Efficiently Generating the Ballistic Phase of Human-Like ... - IEEE Xplore

IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 19, NO. 6, DECEMBER 2014

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Efficiently Generating the Ballistic Phase of Human-Like Aimed Movement Jeffrey N. Shelton and George T.-C. Chiu, Member, IEEE

Abstract—Human-like movement can be efficiently generated by driving a damped inertial plant toward a fixed target with positionbased control, then switching to conventional feedback control as the desired endpoint is approached. Key characteristics of human motion, not easily replicated with traditional control architectures, result from tracking a position-based actuation template derived from human trials. Computation of the applied forcing function requires only linear scaling of this template, or displacementnormalized actuation program (DNAP). Simulated ballistic movements generated with the proposed method are shown to be consistent with human subject kinematic trajectories. Index Terms—Human movement, motion control, normalizeddisplacement control, trajectory planning.

I. INTRODUCTION URRENT-GENERATION android robots, such as the Repliee Q2 developed by Osaka University’s Intelligent Robotics Laboratory [1], are quite life-like in appearance, but unable to produce biologically realistic movement. Upon watching android robots make body movements, the temporal, parietal, and frontal brain areas of naive human subjects react to the mismatch between biological appearance and mechanical motion. This reaction is not seen when the actor is human (biological appearance and motion), nor when the actor is clearly not human (mechanical appearance and motion) [2]. Thus, humans are able to distinguish between their own movement and that of today’s best mechanical imitations. While conventional motion control technology allows machines to accurately repeat a trajectory on a nearly indefinite basis, biological creatures cannot precisely repeat aiming movements [3]–[5]. Android robots using standard motion controls therefore produce movements that are too consistent to be biologically generated. Further, the variation seen in human movement exhibits structure; distinct stereotypical patterns are evident in individual and mean trajectories, as well as in kinematic and temporal variances [6]. Specifically, targeted human movement exhibits the following characteristics:

C

C1. Sinusoidal Acceleration Curve: A roughly sinusoidal accelerative trajectory, without discontinuity, is produced in pointing tasks [3], [7], exhibiting approximately equal periods of acceleration and deceleration [8].

Manuscript received October 1, 2012; revised December 13, 2013; accepted February 19, 2014. Date of publication May 2, 2014; date of current version June 13, 2014. Recommended by Technical Editor Y. Li. The authors are with the School of Mechanical Engineering, Purdue University, West Lafayette, IN 47907 USA (e-mail: [email protected]; [email protected]). Digital Object Identifier 10.1109/TMECH.2014.2316156

C2. Kinematic Uniqueness: Each discrete movement follows a unique trajectory, even if the reaching task remains unchanged [3], [4]. C3. Bell-Shaped Positional Variance: In spatially constrained tasks, positional variance rises and falls in a bell-shaped manner, with a mid-point peak that is significantly larger than the endpoint variance [9], [10]. C4. Bell-Shaped Velocity Curve: Velocity in aimed movement rises and falls along a bell-shaped curve [3], [4]. C5. Fitts’ Law: For spatially constrained tasks, the average movement time (MT ) rises with task difficulty (ID), which is a function of target displacement (D) divided by target width (W ) [11]. Thus, movement time is highly dependent upon the ratio (D/W ), rather than on movement size alone. C6. Temporal Distribution: Individual trial movement durations vary noticeably, even for tasks of the same configuration [9], [12]. Conventional motion control methods are not intended to produce human-like movement, and their shortcomings in this regard generally fall into one of three categories. First, the resulting mean kinematic trajectories often fail to agree with characteristics C1, C4, and C5. Second, trial-to-trial variations may be insufficient to accommodate characteristics C2, C3, and C6. Finally, those few control methods that might meet most (or all) of the aforementioned movement characteristics do so at a high computational cost. A new approach is introduced below for producing humanlike trajectories when rotating a rigid body about a fixed axis. Despite its current restriction to a single degree-of-freedom, our approach is not without useful application. Early explorations into the connection between machine motion and human reaction are being conducted elsewhere [13]; manipulation of a rigid “stick” from one end produces abstract motions that reveal underlying emotional responses in human subjects. “Humanlike” single-axis motions may also prove useful in describing end effector paths that can be invoked through inverse kinematics for multilink actuator arms. System linearization has been accomplished elsewhere for multilink manipulators through the use of software compensation [14]; future studies will examine the possibility of extending the proposed method in a similar manner. We first examine, in Section II, the shortcomings of existing motion control methods with respect to generating biologically accurate movement. In Section III, we describe the construction of actuation programs from experimental trials involving human subjects. Section IV covers how “human-like” movement can be generated from these forcing function templates, while the

