Motion Cue Tracking Performance

The second element of measuring this system’s performance is the precision with which the user tracks the intended (motion cues) as the trajectory evolves. Even though the time interval between DES-controller symbols ([τc[n], τc[n + 1]) is ap-

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proximately constant, with no control applied to the velocity and acceleration of end-effector movement, there is no reference displacement available each time the end-effector position changes to compare magnitudes. The DES-plant (user) au- tonomously determines the end-effector velocity throughout the course of the reaching task. There is also no available mechanism to maintain consistency of that control from one task to another, either upon the same user or across different users. How- ever, what is consistent is the egocentric directional reference frame. Thus we choose to measure the directional precision of movement for each trajectory segment. To quantify the directional precision of tracking we defined q[n]_R3 to be the the nth segment of the user’s actual trajectory through the reaching task in image space. With the unit vectors in the direction of motion cues axes we can define three scalar quantities, ρ_x1, ρ_x2, ρ_x3R, for each trajectory segment given by

ρ_x1[n] = q[n]·ˆx1 ||q[n]|| ρx2[n] = q[n]·ˆx2 ||q[n]|| (3.57) ρ_x3[n] = q[n]·ˆx3 ||q[n]||

Eachρx[n] gives the directional error fraction (DEF) within the range [−1. . .1] per trajectory segment. Similar to the equation (3.55), ρx is the segment based sequence of movement precision scores. The sequence is constructed from the appro- priate ρx[n] component corresponding to the current ˜r[n]. Essentially it measures quality of the expended effort in the intended direction of motion. If ρx[n] = 1 then entirety of the displacement of the end-effector was along the intended direction of the motion cue, whereasρx[n] = 0 would show orthogonal movement and ρx[n] =−1 would show opposing movement during the n-th trajectory segment.

To quantify the movement precision for the entire trajectory we define the directional tracking error as the average DEF, given by

θe= 1 N −1 N−1 X n=1 ρx[n] (3.58)

Chapter 3: Hybrid System Model for Visual Control 72

directed away from the intended direction indicated by the DES-controller output symbols. Values of θe = 0 indicates that the movements were equivalent to an essentially random sequence, which is considered ill-behaved as there should be some bias towards movement corresponding to the motion cues given. Values of 0< θe ≤β are considered to be well-behaved andβ < θe≤1 are ideally-behaved. The threshold value ofβ allows for the realistic notion that ideally-behaved precision is not actually perfect. From a practical perspective, β is also likely to be unique to each individual and converges after successive sessions of practice.

The overall tracking response metric which quantifies performance of the reach- ing task is given by

ψ = 1 N −1 N−1 X n=1 δ[n]·ρx[n] (3.59)

Referring back to the example depicted in Figure 3.15 we can then quantify the tracking response of the three sample trajectories as:

Q_A: δ[n] ={1,1,1,1}, ρx[n] ={0.707,0.707,0.707,0.707}, and ψ = 0.707

Q_B δ[n] ={1,1,0,1}, ρx[n] ={0.707,0.707,0.707,0.707}, and ψ = 0.530

Ideal: δ[n] ={1,1,1,1},ρx[n] ={1,1,1,1}, and ψ = 1

The scoring assignments above maintain the assumption that all motion is within the plane shown and the sample trajectories are subdivided into four segments, each only spanning one triggering event.

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Chapter 4 Human Motor Control and

Performance

Reaching task movements can be described in a number of different internal and/or external coordinate frames representing the trajectory of the hand from initial position and orientation to final position and orientation at the target. From a neurophysio- logical perspective, internal coordinate frames can be specified in terms of joint-space kinematics, joint-space dynamics, or vectorized patterns of actuating muscle activity at each joint [108]. Measurement of vector quantities for internal coordinate frame based control can range from moderately to extremely cumbersome (possibly inva- sive), and external coordinate frame representation is best suited for the work pro- posed here. We presented a set of proposed metrics in the later portion of the previous chapter derived from our hybrid control model for guidance of the reaching task. How- ever there exists a well established metric within the context of natural human motor performance research which should be examined for completeness. Studies in human motor performance for pointing, or reaching tasks often employ Fitts’ Law[32, 109] to quantify the degree of success relative to a subject’s ability track in on a target. As a metric it quantifies an inherent speed-accuracy trade off that exists in human motor performance. The trade off manifests through a comparison between the predicted movement time required to complete the task and the difficulty of performing the task. The speed-accuracy trade off exhibits a linear proportionality between move- ment time and task difficulty. For the one dimensional case, the movement time,M T, is given by equation (4.1)

M T =a+b·ID (4.1)

whereID is the index of difficulty for the reaching/pointing task, witha and b being experimentally derived constants. Fitts’ Law uses an information theoretic approach to establish a linear relationship between the time required to preform the movement and the index of difficulty for that movement. The unit for ID, given in equation (4.2), is ‘bits’. It shows that there is an inversely proportional relationship between

Chapter 4: Human Motor Control and Performance 74

A, the amplitude of movement, and W, the target width along the line of approach when determining the index of difficulty for a particular movement.

ID= log₂ 2A W (4.2)

The diagram in Figure 4.1 illustrates Fitts’ classic reciprocal tapping test [32]. During the test, a subject is required to start at the midpoint of one target region, then move to and tap the corresponding target point within the opposing target region as quickly and accurately as possible. Upon the targeting tap, they are to continue with a reversal of direction and target the previous target region in successive back and forth motions. This difficulty of the a particular tapping test is determined by the ratio of movement amplitude between targets, A, and trying to tap within a region that is W₂ distance on either side of the target line.

Amplitude (A) Target

Width (W)

Target

Width (W)

Figure 4.1: Illustration of the 1-D Fitts’ reciprocal tapping test configuration

A number of variants to Fitts’ Law have been proposed over the years. One that has gained wide spread adoption is the Shannon formulation proposed by Macken- zie [110] which is more closely related to Shannon’s Theorem for the information capacity of a communications channel, given in equation (4.3),

C =Blog₂ S+N N (4.3)

where the channel capacity, C, is a function of the signal power, S, and the noise power, N, given a channel bandwidth, B. The Shannon formulation of Fitts’ Law is given by M T =a+blog₂ A W + 1 (4.4)

Chapter 4: Human Motor Control and Performance 75

This formulation is preferred as it does not produce an erroneously negative value for ID when the amplitude of the movement is less than half the target width, nor an infinite value when the starting position for the task happens to be the target position as well (A = 0). The other notable formulation for targets constrained in one dimension was proposed by Welford [111] prior to MacKenzie’s formulation, but is very similar.

The reader is referred to MacKenzie’s paper [112] and a follow up work authored with Soukoreff [113], which provide an excellent review of the application of Fitts’ Law as a quantitative performance model in the field of human-computer interfacing.