that hand path and hand orientation could be controlled separately, thus simplifying the computational problem of inverse kinematics. Moreover, the observations from the present experiment strongly suggest that a functional segmentation of the proximal and distal joints exists and that the control of wrist motion is dissociated from the rest of joint motions. The contribution of each joint in the control of arm movements could be determined through the principle of minimum energy and minimum discomfort under the constraints of the joint limits. A simplified inverse kinematics model was tested. It shows that these hypotheses can be easily implemented in a geometric algorithm and be used to predict arm prehension postures reasonably well under the constraints of joint limits. Received: 6 August 1998 / Accepted in revised form: 16 December 1998     

Reconstruction of equilibrium trajectories and joint stiffness patterns during single-joint voluntary movements under different instructions

 A method for reconstructing joint compliant characteristics during voluntary movements was applied to the analysis of oscillatory and unidirectional elbow flexion movements. In different series, the subjects were given one of the following instructions: (1) do not intervene voluntarily; (2) keep the trajectory; (3) in cases of perturbations, return back to the starting position as quickly as possible (only during unidirectional movements). Under the instruction ‘keep trajectory’, the apparent joint stiffness increased by 50% to 250%. During oscillatory movements, this was accompanied by a decrease in the maximal difference between the actual and equilibrium joint trajectories and, in several cases, led to a change in the phase relation between the two trajectories. The coefficients of correlation between joint torque and angle were very high (commonly, over 0.9) under the ‘do not intervene’ instruction. They dropped to about 0.6 under the ‘keep trajectory’ and to about 0.3 under the ‘return back’ instructions. Under these two instructions, the low values of the coefficients of correlation did not allow reconstruction of segments of equilibrium trajectories and joint stiffness values in all the subjects. The results provide further support for the λ-version of the equilibrium-point hypothesis and for using the instruction ‘do not intervene voluntarily’ to obtain reproducible time patterns of the central motor command. Received: 14 December 1993/Accepted in revised form: 16 April 1994     

The influence of microgravity on memorized arm movements

To investigate sensory and motor functions in microgravity, goal-oriented arm movements were performed by 9 cosmonauts in weightlessness. The ability to reproduce predefined motor patterns was examined pre-, in-, and post-flight under two different paradigms: In a first test, the cosmonaut had to reproduce passively learned movements with eyes closed, while in the second test, the cosmonaut learned the pattern with eyes open. The different learning paradigms effected the metric parameters of the memorized stimulus pattern while the influence of the different gravity levels resulted in significant offsets and torsions of the reproduced figures. In comparing the inflight condition with preflight, intact proprioceptive afference seemed to play an important role for reproducing movements from motor short-time memory correctly.     

Intermediate representations in the formation of arm trajectories

Psychophysical evidence shows that the planning of an arm trajectory is specified by the central nervous system in extrinsic coordinates. The complex issue of translating the planning of arm movements into muscle forces is discussed in relation to the recent discovery of structures in the brainstem and in the spinal cord. These structures represent discrete maps of motor behavior. Remarkably, the force outputs, produced by activating different zones of the map, sumvectorially. This vectorial combination of motor outputs is a mechanism for producing a vast repertoire of motor behaviors in a simple fashion.     

Octopuses use a human-like strategy to control precise point-to-point arm movements

One of the key problems in motor control is mastering or reducing the number of degrees of freedom (DOFs) through coordination. This problem is especially prominent with hyper-redundant limbs such as the extremely flexible arm of the octopus. Several strategies for simplifying these control problems have been suggested for human point-to-point arm movements. Despite the evolutionary gap and morphological differences, humans and octopuses evolved similar strategies when fetching food to the mouth. To achieve this precise point-to-point-task, octopus arms generate a quasi-articulated structure based on three dynamic joints. A rotational movement around these joints brings the object to the mouth . Here, we describe a peripheral neural mechanism-two waves of muscle activation propagate toward each other, and their collision point sets the medial-joint location. This is a remarkably simple mechanism for adjusting the length of the segments according to where the object is grasped. Furthermore, similar to certain human arm movements, kinematic invariants were observed at the joint level rather than at the end-effector level, suggesting intrinsic control coordination. The evolutionary convergence to similar geometrical and kinematic features suggests that a kinematically constrained articulated limb controlled at the level of joint space is the optimal solution for precise point-to-point movements.     