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kinematics of such movements are analyzed in Section V. We offer concluding remarks in the final section of this paper. II. SHORTCOMINGS OF EXISTING METHODS Mechanized point-to-point motion is often achieved by selecting an appropriate velocity profile (trapezoidal, s-curve, etc.), then designing a feedback control to ensure that the reference trajectory is accurately tracked [15]. Neither a trapezoidal velocity profile, nor an s-curve profile, conforms with characteristic 1, as they both produce accelerative discontinuities. The desired sinusoidal acceleration may be generated with a “super” s-curve [16], but this approach, like most other profile methods, seeks to reach the system’s upper acceleration limit during all movements of sufficient duration. Under such control, short movements conclude faster than longer movements, regardless of target width; this runs contrary to characteristic C5. Time-based measurements of human movement, gathered through motion capture [17], are sometimes directly tracked in an attempt to generate human-like kinematics. However, this approach falls short on at least two fronts. First, linear scaling of any single time-based kinematic template can only take place if the movement duration remains unchanged. That is, doubling the acceleration requires that both the velocity and displacement also double if the resulting kinematic trajectory is to remain geometrically similar with the original template— but all movements scaled in this manner must conclude in the same amount of time. Since, per Fitts’ law (C5), a subject’s average movement duration varies continuously with the ratio (D/W ), an infinite library of trajectories is required to replicate the limitless number of mean kinematic profiles necessitated by all possible movement durations. Second, since no two trials follow precisely the same trajectory, accurate replication of human motion requires the ability to generate new trajectories that remain consistent with the stereotypical characteristics listed above; simply tracking previously executed movements is not sufficient. Trajectory generation for noncaptured aiming tasks has been addressed by breaking measured movement down into primitives for later synthesis [18], and by using neural networks to interpolate between known movement trajectories [19]. Another method of trajectory generation involves capturing the step response of linear filters designed to reproduce human kinematic paths. However, each of these methods produce mean trajectories, and do not account for the kinematic and temporal variations seen on a trial-to-trial basis. Path interpolation methods can, of course, be constructed to accommodate stochastically distributed trajectories [20], but we are not aware of any such approach that does so while dealing with velocity variations that accompany changes in aiming task difficulty. Within the human movement research community, numerous open-loop models have been developed to describe the mean kinematic trajectories of aimed movement [21], [22]. These models usually minimize a particular movement property, such as energy, jerk, or torque change [23], [24]. Such time-based models are unable to produce a rise and fall in positional variance, as the lack of feedback precludes any corrective ac-

tions capable of lowering the positional spread; this runs afoul of characteristic C3. A pure feedback approach, such as stochasticoptimal control [25], complies with many of the human movement characteristics, including the production of appropriate variance curves [26]. However, this ability comes with a high computational cost. Such models also generally require movement time to be specified in advance. Therefore, to accommodate the duration changes mandated by characteristic C6, a new optimal trajectory needs to be computed for each trial. No advance knowledge of movement duration is required using the method outlined below. It has been previously noted that kinematic features of human reaching appear largely repetitive, especially over the leading half of a targeted movement. The two-component model [27] proposes that an initial ballistic phase brings the limb close to its intended target, and a concluding fine-adjustment phase produces small corrections needed to place a limb within the presented target bounds. This paper focuses on the ballistic phase, as the corrective phase can be achieved with conventional feedback techniques. III. ACTUATION PROGRAM CONSTRUCTION We define an actuation program as the forcing function required to move a single-DOF mechanism in a manner that closely matches the kinematic trajectories of a human subject. To construct such a template, a human subject is asked to repeat a specified aiming task n times, while displacement, velocity, and acceleration data is collected at uniform time intervals of Δ. This operation is repeated across p task configurations, with each configuration presenting the subject with a different pairing of target distance and width; this results in a total of (n · p) subject trials. N samples are collected during each trial window of N Δ seconds. Allow i = 1, 2, . . . , p to identify the task configuration, j = 1, 2, . . . , n to track the trial number, and k = 1, 2, . . . , N to mark the sample index. Let di (j, k) represent the displacement sampled at time index k for trial j in configuration i, with di (j, 1) = 0. Similarly, allow vi (j, k) and ai (j, k) to represent the velocity and acceleration data. Assume we wish to mimic human forearm movement using a single-DOF robot that rotates a rigid link about a fixed axis in the horizontal plane. The resulting equation of motion is τ = J θ¨ + β θ˙

(1)

where τ is torque applied at the rotational pivot, θ is the rotational displacement, J is rotational inertia about the fixed axis, and β is rotational damping. Dot notation represents differentiation with respect to time. Based on (1), the torque necessary to cause the robot limb to match the human kinematic profile is τi (j, k) = J ai (j, k) + β vi (j, k).

