Measuring the Allocation of Control Across Degrees-of-Freedom
Maurice R. Masliah, Paul Milgram
Ergonomics in Teleoperation and Control (ETC) Lab
Department of Mechanical and Industrial Engineering
University of Toronto, Ontario Canada M5S 3G8
We propose a new metric, theM-metric, that quantifies how a user is allocating control across multiple degrees-of-freedom (dofs) while performing a docking task using an input device. The development of the M-metric is based on two previous metrics from the literature, integrality  and inefficiency . Integrality is a measurement in the time dimension only, and says nothing about the magnitude or direction of movement. Inefficiency is a measurement in the spatial dimension only, and says nothing about the time course of movements. The M-metric incorporates measurement in both the time and space dimensions. Analysis of pilot data from a 6 dof virtual docking task indicates that the allocation of control may be higher within translational and rotational dofs than between translational and rotations dofs.
Key words:M-metric, allocation of control, integrality, inefficiency, multiple degrees of freedom, input devices.
In a multiple degree-of-freedom (dof) continuous movement task, it is possible to have two movements with equal performance scores, but with very different time-space trajectories. This begs the research question: how is control being allocated across the different dofs? Understanding the allocation of control is one step towards understanding human coordination while using input devices in HCI. The goal of this research is not just to say some something qualitative about human coordination, but to quantify the allocation of control with respect to a useful metric.
Jacob et al.  discussed the degree to which input devices with multiple dofs can be characterized as integral, based on diagonal movement across dofs. In Figure 1, Device A is showing evidence of asynchronous "city-block", or separable stair-step, movement between the x and y dofs. Device B exhibits what Jacob et al. termed "Euclidean" movement, i.e. movement that cuts diagonally across the dofs.
Integrality is measured by first segmenting into equal time units the trajectory of each dof . Each segment is then checked for presence or absence of movement above an arbitrarily chosen magnitude threshold. For each time segment, trajectory movement in all dofs classifies that segment as Euclidean. The end result is a ratio of Euclidean to city-block movements for a given task. This ratio is a measure of the integrality of a given input device, which can then be compared to the integrality ratio of other input devices for the same task.
A coordinated movement is generally recognized as being an efficient movement, where the shortest possible trajectory is considered to be the most efficient. Zhai & Milgram  have proposed using the ratio of the actual trajectory to the optimal trajectory as an inverse measure of coordination, i.e. inefficiency. As long as there exists an ideal or an optimal trajectory, as with for example docking tasks, the userís actual trajectory can be compared to the optimal.
Essentially, integrality is a measure of the simultaneity of motion among multiple dofs. Integrality is a measurement in the time domain only, and says nothing about magnitude or direction of movement. Inefficiency, on the other hand, is a measurement in the spatial domain only, and says nothing about the time course of movements made. That is, Zhai & Milgramís definition of inefficiency does not provide any explicit information describing what is happening in one dof as the other dofs are being manipulated. This poster proposes a new, unified metric, the
TheM-metric consists of the product of two components. The first component encompasses three principles; error reduction, normalization, and area. Error reduction is computed for each dof separately, and represents the instantaneous value of the derivative of the error term, but only for positive values, i.e. while error is being reduced. Error reduction is graphed against time and normalized so that the area under each dof curve is constant. Normalization is carried out by dividing the error reduction values by the total distance moved toward the goal, for each dof. In other words, we define the Normalized Error Reduction Function, FX1(t),(where X1, X2,Ö Xn are the dofs being controlled, ACT = length of actual trajectory, t = time) as:
Figure 2 shows the normalized error reduction curves for two dofs. The area of overlap between the curves tells us how control has been allocated between the different dofs during the task, and always has a value between 0 and 1. A value closer to 1 indicates essentially synchronous control across dofs, while a value closer to 0 indicates a switching of control between the dofs. Any number of dofs may be analyzed by computing the overlap between the normalized error reduction curves. The first component of the M-metric is therefore defined as:
Figure 2. A normalized error reduction graph.
Error reduction refers to movements which reduce the distance to the goal. This means that, for example, two or more dofs moving in sync but not in the appropriate manner, according to equation (1), will not generate a high score. In this respect, in contrast to Jacob et al.'s integrality, the M-metric is a task dependent measured.
The second component of the M-metric computes the ratio of the length of the optimal trajectory relative to the length of the actual trajectory for each dof, and then takes the average of these ratios across all the dofs. There are two major differences between the second component and inefficiency as defined by Zhai & Milgram . The first is that the ratio is optimal/actual instead of actual/optimal. This gives ratios between 0 and 1, instead of between 1 and º . The second difference is that the inefficiency ratio is computed for each dof separately, and then averaged by weighted importance. Weights are put on each dof corresponding to its "relative weight" in the task. For example, in a 2 dof task, if x requires 2 units of translation and y requires 5 units of translation, then x has a weight of 2/7 and y has weight of 5/7. The second component of the M-metric, is therefore defined as (where OPT= length of optimal trajectory):
Pilot data from a 6 dof virtual docking task have provided some evidence that novices find it difficult to simultaneously control a large number of dofs concurrently. To reduce the complexity of the task, subjects control subsets of the dofs, with translation and rotation subsets being the natural groupings. Higher M-metric values have been measured within translational and rotational dofs, than between translational and rotational dofs.
 Jacob, R. J. K., Sibert, L. E., McFarlane, D. C., and Mullen, M. P. JR. Integrality and Separability of Input Devices, ACM Transactions on Computer-Human Interaction 1, 1 (1994), 3-26.
 Zhai, S. and Milgram, P. Quantifying Coordination in Multiple DOF Movement and Its Application to Evaluating 6 DOF Input Devices, In Proceedings of the Conference on Human Factors in Computing Systems CHI '98. ACM, (1998), 320-327.