This papers develops a number of mathematical models for measuring human performance. This is done by modeling the arm like a robot controller and using the DH representation. From this, they define a number of measures, such as reachability, joint functionality, dexterity, effort, joint stress, potential and kinetic energy, force, work and power.

The difference between the surface that can be fitted to rotation vectors during movement and when stationary is considered. They found that a second order surface could be fitted to both types of movements, but that there was less scatter for stationary movements. For upper arm movements (they also considered head movements) they found a correlation between the velocity and the distance from the plane.

As opposed to point to point movements in the horizontal plane, such movements in the vertical plane are not always straight line, they are sometimes curved. However, the path is invariant and velocity has a similar bell shaped curve.

This review looks at the combination of visual and proprioception is motor movements. He considers the activity of neurons in motor systems in the brain. In cases where contradictory information is presented from the two, a decision is made about which to accept more. Either of the two alone seems to be insufficient to specify completely the arm position, however the combination of the two specify it well.

A method is presented for accurately reconstructing 7 joint angles of the arm from placing 4 Fastrak markers (each of which determines the position and orientation of the marker based on measuring a magnetic field).

An experiment was performed when subjects had to move a rod towards a target in a dark room. After movement onset, either the target or the light showing the position of the hand moved slightly. The results showed that while subjects correctly accounted for a change of target location, they used the action hand location rather than the visually defined hand movement. They conclude that this suggests visual feedback of the hand is not used in control of the movement.

Different results are seen for double step hand movements in a plane when there is a gap between when the first disappears and the second target appears. When there is a gap, it takes longer, meaning that a different process is probably used because it has less information (it cannot use retinal error.

In this experiment subjects had to `hit' with a rod a target projected using stereo shutters, that sometimes followed the double step paradigm. By varying the movement of the background, it was ascertained that the movement was adjusted in accordance with the change in position rather than the change in velocity.

A review of types of virtual reality and the tools for interacting with the virtual environment

For straight arm pointing, the arm was found to assume a similar position at a given location regardless of the starting position, similar to Donder's law. Also, the angular position vectors lay in a surface similar to a Fick gimbal (a fixed vertical axis with a horizontal axis that `twists'). The results were only shown for straight arm pointing.

When performing a reaching and grasping task for a cylinder, the final posture of the arm is not invariant, but does depend, in a systematic way on the starting position. They suggest that such findings that show that posture is invariant are due to specific constraints. However they found that the strategies used were similar between subjects.

Prehension movements during reach to grasp a cylinder at a range of orientations were studied. Sometimes the cylinder orientation was perturbed at movement onset. The final limb joint angles were highly predictive despite redundancy, and the same final posture was found also after the perturbation. during the movements, a generalized synergy was observed, that is, a linear relationship was seen between most of the joint angles. They suggest that movement are planned in joint angles using postural transitions.

An experiment was performed to test the hypothesis that 3D upper arm movements can be planned using a mechanism based on comparing the current joint angle with the desired joint angle. They claim that such a mechanism should predict invariant posture despite kinematic redundancy and invariant joint coarticulation in joint space and variable curvature in task space. They tested this using perturbed trials where a cylindrical object which the subject had to grasp was sometimes rotated at movement onset. They observed these predictions, although the elbow joint angles passed through a via point (intermediate posture). No second velocity peak was seen for the perturbed movements, rather a smooth transition.

Donder's law, in general, for arm movements does not hold. That is you cannot predict the orientation of the arm in the final position just from knowing the position. They did find that instead the final position is one that required minimum work from the initial position

If while executed one point to point movement in the horizontal plane and the plan is changed to move to a different location, then the original plan is not aborted, rather a superposition of the original plan and a movement from the first to the second target are combined

In the horizontal plane, point to point movements have a straight line path. In curved and straight movements, the smoothness is ensured by a trajectory that minimizes jerk. This is shown under a variety of curved paths, modeled using via points, where the optimal path matches well to the experimental data.