(2)

Thus, kinematic data from each trial produce a torque curve capable of matching the associated human-generated trajectory when applied to the robot arm. We normalize each trace, creating a unitless vector for trial j ∗ in configuration i∗ , hereafter referred to as the normalized actuation function, that has a peak value of

SHELTON AND CHIU: EFFICIENTLY GENERATING THE BALLISTIC PHASE OF HUMAN-LIKE AIMED MOVEMENT

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unity, and is defined as τ¯i ∗ (j ∗ , k) =

τi ∗ (j ∗ , k) max {τi ∗ (j ∗ , k)}

∀k ∈ {1 . . . N } .

Target

(3)

For each trial, let kf represent the sample index where the movement makes its first upward pass of acceleration through zero. This marks the outer limit of the ballistic phase. We identify the end-of-ballistic-phase displacement as θi (j, kf ). Then, normalized displacement during each trial is

Swing Arm Displacement, θ

Start Post

θi ∗ (j ∗ , k) θ¯i ∗ (j ∗ , k) = θi ∗ (j ∗ , kf )

∀k ∈ {1 . . . N } .

(4)

Note that θ¯i ∗ (j ∗ , k) exceeds unity as the movement overshoots the ballistic endpoint displacement of θi (j, kf ); this occurs for virtually every movement trial. To accommodate overshoot during the interpolation step that follows, create a uniformly spaced vector ζ  that takes on real values between 0 and at least 1.5 (a value chosen to surpass all actual overshoot ratios), at intervals of δζ  . Chose integer values L and Lex , along with vector spacing δζ  , such that L · δζ  = 1 and Lex · δζ  ≥ 1.5. Elements of the resulting vector are referenced as ζ  (q), where q = 1, . . . , Lex . Interpolating τ¯i ∗ (j ∗ , k) and θ¯i ∗ (j ∗ , k) against ζ  produces a normalized actuation profile, ψˆi ∗,j ∗ (q) for each of the sampled trials. Now arrange the (n·p) normalized actuation profiles (assumed to be row vectors) to produce a 2-D matrix, sized (n·p × Lex ). Eliminate data associated with ballistic overshoot by removing columns q = L + 1 through q = Lex ; this causes the resulting matrix to be sized (n·p × L). Average each column to generate a discretized mean actuation trajectory, which can be expressed as  ˆ  ) = ψˆ (ζ  (q)) = 1 ψˆi,j (ζ  (q)) . ψ(ζ n·p i=1 j =1 p

Fig. 1. Experimental configuration, with subjects rapidly moving swing arm from start post to target.

TABLE I NINE TASK CONFIGURATIONS FOR TARGETED FOREARM MOVEMENT

n

(5)

Although the normalized actuation function for each individual ˆ  ), will have a maxtrial peaks at one, the mean of all trials, ψ(ζ imum value slightly less than one (unless all of the individual peaks happen to align perfectly in normalized displacement). Therefore, the prior result is uniformly rescaled in amplitude to ˆ  peak at unity, such that ψ(ζ  ) = m axψ (ζψˆ (ζ)  ) . We assume func{ } tion ψ to move smoothly between the discrete normalized positions of ζ  , and hereafter reference the continuous function ψ(ζ) as a displacement-normalized actuation program, or DNAP. To demonstrate DNAP construction, we collected kinematic data from seven subjects as they extended their forearms in the horizontal plane, about the elbow, in performing a targeting task that involved a single degree of freedom. Grasping a vertical handle, subjects rapidly positioned a pointer, located at the end of a rotating swing arm, within the bounds of a well-marked target zone. This is depicted schematically in Fig. 1. Chair height adjustment allowed each subject’s right elbow to be horizontally aligned with their right shoulder, effectively tying swing arm rotation to the subject’s forearm movement about the elbow. Each subject was presented with nine task configurations, resulting from combining three target distances with three target widths, as listed in Table I. Fitts [11] defined an index of difficulty, shown

in the rightmost column, to be ID = log2 (2D/W ), where D is the target displacement and W the target width. Performing 60 trials in each of the nine configurations, every subject executed a total of 540 trials. Our experiment captured 2 s of kinematic information, sampled at 500 Hz, during each trial. Rotational displacement was gathered with a BEI (Goleta, CA, USA) Model HS-25 Incremental Encoder having an effective resolution of 0.025◦ . Accelerative data were collected using a model 3713 triaxial capacitive accelerometer from PCB Piezotronics (Depew, NY, USA) that was secured to the underside of the swing arm (27.4 cm from the pivot axis) with its measurement axes square to the swing arm’s longitudinal axis. The tangential axis (pointing in the direction of swing arm movement) provided a voltage proportional to angular acceleration; the two remaining axes were not used in this study. Accelerations for each trial were offset to have a zero mean, thus combating voltage drift in the instrumentation; this seemed reasonable given that aimed movement accelerations begin and conclude near zero, are relatively short in duration (compared to the 2 s data window), and are themselves fairly symmetrical about zero. A ninth-order Butterworth low-pass filter, with a cutoff frequency of 18 Hz, was applied offline to remove instrumentation noise from both the displacement and

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(a)

(b)

(c)

Fig. 3. Sequence for transforming unscaled DNAP curve (a) into a preparatory DNAP curve (b), and then into a scaled DNAP function (c).