The representation of arm movements in postcentral and parietal cortex

Considerable experimental evidence supports the hypothesis that the neocortical processes underlying kinesthetic sensation form a hierarchical series of cells signalling increasingly complex patterns of movement of the body. However, this view has been criticized and the data lack quantitative verification under controlled conditions. These studies have also typically used one-dimensional (reciprocal) movements, even of multiple degree-of-freedom joints such as the wrist or shoulder, and have been restricted to passive movements. This latter limitation is particularly critical, since the response of many muscle receptors is affected by fusimotor activity while that of many articular receptors is sensitive to the level of muscle contractile activity. Both factors introduce significant kinesthetic ambiguity to the signals arising from these receptors during active movement. This ambiguity is evident in the discharge of primary somatosensory cortex proprioceptive cells. Studies in area 5 show that single cells signal shoulder joint movements in the form of broad directional tuning curves. The pattern of activity of the entire population encodes movement direction. The cells appear to encode spatial aspects of movement unambiguously, since their discharge is relatively insensitive to the changes in muscle activity required to produce the same movements under different load conditions. It is not yet certain whether the somesthetic activity in area 5 is a kinesthetic representation that is sequential to and hierarchically superior to that in SI, or whether it is a parallel representation with separate and distinct function.     

Planning and execution of hand movements

The presented approach focuses on the attempt to specify strategies of visually organizing sequences of different hand movements and the resulting fine-control of movement close to the target, since in skilled activities our ability to sequence a number of separate movements each having different spatiotemporal characteristics is of central importance. Sequences of different moves have been analysed here. After a gross distance covering part of the move, small correction movements are performed close to the target to reduce the position error. The length of the required correction movements and the corresponding positioning time increase with target distance. In order to investigate motor control strategies two different tasks, with and without time pressure, have been designed. Time pressure forces the subjects to finish the previous move and to prepare the next move simultaneously. Absolute, constant and variable errors revealed that under time pressure the subject changes the control strategy by increasing the constant and reducing the variable error.     

Stability and motor adaptation in human arm movements

In control, stability captures the reproducibility of motions and the robustness to environmental and internal perturbations. This paper examines how stability can be evaluated in human movements, and possible mechanisms by which humans ensure stability. First, a measure of stability is introduced, which is simple to apply to human movements and corresponds to Lyapunov exponents. Its application to real data shows that it is able to distinguish effectively between stable and unstable dynamics. A computational model is then used to investigate stability in human arm movements, which takes into account motor output variability and computes the force to perform a task according to an inverse dynamics model. Simulation results suggest that even a large time delay does not affect movement stability as long as the reflex feedback is small relative to muscle elasticity. Simulations are also used to demonstrate that existing learning schemes, using a monotonic antisymmetric update law, cannot compensate for unstable dynamics. An impedance compensation algorithm is introduced to learn unstable dynamics, which produces similar adaptation responses to those found in experiments.     

Measuring multi-joint stiffness during single movements: numerical validation of a novel time-frequency approach

This study presents and validates a Time-Frequency technique for measuring 2-dimensional multijoint arm stiffness throughout a single planar movement as well as during static posture. It is proposed as an alternative to current regressive methods which require numerous repetitions to obtain average stiffness on a small segment of the hand trajectory. The method is based on the analysis of the reassigned spectrogram of the arm's response to impulsive perturbations and can estimate arm stiffness on a trial-by-trial basis. Analytic and empirical methods are first derived and tested through modal analysis on synthetic data. The technique's accuracy and robustness are assessed by modeling the estimation of stiffness time profiles changing at different rates and affected by different noise levels. Our method obtains results comparable with two well-known regressive techniques. We also test how the technique can identify the viscoelastic component of non-linear and higher than second order systems with a non-parametrical approach. The technique proposed here is very impervious to noise and can be used easily for both postural and movement tasks. Estimations of stiffness profiles are possible with only one perturbation, making our method a useful tool for estimating limb stiffness during motor learning and adaptation tasks, and for understanding the modulation of stiffness in individuals with neurodegenerative diseases.     