Here a real-time implementation of the superposition strategy, as found for human horizontal planar movements, is provided for robotic manipulators for use when trajectory modification takes place. The generated trajectories satisfy the minimum jerk model. An algorithm for orientation superposition is also presented, dealing with the problem of non-commutation of rotations

This paper describes several techniques for dealing with kinematically redundant manipulators in biological cases, and also considers robotic manipulators, while noting that the constraints applied can be quite different for the two. They consider the equilibrium point hypothesis, generalized inverse methods, constraints to reduce the number of degrees of freedom, reduction of 3D rotational degrees of freedom (eg Donder's law), Neural network techniques and estimating the joint torque contribution of muscles. They note that each of these methods has inherent problems.

This paper shows that the combined eye-head system does not follow Listing's law but does still follows Donder's law. The surface that it is constrained to is similar to a Fick gimbal (although a second-order surface gives a better fit). These results were not shown for earlier studies probably because they used a smaller gaze range. They speculate that the reason for a Fick gimbal is because the eye in head is anatomically like a gimbal. Such a system also prevents the accumulation of torsion.

In previous experiments with grasping a cylinder, the findings of invariance of hand configuration were damaged because of the fact the grasping a cylinder largely coerces the arm to a particular configuration. In this experiment, they needed to grasp a small sphere at different locations, that sometimes jumped after movement began. The results suggested that the final arm posture is invariant for a given final object location even when they have to reorganize movements in response to a sudden change in location. ie a form of Donder's law holds.

Previous studies have shown that eye saccades and arm pointing movement are coupled, and that the saccade begins prior to arm movement. However, by examining the electromyographic (EMG) activity of the shoulder muscles, it was found that the arm movement is actually initiated earlier than the saccade - the delay is due to factors such as the relatively high inertia of the arm. The suggest based on their findings that the direction and distance of the arm movement is specified prior to the saccade onset - feedback from the eye movements may be used for corrections on a trial-to-trial basis.

Eye rotations are analyzed in terms of a Lie algebra group, and conversions are made between different systems (between canonical (about some axis), Fick and Helmholtz). He explains the elegance in using a Lie group to represent rotations due to their non-commuting properties, and expresses Listing's and Donder's law in these terms.

It is proposed that trajectories are selected to minimize the variance of the final eye or arm position. This is based on the assumption of signal-dependent noise in the neural control signals. This theory fits well with experimental observations, predicting bell shaped velocity profiles, the speed-accuracy trade-off, as well as the two-thirds power law. They comment that this theory is more general than theories such as the minimum jerk or torque, and is more biologically plausible.

An extension of the minimum jerk model is made to incorporate duration. The cost function now includes time and a constant that is a relative weighting factor between time and jerk. This constant would change depending on the desired speed and accuracy of the movement (ie the speed-accuracy trade off).

A review of the minimum jerk strategy and of considerations in modeling (such as joint vs hand motion) is presented.

In a double step target task, subjects were told sometimes to correct to the new target (pointing), and sometimes to move in the opposite direction (anti-pointing). While in pointing tasks there was a rapid correction, and delayed awareness, in anti-pointing tasks sometimes the subjects began as in a pointing task, the correction times were longer, and there was not a dissociation between awareness and the action. From these findings they suggest the a different, slower system is used for anti-point compared to point movements.

The predictions from a model that uses local feedback (ie an efference copy, not proprioception) compared to an ``open loop'' (no feedback) are compared in the situation where drugs are used to reduce saccade velocity. The results showed that despite the slowing in velocity, amplitude remains constant, weighing in favour of a closed loop theory

For a given final hand position and orientation, there is an arc of 360 degrees for where the elbow could be. However, after applying the physiological constraints, this is reduced to a sixth of this size. This suggests that the orientation and position are not planned independently, but rather together.

The minimum-jerk and minimum torque-change models are compared in modified forms where the approach angle and speed is specified. They found that the minimum-torque change predicted trajectories were closer than those for the minimum-jerk model. They concluded that some optimization took place in intrinsic parameters but did not rule out that optimization also included variables in extrinsic space.