Fig. 2. Normalized actuation profiles extracted from experimental data for Subjects 1 and 5.

acceleration data. Velocity values were derived from the encoder and offset accelerometer data. Acceleration values were integrated to produce velocity estimate vˆa , and displacement values were passed through a differentiating filter to generate velocity estimate vˆs . The two velocity estimates were averaged (ˆ v = mean{ˆ va , vˆd }) to produce a final velocity estimate for each trial. Thick solid lines are the resultant DNAP curves for two of our subjects as shown in Fig. 2. Depicted as thinner dashed lines are a subject’s mean actuation programs in each of the nine task configurations. Error bars in Fig. 2 extend one standard deviation above and below the overall mean, and reflect the variation exhibited by a subject across all 540 trials. The curves for Subject 1, shown in the top pane of Fig. 2, are typical of the seven subjects we tested. Shown in the bottom pane are the results for Subject 5, who displayed the widest separation in task-based mean trajectories. One important feature of a DNAP curve, given that it is a function of normalized position, is that the initial value must be positive. If it were negative, movement would start in the wrong direction—and no movement whatsoever would occur if the initial values were zero. We deem experimental trials to start when angular acceleration exceeds 5% of its peak value, thus producing DNAP templates with positive initial values. Since the DNAP shape is directly dependent upon the rotational inertia and damping of the mechanism to be put in motion, as evidenced by (2), human kinematic data are required to produce new DNAP curves for devices that differ in these mechanical characteristics. However, for a given mechanism, it

is possible to generate an infinite number of synthetic DNAP curves based on data from a limited number of subjects. Spline curves that accurately approximate DNAP templates can be computed from a small number of “knots,” or points along the spline curve. We have had reasonable success replicating DNAP appearance using eight knots, each associated with readily identifiable function landmarks on real DNAP templates (such as initial and final amplitude, peak value, and zero-axis crossing). Mean and variance information for the knot coordinates, as derived from a reasonable number of subjects, allows a multitude of synthetic DNAP curves to be produced, each capable of generating “human-like” movements that are unique from those produced by other nonidentical DNAPs. IV. GENERATING DNAP-BASED MOVEMENT We next consider the production of human-like movement given a single DNAP profile. During the remainder of this paper, we identify ways in which our technique matches the human movement characteristics enumerated in Section I. To begin with, as long as the DNAP is based on human data, any linearly scaled version of ψ(ζ) applied as an input torque to the rotating link naturally complies with movement characteristic C1. A. DNAP Scaling For the DNAP to direct physical movement, it must undergo a conversion from unitless shape into a mapping that relates torque to specific displacements. This is accomplished through two linear transformations; one for ballistic movement size, and a second for task difficulty. The original DNAP function is schematically depicted in Fig. 3(a).

SHELTON AND CHIU: EFFICIENTLY GENERATING THE BALLISTIC PHASE OF HUMAN-LIKE AIMED MOVEMENT

The first scaling factor, ballistic limit α, describes the maximum distance over which a simulated movement follows the DNAP trajectory. During the ballistic phase, movement progresses from an initial normalized position of ζ = 0 to a maximum normalized position of ζ = 1. Thus, to adjust the DNAP curve for the size of the movement to be executed, both axes of the DNAP are scaled by α, which has units of length. The resulting function is deemed to be a preparatory DNAP, and is expressed as ψα (θ), where variable θ = αζ represents the system displacement. A preparatory DNAP is depicted in Fig. 3(b). Note that function ψα (θ) has a maximum value of α, as the underlying mapping ψ(ζ) is undefined when ζ > 1. A second scaling, dependent primarily upon the difficulty of the aiming task, is next applied to only the vertical axis of the preparatory DNAP. This difficulty factor, denoted as λ, has units of force (or torque) per unit displacement. When modified in this manner, the resulting scaled DNAP relates displacement to force (or torque), and is denoted as Ψα ,λ (θ). Fig. 3(c) depicts a scaled DNAP. Its relationship to the original DNAP template can be mathematically expressed as   θ (6) Ψα ,λ (θ) = λ ψα (θ) = λαψ = λα ψ (ζ) .       α    preparatory DNAP scaled DNAP DNAP

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Fig. 4. Plot of experimentally determined ballistic limits (α) versus target displacements (D) for odd-numbered trials, gathered from Subject 1.