Simplified dynamics model of planar two-joint arm movements

A theoretical framework is presented that describes a way in which the inverse dynamics equations of motion of planar two-joint arm movements (EX-model) are reformulated in a simple form. A single point was assumed to define both the wrist and elbow joint centers, and thus the motion of two points in extrinsic space was represented by second-order differential equations to provide the variables in the reformulation (RE-) model. Through an analytical processes, it was shown that the RE-model for reproducing the shoulder joint torque consists of the linearly scaled moment per unit mass responsible for accelerating the wrist and elbow points about the shoulder joint, while that for reproducing the elbow joint torque consists of the linearly scaled moment per unit mass responsible for accelerating the wrist point about the elbow. The scaling factors for variables in the RE-model were based solely on the values for segment lengths, while in the EX-model the inertial parameter data for the segments are involved in its representation. The inertial parameter data of six-arm specimens from the cadaver experiment of Chandler et al. (1975, AMRL Technical Report, Wright-Patterson Air Force Base, OH) were used to develop and verify the numeric solutions of the RE-model. The adequacy of the model varied somewhat among subjects, but minor changes of the physical parameters of the arm segments enabled perfect reformulation, regardless of the specimens. The potential abilities of the RE-model to deal with the complexities in motor control with more simple control schemes are discussed.     

Physiological effects of using the low intensity strength training without relaxation in single-joint and multi-joint movements

The effects of classical strength training (CT) and low intensity strength training without relaxation (TwR) upon size, strength and fatigability of leg muscles in men were compared. A 8-10-week strength training led to an increase of size and maximal voluntary contraction of trained muscles. After the CT, the increment of strength was higher; on the other hand, strength increments related to total work performed increased after the TwR noticeably higher than after the CT. Two training programs influenced the size of total muscle and of muscle fibers (MF) differently: the volume of m. quadriceps femoris increased more after the CT than after the TwR. The CT induced a significant increase of cross sectional area (CSA) of fast MF, and the TwR induced an increase of CSA of slow MF. Resistance to fatigue after the TwR was higher than after the. The effects of TwR were more pronounced in single-joint movements training than in multi-joint movement.     

A model of force and impedance in human arm movements

This paper describes a simple computational model of joint torque and impedance in human arm movements that can be used to simulate three-dimensional movements of the (redundant) arm or leg and to design the control of robots and human-machine interfaces. This model, based on recent physiological findings, assumes that (1) the central nervous system learns the force and impedance to perform a task successfully in a given stable or unstable dynamic environment and (2) stiffness is linearly related to the magnitude of the joint torque and increased to compensate for environment instability. Comparison with existing data shows that this simple model is able to predict impedance geometry well.     

The role of the plant properties in point-to-point arm movements: a robustness study

In recent papers we demonstrated by means of a modeling study that the smoothness of hand paths and the bell-shaped character of hand velocity profiles which have been experimentally observed in point-to-point arm movements can be largely attributed to the biomechanical properties of the arm rather than to specific planning by the central nervous system. In this paper we present a study of the robustness of our earlier results comprising two goals: (i) the determination of the range of model parameters for which such observations remain valid, (ii) the identification of possible relationships between model parameters and kinematic variables. The results of this study imply three conclusions: (i) the valid range of the tested model parameters (namely the main muscle parameters) is large, (ii) the modeled phenomena are well behaved in that parametric changes do not give rise to bifurcations or other behavioral discontinuities in the analyzed ranges, (iii) there exist precise relationships between certain muscle parameters and the time course of the hand velocity. These results point out that the phenomena observed in our previous work are indeed robust and can lead to useful insights into the mechanisms comprising the regulatory action of the central nervous system as well as into the design principles for biologically inspired artificial arms. Received: 4 December 1995 / Accepted in revised form: 6 November 1996