A comparison is made between the minimum-jerk and minimum torque-change model both in terms of path, as a function of time and in terms of cost (ie total jerk or torque-change). The movements considered were nearly horizontal planar movements with the arm supported to prevent gravitational forces and the targets and feedback of the current movement were displayed on a screen. In terms of the path, the minimum-torque change model was a good model whereas the minimum jerk was not. However, when the timing was considered, the observed trajectories were more similar to the predictions of the minimum jerk model. It was also found that in terms of movement cost (ie total jerk), for slower movements the time of peak velocity is less critical than for faster movements and so asymmetrical velocity profiles are less of an issue when the movement is slow.

Coarticulation is considered in three dimensional arm movements. It was not seen in two-segment movements, but was observed in the angle of the elbow when drawing triangles. It was claimed that this coarticulation improves the efficiency of the second movement. Donders' law was not observed, rather an accumulation of changes in arm posture was seen.

An experiment involving movement from a horizontal table to a vertical screen were analyzed to try and determine whether they are planning in workspace or in joint space. They concluded that a trade-off is made between optimizing the two, and that the movements tends to be within a plane in both workspace and joint space

The Superior Colliculus (SC) provides an input for a desired saccade movement. It has been debated whether this is gaze displacement (independent of current orientation) or gaze direction. It was concluded from an experiment with microstimulation of the SC that these are actually encoded as displacements, but in a retinal frame. Downstream structures must then transform this into gaze displacement commands.

Patients recovering from a stroke showed movements that were clearly segmented, but became more `smooth' as they recovered. It is suggested that these movements are the primitive units used during unloaded movements, and these movement primitives each have a bell-shaped speed profile.

Marr proposed that the visual system uses a 2 1/2-D sketch, where there is an accurate representation of the horizontal and vertical position (from a viewer centred perspective) of objects, and of surface orienation, but only a rough representation of the depth.

Experiments were performed on single and double-step eye saccades. It was found that Listing's law is implemented in all these cases by tilting of the angular velocity axis. However, in 3D analysis, it was found that in double-step saccades with mid-flight change in trajectory, a linear superposition of angular velocities can not be used because a tilt in the angular velocity vector is necessary to maintain Listing's law. So the superposition would have to be done before the angular velocity is computed. They suggest perhaps a superposition of velocity commands at the 2D level or some sort of local feedback control

It is found that Listing's law is held much better for eye in head, than eye in space. This was tested by comparing head fixed and head free saccades. They also had a task involving eye fixed in head gazes, which were impossible, except to a small degree. By will, the effective oculomotor range (EOMR) can be reduced, it can be further reduced by restricting vision (slit glasses or pinhole glasses), although the EOMR does not become as small as the visible range. These effects can be explained by a model for eye-head movements.

The tangential hand velocity for different point to point movements in the horizontal plane has a single peaked curved that is invariant under different movement times. This may lead to the hypothesis of control in ``spatial'' coordinates, rather than joint coordinates.

Previous experiments with spinalized frogs showed that micro-stimulation of the premotor areas in the grey matter of the spinal cord resulted in convergent field of forces being applied, due to a balanced recruitment of agonist and antagonist muscles. Different regions produced different force fields, and if two stimulations were applied simultaneously, then the field added vectorially. These force fields may hence be combined to create a repertoire of motor actions.

A comparison is made between the minimum jerk, minimum angle-jerk, minimum torque-change, and the minimum commanded torque change models. They found that the best predictions were provided by the minimum commanded torque change model, which is similar (but computable) to the minimum motor command model.

Arm movements performed at different speeds are compared in order to determine what sort of model is used for arm movements. If a dynamic model is used, then the trajectory would be expected to change with different speeds. However, the trajectories seen in this paper were speed-invariant, implying that a kinematic scheme is used, for example, planning using velocity constraints.

This paper explains how to calculate the mean of a curve and estimate the variance. This is for use in cases where you don't know the equation of the curve. They suggest calculating the mean using penalized least squares (so it will be smooth), and to analyze the variance by looking at the eigenvalues.