At the point where system displacement θ exceeds ballistic limit α, an alternative means of control must be utilized, as the ballistic phase of the movement is necessarily concluded. B. Nominal Scaling Factors α and λ Simulations of DNAP-driven movements are needed to determine nominal scaling factors α and λ. We simulate the response of system (1) using an ordinary-differential-equation (ODE) solver. Actuation function Ψα ,λ (θ) provides the control torque. With each call of the differential function, an updated value for θ is calculated, and the corresponding torque Ψα ,λ (θ) is applied as an input. Our two scaling factors influence the ballistic phase independently, with α modifying movement size and λ controlling its duration. In the analysis that follows, we use only data from even-numbered trials to determine simulation parameters, reserving odd-numbered trials for validation. Ballistic limits for Subject 1, computed in the manner described above, are shown in Fig. 4. Although a linear regression would fit the shown data, it makes little sense to have a positive ballistic limit when D = 0; this results if a linear approximation is used. Thus, our estimate of ballistic limit (r2 = 0.99) takes the form α ˆ = aDb + c ID, where a, b, and c are real constants. Dash-dot lines in Fig. 4 estimate α for three of the nominal task difficulties used in our experimental study; the center line represents a task difficulty of ID = 4.85. From Table I, it can be seen that this is the mean task difficulty for our experimental study. Ideally, easier movements should be positioned above, and more difficult movements below this center line. In a similar manner, difficulty scaling factors for Subject 1 are ˆ= shown in Fig. 5. Best-fit linear regression curves, of the form λ d ID e + f D, are displayed as dash-dot lines, with values for

Fig. 5. Plot of experimentally determined normalized difficulty factors (λ) versus task difficulty (ID) for odd-numbered trials of Subject 1.

real constants d, e, and f given in the legend. Although a linear regression could be used for this dataset, Fitts’ law requires that movement time increase linearly with task difficulty. It can √ be shown that, in the undamped case, MT ∝ 1/ λ. Damping alters the rate of change, but Fitts’ law implies that λ must nonetheless become infinite as ID → 0 to ensure that MT → 0. C. Simulated Kinematics ˆ available, nominal aimWith scaling factor estimates α ˆ and λ ing movements can be simulated. Such trajectories do not display trial-to-trial variations, but map out mean kinematic profiles for the ballistic phase of rapidly executed aiming movements. Simulated trials are generated by replacing τ (t) in (1) with a ˆ such that DNAP scaled by α ˆ and λ, ˙ Ψαˆ ,ˆλ (θ) = J θ¨ + β θ.

(7)

For this study, we used a rotational inertia of J = 0.32 kg · m2 , and a rotational damping rate of β = 0.20 kg·m2 /s. Fig. 6 shows simulated displacement paths based on a DNAP template extracted from the odd-numbered trials of Subject 1,

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Fig. 6. Mean experimental trajectories (dashed lines) in each of nine task configurations compared with nominal simulated traces (solid lines). White boxes indicate the maximum extent of ballistic control, where θ = α. ˆ Visual feedback in humans is delayed by approximately 200 ms (dash-dot line).

using scaling factor estimates given in the legends of Figs. 4 and 5. Dashed lines in Fig. 6 represent the mean displacement trajectories produced by Subject 1 during even-numbered trials. While there exists some divergence away from the experimental trajectories at the end of the ballistic movements, we note a close agreement between simulated and actual traces during the first 200 ms. Trial-to-trial variability, characteristic of human movement, is induced by replacing the nominal scaling factors with “noisy” ˆ  . We denote a two-sided truncated normal disestimates α ˆ  and λ tribution, with mean μ, variance σ 2 , and lower and upper bounds ˆ = α ˆ (1 + εα ) and μ− and μ+ as N−+ (μ, μ− μ+ , σ 2 ). Assign α +  2 ˆ (1 + ελ ), where εα ∼ N− (0, σα , −2σα , 2σα ) and ελ ∼ ˆ =λ λ N−+ (0, σλ2 , −1.5σλ , 2.5σλ ). Human movements do not radically diverge from a mean path, so it is necessary to truncate the range of noise effects εα and ελ . Asymmetry in the ελ distribution compensates for the fact that it takes more energy to speed a movement by interval δt, than is saved by slowing a movement by the same time interval. Values σα = 0.07 and σλ = 0.27 are used in the simulations below. Since the probability of twice drawing the same noise factors from the distributions for εα and ελ is nil, each individual trial takes on a unique path, per movement characteristic C2. Simulating 2 000 trials to generate positional variance values, the plot of Fig. 7 is created. The rise and fall in positional variance occurs naturally, as faster movements reach the target sooner, slowing in velocity and allowing slower movements to catch up. This is in agreement with movement characteristic C3. Reducing the number of simulated trials to 10, so as to permit identification of individual traces, the velocity responses of Fig. 8 are generated. These bell-shaped velocity curves are in keeping with characteristic C4. To display the full range of generated velocities, the highest peaking velocity curve in Fig. 8 is associated with values εα = ελ = +2.0, while the lowest peaking curve results from values εα = ελ = −2.0. The eight remaining velocity curves were randomly generated as detailed above.

Fig. 7. Comparison of actual positional variance (30 even-numbered trials by Subject 1 in Configuration 5) with the positional variance generated by 2,000 simulated trials. Variance values beyond the first (fastest) simulated movement reaching a displacement of θ = α ˆ  are identified by a dash-dot line. The simulated variance line ends when two-thirds of computed trials reach normalized displacement ζ = 1.

Fig. 8. Comparison of actual velocity traces (30 even-numbered trials by Subject 1 in Configuration 5) with 10 simulated trials.

V. DNAP-DRIVEN KINEMATICS To reveal the dynamic behavior of a DNAP-driven movement, an energy balance is next derived. Let γ be a positive real scalar. Then, scaling the left-hand side of (1) by γ produces a corresponding linear change in velocity and acceleration



¨ ˙ γτ (t) = J γ θ(t) + β γ θ(t) . (8) Assume all movements driven by τ (t) to start at an initial time of t0 and an initial displacement of zero. Then, since θ(t) = t ˙   t 0 θ(t )dt , the displacement trajectory for (8) is also scaled by γ. Choose any forcing function τ (t) that causes a monotonic rise in displacement θ(t), and identify that function as τ ∗(t). Assign τ (t) = τ ∗(t), and allow the resulting movement to terminate at time t∗f , with displacement θf∗ . Consequently, as indicated by (8), all movements driven by γτ ∗(t) will monotonically increase

SHELTON AND CHIU: EFFICIENTLY GENERATING THE BALLISTIC PHASE OF HUMAN-LIKE AIMED MOVEMENT

from an initial displacement of zero to a final displacement of γθf∗ in time t∗f . Define the normalized displacement for an τ ∗ -driven movement to be ζ(t) = θ(t)/θf∗ . This results in ζ(t) increasing monotonically from zero to one. It also provides the option of expressing the displacement as θ(t) = θf∗ ζ(t). Since both movement time t and normalized displacement ζ increase continuously over the interval t0 ≤ t ≤ t∗f , their relationship is bijective. Thus, there exists an invertible function f : t → ζ that accommodates ζ = f (t) and t = f −1 (ζ). For notational purposes, if time-domain variable z(t) is known, then z(t) = z f −1 (ζ(t)) = z(ζ). Making appropriate substitutions into (8),



¨ ˙ + β γ θ(ζ) . (9) γ τ ∗(ζ) = J γ θ(ζ) Let ψ ∗(ζ) represent a normalized version of forcing function τ ∗(ζ)}, and assign posiτ (ζ), defined as ψ ∗(ζ) = τ∗(ζ)/max { ∗ τ ∗(ζ)}. This allows the tive real value λ such that λθf = max { ∗ forcing function to be expressed as τ (ζ) = λθf∗ ψ ∗(ζ). Restating (9) in terms of ψ ∗(ζ),



¨ ˙ + β γ θ(ζ) . (10) γλθf∗ ψ ∗(ζ) = J γ θ(ζ) ∗

Given that γ and θf∗ have been arbitrarily chosen, these parameters can be combined into a single variable; say α = γ θf∗ . Thus, α represents the final displacement of any movement driven by a linearly scaled version of τ∗(ζ). With θ(t) = θf∗ ζ(t) = γ −1 αζ(t), we find that



¨ ˙ + β αζ(ζ) . (11) λαψ ∗(ζ) = J αζ(ζ) Since DNAP curves have one positive-to-negative crossing of the horizontal axis, and greater positive lobe area than negative lobe area, they cause monotonic forward movement when applied as a forcing function. Integrating both sides of the above equation, after substituting DNAP function ψ(ζ) for ψ ∗ (ζ), and multiplying through by α,  ζ  ζ  ζ ¨ ˙ 2 2 2 ψ(ζ)dζ = α J ζ(ζ)dζ + α β ζ(ζ)dζ . (12) λα 0     0    0   energy added

kinetic energy

dissipated energy

The first term contains an integral of ψ(ζ) between zero and ζ, representing area under a DNAP curve. Since mechanical work is defined as force times distance, this integral captures normalized energy added to the system. Multiplying both axes of the unscaled DNAP function by α, so as to produce a preparatory DNAP, increases the total energy by a factor of α2 . Scaling the vertical axis to account for movement difficulty further increases applied energy by λ. Thus, the total energy applied by a scaled DNAP is captured in the first term of (12). Normalized kinetic energy is captured by the first integral on 2 the right-hand side, which simplifies to 1/2 J ζ˙ (ζ), and has a time domain equivalent of 12 J ζ˙ 2 (t). Multiplying by α2 provides a total kinetic energy of 12 J θ˙2 . The final integral reflects normalized dissipated energy, simplifying to a time-domain equivalent t of β 0 ζ˙ 2 dt when the damping is fixed. Multiplying this term