The generalized form of the minimum jerk law is examined (MSD - mean squared derivative cost functions), where the derivative of the coordinates may be of order n up to infinity, and its relation to the two-thirds power law. In the case of reaching movements, only minimum jerk or snap reasonably match experimental data. However, for ellipses, the law holds for the entire class of MSD cost functions. In more complex shapes, the plot of curvature against velocities looks like segmented control although this was not specified in the cost function. This case doubts on the use of this feature as an argument for segmented control.

This is an approach for trajectory planning based on the notion of stored postures. Once a final location to move to has been decided, a number of postures are examined, and assigned weights depending on how close they are to the desired location and the travel costs. These postures are then `summed' vectorially, to give the target posture. The trajectory between the starting and target posture are generated by allowing each degree of freedom to move continuously from its initial to the final posture. Hence these trajectories are dependent on the starting location hence contradict the existence of a form of Donder's law, or Listing's law.

This review looks at the use of internal models in planning and controlling reaching movements. He suggests that a mix of feed-forward (the movement planning before execution) and feedback (using a combination of proprioception and visual feedback) is used to control the arm.

In multijoint tasks, there are generally many possible joint combinations. It is shown that rather than always adopting a particular joint combination during such a task, it is possible to define a set of relevant task variables. These can be combined to form a UCM ("uncontrolled manifold"). The different joint configurations were decomposed into those parallel to the set (essential parameters) of task variables, and those perpendicular (unessential). For the example, with pistol shooting, the orientation of the guns barrel relative to a vector from the gun to the target was a task variable whose UCM showed a significant difference between the joint configurations.

Coding of coordinates could be done in eye-centered or hand-centered coordinates. For example, following a target with no arm movement would be in eye-centered, while moving the arm without vision would be in arm-centered. This review finds that it seems that both representations are used, perhaps at the same time, for different uses, and examines where in the brain the coordinate transformation takes place

A method is presented for determining the translation and rotation of a frame by means of a number of landmarks (markers). The method uses singular value decomposition.

Presents the quaternion model which requires eye movements to be planned in three dimensions and shows experimental data supporting this model.

A model of a 3D saccade generator is presented. It receives as input from the Superior Colliculus (SC) a required displacement signal (the error). The current 3D angular velocity required is generated from this error, and the velocity is integrated with a resettable integrator to update the error. Note that the error here is fed back, not the current position. The angular velocity at the end passes through a neural integrator to give the current orientation. This model accurately predicts that faster (rather than larger) saccades are produced with stronger SC stimulation (the magnitude of the error is set spatially). In this model, the Listing's law operator is applied before the required displacement is calculated, i.e. it is an orientation constraint.

A comparison is made of two Oculomotor Velocity-to-Position transforms, one which is a quaternion, non-commutative model, another is a commutative approximation. It was found that the non-commutative model is a poor approximation, and much better results are found using the quaternion model.

The different representations of rotations are considered, and their advantages and relation to each other.

By a simple test, it is shown that a non-commutative model is required to explain the VOR reflex - a commutative model will predict the same path for two rotation regardless of the order. This is not found experimentally, hence a non-commutative model is required.

From a pointing experiment with no visual feedback performed in two conditions (where the hand was visible before the task, and when in was not), it was found that the error in locating the target depended on the starting location and was consistenly biased. The errors where much less when the hand could be viewed before the movement. They suggest that their results support the theory that movements are planned vectorially rather than in absolute positions in space.

An experiment was performed where subjects pointed, in darkness, in a plane to a target briefly lit by a laser from different starting positions. The errors in locating the final point were dependent on the starting point, and the major axis of the ellipse of the variables errors was aligned with the movement direction, suggesting a coordinate system based on the hand, and that direction and amplitude are planned independently. The errors became progressively worse during the experiment, which they suggest is due to incorrect assumptions made by the subject about the end point location due to a lack of visual feedback. They suggest their findings are incompatible with theories that say the final joint angles are input to the motor program, or on selecting postures.

Here is it suggested that there is a segmentation in the planning of the orientation and position of the arm in trajectory planning. This was found from experiments where for different orientations, the variation in hand position was small compared to the possible range. He also looked at the accuracy of planning in the different phases of the movement