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by α2 provides the textbook value for rotational damper work t β 0 θ˙2 dt. Ignoring exogenous disturbances, the energy balance of (12) must be obeyed throughout a DNAP-driven ballistic movement. We next use this relationship to explain why α and λ have distinctly different influences on such movements. As (12) depicts, a common α2 term can be dropped from each of the terms. This reveals that ζ-mapped relative acceleration ¨ ˙ ζ(ζ) and relative velocity ζ(ζ) are independent of movement size α. Since λ represents a difficulty factor, we hold it constant for all aiming tasks exhibiting the same (D/W ) ratio. When λ is held constant, the displacement, velocity, and acceleration curves must be linearly scaled versions of those from any other movement generated by a similarly scaled DNAP function. Also, as a consequence of the one-to-one mapping between ζ and t, the movement time becomes wholly dependent upon the shape of DNAP ψ(ζ). In other words, doubling α causes the kinematic components of position, velocity, and acceleration to likewise double, while the movement duration remains fixed. This is in agreement with the prediction of Fitts’ law (see C5) that rapidly executed aiming movements of equal difficulty, but differing target distance, should nonetheless be of equal duration. Now consider α to be held constant, while λ varies. Given that ballistic limit α is nearly proportional to target distance D (see Fig. 4), this arrangement approximates holding the target distance fixed, but varying precision ratio (D/W ). With difficulty factor λ permitted to vary, the resulting kinematic profiles are no longer self-similar. Raising λ increases the scaled forcing function magnitude, thus increasing the movement’s mean normalized velocity. As evidenced from Fig. 5, difficulty factor λ declines with increasing task difficulty. This means that, regardless of the target displacement, simulated trials will take longer to complete as the index of difficulty increases, in agreement with characteristic C5. Further, with repeated trials scaled ˆ  , each discrete trial will exhibit a by “noisy” difficulty factor λ unique duration in completing the ballistic phase, in agreement with characteristic C6. VI. CONCLUSION When executing aiming tasks, humans adjust their kinematic trajectory in response to the ratio between target width and the movement amplitude. This is true whether the movement takes place along a single axis, or is a complex multidimensional motion. While traditional methods for controlling point-to-point movement do not account for this characteristic in human behavior, the DNAP methodology makes appropriate adjustments in single-axis applications. Beyond adjusting its mean trajectory in response to the task configuration, human movement also displays substantial positional and temporal variance, even after a particular reaching task has been learned through thousands of repetitions [28]. Such variations are characteristic of all limb movements, not just those of the arm [21]. The DNAP approach allows human-like deviations from a mean path to be easily incorporated through the optional addition of white noise to scaling factors α and λ. We believe it feasible to extend the DNAP method to multiple dimensions by focusing on its inherent energy balance. Consider

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parameter θM to be a function of one or more displacement angles in a multilink mechanism. If θM increases monotonically, then normalized measure ζM can grow from zero to one, without reversal, during mechanism motion. Let system energy be plotted against ζM . With each link assigned a fraction of this energy (likely varying with ζM to produce useful movement of the outermost link, and with all fractions totaling unity), appropriate joint torques can be computed so as to maintain the desired energy balance as movement takes place. Work is ongoing to determine how this energy balance might be practically realized. REFERENCES [1] H. Ishiguro and S. Nishio, “Building artificial humans to understand humans,” J. Artif. Organs, vol. 10, no. 3, pp. 133–42, 2007. [2] H. I. J. D. Ayse Pinar Saygin, Thierry Chaminade, and C. Frith, “The thing that should not be: Predictive coding and the uncanny valley in perceiving human and humanoid robot actions,” Soc. Cogn. Affect Neurosci., vol. 7, no. 4, pp. 413–422, Apr. 2012. [3] P. Morasso, “Spatial control of arm movements,” Exp. Brain Res., vol. 42, no. 2, pp. 223–227, 1981. [4] C. G. Atkeson and J. M. Hollerbach, “Kinematic features of unrestrained vertical arm movements,” J. Neurosci., vol. 5, no. 9, pp. 2318–2330, 1985. [5] M. Saerbeck and A. Van Breemen, “Design guidelines and tools for creating believable motion for personal robots,” in Proc. 16th IEEE Int. Symp. Robot Human Interactive Commun., Aug. 2007, pp. 386–391. [6] S. C. Lai, G. Mayer-Kress, J. J. Sosnoff, and K. M. Newell, “Information entropy analysis of discrete aiming movements,” Acta Psych., vol. 119, no. 3, pp. 283–304, Jul. 2005. [7] D. M. Corcos, S. Jaric, G. C. Agarwal, and G. L. Gottlieb, “Principles for learning single-joint movements. 1. Enhanced performance by practice,” Exp. Brain Res., vol. 94, no. 3, pp. 499–513, Jun. 1993. [8] E. R. F. W. Crossman and P. J. Goodeve, “Feedback-control of handmovement and Fitts law,” Q. J. Exp. Psych., Human Exp. Psych., vol. 35A, pp. 251–278, May 1983. [9] L. P. J. Selen, P. J. Beek, and J. H. Van Dieen, “Impedance is modulated to meet accuracy demands during goal-directed arm movements,” Exp. Brain Res., vol. 172, no. 1, pp. 129–138, Jun. 2006. [10] E. Todorov, W. W. Li, and X. C. Pan, “From task parameters to motor synergies: A hierarchical framework for approximately optimal control of redundant manipulators,” J. Robot. Syst., vol. 22, no. 11, pp. 691–710, Nov. 2005. [11] P. M. Fitts, “The information capacity of the human motor system in controlling the amplitude of movement,” J. Exp. Psych., vol. 47, pp. 381– 391, 1954. [12] S. Young, J. Pratt, and T. Chau, “Target-directed movements at a comfortable pace: Movement duration and Fitts’s law,” J. Motor Behavior, vol. 41, no. 4, pp. 339–346, Jul. 2009. [13] J. Harris and E. Sharlin, “Exploring the affect of abstract motion in social human-robot interaction,” in Proc. IEEE RO-MAN, 2011, pp. 441–448. [14] P. H. Chang, K. Park, S. H. Kang, H. I. Krebs, and N. Hogan, “Stochastic estimation of human arm impedance using robots with nonlinear frictions: An experimental validation,” IEEE/ASME Trans. Mechatronics, vol. 18, no. 2, pp. 775–786, Apr. 2013. [15] H. Ding and J. Wu, “Point-to-point motion control for a high-acceleration positioning table via cascaded learning schemes,” IEEE Trans. Ind. Electron., vol. 54, no. 5, pp. 2735–2744, Oct. 2007. [16] H. Li, M. Le, Z. Gong, and W. Lin, “Motion profile design to reduce residual vibration of high-speed positioning stages,” IEEE/ASME Trans. Mechatronics, vol. 14, no. 2, pp. 264–269, Apr. 2009. [17] M. Field, Z. Pan, D. Stirling, and F. Naghdy, “Human motion capture sensors and analysis in robotics,” Ind. Robot, Int. J., vol. 38, no. 2, pp. 163– 171, 2011. [18] O. Jenkins and M. Mataric, “Deriving action and behavior primitives from human motion data,” in Proc. IEEE/RSJ Int. Intell. Robots Syst. Conf., 2002, vol. 3, pp. 2551–2556.

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Jeffrey N. Shelton received the B.S. degree in mechanical engineering from Purdue University, West Lafayette, IN, USA, in 1981, and the M.S. degree in mechanical engineering from Stanford University, Palo Alto, CA, USA, in 1983. He received the M.B.A. degree from Indiana University, Indianapolis, IN, USA, in 1992, with a concentration in marketing. He received the Ph.D. degree from Purdue University, in 2013, with a focus on the control aspects of human movement.

George T.-C. Chiu received the B.S. degree in mechanical engineering from National Taiwan University, Taipei, Taiwan, in 1985 and the M.S. and Ph.D. degrees in mechanical engineering from the University of California at Berkeley, Berkeley, CA, USA, in 1990 and 1994, respectively. He is currently a Professor in the School of Mechanical Engineering with courtesy appointments in the School of Electrical and Computer Engineering and Department of Psychological Sciences, Purdue University, West Lafayette, IN, USA. Before joining Purdue in 1996, he was an R&D Engineer at the Hewlett-Packard Company developing high-performance color inkjet printers and multifunction machines. His current research interests include digital imaging and printing systems, digital fabrication and functional printing, human motor control, motion and vibration control, and perception. Dr. Chiu is the Editor of the Journal of Imaging Science and Technology. He received the 2010 IEEE TRANSACTION ON CONTROL SYSTEMS TECHNOLOGY Outstanding Paper Award. He is a Fellow of the American Society of Mechanical Engineers (ASME) and the Society for Imaging Science and Technology.